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proofpile-arXiv_065-46 | \section{Introduction}
\label{sec:intro}
Few-shot object detection (FSOD) \cite{yan2019meta,wang2020frustratingly,xiao2020few,wu2020multi,zhu2021semantic} aims at detecting objects out of base training set with few support examples per class.
It has received increasing attention from the robotics community due to its vital role in autonomous exploration, since robots are often expected to detect novel objects in an unknown environment but only a few examples can be provided online.
For example, in a mission of rescue shown in \fref{fig:1} (a), the robots are required to detect some uncommon objects such as drill, rope, helmet, vent.
\begin{figure}[!t]
\centering
\includegraphics[width=1\columnwidth]{fig1.pdf}
\caption{Representative images from robots’ exploration and performance comparison of the state-of-the-arts \cite{fan2020few,xiao2020few,wu2020multi,wang2020frustratingly,faster} and proposed AirDet. Solid lines denote results with no fine-tuning and dashed lines indicate results fine-tuned on few-shot data. Without further updating, AirDet can outperform prior work. Besides, unlike the fine-tuned models meeting bottleneck in small objects, AirDet has set an outstanding new level.}
\label{fig:1}
\end{figure}
Despite its recent promising developments, most of existing methods \cite{kang2019few,sun2021fsce,zhang2021accurate,wang2019meta,wang2020frustratingly,wu2020multi,qiao2021defrcn,cao2021nips,li2021few,fan2021generalized} require a careful \textit{offline} fine-tuning stage on novel images before inference. However, the requirement of the fine-tuning process is infeasible for robotic \textit{online} applications, since
(\textbf{1}) new object categories can be dynamically added during exploration, thus re-fine-tuning the model with limited onboard computational resources for novel classes is extremely inefficient for the time-starved tasks such as search and rescue \cite{tariq2018dronaid,farooq2018ground,wang2020visual,chen_tro}.
(\textbf{2}) to save human effort, only very few samples can be provided online\footnote{Since online annotation is needed during mission execution, only 1-5 samples can be provided in most of the robotic applications, which is the main focus of this paper.}, thus the fine-tuning stage \cite{kang2019few,sun2021fsce,zhang2021accurate,wang2019meta,wang2020frustratingly,wu2020multi,qiao2021defrcn,fan2021generalized,li2021few} needs careful \textit{offline} hyper-parameter tuning to avoid over-fitting, which is infeasible for \textit{online} exploration, and (\textbf{3}) fine-tuned models usually perform well for in-domain test \cite{wang2020frustratingly,wu2020multi,xiao2020few,faster,qiao2021defrcn,li2021few}, while suffer from cross-domain test, which is unfavourable for robotic applications.
Therefore, we often expect a few-shot detector that is able to inference without fine-tuning such as \cite{fan2020few}. However, the performance of \cite{fan2020few} is still severely hampered for challenging robotics domain due to (\textbf{1}) ineffective multi-scale detection;
(\textbf{2}) ineffective feature aggregation from multi-support images; and (\textbf{3}) inaccurate bounding box location prediction. Surprisingly, in this paper, we find that all three problems can be effectively solved by learning \textit{class-agnostic relation} with support images. We name the new architecture AirDet, which can produce promising results even without abominable fine-tuning as shown in \fref{fig:1}, which, to the best of our knowledge, is the first feasible few-shot detection model for autonomous robotic exploration. Specifically, the following three modules are proposed based on \textit{class-agnostic relation}, respectively.
\myparagraph{Support-guided Cross-Scale Fusion (SCS) for Object Proposal} One reason for performance degradation in multi-scale detection is that the region proposals are not effective for small scale objects, even though some existing works adopt multiple scale features from query images \cite{zhang2021accurate,zhu2021semantic}. We argue that the proposal network should also include cross-scale information from the support images. To this end, we present a novel SCS module, which explicitly extracts multi-scale features from cross-scale relations between support and query images.
\myparagraph{Global-Local Relation (GLR) for Shots Aggregation} Most prior work \cite{fan2020few,yan2019meta,wu2020multi} simply average multi-shot support feature to obtain a class prototype for detection head. However, this cannot fully exploit the little but valuable information from every support image. Instead, we construct a shots aggregation module by learning the relationship between the multi-support examples, which achieves significant improvements with more shots.
\myparagraph{Prototype Relation Embedding (PRE) for Location Regression} Some existing works \cite{fan2020few,zhang2021accurate} introduced a relation network \cite{sung2018learning} into the classification branch; however, the location regression branch is often neglected. To settle this, we introduce cross-correlation between the proposals from SCS and support features from GLR into the regression branch.
This results in the PRE module, which explicitly utilize support images for precise object localization.
In summary, AirDet is a fully relation-based few-shot object detector, which can be applied directly to the novel classes without fine-tuning. It surprisingly produces comparable or even better results than exhaustively fine-tuned SOTA methods \cite{wang2020frustratingly,faster,xiao2020few,wu2020multi,fan2020few}, as shown in \fref{fig:1} (b).
Besides, as shown in \fref{fig:1} (c), AirDet maintains high robustness in small objects due to the SCS module, which fully takes advantage of the multi-scale support feature.
Note that in this paper, fine-tuning is undesired because it cannot satisfy the online responsive requirement for robots, but it can still improve the performance of AirDet.
\section{Related Works}
\subsection{General Object Detection}
The task of object detection \cite{faster, yolo, liu2016ssd, RCNN,fast,mask} is to find out all the pre-defined objects in an image, predicting their categories and locations, which is one of the core problems in the field of computer vision.
Object detection algorithms are mainly divided into: two-stage approaches \cite{RCNN,fast,faster,mask} and one-stage approaches \cite{liu2016ssd,yolo,yolo2,yolo3}. R-CNN \cite{RCNN}, and its variants \cite{RCNN,fast,faster,mask} serve as the foundation of the former branch; among them, Faster R-CNN \cite{faster} used region proposal network (RPN) to generate class-agnostic proposals from the dense anchors, which greatly improved the speed of object detection based on R-CNN \cite{fast}. On the other hand, YOLO series \cite{yolo,yolo2,yolo3} fall into the second branch, which tackles object detection as an end-to-end regression problem. Besides, the known SSD series \cite{liu2016ssd,li2017fssd} propose to utilize pre-defined bounding boxes to adjust to various object scales inspired by \cite{faster}.
One shortcoming of the above methods is that they require abundant labeled data for training. Moreover, the types and number of object categories are fixed after training (80 classes in COCO, for instance), which is not applicable to robot's autonomous exploration, where unseen, novel objects often appear online.
\subsection{Few-shot Object Detection}
Trained with abundant data for base classes, few-shot object detectors can learn to generalize only using a few labeled novel image shots. Two main branches leading in FSOD are meta-learning-based approaches \cite{yan2019meta,xiao2020few,wu2020multi,fan2020few,han2021query} and transfer-learning-based approaches \cite{wang2020frustratingly,zhu2021semantic,sun2021fsce,wu2021universal,qiao2021defrcn}.
Transfer-learning approaches seek for the best learning strategy of general object detectors \cite{faster} on a few novel images. Wang \textit{et al.}~ \cite{wang2020frustratingly} proposed to fine-tune only the last layer with a cosine similarity-based classifier. Using manually defined positive refinement branch, MPSR \cite{wu2020multi} mitigated the scale scarcity issue. Recent works have introduced semantic relations between novel-base classes \cite{zhu2021semantic} and contrastive proposal encoding \cite{sun2021fsce}.
Aiming at training meta-models on episodes of individual tasks, meta-learning approaches \cite{yan2019meta,xiao2020few,fan2020few,Hu2021CVPR,Zhang2021CVPR,han2021query} generally contain two branches, one for extracting support information and the other for detection on the query image. Among them, Meta R-CNN \cite{yan2019meta}, and FSDet \cite{xiao2020few} target at support guided query channel attention. With novel attention RPN and multi-relation classifier, A-RPN \cite{fan2020few} has set the current SOTA. Very recent works also cover support-query mutual guidance \cite{Zhang2021CVPR}, aggregating context information \cite{Hu2021CVPR}, and constructing heterogeneous graph convolutional networks on proposals \cite{han2021query}.
\subsection{Relation Network for Few-shot Learning}
In few-shot image classification, relation network \cite{sung2018learning}, also known as learning to compare, has been introduced to train a classifier by modeling the class-agnostic relation between a query image and the support images. Once trained and provided with a few novel support images, inference on novel query images can be implemented without further updating.
For few-shot object detection, such relation has only been utilized for the classification branch so far in very few works. For example, Fan \textit{et al.}~ proposed a multi-relation classification network, which consists of global, local, and patch relation branches \cite{fan2020few}. Zhang \textit{et al.}~ leveraged general relation network \cite{sung2018learning} architecture to build multi-level proposal scoring, and support weighting modules \cite{Zhang2021CVPR}. In this work, we thoroughly explore such relation in few-shot detection and propose a fully relation-based architecture.
\subsection{Multi-Scale Feature Extraction}
Multi-scale features have been exhaustively exploited for multi-scale objects in general object detection \cite{liu2016ssd,Shen2017dsod,Kong2016HyperNet,yolo2,lin2017fpn,li2017fssd}. For example, FSSD \cite{li2017fssd} proposed to fuse multi-scale feature and implement detection on the fused feature map. Lin \textit{et al.}~ constructed the feature pyramid network (FPN) \cite{lin2017fpn}, which builds a top-down architecture and employs multi-scale feature map for detection.
For few-shot detection, standard FPN \cite{lin2017fpn} has been widely adopted in prior transfer-learning-based methods \cite{wang2020frustratingly,zhu2021semantic,sun2021fsce,wu2020multi}. In meta-learning, existing meta-learner \cite{Zhang2021CVPR} employs all scales from FPN and implements detection on each scale in parallel, which is computationally inefficient.
\section{Preliminary}
In few-shot object detection \cite{yan2019meta,xiao2020few,wu2020multi,Deng2009imagenet}, the classes are divided into $B$ base classes $\mathcal{C}_{\rm{b}}$ and $N$ novel ones $\mathcal{C}_{\rm{n}}$, satisfying that $\mathcal{C}_{\rm{b}}\cap\mathcal{C}_{\rm{n}}=\varnothing$.
The objective is to train a model that can detect novel classes in $\mathcal{C}_{\rm{n}}$ by only providing $k$-shot labeled samples for $\mathcal{C}_{\rm{n}}$ and abundant images from base classes $\mathcal{C}_{\rm{b}}$.
During training, we adopt the episodic paradigm \cite{yan2019meta}.
Basically, images from the base classes $\mathcal{C}_{\rm{b}}$ are split into query images $\mathbf{Q}_{{\rm{b}}}$ and support images $\mathbf{S}_{{\rm{b}}}$.
Given all support images $\mathbf{S}_{{\rm{b}}}$, the model learns to detect objects in query images $\mathbf{Q}_{{\rm{b}}}$. During test, the model is to detect objects in novel query images $\mathbf{Q}_{{\rm{n}}}$ by only providing few (1-5) labeled novel support images $\mathbf{S}_{{\rm{n}}}$.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: Most existing methods \cite{wang2020frustratingly,zhu2021semantic,sun2021fsce,yan2019meta,xiao2020few,wu2020multi,wu2021universal,cao2021nips,qiao2021defrcn} have to be fine-tuned on $\mathbf{S}_{{\rm{n}}}$ due to the class-specific model design, while AirDet can be applied directly to $\mathbf{Q}_{{\rm{n}}}$ by providing $\mathbf{S}_{{\rm{n}}}$ without fine-tuning.
\section{Methodology}
\begin{figure*}[!t]
\centering
\includegraphics[width=1\textwidth]{main_fig.pdf}
\caption{The pipeline of the autonomous exploration task and the framework of AirDet. During exploration, a few prior raw images that potentially contain novel objects (helmet) are sent to a human user first. Provided with online annotated few-shot data, the robot explorer is able to detect those objects by observing its surrounding environment. AirDet includes 4 modules, \textit{i.e.}, the shared backbone, support-guided cross-scale (SCS) feature fusion module for region proposal, global-local relation (GLR) module for shots aggregation, and relation-based detection head, which are visualized by different colors. }
\label{fig:main}
\end{figure*}
Since only a few shots are given during model test, information from the support images is little but valuable.
We believe that the major limitation of the existing algorithms is that such information from support images is not fully exploited.
Therefore, we propose to learn \textit{class-agnostic relation} with the support images in all the modules of AirDet.
As exhibited in \fref{fig:main}, the structure of AirDet is simple: except for the shared backbones, it only consists of three modules, \textit{i.e.}, a support-guided cross-scale fusion (SCS) module for regional proposal, a global-local relation (GLR) module for shots aggregation, and a relation-based detection head, containing prototype relation embedding (PRE) module for location regression and a multi-relation classifier \cite{fan2020few}. We next introduce two kinds of \textit{class-agnostic relation}, which will be used by the three modules.
\subsection{Class-Agnostic Relation}
To exploit the relation between two features from different aspects, we define two relation modules, \textit{i.e.}, spatial relation $\mathcal{R}_{\rm{s}}(\cdot, \cdot)$ and channel relation $\mathcal{R}_{\rm{c}}(\cdot, \cdot)$.
\myparagraph{1. Spatial Relation:}
Object features from the same category are often correlated along the spatial dimension, thus we define the spatial relation features $\mathcal{R}_{\rm{s}}$ in \eqref{eqn:inner} leveraging on the regular and depth-wise convolution.
\begin{equation}\label{eqn:inner}
\mathcal{R}_{\rm{s}}(\mathbf{A}, \mathbf{B}) = \mathbf{A} \odot \mathrm{MLP}\Big(\mathrm{Flatten}\big(\mathrm{Conv}(\mathbf{B})\big)\Big),
\end{equation}
where inputs $\mathbf{A}, \mathbf{B}\in\mathbb{R}^{C\times W\times H}$ denote 2 general tensors. $\rm{Flatten}$ means flatten the features in spatial (image) domain and $\mathrm{MLP}$ denotes multilayer perceptron (MLP), so that $\rm{MLP}\Big(\rm{Flatten}\big(\rm{Conv}(\mathbf{B})\big)\Big) \in \mathbb{R}^{C\times 1\times 1}$. $\odot$ indicates depth-wise convolution \cite{fan2020few}. Note that we use convolution to calculate correlation since both operators are composed of inner products.
\myparagraph{2. Channel Relation:}
Inspired by the phenomenon that features of different classes are often stored in different channels \cite{li2019siamrpn}, we propose a simple but effective channel relation $\mathcal{R}_{\rm{c}}(\cdot, \cdot)$ in \eqref{eqn:channel} to extract the cross-class relation features.
\begin{equation}\label{eqn:channel}
\mathcal{R}_{\rm{c}}(\mathbf{A}, \mathbf{B}) = \mathrm{{Conv}}\big(\mathrm{Cat}(\mathbf{A}, \mathbf{B})\big) + \mathrm{Cat}\big(\mathrm{{Conv}}(\mathbf{A}), \mathrm{{Conv}}(\mathbf{B})\big),
\end{equation}
where $\mathrm{Cat(\cdot, \cdot)}$ is to concatenate features along the channel dimension.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: The two simple but effective \textit{class-agnostic relation} learners are fundamental building blocks of AirDet, which, to the best of our knowledge, is the first attempt towards a fully relation-based structure in few-shot detection.
\subsection{Support-guided Cross-Scale Fusion (SCS) for Object Proposal}
As mentioned earlier, existing works generate object proposals only using single scale information from query images \cite{kang2019few,xiao2020few,wang2019meta,wu2020multi}, while such strategy may not be effective for small scale novel objects.
Differently, we propose support-guided cross-scale fusion (SCS) in AirDet to introduce multi-scale features and take the relation between query and support images for region proposal.
As shown in \fref{fig:scs} (a), SCS takes support and query features from different backbone blocks (ResNet2, 3, and 4 block) as input.
We first apply \textit{spatial relation}, where the query and support features from the same backbone block are $\mathbf{A}, \mathbf{B}$ in \eqref{eqn:inner}, respectively.
Then we use \textit{channel relation} to fuse the ResNet2 and ResNet3 block features, which are $\mathbf{A}, \mathbf{B}$ in \eqref{eqn:channel}, respectively.
The fused channel relation feature is later merged with the spatial relation feature from ResNet4 block.
The final merged feature is sent to the region proposal network (RPN) \cite{faster} to generate region proposals.
\begin{figure}[ht]
\centering
\includegraphics[width=1\columnwidth]{SCS_GLR.pdf}
\caption{Network architecture of SCS for region proposal and GLR for shots aggregation.}
\label{fig:scs}
\end{figure}
\subsection{Global-Local Relation (GLR) for Shots Aggregation}
In prior attempts \cite{yan2019meta,kang2019few,xiao2020few,wu2020multi}, the support object features from multiple shots are usually averaged to represent the class prototype, which is then used for regression and classification.
Although it can be effective with fine-tuning, we argue that a simple average cannot fully utilize the information from few-shot data. To this end, we built global-local relation (GLR) module in \fref{fig:scs} (b), which leverages the features from every shot to construct the final prototype.
Suppose the $k$-shot deepest support features are $\phi({\mathbf{s}^i})$,~$i=1,\cdots,k$, our final class prototype $\mathbf{e}$ can be expressed as a weighted average of the features:
\begin{equation}
\mathbf{e}=\sum_{i=1}^{k}(\phi({\mathbf{s}^i})\otimes\mathbf{M}^i),
\end{equation}
where $\otimes$ is the element-wise multiplication, and $\mathbf{M}^i$ is a confidence map:
\begin{equation}
\mathbf{M}^{i}=\mathrm{SoftMax}\Big(\mathrm{MLP}\big(\mathbf{f}^{i}\big)\Big),
\end{equation}
where $\mathbf{f}^{i}$ is the output from the channel relation extractor:
\begin{equation}\label{eq:shot-relation}
\mathbf{f}^{i} = \mathcal{R}_{\rm{c}}\left(\mathrm{Conv}(\phi(\mathbf{s}^i)), \frac{1}{k}\sum_{i=1}^{k}\mathrm{Conv}(\phi(\mathbf{s}^i))\right).
\end{equation}
Note that to include both ``global" (all shots) and ``local" (single shot) features, the inputs of the channel relation extractor in \eqref{eq:shot-relation} include both the feature from that shot and the averaged features over all shots.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: Unlike prior work \cite{yan2019meta,kang2019few,xiao2020few,wu2020multi} relying on fine-tuning with more support data for performance gain, AirDet can extract stronger prototype to achieve improvement on more shots without fine-tuning.
\subsection{Prototype Relation Embedding (PRE) for Location Regression}
It has been demonstrated that a multi-relation network \cite{fan2020few} is effective for the classification branch.
Inspired by its success, we further build a prototype relation embedding (PRE) network for the location regression branch.
Given a prototype exemplar $\mathbf{e}\in\mathbb{R}^{C\times a\times a}$, we utilize the spatial relation \eqref{eqn:inner} to embed information from the exemplar to the proposal features $\mathbf{p}^j$,~$j= 1,2,\cdots,p$ as:
\begin{equation}
\mathbf{l}^j = \mathbf{p}^j + \mathcal{R}_{\rm{s}}(\mathbf{p}^j, \mathbf{e}),
\end{equation}
where we take $3\times 3$ convolution layer in \eqref{eqn:inner} for spatial feature extraction.
The proposal features $\mathbf{l}^j$ is then used for bounding box regression through an MLP module following Faster-RCNN \cite{faster}.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: Class-related feature $\mathbf{l}^j$ contains information from support objects, which turns out more effective for location regression even if the objects have never been seen in the training set.
\section{Experiments}
\subsection{Implementation}\label{sec:Imple}
We adopt the training pipeline from \cite{fan2020few}.
To maintain a fair comparison with other methods \cite{wang2020frustratingly,xiao2020few,wu2020multi,faster}, we mainly adopt ResNet101 \cite{He2016res} pre-trained on ImageNet \cite{Deng2009imagenet} as backbone.
The performance of other backbones is presented in \appref{sec:backbone}.
For fair comparison \cite{wang2020frustratingly,xiao2020few,wu2020multi,fan2020few,faster}, we utilized their official implementation, support examples, and models (if provided) in all the experiments. AirDet and the baseline \cite{fan2020few} take the \textbf{same} supports in all the settings.
We use 4 NVIDIA GeForce Titan-X Pascal GPUs for experiments.
Detailed configuration of AirDet can be found in \appref{sec:config} and the source code.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: To save human effort, only very few support examples (1-5 samples per class) can be provided during online exploration. Therefore, we mainly focused on $k=1, 2, 3, 5$-shot evaluation. Since the objects from exploration are usually unseen, we only test novel classes throughout the experiments.
\subsection{In-domain Evaluation}\label{sec:indomain}
We first present the in-domain evaluation on COCO benchmark \cite{lin2014microsoft}, where the models are trained and tested on the same dataset.
Following prior works \cite{yan2019meta,kang2019few,wu2020multi,fan2020few,wang2020frustratingly,xiao2020few,wu2021universal,cao2021nips,fan2021generalized,sun2021fsce,zhu2021semantic}, the 80 classes are split into 60 non-VOC base classes and 20 novel classes. During training, the base class images from COCO trainval2014 are considered available. With few-shot samples per novel class, the models are evaluated on 5,000 images from COCO val2014 dataset.
\begin{table}[!t]
\centering
\setlength{\tabcolsep}{0.2mm}
\fontsize{5.5}{6.5}\selectfont
\caption{Performance comparison on COCO validation dataset. In each setting, \red{red} and \green{green} fonts denote the best and second-best performance, respectively. AirDet achieves significant performance gain on baseline without fine-tuning. With fine-tuning, AirDet sets a new SOTA performance. $^\dag$We randomly sampled 3-5 different groups of support examples and reported the average performance and their standard deviation.}
\begin{threeparttable}
\begin{tabular}{cc|ccc|ccc|ccc|ccc}
\toprul
\multicolumn{2}{c|}{Shots} & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c}{5} \\
\midrule
Method & Fine-tune & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ \\
\multirow{2}{*}{A-RPN \cite{fan2020few}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & 4.32 & 7.62 & 4.3 & 4.67 & 8.83 & 4.49 & 5.28 & 9.95 & 5.05 & 6.08 & 11.17 & 5.88 \\
& & $\pm$0.7 &$\pm$1.3 & $\pm$0.7 & $\pm$0.3 & $\pm$0.5 & $\pm$0.3 & $\pm$0.6 & $\pm$0.8 & $\pm$0.6 & $\pm$0.3 & $\pm$0.4 & $\pm$0.3 \\
\cmidrule{1-14}
\multirow{2}{*}{\textbf{AirDet (Ours)}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & \red{\textbf{5.97}} & \red{\textbf{10.52}} & \red{\textbf{5.98}} & \red{\textbf{6.58}} & \red{\textbf{12.02}} & \red{\textbf{6.33}} & \red{\textbf{7.00}} & \red{\textbf{12.95}} & \red{\textbf{6.71}} & \red{\textbf{7.76}} & \red{\textbf{14.28}} & \red{\textbf{
7.31}} \\
& & \textbf{$\pm$0.4} &\textbf{$\pm$0.9} &\textbf{ $\pm$0.2} & \textbf{$\pm$0.2} & \textbf{$\pm$0.4} & \textbf{$\pm$0.2} & \textbf{$\pm$0.5} & \textbf{$\pm$0.8} & $\pm$0.7& \textbf{$\pm$0.3} & \textbf{$\pm$0.4} & $\pm$0.4 \\
\midrule
FRCN \cite{faster} & \checkmark & 3.26 & 6.66 & 3.04 & 3.73 & 7.79 & 3.22 & 4.59 & 9.52 & 4.07 & 5.32 & 11.20 & 4.54 \\
TFA$_{\mathrm{fc}}$ \cite{wang2020frustratingly} & \checkmark & 2.78 & 5.39 & 2.36 & 4.14 & 7.98 & 4.01 & 6.33 & 12.10 & 5.94 & 7.92 & 15.58 & 7.29 \\
TFA$_{\mathrm{cos}}$ \cite{wang2020frustratingly} & \checkmark & 3.09 & 5.24 & 3.21 & 4.21 & 7.70 & 4.35 & 6.05 & 11.48 & 5.93 & 7.61 & 14.56 & 7.17 \\
FSDetView \cite{xiao2020few} & \checkmark & 2.20 & 6.20 & 0.90 & 3.40 & 10.00 & 1.50 & 5.20 & 14.70 & 2.10 & 8.20 & \red{21.60} & 4.70 \\
MPSR \cite{wu2020multi} & \checkmark & 3.34 & 6.11 & 3.25 & 5.41 & 9.68 & 5.52 & 5.70 & 10.54 & 5.50 & 7.20 & 13.55 & 6.89 \\
A-RPN \cite{fan2020few} & \checkmark & {4.59} & {8.85} & {4.37} & {6.15} & {12.05} & {5.76} & {8.24} & {15.52} & {7.92} & {9.02} & 17.29 & {8.53} \\
W. Zhang \textit{et al.}~ \cite{zhang2021hallucination} & \checkmark & {4.40} & {7.50} & {4.90} & {5.60} & {9.90} & {5.90} & {7.20} & {13.30} & 7.40 & - & - & - \\
FADI \cite{cao2021nips} & \checkmark & \green{5.70} & \green{10.40} & \green{6.00} & \green{7.00} & \green{13.01} & \green{7.00} & \green{8.60} & \green{15.80} & \green{8.30} & \green{10.10} & 18.60 & \red{11.90} \\
\textbf{AirDet (Ours)} & \checkmark & \textbf{\red{6.10}} & \textbf{\red{11.40}} & \textbf{\red{6.04}} & \textbf{\red{8.73}} & \textbf{\red{16.24}} & \textbf{\red{8.35}} & \textbf{\red{9.95}} & \textbf{\red{19.39}} & \textbf{\red{9.09}} & \textbf{\red{10.81}} &\green{\textbf{20.75}} & \textbf{\green{10.27}} \\
\bottomrul
\end{tabular}\label{tab:coco}%
\end{threeparttable}
\end{table}%
\myparagraph{Overall Performance}
As shown in \tref{tab:coco}, AirDet achieves significant performance gain on the baseline \cite{fan2020few}.
AirDet without fine-tuning amazingly also achieves comparable or even better results than many fine-tuned methods.
With fine-tuning, AirDet outperformed existing SOTAs \cite{fan2020few,wang2020frustratingly,xiao2020few,wu2020multi,faster,cao2021nips,zhang2021hallucination}.
Since the results without fine-tuning may be sensitive to support images, we report the averaged performance, and the standard deviation on 3-5 randomly sampled support images, where we surprisingly find AirDet more robust to the variance of support images compared with the baseline \cite{fan2020few}.
\myparagraph{Multi-scale Objects}
\begin{table*}[!t]
\centering
\setlength{\tabcolsep}{0.2mm}
\fontsize{5.5}{6.5}\selectfont
\caption{Performance evaluation on multi-scale objects from COCO. Highest-ranking and second-best scores are marked out with \red{red} and \green{green}, respectively. Without fine-tuning, AirDet can avoid over-fitting and shows robustness on small-scale objects. By virtue of the SCS module, AirDet can achieve higher results than those with FPN.}
\begin{threeparttable}
\begin{tabular}{ccc|ccc|ccc|ccc|ccc}
\toprul
\multicolumn{3}{c|}{Shots} & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c}{5} \\
\midrule
Method & FPN & Fine-tune & AP$_s$ & AP$_m$ & AP$_l$ & AP$_s$ & AP$_m$ & AP$_l$ & AP$_s$ & AP$_m$ & AP$_l$ & AP$_s$ & AP$_m$ & AP$_l$ \\
\multirow{2}{*}{A-RPN \cite{fan2020few}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & \multirow{2}{*}{\text{\ding{55}}} & 2.43 & 5.00 & 6.74 & 2.67 & 5.01 & 7.18 & 3.42 & 6.15 & 8.77 & 3.54 & 6.73 & 9.97 \\
& & & $\pm$0.4 &$\pm$1.0 & $\pm$1.1 & $\pm$0.3 & $\pm$0.3 & $\pm$0.4 & $\pm$0.2 & $\pm$0.5 & $\pm$0.8 & $\pm$0.3 & $\pm$0.03 & $\pm$0.2 \\
\midrule
\multirow{2}{*}{\textbf{AirDet (Ours)}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & \multirow{2}{*}{\text{\ding{55}}} & \red{\textbf{2.85}} & \red{\textbf{6.33}} & \red{\textbf{9.00}} & \red{\textbf{4.00}} & \red{\textbf{6.84}} & \red{\textbf{9.94}} & \red{\textbf{4.13}} & \red{\textbf{7.95}} & \red{\textbf{11.30}} & \red{\textbf{4.22}} & \red{\textbf{8.24}} & \red{\textbf{12.90}} \\
& & & \textbf{$\pm$0.3} &\textbf{$\pm$0.7} &\textbf{ $\pm$0.8} & \textbf{$\pm$0.3} & \textbf{$\pm$0.1} & \textbf{$\pm$0.3} & \textbf{$\pm$0.1} & \textbf{$\pm$0.5} & $\pm$0.9 & \textbf{$\pm$0.2} & $\pm$0.04 & $\pm$0.5 \\
\midrule
FRCN \cite{faster} & \checkmark & \checkmark & 1.05 & 3.68 & 5.41 & 0.94 & 4.39 & 6.42 & 1.12 & 5.11 & 7.83 & 1.99 & 5.30 & 8.84 \\
TFA$_{\mathrm{fc}}$ \cite{wang2020frustratingly} & \checkmark & \checkmark & 1.06 & 2.71 & 4.38 & 1.17 & 4.02 & 7.05 & 1.97 & 5.48 & 11.09 & 2.40 & 6.86 & 12.86 \\
TFA$_{\mathrm{cos}}$ \cite{wang2020frustratingly} & \checkmark & \checkmark & 1.07 & 2.78 & 5.12 & 1.64 & 4.12 & 7.27 & 2.34 & 5.48 & 10.43 & 2.82 & 6.70 & 12.21 \\
FSDetView \cite{wang2020frustratingly} & \text{\ding{55}} & \checkmark & 0.70 & 2.70 & 3.70 & 0.60 & 4.00 & 4.20 & 1.80 & 5.10 & 8.00 & \green{3.00} & 9.70 & 12.30 \\
MPSR \cite{wu2020multi} & \checkmark & \checkmark & 1.23 & 3.82 & 5.58 & 1.89 & 5.69 & 8.73 & 0.86 & 4.60 & 9.96 & 1.62 & 6.78 & 11.66 \\
A-RPN \cite{fan2020few} & \text{\ding{55}} & \checkmark & \green{1.74} & \green{5.17} & \green{6.96} & \green{2.20} & \green{7.55} & \green{10.49} & \green{2.72} & \green{9.51} & \green{14.74} & 2.92 & \green{10.67} & \green{16.08} \\
\textbf{AirDet (Ours)} & \text{\ding{55}} & \checkmark & \red{\textbf{3.05}} & \red{\textbf{6.40}} & \red{\textbf{10.03}} & \red{\textbf{4.00}} & \red{\textbf{9.65}} & \red{\textbf{13.91}} & \red{\textbf{3.46}} & \red{\textbf{11.44}} & \red{\textbf{16.04}} & \red{\textbf{3.27}} & \red{\textbf{11.20}} & \red{\textbf{18.64}} \\
\bottomrul
\end{tabular}\label{tab:coco_scale}%
\end{threeparttable}
\end{table*}%
We next report the performance of methods \cite{wang2020frustratingly,xiao2020few,wu2020multi,fan2020few,faster} and AirDet on multi-scale objects in \tref{tab:coco_scale}.
Thanks to SCS, AirDet achieves the highest performance for multi-scale objects among all the SOTAs.
Especially for small objects, given 5-shots, AirDet can achieve a surprising \textbf{4.22} AP$_s$, nearly doubling the fine-tuned methods with multi-scale FPN features \cite{wang2020frustratingly,wu2020multi}.
\myparagraph{Comparison of 10-Shot}
\begin{table*}[!t]
\centering
\setlength{\tabcolsep}{0.1mm}
\fontsize{5}{6.5}\selectfont
\caption{Performance comparison with 10-shot on COCO validation dataset. \red{Red} and \green{green} fonts indicate best and second-best scores, respectively. AirDet achieves comparable results without fine-tuning and outperforms most methods with fine-tuning, which strongly demonstrates its effectiveness.}
\begin{threeparttable}
\begin{tabular}{ccccccccccccccc}
\toprule[1pt]
Method & Venue & Fine-tune & AP & AP$_{50}$ & AP$_{75}$ & AP$_s$ & AP$_m$ & AP$_l$ & AR$_{1}$ & AR$_{10}$ & AR$_{100}$ & AR$_s$ & AR$_m$ & AR$_l$ \\
\midrule
LSTD \cite{chen2018lstd} & AAAI 2018 & \checkmark & 3.2 & 8.1 & 2.1 & 0.9 & 2.0 & 6.5 & 7.8 & 10.4 & 10.4 & 1.1 & 5.6 & 19.6 \\
MetaDet \cite{wang2019meta} & ICCV 2019 & \checkmark & 7.1 & 14.6 & 6.1 & 1.0 & 4.1 & 12.2 & 11.9 & 15.1 & 15.5 & 1.7 & 9.7 & 30.1 \\
FSRW \cite{kang2019few} & ICCV 2019 & \checkmark & 5.6 & 12.3 & 4.6 & 0.9 & 3.5 & 10.5 & 10.1 & 14.3 & 14.4 & 1.5 & 8.4 & 28.2 \\
Meta RCNN \cite{yan2019meta}& ICCV 2019 & \checkmark & 8.7 & 19.1 & 6.6 & 2.3 & 7.7 & 14.0 & 12.6 & 17.8 & 17.9 & 7.8 & 15.6 & 27.2 \\
TFA$_{\mathrm{fc}}$ \cite{wang2020frustratingly} & ICML 2020 & \checkmark & 9.1 & 17.3 & 8.5 & - & - & - & - & - & - & - & - & - \\
TFA$_{\mathrm{cos}}$ \cite{wang2020frustratingly} & ICML 2020 & \checkmark & 9.1 & 17.1 & 8.8 & - & - & - & - & - & - & - & - & - \\
FSDetView \cite{xiao2020few}& ECCV 2020 & \checkmark & 12.5 & \red{27.3} & 9.8 & 2.5 & 13.8 & 19.9 & 20.0 & 25.5 & 25.7 & 7.5 & 27.6 & 38.9 \\
MPSR \cite{wu2020multi} & ECCV 2020 & \checkmark & 9.8 & 17.9 & 9.7 & 3.3 & 9.2 & 16.1 & 15.7 & 21.2 & 21.2 & 4.6 & 19.6 & 34.3 \\
A-RPN \cite{fan2020few} & CVPR 2020 & \checkmark & 11.1 & 20.4 & 10.6 & - & - & - & - & - & - & - & - & - \\
SRR-FSD \cite{zhu2021semantic}& CVPR 2021 & \checkmark & 11.3 & 23.0 & 9.8 & - & - & - & - & - & - & - & - & - \\
FSCE \cite{sun2021fsce} & CVPR 2021 & \checkmark & 11.9 & - & 10.5 & - & - & - & - & - & - & - & - & - \\
DCNet \cite{Hu2021CVPR} & CVPR 2021 & \checkmark & \green{12.8} & 23.4 & 11.2 & 4.3 & \green{13.8} & \green{21.0} & 18.1 & 26.7 & 25.6 & 7.9 & 24.5 & 36.7 \\
Y. Li \textit{et al.}~ \cite{li2021few} & CVPR 2021 & \checkmark & 11.3 & 20.3 & - & - & - & - & - & - & - & - & - & - \\
FADI \cite{cao2021nips} & NIPS 2021 & \checkmark & 12.2 & 22.7 & \green{11.9} & - & - & - & - & - & - & - & - & - \\
QA-FewDet \cite{han2021query} & ICCV 2021 & \checkmark & 11.6 & 23.9 & 9.8 & - & - & - & - & - & - & - & - & - \\
FSOD$^{up}$ \cite{wu2021universal} & ICCV 2021 & \checkmark & 11.0 & - & 10.7 & \red{4.5} & 11.2 & 17.3 & - & - & - & - & - & - \\
\midrule
\textbf{AirDet} & \textbf{Ours} & \text{\ding{55}} & 8.7 & 15.3 & 8.8 & 4.3 & 9.7 & 14.8 & \green{\textbf{19.1}} & \red{\textbf{33.8}} & \red{\textbf{34.8}} & \red{\textbf{13.0}} & \red{\textbf{37.4}} & \green{\textbf{52.9}} \\
\textbf{AirDet} & \textbf{Ours} & \checkmark & \red{\textbf{13.0}} & \green{\textbf{23.9}} & \red{\textbf{12.4}} & \red{\textbf{4.5}} & \red{\textbf{15.2}} & \red{\textbf{22.8}} & \red{\textbf{20.5}} & \green{\textbf{33.7}} & \green{\textbf{34.4}} & \green{\textbf{9.6}} & \green{\textbf{36.4}} & \red{\textbf{55.0}} \\
\bottomrule[1pt]
\end{tabular}\label{tab:10shot}%
\end{threeparttable}
\end{table*}%
For a more thorough comparison, we present the 10-shot evaluation on the COCO validation dataset in \tref{tab:10shot}. Without fine-tuning, AirDet can surprisingly achieve comparable performance against recent work \cite{wu2020multi,wang2020frustratingly,yan2019meta}, while all of them require a careful fine-tuning stage. Moreover, our fine-tuned model outperforms most prior methods \cite{li2021few,cao2021nips,han2021query,wu2021universal,Hu2021CVPR,sun2021fsce,zhu2021semantic,fan2020few,wu2020multi,xiao2020few,wang2020frustratingly,yan2019meta,kang2019few,wang2019meta,chen2018lstd} in most metrics, especially average recall rate (AR). Besides, the performance superiority on small objects (AP$_s$ and AR$_s$) further demonstrates the effectiveness of AirDet on multi-scale, especially the small-scale objects.
\myparagraph{Efficiency Comparison}
We report the fine-tuning and inference time of AirDet, and the SOTA methods \cite{fan2020few,wang2020frustratingly,xiao2020few,wu2020multi} in a setting of 3-shot one class in \tref{tab:time}, in which the official code and implementation with ResNet101 as the backbone are adopted.
Without fine-tuning, AirDet can make direct inferences on novel objects with a comparable speed, while the others methods \cite{fan2020few,xiao2020few,wu2020multi,wang2020frustratingly} require a fine-tuning time of about 3-30 minutes, which cannot meet the requirements of online exploration.
Note that the fine-tuning time is measured on TITAN X GPU, while such computational power is often unavailable on robots.
\noindent\textbf{Remark}~\refstepcounter{RNum}\textbf{\theRNum}: Many methods \cite{li2021few,cao2021nips,han2021query,wu2021universal,Hu2021CVPR,sun2021fsce,zhu2021semantic,fan2020few,wu2020multi,xiao2020few,wang2020frustratingly,yan2019meta,kang2019few,wang2019meta,chen2018lstd} also require an offline process to fine-tune hyper-parameters for different shots.
While such \textit{off-line} tuning is infeasible for robotic \textit{online} exploration.
Instead, AirDet can adopt \textbf{the same} base-trained model without fine-tuning for implementation.
\begin{table}[!t]
\centering
\setlength{\tabcolsep}{0.1mm}
\caption{Efficiency comparison with official source code. We adopt the pre-trained models provided by \cite{wang2020frustratingly}, so their fine-tuning time is unavailable.}
\fontsize{5}{7.5}\selectfont
\begin{tabular}{cccccccc}
\toprule
Method & \textbf{AirDet} & \multicolumn{1}{l}{A-RPN} \cite{fan2020few} & \multicolumn{1}{l}{FSDet} \cite{xiao2020few}& \multicolumn{1}{l}{MPSR} \cite{wu2020multi} & \multicolumn{1}{l}{TFA$_{\mathrm{fc}}$} \cite{wang2020frustratingly} & \multicolumn{1}{l}{TFA$_{\mathrm{cos}}$} \cite{wang2020frustratingly} & \multicolumn{1}{l}{FRCN$_{\mathrm{ft}}$} \cite{wang2020frustratingly} \\
\midrule
Fine-tuning (min) & \textbf{0} & 21 & 11 & 3 & - & - & - \\
Inference (s/img) & \textbf{0.081} & 0.076 & 0.202 & 0.109 & 0.085 & 0.094 & 0.091 \\
\bottomrule
\end{tabular}%
\label{tab:time}%
\end{table}%
\begin{table}[!t]
\centering
\setlength{\tabcolsep}{0.2mm}
\fontsize{5.5}{6.5}\selectfont
\caption{Cross-domain performance on VOC-2012 validation dataset. \red{Red} and \green{green} fonts denote the first and second place, respectively. AirDet has been demonstrated strong generalization capability, maintaining obvious superiority against others.}
\begin{threeparttable}
\begin{tabular}{cc|ccc|ccc|ccc|ccc}
\toprul
\multicolumn{2}{c|}{Shots} & \multicolumn{3}{c|}{1} & \multicolumn{3}{c|}{2} & \multicolumn{3}{c|}{3} & \multicolumn{3}{c}{5} \\
\midrule
Method & Fine-tune & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ & AP & AP$_{50}$ & AP$_{75}$ \\
\multirow{2}{*}{A-RPN \cite{fan2020few}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & 10.45 & 18.10 & 10.32 & 13.10 & 22.60 & 13.17 & 14.05 & 24.08 & 14.24 & 14.87 & 25.03 & 15.26 \\
& & $\pm$0.1 &$\pm$0.1 & $\pm$0.1 & $\pm$0.2 & $\pm$0.4 & $\pm$0.2 & $\pm$0.2 & $\pm$0.2 & $\pm$0.2 & $\pm$0.08 & $\pm$0.07 & $\pm$0.1 \\
\midrule
\multirow{2}{*}{\textbf{AirDet (Ours)}}$\dag$ & \multirow{2}{*}{\text{\ding{55}}} & \red{\textbf{11.92}} & \red{\textbf{21.33}} & \red{\textbf{11.56}} & \red{\textbf{15.80}} & \red{\textbf{26.80}} & \red{\textbf{16.08}} & \red{\textbf{16.89}} & \red{\textbf{28.61}} & \red{\textbf{17.36}} & \red{\textbf{17.83}} & \red{\textbf{29.78}} & \red{\textbf{
18.38}} \\
& & \textbf{$\pm$0.06} &\textbf{$\pm$0.08} &\textbf{ $\pm$0.08} & \textbf{$\pm$0.08} & \textbf{$\pm$0.3} & \textbf{$\pm$0.05} & \textbf{$\pm$0.1} & \textbf{$\pm$0.1} & \textbf{$\pm$0.1} & \textbf{$\pm$0.03} & \textbf{$\pm$0.03} & \textbf{$\pm$0.1} \\
\midrule
FRCN \cite{faster} & \checkmark & \multicolumn{1}{c}{4.49} & \multicolumn{1}{c}{9.44} & \multicolumn{1}{c|}{3.85} & \multicolumn{1}{c}{5.20} & \multicolumn{1}{c}{11.92} & \multicolumn{1}{c|}{3.84} & \multicolumn{1}{c}{6.50} & \multicolumn{1}{c}{14.39} & \multicolumn{1}{c|}{5.11} & \multicolumn{1}{c}{6.55} & \multicolumn{1}{c}{14.48} & \multicolumn{1}{c}{5.09} \\
TFA$_{\mathrm{cos}}$ \cite{wang2020frustratingly} & \checkmark & \multicolumn{1}{c}{4.66} & \multicolumn{1}{c}{7.97} & \multicolumn{1}{c|}{5.14} & \multicolumn{1}{c}{6.59} & \multicolumn{1}{c}{11.91} & \multicolumn{1}{c|}{6.49} & \multicolumn{1}{c}{8.78} & \multicolumn{1}{c}{17.09} & \multicolumn{1}{c|}{8.15} & \multicolumn{1}{c}{10.46} & \multicolumn{1}{c}{20.93} & \multicolumn{1}{c}{9.53} \\
TFA$_{\mathrm{fc}}$ \cite{wang2020frustratingly} & \checkmark & \multicolumn{1}{c}{4.40} & \multicolumn{1}{c}{8.60} & \multicolumn{1}{c|}{4.21} & \multicolumn{1}{c}{7.02} & \multicolumn{1}{c}{13.80} & \multicolumn{1}{c|}{6.21} & \multicolumn{1}{c}{9.24} & \multicolumn{1}{c}{18.48} & \multicolumn{1}{c|}{8.03} & \multicolumn{1}{c}{11.11} & \multicolumn{1}{c}{22.83} & \multicolumn{1}{c}{9.78} \\
FSDetView \cite{xiao2020few} & \checkmark & 4.80 & 14.10 & 1.40 & 3.70 & 11.60 & 0.60 & 6.60 & 22.00 & 1.20 & 10.80 & 26.50 & 5.50 \\
MPSR \cite{wu2020multi} & \checkmark & \multicolumn{1}{c}{6.01} & \multicolumn{1}{c}{11.23} & \multicolumn{1}{c|}{5.74} & \multicolumn{1}{c}{8.20} & \multicolumn{1}{c}{15.08} & \multicolumn{1}{c|}{8.22} & \multicolumn{1}{c}{10.08} & \multicolumn{1}{c}{18.29} & \multicolumn{1}{c|}{9.99} & \multicolumn{1}{c}{11.49} & \multicolumn{1}{c}{21.33} & \multicolumn{1}{c}{11.06} \\
A-RPN \cite{fan2020few} & \checkmark & \green{9.49} & \green{17.41} & \green{9.42} & \green{12.71} & \green{23.66} & \green{12.44} & \green{14.89} & \green{26.30} & \green{14.76} & \green{15.09} & \green{28.08} & \green{14.17} \\
\textbf{AirDet (Ours)} & \checkmark & \textbf{\red{13.33}} & \textbf{\red{24.64}} & \textbf{\red{12.68}} & \textbf{\red{17.51}} & \textbf{\red{30.35}} & \textbf{\red{17.61}} & \textbf{\red{17.68}} & \textbf{\red{32.05}} & \textbf{\red{17.34}} & \textbf{\red{18.27}} & \textbf{\red{33.02}} & \textbf{\red{17.69}} \\
\bottomrule[1.2pt]
\end{tabular}\label{tab:voc}%
\end{threeparttable}
\end{table}%
\subsection{Cross-domain Evaluation}\label{sec:cross}
Robots are often deployed to novel environments that have never been seen during training, thus cross-domain test is crucial for robotic applications.
In this section, we adopt the same model trained on COCO, while test on PASCAL VOC \cite{everingham2010pascal} and LVIS \cite{gupta2019lvis} to evaluate the model generalization capability.
\myparagraph{PASCAL VOC}
We report the overall performance on PASCAL VOC-2012 \cite{everingham2010pascal} for all methods in \tref{tab:voc}.
In the cross-domain setting, even without fine-tuning, AirDet achieves better performance than methods \cite{wu2020multi,fan2020few,faster,xiao2020few,wang2020frustratingly} that perform relatively well in in-domain test.
This means AirDet has a much stronger generalization capability than most fine-tuned prior methods.
\myparagraph{LVIS}
We randomly sample LVIS \cite{gupta2019lvis} to form 4 splits of classes, each of which contains 16 different classes.
To provide valid evaluation, the classes that have 20 to 200 images are taken for the test.
More details can be found in \appref{sec:lvis}.
The averaged performance with 5-shot without fine-tuning is presented in \tref{tab:lvis-cross}, where AirDet outperforms the baseline \cite{fan2020few} in every split under all metrics. Since the novel categories in the 4 LVIS splits are more (64 classes in total) and rarer (many of them are uncommon) than the VOC 20 classes, the superiority of AirDet in \tref{tab:lvis-cross} highly demonstrate its robustness under class variance.
\begin{table}[!t]
\setlength{\tabcolsep}{.6mm}
\centering
\fontsize{5.5}{6.5}\selectfont
\caption{Cross-domain performance of A-RPN \cite{fan2020few} and AirDet on LVIS dataset. We report the results for 5-shot without fine-tuning on 4 random splits.}
\begin{tabular}{c|cccc|cccc|cccc|cccc}
\toprul
\multicolumn{1}{c|}{Split} & \multicolumn{4}{c|}{1} & \multicolumn{4}{c|}{2} & \multicolumn{4}{c|}{3} & \multicolumn{4}{c}{4} \\
\midrule
Metrict & AP & AP$_{50}$ & AP$_{75}$ & AR$_{10}$ & AP & AP$_{50}$ & AP$_{75}$ & AR$_{10}$ & AP & AP$_{50}$ & AP$_{75}$ & AR$_{10}$ & AP & AP$_{50}$ & AP$_{75}$ & AR$_{10}$ \\
\textbf{AirDet} & \textbf{6.71} & \textbf{12.31} & \textbf{6.51} & \textbf{27.57} & \textbf{9.35} & \textbf{14.23} & \textbf{9.98} & \textbf{25.42} & \textbf{9.09} & \textbf{15.64} & \textbf{8.82} & \textbf{34.64} & \textbf{11.07} & \textbf{16.90} & \textbf{12.30} & \textbf{25.76} \\
A-RPN & 5.49 & 10.04 & 5.27 & 26.59 & 8.85 & 13.41 & 9.46 & 24.45 & 7.49 & 12.34 & 8.13 & 33.85 & 10.80 & 15.46 & 12.24 & 25.05 \\
\bottomrul
\end{tabular}%
\label{tab:lvis-cross}%
\end{table}%
\subsection{Ablation Study and Deep Visualization}\label{sec:abla}
In this section, we address the effectiveness of the proposed three modules via quantitative results and qualitative visualization using Grad-Cam \cite{gradcam}.
\myparagraph{Quantitative Evaluation}
We report the overall performance on 3-shot and 5-shot for the baseline \cite{fan2020few} and AirDet by enabling the three modules, respectively.
It can be seen in \tref{tab:ABLA} that AirDet outperforms the baseline in all cases. With the modules enabled one by one, the results get gradually higher, which strongly demonstrates the necessity and effectiveness of SCS, GLR, and PRE.
\begin{table}[!t]
\centering
\setlength{\tabcolsep}{0.2mm}
\fontsize{5.5}{6.5}\selectfont
\caption{Ablation study of the three modules, \textit{i.e.}, PRE, GLR, and SCS in AirDet. With each module enabled, the performance is improved step by step on our baseline. With the full modules, AirDet can amazingly achieve up to \textbf{35\%} higher results.}
\begin{tabular}{ccc|cccccc|cccccc}
\toprul
\multicolumn{3}{c|}{Module} & \multicolumn{5}{c}{3} & & \multicolumn{6}{c}{5} \\
\midrule
PRE & GLR & SCS & \multicolumn{1}{c}{AP} & $\Delta\%$ & \multicolumn{1}{c}{AP$_{50}$} & $\Delta\%$ & \multicolumn{1}{c}{AP$_{75}$} & $\Delta\%$ & \multicolumn{1}{c}{AP} & $\Delta\%$ & \multicolumn{1}{c}{AP$_{50}$} & $\Delta\%$ & \multicolumn{1}{c}{AP$_{75}$} & $\Delta\%$ \\
\multicolumn{3}{c|}{Baseline \cite{fan2020few} } & 4.80 & 0.00 & 9.24 & 0.00 & 4.49 & 0.00 & 5.73 & 0.00 & 10.68 & 0.00 & 5.53 & 0.00 \\
\checkmark & \multicolumn{1}{c}{} & \multicolumn{1}{c|}{} & 5.15 & +7.29 & 10.11 & +9.41 & 4.71 & +4.90 & 5.94 & +3.66 & 11.54 & +8.05 & 5.34 & -3.43 \\
\checkmark & \checkmark & \multicolumn{1}{c|}{} & 5.59 & +16.46 & 10.61 & +14.83 & 5.12 & +14.03 & 6.44 & +12.39 & 12.08 & +13.11 & 6.06 & +9.58 \\
\midrule
\checkmark & \checkmark & \checkmark & \textbf{6.50} & \textbf{+35.41} & \textbf{12.30} & \textbf{+33.12} & \textbf{6.11} & \textbf{+36.08} & \textbf{7.27} & \textbf{+26.78} & \textbf{13.63} & \textbf{+27.62} & \textbf{6.71} & \textbf{+21.34} \\
\bottomrul
\end{tabular}%
\label{tab:ABLA}%
\end{table}%
\begin{figure}[!t]
\centering
\includegraphics[width=1\columnwidth]{deep.pdf}
\caption{Deep visualization comparison between AirDet and baseline \cite{fan2020few}. In (a), By virtue of SCS, AirDet is capable of finding given support objects effectively. In (b), with similar proposals (\textcolor[rgb]{1,0,0}{red} boxes), AirDet can focus on the entire object (aeroplane) and notice the most representative parts (dog), resulting in more precise regression box and correct classification results. More examples are presented in \appref{sec:more_deep}.}
\label{fig:deep_rpn}
\end{figure}
\myparagraph{How effective is SCS?}
Given 2-shot per class, we first take the highest ranking proposal from RPN \cite{faster} to backpropagate the objectiveness score and resize the gradient map to the original image.
\fref{fig:deep_rpn} (a) exhibits the heat map from both AirDet and the baseline. We observe that AirDet generally concentrates on objects more precisely than the baseline.
Moreover, AirDet can focus better on objects belonging to the support class and is not distracted by other objects (2nd and 3rd row).
This means that AirDet can generate novel object proposals more effectively.
\myparagraph{How effective is GLR and detection head?}
In \fref{fig:deep_rpn} (b), we observe that with similar proposal boxes, AirDet head can better focus on the entire object, \textit{e.g.}, aeroplane is detected with a precise regression box, \textit{e.g.}, the dog is correctly classified with high score. This again demonstrates the effectiveness of our GLR and detection head.
\subsection{Real-World Test}\label{sec:real}
\begin{table*}[!t]
\centering
\setlength{\tabcolsep}{0.6mm}
\fontsize{5.5}{6.5}\selectfont
\caption{3-shot real-world exploration test of AirDet and baseline \cite{fan2020few}. AirDet can be directly applied without fine-tuning and performs considerably more robust than the baseline by virtue of the newly proposed SCS, GLR, and PRE modules.}
\begin{tabular}{ccc|cc|cc|cc|cc|cc}
\toprul
\multicolumn{13}{c}{Real-world Exploration Test} \\
\midrule
\multicolumn{1}{l}{Test/\#Frames} & \multicolumn{2}{c|}{1/\#248} & \multicolumn{2}{c|}{2/\#146} & \multicolumn{2}{c|}{3/\#127} & \multicolumn{2}{c|}{4/\#41} & \multicolumn{2}{c|}{5/\#248} & \multicolumn{2}{c}{6/\#46} \\
\midrule
Metric & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ \\
\textbf{AirDet (Ours)} & \textbf{17.10} & \textbf{54.10} & \textbf{17.90} & \textbf{47.40} & \textbf{24.00} & \textbf{57.50} & \textbf{26.94} & \textbf{48.20} & \textbf{11.28} & \textbf{38.17} & \textbf{20.40} & \textbf{70.63} \\
A-RPN \cite{fan2020few} & 13.56 & 40.40 & 14.30 & 38.80 & 20.20 & 47.20 & 22.41 & 40.14 & 6.75 & 24.10 & 14.70 & 59.38 \\
\midrule
Test/\#Frames & \multicolumn{2}{c|}{7/\#212} & \multicolumn{2}{c|}{8/\#259} & \multicolumn{2}{c|}{9/\#683} & \multicolumn{2}{c|}{10/\#827} & \multicolumn{2}{c|}{11/\#732} & \multicolumn{2}{c}{12/\#50} \\
\midrule
Metric & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ & AP & AP$_{50}$ \\
\textbf{AirDet (Ours)} & \textbf{5.90} & \textbf{16.00} & \textbf{15.26} & \textbf{43.31} & \textbf{7.63} &\textbf{27.88} & \textbf{13.55} & \textbf{23.92} & \textbf{15.74} & \textbf{34.43} & \textbf{21.45} & \textbf{45.83} \\
A-RPN \cite{fan2020few} & 2.39 & 7.60 & 11.27 & 25.24 & 6.16 & 23.40 & 8.10 & 14.85 & 11.54 & 27.28 & 18.20 & 33.98 \\
\bottomrul
\end{tabular}%
\label{tab:subt}%
\end{table*}%
Real-world tests are conducted for AirDet and our baseline \cite{fan2020few} with 12 sequences that were collected from the DARPA Subterranean (SubT) challenge \cite{subtchallenge}.
Due to the requirements of \textit{online} response during the mission, the models can only be evaluated \textbf{without fine-tuning}, which makes existing methods \cite{li2021few,cao2021nips,han2021query,wu2021universal,Hu2021CVPR,sun2021fsce,zhu2021semantic,fan2020few,wu2020multi,xiao2020few,wang2020frustratingly,yan2019meta,kang2019few,wang2019meta,chen2018lstd} impractical.
The environments of SubT challenge also poses extra difficulties, \textit{e.g.}, a lack of lighting, thick smoke, dripping water, and cluttered or irregularly shaped environments, \textit{etc.}~
To test the generalization capabilities, we adopt the same models of AirDet and the baseline as those evaluated in \sref{sec:indomain} and \sref{sec:cross}. The performance of 3-shot for each class is exhibited in \tref{tab:subt}, where AirDet is proved better.
The robot is equipped with an NVIDIA Jetson AGX Xavier, where our method runs at 1-2 FPS without TensorRT acceleration or other optimizations.
\begin{table}[t]
\centering
\setlength{\tabcolsep}{2mm}
\fontsize{6}{6}\selectfont
\caption{Per class results of the real-world tests. We report the instance number of each novel class along with the 3-shot AP results from AirDet and A-RPN \cite{fan2020few}. Compared with the baseline, AirDet achieves higher results for all classes.}
\begin{tabular}{cccccccc}
\toprule[1.2pt]
Class & Backpack & Helmet & Rope & Drill & Vent & Extinguisher & Survivor \\
\midrule
Instances & 626 & 674 & 723 & 587 & 498 & 1386 & 205 \\
AirDet & \textbf{32.3} & \textbf{9.7} & \textbf{13.9} &\textbf{ 10.8} & \textbf{16.2} & \textbf{10.5} & \textbf{10.7} \\
Baseline \cite{fan2020few} & 26.6 & 9.7 & 6 & 9 & 14.4 & 5.6 & 9.1 \\
\bottomrule[1.2pt]
\end{tabular}%
\label{tab:subt_cls}%
\end{table}%
In \tref{tab:subt_cls}, we present the number of instances and the performance on each novel class. To our excitement, AirDet shows smaller variance and higher precision cross different classes.
We also present the support images and representative detected objects in \fref{fig:subt}.
Note that AirDet can detect the novel objects accurately in the query images even if they have distinct scales and different illumination conditions with the supports. We regard this capability to the carefully designed SCS in AirDet.
More visualization are presented in \appref{sec:quali}.
The robustness and strong generalization capability of AirDet in the real-world tests demonstrated its promising prospect and feasibility for autonomous exploration.
\begin{figure}[t]
\centering
\includegraphics[width=1\columnwidth]{SUBT.pdf}
\caption{The provided support images and examples of detection results in the real-world tests. AirDet is robust to distinct object scales and different illumination conditions.}
\label{fig:subt}
\end{figure}
\section{Limitation and Future Work}
Despite the promising prospect and outstanding performance, AirDet still has several limitations.
(1) Since abundant base classes are needed to generalize, AirDet needs a relatively large base dataset to train before inference on novel classes.
(2) Second, AirDet relies on the quality of support images to work well without fine-tuning. This is because the provided few support images are the only information for the unseen classes.
(3) We observe that the failure cases of AirDet are mainly due to false classification, resulting in a high result variance among different classes in COCO and VOC.
(4) Since SCS and the detection head run in loops for multiple novel classes, the efficiency of AirDet will suffer from a large number of novel classes.
We provide quantitative results for limitation (1), (2), and (3) in \appref{sec:de_limi}.
\section{Conclusion}
This paper presents a brand new few-shot detector, AirDet, which consists of 3 newly proposed \textit{class-agnostic relation}-based modules and is free of fine-tuning.
Specifically, with proposed spatial relation and channel relation, we construct support guided cross-scale feature fusion for region proposals, global-local relation network for shots aggregation, and prototype relation embedding for precise localization. With the strong capability to extract \textit{class-agnostic relation}, AirDet can work comparably or even better than those exhaustively fine-tuned methods in both in-domain and cross-domain evaluation.
AirDet is also tested on real-world data with a robotic platform, where its feasibility for autonomous exploration is demonstrated.
\\
\par\noindent
\myparagraph{Acknowledgement}
This work was sponsored by ONR grant \#N0014-19-1-2266 and ARL DCIST CRA award W911NF-17-2-0181. The work was done when Bowen Li and Pranay Reddy were interns at The Robotics Institute, Carnegie Mellon University. The authors would like to thank all members of the Team Explorer for providing data collected from the DARPA Subterranean Challenge.
\bibliographystyle{splncs04}
| 2024-02-18T23:39:39.986Z | 2022-07-26T02:09:22.000Z | algebraic_stack_train_0000 | 7 | 8,522 |
|
proofpile-arXiv_065-110 | \section{Introduction}
The numerical analysis of elastic shells is a
vast field with important applications in physics and engineering. In most
cases, it is carried out via the finite element method. In the physics and computer
graphics literature, there have been suggestions to use simpler methods based
on discrete differential geometry \cite{meyer2003discrete,bobenko2008discrete}. Discrete differential geometry of surfaces is the study of triangulated polyhedral surfaces. (The epithet ``simpler'' has to be understood as ``easier to implement''.) We mention in passing that models based on triangulated polyhedral surfaces have applications in materials science beyond the elasticity of thin shells. E.g., recently these models have been used to describe defects in nematic liquids on thin shells \cite{canevari2018defects}. This amounts to a generalization to arbitrary surfaces of the discrete-to-continuum analysis for the XY model in two dimensions that leads to Ginzburg-Landau type models in the continuum limit \cite{MR2505362,alicandro2014metastability}.
\medskip
Let us describe some of the methods mentioned above in more detail.
Firstly, there are the so-called
\emph{polyhedral membrane models} which in fact can be used for a whole
array of physical and engineering problems (see e.g.~the review
\cite{davini1998relaxed}). In the context of plates and shells, the
so-called Seung-Nelson model \cite{PhysRevA.38.1005} is widely used.
This associates membrane and bending energy to a piecewise affine map $y:\R^2\supset U\to \R^3$, where the pieces are determined by a triangulation $\mathcal T$ of the polyhedral domain $U$. The bending energy is given by
\begin{equation}
E^{\mathrm{SN}}(y)= \sum_{K,L} |n(K)-n(L)|^2\,,\label{eq:1}
\end{equation}
where the sum runs over those unordered pairs of triangles $K,L$ in $\mathcal T$ that share an edge, and $n(K)$ is the surface normal on the triangle $K$. In \cite{PhysRevA.38.1005}, it has been argued that for a fixed
limit deformation $y$, the energy \eqref{eq:1} should approximate the Willmore energy
\begin{equation}
E^{\mathrm{W}}(y)=\int_{y(U)} |Dn|^2\; \mathrm{d}{\mathscr H}^2\label{eq:2}
\end{equation}
when the grid size of the triangulation $\mathcal T$ is sent to 0, and the argument of the discrete energy \eqref{eq:1} approximates the (smooth) map $y$. In \eqref{eq:2} above, $n$ denotes the surface normal and $\H^2$ the two-dimensional Hausdorff measure.
These statements have been made more precise in \cite{schmidt2012universal}, where it has been shown that the result of the limiting process depends on the used triangulations. In particular, the following has been shown in this reference: For $j\in\N$, let $\mathcal T_j$ be a triangulation of $U$ consisting of equilateral triangles such that one of the sides of each triangle is parallel to the $x_1$-direction, and such that the triangle size tends 0 as $j\to\infty$. Then the limit energy reads
\[
\begin{split}
E^{\mathrm{FS}}(y)=\frac{2}{\sqrt{3}}\int_U &\big(g_{11}(h_{11}^2+2h_{12}^2-2h_{11}h_{22}+3h_{22}^2)\\
&-8g_{12}h_{11}h_{12}+2 g_{22}(h_{11}^2+3h_{12}^2)\big)(\det g_{ij})^{-1}\; \mathrm{d} x\,,
\end{split}
\]
where
\[
\begin{split}
g_{ij}&=\partial_i y\cdot\partial_j y\\
h_{ij}&=n\cdot \partial_{ij} y \,.
\end{split}
\]
More precisely, if $y\in C^2(U)$ is given, then the sequence of maps $y_j$ obtained by piecewise affine interpolation of the values of $y$ on the vertices of the triangulations $\mathcal T_j$ satisfies
\[
\lim_{j\to \infty}E^{\mathrm{SN}}(y_j)=E^{\mathrm{FS}}(y)\,.
\]
Secondly, there is the more recent approach to using discrete differential
geometry for shells pioneered by Grinspun et al.~\cite{grinspun2003discrete}.
Their energy does not depend on an immersion $y$ as above, but is defined directly on triangulated surfaces. Given such a surface $\mathcal T$, the energy is given by
\begin{equation}
E^{\mathrm{GHDS}}(\mathcal T)=\sum_{K,L} \frac{l_{KL}}{d_{KL}} \alpha_{KL}^2\label{eq:3}
\end{equation}
where the sum runs over unordered pairs of neighboring triangles $K,L\in\mathcal T$, $l_{KL}$ is the length of the interface between $K,L$, $d_{KL}$ is the distance between the centers of the circumcircles of $K,L$, and $\alpha_{KL}$ is the difference of the angle between $K,L$ and $\pi$, or alternatively the angle between the like-oriented normals $n(K)$ and $n(L)$, i.e. the \emph{dihedral angle}.
In \cite{bobenko2005conformal}, Bobenko has defined an energy for piecewise affine surfaces $\mathcal T$ that is invariant under conformal transformations. It is defined via the circumcircles of triangles in $\mathcal T$, and the external intersection angles of circumcircles of neighboring triangles. Denoting this intersection angle for neighboring triangles $K,L$ by $\beta_{KL}$, the energy reads
\begin{equation}\label{eq:4}
E^\mathrm{B} (\mathcal T) = \sum_{K,L}\beta_{KL}-\pi\, \#\,\text{Vertices}(\mathcal T)\,.
\end{equation}
Here $\text{Vertices}(\mathcal T)$ denotes the vertices of the triangulation $\mathcal T$, the sum is again over nearest neighbors.
It has been shown in \cite{bobenko2008surfaces} that this energy is the same as
\eqref{eq:3} up to terms that vanish as the size of triangles is sent to zero
(assuming sufficient smoothness of the limiting surface). The reference
\cite{bobenko2008surfaces} also contains an analysis of the energy for this
limit. If the limit surface is smooth, and it is approximated by triangulated
surfaces $\mathcal T_\varepsilon$ with maximal triangle size $\varepsilon$ that satisfy
a number of technical assumptions, then the Willmore energy of the limit surface
is smaller than or equal to the limit of the energies \eqref{eq:3} for the approximating surfaces, see Theorem 2.12 in \cite{bobenko2008surfaces}. The technical assumptions are
\begin{itemize}
\item each vertex in the triangulation $\mathcal T_\varepsilon$ is connected to six other vertices by edges,
\item the lengths of the sides of the hexagon formed by six triangles that share one vertex differ by at most $O(\varepsilon^4)$,
\item neighboring triangles are congruent up to $O(\varepsilon^3)$.
\end{itemize}
Furthermore, it is stated that the limit is achieved if additionally the triangulation approximates a ``curvature line net''.
\medskip
The purpose of this present paper is to generalize this convergence result, and put it into the framework of $\Gamma$-convergence \cite{MR1968440,MR1201152}. Instead of fixing the vertices of the polyhedral surfaces to lie on the limiting surfaces, we are going to assume that the convergence is weakly * in $W^{1,\infty}$ as graphs. This approach allows to completely drop the assumptions on the connectivity of vertices in the triangulations, and the assumptions of congruence -- we only need to require a certain type of regularity of the triangulations that prevents the formation of small angles. %
\medskip
We are going to work with the energy
\begin{equation}\label{eq:5}
E(\mathcal T)=\sum_{K,L} \frac{l_{KL}}{d_{KL}} |n(K)-n(L)|^2\,,
\end{equation}
which in a certain sense is equivalent to \eqref{eq:3} and \eqref{eq:4} in the limit of vanishing triangle size, see the arguments from \cite{bobenko2008surfaces} and Remark \ref{rem:main} (ii) below.
\medskip
To put this approach into its context in the mathematical literature, we point out that it is another instance of a discrete-to-continuum limit, which has been a popular topic in mathematical analysis over the last few decades. We mention the seminal papers \cite{MR1933632,alicandro2004general} and the fact that a variety of physical settings have been approached in this vein, such as spin and lattice systems \cite{MR1900933,MR2505362}, bulk elasticity \cite{MR2796134,MR3180690}, thin films \cite{MR2429532,MR2434899}, magnetism \cite{MR2186037,MR2505364}, and many more.
\medskip
The topology that we are going to use in our $\Gamma$-convergence statement is much coarser than the one that corresponds to Bobenko's convergence result; however it is not the ``natural'' one that would yield compactness from finiteness of the energy \eqref{eq:5} alone. For a discussion of why we do not choose the latter see Remark \ref{rem:main} (i) below. Our topology is instead defined as follows:
Let $M$ be some fixed compact oriented two-dimensional $C^\infty$ submanifold of
$\R^3$ with normal $n_M:M\to S^2$. Let $h_j\in W^{1,\infty}(M)$, $j=1,2,\dots$,
such that $\|h_j\|_{W^{1,\infty}}<C$ and $\|h_j\|_{\infty}<\delta(M)/2$ (where $\delta(M)$ is the \emph{radius of injectivity} of $M$, see Definition \ref{def:radius_injectivity} below) such that $
T_j:= \{x+h_j(x)n_M(x):x\in M\}$ are triangulated surfaces (see Definition \ref{def:triangular_surface} below). We say
$\mathcal T_j\to \mathcal S:=\{x+h(x)n_M(x):x\in M\}$ if $h_j\to h$ in
$W^{1,p}(M)$ for all $1\leq p<\infty$. Our main theorem, Theorem \ref{thm:main}
below, is a $\Gamma$-convergence result in this topology.
The regularity assumptions that we impose on the triangulated surfaces under
considerations are ``$\zeta$-regularity'' and the ``Delaunay property''. The
definition of these concepts can be found in Definition
\ref{def:triangular_surface} below.
\begin{thm}
\label{thm:main}
\begin{itemize}
\item[(o)] Compactness: Let $\zeta>0$, and let $h_j$ be a bounded sequence in
$W^{1,\infty}(M)$ such that $\mathcal T_j=\{x+h_j(x)n_M(x):x\in M\}$ is a
$\zeta$-regular triangulated surface and $\|h_j\|_\infty\leq\delta(M)/2$ for $j\in \N$ with $\limsup_{j\to\infty}E(\mathcal T_j)<\infty$. Then there exists a subsequence $h_{j_k}$ and $h\in W^{2,2}(M)$ such that $h_{j_k}\to h $ in $W^{1,p}(M)$ for every $1\leq p < \infty$.
\item[(i)] Lower bound: Let $\zeta>0$. Assume that for $j\in\N$, $h_j\in W^{1,\infty}(M)$ with $\|h_j\|\leq \delta(M)/2$, $\mathcal T_j:=\{x+h_j(x)n_M(x):x\in M\}$ is a
$\zeta$-regular triangulated surface fulfilling the Delaunay
property, and that $\mathcal T_j\to S=\{x+h(x)n_M(x):x\in M\}$ for $j\to\infty$. Then
\[
\liminf_{j\to\infty} E(\mathcal T_j)\geq \int_{S} |Dn_S|^2\; \mathrm{d}\H^2\,.
\]
\item[(ii)] Upper bound: Let $h\in W^{1,\infty}(M)$ with $\|h\|_\infty\leq \delta(M)/2$ and
$S=\{x+h(x)n_M(x):x\in M\}$. Then there exists $\zeta>0$ and a sequence $(h_j)_{j\in\N}\subset W^{1,\infty}(M)$ such that $\mathcal T_j:=\{(x+h_j(x)n_M(x):x\in M\}$ is a
$\zeta$-regular triangulated surface satisfying the
Delaunay property for each $j\in \N$, and we have $\mathcal T_j\to S$ for $j\to \infty$ and
\[
\lim_{j\to\infty} E(\mathcal T_j)= \int_{S} |Dn_S|^2\; \mathrm{d}\H^2\,.
\]
\end{itemize}
\end{thm}
\begin{rem}\label{rem:main}
\begin{itemize}
\item[(i)] We are not able to derive a convergence result in a topology that
yields convergence from boundedness of the energy \eqref{eq:5} alone. Such an
approach would necessitate the interpretation of the surfaces as varifolds or
currents. To the best of our knowledge, the theory of
integral functionals on varifolds (see e.g.~\cite{menne2014weakly,hutchinson1986second,MR1412686}) is not
developed to the point to allow for a treatment of this question. In particular, there does not exist a sufficiently
general theory of lower semicontinuity of integral functionals for varifold-function pairs.
\item[(ii)] We can state
analogous results based on the energy functionals \eqref{eq:3},
\eqref{eq:4}. To do so, our proofs only need to be modified slightly: As soon as we have reduced the situation to the graph case
(which we do by assumption), the upper bound construction can be carried out
as here; the smallness of the involved dihedral angles assures that the
arguments from \cite{bobenko2005conformal} suffice to carry through the proof.
Concerning the lower bound, we also reduce to the case of small dihedral angles by a blow-up procedure around Lebesgue points of the derivative of the surface normal of the limit surface. (Additionally, one can show smallness of the contribution of a few pairs of triangles whose dihedral angle is not small.) Again, the considerations from \cite{bobenko2005conformal} allow for a translation of our proof to the case of the energy functionals \eqref{eq:3},
\eqref{eq:4}.
\item[(iii)] As we will show in Chapter \ref{sec:necess-dela-prop}, we need to require the Delaunay property in order to obtain the lower bound statement. Without this requirement, we will show that a hollow cylinder can be approximated by triangulated surfaces with arbitrarily low energy, see Proposition~\ref{prop: optimal grid}.
\item[(iv)] Much more general approximations of surfaces by discrete geometrical objects have recently been proposed in \cite{buet2017varifold,buet2018discretization,buet2019weak}, based on tools from the theory of varifolds.
\end{itemize}
\end{rem}
\subsection*{Plan of the paper}
In Section \ref{sec:defin-prel}, we will fix definitions and make some preliminary observations on triangulated surfaces. The proofs of the compactness and lower bound part will be developed in parallel in Section \ref{sec:proof-comp-lower}. The upper bound construction is carried out in Section \ref{sec:surf-triang-upper}, and in Section \ref{sec:necess-dela-prop} we demonstrate that the requirement of the Delaunay property is necessary in order to obtain the lower bound statement.
\section{Definitions and preliminaries}
\label{sec:defin-prel}
\subsection{Some general notation}
\begin{notation}
For a two-dimensional submanifold $M\subset\R^3$, the tangent space of $M$ in $x\in M$ is
denoted by $T_{x}M$. For functions $f:M\to\R$, we denote their gradient by $\nabla f\in T_xM$; the norm $|\cdot|$ on $T_xM\subset\R^3$ is the Euclidean norm inherited from $\R^3$. For $1\leq p\leq \infty$, we denote by $W^{1,p}(M)$ the space of functions $f\in L^p(M)$ such that $\nabla f\in L^p(M;\R^3)$, with norm
\[
\|h\|_{W^{1,p}(M)}=\|f\|_{L^p(M)}+\|\nabla f\|_{L^p(M)}\,.
\]
For $U\subset\R^n$ and a function
$f:U\to\R$, we denote the graph of $f$ by
\[
\mathrm{Gr}\, f=\{(x,f(x)):x\in U\}\subset\R^{n+1}\,.
\]
For $x_1,\dots,x_m\subset \R^k$, the convex hull of $\{x_1,\dots,x_m\}$ is
denoted by
\[
[x_1,\dots,x_m]=\left\{\sum_{i=1}^m \lambda_ix_i:\lambda_i\in [0,1] \text{
for } i=1,\dots,m, \, \sum_{i=1}^m\lambda_i=1\right\}\,.
\]
We will identify $\R^2$ with the subspace $\R^2\times\{0\}$ of $\R^3$. The $d-$dimensional Hausdorff measure is denoted by $\H^d$, the $k-$dimensional Lebesgue measure by $\L^k$. The
symbol ``$C$'' will be used as follows: A statement such as
``$f\leq C(\alpha)g$'' is shorthand for ``there exists a constant $C>0$ that
only depends on $\alpha$ such that $f\leq Cg$''. The value of $C$ may change
within the same line. For $f\leq C g$, we also write
$f\lesssim g$.
\end{notation}
\subsection{Triangulated surfaces: Definitions}
\begin{defi}
\label{def:triangular_surface}
\begin{itemize}
\item [(i)] A \textbf{triangle} is the convex hull $[x,y,z]\subset \R^3$ of three points $x,y,z \in \R^3$. A \textbf{regular} triangle is one where $x,y,z$ are not colinear, or equivalently ${\mathscr H}^2([x,y,z])>0$.
\item[(ii)]
A \textbf{triangulated surface} is a finite collection
${\mathcal T} = \{K_i\,:\,i = 1,\ldots, N\}$ of regular triangles
$K_i = [x_i,y_i,z_i] \subset \R^3$ so that
$\bigcup_{i=1}^N K_i \subset \R^3$ is a topological two-dimensional manifold
with boundary; and the intersection of two different triangles $K,L\in {\mathcal T}$ is either empty, a common vertex, or a common edge.
We identify ${\mathcal T}$ with its induced topological manifold
$\bigcup_{i=1}^N K_i \subset \R^3$ whenever convenient. We say that ${\mathcal T}$ is \textbf{flat} if
there exists an affine subplane of $\R^3$ that contains ${\mathcal T}$.
\item[(iii)] The \textbf{size} of the triangulated surface, denoted $\size({\mathcal T})$, is the
maximum diameter of all its triangles.
\item[(iv)] The triangulated surface ${\mathcal T}$ is called $\zeta$\textbf{-regular}, with
$\zeta > 0$, if the minimum angle in all triangles is at least $\zeta$ and
$\min_{K\in {\mathcal T}} \diam(K) \geq \zeta \size({\mathcal T})$.
\item[(v)] The triangulated surface satisfies the \textbf{Delaunay}
property if for every triangle
$K = [x,y,z] \in {\mathcal T}$ the following property holds: Let $B(q,r)\subset \R^3$ be the
smallest ball such that $\{x,y,z\}\subset \partial{B(q,r)}$. Then $B(q,r)$ contains
no vertex of any triangle in ${\mathcal T}$. The point $q = q(K)\in \R^3$ is called
the \textbf{circumcenter} of $K$, $\overline{B(q,r)}$ its
\textbf{circumball} with circumradius $r(K)$, and $\partial B(q,r)$ its \textbf{circumsphere}.
\end{itemize}
\end{defi}
Note that triangulated surfaces have normals defined on all triangles and are compact
and rectifiable. For the argument of the circumcenter map
$q$, we do not distinguish between triples of points $(a,b,c)\in \R^{3\times
3}$ and the triangle $[a,b,c]$ (presuming $[a,b,c]$ is a regular triangle).
\begin{notation}
If ${\mathcal T}=\{K_i:i=1,\dots,N\}$ is a triangulated surface, and
$g:{\mathcal T}\to \R$,
then we identify $g$ with the function
$\cup_{i=1}^N K_i\to \R$ that is
constant on the (relative) interior of each triangle $K$, and equal to
$0$ on $K\cap L$ for $K\neq L\in {\mathcal T}$. In particular we may write in this case
$g(x)=g(K)$ for $x\in \mathrm{int}\, K$.
\end{notation}
\begin{defi}
Let ${\mathcal T}$ be a triangulated surface and $K,L \in {\mathcal T}$. We set
\[
\begin{split}
\l{K}{L} &:= \H^1(K\cap L)\\
d_{KL} &:= |q(K) - q(L)|
\end{split}
\]
If $K,L$ are \textbf{adjacent}, i.e. if $\l{K}{L} > 0$, we may define $|n(K) - n(L)|\in \R$ as the norm of the difference of the normals $n(K),n(L)\in S^2$ which share an orientation, i.e. $2\sin \frac{\alpha_{KL}}{2}$, where $\alpha_{KL}$ is the dihedral angle between the triangles, see Figure \ref{fig:dihedral}. The discrete bending energy is then defined as
\[
E({\mathcal T}) = \sum_{K,L\in {\mathcal T}} \frac{\l{K}{L}}{d_{KL}} |n(K) - n(L)|^2.
\]
Here, the sum runs over all unordered pairs of triangles. If $|n(K) - n(L)| = 0$ or $\l{K}{L} = 0$, the energy density is defined to be $0$ even if $d_{KL}=0$. If $|n(K) - n(L)| > 0$, $\l{K}{L} > 0$ and $d_{KL} = 0$, the energy is defined to be infinite.
\end{defi}
\begin{figure}[h]
\begin{subfigure}{.45\textwidth}
\begin{center}
\includegraphics[height=5cm]{dihedral_triangles_v2.pdf}
\end{center}
\caption{ \label{fig:dihedral}}
\end{subfigure}
\hspace{5mm}
\begin{subfigure}{.45\textwidth}
\includegraphics[height=5cm]{d_KL_l_KL.pdf}
\caption{\label{fig:dkllkl}}
\end{subfigure}
\caption{($\mathrm{A}$) The dihedral angle $\alpha_{KL}$ for triangles $K,L$. It is related to the norm of the difference between the normals via $|n(K)-n(L)|=2\sin\frac{\alpha_{KL}}{2}$. ($\mathrm{B}$) Definitions of $d_{KL}$, $l_{KL}$.}
\end{figure}
\begin{notation}
\label{not:thetaKL}
Let $H$ be an affine subplane of $\R^3$.
For triangles $K,L\subset H$ that share an edge and $v\in\R^3$ parallel to
$H$, we define the function
$\mathds{1}^v_{KL}:H \to \{0,1\}$ as $\mathds{1}_{KL}^v(x) = 1$ if and
only if
$[x,x+v]\cap (K\cap L) \neq \emptyset$. If the intersection $K\cap L$ does
not consist of a single edge, then $\mathds{1}_{KL}^v\equiv
0$. Furthermore, we let $\nu_{KL}\in \R^3$ denote the unit vector parallel to $H$
orthogonal to the shared edge of $K,L$ pointing from $K$ to $L$ and
\[
\theta_{KL}^v=\frac{|\nu_{KL}\cdot v|}{|v|}\,.
\]
\end{notation}
See Figure \ref{fig:parallelogram} for an illustration of Notation \ref{not:thetaKL}.
\begin{figure}
\includegraphics[height=5cm]{char_fun_1.pdf}
\caption{Definition of $\theta_{KL}^v$: The parallelogram spanned by $v$ and the shared side $K\cap L$ has area $\theta^v_{KL}l_{KL}|v|$. This parallelogram translated by $-v$ is the support of $\mathds{1}_{KL}^v$. \label{fig:parallelogram}}
\end{figure}
\medskip
We collect the notation that we have introduced for triangles and triangulated
surfaces for the reader's convenience in abbreviated form: Assume that $K=[a,b,c]$ and $L=[b,c,d]$
are two regular triangles in $\R^3$. Then we have the following notation:
\begin{equation*}
\boxed{
\begin{split}
q(K)&: \text{ center of the smallest circumball for $K$}\\
r(K)& :\text{ radius of the smallest circumball for $K$}\\
d_{KL}&=|q(K)-q(L)|\\
l_{KL}&:\text{ length of the shared edge of $K,L$}\\
n(K)&: \text{ unit vector
normal to $K$ }
\end{split}
}
\end{equation*}
The following are defined if $K,L$
are contained in an affine subspace $H$ of $\R^3$, and $v$ is a vector
parallel to $H$:
\begin{equation*}
\boxed{
\begin{split}
\nu_{KL}&:\text{ unit vector parallel to $H$
orthogonal to}\\&\quad\text{ the shared edge of $K,L$ pointing from $K$ to $L$}\\
\theta_{KL}^v&=\frac{|\nu_{KL}\cdot v|}{|v|}\\
\mathds{1}_{KL}^v&: \text{ function defined on $H$, with value one if}\\
&\quad\text{ $[x,x+v]\cap (K\cap L)\neq \emptyset$, zero otherwise}
\end{split}
}
\end{equation*}
\subsection{Triangulated surfaces: Some preliminary observations}
For two adjacent triangles $K,L\in {\mathcal T}$, we have $d_{KL} = 0$ if and only if the vertices of $K$ and $L$ have the same circumsphere. The following lemma states that for noncospherical configurations, $d_{KL}$ grows linearly with the distance between the circumsphere of $K$ and the opposite vertex in $L$.
\begin{lma}\label{lma: circumcenter regularity}
The circumcenter map $q:\R^{3\times 3} \to \R^3$ is $C^1$ and Lipschitz when
restricted to $\zeta$-regular triangles. For two adjacent triangles $K =
[x,y,z]$, $L = [x,y,p]$, we have that
\[d_{KL} \geq \frac12 \big| |q(K)-p| -r(K) \big|\,.
\]
\end{lma}
\begin{proof}
The circumcenter $q = q(K)\in \R^3$ of the triangle $K = [x,y,z]$ is the solution to the linear system
\begin{equation}
\begin{cases}
(q - x)\cdot (y-x) = \frac12 |y-x|^2\\
(q - x)\cdot (z-x) = \frac12 |z-x|^2\\
(q - x)\cdot ((z-x)\times (y-x)) = 0.
\end{cases}
\end{equation}
Thus, the circumcenter map $(x,y,z)\mapsto q$ is $C^1$ when restricted to $\zeta$-regular $K$. To see that the map is globally Lipschitz, it suffices to note that it is $1$-homogeneous in $(x,y,z)$.
For the second point, let $s=q(L)\in \R^3$ be the circumcenter of $L$. Then by the triangle inequality, we have
\begin{equation}
\begin{aligned}
|p-q|\leq |p-s| + |s-q| = |x-s| + |s-q| \leq |x-q| + 2|s-q| = r + 2d_{KL},\\
|p-q| \geq |p-s| - |s-q| = |x-s| - |s-q| \geq |x-q| - 2 |s-q| = r - 2d_{KL}.
\end{aligned}
\end{equation}
This completes the proof.
\end{proof}
\begin{lma}
\label{lem:char_func}
Let $\zeta>0$, and $a,b,c,d\in \R^2$ such that $K=[a,b,c]$ and $L=[b,c,d]$ are $\zeta$-regular.
\begin{itemize}\item[(i)]
We have that
\begin{equation*}
\int_{\R^2} \mathds{1}_{KL}^v(x)\d x = |v|l_{KL}\theta_{KL}\,.
\end{equation*}
\item[(ii)] Let $\delta>0$, $v,w\in\R^2$, $\bar v=(v,v\cdot w)\in \R^3$,
$\bar a=(a,a\cdot w)\in \R^3$ and $\bar b, \bar c,\bar d\in \R^3$ defined
analogously. Let
$\bar K=[\bar a,\bar b,\bar c]$, $\bar L=[\bar b,\bar c,\bar d]$.
Then
\[
\int_{\R^2} \mathds{1}_{KL}^v(x)\, \d x = \frac{|\bar v|}{\sqrt{1+|w|^2}}
\theta_{\bar K\bar L}^{\bar v}l_{\bar K\bar L}\,.
\]
\end{itemize}
\end{lma}
\begin{proof}
The equation (i) follows from the fact that $\mathds{1}_{KL}^v$ is the
characteristic function of a parallelogram, see Figure \ref{fig:parallelogram}.
To prove (ii) it suffices to observe that
$\int_{\R^2} \mathds{1}_{KL}^v(x)\sqrt{1+w^2}\d x$ is the volume of the
parallelogram from (i) pushed forward by the map $\tilde h(x)= (x,x\cdot
w)$, see Figure \ref{fig:char_fun_2}.
\end{proof}
\begin{figure}[h]
\includegraphics[height=5cm]{char_fun_2.pdf}
\caption{The parallelogram pushed forward by an affine map $x\mapsto (x,x\cdot w)$. \label{fig:char_fun_2}}
\end{figure}
\subsection{Graphs over manifolds}
\begin{assump}
\label{ass:Mprop}
We assume $M\subset\R^3$
is
an oriented compact two-dimensional $C^\infty$-submanifold of $\R^3$.
\end{assump}
This manifold will be fixed in the following. We denote the normal of $M$ by $n_M:M\to S^2$, and the second fundamental form at $x_0\in M$ is denoted by $S_M(x_0):T_{x_0}M\to T_{x_0}M$.
\medskip
\begin{defi}
\label{def:radius_injectivity}
The \emph{radius of injectivity} $\delta(M)>0$ of $M$ is the largest number such that the map $\phi:M\times (-\delta(M),\delta(M))\to \R^3$, $(x,h) \mapsto x + h n_M(x)$ is injective and the operator norm of $\delta(M)S_M(x)\in\mathcal{L}(T_xM)$ is at most $1$ at every $x\in M$.
\end{defi}
We define a graph over $M$ as follows:
\begin{defi}
\label{def:Mgraph}
\begin{itemize}
\item[(i)] A set $M_h = \{x+ h(x)n_M(x)\,:\,x\in M\}$ is called a \emph{graph} over $M$ whenever $h:M\to \R$ is a continuous function with $\|h\|_\infty \leq \delta(M)/2$.
\item[(ii)] The graph $M_h$ is called a ($Z$-)Lipschitz graph (for $Z > 0$) whenever $h$ is ($Z$-)Lipschitz, and a smooth graph whenever $h$ is smooth.
\item[(iii)] A set $N\subset B(M,\delta(M)/2)$ is said to be locally a tangent Lipschitz
graph over $M$ if for every $x_0\in M$ there exists $r>0$ and a Lipschitz
function $h:(x_0 +T_{x_0}M)\cap B(x_0,r)\to \R$ such that the intersection of $N$
with the cylinder $C(x_0,r,\frac{\delta(M)}{2})$ over $(x_0 +T_{x_0}M)\cap B(x_0,r)$ with height $\delta(M)$ in both
directions of $n_M(x_0)$, where
\[
C(x_0,r,s) := \left\{x + tn_M(x_0)\,:\,x\in (x_0 + T_{x_0}M) \cap B(x_0,r), t\in [-s,s] \right\},
\]
is equal to the graph of $h$ over $T_{x_0}M\cap B(x_0,r)$,
\[
N \cap C\left(x_0,r,\frac{\delta(M)}{2}\right) =\{x+h(x)n_M(x_0):x\in (x_0+T_{x_0}M)\cap B(x_0,r)\}\,.
\]
\end{itemize}
\end{defi}
\begin{lma}\label{lma: graph property}
Let $N\subset B(M,\delta(M)/2)$ be locally a tangent Lipschitz graph over $M$.Then $N$ is a Lipschitz graph over $M$.
\end{lma}
\begin{proof}
By Definition \ref{def:Mgraph} (iii), we have that for every $x\in M$, there
exists exactly one element
\[
x'\in
N\cap \left( x+n_M(x_0)[-\delta(M),\delta(M)]\right)\,.
\]
We write $h(x):=(x'-x)\cdot n_M(x)$, which obviously
implies $N=M_h$. For every $x_0\in
M$ there exists a neighborhood of $x_0$ such that $h$ is Lipschitz continuous in
this neighborhood by
the locally tangent Lipschitz property and the regularity of $M$. The global
Lipschitz property for $h$ follows from the local one by a standard covering argument.
\end{proof}
\begin{lma}
\label{lem:graph_rep}
Let $h_j\in W^{1,\infty}(M)$ with $\|h_j\|_{\infty}\leq\delta(M)/2$ and $h_j\weakstar h\in W^{1,\infty}(M)$ for $j\to \infty$. Then
for every point $x\in M$, there exists a neighborhood $V\subset x+T_xM$,
a Euclidean motion $R$ with $U:=R(x+T_xM)\subset \R^2$, functions $\tilde
h_j:U\to\R$ and $\tilde h:U\to \R$ such that $\tilde h_j\weakstar \tilde h$
in $W^{1,\infty}(U)$ and
\[
\begin{split}
R^{-1}\mathrm{Gr}\, \tilde h_j&\subset M_{h_j} \\
R^{-1}\mathrm{Gr}\, \tilde h&\subset M_{h} \,.
\end{split}
\]
\end{lma}
\begin{proof}
This follows immediately from our assumption that $M$ is $C^2$ and the
boundedness of $\|\nabla h_j\|_{L^\infty}$.
\end{proof}
\section{Proof of compactness and lower bound}
\label{sec:proof-comp-lower}
\begin{notation}
\label{not:push_gen}
If $U\subset\R^2$, ${\mathcal T}$ is a flat triangulated surface ${\mathcal T}\subset U$, $h:U\to\R$ is Lipschitz, and
$K=[a,b,c]\in{\mathcal T}$, then we write
\[
h_*K=[(a,h(a)),(b,h(b)),(c,h(c))]\,.
\]
We denote by $h_*{\mathcal T}$ for the triangulated surface defined by
\[
K\in{\mathcal T}\quad\Leftrightarrow \quad h_*K\in
h_*{\mathcal T}\,.
\]
\end{notation}
For an illustration Notation \ref{not:push_gen}, see Figure \ref{fig:push_gen}.
\begin{figure}[h]
\includegraphics[height=5cm]{pushforward_general.pdf}
\caption{Definition of the push forward of a triangulation $\mathcal T\subset \R^2$ by a map $h:\R^2 \to \R$. \label{fig:push_gen}}
\end{figure}
\begin{lma}
\label{lem:CS_trick}
Let $U\subset\R^2$, let ${\mathcal T}$ be a flat triangulated surface with $U\subset
{\mathcal T}\subset\R^2$, let $h$ be a Lipschitz function $U\to \R$ that is
affine on each triangle of ${\mathcal T}$, ${\mathcal T}^*=h_*{\mathcal T}$, let $g$ be a function that is constant on each
triangle of ${\mathcal T}$, $v\in \R^2$, $U^v=\{x\in\R^2:[x,x+v]\subset U\}$, and $W\subset U^v$.
\begin{itemize}\item[(i)]
Then
\[
\begin{split}
\int_{W}& |g(x+v)-g(x)|^2\d x\\
&\leq | v| \left(\sum_{K,L\in{\mathcal T}}
\frac{l_{K^*L^*}}{d_{K^*L^*}} |g(K)-g(L)|^2\right) \max_{x\in W}
\sum_{K,L\in{\mathcal T}} \mathds{1}^v_{KL}(x) \frac{\theta_{KL}^vl_{KL}d_{K^*L^*}}{l_{K^*L^*}}
\,,
\end{split}
\]
where we have written $K^*=h_*K$, $L^*=h_*L$ for $K,L\in {\mathcal T}$.
\item[(ii)]
Let $w\in\R^2$, and denote by
$\bar K$, $\bar L$ the triangles $K,L$ pushed forward by the map
$x\mapsto (x,x\cdot w)$.
Then
\[
\begin{split}
\int_{W}& |g(x+v)-g(x)|^2\d x\\
&\leq \frac{|\bar v|}{\sqrt{1+|w|^2}} \left(\sum_{K,L\in{\mathcal T}}
\frac{l_{K^*L^*}}{d_{K^*L^*}} |g(K)-g(L)|^2\right) \max_{x\in W}
\sum_{K,L\in{\mathcal T}} \mathds{1}^v_{KL}(x) \frac{\theta_{\bar K\bar
L}^{\bar v}l_{\bar K\bar L}d_{K^*L^*}}{l_{K^*L^*}}\,.
\end{split}
\]
\end{itemize}
\end{lma}
\begin{proof}
By the Cauchy-Schwarz inequality, for $x\in W$, we have that
\[
\begin{split}
| g(x+v)- g(x)|^2&\leq \left(\sum_{K,L\in {\mathcal T}} \mathds{1}_{K
L}^v(x)| g(K)- g(L)|\right)^2\\
&\leq \left(\sum_{K,L\in {\mathcal T}}
\frac{l_{K^*L^*}}{\theta_{KL}^vl_{KL}d_{K^*L^*}}\mathds{1}_{K
L}^v(x)| g(K)- g(L)|^2\right)\\
&\qquad \times\left(\sum_{K,L\in {\mathcal T}}
\mathds{1}^v_{KL}(x)\frac{\theta^v_{KL}l_{KL}d_{K^*L^*}}{l_{K^*L^*}}\right)\,.
\end{split}
\]
Using these estimates and Lemma \ref{lem:char_func} (i), we obtain
\begin{equation}
\begin{aligned}
&\int_{U^v} | g(x+v) - g(x)|^2\,\d x\\
\leq & \int_{U^v} \left( \sum_{K,L\in {\mathcal T}}
\mathds{1}^v_{KL}(x)\frac{l_{K^*L^*}}{\theta^v_{KL}l_{KL}d_{K^*L^*}} | g(K) - g(L)|^2 \right)\\
&\quad \times
\left( \sum_{K,L\in {\mathcal T}} \mathds{1}^v_{KL}(x) \frac{\theta^v_{KL}l_{KL}d_{K^*L^*}}{l_{K^*L^*}} \right) \,\d x\\
\leq & |v|\left( \sum_{K,L\in {\mathcal T}}
\frac{l_{K^*L^*}}{d_{K^*L^*}} | g(K) - g(L)|^2 \right) \max_{x\in U^v} \sum_{K,L\in {\mathcal T}} \mathds{1}^v_{KL}(x) \frac{\theta^v_{KL}l_{KL}d_{K^*L^*}}{l_{K^*L^*}}\,.
\end{aligned}
\end{equation}
This proves (i). The claim (ii) is proved analogously, using $
\frac{\theta_{\bar K\bar L}^{\bar v}l_{\bar K\bar L}d_{K^*L^*}}{l_{K^*L^*}}$ instead of $
\frac{\theta_{ K L}^{v}l_{ K L}d_{K^*L^*}}{l_{K^*L^*}}$ in the
Cauchy-Schwarz inequality, and then Lemma \ref{lem:char_func} (ii).
\end{proof}
In the following proposition, we will consider sequences of flat triangulated surfaces
${\mathcal T}_j$ with $U\subset{\mathcal T}_j\subset\R^2$ and sequences of Lipschitz functions
$h_j:U\to \R$. We write ${\mathcal T}_j^*=(h_j)_*{\mathcal T}_j$, and for $K\in {\mathcal T}_j$, we write
\[
K^*=(h_j)_*K\,.
\]
\begin{prop}\label{prop:lower_blowup}
Let $U,U'\subset\R^2$ be open, $\zeta>0$, $({\mathcal T}_j)_{j\in \N}$
a sequence of flat $\zeta$-regular triangulated surfaces with $U\subset{\mathcal T}_j\subset
U'$ and $\mathrm{size} ({\mathcal T}_j) \to
0$. Let $(h_j)_{j\in\N}$ be a sequence of Lipschitz functions $U'\to \R$ with
uniformly bounded gradients such that $h_j$ is affine
on each triangle of ${\mathcal T}_j$ and the triangulated surfaces
${\mathcal T}_j^*=(h_j)_*{\mathcal T}_j$ satisfy the Delaunay property.
\begin{itemize}
\item[(i)] Assume that
\[
\begin{split}
h_j&\weakstar h\quad \text{ in } W^{1,\infty}(U')\,,
\end{split}
\]
and $\liminf_{j\to \infty} \sum_{K,L\in{\mathcal T}_j} \frac{l_{K^*L^*}}{d_{K^*L^*}} |n(K^*) - n(L^*)|^2<\infty$. Then $h\in W^{2,2}(U)$.
\item[(ii)] Let $U=Q=(0,1)^2$, and let $(g_j)_{j\in\N}$ be a sequence of functions $U'\to\R$ such that $g_j$ is constant on
each triangle in ${\mathcal T}_j$. Assume that
\[
\begin{split}
h_j&\to h\quad \text{ in } W^{1,2}(U')\,,\\
g_j&\to g \quad\text{ in }
L^2(U')\,,
\end{split}
\]
where $h(x)=w\cdot x$ and $g(x)=u\cdot x$ for some $u,w\in \R^2$.
Then we have
\[
u^T\left(\mathds{1}_{2\times 2}+w\otimes w\right)^{-1}u \sqrt{1+|w|^2}\leq \liminf_{j\to \infty} \sum_{K,L\in{\mathcal T}_j} \frac{l_{K^*L^*}}{d_{K^*L^*}} |g_j(K) - g_j(L)|^2\,.
\]
\end{itemize}
\end{prop}
\begin{proof}[Proof of (i)]
We write
\[
E_j:= \sum_{K,L\in {\mathcal T}_j} \frac{l_{K^*L^*}}{d_{K^*L^*}} |n(K^*)
- n(L^*)|^2 \,.
\]
Fix $v\in B(0,1)\subset\R^2$, write $U^v=\{x\in\R^2:[x,x+v]\subset U\}$,
and
fix $k\in \{1,2,3\}$.
Define the function $N_j^k:U\to \R^3$ by requiring $N_j^k(x)=n(K^*)\cdot e_k$ for $x\in K\in{\mathcal T}_j$.
By Lemma \ref{lem:CS_trick} with $g_j=N_j^k$, we
have that
\begin{equation}
\label{eq:11}
\int_{U^v} |N_{j}^k(x+v) - N_j^k(x)|^2\,\d x
\leq |v|
\left(\max_{x\in U^v} \sum_{K,L\in{\mathcal T}_j}\mathds{1}^v_{KL}(x) \frac{\theta^v_{KL}l_{KL}d_{K^*L^*}}{l_{K^*L^*}}
\right) E_j\,.
\end{equation}
Since
$h_j$ is uniformly Lipschitz, there exists a constant $C>0$ such that
\[
\frac{l_{KL}}{l_{K^*L^*}}
d_{K^*L^*}<C d_{KL}\,.
\]
We claim that
\begin{equation}\label{eq:15}
\begin{split}
\max_{x\in U^v} \sum_{K,L\in {\mathcal T}_j} \mathds{1}_{KL}^v(x) \theta_{KL}d_{KL}
&\lesssim |v|+C\size({\mathcal T}_{j})\,.
\end{split}
\end{equation}
Indeed, let $K_0,\ldots,K_N\in {\mathcal T}_{j}$ be the sequence of triangles so that there is $i:[0,1]\to \{1,\ldots,N\}$ non-decreasing with $x+tv\in K_{i(t)}$.
We have that for all pairs $K_i,K_{i+1}\in {\mathcal T}_{j}$,
\begin{equation}
\label{eq:12}
\theta_{K_iK_{i+1}} d_{K_iK_{i+1}} = \left|(q(K_{i+1})-q(K_i)) \cdot \frac{v}{|v|}\right| \,,
\end{equation}
which yields the last estimate in \eqref{eq:15}.
Inserting in \eqref{eq:11} yields
\begin{equation}
\int_{U^v} |N_{j}^k(x+v) - N_j^k(x)|^2\,\d x
\leq
C|v|(|v|+C\size({\mathcal T}_{j})) E_j\,.
\end{equation}
By passing to the limit $j\to\infty$ and standard difference quotient arguments,
it then follows that the limit $N^k=\lim_{j\to\infty} N_j^k$ is in
$W^{1,2}(U)$. Since $h$ is also in $W^{1,\infty}(U)$ and $(N^k)_{k=1,2,3}=(\nabla
h,-1)/\sqrt{1+|\nabla h|^2}$ is the normal to the graph of $h$, it follows that $h\in W^{2,2}(U)$.
\end{proof}
\bigskip
\begin{proof}[Proof of (ii)]
We write
\[
E_j:= \sum_{K,L\in {\mathcal T}_j} \frac{l_{K^*L^*}}{d_{K^*L^*}} |g_j(K)
- g_j(L)|^2
\]
and may assume without loss of generality that $\liminf_{j\to \infty}
E_j<\infty$.
Fix $\delta > 0$. Define the set of bad triangles as
\[
{\mathcal B}_j^\delta := \{K \in {\mathcal T}_{j}\,:\,\left|\nabla h_j(K)- w\right| > \delta\}.
\]
Fix $v\in B(0,1)$, and write $Q^v=\{x\in \R^2:[x,x+v]\subset Q\}$. Define the set of good points as
\[
A_j^{\delta,v} := \left\{x\in Q^v: \#\{K\in {\mathcal B}_j^\delta\,:
\,K \cap [x,x+v] \neq \emptyset\} \leq
\frac{\delta|v|}{\size({\mathcal T}_{j})}\right\}.
\]
We claim that
\begin{equation}\label{eq:17}
\L^2(Q^v \setminus A_j^{\delta,v}) \to 0\qquad\text{ for } j\to\infty\,.
\end{equation}
Indeed,
let $v^\bot=(-v_2,v_1)$, and let $P_{v^\bot}:\R^2\to v^\bot \R$ denote the projection onto the linear subspace parallel to $v^\bot$. Now by the definition of $A_j^{\delta,v}$, we may estimate
\[
\begin{split}
\int_{Q^v}|\nabla h_j-w|^2\d x \gtrsim & \# \mathcal B_j^{\delta} \left(\size {\mathcal T}_j \right)^2 \delta\\
\gtrsim & \frac{\L^2(Q\setminus A_j^{\delta,v})}{|v|\size{\mathcal T}_j}\frac{\delta|v|}{\size {\mathcal T}_j} \left(\size {\mathcal T}_j \right)^2 \delta\\
\gtrsim &\L^2(Q^v \setminus A_j^{\delta,v})\delta^2|v|\,,
\end{split}
\]
and hence \eqref{eq:17} follows by
$h_j\to h$ in $W^{1,2}(Q)$.
For the push-forward of $v$ under the affine map $x\mapsto (x,h(x))$,
we write
\[
\bar v= (v,v\cdot w)\in\R^3\,.
\]
Also, for $K=[a,b,c]\in {\mathcal T}_j$, we write
\[
\bar K=[(a,a\cdot w),(b,b\cdot w),(c,c\cdot w)]=h_*K\,.
\]
By Lemma \ref{lem:CS_trick}, we have that
\begin{equation}
\label{eq: difference quotient estimate}
\begin{split}
\int_{A_j^{\delta, v}} &| g_{j}(x+v) - g_j(x)|^2\d x \\
& \leq \frac{|\bar v|}{\sqrt{1+|w|^2}} \left(\max_{x\in A_j^{\delta, v}}
\sum_{K,L\in {\mathcal T}_j} \mathds{1}^v_{KL}(x) \frac{\theta_{\bar K\bar L}^{\bar
v}l_{\bar K\bar L}d_{K^*L^*}}{l_{K^*L^*}}\right) E_j\,.
\end{split}
\end{equation}
We claim that
\begin{equation}
\max_{x\in A_j^{\delta, v}} \sum_{K,L\in {\mathcal T}_j} \mathds{1}_{KL}(x)
\frac{\theta_{\bar K \bar L}^{\bar v}l_{\bar K\bar L}d_{K^*L^*}}{l_{K^*L^*}} \leq
(1+C\delta)\left(|\bar v|+C\size({\mathcal T}_j)\right)\,.\label{eq:16}
\end{equation}
Indeed,
Let $K_0,\ldots,K_N\in {\mathcal T}_{j}$ be the sequence of triangles so that there is $i:[0,1]\to \{1,\ldots,N\}$ non-decreasing with $x+tv\in K_{i(t)}$.
For all pairs $K_i,K_{i+1}\in {\mathcal T}_{j} $ we have
\begin{equation}
\theta_{\bar K_{i}\bar K_{i+1}}^{\bar v}d_{\bar K_i\bar K_{i+1}} = (q(\bar K_{i+1})-q(\bar K_i)) \cdot \frac{\bar v}{|\bar v|} \,.
\end{equation}
Also, we have that for $K_i,K_{i+1}\in {\mathcal T}_{j} \setminus
{\mathcal B}_j^\delta$,
\begin{equation*}
\begin{split}
\frac{l_{K_i^*K_{i+1}^*}d_{\bar K_i\bar K_{i+1}}}{l_{\bar K_i\bar K_{i+1}}d_{K_i^*K_{i+1}^*}}&\leq 1+C\delta\,.
\end{split}
\end{equation*}
Hence
\begin{equation}\label{eq: good triangles}
\begin{split}
\sum_{i\,:\,\{K_i,K_{i+1}\}\cap {\mathcal B}_k^\delta = \emptyset}&
\frac{\theta_{\bar K_{i}\bar K_{i+1}}^{\bar v}l_{\bar K_i\bar K_{i+1}}d_{K_i^*K_{i+1}^*}}{l_{K^*_iK_{i+1}^*}}\\
& \leq (1+C\delta)\sum_{i\,:\,\{K_i,K_{i+1}\}\cap {\mathcal B}_k^\delta = \emptyset}
\left(\left(q(\bar K_{i+1})-q(\bar K_i)\right)\cdot
\frac{\bar v}{|\bar v|}\right)\,
\end{split}
\end{equation}
If one of the triangles $K_i,K_{i+1}$ is in ${\mathcal B}_j^\delta$, then we may estimate
\[
\left|\left(q(\bar K_{i+1})-q(\bar K_i)\right) \cdot \frac{\bar v}{|\bar v|}\right|\leq C\size{\mathcal T}_j\,.
\]
Since there are few bad triangles along $[x,x+v]$, we have, using $x\in A_j^{\delta,v}$,
\begin{equation}\label{eq: bad triangles}
\begin{split}
\sum_{i\,:\,\{K_i,K_{i+1}\}\cap {\mathcal B}_k^\delta \neq \emptyset}&
\frac{\theta_{\bar K_{i}\bar K_{i+1}}^{\bar v}l_{\bar K_i\bar K_{i+1}}d_{K_i^*K_{i+1}^*}}{l_{K^*_iK_{i+1}^*}}-(q(\bar K_{i+1})-q(\bar K_i)) \cdot \frac{\bar v}{|\bar v|}\\
&\leq C\#\{K\in
{\mathcal B}_j^\delta\,:\,K \cap [x,x+v] \neq \emptyset\} \size({\mathcal T}_j)
\\
&\leq C\delta|\bar v|\,.
\end{split}
\end{equation}
Combining \eqref{eq: good triangles} and \eqref{eq: bad triangles} yields
\begin{equation*}
\begin{split}
\sum_{i = 0}^{N-1}\frac{\theta_{\bar K_{i}\bar K_{i+1}}^{\bar v}l_{\bar K_i\bar K_{i+1}}d_{K_i^*K_{i+1}^*}}{l_{K^*_iK_{i+1}^*}}&
\leq (1+C\delta)\sum_{i = 0}^{N-1}(q(\bar K_{i+1})-q(\bar K_i)) \cdot \frac{\bar v}{|\bar v|}+C\delta|\bar v|\\
&= (1+C\delta)(q(\bar K_N) -
q(\bar K_0)) \cdot \frac{\bar v}{|\bar v|} + C \delta |\bar v| \\
&\leq (1+C\delta)\left(|\bar v|
+ C\size({\mathcal T}_{j})\right).
\end{split}
\end{equation*}
This proves \eqref{eq:16}.
\medskip
Inserting \eqref{eq:16} in \eqref{eq: difference quotient estimate}, and
passing to the limits $j\to\infty$ and $\delta\to 0$, we obtain
\[|v\cdot u |^2
\leq \frac{|\bar v|^2}{\sqrt{1+|w|^2}}\liminf_{j\to \infty}E_j\,.
\]
Now let
\[
\underline{u}:=\left(\mathds{1}_{2\times 2},w\right)^T \left(\mathds{1}_{2\times 2}+w\otimes w\right)^{-1}u\,.
\]
Then we have $|\underline{u}\cdot \bar v|=|u\cdot v|$ and hence
\[
\begin{split}
|\underline{u}|^2&=\sup_{v\in \R^2\setminus \{0\}}\frac{|\underline{u}\cdot \bar v|^2}{|\bar v|^2}\\
&\leq \frac{1}{\sqrt{1+|w|^2}}\liminf_{j\to \infty}E_j\,.
\end{split}
\]
This proves the proposition.
\end{proof}
\subsection{Proof of compactness and lower bound in Theorem \ref{thm:main}}
\begin{proof}[Proof of Theorem \ref{thm:main} (o)]
For a subsequence (no relabeling), we have that $h_j\weakstar h$ in
$W^{1,\infty}(M)$. By Lemma \ref{lem:graph_rep}, ${\mathcal T}_j$ may be locally
represented as the graph of a Lipschitz function $\tilde h_j:U\to \R$, and $M_h$
as the graph of a Lipschitz function $\tilde h:U\to \R$, where $U\subset\R^2$
and $\tilde h_j\weakstar \tilde h$ in $W^{1,\infty}(U)$
\medskip
It remains to prove that $\tilde h\in W^{2,2}(U)$. Since the norm of the
gradients are uniformly bounded,
$\|\nabla \tilde h_j\|_{L^\infty(U)}<C$, we have that the projections of ${\mathcal T}_j$ to
$U$ are (uniformly) regular flat triangulated surfaces. Hence by Proposition
\ref{prop:lower_blowup} (i), we have that $\tilde h\in W^{2,2}(U)$.
\end{proof}
\begin{proof}[Proof of Theorem \ref{thm:main} (i)]
Let $\mu_j = \sum_{K,L\in {\mathcal T}_j} \frac{1}{d_{KL}} |n(K) - n(L)|^2
\H^1|_{K\cap L}\in {\mathcal M}_+(\R^3)$. Note that either a subsequence of $\mu_j$ converges
narrowly to some $\mu \in {\mathcal M}_+(M_h)$ or there is nothing to show. We will show in
the first case that
\begin{equation}
\frac{d\mu}{d\H^2}(z) \geq |Dn_{M_h}|^2(z)\label{eq:7}
\end{equation}
at $\H^2$-almost every point $z\in M_h$ which implies in particular the lower bound.
By Lemma \ref{lem:graph_rep}, we may reduce the proof to the situation that $M_{h_j}$, $M_h$ are given as
graphs of Lipschitz functions $\tilde h_j:U\to \R$, $\tilde h:U\to \R$
respectively, where $U\subset \R^2$ is some open bounded set.
We have that $\tilde h_j$ is piecewise
affine on some (uniformly in $j$) regular triangulated surface $\tilde {\mathcal T}_j$ that satisfies
\[
(\tilde h_j)_*\tilde {\mathcal T}_j={\mathcal T}_j\,.
\]
Writing down the surface normal to $M_h$ in the coordinates of $U$,
\[N(x)=\frac{(-\nabla \tilde h, 1)}{\sqrt{1+|\nabla \tilde h|^2}}\,,
\]
we have that almost every $x\in U$ is a Lebesgue point of $\nabla N$.
We write $N^k=N\cdot e_k$ and note that \eqref{eq:7} is equivalent to
\begin{equation}
\label{eq:8}
\frac{\d\mu}{\d\H^2}(z)\geq \sum_{k=1}^3\nabla N^k(x)\cdot \left(\mathds{1}_{2\times
2}+\nabla \tilde h(x)\otimes\nabla \tilde h(x)\right)^{-1}\nabla
N^k(x)\,,
\end{equation}
where $z=(x,\tilde h(x))$.
Also, we define $N_j^k:U\to\R^3$ by letting $N_j^k(x)=n((\tilde h_j)_*K)\cdot
e_k$ for $x\in K\in
\tilde {\mathcal T_j}$. (We recall that $n((\tilde h_j)_*K)$ denotes the
normal of the triangle $(\tilde h_j)_*K$.)
\medskip
Let now $x_0\in U$ be a Lebesgue point of $\nabla \tilde h$ and $\nabla N$.
We write $z_0=(x_0,\tilde h(x_0))$.
Combining the narrow convergence $\mu_j\to\mu$ with the Radon-Nikodym differentiation Theorem, we may choose a sequence $r_j\downarrow 0$ such that
\[
\begin{split}
r_j^{-1}\size{{\mathcal T}_j}&\to 0\\
\liminf_{j\to\infty}\frac{\mu_j(Q^{(3)}(x_0,r_j))}{r_j^2}&= \frac{\d\mu}{\d\H^2}(z_0)\sqrt{1+|\nabla \tilde h(x_0)|^2}\,,
\end{split}
\]
where $Q^{(3)}(x_0,r_j)=x_0+[-r_j/2,r_j/2]^2\times \R$ is the cylinder over $Q(x_0,r_j)$.
Furthermore, let $\bar N_j,\bar h_j,\bar N,\bar h: Q\to \R$ be defined by
\[
\begin{split}
\bar N_j^k(x)&=\frac{N_j(x_0+r_j x)-N_j(x_0)}{r_j}\\
\bar N^k(x)&=\nabla N^k(x_0)\cdot (x-x_0)\\
\bar h_j(x)&=\frac{\tilde h_j(x_0+r_j x)-\tilde h_j(x_0)}{r_j}\\
\bar h(x)&=\nabla \tilde h(x_0)\cdot (x-x_0)\,.
\end{split}
\]
We recall that by assumption we have that $N^k\in W^{1,2}(U)$. This
implies in particular that (unless $x_0$ is contained in a certain set of
measure zero, which we discard), we have that
\begin{equation}\label{eq:9}
\bar N_j^k\to \bar N^k\quad\text{ in } L^2(Q)\,.
\end{equation}
Also, let $T_j$ be the blowup map
\[
T_j(x)=\frac{x-x_0}{r_j}
\]
and let ${\mathcal T}_j'$ be the triangulated surface one obtains by blowing up $\tilde{\mathcal T}_j$,
defined by
\[
\tilde K\in \tilde {\mathcal T}_j\quad \Leftrightarrow \quad T_j\tilde K \in {\mathcal T}_j'\,.
\]
Now let $\mathcal S_j$ be the smallest subset of ${\mathcal T}_j'$ (as sets of
triangles) such that
$Q\subset\mathcal S_j$ (as subsets of $\R^2$).
Note that $\size\mathcal S_j\to 0$, $\bar N_j^k$ is constant and $\bar h_j$ is
affine on each $K\in \mathcal S_j$. Furthermore, for $x\in K\in \tilde {\mathcal T}_j$, we have that
\[
\nabla \tilde h_j(x)=\nabla \bar h_j(T_jx)
\]
This implies in particular
\begin{equation}
\bar h_j\to \bar h\quad \text{ in } W^{1,2}(Q)\,.\label{eq:6}
\end{equation}
Concerning the discrete energy functionals, we have for the rescaled
triangulated surfaces $({\mathcal T}_j')^*=(\bar h_j)_* {\mathcal T}_j'$, with $K^*=(\bar h_j)_*K$
for $K\in {\mathcal T}_j'$,
\begin{equation}\label{eq:10}
\liminf_{j\to\infty} \sum_{K,L\in
{\mathcal T}_j'}\frac{l_{K^*L^*}}{d_{K^*L^*}} |\bar N_j(K)-\bar N_j(L)|^2\leq \liminf_{j\to\infty}r_j^{-2}\mu_j(Q^{(3)}(x_0,r_j)) \,.
\end{equation}
Thanks to \eqref{eq:9}, \eqref{eq:6}, we may apply Proposition
\ref{prop:lower_blowup} (ii) to the sequences of functions $(\bar h_j)_{j\in\N}$,
$(\bar N_j^k)_{j\in\N}$. This yields (after summing over $k\in\{1,2,3\}$)
\[
\begin{aligned}
|Dn_{M_h}|^2(z_0)&\sqrt{1+|\nabla \tilde
h(x_0)|^2}\\
& = \nabla N(x_0)\cdot \left(\mathds{1}_{2\times 2} +\nabla \tilde h(x_0)\otimes
\nabla \tilde h(x_0)\right)^{-1}\nabla N(x_0)\sqrt{1+|\nabla \tilde
h(x_0)|^2} \\
& \leq \liminf_{j\to\infty} \sum_{K, L\in
{\mathcal T}_j'}\frac{l_{K^*L^*}}{d_{K^*L^*}} |\bar N_j(K)-\bar N_j(L)|^2\,,
\end{aligned}
\]
which in combination with \eqref{eq:10} yields \eqref{eq:8} for $x=x_0$, $z=z_0$
and completes the proof
of the lower bound.
\end{proof}
\section{Surface triangulations and upper bound}
\label{sec:surf-triang-upper}
Our plan for the construction of a recovery sequence is as follows: We shall construct optimal sequences of triangulated surfaces first locally around a point $x\in M_h$. It turns out the optimal triangulation must be aligned with the principal curvature directions at $x$. By a suitable covering of $M_h$, this allows for an approximation of the latter in these charts (Proposition \ref{prop: local triangulation}). We will then formulate sufficient conditions for a vertex set to supply a global approximation (Proposition \ref{prop: Delaunay existence}). The main work that remains to be done at that point to obtain a proof of Theorem \ref{thm:main} (ii) is to add vertices to the local approximations obtained from Proposition \ref{prop: local triangulation} such that the conditions of Proposition \ref{prop: Delaunay existence} are fulfilled.
\subsection{Local optimal triangulations}
\begin{prop}\label{prop: local triangulation}
There are constants $\delta_0, C>0$ such that for all $U \subset \R^2$ open, convex, and bounded; and $h\in C^3(U)$ with $\|\nabla h\|_\infty \eqqcolon \delta \leq \delta_0$, the following holds:
Let $\varepsilon > 0$, $ C\delta^2 < |\theta| \leq \frac12$, and define $X \coloneqq \{(\varepsilon k + \theta \varepsilon l , \varepsilon l, h(\varepsilon k + \theta \varepsilon l, \varepsilon l))\in U\times \R\,:\,k,l\in \Z\}$. Then any Delaunay triangulated surface ${\mathcal T}$ with vertex set $X$ and maximum circumradius $\max_{K\in {\mathcal T}} r(K) \leq \varepsilon$ has
\begin{equation}\label{eq: local error}
\begin{aligned}
\sum_{K,L\in {\mathcal T}}& \frac{\l{K}{L}}{d_{KL}}|n(K) - n(L)|^2\\
\leq &\left(1+ C(|\theta|+\delta+\varepsilon)\right) \L^2(U) \times\\
&\times\left(\max_{x\in U} |\partial_{11} h(x)|^2 + \max_{x\in U} |\partial_{22} h(x)|^2 + \frac{1}{|\theta|} \max_{x\in U} |\partial_{12} h(x)|^2 \right)+C\varepsilon\,.
\end{aligned}
\end{equation}
\end{prop}
\begin{proof}
We assume without loss of generality that $\theta > 0$.
We consider the projection of $X$ to the plane,
\[
\bar X:=\{(\varepsilon k + \theta \varepsilon l , \varepsilon l)\in U:k,l\in\Z\}\,.
\]
Let $\bar{\mathcal T}$ be the flat triangulated surface that consists of the triangles of the form
\[
\begin{split}
\varepsilon[ ke_1+l(\theta e_1+e_2),(k+1)e_1+l(\theta e_1+e_2),ke_1+(l+1)(\theta e_1+e_2)]\\
\text{ or } \quad \varepsilon[ ke_1+l(\theta e_1+e_2),(k+1)e_1+l(\theta e_1+e_2),ke_1+(l-1)(\theta e_1+e_2)]\,,
\end{split}
\]
with $k,l\in \Z$ such that the triangles are contained in $U$, see Figure \ref{fig:upper2d_barT}.
\begin{figure}[h]
\centering
\includegraphics[height=5cm]{upper2d_barT.pdf}
\caption{The flat triangulated surface $\bar {\mathcal T}$. \label{fig:upper2d_barT}}
\end{figure}
Obviously the flat triangulated surface $\bar{\mathcal T}$
has vertex set $\bar X$. Also, we have that
\begin{equation}\label{eq:19}
|x-y|\leq |(x,h(x))-(y,h(y))|\leq (1+C\delta)|x-y|
\end{equation}
for all $x,y \in \bar X$. We claim that for $\delta$ chosen small enough, we have the implication
\begin{equation}\label{eq:18}
h_*K=[(x,h(x)),(y,h(y)),(z,h(z))]\in {\mathcal T}\quad \Rightarrow \quad K= [x,y,z]\in \bar{\mathcal T} \,.
\end{equation}
Indeed, if $K\not\in \bar {\mathcal T}$, then either $r(K)>\frac32\varepsilon$ or there exists $w\in X$ with $|w-q(K)|<(1 -C\theta)r(K)$. In the first case, $r(h_*K)>(1-C\delta)\frac32\varepsilon$ by \eqref{eq:19} and hence $h_*K\not\in {\mathcal T}$ for $\delta$ small enough. In the second case, we have by \eqref{eq:19} and Lemma \ref{lma: circumcenter regularity} that
\[
|(w,h(w))-q(h_*K)|<(1+C\delta)(1 -C\theta)r(h_*K)\,,
\]
and hence $h_*K$ does not satisfy the Delaunay property for $\delta$ small enough. This proves \eqref{eq:18}.
Let $[x,y]$ be an edge with either $x,y \in X$ or $x,y \in \bar X$. We call this edge \emph{horizontal} if $(y-x) \cdot e_2 = 0$, \emph{vertical} if $(y-x) \cdot (e_1 - \theta e_2)= 0$, and \emph{diagonal} if $(y-x) \cdot (e_1 + (1-\theta) e_2) = 0$.
By its definition, $\bar {\mathcal T}$ consists only of triangles with exactly one horizontal, vertical, and diagonal edge each. By what we have just proved,
the same is true for ${\mathcal T}$.
\medskip
To calculate the differences between normals of adjacent triangles, let us consider one fixed triangle $K\in {\mathcal T}$ and its neighbors $K_1,K_2,K_3$, with which $K$ shares a horizontal, diagonal and vertical edge respectively, see Figure \ref{fig:upper2d}.
\begin{figure}[h]
\includegraphics[height=5cm]{upper2d.pdf}
\caption{Top view of a triangle $K\in{\mathcal T}$ with its horizontal, diagonal and vertical neighbors $K_1,K_2,K_3$. \label{fig:upper2d}}
\end{figure}
We assume without loss of generality that one of the vertices of $K$ is the origin. We write
$x_0=(0,0)$, $x_1=\varepsilon(1-\theta,-1)$, $x_2=\varepsilon(1,0)$, $x_3=\varepsilon(1+\theta,1)$, $x_4=\varepsilon(\theta,1)$, $x_5=\varepsilon(\theta-1,1)$, and $y_i=(x_i, h(x_i))$ for $i=0,\dots,5$. With this notation we have $K=[y_0,y_2,y_4]$, $K_1=[y_0,y_1,y_2]$, $K_2=[y_2,y_3,y_4]$ and $K_3=[y_4,y_5,y_0]$. See Figure \ref{fig:upper2d_barT}. As approximations of the normals, we define
\[
\begin{split}
v(K)&=\varepsilon^{-2}y_2\wedge y_4\,\\
v(K_1)&=\varepsilon^{-2} y_1\wedge y_2\\
v(K_2)&= \varepsilon^{-2}(y_3-y_2)\wedge(y_4-y_2)\\
v(K_3)&= \varepsilon^{-2} y_4\wedge y_5\,.
\end{split}
\]
Note that $v(L)$ is parallel to $n(L)$ and $|v(L)|\geq 1$ for $L\in \{K,K_1,K_2,K_3\}$.
Hence for $i=1,2,3$, we have that
\[
|n(K)-n(K_i)|^2\leq |v(K)-v(K_i)|^2\,.
\]
For each $x_i$, we write
\[
h(x_i)= x_i \cdot \nabla h(0) + \frac12 x_i \nabla^2 h(0) x_i^T+O(\varepsilon^3)\,,
\]
where $O(\varepsilon^3)$ denotes terms $f(\varepsilon)$ that satisfy
$\limsup_{\varepsilon\to 0}\varepsilon^{-3}|f(\varepsilon)|<\infty$.
By an explicit computation we obtain that
\[
\begin{split}
\left|v(K)-v(K_1)\right|^2&= \varepsilon^2\left|(\theta-1)\theta \partial_{11} h+2(\theta-1)\partial_{12}h+\partial_{22}h\right|^2+O(\varepsilon^3)\\
\left|v(K)-v(K_2)\right|^2&= \varepsilon^2\left(\left|\theta\partial_{11} h+\partial_{12}h\right|^2+\left|(\theta-1)\theta\partial_{11} h+(\theta-1)\partial_{12}h\right|^2\right)+O(\varepsilon^3)\\
\left|v(K)-v(K_3)\right|^2&=\varepsilon^2\left( \theta^2\left|(\theta-1)\partial_{11} h+\partial_{12}h\right|^2+\left|(\theta-1)\partial_{11} h+\partial_{12}h\right|^2\right)+O(\varepsilon^3)\,,
\end{split}
\]
where all derivatives of $h$ are taken at $0$.
Using the Cauchy-Schwarz inequality and $|1-\theta|\leq 1$, we may estimate the term on the right hand side in the first line above,
\[
\left|(\theta-1)\theta\partial_{11} h+2(\theta-1)\partial_{12}h+\partial_{22}h\right|^2
\leq (1+\theta) |\partial_{22}h|^2+ \left(1+\frac{C}{\theta}\right)\left(\theta^2 |\partial_{11} h|^2+|\partial_{12}h|^2\right)\,.
\]
In a similar same way, we have
\[
\begin{split}
\left|\theta\partial_{11} h+\partial_{12}h\right|^2+\left|(\theta-1)\theta\partial_{11} h+(\theta-1)\partial_{12}h\right|^2&\leq C(|\partial_{12}h|^2
+\theta^2|\partial_{11} h|^2)\\
\theta^2\left|(\theta-1)\partial_{11} h+\partial_{12}h\right|^2+\left|(\theta-1)\partial_{11} h+\partial_{12}h\right|^2&\leq (1+\theta)|\partial_{11} h|^2+\frac{C}{\theta}|\partial_{12}h|^2\,,
\end{split}
\]
so that
\[
\begin{split}
\left|n(K)-n(K_1)\right|^2&\leq \varepsilon^2(1+\theta) |\partial_{22}h|^2+ C\varepsilon^2 \left(\theta |\partial_{11} h|^2+ \frac1\theta |\partial_{12}h|^2\right)+O(\varepsilon^3)\\
\left|n(K)-n(K_2)\right|^2&\leq C\varepsilon^2 (|\partial_{12}h|^2
+\theta^2|\partial_{11} h|^2)+O(\varepsilon^3)\\
\left|n(K)-n(K_3)\right|^2&\leq \varepsilon^2(1+\theta)|\partial_{11} h|^2+\frac{C}{\theta}\varepsilon^2|\partial_{12}h|^2+O(\varepsilon^3)\,,
\end{split}
\]
Also, we have by Lemma \ref{lma: circumcenter regularity} that
\[
\begin{split}
\frac{l_{KK_1}}{d_{KK_1}}&\leq 1+C(\delta+\varepsilon+\theta)\\
\frac{l_{KK_2}}{d_{KK_2}}&\leq \left(1+C(\delta+\varepsilon+\theta)\right) \frac{C}{\theta}\\
\frac{l_{KK_3}}{d_{KK_3}}&\leq 1+C(\delta+\varepsilon+\theta)\,.
\end{split}
\]
Combining all of the above, and summing up over all triangles in ${\mathcal T}$, we obtain the statement of the proposition.
\end{proof}
\begin{comment}
\begin{proof}[Old proof]
Consider the points $a,b,c,d,e,f \in U$ defined, after translation, as $a \coloneqq (0,0)$, $b \coloneqq (\varepsilon,0)$, $c \coloneqq (\theta \varepsilon, \varepsilon)$, $d \coloneqq ((1+\theta)\varepsilon, \varepsilon)$, $e \coloneqq ((1-\theta)\varepsilon, -\varepsilon)$, and $f \coloneqq ((\theta-1)\varepsilon,\varepsilon)$. We consider the lifted points $A = (a,h(a))\in X$, and likewise $B,C,D,E,F\in X$.
First, we note that ${\mathcal T}$ only contains translated versions of the triangles $K \coloneqq [A,B,C]$, $L \coloneqq [B,C,D]$, $P \coloneqq [A,B,D]$, and $S \coloneqq [A,C,D]$, as all other circumballs $\overline{B(q,r)}$ with $r\leq \varepsilon$ contain some fourth point in $X$.
We note that ${\mathcal T}$ contains the triangles $K \coloneqq [A,B,C]$, $L \coloneqq [B,C,D]$, $N \coloneqq [A,B,E]$, and $O \coloneqq [A,C,F]$. We then define
\begin{equation}
\begin{aligned}
v(K) \coloneqq \frac1{\varepsilon^2}(B-A)\times(C-A), v(L) \coloneqq \frac1{\varepsilon^2}(D-C)\times (D-B),\\
v(N) \coloneqq \frac1{\varepsilon^2}(B-A) \times (B-E), v(O) \coloneqq \frac1{\varepsilon^2}(C-F)\times(C-A).
\end{aligned}
\end{equation}
Then by the convex projection theorem
\begin{equation}
\begin{aligned}
|n(K) - n(L)| \leq &|v(K) - v(L)| \leq \frac{1}{\varepsilon} (2+\theta) |h(d)-h(c)-h(b)+h(a)|,\\
|n(K) - n(N)| \leq &|v(K) - v(N)| = \frac{1}{\varepsilon} |h(c)-h(a)-h(b)+h(e)|,\\
|n(K) - n(O)| \leq &|v(K) - v(O)| \leq \frac{1}{\varepsilon} (1+\theta) |h(b)-h(a)-h(c)+h(f)|.
\end{aligned}
\end{equation}
By the fundamental theorem of calculus, we may rewrite these second differences of the parallelogram by e.g.
\begin{equation}
h(d)-h(c)-h(b)+h(a) = \frac{1}{|(b-a)\wedge (c-a)|}\int_{[a,b,c,d]} (b-a) \cdot D^2 h(x) (c-a)\,dx
\end{equation}
and thus estimate
\begin{equation}
\begin{aligned}
|h(d)-h(c)-h(b)+h(a)| \leq &(1+C\theta) \int_{[a,b,c,d]} |\partial_{12} h(x)| + \theta |\partial_{11} h(x)|\,dx\\
|h(c)-h(a)-h(b)+h(e)| \leq &(1+C\theta) \int_{[a,b,c,e]}|\partial_{22} h(x)| + |\partial_{12} h(x)| + \theta |\partial_{11} h(x)|\,dx\\
|h(b)-h(a)-h(c)+h(f)| \leq &(1+C\theta)\int_{[a,b,c,f]} |\partial_{12} h(x)| + |\partial_{11} h(x)|\,dx.
\end{aligned}
\end{equation}
We note that all interactions appearing in the left-hand side of \eqref{eq: local error} are translations of the diagonal $(K,L)$, vertical $(K,N)$, and horizontal $(K,O)$ cases above. We estimate the prefactors using Lemma \ref{lma: circumcenter regularity}:
\begin{equation}
\frac{\l{K}{L}}{d_{KL}} \leq \frac{C}{\theta},\,\frac{\l{K}{N}}{d_{KN}} \leq 1 + C\theta + C\delta,\,\frac{\l{K}{O}}{d_{KO}} \leq 1 + C\theta + C\delta.
\end{equation}
This allows us to bound all diagonal interactions, using H\"older's inequality, by
\begin{equation}
\frac12 \sum_{\tilde K, \tilde L\text{ diagonal}} \frac{\l{\tilde K}{\tilde L}}{d_{\tilde K \tilde L}} |n(\tilde K) - n(\tilde L)|^2 \leq \frac{C}{\theta}\left(C\delta^2 + C\theta^2\int_U |\partial_{11} h(x)|^2\,dx\right).
\end{equation}
On the other hand, we may bound the vertical interactions by
\begin{equation}
\begin{aligned}
&\frac12 \sum_{\tilde K, \tilde N\text{ vertical}} \frac{\l{\tilde K}{\tilde N}}{d_{\tilde K \tilde N}} |n(\tilde K) - n(\tilde N)|^2\\
\leq &(1+C\theta+C\delta) \left((1+\theta)\int_U |\partial_{22} h(x)|^2\,dx + \frac{C}{\theta} \int_U|\partial_{12} h(x)|^2 + \theta^2 |\partial_{11}h(x)|^2\,dx \right),
\end{aligned}
\end{equation}
and similarly all horizontal interactions by
\begin{equation}
\begin{aligned}
&\frac12 \sum_{\tilde K, \tilde O\text{ horizontal}} \frac{\l{\tilde K}{\tilde O}}{d_{\tilde K \tilde O}} |n(\tilde K) - n(\tilde O)|^2\\
\leq &(1+C\theta+C\delta) \left((1+\theta)\int_U |\partial_{11} h(x)|^2\,dx + \frac{C}{\theta} \int_U|\partial_{12} h(x)|^2\,dx \right),
\end{aligned}
\end{equation}
Combining these three estimates yields the result.
\end{proof}
\end{comment}
\subsection{Global triangulations}
We are going to use a known fact about triangulations of point sets in $\R^2$, and transfer them to $\R^3$. We first cite a result for planar Delaunay triangulations, Theorem \ref{thm: planar Delaunay} below, which can be found in e.g. \cite[Chapter 9.2]{berg2008computational}. This theorem states the existence of a Delaunay triangulated surface associated to a \emph{protected} set of points.
\begin{defi}
Let $N\subset\R^3$ be compact, $X\subset N$ a finite set of points and
\[
D(X,N)=\max_{x\in N}\min_{y\in X}|x-y|\,.
\]
We say that $X$ is $\bar \delta$-protected if whenever $x,y,z \in X$ form a regular triangle $[x,y,z]$ with circumball $\overline{B(q,r)}$ satisfying $r \leq D(X,N)$, then $\left| |p-q| - r \right| \geq \bar\delta$ for any $p\in X \setminus \{x,y,z\}$.
\end{defi}
\begin{thm}\label{thm: planar Delaunay}[\cite{berg2008computational}]
Let $\alpha > 0$. Let $X\subset \R^2$ be finite and not colinear. Define $\Omega := \conv(X)$. Assume that
\[\min_{x\neq y \in X} |x-y| \geq \alpha D(X,\Omega)\,,
\]
and that $X$ is $\delta D(X,\Omega)$-protected for some $\delta>0$. Then there exists a unique maximal Delaunay triangulated surface ${\mathcal T}$ with vertex set $X$, given by all regular triangles $[x,y,z]$, $x,y,z\in X$, with circumdisc $\overline{B(q,r)}$ such that $B(q,r) \cap X = \emptyset$.
The triangulated surface ${\mathcal T}$ forms a partition of $\Omega$, in the sense that
\[
\sum_{K\in {\mathcal T}} \mathds{1}_K = \mathds{1}_\Omega\quad {\mathscr H}^2\text{almost everywhere}\,,
\]
where $\mathds{1}_A$ denotes the characteristic function of $A\subset \R^3$.
Further, any triangle $K\in {\mathcal T}$ with $\dist(K,\partial \Omega) \geq 4D(X,\Omega)$ is $c(\alpha)$-regular, and $d_{KL} \geq \frac{\delta}{2} D(X,\Omega)$ for all pairs of triangles $K \neq L \in {\mathcal T}$.
\end{thm}
We are now in position to formulate sufficient conditions for a vertex set to yield a triangulated surface that serves our purpose.
\begin{prop}\label{prop: Delaunay existence}
Let $N\subset\R^3$ be a 2-dimensional compact smooth manifold, and let $\alpha, \delta > 0$. Then there is $\varepsilon = \varepsilon(N,\alpha,\delta)>0$ such that whenever $X\subset N$ satisfies
\begin{itemize}
\item [(a)]$D(X,N) \leq \varepsilon$,
\item [(b)] $\min_{x,y\in X} |x-y| \geq \alpha D(X,N)$,
\item [(c)] $X$ is $\delta D(X,N)$-protected
\end{itemize}
then there exists a triangulated surface ${\mathcal T}(X,N)$ with the following properties:
\begin{itemize}
\item [(i)] $\size({\mathcal T}(X,N)) \leq 2D(X,N)$.
\item [(ii)] ${\mathcal T}(X,N)$ is $c(\alpha)$-regular.
\item [(iii)] ${\mathcal T}(X,N)$ is Delaunay.
\item [(iv)] Whenever $K\neq L \in {\mathcal T}(X,N)$, we have $d_{KL} \geq \frac{\delta}{2} D(X,N)$.
\item [(v)] The vertex set of ${\mathcal T}(X,N)$ is $X$.
\item [(vi)] ${\mathcal T}(X,N)$ is a $C(\alpha, N)D(X,N)$-Lipschitz graph over $N$. In particular, ${\mathcal T}(X,N)$ is homeomorphic to $N$.
\end{itemize}
\end{prop}
The surface case we treat here can be viewed as a perturbation of Theorem \ref{thm: planar Delaunay}. We note that the protection property (c) is vital to the argument. A very similar result to Proposition \ref{prop: Delaunay existence} was proved in \cite{boissonnat2013constructing}, but we present a self-contained proof here.
\begin{proof}[Proof of Proposition \ref{prop: Delaunay existence}]
We construct the triangulated surface ${\mathcal T}(X,N)$ as follows: Consider all regular triangles $K=[x,y,z]$ with $x,y,z\in X$ such that the Euclidean Voronoi cells $V_x,V_y,V_z$ intersect in $N$, i.e. there is $\tilde q \in N$ such that $|\tilde q - x| = |\tilde q - y| = |\tilde q - z| \leq |\tilde q - p|$ for any $p\in X\setminus \{x,y,z\}$.
\emph{Proof of (i):} Let $[x,y,z]\in {\mathcal T}(X,N)$. Let $\tilde q \in V_x \cap V_y \cap V_z \cap N$, set $\tilde r := |\tilde q - x|$. Then $\tilde r = \min_{p\in X} |\tilde q - p| \leq D(X,N)$, and because $[x,y,z]\subset \overline{B(\tilde q, \tilde r)}$ we have $\diam([x,y,z])\leq 2 \tilde r \leq 2D(X,N)$.
\emph{Proof of (ii):} Let $\overline{B(q,r)}$ denote the Euclidean circumball of $[x,y,z]$. Then $r\leq \tilde r$ by the definition of the circumball. Thus $\min(|x-y|,|x-z|,|y-z|) \geq \alpha r$, and $[x,y,z]$ is $c(\alpha)$-regular by the following argument: Rescaling such that $r = 1$, consider the class of all triangles $[x,y,z]$ with $x,y,z \in S^1$, $\min(|x-y|,|x-z|,|y-z|) \geq \alpha$. All these triangles are $\zeta$-regular for some $\zeta>0$, and by compactness there is a least regular triangle in this class. That triangle's regularity is $c(\alpha)$.
\emph{Proof of (iii):} Because of (ii), $N\cap \overline{B(q,r)}$ is a $C(\alpha, N)\varepsilon$-Lipschitz graph over a convex subset $U$ of the plane $ x + \R(y-x) + \R(z-x)$, say $N\cap \overline{B(q,r)} = U_h$. It follows that $\tilde q - q = h(\tilde q) n_U$. Because $h(x)= 0$, it follows that $|\tilde q - q| = |h(\tilde q)| \leq C(\alpha, N) D(X,N)^2$.
Thus, for $D(X,N) \leq \delta(2C(\alpha,N))^{-1}$, we have that $|\tilde q - q| \leq \frac{\delta}{2}D(X,N)$. This together with (c) suffices to show the Delaunay property of ${\mathcal T}(X,N)$: Assume there exists $p\in X \setminus \{x,y,z\} \cap B(q,r)$. Then by (c) we have $|p-q| \leq r - \delta D(X,N)$, and by the triangle inequality $|p-\tilde q| \leq |p- q| + \frac{\delta}{2}D(x,N) < \tilde r$, a contradiction.
\emph{Proof of (iv):}
It follows also from (c) and Lemma \ref{lma: circumcenter regularity} that for all adjacent $K,L\in {\mathcal T}(X,N)$ we have $d_{KL} \geq \frac{\delta}{2} D(X,N)$.
\emph{Proof of (v) and (vi):} Let $\eta>0$, to be fixed later. There is $s>0$ such that for every $x_0\in N$, the orthogonal projection $\pi:\R^3 \to x_0 + T_{x_0}N$ is an $\eta$-isometry when restricted to $N\cap B(x_0,s)$, in the sense that that $|D\pi - \mathrm{id}_{TN}|\leq \eta$.
Let us write $X_\pi=\pi(X\cap B(x_0,s))$. This point set fulfills all the requirements of Theorem \ref{thm: planar Delaunay} (identifying $x_0+T_{x_0}N$ with $\R^2$), except for possibly protection.
We will prove below that
\begin{equation}\label{eq:23}
X_\pi\text{ is } \frac{\delta}{4}D(X,N) \text{-protected}.
\end{equation}
We will then consider the planar Delaunay triangulated surface ${\mathcal T}' \coloneqq {\mathcal T}(X_\pi, x_0 + T_{x_0}N)$, and show that for $x,y,z\in B(x_0,s/2)$ we have
\begin{equation}\label{eq:22}
K:=[x,y,z]\in {\mathcal T}(X,N)\quad \Leftrightarrow \quad K_\pi:=[\pi(x),\pi(y),\pi(z)]\in {\mathcal T}'\,
\end{equation}
If we prove these claims, then (v) follows from Theorem \ref{thm: planar Delaunay}, while (vi) follows from Theorem \ref{thm: planar Delaunay} and Lemma \ref{lma: graph property}.
\medskip
We first prove \eqref{eq:23}: Let $\pi(x),\pi(y),\pi(z)\in X_\pi$, write $K_\pi= [\pi(x),\pi(y),\pi(z)]$, and assume $r(K_\pi)\leq D(X_\pi,\mathrm{conv}(X_{\pi}))$. For a contradiction, assume that $\pi(p)\in X_\pi\setminus \{\pi(x),\pi(y),\pi(z)\}$ such that
\[
\left||q(K_\pi)-\pi(p)|-r(K_\pi)\right|<\frac{\delta}{4}D(X,N)\,.
\]
Using again $|D\pi-\mathrm{id}_{TN}|<\eta$ and Lemma \ref{lma: circumcenter regularity},
we obtain, with $K=[x,y,z]$,
\[
\left||q(K)-p|-r(K)\right|<(1+C\eta)\frac{\delta}{4}D(X,N)\,.
\]
Choosing $\eta$ small enough, we obtain a contradiction to (c). This completes the proof of \eqref{eq:23}.
\medskip
Next we show the implication $K\in {\mathcal T}\Rightarrow K_\pi\in {\mathcal T}'$: Let $p\in X\cap B(x_0,s) \setminus \{x,y,z\}$.
Assume for a contradiction that $\pi(p)$ is contained in the circumball of $K_\pi$,
\[
|\pi(p) - q(K_\pi)|\leq r(K_\pi)\,.
\] Then by $|D\pi-\mathrm{id}_{TN}|<\eta$ and Lemma \ref{lma: circumcenter regularity}\,,
\[
|p-q(K)|\leq r(K) + C(\alpha)\eta D(X,N)\,.
\]
Choosing $\eta<\delta/(2C(\alpha))$, we have by (c) that
\[|p-q(K)| \leq r(K) - \delta D(X,N)\,,
\]
which in turn implies $|p-\tilde q| < \tilde r$.
This is a contradiction to $\tilde q \in V_x \cap V_y \cap V_z$, since $p$ is closer to $\tilde q$ than any of $x,y,z$. This shows $K_\pi\in {\mathcal T}'$.
Now we show the implication $K_\pi\in {\mathcal T}'\Rightarrow K\in {\mathcal T}$: Let $x,y,z\in X\cap B(x_0,s/2)$ with $[\pi(x),\pi(y),\pi(z)]\in {\mathcal T}'$. Let $p\in X\cap B(x_0,s) \setminus \{x,y,z\}$. Assume for a contradiction that $|p-\tilde q| \leq \tilde r$. Then again by Lemma \ref{lma: circumcenter regularity} we have
\[
|p - \tilde q| < \tilde r \Rightarrow |p-q| < r + \delta D(X,N) \Rightarrow |p-q| \leq r - \delta D(X,N) \Rightarrow |\pi(p) - q'| < r'.
\]
Here again we used (c) and the fact that $D(X,N)$ is small enough. The last inequality is a contradiction, completing the proof of \eqref{eq:22}, and hence the proof of the present proposition.
\end{proof}
\begin{rem}
A much shorter proof exists for the case of the two-sphere, $N = \mathcal{S}^2$. Here, any finite set $X\subset \mathcal{S}^2$ such that no four points of $X$ are coplanar and every open hemisphere contains a point of $X$ admits a Delaunay triangulation homeomorphic to $\mathcal{S}^2$, namely $\partial \conv(X)$.
Because no four points are coplanar, every face of $\partial \conv(X)$ is a regular triangle $K = [x,y,z]$. The circumcircle of $K$ then lies on $\mathcal{S}^2$ and $q(K) = n(K)|q(K)|$, where $n(K)\in \mathcal{S}^2$ is the outer normal. (The case $q(K)=-|q(K)|n(K)$ is forbidden because the hemisphere $\{x\in \mathcal{S}^2\,:\,x \cdot n(K)>0\}$ contains a point in $X$.) To see that the circumball contains no other point $p\in X\setminus \{x,y,z\}$, we note that since $K\subset \partial \conv(X)$ we have $(p-x)\cdot n(K)< 0$, and thus $|p-q(K)|^2 = 1 + 1 - 2p \cdot q(K) > 1 + 1 - 2x \cdot q(K) = |x-q(K)|^2$.
Finally, $\partial \conv(X)$ is homeomorphic to $\mathcal{S}^2$ since $\conv(X)$ contains a regular tetrahedron.
\end{rem}
We are now in a position to prove the upper bound of our main theorem, Theorem \ref{thm:main} (ii).
\begin{figure}
\includegraphics[height=5cm]{upper_global.pdf}
\caption{The global triangulation of a smooth surface is achieved by first covering a significant portion of the surface with the locally optimal triangulation, then adding additional points in between the regions, and finally finding a global Delaunay triangulation. \label{fig:upper bound}}
\end{figure}
\begin{proof}[Proof of Theorem \ref{thm:main} (ii)]
We first note that it suffices to show the result for $h\in C^3(M)$ with $\|h\|_\infty < \frac{\delta(M)}{2}$. To see this, we approximate in the general case $h\in W^{2,2}(M)\cap W^{1,\infty}(M)$, $\|h\|_\infty \leq \frac{\delta(M)}{2}$ by smooth functions $h_{\beta} := H_\beta h$, where $(H_\beta)_{\beta >0 }$ is the heat semigroup. Clearly $H_\beta h \in C^\infty(M)$, and $\nabla H_\beta h \to \nabla h$ uniformly, so that $\|h\|_{\infty}\leq \frac{\delta}{2}$ and $\|\nabla h_{\beta}\|_\infty <\|\nabla h\|_{\infty}+1$ for $\beta$ small enough.
Then
\[
\int_M f(x,h_\beta(x),\nabla h_\beta(x), \nabla^2 h_\beta) \,\d{\mathscr H}^2 \to \int_M f(x,h(x),\nabla h(x), \nabla^2 h) \,\d{\mathscr H}^2
\]
for $\beta\to 0$ whenever
\[f:M \times [-\delta(M)/2, \delta(M)/2] \times B(0,\|\nabla h\|_{\infty}+1) \times (TM \times TM) \to \R
\]
is continuous with quadratic growth in $\nabla^2 h$. The Willmore functional
\[
h\mapsto \int_{M_h} |Dn_{M_h}|^2\d{\mathscr H}^2\,,
\]
which is our limit functional, may be written in this way. This proves our claim that we may reduce our argument to the case $h\in C^3(M)$, since the above approximation allows for the construction of suitable diagonal sequences in the strong $W^{1,p}$ topology, for every $p<\infty$.
\medskip
For the rest of the proof we fix $h\in C^3(M)$. We choose a parameter $\delta>0$. By compactness of $M_h$, there is a finite family of pairwise disjoint closed
open sets $(Z_i)_{i\in I}$ such that
\[
{\mathscr H}^2\left(M_h \setminus \bigcup_{i\in I} Z_i\right) \leq \delta
\]
and such that, after applying a rigid motion $R_i:\R^3\to \R^3$, the surface $R_i(M_h \cap Z_i)$ is the graph of a function $h_i\in C^2(U_i)$ for some open sets $(U_i)_{i\in I}$ with $\|\nabla h_i\|_\infty \leq \delta$ and $\|\nabla^2 h_i - \diag(\alpha_i,\beta_i)\|_\infty \leq \delta$.
\medskip
We can apply Proposition \ref{prop: local triangulation} to $R_i(M_h \cap Z_i)$ with global parameters $\theta := \delta$ and $\varepsilon>0$ such that $\dist(Z_i,Z_j)>2\varepsilon$ for $i\neq j$, yielding point sets $X_{i,\varepsilon}\subset M_h \cap B_i$. The associated triangulated surfaces ${\mathcal T}_{i,\varepsilon}$ (see \ref{fig:upper bound}) have the Delaunay property, have vertices $X_{i,\varepsilon}$ and maximum circumball radius at most $\varepsilon$. Furthermore, we have that
\begin{equation}\label{eq: sum local interactions}
\begin{aligned}
\sum_{i\in I} &\sum_{K,L\in {\mathcal T}_{i,\varepsilon}} \frac{l_{KL}}{d_{KL}} |n(K) - n(L)|^2\\
& \leq (1+C(\delta+\varepsilon)) \sum_{i\in I} \L^2(U_i)\times\\
&\quad \times \left(\max_{x\in U_i}|\partial_{11}h_i(x)|^2+\max_{x\in U_i}|\partial_{22}h_i(x)|^2+\delta^{-1}\max_{x\in U_i}|\partial_{12}h_i(x)|^2\right)+C\varepsilon\\
&\leq (1+C(\delta+\varepsilon)) \sum_{i\in I} \int_{M_h \cap Z_i} |Dn_{M_h}|^2 \,\d{\mathscr H}^2+C(\varepsilon+\delta)\,,
\end{aligned}
\end{equation}
where in the last line we have used $\|\nabla h_i\|_{\infty}\leq \delta$, $\|\dist(\nabla^2h_i,\diag(\alpha_i,\beta_i)\|_{\infty}\leq \delta$, and the identity
\[
\begin{split}
\int_{M_h \cap Z_i} |Dn_{M_h}|^2 \,\d{\mathscr H}^2&=
\int_{(U_i)_{h_i}}|Dn_{(U_i)_{h_i}}|^2\d\H^2\\
&=\int_{U_i}\left|(\mathbf{1}_{2\times 2}+\nabla h_i\otimes \nabla h_i)^{-1}\nabla^2 h_i\right|^2(1+|\nabla h_i|^2)^{-1/2}\d x\,.
\end{split}
\]
We shall use the point set $Y_{0,\varepsilon} := \bigcup_{i\in I} X_{i,\varepsilon}$ as a basis for a global triangulated surface. We shall successively augment the set by a single point $Y_{n+1,\varepsilon} := Y_{n,\varepsilon} \cup \{p_{n,\varepsilon}\}$ until the construction below terminates after finitely many steps. We claim that we can choose the points $p_{n,\varepsilon}$ in such a way that for every $n\in\N$ we have
\begin{itemize}
\item [(a)] $\min_{x,y\in Y_{n,\varepsilon}, x\neq y} |x-y| \geq \frac{\varepsilon}{2}$.
\item [(b)] Whenever $x,y,z,p\in Y_{n,\varepsilon}$ are four distinct points such that the circumball $\overline{B(q,r)}$ of $[x,y,z]$ exists and has $r\leq \varepsilon$, then
\[
\left| |p-q| - r \right| \geq \frac{\delta}{2} \varepsilon.
\]
If at least one of the four points $x,y,z,p$ is not in $Y_{0,\varepsilon}$, then
\begin{equation}\label{eq:21}
\left| |p-q| - r \right| \geq c \varepsilon,
\end{equation}
where $c>0$ is a universal constant.
\end{itemize}
First, we note that both (a) and (b) are true for $Y_{0,\varepsilon}$.
Now, assume we have constructed $Y_{n,\varepsilon}$. If there exists a point $x\in M_h$ such that $B(x,\varepsilon) \cap Y_{n,\varepsilon} = \emptyset$, we consider the set $A_{n,\varepsilon}\subset M_h \cap B(x,\frac{\varepsilon}{2})$ consisting of all points $p\in M_h \cap B(x,\frac{\varepsilon}{2})$ such that for all regular triangles $[x,y,z]$ with $x,y,z\in Y_{n,\varepsilon}$ and circumball $\overline{B(q,r)}$ satisfying $r\leq 2\varepsilon$, we have $\left||p-q| - r\right| \geq c \varepsilon$.
Seeing as how $Y_{n,\varepsilon}$ satisfies (a), the set $A_{n, \varepsilon}$ is nonempty if $c>0$ is chosen small enough, since for all triangles $[x,y,z]$ as above we have
\[
{\mathscr H}^2\left(\left\{ p\in B(x,\frac{\varepsilon}{2})\cap M_h\,:\,\left||p-q| - r\right| < c \varepsilon \right\}\right) \leq 4c \varepsilon^2,
\]
and the total number of regular triangles $[x,y,z]$ with $r\leq 2\varepsilon$ and $\overline{B(q,r)}\cap B(x,\varepsilon)\neq \emptyset$ is universally bounded as long as $Y_{n,\varepsilon}$ satisfies (a).
We simply pick $p_{n,\varepsilon}\in A_{n,\varepsilon}$, then clearly $Y_{n+1,\varepsilon} \coloneqq Y_{n,\varepsilon} \cup \{p_{n,\varepsilon}\}$ satisfies (a) by the triangle inequality. We now have to show that $Y_{n+1,\varepsilon}$ still satisfies (b).
This is obvious whenever $p = p_{n,\varepsilon}$ by the definition of $A_{n,\varepsilon}$. If $p_{n,\varepsilon}$ is none of the points $x,y,z,p$, then (b) is inherited from $Y_{n,\varepsilon}$. It remains to consider the case $p_{n,\varepsilon} = x$. Then $x$ has distance $c\varepsilon$ to all circumspheres of nearby triples with radius at most $2\varepsilon$. We now assume that the circumball $\overline{B(q,r)}$ of $[x,y,z]$ has radius $r \leq \varepsilon$ and that some point $p\in Y_{n,\varepsilon}$ is close to $\partial B(q,r)$. To this end, define
\[
\eta \coloneqq \frac{\left||p-q| - r \right|}{\varepsilon}\,.
\]
We show that $\eta \geq \eta_0$ for some universal constant. To this end, we set
\[
p_t \coloneqq (1-t)p + t\left(q+r\frac{p-q}{|p-q|}\right)
\]
(see Figure \ref{fig:pt}) and note that if $\eta\leq \eta_0$, all triangles $[y,z,p_t]$ are uniformly regular.
\begin{figure}[h]
\centering
\includegraphics[height=5cm]{pt.pdf}
\caption{The definition of $p_t$ as linear interpolation between $p_0$ and $p_1$. \label{fig:pt}}
\end{figure}
Define the circumcenters $q_t \coloneqq q(y,z,p_t)$, and note that $q_1 = q$. By Lemma \ref{lma: circumcenter regularity}, we have $|q_1 - q_0| \leq C|p_1 - p_0| = C\eta \varepsilon$ if $\eta\leq \eta_0$. Thus the circumradius of $[y,z,p_0]$ is bounded by
\[
|y-q_0| \leq |y-q| + |q-q_0| \leq (1+C\eta)\varepsilon \leq 2\varepsilon
\]
if $\eta\leq \eta_0$. Because $x\in Y_{n+1,\varepsilon} \setminus Y_{n,\varepsilon} \subset A_{n,\varepsilon}$, we have, using \eqref{eq:21},
\[
c\varepsilon \leq \left| |x-q_0| - |p-q_0|\right| \leq \left| |x-q| - |p-q| \right| + 2 |q - q_0| \leq (1+2C)\eta\varepsilon,
\]
i.e. that $\eta \geq \frac{c}{1+2C}$. This shows (b).
\begin{comment}
We note that by (a) we have $r\geq \frac{\varepsilon}{4}$. We set $p_t \coloneqq (1-t) p + t\left((q + r\frac{p-q}{|p-q|}\right)$ for $t\in[0,1]$. If $\eta<\eta_0$, then the triangles $[y,z,p_t]$ are all $\zeta_0$-regular triangles for some universal constants $\zeta_0,\eta_0>0$. By Lemma \ref{lma: circumcenter regularity}, then $|q(y,z,p_0) - q(y,z,p)|\leq C \eta \varepsilon$. However, $q(y,z,p_0) = q$, and $|p-q(y,z,p)| \leq 2\varepsilon$ for $\eta<\eta_0$. By the choice $x\in A_{n,\varepsilon}$ then
\[
\left| |x-q(y,z,p)| - |p-q(y,z,p)| \right| \geq c\varepsilon,
\]
which implies that
\[
\left||x-q| - |p-q| \right| \geq c \varepsilon - C \eta \varepsilon,
\]
i.e. that $\eta \geq \min\left( \frac{1}{1+C}, \eta_0, \frac14 \right)$, which is a universal constant. This shows (b).
\end{comment}
Since $M_h$ is compact, this construction eventually terminates, resulting in a set $X_\varepsilon := Y_{N(\varepsilon),\varepsilon} \subset M_h$ with the properties (a), (b), and $D(X_\varepsilon,M) \leq \varepsilon$.
\medskip
Consider a Lipschitz function $g:M_h\to \R$. Since $M_h$ is a $C^2$ surface, we have that for $\|g\|_{W^{1,\infty}}$ small enough, $(M_h)_g$ is locally a tangent Lipschitz graph over $M$, see Definition \ref{def:Mgraph} (iii). By Lemma \ref{lma: graph property}, this implies that $(M_h)_g$ is a graph over $M$.
Invoking Proposition \ref{prop: Delaunay existence} yields a Delaunay triangulated surface ${\mathcal T}_\varepsilon \coloneqq {\mathcal T}(X_\varepsilon, M_h)$ with vertex set $X_\varepsilon$ that is $\zeta_0$-regular for some $\zeta_0>0$, and $\bigcup_{K\in {\mathcal T}_\varepsilon} = (M_h)_{g_\varepsilon}$ with $\|g_\varepsilon\|_{W^{1,\infty}}\leq C(\delta)\varepsilon$.
By the above, there exist Lipschitz functions $h_\varepsilon:M\to \R$ such that
$(M_h)_{g_\varepsilon} = M_{h_\varepsilon}$, with $h_\varepsilon \to h$ in $W^{1,\infty}$, $\|h_\varepsilon\|_\infty \leq \frac{\delta(M)}{2}$ and $\|\nabla h_\varepsilon\|\leq \|\nabla h\|_{\infty}+1$.
\medskip
It remains to estimate the energy. To do so, we look at the two types of interfaces appearing in the sum
\[
\sum_{K,L\in {\mathcal T}_\varepsilon} \frac{l_{KL}}{d_{KL}} |n(K) - n(L)|^2.
\]
First, we look at pairwise interactions where $K,L\in {\mathcal T}(X_{i,\varepsilon})$ for some $i$. These are bounded by \eqref{eq: sum local interactions}.
Next, we note that if $\varepsilon < \min_{i\neq j \in I} \dist(B_i,B_j)$, it is impossible for $X_{i,\varepsilon}$ and $X_{j,\varepsilon}$, $i\neq j$, to interact.
Finally, we consider all interactions of neighboring triangles $K,L\in {\mathcal T}_\varepsilon$ where at least one vertex is not in $Y_{0,\varepsilon}$. By \eqref{eq:21}, these pairs all satisfy $\frac{l_{KL}}{d_{KL}} \leq C$ for some universal constant $C$ independent of $\varepsilon,\delta$, and $|n(K) - n(L)|\leq C\varepsilon$ because ${\mathcal T}$ is $\zeta_0$-regular and $M_h$ is $C^2$. Further, no points were added inside any $B_I$. Thus
\[
\begin{split}
\sum_{\substack{K,L\in {\mathcal T}_\varepsilon\,:\,\text{at least}\\\text{ one vertex is not in }Y_{0,\varepsilon}}}& \frac{l_{KL}}{d_{KL}} |n(K) - n(L)|^2 \\
&\leq C{\mathscr H}^2\left(M_h \setminus \bigcup_{i\in I}B(x_i, r_i - 2\varepsilon)\right)\\
&\leq C \delta + C(\delta)\varepsilon.
\end{split}
\]
Choosing an appropriate diagonal sequence $\delta(\varepsilon) \to 0$ yields a sequence ${\mathcal T}_\varepsilon = M_{h_\varepsilon}$ with $h_\varepsilon\to h$ in $W^{1,\infty}(M)$ with
\[
\limsup_{\varepsilon \to 0} \sum_{K,L\in {\mathcal T}_\varepsilon} \frac{l_{KL}}{d_{KL}} |n(K) -n(L)|^2 \leq \int_{M_h} |Dn_{M_h}|^2\,d{\mathscr H}^2.
\]
\end{proof}
\section{Necessity of the Delaunay property}
\label{sec:necess-dela-prop}
We now show that without the Delaunay condition, it is possible to achieve a lower energy. In contrast to the preceding sections, we are going to choose an underlying manifold $M$ with boundary (the ``hollow cylinder'' $S^1\times[-1,1]$). By ``capping off'' the hollow cylinder one can construct a counterexample to the lower bound in Theorem \ref{thm:main}, where it is assumed that $M$ is compact without boundary.
\begin{prop}\label{prop: optimal grid}
Let $M =S^1\times[-1,1] \subset \R^3$ be a
hollow cylinder and $\zeta>0$. Then there are $\zeta$-regular triangulated
surfaces ${\mathcal T}_j\subset \R^3$ with $\size({\mathcal T}_j) \to 0$ and ${\mathcal T}_j \to M$ for
$j\to\infty$ with
\[
\limsup_{j\to\infty} \sum_{K,L\in {\mathcal T}_j} \frac{\l{K}{L}}{d_{KL}}
|n(K)-n(L)|^2 < c(\zeta) \int_M |Dn_M|^2\,d\H^2\,,
\]
where the positive constant $c(\zeta)$ satisfies
\[
c(\zeta)\to 0 \quad \text{ for } \quad\zeta\to 0\,.
\]
\end{prop}
\begin{figure}
\includegraphics[height=5cm]{cylinder2.pdf}
\caption{A non-Delaunay triangulated cylinder achieving a low energy . \label{fig:cylinder}}
\end{figure}
\begin{proof}
For every $\varepsilon = 2^{-j}$ and $s\in\{2\pi j^{-1}:j=3,4,5,\dots\}$, we
define a flat triangulated surface ${\mathcal T}_j\subset \R^2$ with
$\size({\mathcal T}_j) \leq \varepsilon$ as follows: As manifolds with boundary, ${\mathcal T}_j=[0,2\pi]\times
[-1,1]$ for all $j$; all triangles are isosceles,
with one side a translation of $[0,\varepsilon]e_2$ and height $s\varepsilon$ in
$e_1$-direction. We neglect the triangles close to the boundary
$[0,2\pi]\times\{\pm 1\}$, and leave it to
the reader to verify that their contribution will be negligeable in the end.
\medskip
We then wrap this triangulated surface around the cylinder, mapping the
corners of triangles onto the surface of the cylinder via $(\theta,t) \mapsto
(\cos\theta, \sin\theta, t)$, to obtain a triangulated surface $\tilde {\mathcal T}_j$.
Obviously, the topology of $\tilde {\mathcal T}_j$ is $S^1\times[-1,1]$.
Then we may estimate all terms $\frac{\l{K}{L}}{d_{KL}} |n(K) - n(L)|^2$. We
first find the normal of the reference triangle $K\in \tilde {\mathcal T}_j$ spanned by the points $x = (1,0,0)$, $y = (1,0,\varepsilon)$, and $z = (\cos(s\varepsilon),\sin(s\varepsilon),\varepsilon/2)$. We note that
\[
n(K) = \frac{(y-x) \times (z-x)}{|(y-x) \times (z-x)|} = \frac{(-s\varepsilon\sin(s\varepsilon), s\varepsilon(\cos(s\varepsilon)-1),0)}{s\varepsilon(2-2\cos(s\varepsilon))} = (1,0,0) + O(s\varepsilon).
\]
We note that the normal is the same for all translations $K+te_3$ and for all triangles bordering $K$ diagonally. The horizontal neighbor $L$ also has $n(L) = (1,0,0) + O(s\varepsilon)$. However, we note that the dimensionless prefactor satisfies $\frac{\l{K}{L}}{d_{KL}} \leq \frac{2\varepsilon}{\varepsilon/s} = s$. Summing up the $O(s^{-1}\varepsilon^{-2})$ contributions yields
\[
\sum_{K,L\in {\mathcal T}_j} \frac{\l{K}{L}}{d_{KL}} |n(K) - n(L)|^2 \leq C \frac{s^3\varepsilon^2}{s\varepsilon^2} = Cs^2.
\]
This holds provided that $\varepsilon$ is small enough. Letting $s\to 0$, we see that this energy is arbitrarily small.
\end{proof}
\bibliographystyle{alpha}
| 2024-02-18T23:39:40.213Z | 2021-06-14T02:22:34.000Z | algebraic_stack_train_0000 | 19 | 15,219 |
|
proofpile-arXiv_065-129 | \section{Introduction}
\begin{figure*}[t!]
\centering
\includegraphics[width=0.75\textwidth]{introduction_modified.pdf}
\caption{(a) Each angle of illumination, here labelled as angular axis, corresponds to a time step in an analogous temporal axis. (b) The raw intensity diffraction pattern $\mathbf{g}_n,\: n\!=\!1,\ldots,N\!\!=\!\!42$\ \ of the at $n$-th angular sequence step is followed by gradient descent and moving average operations to construct a shorter Approximant sequence $\mathbf{\tilde{f}}_m{}^{[1]},\: m\!=\!1,\ldots,M\!\!=\!\!12$. The Approximants $\mathbf{\tilde{f}}_m{}^{[1]}$ are encoded to $\xi_m$ and fed to the recurrent dynamical operation whose output sequence $\mathbf{h}_m, m\!=\!1,\ldots,12$\ \ the angular attention scheme merges into a single representation $a$, and that is finally decoded to produce the 3D reconstruction $\mathbf{\hat{f}}$. Training adapts the weights of the learned operators in this architecture to minimize the training loss function $\mathcal{E}(\mathbf{f},\hat{\mathbf{f}})$ between $\mathbf{\hat{f}}$ and the ground truth object $\mathbf{f}$.}
\label{fig:introduction}
\end{figure*}
Optical tomography reconstructs the three-dimensional (3D) internal refractive index profile by illuminating the sample at several angles and processing the respective raw intensity images. The reconstruction scheme depends on the scattering model that is appropriate for a given situation. If the rays through the sample can be well approximated as straight lines, then accumulation of absorption and phase delay along the rays is an adequate forward model, {\it i.e.} the projection or Radon transform approximation applies. This is often the case with hard x-rays through most materials including biological tissue; for that reason, Radon transform inversion has been widely studied\ \cite{radon1986determination,radon1917determination,bracewell1967inversion,feldkamp1984practical,dreike1976convolution,wang1993general,kudo1991helical,grangeat1991mathematical,katsevich2002analysis,choi2007tomographic}.
The next level of complexity arises when diffraction and multiple scattering must be taken into account in the forward model; then, the Born or Rytov expansions and the Lippmann-Schwinger integral equation \cite{ishimaru2017electromagnetic,tatarski2016wave,wolf1969three,devaney1981inverse,pham2020three} are more appropriate. These follow from the scalar Helmholtz equation using different forms of expansion for the scattered field \cite{marks2006family}. In all these approaches, weak scattering is obtained from the first order in the series expansion. Holographic approaches to volumetric reconstruction generally rely on this first expansion term\ \cite{milgram2002computational,tian2010quantitative,hahn2008wide,park2009recent,nehmetallah2012applications,williams2013digital,brady2009compressive,choi2010compressive,rivenson2018phase,wu2019bright,rivenson2019deep,zhang2018twin}. Often, solving the Lippmann-Schwinger equation is the most robust approach to account for multiple scattering, but even then the solution is iterative and requires excessive amount of computation especially for complex 3D geometries. The inversion of these forward models to obtain the refractive index in 3D is referred to as inverse scattering, also a well studied topic \cite{kamilov2016recursive,kamilov2016optical,giorgi2013application,chew1990reconstruction,sun2018efficient,lu1985multidimensional,lu1986jkm,tsihrintzis2000higher}.
An alternative to the integral methods is the beam propagation method (BPM), which sections the sample along the propagation distance $z$ into slices, each slice scattering according to the thin transparency model, and propagates the field from one slice to the next through the object\ \cite{feit1980computation}. Despite some compromise in accuracy, BPM offers comparatively light load of computation and has been used as forward model for 3D reconstructions\ \cite{pham2020three}. The analogy of the BPM computational structure with a neural network was exploited, in conjunction with gradient descent optimization, to obtain the 3D refractive index as the ``weights'' of the analogous neural network in the learning tomography approach \cite{kamilov2015learning,shoreh2017optical,lim2018learning}. BPM has also been used with more traditional sparsity-based inverse methods\ \cite{kamilov2016optical,chowdhury2019high}. Later, a machine learning approach with a convolutional neural network (CNN) replacing the iterative gradient descent algorithm exhibited even better robustness to strong scattering for layered objects, which match well with the BPM assumptions \cite{goy2019high}. Despite great progress reported by these prior works, the problem of reconstruction through multiple scattering remains difficult due to the extreme ill-posedness and uncertainty in the forward operator; residual distortion and artifacts are not uncommon in experimental reconstructions.
Inverse scattering, as inverse problems in general, may be approached in a number of different ways to regularize the ill-posedness and thus provide some immunity to noise \cite{bertero1998introduction,candes2006robust}. Recently, thanks to a ground-breaking observation from 2010 that sparsity can be learnt by a deep neural network \cite{gregor2010learning}, the idea of using machine learning to approximate solutions to inverse problems also caught on \cite{barbastathis2019use}. In the context of tomography, in particular, deep neural networks have been used to invert the Radon transform \cite{jin2017deep} and recursive Born model \cite{kamilov2016recursive}, and were also the basis of some of the papers we cited earlier on holographic 3D reconstruction\ \cite{wu2019bright,rivenson2018phase,rivenson2019deep}, learning tomography\ \cite{kamilov2015learning,shoreh2017optical,lim2018learning}, and multi-layered strongly scattering objects\ \cite{goy2019high}. In prior work on tomography using machine learning, generally, the intensity projections are all fed as inputs to a computational architecture that includes a neural network, and the output is the 3D reconstruction of the refractive index. The role of the neural network is to learn the priors that apply to the particular class of objects being considered and the relationship of these priors to the forward operator (Born, BPM, etc.) so as to produce a reasonable estimate of the inverse.
Here we propose a rather distinct approach to exploit machine learning for 3D refractive index reconstruction under strong scattering conditions. Our motivation is that, as the angle of illumination is changed, the light goes through {\em the same scattering volume,} but the scattering events follow a different sequence. At the same time, the intensity diffraction pattern obtained from a new angle of illumination adds information to the tomographic problem, but that information is constrained by ({\it i.e.}, is not orthogonal to) the previously obtained patterns. We interpret this as similar to a dynamical system, where as time evolves and new inputs arrive, the output is constrained by the history of earlier inputs. (The convolution integral is the simplest and best known expression of this relationship between the output of a system and the history of the system's input.)
The analogy between strong scattering tomography and a dynamical system suggests the recurrent neural network (RNN) architecture as a strong candidate to process intensity diffraction patterns in sequence, as they are obtained one after the other; and process them recurrently so that each intensity diffraction pattern from a new angle improves over the reconstructions obtained from the previous angles. Thus, we treat multiple diffraction patterns under different illumination angles as a temporal sequence, as shown in Figure~\ref{fig:introduction}. The angle index $\theta$ replaces what in a dynamical system would have been the time $t$. This idea is intuitively appealing; it also leads to considerable improvement in the reconstructions, removing certain artifacts that were visible in \cite{goy2019high}, as we will show in section~\ref{sec:results}.
The way we propose to use RNNs in this problem is quite distinct from the recurrent architecture proposed first in \cite{gregor2010learning} and subsequently implemented, replacing the recurrence by a cascade of distinct neural networks, in \cite{jin2017deep,inv:mardani2017a,inv:mardani2017b}, among others. In these prior works, the input to the recurrence can be thought of as clamped to the raw measurement, as in the proximal gradient \cite{inv:daubechies04} and related methods; whereas, in our case, the input to the recurrence is itself dynamic, with the raw intensity diffraction patterns from different angles forming the input sequence. Moreover, by utilizing a modified gated recurrent unit (more on this below) rather than a standard neural network, we do not need to break the recurrence up into a cascade.
Typical applications of RNNs \cite{williams1989learning,hochreiter1997long} are in temporal sequence learning and identification. In imaging and computer vision, RNN is applied in 2D and 3D: video frame prediction \cite{xingjian2015convolutional,wang2018eidetic,wang2017predrnn,wang2018predrnn++}, depth map prediction \cite{cs2018depthnet}, shape inpainting \cite{wang2017shape}; and stereo reconstruction \cite{liu2020novel,choy20163d} or segmentation \cite{le2017multi,stollenga2015parallel} from multi-view images, respectively. Stereo, in particular, bears certain similarities to our tomographic problem here, as sequential multiple views can be treated as a temporal sequence. To establish the surface shape, the RNNs in these prior works learn to enforce consistency in the raw 2D images from each view and resolve the redundancy between adjacent views in recursive fashion through the time sequence ({\it i.e.}, the sequence of view angles). Non-RNN learning approaches have also been used in stereo, e.g. Gaussian mixture models\ \cite{hou2019multi}.
In this work, we replaced the standard long-short term memory (LSTM)\ \cite{hochreiter1997long} implementation of RNNs with a modified version of the newer gated recurrent unit (GRU) \cite{cho2014learning}. The GRU has the advantage of fewer parameters but generalizes comparably with the LSTM. Our GRU employs a split convolutional scheme to explicitly account for the asymmetry between the lateral and axial axes of propagation, and an angular attention mechanism that learns how to reward specific angles in proportion to their contribution to reconstruction quality. For isotropic (in the ensemble sense) samples as we consider here, it turns out that the attention mechanism treats all angles equally, yet we found that its presence still improves the quality of the training algorithm. For more general sample classes with spatially anisotropic structure, angular attention may be expected to treat different angles of illumination with more disparity.
Details in experiments are delineated in Section~\ref{sec:experiment}. The computational elements are all described in Section~\ref{sec:comput_arch}, while training and testing procedures are illustrated in Section~\ref{sec:train_and_test}. The results of our experimental study are in Section~\ref{sec:results}, showing significant improvement over static neural network-based reconstructions of the same data both visually and in terms of several quantitative metrics. We also include results from an ablation study that indicates the relative significance of the new components we introduced to the quality of the reconstructions.
\iffalse Performance of our RNN architecture is qualitatively and quantitatively compared with the baseline from earlier works in Section~\ref{sec:results}.\ref{subsec:comparison_baseline}. An ablation study to quantify contribution of each element to the overall performance is given in Section~\ref{sec:results}.\ref{subsec:ablation}. Section~\ref{sec:results}.\ref{subsec:number_of_patterns} shows how quality of reconstructions is incrementally enhanced as the number of patterns that enters the trained network for testing increases. Finally, in Section~\ref{sec:conclusion} we share some concluding thoughts and suggestions for future work.
\fi
\section{Experiment} \label{sec:experiment}
\begin{figure}[t!]
\centering
\includegraphics[width=\linewidth]{optical_apparatus.pdf}
\caption{Optical apparatus used for experimental data acquisition\ \protect{\cite{goy2019high}}. L$1-4$: lenses, F$1$: pinhole, A$1$: aperture, EM-CCD: electron-multiplying charge coupled device. $f_{\text{L}_3}:f_{\text{L}_4} = 2:1$. The object is rotated along both $x$ and $y$ axes. The defocus distance between the conjugate plane to the exit object surface and the EM-CCD is $\Delta z = 58.2\:\text{mm}$.}
\label{fig:optical_apparatus}
\end{figure}
The experimental data are the same as in \cite{goy2019high}, whose experimental apparatus is summarized in Figure~\ref{fig:optical_apparatus}. We repeat the description here for the readers' convenience. The He-Ne laser (Thorlabs HNL210L, power: $20\:\text{mW}$, $\lambda = 632.8\:\text{nm}$) illuminated the sample after spatial filtering and beam expansion. The illumination beam was then de-magnified by the telescope ($f_{\text{L}_3} : f_{\text{L}_4} = 2:1$), and the EM-CCD (Rolera EM-C$2$, pixel pitch: $8\:\mu\text{m}$, acquisition window dimension: $1002\:\times\:1004$) captured the experimental intensity diffraction patterns. The integration time for each frame was $2\:\text{ms}$, and the EM gain was set to $\times 1$. The optical power of the laser was strong enough for the captured intensities to be comfortably outside the shot-noise limited regime.
Each layer of the sample was made of fused silica slabs ($n=1.457$ at $632.8$ nm and at $20\:^\circ$C). Slab thickness was $0.5\text{mm}$, and patterns were carefully etched to the depth of $575\pm 5$ nm on the top surface of each of the four slabs. To reduce the difference between refractive indices, gaps between adjacent layers were filled with oil ($n = 1.4005\pm 0.0002$ at $632.8$ nm and at $20^\circ$C), yielding binary phase depth of $-0.323\pm 0.006\:\text{rad}$. The diffraction patterns used for training were prepared with simulation precisely matched to the apparatus of Figure~\ref{fig:optical_apparatus}. For testing, we used a set of diffraction patterns that was acquired experimentally.
Objects used for both simulation and experiment are dense-layered, transparent, \textit{i.e.} of negligible amplitude modulation, and of binary refractive index. They were drawn from a database of IC layout segments\ \cite{goy2019high}. The feature depth of $575\pm 5\:\text{nm}$ and refractive index contrast $0.0565\pm0.0002$ at $632.8$ nm and at $20\:^\circ$C were such that weak scattering assumptions are invalid and strong scattering has to be necessarily taken into account. The Fresnel number ranged from $0.7$ to $5.5$ for the given defocus amount $\Delta z=58.2\:\text{mm}$ for the range of object feature sizes.
To implement the raw image acquisition scheme, the sample was rotated from $-10$ degree to $10$ degree with a $1$-degree increment along both the $x$ and $y$ axes, while the illumination beam and detector remained still. This resulted in $N=42$ angles and intensity diffraction patterns in total (see Section~\ref{sec:comput_arch}.\ref{subsec:conv_enc_dec}). Note that \cite{goy2019high} only utilized $22$ patterns out of with a $2$-degree increment along both $x$ and $y$ axes. The comparisons we show later are still fair because we retrained all the algorithms of \cite{goy2019high} for the $42$ angles and $1^\circ$ increment.
\section{Computational architecture}\label{sec:comput_arch}
\begin{figure*}[t!]
\centering
\includegraphics[width=\textwidth]{architecture.pdf}
\caption{Details on implementing the dynamical scheme of Figure~\protect{\ref{fig:introduction}}. (a) Overall network architecture; (b) tensorial dimensions of each layer; (c) down-residual block (DRB); (d) up-residual block (URB); and (e) residual block (RB). $K$ and $S$ indicate the sizes of kernel and stride, respectively, and the values shown apply only to the row and column axes. For the layer axis, $K=4$ and $S=1$ always. The disparities are to implement the split convolution scheme; please see Section~\protect{\ref{sec:comput_arch}}.\protect{\ref{subsec:sc_gru}} and Figure~\protect{\ref{fig:split_convolution}}.}
\label{fig:architecture}
\end{figure*}
The proposed RNN architecture is shown in detail in Figure~\ref{fig:architecture}. The forward model and gradient descent Approximant (pre-processing) algorithm are described in Section~\ref{subsec:approximants}. The split-convolutional GRU, convolutional encoder and decoder, and the angular attention mechanism are described in Sections~\ref{subsec:sc_gru}, \ref{subsec:conv_enc_dec}, and \ref{subsec:angular_att}, respectively. The total number of parameters in this computational architecture is $\sim 21\text{M}$ (more on this topic in section~\ref{sec:train_and_test}.\ref{subsec:training_rnn}).
\subsection{Approximant computations}\label{subsec:approximants}
The dense-layered, binary-phase object is illuminated at a sequence of angles, and the corresponding diffraction intensity patterns are captured by a detector. At the $n$-th step of the sequence, the object is illuminated by a plane wave at angles $\left(\theta_{nx},\theta_{ny}\right)$ with respect to the propagation axis $z$ on the $xz$ and $yz$ planes, respectively. Beyond the object, the scattered field propagates in free space by distance $\Delta z$ to the digital camera (the numerical value is $\Delta z=58.2$mm, as we saw in section~\ref{sec:experiment}). Let the forward model under the $n$-th illumination angle be denoted as $H_n$, $n=1,2,\ldots, N$; that is, the $n$-th intensity diffraction pattern at the detector plane produced by the phase object $\mathbf{f}$ is $\mathbf{g}_n\equiv H_n(\mathbf{f})$.
In the simulations, the forward operators $H_n$ are obtained from the non-paraxial beam propagation method (BPM) \cite{feit1980computation,goy2019high,kamilov2016optical}. Let the $j$-th cross-section of the computational window perpendicular to $z$ axis be $f^{[j]} = \exp\left(i\varphi^{[j]}\right),\: j=1,\ldots,J$ where $J$ is the number of slices the we divide the object into, each of axial extent $\delta z$. At the $n$-th illumination angle, the BPM is initialized as $f_n^{[0]}=\text{exp}\left[ik\left(x\sin\theta_{nx}+y\sin\theta_{ny}\right)\right]$, where $k$ is the wavenumber. The optical field at the $(j+1)$-th slice is
\begin{equation}\label{eq:BPM-iteration}
\begin{split}
\psi_n^{[j+1]} = \mathcal{F}^{-1}&\bigg[\mathcal{F}\left[\psi_n^{[j]}\circ f_n^{[j]}\right](k_x,k_y)\\
&\cdot\exp\left(-i\left(k-\sqrt{k^2-k_x^2-k_y^2}\right)\delta z\right)\bigg],
\end{split}
\end{equation}
where $\delta z$ is equal to the slab thickness, \textit{i.e.} $0.5\:\text{mm}$; ${\cal F}$ and ${\cal F}^{-1}$ are the Fourier and inverse Fourier transforms, respectively; and $\chi_1\circ\chi_2$ denotes the Hadamard (element-wise) product of the functions $\chi_1$, $\chi_2$. The Hadamard product is the numerical implementation of the thin transparency approximation, which is inherent in the BPM. To obtain the intensity at the detector, we define the $(J+1)$-th slice displaced by $\Delta z$ from the $J$-th slice (the latter is the exit surface of the object) to yield
\begin{equation}\label{eq:forw}
\mathbf{g}_n\equiv H_n(\mathbf{f})=\left|\psi_n^{[J+1]}\right|^2.
\end{equation}
The purpose of the Approximant, in general, is to produce a crude estimate of the volumetric reconstruction using the forward operator alone. This has been well established as a helpful form of preprocessing for subsequent treatment by machine learning algorithms\ \cite{goy2018low,goy2019high}. Previous works constructed the Approximant as a single-pass gradient descent algorithm \cite{kamilov2016optical,goy2019high}. Here, due to the sequential nature of our reconstruction algorithm, as each intensity diffraction pattern from a new angle of illumination $n$ is received, we instead construct a sequence of Approximants, indexed by $n$, by minimizing the functionals
\begin{equation}\label{eq:new_loss_function}
\mathcal{L}_n(\mathbf{f}) = \frac{1}{2}||H_n(\mathbf{f})-\mathbf{g}_n||_2^2,\quad n=1,2,\ldots,N.
\end{equation}
The gradient descent update rule for this functional is
\begin{multline}\label{eq:new_approximants}
\mathbf{f}_n^{[l+1]} = \mathbf{f}_n^{[l]} -s\left(\nabla_\mathbf{f}\mathcal{L}_n\left(\mathbf{f}_n^{[l]}\right)\right)^\dagger = \\
= \mathbf{f}_n^{[l]} -s\left(H_n^T\left(\mathbf{f}^{[l]}\right)\nabla_\mathbf{f}
H_n\left(\mathbf{f}_n^{[l]}\right)-\mathbf{g}_n^T\nabla_\mathbf{f}H_n\left(\mathbf{f}_n^{[l]}\right)\right)^\dagger,
\end{multline}
where $\mathbf{f}_n^{[0]}=\mathbf{0}$ and $s$ is the descent step size and in the numerical calculations was set to $0.05$ and the superscript $\dagger$ denotes the transpose. The single-pass, gradient descent-based Approximant was used for training of the RNN but with an additional pre-processing step that will be explained in (\ref{eq:moving_window}).
We also implemented a denoised Total Variation (TV) based Approximant, to be used only at the testing stage of the RNN. In this case, the functional to be minimized is
\begin{equation}\label{eq:TV_Approx}
\mathcal{L}^{\text{TV}}_n(\mathbf{f}) = \frac{1}{2} ||H_n(\mathbf{f})-\mathbf{g}||_2^2 + \kappa\text{TV}_{l_1}(\mathbf{f}),\quad n=1,2,\ldots,N,
\end{equation}
where the TV-regularization parameter was chosen as $\kappa=10^{-3}$, and for $\mathbf{x}\in \mathcal{R}^{P\times Q}$ the anisotropic $l_1$-TV operator is
\begin{equation}
\begin{split}
\text{TV}_{l_1}(\mathbf{x}) = &\sum_{p=1}^{P-1}\sum_{q=1}^{Q-1} \Big(\left|x_{p,q} - x_{p+1,q}\right| + \left|x_{p,q} - x_{p,q+1}\right|\Big)\\
& + \sum_{p=1}^{P-1} \left|x_{p,Q}-x_{p+1,Q}\right| +\sum_{q=1}^{Q-1} \left|x_{P,q}-x_{P,q+1}\right|
\end{split}
\end{equation}
with reflexive boundary conditions \cite{beck2009fast,chambolle2004algorithm}. To produce the Approximants for testing from this functional, we first ran $3$ iterations of the gradient descent and ran $2$ iterations of the FGP-FISTA (Fast Gradient Projection with Fast Iterative Shrinkage Thresholding Algorithm)\ \cite{beck2009fast,beck2009fista}.
The sequence of $N$ Approximants for either training or testing procedure is a $4$D spatiotemporal sequence $\mathbf{F}=\left(\mathbf{f}_1^{[1]},\mathbf{f}_2^{[1]},\ldots,\mathbf{f}_N^{[1]}\right)$. As an additional processing step, to suppress unwanted artifacts in the Approximants of the experimentally captured intensities $\mathbf{g}_n$, we reduce the sequence size to $M$ by applying a moving average window as
\begin{equation}\label{eq:moving_window}
\tilde{\mathbf{f}}_m^{[1]} =
\begin{dcases}
\frac{1}{N_{\text{w}}+1}\sum_{n=m}^{m+N_{\text{w}}} \mathbf{f}_n^{[1]}, & 1\leq m\leq N_{\text{h}}\\
\frac{1}{N_{\text{w}}+1}\sum_{n=m}^{m+N_{\text{w}}} \mathbf{f}_{n+N_{\text{w}}}^{[1]}, & N_{\text{h}}+1\leq m\leq M.
\end{dcases}
\end{equation}
To be consistent, the moving average window was applied to the Approximants for both training and testing. In this study, $N_{\text{w}}=15$, $N_{\text{h}}=6$ and $M=12$. These choices follow from the following considerations. We have $N=42$ diffraction patterns for each sequence: $21$ captured along the $x$ axis ($1-21$) and the remaining ones along the $y$ axis ($22-42$). The window is first applied to $21$ patterns from $x$-axis rotation, which thus generates $6$ averaged diffraction patterns, and then the window is applied to the remaining $21$ patterns from $y$-axis rotation, resulting in the other $6$ patterns. Therefore, the input sequence to the next step in the architecture of Figure~\ref{fig:architecture}, {\it i.e.} to the encoder (Section~\ref{subsec:conv_enc_dec}), consists of a sequence of $M=12$ averaged Approximants~$\tilde{\mathbf{f}}_m^{[1]}$.
\subsection{Split-convolutional gated recurrent unit (SC-GRU)}\label{subsec:sc_gru}
Recurrent neural networks involve a recurrent unit that retains memory and context based on previous inputs in a form of latent tensors or hidden units. It is well known that the Long Short-Term Memory (LSTM) is robust to instabilities in the training process. Moreover, in the LSTM, the weights applied to past inputs are updated according to usefulness, while less useful past inputs are forgotten. This encourages the most salient aspects of the input sequence to influence the output sequence\ \cite{hochreiter1997long}. Recently, the Gated Recurrent Unit (GRU) was proposed as an alternative to LSTM. The GRU effectively reduces the number of parameters by merging some operations inside the LSTM, without compromising quality of reconstructions; thus, it is expected to generalize better in many cases\ \cite{cho2014learning}. For this reason, we chose to utilize the GRU in this paper as well.
The governing equations of the standard GRU are as follows:
\begin{equation}\label{eq:gru_equations}
\begin{gathered}
r_m = W_r \xi_m + U_r h_{m-1}+b_r\\
z_m = W_z \xi_m + U_zh_{m-1} + b_z\\
\Tilde{h}_m = \text{tanh}\left(W\xi_m+U\left(r_m\circ h_{m-1}\right)+b_h\right)\\
h_m = (1-z_m)\circ \Tilde{h}_m + z_m\circ h_{m-1},
\end{gathered}
\end{equation}
where $\xi_m$, $h_m$, $r_m$, $z_m$ are the inputs, hidden features, reset states, and update states, respectively. Multiplication operations with weight matrices are performed in a fully connected fashion.
We modified this architecture so as to take into account the asymmetry between the lateral and axial dimensions of optical field propagation. This is evident even in free-space propagation, where the lateral components of the Fresnel kernel
\[
\expb{i\pi\frac{x^2+y^2}{\lambda z}}
\]
are shift invariant and, thus, convolutional, whereas the longitudinal axis $z$ is not. The asymmetry is also evident in nonlinear propagation, as in the BPM forward model (\ref{eq:BPM-iteration}) that we used here. This does not mean that space is anisotropic --- of course space is isotropic! The asymmetry arises because propagation and the object are 3D, whereas the sensor is 2D. In other words, the orientation of the image plane breaks the symmetry in object space so that the scattered field from a certain voxel within the object {\em apparently} influences the scattered intensity from its neighbors at the detector plane differently in the lateral direction than in the axial direction. To account for this asymmetry in a profitable way for our learning task, we first define the operators $W_r$, $U_r$, etc. as convolutional so as to keep the number of parameters down (even though in free space propagation the axial dimension is not convolutional and under strong scattering neither dimension is nonlinear); and we constrain the convolutional kernels of the operators to be the same in the lateral dimensions $x$ and $y$, and allow the axial $z$ dimension kernel to be different. This approach justifies the term Split-Convolutional, and we found it to be a good compromise between facilitating generalization and adhering to the physics of the problem.
\begin{figure}[htbp!]
\centering
\includegraphics[width=\linewidth]{split_convolution_figure.pdf}
\caption{Split convolution scheme: different convolution kernels are applied along the lateral $x,y$ axes {\it vs.} the longitudinal $z$ axis. In our present implementation, the kernels' respective dimensions are $3 \times 3 \times 1$ (or $1 \times 1 \times 1$) and $1 \times 1 \times 4$. The lateral and longitudinal convolutions are computed separately and the results are then added element-wise. The split convolution scheme is used in both the gated recurrent unit (Section~\protect{\ref{sec:comput_arch}}.\protect{\ref{subsec:sc_gru}}) and the encoder/decoder (Section~\protect{\ref{sec:comput_arch}}.\protect{\ref{subsec:conv_enc_dec}}).}
\label{fig:split_convolution}
\end{figure}
We also replaced the tanh activation function of the standard GRU with a rectified linear unit (ReLU) activation \cite{dey2017gate} as the ReLU is computationally less expensive and helpful to avoid local minima with fewer vanishing gradient problems \cite{nair2010rectified,glorot2011deep}. The final form of our SC-GRU dynamics is
\begin{equation}\label{eq:new_gru_equations}
\begin{gathered}
r_m = W_r*\xi_m + U_r*h_{m-1}+b_r\\
z_m = W_z*\xi_m + U_z*h_{m-1} + b_z\\
\Tilde{h}_m = \text{ReLU}\left(W*\xi_m+U*\left(r_m\circ h_{m-1}\right)+b_h\right)\\
h_m = (1-z_m)\circ \Tilde{h}_m + z_m\circ h_{m-1},
\end{gathered}
\end{equation}
where $*$ denotes our split convolution operation.
\subsection{Convolutional encoder and decoder}\label{subsec:conv_enc_dec}
Convolutional neural networks (CNNs) are placed before and after the SC-GRU as encoder and decoder, respectively. This architectural choice was inspired by \cite{sinha2017lensless,gehring2016convolutional,hori2017advances,zhao2017learning}. The encoder and decoder also utilize split convolution, as shown in Figure~\ref{fig:split_convolution}, in conjunction with residual learning, which is known to improve generalization in deep networks\ \cite{he2016deep}. As in \cite{sinha2017lensless}, the encoder and decoder utilize down-residual blocks (DRB), up-residual blocks (URB), and residual blocks (RB); however, there are no skip connections in our case, {\it i.e.} this is not a U-net\ \cite{ronneberger2015u} architecture. The encoder learns how to map its input ({\it i.e.} the $\tilde{\mathbf{f}}_m^{[1]}$ sequence) onto a low-dimensional nonlinear manifold. The compression factor is $16$ for the lateral input dimensions, but the axial dimension is left intact, as shown in Figure~\ref{fig:architecture}. This eases the burden on the training process as the number of parameters is reduced; more importantly, encoding abstracts features out of the high-dimensional inputs, passing latent tensors over to the recurrent unit. Letting the encoder for the $m$-th angle Approximant be symbolized as $\text{Enc}_m\left(\cdot\right)$, $\xi_m = \text{Enc}_m\left(\tilde{\mathbf{f}}_m^{[1]}\right)$ in (\ref{eq:new_gru_equations}). The decoder restores the output of the RNN to the native dimension of the object we are reconstructing.
\subsection{Angular attention mechanism}\label{subsec:angular_att}
Each intensity diffraction pattern from a new angle of illumination is combined at the SC-GRU input with the hidden feature $h_m$ from the same SC-GRU's previous output. After $M$ iterations, there are $M$ different hidden features resulting from $N$ illumination angles, as seen in (\ref{eq:moving_window}). Since the forward operator $H_n(\mathbf{f})$ is object dependent, the qualitative information that each such new angle conveys will vary with the object. It then becomes interesting to consider whether some angles of illumination convey more information than others.
The analogue in temporal dynamical systems, the usual domain of application for RNNs, is the {\em attention} mechanism. It decides which elements of the system's state are the most informative. In our case, of course, time has been replaced by the angle of illumination, so we refer to the same mechanism as {\em angular attention:}\ it evaluates the contents of the previously received intensity diffraction patterns from different angles of illumination and assigns to each a compatibility function $e_m$, essentially a weight that is relevant to that illumination's importance for the overall reconstruction.
Following the summation style attention mechanism\ \cite{bahdanau2014neural}, we compute the compatibility function $e_m$ as output of a neural network with hidden units (layers) $V_e$, $W_e$ and the weights $\alpha_m$ from the compatibility function as
\begin{equation}\label{eq:attention-VeWe}
\begin{gathered}
e_m = V_e\:\text{tanh}\left(W_e h_m\right), \\
\alpha_m = \text{softmax}\left(e_m\right) = \frac{\text{exp}(e_m)}{\sum_{m=1}^{M} \text{exp}(e_m)}, \\
\quad m = 1,2,\ldots, M.
\end{gathered}
\end{equation}
The final angular attention output $a$ is then computed from a linear combination of the hidden features as
\begin{equation}\label{eq:attention}
a=\sum_{m=1}^{M} \alpha_m h_m.
\end{equation}
For the ablation study of Section~\ref{sec:results}, only the last hidden feature $h_M$ is passed on to the decoder, {\it i.e.} the angular attention mechanism is not used. There is an alternative, dot-product attention mechanism\ \cite{vaswani2017attention}, but we chose not to implement it here.
\section{Training and testing procedures}\label{sec:train_and_test}
\subsection{Training the recurrent neural network}\label{subsec:training_rnn}
For training and validation, $5000$ and $500$ layered objects were used, respectively. For each object, a sequence of intensity diffraction patterns from the $N=42$ angles of illumination was produced by BPM, as described earlier. The Approximants were obtained each as a single iteration of the gradient descent. All of the architectures were trained for $100$ epochs with a training loss function (TLF) of negative Pearson correlation coefficient (NPCC) \cite{li2018imaging}, defined as
\begin{equation}
\sst{\mathcal{E}}{NPCC}\big(f,\hat{f}\big) \equiv -\:\frac{\displaystyle{\sum_{x,y}}\Big(f(x,y)-\big<f\big>\Big)\Big(\hat{f}(x,y)-\big<\hat{f}\big>\Big)}{\sqrt{\displaystyle{\sum_{x,y}}\Big(f(x,y)-\big<f\big>\Big)^2}\sqrt{\displaystyle{\sum_{x,y}}\Big(\hat{f}(x,y)-\big<\hat{f}\big>\Big)^2}}, \label{eq:tlf-npcc}
\end{equation}
where $f$ and $\hat{f}$ are a ground truth image and its corresponding reconstruction. In this article, our NPCC function was defined to perform computation in $3$D. We used a stochastic gradient descent scheme with the \textit{Adam} optimizer \cite{kingma2014adam}. The learning rate was set to be $10^{-3}$ initially and halved whenever validation loss plateaued for $5$ consecutive epochs. Batch size was set to be $10$. The desktop computer used for training has Intel Xeon W-$2295$ CPU at $3.00$ GHz with $24.75$ MB cache, $128$ GB RAM, and dual NVIDIA Quadro RTX $8000$ GPUs with $48$ GB VRAM.
For comparison, we also re-trained the $3$D-DenseNet architecture with skip connections in \cite{goy2019high} with the same training scheme above, \textit{i.e.} on \textit{Adam} for $100$ epochs and with batch size of $10$ and the same learning rate initial value and halving strategy. This serves as baseline; however, the number of parameters in this network is $0.5\:\text{M}$, whereas in our RNN architecture the number of parameters is $21\:\text{M}$. We also trained an enhanced version of the $3$D-DenseNet by tuning the number of dense blocks, the number of layers inside each dense block, filter size, and growth rate to match the total number of parameters with that of the RNN, {\it i.e.} $21\:\text{M}$. In the next section, we refer to these two versions of the $3$D-DenseNet as Baseline ($0.5\:\text{M}$) and Baseline ($21\:\text{M}$), respectively.
\subsection{Testing procedures and metrics}
A simple affine transform is first applied to the raw experimentally obtained intensity diffraction patterns to correct slight misalignment. Then we run the gradient descent method up to $3$ iterations of the gradient descent (\ref{eq:new_approximants}) and the FGP-FISTA up to $2$ iterations, to test the trained network using the TV-based Approximants (\ref{eq:TV_Approx}).
Even though training used NPCC as in (\ref{eq:tlf-npcc}), we investigated two additional metrics for testing: probability of error (PE), the Wasserstein distance \cite{villani2003topics,kolouri2017optimal}. We also quantified test performance using the SSIM (Structural Similarity Index Metric) \cite{wang2004image}, shown in the Supplementary material.
PE is the mean absolute error between two binary objects; in the digital communication community it is instead referred to as Bit Error Rate (BER). To obtain the PE, we first threshold the reconstructions and then define
\begin{equation}
\text{PE} = \frac{\left(\text{\# false negatives}\right)\: + \:\left(\text{\# false positives}\right)}{\text{total \# pixels}}.
\end{equation}
We found that it oftentimes helps to accentuate the differences between a binary phase ground truth object and its binarized reconstruction as even small residual artifacts, if they are above the threshold, are thresholded to be one, and thus they are taken into account to the probability of error calculation more than they would have been to other metrics. With these procedures, PE is a particularly suitable error metric for the kind of objects we consider in this paper.
PE is also closely related to the two-dimensional Wasserstein distance as we will now show through an analytical derivation. The latter metric involves an optimization process in terms of a transport plan to minimize the total cost of transport from a source distribution to a target distribution. The two-dimensional Wasserstein distance is defined as
\begin{equation}
\begin{gathered}
W_{p=1} = \min_P \langle P,C\rangle = \min_P\sum_{ij}\sum_{kl}\gamma_{ij,kl}C_{ij,kl},\\
\text{s.t.}\:\: \sum_{kl}\gamma_{ij,kl} = f_{ij},\: \sum_{ij}\gamma_{ij,kl}=g_{kl},\:\gamma_{ij,kl}\geq 0,
\end{gathered}
\end{equation}
where $f_{ij}$ and $g_{kl}$ are a ground truth binary object and its binary reconstruction, \textit{i.e.} $f_{ij}, g_{kl}, \gamma_{ij,kl} \in\{0,1\}$, a coupling tensor $P=\left(\gamma_{ij,kl}\right)$, and a cost tensor $C_{ij,kl}=\left|x_{ij}-x_{kl}\right|$. PE can be reduced to have a similar, but not equivalent, form to that of the Wasserstein distance. For $i,j,k,l$ where $\gamma_{ij,kl}\neq 0$,
\begin{equation}\label{eq:prob}
\begin{split}
\text{PE} &= \frac{1}{N^2}\sum_{ij}\left|f_{ij}-g_{ij}\right|\\
&= \frac{1}{N^2}\sum_{ij}\left|f_{ij}-\sum_{kl}g_{kl}\:\delta\left[i-k,j-l\right]\right|\\
&= \frac{1}{N^2}\sum_{ij}\left|\sum_{kl}\gamma_{ij,kl}\left(1-\frac{g_{kl}\:\delta\left[i-k,j-l\right]}{\gamma_{ij,kl}}\right)\right|\\
&\equiv \sum_{ij}\left|\sum_{kl}\gamma_{ij,kl}\Tilde{C}_{ij,kl}\right|\\
&= \sum_{ij,kl;\gamma_{ij,kl}\neq 0} \gamma_{ij,kl}\Tilde{C}_{ij,kl}, \qquad \text{where}
\end{split}
\end{equation}
\begin{equation}
N^2\Tilde{C}_{ij,kl} = 1\!-\!\frac{g_{kl}\:\delta\left[i-k,j-l\right]}{\gamma_{ij,kl}}=
\begin{dcases}
\:1, & \:\text{if}\:\: ij \neq kl\\
\:1\! -\! g_{kl}, & \:\text{if}\:\: ij = kl.
\end{dcases}
\end{equation}
This shows that the PE is a version of the Wasserstein distance with differently defined cost tensor.
\section{Results}\label{sec:results}
\begin{figure*}[htbp]
\centering
\includegraphics[width=0.55\textwidth]{number_of_patterns.pdf}
\caption{Progress of 3D reconstruction performance as new windowed Approximants $m=1,\ldots, M\!\!=\!\!12$ according to (\protect{\ref{eq:moving_window}}) applied on experimental data are presented to the recurrent scheme. The same progression can be found in the Online Materials as a movie.}
\label{fig:number_of_patterns}
\end{figure*}
\begin{table*}[htbp!]
\begin{center}
\begin{tabular}{c||c c c c|c}
\hline
\textbf{Probability of error ($\%$)} ($\downarrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Baseline (0.5 M)} & 6.604 & 5.255 & 7.837 & 3.204 & 5.725\\
\text{Baseline (21 M)} & 6.604 & 5.725 & 5.652 & 2.856 & 5.209\\
\hdashline
\text{Proposed RNN (21 M)} & \textbf{5.408} & \textbf{4.828} & \textbf{2.332} & \textbf{1.660} & \textbf{3.557}\\
\hline\hline
\textbf{Wasserstein distance} ($\times\:10^{-2}$) ($\downarrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Baseline (0.5 M)} & 2.854 & 1.466 & 2.783 & 0.9900 & 2.023\\
\text{Baseline (21 M)} & 2.703 & 1.171 & 2.475 & 0.8112 & 1.790\\
\hdashline
\text{Proposed RNN (21 M)} & \textbf{1.999} & \textbf{1.093} & \textbf{1.749} & \textbf{0.6403} & \textbf{1.370}\\
\hline\hline
\textbf{PCC} ($\uparrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Baseline (0.5 M)} & 0.8818 & 0.6426 & 0.8658 & 0.6191 & 0.7523\\
\text{Baseline (21 M)} & 0.8859 & 0.6430 & 0.9021 & 0.6132 & 0.7611\\
\hdashline
\text{Proposed RNN (21 M)} & \textbf{0.8943} & \textbf{0.6612} & \textbf{0.9551} & \textbf{0.7039} & \textbf{0.8036}\\
\hline
\iffalse
\hline
\textbf{SSIM} ($\uparrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Baseline (0.5 M)} & 0.7606 & 0.7409 & 0.7299 & 0.8046 & 0.7590\\
\text{Baseline (21 M)} & 0.7702 & 0.7557 & 0.7978 & 0.8357 & 0.7899\\
\hdashline
\text{Proposed RNN (21 M)} & \textbf{0.7987} & \textbf{0.8128} & \textbf{0.8652} & \textbf{0.9154} & \textbf{0.8480}\\
\hline
\fi
\end{tabular}
\end{center}
\caption{Quantitative comparison between the baseline (static) and dynamic reconstruction from testing on experimental data, according to PE, Wasserstein distance ($p=1$), and PCC. SSIM comparisons are in the Supplementary materials.}
\label{tab:quantitative_comparison}
\end{table*}
Our RNN is first trained as described in Section~\ref{sec:train_and_test}, and then tested with the TV-based Approximants (\ref{eq:TV_Approx}) applied to the experimentally obtained diffraction patterns. The evolution of the RNN output as more input patterns are presented is shown in Figure~\ref{fig:number_of_patterns}. When the recurrence starts with $m=1$, the volumetric reconstruction is quite poor; as more orientations are included, the reconstruction improves as expected. A movie version of this evolution for $m=1,\ldots, M$ is included in the online materials.
\begin{figure*}[htbp!]
\centering
\includegraphics[width=0.72\textwidth]{qualitative_comparison.pdf}
\caption{Qualitative comparison on test performance between the baseline and proposed architectures using experimental data. The baseline architectures are $3$D-DenseNet CNN architectures with $0.5$ M and $21$ M parameters. Our proposed architecture is a recurrent neural network with elements described in Section~\ref{sec:comput_arch}.}
\label{fig:qualitative_comparison}
\end{figure*}
\begin{table*}[htbp!]
\begin{center}
\begin{tabular}{c||c c c c|c}
\hline
\textbf{Probability of error ($\%$)} ($\downarrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Proposed RNN (21 M)} & \textbf{5.408} & 4.828 & \textbf{2.332} & \textbf{1.660} & \textbf{3.557}\\
\hdashline
\text{-- ReLU activation (21 M)} & 6.262 & \textbf{4.718} & 3.241 & 1.904 & 4.031\\
\text{-- angular attention (21 M)} & 9.399 & 5.566 & 11.64 & 3.375 & 7.495\\
\text{-- split convolution (43 M)} & 9.674 & 6.342 & 14.43 & 2.405 & 8.212\\
\hline\hline
\textbf{Wasserstein distance} ($\times\:10^{-2}$) ($\downarrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Proposed RNN (21 M)} & \textbf{1.999} & \textbf{1.093} & \textbf{1.749} & \textbf{0.6403} & \textbf{1.370}\\
\hdashline
\text{-- ReLU activation (21 M)} & 2.291 & 1.156 & 1.886 & 0.6692 & 1.501\\
\text{-- angular attention (21 M)} & 3.016 & 1.587 & 3.672 & 1.063 & 2.335\\
\text{-- split convolution (43 M)} & 4.005 & 2.863 & 3.651 & 2.233 & 3.188\\
\hline\hline
\textbf{PCC} ($\uparrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Proposed RNN (21 M)} & \textbf{0.8943} & 0.6612 & \textbf{0.9551} & \textbf{0.7039} & \textbf{0.8036}\\
\hdashline
\text{-- ReLU activation (21 M)} & 0.8832 & \textbf{0.6836} & 0.9406 & 0.6725 & 0.7950\\
\text{-- angular attention (21 M)} & 0.8281 & 0.6252 & 0.8145 & 0.4657 & 0.6834\\
\text{-- split convolution (43 M)} & 0.8005 & 0.4525 & 0.7313 & 0.4910 & 0.6188\\
\hline
\iffalse
\hline
\textbf{SSIM} ($\uparrow$) & \text{Layer} 1 & \text{Layer} 2 & \text{Layer} 3 & \text{Layer} 4 & \text{Overall}\\
\hline
\text{Proposed RNN (21 M)} & \textbf{0.7987} & \textbf{0.8128} & \textbf{0.8652} & \textbf{0.9154} & \textbf{0.8480}\\
\hdashline
\text{-- ReLU activation (21 M)} & 0.7787 & 0.8088 & 0.8459 & 0.8971 & 0.8326\\
\text{-- angular attention (21 M)} & 0.6876 & 0.7175 & 0.6612 & 0.7826 & 0.7122\\
\text{-- split convolution (43 M)} & 0.6205 & 0.4740 & 0.5953 & 0.5274 & 0.5543\\
\hline
\fi
\end{tabular}
\end{center}
\caption{Quantitative assessment of ablation effects. Values inside the parentheses in the first column indicate the number of parameters. When we ablate split convolution, we rather choose $3\times 3\times 3$ being the uniform kernel, and, hence, the number of parameters increases. SSIM comparisons are in the Supplementary materials.}
\label{tab:ablation_study_quantitative}
\end{table*}
Visual comparisons with the baseline $3$D-DenseNets with $0.5$ M and $21$ M parameters are shown in Figure~\ref{fig:qualitative_comparison}. The RNN results show substantial visual improvement, with fewer artifacts and distortions compared to static approaches, e.g. \cite{goy2019high}. Quantitatively comparisons in terms of our chosen metrics PE, Wasserstein, and PCC are in Table~\ref{tab:quantitative_comparison}.
\begin{figure*}[t!]
\centering
\includegraphics[width=0.61\textwidth]{ablation_study_qualitative.pdf}
\caption{Visual quality assessment from the ablation study on elements described in Section~\ref{sec:comput_arch}. Rows $3-5$ show reconstructions based on experimental data for each layer upon ablation of ReLU activation (\ref{eq:new_gru_equations}), {\it i.e.}, using the more common tanh activation function instead (row 3); angular attention mechanism (row 4); and split convolution (row 5). The rows are ordered by increasing severity of the ablation effect.}
\label{fig:ablation_study_qualitative}
\end{figure*}
We conducted an ablation study, and its purpose is to isolate and compare quantitatively the contribution to the reconstruction of each element described in Figure~\ref{fig:architecture} and Section~\ref{sec:comput_arch}. We remove, one at a time, the split convolution, angular attention mechanism, and ReLU activation, and quantify performance again. Ablation in the case of ReLU activation means that we replace it with the tanh activation function, which is more usual. The ablated architectures are also trained under the same training scheme in Section~\ref{sec:train_and_test}.\ref{subsec:training_rnn} and tested with the same TV-based Approximants.
Visually, the ablation of the split convolution affects and degrades the testing performance worst, followed by the ablation of the angular attention mechanism and the ReLU activation. These findings are supported quantitatively as well in Table~\ref{tab:ablation_study_quantitative}. Note that the substitution of the ReLU with the tanh does not bring a large increase compared to others, but even slightly better in some case (see the probability of error of Layer $2$ in Table~\ref{tab:ablation_study_quantitative}). Thus, we find that (1) the split convolution should be considered to replace a general $3$D convolution when designing a recurrent unit and a convolutional encoder/decoder; (2) the angular attention mechanism is helpful when the inputs are formulated into temporal sequences; and (3) the choice of ReLU over tanh is still helpful but somewhat less significant and may be application-dependent. With respect to attention, in particular, even though the module's presence clearly contributes to good training quality, we found that the coefficients converge to $\alpha_m\approx 1/M$ for all $m$, consistent with the more-or-less angularly invariant class of samples---at least in the statistical sense, and for the small range of illumination angles that we used. A more detailed study of the angular attention module can be found in the Supplementary Material.
\section{Conclusions and discussion}\label{sec:conclusion}
We have proposed a radically new recurrent neural network scheme for processing raw inputs from different angles of illumination dynamically, {\it i.e.} as a sequence, with each new angle improving the 3D reconstruction. We have found this scheme to offer significant qualitative and quantitative improvement over static machine learning schemes, where the raw inputs from all angles are processed at once by a neural network. Through an ablation study, we found that sandwiching the recurrent structure between a convolutional encoder/decoder helps improve the reconstructions. Even more interestingly, an angular attention mechanism, rewarding raw inputs from certain angles as more informative and penalizing others, also contributes significantly to improving reconstruction fidelity albeit less than the encoder/decoder pair.
Even though we used the dynamic machine learning approach in the most difficult case of 3D reconstruction when strong scattering is present, there is no reason to doubt that it would be applicable to less ill-posed cases as well, e.g. optical diffraction tomography and Radon inverse. Also possible are alternative implementations of the RNN, e.g. with LSTMs or Reservoir Computing \cite{lukovsevivcius2009reservoir,lukovsevivcius2012reservoir,schrauwen2007overview}, and further exploration of split convolutional variants or DenseNet variants for the encoder/decoder and dynamical units; we leave these investigations to future work.
\section{Funding}
\noindent Southern University of Science and Technology (6941806); Intelligence Advanced Research Projects Activity (FA8650-17-C-9113); Korea Foundation for Advanced Studies.~\\
\section{Acknowledgments}
\noindent I. Kang acknowledges partial support from KFAS (Korea Foundation for Advanced Studies) scholarship. We are grateful to Jungmoon Ham for her assistance with drawing Figures~\ref{fig:introduction} and \ref{fig:split_convolution}, and to Subeen Pang, Mo Deng and Peter So for useful discussions and suggestions.~\\
\noindent\textbf{Disclosures.} The authors declare no conflicts of interest.
| 2024-02-18T23:39:40.296Z | 2020-07-22T02:14:35.000Z | algebraic_stack_train_0000 | 21 | 7,788 |
|
proofpile-arXiv_065-210 | \section{Introduction} \label{sec:intro}
Linear waves in an inviscid, perfectly-conducting fluid
permeated by a uniform magnetic field $\mib{B}_0$
in a frame rotating with rate $\mib{\Omega}$ satisfy the dispersion relation \citep{L54}
\begin{equation}
\omega = \pm \frac{\mib{\Omega}\cdot\mib{k}
\pm \sqrt{(\mib{\Omega}\cdot\mib{k})^2
+ |\mib{k}|^2 (\mib{B}_0 \cdot\mib{k})^2/\rho\mu_0} }
{|\mib{k}|} \; , \label{eq:MC_dispersion_relation_general}
\end{equation}
where $\omega$ is the frequency, $\mib{k}$ is the wavenumber vector,
$\rho$ is the density, and $\mu_0$ the magnetic permeability.
This yields a wide variety of magnetic Coriolis (MC) waves,
including fast (modified inertial) and slow (magnetostrophic) waves; the latter being
unique to rotating magnetohydrodynamics (MHD).
In this manuscript
we consider magnetostrophic waves
for which $(\mib{\Omega}\cdot\mib{k})^2/|\mib{k}|^2 \gg (\mib{B}_0\cdot\mib{k})^2/(\rho\mu_0)$:
In particular, one class which has the relation
\begin{equation}
\omega \approx - \frac{ (\mib{B}_0 \cdot \mib{k})^2 |\mib{k}|^2 }{ \rho\mu_0\beta k } \; .
\label{eq:slowMR}
\end{equation}
Here $\beta$ denotes the beta parameter, $k$ is the azimuthal wavenumber, and the minus sign
indicates waves travel opposite to the hydrodynamic Rossby wave, $\omega = \beta k/|\mib{k}|^2$.
This class is sometimes referred to as slow hydromagnetic-planetary or magnetic-Rossby (MR) waves \citep{H66}.
Relation (\ref{eq:slowMR}) indicates they are dispersive, and depend on the background field and the wavelength;
these waves have been suggested to be important
in Earth's fluid core and for the geomagnetic westward drift \citep[e.g.][]{H66,M67,CFF14,HJT15,NSKHH20}.
Other classes of MC waves include torsional Alfv\'{e}n waves
for which $\mib{\Omega}\cdot\mib{k} \approx 0$
and $(\mib{\Omega}\cdot\mib{k})^2/|\mib{k}|^2 \ll (\mib{B}_0\cdot\mib{k})^2/(\rho\mu_0)$
\citep{Bra70,RA12,GJF15}.
\revthree{More recently inertial-Alfv\'{e}n waves \citep{BD16} have been claimed
to account for the geomagnetic jerks \citep{AF19}.}
Laboratory experiments have identified several types of magnetostrophic waves
in spherical Couette flows with a dipolar magnetic field being applied \citep{SAetal08}.
We note the wave dynamics relies on both the direction and the morphology
of the background magnetic field,
as illustrated in the simple planar model (\ref{eq:slowMR}).
Here we focus on the problem with a purely azimuthal basic field;
for this case (\ref{eq:slowMR}) reduces to $\omega \propto k |\mib{k}|^2$\revthree{,
indicating its linear and cubic relationship to the azimuthal wavenumber.}
The linear theory for MC waves in stably-stratified, thin layers is well-studied
\citep[e.g.][]{Bra67,G00,ZOBS07,MJT17}
as observational exploration of the geomagnetic field and the solar corona has
developed to reveal periodic patterns \citep{CAM15,MCML17}.
Stratification in general introduces a correction term to
the dispersion relations of MC waves,
whilst in a thin layer the direction of
travel is usually reversed;
however, this is not always true in spherical geometries.
The unstratified thick shell problem considered here
is sufficient to provide some fundamental
understanding of the nonlinear problem.
Theoretical investigation is expanding
to consider their nonlinear properties such as
turbulence \citep{TDH07} and
triadic resonances \citep{RR15}.
\citet{L17} found a couple of cases in which
nonlinear equatorial waves in the shallow water MHD should be governed
by Korteweg-de Vries (KdV) equations and so behave like solitary waves.
They were mostly fast MR modes, recovering the equatorial Rossby wave soliton \citep{Boy80}
in the nonmagnetic limit,
but he reported one case in which the wave would slowly travel in the opposite azimuthal direction.
\citet{H19} investigated magnetostrophic MR waves in a Cartesian quasi-geostrophic (QG) model.
The slow, weakly-nonlinear waves
led to evolution obeying the KdV equation unless
the basic state -- all the magnetic field, topography, and zonal flow -- is uniform.
Slow MR waves have been seen in spherical dynamo DNS
travelling with crests/troughs that were isolated and sharp,
unlike the continuous wave trains
that might be expected \citep{HJT15,HTJ18}.
Hydrodynamic Rossby wave solitons have been extensively studied,
motivated by atmosphere and ocean dynamics \citep[e.g.][]{C71,R77,Boy80}.
In the long wave limit it has been demonstrated that the QG soliton relies on the presence of a shear in the basic flow or topography.
\citet{R77} further analysed nonlinear critical layers arising from singularities
as the wave speed approaches the basic flow speed,
and discussed their relevance for the persistence of Jupiter's Great Red Spot.
The present manuscript demonstrates that weakly nonlinear slow MR waves in spherical containers
yield soliton solutions.
We adopt simple QG MHD models
and asymptotically derive the evolution equation for the long wave
when the basic magnetic field and flow are both azimuthal.
We demonstrate:
(i) the amplitude at the first order is described by the KdV equation
for the chosen basic states,
(ii) the problem is dictated by an ODE, which
has no singularities as the wave speed approaches the basic flow speed,
and (iii) the single soliton (solitary wave) solution to the KdV equation
implies an isolated eddy that progresses in a stable permanent form
on magnetostrophic timescales.
\section{Theoretical foundations}
We consider an inviscid, incompressible, ideal quasi-geostrophic (QG) model of electrically conducting fluid
within a rapidly rotating shell,
bounded by inner and outer spheres of radii $r_\tx{i}$ and $r_\tx{o}$, respectively
\citep[e.g.][]{Bus70,GJ06}.
We use polar coordinates $(s,\varphi,z)$ with rotation $\Omega \hat{\mib{z}}$.
For rapid rotation, the incompressible horizontal QG fluid motion can be expressed
as $\mib{u} \approx \nabla \times \psi (s, \varphi) \hat{\mib{z}}$
with $\psi$ a streamfunction,
so it is independent of $z$.
When the magnetic field is not too strong to violate the QG approximation,
we further assume the magnetic field may be written as
$\mib{B} \approx \nabla \times g (s, \varphi) \hat{\mib{z}}$
with $g$ being the potential
\citep[e.g.][]{Bus76,AJPJ00,TDH07,CFF14}.
No penetration on
the spherical boundaries
at $z=\pm H = \pm \sqrt{r_\tx{o}^2 - s^2}$
enables us to represent the Coriolis term of the axial vorticity equation
in terms of the topography-induced beta parameter. The equations for the $z$-components of the vorticity
and the magnetic potential in dimensionless form are then:
\begin{eqnarray}
\frac{\partial}{\partial t} \Delta_\tx{H} \psi
- \mathcal{J} [ \psi, \Delta_\tx{H} \psi ]
- \frac{1}{Le^2}
\frac{\beta}{s} \frac{\partial \psi}{\partial \varphi}
&=& - \frac{1}{Le^2}
\mathcal{J} [ g, \Delta_\tx{H} g ] \\
\mbox{and\ } \quad
\frac{\partial}{\partial t} g
&=& \mathcal{J} [ \psi, g ] \; , \label{eq:current_sphere}
\end{eqnarray}
where $\Delta_\tx{H} = (1/s) \partial/\partial s (s \partial/\partial s) + ( 1/s^2 ) \partial^2/\partial \varphi^2$,
and
$\mathcal{J} [f_1,f_2]
= ( \partial f_1/\partial s \; \partial f_2/\partial \varphi
-\partial f_2/\partial s \; \partial f_1/\partial \varphi )/s$
for any functions $f_1$ and $f_2$.
Here the length, the magnetic field, and the velocity
are, respectively, scaled by the radius of the outer sphere $r_\tx{o}$,
the mean field strength $B_0$
and the MC wave speed
$B_0^2/(2\Omega r_\tx{o} \rho \mu_0) = c_\tx{M}^2/c_\tx{C}$;
$c_\tx{M}^2 = B_0^2/(\rho\mu_0)$ and $c_\tx{C} = 2\Omega r_\tx{o}$.
The Lehnert number $Le
= c_\tx{M}/c_\tx{C}$, whilst the beta parameter is given by $\beta = s/(1 - s^2)$.
Impermeable boundary conditions are applied so that
\begin{eqnarray}
\frac{1}{s} \frac{\partial \psi}{\partial \varphi}= 0
\quad &\mbox{at\ }& \quad s = \eta, 1 ,
\label{eq:bc_sphere}
\end{eqnarray}
where the aspect ratio $\eta = r_\tx{i}/r_\tx{o}$.
As $\beta \to \infty$ at $s=1$, the governing equations are singular there; these boundary conditions ensure that the regular solution is selected.
Of particular interest is the regime when $Le^{-1}$ is large.
Taking the limit leads to a balance between the vortex stretching and the Lorentz term in the vorticity equation:
\begin{eqnarray}
\beta \frac{1}{s} \frac{\partial \psi}{\partial \varphi}
&=& \mathcal{J} [ g, \Delta_\tx{H} g ]
\; , \label{eq:vorticity_sphere_magnetostrophic}
\end{eqnarray}
whilst (\ref{eq:current_sphere}) retains its same form.
The nonlinear problems have two source terms acting on the magnetostrophic wave:
below we asymptotically solve the weakly nonlinear cases.
To seek solitary long-wave solutions
we introduce slow variables with a small parameter $\epsilon$ ($\ll 1)$ and a real constant $c$:
\begin{equation}
\tau = \epsilon^{3/2} t \;, \qquad
\zeta = \epsilon^{1/2} (\varphi - c t) \; .
\end{equation}
Note that this assumes a long spatial scale in the azimuthal direction compared with the radial direction. This is reasonable for small $m$. We then expand variables with $\epsilon$ as
\begin{eqnarray}
\psi = \psi_0 (s) + \epsilon \psi_1 (s,\zeta,\tau)
+ ... \; , \quad
g = g_0 (s) + \epsilon g_1 (s,\zeta,\tau)
+ ... \;,
\end{eqnarray}
for the basic state satisfying
\begin{equation}
- D \psi_0
= \overline{U}(s) \; , \quad
- D g_0
= \overline{B}(s),
\end{equation}
where $D = d/ds$.
At zeroth order the equations of vorticity (\ref{eq:vorticity_sphere_magnetostrophic})
and of electric potential (\ref{eq:current_sphere}), and the boundary condition (\ref{eq:bc_sphere}) are all trivial.
At $\mathcal{O}(\epsilon)$,
(\ref{eq:vorticity_sphere_magnetostrophic}) and (\ref{eq:current_sphere}) become
\begin{equation}
\beta \frac{\partial \psi_1}{\partial \zeta}
= - \left[ \overline{B} \mathcal{D}^2 - \overline{J} \right] \frac{\partial g_1}{\partial \zeta},
\qquad \textrm{where} \qquad
\mathcal{D}^2 = \frac{1}{s} \frac{\partial}{\partial s} s \frac{\partial}{\partial s} \quad \textrm{and} \quad
\overline{J} = D \frac{1}{s} D (s \overline{B})
\label{eq:psi1_sphere}
\end{equation}
\begin{equation}
\textrm{and} \qquad
\left( \frac{\overline{U}}{s} - c \right)
\frac{\partial g_1}{\partial \zeta}
= \frac{\overline{B}}{s}
\frac{\partial \psi_1}{\partial \zeta}
\; , \label{eq:g1_sphere}
\end{equation}
respectively.
Substituting (\ref{eq:psi1_sphere}) into (\ref{eq:g1_sphere}) gives
a homogeneous PDE with respect to $g_1$:
\begin{equation}
\mathcal{L} \frac{\partial g_1}{\partial \zeta} \equiv
\left\{
\frac{\overline{B}}{ \beta s} \left[ \overline{B} \mathcal{D}^2
- \overline{J} \right]
+ \left( \frac{\overline{U} }{s} - c \right)
\right\} \frac{\partial g_1}{\partial \zeta}
= 0
\label{eq:g1_pde_sphere}
\end{equation}
where $\mathcal{L}$ represents the linear differential operator
comprising of $s, {\partial/\partial s}$ or $D, \overline{B},\beta, \overline{U}$, and $c$.
Inserting the boundary conditions (\ref{eq:bc_sphere}) at this order into (\ref{eq:g1_sphere}) yields
\begin{eqnarray}
\frac{\partial g_1}{\partial \zeta}
= 0
\quad \mbox{at\ } \quad s = \eta , 1 \; .
\end{eqnarray}
We then seek a solution in the form of $g_1 = \Phi(s) G(\zeta,\tau)$, so that
\begin{equation}
\mathcal{L} \Phi = 0
\qquad \mbox{and\ } \qquad
\Phi = 0 \quad\mbox{at\ }\quad s = \eta, 1
\; .
\label{eq:g1_ode_sphere}
\end{equation}
Now the linear operator $\mathcal{L}$ is the ordinary differential operator
with the partial derivatives with respect to $s$ replaced by $D$.
Given a basic state,
the ODE (\ref{eq:g1_ode_sphere}) together with the boundary conditions is
an eigenvalue problem to determine the eigenfunction $\Phi$ with eigenvalue $c$;
it can have many eigensolutions.
We note that the second-order ODE (\ref{eq:g1_ode_sphere}) remains non-singular
as $\overline{U}/s \rightarrow c$,
but not as $\overline{B}^2/\beta \rightarrow 0$ unless $s = 0$.
Below we concentrate on cases in which (\ref{eq:g1_ode_sphere}) has no internal singularities,
i.e. there is a discrete spectrum.
We consider cases where the $z$-averaged
toroidal magnetic fields do not pass through zero
\revthree{(e.g. figure 3 of \citet{SJNF17}; figures 1-2 of \citet{HTJ18})}.
We proceed to the next order to obtain the amplitude function.
Eqs.~(\ref{eq:vorticity_sphere_magnetostrophic}) and (\ref{eq:current_sphere}) at $\mathcal{O}(\epsilon^2)$
yield
\begin{equation}
\beta \frac{\partial \psi_2}{\partial \zeta}
= - \left[ \overline{B} \mathcal{D}^2 - \overline{J}
\right] \frac{\partial g_2}{\partial \zeta}
- \frac{\overline{B}}{s^2} \frac{\partial^3 g_1}{\partial \zeta^3}
+ \left( \frac{\partial g_1}{\partial s} \frac{\partial}{\partial \zeta}
- \frac{\partial g_1}{\partial \zeta} \frac{\partial}{\partial s}
\right) \mathcal{D}^2 g_1
\label{eq:psi2_sphere}
\end{equation}
\begin{equation}
\textrm{and} \qquad
\left( \frac{\overline{U}}{s} - c \right) \frac{\partial g_2}{\partial \zeta}
- \frac{\overline{B}}{s} \frac{\partial \psi_2}{\partial \zeta}
= - \frac{\partial g_1}{\partial \tau}
+ \frac{1}{s} \left( \frac{\partial \psi_1}{\partial s} \frac{\partial g_1}{\partial \zeta}
- \frac{\partial \psi_1}{\partial \zeta}\frac{\partial g_1}{\partial s}
\right)
\; . \qquad \label{eq:g2_sphere}
\end{equation}
Eliminating $\psi_2$ using (\ref{eq:psi2_sphere}) and $\psi_1$ using (\ref{eq:psi1_sphere}),
(\ref{eq:g2_sphere}) becomes the inhomogeneous PDE
\begin{eqnarray}
\mathcal{L} \frac{\partial g_2}{\partial \zeta}
&=& - \frac{\overline{B}^2}{s^3 \beta} \frac{\partial^3 G}{\partial \zeta^3}
\Phi
- \frac{\partial G}{\partial \tau} \Phi \nonumber \\
&+& G \frac{\partial G}{\partial \zeta} \left\{ \frac{2\overline{B}}{\beta s}
\left[ (D \Phi) D^2 \Phi - \Phi D (D^2 \Phi) \right]
- \frac{\Phi D^2 \Phi}{s} D \left(\frac{\overline{B}}{\beta} \right)
+ \frac{\Phi^2}{s} D \left(\frac{\overline{J}}{\beta} \right)
\right\}
\;
\quad \qquad \label{eq:g2_pde_sphere}
\end{eqnarray}
where $D^2 = (1/s) D s D$. The boundary conditions here are
\begin{equation}
\frac{\partial g_2}{\partial \zeta} = 0
\quad \mbox{at\ } \quad s = \eta , 1 \; .
\end{equation}
The adjoint linear problem corresponding to (\ref{eq:g1_pde_sphere}) is
\begin{equation}
\mathcal{L}^\dag \Phi^\dag \equiv
\left\{ \left[
D^2 \overline{B}
- \overline{J} \right] \frac{\overline{B}}{\beta s }
+ \left( \frac{\overline{U}}{s} - c \right) \right\} \Phi^\dag = 0
\; . \label{eq:g1_ode_adjoint_sphere}
\end{equation}
The adjoint boundary conditions are
\begin{equation}
\frac{\overline{B}^2}{s\beta} \Phi^\dag = 0
\quad \mbox{at\ } \quad s = \eta , 1 \; .
\label{eq:g1_bc_adjoint_sphere}
\end{equation}
Note that the substitution $ \overline{B}^2 \Phi^\dag / s \beta = \Phi$ reduces the adjoint problem to the
ordinary linear problem (\ref{eq:g1_pde_sphere}) so, provided $\overline{B}^2 \Phi^\dag /s\beta$ is non-zero
in the sphere, the adjoint eigenfunction $\Phi^\dag$ can simply be found by dividing the solution of
(\ref{eq:g1_pde_sphere})
by $\overline{B}^2/s\beta$.
The solvability condition to (\ref{eq:g2_pde_sphere}) is thus given by
\begin{equation}
\frac{\partial G}{\partial \tau}
+ \alpha \; G \frac{\partial G}{\partial \zeta}
+ \gamma \; \frac{\partial^3 G}{\partial \zeta^3} = 0,
\label{eq:KdV}
\end{equation}
where $\alpha = \alpha_0/\delta_0$, $\gamma = \gamma_0/\delta_0$,
\begin{eqnarray}
&&\alpha_0 = \int _{\eta}^1 \Phi^\dag \left\{ \frac{2\overline{B}}{\beta}
\left[ \Phi D (D^2 \Phi) - (D \Phi) D^2 \Phi \right]
+ \Phi (D^2 \Phi) D \left(\frac{\overline{B}}{\beta} \right)
- \Phi^2 D \left(\frac{\overline{J}}{\beta} \right)
\right\} \,ds
, \nonumber \\
&&\gamma_0 =
\int_{\eta}^{1} \Phi^\dag \frac{\overline{B}^2}{s^2 \beta} \Phi
\, ds
,
\quad \mbox{and\ } \quad
\delta_0 = \int_{\eta}^{1} {\Phi^\dag \Phi} \ s \, ds . \qquad \quad
\label{eq:g2_KdV_sphere}
\end{eqnarray}
Eq.~(\ref{eq:KdV}) is the Korteweg-de Vries equation
if the coefficients, $\alpha$ and $\gamma$, are both nonzero.
In the following section we examine the coefficients
for different choices of the basic state.
We note that the presence of $\overline{U}$ does not directly impact either $\alpha$ or $\gamma$.
It however dictates $\Phi$ and $\Phi^\dag$ through the linear problems
at $\mathcal{O}(\epsilon)$
and then may contribute to the terms at $\mathcal{O}(\epsilon^2)$.
This is in contrast with the hydrodynamic case \citep[e.g.][]{R77},
where the basic flow enters the nonlinear term at $\mathcal{O}(\epsilon^2)$ too.
The mean-flow effect on the magnetostrophic wave arises from the equation for the
magnetic potential (\ref{eq:current_sphere}).
Solutions to (\ref{eq:KdV}) may take the form of solitary (single or multiple soliton),
cnoidal, similarity, and rational waves \citep[e.g.][]{W74,DJ89}.
For instance, for a single soliton the asymptotic solution up to $\mathcal{O}(\epsilon)$ is
\begin{eqnarray}
g (s,\varphi, t)
&
= -\int_{\eta}^{s} \overline{B} ds + \epsilon \; \mathrm{sgn}(\alpha \gamma) \;
\Phi \;{\mathrm{sech}^2 F},
\label{eq:single-soliton-g_sphere} \\
\psi (s,\varphi, t)
&
= -\int_{\eta}^{s} \overline{U} ds
- \epsilon \; \mathrm{sgn}(\alpha \gamma) \;
\left( \frac{\overline{B}}{\beta} D^2 \Phi - \frac{\overline{J}}{\beta} \Phi \right) \;
{\mathrm{sech}^2 F},
\label{eq:single-soliton-psi_sphere}
\end{eqnarray}
where
\begin{equation}
F (\varphi, t)
= \sqrt{ \frac{\alpha }{12 \gamma} \mathrm{sgn}(\alpha \gamma) }
\left[ \epsilon^{1/2} (\varphi -ct)
- \epsilon^{3/2} \; \mathrm{sgn}(\alpha \gamma)
\; \frac{\alpha t}{3}
\right] \; .
\end{equation}
This is an eddy that has the solitary characteristics in azimuth,
riding on the basic state with the linear wave speed.
The finite-amplitude effect $\alpha$ accelerates the retrograde propagation
if $\gamma < 0$, but decelerates it when $\gamma > 0$.
The characteristic waveform is clearly visible in the magnetic potential.
\section{Illustrative examples}
We solve the eigenvalue problem (\ref{eq:g1_ode_sphere})
and the adjoint problem (\ref{eq:g1_ode_adjoint_sphere})-(\ref{eq:g1_bc_adjoint_sphere})
for different basic states
and calculate the respective coefficients of the evolution equation (\ref{eq:KdV})
in a spherical cavity,
with $\eta = 0.35$.
We consider \revthree{three cases investigated in \citet{CFF14}};
the first has a $\overline{B}$ that is a linearly increasing function of $s$
(referred to as a Malkus field hereafter),
the second $\overline{B}$ is inversely proportional to $s$ (an electrical-wire field)\revthree{,
and the third one is $(3/2) \cos{\{ \pi(3/2 - 50 s/19) \} } + 2$,
which was adoped by \citet{CFF14} to model a profile of the radial magnetic field $B_s$ within Earth's core
(a CFF field)}.
For
\revthree{the Malkus and wire}
fields the terms $\overline{J}$
in (\ref{eq:g1_ode_sphere}), (\ref{eq:g1_ode_adjoint_sphere}) and (\ref{eq:g2_KdV_sphere})
all vanish\revthree{, whereas this is not the case for the CFF field}.
The Malkus field
case has been extensively studied in the literature \citep[e.g.][]{M67,RL79,ZLS03,MJT17}.
We also consider the inclusion of a basic zonal flow $\overline{U}$
that is prograde with either a linear or quadratic dependence on $s$.
Table \ref{table:cases_spheres} summarises the results,
listing the eigenvalue $\lambda = \sqrt{|c|}/2$ (see below) and $c$ for the $n$-th mode,
the coefficients $\alpha$, $\gamma$, and $\delta_0$ as calculated from the eigenfunction $\Phi$,
the adjoint eigensolution $\Phi^\dag$ and (\ref{eq:g2_KdV_sphere}),
and whether/at which $s$ the wave speed $c$ approaches the basic angular velocity $\overline{U}/s$.
Here the $n$-th mode has $(n-1)$ zeros within the explored interval.
Negative values of $c$ indicate retrograde waves.
More notably, in the all cases we obtain
nonzero $\alpha$ and $\gamma$ for all $n$ examined and so the KdV equations are appropriate.
\revtwo{The fraction $|\alpha/\gamma|$ and their signs characterise the solitons.}
\begin{table}
\begin{center}
\def~{\hphantom{0}}
\begin{tabular}{ccc cccccc}
$\overline{B}$ & $\overline{U}$ & $n$ & $\lambda$ & $c$ & $\alpha$ & $\gamma$ & $\delta_0 \times 10^{2}$ & $s$ at which $c= \overline{U}/s$ \\[3pt]
$s$ & 0 & 1 & ~1.56402 & ~~-9.7847 & -12.854~ & 0.87465 & 4.9020~~ & --- \\
& & 2 & ~2.88117 & ~-33.2045 & -14.639~ & 1.0480~ & 0.79920~ & --- \\
& & 3 & ~4.18526 & ~-70.0655 & -26.422~ & 1.1156~ & 0.26204~ & --- \\
$1/s$& 0 & 1 & ~2.34412 & ~-21.9795 & -36.930~ & 1.2464~ & 0.92993~ & --- \\
& & 2 & ~4.41698 & ~-78.0389 & -31.920~ & 2.1442~ & 0.14054~ & --- \\
& & 3 & ~6.47665 & -167.788~ & -70.056~ & 2.8417~ & 0.044739 & --- \\
$^{\circ}s$ & 0 & 1 & & ~~-9.7847 & -12.854~ & 0.87465 & 4.9023~~ & --- \\
$^\circ 1/s$& 0 & 1 & & ~-21.9795 & -36.865~ & 1.2464~ & 0.92800~ & --- \\
\revthree{
$^\circ$CFF}
& 0 & 1 & & ~-11.0427 & -11.493~ & 2.8531~ & 0.51035~ & --- \\
& & 2 & & ~-32.2790 & -19.611~ & 4.7250~ & 0.12427~ & --- \\
& & 3 & & ~-71.6553 & -43.375~ & 4.4968~ & 0.053649 & --- \\
$^{\circ}s$ &$s$ & 1 & & ~~-8.7847 & -12.854~ & 0.87465 & 4.9023~~ & none \\
$^\circ 1/s$&$s$ & 1 & & ~-20.9795 & -36.865~ & 1.2464~ & 0.92800~ & none \\
\revthree{
$^\circ$CFF}
&$s$ & 1 & & ~-10.0427 & -11.493~ & 2.8531~ & 0.51035~ & none \\
$^{\circ}s$ &$4s(1-s)$
& 1 & & ~~-8.8379 & ~-9.5075 & 0.90339 & 4.9193~~ & none \\
$^\circ 1/s$&$4s(1-s)$
& 1 & & ~-21.4523 & -35.429~ & 1.2659~ & 0.92748~ & none \\
\revthree{
$^\circ$CFF} &$4s(1-s)$
& 1 & & ~-10.6163 & ~-9.8441 & 2.9722~ & 0.50834~ & none \\
$^{\circ}s$ &$80s(1-s)$
& 1 & & ~~12.9242 & ~31.273~ & 1.4622~ & 4.0079~~ & 0.8384 \\
$^\circ 1/s$&$320s(1-s)$
& 1 & & ~~33.1890 & ~10.307~ & 3.4093~ & 1.3187~~ & 0.8963 \\
\revthree{
$^\circ$CFF} &$320s(1-s)$
& 1 & & ~~44.4360 & ~41.749~ & 13.789~~ & 0.67936~ & 0.8611 \\
\end{tabular}
\caption{Values of $\lambda$, $c$, $\alpha$, $\gamma$, and $\delta_0$ of the $n$-th mode for the
basic magnetic field $\overline{B}$ and flow $\overline{U}$ in the spherical model $\beta = s/(1-s^2)$.
\revthree{The CFF field $\overline{B}$ is given as $(3/2) \cos{\{ \pi(3/2 - 50 s/19) \} } + 2$.}
Cases indicated by $^\circ$ are evaluated with the routine bvp4c and the modified outer boundary condition.}
\label{table:cases_spheres}
\end{center}
\end{table}
For the Malkus field ($\overline{B} = s$) and no mean flow $\overline{U}$,
we let $x = 1-s^2$ and $\Phi(x) = x y(x)$ to rewrite the ODE (\ref{eq:g1_ode_sphere}) as
\begin{equation}
x(1-x) \frac{d^2 y}{dx^2} + (2-3x) \frac{dy}{dx} + (\lambda^2 -1) y =0
\end{equation}
where $\lambda^2 = - c/4$.
This is a hypergeometric equation, which has a solution
\begin{equation}
\Phi (s) = (1-s^2) F(1+\lambda, 1-\lambda ; 2; 1-s^2) ,
\quad \mbox{and\ } \quad
\Phi^\dag
= \frac{\Phi}{1-s^2} , \label{eq:phi1_sol_malkus}
\end{equation}
where $F$ denotes the hypergeometric function \citep[e.g.][]{AS65}.
The eigenvalue $\lambda$ is determined by the condition $\Phi=0$ at $s=\eta$.
The adjoint solution is related to the axial electrical current generated at this order
as $-D^2 \Phi = -c \Phi s\beta/\overline{B}^2 = -c \Phi^\dag$,
implying the current is nonzero at $s=1$.
Figure \ref{fig:sphere_Malkus} shows the solutions in the Malkus case.
\revone{Figure \ref{fig:sphere_Malkus}(a)} shows profiles of $\overline{B}(s)$, the topography $\beta$,
the eigenfunctions $\Phi$ for $n = 1$ and $2$,
and their adjoint eigenfunctions $\Phi^\dag$ (\ref{eq:phi1_sol_malkus}).
This yields
$\alpha \approx -12.85$ and $\gamma \approx 0.87$
for $n=1$; the nonlinear effect is more significant than the dispersive one.
\revone{Figure \ref{fig:sphere_Malkus}(b)} illustrates a single soliton solution
(\ref{eq:single-soliton-psi_sphere}) of $\psi$ for $n=1$.
If the amplitude $\epsilon$ is too large, neglected higher order terms will be significant;
if $\epsilon$ is too small the azimuthal scale of the solitary wave is too large to fit in,
so we choose $\epsilon = 0.1$ as a reasonable compromise.
The streamfunction $\psi$ is negative,
indicating a clockwise solitary eddy.
The retrogradely propagating vortex $\psi_1$ is slightly more concentrated at the outer shell
than the magnetic potential $g_1$ (not shown).
As $c < 0$ and $\gamma > 0$,
the dispersion term reduces the retrograde propagation speed.
We note that
a clockwise vortex is observed
in Earth's core \citep{PJ08} and geodynamo simulations \citep{SJNF17}:
its implications are discussed in the final section.
The same basic states admit high-$n$ modes with more isolated structure
to have the KdV equations with nonzero $\alpha$ and $\gamma$ (Table \ref{table:cases_spheres}).
The speed $|c|$ increases with $n$, confirming the dispersivity of the wave.
The eigenfunction $\Phi$ for $n=2$
is negative at small $s$,
and then turns positive when $s \gtrsim 0.787$ (dashed-dotted curve in \revone{figure \ref{fig:sphere_Malkus}a}),
so the eddy is clockwise in the outer region and anticlockwise in the inner region (\revone{figure \ref{fig:sphere_Malkus}c}).
\begin{figure}
\centerline{
\includegraphics[height=32mm]{./figure1} }
\caption{Spherical case for the Malkus field $\overline{B}=s$ and $\overline{U}=0$.
(a) Profiles of $\overline{B}$ [red solid curve], $\beta$ [green solid],
$\Phi$ for $n=1$ [black dashed] and $n=2$ [black dashed-dotted],
and $\Phi^\dag$ for $n=1$ [blue dashed] and $n=2$ [blue dashed-dotted].
Streamfunctions $\psi$ of the single soliton solution for (b) $n=1$ and (c) $n=2$,
provided $\epsilon = 0.1$.
The dashed (solid) contour lines represent its negative (positive) value, i.e. clockwise (anti-clockwise).}
\label{fig:sphere_Malkus}
\end{figure}
We next consider the basic field given by the wire field, $\overline{B} = 1/s$,
whilst $\overline{U} = 0$.
By using $\Phi(x) = x e^{\lambda x} y(x)$, (\ref{eq:g1_ode_sphere})
may be reduced to a confluent Heun equation
\begin{equation}
x(1-x) \frac{d^2 y}{dx^2}
+ \{2 + (2 \lambda -3)x -2 \lambda x^2 \} \frac{dy}{dx}
+ \{ (\lambda^2 +2 \lambda -1) - (\lambda^2+3 \lambda)x\}y =0 .
\end{equation}
The solution regular at $s=1$ corresponding to the eigenvalue $\lambda$ is
\begin{equation}
\Phi = (1-s^2) e^{\lambda(1-s^2)} H_\tx{c}(q_\tx{c}, \alpha_\tx{c}, \gamma_\tx{c},\delta_\tx{c},\epsilon_\tx{c}; 1-s^2),
\quad \mbox{and\ } \quad
\Phi^\dag
= \frac{s^4}{1-s^2} \Phi ,
\label{eq:phi1_sol_wire}
\end{equation}
where $H_\tx{c}$ represents the confluent Heun function with
the accessory parameter $q_\tx{c} =\lambda^2 +2 \lambda - 1$ and exponent parameters
$\alpha_\tx{c} = \lambda^2 + 3 \lambda, \gamma_\tx{c} = 2, \delta_\tx{c} = 1$ and $\epsilon_\tx{c} = 2 \lambda$ \citep{OLBC10}.
\revone{
This case admits a simple form of the coefficients (\ref{eq:g2_KdV_sphere}) such that
\begin{eqnarray}
&&\alpha_0
= -4 \lambda^2 \int^{1-\eta^2}_0
x (2x+1) (1-x)^2 e^{3\lambda x} H_\tx{c}^3 \,dx , \quad
\gamma_0
= \frac{1}{2} \int^{1-\eta^2}_{0} \frac{x^2}{1-x} e^{2\lambda x} H_\tx{c}^2 \, dx
, \nonumber \\
&&\quad \textrm{and} \quad
\delta_0
= \frac{1}{2} \int^{1-\eta^2}_{0} x(1-x)^2 e^{2\lambda x} H_\tx{c}^2 \,dx .
\end{eqnarray}
}
To evaluate the function we use the algorithm of \citet{Mot18} below.
Figure \ref{fig:sphere_invB}(a) gives profiles of the basic state
and eigenfunctions.
The figure shows that $\Phi$ for $n=1$
has a peak nearer the outer boundary, compared with that for the Malkus field;
it is still propagating retrogradely and is dispersive.
This case yields
$\alpha \approx -36.9$ and $\gamma \approx 1.25$
for $n=1$ and with $\epsilon=0.1$ the soliton is a more compact, clockwise eddy (\revone{figure \ref{fig:sphere_invB}b}).
Analysis of the individual terms of the coefficient $\alpha_0$ in (\ref{eq:g2_KdV_sphere})
implies that the presence of high order derivatives is favourable for nonlinear effects.
For $n=2$,
dispersive effects are enhanced compared to nonlinear ones.
The solitary eddy is clockwise in the outer region when $s \gtrsim 0.894$
and anticlockwise in the inner region (\revone{figure \ref{fig:sphere_invB}c}).
\begin{figure}
\centerline{
\includegraphics[height=32mm]{./figure2} }
\caption{Spherical case for the wire field $\overline{B}=1/s$ and $\overline{U}=0$.
(a) Profiles of $\overline{B}$ [red solid curve], $\beta$ [green solid],
$\Phi$ for $n=1$ [black dashed] and $n=2$ [black dashed-dotted],
and $\Phi^\dag$ for $n=1$ [blue dashed] and $n=2$ [blue dashed-dotted].
Streamfunctions $\psi$ of the single soliton solution for (b) $n=1$ and (c) $n=2$,
provided $\epsilon = 0.1$.}
\label{fig:sphere_invB}
\end{figure}
To explore more general cases
we implement the Matlab routine bvp4c to solve the eigenvalue problems.
We retain the boundary condition $\Phi = 0$ at $s = \eta = 0.35$,
but use the modified condition $\Phi + (1-s) {D\Phi} = 0$ close to the outer boundary $s=0.99999$
to avoid the numerical issue arising from singularities when $s \rightarrow 1$.
We also impose a normalising condition $D\Phi$ at the inner boundary:
\revthree{
the values for the Malkus field and the CFF field are given by (\ref{eq:phi1_sol_malkus}),
whereas the one for the wire field is by (\ref{eq:phi1_sol_wire}).
}
The number of gridpoints in $s$ is $500$ in all cases.
Given the obtained $c$, the same routine is adopted
to solve the boundary value problems for $\Phi^\dag$.
For consistency with the earlier cases we set $\Phi^\dag = 1$ at the outer boundary.
The codes are benchmarked with the exact solutions.
With modified boundary condition,
our computational results match the expected eigenvalues $\lambda = \sqrt{|c|}/2$
and eigenfunctions $\Phi$ for $1 \le n \le 3$
with errors less than 0.01 \% and 0.2 \%, respectively.
\revthree{
Now the third basic field, $\overline{B} = (3/2) \cos{\{ \pi(3/2 - 50 s/19) \} } + 2$, is examined.
Figure \ref{fig:sphere_CFF}(a) depicts the basic state, the eigenfunctions for $n=1$,
and additionally $\overline{J}$ (represented by the red dotted curve).
It is nonzero except at $s \approx 0.40$ and $0.78$
and is negatively peaked at $s \approx 0.59$.
The eigenvalues $c$ do not differ from those in the Malkus case very much (Table \ref{table:cases_spheres}).
For $n=1$,
$\Phi$ has a peak at $s \approx 0.61$ (blue dashed curve), as so does the basic field.
This case gives $\alpha \approx -11.5$ and $\gamma \approx 2.85$.
Indeed the term including $\overline{J}$ dominates over the ODE (\ref{eq:g1_ode_sphere})
and also over $\alpha_0$ (\ref{eq:g2_KdV_sphere});
if the term $\Phi^2 D(\overline{J}/\beta)$ were absent, $\alpha$ would become $\approx 1.68$.
Figure \ref{fig:sphere_CFF}(b) illustrates the magnetic potential $g_1$ (\ref{eq:single-soliton-g_sphere}),
where the basic state is excluded for visualisation,
It is clockwise and centred at the $s \approx 0.61$.
Similarly the streamfunction $\psi$ (\ref{eq:single-soliton-psi_sphere}) is displayed
in figure \ref{fig:sphere_CFF}(c):
now the distinction from the magnetic component is evident.
The solitary eddy is more confined nearer the outer boundary,
as $\overline{B} D^2 \Phi - \overline{J}\Phi$ in (\ref{eq:single-soliton-psi_sphere}) becomes significant
only when $s \gtrsim 0.8$ (not shown).
}
\begin{figure}
\centerline{
\includegraphics[height=32mm]{./figure3} }
\caption{\revthree{Spherical case for the CFF field $\overline{B}=(3/2)\cos{\{ \pi(3/2 - 50s/19) \} }+2$,
$\overline{U}=0$, and $n=1$.
(a) Profiles of $\overline{B}$ [red solid curve], $\beta$ [green solid],
$\overline{J}$ [red dotted; normalised for visualisation], $\Phi$ [black dashed], and $\Phi^\dag$ [blue dashed].
(b) Magnetic potential $g_1$ of the single soliton solution, where the basic state $g_0$ is excluded to help visualisation.
(c) Streamfunctions $\psi$ of the solution, provided $\epsilon = 0.1$.}
}
\label{fig:sphere_CFF}
\end{figure}
Including a basic flow $\overline{U} = s$
is equivalent to the addition of solid body rotation.
Therefore it affects the speed $c$ of propagation of the mode,
whilst leaving its other properties unchanged (Table \ref{table:cases_spheres}).
For a more realistic flow, $\overline{U} = 4 s(1-s)$, with the Malkus field,
the structures of $\Phi$ and $\Phi^\dag$ are not drastically altered
(leading to $\delta_0 \approx 0.049$).
The dominance of the nonlinearity over the dispersion, $|\alpha/\gamma|$, is however weakened.
The presence of the same basic flow in the wire field case also exhibits this property.
Finally, we comment on the behaviour of solutions in the vicinity of the point $s$
at which $\overline{U}/s$ equals $c$,
the location of a critical layer for the hydrodynamic Rossby wave soliton \citep[e.g.][]{R77}.
We impose a fast mean zonal flow, $\overline{U} = 80 s(1-s)$, in the Malkus field case;
figure \ref{fig:sphere_Malkus_shear}(a) shows the basic state
and additionally the deviation from the wave speed, $\overline{U}/s - c$ (blue {dotted} curve).
The curve shows this case has such a critical point at $s \approx 0.838$.
Nevertheless the impact is hardly seen in the eigenfunctions $\Phi$ and $\Phi^\dag$:
there are no discontinuities in the derivative $D\Phi$ (\revone{figure \ref{fig:sphere_Malkus_shear}b})
and hence in the solitary wave solutions (\revone{figure \ref{fig:sphere_Malkus_shear}c}).
This remains true for the wire field case with $\overline{U}=320s(1-s)$.
\begin{figure}
\centerline{
\includegraphics[height=32mm]{./figure4} }
\caption{Spherical case for the Malkus field $\overline{B}=s$, the basic flow $\overline{U} = 80s(1-s)$,
and $n=1$.
(a) Profiles of $\overline{B}$ [red solid curve], $\beta$ [green solid],
$\overline{U}/10$ [{blue solid}; scaled for visualisation],
and the deviation $\overline{U}/s - c$ [blue dotted].
(b) Profiles of $\Phi$ [black dashed], $\Phi^\dag$ [blue dashed], and $D \Phi$ \revone{[black dotted]}.
(c) Streamfunction $\psi_1$ of the single soliton solution,
where the basic state $\psi_0$ is excluded to help visualisation.}
\label{fig:sphere_Malkus_shear}
\end{figure}
\section{Concluding remarks}
In this paper we have performed a weakly nonlinear analysis of magnetostrophic waves
in QG spherical models with azimuthal magnetic fields and flows.
The model we considered is an annulus model \citep{Bus76,CFF14} of the form
utilised by \citet{H66} for linear magnetic Rossby (MR) waves.
We found that
the evolution of the long-wavelength, slow-MR waves in the spherical shells obeyed the KdV equation,
whether the toroidal magnetic field and/or the zonal flow were sheared or not.
The model we consider here is formally valid for cases where the azimuthal lengthscale is
much longer than that in radius; the most obvious application of which is for thin spherical shells.
For thicker spherical shells like those representative of Earth's fluid outer core,
the ratio of these lengthscales is of the order ten.
For thinner shells relevant to other astrophysical objects one might expect the asymptotic procedure
to give a better approximation to the true behaviour.
We find that solutions may take the form of a single soliton solution (for $n=1$)
which is a clockwise, solitary eddy when basic state magnetic field is
\revthree{any of} a Malkus field ($\overline{B} \propto s$),
a magnetic wire field ($\overline{B} \propto 1/s$)\revthree{, and
a CFF field (comprising of a trigonometric function)}.
In addition to these steadily progressing single-solitons we also find $N$-soliton solutions;
as these satisfy the KdV equation we know that these may have peculiar interactions
including a phase shift after a collision and FPU recurrence \citep[e.g.][]{DJ89}.
We conclude by noting that inversion of the geomagnetic secular variation appears
to detect an anticyclonic gyre in Earth's core \citep{PJ08,GJF15,BHFMG18};
it is off-centred with respect to the rotation axis
and is believed to have existed for more than a hundred years.
Moreover, DNS of dynamos driven by convection in rapidly-rotating spherical shells
have exhibited the emergence of a large vortex
which circulated clockwise and modulated very slowly \citep{SJNF17}; in these simulations
the averaged toroidal magnetic field tended to strengthen beneath the outer boundary.
Our solution tentatively supports the idea that such an isolated single eddy should persist,
while drifting on MC timescales of
\revthree{$\mathcal{O}(10^{2\textrm{-}4})$ years}.
The long wave can be initiated through instabilities
due to differentially rotating flows \citep{SAetal08},
due to thermally insulating boundaries \citep{HTS14},
and due to the magnetic diffusivity \citep{RL79,ZLS03}.
The steadily drifting feature of the solitons should of course be altered during the long-term evolution
when dissipation plays a role in the dynamics.
\revthree{The presence of dissipation may also alter the eigenfunction \citep{CFF14}
and thus the detailed morphology of the soliton too.}
\revthree{
We note an alternative to account for the eccentric gyre is
a flow induced by, for example, the coupling with the rocky mantle and the solid inner core,
as DNS by \citet{AFF13} had demonstrated.
The issue ends up in a debate which has lasted for decades:
does the geomagnetic westward drift represent the advection due to a large scale fluid motion
\citep{BFGN50} or hydromagnetic wave motion \citep{H66}.}
We shall investigate these issues further, as well as the role of critical layers,
by solving initial value problems in a future study.
\begin{acknowledgments}
\section*{Acknowledgments}
The authors are grateful to
Andrew Soward, Anna Kalogirou, Adrian Barker,
and Yoshi-Yuki Hayashi for discussion and comments.
K.~H. was supported by the Japan Science and Technology under
the Program to Supporting Research Activities of Female Researchers.
\end{acknowledgments}
\bibliographystyle{jfm}
| 2024-02-18T23:39:40.688Z | 2020-07-22T02:14:48.000Z | algebraic_stack_train_0000 | 39 | 6,021 |
|
proofpile-arXiv_065-296 | \section{Introduction}
Molecular descriptors \cite{ref1} are mathematical quantities that describe the structure or shape of molecules, helping to predict the activity and properties of molecules in complex experiments. In the last few years, several new molecular structure descriptors have been conceived \cite{ref2,ref3,ref4,ref5}. Molecular descriptors play a significant role in chemistry, pharmacology, etc. Topological indices have a prominent place among molecular structure descriptors. Modeling physical and chemical properties of molecules, designing pharmacologically active compounds and recognizing environmentally hazardous materials are just a few applications of topological indices, see~\cite{ref6}. One of the most important topological indices is the \emph{Szeged index}. The Szeged index of a connected graph $G$, $Sz(G)$, is defined as
$$Sz(G) = \sum_{e = uv\in E(G)} n_u^G(e)n_v^G(e),$$
where $n_u^G(e)$ is the number of vertices in $G$ that are closer to $u$ than $v$.
Similarly, the \emph{weighted Szeged index} of a connected graph $G$, $wSz(G)$, introduced
by Ili\'{c} and Milosavljevi\'{c} in \cite{ilic}, is defined as
$$wSz(G) = \sum_{e = uv\in E(G)} \left(d_G(u)+d_G(v)\right)n_u^G(e)n_v^G(e),$$
where $d_G(u)$ is the degree of $u$ in $G$.
Recently some researches have studied the weighted
Szeged index of special classes of graphs and graph
operations \cite{jan,ref10,ref11,ref12}. One of the
main related open problems is characterizing a graph
of fixed order with minimum weighted Szeged index.
Regarding this problem, if true, the following
conjecture would substantially restrict the structure
of such graphs.
\begin{conjecture}\cite{jan} \label{min-tree}
For $n$-vertex graphs the minimum weighted Szeged
index is attained by a tree.
\end{conjecture}
Unlike many other topological indices, for example
ABC-index \cite{ABCJCTB,krag_tree}, it is not easy
to prove if the graph with minimum weighted Szeged
index on a fixed number of vertices is a tree.
In \cite{jan} some properties of minimum weighted
Szeged index trees are described and the list of
minimum trees on at most $25$ vertices is presented.
We refer the reader to \cite{bondy2008} for basic
graph theory terminology. Throughout this work,
we will use $n$ to denote the number of vertices
of a graph. The rest of the work is organized
as follows. In Section \ref{sec:ending} we
introduce the concept of ending branch, present
some basic results about their structure and
use these results to determine the structure of
ending branches for some orders. We present
computational results on the structure of optimal
trees for the weighted Szeged index, as well as
the structure of optimal ending branches in
Section \ref{sec:computational}. We cover
trees up to 81 vertices; it is not hard to
computationally push this bound, but we only
intend to present the readers with a dataset
that might give a fair idea of what is happening
with these structures, and not giving the most
complete list possible. Finally, we present
conclusions and some conjectures based on our
observations in Section \ref{sec:conc}.
\section{Ending Branches}
\label{sec:ending}
Let $T$ be a tree with the smallest weighted Szeged index on a fixed number of vertices and let $R$ be a vertex of highest degree in $T$. Note that removing any edge from $T$ results in two connected components. We will call the the one that does not contain $R$ an \emph{ending branch}. In this section we will study the ending branches that result in small weighted Szeged index trees. By the weighted Szeged index of an ending branch we mean the sum of the weighted Szeged index of all edges in the ending branch along with the half-edge connecting the ending branch to the rest of the tree. Note that to calculate the weighted Szeged index of an ending branch, it is sufficient to have the total number of vertices of the graph besides the structure of the ending branch. The following proposition clarifies the idea.
\begin{proposition}
Let $n$ be a positive integer and let $T$ be a tree with the
smallest weighted Szeged index on $n$ vertices. If $uv$ is
an edge of $T$ and $v$ is the root of the associated
ending branch, then the weighted Szeged index of the ending
branch does not depend on the degree $d_u$ of $u$.
\end{proposition}
\begin{proof}
Consider two edges $uv$ and $uv'$ of $T$ and
suppose that the associated ending branches with roots $v$
and $v'$ have the same number of vertices.
The difference between the weighted Szeged index of these two graphs will depend on $d_u$ only on the edge $uv$ which is
$$(d_u+d_v)n_u^v n_v^u - (d_u+d_{v'})n_u^{v'} n_{v'}^u,$$
and if the number of vertices in the ending branch is the same then it simplifies to
$$(d_v-d_{v'})n_u^v n_v^u,$$
which does not depend on $d_u$, as claimed.
\end{proof}
Now we can compare the weighted Szeged index of different ending branches. By a {\em minimal ending branch on $k$ vertices} we mean the ending branch that has the smallest weighted Szeged index among all possible ending branches of $k$ vertices.
\begin{corollary}\label{cor:end}
In any minimum weighted Szeged index tree all ending branches are also minimal.
\end{corollary}
This allows us to focus on the best ending branches of a given size in a tree.
\begin{observation}
There is only one possible ending branch of size 1 and one of size 2.
\end{observation}
\begin{figure}[htb]
\centering
\includegraphics{end1-2.pdf}
\caption{Ending branches of size 1 and 2.}
\label{fig:1-2}
\end{figure}
Let $T_i$ be a best ending branch of size $i$. For $T_1$ and $T_2$ we have:
$$wSz(T_1)=n-1,$$
$$wSz(T_2)=(n-1) + 2\times 2 \times (n-2) = 5n-9.$$
Figure \ref{fig:3} shows the two possible ending branches of size 3.
\begin{figure}[htb]
\centering
\includegraphics{end3.pdf}
\caption{Possible ending branches of size 3.}
\label{fig:3}
\end{figure}
\begin{proposition}
The 3-ray is a better ending branch than a vertex of degree 3 with 2 leaves attached to it, i.e. in Figure \ref{fig:3}, (a) is smaller than (b).
\end{proposition}
\begin{proof}
Let $n$ be the number of vertices of $T$, then we have:
$$wSz(T_a)=(1+2)(n-1)(1)+(2+2)(n-2)(2) + 2(n-3)(3)=17n-37,$$
$$wSz(T_b)=2(1+3)(n-1)(1)+3(n-3)(3)=17n-35.$$
Therefore for any $n$ we have that $wSz(T_a)<wSz(T_b)$.
\end{proof}
Note that $wSz(T_a)$ and $wSz(T_b)$ can be rephrased as
$$wSz(T_a)=2\times3\times(n-3)+ 2\times 2 \times (n-2) + wSz(T_2),$$
$$wSz(T_b)=3\times3\times(n-3) + 2\times 3\times (n-1) + 2wSz(T_1).$$
This idea can be generalized as follows. Let $v$ be the root of an ending branch and let $x_1, x_2, \ldots, x_k$ be its children. Also let $n_i$ be the size of the ending branch with root $x_i$ (for $i \in \{ 1, \ldots, k \}$). Then the best ending branch with root $v$ has the following weighted Szeged index:
$$wSz(T_v)=d_v\times n_v \times (n-n_v) + \sum_{i=1}^k d_v\times n_i\times(n-n_i) + \sum_{i=1}^k wSz(T_{x_i}),$$
where $n_v$ is the total number of vertices in this ending branch.
Observe that $\sum_{i=1}^k n_i =n-1$, and for every partition of $n-1$ into integers we will get a new ending branch. We need to compare all these branches to find the best ending branch on $n_v$ number of vertices.
\begin{figure}[htb]
\centering
\includegraphics{end4.pdf}
\caption{Possible ending branches of size 4.}
\label{fig:4}
\end{figure}
As an example, to find the best ending branch on 4 vertices we need to find all partitions of 3 and we have $3=3 , 3=2+1, 3=1+1+1$. See Figure \ref{fig:4}. We have:
$$wSz(T_{3})=2\times 4\times (n-4) + 2\times 3\times (n-3) + 17n-35 = 31n-85,$$
$$wSz(T_{2+1}) = 3\times 4\times (n-4) + 3\times 2\times (n-2) + 3\times (n-1) + 5n-9 + n-1 = 27n - 73,$$
$$wSz(T_{1+1+1})=4\times4\times(n-4) + 3\times 4\times(n-1) + 3\times (n-1) = 31n-79.$$
Now it is easy to see that $wSz(T_{2+1})<wSz(T_3)<wSz(T_{1+1+1})$ for $n>6$. Therefore, in any minimal weighted Szeged index tree of size more than 6, if an ending branch of size 4 occurs, it is going to be $T_{2+1}$.
\section{Computational Results}
\label{sec:computational}
In this section we will use Corollary \ref{cor:end} to find minimum ending branches of higher order, say $n_v$, when the tree has $n$ vertices. Let us assume that we know all the minimum ending branches of order up to $n_v-1$.
Any ending branch has a root, say $v$, and the sum of the order of its children is $n_v-1$.
Basically, the order of children of $v$ is a partition of $n_v-1$. So we need to go over all possible partitions of $n_v-1$ and since we know the best ending branches of order up to $n_v-1$ we can just calculate and compare the best ending branch of order $n_v$.
This is when we have a fixed $n$. In general, when $n$ is a variable, for each partitioning of $n_v-1$ into integers we get a linear equation and to compare linear equations we sometimes need a bound.
Here we present the result of our computations. Table \ref{tab:branch} shows the best ending branches of size $n_v$. When $n$ is at least a specific value, then $v$ has $d_v$ children of shown order. Using the same approach we can find the minimal weighted Szeged index trees as well. Table \ref{tab:tree} shows the minimum weighted Szeged index trees of order up to 81.
Based on this calculation and as an example, the minimum weighted Szeged index tree on 67 vertices has a root of degree 4 and the root has three children of order 16, one child of order 18. A minimum ending branch on 16 vertices (when there are more than 18 vertices in the tree) has three children of order 5. A minimal ending branch of order 5 (when there are more than 6 vertices in the tree) has two children of order 2 and there is one possible ending branch of order 2, shown in Figure \ref{fig:1-2}. A drawing of the minimum weighted Szeged index tree on 67 vertices is shown in Figure \ref{fig:5}.
\small{
\begin{longtable}{| c | c | c | l |}
\hline
$n_v$ & $n\geq$ & $d_v$ & children \\ [0.5ex]
\hline\hline
2 & 2 & 1 & 1 \\
\hline
3 & 3 & 1 & 2 \\
\hline
4 & 6 & 2 & 1, 2 \\
\hline
5 & 6 & 2 & 2, 2 \\
\hline
6 & 8 & 2 & 2, 3 \\
\hline
7 & 9 & 3 & 2, 2, 2 \\
\hline
8 & 14 & 3 & 2, 2, 3 \\
\hline
9 & 11 & 2 & 3, 5 \\
\hline
10 & 14 & 3 & 3, 3, 3 \\
\hline
10* & 14 & 2 & 4, 5 \\
\hline
11 & 13 & 2 & 5, 5 \\
\hline
12 & 15 & 3 & 3, 3, 5 \\
\hline
13 & 18 & 3 & 3, 4, 5 \\
\hline
14 & 17 & 3 & 3, 5, 5 \\
\hline
15 & 18 & 3 & 4, 5, 5 \\
\hline
16 & 18 & 3 & 5, 5, 5 \\
\hline
17 & 20 & 3 & 5, 5, 6 \\
\hline
18 & 22 & 3 & 5, 5, 7 \\
\hline
19 & 25 & 3 & 5, 6, 7 \\
\hline
20 & 32 & 3 & 5, 7, 7 \\
\hline
21 & 70 & 3 & 6, 7, 7 \\
\hline
22 & 45 & 3 & 7, 7, 7 \\
\hline
23 & 51 & 3 & 7, 7, 8 \\
\hline
24 & 48 & 3 & 7, 7, 9 \\
\hline
25 & 49 & 3 & 7, 7, 10 \\
\hline
26 & 39 & 3 & 7, 7, 11 \\
\hline
27 & 48 & 3 & 7, 8, 11 \\
\hline
28 & 47 & 3 & 7, 9, 11 \\
\hline
29 & 50 & 3 & 7, 10, 11 \\
\hline
30 & 41 & 3 & 7, 11, 11 \\
\hline
31 & 45 & 3 & 8, 11, 11 \\
\hline
32 & 44 & 3 & 9, 11, 11 \\
\hline
33 & 44 & 3 & 10, 11, 11 \\
\hline
34 & 42 & 3 & 11, 11, 11 \\
\hline
35 & 54 & 3 & 11, 11, 12 \\
\hline
36 & 55 & 3 & 11, 11, 13 \\
\hline
37 & 54 & 3 & 11, 11, 14 \\
\hline
38 & 55 & 3 & 11, 11, 15 \\
\hline
39 & 45 & 3 & 11, 11, 16 \\
\hline
40 & 54 & 3 & 11, 12, 16 \\
\hline
41 & 55 & 3 & 11, 13, 16 \\
\hline
42 & 54 & 3 & 11, 14, 16 \\
\hline
43 & 55 & 3 & 11, 15, 16 \\
\hline
44 & 50 & 3 & 11, 16, 16 \\
\hline
45 & 66 & 3 & 12, 16, 16 \\
\hline
46 & 56 & 3 & 13, 16, 16 \\
\hline
47 & 55 & 3 & 14, 16, 16 \\
\hline
48 & 55 & 3 & 15, 16, 16 \\
\hline
49 & 56 & 3 & 16, 16, 16 \\
\hline
50 & 65 & 3 & 16, 16, 17 \\
\hline
51 & 64 & 3 & 16, 16, 18 \\
\hline
52 & 72 & 3 & 16, 16, 19 \\
\hline
53 & 83 & 3 & 16, 16, 20 \\
\hline
54 & 98 & 3 & 16, 16, 21 \\
\hline
55 & 121 & 3 & 16, 16, 22 \\
\hline
56 & 125 & 3 & 16, 17, 22 \\
\hline
57 & 165 & 3 & 16, 18, 22 \\
\hline
58 & 254 & 3 & 16, 19, 22 \\
\hline
59 & 506 & 3 & 16, 20, 22 \\
\hline
60 & 66 & 4 & 11, 16, 16, 16 \\
\hline
61 & 67 & 4 & 12, 16, 16, 16 \\
\hline
62 & 68 & 4 & 13, 16, 16, 16 \\
\hline
63 & 70 & 4 & 14, 16, 16, 16 \\
\hline
64 & 71 & 4 & 15, 16, 16, 16 \\
\hline
65 & 71 & 4 & 16, 16, 16, 16 \\
\hline
66 & 72 & 4 & 16, 16, 16, 17 \\
\hline
67 & 73 & 4 & 16, 16, 16, 18 \\
\hline
68 & 77 & 4 & 16, 16, 16, 19 \\
\hline
69 & 78 & 4 & 16, 16, 16, 20 \\
\hline
70 & 79 & 4 & 16, 16, 16, 21 \\
\hline
71 & 80 & 4 & 16, 16, 16, 22 \\
\hline
72 & 82 & 4 & 16, 16, 17, 22 \\
\hline
73 & 84 & 4 & 16, 16, 18, 22 \\
\hline
74 & 87 & 4 & 16, 16, 19, 22 \\
\hline
75 & 93 & 4 & 16, 16, 20, 22 \\
\hline
76 & 100 & 4 & 16, 16, 21, 22 \\
\hline
77 & 99 & 4 & 16, 16, 22, 22 \\
\hline
78 & 107 & 4 & 16, 17, 22, 22 \\
\hline
79 & 117 & 4 & 16, 18, 22, 22 \\
\hline
80 & 128 & 4 & 16, 19, 22, 22 \\
\hline
\caption{Minimal Ending branches}
\label{tab:branch}
\end{longtable}
}
\begin{longtable}{| c | c | l |}
\hline
$n$ & $d_R$ & children \\ [0.5ex]
\hline\hline
2 & 1 & 1 \\
\hline
3 & 1 & 2 \\
\hline
4 & 2 & 1, 2 \\
\hline
5 & 2 & 2, 2 \\
\hline
6 & 3 & 1, 2, 2 \\
\hline
7 & 3 & 2, 2, 2 \\
\hline
8 & 3 & 2, 2, 3 \\
\hline
9 & 4 & 2, 2, 2, 2 \\
\hline
10 & 3 & 2, 2, 5 \\
\hline
11 & 3 & 2, 3, 5 \\
\hline
12 & 4 & 2, 2, 2, 5 \\
\hline
13 & 3 & 2, 5, 5 \\
\hline
14 & 3 & 3, 5, 5 \\
\hline
15 & 4 & 3, 3, 3, 5 \\
\hline
15* & 3 & 4, 5, 5 \\
\hline
16 & 3 & 5, 5, 5 \\
\hline
17 & 4 & 3, 3, 5, 5 \\
\hline
18 & 4 & 2, 5, 5, 5 \\
\hline
18* & 4 & 3, 4, 5, 5 \\
\hline
19 & 4 & 3, 5, 5, 5 \\
\hline
20 & 4 & 4, 5, 5, 5 \\
\hline
21 & 4 & 5, 5, 5, 5 \\
\hline
22 & 4 & 5, 5, 5, 6 \\
\hline
23 & 4 & 5, 5, 5, 7 \\
\hline
24 & 5 & 3, 5, 5, 5, 5 \\
\hline
25 & 5 & 4, 5, 5, 5, 5 \\
\hline
26 & 5 & 5, 5, 5, 5, 5 \\
\hline
27 & 5 & 5, 5, 5, 5, 6 \\
\hline
28 & 5 & 5, 5, 5, 5, 7 \\
\hline
29 & 5 & 5, 5, 5, 6, 7 \\
\hline
30 & 4 & 3, 5, 5, 16 \\
\hline
31 & 4 & 4, 5, 5, 16 \\
\hline
32 & 4 & 5, 5, 5, 16 \\
\hline
33 & 4 & 5, 5, 6, 16 \\
\hline
34 & 4 & 5, 5, 7, 16 \\
\hline
35 & 4 & 5, 6, 7, 16 \\
\hline
36 & 5 & 5, 5, 5, 5, 15 \\
\hline
37 & 5 & 5, 5, 5, 5, 16 \\
\hline
38 & 5 & 5, 5, 5, 6, 16 \\
\hline
39 & 5 & 5, 5, 5, 7, 16 \\
\hline
40 & 5 & 5, 5, 6, 7, 16 \\
\hline
41 & 5 & 5, 5, 7, 7, 16 \\
\hline
42 & 4 & 7, 7, 11, 16 \\
\hline
43 & 5 & 5, 7, 7, 7, 16 \\
\hline
44 & 4 & 10, 11, 11, 11 \\
\hline
45 & 4 & 11, 11, 11, 11 \\
\hline
46 & 4 & 7, 11, 11, 16 \\
\hline
47 & 4 & 8, 11, 11, 16 \\
\hline
48 & 4 & 9, 11, 11, 16 \\
\hline
49 & 4 & 10, 11, 11, 16 \\
\hline
50 & 4 & 11, 11, 11, 16 \\
\hline
51 & 4 & 7, 11, 16, 16 \\
\hline
52 & 5 & 7, 11, 11, 11, 11 \\
\hline
53 & 4 & 9, 11, 16, 16 \\
\hline
54 & 4 & 10, 11, 16, 16 \\
\hline
55 & 4 & 11, 11, 16, 16 \\
\hline
56 & 5 & 11, 11, 11, 11, 11 \\
\hline
57 & 4 & 11, 13, 16, 16 \\
\hline
58 & 4 & 11, 14, 16, 16 \\
\hline
59 & 4 & 11, 15, 16, 16 \\
\hline
60 & 4 & 11, 16, 16, 16 \\
\hline
61 & 5 & 11, 11, 11, 11, 16 \\
\hline
62 & 4 & 13, 16, 16, 16 \\
\hline
63 & 4 & 14, 16, 16, 16 \\
\hline
64 & 4 & 15, 16, 16, 16 \\
\hline
65 & 4 & 16, 16, 16, 16 \\
\hline
66 & 4 & 16, 16, 16, 17 \\
\hline
67 & 4 & 16, 16, 16, 18 \\
\hline
68 & 5 & 11, 11, 13, 16, 16 \\
\hline
69 & 5 & 11, 11, 14, 16, 16 \\
\hline
70 & 5 & 11, 11, 15, 16, 16 \\
\hline
71 & 5 & 11, 11, 16, 16, 16 \\
\hline
72 & 5 & 11, 12, 16, 16, 16 \\
\hline
73 & 5 & 11, 13, 16, 16, 16 \\
\hline
74 & 5 & 11, 14, 16, 16, 16 \\
\hline
75 & 5 & 11, 15, 16, 16, 16 \\
\hline
76 & 5 & 11, 16, 16, 16, 16 \\
\hline
77 & 5 & 12, 16, 16, 16, 16 \\
\hline
78 & 5 & 13, 16, 16, 16, 16 \\
\hline
79 & 5 & 14, 16, 16, 16, 16 \\
\hline
80 & 5 & 15, 16, 16, 16, 16 \\
\hline
81 & 5 & 16, 16, 16, 16, 16 \\
\hline
\caption{Minimum weighted Szeged index trees.}
\label{tab:tree}
\end{longtable}
\begin{figure}[htb]
\centering
\includegraphics[width=0.4\textwidth]{67_vertices.pdf}
\caption{Best weighted Szeged index on 67 vertices.}
\label{fig:5}
\end{figure}
\subsection{Regular Trees}
By a regular ending branch we mean a minimal (weighted Szeged index) ending branch whose children
are of the same order. In the first 80 minimal ending branches 1, 2, 3, 5, 7, 10, 11, 16, 22, 34, 49, 65
are the regular ones. Based on our observations we have the following conjecture. By a main branch we mean an ending branch that is directly connected to the root of the tree.
\begin{conjecture}
In a minimum weighted Szeged index tree all but at most one main ending branches are regular ending branches.
\end{conjecture}
The same definition applies to regular trees and the first regular minimal trees are
1, 2, 3, 5, 7, 9, 16, 21, 26, 45, 56, 65 and 81.
The simplified degree sequence of a regular ending branch is the degree sequence of vertices on a path from the root of the ending branch to a leaf. For example, the simplified degree sequence of a minimal ending branch of order 65 is 4, 3, 2, 1.
Our calculation shows that a minimal ending branch of order 326 (again our expectation) is not a regular ending branch with simplified degree sequence of 5, 4, 3, 2, 1. Actually an ending branch on 326 vertices with 3 children of order 103, 103, 119 works better than the regular ending branch.
We note that there seem to be no vertices of degree greater than 6 in the optimal trees.
In order to understand this phenomenon, we offer the following thought experiment.
We will calculate an approximation of the weighted Szeged index for complete $k$-ary
trees with a very high number $n$ of vertices.
Note that the root has degree $k$, while the other vertices have degree $k+1$. We will ignore this
complication in order to simplify the expressions, and treat the every edge as having degree sum
$2k+2$, so we can just calculate the ordinary Szeged index and multiply it by $2k+2$. The Szeged
index of a complete $k$-ary tree is estimated as follows.
Each edge from the root to one of its $k$ children contributes the product $\frac{n-1}{k} \left( n - \frac{n-1}{k} \right)$ because
the child has $\frac{n-1}{k}$ descendants. In this expression the $n^2$ term dominates, since we assumed that
$n$ is very large; its term is $\frac{k-1}{k^2} n^2$. There are $k$ edges like this, so their overall contribution to
the Szeged index is $\frac{k-1}{k} n^2 + o(n^2)$. Similarly, $k^2$ edges between the children and grandchildren
of the root contribute collectively $k^2 \left( \frac{n-k-1}{k^2} \right) \left( n - \frac{n-k-1}{k^2} \right)$ which equals
$\frac{k^2 - 1}{k^2} n^2 + o(n^2)$. A similar calculation for the remaining levels reveals that the Szeged
index is $n^2 \left( \frac{k-1}{k} + \frac{k^2 - 1}{k^2} + \frac{k^3 - 1}{k^3} + \dots \right) + o(n^2)$. Since there are $\log_k n$
levels, we can say roughly that the Szeged index of a complete $k$-ary tree is about $n^2 \log_k n$, and the
weighted Szeged index is about $n^2 (2k+2) \log_k n$. The expression can be analyzed, but since it only represents a rough approximation we can just look at a large value. When $n=10^6$, the expression
appears to be minimized around $k=4$; we feel this may explain why we don't see vertices of degree more than
6. In any event, a large $k$, say $k=10$, would be disadvantageous. We also see from the calculation that the
contributions of the lower levels of a complete $k$-ary tree are slightly increasing as we go down the tree, and
perhaps this explains why the degrees in our optimal trees are decreasing. In large optimal
trees we have found that the degrees are decreasing just enough to make contributions of all levels roughly the
same (with the exception of the bottom three or four levels).
\section{Conclusions}
\label{sec:conc}
In view of Conjecture \ref{min-tree}, it is a good idea to
understand the structure of minimum weighted Szeged index
trees. Even if it turns out to be false, knowing the
structure of such trees could give us some insight to
understand the structure of minimum weighted Szeged index
graphs in the general case.
In this work we introduced the concept of ending branch,
which we used to analyze the structure of minimum weight
Szeged index trees in a recursive fashion. Our observations
were useful to computationally construct the trees on at most
81 vertices, extending the list of 25 trees given in
\cite{jan}.
Based on our results and experimental observations,
we finalize the section with some conjectures that,
if true, can give insights on the structure of minimum
weighted Szeged index trees. Also, they represent well
determined future lines of work that might be explored.
\begin{conjecture}
In a minimum weighted Szeged index tree, the degree sequence from the root to any leaf is non-increasing.
\end{conjecture}
\begin{conjecture}
Vertices of degree 1 are attached to vertices of degree at most 3.
\end{conjecture}
\begin{conjecture}
There are no vertices of degree greater than 6 in a minimum weighted Szeged index tree.
\end{conjecture}
| 2024-02-18T23:39:40.991Z | 2020-07-22T02:05:21.000Z | algebraic_stack_train_0000 | 57 | 3,932 |
|
proofpile-arXiv_065-339 | \subsection{Detection Statistics}
Table 2 lists the 150 H$_2$O maser sites
observed at IRAM in the CS J = 5\to4, 3\to2, and 2\to1 transitions.
Tables 1 and 2 of Paper I list the positions of the masers and
the CS J=7\to6 line parameters or upper limits.
Table 2 of the present paper lists the source names in order of
increasing galactic longitude,
the radiation temperature (T$_R^*$), integrated
intensity ($\int{T_R^*dv}$), velocity centroid (V$_{LSR}$), and
full width at half maximum (FWHM) for the three transitions.
For CS data obtained in 1990 June, we list the line parameters at the
position in the nine-point map with the strongest emission in the
J=5\to4 line. This choice is based on the results of \S 3.2, where we
find that the J=5\to4 emission almost always peaks at the maser position.
While the line parameters for 1990 June are useful in detection statistics and
as a guide for follow-up work, we have found that the position correction
was inadequate for them to be used together with the J=7\to6 data to
determine densities; therefore we do not use the 1990 June data in \S 4.
For undetected lines, the upper limits to T$_R^*$ are 3$\sigma$.
For CS J = 3\to2 and 2\to1,
we have tabulated only the data with the highest
spectral resolution. We also observed the C$^{34}$S lines in 49 of
the strongest CS emitters.
The results for C$^{34}$S are presented in Table 3. Transitions
listed with dashes (--) instead of values or
upper limits to T$_R^*$ were not observed.
Table 4 has the results for J=10\to9 and 14\to13.
Usually, we obtained the line parameters from Gaussian
fits to the lines but
some sources listed in Table 2 had spectra with more
than one peak.
To determine the line parameters in these cases,
we took the following approach.
First, if the profiles of the higher J (i.e., 7\to6 or 5\to4) lines or
C$^{34}$S
lines (where available)
matched one or more of the peaks seen in the lower
J transitions, we assumed that
the source was composed of distinct cloud
components (e.g., Figure 1a); and we derived
the line parameters by performing a multiple
Gaussian fit to the whole profile.
Each Gaussian component is listed individually in Table 2.
Three sources have 2 velocity components and one has 3 components; these
are identified in Tables 2 and 3 by the notation `` C\#'' (where
\# is the component number). With the inclusion
of all the separate components, Table 2 displays results for 155
cloud components.
Second, if comparison of CS data with
C$^{34}$S data indicated that the
CS line was self-absorbed (Figure 1b shows an example of this situation),
we calculated the line parameters ($\int T_R^* dV$, V$_{LSR}$, and FWHM)
from moment integrals over the profile. Then $T_R^*$ was calculated from
$\int T_R^* dV$ /FWHM (values given in parentheses in Table 2).
Only 18 of the 150 spectra
were obviously self-absorbed in CS 2\to1, with smaller numbers showing obvious
self-absorption in the higher-J lines.
Of course, self-absorption may exist at a less obvious level in other sources.
Figure 2 illustrates the detection rate for the observed CS transitions.
The distribution counts as detected
only those sources with observed T$_R^* \geq 0.5$K .
Because the sensitivity achieved for
the CS J = 7\to6 line (Paper I) was similar to that for the lower
J transitions, the drop in the detection rate towards higher rotational
levels reflects a real drop in the number of
sources exhibiting emission at the same level in the higher J lines.
\subsection{Extent of the Dense Gas: CS J = 5\to4 Maps}
To determine the effect that very dense gas has upon star formation,
we need to know the extent of the gas and its location within the star-forming
regions. We have observed 21 of our sources in the
CS 5\to4 line with the CSO. For each source, we made a cross-scan in R.A.
and Dec., typically consisting of 9 points. For most of the sources, the
separation of the observed points was 30\arcsec. For a few of the smaller
sources, we made the observations at 15\arcsec\ intervals.
In addition, we have assembled from the literature
data taken with the same equipment for four
other sources from our survey.
Table 5 lists the mapping results
for all 25 sources. The integrated intensities
listed in Table 5 are for the interpolated maximum along each cross scan.
{}From the maps we derived diameters and beam correction factors,
$F_c = (\Omega_{source}+\Omega_{beam}$)/$\Omega_{beam}$.
The beam correction factors were calculated assuming that
a Gaussian was a good representation of both the beam shape and the source
intensity distribution. Using the integrated intensity, the $F_c$, and
the distances, $d$(kpc), we calculated the luminosity in the CS J=5\to4
line from
\begin{equation}
L({\rm CS \ 5-4}) = 1.05 \times 10^{-5} L_{\sun} d^2 F_c \int{T_R^*dv}.
\end{equation}
Table 5 also lists the offsets of the CS 5\to4 peaks from the maser
positions in arcseconds.
With the exception of a few of the larger sources, differences
in the peak position of
the CS 5\to4 distribution and the H$_2$O maser position are smaller than
the combined pointing uncertainties and maser positional uncertainties
($\pm$3\arcsec\ and $\leq\pm$8\arcsec, respectively).
Jenness et al. (1995) have also found a very good correlation between the
peak of the submillimeter emission and the maser position.
The mean diameter of the sources listed in Table 3 is 1.0 pc.
The dispersion about this mean, however, is large (0.7 pc). If one
examines sources at $d\leq 3.0$ pc, the mean diameter is 0.5 pc with
a dispersion of 0.4 pc. This difference, while significant, probably
does not arise from observational biases in the CS data. Most
of the more distant sources are well resolved and bright. It is more
likely that the differences arise from selection biases in the original
samples used to search for H$_2$O masers.
Complete mapping of the CS 5\to4 line in several sources gives similar
sizes. The emission in NGC2024 has a diameter of 0.4 pc, while S140 has
a diameter of 0.8 pc (Snell et al. 1984). The emission in M17 is more
extensive:
2.3 pc in 5\to4 (Snell et al.); 2.1 pc in 7\to6 (Wang et al. 1993).
\section{Analysis}
With the addition of the lower J transitions in the present study to the
CS J = 7\to6 data from Paper I, we can
determine densities in a large sample of star-forming regions.
In \S 4.1, we discuss the calculations and examine the effects
of opacity and uncertainties in kinetic temperature
on density and column density determinations.
In \S 4.2, we consider the effects of density inhomogeneities, and
we compute masses in \S 4.3.
\subsection{Densities and Column Densities}
To determine densities and column densities,
we used a large velocity gradient (LVG) code to solve
the coupled equations of statistical equilibrium and radiative transfer,
including the first 20 rotational levels of CS in the calculation.
We assume that the gas has a constant density and temperature and that it
uniformly fills all the beams used in this study.
We calculated a 20$\times$20 grid of radiation temperatures
in column density per velocity interval -- density space
for a kinetic temperature of 50 K.
The CS densities in the LVG model grid ran from
$10^4$ to $10^8$ cm$^{-3}$, and the column densities per velocity interval
(N/$\Delta$v) ranged from $10^{11}$ to $10^{16}$
cm$^{-2}$/km s$^{-1}$. These ranges span the parameter space of
all solutions which fit our data. All the models converged to a solution.
Using a $\chi^2$ minimization routine, we fit the LVG models to the
observed CS line intensities.
Table 6 lists the densities for 71 sources.
We have not included fits for the CS data obtained in 1990 June, for reasons
discussed below. We have listed the log of the density and column density,
along with the value of $\chi^2$ and a key to which transitions were used
and whether the lines were self-absorbed. The values of density and column
density apply to the region selected by the typical beam used for
the observations (about 20\arcsec ). The $\chi^2$ values allow us
to assess whether the models (any of the points in the LVG grid) are
a good representation of the data.
The distribution of $\chi^2$ values for sources with 4 transitions
(40 sources) is similar to
what is expected theoretically if the model is a reasonable fit to the data,
as is the distribution for sources with only three transitions (31 sources).
These facts suggest that our estimates of the calibration uncertainties
are reasonable. We originally included the 1990 June data in the fits, but
they had a very high percentage of bad fits, leading us to conclude that
the uncertain pointing made them unsuitable for combining with the
CSO J=7\to6 data. The 8 self-absorbed sources with fits in Table 6
(marked by a flag) do not have $\chi^2$\ significantly worse than the other
sources. One source with 3 transitions (212.25-1.10) produced a very uncertain
density, and we have excluded it from the statistics that follow.
The mean logarithmic density for sources with detected emission from
all 4 CS transitions is $\langle$log(n)$\rangle$ $= 5.93 \pm 0.23$, where 0.23 represents
the standard deviation of the distribution.
The mean logarithmic column density is $\langle$log(N)$\rangle$ $= 14.42 \pm 0.49$.
The results for the sources undetected in J=7\to6 are
$\langle$log(n)$\rangle$ $= 5.59 \pm 0.39$; $\langle$log(N)$\rangle$ $ = 13.57 \pm 0.35$.
Figure 3 shows histograms of the densities and column densities.
The solid line plots the densities determined from
all 4 CS transitions and the dashed line is the density distribution
for sources without J= 7\to6 detections.
These results show that the difference between a CS 7\to6 detection
and non-detection is more related to column density than to volume density.
Therefore, the detectability of lines of
high critical density is more affected by the quantity of dense gas
present than by its density.
To check whether the difference was solely a result of having a J=7\to6 line
to fit, we re-fit the sources with 7\to6 detections,
forcing the $\chi ^2$ fitting routine
to ignore the CS 7\to6 line and to fit only the 3 lower transitions.
The resulting $\langle$log(n)$\rangle$\ is $5.71 \pm 0.19$, and $\langle$log(N)$\rangle$\ is $14.36 \pm 0.49$.
This result confirms our conclusion that the most significant difference
between
a J=7\to6 detection and a non-detection is the column density.
What effect would high opacity in the CS lines have on the
derived densities and column densities?
Eighteen of the sources in this survey have noticeable self-absorption in
at least one transition.
In addition, an LVG model run for the mean density, column density, and
linewidth results in CS line opacities that are roughly unity.
Thus, self-absorption may affect the fits, even if it is not apparent
in the line profiles.
Since the C$^{34}$S transitions will usually be optically thin,
we independently fit the C$^{34}$S transitions to an LVG model grid,
with a range of parameters identical to those used in the original CS grid.
Table 6 lists the densities, column densities, and $\chi{^2}$ derived from
fits to the C$^{34}$S data.
Problems with the receivers during the C$^{34}$S observations meant that
we have various combinations of lines to fit, as indicated by
the key in Table 6.
There are few sources with both adequate CS and acceptable C$^{34}$S data.
The fits to the sources with three transitions of C$^{34}$S give
$\langle$log(n)$\rangle$ $= 5.95 \pm 0.20$,
essentially identical to the $\langle$log(n)$\rangle$\ derived from 4 transitions of CS.
The mean difference between CS and C$^{34}$S in log(n) is $0.07\pm0.24$,
indicating no significant difference in the derived densities.
It is unlikely that the densities calculated for sources in our survey from the
CS lines alone are seriously affected by CS optical depth.
The average isotope ratio, $N(CS)/ N(C^{34}S)$, is $5.1\pm 2.2$,
clearly less than the terrestrial ratio, and
lower than the isotope ratios of 9--17 found by Mundy et al. (1986)
and 13 (Wang et al. 1993). Chin et al. (1996) have recently found
evidence for low values of this ratio in the inner Galaxy, but our
values are lower still. It is likely that our procedure has underestimated
$N(CS)$ to some extent. For this reason, and also because
these ratios are not very well determined
for individual sources, we have adopted an isotopic abundance
ratio of 10 in what follows.
By increasing the number of transitions,
simultaneous fitting of the CS and C$^{34}$S data should,
in principle, allow us to determine the
densities and column densities more accurately.
Using the LVG model grid for CS and constraining the isotope ratio to be
10, we fit CS and C$^{34}$S transitions simultaneously.
The results are listed in Table 6.
While neither the densities nor the column densities
are significantly different from those determined from
fits to the CS data alone, $\chi ^2$ is considerably larger.
The poor fits probably result from assuming a fixed isotopic
abundance ratio for all sources.
It is likely that many of the regions of massive star formation
contained within this study have temperatures in excess of 50 K.
At the densities
implied by the CS observations, the gas kinetic temperature will be coupled
to the dust temperature. For grains with opacity decreasing linearly
with wavelength, one can write
\begin{equation}
T_D = C [{{L}\over{\theta^2 d^2}}]^{0.2},
\end{equation}
where $L$ is the luminosity in solar units, $d$ is the distance in kpc,
and $\theta$ is the angular separation from the heating source in arcseconds.
Using these units, $C = 15$ to 40 (Makinen et al. 1985, Butner et al. 1990).
We can estimate the range of temperatures in our sources from the
luminosities in Table 7 and distances in Table 5; $\langle (L/d^2)^{0.2}
\rangle
= 7.5\pm 1.6$. At a radius of 10$\arcsec$, characteristic of the beams in
Table 1 and the beam of the J = 7\to6 observations, $T_D = 50$ to 100 K.
To assess the effects of temperature uncertainties on the derived
source properties, we also fit the sources with 4 transitions to a grid
of models run for a temperature of 100 K. The value of $\langle$log(n)$\rangle$\ decreased by
0.3 and the value of $\langle$log(N)$\rangle$\ was essentially unchanged. Regardless of the
assumed temperature, our data imply a thermal pressure, $nT \sim 4 \times 10^7$
K cm$^{-3}$, which is much higher than found in regions not forming
massive stars.
Within the limitations of a single-density model, we conclude that
the effects of opacity and temperature on the determinations of density
are not severe (about at the factor of 2 level). Typical densities in regions
detected in the J=7\to6 survey are $10^6$ cm$^{-3}$. Toward water masers not
detected in the J=7\to6 survey, the densities are about a factor of 2 less,
but the column densities of CS are about a factor of 7 less, on average,
than the values found for regions detected in the J=7\to6 line.
The densities for both groups of sources are considerably
less than the critical density of the
CS J=7\to6 line ($2 \times 10^7$ cm$^{-3}$), reminding us that detection
of emission from a hard-to-excite line does not imply the existence of gas at
the critical density. Molecules can emit significantly in high-J transitions
with critical densities considerably above the actual density because
of trapping and multilevel effects (see also Evans 1989).
For example, levels with J$>>0$ have many possible routes for excitation
by collisions, but only one radiative decay path.
The high densities found in this survey of regions forming massive stars
are similar to those obtained from other, more detailed, studies
of individual, luminous, star-forming regions (see ref. in \S 1).
Consequently, the results found from studies of a few clouds can be applied,
in a statistical sense, to the broader sample of massive star-forming
regions.
\subsection{ Multiple Density Models}
Our LVG analysis assumes that the density is uniform and that
the emitting gas fills the beam.
How good are these assumptions? Figure 4 gives examples of LVG
model fits to several of the sources: three with good fits and three
with bad fits, as measured by the $\chi^2$\ value.
While the LVG models generally fit the data within the uncertainties,
a closer look reveals that the discrepancies between model and observation
are very consistent, even for the good fits.
Almost all fits overpredict the 3\to2 and 5\to4 lines
and underpredict the 2\to1 and 7\to6 lines.
Thus, the data have, on average, a smaller variation of intensity with
J than do the best-fit
LVG models, as would be expected for a source with
a mixture of gas at different densities.
In this section, we examine models with varying densities to
see how well they explain the intensity versus J distribution.
\markcite{Snell et al. (1984) } and Wang et al. (1993) have discussed
the effects of fitting a single density to the CS emission from
a mixture of gas at about $10^6$ cm$^{-3}$\ and gas lower in density by about a
factor of 10. They showed that, until the filling factor of the high
density gas becomes very small (i.e., $f< 0.2$), the density derived
from fitting a single density model matches that of the high density
component to within a factor of two.
The CS transitions we have observed should behave
in a similar way in that they are biased toward measuring gas with
densities close to $10^6$ cm$^{-3}$.
We now ask a more radical question. Could the apparent density near
$10^6$ cm$^{-3}$\ be an artifact of fitting to a single density a mixture of
ultra-dense gas (n = $10^8$ cm$^{-3}$) and gas at a much lower (n = $10^4$ cm$^{-3}$)
density?
In this picture, the histogram of densities (Figure 3) would be produced by
varying the filling factor of the dense component.
We chose a value of $10^8$ cm$^{-3}$\ for the
density of the ultra-dense gas because the 7\to6 transition
becomes completely thermalized at that density. Thus, the component
with n$= 10^8$ cm$^{-3}$\ represents any gas with n$\geq 10^8$ cm$^{-3}$.
We synthesized clouds from a mixture of these two components
at 20 values of N/$\Delta$v between
$10^{12}$ and 10$^{16}$ cm$^{-2}$/km s$^{-1}$.
For each density and column density, we used the LVG code to calculate the
expected emission. We then
varied the filling factor of the ultra-dense gas ($f$) and
the low-density gas ($1-f$), with $0 \leq f \leq 1$ in steps of 0.05,
and summed the contributions to each transition for each possible combination
of $f$, column density of the gas at n$= 10^4$ cm$^{-3}$\ (N$_{\rm low}$), and
column density of the gas at n$= 10^8$ cm$^{-3}$\ (N$_{\rm high}$).
These results then formed
a grid of models which could be fitted to the data, just as the single-density
models had been fitted. We found that the $\chi^2$\ value worsened,
despite the extra free parameter, for sources where the single-density
fit had been good ($\chi^2$ $\leq 1$). On the other hand, the sources
which were poorly fitted ($\chi^2$ $> 1$) with the single-density model
were better fitted with the two-density model. The two-density fits
typically required very high column densities ($\langle$log(N)$\rangle$ $= 16.16$) of
the low-density gas compared to those of the ultra-dense gas
($\langle$log(N)$\rangle$ $= 13.85$).
To see if we could constrain the amount of ultra-dense gas in the sources
with poor single-density fits, we followed a similar, but
less restrictive, procedure.
We started by assuming that the CS J = 2\to1 and 3\to2
transitions effectively probe the low density gas in the
beam, and we used them to fit the density (n$_{\rm low}$)
and column density (N$_{\rm low}$) of the
low-density component. We then used the LVG code
to obtain the expected emission from each rotational
transition for a gas at this density and column density at a
temperature of 50K. These intensities, multiplied by ($1-f$),
were used to represent the
lower density component. We then searched a parameter space of $f$ and
log(N/$\Delta$v) for the best values for the ultra-dense component
(density once again fixed at 10$^8$ cm$^{-3}$). We summed ($1-f$) times
the lower density intensities and $f$ times the ultra-dense gas intensities
and compared this sum to the observations.
This method has a large number of free parameters: $f$, n$_{\rm low}$,
N$_{\rm low}/ \Delta$v, and N$_{\rm high}/ \Delta$v, which are constrained
by only 4 transitions. Furthermore, it
does not correct the properties of the lower
density component for the contributions of the high
density gas to the J = 2\to1 and 3\to2 emission. We use it for
illustrative purposes only. We show the two-density fits as dashed
lines in Figure 4, but we do not tabulate the results. The mean
properties of these solutions for the sources with single-density $\chi^2$ $> 1$
are as follows: $f = 0.22$, log(n$_{\rm low})
= 5.4 \pm 0.3$, log(N$_{\rm low}) =
14.39$, and log(N$_{\rm high}) = 14.39$ (equal column densities in the two
components). Thus,
in general, the filling factor of ultra-dense gas is small (less than
25\%), and the data still favor a large amount of gas at $n > 10^5$ cm$^{-3}$.
Another possible source model is a continous density gradient, such
as a power law. Power-law density distributions have been proposed for
regions of low-mass star formation on theoretical grounds (Shu 1977)
and seem to fit the observations well in some cases (e.g., Zhou et al. 1991).
They have also been applied to some regions forming stars of higher mass
(e.g., Zhou et al. 1994; Carr et al. 1995). The latter reference is
particularly relevant here, as it included a more complete
analysis of GL2591 (called CRL2591 in this paper),
including data from this paper, but adding other data. While Table 6
indicates a good fit to the data for that source with a single-density model,
Carr et al. found that a single density cannot fit all the data,
when other data are included, particularly J = 5\to4
and 10\to9 data from the CSO.
They developed models with power-law density and temperature gradients
that fit all the data. We can use the example of CRL2591 to explore the meaning
of the densities in Table 6 if the actual density distribution is
a power law. If
$n(r) = n_1 r_{pc}^{-\alpha}$, with $n_1$ (the density at 1 pc) set
by matching the line profiles (Carr et al. 1995),
the density in Table 6 is reached at
radii of 18\arcsec\ to 7\arcsec\ for $1 \leq \alpha \leq 2$, corresponding to filling
factors of 0.3 to 0.6 in our largest beam. We conclude that, in this source,
the densities derived in this study characterize gas on scales somewhat
smaller than our beams, if the source has a density gradient. Similar studies
of other sources are needed to see if this conclusion can be generalized.
Further evidence for a range of densities is that J=10\to9 emission has
been seen in a number of sources (Hauschildt et al. 1993 and our Table 4).
The data do not warrant detailed source-by-source modeling, but we have
predicted the expected J=10\to9 emission from a source with the mean
properties found in \S 4.1: log(n) = 5.93 and log(N) = 14.42. We assumed
a linewidth of 5.0 km s$^{-1}$, about the mean for our sample, and T$_K$ = 50 K.
The predicted T$_R$ of the J=10\to9 line is 0.2 K for this average cloud,
weaker than any of the detections.
If we use the conditions for the cloud with properties at the high end of the
1 $\sigma$ spread, we can produce T$_R$ = 1.6 K, about the weakest detection.
Increasing T$_K$\ to 100 K raises the prediction to 7 K, similar to many of
the detections. Detection of a J=10\to9 line therefore implies a cloud with
higher density, column density, and/or temperature than the average cloud
in our sample of sources detected at J=7\to6.
\subsection{Masses }
Table 7 contains mass estimates for the regions for which
we have determined cloud sizes. We have computed three different
estimates. The first estimate assumes that the volume density
fills a spherical volume with the diameter of the J=5\to4 emission:
\begin{equation}
M_n = {{4}\over{3}}\pi{r^3}{n}{\mu},
\end{equation}
where r is the radius of the cloud
and $\mu=2.34m_H$ is the mean mass per particle.
The second estimate uses the CS column densities (N) and the formula:
\begin{equation}
M_N = \pi{r^2}{{N}\over{X}}{\mu},
\end{equation}
where X is the abundance of CS. We have used $X = 4 \times 10^{-10}$,
based on a more detailed analysis of one of the sources in this
study (Carr et al. 1995).
Finally, we estimated masses from the virial theorem:
\begin{equation}
M_{V} = {{5}\over{3}}{{R V^2_{rms}}\over{G}},
\end{equation}
for a spherical, non-rotating cloud. Assuming
that the velocity profile is Gaussian, $V_{rms}$
is related to the FWHM ($\Delta{v}$) of the line by
$V_{rms} = \sqrt{3} \Delta{v}/2.35$.
We used the average $\Delta{v}$ of the CS lines.
The value of $M_n$ for GL490 is probably underestimated
substantially because the maser position is quite far from
the peak of a very compact source. Zhou et al. (1996) have
analyzed this source in more detail and found considerably higher
densities from spectra on the peak. Consequently, we ignore this
source in the following discussion.
The average ratio of $M_N/M_n$ is $0.84\pm 0.73$.
The agreement is gratifying, but the poorly known abundance of CS makes $M_N$
quite uncertain. In contrast, the agreement between $M_n$ and $M_V$ is worse,
with $M_n$ almost always considerably larger than $M_V$.
A likely explanation is that the gas is distributed inhomogeneously
within the beam, whereas the calculation of
$M_n$ assumes that the density is uniformly distributed.
We have used the ratio of $M_V$ to $M_n$ to estimate the volume
filling factor ($f_v$) of the gas, also listed in Table 7.
The filling factors have a large range (0.02 to 2.3) and a mean
value of $0.33\pm 0.59$.
The virial mass estimate is susceptible to
error because the linewidth may be affected by unbound
motions, such as outflows, and it ignores effects of external
pressure. Least certain is $M_n$, which depends on the cube
of the size (and hence distance). Each mass estimate depends on
a different power of the size, making their ratio strongly dependent
on uncertainties in the distance.
In view of the problems inherent in each of the
different mass calculations, the masses agree reasonably well.
Because the virial mass estimates have the fewest potential
problems, we will use them in what follows.
The average $M_V = 3800$ M$_{\sun}$.
\section{Implications }
\subsection{Comparison to Other Star-Formation Regions }
Are the high densities seen in this survey
peculiar to regions of massive star formation or are they
a feature of star formation in general?
Lada, Evans, \& Falgarone (1996) have found that
the density in the most active star-forming cores in L1630
is about log(n) = 5.8, very similar to what we find.
We also compared the results of our
study with surveys of regions forming low-mass stars.
\markcite{Zhou et al. (1989)}
observed a sample of low-mass cores in CS transitions up to J=5\to4
and derived densities of $\langle$log(n)$\rangle$ $ = 5.3\pm 1.1$.
These densities are about a factor of 4 lower than the densities
we find in this study (and in other studies of regions of massive
star formation). Since Zhou et al. (1989) did not have J=7\to6 data,
it may be more appropriate to compare with our fits to sources without
J=7\to6 detections; in that case, our densities are larger by
a factor of about 2. The net result is that regions forming massive
stars do seem to have larger densities when similar techniques are
used, but the difference is not an order of magnitude.
The ability to form low-mass stars in regions of massive star formation
may depend on whether the Jeans mass remains low as the cloud is heated.
We can calculate the Jeans mass from
\begin{equation}
M_J(\rm{M}_{\sun}) = 18T^{{3}\over{2}}n^{-{{1}\over{2}}}.
\end{equation}
Using the mean logarithmic densities and the assumed temperatures
(10 K for the low-mass cores, 50 K for our sample),
we compute $\langle M_J\rangle = 1.3$ M$_{\sun}$
for the clouds forming low-mass stars
and $\langle M_J\rangle = 7 $M$_{\sun}$ for clouds in this
study with J=7\to6 emission.
The assumed temperatures make $M_J$ higher in regions forming massive stars
even though they are denser. However, the strong dependence of $M_J$
on temperature means that statements about average properties
should not be taken too literally until the temperatures are
known better. In addition, the
fragmentation spectrum may have been established early in the evolution
of the core, before the temperatures were raised by the formation
of massive stars.
\subsection {Do Larson's Laws Apply to Massive Cores?}
Most studies of the global properties of molecular clouds deal with
the usual linewidth--size--density relations, as proposed by Larson (1981)
and confirmed by others (e.g., Fuller \& Myers 1992; Solomon et al. 1987;
Caselli \& Myers 1995). These relations were generally found by comparing
properties of whole clouds; similar relations were found within single
clouds by comparing map sizes in transitions of different molecules.
A recent paper by Caselli \& Myers (1995) includes
information on both low mass cores and more massive cores within
the Orion molecular cloud. They fit the non-thermal linewidth (the
observed linewidth after correction for the thermal contribution) and cloud
radius for these types of regions separately to this relation:
\begin{equation}
{\rm log} \Delta v ({\rm km s^{-1}}) = b + q {\rm log} R({\rm pc}).
\end{equation}
They found a strong relationship (correlation coefficient, $r = 0.81$)
in low-mass cores with $b= 0.18 \pm 0.06$ and $q= 0.53 \pm 0.07$.
The relation was considerably weaker ($r = 0.56$) and
flatter ($q = 0.21 \pm 0.03$) in the massive cores.
In Figure 5, we plot log($\Delta v$) versus log$R$ for the sources
in Table 5, which are generally denser and more massive than the
cores studied by Caselli \& Myers. No relationship is apparent
(the correlation coefficient is only $r = 0.26$),
despite the fact that our sample covers a range of 30 in source size.
Nevertheless, we fitted the data to equation 7 using least squares
and considering uncertainties in both variables (we assumed 20\%
uncertainties in size and used the standard deviation of the linewidths
of the different lines for the uncertainty in $\Delta v$).
The result was $b = 0.92 \pm 0.02$ and $q= 0.35\pm 0.03$, but
the goodness of fit parameter, $Q = 2.8 \times 10^{-8}$,
whereas a decent fit should have $Q>0.001$. Alternatively, we minimized
the mean absolute deviation (robust estimation; see Press et al. 1992).
The result was $b = 0.80$ and $q= 0.08$, indicating essentially no
size--linewidth relation.
Thus our data confirm
the trend discernable in the analysis of Caselli \& Myers: the
$\Delta v - R$ relation tends to break down in more massive cores.
We have plotted the Caselli \& Myers relations in Figure 5, along
with Larson's original relation.
It is clear that our sources have systematically higher linewidths
at a given radius than sources in other studies. For the radii we
are probing, most other studies were considering linewidths from CO or
its isotopes and may thus have included a larger contribution from low-density
envelopes. The usual relations would predict larger $\Delta v$ in these
regions,
which would make the discrepancy worse. However, our sources are regions of
massive star formation,
and Larson (1981) noted that such regions (Orion and M17 in his study) tended
to have larger $\Delta v$ for given size and not to show a size--linewidth
correlation.
Most previous studies have found an inverse relation between $mean$ density
and size, corresponding to a constant column density. However, Scalo (1990)
and Kegel (1989) have
noted that selection effects and limited dynamic range may have produced this
effect, and Leisawitz (1990) found no relationship between density and size
in his study of clouds around open clusters. In previous studies,
the $mean$ densities were found by dividing a column density by a size,
which might be expected to introduce an inverse correlation if the column
density tracer has a limited dynamic range. Since our
densities were derived from an excitation analysis, it may be interesting to
see if any correlation exists in our data.
We plot log(n) versus log$R$ in Figure 5. Again, no correlation
is evident ($r = -0.25$), and our densities all lie well above (factors of
100!)
predictions from previous relations (e.g., Myers 1985). Again, Larson
(1981) noted a similar, though much less dramatic, tendency for regions
of massive star formation in his analysis. For a recent theoretical
discussion of these relations, see V\'azquez-Semadeni, Ballesteros-Paredes,
\& Rodr\'iguez (1997).
To use data on sources without
size information, we plot (in the bottom panel of Figure 5)
log($\Delta v$) versus log(n). The previous relations would
predict a negative slope (typically $-0.5$) in this relation.
In contrast to the predictions, our data show a positive, but small,
correlation coefficient ($r= 0.40$). The slope from a least squares fit
is quite steep ($1.3\pm 0.2$), but robust estimation gives a slope of
only 0.39. In addition, the linewidths are much larger than would have
been predicted for these densities from previous relations.
These results suggests that an uncritical application of scaling
relations based on {\it mean} densities to actual densities, especially in
regions of massive star formation, is likely to lead to errors.
The fact that Larson's laws are not apparent in our data indicate
that conditions in these very dense cores with massive star formation
are very different from those in more local regions of less massive
star formation. The linewidths may
have been affected by star formation (outflows, expanding HII regions,
etc.); the higher densities are probably caused by gravitational
contraction, which will also increase the linewidths.
While the regions in this study may not be typical of most molecular
gas, they are typical of regions forming most of the massive stars
in the Galaxy. These conditions (denser, more turbulent than usually
assumed) may be the ones relevant for considerations of initial mass
functions.
\subsection{Luminosity, Star Formation Efficiency, and Gas Depletion Time}
We have collected data from the literature (or our own
unpublished data) on the luminosity of the sources in Table 7.
The ratio of the luminosity to the virial mass
($L/M$), roughly proportional to the star formation rate per
unit mass, ranges from 24 to 490 in solar units (see Table 7)
with a mean of $190 \pm 43$, where 43 is the standard deviation of the mean
(all other uncertainties quoted in the text are standard
deviations of the distribution). Previous studies, using masses
determined from CO luminosity, have
found much lower average values of $L/M$: 4.0 for the inner galaxy
(Mooney \& Solomon 1988); 1.7 for the outer galaxy (Mead,
Kutner, \& Evans 1990). In fact, the maximum values in those
samples were 18 and 5, respectively, smaller than any of our
values. The enormous difference is caused by
the fact that we are calculating the mass of the dense gas,
which is much less than the mass computed from the CO luminosity.
While we have also tried to use luminosities measured with
small beams, the main difference is in the mass. One
way to interpret this result is that the star formation rate
per unit mass
rises dramatically (a factor of 50) in the part of the cloud with dense gas.
The star formation rate per unit mass of very dense gas may be more relevant
since stars do not seem to
form randomly throughout molecular clouds (Lada et al. 1991).
Instead, the 4 most massive CS cores in L1630,
which cover only 18\% of the surveyed area,
contain 58\% to 98\% of all the forming stars, depending on
background correction. Li et al. (1996) have found that there
is little evidence for any recent star formation outside the clusters,
suggesting that the 98\% number is closer to correct.
The star formation efficiency in the clusters can be
quite high (e.g., 40\%) compared to that of the cloud
as a whole (4\%) (Lada et al. 1991).
The gas depletion time ($\tau$) is the time to turn all the molecular gas into
stars. Considering only stars of $M > 2$ M$_{\sun}$, the star
formation rate can be written as
$dM/dt$ (M$_{\sun}$ yr$^{-1}$) = $4 \times 10^{-10} L$
(Gallagher \& Hunter 1986; Hunter et al. 1986). The coefficient differs
by only 20\% if the lower mass cutoff is 10 M$_{\sun}$.
The gas depletion time can then be written as
$\tau\ = 2.5 \times 10^{9} (M/L)$ yr. Using our value of average $L/M = 190$,
$\tau = 1.3 \times 10^7$ yr. This time is comparable to that for
dispersal of clouds surrounding open clusters; clusters with ages
in excess of $1.0\times 10^7$ yr do not have associated molecular clouds
with masses as large as $10^3 M_{\sun}$ (Leisawitz, Bash, \& Thaddeus
1989).
\subsection{ Luminosity of the Galaxy in CS J = 5\to4}
CS J = 5\to4 emission has been seen toward the centers of
NGC 253, M82, IC 342, Maffei 2, and NGC 6946
(\markcite{Mauersberger and Henkel 1989; }
\markcite{Mauersberger et al. 1989)}.
For comparison to studies of other galaxies, we will estimate the luminosity
of the Milky Way in CS 5\to4 [$L_G ({\rm CS \ 5-4})$] from the mean $L ({\rm CS \ 5-4})$\
per cloud in Table 5
and an estimate of the number of such clouds ($n_{cl}$) in the Galaxy.
{}From Table 5
we find $\langle L({\rm CS \ 5-4}) \rangle = 4 \times 10^{-2}$ L$_{\sun}$ and
$\langle\int{T_R^*dv}\rangle = 34$ K km s$^{-1}$, whereas
$\langle\int{T_R^*dv}\rangle = 42$ K km s$^{-1}$ for the whole sample in
Table 2. If we correct for the fact that the mean integrated intensity
of the sources in Table 5 is less than the mean of the whole sample,
we would get $5 \times 10^{-2}$ L$_{\sun}$ for the typical core.
We do not have
a direct measurement of $n_{cl}$ because our survey is incomplete.
The most recent update to the H$_2$O\ maser catalog (Brand et al. 1994)
brings the total
number of masers with IRAS colors characteristic of star formation regions
(see Palagi et al. 1993) to 414. If we assume
that our CS 5\to4 detection rate of 75\% applies equally to the other
sources, we would expect 311 regions of CS J = 5\to4 emission
in a region which covers two thirds of the galaxy. If we correct for the
unsurveyed third of the galaxy, we would estimate the total number
of cloud cores emitting CS J =5\to4 to be 466.
Consequently, we will assume $n_{cl} = 311 - 466$,
with the larger values probably being more likely.
Using these numbers, we calculate $L_G ({\rm CS \ 5-4}) = 15 - 23$ L$_{\sun}$.
Even though we have made some completeness corrections, we expect these
estimates to be underestimates because of our limited sensitivity and the
likelihood of CS emission from dense regions without H$_2$O\ masers.
These values can be compared with the luminosities of other galaxies
in Table 8. However, our estimate applies to the entire Galaxy excluding
the inner 400 pc,
while the $L ({\rm CS \ 5-4})$\ for other galaxies are derived from
a single beam, centered on the nucleus, with a radius given in the
Table. The inner 100 pc of M82 and NGC 253 emit more CS J = 5\to4
than does our entire Galaxy, excluding the inner 400 pc.
We can also compare our Galaxy to others in terms of its star formation rate
per unit mass. In \S 5.3, we used $L/M$, with $M$ as the virial mass, to
measure this quantity. Because linewidths in galaxy observations are likely to
reflect the total mass, rather than the gaseous mass, we will use
$L$/ $L ({\rm CS \ 5-4})$\ as a stand-in for the star formation rate per unit mass of dense
gas.
We have tabulated the far-infrared luminosity of the galaxies in Table 8,
using the data with the smallest available beam, to provide the best
match to the CS J = 5\to4 observations, which were mostly done with the
IRAM 30-m telescope ($11\arcsec$ beam).
The resulting $L$/$L ({\rm CS \ 5-4})$\ values range from $5.0 \times 10^7$ (NGC 253) to
$1.7 \times 10^9$ (M82). These numbers apply to regions typically 100 pc in
radius.
For our Galaxy, we have only the total $L ({\rm CS \ 5-4})$, so we compare to the total
$L = 1.8 \times 10^{10}$ L$_{\sun}$ (Wright et al. 1991). The result is
$8 - 13 \times 10^8$, nominally similar to M82; however, much of the
far-infrared emission of our Galaxy is likely to result from heating by
older stars.
Probably a more useful comparison is to the values of $L$/$L ({\rm CS \ 5-4})$\ in individual
clouds (Table 7). No individual cloud approaches the value in M82.
The highest value in Table 7 is about twice that of NGC 253 and half that
of IC 342.
\section{Summary}
\begin{enumerate}
\item Very dense gas is common in regions of massive star formation.
The gas density for the regions selected by having a water maser
is $\langle$log(n)$\rangle$ $ = 5.93$ and the CS column density is $\langle$log(N)$\rangle$ $ = 14.42$.
For regions without CS J = 7\to6 emission
the mean density is half as large and the mean column
density is about 7 times smaller.
These results are relatively insensitive to both CS optical
depth and to changes in the kinetic temperature of the region.
The mean density is an order of magnitude less than the critical
density of the J = 7\to6 line because of trapping and multilevel excitation
effects.
\item
In many regions forming massive stars, the
CS emission is well modeled by a single density gas component, but
many sources also show evidence for a range of densities.
{}From simulations of emission from gas composed of two different
densities ($10^4$ and $10^8$ cm$^{-3}$), we conclude that there are
few clouds with filling factors of ultra-dense gas (n$ = 10^8$ cm$^{-3}$)
exceeding 0.25.
\item
The densities calculated for the sources in this survey are comparable
to the densities seen from detailed studies of a few individual
regions forming massive stars.
Therefore, it is likely that very dense gas is
a general property of such regions.
The average density of regions forming massive stars
is at least twice the average in regions forming only low-mass
stars.
\item
Using a subsample of sources whose CS 5\to4 emission was
mapped at the CSO, the average cloud diameter is 1.0 pc and the average
virial mass is 3800 M$_{\sun}$.
\item
We see no evidence for a correlation between linewidth and size or density
and size in our sample.
Our linewidths and densities are systematically larger at a given size
than those predicted by previous relations.
There is, however, a positive correlation between linewidth and density,
the opposite of predictions based on the usual arguments.
\item
The ratio $L/M$, which is a measure of star formation
rate per unit mass for the dense gas probed by CS J=5\to4 emission,
ranges from 24 to 490, with an average value of 190.
\item
The dense gas depletion time, $\tau \sim 1.3 \times 10^7$ yr,
comparable to the dispersal time of gas around
clusters and OB associations.
\item
The estimated Galactic luminosity in the CS J = 5\to4 line is
$14-23$ L$_{\sun}$. This range of values is considerably less than
what is seen in the inner 100 pc of starburst galaxies. In addition,
those galaxies have a higher ratio of far-infrared luminosity to
CS J = 5\to4 luminosity than any cloud in our sample.
\end{enumerate}
\acknowledgements
We are grateful to the staff of the IRAM 30-m telescope for assistance
with the observations. We also thank T. Xie, C. M. Walmsley, and J. Scalo
for helpful discussion.
This research was supported in part by NSF Grant AST-9317567 to the
University of Texas at Austin.
\clearpage
\begin{table}[h]
\caption{Observing Parameters}
\vspace {3mm}
\begin{tabular}{l c c c c c c c }
\tableline
\tableline
Line & $\nu$ & Telescope & $\eta_{mb}^a$ & $\theta_b^a$ & $\langle
T_{sys}\rangle^b$ & $\delta v$ & $\delta v$ \cr
& (GHz) & & & ($\arcsec$) &(K) & (km s$^{-1}$) &(km s$^{-1}$) \\ \tableline
CS 2\to1 & 97.980968 & IRAM & 0.60 & 25\arcsec & 675 & 0.31$^c$
& 3.06$^d$ \cr
CS 3\to2 & 146.969049 & IRAM & 0.60 & 17\arcsec & 990 & 0.32$^e$ &
2.04$^d$ \cr
CS 5\to4 & 244.935606 & IRAM & 0.45 & 10\arcsec & 2500 & 1.22$^f$
& \nodata \cr
C$^{34}$S 2\to1 & 96.412982 & IRAM & 0.60 & 25\arcsec & 620 &
0.31$^{c,g}$ & 3.11$^d$ \cr
C$^{34}$S 3\to2 & 144.617147 & IRAM & 0.60 & 17\arcsec & 835 &
0.32$^{e,h}$ & 2.07$^d$ \cr
C$^{34}$S 5\to4 & 241.016176 & IRAM & 0.45 & 10\arcsec & 2700 & 1.24$^f$
& \nodata \cr
CS 5\to4 & 244.935606 & CSO & 0.71 & 30\arcsec & 445 & 0.17$^i$
& 1.2$^j$ \cr
C$^{34}$S 7\to6 & 337.396602 & CSO & 0.55 & 20\arcsec & 1000 & 0.12$^i$
& 0.89$^j$ \cr
CS 10\to9 & 489.75104 & CSO & 0.39 & 14\arcsec & 4300 & 0.09$^i$
& 0.61$^j$ \cr
CS 14\to13 & 685.434764 & CSO & 0.31 & 11\arcsec & 2050 & 0.06$^i$
& 0.44$^j$ \cr
\end{tabular}
\tablecomments{(a) Efficiency and beam size; (b) average $T_{sys}$ during
observing;
(c) 100 kHz filterbank; (d) Split 1 MHz filterbank; (e) Autocorrelator; (f) 1
MHz filterbank;
(g) $\Delta{V} = 0.486$ km s$^{-1}$\ for C$^{34}$S 2-1 in autocorrelator;
(h) $\Delta{V} = 0.207$ km s$^{-1}$\ for C$^{34}$S 3-2 in 100 KHz filterbank; (i) 50
MHz AOS;
(j) 500 MHz AOS.}
\end{table}
\clearpage
\begin{table}[h]
\caption{Standin for table 2 ps file. Discard this page.}
\vspace {3mm}
\begin{tabular}{l r r r r r }
\tableline
\tableline
Source & $\int$T$_{R^*}$dV & $V_{LSR}$ & FWHM & $T_R^*(10-9)$ & $T_R^*$(14-13)
\cr
& K km s$^{-1}$ & km s$^{-1}$ & km s$^{-1}$ & K & K \\ \tableline
\end{tabular}
\tablerefs{
(a) Carr et al. (1995)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Standin for table 3 ps file. Discard this page.}
\vspace {3mm}
\begin{tabular}{l r r r r r }
\tableline
\tableline
Source & $\int$T$_{R^*}$dV & $V_{LSR}$ & FWHM & $T_R^*(10-9)$ & $T_R^*$(14-13)
\cr
& K km s$^{-1}$ & km s$^{-1}$ & km s$^{-1}$ & K & K \\ \tableline
\end{tabular}
\tablerefs{
(a) Carr et al. (1995)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Results for CS $J = 10\rightarrow 9$ and $J= 14\rightarrow 13$ Lines}
\vspace {3mm}
\begin{tabular}{l c c c c c }
\tableline
\tableline
Source & $\int$T$_{R^*}$dV & $V_{LSR}$ & FWHM & $T_R^*(10-9)$ & $T_R^*$(14-13)
\cr
& (K km s$^{-1}$) & (km s$^{-1}$) & (km s$^{-1}$) & (K) & (K) \\ \tableline
GL2591$^a$ & 2.7 & -5.3 & 1.6 & 1.6 & \cr
S158A & 22.6 & -57.2 & 2.9 & 7.2 & \nodata \cr
W3(2) & 6.4 & -38.49 & 2.28 & 2.6 & \nodata \cr
W3(OH) & \nodata & \nodata &\nodata & \nodata & $<1.6$ \cr
S255 & 10.3 & 8.2 & 2.3 & 4.4 & $<0.7$ \cr
\end{tabular}
\tablerefs{
(a) Carr et al. (1995)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Diameters, Offsets and Luminosities from CS J = 5$\rightarrow$4 Maps}
\vspace {3mm}
\begin{tabular}{l c c c c c c c }
\tableline
\tableline
Source & ref & Dist. & $\int$T$_{R^*}$dV & $L ({\rm CS \ 5-4})$\ & Beam Corr. & Diameter &
Offset \cr
& & (kpc) & (K km s$^{-1}$) & (10$^{-2}$ L$_{\sun}$) & & (pc) & (arcsec) \\
\tableline
W43S & & 8.5 & 52.8 & 6.1 & 1.5 & 0.9 & (0,5) \cr
W43Main1 & & 7.5 & 22.1 & 5.2 & 4.0& 1.9 & (20,-36) \cr
W43Main3 & & 6.8 & 32.4 & 4.6 & 2.9 & 1.4 & (-8,2) \cr
31.25-0.11 & & 13 & 9.0 & 5.7 & 3.6 & 3.0 & (-12,-15) \cr
31.44-0.26 & & 9.4 & 23.0 & 8.6 & 4.0 & 2.4 & (-2,-4) \cr
32.8+0.2A & & 15 & 64.1 & 15 & 1.0 & $<$1.1 & (-5,-4) \cr
W44 & & 3.7 & 87.9 & 3.1 & 2.5 & 0.7 & (-3,0) \cr
W51W & & 7 & 12.0 & 1.6 & 2.6 & 1.3 & (0,-7) \cr
W51N & & 7 & 79.3 & 17 & 4.2 & 1.8 & (0,-5) \cr
W51M & & 7 & 152 & 19 & 2.4 & 1.2 & (-3,-2) \cr
ON1 & & 6 & 24.4 & 1.6 & 1.7 & 0.7 & (0,0) \cr
K3-50 & & 9 & 11.3 & 1.9 & 2.0 & 1.3 & (-5,5) \cr
ON3 & & 9 & 11.0 & 1.8 & 2.0 & 1.3 & (0,-4) \cr
ON2S & & 5.5 & 22.3 & 1.5 & 2.2 & 0.9 & (-6,0) \cr
ON2N & & 5.5 & 15.4 & 1.0 & 2.1 & 0.8 & (6,5) \cr
S106 & & 0.6 & 5.4 & 0.004 & 2.2 & 0.1 & (20,0) \cr
CRL 2591 & 1& 1.0 & 7.9 & 0.024 & 3.3 & 0.22 & (0,0) \cr
DR21 S & & 3 & 44.8 & 1.0 & 2.3 & 0.5 & (-6,5) \cr
W75(OH) & & 3 & 47.6 & 1.1 & 2.4 & 0.5 & (-6,-5) \cr
W75S1 & & 3 & 9.4 & 0.9 & 9.7 & 1.3 & (-3,7) \cr
W75S3 & & 3 & 6.8 & 0.2 & 3.2 & 0.7 & (0,0) \cr
W75N & & 3 & 35.2 & 0.8 & 2.5 & 0.5 & (-5,6) \cr
CepA & 2 & 0.73 & 30.0 & 0.1 & 5.5 & 0.2 & (10,12) \cr
W3(2) & 2 & 2.3 & 26.3 & 0.8 & 5.5 & 0.7 & (0,12) \cr
GL 490 & 2 & 0.9 & 7.5 & 0.01 & 1.8 & 0.12 & (-14,-12) \cr
\end{tabular}
\tablerefs{
(1) Carr et al. (1995); (2) Zhou et al. (1996)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Standin for table 6 ps file. Discard this page.}
\vspace {3mm}
\begin{tabular}{l r r r r r }
\tableline
\tableline
Source & $\int$T$_{R^*}$dV & $V_{LSR}$ & FWHM & $T_R^*(10-9)$ & $T_R^*$(14-13)
\cr
& K km s$^{-1}$ & km s$^{-1}$ & km s$^{-1}$ & K & K \\ \tableline
\end{tabular}
\tablerefs{
(a) Carr et al. (1995)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Masses and Luminosities}
\vspace {3mm}
\begin{tabular}{l c c c c c c c c c}
\tableline
\tableline
Source & Flag & $M_n$ & $M_N$ & $M_V$ & $f_v$ & $L$ & Ref. & $L/M_V$ & $L$/
$L ({\rm CS \ 5-4})$\ \cr
& & ($M_{\sun}$) & ($M_{\sun}$) & ($M_{\sun}$) & & ($L_{\sun}$) &
&($L_{\sun}/M_{\sun}$) &($10^7$) \\ \tableline
W43 S& C$^{34}$S& $2.3\times10^4$& $2.8\times10^4$& $1.8\times10^3$& 0.08&
\nodata & \nodata & \nodata & \nodata\cr
31.44$-$0.26& C$^{34}$S& $3.9\times10^5$& $1.2\times10^5$& $6.3\times10^3$&
0.02& \nodata & \nodata & \nodata & \nodata \cr
32.8+0.20 A& C$^{34}$S& $5.6\times10^4$& $1.3\times10^4$& $7.0\times10^3$&
0.13& \nodata & \nodata & \nodata & \nodata \cr
W44 & C$^{34}$S& $1.6\times10^4$& $3.9\times10^4$& $4.5\times10^3$& 0.27&
$3.0\times10^5$ & 4 & 67 & 1.0 \cr
W51 W& C$^{34}$S& $5.9\times10^4$& $1.4\times10^4$& $1.5\times10^3$& 0.03&
\nodata & \nodata & \nodata & \nodata \cr
W51 N C2& C$^{34}$S& $6.2\times10^5$& $1.2\times10^5$& $1.3\times10^4$& 0.02&
$4.0\times10^6$ & 4 & 310 & 2.4 \cr
W51M& CS & $7.2\times10^4$& $8.8\times10^4$& $1.6\times10^4$& 0.23&
$2.8\times10^6$& 3 & 170 & 1.5 \cr
K3$-$50& C$^{34}$S& $5.9\times10^4$& $9.4\times10^3$& $6.1\times10^3$& 0.10&
$2.1\times10^6$& 5 & 340 & 11 \cr
ON3& C$^{34}$S& $1.7\times10^4$& $7.3\times10^3$& $2.3\times10^3$& 0.13&
$3.7\times10^5$& 5 & 160 & 2.1 \cr
ON2S&C$^{34}$S& $3.3\times10^4$& $8.0\times10^3$& $9.1\times10^2$& 0.03 &
\nodata & \nodata & \nodata & \nodata \cr
CRL2591& CS& $3.0\times10^2$& $5.0\times10^2$& $3.2\times10^2$& 1.1&
$2.0\times10^4$& 2& 63 & 8.3 \cr
DR21 S& C$^{34}$S& $3.6\times10^3$& $3.5\times10^3$& $1.1\times10^3$& 0.31&
$5.0\times10^5$& 6 & 460 & 5.0 \cr
W75(OH)& C$^{34}$S& $5.6\times10^3$& $9.6\times10^3$& $1.6\times10^3$& 0.27&
$5.4\times10^4$& 8 & 35 & 0.5 \cr
W75 N& C$^{34}$S& $6.6\times10^3$& $3.8\times10^3$& $1.4\times10^3$& 0.22&
$1.8\times10^5$& 3 & 130 & 2.3 \cr
Cep A& CS& $2.5\times10^2$& $4.3\times10^2$& $5.9\times10^2$& 2.3&
$2.5\times10^4$& 1 &42 & 2.5 \cr
W3(2)& C$^{34}$S& $1.9\times10^4$& $2.6\times 10^3$& $6.1\times10^2$& 0.03&
$3.0\times10^5$& 7 & 490 & 3.8 \cr
GL490& CS & $6.2$& $2.8\times10^1$& $9.1\times10^1$& 15& $2.2\times10^3$& 2& 24
& 2.2 \cr
\end{tabular}
\tablerefs{
(1) Evans et al. (1981); (2) Mozurkewich et al. (1986); (3) Jaffe, unpublished;
(4) Jaffe et al. (1985);
(5) Thronson \& Harper (1979); (6) Colom\'e et al. (1995); (7) Campbell et al.
(1995)}
\end{table}
\clearpage
\begin{table}[h]
\caption{Comparison to Other Galaxies}
\vspace {3mm}
\begin{tabular}{l c c c c c c c c }
\tableline
\tableline
Source & Distance & Radius & $\int{T^*_Rdv}$ & $L ({\rm CS \ 5-4})$\ & Ref. & $L$ & Ref. &
$L$/ $L ({\rm CS \ 5-4})$\ \cr
& (Mpc) & (pc) & (K km s$^{-1}$) & (L$_{\sun}$) & & ($10^9$ L$_{\sun}$) & &
($10^7$) \\ \tableline
NGC 253 & 2.5$^a$ & 67 & 23.5 & 154 & 1 & 8 & 3 & 5 \cr
Maffei 2 & 5 & 133 & $<$2 & $<$53 & 2 & 9.5 & 2 & $>18$ \cr
IC 342 & 1.8$^b$ & 48 & 0.76 & 3 & 1 & 0.64 & 4 & 21 \cr
M82 & 3.2$^c$ & 85 & 2.6 & 28 & 1 & 47 & 3 & 170 \cr
NGC 6946 & 5 & 133 & $<$2.8 & $<$74 & 2 & 1.2 & 3 & $>1.7$ \cr
\end{tabular}
\tablerefs{ (a) de Vaucouleurs (1978); (b) McCall (1987); (c) Tammann \&
Sandage (1968);
(1) Mauersberger \& Henkel (1989); (2) Mauersberger et al. (1989);
(3) Smith \& Harvey 1996;
(4) Becklin et al. (1980) for flux, McCall (1987) for distance. }
\end{table}
\clearpage
| 2024-02-18T23:39:41.182Z | 1996-09-07T01:12:47.000Z | algebraic_stack_train_0000 | 67 | 9,574 |
|
proofpile-arXiv_065-428 | \section{Guidelines}
It is well known that the effects on the physics of a field,
due to a much heavier field coupled to the former, are not detectable at
energies comparable to the lighter mass. More precisely the
Appelquist-Carazzone (AC) theorem~\cite{ac} states that for a Green's function
with only light external legs, the effects of the heavy loops
are either absorbable in a redefinition of the bare couplings
or suppressed by powers of $k/M$ where $k$ is the energy scale
characteristic of the Green's function
(presumably comparable to the light mass),
and $M$ is the heavy mass. However the AC theorem does not allow
to make any clear prediction when $k$ becomes close to $M$ and, in this
region one should expect some non-perturbative effect due to the onset of new
physics.
\par
In the following we shall make use of the Wilson's renormalization
group (RG) approach to discuss the physics of the light field from the
infrared region up to and beyond the mass of the heavy field.
Incidentally, the RG
technique has been already employed to proof the AC theorem~\cite{girar}.
The RG establishes the flow equations of the various coupling constants of
the theory for any change in the observational energy scale; moreover
the improved RG equations, originally derived by F.J. Wegner
and A. Houghton~\cite{weg},
where the mixing of all couplings (relevant and irrelevant) generated
by the blocking procedure is properly taken into account, should
allow to handle the non-perturbative features arising
when the heavy mass threshold is crossed.
\par
We shall discuss the simple case of two coupled scalar fields and since
we are interested in the modifications of the parameters
governing the light field, due to the heavy loops, we shall consider
the functional integration of the heavy field only. The action at a given
energy scale $k$ is
\begin{equation}
S_k(\phi,\psi)=\int d^4 x~\left ({1\over 2} \partial_\mu \phi
\partial^\mu \phi+
{1\over 2} W(\phi,\psi) \partial_\mu \psi \partial^\mu \psi+
U(\phi,\psi) \right ),
\label{eq:acteff}
\end{equation}
with polynomial $W$ and $U$
\begin{equation}
U(\phi,\psi)=\sum_{m,n}{{G_{2m,2n} \psi^{2m}\phi^{2n}}\over {(2n)!(2m)!}};
~~~~~~~~~~~~~~~~~~~~
W(\phi,\psi)=\sum_{m,n}{{H_{2m,2n}\psi^{2m}\phi^{2n}}
\over {(2n)!(2m)!}}.
\label{eq:svil}
\end{equation}
Since we want to focus on the light field, which we choose to be $\psi$,
we have simply set to 1 the wave function renormalization of $\phi$.
In the following we shall analyse the symmetric phase of the theory
with vanishing vacuum energy $G_{0,0}=0$.
\par
We do not discuss here the procedure employed~\cite{jan}
to deduce the RG coupled equations for the
couplings in Eq.~\ref{eq:svil}, because it is thoroughly explained in the
quoted reference. Since it is
impossible to handle an infinite set of equations and a truncation in the
sums in Eq.~\ref{eq:svil} is required, we keep in the action
only terms that do not exceed the sixth power in the fields and their
derivatives. Moreover we choose the initial condition for the RG equations
at a fixed ultraviolet scale $\Lambda$
where we set $H_{0,0}=1$, $G_{0,4}=
G_{2,2}=G_{4,0}=0.1$ and
$G_{0,6}=G_{2,4}=G_{4,2}=G_{6,0}=
H_{2,0}=H_{0,2}=0$,
and the flow of the various couplings is determined as a function of
$t=ln \left (k/ \Lambda\right )$, for negative $t$.
\begin{figure}
\psfig{figure=fig1.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{0,2}(t)/\Lambda^2$ (curve (1)) and $10^6\cdot
G_{2,0}(t)/\Lambda^2$ (curve (2)) vs $t=log\left ({{k}/{\Lambda}}\right )$.
\break
(b): $G_{2,2}(t)$ (1), $G_{0,4}(t)$ (2),
$G_{4,0}(t)$ (3) vs $t$.
\label{fig:funo}}
\end{figure}
\par
In Fig.~\ref{fig:funo}(a) $G_{0,2}(t)/\Lambda^2$ (curve (1) ) and
$10^6 \cdot G_{2,0}(t)/\Lambda^2$ (curve (2)) are plotted. Clearly the heavy
and the light masses become stable going toward the IR region and their value
at $\Lambda$ has been chosen in such a way that the stable IR values are,
$M\equiv\sqrt {G_{0,2}(t=-18)}\sim 10^{-4}\cdot \Lambda$
and $m\equiv\sqrt{G_{2,0}(t=-18)}\sim 2\cdot 10^{-7}\cdot\Lambda$.
So, in principle, there are three scales:
$\Lambda$, ($t=0$), the heavy mass $M$,
($t\sim -9.2$), the light mass $m$, ($t\sim -16.1$).
In Fig.~\ref{fig:funo}(b) the three renormalizable dimensionless
couplings are shown; the neat change around $t=-9.2$, that is $k \sim M$,
is evident and the curves become flat below this value.
The other four non-renormalizable couplings included in $U$ are
plotted in Fig.~\ref{fig:fdue}(a), in units of $\Lambda$.
Again everything is flat below $M$, and the values of the couplings
in the IR region coincide with their perturbative Feynman-diagram
estimate at the one loop level; it is easy to realize that
they are proportional to $1/M^2$, which, in units of $\Lambda$,
is a big number. Thus the large values in
Fig.~\ref{fig:fdue}(a) are just due to the scale employed and, since
these four couplings for any practical purpose, must
be compared to the energy scale at which they are calculated, it is
physically significant to plot them in units of
the running cutoff $k$:
the corresponding curves are displayed in Fig.~\ref{fig:fdue}(b);
in this case the couplings are strongly suppressed below $M$.
\begin{figure}
\psfig{figure=fig2.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{6,0}(t)\cdot \Lambda^2$ (1), $G_{0,6}(t)\cdot \Lambda^2$ (2),
$G_{4,2}(t)\cdot \Lambda^2$ (3) and
$G_{2,4}(t)\cdot \Lambda^2$ (4) vs $t$.\break
(b): $G_{6,0}(t)\cdot k^2$ (1), $G_{0,6}(t)\cdot k^2$ (2),
$G_{4,2}(t)\cdot k^2$ (3) and
$G_{2,4}(t)\cdot k^2$ (4) vs $t$.
\label{fig:fdue}}
\end{figure}
\par
It must be remarked that there is no change in the couplings when
the light mass threshold is crossed. This is a consequence of having
integrated the heavy field only: in this case one could check directly
from the equations ruling the coupling constants flow, that
a shift in the
initial value $G_{2,0}(t=0)$ has the only effect
(as long as one remains in the symmetric phase)
of modifying $G_{2,0}(t)$, leaving the other curves unchanged.
Therefore the results obtained are independent of $m$ and
do not change even if $m$ becomes much larger than $M$.
An example of the heavy mass dependence is shown in Fig.~\ref{fig:ftre}(a),
where $G_{6,0}(t)$ is plotted, in units of the running cutoff $k$, for three
different values of $G_{0,2}(t=0)$, which correspond respectively to
$M/\Lambda\sim 2\cdot 10^{-6}$, (1),
$M/\Lambda\sim 10^{-4}$, (2) and
$M/\Lambda\sim 0.33$, (3).
Note, in each curve, the change of slope when the $M$ scale is crossed.
$H_{0,0}=1,~~H_{0,2}=0$ is a constant solution of the corresponding equations
for these two couplings; on the other hand $H_{2,0}$ is not constant
and it is plotted in units of the running cutoff $k$ in Fig.~\ref{fig:ftre}(b),
for the three values of $M$ quoted above.
\begin{figure}
\psfig{figure=fig3.ps,height=4.3cm,width=12.cm,angle=90}
\caption{
(a): $G_{6,0}(t)\cdot k^2$ vs $t$
for $M/\Lambda \sim 2\cdot 10^{-6}$ (1),
$\sim 10^{-4}$ (2),
$\sim 0.33$ (3).\break
(b): $H_{2,0}(t)\cdot k^2$ for the three values of $M/\Lambda$
quoted in (a).
\label{fig:ftre}}
\end{figure}
\par
In conclusion, according to the AC theorem all couplings are constant at low
energies
and a change in the UV physics can only shift their values in the IR region.
Remarkably, for increasing $t$,
no trace of UV physics shows up until one reaches $M$,
that acts as a UV cut-off for the low energy physics.
Moreover, below $M$, no non-perturbative effect appears due to the
non-renormalizable couplings that vanish fastly in units of $k$.
Their behavior above $M$ is somehow constrained by the
renormalizability condition fixed at $t=0$, as clearly shown in
Fig.~\ref{fig:ftre}(a) (3). Finally, the peak of $H_{2,0}$ at $k\sim M$
in Fig.~\ref{fig:ftre}(b), whose width and
height are practically unchanged in the three examples,
is a signal of non-locality of the theory limited to the region
around $M$.
\section*{Acknowledgments}
A.B. gratefully acknowledges Fondazione A. Della Riccia and INFN
for financial support.
\section*{References}
| 2024-02-18T23:39:41.423Z | 1996-09-19T12:32:13.000Z | algebraic_stack_train_0000 | 84 | 1,524 |
|
proofpile-arXiv_065-463 | \section{Introduction}
\begin{figure}
\vspace*{13pt}
\vspace*{6.7truein}
\special{psfile=figa.ps
voffset= 240 hoffset= -40 hscale=50 vscale=50 angle = 0}
\special{psfile=figb.ps
voffset= 240 hoffset= 220 hscale=50 vscale=50 angle = 0}
\special{psfile=figc.ps
voffset= -40 hoffset= 90 hscale=50 vscale=50 angle = 0}
\caption{Histogram
of the number of models that yield a particular prediction
for $m_{\nu_{\mu}}^2- m_{\nu_{e}}^2$
assuming (a) small angle and (b) large angle solution to solar
neutrino problem. In (c) we
solve the solar neutrino problem via small angle $e$--$\tau$
oscillations and check whether this is compatible with
the LSND result.
}
\label{fig}
\end{figure}
In the Standard Model of elementary particles (SM)
both lepton number ($L$) and baryon number ($B$) are
conserved due to an accidental symmetry,
{\sl i.e.} there is no renormalizable, gauge-invariant
term that would break the symmetry.
In the minimal supersymmetric extension of the SM (MSSM)
the situation is different. Due to a the variety
of scalar partners the MSSM allows for a host of new
interactions many of which violate $B$ or $L$.
Since neither $B$ nor $L$ violation has been
observed in present collider experiments
these couplings are constrained from above.
More constraints arise from neutrino
physics
or cosmology.
Thus, all lepton and baryon number violating
interaction are often eliminated by imposing
a discrete, multiplicative symmetry called
$R$-parity,\cite{r-parity}
$R_p \equiv (-1)^{2S+3B+L}$, where $S$ is the spin.
One very attractive feature of
$R_p$ conserving models is that
the lightest supersymmetric particle (LSP)
is stable and a good cold dark matter candidate.\cite{cdm}
However, while the existence of a dark matter candidate
is a very desirable prediction, it does not prove
$R_p$ conservation and
one should consider more general models.
Here, we will investigate
the scenario where $R_p$ is broken explicitly via
the terms\cite{suzuki} $W = \mu_i L_i H$,
where $H$ is the Higgs coupling to up-type fermions
and $L_i$ ($i = 1,2,3$) are the left-handed lepton doublets.
Clearly, these Higgs-lepton mixing terms violate
lepton-number. As a result, majorana masses will be generated for
one neutrino at tree-level and for the remaining
two neutrinos at the one-loop level.
These masses were calculated in the frame-work of minimal supergravity
in ref.~\citenum{npb} and the numerical results will be
briefly summarized here.
There are three $R_P$ violating parameters which can be used to fix
1) the tree-level neutrino mass,
2) the $\mu$--$\tau$ mixing angle and
3) the $e$--$\mu$ mixing angle.
The question of whether e.g. the solar\cite{solarn}
and the atmospheric\cite{atmosphericn} neutrino puzzle
can be solved simultaneously depends on the prediction of
$m_{\nu_\mu}^2-m_{\nu_e}^2$.
In fig.~1 we have scanned the entire SUSY parameter space
consisting of the Higgsino (gaugino) mass parameter,
$\mu$ ($m_{1/2}$), the trilinear scalar interaction parameter $A_0$,
and the ratio of Higgs VEVs, $\tan\beta$. The universal
scalar mass parameter $m_0$ is fixed by minimizing the potential.
Plotted is the number of models yielding a particular prediction for
$m_{\nu_\mu}^2-m_{\nu_e}^2$ for
(a) sin$^2 2 \theta_{e \nu_\mu} = 0.008$ and
(b) sin$^2 2 \theta_{e \nu_\mu} = 1$.
We fix $m_{\nu_\tau}=0.1$~eV and
sin$^2 2 \theta_{\mu \nu_\tau} = 1$ in order to solve the
atmospheric neutrino problem.
We see that both
long wave-length oscillation (LWO)\cite{lwo}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-10}$~eV$^2$)
and MSW effect\cite{msw-effect}
($m_{\nu_\mu}^2-m_{\nu_e}^2=10^{-5}$~eV$^2$)
can be accommodated.
In fig.~1(c) we solve the solar neutrino problem via $e$--$\tau$
oscillations and we fix sin$^2 2 \theta_{e \nu_\mu} = 0.004$
in order to accommodate the LSND result.\cite{lsnd}
We see that most models are already ruled out
by collider constraints and even more by dark matter (DM) constraints.
However, a very small (but non-zero) number of models
yields a prediction compatible with the LSND
result (the dotted line is lower limit of LSND).
\noindent{\bf Acknowledgements}
This work was supported in parts by the DOE under
Grants No. DE-FG03-91-ER40674 and by the
Davis Institute for High Energy Physics.
\vskip0.3cm
\noindent{\bf References}
| 2024-02-18T23:39:41.543Z | 1996-09-30T23:32:09.000Z | algebraic_stack_train_0000 | 90 | 736 |
|
proofpile-arXiv_065-679 | \subsection*{1. Introduction}
The connection of positive knots with transcendental numbers, via the
counterterms of quantum field theory, proposed in~\cite{DK1} and
developed in~\cite {DK2}, and has been vigorously tested against
previous~\cite{GPX,DJB} and new~\cite{BKP} calculations,
entailing knots with up to 11 crossings, related by counterterms with
up to 7 loops to numbers that are irreducible
multiple zeta values (MZVs)~\cite{DZ,LM}.
Cancellations of transcendentals in gauge
theories have been illuminated by knot theory~\cite{BDK}. All-order results,
from large-$N$ analyses~\cite{BGK} and Dyson-Schwinger methods~\cite{DKT},
have further strengthened the connection of knot theory and number theory,
via field theory. A striking feature of this connection is that the
first irreducible MZV of depth 2 occurs at weight 8~\cite{DJB,BBG}, in
accord with the appearance of the first positive 3-braid knot at crossing
number 8. Likewise the first irreducible MZV of depth 3 occurs at weight
11~\cite{BG}, matching the appearance of the first positive 4-braid
at 11 crossings, obtained from skeining link diagrams that encode momentum
flow in 7-loop counterterms~\cite{BKP}. Moreover, the
investigations in~\cite{BGK} led to a new discovery at weight 12,
where it was found that the reduction of MZVs first entails alternating
Euler sums. The elucidation of this phenomenon
resulted in an enumeration~\cite{EUL} of irreducible Euler sums
and prompted intensive searches for evaluations of sums of
arbitrary depth~\cite{BBB}. A review of all these developments is
in preparation~\cite{DK}.
This paper pursues the connection to 8 and 9 loops, entailing knots
with up to 15 crossings.
In Section~2, we enumerate irreducible MZVs by weight.
Section~3 reports calculations of Feynman diagrams that yield
transcendental knot-numbers entailing MZVs up to weight 15.
In Section~4 we enumerate positive knots, up to 15 crossings,
and give the braid words and HOMFLY polynomials~\cite{VJ} for
all knots associated with irreducible MZVs of weight $n<17$.
Section~5 gives our conclusions.
\subsection*{2. Multiple zeta values}
We define $k$-fold Euler sums~\cite{BBG,BG} as in~\cite{EUL,BBB},
allowing for alternations of signs in
\begin{equation}
\zeta(s_1,\ldots,s_k;\sigma_1,\ldots,\sigma_k)=\sum_{n_j>n_{j+1}>0}
\quad\prod_{j=1}^{k}\frac{\sigma_j^{n_j}}{n_j^{s_j}}\,,\label{form}
\end{equation}
where $\sigma_j=\pm1$, and the exponents $s_j$ are positive integers,
with $\sum_j s_j$ referred to as the weight (or level) and $k$ as the depth.
We combine the strings of exponents and signs
into a single string, with $s_j$ in the $j$th position when $\sigma_j=+1$,
and $\overline{s}_j$ in the $j$th position when $\sigma_j=-1$.
Referring to non-alternating sums as MZVs~\cite{DZ},
we denote the numbers of irreducible Euler sums
and MZVs by $E_n$ and $M_n$, at weight $n$, and find that
\begin{equation}
1-x -x^2=\prod_{n>0}(1-x^n)^{E_n}\,;\quad
1-x^2-x^3=\prod_{n>0}(1-x^n)^{M_n}\,,\label{EM}
\end{equation}
whose solutions, developed in Table~1, are given in closed form by
\begin{eqnarray}
E_n=\frac{1}{n}\sum_{d|n}\mu(n/d)L_d\,;
&&L_n=L_{n-1}+L_{n-2}\,;\quad L_1=1\,;\quad L_2=3\,,\label{Es}\\
M_n=\frac{1}{n}\sum_{d|n}\mu(n/d)P_d\,;
&&P_n=P_{n-2}+P_{n-3}\,;\quad P_1=0\,;\quad P_2=2;\quad P_3=3\,,\label{Ms}
\end{eqnarray}
where $\mu$ is the M\"obius function, $L_n$ is a Lucas number~\cite{EUL} and
$P_n$ is a Perrin number~\cite{AS}.
\noindent{\bf Table~1}:
The integer sequences\Eqqq{Es}{Ms}{Kn} for $n\leq20$.
\[\begin{array}{|r|rrrrrrrrrrrrrrrrrrrr|}\hline
n&1&2&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17&18&19&20\\\hline
E_n&1&1&1&1&2&2&4&5&8&11&18&25&40&58&90&135&210&316&492&750\\
M_n&0&1&1&0&1&0&1&1&1&1&2&2&3&3&4&5&7&8&11&13\\
K_n&0&0&1&0&1&1&1&2&2&3&4&5&7&9&12&16&21&28&37&49\\\hline
\end{array}\]
In~\cite{EUL}, $E_n=\sum_k E_{n,k}$ was apportioned, according to
the minimum depth $k$ at which irreducibles of weight $n$ occur.
Similarly, we have apportioned $M_n=\sum_k M_{n,k}$.
The results are elements of Euler's triangle~\cite{EUL}
\begin{equation}
T(a,b)=\frac{1}{a+b}\sum_{d|a,b}\mu(d)\,P(a/d,b/d)\,,
\label{ET}
\end{equation}
which is a M\"obius transform of Pascal's triangle, $P(a,b)={a+b\choose a}
=P(b,a)$.
We find that
\begin{equation}
E_{n,k}=T(\df{n-k}{2},k)\,;\quad
M_{n,k}=T(\df{n-3k}{2},k)\,,
\label{EMb}
\end{equation}
for $n>2$ and $n+k$ even.
There is a remarkable feature of the result for $M_{n,k}$: it gives
the number of irreducible Euler sums of weight $n$ and depth $k$ that occur
in the reduction of MZVs, which is {\em not\/} the same as the
number of irreducible MZVs of this weight and depth. It was shown
in~\cite{BGK,EUL} that alternating multiple sums occur in the reduction of
non-alternating multiple sums. For example, $\zeta(4,4,2,2)$
cannot be reduced to MZVs of lesser depth, but it can~\cite{EUL} be reduced
to the alternating Euler sum $\zeta(\overline9,\overline3)$.
Subsequently we found
that an analogous ``pushdown'' occurs at weight 15, where depth-5 MZVs,
such as $\zeta(6,3,2,2,2)$, cannot be reduced to MZVs of lesser depth,
yet can be reduced to alternating Euler sums, with
$\zeta(9,\overline3,\overline3)-\frac{3}{14}\zeta(7,\overline5,\overline3)$
serving as the corresponding depth-3 irreducible. We conjecture
that the number, $D_{n,k}$, of MZVs of weight $n$ and depth $k$
that are not reducible to MZVs of lesser depth
is generated by
\begin{equation}
1-\frac{x^3 y}{1-x^2}+\frac{x^{12}y^2(1-y^2)}{(1-x^4)(1-x^6)}
=\prod_{n\ge3} \prod_{k\ge1} (1-x^n y^k)^{D_{n,k}},\label{Pd}
\end{equation}
which agrees with~\cite{BBG,BG} for $k<4$ and all weights $n$,
and with available data on MZVs, obtained from binary
reshuffles~\cite{LM} at weights $n\leq20$ for $k=4$, and $n\leq18$
for $k>4$. Further tests of\Eq{Pd} require very large scale
computations, which are in progress, with encouraging results.
However, the work reported here does not rely on this conjecture;
the values of $\{M_n\mid n\le15\}$ in Table~1 are sufficient for present
purposes, and these are amply verified by exhaustive use of
the integer-relation search routine MPPSLQ~\cite{DHB}.
Finally, in this section, we remark on the simplicity of the prediction
of\Eq{Ms} for the dimensionality, $K_n$, of the search space
for counterterms that evaluate to MZVs of weight $n$.
Since $\pi^2$, with weight $n=2$, does
not occur in such counterterms, it follows that they
must be expressible in terms of transcendentals that are enumerated by
$\{M_n\mid n\geq3\}$, and products of such knot-numbers~\cite{DK1,BGK,EUL}.
Thus $K_n$ is given by a Padovan sequence:
\begin{equation}
\sum_n x^n K_n=\frac{x^3}{1-x^2-x^3}\,;\quad
K_n=K_{n-2}+K_{n-3}\,;\quad K_1=0\,;\quad K_2=0\,;\quad K_3=1\,,\label{Kn}
\end{equation}
which is developed in Table~1. Note that the dimension of
the search space for a general MZV of weight $n$ is $K_{n+3}$~\cite{DZ},
which exceeds $K_n$ by a factor~\cite{AS} of $2.324717957$, as $n\to\infty$.
\subsection*{3. Knot-numbers from evaluations of Feynman diagrams}
The methods at our disposal~\cite{DK1,DK2,BKP} do not yet permit us to
predict, {\em a priori\/}, the transcendental knot-number assigned to
a positive knot by field-theory counterterms; instead we need
a concrete evaluation of at least one diagram
which skeins to produce that knot.
Neither do they allow us to predict the rational coefficients with which
such knot-numbers, and their products, corresponding to factor knots,
occur in a counterterm; instead we must, at present, determine these
coefficients by analytical calculation, or by applying a lattice method,
such as MPPSLQ~\cite{DHB}, to (very) high-precision numerical data.
Nonetheless, the consequences of~\cite{DK1,DK2} are highly predictive
and have survived intensive testing with amazing fidelity. The origin
of this predictive content is clear: once a knot-number is determined
by one diagram, it is then supposed, and indeed found, to occur in the
evaluation of all other diagrams which skein to produce that knot.
Moreover, the search
space for subdivergence-free counterterms that evaluate to MZVs
is impressively smaller than that for the MZVs themselves,
due to the absence of any knot associated with $\pi^2$,
and again the prediction is borne out by a wealth of data.
We exemplify these features by considering diagrams that
evaluate to MZVs of depths up to 5, which is the greatest depth
that can occur at weights up to 17, associated with knots up to
crossing-number 17, obtained from diagrams with up to 10 loops.
We follow the economical notation of~\cite{GPX,DJB,BKP}, referring
to a vacuum diagram by a so-called angular diagram~\cite{GPX}, which
results from choosing one vertex as origin, and indicating all vertices
that are connected to this origin by dots, after removing the origin
and the propagators connected to it.
{From} such an angular diagram one may uniquely reconstruct
the original Feynman diagram. The advantage of this notation is that
the Gegenbauer-polynomial $x$-space technique~\cite{GPX} ensures
that the maximum depth of sum which can result is the smallest
number of propagators in any angular diagram that characterizes
the Feynman diagram. Fig.~1 shows a naming convention for log-divergent
vacuum diagrams with angular diagrams that yield up to 5-fold sums.
To construct, for example, the 7-loop diagram $G(4,1,0)$ one places
four dots on line 1 and one dot on line 2.
Writing an origin at any point disjoint from the angular diagram,
and joining all 6 dots to that origin, one recovers the
Feynman diagram in question.
Using analytical techniques developed
in~\cite{GPX,DJB,BG,EUL}, we find that all subdivergence-free diagrams
of $G$-type, up to 13 loops (the highest number computable in
the time available), give counterterms that evaluate
to $\zeta(2n+1)$, their products, and depth-3 knot-numbers
chosen from the sets
\begin{eqnarray}
N_{2m+1,2n+1,2m+1}&=&
\zeta(2m+1,2n+1,2m+1)-\zeta(2m+1)\,\zeta(2m+1,2n+1)\nonumber\\&&{}
+\sum_{k=1}^{m-1}{2n+2k\choose2k}\zeta_P(2n+2k+1,2m-2k+1,2m+1)\nonumber\\&&{}
-\sum_{k=0}^{n-1}{2m+2k\choose2k}\zeta_P(2m+2k+1,2n-2k+1,2m+1)
\,,\label{K3o}\\
N_{2m,2n+1,2m}&=&
\zeta(2m,2n+1,2m)+\zeta(2m)\left\{\zeta(2m,2n+1)+\zeta(2m+2n+1)\right\}
\nonumber\\&&{}
+\sum_{k=1}^{m-1} {2n+2k\choose2k }\zeta_P(2n+2k+1,2m-2k,2m)\nonumber\\&&{}
+\sum_{k=0}^{n-1} {2m+2k\choose2k+1}\zeta_P(2m+2k+1,2n-2k,2m)
\,,\label{K3e}
\end{eqnarray}
where $\zeta_P(a,b,c)=\zeta(a)\left\{2\,\zeta(b,c)+\zeta(b+c)\right\}$.
The evaluation of a 9-loop non-planar example, $G(3,2,2)$,
is given in~\cite{EUL}: it evaluates in terms of MZVs of weights
ranging from 6 to 14. Choosing from\Eqq{K3o}{K3e}
one knot-number at 11 crossings and two at 13 crossings,
one arrives at an expression from which all powers of $\pi^2$ are banished,
which is a vastly more specific result than for a generic collection of MZVs
of these levels, and is in striking accord with what is required by the
knot-to-number connection entailed by field theory. Moreover, all planar
diagrams that evaluate to MZVs have been found to contain terms purely of
weight $2L-3$ at $L$ loops, matching the pattern of the zeta-reducible
crossed ladder diagrams~\cite{DK1,DK2}.
Subdivergence-free counterterms obtained from the $M$-type angular diagrams of
Fig.~1 evaluate to MZVs of weight $2L-4$, at $L$-loops, with depths up to 4.
Up to $L=8$ loops, corresponding to 12 crossings, the depth-4 MZVs can be
reduced to the depth-2 alternating sums~\cite{EUL}
$N_{a,b}\equiv\zeta(\overline{a},b)-\zeta(\overline{b},a)$. The knot-numbers
for the $(4,3)$ and $(5,3)$ torus knots may be taken as $N_{5,3}$
and $N_{7,3}$, thereby banishing $\pi^2$ from the associated 6-loop
and 7-loop counterterms. In general, $N_{2k+5,3}$ is a $(2k+8)$-crossing
knot-number at $(k+6)$ loops. Taking the second knot-number at 12 crossings
as~\cite{BGK,EUL} $N_{7,5}-\frac{\pi^{12}}{2^5 10!}$,
we express all 8-loop $M$-type counterterms in a $\pi$-free form.
At 9 loops, and hence 14 crossings, we encounter the first depth-4 MZV
that cannot be pushed down to alternating Euler sums of depth 2.
The three knot-numbers are again highly specific: to $N_{11,3}$
we adjoin
\begin{equation}
N_{9,5}+\df{5\pi^{14}}{7032946176}\,;\quad
\zeta(5,3,3,3)+\zeta(3,5,3,3)-\zeta(5,3,3)\zeta(3)
+\df{24785168\pi^{14}}{4331237155245}\,.\label{k14}
\end{equation}
Having determined these knot-numbers by applying MPPSLQ to
200 significant-figure evaluations of two counterterms, in a search space
of dimension $K_{17}=21$, requisite for generic MZVs of weight 14,
knot theory requires that we find the remaining five $M$-type counterterms
in a search space of dimension merely $K_{14}=9$. This prediction is totally
successful. The rational coefficients are too cumbersome to write here;
the conclusion is clear: when counterterms evaluate to MZVs they live
in a $\pi$-free domain, much smaller than that inhabited by a generic
MZV, because of the apparently trifling circumstance that a knot with
only two crossings is necessarily the unknot.
Such wonders continue, with subdivergence-free diagrams of types
$C$ and $D$ in Fig.~1
Up to 7 loops we have obtained {\em all\/} of them in terms of the
established knot-numbers $\{\zeta(3),\zeta(5),\zeta(7),N_{5,3},\zeta(9),
N_{7,3},\zeta(11),N_{3,5,3}\}$,
associated in~\cite{BKP,BGK} with the positive knots
$\{(3,2),(5,2),(7,2),8_{19}=(4,3),(9,2),10_{124}=(5,3),(11,2),11_{353}
=\sigma_1^{}\sigma_2^{3}\sigma_3^{2}\sigma_1^{2}\sigma_2^{2}\sigma_3^{}\}$,
and products of those knot-numbers, associated with the corresponding
factor knots.
A non-planar $L$-loop diagram may have terms of different weights,
not exceeding $2L-4$.
Invariably, a planar $L$-loop diagram evaluates purely
at weight $2L-3$.
Hence we expect the one undetermined MZV knot-number at
15 crossings to appear in, for example, the planar 9-loop diagram
$C(1,0,4,0,1)$. To find the precise combination of
$\zeta(9,\overline3,\overline3)-\frac{3}{14}\zeta(7,\overline5,\overline3)$
with other Euler sums would require a search in a space of dimension
$K_{18}=28$. Experience suggests that would require an evaluation
of the diagram to about 800 sf, which is rather ambitious,
compared with the 200 sf which yielded\Eq{k14}. Once the number is found,
the search space for further counterterms shrinks to dimension $K_{15}=12$.
\subsection*{4. Positive knots associated with irreducible MZVs}
Table~2 gives the braid words~\cite{VJ} of 5 classes of positive knot.
For each type of knot, ${\cal K}$,
we used the skein relation to compute the HOMFLY polynomial~\cite{VJ},
$X_{\cal K}(q,\lambda)$, in terms of
$p_n=(1-q^{2n})/(1-q^2)$, $r_n=(1+q^{2n-1})/(1+q)$,
$\Lambda_n=\lambda^n(1-\lambda)(1-\lambda q^2)$.
\noindent{\bf Table~2}:
Knots and HOMFLY polynomials associated with irreducibles MZVs.
\[\begin{array}{|l|l|l|}\hline{\cal K}&X_{\cal K}(q,\lambda)\\\hline
{\cal T}_{2k+1}=\sigma_1^{2k+1}&T_{2k+1}=\lambda^k(1+q^2(1-\lambda)p_k)\\
{\cal R}_{k,m}=\sigma_1^{}\sigma_2^{2k+1}\sigma_1^{}\sigma_2^{2m+1}&
R_{k,m}= T_{2k+2m+3}+q^3p_k p_m\Lambda_{k+m+1}\\
{\cal R}_{k,m,n}=
\sigma_1^{}\sigma_2^{2k}\sigma_1^{}\sigma_2^{2m}\sigma_1^{}\sigma_2^{2n+1}&
R_{k,m,n}=R_{1,k+m+n-1}+q^6p_{k-1}p_{m-1}r_n\Lambda_{k+m+n+1}\\
{\cal S}_{k}=
\sigma_1^{}\sigma_2^{3}\sigma_3^{2}\sigma_1^{2}\sigma_2^{2k}\sigma_3^{}&
S_{k}= T_3^2T_{2k+3}+q^2p_k r_2(q^2(\lambda-2)+q-2)\Lambda_{k+3}\\
{\cal S}_{k,m,n}=
\sigma_1^{}\sigma_2^{2k+1}\sigma_3^{}\sigma_1^{2m}\sigma_2^{2n+1}\sigma_3^{}
&S_{k,m,n}=T_{2k+2m+2n+3}+q^3(p_k p_m+p_m p_n+p_n p_k\\&\phantom{S_{k,m,n}=}
\quad{}+(q^2(3-\lambda)-2q)p_k p_m p_n)\Lambda_{k+m+n+1}\\\hline
\end{array}\]
Noting that ${\cal S}_{1,1,1}={\cal S}_{1}$ and
${\cal S}_{m,n,0}={\cal R}_{m,n,0}={\cal R}_{m,n}$,
one easily enumerates the knots of Table~2. The result is given,
up to 17 crossings, in the last row of Table~3,
where it is compared with the enumeration of all prime knots, which is known
only to 13 crossings, and with the enumeration of positive knots,
which we have
achieved to 15 crossings, on the assumption that the HOMFLY polynomial
has no degeneracies among positive knots. It is apparent
that positive knots are sparse, though they exceed the irreducible
MZVs at 10 crossings and at all crossing numbers greater than 11.
The knots of Table 2 are equal in number to the irreducible MZVs up to
16 crossings; thereafter they are deficient.
Table~4 records a finding that may be new: the Alexander
polynomial~\cite{VJ}, obtained by setting $\lambda=1/q$ in the HOMFLY polynomial,
is not faithful for positive knots. The Jones polynomial~\cite{VJ}, with
$\lambda=q$, was not found to suffer from this defect.
Moreover, by using REDUCE~\cite{RED}, and assuming the fidelity of the
HOMFLY polynomial in the positive sector, we were able to prove,
by exhaustion, that none of the $4^{14}\approx2.7\times10^8$ positive
5-braid words of length 14 gives a true 5-braid 14-crossing knot.
\noindent{\bf Table~3}:
Enumerations of classes of knots by crossing number, $n$,
compared with\Eq{Ms}.
\[\begin{array}{|r|rrrrrrrrrrrrrrr|}\hline
n&3&4&5&6&7&8&9&10&11&12&13&14&15&16&17\\\hline
\mbox{prime knots}&1&1&2&3&7&21&49&165&552&2176&9988&?&?&?&?\\
\mbox{positive knots}&1&0&1&0&1&1&1&3&2&7&9&17&47&?&?\\
M_n&1&0&1&0&1&1&1&1&2&2&3&3&4&5&7\\
\mbox{Table~2 knots}&1&0&1&0&1&1&1&1&2&2&3&3&4&5&5\\\hline
\end{array}\]
\noindent{\bf Table~4}:
Pairs of positive knots with the same Alexander polynomial,
$X_{\cal K}(q,1/q)$.
\[\begin{array}{|l|l|l|}\hline{\cal K}_1&{\cal K}_2&
X_{{\cal K}_1}(q,\lambda)-X_{{\cal K}_2}(q,\lambda)\\\hline
{\cal S}_{2,1,2}=
\sigma_1^{}
\sigma_2^{5}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{5}
\sigma_3^{}&
\sigma_1^{3}
\sigma_2^{4}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{2}
\sigma_3^{2}
\sigma_2^{}&
q^4(1-\lambda q)p_2r_2\Lambda_6\\
(\sigma_1^{}
\sigma_2^{2}
\sigma_3^{})^2
\sigma_1^{}
\sigma_2^{5}
\sigma_3^{}
&
(\sigma_1^{}
\sigma_2^{2}
\sigma_3^{})^2
\sigma_1^{3}
\sigma_2^{}
\sigma_1^{2}
\sigma_3^{}&
q^5(1-\lambda q)p_2\Lambda_6\\
\sigma_1^{5}
\sigma_2^{}
\sigma_3^{}
\sigma_1^{2}
\sigma_2^{3}
\sigma_3^{2}
\sigma_2^{}&
\sigma_1^{2}
\sigma_2^{2}
\sigma_1^{3}
\sigma_2^{7}&
q^5(1-\lambda q)p_2\Lambda_6\\\hline
\end{array}\]
The association~\cite{DK1,DK2} of the 2-braid torus knots
$(2k+1,2)={\cal T}_{2k+1}$
with the transcendental numbers $\zeta(2k+1)$ lies at the heart of our work.
In~\cite{DK2,BKP}, we associated the 3-braid torus knot
$(4,3)=8_{19}={\cal R}_{1,1}$ with the unique irreducible MZV
at weight 8, and in~\cite{BKP} we associated $(5,3)=10_{124}={\cal R}_{2,1}$
with that at weight 10. The 7-loop counterterms of $\phi^4$-theory
indicate that the knot-numbers associated with
$10_{139}=\sigma_1^{}\sigma_2^{3}\sigma_1^{3}\sigma_2^{3}$
and
$10_{152}=\sigma_1^{2}\sigma_2^{2}\sigma_1^{3}\sigma_2^{3}$
are not~\cite{BKP} MZVs.
At 11 crossings, the association of the knot-number $N_{3,5,3}$
with ${\cal S}_1={\cal S}_{1,1,1}\equiv11_{353}$ is rock solid:
we have obtained
this number analytically from 2 diagrams and numerically from another 8,
in each case finding it with different combinations of $\zeta(11)$
and the factor-knot transcendental $\zeta^2(3)\zeta(5)$.
In~\cite{BGK} we associated the family of knots ${\cal R}_{k,m}$
with the knot-numbers $N_{2k+3,2m+1}$, modulo multiples of $\pi^{2k+2m+4}$
that have now been determined up to 14 crossings.
It therefore remains to explain here how:
(a) two families of 4-braids, ${\cal S}_{k}$ and ${\cal S}_{k,m,n}$,
diverge from their common origin at 11 crossings, to give two knots
at 13 crossings, and three at 15 crossings, associated with triple Euler sums;
(b) a new family, ${\cal R}_{k,m,n}$, begins at 14 crossings, giving
the $(7,3)$ torus knot, $(\sigma_1^{}\sigma_2^{})^7
=(\sigma_1^{}\sigma_2^4)^2\sigma_1^{}\sigma_2^3
={\cal R}_{2,2,1}$, associated with a truly irreducible four-fold sum.
To relate the positive knots of Table 2 to Feynman diagrams
that evaluate to MZVs we shall
dress their braid words with chords. In each of Figs.~2--8, we
proceed in two stages: first we extract, from a braid word,
a reduced Gauss code that defines a trivalent chord diagram;
then we indicate how to shrink propagators to obtain a scalar diagram that
is free of subdivergences and has an overall divergence
that evaluates to MZVs. Our criterion for reducibility to MZVs is
that there be an angular diagram, obtained~\cite{GPX,DJB}
by choosing one vertex as an origin, such that the angular
integrations may be performed without encountering
6--j symbols, since these appeared in all the diagrams involving the
non-MZV knots $10_{139}$ and $10_{152}$ at 7 loops~\cite{BKP}, whereas
all the MZV-reducible diagrams could be cast in a form free of 6--j
symbols.
The first step -- associating a chord diagram with a knot --
allows considerable freedom: each
chord is associated with a horizontal propagator connecting vertical
strands of the braid between crossings, and there are almost
twice as many crossings as there are chords in the corresponding
diagram. Moreover, there are several braid words representing the same knot.
Thus a knot can be associated with several chord diagrams. Figs.~3b and~4b
provide an example: each diagram obtained from the $(5,2)$ torus knot
yields a counterterm involving $\zeta(5)$,
in a trivalent theory such as QED or Yukawa theory.
In Table 2 we have five families of braid words: the 2-braid torus knots,
two families of 3-braids, and two families of 4-braids. We begin with
the easiest family, ${\cal T}_{2k+1}$.
Consider Fig.~2a. We see the braid $\sigma_1^3$, dressed with two
horizontal propagators. Such propagators will be called chords,
and we shall refer to Figs.~2a--8a as chorded braid diagrams.
In Fig.~2a the two chords are labelled 1 and 2.
Following the closed braid, starting from the upper end
of the left strand, we encounter each chord twice, at vertices
which we label $\{1,{1^\prime}\}$ and $\{2,{2^\prime}\}$. These
occur in the order
$1,2,{1^\prime},{2^\prime}$ in Fig.~2a. This is the same order as they
are encountered on traversing the circle of Fig.~2b, which
is hence the same diagram as the chorded braid of Fig.~2a.
As a Feynman diagram, Fig.~2b is indeed associated with
the trefoil knot, by the methods of~\cite{DK1}.
We shall refer to the Feynman diagrams of Figs.~2b--8b as
chord diagrams\footnote{The reader familiar with
recent developments in knot theory and topological field theory might
notice that our notation is somewhat motivated by the connection between
Kontsevich integrals~\cite{LM} and chord diagrams. In~\cite{DK}
this will be discussed in detail and related to the work in~\cite{DK1}.}.
Each chord diagram is merely
a rewriting of the chorded braid diagram that precedes it,
displaying the vertices
on a hamiltonian circuit that passes through all the vertices.
The final step is trivial in this example:
the scalar tetrahedron is already log-divergent in 4 dimensions, so no
shrinking of propagators is necessary. Fig.~2c records the trivial
angular diagram, obtained~\cite{GPX} by choosing ${2}$ as an origin
and removing the propagators connected to it:
this merely represents a wheel with 3 spokes. In general~\cite{DJB}
the wheel with $n+1$ spokes delivers $\zeta(2n-1)$.
In Fig.~3a we give a chording of the braid $\sigma_1^{2n-1}$,
which is the simplest representation of the $(2n-1,2)$ torus knot,
known from previous work~\cite{DK1,DK2} to be associated with
a $(n+1)$-loop diagram, and hence with a hamiltonian circuit that has $n$
chords. Thus each addition of $\sigma_1^2$ involves adding a
single chord, yielding the chord diagram of Fig.~3b. To obtain
a logarithmically divergent scalar diagram, we shrink the propagators
connecting vertex ${2^\prime}$ to vertex ${n^\prime}$, drawn with a thick
line on the hamiltonian circuit of Fig.~3b, and hence obtain
the wheel with $n+1$ spokes, represented by the angular diagram of Fig.~3c.
To show how different braid-word representations of the
same knot give different chord diagrams, yet the same transcendental,
we consider Fig.~4.
In Fig.~4a we again have a chorded braid with $n$ chords, which this time
is obtained by combining $\sigma_1\sigma_2\sigma_1\sigma_2$
with $n-2$ powers of $\sigma_2^2$.
The resultant braid word,
$\sigma_1^{}\sigma_2^{}\sigma_1^{}\sigma_2^{2n-3}$,
is the $(2n-1,2)$ torus knot written as a 3-braid.
Labelling the pairs of vertices of Fig.~4b, one sees that it is identical
to the closure of the braid of Fig.~4a.
Shrinking together the vertices
$\{{2^\prime},{n^\prime},\ldots,{3^\prime}\}$
gives the angular diagram of Fig.~4c, which is
the same as Fig.~3c and hence delivers $\zeta(2n-1)$.
This ends our consideration of the 2-braid torus knots. We now turn
to the class ${\cal R}_{k,m}$ in Fig.~5.
The first member ${\cal R}_{1,1}=8_{19}=(4,3)$ appears at 6 loops,
with five chords.
It was obtained from Feynman diagrams in \cite{DK2}, and found
in~\cite{BKP} to be associated with an MZV of depth 2.
In Fig.~5a we add singly-chorded powers of $\sigma_2^2$
to a chorded braid word that delivers a Feynman diagram for which the
procedures of~\cite{DK1} gave $8_{19}$ as one of its skeinings.
In general, we have $k+m+3$ chords and thus $k+m+4$ loops.
The resulting chord diagram
is Fig.~5b, whose 7-loop case was the basis for associating
$10_{124}$ with the MZV $\zeta(7,3)$~\cite{BKP}.
Shrinking the propagators indicated by thickened lines in Fig.~5b,
we obtain diagram $M(k,1,1,m)$, indicated by the angular diagram of
Fig.~5c. Explicit computation of all such diagrams, to 9
loops, shows that this family is indeed MZV-reducible, to 14 crossings.
In Fig.~6 we repeat the process of Fig.~5 for the knot class
${\cal R}_{k,m,n}$. Marked boxes, in Fig.~6a,
indicate where we increase the number of chords.
Fig.~6b shows the highly non-planar chord diagram
for this knot class. This non-planarity is maintained
in the log-divergent diagram obtained by shrinking the thickened lines
in Fig.~6b. The parameters $m$ and $k$ correspond to the
series of dots in the corresponding angular diagram of Fig.~6c.
Non-planarity is guaranteed by the two remaining dots,
which are always present.
For $n>1$, we see even more propagators in the angular diagram.
The absence of 6--j symbols from angular integrations leads us to believe
that the results are reducible to MZVs; the non-planarity entails
MZVs of even weight, according to experience up to 7 loops~\cite{BKP}.
We now turn to the last two classes of knots: the 4-braids of Table~2.
In Fig.~7a we give a chorded braid diagram for knots
of class ${\cal S}_k$. Again, the marked box indicates
how we add chords to a chorded braid diagram
that corresponds to a 7-loop Feynman diagram, already known~\cite{BKP}
to skein to produce ${\cal S}_1=11_{353}$.
Shrinking the thickened lines in Fig.~7b, we obtain a log-divergent
planar diagram containing: a six-point coupling,
a $(k+3)$-point coupling, and $k+5$ trivalent vertices.
This is depicted in Fig.~7c as an angular diagram obtained by
choosing the $(k+3)$-point coupling as an origin.
Choosing the 6-point coupling as an origin for
the case $k=1$ confirms that ${\cal S}_1=11_{353}$ is associated
with $\zeta(3,5,3)$ via the 7-loop diagram $G(4,1,0)$.
However, for $k=3$ there is no way of obtaining MZVs of depth
3 from either choice of 6-point origin. Hence we expect a depth-5
MZV to be associated with the 15-crossing knot ${\cal S}_3$, with the
possibility of depth-7 MZVs appearing at higher crossings.
Finally we show that the three-parameter class
${\cal S}_{k,m,n}$, also built on $11_{353}={\cal S}_{1,1,1}$,
is associated with depth-3 MZVs.
The chorded braid of Fig.~8a indicates
the three places where we can add further chords.
Fig.~8b gives the chord diagram associated with it,
and indicates how to shrink propagators to obtain a log-divergent
diagram, represented by the angular diagram $G(m+n+2,k,0)$
of Fig.~8c, which evaluates in terms of depth-3 MZVs
up to 13 loops, and presumably beyond.
\subsection*{5. Conclusions}
In summary, we have
\begin{enumerate}
\item enumerated in\Eqq{Es}{Ms} the irreducibles entailed by Euler sums
and multiple zeta values at weight $n$; apportioned them by depth in\Eq{EMb};
conjectured the generator\Eq{Pd} for the number, $D_{n,k}$, of MZVs of weight
$n$ and depth $k$ that are irreducible to MZVs of lesser depth;
\item determined all MZV knot-numbers to 15 crossings, save one,
associated with a 9-loop diagram that evaluates to MZVs of depth 5
and weight 15;
\item enumerated positive knots to 15 crossings, notwithstanding
degenerate Alexander polynomials at 14 and 15 crossings;
\item developed
a technique of chording braids so as to generate families of knots
founded by parent knots whose relationship to Feynman diagrams
was known at lower loop numbers;
\item combined all the above to identify,
in Table~2, knots whose enumeration, to 16 crossings, matches that of MZVs.
\end{enumerate}
Much remains to be clarified in this rapidly developing area.
Positive knots, and hence the transcendentals associated with them
by field theory, are richer in structure than MZVs: there are more of them
than MZVs; yet those whose knot-numbers are MZVs evaluate in search
spaces that are significantly smaller than those for the MZVs, due to
the absence of a two-crossing knot. After 18 months of intense collaboration,
entailing large scale computations in knot theory, number theory, and field
theory, we are probably close to the boundary of what can be discovered
by semi-empirical methods. The trawl, to date, is impressive, to our minds.
We hope that colleagues will help us to understand it better.
\noindent{\bf Acknowledgements}
We are most grateful to Don Zagier for his generous comments,
which encouraged us to believe in the correctness of our discoveries\Eq{EMb},
while counselling caution as to the validity of\Eq{Pd}
in so far uncharted territory with depth $k>4$.
David Bailey's MPPSLQ~\cite{DHB}, Tony Hearn's REDUCE~\cite{RED}
and Neil Sloane's superseeker~\cite{NJAS} were instrumental in this work.
We thank Bob Delbourgo for his constant encouragement.
\newpage
| 2024-02-18T23:39:42.282Z | 1996-11-18T11:27:03.000Z | algebraic_stack_train_0000 | 131 | 5,534 |
|
proofpile-arXiv_065-929 | \section{Introduction}\label{intro}
Spiral waves are spatio-temporal patterns typically found in
distributed media with active elements. They have been
studied extensively for excitable and oscillatory media.
\cite{gen,act} For both types of media, it is conventional
to consider systems with two dynamical variables.
Activator-inhibitor or propagator-controller systems are
often used to analyse spiral dynamics in excitable media
\cite{act,fife}, while the complex Ginzburg--Landau equation
is the prototypical model describing spatially-distributed
oscillatory media near the Hopf bifurcation point \cite{kur}.
Spiral waves may also exist in media where the local
dynamics supports complex periodic or even chaotic motion
that cannot be represented in a two-dimensional phase plane.
Various patterns involving rotating spiral waves have been
observed in coupled map lattices or reaction-diffusion
dynamics based on the R\"{o}ssler chaotic attractor
\cite{ma}. The three-variable reaction-diffusion system with
chaotic local reaction kinetics given by the
Willamowski-R\"{o}ssler rate law \cite{wr} has been studied
in \cite{prl}. Stable spiral waves exist in this system and
the nucleation and annihilation of spiral pairs leading to
spiral turbulence have been observed. The change of
dimensionality of phase space from two to three
significantly complicates the description of the dynamics.
Descriptions in terms of phase and amplitude, well
established for two-variable models, cannot be directly
generalized. Although several definitions have been proposed
for the phase of chaotic oscillations, all of them suffer
from some degree of ambiguity (see \cite{pik} for a
discussion). Similar difficulties arise in the consideration
of nonchaotic oscillatory dynamics which is nevertheless
more complex than a single loop in phase space; for example,
in the oscillations that appear in the period doubling
cascade to chaos or in the mixed-mode oscillations observed
in experiments in chemical systems. \cite{mix}
In this paper we study the spatiotemporal organization of a
reacting medium which supports a single spiral wave and
where the local rate law exhibits period-$2^n$ or chaotic
oscillations. Through an analysis of the dynamics at
different spatial points in the medium we show that a number
of phenomena arise for $n>0$ which are nonexistent in
period-1 oscillatory media. Section~\ref{spiral} introduces
the model and presents some features of the spiral wave
behavior in a chaotic medium. The local dynamics in the
medium is considered in detail in Sec.~\ref{local}. The
analysis allows one to identify the loop exchange process
for local trajectories and the complicated pattern of the
distribution of different types of local dynamics in the
medium. A characteristic feature of this distribution is the
existence of a curve where the local dynamics is effectively
period-1. Section~\ref{topo} introduces a coarse-grained
description of $2^n$-periodic local orbits which allows one
to characterize the local dynamics that is observed in the
medium. The topological conflict between the phase space
structure of local trajectories and the constraints imposed
on the medium by the existence of a spiral wave is
considered in Sec.~\ref{glob}. We show that the observed
changes of the local orbits are necessary to maintain the
global coherence of the medium. The conclusions of the study
are presented in Sec.~\ref{conc}.
\section{Spiral Waves in Periodic and Chaotic Media}\label{spiral}
While many aspects of the phenomena we describe in this
paper are general and apply to systems in which complex
periodic or chaotic orbits exit, we consider situations
where a chaotic attractor arises by a period-doubling
cascade and confine our simulations to the
Willamowski-R\"{o}ssler (WR) model \cite{wr},
\begin{eqnarray}
A_1 +X &\mathrel{
\mathop{\kern0pt {\rightleftharpoons}}\limits^{{k_1}}_{k_{-1}}}& 2X,\;\;
X+Y
\mathrel{\mathop{\kern0pt {\rightleftharpoons}}\limits^{{k_2}}_{k_{-2}}}
2Y,\nonumber \\
A_5 +Y &
\mathrel{\mathop{\kern0pt {\rightleftharpoons}}\limits^{{k_3}}_{k_{-3}}}&
A_2,\;\;
X+Z
\mathrel{\mathop{\kern0pt {\rightleftharpoons}}\limits^{{k_4}}_{k_{-4}}}
A_3, \label{eq_mechanism} \\
A_4 +Z &
\mathrel{\mathop{\kern0pt {\rightleftharpoons}}\limits^{{k_5}}_{k_{-5}}}&
2Z\;. \nonumber
\end{eqnarray}
Only the $X$, $Y$ and $Z$ species vary with time; all others
are assumed fixed by flows of reagents. Study of this model
allows us to illustrate most features of the structure of a
spatially distributed medium supporting spiral waves. In
addition, it is useful to deal with a specific example since
certain aspects of the analysis of periodic and chaotic
orbits in high-dimensional concentration phase spaces rely
on geometrical constructions that pertain to a specific
class of attractors.
The rate law that follows from the mechanism
(\ref{eq_mechanism}) is
\begin{eqnarray}
\label{mass}
{ d c_x(t) \over d t} &=&\kappa_1 c_x -\kappa_{-1}
c_x^2 -\kappa_2 c_x c_y
+\kappa_{-2} c_y^2 -\kappa_4 c_x c_z \nonumber \\ & & +\kappa_{-4}
=R_x({\bf c}(t))\;, \nonumber \\
{ d c_y(t) \over d t} &=&\kappa_2c_x c_y
-\kappa_{-2} c_y^2 -\kappa_{3} c_y + \kappa_{-3}
\\ & & =R_y({\bf c}(t))\;, \nonumber \\
{ d c_z(t) \over d t} &=&-\kappa_4 c_x c_z +
\kappa_{-4} +\kappa_5 c_z -\kappa_{-5} c_z^2
=R_y({\bf c}(t))\;, \nonumber
\end{eqnarray}
where the rate coefficients $\kappa_i$ include the
concentrations of any species held fixed by constraints. We
take $\kappa_2$ to be the bifurcation parameter while all
other coefficients are fixed: ($\kappa_1=31.2,
\kappa_{-1}=0.2, \kappa_{-2}=0.1, \kappa_{3}=10.8,
\kappa_{-3}=0.12, \kappa_{4}=1.02, \kappa_{-4}=0.01,
\kappa_{5}=16.5, \kappa_{-5}=0.5$). In this parameter region
the WR model has been shown \cite{ka} to possess a chaotic
attractor arising from a period-doubling cascade as
$\kappa_2$ is varied in the interval [1.251,1.699].
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f1.eps}
\end{center}
\caption{Chaotic attractor for the Willamowski-R\"{o}ssler
model at $\kappa_2=1.567$. }
\label{cha}
\end{figure}
Figure~\ref{cha} shows the four-banded chaotic attractor at
$\kappa_2 = 1.567$. Throughout the entire parameter domain
$\kappa_2\in [1.251,1.699]$ the system's attractor is
oriented so that its projection onto the $(c_x,c_y)$ plane
exhibits a folded phase space flow circulating around the
unstable focus ${\bf c^*}$. This allows one to introduce a
coordinate system in the Cartesian $(c_x,c_y,c_z)$ phase
space which is appropriate for the description of the
attractor. We take the origin of a cylindrical coordinate
system $(\rho,\phi,z)$ at ${\bf c^*}$ so that the $z$ and
zero-phase-angle ($\phi=0$) axes are directed along the
$c_z$ and $c_y$ axes, respectively. The phase angle $\phi$
increases along the direction of flow as shown in
Fig.~\ref{frame}.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f2.eps}
\end{center}
\caption{Cylindrical coordinate frame $(\rho,\phi,z)$ with
origin at ${\bf c}^*$ in the $(c_x,c_y,c_z)$ phase space. A
period-2 orbit is shown in this coordinate frame.}
\label{frame}
\end{figure}
For a period-1 oscillation $\phi$ coincides with the usual
definition of the phase and uniquely parametrizes the
attractor $\rho_a=\rho_a(\phi), z_a=z_a(\phi), \phi\in
[0,2\pi)$. After the first period-doubling this
parametrization is no longer unique since the periodic orbit
does not close on itself after $\phi$ changes by $2\pi$. For
a period-$2^n$ orbit $2^n$ of its points lie in any
semi-plane $\phi=\phi_0$. The angle variable $\Phi\in
[0,2^n\cdot 2\pi)$ may be used to parametrize the
period-$2^n$ attractor if one acknowledges that all $\Phi$
from the interval $[0,2^n\cdot 2\pi)$ are different but any
two values of $\Phi$, $\Phi_1$ and $\Phi_2$, with
$\Phi_2=\Phi_1+ 2^n\cdot 2\pi$, are equivalent. For a
chaotic orbit $(n \rightarrow \infty)$ all angles $\Phi \in
[0,\infty)$ are non-degenerate. When $\Phi$ is defined in
this way it is no longer an observable. Indeed, any $\Phi\in
[0,2^n\cdot 2\pi)$ can be represented as $\Phi=\phi +m\cdot
2\pi$ where $\phi\in [0,2\pi)$ and $m\in {\bf N}$. While
$\phi$ is just the angle coordinate in the $(\rho,\phi,z)$
system and is a single-valued function of the instantaneous
concentrations $\phi=\phi(c_x(t),c_y(t),c_z(t))$, the
integer number of turns $m$ can be calculated only if the
entire attractor is known.
The spatially-distributed system is described by the
reaction-diffusion equation,
\begin{equation}
\label{rds}
{\partial {\bf c}({\bf x},t) \over \partial t} = {\bf R}({\bf c}({\bf x},t))
+ D \nabla^2 {\bf c}({\bf x},t)\;,
\end{equation}
where we have assumed the diffusion coefficients of all
species are equal. If the rate law parameters correspond to
a period-1 limit cycle, we may initiate a spiral wave in the
medium and describe its dynamics and structure using
well-developed methods. The core of such a spiral wave is a
topological defect which is characterized by the topological
charge \cite{me}
\begin{equation}
\label{charge}
{1 \over 2 \pi} \oint \nabla \phi({\bf r})\cdot d{\bf l}=n_t\;,
\end{equation}
where $\phi({\bf r})$ is the local phase and the integral is
taken along a closed curve surrounding the defect. To obtain
additional insight into the organization of the medium
around the defect the local dynamics may be considered. For
this purpose we introduce a polar coordinate system ${\bf
r}={\bf x}-{\bf r}_d(t)=(r,\theta)$ centered at the defect
whose (possibly time-dependent position) is ${\bf r}_d(t)$.
Let ${\bf c}({\bf r},t)$ be a vector of local concentrations
at space point ${\bf r}=(r,\theta)$. A local trajectory in
the concentration phase space from $t=t_0$ to $t=t_0+\tau$
at point ${\bf r}$ in the medium will be denoted by
\begin{equation}
C({\bf r}|t_0,\tau) = \{ {\bf c}({\bf r},t)| t\in
[t_0,t_0+\tau]\}\;.
\end{equation}
Figure~\ref{p-1tr} shows a number of local trajectories
$C(r,\theta| t_0,\tau)$ at points with increasing separation
$r$ from the defect for a period-1 oscillation at $\kappa_2
=1.420$. One sees that as $r \rightarrow 0$ the oscillation
amplitude decreases and the limit cycle shrinks to the phase
space point ${\bf c}^*_d$ corresponding to the spiral core.
The results of our simulations show that the value of ${\bf
c}^*_d$ differs only slightly from ${\bf c^*}$ which is
chosen as the origin of the coordinate frame
$(\rho,\phi,z)$. Thus, the angle $\phi$ can serve as a phase
that characterizes all points in the period-1 oscillatory
medium except for a small neighborhood of the defect with
radius $r\approx 1$. \cite{amend} The concentration field
${\bf c}({\bf r},t)$ is organized so that the instantaneous
$(c_x,c_y,c_z)$ phase space representation of the local
concentration on any closed path in the medium surrounding
the defect is a simple closed curve encircling ${\bf c^*}$.
For large $r$, $r\geq r_{max}$ (in Fig.~\ref{p-1tr}
$r_{max}\approx 40$), one finds that $C(r,\theta| t_0,\tau)$
ceases to change shape and is indistinguishable from the
period-1 attractor of (\ref{mass}) on the scale of the figure.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f3.eps}
\end{center}
\caption{Local trajectories calculated for the period-1
oscillatory medium ($\kappa_2=1.420$) at radii 5, 10, 20,
30, 40, 56, and fixed $\theta$. The periodic orbits grow
monotonically in size with $r$; the difference between
trajectories corresponding to $r=40$ and $r=56$ is not
resolved on the scale of the figure. Local orbits appear to
be independent on angle $\theta$. The location of ${\bf
c}^*$ is designated by a diamond.}
\label{p-1tr}
\end{figure}
One may initiate the analog of a defect in $2^n$-periodic
and chaotic media. The defect serves as the core of a spiral
wave which may exist even if the oscillation is not simply
period-1. A defect was introduced in the center of the
medium by fixing $c_z({\bf r})=c^{*}_{z}$ and choosing
initial concentrations $ (c_x({\bf r}),c_y({\bf r}))$ to
produce orthogonal spatial gradients. The influence of the
symmetry of the spatial domain on the dynamics was
investigated by performing simulations on square $(L \times
L)$ arrays as well as on disk-shaped domains with radius
$R$. No-flux boundary conditions were used to prevent the
formation of defects with opposite topological charge within
the medium and to minimize effects arising from the
self-interaction of spiral waves. The implementation of
these initial and boundary conditions does not guarantee the
formation of a solitary stable spiral wave; new spiral pairs
and other patterns (e.g. pacemakers) may appear as a result
of instabilities of the spiral arm and lead to spiral
turbulence. The ability to maintain a stationary spiral wave
in the center of the medium is sensitive to the parameters.
For various values of the system size and rate constants the
defect can move along expanding or contracting spiral
trajectories or trajectories with complex ``daisy"-like
forms \cite{daisy}. Simulations show that the stability of a
spiral wave with a stationary core located at the center of
the medium increases with the system size and for rate
constants lying close to the chaotic regime within the
period-doubling cascade. In the following we restrict our
considerations to parameters that lead to the formation of a
single spiral wave whose core is stationary and lies in the
center of the domain. Long transient times ($\approx10^2$
spiral revolutions) are often necessary to reach this
attracting state.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f4.eps}
\end{center}
\caption{Frames showing a rotating spiral wave in the
chaotic ($\kappa_2=1.567$) disk-shaped medium with $R=80$.
The local angle variable $\phi(r,\theta,t)$ is shown as grey
shades. Time increases from left to right and from top to
bottom. The frames are separated by one period of spiral
revolution $T_r$. The integration time step is $\Delta t=
10^{-4}$ and the scaled diffusion coefficient is $D \Delta
t/ (\Delta x)^2 = 10^{-2}$.}
\label{4sp}
\end{figure}
Figure~\ref{4sp} shows four consecutive states of the
disk-shaped medium with $R=80$, separated by one period of
the spiral rotation, $T_r$, for $\kappa_2 =1.567$ where the
rate law supports a chaotic attractor. Only within a
sufficiently small region with radius $r \approx 20$
centered on the defect does the medium return to the same
state after one period of spiral rotation. At points farther
from the defect the system appears to return to the same
state only after two spiral rotation periods. The transition
from period-1 to period-2 behavior occurs smoothly along any
ray emanating from the defect.
\section{Analysis of Local Dynamics} \label{local}
More detailed information may be obtained from an
investigation of the local dynamics of the medium supporting
a spiral wave. Local trajectories $C({\bf r}|t_0,\tau)$ were
computed along rays emanating from the defect at various
angles $\theta$. Figure~\ref{6traj} (left column) shows
short-time trajectories ($\tau \approx 10T_r$) at different
radii $r$ and arbitrary but large $t_0$. These trajectories
clearly demonstrate that the local dynamics undergoes
transformation from small-amplitude period-1 oscillations in
the neighborhood of the defect to period-4 oscillations near
the boundary.\cite{f3}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f5.eps}
\end{center}
\caption{Local trajectories $C(r|\,t_0,\tau)$ for the
disk-shaped medium ($\kappa_2=1.567,\;R=80$): (a, d) $r=10$;
(b, e) $r=35$; (c, f) $r=76$. The observation times are
$\tau\approx10T_r$ for left column and $\tau =80T_r$ for
right column. All the trajectories are shown on the same
scale.}
\label{6traj}
\end{figure}
The well-resolved period-doubling structure of $C({\bf
r}|t_0,\tau)$ is destroyed if the time of observation $\tau$
becomes sufficiently large. The right column of
Fig.~\ref{6traj} shows trajectories sampled at the same
spatial locations but with the time of observation $\tau =
80 T_r$. These long-time trajectories appear to be ``noisy"
period-1 and period-2 orbits: the trajectory in panel (d) is
a thickened period-1 orbit while both the period-2 (panel
(b)) and period-4 (panel (c)) orbits now appear as thickened
period-2 orbits in panels (e) and (f) with trajectory
segments lying between the period-2 bands. As $\tau$ tends
to infinity the resulting local attractor, $C(r)$, is
independent of $t_0$ and the angle $\theta$.
\subsection{First-return maps}
An analysis of the local trajectories shows that the
period-doubling phenomenon is not a monotonic function of
$r$. Consider the first return map constructed from a
Poincar\'{e} section of a local trajectory $C({\bf
r}|t_0,\tau)$ in the following way: choose the plane $c_y =
c_y^{*}$ with normal ${\bf n}$ along the $c_y$ axis as the
surface of section and select those intersection points
where ${\bf n}$ forms a positive angle with the flow. This
yields a set $\{(c_x({\bf r},t_n),c_z({\bf r},t_n))| n\in
[1,N]\}$ where $t_0 <t_1<t_2<\ldots <t_N<t_0+\tau$ is a
sequence of times at which the trajectory crosses the
surface of section. For the WR model the points $(c_x({\bf
r},t_n),c_z({\bf r},t_n))$ lie on a curve which deviates
only slightly from a straight line. Consequently, we may
choose either $c_x$ or $c_z$ to construct the first return
map. Let $\xi_n({\bf r})=c_x({\bf r},t_n)$ denote a point in
the Poincar\'{e} section. The relation $\xi_{n+1}({\bf
r})=f(\xi_{n}({\bf r}))$ between the successive
intersections of the Poincar\'{e} surface defines the local
first return map,
\begin{eqnarray}
g({\bf r}|t_0,\tau) & = &\{ (\xi_n({\bf r}),\xi_{n+1}({\bf
r})) \nonumber \\ & & |\,t_n\in [t_0,t_0+\tau], n\in [1,N]
\}\;.
\end{eqnarray}
Combining such maps for all $r$ along some ray emanating
from the defect at an angle $\theta$, we obtain the
cumulative first return map,
\begin{equation}
G(\theta|t_0,\tau)= \bigcup_{r\in (0,R)} g(r,\theta |t_0,\tau)\;.
\end{equation}
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f6.eps}
\end{center}
\caption{Cumulative first return map $G$ constructed for the
disk-shaped array ($\kappa_2=1.567,\;R=80$). The letters
indicate the $r$ values discussed in the text: (a) 20, (b)
31 and (c) 43.}
\label{r-frm}
\end{figure}
For sufficiently long times $\tau$, $g$ is independent of
$\theta$ and $t_0$. Letting $\lim_{\tau \to \infty}
g(r,\theta |t_0,\tau)= g(r)$, we may write the corresponding
cumulative first return map as $G=\lim_{\tau \to \infty}
G(\theta|t_0,\tau)$. Figure~\ref{r-frm} shows $G$ for the
disk-shaped medium under consideration. The first return map
is comprised of several branches which can be identified as
thread-like maxima of the first return map point density.
These branches are parametrized by the spatial coordinate
with $r$ increasing from the bottom left corner to the ends
of the wide-spread arms of $G$ (cf. Fig.~\ref{r-frm}).
Generally for $r\leq 40$ points lying on lines $\xi_n(r) +
\xi_{n+1}(r) = {\rm const}$ belong to the same $g(r)$ though
overlaps of neighboring $g$-map points are common. Thus,
measuring the separation between branches of $G$ in the
direction perpendicular to the bisectrix one can determine
the character of $C(r)$. In spite of some evidence of fine
structure, from the fact that map points are located along
the bisectrix in Fig.~\ref{r-frm} one can infer that up to
$r=20$ the local dynamics is predominantly period-1.
Starting from $r=21$ (labeled by $a$ in Fig.~\ref{r-frm}),
$G$ splits into two branches which diverge from the
bisectrix indicating a period-2 structure of $C(r)$. As $r$
increases these branches bend and cross the bisectrix at
$r=31$ (labeled by $b$ in Fig.~\ref{r-frm}), indicating a
return of the local dynamics to the period-1-like pattern.
After this crossing the separation between the branches
grows rapidly reflecting the development of period-2
structure. An examination of the main branches of $G$
reveals period-4 fine structure. This period-4 structure is
visible for $r>28$ and beyond $r\approx 43$ (labeled by $c$
in Fig.~\ref{r-frm}) it becomes prominent and can be easily
seen in the structure of $C(r)$ (cf. Fig.~\ref{6traj}).
\subsection{Loop exchange and $\Omega$ curve}
~From the analysis of the time series of the local
concentration one may determine the processes responsible
for the differences between the local trajectories $C({\bf
r}|t_0,\tau)$ for short and long time intervals $\tau$ (cf.
Fig.~\ref{6traj}). Figure~\ref{p2ex} shows the signature of
this phenomenon for $c_x(r,t)$ at $r=50$ in a disk-shaped
array with $R=80$ and $\kappa_2 = 1.544$, a parameter value
corresponding to period-4 dynamics in the rate law. Every
second maximum of $c_x(r,t)$ is indicated by diamond or
cross symbols. The envelope curves obtained by joining like
symbols cross at $t=t_{ex}$, thus the curve which connected
large-amplitude maxima at $t<t_{ex}$ joins low-amplitude
maxima at $t>t_{ex}$ and vice-versa. This implies that if at
some $t_0<t_{ex}$ the representative point ${\bf c}(r,t_0)$
was found on the small-amplitude band of period-2, then at
$t =t_0+nT_2>t_{ex}$, where $T_2$ is the period of the
period-2 oscillation, it will be found on the
larger-amplitude band.\cite{f4} This phenomenon can be
interpreted as an exchange of the local attractor's bands.
Indeed, approaching $t_{ex}$ from the left one finds that
with each period of oscillation the small-amplitude band
grows while large-amplitude band shrinks. At $t=t_{ex}$ both
bands reach and pass each other.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f7.eps}
\end{center}
\caption{Concentration time series $c_x(r,t)$ at $r=50$ for
the disk-shaped array ($\kappa_2=1.544,\;R=80$) showing the
loop exchange process. Time unit equals $10^5$ $\Delta t$.}
\label{p2ex}
\end{figure}
For a short period of time near $t_{ex}$ the bands are
indistinguishable in phase space and the oscillation is
effectively period-1. It is this exchange phenomenon that
produces loops that fill the gap between the period-2 bands
in the long-time local trajectories (cf. Fig.~\ref{6traj})
and contribute to a sparsely scattered ``gas"-like density
in $G$ (cf. Fig.~\ref{r-frm}).
An examination of the loop exchanges at different locations
in the medium revealed the existence of the following
spatio-temporal pattern. At any fixed location the exchange
occurs periodically, with period $T_{ex}\approx 55T_r$,
independent of the position $(r,\theta)$ in the medium. For
sufficiently large radii $(r\geq 35)$ this periodicity takes
an even stronger form: the entire oscillation pattern,
however complex, returns with period $T_{ex}$ to the same
configuration.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f8.eps}
\end{center}
\caption{Sketch of the $\Omega$ curve for the disk-shaped
array ($\kappa_2=1.567,\;R=80$). Points where the period-2
band exchange was observed are indicated by diamonds. }
\label{r-om}
\end{figure}
This property smoothly disappears as the defect is
approached. For two locations ${\bf r}_1=(r_0,\theta_1)$ and
${\bf r}_2=(r_0,\theta_2)$ at the same radius $r_0$ from the
defect but at different angles, the oscillation pattern at
one of them, say ${\bf r}_2$, can be obtained from the
corresponding pattern at ${\bf r}_1$ through translation in
time by $T_{ex}(\theta_2-\theta_1)/2\pi$, the sign of the
translation being defined by ${\rm
sign}(\theta_2-\theta_1)$. In view of this observation it is
convenient to introduce a coordinate system $(r',\theta')$
rotating with angular velocity $2\pi/T_{ex}$ relative to the
laboratory-fixed coordinate system $(r,\theta)$. In this
rotating frame the local dynamics is described by a
time-homogeneous pattern, unique for every spatial point
${\bf r'}$, and the locations in the medium where loop
exchange occurs correspond to points where the local
dynamics always has a period-1-like character. The set of
loop exchange points constitute a curve $\Omega$ with spiral
symmetry which winds twice around the defect (see
Fig.~\ref{r-om}). The two convolutions of $\Omega$ lie close
to circular arcs with radii 19 and 32. This result may be
compared with the data obtained from an examination of $G$
(cf. Fig~\ref{r-frm}). The crossings of the bands of $G$
occur at loci lying on $\Omega$.
Close to the defect the resolution of the loop exchange
event is difficult. At $r<18$ the difference between the
period-2 bands is comparable to the band thickness and the
determination of $\Omega$ for smaller radii becomes
impractical. Variation of the system parameters results in a
change of the characteristics of $\Omega$; for example, the
radius of the domain $R$ does not affect the shape of the
$\Omega$ but does change the angular velocity with which the
coordinate frame $(r',\theta')$ in which $\Omega$ is
immobile rotates relative to the laboratory-fixed frame
$(r,\theta)$. The angular velocity is higher for smaller
system sizes: a decrease in $R$ from 80 to 60 reduces the
period $T_{ex}$ by a factor of 0.42. A change in the rate
constants $\kappa_i$ leads to a deformation of $\Omega$,
although the identification of $\Omega$ as a set of exchange
points remains and it retains the topology of a curve
passing from the defect to the boundary. In Sec.~\ref{glob}
we shall show that the existence of $\Omega$ is essential
for the maintenance of spatial continuity in media composed
of $2^n$-periodic oscillators.
Simulations on a square array with dimension $80\times 80$
(all parameters were the same as for the disk) show that a
rotating frame is not necessary to observe the
time-homogeneous local dynamics of $C({\bf
r}|t_0,\tau)=C({\bf r}|\tau)$. For this system geometry the
$\Omega$ curve is fixed in the medium,
\end{multicols}
\widetext
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f9.eps}
\end{center}
\caption{Local trajectories calculated on circle with $r=55$
for the square array. All the trajectories are shown on the
same scale.}
\label{circ}
\end{figure}
\begin{multicols}{2}
\narrowtext
\noindent a slight wobbling of the defect (frame origin)
being neglected. Figure~\ref{circ} shows a number of
long-time $(\tau \gg T_r)$ local trajectories on a circle
with radius $r_0=55$ surrounding the defect in the square
domain. One sees a significant dependence of the shape of
$C(r_0,\theta|\tau)$ on the angle $\theta$. The local
trajectories range from a period-1 orbit at the intersection
with $\Omega$ to the well-established period-4 orbit
observed in a certain range of $\theta$. To highlight the
loop exchange phenomenon, a particular time instant $t=t^*$
is marked on all the trajectories (see Fig.~\ref{circ}).
Compare the two $C(r_0,\theta|\tau)$ at the locations
$\theta_1$ and $\theta_2$ chosen symmetrically on either
side of the point $\theta=\theta_{\Omega}$ where the circle
intersects $\Omega$. Visual inspection of these orbits shows
that their shapes are essentially identical but
representative points ${\bf c}(\theta_1,t^*)$ and ${\bf
c}(\theta_2,t^*)$ appear on different period-2 bands of the
corresponding orbits. This clearly demonstrates that the
period-2 bands do not just approach but indeed pass each
other at $\theta=\theta_{\Omega}$, exchanging their
positions in phase space. Since it is not necessary to work
in a rotating coordinate system in the case of a square
domain, one may resolve the fine structure of the local
trajectories to a greater degree as can be seen
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f10.eps}
\end{center}
\caption{Cumulative first return map $G(\theta)$ for the
square array ($\kappa_2=1.567,\;L=80,\; \theta=0$) (top
panel) and a magnification of a portion of its structure
(bottom panel). Letters on the bottom panel denote radii for
which corresponding portions of $G(\theta)$ are constructed:
(a) 9; (b) 19; (c) 25, and (d) 31.}
\label{s-frm}
\end{figure}
\noindent in Fig.~\ref{s-frm} (a,b) which shows the
cumulative first return map $G(\theta)$ and a magnification
of a portion of its structure (compare with
Fig.~\ref{r-frm}). The results show that $G$ is comprised of
four branches with the fine structure of period-4 resolved
even in the vicinity of the defect ($r=5$ is the closest
distance to the defect for which $g(r)$ is shown). Any
perturbation of the self-organized pattern of local
oscillator synchronization due to irregular motion of the
defect, influence of the boundary or the presence of another
defect may obliterate subtle fine structure of the local
trajectories. In such a circumstance one is able to observe
only two gross branches of $G$ and their split nature is not
resolvable except for very large $r$. These observations
allow one to suppose that the local trajectories may have
the same number of fine structure levels everywhere in the
medium but the degree to which different levels are resolved
in their phase portraits strongly depends on the position in
the medium relative to the defect. In view of this
hypothesis the phenomenon of spatial period-doubling should
not be understood in the literal sense but rather as an
enhanced ability to resolve the fine structure with the
increase of separation from the defect.
The stationary rotating spiral wave arises from the complex
defect-organized cooperation of local oscillators. Each
location in the medium develops some site-specific pattern
of oscillation which often differs significantly from that
of the corresponding rate law attractor and varies
substantially from one space point to another. There exists
a (possibly rotating) reference frame $(r',\theta')$,
centered on the moving defect, in which local dynamics takes
a simple, time-homogeneous form. Each point of the medium in
this frame can be assigned a unique oscillatory pattern,
different for different spatial points. This allows one to
introduce the notion of a defect-organized field associated
with $(r',\theta')$ which specifies the pattern of dynamics
in every spatial point of the medium. This field exhibits a
complicated architecture lacking of any simple symmetries
(which can be easily seen from the shape of the $\Omega$
curve). The slow rotation of this field in disk-shaped
arrays restores the circular symmetry of the solution.
Although the manner in which different types of local
dynamics are distributed in the medium is complex, it is not
disordered. Due to the continuity of the medium maintained
by the diffusion, it obeys certain topological principles
studied in the subsequent sections.
\section{Coarse-grained description of local trajectories} \label{topo}
In the previous section the phase space shapes of the local
trajectories were shown to vary considerably but smoothly
from one point in the medium to another. To describe the
transformations of these orbits into each other, it is
useful to introduce a description which captures only
topologically significant changes of phase portraits and
disregards unimportant details. To understand the
topological principles which determine the global
organization of the defect-organized field one also needs a
means to compare the time dependence of local trajectories.
In this section we present a scheme that allows one to
partition the continuum of all the observed local
trajectories into a finite number of discrete classes
according to their phase space shape and time dependence.
\subsection{Representation of attractors by closed braids}
Consider a period-$2^n$ attractor, $P_{2^n}$, consisting of
$2^n$ loops in the concentration phase space ${\cal P} =
(c_x,c_y,c_z)$. Using the cylindrical coordinate system
introduced earlier, we may project $P_{2^n}$ on the
$(\rho,\phi)$ plane preserving its original orientation and
3D character by explicitly indicating whether
self-intersections correspond to over or under crossings.
Such a projection shows a span of $\phi$ free from crossings
where loops are essentially parallel to each other. This
span can be used to number loops, say, in the order of their
separation from the origin. This procedure maps $P_{2^n}$
onto a closed braid ${\bar B}_{2^n}$ \cite{bir}.
Figure~\ref{p-4toB} illustrates the construction of the
braid representation for the $P_4$ attractor of the WR model.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f11.eps}
\end{center}
\caption{Projection of the $P_4$ attractor on the
$(c_x,c_y)$ plane (top panel) and the corresponding closed
braid ${\bar B}_4$ (bottom panel). }
\label{p-4toB}
\end{figure}
It is convenient to subdivide the closed braid ${\bar
B}_{2^n}$ into the open braid $B_{2^n}$ (separated by dashed
lines in Fig.~\ref{p-4toB}) and its closure where threads
run parallel to each other. The direction of the flow on the
attractor is indicated by the arrows. Each crossing on the
projection of $P_4$ corresponds to an elementary braid
$\sigma_i$ which refers to the fact that thread $i$
overcrosses thread $i+1$ (cf. Fig.~\ref{brth} in Appendix
for notation rule). An under crossing will be designated by
${\sigma}_i^{-1}$. A braid may be described by a braid word
that gives the order and types of crossings of braid
threads. For example, for the closed braid corresponding to
$P_4$ (cf. Fig.~\ref{p-4toB}) $P_4 \mapsto {\bar B}_4 =
\overline{\sigma_3\sigma_2\sigma_1\sigma_3\sigma_2}$. The
closed braid${\bar B}_{2^n}$ corresponding to $P_{2^n}$ can
be represented by several braid words, which can be
transformed into one another by a set of allowed moves (see
Appendix).
Any braid word representing $P_{2^n}$ induces a permutation
$\pi^{(n)}_i$ describing the order in which loops of
$P_{2^n}$ are visited during one oscillation period
$T_{2^n}$. In general, each $P_{2^n}$ attractor is
represented by several possible $\pi^{(n)}_i$, their number
growing with $n$; for example, for $P_2$ there is only one
permutation $\pi^{(1)}_1= (^{12}_{21})$ while two
permutations $\pi^{(2)}_1 = (^{1234}_{3421})$ (which
corresponds to the braid shown in Fig.~\ref{p-4toB}) and
$\pi^{(2)}_2 = (^{1234}_{4312})$ exist for $P_4$. With a
given loop numbering convention each braid word represents a
unique permutation while one permutation can be induced by
many braid words.
\subsection{Symbolic representation of periodic orbits}
Take two period-$2^n$ oscillators whose trajectories ${\bf
c}_1(t),{\bf c}_2(t)$ lie on the same attractor, but which
are nevertheless non-identical since at any given time $t$
their dynamical variables are different ${\bf c}_1(t) \neq
{\bf c}_2(t)$. Since the orbits are periodic there is a time
$\delta t$ such that ${\bf c}_1(t + \delta t) = {\bf
c}_2(t)$ for any $t$. This operation can be formally
considered as an action of translation operator ${\cal
T}_{\delta t}$ on the trajectory of the first oscillator:
\begin{equation}
{\cal T}_{\delta t}\; {\bf c}_1(t) = {\bf c}_1(t + \delta t) = {\bf c}_2(t).
\end{equation}
The concentration time series ${\bf c}(t)$ of the first
oscillator then appears to be shifted backward by $\delta t$
relative to that of the time series of the second oscillator
if $\delta t > 0$ and forward otherwise. Of course,
trajectories corresponding to different attractors cannot be
made to correspond by such time translations, e.g. $P_{2^n}$
attractors described by different permutations $\pi^{(n)}_i$
have different patterns of oscillation, but even if two
$P_{2^n}$ lie in the same $\pi^{(n)}_i$ class their actual
shapes in ${\cal P}$ may differ significantly.
To compare the local dynamics at different points in the
medium one needs to single out the most important
characteristic features of the oscillation pattern while
discarding unnecessary details. A coarse-grained symbolic
description of trajectories appears to be useful for this
purpose. We assume that the times $t_1,t_2,\ldots,t_{2^n}$
at which the trajectory crosses a surface of section
$\phi=\phi_0$ (see Sec.~2) are approximately equally spaced,
independent of the choice of $\phi_0$. Thus, the phase point
${\bf c}(t)$ moving along $P_{2^n}$ takes approximately the
same time $T_{2^n}/2^n$ to traverse each loop of the
attractor.\cite{f5} At $t=t_0$ let the phase point of the
period-$2^n$ orbit be on the $j_0$-th loop of $P_{2^n}$, at
$t=t_0+T_{2^n}/2^n$ on the $j_1$-th loop, and so on (where
$j_l\in[1,2^n],\; l\in[1,2^n],\; j\equiv\!\!\!\!\!\!/ \; l$)
until at $t=t_0+T_{2^n}$ the phase point returns to the
$j_0$-th loop and the pattern $(j_0,j_1,\ldots j_{2^n})$
repeats. The symbolic string $s_j=(j_0,j_1,\ldots j_{2^n})$
constructed in this way captures the most significant gross
features of the oscillation pattern it describes. In this
coarse-grained representation the number of possible
non-identical trajectories corresponding to a particular
$\pi^{(n)}_i$ of $P_{2^n}$ is finite and the different
trajectories are simply given by the $2^n$ cyclic
permutations of $s_j$. Likewise the time translation
operators constitute a finite group ${\cal T}_{l}, \;
l\in[-2^{n-1},2^{n-1})$. They act on the symbolic string
representing the orbit to give one of its cyclic
permutations. ~From its definition it can be easily seen that
$\pi^{(n)}_i$ serves as a symbolic permutation
representation of ${\cal T}_{+1}$ for the corresponding
$i$-th permutation class of $P_{2^n}$. Indeed, consider as
an example a period-4 oscillation whose representative point
lies on loop 3 at the reference moment of time $t=t_0$. Then
for the pattern of oscillation determined by $\pi^{(2)}_1$
the state reads $s_1=(3241)$. To obtain the new state
translated by $T_4/4$ backward one acts on $s_1$ by the
permutation representation $\pi^{(2)}_1$ of the ${\cal
T}_{+1}$ operator to get
\begin{equation}
{\cal T}_{+1}\: s_1=
\left(^{{\displaystyle 1234}}_{{\displaystyle 3421}}\right)
(3241)=(2413)=s_2,
\end{equation}
which correctly describes the result of the shift of the
initial state $s_1$.
\section{ Global organization of medium}\label{glob}
\subsection{Period-1 regime}
We now return to the spatially distributed medium and begin
by reviewing some properties of the local dynamics in the
vicinity of a stable defect with topological charge $n_t=\pm
1$ in a period-1 oscillatory medium. Consider a cyclic path
$\Gamma=\{r=r_0>r_{max},\theta\in [0,2\pi)\}$ surrounding
the defect. Here $r_{max}$ is a radius such that for all
$(r,\theta), r>r_{max},\: \theta\in [0,2\pi)$ the shape of
the local orbit in phase space ${\cal P}$ is independent of
$(r,\theta)$ and closely approximates that of the period-1
attractor of the mass action rate law (see
Sec.~\ref{spiral}). If one starts at an arbitrary point
$(r_0,\theta_0)\in \Gamma$ one finds that the instantaneous
local phase $\phi({\bf r},t)$ changes by $2\pi$ or $-2\pi$
(depending on the sign of the topological charge) along
$\Gamma$. Let us now fix a particular time instant $t=t^*$
and construct the set of points ${\cal S}=\{{\bf c}({\bf
r},t^*),r\in\Gamma\}$ as a phase space image of
instantaneous concentrations at points lying on $\Gamma$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f12.eps}
\end{center}
\caption{${\cal S}$-curves (shown by diamonds) constructed
for $\Gamma$ with $r_0=55$ in period-1 oscillatory
($\kappa_2=1.420$) (a) and chaotic ($\kappa_2=1.567$) (b)
media. Solid curves represent short time local trajectories
on $\Gamma$.}
\label{s-curve}
\end{figure}
The property of a defect (\ref{charge}) and the continuity
of the medium insure that ${\cal S}$ is a simple closed
curve winding once around ${\bf c}^*$.
Figure~\ref{s-curve}(a) shows the ${\cal S}$-curve
constructed for the contour $\Gamma$ with radius $r_0=55,\,
r_0>r_{max}$ in a period-1 oscillatory medium with
$\kappa_2=1.420$. Since all the points on the ${\cal
S}$-curve lie at the same time on the local trajectories
$C({\bf r}|t_0,\tau),\, {\bf r}\in\Gamma$ with $t^*\in
[t_0,t_0+\tau]$, and for $\Gamma$ with $r_0>r_{max}$ all the
local trajectories are the same and approximated by the
period-1 attractor of the system (\ref{mass}), the ${\cal
S}$-curve simply coincides with this attractor for any $t^*$
(cf. Fig.~\ref{s-curve}(a)). The ${\cal S}$-curve
constructed for an arbitrary simple closed path encircling
the defect in the medium possesses the same property as long
as the path lies in the open region $r>r_{max}$.
This result can be reformulated in terms of time
translations of local trajectories as follows. Let the local
trajectory $C(r_0,\theta_0 |t_0,\tau)$ at the point
$(r_0,\theta_0)\in\Gamma$ be taken as a reference, then all
of the local trajectories on $\Gamma$ can be obtained
through the translation of $C(r_0,\theta_0 |t_0,\tau)$ by
some time $\delta t(\theta-\theta_0)$ (see Sec.~\ref{topo}).
The condition (\ref{charge}) implies that $\delta
t(\theta-\theta_0)$ is a monotonically increasing
(decreasing) function such that $\delta t(2\pi)=\pm T_1$
where $T_1$ is the period of oscillation and the sign is
that of $n_t$. Thus, the oscillation pattern is continuously
time shifted along $\Gamma$ such that upon return to the
initial point it has experienced translation by the period.
\subsection{Period-$2^n$ regime}
For $2^n$-periodic and chaotic media property (\ref{charge})
holds where $\phi({\bf r},t)$ should be understood as the
angle variable introduced in Sec.~\ref{spiral}. This can be
seen from the following argument. Take a period-2 medium
with rate constants chosen in the vicinity of the
bifurcation from period-1 to period-2 such that the
attractor $P_2$ of (\ref{mass}) lies infinitesimally close
to $P_1$ from which it bifurcated. Due to the continuity of
the solutions of the reaction-diffusion equation
(\ref{rds}), the value of $\oint \phi({\bf r},t)\;d{\bf l}$
cannot change abruptly when the bifurcation parameter is
changed through the period-doubling bifurcation. This
implies that the ${\cal S}$-curve constructed for a contour
$\Gamma$ in a period-$2^n$ medium, as in case of a simple
period-1 medium, is a closed curve which loops once around
${\bf c}^*$ in phase space. This is illustrated in panel (b)
of Fig.~\ref{s-curve} which shows ${\cal S}$ for contour
$\Gamma$ with radius $r_0=55$ in medium with
$\kappa_2=1.567$ and time $t=t^*$. Recall again that the
points of the ${\cal S}$ curve have to lie on the local
trajectories $C({\bf r}|t_0,\tau),\, {\bf r}\in\Gamma$ (cf.
Fig.~\ref{circ} where points designated by diamonds lie on
${\cal S}$ for the chosen time moment and contour shown in
the figure). Since the local trajectories in a period-$2^n$
medium loop several times around ${\bf c}^*$, the curve
${\cal S}$ which winds only once $(n_t=\pm1)$ around ${\bf
c}^*$ cannot span the entire local trajectory as is the case
for a period-1 medium. As one sees from
Fig.~\ref{s-curve}(b) ${\cal S}$ follows the larger loop of
the local trajectory, which for $\Gamma$ with $r_0=55$ is
typically a period-2 orbit (cf. Fig.~\ref{circ}), and
instead of making the second turn on the smaller loop, it
crosses the gap between the loops and closes on itself.
Although the shape of ${\cal S}$ changes with time (see
\cite{prl} for details), for any $t^*$ there exist segments
of ${\cal S}$ which connect different loops of local
trajectories. This behavior of the ${\cal S}$ curves would
be impossible if loop exchanges were nonexistent. The
analysis shows that the segments of ${\cal S}$ covering the
gaps between the loops of the local trajectories are images
of points on $\Gamma$ which lie close to the intersection
with the $\Omega$ curve. Thus, the loop exchanges observed
in period-$2^n$ media are necessary to reconcile the
contradiction between the one-loop topology of the ${\cal
S}$ curves determined by the presence of a defect and the
multi-loop topology of the local trajectories determined by
the local rate law.
The change of the local trajectories along the contour
$\Gamma$ in period-$2^n$ media can be considered in terms of
time translations if one adopts a generalization of the
translation operation in the following way. In a period-2
medium let the contour $\Gamma$ and the reference point
$(r_0,\theta_0)\in\Gamma$ be chosen so that $\Gamma$
intersects the $\Omega$ curve in the single point
$(r_0,\theta_{\Omega})$ and suppose that these points are
sufficiently separated from each other. Since the shapes of
the local orbits change significantly along any closed path
surrounding a defect (cf. Fig.~\ref{circ}) these
trajectories cannot be made to coincide by time translation
as this operation is defined in Sec.~\ref{topo}.
Nevertheless, the general features of the temporal pattern
of the trajectories are preserved (e.g. sharp maxima in
$c_i(t)$ time series) and for two
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f13.eps}
\end{center}
\caption{Period-2 local concentration time series $c_x({\bf
r},t)$ calculated on a cyclic path $\Gamma$ surrounding the
defect: (a) series sampled at four consecutive locations
separated by $\delta\theta=30^o$; (b) two series sampled at
locations chosen symmetrically on either side of the
intersection with the $\Omega$ curve. }
\label{gamma}
\end{figure}
\noindent locations $(r_0,\theta_1)$ and $(r_0,\theta_2)$
one is able to find a time shift $\Delta
t(\theta_1,\theta_2)$ such that some measure of the
deviation between the trajectories, say,
\begin{equation}
\label{match}
M(\Delta t(\theta_1,\theta_2))= \int_{t_0}^{t_0+\tau}
|{\bf c}^{(1)}(t+\Delta t) - {\bf c}^{(2)}(t)|\: dt ,
\end{equation}
is minimized. Choosing the local trajectory
$C(r_0,\theta_0|t_0,\tau)$ as a reference and comparing it
to all the other local orbits on $\Gamma$ one is able to
define the time shift function $\delta
t(\theta-\theta_0)\equiv\Delta t(\theta,\theta_0)$. The
shift function $\delta t(\theta-\theta_0)$ increases (or
decreases) monotonically and almost linearly (see
Fig.~\ref{gamma}(a)) with $d (\delta t)/d \theta \approx
T_2/2\cdot 2\pi$ everywhere on $\Gamma$ except for a small
neighborhood of $\theta=\theta_{\Omega}$ where it exhibits
break. Indeed, the loop exchange at $\theta=\theta_{\Omega}$
causes the discontinuity of $\delta t(\theta-\theta_0)$. At
$\theta=\theta_{\Omega}$ both loops of the local orbit
become equivalent and the oscillation is effectively
period-1 with period $T_1=T_2/2$. Since the loops exchange
at $\theta=\theta_{\Omega}$, to find the best match
(\ref{match}) between local trajectories sampled at points
$(r_0,\theta_{\Omega} -\varepsilon)$ and
$(r_0,\theta_{\Omega}+\varepsilon)$, one needs to translate
one of the trajectories by $\delta t = T_1 +
O(\varepsilon)$. This can be easily seen in
Fig.~\ref{gamma}(b) which displays two $c_x(t)$ series
calculated at spatial points lying
$\theta-\theta_{\Omega}=\pm 10^o$ on either side of
$\theta_{\Omega}$ on $\Gamma$.
\subsection{ Trajectory transformations along $\Gamma$}
The transformation of local trajectories along $\Gamma$ can
be imagined to occur a result of two separate processes.
Suppose everywhere on $\Gamma$ except
$\theta=\theta_{\Omega}$ the shape of the local trajectories
in ${\cal P}$ is the same and is equivalent to that of
$C(r_0,\theta_0|t_0,\tau)$. Then all the other local
trajectories $C(r_0,\theta|t_0,\tau), \; \theta\in[0,2\pi),
\theta\neq \theta_{\Omega}$ can be found by time translation
of $C(r_0,\theta_0|t_0,\tau)$ by $\delta
t(\theta-\theta_0)=T_2(\theta-\theta_0)/2\cdot 2\pi$. Assume
that all the deformations of the phase space portrait of the
local trajectory which take place along $\Gamma$, including
the exchange of loops, occur at the point
$\theta=\theta_{\Omega}$ so that the passage through
$\theta_{\Omega}$ shifts the oscillation by $\delta
t_p=T_1=T_2/2$. Then the result of the continuous time
translation that occurs during $2\pi$ circulation along
$\Gamma$ may be described by the action of the ${\cal
T}_{n_t}$ operator $(n_t=\pm 1)$, while the result of the
loop exchange is described by the operator ${\cal
T}_{-n_t}$.~\cite{f9} The total transformation of the local
oscillation after a complete cycle over $\Gamma$ is
equivalent to the identity transformation and thus the
result is in accord with continuity of the medium. If one
makes the assumption that loop exchange does not occur on
some contour $\Gamma$ encircling a defect with $|n_t|=1$,
the time shift function $\delta t(\theta-\theta_0)$ becomes
monotonic and continuous everywhere on $\Gamma$. As a result
one arrives at the incorrect conclusion that starting from
the point $(r_0,\theta_0)$ with the oscillation pattern
symbolically represented by the string $s_1$, say
$s_1=(12)$, and moving along $\Gamma$ in the clockwise
direction one returns to the same point $(r_0,\theta_0+
2\pi)\equiv (r_0,\theta_0)$ but with the oscillation pattern
shifted by $T_2/2$ and given by $s_2=(21)\neq s_1$. Note
that this contradiction does not arise in the period-1
oscillatory medium where circulation over any closed path
encircling a defect results in the translation by the entire
period which automatically satisfies the continuity
principle. Thus the necessity of loop exchanges in
period-$2^n$, $n>0$ media with a topological defect
demonstrated earlier in this section in terms of ${\cal
S}$-curves is now explained in terms of time translations.
The results for the period-2 medium can be generalized for
any $n>1$ using the following hypothesis. ~From the main
property of a topological defect (\ref{charge}) it follows
that integration of an infinitesimal continuous shift
$d(\delta t)$ over any closed path surrounding a defect
results in a total shift by $\pm T_{2^n}/2^n$ and can be
symbolically described by the ${\cal T}_{n_t}$ operator.
Numerical simulations demonstrate the existence of time
translation discontinuity points such that sum of $\delta t$
jumps over these points amounts to a shift of $\mp
T_{2^n}/2^n$ described by the ${\cal T}_{-n_t}$ operator.
The locations of these points in the medium can be
identified with the $\Omega$ curve and the origin of the
time translation discontinuities with the loop exchange
phenomenon. The relation (\ref{prod}) of the Appendix
connects translations and loop exchanges and allows one to
predict the number and the kind of loop exchanges necessary
to perform the required ${\cal T}_{-n_t}$ translation.
\subsection{Examples}
Consider again the square $80\times80$ array with rate
constants corresponding to chaotic regime
($\kappa_2=1.567$). As period-4 fine structure is the
highest level of local organization resolved in the medium,
it is sufficient to use the formalism developed above for
$P_4$ to describe the local dynamics. The analysis shows
that in the bulk of the medium the oscillation is given by
the $\pi^{(2)}_1=(^{1234}_{3421})$ pattern.~\cite{f10} Using
this data and the results presented in the Appendix one can
easily enumerate all the sequences of exchanges resulting in
${\cal T}_{+1}$ translation. Indeed, one should expect
either exchange of loops 3 and 4 followed by the exchange of
period-2 bands $(^{1234}_{3412})$ or first the period-2
bands exchange followed by exchange of loops 1 and 2.
Figure~\ref{s-om} is a schematic representation of the
medium with a negatively charged $(n_t=-1)$ defect in the
center and the $\Omega$ curve displayed.
Consider the change of the oscillation pattern along ray
$ABC$ emanating from the defect as the value of $r$
increases (see Fig.~\ref{s-frm} for the cumulative
first-return map constructed for this ray). The pattern of
oscillation $s_A=(4132)$ corresponding to permutation
$\pi^{(2)}_1=(^{1234}_{3421})$ can be followed from $r=5$ to
$r=19$ where the period-2 bands undergo exchange. This
results in the switch to the oscillation pattern described
by $\pi^{(2)}_2=(^{1234}_{4312})$ seen at $r=21$. The
pattern $\pi^{(2)}_1$ is restored after loops 1 and 2
exchange at $r=22$ and and this pattern persists until
another exchange occurs at $r=28$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f14.eps}
\end{center}
\caption{Sketch of the $\Omega$ curve for the square array
($\kappa_2=1.567,\;L=80$). The points were obtained from
simulations. The ray $ABC$ intersects $\Omega$ at locations
with radii 20 and 31.}
\label{s-om}
\end{figure}
Using translation operator ${\cal T}_{+1}$ one can express
the transition of the state $s_A$ ($r<20$) through the
sequence of loop exchanges described above to the state
$s_B=(1324)$ (for $22<r<28$) as $s_B={\cal T}_{+1} s_A$. The
same shift can be achieved by continuous translation along
the path $ADEFB$ which does not intersect $\Omega$ but winds
once counter-clockwise around the defect. The $c_x(t)$ time
series at points $A,D,E,F$ and $B$ are displayed in
Fig.~\ref{abc} and demonstrate that this is the case.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f15.eps}
\end{center}
\caption{Concentration time series $c_x({\bf r},t)$
calculated at points $A,D,E,F,B$ of the square array shown
in Fig.~15. }
\label{abc}
\end{figure}
Continuing to advance along the ray $ABC$, one finds that at
$r=28$ loops 3 and 4 exchange and oscillation switches once
more to the state corresponding to $\pi^{(2)}_2$. After the
period-2 band exchange at $r=32$ the pattern corresponding
to $\pi^{(2)}_1$ is reinstated and remains unchanged for all
$r>32$. Again the oscillation at $r>32$, described
symbolically by $s_C=(3241)$, appears to be shifted by
$T_4/4$ relative to $s_B$ and by $T_4/2$ relative to $s_A$.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f16.eps}
\end{center}
\caption{ Segment of the concentration time series
$c_x(r,t)$ calculated for the disk-shaped array
($\kappa_2=1.567,\;R=80$) at $r=76$ showing the $T_4/4$ time
shift of the oscillation pattern (see explanation in the
text).}
\label{p4ex}
\end{figure}
The existence of a $T_4/4$ shift after crossing $\Omega$ can
also be seen from the results for the disk-shaped array with
$R=80$. Figure~\ref{p4ex} shows a segment of the $c_x({\bf
r},t)$ time series sampled in a fixed frame $(r,\theta)$ at
$r=76$. In this coordinate system $\Omega$ slowly rotates
clockwise (again $n_t=-1$) with period $T_{ex}$. Two time
windows each of length $T_4$ marked by dotted lines and
separated by $\Delta t = 8T_4$ allow one to see how the
oscillation state (4132) is substituted by its forward
$T_4/4$ translation (2413) after the $\Omega$ curve passes
the observation point at $t = t_{ex}$.
\section{Conclusions} \label{conc}
General principles underlie the organization of
$2^n$-periodic or chaotic media supporting spiral waves. As
in simple oscillatory media, the core of a spiral is a
topological defect which acts as an organizing center
determining dynamics in its vicinity; however, the
structural organization of the medium that arises from the
existence of the defect is far more complicated. Due to the
absence of a conventional definition of phase for
oscillations more complex than period-1, the identification
of a defect in terms of the relation (\ref{charge}) is not
obvious and requires the introduction of (often
model-dependent) phase substitutes which for some systems
may be provided by angle variables. Despite the
complications with the definition of phase, one can identify
a defect in terms of local trajectories. Indeed, as one
moves away from the defect the local dynamics takes the form
of a progression of period-doubled orbits, from near
harmonic, small-amplitude, period-1 orbits to ``noisy''
period-$2^l$ orbits, where $l$ is a function of variables
such as diffusive coupling and the system size and shape.
The presence of a defect imposes topological constraints on
the global organization of medium as well. As was shown
above, when $2^n>n_t$ the $2^n$-loop structure of the local
trajectories conflicts with the period-$n_t$ structure of
the ${\cal S}$-curve and a complex, asymmetric, spatial
pattern of local dynamics, the defect-organized field,
arises as a result of the necessity to maintain the
continuity of the medium. The most prominent characteristic
feature of this field is the $\Omega$ curve defined as the
set of points where the local dynamics most closely
resembles period-1. This signals the exchange of period-2
bands. If the local trajectories possess structure finer
than period-2, other loop exchanges leading to more subtle
changes in the local orbits can be found in the vicinity of
$\Omega$. The net result of these exchanges is to produce a
time shift of the trajectories which compensates for the
smooth time translation accumulated on continuous paths.
Since the topological continuity must be observed on any
arbitrarily large closed path encircling a defect and,
therefore, this contour has a point of intersection with
$\Omega$, a single defect in a period-$2^n$ $(n>0)$ medium
cannot be localized.
We point out again that many of the phenomena we have
discussed above are not dependent on the existence of a
period-doubling cascade or chaotic local dynamics, although
this is the case we have analysed in detail.
Reaction-diffusion systems with local complex periodic
orbits in phase space dimensions higher than two should
exhibit similar features when they support spiral waves. It
should be possible to experimentally probe the phenomena
described in this paper. The appropriate parameter regime
can be determined from investigations of well-stirred
systems. For example, period-doubling and chaotic attractors
have been observed in the Belousov-Zhabotinsky reaction.
\cite{BZchaos} If the spiral wave dynamics is then studied
in a continuously-fed-unstirred reactor \cite{cfur}, one
should be able to observe the characteristics of the spiral
dynamics and the loop exchange process that serve as
signatures of the phenomena described above.
\section{Acknowlegements}
We thank Peter Strizhak for his interest in this work and
helpful comments. This work was supported in part by a grant
from the Natural Sciences and Engineering Research Council
of Canada and by a Killam Research Fellowship (R.K.).
\begin{appendix}
\section{Braid moves and loop exchange operators}
In this appendix we make use of the projection of the
period-doubled attractors $P_{2^n}$ onto closed braids
${\bar B}_{2^n}$ (see Sec.~\ref{topo}) to show how loop
exchanges affect the pattern of oscillation. We demonstrate
that those combinations of loop exchanges that produce
identity transformations of $P_{2^n}$ result in nontrivial
time translations of trajectories.
Each closed braid ${\bar B}_{2^n}$ is represented by a set
of non-identical braid words with their number rapidly
growing with $n$. Without violation of the topology of
${\bar B}_{2^n}$ they can be transformed into one another by
the following set of moves (see, e.g. \cite{bir}) :
\begin{enumerate}
\item commutation relation, $\sigma_i\sigma_j =
\sigma_j\sigma_i, \;\; |i-j| \geq 2$;
\item type 2 Reidemeister move, $\sigma_i\sigma_i^{-1} =
\sigma_i^{-1}\sigma_i = {\bf 1}$;
\item type 3 Reidemeister move,
$\sigma_i\sigma_{i+1}\sigma_i =
\sigma_{i+1}\sigma_i\sigma_{i+1}$;
\item first Markov move, $\sigma_i\Sigma\sigma_i^{-1} =
\sigma_i^{-1}\Sigma\sigma_i = \Sigma, \; \Sigma \in B$;
\end{enumerate}
where $B$ is a set of open braids. While the first three
rules are common for all braids, rule 4 is specific for
closed braids. Indeed, it can be written in the form
$\sigma_i\Sigma = \Sigma\sigma_i$ which, for elementary
braids $\sigma_i$, corresponds to moving $\sigma_i$ around
${\bar B}_{2^n}$ resulting in the exchange of the closed
braid loops (cf. Fig.~\ref{brth}(d)). Type 1 Reidemeister
(or second Markov) moves are not allowed since they do not
preserve the number of loops, an essential feature of
$P_{2^n}$
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f17.eps}
\end{center}
\caption{Conventional designations and basic braid moves:
(a) definition of elementary braids $\sigma_i$ and
$\sigma_i^{-1}$; (b) type 2 Reidemeister move; (c) type 3
Reidemeister move, and (d) first Markov move ($\Sigma$
represents arbitrary braid). }
\label{brth}
\end{figure}
\noindent attractors. While rules 1, 2 and 3 do not affect
$\pi^{(n)}_i$, the first Markov move does (except for the
degenerate case of $P_2$ which is represented by the single
permutation $\pi^{(1)}_1$). Thus any number of
rearrangements affecting only the braid $B_{2^n}$ of the
closed braid ${\bar B}_{2^n}$ leave the braid word in the
same permutation class $\pi^{(n)}_i$, while each application
of the first Markov move yields a new permutation class.
\subsection{Loop exchanges for $P_2$ and $P_4$}
We now examine how the loop exchanges influence the patterns
of oscillation for the period-2 and period-4 attractors. For
$P_2$ one has only the single braid word $\sigma_1
(\sigma_1^{-1})$ and the single permutation $\pi^{(1)}_1 =
(^{12}_{21})$ induced by $\sigma_1$. Two different symbolic
states $s_1 = (12)$ and $s_2 = (21)$ are possible for the
period-2 oscillation with respect to some fixed time frame.
We introduce an operator $A^{(1)}_1$ whose action on the
closed braid representing $P_2$ is to move $\sigma_1$ by
$2\pi$ in a direction opposite to the flow. The result of
the action of this operator, which is the first Markov move
for $\Sigma = {\bf 1}$, is to leave the attractor $P_2$
unchanged; however, one finds that loops 1 and 2 have
exchanged their locations in phase space. In the time series
for the dynamical variable $c_i(t)$ the exchange can be seen
as a substitution of taller maxima (2) by shorter maxima (1)
and vice-versa. If this process is followed in time it
produces the characteristic pattern shown in
Fig.~\ref{p2ex}. Thus, application of $A^{(1)}_1$ to $P_2$
induces a transformation of the oscillation state $s_1$ into
$s_2$ and vice-versa. This can be symbolically described as
an action of an exchange operator ${\cal A}^{(1)}_1$
represented by the permutation $(^{12}_{21})$ :
\begin{eqnarray}
{\cal A}^{(1)}_1\: s_1 & = &
\left(^{{\displaystyle 12}}_{{\displaystyle 21}}\right)
(12) = (21) = s_2, \nonumber \\
{\cal A}^{(1)}_1\: s_2 & = &
\left(^{{\displaystyle 12}}_{{\displaystyle 21}}\right)(21) = (12) = s_1.
\end{eqnarray}
One sees that the action of ${\cal A}^{(1)}_1$ is equivalent
to that of ${\cal T}_{+1}$ which translates the oscillation
pattern by half a period. The inverse of the braid operator
$A^{(1)}_1$ can be introduced in an analogous way as an
operator moving $\sigma_1$ {\it along} the direction of the
flow. It corresponds to an exchange operator $({\cal
A}^{(1)}_1)^{-1} = {\cal T}_{-1}$ acting on strings. Since
for $P_2$ application of ${\cal T}_{+1}$ or ${\cal T}_{-1}$
results in essentially the same states the sign of the shift
is chosen to maintain consistency with corresponding
operators for $P_{2^n}$, $n>1$. Double application of ${\cal
A}^{(1)}_1$ results in translation by a full period and thus
in the identity operator
\begin{equation}
({\cal A}^{(1)}_1)^2 = ({\cal A}^{(1)}_1)^{-2} = {\bf 1}.
\end{equation}
The $P_4$ attractor possesses a richer set of
transformations. Since under coarse-graining braid words
which induce the same $\pi^{(n)}_i$ are indistinguishable,
we can single out two essential representatives $\Sigma_1 =
\sigma_3\Sigma$ for $\pi^{(2)}_1=(^{1234}_{3421})$ and
$\Sigma_2 = \sigma_1\Sigma$ for $\pi^{(2)}_2=
(^{1234}_{4312})$, where $\Sigma$ stands for
$\sigma_2\sigma_1\sigma_3\sigma_2$. Let $A^{(2)}_1$ be an
operator on ${\bar B_2}$ which moves the double-thread
crossing (cf. large dashed box in Fig.~\ref{p-4toB})
$\Sigma$ by $2\pi$ in the direction opposite to the flow,
analogous to the action of $A^{(1)}_1$ on $\sigma_1$ for
${\bar B_2}$. In fact, the action of $A^{(2)}_1$ can be seen
as an exchange of the period-2 bands of $P_4$, each
consisting of two period-4 loops. Unlike $A^{(1)}_1$ the
operator $A^{(2)}_1$ alternates braid words and the
corresponding pattern-defining permutations
$\Sigma_1,\pi^{(2)}_1
\stackrel{A^{(2)}_1}{\longleftrightarrow}
\Sigma_2,\pi^{(2)}_2$. The application of $A^{(2)}_1$
induces transformations of symbolic strings $s_j$ which
again can be described by the action of an exchange operator
${\cal A}^{(2)}_1$ represented by the permutation
$(^{1234}_{3412})$. Due to the apparent similarity of action
of $A^{(1)}_1$ and $A^{(2)}_1$ on the corresponding
attractors, ${\cal A}^{(2)}_1$ inherits the algebraic
properties of ${\cal A}^{(1)}_1$. Indeed, ${\cal A}^{(2)}_1$
produces identity operator when applied twice and, thus, is
equal to its inverse.
Finer rearrangement of the $P_4$ loop structure is provided
by the action of $A^{(2)}_2$ defined as an operator which
moves the single crossing (enclosed in the smaller box in
Fig.~\ref{p-4toB}) by $2\pi$ in the direction opposite to
flow. ~From the structure of ${\bar B_4}$ one sees that after
application of $A^{(2)}_2$ the single crossing does not
return to the same location in ${\cal P}$ but appears on the
other period-2 band; thus, braid words and $\pi^{(n)}_i$
permutations alternate. Depending on the initial state, the
application of $A^{(2)}_2$ results in different loop
exchanges. In the case of $\Sigma_1 = \sigma_3\Sigma$ action
of $A^{(2)}_2$ leads to the exchange of loops $3$ and $4$
and results in the string transformation described by the
exchange operator ${\cal A}^{(2)}_2$ with symbolic
representation $(^{1234}_{1243})$. When it acts on $\Sigma_2
= \sigma_1\Sigma$ it exchanges loops 1 and 2 and the
permutation representation of ${\cal A}^{(2)}_2$ changes to
$(^{1234}_{2134})$. The inverse of $A^{(2)}_2$ moves the
single crossing along the flow and produces opposite
results; i.e., acting on $\Sigma_1$ it leads to $1
\leftrightarrow 2$ exchange and when applied to $\Sigma_2$
it results in the exchange $3 \leftrightarrow 4$.
Note the difference between action of $A^{(n)}_i$ on braids
and the action of exchange operators ${\cal A}^{(n)}_i$ on
symbolic strings. While several operations $A^{(n)}_i$
applied to the same initial braid word lead to equivalent
final words (e.g. $A^{(2)}_1$ and $A^{(2)}_2$) the resulting
loop exchanges and, thus their permutation descriptions, can
be quite different ($(^{1234}_{3412})$ and
$(^{1234}_{1243})$ for the example chosen). Consequently,
compositions of braid operations returning the braid ${\bar
B}_{2^n}$ to its initial state (e.g. $A^{(2)}_2\circ
A^{(2)}_2$) may induce nontrivial translations of $s_j$. To
demonstrate this let $({\cal A}^{(2)}_2)^2$ act on the trial
state $s_1 = (3241)$. Applying the rules one obtains
\begin{equation}
({\cal A}^{(2)}_2)^2 \: s_1=
\left(^{{\displaystyle 1234}}_{{\displaystyle 2134}}\right)
\left(^{{\displaystyle 1234}}_{{\displaystyle 1243}}\right)
(3241)=(4132)={\cal T}_{+2}\:s_1,
\end{equation}
thus relating the simultaneous loop exchange
$(13)\leftrightarrow (24)$ with translation of the period-4
oscillation by half of a period. This implies as well that
application of ${\cal A}^{(2)}_2$ four times results in the
identity string transformation $ ({\cal A}^{(2)}_2)^4={\cal
T}_{+4}={\bf 1}$ and, therefore, $({\cal
A}^{(2)}_2)^{-1}=({\cal A}^{(2)}_2)^3$. Compositions of
braid operators $A^{(2)}_1$ and $A^{(2)}_2$ provide another
example of how identity braid operators induce nontrivial
string transformations. Since both operators and their
inverses alternate braid words $\Sigma_1 \leftrightarrow
\Sigma_2$ the application of the composition of any two of
them returns the braid to the same $\pi^{(2)}_i$ permutation
class and, thus, the resulting string transformation is
equivalent to some translation. The relations for the
compositions of the ${\cal A}^{(2)}_1$ and ${\cal
A}^{(2)}_2$ operators can be obtained directly from their
symbolic representations :
\begin{eqnarray}
\label{exch}
{\cal A}^{(2)}_1 &\circ& {\cal A}^{(2)}_2 =
\left(^{{\displaystyle 1234}}_{{\displaystyle 1243}}\right)
\left(^{{\displaystyle 1234}}_{{\displaystyle 3412}}\right) =
{\cal A}^{(2)}_2\circ {\cal A}^{(2)}_1
\nonumber \\ & &
=
\left(^{{\displaystyle 1234}}_{{\displaystyle 3412}}\right)
\left(^{{\displaystyle 1234}}_{{\displaystyle 2134}}\right) =
\left(^{{\displaystyle 1234}}_{{\displaystyle 3421}}\right)
= \pi^{(2)}_1 = {\cal T}_{+1} \; ,
\\
{\cal A}^{(2)}_1 &\circ& ({\cal A}^{(2)}_2)^{-1} =
\left(^{{\displaystyle 1234}}_{{\displaystyle 2134}}\right)
\left(^{{\displaystyle 1234}}_{{\displaystyle 3412}}\right)=
({\cal A}^{(2)}_2)^{-1}\circ {\cal A}^{(2)}_1
\nonumber \\ & &
= \left(^{{\displaystyle 1234}}_{{\displaystyle 3412}}\right)
\left(^{{\displaystyle 1234}}_{{\displaystyle 1243}}\right)=
\left(^{{\displaystyle 1234}}_{{\displaystyle 4312}}\right)=
\pi^{(2)}_2 = {\cal T}_{-1} \; .
\nonumber
\end{eqnarray}
These relations are constructed using the assumption that
the initial state of the braid is $\Sigma_1$. Although
application to an alternative initial condition changes
actual permutation representations of the exchange operators
it yields algebraically equivalent results. ~From
(\ref{exch}) one sees that all the exchange operators
commute and their compositions provide operators which
translate the oscillation by all the allowed multiples of
$T_4/4$.
\subsection{Loop exchanges for $P_{2^n}$ attractors}
A generalization of the phenomena discussed above to
arbitrary $n$ may be inferred from the observation of the
structural organization of the closed braids ${\bar
B_{2^n}}$ corresponding to period-doubled attractors
$P_{2^n}$. Indeed, ${\bar B_{2^{n+1}}}$ can be obtained from
${\bar B_{2^n}}$ by doubling each thread of ${\bar B_{2^n}}$
and adding a single crossing on top to preserve simple
connectivity of the construction. The braid ${\bar B_{2^n}}$
arising as a result of $n$ successive iterations of this
procedure can be subdivided into $n$ non-overlapping
structurally similar blocks of braids $\Sigma^{(n)}_m, \;
m=\overline{1,n}$. This principle of structural organization
is illustrated in Fig.~\ref{p-8} representing $B_8$ and its
three crossing blocks shown in a series of boxes with
decreasing sizes. The analysis shows that these blocks can
be moved as whole entities along ${\bar B_{2^n}}$ without
interference from each other resulting in the exchange of
those loops along which they move. The essential parts of
these moves can be represented by a set $A^{(n)}_m$ of
$2\pi$ movements of structural blocks $\Sigma^{(n)}_m$ so
that $A^{(n)}_1$ corresponds to the largest block and
results in an exchange involving all the $2^n$ loops,
$A^{(n)}_2$ corresponds to movement of next-smaller braid
block and results in the exchange of $2^{n-1}$ loops, and so
on. The transformations of time trajectories resulting from
exchanges of loops can be again described by the action of
permutation operators ${\cal A}^{(n)}_m$ on symbolic strings
$s_j$. The fact that the crossing blocks move independently
results in the commutivity of operators ${\cal A}^{(n)}_m$
with each other. The geometry of ${\bar B_{2^n}}$ also
defines the basic algebraic property of ${\cal A}^{(n)}_m$
demonstrated above for the $n=1,2$ examples
\begin{equation}
({\cal A}^{(n)}_m)^{2^m} = {\bf 1}, \; m\in[1,n] .
\end{equation}
Some compositions of the exchange operators yield
translation operators ${\cal T}_{l}$ where
$l\in[-2^{n-1},2^{n-1})$. For the discussion of the
phenomena described in Sec.~\ref{local} only the operator
${\cal T}_{+1}$ and its inverse are of particular interest.
\begin{figure}[htbp]
\begin{center}
\leavevmode
\epsffile{f18.eps}
\end{center}
\caption{Braid $B_8$ constructed for the $P_8$ attractor. }
\label{p-8}
\end{figure}
Using induction from the analysis of cases with small $n$
one may infer the general expression for the ${\cal T}_{+1}$
translation operator :
\begin{equation}
\label{prod}
{\cal T}_{+1} = \prod_{m=1}^n {\cal A}^{(n)}_m \;.
\end{equation}
\end{appendix}
| 2024-02-18T23:39:43.058Z | 1996-09-09T22:09:39.000Z | algebraic_stack_train_0000 | 179 | 11,992 |
|
proofpile-arXiv_065-1195 | \section{Introduction}
Recently, experiments at the Tevatron collider at Fermilab and
the Large Electron Positron (LEP) collider at CERN have greatly
extended our understanding of the Standard Model (SM).
Several LEP energy upgrades
have been staged on a migration towards LEP190, which will be
replaced by the Large Hadron Collider (LHC).
Prior to the operation of the LHC, the Tevatron collider will begin the
Main Injector era, possibly with an extended running time,
perhaps even at a higher luminosity than originally planned.
Despite its phenomenological
successes, the SM suffers from various theoretical problems which
make it seem unlikely to be a complete theory.
The rich physics program outline above is expected to definitively probe
the energy regime responsible for Electroweak Symmetry Breaking (EWSB).
One promising extension of the SM, which successfully addresses
several of its deficiencies, is the
the Minimal Supersymmetric Standard Model (MSSM) \cite{baer}.
Monte Carlo simulations are useful tools for studying
the phenomenological implications of a new theory, particularly
for determining search strategies and optimizing detector design.
{\tt SPYTHIA} is an event level Monte Carlo program which simulates
particle production and decay at lepton and hadron colliders in
the MSSM. It is an extension
of {\tt PYTHIA 5.7} \cite{pyt57}, with all of its previous capabilities,
and additional particles, hard scattering processes, and decays.
Furthermore, the simulation of the MSSM Higgs sector already
present in {\tt PYTHIA} is extended to include the decay of Higgs bosons
into MSSM particles and the decay of MSSM particles into Higgs boson.
The are already a few similar programs available.
Most of the processes included in {\tt SPYTHIA}
are already present in the {\tt ISAJET/ISASUSY} \cite{isajet}
event generator. In addition, a number of processes and decays relevant for
lepton colliders are included in the generator {\tt
SUSYGEN} \cite{susygen}, which
is interfaced to {\tt JETSET} \cite{jetset} and includes initial
state photon
radiation. The development of {\tt SPYTHIA} arose from the desire
to study the phenomenology of the MSSM at lepton--lepton,
lepton--hadron, and hadron--hadron colliders using the
initial and final state radiation and fragmentation models of
{\tt PYTHIA/JETSET}.
Additionally, one can perform cross checks and estimate model
uncertainties with other generators.
Like the SM, the MSSM contains a number of parameters with the
dimension of mass which are not fixed by the theory. Supergravity
(SUGRA) inspired models reduce the number of free parameters by
imposing universality and exploiting the apparent unification of
gauge couplings. Five parameters fixed at the gauge coupling
unification scale,
$\tan\beta, M_0, m_{1/2}, A_0,$ and $sign(\mu)$, are then related
to the mass parameters at the scale of EWSB by renormalization
group equations \cite{pierce}. {\tt ISASUSY} and {\tt SUSYGEN}
numerically solve these equations to determine the mass
parameters. Alternatively, they can input a general set of
parameters. {\tt SPYTHIA} operates in the second manner,
with a slightly more general set of input parameters. There are
three reasons for this: (1) programs already exist to calculate
the SUGRA inspired mass parameters, (2) approximate analytic
formulae \cite{drees} also exist which reproduce the output of
{\tt ISASUSY} within $\simeq 10\%$, and (3) we desire to study a
much richer phenomenology than that possible in SUGRA inspired
models. The {\tt SPYTHIA} input parameters are described in detail
later.
In the following, it is assumed that the reader has a working knowledge of
{\tt PYTHIA/JETSET}\footnote{A postscript version of the user manual
for {\tt PYTHIA 5.7/JETSET 7.4}
is available at the {\tt URL} address
http://thep.lu.se/tf2/staff/torbjorn/manual.ps.}.
The modifications made in {\tt SPYTHIA} are the sole product
and responsibility of this author and not H.U.~Bengtsson or
T.~Sj\"ostrand.
However, these modifications will be included in the future release
of {\tt PYTHIA 6.1}.
Sec. 2 explains the physics assumptions behind the implementation of
supersymmetry in {\tt PYTHIA} and catalogs the new particles,
processes and input parameters.
The parameters and routines used in the MSSM simulation are described in Sec. 3.
Availability and setup of the {\tt SPYTHIA} package are
detailed in Sec.~4, along with some examples of its use.
Conclusions are presented in Sec.~5.
\section{Simulation of Supersymmetry}
\subsection{Particle Spectrum}
{\tt SPYTHIA} assumes the particle content of the MSSM.
Each SM fermion has a scalar partner with the same quantum
numbers. For counting purposes, the fermion states
$\psi_L$ and $\psi_R$ are separate particles
with scalar partners $\phi_L$ and $\phi_R$. Each SM gauge boson has
a fermion partner with the same quantum numbers. The Higgs sector is
extended to two complex scalar doublets, leading to 3 additional
physical Higgs bosons ${\rm H,A}$ and ${\rm H}^\pm$ to complement the SM
Higgs ${\rm h}$.
Fermion higgsino partners are added for the two scalar doublets.
Additionally, a light gravitino $\tilde{\rm G}$ is included to allow
studies of gauge mediated dynamical SUSY breaking \cite{dine}.
The particle partners and {\tt SPYTHIA KF} codes are listed in
Table~1. Note that antiparticles of scalar particles are denoted
by $^*$ in the text.
Because of the large Yukawa
couplings and the running masses for colored particles, the interaction and
mass eigenstates for the third generation sfermions can be
significantly mixed. We denote the top and bottom squark mass
eigenstates $\tilde{\rm t}_1$,$\tilde{\rm t}_2$,$\tilde{\rm b}_1$, and $\tilde{\rm b}_2$ to distinguish them from
the nearly degenerate eigenstates for the lighter squarks. In addition,
the tau slepton mass eigenstates are $\tilde\tau_1$ and $\tilde\tau_2$.
The subscript $1$ or $2$ refers to the lightest and heaviest state
respectively. In SUGRA inspired models, in the absence of mixing
so that interaction eigenstates are the same as mass eigenstates, the
right eigenstate is lighter than the left. In this case, for example,
$\tilde{\rm b}_1 = \tilde{\rm b}_R$ and $\tilde{\rm b}_2 = \tilde{\rm b}_L$.
For completeness, we include the sterile $\tilde{\nu}_R$ particles.
The mixing of the partners to
the electroweak gauge bosons (gauginos) and the two Higgs doublets
(higgsinos) lead to the mass
eigenstates for neutralino $\tilde{\rm N}_i$ and
chargino $\tilde{\rm C}_i$ particles.
A complete listing of the full particle spectrum,
including decay channels, branching ratios, and the total width
is available (as usual) through the subroutine {\tt LULIST}.
\begin{table}[pt]
\caption{{\tt SPYTHIA} particle {\tt KF} codes.}
\begin{center}
\begin{tabular}{|c|c|c||c|c|c|}
\hline
{\tt KF} & Name & Printed & {\tt KF} & Name & Printed \\
\hline
41 & $\tilde{\rm d}_L$ & \~{}{\tt d\_L} & 59 & $\tilde{\nu}_{\mu}{}_L$ & \~{}{\tt numu\_L} \\
42 & $\tilde{\rm d}_R$ & \~{}{\tt d\_R} & 60 & $\tilde{\nu}_{\mu}{}_R$ & \~{}{\tt numu\_R} \\
43 & $\tilde{\rm u}_L$ & \~{}{\tt u\_L} & 61 & $\tilde{\tau}{}_1$ & \~{}{\tt tau\_1} \\
44 & $\tilde{\rm u}_R$ & \~{}{\tt u\_R} & 62 & $\tilde{\tau}{}_2$ & \~{}{\tt tau\_2} \\
45 & $\tilde{\rm s}_1$ & \~{}{\tt s\_L} & 63 & $\tilde{\nu}_{\tau}{}_L$ & \~{}{\tt nutau\_L} \\
46 & $\tilde{\rm s}_2$ & \~{}{\tt s\_R} & 64 & $\tilde{\nu}_{\tau}{}_R$ & \~{}{\tt nutau\_R} \\
47 & $\tilde{\rm c}_L$ & \~{}{\tt c\_L} & 65 & $\tilde{\rm g}$ & \~{}{\tt g} \\
48 & $\tilde{\rm c}_R$ & \~{}{\tt c\_R} & 66 & $\tilde{\rm N}_1$ & \~{}{\tt {N}$_1$} \\
49 & $\tilde{\rm b}_1$ & \~{}{\tt b\_1} & 67 & $\tilde{\rm N}_2$ & \~{}{\tt {N}$_2$} \\
50 & $\tilde{\rm b}_2$ & \~{}{\tt b\_2} & 68 & $\tilde{\rm N}_3$ & \~{}{\tt {N}$_3$} \\
51 & $\tilde{\rm t}_1$ & \~{}{\tt t\_1} & 69 & $\tilde{\rm N}_4$ & \~{}{\tt {N}$_4$} \\
52 & $\tilde{\rm t}_2$ & \~{}{\tt t\_2} & 70 & $\tilde{\rm C}^\pm_1$ & \~{}{\tt C$_1$} \\
53 & $\tilde{\rm e}_L$ & \~{}{\tt e\_L} & 71 & $\tilde{\rm C}^\pm_2$ & \~{}{\tt C$_2$} \\
54 & $\tilde{\rm e}_R$ & \~{}{\tt e\_R} & & & \\
55 & $\tilde{\nu}_{e}{}_L$ & \~{}{\tt nue\_L} & & & \\
56 & $\tilde{\nu}_{e}{}_R$ & \~{}{\tt nue\_R} & & & \\
57 & $\tilde{\mu}{}_L$ & \~{}{\tt mu\_L} & & & \\
58 & $\tilde{\mu}{}_R$ & \~{}{\tt mu\_R} & 76 & $\tilde{\rm G}$ & \~{}{\tt grvtno} \\
\hline
\end{tabular}
\end{center}
\end{table}
\subsection{Particle Decays}
In {\tt PYTHIA}, resonances are decayed so
that color flow information is correctly passed from unstable particles
to their decay products and no unstable colored
particles are passed on to {\tt JETSET}.
In {\tt SPYTHIA}, all Supersymmetric partners are
treated as resonances.
{\tt SPYTHIA} assumes R--symmetry, which has two major phenomenological
consequences:
(1) Supersymmetric particles are produced in pairs,
and (2) there is one and only one stable Supersymmetric particle.
By default, the lightest
superpartner (LSP) is $\tilde{\rm N}_1$, unless $\tilde{\rm N}_1$ is allowed to decay to
a gravitino. In that case, $\tilde{\rm N}_1$ is the next to lightest
superpartner (NLSP) and $\tilde{\rm G}$ is the LSP.
The decays of superpartners are calculated using the formulae of
Refs.~\cite{gunion1,bartl1,bartl2,bartl3}.
All decays are spin averaged.
For simplicity, the kinematics of
three body decays of neutralinos, charginos, and the gluino are sampled
using only the phase space weight. Decays involving $\tilde{\rm b}$ and $\tilde{\rm t}$ use the
formulae of \cite{bartl3}, so they are valid for large values of
$\tan\beta$. The one loop decays $\tilde{\rm N}_j\to\tilde{\rm N}_i\gamma$ and
$\tilde{\rm t}\to c\tilde{\rm N}_1$ are also included.
\subsection{Processes}
\indent Tables~2--5 are meant to update
Tables~11--14 in the {\tt Pythia 5.7}
manual. The number of available processes in {\tt SPYTHIA}
is extended to 400, with processes 201--280 reserved for
the MSSM. In processes 210 and 213, $\tilde{\ell}$ refers to both $\tilde{\rm e}$ and
$\tilde{\mu}$. For ease of readability, we have removed the subscript $L$ on $\tilde{\nu}$.
$\tilde{\rm t}_i\tilde{\rm t}^*_i, \tilde{\tau}_i\tilde{\tau}_j^*$ and $\tilde{\tau}_i\tilde{\nu}_\tau^*$
production correctly account for sfermion mixing. Several processes
are conspicuously absent from the table. For example, processes
{\tt 255} and {\tt 257} would simulate the associated production
of right handed squarks with charginos. Since the right handed squark
only couples to the higgsino component of the chargino, the interaction
strength is proportional to the quark mass, so these processes can
be ignored.
\begin{table}[hpt]
\caption{Subprocess codes, part 1. First column is `+' for processes
implemented and blank for those that are only foreseen. Second is
the subprocess number {\tt ISUB}, and third the description of the
process. The final column gives references from which the
cross sections have been obtained.}
\begin{center}
\begin{tabular}{|c|r|l|l|}
\hline
In & No. & Subprocess & Reference \\
\hline
+ & 201 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm e}_L \tilde{\rm e}_L^*$ & \cite{bartl4,dawson1} \\
+ & 202 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm e}_R \tilde{\rm e}_R^*$ & \cite{bartl4,dawson1} \\
+ & 203 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm e}_L \tilde{\rm e}_R^*+\tilde{\rm e}_L^* \tilde{\rm e}_R$ &
\cite{bartl4} \\
+ & 204 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\mu}_L \tilde{\mu}_L^*$ & \cite{bartl4,dawson1} \\
+ & 205 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\mu}_R \tilde{\mu}_R^*$ & \cite{bartl4,dawson1} \\
+ & 206 & ${\rm f}_i \overline{\rm f}_i\to\tilde{\mu}_L \tilde{\mu}_R^*+\tilde{\mu}_L^* \tilde{\mu}_R$ &
\cite{bartl4} \\
+ & 207 & ${\rm f}_i \overline{\rm f}_i\to\tilde{\tau}_1 \tilde{\tau}_1^*$ & \cite{bartl4,dawson1} \\
+ & 208 & ${\rm f}_i \overline{\rm f}_i\to\tilde{\tau}_2 \tilde{\tau}_2^*$ & \cite{bartl4,dawson1} \\
+ & 209 & ${\rm f}_i \overline{\rm f}_i\to\tilde{\tau}_1
\tilde{\tau}_2^*+\tilde{\tau}_1^*\tilde{\tau}_2$&\cite{bartl4} \\
+ & 210 & ${\rm f}_i \overline{\rm f}_j\to \tilde{\ell}_L {\tilde{\nu}}_\ell^*+
\tilde{\ell}_L^* \tilde{\nu}_\ell$&\cite{dawson1} \\
+ & 211 & ${\rm f}_i \overline{\rm f}_j\to \tilde{\tau}_1
\tilde{\nu}_\tau^*+\tilde{\tau}_1^*\tilde{\nu}_\tau$ & \cite{dawson1} \\
+ & 212 & ${\rm f}_i \overline{\rm f}_j\to \tilde{\tau}_2
\tilde{\nu}_\tau{}^*+\tilde{\tau}_2^*\tilde{\nu}_\tau$
& \cite{dawson1} \\
+ & 213 & ${\rm f}_i \overline{\rm f}_i\to \tilde{\nu_\ell} \tilde{\nu_\ell}^*$ &
\cite{bartl4,dawson1} \\
+ & 214 & ${\rm f}_i \overline{\rm f}_i\to \tilde{\nu}_{\tau} \tilde{\nu}_{\tau}^*$
& \cite{bartl4,dawson1} \\
+ & 216 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_1 \tilde{\rm N}_1$ & \cite{bartl1} \\
+ & 217 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_2 \tilde{\rm N}_2$ & \cite{bartl1} \\
+ & 218 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_3 \tilde{\rm N}_3$ & \cite{bartl1} \\
+ & 219 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_4 \tilde{\rm N}_4$ & \cite{bartl1} \\
+ & 220 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_1 \tilde{\rm N}_2$ & \cite{bartl1} \\
+ & 221 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_1 \tilde{\rm N}_3$ & \cite{bartl1} \\
+ & 222 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_1 \tilde{\rm N}_4$ & \cite{bartl1} \\
+ & 223 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_2 \tilde{\rm N}_3$ & \cite{bartl1} \\
+ & 224 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_2 \tilde{\rm N}_4$ & \cite{bartl1} \\
+ & 225 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm N}_3 \tilde{\rm N}_4$ & \cite{bartl1} \\
+ & 226 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm C}^\pm_1 \tilde{\rm C}^\mp_1$ & \cite{bartl2} \\
+ & 227 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm C}^\pm_2 \tilde{\rm C}^\mp_2$ & \cite{bartl2} \\
+ & 228 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm C}^\pm_1 \tilde{\rm C}^\mp_2$ & \cite{bartl2} \\
+ & 229 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_1 \tilde{\rm C}^\pm_1$ & \cite{bartl1,bartl2} \\
+ & 230 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_2 \tilde{\rm C}^\pm_1$ & \cite{bartl1,bartl2} \\
+ & 231 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_3 \tilde{\rm C}^\pm_1$ & \cite{bartl1,bartl2} \\
+ & 232 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_4 \tilde{\rm C}^\pm_1$ & \cite{bartl1,bartl2} \\
+ & 233 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_1 \tilde{\rm C}^\pm_2$ & \cite{bartl1,bartl2} \\
+ & 234 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_2 \tilde{\rm C}^\pm_2$ & \cite{bartl1,bartl2} \\
+ & 235 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_3 \tilde{\rm C}^\pm_2$ & \cite{bartl1,bartl2} \\
+ & 236 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm N}_4 \tilde{\rm C}^\pm_2$ & \cite{bartl1,bartl2} \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{table}[hpt]
\caption{Subprocess codes, part 2. First column is `+' for processes
implemented and blank for those that are only foreseen. Second is
the subprocess number {\tt ISUB}, and third the description of the
process. The final column gives references from which the
cross sections have been obtained.}
\begin{center}
\begin{tabular}{|c|r|l|l|}
\hline
In & No. & Subprocess & Reference \\
\hline
+ & 237 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm g} \tilde{\rm N}_1$ & \cite{dawson1} \\
+ & 238 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm g} \tilde{\rm N}_2$ & \cite{dawson1} \\
+ & 239 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm g} \tilde{\rm N}_3$ & \cite{dawson1} \\
+ & 240 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm g} \tilde{\rm N}_4$ & \cite{dawson1} \\
+ & 241 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm g} \tilde{\rm C}^\pm_1$ & \cite{dawson1} \\
+ & 242 & ${\rm f}_i \overline{\rm f}_j \to \tilde{\rm g} \tilde{\rm C}^\pm_2$ & \cite{dawson1} \\
+ & 243 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm g} \tilde{\rm g}$ & \cite{dawson1} \\
+ & 244 & ${\rm g} {\rm g} \to \tilde{\rm g} \tilde{\rm g}$ & \cite{dawson1} \\
+ & 246 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_L \tilde{\rm N}_1$ & \cite{dawson1} \\
+ & 247 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_R \tilde{\rm N}_1$ & \cite{dawson1} \\
+ & 248 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_L \tilde{\rm N}_2$ & \cite{dawson1} \\
+ & 249 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_R \tilde{\rm N}_2$ & \cite{dawson1} \\
+ & 250 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_L \tilde{\rm N}_3$ & \cite{dawson1} \\
+ & 251 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_R \tilde{\rm N}_3$ & \cite{dawson1} \\
+ & 252 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_L \tilde{\rm N}_4$ & \cite{dawson1} \\
+ & 253 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_R \tilde{\rm N}_4$ & \cite{dawson1} \\
+ & 254 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_j}{}_L \tilde{\rm C}^\pm_1$ & \cite{dawson1} \\
+ & 256 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_j}{}_L \tilde{\rm C}^\pm_2$ & \cite{dawson1} \\
+ & 258 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_L \tilde{\rm g}$ & \cite{dawson1}\\
+ & 259 & ${\rm f}_i {\rm g} \to {\tilde{\rm q}_i}{}_R \tilde{\rm g}$ & \cite{dawson1}\\
+ & 261 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm t}_1 \tilde{\rm t}^*_1$ & \cite{dawson1} \\
+ & 262 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm t}_2 \tilde{\rm t}^*_2$ & \cite{dawson1} \\
+ & 263 & ${\rm f}_i \overline{\rm f}_i \to \tilde{\rm t}_1 \tilde{\rm t}^*_2+\tilde{\rm t}^*_1 \tilde{\rm t}_2$ & \cite{dawson1} \\
+ & 264 & ${\rm g} {\rm g} \to \tilde{\rm t}_1 \tilde{\rm t}^*_1$ & \cite{dawson1} \\
+ & 265 & ${\rm g} {\rm g} \to \tilde{\rm t}_2 \tilde{\rm t}^*_2$ & \cite{dawson1} \\
+ & 271 & ${\rm f}_i {\rm f}_j \to {\tilde{\rm q}_i}{}_L {\tilde{\rm q}_j}{}_L$ & \cite{dawson1} \\
+ & 272 & ${\rm f}_i {\rm f}_j \to {\tilde{\rm q}_i}{}_R {\tilde{\rm q}_j}{}_R$ & \cite{dawson1} \\
+ & 273 & ${\rm f}_i {\rm f}_j \to {\tilde{\rm q}_i}{}_L {\tilde{\rm q}_j}{}_R+
{\tilde{\rm q}_i}{}_R {\tilde{\rm q}_j}{}_L$ & \cite{dawson1} \\
+ & 274 & ${\rm f}_i \overline{\rm f}_j \to {\tilde{\rm q}_i}{}_L {\tilde{\rm q}^*_j}{}_L$ & \cite{dawson1} \\
+ & 275 & ${\rm f}_i \overline{\rm f}_j \to {\tilde{\rm q}_i}{}_R {\tilde{\rm q}^*_j}{}_R$ & \cite{dawson1} \\
+ & 276 & ${\rm f}_i \overline{\rm f}_j \to {\tilde{\rm q}_i}{}_L {\tilde{\rm q}^*_j}{}_R+
{\tilde{\rm q}_i}{}_R {\tilde{\rm q}^*_j}{}_L$ & \cite{dawson1} \\
+ & 277 & ${\rm f}_i \overline{\rm f}_i \to {\tilde{\rm q}_j}{}_L {\tilde{\rm q}^*_j}{}_L$ & \cite{dawson1} \\
+ & 278 & ${\rm f}_i \overline{\rm f}_i \to {\tilde{\rm q}_j}{}_R {\tilde{\rm q}^*_j}{}_R$ & \cite{dawson1} \\
+ & 279 & ${\rm g} {\rm g} \to {\tilde{\rm q}_i}{}_L {\tilde{\rm q}^*_i}{}_L$ & \cite{dawson1} \\
+ & 280 & ${\rm g} {\rm g} \to {\tilde{\rm q}_i}{}_R {\tilde{\rm q}^*_i}{}_R$ & \cite{dawson1} \\
\hline
\end{tabular}
\end{center}
\end{table}
Because there are so many processes involved, there are shortcuts
to allow the simulation of various classes of signals. These classes are
accessible through the parameter {\tt MSEL}, and are listed in Table~6.
\begin{table}[hpt]
\caption{Classes of processes accessible through the parameter {\tt MSEL}
and the individual processes codes {\tt ISUB}.}
\begin{center}
\begin{tabular}{|c|l|}
\hline
{\tt MSEL} & Description \\
\hline
39 & All MSSM processes except Higgs production \\
40 & Squark and gluino production, {\tt ISUB} = 243, 244, 258, 259, 271--280 \\
41 & Stop pair production, {\tt ISUB} = 261--265 \\
42 & Slepton pair production, {\tt ISUB} = 201--214 \\
43 & Squark or gluino with chargino or neutralino, {\tt ISUB} = 237--242, 246--256 \\
44 & Chargino--neutralino pair production, {\tt ISUB} = 216--236 \\
\hline
\end{tabular}
\end{center}
\end{table}
\vskip 10cm
\section{The Parameters and Routines of the MSSM Simulation}
All of the Supersymmetric extensions of the code are included
in the standard {\tt PYTHIA} library.
By default, Supersymmetry is not simulated.
However, by setting various parameters, a rich MSSM phenomenology is
available.
The parameters available to the user are stored in the
{\tt FORTRAN} common block \\
{\tt COMMON/PYMSSM/IMSS(0:99),RMSS(0:99)}.
In general, options are set by the {\tt IMSS} array, while
real valued parameters are set by {\tt RMSS}. The entries
{\tt IMSS(0)} and {\tt RMSS(0)} are not used, but are available
for compatibility with the {\tt C} programming language.
The arrays are described below. The default values are denoted
by (D). \\
\subsection{The MSSM Parameters}
\drawbox{COMMON/PYMSSM/IMSS(0:99),RMSS(0:99)}%
\begin{entry}
\itemc{Purpose:} to give access to parameters that allow the
simulation of the MSSM.
\iteme{IMSS(1) :} (D=0) level of MSSM simulation.
\begin{subentry}
\iteme{= 0 :} No MSSM simulation.
\iteme{= 1 :} A general MSSM simulation. The parameters of the model
are set by the array {\tt RMSS}.
\iteme{= 2 :} An approximate SUGRA simulation using the analytic
formulate of \cite{drees} to reduce the number of free parameters.
In this case, only five input parameters are used.
{\tt RMSS}{\tt(1)} is the common gaugino mass
$m_{1/2}$, {\tt RMSS}{\tt(8)} is the common scalar mass $m_0$,
{\tt RMSS}{\tt(4)} fixes the sign of the higgsino mass $\mu$,
{\tt RMSS}{\tt(16)} is the common trilinear coupling $A$, and
{\tt RMSS}{\tt(5)} is $\tan\beta=v_2/v_1$.
\end{subentry}
\iteme{IMSS(2) :} (D=0) treatment of $U(1), SU(2),$ and $SU(3)$ gaugino
mass parameters.
\begin{subentry}
\iteme{= 0 :} The gaugino parameters $M_1, M_2$ and $M_3$ are
set by {\tt RMSS(1), RMSS(2),} and {\tt RMSS(3)}, i.e. there is
no forced relation between them.
\iteme{= 1 :} The gaugino parameters are fixed by the relation
$M_1/\alpha_1 3/5=M_2/\alpha_2=M_3/\alpha_3=X$ and the
parameter {\tt RMSS(1)}. If {\tt IMSS(1)=2}, then
{\tt RMSS(1)} is treated as the common gaugino mass $m_{1/2}$
and {\tt RMSS(20)} is the GUT scale coupling constant
$\alpha_{GUT}$, so that $X=m_{1/2}/\alpha_{GUT}$.
\iteme{= 2 :} $M_1$ is set by {\tt RMSS(1)}, $M_2$ by {\tt RMSS(2)}
and $M_3 = M_2\alpha_3/\alpha_2$. In such a scenario,
the U(1) gaugino mass behaves anomalously.
\end{subentry}
\iteme{IMSS(3) :} (D=0) treatment of the gluino mass parameter.
\begin{subentry}
\iteme{= 0 :} The gluino mass parameter $M_3$ is used to calculate
the gluino pole mass with the formulae of \cite{martin1}. The effects of
squark loops can significantly shift the mass.
\iteme{= 1 :} $M_3$ is the gluino pole mass. The effects of squark
loops are assumed to have been included in this value.
\end{subentry}
\iteme{IMSS(4) :} (D=1) treatment of the Higgs sector.
\begin{subentry}
\iteme{= 0 :} The Higgs sector is determined by
the approximate formulae of \cite{carena} and the pseudoscalar mass $M_A$
set by {\tt RMSS(19)}.
\iteme{= 1 :} The Higgs sector is determined by the exact
formulae of \cite{carena} and the pseudoscalar mass $M_{\rm A}$ set by
{\tt RMSS(19)}. The pole mass for $M_{\rm A}$ is not the same as the input
parameter .
\iteme{= 2 :} The Higgs sector is fixed by the mixing angle $\alpha$
set by {\tt RMSS(18)} and the mass values {\tt PMAS(I,1)}, where
{\tt I=25,35,36,} and {\tt 37}.
\end{subentry}
\iteme{IMSS(7) :} (D=0) treatment of the scalar masses in an
extension of SUGRA models. The presence of additional $U(1)$
symmetries at high energy scales can modify the boundary
conditions for the scalar masses at the unification scale.
\begin{subentry}
\iteme{= 0 :} No additional $D$--terms are included. In
SUGRA models, all scalars have the mass $m_0$ at the
unification scale.
\iteme{= 1 :} {\tt RMSS}{\tt(23--25)} are the values of $D_X, D_Y$ and
$D_S$ at the unification scale in the model of \cite{martin2}.
The boundary conditions for the scalar masses are shifted based
on their quantum numbers under the additional $U(1)$ symmetries.
\end{subentry}
\iteme{IMSS(8) :} (D=1) treatment of the $\tilde{\tau}$ mass eigenstates.
\begin{subentry}
\iteme{= 0 :} The $\tilde{\tau}$ mass eigenstates are calculated
using the parameters {\tt RMSS(13,14,17)}.
\iteme{= 1 :} The $\tilde{\tau}$ mass eigenstates are identical to
the interaction eigenstates, so they are treated
identically to $\tilde{\rm e}$ and $\tilde{\mu}$ .
\end{subentry}
\iteme{IMSS(9) :} (D=0) treatment of the right handed squark mass
eigenstates for the first two generations.
\begin{subentry}
\iteme{= 0 :} The $\tilde{\rm q}_R$ masses are fixed by {\tt RMSS}{\tt(9)}. $\tilde{\rm d}_R$ and
$\tilde{\rm u}_R$ are identical except for Electroweak $D$--term contributions.
\iteme{= 1 :} The masses of $\tilde{\rm d}_R$ and $\tilde{\rm u}_R$
are fixed by {\tt RMSS(9)} and {\tt RMSS(22)} respectively.
\end{subentry}
\iteme{IMSS(10) :} (D=0) allowed decays for $\tilde{\rm N}_2$.
\begin{subentry}
\iteme{= 0 :} The second lightest neutralino $\tilde{\rm N}_2$ decays
with a branching ratio calculated from the MSSM parameters.
\iteme{= 1 :} $\tilde{\rm N}_2$ is forced to decay only to $\tilde{\rm N}_1 \gamma$,
regardless of the actual branching ratio.
This can be used for detailed studies of this particular final state.
\end{subentry}
\iteme{IMSS(11) :} (D=0) choice of the lightest superpartner (LSP).
\begin{subentry}
\iteme{= 0 :} $\tilde{\rm N}_1$ is the LSP.
\iteme{= 1 :} $\tilde{\rm N}_1$ is the next to lightest superparter (NLSP)
and the gravitino is the LSP. The gravitino
decay length is calculated from the gravitino
mass set by {\tt RMSS(21)} and the $\tilde{\rm N}_1$ mass and mixings.
\end{subentry}
\iteme{RMSS(1) :} If {\tt IMSS}{\tt(1)=1} $M_1$, the U(1) gaugino mass.
If {\tt IMSS}{\tt(1)=2}, then the common gaugino mass $m_{1/2}$.
\iteme{RMSS(2) :} $M_2$, the SU(2) gaugino mass.
\iteme{RMSS(3) :} $M_3$, the SU(3) (gluino) mass parameter.
\iteme{RMSS(4) :} $\mu$, the higgsino mass parameter.
If {\tt IMSS(1)=2}, only the sign of $\mu$ is used.
\iteme{RMSS(5) :} $\tan\beta$, the ratio of Higgs expectation
values.
\iteme{RMSS(6) :} Left slepton mass $M_{\tilde \ell_L}$.
The sneutrino mass is fixed by a sum rule.
\iteme{RMSS(7) :} Right slepton mass $M_{\tilde \ell_R}$.
\iteme{RMSS(8) :} Left squark mass $M_{\tilde{\rm q}_L}$. If {\tt IMSS(1)=2},
the common scalar mass $m_0$.
\iteme{RMSS(9) :} Right squark mass $M_{\tilde{\rm q}_R}$. $M_{\tilde{\rm q}_R}$
when {\tt IMSS(9)=1}.
\iteme{RMSS(10) :} Left squark mass for the third
generation $M_{\tilde{\rm q}_L}$.
\iteme{RMSS(11) :} Right sbottom mass $M_{\tilde{\rm b}_R}$.
\iteme{RMSS(12) :} Right stop mass $M_{\tilde{\rm t}_R}$ If
negative, then it is assumed that $M_{\tilde{\rm t}_R}^2 < 0$.
\iteme{RMSS(13) :} Left stau mass $M_{\tilde \tau_L}$.
\iteme{RMSS(14) :} Right stau mass $M_{\tilde \tau_R}$.
\iteme{RMSS(15) :} Bottom trilinear coupling $A_{\rm b}$.
\iteme{RMSS(16) :} Top trilinear coupling $A_{\rm t}$. If {\tt IMSS(1)=2},
the common trilinear coupling $A$.
\iteme{RMSS(17) :} Tau trilinear coupling $A_\tau$.
\iteme{RMSS(18) :} Higgs mixing angle $\alpha$. This is only used when
all of the Higgs parameters are set by the user, i.e {\tt IMSS(4)=2}.
\iteme{RMSS(19) :} Pseudoscalar Higgs mass parameter $M_{\rm A}$.
\iteme{RMSS(20) :} (D=.041) GUT scale coupling constant $\alpha_{GUT}$.
\iteme{RMSS(21) :} (D=1.0) The gravitino mass (eV).
\iteme{RMSS(22) :} $\tilde{\rm u}_R$ mass when {\tt IMSS(9)=1}.
\iteme{RMSS(23) :} $D_X$ contribution to scalar masses when
{\tt IMSS(7)=1} (GeV$^2$).
\iteme{RMSS(24) :} $D_Y$ contribution to scalar masses when
{\tt IMSS(7)=1} (GeV$^2$).
\iteme{RMSS(25) :} $D_S$ contribution to scalar masses when
{\tt IMSS(7)=1} (GeV$^2$).
\end{entry}
\drawbox{COMMON/PYSSMT/ZMIX(4,4),UMIX(2,2),VMIX(2,2),SMZ(4),SMW(2),SFMIX(16,4)}%
\begin{entry}
\itemc{Purpose:} to provide information on the neutralino, chargino,
and sfermion mixing parameters. The variables should not be changed
by the user.
\iteme{ZMIX(4,4) :} the neutralino mixing matrix in the Bino--neutral
Wino--Up higgsino--Down higgsino basis.
\iteme{UMIX(2,2) :} the chargino mixing matrix in the charged Wino--charged
higgsino basis.
\iteme{VMIX(2,2) :} the charged conjugate chargino
mixing matrix in the wino--charged higgsino basis.
\iteme{SMZ(4) :} the signed masses of the neutralinos.
\iteme{SMW(2) :} the signed masses of the charginos.
\iteme{SFMIX(16,4) :} the sfermion mixing matrices $\bf T$ in the L--R basis,
identified by
the corresponding fermion, i.e. {\tt SFMIX(6,I)} is the stop mixing matrix.
The four entries for each sfermion are ${\rm T}_{11}, {\rm T}_{12},
{\rm T}_{21},$ and
${\rm T}_{22}$.
\end{entry}
\subsection{The MSSM Physics Routines}
The following subroutines and functions need not be accessed by the
user, but are described for completeness.
\begin{entry}
\iteme{SUBROUTINE PYAPPS :} uses approximate analytic formulae to
determine the full set of MSSM parameters from SUGRA inputs.
\iteme{SUBROUTINE PYGLUI :} calculates gluino decay modes.
\iteme{SUBROUTINE PYGQQB :} calculates three body decays of gluinos
into neutralinos or charginos and third generation fermions. These
routines are valid for large values of $\tan\beta$.
\iteme{SUBROUTINE PYCJDC :} calculates the chargino decay modes.
\iteme{SUBROUTINE PYHEXT :} calculates the non--Standard Model decay
modes of the Higgs bosons.
\iteme{SUBROUTINE PYHGGM :} determines the Higgs boson mass spectrum
using several inputs.
\iteme{SUBROUTINE PYINOM :} finds the mass eigenstates and mixing
matrices for the charginos and neutralinos.
\iteme{SUBROUTINE PYMSIN :} initializes the MSSM simulation.
\iteme{SUBROUTINE PYNJDC :} calculates neutralino decay modes.
\iteme{SUBROUTINE PYPOLE :} computes the Higgs boson masses using
a renormalization group improved leading--log approximation and
two loop leading--log corrections.
\iteme{SUBROUTINE PYRNMT :} determines the running mass of the top
quark.
\iteme{SUBROUTINE PYSFDC :} calculates sfermion decay modes.
\iteme{SUBROUTINE PYSUBH :} computes the Higgs boson masses using
only renormalization group improved formulae.
\iteme{SUBROUTINE PYTBDY :} samples the phase space for three body
decays of neutralinos, charginos, and the gluino.
\iteme{SUBROUTINE PYTHRG :} computes the masses and mixing matrices of
the third generation sfermions.
\end{entry}
\section{Setup and Use of the {\tt SPYTHIA} Generator}
The {\tt SPTYHIA} code is available as a uuencoded file on the
World Wide
Web\footnote{http://www.hep.anl.gov/theory/mrenna/spythia.hmtl}.
It operates in the same fashion as the standard {\tt PYTHIA}
distribution. See the {\tt PYTHIA} manual for a detailed
example of its usage.
Below, we present several examples of how to simulate MSSM
phenomenology with {\tt SPYTHIA}.
\subsection{Example 1: Light Stop}
The first example is an MSSM model with a light neutralino $\tilde{\rm N}_1$ and a
light stop $\tilde{\rm t}_1$, so that ${\rm t}\to \tilde{\rm t}_1\tilde{\rm N}_1$ can occur.
The input parameters are
{\tt IMSS}{\tt(1)=1}, {\tt RMSS}{\tt(1)=70.},
{\tt RMSS}{\tt(2)=70.}, {\tt RMSS}{\tt(3)=225.},
{\tt RMSS}{\tt(4)=-40.}, {\tt RMSS}{\tt(5)=1.5}, {\tt RMSS}{\tt(6)=100.},
{\tt RMSS}{\tt(7)=125.}, {\tt RMSS}{\tt(8)=250.}, \\
{\tt RMSS}{\tt(9)=250.}, {\tt RMSS}{\tt(10)=1500.},
{\tt RMSS}{\tt(11)=1500.}, {\tt RMSS}{\tt(12)=-128.},
{\tt RMSS}{\tt(13)=100.}, \\
{\tt RMSS}{\tt(14)=125.},
{\tt RMSS}{\tt(15)=800.}, {\tt RMSS}{\tt(16)=800.},
{\tt RMSS}{\tt(17)=0.}, and {\tt RMSS}{\tt(19)=400.0.}
The top mass is fixed at 175 GeV, {\tt PMAS(6,1)=175.0}.
The resulting model has $M_{\tilde{\rm t}_1}=55$ GeV and $M_{\tilde{\rm N}_1}=38$ GeV.
{\tt IMSS(1)=1} turns on the MSSM simulation.
By default, there are no intrinsic relations between the gaugino masses,
so $M_1=70$ GeV, $M_2=70$ GeV, and $M_3=225$ GeV. The pole mass of
the gluino is slightly higher than the parameter $M_3$, and the
decay $\tilde {\rm g}\to\tilde{\rm t}_1^*{\rm t}+\tilde{\rm t}_1\bar{\rm t}$
occurs almost 100\% of the time.
\subsection{Example 2: Approximate SUGRA}
The second example is an approximate SUGRA model. The input
parameters are {\tt IMSS}{\tt(1)=2}, {\tt RMSS}{\tt(1)=200.},
{\tt RMSS}{\tt(4)=1.,}
{\tt RMSS}{\tt(5)=10.},
{\tt RMSS}{\tt(8)=800.}, and {\tt RMSS}{\tt(16)=0.0}.
The resulting model has
$M_{\tilde{\rm d}_L}=901$ GeV, $M_{\tilde{\rm u}_R}=890$ GeV, $M_{\tilde{\rm t}_1}=538$ GeV,
$M_{\tilde{\rm e}_L}=814$ GeV, $M_{\tilde{\rm g}}=560$ GeV, $M_{\tilde{\rm N}_1}=80$ GeV,
$M_{\tilde{\rm C}^\pm_1}=151$ GeV, $M_{\rm h}=110$ GeV, and $M_{\rm A}=883$ GeV. It
corresponds to the choice $M_0$=800 GeV, $M_{1/2}=$200 GeV,
$\tan\beta=10$, $A_0=0$, and $sign(\mu)>0$. The output is similar
to an {\tt ISASUSY} run, but there is not exact agreement.
\subsection{Example 3: {\tt ISASUSY} Model}
The final example demonstrates how to convert the output of
an {\tt ISASUSY}\footnote{Information on the ISAJET and ISASUSY
programs can be found on the World Wide Web at
http://wwwcn1.cern.ch/asd/cernlib/mc/isajet.html.}
run using the same SUGRA inputs into the {\tt SPYTHIA} format.
This requires a general model with the parameters
{\tt IMSS}{\tt(1)=1}, {\tt IMSS}{\tt(3)=1}, {\tt RMSS}{\tt(1)=83.81},
{\tt RMSS}{\tt(2)=168.90}, {\tt RMSS}{\tt(3)=581.83}, {\tt RMSS}{\tt(4)=283.37},
{\tt RMSS}{\tt(5)=10.}, {\tt RMSS}{\tt(6)=813.63}, {\tt RMSS}{\tt(7)=804.87},
{\tt RMSS}{\tt(8)=917.73}, \\ {{\tt RMSS}{\tt(9)=909.89},
{\tt RMSS}{\tt(10)=772.87},
{\tt RMSS}{\tt(11)=901.52}, {\tt RMSS}{\tt(12)=588.33},
{\tt RMSS}{\tt(13)=813.63}, \\
{\tt RMSS}{\tt(14)=804.87}, {\tt RMSS}{\tt(15)=610.54},
{\tt RMSS}{\tt(16)=422.35},
{\tt RMSS}{\tt(17)=600.}, and \\ {\tt RMSS}{\tt(19)=858.412}.
\section{Conclusions}
{\tt SPYTHIA} simulates the physics of the MSSM using the
{\tt PYTHIA/JETSET} platform. References to the underlying
physics that is simulated have been provided, as well as an
overview of the user interface to the program. With the
appropriate choice of parameters, a wide range of MSSM
models can be studied, including those with a light
gravitino and extra $D$--term contributions to scalar masses.
\newpage
| 2024-02-18T23:39:43.855Z | 1996-09-15T21:52:27.000Z | algebraic_stack_train_0000 | 227 | 5,825 |
|
proofpile-arXiv_065-1530 | \section{Introduction and motivation}
Over the past two decades it has become clear that our universe today is in a phase of accelerated expansion driven by an unknown component---the dark energy. Although the cosmological constant provides an almost perfect fit to the observational data and is theoretically the most economic solution to this problem, its unnaturally small value has forced us in the quest of finding other, more satisfying dark energy models. A large fraction of the recent proposals rely on modifying gravity at large distances and fall in the class of the effective field theories (EFTs) of dark energy~\cite{Gubitosi:2012hu}. This framework unifies competing models of modified gravity that in a given window of energy scales involve only one additional scalar degree of freedom with respect to General Relativity (GR). Written directly for the cosmological perturbations, the EFT of dark energy can be interpreted as a low energy effective theory for the Goldstone boson of broken time translations. In the spirit it is very similar to the EFT of inflation where the scalar degree of freedom is usually the inflaton field breaking the de Sitter invariance of the early universe \cite{Cheung:2007st}. This approach captures most of the various phenomenological effects arising in the different dark energy models and allows to discriminate between them.
A typical feature of the dark energy models involving a light scalar degree of freedom is to introduce order one modifications of gravity on large scales. While it is an acceptable and seeked for effect on cosmological scales, these theories are highly constrained by short distance observations. In particular, any such modified gravity theory needs to have a working screening mechanism, such as the Vainshtein mechanism \cite{Vainshtein:1972sx,Babichev:2013usa}, in order to recover the standard GR results at short scales. An important subclass of the EFTs of dark energy implementing the Vainshtein mechanism is the Horndeski theories, originally formulated in \cite{Horndeski:1974wa} and later rediscovered in \cite{Deffayet:2009wt,Deffayet:2011gz}. These are the most general scalar--tensor theories with second order metric and scalar field equations of motion and have been extensively studied in the recent years.
A novel avenue for testing the modified gravity models has opened up due to the recent observation of the gravitational wave emission during the black hole and neutron star merger events \cite{Abbott:2016blz,TheLIGOScientific:2017qsa,Monitor:2017mdv}. The significance of these observations lies in the fact that it offers the possibility to test gravity in the previously inaccessible highly relativistic, strong-field regime. For instance, the observation of gravitational waves from the binary neutron star merger, arriving simultaneously with their optical counterpart, allowed to constrain the speed of gravitational waves to coincide with the speed of light up to deviations of order $10^{-15}$~\cite{TheLIGOScientific:2017qsa,Monitor:2017mdv}. This observation alone ruled out a wide range of operators within the EFTs for dark energy that were predicting subluminal propagation speeds for the gravity waves \cite{Creminelli:2017sry, Sakstein:2017xjx,Baker:2017hug,Ezquiaga:2017ekz}. A possible caveat of this conclusion might lie in the fact that the energy scale at which the gravitational waves were observed lies very close to the strong coupling scale of many of the dark energy models in question. Being only effective field theories they would naturally receive corrections above the strong coupling scale that could render the speed of propagation of the gravity waves to lie close to the speed of light \cite{deRham:2018red}. We shall comment more on this point below.
Another possible way of testing GR with the help of the gravitational wave observations is by studying the strong-field effects. These include, in particular, the quasi-normal mode spectrum \cite{Dong:2017toi,Tattersall:2018nve,Franciolini:2018uyq,Witek:2018dmd,Mirbabayi:2018mdm} and the violations of the strong equivalence principle leading to the emission of the dipolar radiation \cite{Berti:2015itd}.
Motivated by these advances in observational cosmology, in this work we shall focus our attention on the possible violations of the equivalence principle within the modified theories of gravity. Equivalence principle in one of its simplest forms can be stated as the equality between the inertial and gravitational masses of a given object \cite{Will:2005va}. When referring to objects with small self-gravitational energy one talks about the \emph{weak} equivalence principle. In turn, the \emph{strong} equivalence principle requires it to hold also for objects with a large fraction of their mass consisting of the gravitational binding energy, like black holes and neutron stars.
The violations of the equivalence principle in modified gravity theories arise due to the coupling of the new gravitational degrees of freedom---a single scalar field in our case---to matter. It is in general believed that in theories where the extra fields are only coupled to the metric and bear no direct coupling to the matter fields the weak equivalence principle is obeyed (for exceptions, see \cite{Hui:2009kc}). However, for gravitational bodies with strong self-gravity the additional fields due to the modification of gravity will still effectively couple to matter \cite{Will:2005va}. Even though at leading order the coupling to matter is absent, at higher orders in perturbations it will arise because of the non-minimal coupling to the metric, which in turn is coupled to matter. As a result the body's gravitational mass will depend on the additional fields. This leads to the fact that strongly gravitating bodies follow trajectories that depend on their internal structure and composition, known as the Nordtvedt effect \cite{Nordtvedt:1968qr}.
The physical consequence of the Nordtvedt effect is the appearance of the fifth forces and this has profound implications on the gravitational wave emission. In particular, these fifth forces increase the total power of the gravitational wave emission as a result of which the orbits of inspiraling binaries shrink faster than in GR. One can show that this happens due to the dipole emission of the gravitational radiation (see \cite{Barausse:2017gip} and references therein). The monopole and dipole emission are therefore known as the red flags for scalar-tensor theories. In fact, even before the observation of the gravitational waves, the previous data on the observed mildly relativistic binary pulsars was providing the most stringent tests~\cite{Freire:2012mg} on the `classic' scalar-tensor theories of the Brans--Dicke type~\cite{Brans:1961sx}.
The recent observations by LIGO and Virgo will provide an unseen level of precision for constraining the violations of the equivalence principle. It is therefore timely to review our understanding of the extent up to which such violations occur in modified gravity theories. The focus of this work will be to investigate the effects due to the cosmological expansion---an issue that up to the best of our knowledge has not been explored in this context in the previous literature.
Let us also emphasize that we will not consider explicit violations of the EP. Such violations usually arise in theories where different matter fields are coupled to gravity with different strength already at the level of action. Instead we shall assume a universal coupling of our scalar-tensor theory to all matter fields.
There are a several earlier works on the equivalence principle in scalar-tensor theories that need to be mentioned before we present our results.
The first class of works is related to the no-hair theorem for black holes in scalar-tensor theories. It is known that in GR black holes obey a no-hair theorem that can be extended also to the simplest of scalar-tensor theories with canonical derivative interactions for the scalar field \cite{Bekenstein:1971hc,Bekenstein:1995un}. In \cite{Hui:2012qt}, the proof of the no-hair theorem was further extended to the case of Galileons on asymptotically flat backgrounds \cite{Nicolis:2008in}. The main assumptions needed for the proof were the shift symmetry of the scalar field action, as well as the spherical symmetry and time independence of the black hole and scalar field solution. As a consequence of the theorem the black hole does not admit any non-trivial profile of the Galileon field. On the other hand ordinary matter does source a scalar field profile and thus experiences the fifth force. Hence, a test particle and a black hole would move on different geodesics in any scalar-tensor theory that can be reduced to Galileons in some relevant limit. This suggests the violation of the strong equivalence principle in these theories. It was later shown in \cite{Sotiriou:2013qea,Sotiriou:2014pfa} that the no-hair theorem of \cite{Hui:2012qt} is violated in a more general set-up by also including a coupling of the scalar field to the Gauss-Bonnet term. Such a coupling appears generically in Horndeski theories \cite{Horndeski:1974wa,Deffayet:2009wt,Deffayet:2011gz}.
The situation turns out to be different for static spherically symmetric and asymptotically flat \emph{star} solutions (\emph{i.e.} solutions that are regular everywhere in the spacetime and possess no horizon). In \cite{Lehebel:2017fag,Barausse:2017gip} it was shown that stars in shift-symmetric Horndeski theories obey a no-hair theorem and thus have a zero scalar charge. A similar conclusion was made earlier in \cite{Barausse:2015wia} where it was found that there is no emission of dipolar radiation in such theories. Let us emphasize that all the works mentioned in the context of no-hair theorems work on asymptotically flat backgrounds and make use of \emph{static} scalar field profiles. Relaxing one of these assumptions allows one to evade the no-hair theorems of \cite{Hui:2012qt,Lehebel:2017fag}. For instance, allowing for (Anti) de Sitter asymptotics can lead to hairy black hole and star solutions \cite{Rinaldi:2012vy,Minamitsuji:2013ura} in the Fab Four \cite{Charmousis:2011bf} subclass of the Horndeski theories. A review on asymptotically flat black hole solutions with non-trivial scalar hair can be found in \cite{Herdeiro:2015waa}.
A line of research closely related to the no-hair theorems is the existence of static spherically symmetric solutions in scalar-tensor theories. There is a lot of literature on such solutions by using both time independent and time dependent scalar field profiles leading to both asymptotically flat and asymptotically (Anti-)de Sitter solutions. In this work we shall focus on static spherically symmetric field configurations on top of cosmological backgrounds and \emph{time dependent} scalar field profiles. In the context of black hole solutions in scalar-tensor theories these were first used in \cite{Babichev:2012re,Babichev:2013cya} giving both asymptotically flat and asymptotically de Sitter solutions in the vacuum. Phenomenologically viable asymptotically flat neutron star solutions in the presence of matter were found in \cite{Cisterna:2015yla}. Of particular relevance for this paper is the work by Babichev et al. \cite{Babichev:2016fbg} where static spherically symmetric asymptotically de Sitter solutions in the cubic Galileon theory were studied. We shall discuss these in more detail in Section~\ref{sec:equiv_dS}.
The second class of works related to the equivalence principle in modified gravity theories is concerned with objects with negligible gravitational self-energy \cite{Hui:2009kc,Hui:2010dn}. These are only dealing with the question of possible violations of the \emph{weak} equivalence principle. In particular, it was shown in \cite{Hui:2009kc} that in scalar-tensor theories with Chameleon screening mechanism \cite{Khoury:2003aq,Khoury:2003rn}, such as the Brans--Dicke theory \cite{Brans:1961sx}, there are $\mathcal O(1)$ violations to the weak equivalence principle. On the other hand, it was argued that in theories with Vainshtein screening mechanisms such violations are absent. These results were obtained by studying the motion of an extended object in an external gravitational field by the method of Einstein, Infeld, Hofman \cite{Einstein:1938yz,damour}. We shall review this approach in great detail in Section~\ref{sec:geodesics}. Finally, let us also mention that in \cite{deRham:2012fw} the Vainshtein mechanism in conformally coupled cubic Galileon theory was studied for binary systems. It was found that both the monopole and dipole radiation vanishes at the leading order due to successful Vainshtein screening, thus confirming that the weak equivalence principle is only violated once the relativistic corrections are included.
The paper is organized as follows. In Section~\ref{sec:general} we shall present the effective field theory action for dark energy that we shall be using in the rest of this work and derive the equations of motion for the relevant degrees of freedom. We shall also introduce the various types of objects studied: test particles, extended objects and black holes. In Section~\ref{sec:shift} we shall then discuss the extent up to which the shift symmetry is preserved in the scalar--tensor theories under question. We shall review the no-hair theorem for the black holes in asymptotically flat Galileon theories relying on the conservation of the shift symmetry current. In~\ref{sec:currentfrw} we shall then show how the argument is modified on FRW backgrounds and argue that the weak equivalence principle holds for screened extended objects in Sec.~\ref{extended}.
An argument that both the strong and the weak equivalence principles hold in the special case of de Sitter universe is presented in Section~\ref{sec:equiv_dS}. By an appropriate choice of the coordinate system, we derive the solutions for the asymptotic field profiles for the fields far away from the object in Sec.~\ref{sec:solution}. In particular, we show that the scalar field profile is non-trivial and only depends on the ADM mass of the object, thus corresponding to a secondary hair. We present the explicit asymptotic solution and show that it approaches the Schwarzschild-de Sitter solution in the subhorizon limit. In Sec.~\ref{sec:generalize} we present a framework for studying the violations of the strong equivalence principle due to FRW-type of deviations from de Sitter spacetime.
Finally, in Section~\ref{sec:geodesics} we study the motion of extended objects on FRW background by the method of Einstein, Infeld and Hoffmann \cite{Einstein:1938yz,damour}. As in Sec.~\ref{extended} we arrive at the conclusion that the weak equivalence principle holds for extended objects. We compute the violations due to the departures from the full Vainshtein screening regime in Sec.~\ref{sec:violations}. We conclude in Sec.~\ref{conclusions}.
\section{Generalities}\label{sec:general}
\subsection{The action}
We shall consider an effective field theory for cosmological perturbations around an FRW background driven by a time-dependent scalar field. The constant time hypersurfaces are parametrized by the scalar field and in the so-called unitary gauge the time and the scalar coincide. Around the metric $ds^2= -dt^2+a(t)^2d\vec x^2$ the EFT action written in this gauge reads \cite{Gubitosi:2012hu,Gleyzes:2014dya}:
\begin{equation}
\begin{split}
\label{act_unit}
S&= \int d^4x\,\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}f(t)R-\Lambda(t)-c(t)g^{00}\right.\\
&\left.+\frac{m_2^4(t)}{2}\left(\delta g^{00}\right)^2-\frac{m_3^3(t)}{2}\delta K\delta g^{00}\right
]\\&+\int d^4x\,\sqrt{-g}\,\mathcal L_{\text m}(\psi_{\text m},g_{\mu\nu})\,.
\end{split}
\end{equation}
Here $\delta g^{00}=g^{00}+1$, $\delta K=K-3H$, $H=\dot a/a$ is the Hubble parameter, and $K=\nabla_\alpha n^{\alpha}$ is the extrinsic curvature of surfaces with $t=\text{const}$. The unit vector $n_\mu$ is perpendicular to the constant time slicing, parametrized through the scalar field as $n_\mu\equiv -\partial_\mu\phi/\sqrt{-(\partial\phi)^2}$. The above action does not contain an explicit dependence on the scalar field perturbations $\pi$ since these are set to zero in the unitary gauge where $\phi=M_{\rm Pl}^2t$. The function $\mathcal L_{\text m}$ stands for the Lagrangian density for some matter fields collectively denoted by $\psi_{\text m}$, that are assumed to only be coupled to the metric $g_{\mu \nu}$, and not to the scalar field. In other words, the action \eqref{act_unit} is written in the Jordan frame.
The action \eqref{act_unit} contains contributions to all orders in perturbations. However, only the first three terms contribute to the background equations. The first non-zero contribution from the $m_2^4,m_3^3$ terms on the second line appears at quadratic order.
In fact, the action~\eqref{act_unit} is not the most general quadratic low-energy effective action preserving the spatial diffeomorphisms, and a couple of additional terms should be included \cite{Gleyzes:2014dya}:
\begin{equation}\label{m4}
+m_4^2(t)(\delta K^\mu_\nu\delta K^\nu_\mu-\delta K^2)+\frac{\tilde m_4^2(t)}{2}R\,\delta g^{00}\,.
\end{equation}
However, these terms have lately been in an increasing tension with observations. In particular, the observation of the gravitational waves from the neutron star--black hole binary merger has constrained the speed of gravitational waves to coincide with the speed of light \cite{TheLIGOScientific:2017qsa,Monitor:2017mdv}. This observation then enforces the parameter choice $m_4=0$ \cite{Creminelli:2017sry,Sakstein:2017xjx,Baker:2017hug,Ezquiaga:2017ekz}. In Horndeski theories, $m_4=\tilde m_4$, and thus both of the terms above are excluded. In principle, the operator $R\,\delta g^{00}$ is still allowed in beyond Horndeski theories where $m_4\neq\tilde m_4$ \cite{Gleyzes:2014dya}. However, it was pointed out in \cite{Creminelli:2018xsv} that the presence of these operators would result in a graviton decay into the scalar field particles. The fact that we do observe the gravitational waves then puts a bound on the decay rate, setting $\tilde m_4=0$. It should, however, be kept in mind that the frequency of that particular neutron star merger event lies very close to the strong coupling scale of the dark energy models \eqref{act_unit}, \eqref{m4}. This means that, in principle, new operators, coming from a hypothetic UV completion of these theories can affect the speed of gravity waves and the decay rate on that scale \cite{deRham:2018red}. Hence, ruling out the operators \eqref{m4} might be somewhat premature; we will however stick to the more conservative attitude and not consider them in this work.
We also note that the function $f(t)$ typically arises in Brans-Dicke type theories \cite{Brans:1961sx}. Since these are known to introduce $\mathcal O(1)$ violations to both the strong \cite{Hawking:1972qk} and the weak equivalence principle \cite{Hui:2009kc}, we shall not consider it in the following and set $f(t)\equiv 1$.
In the remainder of this work we shall be working with the covariant form of the unitary gauge action~\eqref{act_unit}. Its diffeomorphism invariance can be restored by the usual St\"uckelberg trick: we perform an infinitesimal time diffeomorphism $t\to t+\pi(t,\vec x)$ and introduce a scalar field
\begin{equation}\label{phi}
\frac{\phi}{M_{\rm Pl}^2}=t+\pi(t,\vec x)\,.
\end{equation}
It is important to emphasize the fact that it is possible to restore the full diffeomorphism invariance by introducing a single scalar field. This shows that the low-energy dynamics of the perturbative action \eqref{act_unit} is entirely determined by one single additional degree of freedom compared to pure GR. This constrains our analysis to this type of models. While it includes, for instance, the Horndeski theories, it does not include models that contain more than one degree of freedom in the IR. A typical example of what these theories do not describe would be massive gravity where~$\pi$ arises as the helicity-0 mode of a massive spin-2 multiplet \cite{ArkaniHamed:2002sp,deRham:2010ik,deRham:2010kj}. This was in fact clear already from the action \eqref{act_unit} since by construction it is invariant under the spatial diffeomorphisms. In contrast, the massive gravity action, even in the decoupling limit where it is dominated by the scalar-tensor interactions, is known to break the spatial diffeomorphisms.
Following \cite{Gubitosi:2012hu} and performing the replacements \eqref{replace1}, \eqref{replace2} we find the covariant action to be:
\begin{align}\label{act_cov}
S&=\int d^4x\,\sqrt{-g}\left[\frac{M_{\rm Pl}^2}{2}R-\Lambda-c\,\frac{(\partial\phi)^2}{M_{\rm Pl}^4}\right.\nonumber\\
&+\frac{m_2^4}{2}\left(1+\frac{(\partial\phi)^2}{M_{\rm Pl}^4}\right)^2\\
&-\left.\frac{m_3^3}{2}\left(1+\frac{(\partial\phi)^2}{M_{\rm Pl}^4}\right)\left(-3H-\frac{3}{2M_{\rm Pl}^2}\Box\phi\right)\right.\nonumber\\
&+\left.\frac{m_3^3}{4M_{\rm Pl}^6}\left(1+\frac{(\partial\phi)^2}{M_{\rm Pl}^4}\right)\left((\partial\phi)^2\Box\phi+\partial^\mu\phi\,\partial_\mu(\partial\phi)^2\right)\right]\nonumber\\
&+\int d^4x\,\sqrt{-g}\,\mathcal L_{\text m}(\psi_{\text m},g_{\mu\nu})\,.\nonumber
\end{align}
When expanded in perturbations it coincides with the unitary gauge action \eqref{act_unit} up to cubic order.
All the quantities $\Lambda, c, m_2, m_3$ that were time dependent in the unitary gauge are now functions of the scalar field, \textit{e.g.} $\Lambda=\Lambda\left(\frac{\phi}{M_{\rm Pl}^2}\right)$ etc. We have suppressed this in \eqref{act_cov} for brevity.
\subsection{Equations of motion}
In order to assess whether the equivalence principle holds in theories described by the action~\eqref{act_cov}, we shall study the gravitational field \emph{created} by a massive object, as well as its motion in some \emph{external} gravitational field. We assume that the expansion of the universe is driven by a background dark energy component due to the scalar field and some other perfect fluid-type matter component with $T^{\text m}\,^\mu_\nu=\text{diag}(-\rho_{\text{m}},p_{\text{m}},p_{\text{m}},p_{\text{m}})$. We describe the object on this background by a stress-energy tensor $\delta T^{\text{m}}_{\mu\nu}$.
We consider metric perturbations around the flat FRW background in the Newtonian gauge:
\begin{equation}\label{gauge}
ds^2=-e^{2\Phi}dt^2+a^2(t)e^{-2\Psi}\delta_{ij}dx^idx^j\,,
\end{equation}
where the potentials $\Phi$ and $\Psi$ are the only gravitational degrees of freedom. In addition, we have the scalar field perturbation $\pi$. The equations of motion, derived from the action.~\eqref{act_cov}, can be written as:
\begin{align}
&M_{\rm Pl}^2 G_{\mu \nu}=T^{\phi}_{\mu\nu} +T^{\rm m}_{\mu\nu}\,, \label{Einsteineqs}\\
&\nabla_\mu (T^{\rm m})^\mu_\nu=0\,, \label{matter_cons}\\
&\nabla_\mu \left[\left(T^{\phi}\right)^\mu_\nu-M_{\rm Pl}^2 G^\mu_\nu\right]=\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta\phi}\partial_\nu\phi\label{eqpi}\,.
\end{align}
The first equation comes from the variation of the total action with respect to the metric $g^{\mu\nu}$. We define the matter stress-energy tensor in the canonical way as $T^{\rm m}_{\mu\nu}\equiv\frac{-2}{\sqrt{-g}}\frac{\delta S_{\text m}}{\delta g^{\mu\nu}}$, and the stress-energy tensor associated to the scalar field, $T^{\phi}_{\mu\nu}$, is implicitly defined through Eq.~\eqref{Einsteineqs}. We give the full covariant expression for the variation of the action~\eqref{act_cov} with respect to the metric in Appendix~\ref{app:emtcov}. The second equation above comes from assuming that the matter action is only coupled to the metric and is invariant under spacetime diffeomorphisms. The last equation is the conservation equation---a consequence of the diffeomorphism invariance of the action. The $\nu=0$ component of it is equivalent to the equation of motion for the scalar field~$\phi$.
The background equations of motion for the FRW metric following from \eqref{Einsteineqs} fix the functions $c$ and $\Lambda$ as
\begin{equation}
\label{backeqcL}
\begin{split}
c(t)&=-\frac{1}{2}(\rho_m+p_m)-M_{\rm Pl}^2\dot H\,,\\
\Lambda(t)&=\frac{1}{2}(p_m-\rho_m)+M_{\rm Pl}^2\left(\dot H+3H^2\right),
\end{split}
\end{equation}
where $\rho_m$ and $p_m$ are the energy density and pressure of the matter component.
For the purpose of this paper it will be convenient to write the Einstein equations for perturbations as:
\begin{align}\label{split}
&M_{\rm Pl}^2 G^{\rm L}_{\mu\nu}-T^{{\rm L}, \phi}_{\mu \nu}= \tau_{\mu \nu}\\\label{deftau}
& \tau_{\mu \nu}\equiv T^{{\rm NL}, \phi}_{\mu \nu}+\delta T^{\rm m}_{\mu \nu}-M_{\rm Pl}^2 G^{\rm NL}_{\mu\nu}\,,
\end{align}
where we have introduced $\tau_{\mu\nu}$---an analogue of the Einstein's pseudo stress-energy tensor. The superscript ``L'' means we only keep the linear part of the expressions, while the ``NL'' stands for everything else except for those linear terms. Note that such a definition does not require the validity of a perturbative expansion, which is not guaranteed close to the source. One can nevertheless always define a linear tensor and arbitrarily remove it from the full tensor. This expression assumes that the cosmological background is already solved for, so that no background quantities remain. Let us note that in GR one defines the pseudo stress tensor as $t_{\mu\nu}=M_{\rm Pl}^2G^{\rm L}_{\mu\nu}=\delta T^{\rm m}_{\mu \nu}-M_{\rm Pl}^2 G^{\rm NL}_{\mu\nu}$. The virtue of this definition is that due to the linearized Bianchi identities for the Einstein tensor, this stress tensor is conserved in flat space, \emph{i.e.} $\partial_\mu t^\mu\,_\nu=0$. In the case when the gravitational theory is modified by an additional degree of freedom and we are on an FRW background, it is necessary to include also $T^{{\rm L}, \phi}_{\mu \nu}$ in the definition above. Only then $\partial_\mu\tau^\mu\,_\nu=0$ at linear level. We shall discuss this in more detail in Section~\ref{sec:method}.
Given Eq.~\eqref{split}, one can view the nonlinearities of the EFT theory as a source term for the linear solution. The virtue of Eq.~\eqref{split} is then that it is a fully nonlinear equation. Although this might appear as a trivial observation this will, in fact, be important when we will want to distinguish between infinitesimal test particles and objects of finite size and/or large self-gravitational energy.
To demonstrate this, let us focus on the quasi-static, subhorizon limit (in Fourier space, this corresponds to considering wave-numbers $k\gg aH$ and frequencies $\omega\ll k$). Then, the leading order equations take the form:
\begin{itemize}
\item \textbf{$00$} component of eq.~\eqref{Einsteineqs}
\begin{equation}
\label{eq00lin}
\frac{1}{a^2}\left[2M_{\rm Pl}^2\Delta\Psi+m_3^3\Delta\pi\right]=-\tau^0\,_0\,;
\end{equation}
\item \textbf{$ij$} component of eq.~\eqref{Einsteineqs}
\begin{equation}
\label{eqijlin}
\frac{\partial_i\partial_j-\delta^i_j\Delta}{a^2}\left[M_{\rm Pl}^2(\Psi-\Phi)\right]=\tau^i\,_j\,;
\end{equation}
\item equation for $\pi$
\begin{equation}
\begin{split}
\label{eqpilin}
\frac{\Delta}{a^2} \bigg[m_3^3\Phi+\left(2c+(m_3^3)^{\hbox{$\cdot$}}+Hm_3^3 \right)\pi\bigg]=-\mathcal{E}_{{\rm NL},\pi}\,,
\end{split}
\end{equation}
\end{itemize}
where in the last equation, we have defined $\mathcal{E}_{{\rm NL},\pi}$ as the nonlinear part of the equation of motion for the scalar. The set of equations given above then fully determines the three fields $\Phi,\Psi,\pi$; in particular, their asymptotic behaviour far away from the object.
\subsection{The ADM mass}\label{sec:adm}
Let us point out that the full nonlinear solution to the $00$ Einstein equation can be expressed in terms of the ADM mass of the object, defined as
\begin{equation}\label{defM}
M=-\int d^3x\,a^3\tau^0\,_0\,.
\end{equation}
Let us demonstrate how it works in the subhorizon limit. Integrating both sides of the equation~\eqref{eq00lin} over some spatial region enclosing the object fixes the relationship between the fields $\Psi$ and $\pi$ in terms of the mass $M$. The solution to the equation \eqref{eqijlin} can be similarly expressed in terms of a volume integral over $\tau^i\,_i$. However, for stationary and virialized systems, as well as non-relativistic sources one can assume that the space integral of $\tau^i\,_i$ is either exactly zero, or is much smaller than the contributions from $\tau^0\,_0$ \cite{Hui:2010dn}. It can thus be neglected from the right-hand side of~\eqref{eqijlin}, giving the usual relationship $\Phi=\Psi$.
Importantly, these observations hold for all types of objects:\\
$(i)$ For a \textit{test particle} all nonlinear contributions to $\tau ^\mu\,_\nu$ are negligible, so that $\tau^\mu\,_\nu=(\delta T^{\rm{m}})^\mu\,_\nu$. In particular, for a non-relativistic point mass located at the origin the only non-zero component of the stress tensor is $(\delta T^{\rm{m}})^0\,_0=-M_0\delta^{(3)}(0)/a(t)^3$ and the ADM mass \eqref{defM} coincides with $M_0$. In the case when the source is spherically symmetric and $\dot m_3=0$, one can solve for the field profiles, to get $\Phi,\Psi,\pi\sim -GM_0/(a(t)r)$. Strictly speaking, the rest mass $M_0$ coincides with the ADM mass only up to post-Newtonian corrections, \emph{i.e.} only up to terms coming from the nonlinear Einstein tensor in \eqref{deftau}. The size of these corrections can be estimated at the quadratic level to be of the order
\begin{equation}
\frac{-\int d^3x\,\tau^0\,_0}{M_0}\sim \frac{M_{\rm Pl}^2\int d^3x\,(\Psi')^2}{M_0}\sim \frac{\frac{M_0^2}{M_{\rm Pl}^2}\frac{1}{r}}{M_0}\sim\frac{r_g}{r}\,,
\end{equation}
where $r_g=M/M_{\rm Pl}^2$ is the Schwarzschild radius of the object. For a point particle, the contribution to the ADM mass from the non-linear stress tensor associated to the scalar field is negligible. We will discuss these in more detail in the context of motion of extended objects in Section~\ref{sec:violations}.
\\
$(ii)$ For an \textit{extended object}---an object of finite size, but negligible self-gravitational energy---the relevant contributions to $\tau^\mu\,_\nu$ are the first two terms in \eqref{deftau}. The nonlinear part of the Einstein tensor can still be neglected in this case.\\
$(iii)$ For a \textit{black hole}, there is no matter stress-tensor, \textit{i.e.} $\delta T^m_{\mu\nu}=0$. The pseudo-tensor $\tau^\mu\,_\nu$ is fully determined by the nonlinear contributions: both due to the gravitational self-interactions and those arising in the stress tensor of the scalar field.
In any of the cases discussed, the equation \eqref{eq00lin} can be solved in terms of the ADM mass defined in \eqref{defM} while the right-hand side of Eq.~\eqref{eqijlin} can be neglected. The only unknown that remains is the solution to the scalar field equation \eqref{eqpilin}. This will be the subject of the next section.
\section{The (approximate) shift invariance of the scalar}\label{sec:shift}
The idea behind this section lies in the observation that, apart from the generic explicit $\phi$-dependence of the coefficients $\Lambda, c, H, m_2$ and $m_3$, the covariant action \eqref{act_cov} is invariant under constant shifts: $\phi(t,\vec x)\to\phi(t,\vec x)+\text{const}$. Moreover, since the unitary gauge action \eqref{act_unit} was introduced as an EFT of broken time translations during the dark energy dominated epoch of our universe, we are interested in a slowly varying Hubble rate and its derivatives. With this in mind, we can assume that also the other generically time dependent parameters in the action, $\Lambda(t),c(t),m_i(t)$, vary only mildly during one Hubble time. More specifically, we shall assume that $\dot m_i\sim Hm_i $ so that terms like $m_i\partial\pi\gg\dot m_i\pi$ in the subhorizon limit.\footnote{We note that the same reasoning would apply for the time dependence of the parameters in the EFT of inflation \cite{Cheung:2007st}.} This would allow us to neglect all the terms in the equation of motion of the scalar field arising from the violation of the shift invariance and instead write it as a current conservation equation
\begin{equation}
\nabla_\mu J^\mu = 0\,,
\end{equation}
where $J^\mu$ is the current due to constant shifts of~$\phi$.
\subsection{Static case}
The presence of the shift symmetry has proven to be very useful when discussing the equivalence principle in asymptotically flat spacetimes. It was used both for showing that the weak equivalence principle holds for theories with Vainshtein screening mechanism \cite{Hui:2009kc} and for the derivation of the no-hair theorem for static black holes in Galileon theories \cite{Hui:2012qt}. For time independent solutions, the divergence of the shift symmetry current in these cases can be reduced to $\nabla_\mu J^\mu=\partial_iJ^i$.
The only shift symmetry breaking term contributing to the scalar fields equation of motion in these models is the scalars coupling to the trace of the matter stress-energy tensor, $\phi T^{\text{m}}\,_\mu\,^\mu$. Integrating this over some spatial volume and using the Gauss's law to convert it into a surface integral yields:
\begin{equation}\label{eom_tens}
\oint dS_i\,J^i=-q\int d^3 x\,T^{\text{m}}\,_\mu\,^\mu\equiv qQ\,,
\end{equation}
where $q = 1$ for ordinary matter and $q = 0$ for black holes. One can then choose the integration surface to lie far away from the moving object to get an asymptotic linear relationship between the fields $\Psi,\Phi,\pi$. Moreover, for objects with weak self-gravity the \emph{scalar charge} $Q$, defined in \eqref{eom_tens}, is given by the total inertial mass of the object, $Q\simeq M$ \cite{Hui:2010dn}.
The scalar field profile, in the region asymptotically far from the object, can then be found to be of the form $\pi\sim -qGQ/r$, giving the interpretation of a scalar charge to the parameter $Q$. The non-vanishing field profile for $\pi$ is also referred to as the \textit{scalar hair}. Due to the couplings to the other gravitational potentials this introduces an additional gravitational force. The equivalence principle is then said to be obeyed in all cases when $Q$ only depends on the total inertial mass~$M$ and the constant $q$ does not vary from object to object. In this case, the modified gravity theory only redefines the effective Newton's constant. Importantly, in Galileon theories $q =0$ for black holes, due to the absence of the coupling $\phi T^{\text{m}}\,_\mu\,^\mu$. This phenomenon is known as the \emph{no-hair} theorem \cite{Hui:2012qt}. Hence, it is only the weak equivalence principle that is obeyed in flat space Galileon theories.
Another terminology often used to make distinction of cases when the equivalence principle is violated is the concepts of \textit{primary} and \textit{secondary} hair~\cite{Coleman:1991ku}. The term `primary hair' is used when talking about a black hole hair that induces new quantum numbers. In the case of `secondary hair' the additional charge is entirely determined by the existing conserved charges, \textit{e.g.} like in the case when the scalar hair only depends on the ADM mass.
Remarkable counterexamples are theories with Chameleon-type screening mechanisms, like $f(R)$ and Brans-Dicke theories. It was shown in~\cite{Hui:2009kc} that there the constant $q$ varies from $q=0$ for screened objects to $q = 1$ for unscreened objects, even when the gravitational self-interactions can be neglected. This manifests when considering the equation of the motion of the object: the difference in the scalar charge leads to a difference in the inertial and gravitational masses of the object. Hence, objects in such theories do not move on geodesics.
\subsection{The current}\label{sec:currentfrw}
Let us analyze the extent up to which the shift symmetry is violated in the action \eqref{act_cov}. In general, given a covariant scalar field action of the form $S=\int d^4x\sqrt{-g}\,\mathcal L(\phi,\nabla_\mu\phi,\nabla_\mu\nabla_\nu\phi)$, the equation of motion for the scalar field can be put in the form
\begin{equation}\label{eom_current}
{\mathcal E_\phi}\equiv\frac{1}{\sqrt{-g}}\frac{\delta S}{\delta \phi}=\frac{\partial\mathcal L}{\partial\phi}-\nabla_\mu J^\mu=0\, .
\end{equation}
The current $J^\mu$ is defined as:
\begin{equation}\label{current}
J^\mu=\frac{\partial\mathcal L}{\partial(\nabla_\mu\phi)}-\nabla_\nu\left(\frac{\partial\mathcal L}{\partial(\nabla_\mu\nabla_\nu\phi)}\right)\,
\end{equation}
and is due to constant shifts of $\phi$; whereas a non-zero $\partial\mathcal L/\partial\phi$ only arises from the breaking of the shift symmetry in the action for $\phi$.\footnote{Let us for a moment discuss the difference between the shift symmetry in $\phi$ versus shift symmetry in $\pi$. Specifically, consider a situation when we had started from some covariant Lagrangian for $\phi$ admitting a more general homogeneous background solution $\phi_0(t)$ with $\dot \phi_0(t)\neq\text{ const}$. The unitary gauge action for perturbations around this solution would then correspond to setting $\phi(t,\vec x)=\phi_0(t)$. As in the previous case the invariance under time diffeomorphisms can be recovered by performing a transformation
\begin{equation}
t\to t+\pi(t,\vec x)\,,\qquad \phi_0(t)\to \phi_0(t)+\dot\phi_0(t)\pi(t,\vec x)\,.
\end{equation}
However in the case when $\dot\phi_0\neq \text{const}$, the symmetry of $\phi$ under constant shifts does not imply a symmetry of $\pi$ under constant shifts. Indeed, under the shift $\phi\to\phi+c$ we get $\pi\to\pi+\dot\phi_0(t)^{-1}c$. Even though it is always possible to perform a field redefinition $\phi\to\tilde\phi(\phi)$ so that $\dot{\tilde\phi}_0=\text{const}$, this does not guarantee that the action for $\tilde\phi$ would remain shift invariant. Since it is the field $\pi$ that encodes the approximate invariance under the time translations then it is the shift invariance of $\pi$ that bears a physical importance. In the rest of this paper we shall assume that $\phi_0(t)=t$ and treat the shift symmetry of $\pi$ and $\phi$ as equivalent.
}
We give the explicit expressions of the current derived from the action \eqref{act_cov} in Appendix~\ref{app:current}.
In the case when the shift invariance is only slightly broken, one would expect the equation of motion for the scalar to reduce to $\nabla_\mu J^\mu=0$ in the quasi-static limit. However, in our EFT of dark energy setup, it is precisely the explicit $\phi$~dependence of the action that drives the cosmological evolution. Hence $\partial\mathcal L/\partial\phi$ does contribute to the background evolution with
\begin{equation}\label{arg1}
M_{\text{Pl}}^2\,\frac{\partial\mathcal L}{\partial\phi}=\dot c(t)-\dot\Lambda(t)\,.
\end{equation}
Another important difference from the current conservation in shift symmetric theories in the static case is that $J_0\neq 0$ on cosmological backgrounds. Indeed, the current components read:
\begin{equation}\label{arg2}
J_0=\frac{2c(t)}{M_{\rm Pl}^2}\,,\qquad J_i=0\,.
\end{equation}
Physically it makes sense---a cosmological background naturally picks out a time direction which consequently leads to a non-vanishing $J^0$.
Combining \eqref{arg1} and \eqref{arg2} in the equation of motion~\eqref{eom_current} gives the standard conservation equation $\dot\rho_m+3H\rho_m=0$ on an FRW background.
We face a more serious obstruction to the shift symmetry argument on FRW background when inspecting the higher order current components.
As we show in Appendix~\ref{sec:current2}, the contribution to $\nabla_\mu J^\mu$ coming from $\nabla_0J^0$ is in general comparable to $\nabla_i J^i$. This, combined with the non-conservation of the current, $\partial\mathcal L/\partial\phi\neq 0$, prevents us from having an equation of the type \eqref{eom_tens} in a general case. In the absence of it, there is no advantage in recasting the scalar equation in the form~\eqref{eom_current}. Instead one has to work with the equation~\eqref{eqpilin} that can only be solved perturbatively and after integration contains an arbitrary integration constant that cannot be fixed by any guiding principle. One is thus unable to make exact non-perturbative arguments needed to discuss black hole hair and the strong equivalence principle. We therefore focus on extended objects in the remainder of this section.
\subsection{Extended objects in the subhorizon limit}\label{extended}
In theories with Vainshtein screening mechanism~\cite{Vainshtein:1972sx}, which is the case of all models described by the action \eqref{act_unit} (or, equivalently, \eqref{act_cov}), extended objects require a special attention. The reason for this is the presence of higher derivative scalar self-interactions, that, for an object of finite size, become dominant below some distance $r_V$ in the vicinity of the object. In these models the interactions responsible for the Vainshtein mechanism at the $n$th order in field perturbations are contained in terms of the type $(\partial\pi)^{2k}(\partial^2\pi)^{l}$ with $n=l+2k$, $k\geq1$. Due to the antisymmetric structure of these higher order self-interactions, in four spacetime dimensions only interactions up to $l=3$ appear in the most general Horndeski theories (see, for instance, \cite{Nicolis:2008in,Deffayet:2011gz}). Hence, in the scalar field equation of motion we expect that for $n=2,3,4$ the equation is dominated by $(\partial^2\pi)^n$ terms.
For the choice of operators in the EFT action~\eqref{act_unit}, only the cubic Vainshtein interactions with $l=1$ are present giving a $(\partial^2\pi)^2$ term in the equation of motion at quadratic order. For extended objects with negligible gravitational self-interactions this is the leading contribution to the quadratic scalar fields equation of motion.
Due to the special derivative structure of the Vainshtein interactions, the equations of motion for perturbations around cosmological solutions on subhorizon scales always take the form of spatial divergences \cite{Kimura:2011dc}. At quadratic order, all the highest derivative contributions to both metric and scalar fields equations of motion take the form
\begin{equation}\label{der_structure2}
\begin{split}
\Delta \mathcal X\Delta \mathcal Y-&\partial_i\partial_j\mathcal X\partial_i\partial_j\mathcal Y\\
&=\partial_i\left(\partial_i \mathcal X\Delta \mathcal Y-\partial_j\mathcal X\partial_i\partial_j \mathcal Y\right)\,,
\end{split}
\end{equation}
where $\{\mathcal X,\mathcal Y\}$ denote any of the fields $\{\Phi,\Psi,\pi\}$.
For the expressions at higher orders, see \cite{Kimura:2011dc}. Indeed, we find that the quasi-static subhorizon contribution to the quadratic equation of motion of the scalar field reads:
\begin{equation}\label{divergence}
\mathcal E_{2,\pi}=-M_{\rm Pl}^2\partial_iJ^i=-\frac{m_3^3}{a^4}\partial_i\left[\partial_i\pi\Delta\pi-\partial_j\pi\partial_i\partial_j\pi\right]\,.
\end{equation}
We also note that it is only the divergence of the spatial current component that contributes to the above equation. As we argue in Appendix~\ref{sec:current2}, both $\nabla_0J^0$ and $\partial\mathcal L/\partial\phi$ are subleading in comparison to the Vainshtein interactions at this order.
At cubic order, only terms like $(\partial\pi)^2\partial^2\pi$ arise which, in the small field approximation, are negligible in comparison to the quadratic term. The same is true for all the other higher order terms that all can be safely neglected due to the fact that, by definition, gravitational self-interactions are negligible for extended objects and all the terms like $\pi,\partial\pi$ can still be treated perturbatively. Hence, the nonlinear equation of motion for $\pi$, for an extend object in subhorizon limit can be written as
\begin{equation}\label{eomNL}
\mathcal{E}_{\text{NL},\pi}=\mathcal E_{2,\pi}=-M_{\rm Pl}^2\partial_i J^i\,.
\end{equation}
Note that for later convenience we have included a factor of $M_{\rm Pl}^2$ in the definition of $\mathcal E_\pi$.
We can now run the argument, similar to that of the static case, for the current conservation for extended objects. In particular, by substituting~\eqref{eomNL} on the right-hand side of the equation \eqref{eqpilin} we see that one can reduce the volume integral to a surface integral. This is true for arbitrary location of the surface, even inside the Vainshtein radius. Choosing the surface to lie far away from the object, \emph{i.e.} for $r\gg r_V$, allows one to treat $\pi$ linearly. In spherically symmetric case this gives the asymptotic relation:
\begin{equation}
\partial_r\bigg[m_3^3\Phi+\left(2c+(m_3^3)^{\hbox{$\cdot$}}+Hm_3^3 \right)\pi\bigg]=0
\end{equation}
In combination with \eqref{eq00lin} and \eqref{eqijlin} this allows one to solve for all the gravitational field profiles solely in terms of the ADM mass. The above solution is only valid for the gravitational field far away from the object, at distances that exceed its Vainshtein radius. Nevertheless there is no fundamental obstacle in determining the gravitational fields also at shorter distances by using the non-linear expression \eqref{eomNL}. These would then have to match the asymptotic solutions found above which, as we have shown, do not contain any new additional charge and are fixed by the ADM mass of the object. Hence the weak equivalence principle holds up to post-Newtonian and \emph{post-Vainshtein} corrections. We shall discuss what we mean by the latter in Section~\ref{sec:violations} where we shall study the motion of the extended object in an \emph{external} gravitational field.
\section{Equivalence principle in de Sitter universe}\label{sec:equiv_dS}
An outstanding example of where the vanishing of the current is an exact statement is the case of de Sitter spacetime. For instance, there are known numerical black hole solutions with de Sitter asymptotics in the cubic Galileon theory \cite{Babichev:2016fbg}. These solutions involve a time dependent scalar field profile and a static solution for the metric, written in the static coordinate patch of de Sitter. In general, their $J^0\neq 0$ while $J^\rho = 0$ coincides with the $0\rho$ component of the Einstein equation (where $\rho= re^{Ht}$ is the radial coordinate in the static de Sitter coordinate system\footnote{The notations for the static coordinates $\rho, \tau$ and the FRW coordinates $r,t$ are exchanged in \cite{Babichev:2016fbg}.}). The fact that a time dependent scalar field profile can be compatible with the vanishing of the spherical component of the Noether current, allowing for static and spherically symmetric metric solutions was already noticed in \cite{Babichev:2015rva}. The findings of \cite{Babichev:2016fbg} also show that there is a non-vanishing scalar field profile associated with their black hole solutions. At small scales the scalar field solution depends on an arbitrary integration constant that can be interpreted as the primary hair of the black hole. The large scale asymptotics however only depend on the black hole mass and thus only give a secondary hair. Overall, it seems to indicate that $(i)$~the black holes do have hair on cosmological backgrounds, even in shift symmetric theories, and that $(ii)$~the strong equivalence principle \textit{might} hold for black holes on de Sitter space.
We shall now show that the equivalence principle does indeed hold for spherically symmetric solutions of the action \eqref{act_cov} in de Sitter space. Restricting to de Sitter universe in our EFT implies setting $\rho_{\text m}=p_{\text{m}}=c = 0$ and $\Lambda = 3M_{\rm Pl}^2H^2 = \text{const}$. Inspired by the ansatz of \cite{Babichev:2016fbg} we shall assume that the scalar fields $\Phi,\Psi,\pi$ are not only spherically symmetric, but also that the dependence on $t,r$ only comes through
\begin{equation}\label{symmetry}
\Phi(r,t) = \Phi(re^{Ht})=\Phi(\rho)\,\quad\text{etc.}
\end{equation}
That it is reasonable to ask for such static solutions to be compatible with the expansion of the spacetime is due to the isometries of de Sitter space. Indeed, the metric \eqref{gauge} on the background with $a(t)=e^{Ht}$, rewritten in spherically symmetric coordinates is invariant under $t\to t+\lambda$, $r\to e^{-H\lambda}$. Since under this transformation $\rho=re^{Ht}\to \rho$ is also invariant, then the static ansatz \eqref{symmetry} is compatible with the isometries of de Sitter background.\footnote{We thank Andrew Tolley for pointing this out.}
In fact, it also has a physically well motivated meaning since $\rho=re^{Ht} = a(t) r$ is just the physical radius as opposed to the comoving one. For instance, also the gravitational potential in GR around a spherically symmetric static source in an FRW universe equals to $\Phi=-GM/(a(t)r)$. We also emphasize that the static ansatz \eqref{symmetry} can only be a valid solution if also the coefficients $m_2$ and $m_3$ are time independent. Therefore, in the remainder of this section we shall set
\begin{equation}
m_2\,,m_3=\text{const}\,.
\end{equation}
It might appear that since the time diffeomorphisms (including constant shifts) are broken by our EFT construction, forbidding these coefficients to bear any time dependence is unnatural. Nevertheless it is justified when constraining the analysis to de Sitter spacetime. Time dependence of $m_2,m_3$ in this case introduce deviations from exact de Sitter and their time derivatives can be regarded as additional slow-roll parameters. We shall come back to this point in Section~\ref{sec:generalize}.
The linearized equations of motion of the field perturbations for the vacuum configuration (\emph{i.e.} in the absence of any additional matter sources besides the metric and the scalar field) can be simplified to:
\begin{align}
\Psi' - \Phi'=0\,,\quad\Phi+\rho\,\Phi'=0\,,\nonumber\\\label{lineomdS}
-\Phi+H\rho\,\pi'=0\,,
\end{align}
where the prime denotes a derivative with respect to the physical radius $\rho$. The above equations are solved by $1/\rho$ profiles for all the three fields. A few remarks are in order. In particular, let us note that the first equation in \eqref{lineomdS} follows from the off-diagonal $ij$ components of the Einstein equations and has been integrated over the radial coordinate. On the other hand, the second and third equation in \eqref{lineomdS} are specific combinations of the $tx$ and $xx$ components of the Einstein equations and only depend on the first order derivatives without the need of integration. This already points towards the fact that the asympotic relationships between the three fields $\Phi, \Psi, \pi$ are independent on the nonlinearities of the system under the symmetry conditions \eqref{symmetry}.
\subsection{The current}
Let us elaborate on this statement by looking at the current components as defined in~\eqref{current}. In particular, we will show that the asymptotic relationship \eqref{lineomdS} can be expressed as the vanishing of one of the current components. Due to the field dependence \eqref{symmetry} it proves useful to switch from the Cartesian coordinate system first to the spherical coordinates and then to the coordinate system $(\tau,\rho,\varphi,\theta)$ with $\tau = t$ and $\rho = a(t)r$. The metric~\eqref{gauge} thus becomes:
\begin{align}\label{metric2}
ds^2 &= \left(-e^{2\Phi(\rho)}+H^2\rho^2e^{-2\Psi(\rho)}\right)d\tau^2+\\
&-2H\rho \,d\rho d\tau +e^{-2\Psi(\rho)}d\rho^2+\rho^2e^{-2\Psi(\rho)}\,d\Omega^2\,.\nonumber
\end{align}
We then find the $J^\rho$ component of the current from Eq.~\eqref{current} by first finding the current components $J^t, J^i$ in the comoving Cartesian coordinates and then transforming them as
\begin{align}\label{jrho}
J^\rho & = a(t)^2\frac{x^iJ^i}{\rho}+\rho HJ^t\\
&=4H\rho \,m_2^4\left(-\Phi+H\rho\pi'\right)\nonumber\\
&-3H^2\rho \,m_3^3\left(\Phi+\rho\Psi'\right)
-{m_3^3}\left(H\,\pi'+\Phi'\right)\,.\nonumber
\end{align}
By comparing the above expression to the equations of motion \eqref{lineomdS} it is straightforward to see that $J^\rho = 0$ on shell. Hence, the radial component of the current vanishes exactly on the equations of motion. Curiously, also the linear $J^\tau$ component vanishes on shell. Indeed,
\begin{align}
J^\tau&=4m_2^4\left(-\Phi+H\rho\pi'\right)\nonumber\\
&-3H \,m_3^3\left(\Phi+\rho\Psi'\right)
-\frac{m_3^3}{\rho^2}\left(2\rho\,\pi'+\rho^2\pi''\right)\,.\nonumber
\end{align}
The vanishing of the radial current component can be understood when considering the variation of the action \eqref{act_cov} under diffeomorphism transformations. This relation was observed earlier in \cite{Babichev:2015rva} for static black holes; we shall follow their analysis closely below.
A variation of the action \eqref{act_cov} can be written in a general form as
\begin{equation}
\delta S = \int d^4 x\,\sqrt{-g}\left(\frac{\delta S}{\delta g^{\mu\nu}}\delta g^{\mu\nu}+\frac{\delta S}{\delta \phi}\delta\phi\right)\,.
\end{equation}
If the variation is due to a diffeomorphism transformation
\begin{equation}
x^\mu\quad\to \quad\tilde x^\mu=x^\mu+\xi^\mu(x)
\end{equation}
the change in the metric and the scalar field is accordingly:
\begin{equation}
\delta g^{\mu\nu}=\xi^{\mu;\nu}+\xi^{\nu;\mu}\,,\quad \delta \phi=-\partial_\mu\phi\,\xi^\mu\,,
\end{equation}
where $\xi^{\mu;\nu}\equiv g^{\nu\alpha}\nabla_\alpha\xi^\mu$. By defining the metric equations of motion as
\begin{equation}
\mathcal E_{\mu\nu}\equiv\frac{2}{\sqrt{-g}}\frac{\delta S}{\delta g^{\mu\nu}}\,
\end{equation}
and the scalar field equation of motion $\mathcal E_\phi$ as in~\eqref{eom_current} we can rewrite the variation as
\begin{align}\label{var_gen}
\delta S &= \int d^4 x\,\sqrt{-g}\left[\frac{1}{2}\mathcal E_{\mu\nu}\left(\xi^{\mu;\nu}+\xi^{\nu;\mu}\right)\right.\nonumber\\
&\left.+\left(\nabla_\alpha J^\alpha-\frac{\partial\mathcal L}{\partial\phi}\right)\partial_\mu\phi\,\xi^\mu\right]\,.
\end{align}
The traditional use of this expression is to obtain the conservation equations (the so-called \emph{Bianchi identities}):
\begin{equation}\label{bianchi}
\nabla^\nu\mathcal E_{\mu\nu}+\mathcal E_\phi\partial_\mu\phi=0\,.
\end{equation}
This equation has a twofold significance. First, it means that on scalar fields equation of motion the stress-energy tensor is conserved. Second, it shows that the equations of motion of the metric and of the scalar field are not all independent, but are related through \eqref{bianchi}. Indeed, for our field configuration \eqref{phi}, \eqref{symmetry}, and \eqref{metric2}, its 0-th component at linear level reads
\begin{equation}\label{offshell}
\partial_\tau\left(\mathcal E^\tau_\tau-M_{\text{Pl}}^2\,J^\tau\right)+\frac{1}{\rho^2}\partial_\rho\left(\rho^2\left(\mathcal E^\rho_\tau- M_{\text{Pl}}^2\,J^\rho\right)\right)=0\,.
\end{equation}
This is an off-shell equation and is exactly satisfied.
There is yet another way how to make use of the variation of the action above. To see this let us consider a diffeomorphism
\begin{equation}\label{ximu}
\xi^\mu(x)=\left(\xi^\tau(\rho),0,0,0\right)\,.
\end{equation}
One can show that for this choice of $\xi^\mu$ we have
\begin{equation}\label{xidS}
\mathcal E_{\mu\nu}\left(\xi^{\mu;\nu}+\xi^{\nu;\mu}\right)=2\mathcal E_\tau^\rho\,\partial_\rho\xi^\tau\,,
\end{equation}
valid up to arbitrary order in perturbations. For the variation of the action we thus obtain
\begin{align}\label{var}
\delta S = \int d^4x\,\sqrt{-g}\left[\partial_\rho\xi^\tau\left(\mathcal E_\tau^\rho-M_{\text{Pl}}^2\,J^\rho\right)-\frac{\partial\mathcal L}{\partial\phi}M_{\text{Pl}}^2\,\xi^\tau\right]\,.\end{align}
Hence, in the absence of an explicit violation of the shift invariance, \textit{i.e.} when $\partial\mathcal L/\partial\phi=0$, one arrives at the relationship:\footnote{We should note that choosing the gauge parameter $\xi^\mu$ to be time independent makes the variational problem somewhat subtle. In general, the relationship \eqref{epsrhotau} should come with an additional time integral $\int_{\tau_{\text{initial}}}^{\tau_{\text{final}}}d\tau$\,. However, in an entirely static in $\tau$ situation, which in our case is equivalent to saying that $\partial\mathcal L/\partial\phi=0$, the integral is trivial. We thank Alberto Nicolis for extended discussions on this.}
\begin{align}\label{epsrhotau}
&\mathcal E_\tau^\rho-M_{\text{Pl}}^2\,J^\rho=0\,.
\end{align}
Given the expression \eqref{non_cons}, we see that this is precisely the case in de Sitter universe with time independent coefficients $m_2,m_3$. From the above equation it thus follows that the radial component of the current coincides with the equation of motion $\mathcal E_\tau^\rho$ at all orders. One can check by explicit computations at linear order that indeed the equation~\eqref{epsrhotau} is satisfied off-shell.
Hence, we conclude that the radial component of the current vanishes on shell at all orders in perturbations:
\begin{equation}\label{jzero}
J^\rho = 0\,.
\end{equation}
In particular, the linearized version of the above equation gives an asymptotic relationship between the fields $\Phi,\Psi,\pi$ and their first derivatives.
\subsection{Nonlinearities, matter and black holes}\label{sec:solution}
The linearized equations \eqref{lineomdS} are valid in vacuum and do not tell what the gravitational fields around an object of a given mass are. To take this into account, let us retain the pseudo stress tensor $\tau^0_0$, as defined in \eqref{deftau}, in our equations of motion. We note that the variation \eqref{var} include $\tau^0_0$ implicitly, since $\mathcal E_{\mu\nu}$ refers to the full metric equations of motion. Hence, also the statement about the vanishing of the current \eqref{jzero} remains unchanged and gives one equation for the three fields $\Phi,\Psi,\pi$.
If $\tau^0_0$ is only due to nonlinearities, \textit{i.e.} $\delta T^{\text{m}}_{\mu\nu}=0$, then we are considering black holes. In turn, a point-like test particle placed at the origin would have $\tau^0_0=-M\delta^{(3)}(0)/a(\tau)^3$. In the following we shall leave the internal decomposition of $\tau^0_0$ unspecified and solve for the gravitational fields sourced by it.
Due to the spherical symmetry of the system, the only non-zero components of the metric equations of motion are $\mathcal E_{\tau\tau}\,,\mathcal E_{\tau\rho}\,,\mathcal E_{\rho\rho}\,,\mathcal E_{\varphi\varphi}\,,\mathcal E_{\theta\theta}$. Moreover, the last three of them only depend on two independent combinations of the orginial Cartesian equations of motion, defined through:
\begin{align}
\mathcal E_{ij}&\equiv\mathcal E_{ij}^{\text{off}}+\delta_{ij}\mathcal E^{\text{diag}}\,,\\
\mathcal E_{ij}^{\text{off}}&\equiv\left(x_ix_j-\frac{1}{3}\delta_{ij}r^2\right)\mathcal E^{\text{off}}\,.
\end{align}
Combined with the equation of motion for the scalar and with the $\tau$ and $\rho$ components of the Bianchi identities \eqref{bianchi} we have in total $4+1-2=3$ equations for the three fields $\Psi,\Phi,\pi$. The most convenient choice is setting $J^\rho,\,\mathcal E_{\tau\tau},\,\mathcal E^{\text{off}}$ to zero. As in the case of the subhorizon limit, the off-diagonal equation allows us to set $\Phi=\Psi$. Using this, vanishing of $J^\rho$ gives the following relationship:
\begin{equation}\label{phipi}
-\Phi+H\rho\pi'=m_3^3\frac{1+3H^2\rho^2}{-m_3^3+4H\rho^2m_2^4}(\rho\Phi)'\,.
\end{equation}
We note that the right-hand side of this equation vanishes for $1/\rho$ profiles, \textit{i.e.} when $\rho\Phi=\text{const}$. Finally, the $\mathcal E_{\tau\tau}$ equation reads:
\begin{align}
0&=m_3^3\Delta\pi+2M_{\text{Pl}}^2\Delta\Phi+\tau^0_0\\
&+\left(4m_2^4+3m_3^3H\right)\left(\Phi-H\rho\pi'\right)\nonumber\\
&+3H\left(m_3^3-2HM_{\text{Pl}}^2\right)(\rho\Phi)'\,.\nonumber
\end{align}
By using the equation \eqref{phipi}, the second line can be expressed in terms of $(\rho\Phi)'$ and thus only the first line survives for $1/\rho$ profiles. It can be solved in terms of the ADM mass, defined in \eqref{defM}, to give:
\begin{equation}\label{sol_ep}
\Phi+\frac{m_3^3}{2M_{\text{Pl}}^2}\pi=-\frac{MG}{\rho}\,,
\end{equation}
where $G$ is the Newton's constant with $2M_{\text{Pl}}^2=(4\pi G)^{-1}$\,. Together with Eq.~\eqref{phipi} and the relation $\Phi=\Psi$ this determines both the scalar field profile and the gravitational potential felt by the object entirely in terms of its ADM mass. This means that the equivalence principle is obeyed in these theories, since the ADM mass bears no distinction between how it is split into rest masses, energy from self-interactions or self-gravitational energy. Hence, it treats all objects on equal footing.
We also observe that it is obvious from the equation above that a time dependent $m_3(\tau)$ would be inconsistent with the solution \eqref{sol_ep} for a conserved ADM mass~\eqref{defM}. Therefore, for the field ansatz~\eqref{symmetry} to be a solution of our system, we must restrict our argument to time independent $m_3$, as argued earlier. At the same time, a time dependent $m_2$ still allows for static gravitational potentials at linear level. Indeed, for $1/\rho$ profiles, the time dependence of $m_2$ would be irrelevant for solving Eq.~\eqref{phipi}. However, at higher orders $\dot m_2\neq 0$ would inevitably introduce time dependence in the system and thus must be set to zero for field configurations \eqref{symmetry}. We therefore conclude that time-dependent $m_2,m_3$ violate the equivalence principle. We discuss it together with the violation due to the time dependence of the Hubble parameter in the next section.
Finally, let us point out that the solution given by equations \eqref{phipi} and \eqref{sol_ep} is a valid spherically symmetric, static (in physical coordinates) star and black hole solution in de Sitter space. Transforming the coordinates $(\tau,\rho)$ to the static patch coordinates $(\tilde\tau,\rho)$ by
\begin{equation}
\tilde \tau=\tau-\frac{1}{2H}\ln \left(1-H^2\rho^2\right)\,,
\end{equation}
we obtain
\begin{align}
&\tilde g_{\tau\tau}=-{\rm e}^{2\Phi}+H^2\rho^2{\rm e}^{-2\Psi}\,,\\
&\tilde g_{\rho\rho}=\frac{{\rm e}^{-2\Psi}-H^2\rho^2{\rm e}^{2\Phi}}{(1-H^2\rho^2)^2}\,,\\
&\tilde g_{\tau\rho}=\frac{ H\rho}{1-H^2\rho^2}\left({\rm e}^{2\Phi}-{\rm e}^{-2\Psi}\right)\,.
\end{align}
In the subhorizon small field limit with $H\rho,\Phi\ll1$ it reduces to the Schwarzschild-de Sitter metric with $f(\rho)=1+2\Phi-H^2\rho^2$.
\subsection{Generalize to FRW?}\label{sec:generalize}
An FRW universe with a slowly changing Hubble rate can be described by the theory \eqref{act_cov} upon including a background stress-energy tensor $T^{\rm m}\,^\mu_\nu=\text{diag}(-\rho_{\text{m}},p_{\text{m}},p_{\text{m}},p_{\text{m}})$. In the following we shall assume a pressureless matter with $p_{\text{m}}=0$. As opposed to the case of de Sitter, now the matter energy density is time dependent and thus, also the Hubble rate and the parameters $\Lambda=\Lambda(\tau)$ and $c=c(\tau)$, as determined by equations~\eqref{backeqcL}.
The complexity of the equations of motion increases considerably and we cannot obtain an analytic solution even in the absence of any additional matter source and nonlinearities. While the solution can in principle be found numerically, the previous argument that allowed us to set $J^\rho=0$ in the case of de Sitter cannot be applied anymore. Let us demonstrate how several of the steps in derivation that lead to \eqref{jzero} cannot be performed anymore.
As before, we wish to consider variation of the action \eqref{var_gen} under a diffeomorphism \eqref{ximu}. For a time dependent system we have to allow for $\xi^\tau=\xi^\tau(\tau,\rho)$. We then find
\begin{align}
\frac{1}{2}\mathcal E_{\mu\nu}&\left(\xi^{\mu;\nu}+\xi^{\nu;\mu}\right)=\nonumber\\
&\mathcal E^\tau_\tau\,\partial_\tau\xi^\tau+\mathcal E_\tau^\rho\,\partial_\rho\xi^\tau-\rho\dot H\mathcal E^\tau_\rho\xi^\tau\,.
\end{align}
In comparison to \eqref{xidS} there is an extra term proportional to $\xi^\tau$. Upon inserting this in the variation of the action we obtain
\begin{align}\label{varfrw}
\delta S = &\int d^4x\,\sqrt{-g}\left[\partial_\rho\xi^\tau\left(\mathcal E_\tau^\rho-M_{\text{Pl}}^2\,J^\rho\right)\right.\\
&\left.+\partial_\tau\xi^\tau\left(\mathcal E_\tau^\tau-M_{\text{Pl}}^2\,J^\tau\right)\right.\nonumber\\
&\left.-\left(\rho\dot H\mathcal E^\tau_\rho+\frac{\partial\mathcal L}{\partial\phi}M_{\text{Pl}}^2\right)\,\xi^\tau\right]\,.\nonumber
\end{align}
We further note that in the presence of $\rho_{\text{m}}$ the background values of $J^\mu$ and $\partial\mathcal L/\partial\phi$ become
\begin{align}
&J^\tau=\frac{2c(\tau)}{M_{\text{Pl}}^2}\,,\quad J^\rho=H\rho J^\tau\,,\\
&M_{\text{Pl}}^2\,\frac{\partial\mathcal L}{\partial\phi}=\dot c(\tau)-\dot\Lambda(\tau)\,,
\end{align}
giving additional contributions when expanding the variation \eqref{varfrw} up to given order in perturbations. Moreover, in distinction from de Sitter case, $\partial\mathcal L/\partial\phi$ is non-zero already at linear level, as is obvious from the non-perturbative expression \eqref{non_cons}. The resulting conservation equation of our interest can then be obtained by varying the action with respect to $\xi^\tau=\xi^\tau(\tau,\rho)$ and gives:
\begin{align}\label{eom_frw}
-\partial_\rho&\left(\sqrt{-g}(\mathcal E_\tau^\rho-M_{\text{Pl}}^2\,J^\rho)\right)\nonumber\\
&-\sqrt{-g}\left(\rho\dot H\mathcal E^\tau_\rho+M_{\text{Pl}}^2\frac{\partial\mathcal L}{\partial\phi}\right)\nonumber\\
&-\partial_\tau\left(\sqrt{-g}\left(\mathcal E^\tau_\tau-M_{\rm Pl}^2J^\tau\right)\right)=0\,.
\end{align}
This is again an off-shell equation. In distinction from \eqref{epsrhotau} the first line appears with a radial derivative and there are new contributions due to the time dependence.
In particular, there is no way of arguing that the term in the second line can be dropped. Moreover, one can explicitly check that it contains contributions of the same order as the term in the first line. Indeed, at linear oder we find:
\begin{align*}
(1^{\text{st}})&\supset4\rho^3H^3m_3^3\left(\frac{\dot H}{H^2}+\frac{(m_3^3)^{\hbox{$\cdot$}}}{Hm_3^3}\right)\pi'\\
(2^{\text{nd}})&\supset\rho^3H^2\left[\rho_{\text{m}}\left(-3+\frac{\dot H}{H^2}\right)\right.\\
&\left.+4m_3^3\frac{\dot H}{H}+2M_{\text{Pl}}^2H^2\left(\left(\frac{\dot H}{H^2}\right)^2+\frac{\ddot H}{H^3}\right)\right]\pi'\,,
\end{align*}
where we use the notations $(1^{\text{st}})$ and $(2^{\text{nd}})$ for the first and second line in \eqref{eom_frw} respectively. We note that all the terms appear due to the presence of matter, $\rho_{\text{m}}$, and the time dependence of the Hubble parameter and $m_3$. Importantly, we see that the $m_3^3$ terms give contributions of the same order.
Thus one cannot neglect the impact of the FRW spacetime and there is no way to argue that an equation of type \eqref{jzero} should exist. Nevertheless, one should keep in mind that in the context of dark energy (or, similiarly, of inflation) one is interested in quasi de Sitter spacetimes. Thus one can work with the result of de Sitter, where we have shown that the equivalence principle holds, and only consider violations of it due to the time dependence of the Hubble rate and parameters $m_2,m_3$. In particular, one could introduce a set of approximately constant parameters corresponding to
\begin{equation}
\rho_{\rm m}\,, \frac{\dot H}{H^2}\,,\frac{\ddot H}{H^3}\,,\frac{(m_2^4)^{\hbox{$\cdot$}}}{Hm_2^4}\,,\frac{(m_3^3)^{\hbox{$\cdot$}}}{Hm_3^3}\,,\dots
\end{equation}
to quantify these deviations. It is beyond the scope of the present paper and will be left for future work.
\section{Geodesics in FRW}\label{sec:geodesics}
In this section we shall compute the acceleration felt by an object moving in an external configuration of the fields $\{\Phi_0,\Psi_0,\pi_0\}$ in the modified gravity theories \eqref{act_unit}. We shall then check whether the equivalence principle holds in these models, \emph{i.e.} whether the object moves on a geodesic. A particular emphasis will be put on describing the motion of finite size (extended) objects. For this we shall use the approach of Einstein, Infeld and Hoffmann \cite{Einstein:1938yz,damour} and will follow closely the procedure of \cite{Hui:2009kc}.
An alternative formalism for describing the dynamics of non-relativistic extended objects coupled to gravity relies on using effective field theory methods \cite{Goldberger:2004jt,Goldberger:2007hy}. In flat space, one can see gravity as a theory of an interacting spin-two particle described by a tensor field $h_{\mu\nu}\equiv g_{\mu\nu}-\eta_{\mu\nu}$. The extended object can in turn be approximated as a point-particle with a worldline coordinate $x^{\mu}(\lambda)$ that depends on an arbitrary affine parameter $\lambda$. It is possible to describe their mutual interactions, as well as their interactions with external fields, by constructing an EFT Lagrangian that is consistent with the symmetries of the long-wavelength physics. In our case the relevant symmetry is the general coordinate invariance. At leading order, the effective action describing the graviton and a point particle of mass $M$ is simply:
\begin{equation}\label{eff_pp}
\begin{split}
S_{\rm eff}&=S_{\rm gravity}+S_{\rm pp}\\
&=\frac{M_{\rm Pl}^2}{2}\int d^4x\sqrt{-g}\,R-M\int d\tau\,,
\end{split}
\end{equation}
where $d\tau^2=g_{\mu\nu}(x(\lambda))dx^\mu dx^\nu$ is the proper time variable. Varying with respect to $x^\mu$ gives the usual geodesic equation of a test particle. The corrections due to the finite size effects then emerge from the higher order operators that can be added to the point particle action \cite{Goldberger:2004jt,Goldberger:2007hy}:
\begin{equation}
S_{\rm pp}=-M\int d\tau+c_R\int d\tau R+c_V\int d\tau\,R_{\mu\nu} \dot x^\mu\dot x^\nu+\dots\,,
\end{equation}
where $\dot x^\mu=dx^\mu/d\tau$ and the dots stand for an infinite series of higher curvature corrections. This EFT is valid at energy scales $kr_s\ll 1$ where $r_s$ is the size of the object. The precise values of the coefficients $c_V, c_R$ can be obtained from a UV matching procedure to some microscopic model of the internal structure of the object. The idea is then to compare the observables computed in the full short distance theory and in the effective theory.
In the case when gravity is modified to a scalar-tensor theory, the additional scalar degree of freedom, $\pi$, also needs to be included in the EFT description \eqref{eff_pp}. In particular, in a shift-invariant theory with Vainshtein screening mechanism the effective action describing the coupling of the scalar field to a localized source would include operators like
\begin{equation}
\begin{split}
\int d\tau\,\left(c^{(1)}\partial_\mu\partial_\nu\pi+c^{(2)}\partial_\mu\partial_\alpha\pi\partial^\alpha\partial_\nu\pi+\dots\right)\dot x^\mu\dot x^\nu\,.
\end{split}
\end{equation}
For an extended object, the dimensionful coefficients $c^{(1)},c^{(2)}$ parametrize our ignorance about the internal structure and finite-size effects of the object. Importantly, inside the Vainshtein region the second order derivatives $\partial^2\pi$ can lead to sizeable effects. The point that we want to emphasize here is that \emph{a~priori} one would therefore expect these operators to modify the geodesic equation of the extended object. The upshot of this section is to show, in an indirect way, that such corrections do not arise in the scalar-tensor theories governed by the action~\eqref{act_cov}.
\subsection{The method}\label{sec:method}
In what follows we shall be working in the Cartesian FRW coordinates \eqref{gauge} and will always take the quasi-static subhorizon limit.
To characterize the object, in addition to its mass $M$, defined in \eqref{defM}, we also define its position and momentum:
\begin{align}
X^i&\equiv -\int d^3x\,a^3\,x^i \,\tau_0\,^0/M\,\label{defXi}\,,\\
P_i&\equiv \int d^3 x\, a^3 \,\tau_i\,^0\label{defPi}\,,
\end{align}
where the volume integral is taken over a sphere of radius $r$ enclosing it. The gravitational force exerted on the moving object can then be found as the time derivative of the momentum $\dot P_i$. In order to test the equivalence principle we shall relate the force to the acceleration of the object, calculated as $\ddot X^i$. In order to take the time derivatives of $P_i$ and $X_i$ and to manipulate the volume integrals in the definitions \eqref{defXi} and \eqref{defPi}, we will need various conservation laws that we present below.
\subsubsection{Conservation laws}
The Bianchi identity of the Einstein tensor, i.e. the fact that $\nabla_\mu G^\mu\,_\nu=0$, guarantees that the linearized Einstein tensor obeys the following linearized conservation laws:
\begin{align}
&\partial_\nu\left(a^3G^{\rm{L}}\,_0\,^\nu\right)=a^3HG^{\rm{L}}\,_i\,^i\,,\\
&\partial_\nu\left(a^3G^{\rm{L}}\,_i\,^\nu\right)=a^3\partial_i\Phi\left(\bar G^0_0-\frac{1}{3}\bar G^i_i\right)\,,
\end{align}
where the bar denotes the background values. We remark that the first relation is only valid in the subhorizon limit. Combining the above expressions with the Einstein equations \eqref{Einsteineqs} and the covariant conservation law \eqref{eqpi} we can derive the conservation laws for the pseudo stress-energy tensor $\tau^\mu\,_\nu$. These read:
\begin{align}\label{cons0}
&\partial_\mu\left(a^3\tau^\mu\,_0\right)=a^3\left[H\tau^i\,_i+\mathcal E_{\rm{NL},\pi}\right]\\\label{consi}
&\partial_\mu\left(a^3\tau^\mu\,_i\right)=-a^3\partial_i\Phi(\rho_{\text{m}}+p_{\text{m}})\,,
\end{align}
where $\mathcal E_{\rm{NL},\pi}$ is defined as the nonlinear part of Eq.~\eqref{eqpilin}. We stress again that the first relation in only valid in the subhorizon limit while the second equation is exact. We see that the $0th$ component of the pseudo stress tensor is not conserved at nonlinear level. In comparison, for an object moving on a flat background in GR, the right-hand side of the first equation vanishes.
Finally, let us note that due to \eqref{cons0} in general also the mass $M$ in \eqref{defM} is not conserved in theories~\eqref{act_unit}. Instead, we obtain
\begin{equation}\label{dotM}
\dot M=\int d^3x\,a^3\left[-\mathcal E_{\rm{NL},\pi}-H\tau^i\,_i+\partial_i\tau^i\,_0\right]\,.
\end{equation}
As was explained in \ref{sec:adm}, the integral over the second term vanishes for stationary and virialized systems. The last term can in turn be transformed into a surface integral $\oint dS_i\,\tau^i\,_0$ that vanishes under the assumption of zero energy flux through the integration surface. Hence, the non-conservation of the ADM mass, only arises due to the first term in \eqref{dotM}.
\subsubsection{Geodesic equation}
In order to write the geodesic equation, we will need to relate the force $\dot P_i$ to the acceleration of the object $\ddot X^i$. Let us first compute $\dot X^i$. Using the defenition~\eqref{defXi} together with the conservation law \eqref{cons0} we can write
\begin{align}\label{dotXi}
\dot X^i=&\frac{1}{a^2}\frac{P_i}{M}\\
&-\frac{1}{M}\int d^3x\,x^i\,a^3\left[H\tau^j\,_j+\mathcal E_{\rm{NL},\pi}\right]-\frac{\dot M}{M}X^i\,.\nonumber
\end{align}
In deriving the above expression we have neglected the surface integral $\int dS_j\,x^i\tau^j\,_0$. For future reference let us define the integral appearing in the above equation as:
\begin{equation}\label{nonlinear}
\mathcal I(t)\equiv\int d^3x\,x^i\,a^3\left[H\tau^j\,_j+\mathcal E_{\rm{NL},\pi}\right]\,.
\end{equation}
By taking another derivative we arrive to the expression for the acceleration
\begin{align}\label{geodesic}
\ddot X+2H\dot X=\frac{1}{a^2M}\partial_0\left( P_i
-a^2\,\mathcal I\right)\,,
\end{align}
where we have omitted the terms arising due to the non-conservation of the ADM mass for reasons that we shall explain below. We note that the additional Hubble friction term on the left-hand side of this equation arises naturally in the geodesic equation for a worldline $X^\mu(\lambda)$ in FRW spacetime, which in the absence of the force would read $\frac{d^2X^\mu}{d\lambda^2}+\Gamma^\mu_{\nu\rho}\frac{dX^\nu}{d\lambda}\frac{dX^\rho}{d\lambda}=0\,$.
The quantity on the right-hand side of \eqref{geodesic} gives the force exerted on the moving object. For this we need to evaluate the time derivative of the momentum, defined in~\eqref{defPi}. By using the conservation law~\eqref{consi} we find
\begin{equation}\label{dotPi}
\dot P_i=-a^3\oint dS_j \tau^j\,_i -a^3\int d^3 x\,(\rho_{\text{m}}+p_{\text{m}})\,\partial_i\Phi\,,
\end{equation}
where the integration measure $dS_j=dA \cdot n_j$ denotes a surface integration with $dA$ being the surface area element with the outgoing unit normal $n_j$. The integration surface can be chosen so that $\delta T^{\text{m}}\,_\mu\,^\nu$ vanishes on it so that one only needs to evaluate the nonlinear contributions to the integrand $\tau^j\,_i$. The advantage of defining the momentum $P_i$ through the pseudo stress tensor $\tau_\mu\,^\nu$ defined as in \eqref{deftau} is apparent now---according to \eqref{consi} it is conserved and thus we were able to reduce the computation of its time derivative to a surface integral instead of a volume integral. This allows us to compute the gravitational force felt by the moving object without specifying its internal structure. Moreover, we only need to specify the fields on the boundary, which we can choose to be far away from the source, where one can use the perturbative expansion.
\subsubsection{Background--object split}
In order to evaluate the surface integral, we decompose the field perturbations $\mathcal X=\{\Phi,\Psi,\pi\}$ on the surface of the sphere, centered around $r=0$ and enclosing the object of interest, in two parts~\cite{Hui:2009kc}:
\begin{equation}\label{ansatz}
\mathcal X=\mathcal X_0+\mathcal X_1(r)\,,\, \mathcal X_0(x^i)\simeq \mathcal X_0(0)+\partial_i\mathcal X_0(0)x^i\,,
\end{equation}
where $\mathcal X_1$ represent the fields due to the object itself while $\mathcal X_0$ amount to some large scale background fields that vary mildly on the scale of the sphere. We do not specify the origin of these background fields, but use the freedom to add such linear gradient fields to any other solution to equations consisting of second order spatial derivatives. This is the case in the quasi-static subhorizon limit. For performing the integrals in \eqref{dotPi} we use the following useful relationships for $\{\mathcal X,\mathcal Y\}$ satisfying the ansatz \eqref{ansatz}:\footnote{We have also made use of the identities $\partial_i \mathcal X_1(r)=n_i\mathcal X_1'$, $\partial_i\partial_j \mathcal X_1(r)=n_i n_j\mathcal X_1''+(\delta_{ij}-n_in_j)\frac{\mathcal X_1'}{r}$, $\oint dA\cdot n_j=0$ and $\oint dA\cdot n_i n_j = \frac{4\pi}{3}r^2\delta_{ij}$.}
\begin{align}
&\oint dS_i\,\partial^i\mathcal X\,\partial_j \mathcal Y=\frac{4\pi}{3}r^2\partial_j \mathcal X_0Y'_1+4\pi r^2\partial_j\mathcal Y_0\mathcal X'_1\,,\nonumber\\
&\oint dS_i\,\mathcal X(\delta_{ij}\Delta\mathcal Y-\partial_i\partial_j\mathcal Y)=\frac{8\pi}{3}r^2\partial_j\mathcal X_0\mathcal Y'_1\,,\nonumber\\
&\oint dS_j\,\partial_k\mathcal X\,\partial^k\mathcal X=\frac{8\pi}{3}r^2\partial_j\mathcal X_0\mathcal X'_1\,.
\end{align}
By performing this split in the last term of \eqref{dotPi} as $\Phi=\Phi_0+\Phi_1(r)$, we see that the only surviving term is the one coming from $\partial_i\Phi_0$. Assuming that inside our volume of integration the mass of the object is larger than the total mass of the background matter fields characterized by $\rho_{\text m}, p_{\text m}$, allows us to neglect the last term in \eqref{dotPi}.
\subsection{Test particles}
It is instructive to start by looking at the case of test particles, even though the result is trivial---their inertial and gravitational masses are equal and they move on geodesics.
The advantage of test particles is that the perturbative expansion is a valid approximation everywhere. In particular, the integral \eqref{nonlinear} can be evaluated at quadratic order in perturbations by making use of the following useful relationship, derived from~\eqref{eqpi}:
\begin{equation}\label{quadratic}
H\tau^{(2)i}\,_i+\mathcal E_{(2,\pi)}=\partial_iT^{(2,\pi)i}\,_0\,.
\end{equation}
The above relationship is derived in subhorizon approximation, \emph{i.e.} in the leading order of derivatives. For the derivative of the ADM mass we thus obtain:
\begin{equation}
\dot M=-M_{\rm Pl}^2\oint dS_ja^3G^{(2)}\,^j\,_0=0\,.
\end{equation}
Similarly, for the geodesic equation \eqref{geodesic} we find:
\begin{align}\label{geodesic_new}
\ddot X^i+&2H\dot X^i=\\
&\frac{1}{a^2M}\,\partial_0\left[P_i-\int d^3x\,a^3\,T^{(2,\pi)0}\,_i\right]\,.\nonumber
\end{align}
We stress that this equation was derived by only assuming the validity of the perturbative expansion without ever referring to the exact form of the action. We have then used the scalar field equation of motion \eqref{eqpi} together with the conservation laws of the pseudo stress tensor in \eqref{cons0} and \eqref{consi}. The equation \eqref{geodesic_new} is, however, important---it states that in the presence of a scalar field the force that is felt by a test particle moving in the gravitational field is not given by the time derivative of momentum as defined in \eqref{defPi}. Instead, the contribution of the scalar field has to be subtracted. Physically this shows that the gravitational force felt by a test particle is not altered by the presence of the scalar. In general, however, the nonlinear effects modify the geodesic equation as in \eqref{geodesic}.
As was said above, for the test particle, all the nonlinear quantities can be evaluated at quadratic order. By explicit calculation for the action \eqref{act_cov} we then find
\begin{equation}\label{dotPiv2}
\begin{split}
\dot P_i-\partial_0\int d^3x\,&a^3T^{(\pi,2)}\,^0_i=\\
&\partial_i\Phi_0\int d^3x\,a^3\left(\tau^0\,_0-\frac{1}{3}\tau^i\,_i\right)\\
+&\partial_i\pi_0\int d^3x\, a^3\mathcal E_{\rm{NL},\pi}\,.
\end{split}\,
\end{equation}
We note that the right-hand side of this expression is completely nonlinear and we have not assumed validity of the perturbative expansion. We have only used the explicit form of the quadratic stress-energy tensor, together with the equations of motion \eqref{eq00lin}--\eqref{eqpilin} written through the nonlinear quantities $\tau^\mu\,_\nu$.
For test particles, the nonlinear term on the last line can be dropped, and for a non-relativistic source $\tau^0\,_0\gg\tau^i\,_i$. We thus get that the only relevant contribution on the right-hand side of the above equation is $-M\partial_i\Phi_0$, where $M$ stands for the ADM mass defined in~\eqref{defM}. Inserting this in \eqref{geodesic_new} we recover the correct geodesic equation
\begin{equation}\label{geodesic0}
\ddot X^i+2\frac{\dot a}{a}\dot X^i=-\frac{1}{a^2}\partial_i\Phi_0\,.
\end{equation}
We present the detailed derivation of Eq.~\eqref{dotPiv2} together with the expressions for the quadratic components of the stress-energy tensor $T^\pi_{\mu\nu}$ in Appendix~\ref{sec:emt}.
\subsection{Extended objects}\label{sec:extended}
In this section we shall extend the derivation of the gravitational force felt by a test particle to the case of extended objects. In this case we assume that the self-gravitational energy is still negligible, however screening effects close to the object need to be taken into account. Technically this means that far from the object, \emph{i.e.} outside its Vainshtein radius, the fields $\Phi,\Psi,\pi$ and their derivatives are still in the perturbative regime. Inside the Vainshtein radius, however, while the fields themselves can still be treated perturbatively, the higher derivative terms $\partial^2\mathcal X$ for $\mathcal X\in\{\Phi,\Psi,\pi\}$ become dominant and need to be taken into account.
Let us start by considering the time derivative of the center of mass coordinate, given in \eqref{dotXi}. In particular, let us discuss the problematic integral $\mathcal I$ defined in~\eqref{nonlinear}. The integrand here is a fully nonlinear expression. In the case of test particles the perturbative expansion was valid everywhere and we could use the quadratic expression \eqref{quadratic} in order to evaluate this integral. The situation is different for sizeable objects---in the close vicinity to the object, \emph{i.e.} inside its Vainshtein radius $r_V$, the nonlinear derivative interactions of the scalar field dominate. However, as was discussed in great detail in Sec.~\ref{extended}, in theories described by the action~\eqref{act_cov}, the leading Vainshtein interactions arise at the cubic order. These result in the following non-trivial contributions in the integrand of \eqref{nonlinear}:
\begin{equation*}
\mathcal I^{\text{lead}}=\int d^3x\,x^i\,a^3\left[H\tau^{(\pi,2)j}\,_j+\mathcal E_{2,\pi}+H\tau^{(\pi,3)j}\,_j\right]\,.
\end{equation*}
All higher order contributions will be subleading. We further note that the derivative structure of these leading terms is $\tau^{(\pi,2)j}\,_j\sim(\partial\pi)^2$, $\mathcal E_{2,\pi}\sim (\partial^2\pi)^2$ and $\tau^{(\pi,3)j}\,_j\sim(\partial\pi)^2\partial^2\pi$. It is known that inside the Vainshtein region $\partial^2\pi\gg(\partial\pi)^2$. Hence, the cubic term above can be neglected and the integrand of \eqref{nonlinear} contains contributions that cannot be neglected for sizable objects with negligible self-gravity only in terms up to quadratic order in perturbations. As for the test particle, it can be evaluated by using the relationship \eqref{quadratic}, giving
\begin{equation}
\mathcal I^{(2)}=\int d^3x\,x^i\,a^3\partial_jT^{(\pi,2)j}\,_0=-\int d^3x\,a^3\,T^{(\pi,2)i}\,_0\,,
\end{equation}
where we have dropped a surface integral over energy flux in the last equality.
In order to derive the geodesic equation we substitute the above result in \eqref{geodesic}, leading to \eqref{geodesic_new}. In distinction from the case of the test particles, however, we have to take into account also the cubic contributions to the time derivative of the momentum $P_i$. Thus, at the leading order in the subhorizon limit instead of Eq.~\eqref{dotPiv2} we get:
\begin{equation}\label{dotPiv3}
\begin{split}
\dot P_i-&\partial_0\int d^3x\,a^3T^{(\pi,2)}\,^0_i=\\
&\partial_i\Phi_0\int d^3x\,a^3\left(\tau^0\,_0-\frac{1}{3}\tau^i\,_i\right)\\
+&\partial_i\pi_0\int d^3x\, a^3\mathcal E_{\rm{2},\pi}-\oint dS_j\,a^3T^{(\pi,3) j}\,_i\,.
\end{split}\,
\end{equation}
At this point the fact that we are dealing with an extended object again becomes relevant. It can still be taken to be non-relativistic, so that the terms on the second line can be expressed in terms of the ADM mass as in the case of the test particles, leading to $-M\partial_i\Phi_0$. However, the cubic terms on the second line are slightly more problematic. In particular, there is a remaining volume integral which we cannot evaluate without knowing the full nonlinear field solutions also within the Vainshtein regime. A resolution to this comes from the approximate shift invariance of the theory. This allows one to write the equation of motion for the scalar field in the subhorizon limit in the form of a spatial divergence as in \eqref{divergence}. In fact, it takes the form of a \emph{twofold} spatial divergence:\footnote{This observation holds also in the case when the quartic Vainshtein interactions are present in the action. See Appendix~\ref{cubic} for the explicit form.}
\begin{align}\label{eq_quadr}
\mathcal E^{\text{lead}}_{2,\pi}&=-\frac{m_3^3}{a^4}\partial_i\partial_j\left[\partial_i\pi\partial_j\pi-\delta_{ij}(\partial\pi)^2\right]\\
&=\partial_iT^{(2,\pi)i}\,_0\,.\nonumber
\end{align}
The last equality follows from the general relation~\eqref{quadratic} (given that in the quasi-static subhorizon limit $\tau^{(2)i}\,_i\ll\mathcal E^{\text{lead}}_{2,\pi}$), but can also be seen to hold from the explicit form \eqref{defTi0}.
There are then two possible ways to deal with the cubic terms on the last line of~\eqref{dotPiv3}. The simplest way of evaluating them is to use the explicit form of the scalar equation of motion to rewrite the cubic terms as surface integrals
\begin{equation}\label{surf_int}
\oint dS_j\,a^3\left(\partial_i\pi_0\, T^{(2,\pi)j}\,_0-T^{(\pi,3) j}\,_i\right)\,.
\end{equation}
We can then choose the integration surface outside the Vainshtein radius of the object, where the field perturbations are in the linear regime and the arising contributions from the integrals are negligible. By substituting the remaining terms of \eqref{dotPiv3} in \eqref{geodesic_new} we then arrive at the geodesic equation
\begin{equation}\label{geodesic_extended}
\begin{split}
\ddot X^i+&2\frac{\dot a}{a}\dot X^i=-\frac{1}{a^2}\partial_i\Phi_0\\
&+\frac{1}{a^2MH}\partial_i(H\pi_0)\int d^3x\, a^3\mathcal E^{\rm{sublead}}_{\rm{NL},\pi}\,,
\end{split}
\end{equation}
where we have added the factors of $H$ in the last term, so that both $\partial_i\Phi_0$ and $\partial_i(H\pi_0)$ have the same dimensionality.
One might be rightly concerned that taking the integration surface so far away from the object the background-object split is not valid anymore. Indeed, for any ordinary object its Vainshtein radius is much larger than its size. Moreover, it overlaps with the Vainshtein radii of other objects, even when the physical distance between the actual objects is large. Thus it is certainly not reasonable to assume that the background fields can be approximated as linear divergence fields over such large scales. In \cite{Hui:2009kc} it was, however, argued that it is sufficient if one can approximate the background field as a linear gradient field in a neighbourhood close to the object.\footnote{For limitations of this assumption in the deeply non-linear regime when the Vainshtein regions of the test object and the object sourcing the background field overlap, see~\cite{Hiramatsu:2012xj}.} Due to the shift symmetry of the derivative, $\partial_\mu\pi\to\partial_\mu\pi+c_\mu$, one can then extrapolate the linear background field close to the object up to distances exceeding the Vainshtein radius. This argument cannot be straightforwardly applied on FRW backgrounds, even though one would expect the argument to hold in the subhorizon limit.
Let us prove that the geodesic equation \eqref{geodesic_extended} is in fact the correct result for the equation of motion of an extended object moving in a linear background field. For this we shall evaluate the integral \eqref{surf_int} exactly, on a surface close to the object where the background fields can safely be assumed to be linear gradient fields. By performing the background-object split in the expression \eqref{cubic_emt} of the cubic stress-energy tensor components, we obtain for its surface integral:
\begin{equation}
\oint dS_j\,T^{(\pi,3)j}\,_i=-\frac{2m_3^3}{a^4}\partial_i\pi_0\frac{(\pi_1')^2}{r}\cdot 4\pi r^2\,,
\end{equation}
where we have used that $\oint dA\cdot n_in_jn_k=0$. Similarly, for the other integral in \eqref{surf_int} we find:
\begin{equation}
\oint dS_j \,T^{(2,\pi)j}\,_0=-\frac{2m_3^3}{a^4}\frac{(\pi_1')^2}{r}\cdot 4\pi r^2\,.
\end{equation}
Combining the two we see that the total cubic contribution \eqref{surf_int} vanishes without specifying the radius of the integration surface $r$. Hence, the geodesic equation \eqref{geodesic_extended} is indeed our final result. We thus conclude that the equivalence principle holds for screened extended objects, up to some violations due to the subleading terms in the equation for $\pi$. These arise from the terms that do not come in the form of a spatial divergence and will be evaluated in the next section.
At last, let us note that starting from Eq.~\eqref{geodesic} we have dropped the contributions of the order $\mathcal O(H\dot M/M)$ to the geodesic equation. Given the expression \eqref{dotM} one can argue, along the same lines as above for the integral $\mathcal I$, that the ADM mass is also conserved for extended objects.
\subsection{Estimate for violation}\label{sec:violations}
In this section we would like to estimate the order of magnitude of the violations of the equivalence principle for extended objects. According to our computations, the only modification to the geodesic equation is of the form \eqref{geodesic_extended}. As we emphasized already, the Vainshtein mechanism takes place due to the cubic order interactions in the EFT action. Hence, the relevant nonlinear contributions to the equations of motion truncate at second order and are given by $\mathcal E^{\text{lead}}_{2,\pi}$ in \eqref{eq_quadr}. We are interested in evaluating the contributions from the next-to-leading terms appearing at quadratic level. These are, for instance,
\begin{equation}\label{nlo}
\mathcal E^{\rm sublead}_{2,\pi}=-\frac{m_3^3}{2a^2}H^2\partial_i\pi\partial_i\pi\,.
\end{equation}
By performing the object-background split \eqref{ansatz} we find that the volume integral over these terms gives
\begin{equation}\label{correction}
\begin{split}
\int d^3x\,\mathcal E^{\rm sublead}_{2,\pi}&=m_3^3 H^2\int d^3x\,\left[\left(\partial_i\pi_0\right)^2+\left(n_i\pi_1'\right)^2\right]\\
&=4\pi m_3^3H^2\int dr\,r^2\,\left(\pi_1'\right)^2\,,
\end{split}
\end{equation}
where in the last equality we have dropped the contributions coming from the background since the integral should be dominated by the contributions from the object itself. We note that we are integrating over a sphere of radius $r$ that is enclosing the object, but is large enough so that the field $\pi_1$ is determined by its asymptotic monopole solution. On the other hand the sphere should be small enough that the external gravitational fields can be approximated as linear gradient fields.
\subsubsection{Linear solution}
Let us first evaluate \eqref{correction} for $\pi_1(r)$ given by the asymptotic solution from the linear equations of motion, obtained by combining eqns. \eqref{eq00lin}-\eqref{eqpilin}:
\begin{equation}
\left[-\frac{2M_{\rm Pl}^2}{m_3^3}c-2M_{\rm Pl}^2H+m_3^3\right]\pi_1'=\frac{M}{a}\frac{1}{4\pi r^2}\,.
\end{equation}
For a generic modification of gravity, all the terms inside the square brackets are of the same order:
\begin{equation}\label{order}
\frac{M_{\rm Pl}^2}{m_3^3}c\simM_{\rm Pl}^2H\sim m_3^3\,.
\end{equation}
We can thus write an order of magnitude estimate
\begin{equation}\label{pi_lin}
\pi_1'\sim\frac{M}{m_3^3}\frac{1}{r^2}\,.
\end{equation}
\subsubsection{Vainshtein solution}
Next, let us estimate the Vainshtein scale for the theory at hand. The only equation in which the Vainshtein nonlinearities play a significant role is the scalar field equation of motion. In the subhorizon limit this reads:
\begin{equation}\label{eom_vainsh}
\begin{split}
\frac{1}{a^2}&\left[2c\Delta\pi+m_3^3\left(H\Delta\pi+\Delta\Phi\right)\right]\\
&=\frac{m_3^3}{a^4}\partial_i\partial_j\left[\partial_i\pi\partial_j\pi-\delta_{ij}(\partial\pi)^2\right]\,.
\end{split}
\end{equation}
We shall fix the Hubble parameter to its present value and set $a(t)=a_0=1$ in this section; this will give correct order of magnitude estimates.
The Vainshtein scale is defined as the scale at which the nonlinear terms on the right-hand side become of the same order as the linear terms on the left-hand side of this equation. We can estimate the scale by schematically comparing
\begin{equation}
m_3^3 \partial^2(\partial\pi)^2\sim c\,\partial^2\pi\sim m_3^3\partial^2\Phi \,,
\end{equation}
where in the spherically symmetric case we can further replace $\partial\to r^{-1}$.
From the first 'equality' in the above relation we then obtain that at the Vainshtein radius $\pi\sim c\,r_V^2/m_3^3$. By equating this with the linear solution \eqref{pi_lin} and using $c\sim m_3^6/M_{\rm Pl}^2$ from \eqref{order}, we get for the Vainshtein scale
\begin{equation}
r_V=\left(\frac{MM_{\rm Pl}^2}{m_3^6}\right)^{1/3}\,.
\end{equation}
We can then find the spherically symmetric solution for $\pi$ deep inside the Vainshtein radius, when the nonlinear terms in \eqref{eom_vainsh} dominate. Simplifying the above expression as $r^{-2}\Phi\sim r^4\pi^2$ and taking the linear solution for $\Phi\sim M/M_{\rm Pl}^2/r_V$, we obtain the screened solution
\begin{equation}\label{inside}
\pi_{\rm screen}\sim \sqrt{\frac{Mr}{M_{\rm Pl}^2}}\,,\qquad r\ll r_V\,.
\end{equation}
\subsubsection{The corrections}
Now we are able to evaluate the correction \eqref{correction} by splitting the integral of the subleading terms in~\eqref{correction} as
\begin{equation}\label{int_split}
\begin{split}
\int dr\,r^2\,\left(\pi_1'\right)^2&=\int_0^{r_V} dr\,r^2\,\left(\partial_r\pi_{\rm screen}\right)^2\\
&+\int_{r_V}^r dr'\,r'^2\,\left(\partial_{r'}\pi_1\right)^2\,.
\end{split}
\end{equation}
Given the functional forms of the solutions \eqref{inside} and \eqref{pi_lin} we see that both integrals in \eqref{int_split} give their main contribution at $r=r_V$ and are of the same order of magnitude:
\begin{equation}
\frac{\int^{r_V} dr\,r^2\,\left(\partial_r\pi_{\rm screen}\right)^2}{\int^{r_V} dr\,r^2\,\left(\partial_r\pi_{1}\right)^2}\sim\frac{\frac{r_V^2M}{M_{\rm Pl}^2}}{\frac{M^2}{m_3^6 r_V}}=\frac{r_V^3m_3^6}{MM_{\rm Pl}^2}=\mathcal O(1)\,.
\end{equation}
To evaluate the contribution of the subleading terms to the geodesic equation \eqref{geodesic_extended} we can therefore approximate it by evaluating one of the integrals at the Vainshtein radius. After recasting the result in a more meaningful form we find
\begin{equation}\label{corr1}
\begin{split}
\frac{H^{-1}}{M}\int d^3x\,\mathcal E^{\rm sublead}_{\rm 2,\pi}\sim\frac{r_g}{r_V}\,,
\end{split}
\end{equation}
where $r_g=M/M_{\rm Pl}^2$ is the Schwarzschild radius of the object and we have used $HM_{\rm Pl}^2\sim m_3^3$.\footnote{Let us remark that assuming $HM_{\rm Pl}^2\sim m_3^3$ is equivalent to relating the Vainshtein radius, Schwarzschild radius and the Hubble scale as $r_VH\sim (r_g/r_V)^{1/2}$. It is, in fact, the case in massive gravity theory where $r_V^3\sim M/(M_{\rm Pl}^2 m^2)$ and the graviton mass is taken to be $m\sim H$. } For an object of mass $M_0$, the ratio $r_g/r_V$ equals to
\begin{equation}
\frac{r_g}{r_V}\sim\left(\frac{HM_0}{M_{\rm Pl}^2}\right)^{2/3}\ll1\,.
\end{equation}
For any astrophysical object this ratio is very small. As an example, for the Sun this ratio is $\sim10^{-15}$. In fact, the mass of the object should coincide with the mass of the whole universe for this ratio to be of order one.
As we have emphasized already, for ordinary objects the Vainshtein radius is much larger than their size. Therefore, for the background-object split to be valid, the integration volume in \eqref{int_split} should only extend up to $r\ll r_V$. In this case we find an additional suppression factor to \eqref{corr1}:
\begin{equation}
\begin{split}
\frac{H^{-1}}{M}\int d^3x\,\mathcal E^{\rm sublead}_{\rm 2,\pi}\sim\left(\frac{r}{r_V}\right)^2\frac{r_g}{r_V}\,,
\end{split}
\end{equation}
leading to a modified geodesic equation of the form:\footnote{Without the assumption $m_3^3\sim HM_{\rm Pl}^2$ the scales $r_V,r_g, H^{-1}$ decouple and the corrections can be recast as $\left(\frac{r}{r_V}\right)^{3/2}\left(\frac{r}{H^{-1}}\right)\left(\frac{r_g}{r}\right)^{1/2}$. We recognize the power $3/2$ as the power of the Vainshtein corrections to the gravitational potential in the DGP theory \cite{Dvali:2002vf}. }
\begin{equation}\label{res_viol}
\ddot X^i=-\partial_i\Phi_0+\partial_i(H\pi_0)\left(\frac{r}{r_V}\right)^2\frac{r_g}{r_V}\,,\quad r\ll r_V\,.
\end{equation}
Again, for the example of the Sun, the ratio $R_{\odot}/r_V\sim 10^{-10}$. This ratio is somewhat larger for galaxies. Combining the two suppression factors we thus conclude that the violation of the equivalence principle is very tiny. In principle, if one could measure deviations from the geodesic motion (if any) up to this level of precision one could use (a more accurate version of) these estimates of the deviations from the equivalence principle to put constraints on the EFTs of dark energy described by the action \eqref{act_cov}.
\section{Conclusions and Discussion}\label{conclusions}
Without doubt the equivalence principle is one of the cornerstone predictions of General Relativity. In this work we have addressed the question of its violation on cosmological backgrounds in scalar-tensor theories of modified gravity. We find that an exactly de Sitter spacetime stands out as a particular gravitational background where both the weak and the strong equivalence principles are obeyed. Any violations of the equivalence principle can therefore be parametrized in terms of the time derivatives of the Hubble rate. For slowly changing Hubble rate, which is the situation in the present day universe, these are small and can thus be treated perturbatively. This provides new ways for characterizing the violations of the equivalence principle that comes particularly timely given the new gravity wave observations opening up the possibility of testing gravity in the strong-field regime.
We have also investigated the motion of weakly gravitating objects on FRW spacetimes with arbitrary time dependence of the Hubble rate. We show that the weak equivalence principle is obeyed for test particles and extended objects moving in some external gravitational potential on these backgrounds.
This result complements the findings of \cite{Barausse:2015wia,Lehebel:2017fag,Barausse:2017gip} that in shift symmetric Horndeski theories in flat space the star solutions bear zero scalar charge and thus have no-hair. Consequently, the gravitational and inertial masses of compact stars are equal, \emph{i.e.} they obey the equivalence principle, and the dipolar gravitational wave emission vanishes \cite{Barausse:2015wia}. However, in distinction from \cite{Barausse:2015wia,Lehebel:2017fag,Barausse:2017gip} we do not claim that the scalar field profile is vanishing. Instead we show that the scalar hair is secondary and is determined entirely in terms of the ADM mass of the object. As a result the extended objects move on geodesics, as in General Relativity.
We note, however, that the proof given in \cite{Barausse:2015wia} relies on the assumption that the Vainshtein radius of the star is much less than the gravitational wavelength, \emph{i.e.} that $r_V\ll\lambda_{\rm GW}$. This is in general not true. For compact stars of masses close to the solar mass, the Vainshtein radius is $r_V\sim 10^{15}$~km while the wavelength of the gravity waves is $\lambda_{\rm GW}\sim 10^4$~km. In this work we have shown that compact objects move on geodesics also on distances less than its Vainshtein radius. The only condition we use is that in the close vicinity of the object the total external gravitational field can be treated as a linear gradient field. For a binary system of two neutron stars this would mean that our conclusion is valid for distances much less than the binary separation, while one is still allowed to treat the two binaries in isolation \cite{Barausse:2017gip}.
Finally, we have also presented an example of how the corrections to the object's geodesic equation can be computed. The result given in~\eqref{res_viol} shows the scaling of the violations of the equivalence principle at a given distance $r$ from the extended object with respect to the various length scales, $r_V, r_g, H^{-1}$. This includes all the assumptions made in our derivation: $(i)$ subhorizon limit, $r\ll H^{-1}$; $(ii)$ weak gravitation, $r\gg r_g$; $(iii)$ distances less than the Vainshtein radius, $r\ll r_V$. By focusing on a specific scalar-tensor model one can derive this result to an arbitrary level of precision that can then be constrained by the gravitational wave measurements. We leave this for future work.
\section*{Acknowledgements}
We are particularly grateful to
Paolo Creminelli, J\'er\^ome Gleyzes
and Filippo Vernizzi
for many useful discussions and for collaboration in the early stages of this project. We thank Lam Hui and Jeremy Sakstein for insightful discussions. We further thank Claudia de Rham, Alberto Nicolis and Andrew Tolley for many useful comments on the manuscript. LA is supported by European Union's Horizon 2020 Research Council grant 724659 MassiveCosmo ERC-2016-COG and by the European Research Council under the European Union's Seventh Framework Programme (FP7/2007-2013),
ERC Grant agreement ADG 339140.
| 2024-02-18T23:39:45.160Z | 2019-01-23T02:26:42.000Z | algebraic_stack_train_0000 | 287 | 17,682 |
|
proofpile-arXiv_065-1592 | \section{Introduction}
In this paper, we work over the field $\ensuremath\mathds{C}$ of complex numbers. Let $X$ be a smooth projective variety, then the cohomology group of $X$ carries a Hodge structure given by
\[
\ensuremath\mathrm{H}^k(X,\ensuremath\mathds{Z})\otimes \ensuremath\mathds{C} = \bigoplus_{p+q=k} \ensuremath\mathrm{H}^{p,q}(X), \qquad \overline{\ensuremath\mathrm{H}^{p,q}(X)} = \ensuremath\mathrm{H}^{q,p}(X).
\]
The group of integral Hodge classes, denoted $\mathrm{Hdg}^{2p}(X,\ensuremath\mathds{Z})$, consists of all the elements $\alpha\in \ensuremath\mathrm{H}^{2p}(X,\ensuremath\mathds{Z})$ such that $\alpha\otimes 1 \in \ensuremath\mathrm{H}^{2p}(X,\ensuremath\mathds{Z})\otimes \ensuremath\mathds{C}$ is in the summand $\ensuremath\mathrm{H}^{p,p}(X)$. One easily sees that the torsion classes are all integral Hodge classes, \textit{i.e.}
\[
\ensuremath\mathrm{H}^{2p}(X,\ensuremath\mathds{Z})_{\mathrm{tor}} \subseteq \mathrm{Hdg}^{2p}(X,\ensuremath\mathds{Z}).
\]
W.~Hodge discovered that the cohomology class $[Z]$ of an algebraic cycle $Z$ on $X$ is always an integral Hodge class.
\begin{conj}[Integral Hodge Conjecture]
Every integral Hodge class is the cohomology class of an algebraic cycle.
\end{conj}
It is known since Atiyah--Hirzebruch \cite{ah} that the integral Hodge conjecture is false. Since then, many theories and techniques were developed to construct more counter-examples. In the recent paper \cite{bo}, Benoist and Ottem used a degeneration argument to show that certain integral Hodge class is not algebraic. In this paper, we generalize their method to produce more counter-examples.
Our method is based on the following simple observation. Let $Y$ be a smooth projective variety and $U\subseteq Y$ a dense open subvariety. If $Z_U$ is an algebraic cycle on $U$, then it extends to an algebraic cycle $Z$ on $Y$ by taking the closure. However, a (locally finite) toplological cycle $z_U$ on $U$ does not necessarily extend to one on $Y$. The main reason is that the closure of $z_U$ might have a nontrivial boundary. Hence a cohomology class being algebraic imposes stronger extension property on the class. We make the following definition to make the discussion easier.
\begin{defn}
Let $\pi: \mathcal{X}\rightarrow B$ be a flat projective morphism between smooth complete varieties. Let $0\in B$ be a closed point such that $X=\mathcal{X}_0:=\pi^{-1}0$ is a smooth fiber. Let $\alpha\in \ensuremath\mathrm{H}^{k}(X,R)$ be a cohomology class with coefficients in a commutative ring $R$. We say that $\alpha$ is \textit{extendable} if the following hold.
\begin{itemize}
\item There exists a smooth complete variety $\tilde{B}$ together with a generically finite morphism $\tilde{B}\rightarrow B$.
\item For some resolution $\tilde{\mathcal{X}}$ of $\mathcal{X}':=\mathcal{X}\times_B\tilde{B}$ and some preimage $\tilde{0}\in\tilde{B}$ of $0$, we have $X = \tilde{\mathcal{X}}_{\tilde{0}}:=\tilde\pi^{-1}(\tilde{0})$ where $\tilde{\pi}: \tilde{\mathcal{X}} \rightarrow \tilde{B}$ is the morphism induced by $\pi$.
\item There exists a cohomology class $\tilde{\alpha}\in \ensuremath\mathrm{H}^k(\tilde{\mathcal{X}}, R)$ such that $\alpha = \tilde{\alpha}|_X$.
\end{itemize}
\end{defn}
\begin{rmk}
Assume that $0\in B$ is a very general point. If $\alpha \in \ensuremath\mathrm{H}^{2p}(X,\ensuremath\mathds{Z})$ is the class of an algebraic cycle $Z$, then $\alpha$ is extendable. Indeed, one can identify $X=\pi^{-1}0$ with the geometric generic fiber $\mathcal{X}_{\bar\eta}$. The algebraic cycle $Z$ can then be defined over a finite extension of $\eta_B$. Then a standard argument shows that there exists some generically finite morphism $\tilde{B}\rightarrow B$ and an algebraic cycle $\mathcal{Z}'$ on ${\mathcal{X}}':=\mathcal{X}\times_B\tilde{B}$ such that $[\mathcal{Z}'_{\tilde{0}}] = \alpha$. The strict transform $\tilde{\mathcal{Z}}$ of $\mathcal{Z}'$ in the resolution $\tilde{\mathcal{X}}$ sastisfies $[\tilde{\mathcal{Z}}_{\tilde{0}}] = \alpha$. Thus we can simply take $\tilde{\alpha} = [\tilde{\mathcal{Z}}]$.
\end{rmk}
This remark gives rise to the following non-algebraicity criterion.
\vspace{3mm}
\noindent \textbf{(Non-algebraicity criterion)} \textit{If an integral Hodge class $\alpha$ is not extendable, then this class is not algebraic on a very general fiber.}
\vspace{3mm}
Our first main result is the following non-extendability of vanishing cycles on odd dimensional smooth hypersurfaces.
\begin{thm}[Theorem \ref{thm non-ext}]
Let $\mathcal{X}\longrightarrow B=\ensuremath\mathds{P}^1$ be a Lefschetz pencil of smooth hypersurfaces of odd dimension $n$. Let $X=\pi^{-1}0$ be a smooth fiber. Then every non-zero element $\alpha\in \ensuremath\mathrm{H}^n(X,R)$ is non-extendable, where $R$ is a nonzero commutative ring.
\end{thm}
This non-extendability can be used to obstruct algebraicity as follows. For simplicity, we take $S$ to be an Enriques surface. Then $\ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z})=\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}$ with a generator $u$.
\begin{cor}[Corollary \ref{cor non-alg}]
Let $X\subset \ensuremath\mathds{P}^{n+2}$ be a very general hypersurface of odd dimension $n$. For every element $\alpha\in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})$ which is not divisible by $2$, the torsion class $\alpha\otimes u\in \ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})$ is not extendable (in a Lefschetz pencil) and hence not algebraic.
\end{cor}
\begin{rmk}
The proof of the corollary reduces to the non-extendability of the image $\bar\alpha$ of $\alpha$ in $\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$; see section 3. In \cite{bo}, Benoist and Ottem considered the case where $X=E$ is a very general elliptic curve. Their method involves an element $\alpha\in \ensuremath\mathrm{H}^1(E,\ensuremath\mathds{Z})$. Instead of considering the topological extendability of $\bar\alpha \in \ensuremath\mathrm{H}^1(E,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$, they consider the degeneration of the double cover $E'\longrightarrow E$ associated to $\bar\alpha$. The obstruction in the Benoist--Ottem example was given an interpretation via unramified cohomology by Colliot-Th\'el\`ene \cite{ct}. It is interesting to see if a similiar interpretation exist for our generalisation.
The counter-examples to the integral Hodge conjecture obtained via the above corollary are all around the range of middle degree cohomology.
\end{rmk}
Our method also works when $X$ is a hyperplane section of a smooth projective variety $Y$. This more general case is treated in Theorem \ref{thm general case}. Our result shows that there exist integral Hodge classes which are not extendable. Given the outstanding Hodge conjecture, it is natural to ask whether every rational Hodge class on a very general fiber is extendable.
\vspace{3mm}
\noindent\textbf{Acknowledgement.} A large part of the computations in Section 2 were carried out in the summer of 2018 when I was visiting University of Science and Technology of China. I thank Mao Sheng for the invitation. I also thank John Ottem for the interesting discussions related to this paper. This research was partially supported by NWO Innovational Research Incentives Scheme 016.Vidi.189.015.
\section{Vanishing cycles under blow-up}
\subsection{An induction process}
Let $r\geq 2$ be an integer and let $X_r$ be a complex analytic space with a unique singular point $P_r$. Assume that $P_r$ has an open neighborhood $U_r$ such that
\[
U_r \cong \{(t,\mathbf{z})\in \ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}: t^r=z_0^2 + \cdots + z_n^2\}.
\]
Let $D_\epsilon^{n+1}:=\{\mathbf{x}\in\mathds{R}^{n+1}: |\mathbf{x}|\leq \epsilon\}$ be the closed disc. We have continuous maps
\begin{equation}\label{eq varphi_r}
\varphi_{r,a} : D^{n+1}_{\epsilon} \longrightarrow U_r, \qquad \varphi_{r,a}(\mathbf{x}) = (\xi_r^a|\mathbf{x}|^{2/r},\mathbf{x}),
\end{equation}
where $\xi_r=\exp(\frac{2\pi i}{r})$ and $a=0,1,\ldots, r-1$. Let $M'$ be the blow-up of $M=\ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}$ at the point $P=(0, \mathbf{0})$. Let $U'_r\subset M'$ be the strict transform of $U_r$ and $\rho: X'_r \rightarrow X_r$ be the resulting blow-up of $X_r$ at the point $P_r$. We write
\[
M\backslash \{(0,\mathbf{0})\}= V \cup V_0\cup \cdots \cup V_n, \quad V=\{t\neq 0\},\;\; V_i = \{z_i\neq 0 \}.
\]
Then $M'$ admits a corresponding open cover
\[
M' = V' \cup V'_0 \cup \cdots \cup V'_n.
\]
Here $V' \cong \ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}$ and the map $V' \rightarrow V\cup\{(0,\mathbf{0})\}$ is given by
\[
(t,w_0,\ldots,w_n) \mapsto (t, tw_0,\ldots,tw_n).
\]
Similarly, we have $V'_i\cong \ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}$ and the map $V'_i\rightarrow V_i\cup\{(0,\mathbf{0})\}$ is given by
\[
(t,w_0,\ldots, w_n) \mapsto (tw_i, w_0w_i, \ldots, w_{i-1}w_i, w_i, w_{i+1}w_i, \ldots, w_nw_i).
\]
The exceptional divisor $E$ of the blow-up $M'\rightarrow M$ is isomorphic to $\ensuremath\mathds{P}^{n+1}$ and the open cover
\[
E= (E\cap V')\cup\bigcup_{i=0}^n (E\cap V'_i)
\]
is the standard affine cover associated to the homogeneous coordinates $[T:Z_0:\cdots : Z_n]$ of $\ensuremath\mathds{P}^{n+1}$.
We have the following commutative diagram
\[
\xymatrix{
U'_r\ar[r]\ar[d]_{\rho} &M'\ar[d]\\
U_r\ar[r] &M
}
\]
Furthermore, $U'_r\cap V' \subset V'$ is defined by the equation
\[
t^{r-2} = w_0^2 + \cdots +w_n^2.
\]
Thus $U'_r\cap V'$ is smooth if $r=2,3$; it is singular at the point $(t,\mathbf{w}) = (0,\mathbf{0})$ if $r\geq 4$. The intersection $U'_r\cap V'_i$ is defined by the equation
\[
t^rw_i^{r-2} = w_0^2+\cdots + w_{i-1}^2 + 1 + w_{i+1}^2 +\cdots + w_n^2
\]
which is always smooth.
If $r=2$, then the exceptional divisor of $U'_r\rightarrow U_r$ is the smooth quadric
\[
Q=\{T^2 = Z_0^2+\cdots Z_n^2\} \subset E=\ensuremath\mathds{P}^{n+1}.
\]
If $r\geq 3$, then the exceptional divisor of $U'_r\rightarrow U_r$ is the singular quadric
\[
Q'=\{0=Z_0^2 + \cdots + Z_n^2\}\subset E=\ensuremath\mathds{P}^{n+1}.
\]
The singular point of $Q'$ is $P'_r:=[1:0:\cdots:0]\in\ensuremath\mathds{P}^{n+1}$.
The map $\varphi_{r,a}$ restricted to $D^{n+1}_\epsilon\backslash\{\mathbf{0}\}$ lifts to $V'$, which is given by
\[
\mathbf{x}=(x_0,\ldots, x_n) \mapsto (\xi_r^a|\mathbf{x}|^{2/r}, \xi_r^{-a}|\mathbf{x}|^{1-2/r}\theta(\mathbf{x})), \qquad\mathbf{x}\in D_{\epsilon}^{n+1}\backslash \{\mathbf{0}\},
\]
where $\theta(\mathbf{x})= \frac{\mathbf{x}}{|\mathbf{x}|}\in S^{n}$. If $r\geq 3$, then the above map extends to
\[
\varphi'_{r,a}: D^{n+1}_{\epsilon} \longrightarrow V'
\]
by the same formula and $\mathbf{0} \mapsto (0,\mathbf{0})\in V'$, which is the singular point of $Q'$. In this case, $X'_r$ is locally defined by the equation
\[
t^{r-2} = w_0^2 + \cdots + w_n^2.
\]
The following lemma implies that the same argument can be repeated on $(X_{r-2}, P_{r-2}) =(X'_r,P'_r)$.
\begin{lem}\label{lem induction}
(1) If $r=2$, then the lifting of $\varphi_{2,a}|_{D^{n+1}_\epsilon\backslash\{\mathbf{0}\}}$ to $X'_r$ can be extended to a continuous map
\[
\varphi'_{2,a} : [0,\epsilon]\times S^n \longrightarrow X'_r
\]
such that $\varphi'_{2,a}(\rho,\mathbf{x}) = \varphi_{2,a}(\rho\mathbf{x})$ for all $(\rho,\mathbf{x})\in (0,\epsilon]\times S^n$. Furthermore,
\[
\varphi'_{2,a}(0,\mathbf{x}) = [1:(-1)^ax_0:\cdots:(-1)^ax_n]\in Q,
\]
which is an $n$-sphere in $Q\subset E\cong\ensuremath\mathds{P}^{n+1}$ that vanishes in the homology of $\ensuremath\mathds{P}^{n+1}$. In this case, $X'_r$ is smooth.
(2) If $r = 3$, then the lifting of $\varphi_{r,a}|_{D^{n+1}_\epsilon\backslash\{\mathbf{0}\}}$ to $X'_r$ can be extended to a continuous map
\[
\varphi'_{r,a} : D^{n+1}_\epsilon \longrightarrow X'_r
\]
such that $\varphi'_{r,a}(\mathbf{0}) = P'_r$ is the singular point of $Q'$. In this case, $X'_r$ is smooth.
(3) If $r\geq 4$, then $X'_r$ is singular at the point $P'_r$ where $X'_r$ is locally defined by an equation
\[
{t'}^{r-2} = {z'}_0^2 + {z'}_1^2+\cdots + {z'}_n^2.
\]
The lifting of $\varphi_{r,a}|_{D^{n+1}_\epsilon\backslash\{\mathbf{0}\}}$ to $X'_r$ can be extended to a continuous map
\[
\varphi_{r-2,0}: D^{n+1}_{\epsilon'} \longrightarrow X'_r,
\]
with $\varphi_{r-2,0}(\mathbf{0}) = P'_r$ being the singular point of $X'_r$ and $\varphi_{r-2,0}(\mathbf{x}) = (|\mathbf{x}|^{\frac{2}{r-2}}, \mathbf{x})$ for general $\mathbf{x}$.
\end{lem}
\begin{proof}
For (1), we note that, in this case, the lifting of $\phi_{2,a}$ restricted to $D^{n+1}_{\epsilon}\backslash\{\mathbf{0}\}$ is given by
\[
\mathbf{x}=(x_0,\ldots,x_n) \mapsto ((-1)^a|\mathbf{x}|, (-1)^a\theta(\mathbf{x})).
\]
Note that $(0,\epsilon]\times S^n\cong D^{n+1}_{\epsilon}\backslash\{\mathbf{0}\}$. It is clear that the above map extends to a continuous map $\varphi'_{2,a}: [0,\epsilon]\times S^n$ as stated.
Statement (2) can be shown similarly.
We show the last statement and assume that $r\geq 4$. We have already seen that $X'_r$ has a unique singular point $P'_r$ such that $X'_r$ is locally defined by
\[
t^{r-2} = w_0^2 + w_1^2 + \cdots + w_n^2
\]
and that there is a lifting $\varphi'_{r,a}$ of $\varphi_{r,a}$ given by
\[
\varphi'_{r,a}(\mathbf{x}) = (\xi_r^a|\mathbf{x}|^{2/r}, \xi_r^{-a}|\mathbf{x}|^{1-2/r}\theta(\mathbf{x})), \quad \mathbf{x}\in D^{n+1}_\epsilon\backslash \{\mathbf{0}\}
\]
and $\varphi'_{r,a}(\mathbf{0}) = (0,\mathbf{0})=P'_r$. We introduce a new set of coordinates
\begin{align*}
t' &= \xi_r^{-a}t,\\
z'_i & = \xi_r^a w_i
\end{align*}
and we see that the local defining equation of $X'_r$ around $P'_r$ becomes
\[
{t'}^{r-2} = {z'}_0^2 + {z'}_1^2+\cdots + {z'}_n^2.
\]
Furthermore, in terms of $(t',\mathbf{z}')$, the map $\varphi'_{r,a}$ becomes
\[
\mathbf{x}\mapsto (|\mathbf{x}|^{2/r},|\mathbf{x}|^{1-2/r}\theta(\mathbf{x}))
\]
for $\mathbf{x}\neq\mathbf{0}$ and $\mathbf{0}\mapsto (0,\mathbf{0})$. Let $\epsilon' = \epsilon^{1-2/r}$ and define a homeomorphism $D^{n+1}_\epsilon \rightarrow D^{n+1}_{\epsilon'}$ by $\mathbf{x}\mapsto \mathbf{x}' = |\mathbf{x}|^{-2/r}\mathbf{x}$ for $\mathbf{x}\neq \mathbf{0}$ and $\mathbf{0}\mapsto \mathbf{x}'=\mathbf{0}$. It follows that the composition $D^{n+1}_{\epsilon'}\rightarrow D^{n+1}_{\epsilon}\rightarrow X'_r$ becomes
\[
\mathbf{x}' \mapsto (|\mathbf{x}'|^{\frac{2}{r-2}}, \mathbf{x}').
\]
This concludes the proof.
\end{proof}
\subsection{Application to vanishing cycles}
\subsection{Local situation}
Let $\Delta\subset\ensuremath\mathds{C}$ be the unit open disc in the complex plane and $\Delta^*=\Delta\backslash \{0\}$. Let $\pi: X\rightarrow \Delta$ be a proper map of complex manifolds such that $X^*\rightarrow \Delta^*$ is smooth, where $X^*=\pi^{-1}\Delta^*$. We write $X_t:=\pi^{-1}t$, $t\in \Delta$. Assume that $X_0=\pi^{-1}(0)$ has one ordinary double point $P$ such that we have local coordinates $(z_0,\ldots, z_n)$ on an open neighborhood $U$ of $P$ and
\[
\pi(z_0,z_1,\ldots, z_n) = z_0^2 + z_1^2 + \cdots + z_n^2.
\]
Let $\psi_r:\Delta \rightarrow \Delta$ be the map $\psi_r(t) = t^r$ and $X_r:= \psi_r^* X$ be the base change of $X$, where $r\geq 2$. Namely, we have the following fiber product quare
\[
\xymatrix{
X_r\ar[r]^{\psi'_r}\ar[d]_{\pi_r} &X\ar[d]^{\pi}\\
\Delta\ar[r]^{\psi_r} &\Delta
}
\]
Let $U_r=\psi_r^* U$ be the corresponding base change of $U$. Thus $U_r$ is an open neighborhood of the point $P_r={\psi'_r}^{-1}(P)$. Hence we have $U_r\subset \ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}$ defined be
\[
U_r= \Set{(t,z_0,\ldots,z_n)\in \ensuremath\mathds{C}\times \ensuremath\mathds{C}^{n+1}}{t^r = z_0^2 + z_1^2 + \cdots + z_n^2}.
\]
Then $P_r$ is the unique singular point of $U_r$ and it has coordinates $(0,0,\ldots,0)$.
For any positive real number $\epsilon \in (0,1)$, let
\[
S_{\epsilon}^{n}=\{(\epsilon^2, x_0,\ldots, x_n)\in U_r: x_0^2+ \cdots + x_n^2=\epsilon^2, x_i\in \mathds{R} \}\subset X_{\epsilon^2}
\]
be a vanishing sphere. Let
\[
D_{\epsilon}^{n+1} \rightarrow {X}, \quad (x_0,\ldots,x_n)\mapsto (\sum x_i^2, x_0, \ldots, x_n)
\]
where $D_{\epsilon}^{n+1}=\{(x_0,\ldots,x_n)\in \ensuremath\mathds{R}^{n+1}: x_0^2+\cdots + x_n^2\leq \epsilon^2\}$ is a small disc whose boundary gives the vanishing shpere $S_{\epsilon}^n$.
Let $\rho: X'_{r}\rightarrow X_r$ be the blow-up of $X_r$ at the point $P_r$ and let $U'_r := \rho^{-1} U_r$. There are $r$ different ways to lift the map $D^{n+1}_{\epsilon}\backslash \{0\}\rightarrow {X}^*$ to $(X'_r)^*$ given by
\begin{equation}
\mathbf{x}=(x_0,\ldots,x_n) \mapsto (\xi_r^a|\mathbf{x}|^{2/r}, x_0,\ldots, x_n),\quad a=0,1,\ldots,r-1.
\end{equation}
We have seen in Lemma \ref{lem induction} that $X'_r$ is again singular if $r\geq 4$ with a single singular point $P'_r$ and the blow-up proess can be repeated.
\begin{lem}\label{lem local vc}
The following statements are true.
(1) The singularity of $X_r$ can be resolved by successively blowing up the singular points
\[
\tilde{X} = X_r^{(b)}\longrightarrow \cdots \longrightarrow X_r^{(2)}\longrightarrow X_r^{(1)}=X'_r\longrightarrow X_r
\]
where $b=[\frac{r}{2}]$ and $X^{(b)}_r$ is smooth.
(2) Let $Q\subseteq \tilde{X}$ be the exceptional divisor of the last blow-up $X^{(b)}_r\longrightarrow X_r^{(b-1)}$. Then $Q$ is a component of $\tilde{X}_0 = \tilde{\pi}^{-1}(0)$, where $\tilde{\pi}:\tilde{X}\longrightarrow \Delta$ is the composition of all the blow-ups together with $\pi_r$. If $r=2b$ is even, then $Q$ is a smooth quardric hypersurface of dimension $n$; if $r=2b+1$ is odd, then $Q$ is a cone over a smooth quadric hypersurface of dimension $n-1$.
(3) If $n$ is odd, then any of the $r$ liftings of the vanishing sphere $S^n_{\epsilon} \subset X_{\epsilon^2}$ to $\tilde{X}$ vanishes in $\ensuremath\mathrm{H}_n(\tilde{X},\ensuremath\mathds{Z})$.
(4) If $n$ is even and $r$ is odd, then any of the $r$ liftings of the vanishing sphere $S^n_{\epsilon} \subset X_{\epsilon^2}$ to $\tilde{X}$ vanishes in $\ensuremath\mathrm{H}_n(\tilde{X},\ensuremath\mathds{Z})$.
(5) If $n$ is even and $r$ is also even, then any of the $r$ the liftings of the vanishing sphere $S^n_{\epsilon} \subset X_{\epsilon^2}$ to $\tilde{X}$ is homologous to some sphere $S^n\subset Q$. Furthermore, the sphere $S^n$ vanishes in $\ensuremath\mathrm{H}_n(\ensuremath\mathds{P}^{n+1},\ensuremath\mathds{Z})$ under the embedding $Q\hookrightarrow \ensuremath\mathds{P}^{n+1}$ of $Q$ as a quadric hypersurface.
\end{lem}
\begin{proof}
Most of the statements are direct application of Lemma \ref{lem induction}. We only need to prove (3) when $r$ is even. In this case, by (1) of Lemma \ref{lem induction}, we know that the lifting of $S^n_{\epsilon}\subset X_{\epsilon^2}$ to $\tilde{{X}}$ is homologous to an $n$-sphere $S^n\subset Q$. Thus the homology class of the lifting of $S^n_\epsilon$ lands in the image of
\[
\ensuremath\mathrm{H}_n(Q,\ensuremath\mathds{Z}) \longrightarrow \ensuremath\mathrm{H}_n(\tilde{X},\ensuremath\mathds{Z}).
\]
When $n$ is odd, we have $\ensuremath\mathrm{H}_n(Q,\ensuremath\mathds{Z}) = 0$ since a smooth quadric has trivial homology goup in odd degree. Thus we obtain the vanishing in (3).
\end{proof}
\subsection{Global situation}
Let $\mathcal{X}$ be a smooth algeraic variety of dimension $n+1$ and $B$ a smooth curve. Let $\pi: \mathcal{X} \longrightarrow B$ be a proper morphism such that the following conditions holds.
\begin{itemize}
\item There exists a set $S=\{b_1,b_2, \ldots, b_m\}\subset B$ of finitely many points such that $X_{b_i}=\pi^{-1}b_i$ contains exactly one isolated singular point $P_i$ which is an ordinary double point.
\item The morphism $\pi$ is smooth over $B\backslash S$.
\end{itemize}
Let $0\in B$ be a point not in $S$ and let $X=X_0$. Thus $X$ is a smooth complete variety over $\ensuremath\mathds{C}$. Let $\Delta_i\subset B$ be a small disc centered at $b_i$. Let $\Delta_i^* := \Delta_i\backslash\{b_i\}$.
\begin{defn}
A sphere $S_\epsilon^n\subset X_{t_i}$, $t_i\in\Delta_i^*$, is called a \textit{vanishing sphere} associated to $P_i$ if the following conditions hold: (1) there exist local coordinates $\mathbf{z}=(z_0,z_1,\ldots,z_n)$ of $\mathcal{X}$ at $P_i=(0,0,\ldots,0)$; (2) there is a local coordinate $t$ on $\Delta_i$ such that $\pi$ is locally given by
$$
t=\pi (\mathbf{z}) = z_0^2+ z_1^2 + \cdots + z_n^2;
$$
(3) with the above coordinates, we have $t_i=\epsilon^2$ and $S^n_{\epsilon}$ is given by all points $\mathbf{z} = (x_0, x_1,\ldots,x_n)$ with $x_i\in\ensuremath\mathds{R}$ and $\sum x_i^2=\epsilon^2$.
\end{defn}
Let $\gamma: [0,1] \longrightarrow B\backslash S$ be a continous path such that $\gamma(0)=0$ and $\gamma(1) = t_i\in \Delta_i^*$ for some $i$. Then
\[
\gamma_* : \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z}) \longrightarrow \ensuremath\mathrm{H}_n(X_{t_i},\ensuremath\mathds{Z})
\]
is an isomorphism.
\begin{defn}
We say that a class $\alpha\in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})$ is a \textit{primitive vanishing class} if there exists a path $\gamma:[0,1]\longrightarrow B\backslash S$ as above such that $\gamma_*\alpha\in \ensuremath\mathrm{H}_n(X_{t_i},\ensuremath\mathds{Z})$ is the class of a vanishing sphere associated to $P_i$. A class $\alpha'\in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})$ is a \textit{vanishing class} (associated to $\mathcal{X}/B$) if it is an integral linear combination of primitive vanishing classes.
\end{defn}
\begin{prop}\label{prop global vc}
Let $\mathcal{X}$ be a smooth algeraic variety and $B$ a smooth curve. Let $\pi: \mathcal{X} \longrightarrow B$ be a proper morphism as above. Let $0\in B\backslash S$ and $X= X_0$. Let $\tilde{B}$ be another smooth curve and let $f: \tilde{B}\rightarrow B$ be a finite morphism. Let $\mathcal{X}':=\mathcal{X}\times_B \tilde{B}$ be the base change of $\mathcal{X}$ and let $\tilde{\mathcal{X}}$ be a resolution of $\mathcal{X}'$. Let $\tilde{\pi}: \tilde{\mathcal{X}}\longrightarrow \tilde{B}$ be the resulting morphism induced from the morphism $\pi$. Let $\tilde{0}\in \tilde{B}$, such that $f(\tilde{0})=0$ and hence $X\cong \tilde{\pi}^{-1}\tilde{0}$. Let $j: X\hookrightarrow \tilde{\mathcal{X}}$ be the embedding. Let $\alpha\in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})$ be a vanishing class associated to $\mathcal{X}/B$.
(1) If $n$ is odd, then $j_*\alpha = 0$ in $\ensuremath\mathrm{H}_n(\tilde{\mathcal{X}},\ensuremath\mathds{Z})$.
(2) If $n$ is even and $\tilde{\mathcal{X}}$ is obtained by successively blowing up the singular points, then $j_*\alpha$ is in the image of
\[
\bigoplus_{l=1}^N \ensuremath\mathrm{H}_n(Q_{l},\ensuremath\mathds{Z})_{\mathrm{van}}\longrightarrow \ensuremath\mathrm{H}_n(\tilde{\mathcal{X}},\ensuremath\mathds{Z}),
\]
where $Q_l$ runs through all smooth qudric hypersufaces appearing as components of the exceptional set of the morphism $\tilde{\mathcal{X}}\longrightarrow \mathcal{X}'$ and $\ensuremath\mathrm{H}_n(Q_l,\ensuremath\mathds{Z})_{\mathrm{van}}$ consists of classes $\beta\in \ensuremath\mathrm{H}_n(Q_l,\ensuremath\mathds{Z})$ that vanish in $\ensuremath\mathrm{H}_n(\ensuremath\mathds{P}^{n+1},\ensuremath\mathds{Z})$ under the natural embedding $Q_l\subset \ensuremath\mathds{P}^{n+1}$.
\end{prop}
\begin{proof}
We first look at the local behaviour of the morphism $f: \tilde{B}\longrightarrow B$ around a point $b_i\in S$. Assume that
\[
f^{-1}b_i = \{b'_{i,1}, b'_{i,2}, \ldots, b'_{i,m_i}\}\subset \tilde{B}.
\]
For each point $b'_{i,l}$, we can find a small disc $\Delta_{i,l}\subset \tilde{B}$ centered at $b'_{i,l}$ such that the morphism $f:\tilde{B}\longrightarrow B$ restricts to the analytic map
\[
f_{i,l}: \Delta_{i,l}\longrightarrow \Delta_i,\qquad z\mapsto z^{r_{i,l}}.
\]
To prove the proposition, we first assume that the resolution $\tilde{\mathcal{X}}\longrightarrow \mathcal{X}'$ is the one obtained by successively blowing up the singular points. Without loss of generality, we may assume that $\alpha$ is a primitive vanishing class. Thus there is a path $\gamma: [0,1]\longrightarrow B\backslash S$ with $\gamma(0) = 0\in B$ and $\gamma(1)=t_i \in \Delta_i^*$ such that $\gamma_* \alpha$ is the class of a vanishing sphere in $X_{t_i}$. We may choose $\gamma$ in such a way that it avoids all the branching points of the morphism $f$. Thus there exists a unique lifting $\tilde{\gamma}: [0,1]\longrightarrow \tilde{B}$ such that $\tilde{\gamma}(0) = \tilde{0}$. Then $\tilde\gamma(1)=\tilde{t}_i\in \Delta_{i,l}^*$ for some $l\in \{1,2,\ldots,m_i\}$. Furthermore, $\tilde\gamma_*\alpha$ is the class of a lifting $\tilde{S}^n_\epsilon$ of the vanishing sphere $S^n_{\epsilon}$ in $X_{t_i}$. If $n$ is odd, then by Lemma \ref{lem local vc} (3) we know that the homology class of $\tilde{S}^n_\epsilon$ vanishes in $\ensuremath\mathrm{H}_n (\tilde\pi^{-1}\Delta_{i,l}, \ensuremath\mathds{Z})$ and hence also in $\ensuremath\mathrm{H}_n(\tilde{\mathcal{X}},\ensuremath\mathds{Z})$. Similarly, if $n$ is even, we conclude from Lemma \ref{lem local vc} (4) and (5).
Now assume $n$ is odd. We still need to establish the vanishing on an arbitrary resolotion $\tilde{\mathcal{X}}_1$ of $\mathcal{X}'$. Let $\tilde{\mathcal{X}}$ be the resolution of $\mathcal{X}'$ obtained by successively blowing up the singular points. Then we can find another resolution $\tilde{\mathcal{X}}_2$ which dominates both $\tilde{\mathcal{X}}$ and $\tilde{\mathcal{X}}_1$, namely we have a diagram
\[
\xymatrix{
&\tilde{\mathcal{X}}_2\ar[ld]_{\tau}\ar[rd]^{\tau'} &\\
\tilde{\mathcal{X}} &&\tilde{\mathcal{X}}_1
}
\]
Let $j_1: X\hookrightarrow \tilde{\mathcal{X}}_1$, $j_2: X\hookrightarrow \tilde{\mathcal{X}}_2$ and $j: X\hookrightarrow \tilde{\mathcal{X}}$ be the inclusion of the fiber over $\tilde{0}\in \tilde{B}$ in the corresponding models. Set $\alpha_1=j_{1,*}\alpha$, $\alpha_2=j_{2,*}\alpha$ and $\tilde{\alpha}=j_*\alpha$ to be the corresponding homology classes. We have already see that $\tilde\alpha=0$. Since these models are isomorhic on an open neighborhood of the fiber $X$. We have
$\alpha_2 = \tau^*\tilde{\alpha}=0$ and $\alpha_1 = \tau'_*\alpha_2=0$.
\end{proof}
\section{Applications to the integral Hodge conjecture}
In this section, we construct a class of new examples of the failure of the integral Hodge conjecture. These generalises the examples of Benoist--Ottem \cite{bo}.
Let $S$ be an Enriques surface. The cohomology groups of $S$ are described as follows.
\begin{align*}
\ensuremath\mathrm{H}^0(S,\ensuremath\mathds{Z}) = \ensuremath\mathds{Z},\qquad &\ensuremath\mathrm{H}^0(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}/2\ensuremath\mathds{Z},\\
\ensuremath\mathrm{H}^1(S,\ensuremath\mathds{Z}) = 0,\qquad &\ensuremath\mathrm{H}^1(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) =\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}, \\
\ensuremath\mathrm{H}^2(S,\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}^{\oplus 10}\oplus \ensuremath\mathds{Z}/2\ensuremath\mathds{Z},\qquad &\ensuremath\mathrm{H}^2(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) = (\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})^{\oplus 12} ,\\
\ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}/2\ensuremath\mathds{Z},\qquad &\ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}/2\ensuremath\mathds{Z},\\
\ensuremath\mathrm{H}^4(S,\ensuremath\mathds{Z}) =\ensuremath\mathds{Z},\qquad &\ensuremath\mathrm{H}^4(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}/2\ensuremath\mathds{Z}.
\end{align*}
\subsection{Special case: hypersurfaces}
Let $X\subset \ensuremath\mathds{P}^{n+1}$ be a smooth hypersurface. Assume that the dimension $n$ of $X$ is odd. By Lefschetz Hyperplane Theorem, we know that
\[
\ensuremath\mathrm{H}^p(\ensuremath\mathds{P}^{n+1},\ensuremath\mathds{Z}) \longrightarrow \ensuremath\mathrm{H}^p(X,\ensuremath\mathds{Z})
\]
is an isomorphism for $p<n$. Thus $\ensuremath\mathrm{H}^{<n}(X,\ensuremath\mathds{Z})$ is torsion free and algebraic. Similarly, $\ensuremath\mathrm{H}_{<n}(X,\ensuremath\mathds{Z})$ is torsion free. By Serre duality, we conclude that $\ensuremath\mathrm{H}^{>n}(X,\ensuremath\mathds{Z})$ is also torsion free and so is $\ensuremath\mathrm{H}_{>n}(X,\ensuremath\mathds{Z})$. Then by the universal coefficient thoerem for cohomology, we see that
\[
\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}) \cong \Hom_{\ensuremath\mathds{Z}}(\ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z}), \ensuremath\mathds{Z})\; \oplus \;\ensuremath\mathrm{H}_{n-1}(X,\ensuremath\mathds{Z})_{\mathrm{tor}}
\]
is also torsion free. Hence we conclude that both $\ensuremath\mathrm{H}^*(X,\ensuremath\mathds{Z})$ and $\ensuremath\mathrm{H}_*(X,\ensuremath\mathds{Z})$ are torsion free.
\begin{lem}
The following equality holds
\[
\ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z}) = \bigoplus_{i=0}^4 \ensuremath\mathrm{H}^{n+3-i}(X,\ensuremath\mathds{Z})\otimes\ensuremath\mathrm{H}^i (S,\ensuremath\mathds{Z}).
\]
\end{lem}
\begin{proof}
The K\"unneth formula applied to this case gives
\[
\ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})=
\left( \bigoplus_{i=0}^4 \ensuremath\mathrm{H}^{n+3-i}(X,\ensuremath\mathds{Z})\otimes\ensuremath\mathrm{H}^i (S,\ensuremath\mathds{Z})\right) \oplus \left( \bigoplus_{i=0}^4 \mathrm{Tor}_1\Big( \ensuremath\mathrm{H}^{n+4-i}(X,\ensuremath\mathds{Z}), \ensuremath\mathrm{H}^i (S,\ensuremath\mathds{Z}) \Big)\right)
\]
Since the cohomology of $X$ is torsion free, we see that in the $\mathrm{Tor}_1$-term vanishes.
\end{proof}
\begin{thm}\label{thm non-ext}
Let $\pi: \mathcal{X} \longrightarrow B=\ensuremath\mathds{P}^1$ be a Lefschetz pencil of smooth hypersurfaces of odd dimension $n$. Let $X=\pi^{-1}0$ be a smooth fiber. Then any non-zero element $\alpha \in \ensuremath\mathrm{H}^n(X,R)$ is non-extendable, where $R$ is a non-zero commutative ring.
\end{thm}
\begin{proof}
Assume that $\alpha$ is extendable. Then there exists a smooth projective curve $\tilde{B}$ and a finite morphism $\tilde{B}\longrightarrow B$ such that a resolution $\tilde{\mathcal{X}}$ of the base change $\mathcal{X}'=\mathcal{X}\times_{B}\tilde{B}$ is obtained by successively blowing up the singular points. Let $\tilde{\pi}:\tilde{\mathcal{X}}\longrightarrow \tilde{B}$ be the induced morphism. Furthermore, we have a cohomology class $\tilde{\alpha} \in \ensuremath\mathrm{H}^n(\tilde{\mathcal{X}}, R)$ such that $\tilde{\alpha}|_X = \alpha$, where $X=\tilde{\pi}^{-1}\tilde{0}$ for some preimage $\tilde{0}\in \tilde{B}$ of $0$. Let $j: X \hookrightarrow \tilde{\mathcal{X}}$ be the inclusion. Let $\beta\in\ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})$. Since $n$ is odd, we know that $\beta$ vahishes in $\ensuremath\mathrm{H}_n(\ensuremath\mathds{P}^{n+1},\ensuremath\mathds{Z})$. By Lefschetz theory (see for example \cite{lam}), we know that $\beta$ is a vanishing class associated to the Lefschetz pencil $\mathcal{X}\longrightarrow B$. Then by (1) of Proposition \ref{prop global vc}, we see that $j_*\beta =0$ in $\ensuremath\mathrm{H}_n(\tilde{\mathcal{X}},\ensuremath\mathds{Z})$. Thus
\[
\langle\alpha,\beta\rangle = \langle j^*\tilde\alpha,\beta \rangle = \langle \tilde{\alpha}, j_*\beta \rangle = 0.
\]
This forces that $\alpha = 0$ since $\ensuremath\mathrm{H}^n(X,R) = \Hom_{\ensuremath\mathds{Z}}(\ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z}), R)$.
\end{proof}
\begin{cor}\label{cor non-alg}
Let $i: X\hookrightarrow \ensuremath\mathds{P}^{n+1}$ be a smooth hypersurface of odd dimension $n$ and let $S$ be an Enriques surface. Let $\alpha\in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})$ be an element not divisible by $2$ and let $u\in \ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z})$ be the unique nonzero element. Then the torsion class $\alpha\otimes u\in \ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})$ is non-extendable in $\mathcal{X}\times S \longrightarrow B$ for any Lefschetz pencil $\mathcal{X}\longrightarrow B$ containing $X$. If $X$ is very genery, then $\alpha\otimes u$ is not algebraic.
\end{cor}
\begin{proof}
Let $\pi: \mathcal{X}\longrightarrow B=\ensuremath\mathds{P}^1$ be a Lefschetz pencil of hypersurfaces of dimension $n$ such that for some point $0\in B$ the corresponding fiber $X_0:=\pi^{-1}0\cong X$. Assume that $\alpha\otimes u$ is extendable. As in the above proof, there exist a smooth projective curve $\tilde{B}$, a fninite morphism $\tilde{B}\rightarrow B$, a resolution $\tilde{\mathcal{X}}$ of $\mathcal{X}':=\mathcal{X}\times_B \tilde{B}$, an identification $X=\tilde{\pi}^{-1}(\tilde{0})$ and a cohomology class $\tau \in \ensuremath\mathrm{H}^{n+3}(\tilde{\mathcal{X}}\times S,\ensuremath\mathds{Z})$ such that $\tau|_{X\times S} = \alpha\otimes u$. Consider the class $\bar{\tau}\in \ensuremath\mathrm{H}^{n+3}(\tilde{\mathcal{X}}\times S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ which is obtained from $\tau$ modulo 2. Let $u'\in \ensuremath\mathrm{H}^1(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ be the unique non-zero element which is associated to the $K3$ covering $\tilde{S}\longrightarrow S$. Then we get
\[
\tilde{\alpha} := \bar{\tau}^* u' \in \ensuremath\mathrm{H}^n(\tilde{\mathcal{X}},\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})
\]
which stisfies the following condition
\[
\tilde{\alpha}|_{X} = (\bar{\tau}^* u')|_{X} = (\bar{\tau}|_{X\times S})^*u' = (\bar\alpha \otimes \bar{u})^*u' = \bar\alpha
\]
where $\bar\alpha$ is the image of $\alpha$ in $\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ and $\bar{u}$ is the image of $u$ in $\ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$. The last equality uses the duality relation $\langle \bar{u}, u'\rangle =1$. It follows that $\bar{\alpha}\in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ is extendable. By the above theorem, we have $\bar{\alpha} = 0$ and hence $\alpha$ is divisible by $2$ in $\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})$. This give a contradiction.
It follows that $\alpha\otimes u$ is not algebraic for a very general memeber $X$ in a Lefschetz pencil. In particular, this holds for a very general $X$.
\end{proof}
\subsection{General case: hyperplane sections}
Let $Y$ be a smooth projective variety with a very ample line bundle $\ensuremath\mathcal{O}_Y(1)$ which gives rise to an embedding $Y\hookrightarrow \ensuremath\mathds{P}^N$. The same argument as above gives the following.
\begin{thm}\label{thm general case}
Let $\pi:\mathcal{X}\rightarrow B=\ensuremath\mathds{P}^1$ be a Lefschetz pencil in $|\ensuremath\mathcal{O}_Y(1)|$. Let $X=\pi^{-1}0$ be a smooth fiber and let $i: X\hookrightarrow Y$ be the embedding. Assume that $\dim Y = n+1$ where $n$ is an odd integer. Let $R$ be a nonzero commutative ring.
(1) If $\alpha\in \ensuremath\mathrm{H}^n(X,R)$ is extendable, then we have
\[
\langle \alpha,\beta \rangle = 0,
\]
for all $\beta\in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})_{\mathrm{van}}:=\ker \{i_*: \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})\longrightarrow \ensuremath\mathrm{H}_n(Y,\ensuremath\mathds{Z})\}$. Furthermore, if $\ensuremath\mathrm{H}_{n}(Y,\ensuremath\mathds{Z})$ vanishes and $\ensuremath\mathrm{H}_{n-1}(Y,\ensuremath\mathds{Z})$ is torsion-free, then every nonzero element $\alpha \in \ensuremath\mathrm{H}^n(X,R)$ is non-extendable.
(2) Let $S$ be an Enriques surface and $u\in \ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z})$ be the unique nonzero element. Let $\alpha \in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})$. If $\alpha\otimes u$, viewed as an element in $\ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})$, is extendable (in the family $\mathcal{X}\times S\rightarrow B$), then $\bar\alpha\in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ is extendable.
(3) Assume that $\ensuremath\mathrm{H}_n(Y,\ensuremath\mathds{Z})$ vanishes and that $\ensuremath\mathrm{H}_{n-1}(Y,\ensuremath\mathds{Z})$ is torsion-free. If $X$ is very general in $|\ensuremath\mathcal{O}_Y(1)|$, then for all $\alpha\in \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})$ the class $\alpha\otimes u \in \ensuremath\mathrm{H}^{n+3}(X,\ensuremath\mathds{Z})$ is not algebraic unless it is zero.
\end{thm}
\begin{proof}
We will use the notations $\tilde{B}$, $\mathcal{X}'$, $\tilde{\mathcal{X}}$, $j: X=\tilde{\pi}^{-1}\tilde{0} \hookrightarrow \tilde{\mathcal{X}}$ as in the previous proofs.
(1) If $\alpha$ is extendable, then there exists $\tilde{\alpha}\in \ensuremath\mathrm{H}^n(\tilde{\mathcal{X}}, R)$ such that $\alpha = j^*\tilde{\alpha}$. Then we again have
\[
\langle \alpha,\beta\rangle = \langle j^*\tilde\alpha,\beta\rangle = \langle \alpha,j_*\beta\rangle =0,
\]
since $j_*\beta = 0$ for all $\beta\in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})_{\mathrm{van}}$ by Lefschetz theory and (1) of Proposition \ref{prop global vc}. Assume that $\ensuremath\mathrm{H}_n(Y,\ensuremath\mathds{Z}) = 0$, then we have
\[
\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})_{\mathrm{van}} = \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z}).
\]
If $\ensuremath\mathrm{H}_{n-1}(Y,\ensuremath\mathds{Z})$ is torsion free, then by Lefschetz hyperplane theorem, we know that $\ensuremath\mathrm{H}_{n-1}(X,\ensuremath\mathds{Z})$ is also torsion free. Then the universal coefficient theorem for cohomology becomes
\[
\ensuremath\mathrm{H}^n(X,R) = \Hom_{\ensuremath\mathds{Z}}(\ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z}), R).
\]
Thus the vanishing of $\langle\alpha,\beta\rangle = 0$ for all $\beta \in \ensuremath\mathrm{H}_n(X,\ensuremath\mathds{Z})$ implies $\alpha = 0$ in $\ensuremath\mathrm{H}^n(X,R)$.
(2) and (3): the proof is the same as that of Corollary \ref{cor non-alg}. One only needs to note that ,under the assumptions of (3), the group $\ensuremath\mathrm{H}^{n+1}(X,\ensuremath\mathds{Z})$ is also torsion free by Poincar\'e duality. Thus the universal coefficient theorem implies
\[
\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z}) = \ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})\otimes \ensuremath\mathds{Z}/2\ensuremath\mathds{Z}.
\]
Thus $\bar\alpha =0 $ in $\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z}/2\ensuremath\mathds{Z})$ if and only if $\alpha\otimes u =0$ in $\ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})$ since the K\"unneth formula gives
\[
\ensuremath\mathrm{H}^n(X,\ensuremath\mathds{Z})\otimes \ensuremath\mathrm{H}^(S,\ensuremath\mathds{Z}) \hookrightarrow \ensuremath\mathrm{H}^{n+3}(X\times S,\ensuremath\mathds{Z})
\]
and $\ensuremath\mathrm{H}^3(S,\ensuremath\mathds{Z}) = \ensuremath\mathds{Z}/2\ensuremath\mathds{Z}$. Then (3) follows from (1) and (2)
\end{proof}
| 2024-02-18T23:39:45.409Z | 2019-01-23T02:27:48.000Z | algebraic_stack_train_0000 | 295 | 6,729 |
|
proofpile-arXiv_065-1811 | \section{Introduction}
For a positive integer $\ell$, we denote by $[\ell]$ the set $\{1,\dots,\ell\}$.
Given $k\geq 2$, for a finite set $V$, we use $[V]^k$ to denote the collection of all subsets of $V$ of size $k$.
A \textit{$k$-uniform hypergraph} $H$ ($k$-graph for short) is a pair $H=(V(H),E(H))$ where $V(H)$ is a finite set of \textit{vertices} and $E(H)\subseteq [V(H)]^k$ is a set of \textit{edges}. As usual $2$-graphs are simply called graphs.
We simply write $v(H)=|V(H)|$ and $e(H)=|E(H)|$. Let $\partial H:=\{S\in [V(H)]^{k-1}: \exists \ e\in E(H), S\subseteq e\}$ be the \emph{shadow} of $H$.
Given a $k$-graph $F$, a \textit{$k$-partite realisation} of $F$ is a partition of $V(F)$ into vertex classes $V_1, \dots, V_k$ so that for any $e\in E(F)$ and $i\in [k]$ we have $|e\cap V_i| =1$. Clearly, we must have $|V_i| \ge 1$ for every $i\in [k]$. In particular, we say that $F$ is \textit{$k$-partite} if it admits a $k$-partite realisation.
Given a $k$-graph $H=(V, E)$ and a set $S$ of $s$ vertices in $V$ with $1\le s\le k-1$, let $\deg_H(S)$ (or $\deg(S)$) denote the number of edges of $H$ containing $S$ as a subset. Let $N_H(S)$ (or $N(S)$) denote the neighbor set of $S$, i.e., $N_H(S)=\{ S' \in [V]^{k-s}\ : S'\cup S\in E\}$.
The \textit{minimum $s$-degree} $\delta_s(H)$ is the minimum of $\deg(S)$ over all $s$-subsets $S$ of $V$. In particular, we call the minimum $1$-degree and the minimum $(k-1)$-degree of $H$ as the \textit{minimum degree} and \textit{minimum codegree} of $H$ respectively. If $S=\{v\}$ is a singleton, then we simply write $\deg(v)$ and $N(v)$ instead.
In this paper, we focus on a classical extremal problem, the perfect tiling problem.
Given two $k$-graphs $F$ and $H$, a \textit{perfect $F$-tiling (or $F$-factor)} in $H$ is a set of vertex disjoint copies of $F$ that together covers the vertex set of $H$.
Furthermore, if $H$ does not contain copies of $F$, we say $H$ is \textit{$F$-free}.
The study of perfect tilings in graph theory has a long and profound history, ranging from classical results back to Corradi--Hajnal~\cite{Corr} and Hajnal--Szemer\'edi~\cite{zbMATH03344609} on $K_k$-factors to the celebrated result of Johansson--Kahn--Vu~\cite{Johansson2008Factors} on perfect tilings in random graphs.
We note that the $F$-factor problem for quasi-random graphs with positive density and minimum degree $\Omega(n)$ has been solved implicitly by Koml\'{o}s--S\'{a}rk\"{o}zy--Szemer\'{e}di~\cite{Koml1997Blow} in the course of developing the famous Blow-up Lemma.
\subsection{Quasi-random graphs and hypergraphs}
The study of quasi-random graphs was launched in late 1980s by Chung, Graham and Wilson~\cite{Chung-quasi-random}.
These are constant density graphs which behave like the random graphs.
There is a list of properties that force a graph to be quasi-random and it was shown that these properties are all equivalent (up to the dependency of constants).
The investigation of quasi-random $k$-graphs was started by Chung and Graham~\cite{chung1990quasi} and is widely popular \cite{quasihyper,Ch90,Ch91,Chung10,wquasi,G06,G07,KNRS,KRS02,LenzMubayi_eig,lenz2015poset,RSk04,Towsner}.
Unlike graphs, there are several non-equivalent quasi-random notions of quai-randomness in $k$-graphs for $k\ge 3$ (see~\cite{Reiher_2017}).
In addition, the subgraph containment problem for quasi-random $k$-graphs, $k\geq3$, is quite different from the case $k=2$ and
has been an interesting topic over decades.
Indeed, for $k\geq 3$, R\"odl noted that by a construction of~\cite{ErdosHajnal}, quasi-random $k$-graphs may not contain a single copy of, say, a $(k+1)$-clique.
In fact below we will consider two different notions of quasi-random $k$-graphs,
which have been studied by Reiher, R\"odl and Schacht~\cite{MR3548293}.
The first and weakest concept that we consider here is $(p, \mu, \ppoints{scale=0.11})$-quasi-randomness.
\begin{defi}
Given integers $n\geq k\ge 2$, let real numbers $p \in [0, 1]$, $\mu >0$, and $H=(V,E)$ be a $k$-graph with $n$ vertices. We say that $H$ is ($p,\mu,\ppoints{scale=0.11}$)\emph{-dense} if for all $X_1,\dots,X_k\subseteq V$,
\begin{equation}\label{eq:count}
e_{\ppoints{scale=0.11}}(X_1,\dots,X_k)\geq p|X_1|\cdots|X_k|-\mu n^k,
\end{equation}
where $e_{\ppoints{scale=0.11}}(X_1,\dots,X_k)$ is the number of $(x_1,\dots,x_k)\in X_1\times \cdots \times X_k$ such that $\{x_1,\dots,x_k\}\in E$.
\end{defi}
Next we come to a stronger concept of quasi-random hypergraphs where rather than ``$k$ sets of vertices" we consider ``a set of vertices and a set of $(k-1)$-elements".
\begin{defi}
Given integers $n\geq k\ge 2$, let real numbers $p \in [0, 1]$, $\mu >0$, and $H=(V,E)$ be a $k$-graph with $n$ vertices. We say that $H$ is ($p,\mu,\pedge{scale=0.14}$)\emph{-dense} if for all $X\subseteq V$ and $Y\subseteq V^{k-1}$,
\begin{equation}
e_{\pedge{scale=0.14}}(X,Y)\geq p|X||Y|-\mu n^k,
\end{equation}
where $e_{\pedge{scale=0.14}}(X,Y)$ is the number of pair $(x,(y_1, \dots, y_{k-1}))\in X\times Y$ such that $\{x,y_1, \dots, y_{k-1}\}\in E$.
\end{defi}
Throughout the paper, we say that $H$ is an \emph{$(n,p,\mu, \mathscr{S})$ $k$-graph} if $H$ has $n$ vertices and is ($p,\mu,\mathscr{S}$)-dense for a symbol $\mathscr{S}\in \{\ppoints{scale=0.11}, \pedge{scale=0.14}\}$.
Given $p\in
[0,1]$ and $\mathscr{S}\in \{\ppoints{scale=0.11}, \pedge{scale=0.14}\}$, we sometimes write a $k$-graph is \emph{$p$-dense $\mathscr{S}$-quasi-random} (for short \emph{$p$-dense}) to mean that it is ($p,\mu,\mathscr{S}$)-dense for some small $\mu$.
Let $\mathcal{C}(n,p,\mu, \mathscr{S})$ denote the class of all ($n,p,\mu, \mathscr{S}$) $k$-graphs and one can observe that
\[
\mathcal{C}(n,p,\mu, \pedge{scale=0.14})\subseteq \mathcal{C}(n,p,\mu, \ppoints{scale=0.11})
\]
holds for all $p\in [0,1]$ and $\mu>0$.
Now for $\mathscr{S}\in \{\ppoints{scale=0.11}, \pedge{scale=0.14}\}$, we give a formal definition of Tur\'an densities for $\mathscr{S}$-quasi-random $k$-graphs, which is an extension of $\mathscr{S}$-quasi-random $3$-graphs in~\cite{MR3548293}. Given a $k$-graph $F$, we define:
\begin{equation*}
\begin{split}
\label{dot-turan-dense}
\pi_{\mathscr{S}}(F) = \sup \{ p\in [0,1] & : \text{for\ every\ } \mu>0 \ \text{and\ } n_0\in \mathbb{N},\ \text{there\ exists\ an\ } F \text{-free} \\
&\quad (n, p,\mu,\mathscr{S})\ k \text{-graph\ } H\ \text{with\ } n\geq n_0 \}.
\end{split}
\end{equation*}
\subsection{$F$-factors in quasi-random $k$-graphs}
Lenz and Mubayi~\cite{Lenz2016Perfect} were the first to study the $F$-factor problems in quasi-random $k$-graphs, for $k\ge 3$.
For $\mathscr{S}\in \{\ppoints{scale=0.11}, \pedge{scale=0.14}\}$, $k\ge 2$ and $1\leq s \leq k-1$, define $\textbf{FACTOR}_k^{\mathscr{S} ,s}$ as the collection of $k$-graphs $F$ such that for all $0< p,\alpha<1$, there is some $n_0\in \mathbb{N}$ and $\mu > 0$ so that if $H$ is an $(n,p,\mu ,\mathscr{S})$ $k$-graph with $\delta _s(H)\geq \alpha n^{k-s}$, $n \geq n_0$ and $v(F)\mid n$, then $H$ has an $F$-factor.
For $k=2$, by the aforementioned result of~\cite{Koml1997Blow}, all graphs belong to $\textbf{FACTOR}_2^{\ppoints{scale=0.11} ,1}$. Thus, Lenz and Mubayi raised the following natural question.
\begin{prob}{\rm(\cite[Problem 2]{Lenz2016Perfect}).}\label{prob}
For $k\ge 3$, which $k$-graphs $F$ belong to $\textbf{FACTOR}_k^{\ppoints{scale=0.11} ,1}$?
\end{prob}
In~\cite{Lenz2016Perfect},
Lenz and Mubayi proved that all linear $k$-graphs belong to $\textbf{FACTOR}_k^{\ppoints{scale=0.11} ,1}$ and provided a family of $3$-graphs not in $\textbf{FACTOR}_k^{\ppoints{scale=0.11} ,1}$. More precisely, by a parity-based construction~\cite[Theorem 5]{Lenz2016Perfect}, they constructed an $(n,\frac{1}{8},\mu, \ppoints{scale=0.11} )$ 3-graph $H^*$ with $\delta _1(H^*)\geq (\frac{1}{8}-\mu )\binom{n}{2}$, $\delta _2(H^*)\geq (\frac{1}{8}-\mu ){n}$ and $H^*$ has no $F$-factor for those $F$ that $F$ has an even number of vertices and admits a partition of $V(F)$ into pairs such that each pair has a common member in their neighbor sets. Let $K_{2,2,2}$ denote the complete 3-partite 3-graph with parts of size two. It is easy to check that $K_{2,2,2}$ satisfies this property.
In our work~\cite{DHSWZ}, we completely solved Problem~\ref{prob} for $k=3$. Moreover, we also characterised all $k$-partite $k$-graphs $F$ with $F\in \textbf{FACTOR}^{\ppoints{scale=0.11},1}_k$.
\begin{thm}{\rm({\cite[Theorem 1.4]{DHSWZ}}).}\label{k-partite}
For $k \geq 3$, a $k$-partite $k$-graph $F\in \textbf{FACTOR}^{\ppoints{scale=0.11},1}_k$ if and only if there exists $v^*\in V(F)$ such that $|e\cap e'|\leq 1$ for any two edges $e,e'$ with $v^*\in e$ and $v^*\notin e'$.
\end{thm}
It is natural to consider the following problem.
\begin{prob}\label{partiteprob}
Given a $k$-partite $k$-graph $F$, what conditions of quasi-randomness and degree for the host $k$-graph $H$ are sufficient for ensuring $F$-factor?
\end{prob}
By the construction of Lenz and Mubayi, we naturally want to know, does $p>1/8$ guarantee that $H$ with the minimum degree $\Omega(n^2)$ has a $K_{2,2,2}$-factor? Unfortunately, this appears to be false by the following result.
\begin{thm}\label{construction}
Let $F$ be a $3$-graph satisfying that for each $v\in V(F)$ there exists a vertex $u$ such that $N(v)\cap N(u)\neq \emptyset$. Then for any $p\in(0,1)$ and $\mu>0$, there exists an $n_0$ and $\alpha>0$ such that for all $n\ge n_0$, there exists an $(n,p,\mu, \ppoints{scale=0.11})$ $3$-graph $H$ with $\delta_1(H)\ge \alpha n^{2}$ such that $H$ has no $F$-factor.
\end{thm}
$K_{2,2,2}$ is an example of $3$-graphs $F$ that satisfy the conditions of Theorem~\ref{construction}. For $p$ close to $1$ this shows that $\ppoints{scale=0.11}$-quasi-randomness and minimum degree conditions of the host $3$-graph $H$ are not sufficient for ensuring $K_{2,2,2}$-factors.
However, if we replace the minimum degree with the minimum codegree condition, then we can prove the following result.
\begin{thm}\label{1/8-dot-codegree}
Given $0<\varepsilon,\alpha <1$, and a $3$-partite $3$-graph $F$ of order $f$,
there exists an $n_0$ and $\mu>0$ such that the following holds for $n\geq n_0$. Let $H$ be an $(n,\frac{1}{8}+\varepsilon,\mu ,\ppoints{scale=0.11} )$ 3-graph with $\delta_2(H)\geq \alpha n$ and $n\in f\mathbb{N}$. Then $H$ has an $F$-factor.
\end{thm}
\subsection{Perfect tilings of $k$-graphs in quasi-random $k$-graphs with positive density}
Note that (see~\cite{Han2017Minimum}) a trivial necessary condition for a $k$-graph $H$ to contain an $F$-factor is that every vertex of $H$ is covered by at least one copy of $F$.
This can be visualized as a set of (not necessarily vertex disjoint or distinct) copies of $F$ whose union covers the vertex set of the host graph, so also called an \emph{$F$-cover}. For the $F$-factor problems, the cover property plays an essential role.
Accordingly, the collection $\textbf{COVER}_k^{\mathscr{S} ,s}$ of $k$-graphs $F$ satisfies that for all $0< p,\alpha<1$, there is some $n_0\in \mathbb{N}$ and $\mu > 0$ so that if $H$ is an $(n,p,\mu ,\mathscr{S})$ $k$-graph with $\delta _s(H)\geq \alpha n^{k-s}$ and $n \geq n_0$, then each vertex $v$ in $V(H)$ is contained in a copy of $F$, namely, $H$ has an $F$-cover.
We proved the following equivalence relation in~\cite{DHSWZ}, which reduced the problem of $F$-factors to the study of ``covering every vertex of the host $k$-graph by a copy of $F$''.
\begin{thm}{\rm({\cite[Theorem 1.3]{DHSWZ}}).}\label{factor-cover1}
For $k \geq 2$, $\textbf{FACTOR}^{\ppoints{scale=0.11},1}_k=\textbf{COVER}^{\ppoints{scale=0.11},1}_k$.
\end{thm}
Actually, the proof of Theorem~\ref{factor-cover1} works for other (stronger) quasi-randomness and we choose not to reproduce the proof here.
\begin{thm}\label{factor-cover2}
For any $k \geq 2$, $\textbf{FACTOR}^{\pedge{scale=0.14},1}_k=\textbf{COVER}^{\pedge{scale=0.14},1}_k$.
\end{thm}
As an application of Theorem~\ref{factor-cover2}, we obtain the following result.
\begin{thm}\label{dot-edge-partite}
For any $k$-partite $k$-graph $F$, $F\in \textbf{FACTOR}_k^{\pedge{scale=0.14},1}$.
\end{thm}
As mentioned above, the construction of Lenz and Mubayi~\cite{Lenz2016Perfect} showed that $K_{2,2,2}\notin \textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$. We naturally want to know, which $3$-partite $3$-graphs belong to $\textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$? Let us recall some definitions introduced in our work~\cite{DHSWZ}.
Given a $k$-graph $F$ with $k\ge 2$, for a bipartition $\mathcal{P}:=\{V_1,V_2\}$ of $V(F)$ and $e\in E(F)$, we write $\textbf{i} _{\mathcal{P},F}:=(|V_1|, |V_2|)$ and $\textbf{i} _{\mathcal{P},F}(e)=(|e\cap V_1|, |e\cap V_2|)$.
Given an integer $2\le s\le k-1$, we say that a bipartition $\mathcal{P}$ of $V(F)$ is \emph{$s$-shadow disjoint} if any two edges $e, e'\in E(F)$ with $\textbf{i} _{\mathcal{P},F}(e)\neq \textbf{i} _{\mathcal{P},F}(e')$ satisfy that $|e\cap e'|<s$.
Let $L^{s}_{F}$ be the lattice generated by all $\textbf{i} _{\mathcal{P},F}$ such that $\mathcal{P}$ is $s$-shadow disjoint and we denote ${\bf TRANS}^{s}_k$ as the collection of $k$-graphs $F$ with $(1,-1)\in L^{s}_F$.
Inspired by the Theorem~\ref{factor-cover2},
we also give an invisible characterization of $\textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$.
\begin{thm}\label{factor3-2}
$\textbf{FACTOR}_3^{\ppoints{scale=0.11} ,2}$ = $\textbf{COVER}_3^{\ppoints{scale=0.11} ,2}\cap {\bf TRANS}^2_3\cap \{F:F\ {\rm is}\ {\rm a}\ {\rm 3}{\text{-}}{\rm graph}\ {\rm and}\ \pi_{\ppoints{scale=0.11}}(F)=0\}$.
\end{thm}
As an application of Theorem~\ref{factor3-2}, the following corollary can be obtained immediately.
\begin{cor}\label{3-partite3-graph}
For a $3$-partite $3$-graph $F$, if $F\in {\bf TRANS}^2_3$, then $F\in \textbf{FACTOR}_3^{\ppoints{scale=0.11} ,2}$.
\end{cor}
For $k\ge 3$,
we give a characterization of all complete $k$-partite $k$-graphs in $\textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$ and prove that a class of $k$-partite $k$-graphs belongs to $\textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$.
Let us recall some definitions introduced by Mycroft~\cite{Mycroft-par}.
Given a $k$-partite $k$-graph $F$, we define
\[\mathcal{S}(F):=\bigcup_{\chi} \{|V_1|, \dots, |V_k|\},\]
where in each case the union is taken over all $k$-partite realisations $\chi$ of $F$ into vertex classes $V_1, \dots, V_k$ of $V(F)$.
The greatest common divisor of $\mathcal{S}(F)$, is denoted by $\gcd(\mathcal{S}(F))$.
For example, let $K_{a_1,\dots ,a_k}$ denote the complete $k$-partite $k$-graph with parts of size $a_1,\dots ,a_k$. Observe that $K_{a_1,\dots ,a_k}$ has only one $k$-partite realisation up to permutations of the vertex classes $V_1, \dots, V_k$, so $\gcd(\mathcal{S}(K_{a_1,\dots ,a_k}))=\gcd(a_1,\dots ,a_k)$.
For all $k$-partite $k$-graphs $F$ with $\gcd(\mathcal{S}(F))=1$, we have the following result.
\begin{thm}\label{gcd}
If a $k$-partite $k$-graph $F$ satisfies $\gcd(\mathcal{S}(F))=1$, then $F\in \textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$.
Moreover, if $F$ is a complete $k$-partite $k$-graph, then $F\in \textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$ if and only if $\gcd(\mathcal{S}(F))=1$.
\end{thm}
Note that for a $k$-partite $k$-graph $F$, $F\in \textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$ does not imply $\gcd(\mathcal{S}(F))=1$ in general. For example, consider $F$ as a matching of size $2$. Clearly, it belongs to $\textbf{FACTOR}_k^{\ppoints{scale=0.11},k-1}$ but $\gcd(\mathcal{S}(F))=2$.
\subsection{Related work.}
As mentioned above, minimum degree conditions forcing perfect $F$-tilings in graphs have a long history and have been well studied.
In the past decade there has been substantial interest in extending these results to $k$-graphs (see a recent survey~\cite{Zhao-survey}).
However, only a handful of optimal minimum degree thresholds are known. There are also several results related to our setting, namely, the recent results on locally dense (hyper)graphs. These are notions posed by Erd\H{o}s and S\'os~\cite{E-S1982} as strengthenings of \emph{having sublinear independence number}. Namely, a $k$-graph $H$ on $n$ vertices is \emph{locally dense} if it satisfies that for every subset $X\subseteq V(H)$, $H[X]$ contains at least $p|X|^k-\mu n^k$ edges, where $1/n\ll \mu \ll p$. Recently, Reiher and Schacht~\cite{locally-clique} gave the minimum degree condition forcing a $K_k$-factor in locally dense graphs. Staden and Treglown~\cite{bandwidth} proved a version of the Bandwidth theorem for locally dense graphs. Similar results on perfect matchings and tilings in $k$-graphs have been considered by Han~\cite{PMtilings}.
\subsection*{Organization}
The rest of this paper is organized as follows.
In the next section, we introduce proof ideas and give a probabilistic construction to prove Theorem~\ref{construction}.
We give a proof of Theorem~\ref{dot-edge-partite} in Section 3.
In Section 4, we prove an absorbing lemma, which is the main proof tool of Theorem~\ref{1/8-dot-codegree}, Theorem~\ref{factor3-2} and Theorem~\ref{gcd}. We review the hypergraph regularity method in Section 5.
The following three sections contain proofs of Theorem~\ref{1/8-dot-codegree}, Theorem~\ref{factor3-2} and Theorem~\ref{gcd} respectively.
The last section contains some remarks.
\section{Proof ideas and a proof of Theorem~\ref{construction}}
\subsection{Proof ideas}
Although our constructions of $F$-factors follow the popular framework of absorption method (indeed a variant developed by the second author~\cite{Han2017Perfect}),
the proof does contain novel components and reveal interesting phenomena in quasi-random $k$-graphs.
For this, Theorem~\ref{factor3-2} is a prime example, which says that for $3$-graphs $F$ with $\pi_{\ppoints{scale=0.11}}(F)=0$, two more variants of the Tur\'an problem need to be considered:
\begin{enumerate}
\item $F\in \textbf{COVER}_3^{\ppoints{scale=0.11} ,2}$, that is, every vertex must be contained in a copy of $F$ in $H$;
\item $F\in {\bf TRANS}^2_3$, from which we will deduce that given any bipartition $(X, Y)$ of $V(H)$, there are certain types of embeddings of $F$ that give rise to the existence of a \emph{transferral}.
\end{enumerate}
The latter point will be made precise in Section 7.
Both these two properties are clearly strengthenings of the Tur\'an problem and are crucial in the construction of absorbing sets.
In our proofs, the verification of (1) is straightforward for $k$-partite $k$-graphs (for general $k$-graphs this is quite challenging and actually the bottleneck to a solution of Problem~\ref{prob} for $k\ge 4$).
For the study of (2) in Theorems~\ref{1/8-dot-codegree} and~\ref{factor3-2},
we use the regularity method and embedding techniques pioneered by Reiher--R\"odl--Schacht~\cite{Reiher2018Hypergraphs} in quasi-random $3$-graphs (see Sections 6 and 7).
\subsection{A proof of Theorem~\ref{construction}}
Now we prove Theorem~\ref{construction} using the following example. \\
\noindent {\bf Construction.} For $n\in \mathbb{N}$, define a probability distribution $H(n)$ on $3$-graphs of order $n$ as follows.
Let $G\in \mathbb{G}(n-1,q)$ be the random graph on $n-1$ vertices with $q\in(0,1)$.
Now let the vertex set of $H(n)$ be $V(G)\cup \{z\}$ where $z$ is a new vertex and we define $E(H(n))$ as follows. For each $3$-set $e\subseteq V(G)$, if $G[e]$ is a triangle in $G$, then add $e$ into $E(H(n))$; for each pair $\{u,v\}\subseteq V(G)$, if $\{u,v\}\notin E(G)$, then
add $\{u,v,z\}$ into $E(H(n))$.
Therefore, for each $3$-set $e$ not containing $z$, we have $e \in E(H(n))$ with probability $q^3$. Let $X_1, X_2, X_3\subset V(H(n)) \setminus \{z\}$, then the expected value of $e_{\ppoints{scale=0.11}}(X_1,X_2, X_3)$ is at least $q^3|X_1|(|X_2|-1)(|X_3|-2)$.
For each vertex $w\in V(H(n))\setminus \{z\}$, the expected value of $\deg(w)$ in $H(n)$ is at least $q^3\binom{n-2}{2}$, and the expected value of $\deg(z)$ is $(1-q)\binom{n-1}{2}$.
Let $\alpha=\min\{\frac{q^3}{4},\frac{1-q}{4}\}$. By concentration inequalities (e.g. Janson's inequality) and the union bound, for every $\mu>0$ and sufficiently large $n$, there exists $H\in H(n)$ such that $H$ is an $(n,q^3,\mu,\ppoints{scale=0.11})$ $3$-graph and $\delta_1(H)\ge \alpha n^2$.
Now let us consider the vertex $z$ in $H$. Suppose that there exists a copy $F'$ of $F$ in $H$ and an isomorphic mapping $g: V(F)\to V(F')$ such that $g(v)=z$. Recalling the condition of $F$, there is a vertex $u\in V(F)$ such that $N_F(u)\cap N_F(v)\neq \emptyset$. Then $N_H(g(u))\cap N_H(z)\neq \emptyset$, which
contradicts with the construction of $H$ that $N(z) \cap N(w)=\emptyset$ for any $w\in V(H)$.
Therefore, $H$ does not have an $F$-factor.
\section{The proof of Theorem~\ref{dot-edge-partite}}
In this section, our aim is to prove Theorem~\ref{dot-edge-partite}.
Throughout the proof, we use the notion introduced by Lo and Markstr\"{o}m~\cite{Lo2015}. Let $H$ be a $k$-graph on $n$ vertices.
Given two vertices $u, v\in V(H)$ and a constant $\eta>0$, a $(k-1)$-set $S\in N(u)\cap N(v)$ is said to be \textit{$\eta$-good} for $\{u,v\}$ if $\deg(S)\ge \eta n$. Otherwise, $S$ is \textit{$\eta$-bad} for $\{u,v\}$. In particular, we say that $\{u, v\}$ is \textit{$\eta$-good} if the number of $\eta$-good $(k-1)$-sets for $\{u, v\}$ is at least $\eta n^{k-1}$. Otherwise, we say $\{u,v\}$ is \textit{$\eta$-bad}.
We have the following proposition.
\begin{prop}\label{x-good}
Given $0<\alpha<1$, there exists an integer $n_0$ such that the following holds for $n\ge n_0$.
Let $H$ be a $k$-graph of order $n$ with $\delta_1(H)\ge \alpha \binom{n-1}{k-1}$. If there exists $\alpha'>0$ such that all but at most $\frac{\alpha}{2}\binom{n-1}{k-1}$ $(k-1)$-sets $S\subset V(H)$ satisfy $\deg_H(S)\ge \alpha'n$, then for any $0<\eta<\frac{\alpha\alpha'}{4k!}$ and every $v\in V(H)$, there exists $u\in V(H)\setminus \{v\}$ such that $\{u,v\}$ is $\eta$-good in $H$.
\end{prop}
\begin{proof}
Suppose that $H$ is a $k$-graph satisfying the above conditions.
Let
\[
\mathcal{S}=\left\{S\in [V]^{k-1} : \deg(S)<\alpha'n \right\}.
\]
Then $|\mathcal{S}|<\frac{\alpha}{2}\binom{n-1}{k-1}$. For every $v\in V(H)$, we have $|N(v)|\ge \alpha \binom{n-1}{k-1}$ since $\delta_1(H)\ge \alpha \binom{n-1}{k-1}$.
Let $N^*(v):=N(v)\setminus \mathcal{S}$. Then $|N^*(v)|\ge \frac{\alpha}{2}\binom{n-1}{k-1}$ and every $(k-1)$-set $S\in N^*(v)$ satisfies $\deg(S)\ge \alpha'n$. Assume that for any $u\in V(H)\setminus \{v\}$, $\{u,v\}$ is \textit{$\eta$-bad}, namely $|N(u)\cap N^*(v)|<\eta n^{k-1}$. By double counting, we have
\[
\frac{\alpha}{2}\binom{n-1}{k-1} \alpha'n \le \sum_{S\in N^{*}(v)}{\deg(S)}< | V(H)\setminus \{v\}|\eta n^{k-1}+|N^{*}(v)|,
\]
which yields $\eta>\frac{\alpha\alpha'}{4k!}$.
Therefore, there exists a vertex $u\in V(H)\setminus \{v\}$ such that $\{u,v\}$ is $\eta$-good in $H$ for all $0<\eta<\frac{\alpha\alpha'}{4k!}$.
\end{proof}
Let $K^k_k(m)$ be a complete $k$-partite $k$-graph with $m$ vertices in each part.
Our proof of Theorem~\ref{dot-edge-partite} is based on the following lemma from~\cite{Lo2015}, which tells that if $\{u,v\}$ is $\eta$-good, then $u$ is in a copy of $K^k_k(m)$.
\begin{lemma}{\rm{(}\cite[Lemma 4.2]{Lo2015}\rm{)}}.\label{good-reach.}
Let $k,m\ge 2$ be integers and $\eta\in (0,1)$. There exists a constant $\beta_0=\beta_0(k,m,\eta)>0$ and an integer $n_0=n_0(k,m,\eta)$ such
that for every $k$-graph $H$ of order $n\ge n_0$, if $\{u,v\}$ is $\eta$-good in $H$ for $u,v\in V(H)$, then for all $0<\beta\leq \beta_0$ there are at least $\beta n^{f-1}$ $(f-1)$-sets $W$ such that both $H[\{u\}\cup W]$ and $H[\{v\}\cup W]$ contain $K^k_k(m)$-factors. In particular, $u$ is in a copy of $K^k_k(m)$.
\end{lemma}
Theorem~\ref{dot-edge-partite} easily follows from Theorem~\ref{factor-cover2}, Proposition~\ref{x-good} and Lemma~\ref{good-reach.}.
\begin{proof}[Proof of Theorem~\ref{dot-edge-partite}]
Suppose that $F$ is a $k$-partite $k$-graph with vertex classes $X_1,\dots,X_k$.
By Theorem~\ref{factor-cover2}, it suffices to show that $F\in \textbf{COVER}^{\pedge{scale=0.14},1}_k$.
Let $m:=\max\{|X_i|: i\in[k]\}$, then $F$ is a sub-hypergraph of $K^k_k(m)$. Next, our goal is to show that given $0<p,\alpha<1$, we take $\mu>0$ small enough and $n$ large enough such that every vertex $v$ in an $(n,p,\mu,\pedge{scale=0.14})$ $k$-graph $H$ with $\delta_1(H)\geq \alpha n^{k-1}$ is contained in a copy of $K^k_k(m)$.
Given $0<p,\alpha<1$, we select $0<\mu<\eta<\frac{\alpha p}{8k!}$. Next we apply Lemma~\ref{good-reach.} and get the returned constant $\beta_0=\beta_0(k,m,\eta)>0$. Let $H$ be an $(n,p,\mu,\pedge{scale=0.14})$ $k$-graph with $\delta_1(H)\geq \alpha n^{k-1}$ and $v(F)\mid n$ and set $V:=V(H)$. Let
\[
\mathcal{S}=\left\{S\in [V]^{k-1} : \deg(S)<\frac{p}{2}n \right\}.
\]
Since $H$ is $(p,\mu, \pedge{scale=0.14})$-dense, we obtain that
\[
p|\mathcal{S}||V|-\mu n^k\le e_{\pedge{scale=0.14}}(\mathcal{S},V)<|\mathcal{S}|\frac{p}{2}n,
\]
which implies that $|\mathcal{S}|<\frac{2\mu}{p}n^{k-1}<\frac{\alpha}{2}n^{k-1}$.
We apply Proposition~\ref{x-good} to $H$ and obtain that for every $v\in V(H)$, there exists $u\in V(H)\setminus \{v\}$ such that $\{u,v\}$ is $\eta$-good in $H$. By Lemma~\ref{good-reach.},
$v$ is contained in a copy of $K^k_k(m)$ and we are done.
\end{proof}
\section{Tools}
In this section, we introduce some tools to prove Theorems~\ref{1/8-dot-codegree}, ~\ref{factor3-2} and~\ref{gcd}.
The key tool is the lattice-based absorption method developed by Han~\cite{Han2017Perfect}, which builds on the absorbing method initiated by R\"{o}dl, Ruci\'{n}ski and Szemer\'{e}di~\cite{R2015A}.
Throughout the paper, we write $a\ll b\ll c$ to mean that we can choose the positive constants $a, b, c$ from right to left. More precisely, there are increasing functions
$f$ and $g$ such that, given $c$, whenever we choose some $b\le f(c)$ and $a\le g(b)$, the subsequent
statement holds. Hierarchies of other lengths are defined similarly.
The absorption approach splits the proofs into the following two parts: one is on finding an almost perfect $F$-tiling in $H$ by an almost perfect tiling lemma (Lemma~\ref{Almost-factor}), and the other is on ``finishing up" the perfect $F$-tiling by an absorbing lemma (Lemma~\ref{Absorbing-Lemma}).
\begin{lemma}{\rm(Almost Perfect Tiling {\cite[Lemma 2.1]{DHSWZ}}).} \label{Almost-factor}
Given $0< p,\alpha <1$ and a $k$-graph $F$ satisfying $\pi_{\ppoints{scale=0.11}}(F)=0$, for any $0 <\omega < 1$, there exists an $n_0$ and $\mu > 0$ such that the following holds for $n\geq n_0$. If $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph, then there exists an $F$-tiling that covers all but at most $\omega n$ vertices of $H$.
\end{lemma}
\subsection{Absorbing lemma}
Before giving the absorbing lemma, we need some definitions introduced by Keevash and Mycroft~\cite{Ke2015}.
Let $r, f, k$ be integers with $f\geq k\geq 2$ and let $F$ be a $k$-graph of order $f$. Suppose that $H$ is a $k$-graph with a partition $\mathcal{P} =\{V_0,V_1,\dots,V_r\}$ of $V (H)$.
The \textit{index vector} $\mathbf{i}_{\mathcal{P}}(S)\in \mathbb{Z}^r$ of a subset $S\subseteq V(H)$ w.r.t. (with respect to) $\mathcal{P}$ is the vector whose coordinates are the sizes of the intersections of $S$ with $V_1,\dots,V_r$, that is, $\mathbf{i}_{\mathcal{P}}(S)=(|S\cap V_1|, \dots, |S\cap V_r|)$.
We call an index vector $\mathbf{i}_{\mathcal{P}}(S)$ an \emph{$s$-vector} if all its coordinates are non-negative and their sum is $s$.
Given $\lambda>0$, an $f$-vector $\mathbf{v} \in \mathbb{Z}^r$ is called a \emph{$\lambda$-robust $F$-vector} if at least $\lambda (v(H))^f$ copies $F'$ of $F$ in $H$ satisfy $\mathbf{i}_{\mathcal{P}}(V(F'))=\mathbf{v}$. Let $I^{\lambda}_{\mathcal{P},F}(H)$ be the set of all $\lambda$-robust $F$-vectors and $L^{\lambda}_{\mathcal{P},F}(H)$ be the lattice generated by the vectors of $I^{\lambda}_{\mathcal{P},F}(H)$. For $j \in [r]$, let $\mathbf{u}_j\in \mathbb{Z}^r$ be the $j^{th}$ unit vector, namely, $\mathbf{u}_j$ has $1$ on the $j^{th}$ coordinate and $0$ on other coordinates.
To ease notation, throughout the paper, for a $k$-graph $H$ if we consider a partition $\mathcal{P}=\{V_0, V_1, V_2\}$ of $V(H)$ satisfying $V_0=\emptyset$, then we write $X=V_1$ and $Y=V_2$, i.e., $\{X,Y\}:=\{ V_1, V_2\}$.
\begin{lemma}[Absorbing Lemma]\label{Absorbing-Lemma}
Suppose that $1/n\ll \mu, \gamma'\ll \lambda \ll \zeta \ll \gamma \ll p,\alpha <1$ and $f,k, n \in \mathbb{N}$ with $f\ge k\ge 3$.
Let $F\in \textbf{COVER}_k^{\ppoints{scale=0.11}, k-1}$ with $f:=v(F)$ and $H$ be an $(n, p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$.
If each partition $\mathcal{P}=\{V_0, V_1, V_2\}$ of $V(H)$ with $|V_i| \ge \zeta n$ for $i=1, 2$ satisfies $(1,-1) \in L^{\lambda}_{\mathcal{P},F}(H)$,
then there exists a vertex set $W \subseteq V (H)$ with $|W| \leq \gamma n$ such that for any vertex set $U\subseteq V(H)\setminus W$ with $|U|\leq \gamma' n$ and $|U|\in f\mathbb{N}$, both $H[W]$ and $H[W \cup U]$ contain $F$-factors.
\end{lemma}
To prove Lemma~\ref{Absorbing-Lemma}, we use the notion of reachability introduced by Lo and Markstr\"{o}m~\cite{Lo2015}. Let $H$ be a $k$-graph on $n$ vertices.
Given a $k$-graph $F$ of order $f$, a constant $\beta> 0$ and an integer $i \ge 1$, we say that two vertices $u, v$ in $H$ are \textit{$(F, \beta, i)$-reachable} (in $H$) if there are at least $\beta n^{if-1}$ $(if-1)$-sets $W$ such that both $H[\{u\}\cup W]$ and $H[\{v\}\cup W]$ contain $F$-factors. Moreover, we call $W$ a \emph{reachable set} for $\{u,v\}$.
Given a vertex set $U\subseteq V(H)$, $U$ is said to be \emph{$(F, \beta, i)$-closed} if every two vertices in $U$ are $(F, \beta, i)$-reachable in $H$.
For $v\in V(H)$, let $\tilde{N}_{F,\beta, i}(v)$ be the set of vertices that are $(F,\beta, i)$-reachable to $v$.
We will use the following result of Han--Zang--Zhao~\cite[Lemma 3.6]{Han2017Minimum}.
Although they proved Lemma~\ref{absorb} for $F$ being a complete $3$-partite $3$-graph, in fact it holds for $F$ being a $k$-graph. Therefore, we choose not to reproduce the proof here.
\begin{lemma}{\rm({\cite[Lemma 2.9]{DHSWZ}}).}~\label{absorb}
Suppose that $1/n \ll \gamma' \ll \gamma , \beta'_0 <1$ and $i'_0, k, f, n \in \mathbb{N}$. Let $F$ be a $k$-graph with $f:=v(F)$.
Suppose $H$ is a $k$-graph on $n$ vertices with the following two properties.
\begin{enumerate}[label=$(\roman*)$]
\item[{\rm ($\clubsuit$)}] For any $v\in V(H)$, there are at least $\gamma n^{f-1}$ copies of $F$ containing it; \label{item:1}
\item[{\rm ($\spadesuit$)}] there exists $V'_0\subseteq V(H)$ with $|V'_0|\leq \gamma^2n$ such that $V (H)\setminus V'_0$ is $(F, \beta'_0, i'_0)$-closed in $H$.\label{item:2}
\end{enumerate}
Then there exists a vertex set $W$ with $V'_0\subseteq W\subseteq V(H)$ and $|W|\leq \gamma n$ such that for any vertex set $U \subseteq V (H)\setminus W$ with $|U| \leq \gamma' n$ and $|U| \in f\mathbb{N}$, both $H[W]$ and $H[U \cup W]$ contain $F$-factors.
\end{lemma}
Note that to prove Lemma~\ref{Absorbing-Lemma}, it suffices to show that ($\clubsuit$) and ($\spadesuit$) hold for $k$-graphs $H$ satisfying the conditions of Lemma~\ref{Absorbing-Lemma}. We first consider the property ($\clubsuit$).
Recall that the family $\textbf{COVER}_k^{\ppoints{scale=0.11},s}$ consists of $k$-graphs $F$ satisfying that for all $0< p, \alpha<1$, there is some $n_0\in \mathbb{N}$ and $\mu > 0$ so that if $H$ is an $(n, p, \mu, \ppoints{scale=0.11})$ $k$-graph $H$ with $\delta_{s}(H)\geq \alpha n^{k-s}$ and $n \geq n_0$, then each vertex $v$ in $V(H)$ is contained in a copy of $F$. In our work~\cite{DHSWZ}, we proved the following result for $s=1$. The proof indeed works for all $s\in [k-1]$.
\begin{lemma}{\rm({\cite[Lemma 2.3]{DHSWZ}}).}\label{cover-supersaturation}
Given $s\in [k-1]$, $F\in \textbf{COVER}_k^{\ppoints{scale=0.11} ,s}$ with $f:=v(F)$, and $0<p,\alpha<1$, there exist $\mu,\gamma>0$ and an $n_0\in \mathbb{N}$ such that the following holds for $n\geq n_0$.
If $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta _s(H)\geq \alpha n^{k-s}$, then for any vertex $v$ in $V(H)$, $v$ is contained in at least $\gamma n^{f-1}$ copies of $F$.
\end{lemma}
Therefore, the property ($\clubsuit$) follows from Lemma~\ref{cover-supersaturation} applied with $s=k-1$.
Next we consider the property ($\spadesuit$). Following the approach in~\cite{Han2017Minimum}, given a $k$-graph $H$, we first find a partition of $V(H)$ such that all parts are $(F, \beta, i)$-closed in $H$ and then study the reachability between different parts. The following lemma provides such a partition.
\begin{lemma}{\rm (\cite[Theorem 6.3]{Han_2020})}.
\label{partation}
Suppose that $1/n\ll \beta_0\ll \beta \ll \delta, 1/c, 1/f <1$ and $c, f,k, n \in \mathbb{N}$ with $f\ge k\ge 3$.
Let $F$ be a $k$-graph with $f:=v(F)$. Assume $H$ is a $k$-graph on $n$ vertices and $S\subseteq V(H)$ is such that $|\tilde{N}_{F, \beta , 1}(v)\cap S|\geq \delta n$ for any $v\in S$. Further, suppose every set of $c+1$ vertices in $S$ contains two vertices that are $(F, \beta , 1)$-reachable in $H$. Then we can find a partition $\mathcal{P}$ of $S$ into $V_1,\dots,V_r$ with $r \leq \min\{c,\lfloor1/{\delta}\rfloor\}$ such that for any $i \in [r]$, $|V_i|\geq (\delta -\beta )n$ and $V_i$ is $(F, \beta_0 , 2^{c-1})$-closed in $H$.
\end{lemma}
In order to use Lemma~\ref{partation} later, we need the following two lemmas.
\begin{lemma}{\rm (\cite[Lemma 2.6]{DHSWZ})}. \label{S-closed}
Given integers $c, k, f\geq 2$ and a constant $0 < \beta <1$, there exist $1/n \ll \delta \ll 1/c$ such that the following holds.
Let $F$ be a $k$-graph on $f$ vertices. Assume that $H$ is a $k$-graph on $n$ vertices satisfying that every set of $c+1$ vertices contains two vertices that are $(F,\beta ,1)$-reachable in $H$. Then there exists $S\subseteq V(H)$ with $|S|\geq (1-c\delta )n$ such that $|\tilde{N}_{F, \beta , 1}(v)\cap S|\geq \delta n$ for any $v\in S$.
\end{lemma}
\begin{lemma}\label{c+1-reachable}
Given integers $k, f\geq 2$ and a constant $0 <\gamma <1$, there exist $1/n \ll \beta \ll \gamma$ such that the following holds. Let $F$ be a $k$-graph on $f$ vertices.
If $H$ is a $k$-graph on $n$ vertices satisfying that every vertex $v$ in $V(H)$ is contained in at least $\gamma n^{f-1}$ copies of $F$, then every set of $\lfloor 1/\gamma \rfloor+1$ vertices in $V(H)$ contains two vertices that are $(F,\beta ,1)$-reachable in $H$.
\end{lemma}
\begin{proof}
Set $c:=\lfloor 1/\gamma \rfloor$ and choose $\beta $ such that $(c+1)\gamma>1+(c+1)^2\beta $. For a vertex $v\in V(H)$, let $C_F(v)$ be the family of copies of $F$ containing $v$.
Given any $c+1$ vertices $v_1, \dots , v_{c+1}$, since each of them is contained in at least
$\gamma n^{f-1}$
copies of $F$, we have $\sum_{i=1} ^{c+1}|C_F(v_i)|\geq (c+1)\gamma n^{f-1}>(1+(c+1)^2\beta )n^{f-1}$. This implies that there exist two vertices $u,w$ such that there are at least $\beta n^{f-1}$ $(f-1)$-sets $W$ such that both $H[\{u\}\cup W]$ and $H[\{w\}\cup W]$ are copies of $F$ in $H$. Namely, they are $(F,\beta,1)$-reachable in $H$.
\end{proof}
By Lemma~~\ref{partation} and the conditions of $H$ in Lemma~\ref{Absorbing-Lemma}, the property ($\spadesuit$) can be obtained by the following lemma directly.
\begin{lemma}{\rm{(\cite[\rm Lemma 3.9]{Han2017Minimum})}.}
\label{closed}
Let $i_0, k, r,f> 0$ be integers and let $F$ be a $k$-graph with $f:=v(F)$. Given constants $\zeta, \beta_0 , \lambda > 0$, there exists $\beta_0'>0$ and integers $i'_0, n_0$ such that the following holds for all $n\geq n_0$. Let $H$ be a $k$-graph on $n$ vertices with a partition $\mathcal{P} =\{V_0,V_1,\dots,V_r\}$ of $V (H)$ such that for each $j\in [r]$, $|V_j|\geq \zeta n$ and $V_j$ is $(F,\beta_0 , i_0)$-closed in $H$. If $\mathbf{u}_j -\mathbf{u}_l \in L^{\lambda}_{\mathcal{P},F}(H)$ for all $1 \leq j <l \leq r$, then $V(H)\setminus V_0$ is $(F, \beta'_0, i'_0)$-closed in $H$.
\end{lemma}
\begin{proof}[Proof of Lemma~\ref{Absorbing-Lemma}]
Let $f\ge k\ge 3$ be integers and $F\in \textbf{COVER}_k^{\ppoints{scale=0.11}, k-1}$ with $f:=v(F)$. We choose constants satisfying the following hierarchy:
\[
1/n \ll \mu, \gamma' \ll \beta'_0 \ll \lambda, \beta_0 \ll \zeta, \beta \ll \delta\ll \gamma \ll p,\alpha, 1/f <1,
\]
$\delta< \gamma^3$ and $\delta-\beta>\zeta $. Let $H$ be an $(n, p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$ satisfying conditions in Lemma~\ref{Absorbing-Lemma}. By Lemma~\ref{absorb}, it suffices to verify that $H$ satisfies properties ($\clubsuit$) and ($\spadesuit$). As mentioned above, $H$ satisfies property ($\clubsuit$) from Lemma~\ref{cover-supersaturation} applied with $s=k-1$.
Since every vertex $v$ in $V(H)$ is contained in at least $\gamma n^{f-1}$ copies of $F$, every set of $\lfloor 1/\gamma \rfloor+1$ vertices in $V(H)$ contains two vertices that are $(F,\beta ,1)$-reachable in $H$ by Lemma~\ref{c+1-reachable}.
Set $c:=\lfloor 1/\gamma \rfloor$. Next by Lemma~\ref{S-closed}, there exists $S\subseteq V(H)$ such that $|S|\geq (1-c\delta )n\ge (1-\gamma^2)n$ and $|\tilde{N}_{F, \beta , 1}(v)\cap S|\geq \delta n$ for any $v\in S$.
Following Lemma~\ref{partation}, we get a partition $\mathcal{P}'$ of $S$ into $V_1,\dots,V_r$ with $r \leq \min\{c,\lfloor1/{\delta}\rfloor\}$ such that for any $i \in [r]$, $|V_i|\geq (\delta -\beta )n>\zeta n $ and $V_i$ is $(F, \beta_0 , 2^{c-1})$-closed in $H$.
Let $\mathcal{P}''=\{V'_0, V_1,\dots,V_r \}$ with $V'_0=V(H)\setminus S$. Clearly, $|V'_0|\le \gamma^2 n$.
Recalling the conditions in Lemma~\ref{Absorbing-Lemma}, for all $1 \leq j <l \leq r$, $\mathcal{P}=\{V_0, V_j, V_l\}$ with $V_0=\bigcup_{i\in [r]\setminus \{j,l\}}V_i$ satisfies $(1,-1)\in L^{\lambda}_{P,F}(H)$. We have $\mathbf{u}_j -\mathbf{u}_l\in L^{\lambda}_{P'',F}(H)$. Finally, we apply Lemma~\ref{closed} with $i_0=2^{c-1}$ and partition $\mathcal{P}''$. Therefore, the property ($\spadesuit$) holds for $H$ by Lemma~\ref{closed}.
\end{proof}
At last, by combining Lemma~\ref{Almost-factor} and Lemma~\ref{Absorbing-Lemma} we obtain the following corollary.
\begin{cor}\label{covertrans}
Suppose that $1/n\ll \mu\ll \lambda' \ll \zeta, p,\alpha, 1/f <1$ and $f,k, n \in \mathbb{N}$ with $f\ge k\ge 3$ and $n\in f\mathbb{N}$.
Let $F\in \textbf{COVER}_k^{\ppoints{scale=0.11}, k-1}$ with $f:=v(F)$ and $\pi_{\ppoints{scale=0.11}}(F)=0$.
Suppose $H$ is an $(n, p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$ and the following property.
\begin{enumerate}[label=$(\roman*)$]
\item[{\rm ($\heartsuit$)}] For each induced subhypergraph $H'$ on $V'\subseteq V(H)$ with $|V'|\ge 2\zeta n$, each partition $\mathcal{P'}=\{X, Y\}$ of $V'$ with $|X|, |Y|\ge \zeta n$ satisfies $(1,-1) \in L^{\lambda'}_{\mathcal{P'},F}(H')$.\label{item:3}
\end{enumerate}
Then $H$ has an $F$-factor.
\end{cor}
\begin{proof}
We choose constants satisfying the following hierarchy:
\[
1/n\ll \mu, \gamma' \ll \lambda' \ll \zeta \ll \gamma \ll \alpha, p, 1/f < 1.
\]
Let $n\in f\mathbb{N}$ and $F\in \textbf{COVER}_k^{\ppoints{scale=0.11}, k-1}$ with $f:=v(F)$ and $\pi_{\ppoints{scale=0.11}}(F)=0$. Suppose that $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$ and $\mathcal{P}=\{V_0,V_1,V_2\}$ is a partition of $V(H)$ with $|V_i|\geq \zeta n$ for $i=1,2$.
Set $H':=H[V_1\cup V_2]$. By the property ($\heartsuit$), the partition $P'=\{V_1, V_2\}$ of $V(H')$ satisfies $(1,-1) \in L^{\lambda'}_{\mathcal{P'},F}(H')$ which implies that $(1,-1)\in L^{\lambda}_{\mathcal{P},F}(H)$, where $\lambda:=\lambda'(2\zeta)^f$.
By Lemma~\ref{Absorbing-Lemma}, we can find an absorbing set $W\subseteq V(H)$ with $|W|\leq \gamma n$.
Let $H'':=H[V(H)\setminus W]$ and $\mu_1=\mu/(1-\gamma)^k$. Note that $H''$ is an $(n-|W|,p,\mu_1, \ppoints{scale=0.11})$ $k$-graph since $|W|\leq \gamma n$ and
\[
\mu n^k \le \frac{\mu}{(1-\gamma)^k}v(H'')^k= \mu_1 v(H'')^k.
\]
Since $\pi_{\ppoints{scale=0.11}}(F)=0$, applying Lemma~\ref{Almost-factor} on $H''$ with $\omega=\gamma'$, we obtain an $F$-tiling that covers all but a set $U$ of at most $\gamma' n$ vertices. By the absorbing property of $W$, $H[W\cup U]$ contains an $F$-factor and thus we obtain an $F$-factor of $H$.
\end{proof}
\section{Hypergraph Regularity Method}
In order to prove Theorem~\ref{1/8-dot-codegree} and Theorem~\ref{factor3-2}, we will use the hypergraph regularity method for $3$-graphs to verify the conditions of Corollary~\ref{covertrans}.
Here we follow the approach from R\"{o}dl and Schacht~\cite{MR2351689,regular-lemmas2007}, combined with results from~\cite{CFKO} and~\cite{K2010Hamilton}.
The central concepts of hypergraph regularity lemma are \emph{regular complex} and \emph{equitable partition}. Before we state the hypergraph regularity lemma, we introduce some necessary notation.
For reals $x,y,z$ we write $x = y \pm z$ to denote that $y-z \leq x \leq y+z$.
\subsection{Regular complexes}
A \emph{hypergraph} $\mathcal{H}$ consists of a vertex set $V(\mathcal{H})$ and an edge set $E(\mathcal{H})$, where every edge $e \in E(\mathcal{H})$ is a non-empty subset of $V(\mathcal{H})$. So a $3$-graph as defined earlier is a $3$-uniform hypergraph in which every edge has size $3$. A hypergraph $\mathcal{H}$ is a \emph{complex} if whenever $e \in E(\mathcal{H})$ and $e'$ is a non-empty subset of $e$ we have that $e'\in E(\mathcal{H})$. All the complexes considered in this paper have the property that all vertices are contained in an edge. A complex $\mathcal{H}^{\leq 3}$ is a \emph{$3$-complex} if all the edges of $\mathcal{H}^{\leq 3}$ consist of at most $3$ vertices. Given a $3$-complex $\mathcal{H}^{\leq 3}$, for each $i \in \{1,2,3\}$, the edges of size $i$ are called \emph{$i$-edges} of $\mathcal{H}^{\leq 3}$ and we denote by $H^{(i)}$ the \emph{underlying $i$-graph} of $\mathcal{H}^{\leq 3}$: the vertices of $H^{(i)}$ are those of $\mathcal{H}^{\leq 3}$ and the edges of $H^{(i)}$ are the $i$-edges of $\mathcal{H}^{\leq 3}$. Note that a $3$-graph $H$ can be turned into a $3$-complex by making every edge into a \emph{complete $i$-graph} $K^{(i)}_3$ (i.e., consisting of all $\binom{3}{i}$ different $i$-tuples on $3$ vertices), for each $ i\in \{1,2,3\}$.
Given positive integers $s\geq 3$ and $i\in \{1,2,3\}$, an \emph{$(s,i)$-graph} $H^{(i)}_s$ is an $s$-partite $i$-graph, by which we mean that the vertex set of $H^{(i)}_s$ can be partitioned into sets $V_1,\dots, V_s$ such that every edge of $H^{(i)}_s$ meets each $V_j$ in at most one vertex for $j\in [s]$. Similarly, an \emph{$(s,i)$-complex} $\mathcal{H}^{\leq i}_s$ is an $s$-partite $i$-complex.
Given $i \in\{2,3\}$, let $H^{(i)}_i$ and $H^{(i-1)}_i$ be on the same vertex set. We denote by $\mathcal{K}_i(H^{(i-1)}_i)$ for the family of $i$-sets of vertices which form a copy of the complete $(i-1)$-graph $K^{(i-1)}_i$ in $H^{(i-1)}_i$. We define the \emph{density} of $H^{(i)}_i$ w.r.t. $H^{(i-1)}_i$ to be
\[
d(H^{(i)}_i|H^{(i-1)}_i):= \begin{cases}
\frac{|E(H^{(i)}_i)\cap \mathcal{K}_i(H^{(i-1)}_i)|}{|\mathcal{K}_i(H^{(i-1)}_i)|} &\text{if\ } |\mathcal{K}_i(H^{(i-1)}_i)|>0,\\
0&\text{otherwise}.
\end{cases}
\]
More generally, if $\mathbf{Q}:= (Q(1), Q(2),\dots, Q(r))$ is a collection of $r$ subgraphs of $H^{(i-1)}_i$, we define $\mathcal{K}_i(\mathbf{Q}):= \bigcup^r_{j=1}\mathcal{K}_i(Q(j))$ and
\[
d(H^{(i)}_i|\mathbf{Q}):= \begin{cases}
\frac{|E(H^{(i)}_i)\cap \mathcal{K}_i(\mathbf{Q})|}{|\mathcal{K}_i(\mathbf{Q})|} &\text{if\ } |\mathcal{K}_i(\mathbf{Q})|>0,\\
0&\text{otherwise}.
\end{cases}
\]
We say that an $H^{(i)}_i$ is \emph{$(d_i, \delta, r)$-regular} w.r.t. an $H^{(i-1)}_i$ if every $r$-tuple $\mathbf{Q}$ with $|\mathcal{K}_i(\mathbf{Q})| \geq \delta|\mathcal{K}_i(H^{(i-1)}_i)|$ satisfies $d(H^{(i)}_i|\mathbf{Q}) = d_i \pm \delta$. Instead of $(d_i, \delta, 1)$-regular, we refer to $(d_i, \delta)$-\emph{regular}.
Moreover, for $i\in \{2,3\}$ and $s \geq i$, we say that $H^{(i)}_{s}$ is \emph{$(d_i, \delta,r)$-regular} w.r.t. $H^{(i-1)}_{s}$ if for every $\Lambda_i\in [s]^i$ the restriction $H^{(i)}_{s}[\Lambda_i]=H^{(i)}_{s}[\cup_{\lambda\in \Lambda_i }V_{\lambda}]$ is \emph{$(d_i, \delta,r)$-regular} w.r.t. the restriction $H^{(i-1)}_{s}[\Lambda_i]=H^{(i-1)}_{s}[\cup_{\lambda\in \Lambda_i }V_{\lambda}]$.
\begin{defi}[$(d_2, d_3, \delta_3, \delta, r)$-regular complexes]
Given $s\ge 3$ and an $(s, 3)$-complex $\mathcal{H}$, we say that $\mathcal{H}$ is \emph{$(d_2, d_3, \delta_3, \delta, r)$-regular} if the following conditions hold:\vspace{3mm}
$\bullet$ every $2$-tuple $\Lambda_2$ of vertex classes, either $H_s^{(2)}[\Lambda_2]$ is $(d_2, \delta)$-regular w.r.t $H_{s}^{(1)}[\Lambda_2]$ or
\\
$d(H_s^{(2)}[\Lambda_2]|H^{(1)}_s[\Lambda_2]) = 0$;
$\bullet$ for every $3$-tuple $\Lambda_3$ of vertex classes either $H^{(3)}_s[\Lambda_3]$ is $(d_3, \delta_3, r)$-regular w.r.t $H^{(2)}_s[\Lambda_3]$ or $d(H^{(3)}_s[\Lambda_3]|H^{(2)}_s[\Lambda_3]) = 0$.\vspace{3mm}
\end{defi}
\subsection{Equitable partition}\label{section-equ-partition}
Suppose that $V$ is a finite set of vertices and $\mathcal{P}^{(1)}$ is a partition of $V$ into sets $V_1, \dots , V_{a_1}$, which will be called \emph{clusters}. For two disjoint sets $V_i$ and $V_j$, we denote by $K(V_i,V_j)$ the \emph{complete bipartite} graph between $V_i$ and $V_j$. We denote by $\mathrm{Cross}_2:=\bigcup_{1\leq i<j\leq a_1}K(V_i,V_j)$ and $\mathrm{Cross}_3$ the set of all $3$-subsets of $V$ that meet each $V_i$ in at most one vertex for $1\leq i\leq a_1$. Let $\mathcal{P}^{ij}$ be a partition of $K(V_i,V_j)$. Then the partition classes of $\mathcal{P}^{ij}$ are some bipartite graphs $P^{(2)}$. Let
$\mathcal{P}^{(2)}=\cup_{1\le i<j\le a_1} \mathcal{P}^{ij}$. So $\mathcal{P}^{(2)}$ is a partition of $\mathrm{Cross}_2$ into several bipartite graphs.
For every $\{x,y\}\in \mathrm{Cross}_2$ with $x\in V_i$ and $y\in V_j$ for $1\le i<j\le a_1$, there exists a unique bipartite graph $P^{(2)}(\{x,y\})\in \mathcal{P}^{(2)}$ so that $\{x,y\}\in P^{(2)}(\{x,y\})$. Now we give the definition of \emph{polyad}. The polyad of $\{x,y\}$ is
\[
\hat{P}^{(1)}=\hat{P}^{(1)}(\{x,y\})=V_i\dot\cup V_j.
\]
Clearly, $\mathcal{K}_2(\hat{P}^{(1)})=K(V_i,V_j)$. For every $3$-set $J\in \mathrm{Cross}_3$ the polyad of $J$ is:
\[
\hat{P}^{(2)}(J):=\bigcup\big\{P^{(2)}(I): I\in [J]^{2}\big\}.
\]
So we can view $\hat{P}^{(2)}(J)$ as a $3$-partite graph (whose vertex classes are clusters intersecting $J$). Let $\mathcal{\hat{P}}^{(2)}$ be the set consisting of all the $\hat{P}^{(2)}(J)$ for all $J\in \mathrm{Cross}_3$. It is easy to verify $\{\mathcal{K}_3(\hat{P}^{(2)}) : \hat{P}^{(2)}\in \mathcal{\hat{P}}^{(2)}\}$ is a partition of $\mathrm{Cross}_3$.
Given a vector of positive integers $\mathbf{a}=(a_1,a_2)$, we say that $\mathcal{P} = \{\mathcal{P}^{(1)},\mathcal{P}^{(2)}\}$ is a \emph{family of partitions} on $V$, if the following conditions hold:\vspace{3mm}
$\bullet$ $\mathcal{P}^{(1)}$ is a partition of $V$ into $a_1$ clusters.
$\bullet$ $\mathcal{P}^{(2)}$ is a partition of $\mathrm{Cross}_2$ satisfying that
for every $P^{(2)}\in \mathcal{P}^{(2)}$, there exists $\{i,j\}\in [a_1]^2$ such that $P^{(2)} \subseteq K(V_i,V_j)$. Moreover,
$|\{P^{(2)}\in \mathcal{P}^{(2)}: P^{(2)}\subseteq K(V_i,V_j)\}|=a_2$ for every $\{i,j\}\in [a_1]^2$.
So for each $J \in \mathrm{Cross}_3$ we can view $\bigcup^{2}_{i=1}\hat{P}^{(i)}(J)$ as a $(3, 2)$-complex.
\vspace{3mm}
\begin{defi}[$(\eta, \delta, t)$-equitable]\label{eq-partition}
Suppose $V$ is a set of $n$ vertices, $t$ is a positive integer and $\eta, \delta>0$. We say a family of partitions $\mathcal{P}$ is \emph{$(\eta, \delta, t)$-equitable} if it satisfies the following:
\begin{enumerate}
\item $\mathcal{P}^{(1)}$ is a partition of $V$ into $a_1$ clusters of equal size, where $1/\eta \leq a_1 \leq t$ and $a_1$ divides $n$;
\item $\mathcal{P}^{(2)}$ is a partition of $\mathrm{Cross}_2$ into at most $t$ bipartite subgraphs;
\item there exists $d_2$ such that $d_2\geq 1/t$ and $1/d_2 \in \mathbb{N}$;
\item for every $3$-set $K \in \mathrm{Cross}_3$, $\hat{P}^{(2)}(K)$ is $(d_2, \delta)$-regular w.r.t. $\bigcup_{I\in [K]^{2}}\hat{P}^{(1)}(I)$.
\end{enumerate}
\end{defi}
Note that the final condition implies that the classes of $\mathcal{P}^{(2)}$ have almost equal size.
\subsection{Statement of the regularity lemma.}
Let $\delta_3 > 0$ and $r \in \mathbb{N}$. Suppose that $H$ is a $3$-graph on $V$ and $\mathcal{P}$ is a family of partitions on $V$. Given a polyad $\hat{P}^{(2)} \in \hat{\mathcal{P}}^{(2)}$, we say that $H$ is \emph{$(\delta_3, r)$-regular} w.r.t. $\hat{P}^{(2)}$ if $H$ is $(d_3, \delta_3, r)$-regular w.r.t. $\hat{P}^{(2)}$ for some $d_3$. Finally, we define that $H$ is \emph{$(\delta_3, r)$-regular} w.r.t. $\mathcal{P}$.
\begin{defi}[\emph{$(\delta_3, r)$-regular} w.r.t. $\mathcal{P}$]
We say that a $3$-graph $H$ is \emph{$(\delta_3, r)$-regular} w.r.t. $ \mathcal{P}$ if
\[
\big|\bigcup\big\{\mathcal{K}_3(\hat{P}^{(2)}) : \hat{P}^{(2)}\in \mathcal{\hat{P}}^{(2)}\\
\text{and\ } H \text{\ is\ not\ } (\delta_3, r)\text{-regular\ w.r.t.\ } \hat{P}^{(2)} \big\}\big| \le \delta_3 |V|^3.
\]
\end{defi}
This means that no more than a $\delta_3$-fraction of the $3$-subsets of $V$ form a $K_3^{(2)}$ that lies within a polyad w.r.t. which $H$ is not regular.
Now we are ready to state the regularity lemma.
\begin{thm}[Regularity lemma \rm{\cite[Theorem 17]{regular-lemmas2007}}] \label{Reg-lem}
For all positive constants $\eta$ and $\delta_3$ and all functions $r:\mathbb{N} \rightarrow \mathbb{N}$ and $\delta:\mathbb{N} \rightarrow (0,1]$, there are integers $t$ and $n_0$ such that the following holds.
For every $3$-graph $H$ of order $n\ge n_0$ and $t!$ dividing $n$, there exists a family of partitions $ \mathcal{P}$ of $V(H)$ such that\vspace{2mm}
$(1)$ $\mathcal{P}$ is $(\eta, \delta(t), t)$-equitable and
$(2)$ $H$ is $(\delta_3, r(t))$-regular w.r.t. $\mathcal{P}$.
\end{thm}
Note that the constants in Theorem~\ref{Reg-lem} can be chosen to satisfy the following hierarchy:
\[
\frac{1}{n_0}\ll \frac{1}{r}=\frac{1}{r(t)}, \delta=\delta(t)\ll \min\{\delta_3,1/t\}\ll \eta.
\]
Given $d\in (0,1)$, we say that an edge $e$ of $H$ is \emph{$d$-useful} if it lies in $\mathcal{K}_3 (\hat{P}^{(2)})$ for some $\hat{P}^{(2)} \in \hat{\mathcal{P}}^{(2)}$ such that $H$ is $(d_3, \delta_3, r)$-regular w.r.t. $\hat{P}^{(2)}$ for some $d_3\geq d$. If we choose $d\gg \eta$, then the following lemma will be helpful in later proofs.
\begin{lemma}\rm{(\cite[Lemma 4.4]{K2010Hamilton}).}\label{useful-edge}
At most $2dn^3$ edges of $3$-graph $H$ are not $d$-useful.
\end{lemma}
To ease notation, we denote by $P^{ij}_{a}=(V_i\dot\cup V_j,E^{ij}_{a})$ a specific bipartite subgraph in $K(V_i,V_j)$ and let $P^{ijk}_{abc}=P^{ij}_{a}\cup P^{ik}_{b}\cup P^{jk}_{c}$ denote a specific polyad over $2$-graphs, i.e. $ P^{ijk}_{abc}=(V_i\dot\cup V_j\dot\cup V_k,E^{ij}_{a}\cup E^{ik}_{b}\cup E^{jk}_{c})$.
\subsection{Statement of a counting lemma.}
In our proofs we shall also use a counting lemma. Before stating this lemma, we need more definitions.
Suppose that $\mathcal{H}$ is an $(s, 3)$-complex with vertex classes $V_1,\dots,V_s$, which all have size $m$. Suppose also that $\mathcal{G}$ is an $(s, 3)$-complex with vertex classes $X_1,\dots, X_s$ of size at most $m$. For $i\in\{2,3\}$, we write $E_i(\mathcal{G})$ for the set of all $i$-edges of $\mathcal{G}$ and $e_i(\mathcal{G}):= |E_i(\mathcal{G})|$. We say that $\mathcal{H}$ \emph{respects the partition} of $\mathcal{G}$ if whenever $\mathcal{G}$ contains an $i$-edge with vertices in $X_{j_1}, \dots , X_{j_i}$, then there is an $i$-edge of $\mathcal{H}$ with vertices in $V_{j_1}, \dots , V_{j_i}$. On the other hand, we say that a labelled copy of $\mathcal{G}$ in $\mathcal{H}$ is \emph{partition-respecting} if for each $i \in[s]$ the vertices corresponding to those in $X_i$ lie within $V_i$. We denote by $|\mathcal{G}|_{\mathcal{H}}$ the number of labelled, partition-respecting copies of $\mathcal{G}$ in $\mathcal{H}$.
\begin{lemma}\label{Counting-lem}{\rm(Counting lemma {\cite[Theorem 4]{CFKO}}).} Let $s,r,t,n_0$ be positive integers and
let $d_2, d_3$, $\delta,\delta_3$ be positive constants such that $1/{d_2},1/{d_3}\in \mathbb{N}$ and
\[
1/{n_0}\ll 1/r, \delta\ll \min\{\delta_3, d_2 \}\ll \delta_3 \ll d_3, 1/s, 1/t.
\]
Then the following holds for all integers $n\ge n_0$. Suppose that $\mathcal{G}$ is an $(s, 3)$-complex on $t$ vertices with vertex classes $X_1,\dots, X_s$. Suppose also that $\mathcal{H}$ is a $(d_2,d_3,\delta_3,\delta, r)$-regular $(s, 3)$-complex with vertex classes $V_1,\dots,V_s$ all of size $n$, which respects the partition of $\mathcal{G}$. Then
\[
|\mathcal{G}|_{\mathcal{H}}\ge \frac{1}{2}n^t d^{e_2(\mathcal{G})}_2d^{e_3(\mathcal{G})}_3.
\]
\end{lemma}
\subsection{Reduced Hypergraphs.}
In order to use the counting lemma, we need to know the distribution of dense and regular polyads. For this purpose we need to introduce the so-called \textit{reduced hypergraphs} for 3-graphs. The terminology below follows \cite[Section~3]{reiher2018some}.
Given a $3$-graph $H$ on $n$ vertices with $n$ sufficiently large, applying Theorem~\ref{Reg-lem} to $H$, we obtain a family of partitions $ \mathcal{P}=\{\mathcal{P}^{(1)},\mathcal{P}^{(2)}\}$ of $V(H)$ such that
$(1)$ $\mathcal{P}$ is $(\eta, \delta(t), t)$-equitable and
$(2)$ $H$ is $(\delta_3, r(t))$-regular w.r.t. $\mathcal{P}$.
By the family of partitions $ \mathcal{P}$, we define an auxiliary $3$-graph $\mathcal{A}$ of $H$ as follows.
For each bipartite subgraphs
$P^{(2)}$ in $\mathcal{P}^{(2)}$, we view $P^{(2)}$ as a vertex of $\mathcal{A}$, namely, the sets $\mathcal{P}^{ij}$ for $\{i,j\}\in [a_1]^2$ as the \textit{vertex classes} of $\mathcal{A}$, and the 3-partite 3-graph $\mathcal{A}^{ijk}$ as the \textit{constituents} of $\mathcal{A}$
for any three distinct indices $i, j, k \in [a_1]$. We define $\mathcal{A}^{ijk}$ with vertex classes $\mathcal{P}^{ij}$, $\mathcal{P}^{jk}$ and $\mathcal{P}^{ik}$, and a triple $\{P^{ij}_a,P^{ik}_b,P^{jk}_c\}$ is defined to be an edge of $\mathcal{A}^{ijk}$ if and only if $H$ is $(d,\delta_3,r)$-regular w.r.t. $P^{ijk}_{abc}$ for some $d\geq d_3$.
Under such circumstances we call the $\binom{|a_1|}{2}$-partite $3$-graph $\mathcal{A}$ defined by
\[
V(\mathcal{A})=\bigcup_{\{i,j\}\in {I}^{2}}\mathcal{P}^{ij} \ \ \text{and}\ \
E(\mathcal{A})=\bigcup_{\{i,j,k\}\in {I}^{3}}E(\mathcal{A}^{ijk})
\]
a \textit{reduced hypergraph} of $H$. For any three vertex subsets $P_1,P_2,P_3\subseteq V(\mathcal{A})$, let $E_{\mathcal{A}}(P_1,P_2,P_3)$ be the set of $(p_1,p_2,p_3)\in P_1\times P_2\times P_3$ such that $\{p_1,p_2,p_3\}\in E(\mathcal{A})$. For $\eta>0$ such a reduced hypergraph $\mathcal{A}$ is said to be \emph{$\eta$-dense} if
\[
|E(\mathcal{A}^{ijk})|\geq \eta |\mathcal{P}^{ij}||\mathcal{P}^{ik}||\mathcal{P}^{jk}|
\]
holds for every triple $\{i,j,k\}\in I^{3}$.
\section{Proof of Theorem~\ref{1/8-dot-codegree}}
In order to prove Theorem~\ref{1/8-dot-codegree}, it suffices to verify the conditions in Corollary~\ref{covertrans}. The subsequent two results serve this purpose.
\begin{prop}\label{cover1}
All $k$-partite $k$-graphs $F$ belong to $\textbf{COVER}_k^{\ppoints{scale=0.11} ,k-1}$.
\end{prop}
\begin{proof}
Given a $k$-partite $k$-graph $F$ with vertex classes $X_1,\dots,X_k$, let $1/n \ll \mu \ll p, \alpha<1$ and $m:=\max\{|X_i|: i\in[k]\}$.
Suppose $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$.
First note that $\delta_{k-1}(H)\geq \alpha n$ implies that $\delta_1(H)\geq \alpha\binom{n-1}{k-1}$. By Proposition~\ref{x-good}, for any $0<\eta<\frac{\alpha^2}{4k!}$ and each $v\in V(H)$, there exists $u\in V(H)\setminus \{v\}$ such that $\{u,v\}$ is $\eta$-good.
By Lemma~\ref{good-reach.},
$v$ is contained in a copy of $K^k_k(m)$ and we are done.
\end{proof}
\begin{lemma}\label{transferral}
Suppose that $1/n\ll \mu \ll \lambda \ll \varepsilon, \zeta <1$ and $f, n \in \mathbb{N}$ with $f\ge 3$.
Let $F$ be a $3$-partite $3$-graph with $f:=v(F)$. If $H$ is an $(n,\frac{1}{8}+\varepsilon,\mu,\ppoints{scale=0.11})$ $3$-graph and $\mathcal{P}=\{X,Y\}$ is a partition of $V(H)$ with $|X|,|Y|\ge \zeta n$, then $(1,-1)\in L_{\mathcal{P},F}^{\lambda }(H)$.
\end{lemma}
\begin{proof}
Suppose that $F$ is a 3-partite 3-graph with vertex classes $X_1,X_2,X_3$ and $|X_i|=f_i$ for $i\in [3]$. We choose constants satisfying the following hierarchy:
\[
1/n\ll \mu \ll \lambda \ll 1/r, \delta \ll 1/t \le d_2 \ll \eta, \delta_3 \ll d_3\ll \varepsilon ,\zeta, 1/f,
\]
where $1/d_2\in \mathbb{N}$. Recall that the hypergraph regularity lemma is proved by iterated refinements starting with an arbitrary initial partition. Hence, applying Theorem~\ref{Reg-lem} to $H$ with an initial partition $\{X,Y\}$, we obtain a vertex partition $\mathcal{P}^{(1)}= \{V_1, V_2, \dots, V_{t_1+t_2}\} $ of $V(H)$ where $1/\eta \le t_1+t_2\le t$ and a $2$-edge partition $\mathcal{P}^{(2)}=\cup_{1\le i<j\le t_1+t_2} \mathcal{P}^{ij}$. In particular, $H$ is $(\delta_3,r)$-regular w.r.t. $\mathcal{P}(2)=\{\mathcal{P}^{(1)}, \mathcal{P}^{(2)}\}$. Without loss of generality, we may assume that $X=\bigcup _{i=1}^{t_1}V_i$, $Y=\bigcup _{i=t_1+1}^{t_2}V_i$ and $t!\mid n$ (by discarding up to $t!$ vertices if necessary).
Let $\mathcal{A}$ be the reduced hypergraph of $H$. Then $V(\mathcal{A})=\cup_{1\le i<j\le t_1+t_2} \mathcal{P}^{ij}$.
We below show that $\mathcal{A}$ is $p^*$-dense for some $p^*>\frac{1}{8}$, i.e. given a triple $\{i,j,k\}\in [t_1+t_2]^3$,
\[
|E(\mathcal{A}^{ijk})|\geq p^*|\mathcal{P}^{ij}||\mathcal{P}^{ik}||\mathcal{P}^{jk}|
\]
holds for some $p^*>\frac{1}{8}$. As $H$ is an $(n,\frac{1}{8}+\varepsilon,\mu,\ppoints{scale=0.11})$ 3-graph, the number of $d_3$-useful edges in $E(V_i,V_j,V_k)$ is at least
\[
(\frac{1}{8}+\varepsilon )|V_i||V_j||V_k|-\mu n^3-2d_3n^3\geq (\frac{1}{8}+\frac{\varepsilon}{2} )|V_i||V_j||V_k|
\]
by our choices of constants.
On the other hand, by the triangle counting lemma, each triad $P_{abc}^{ijk}$ satisfies
\[
\mathcal{K}_3(P_{abc}^{ijk})\leq (d_2^3+3\delta)|V_i||V_j||V_k|,
\]
so we have
\[
(\frac{1}{8}+\frac{\varepsilon}{2} )|V_i||V_j||V_k|\leq |E(\mathcal{A}^{ijk})|\cdot (d_2^3+3\delta)|V_i||V_j||V_k|.
\]
Hence, by our choices of constants,
$|E(\mathcal{A}^{ijk})|\geq p^*|\mathcal{P}^{ij}||\mathcal{P}^{ik}||\mathcal{P}^{jk}|$ for some $p^*>\frac{1}{8}$ holds.
We begin to show that $(1,-1)\in L_{\mathcal{P},F}^{\lambda }(H)$ by contradiction. Choose arbitrary clusters $V_i,V_j,V_k\subseteq X$. As $H$ is an $(n,\frac{1}{8}+\varepsilon,\mu ,\ppoints{scale=0.11})$ 3-graph, we have $e_H(V_i,V_j,V_k)\geq (\frac{1}{8}+\varepsilon)|V_i||V_j||V_k|-\mu n^3$. Therefore, there are at least $(\frac{1}{8}+\varepsilon)|V_i||V_j||V_k|-\mu n^3-2d_3n^3>0$ $d_3$-useful edges in $E_H(V_i,V_j,V_k)$, which implies there is a tripartite graph $P^{ijk}_{abc}$ such that $H$ is $(d,\delta_3,r)$-regular w.r.t. $P^{ijk}_{abc}$ for some $d\geq d_3$. This gives us a $(3,3)$-complex by adding $E(H)\cap \mathcal{K}(P^{ijk}_{abc})$ as the ``3rd level". Here we turn $F$ into a $(3,3)$-complex by making each edge into a complete $i$-graph $K_3^{(i)}$ for $i=1,2$. By Lemma~\ref{Counting-lem}, there are at least $\frac{1}{2}(\frac{n}{t_1+t_2})^fd_3^{e(F)}d_2^{|\partial F|}\geq \lambda n^f$ copies of $F$ in $H[V_i\cup V_j\cup V_k]$. Thus, we have $(f,0)\in L_{\mathcal{P},F}^{\lambda }(H)$. Similarly, we have $(0,f),(f_1+f_2,f_3)\in L_{\mathcal{P},F}^{\lambda }(H)$. We assume that $(1,-1)\notin L_{\mathcal{P},F}^{\lambda }(H)$, so
\begin{gather}
(f-1,1),(1,f-1),(f_1+f_2-1,f_3+1)\notin L_{\mathcal{P},F}^{\lambda }(H). \tag{$\star$}
\end{gather}
For any vertex class $\mathcal{P}^{i_1i_2}\subseteq V(\mathcal{A})$ and any vertex $P^{i_1i_2}_a\in \mathcal{P}^{i_1i_2}$, we color it as follows.
~\\
\noindent \textbf{Case 1.} If $i_1,i_2\in [t_1]$ or $i_1,i_2\in [t_1+t_2]\setminus[t_1]$, then $P^{i_1i_2}_a$ is colored by {\color{red}red} if there is an edge $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}\in E(\mathcal{A})$ for some $i_3\in [t_1+t_2]\setminus[t_1]$ while $P^{i_1i_2}_a$ is colored by {\color{blue}blue} if there is an edge $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}\in E(\mathcal{A})$ for some $i_3\in [t_1]$.
\noindent \textbf{Case 2.} If $i_1\in [t_1]$ and $i_2\in [t_1+t_2]\setminus[t_1]$, then $P^{i_1i_2}_a$ is colored by {\color{red}red} if there is an edge $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}\in E(\mathcal{A})$ for some $i_3\in [t_1]$ while $P^{i_1i_2}_a$ is colored by {\color{blue}blue} if there is an edge $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}\in E(\mathcal{A})$ for some $i_3\in [t_1+t_2]\setminus[t_1]$.
~\\
We claim that any vertex $P^{i_1i_2}_a\in V(\mathcal{A})$ can not receive two colors for $1\leq i_1<i_2\leq t_1+t_2$. For $i_1,i_2\in [t_1]$, if not, there exist two indices $i_3\in [t_1]$ and $i'_3\in [t_1+t_2]\setminus[t_1]$ satisfying both $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}$ and $\{P^{i_1i_2}_a,P^{i_1i'_3}_b,P^{i_2i'_3}_c\}$ form edges. By Lemma~\ref{Counting-lem}, for any fixed $v^*\in X_3$, there are at least $\frac{1}{2}(\frac{n}{t_1+t_2})^fd_3^{e(F)}d_2^{|\partial F|}\geq \lambda n^f$ copies of $F$ satisfying that $X_1$ is embedded into $V_{i_1}$, $X_2$ is embedded into $V_{i_2}$, $X_3\setminus \{v^*\}$ into $V_{i_3}$ and $\{v^*\}$ into $V_{i'_3}$. Therefore $(f-1,1)\in L_{\mathcal{P},F}^{\lambda }(H)$, which contradicts our assumption in ($\star$). Similarly, for $i_1,i_2\in [t_1+t_2]\setminus[t_1]$, any vertex $P^{i_1i_2}_a\in V(\mathcal{A})$ can not receive two colors either. For Case 2, if not, there exist two indices $i_3\in [t_1]$ and $i'_3\in [t_1+t_2]\setminus[t_1]$ satisfying both $\{P^{i_1i_2}_a,P^{i_1i_3}_b,P^{i_2i_3}_c\}$ and $\{P^{i_1i_2}_a,P^{i_1i'_3}_b,P^{i_2i'_3}_c\}$ form edges. By Lemma~\ref{Counting-lem}, for any fixed $v^*\in X_2$, there are at least $\frac{1}{2}(\frac{n}{t_1+t_2})^f d_3^{e(F)}d_2^{|\partial F|}\geq \lambda n^f$ copies of $F$ satisfying that $X_1$ is embedded into $V_{i_1}$, $X_3$ is embedded into $V_{i_2}$, $X_2\setminus \{v^*\}$ into $V_{i_3}$ and $\{v^*\}$ into $V_{i'_3}$. Therefore $(f_1+f_2-1,f_3+1)\in L_{\mathcal{P},F}^{\lambda }(H)$, a contradiction.
Given any vertex class $\mathcal{P}^{i_1i_2}\subseteq V(\mathcal{A})$, let $\mathcal{P}^{i_1i_2}_{{\rm {\color{red}red}}}\subseteq \mathcal{P}^{i_1i_2}$ be the set of vertices colored by {\color{red}red} in $\mathcal{P}^{i_1i_2}$ and $\mathcal{P}^{i_1i_2}_{{\rm {\color{blue}blue}}}\subseteq \mathcal{P}^{i_1i_2}$ be the set of vertices colored by {\color{blue}blue} in $\mathcal{P}^{i_1i_2}$. Note that the following holds.
\begin{itemize}
\item [{\rm (i)}] For any $i_1,i_2\in [t_1]$ and $i_3\in [t_1+t_2]\setminus[t_1]$, $E(\mathcal{A}^{i_1i_2i_3})=E_{\mathcal{A}}(\mathcal{P}^{i_1i_2}_{{\rm {\color{red}red}}},\mathcal{P}^{i_1i_3}_{{\rm {\color{red}red}}},\mathcal{P}^{i_2i_3}_{{\rm {\color{red}red}}})$.
\item [{\rm (ii)}] For any $i_1\in [t_1]$ and $i_2,i_3\in [t_1+t_2]\setminus[t_1]$, $E(\mathcal{A}^{i_1i_2i_3})=E_{\mathcal{A}}(\mathcal{P}^{i_1i_2}_{{\rm {\color{blue}blue}}},\mathcal{P}^{i_1i_3}_{{\rm {\color{blue}blue}}},\mathcal{P}^{i_2i_3}_{{\rm {\color{blue}blue}}})$.
\item [{\rm (iii)}] For any $i_1,i_2,i_3\in [t_1]$, $E(\mathcal{A}^{i_1i_2i_3})=E_{\mathcal{A}}(\mathcal{P}^{i_1i_2}_{{\rm {\color{blue}blue}}},\mathcal{P}^{i_1i_3}_{{\rm {\color{blue}blue}}},\mathcal{P}^{i_2i_3}_{{\rm {\color{blue}blue}}})$.
\item [{\rm (iv)}] For any $i_1,i_2,i_3\in [t_1+t_2]\setminus[t_1]$, $E(\mathcal{A}^{i_1i_2i_3})=E_{\mathcal{A}}(\mathcal{P}^{i_1i_2}_{{\rm {\color{red}red}}},\mathcal{P}^{i_1i_3}_{{\rm {\color{red}red}}},\mathcal{P}^{i_2i_3}_{{\rm {\color{red}red}}})$.
\end{itemize}
First, by {\rm(iii)}, since $\mathcal{A}$ is $p^*$-dense with $p^* >1/8$, there must exist $i'_1,i'_2\in [t_1]$ such that $|\mathcal{P}^{i'_1i'_2}_{{\rm {\color{red}red}}}|<\frac{1}{2}|\mathcal{P}^{i'_1i'_2}|$. By {\rm (i)}, we have that for any $i_3\in [t_1+t_2]\setminus[t_1]$, $|\mathcal{P}^{i'_1i_3}_{{\rm {\color{red}red}}}|>\frac{1}{2}|\mathcal{P}^{i'_1i_3}|$ or $|\mathcal{P}^{i'_2i_3}_{{\rm {\color{red}red}}}|>\frac{1}{2}|\mathcal{P}^{i'_2i_3}|$ as $\mathcal{A}$ is $p^*$-dense with $p^*>\frac{1}{8}$. Let $I_1\subseteq [t_1+t_2]\setminus[t_1]$ be the set of indices satisfying the former inequality and $I_2\subseteq [t_1+t_2]\setminus[t_1]$ be the set of indices satisfying the latter. Without loss of generality, we assume $|I_1|\geq \frac{1}{2}t_2$. For any two indices $i_3,i'_3\in I_1$, we have $|\mathcal{P}^{i'_1i_3}_{{\rm {\color{blue}blue}}}|<\frac{1}{2}|\mathcal{P}^{i'_1i_3}|$ and $|\mathcal{P}^{i'_1i'_3}_{{\rm {\color{blue}blue}}}|<\frac{1}{2}|\mathcal{P}^{i'_1i'_3}|$ by the definition of $I_1$.
Therefore, $|\mathcal{P}^{i_3i'_3}_{{\rm {\color{blue}blue}}}|>\frac{1}{2}|\mathcal{P}^{i_3i'_3}|$ holds by {\rm (ii)} and $p^*$-denseness of $\mathcal{A}$ for some $p^*>\frac{1}{8}$. This implies that for any two indices $i_3,i'_3\in I_1$, $|\mathcal{P}^{i_3i'_3}_{{\rm {\color{red}red}}}|<\frac{1}{2}|\mathcal{P}^{i_3i'_3}|$, which contradicts that $\mathcal{A}$ is $p^*$-dense with $p^*>\frac{1}{8}$ by {\rm (iv)}.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{1/8-dot-codegree}]
Suppose that $1/n\ll \mu\ll \lambda' \ll \zeta, \varepsilon,\alpha, 1/f <1$ and $f, n \in \mathbb{N}$ with $n\in f\mathbb{N}$.
Let $F$ be a $3$-partite $3$-graph $F$ with $f:=v(F)$. By the result of Erd\H{o}s~\cite{E-1964}, $\pi_{\ppoints{scale=0.11}}(F)=0$.
Suppose that $H$ is an $(n,\frac{1}{8}+\varepsilon,\mu,\ppoints{scale=0.11})$ $3$-graph with $\delta_2(H)\ge \alpha n$.
For each induced subhypergraph $H'$ on $V'\subseteq V(H)$ with $|V'|\ge 2\zeta n$, $H'$ is a $(v(H'), \frac{1}{8}+\varepsilon, \mu', \ppoints{scale=0.11})$ $3$-graph with $\mu'=\mu/(2\zeta)^3$. For $H'$ and a partition $\mathcal{P'}=\{X, Y\}$ of $V(H')$ with $|X|, |Y|\ge \zeta n$, we apply Lemma~\ref{transferral} to get $\lambda'$, then the property ($\heartsuit$) in Corollary~\ref{covertrans} holds for $H$.
By Proposition~\ref{cover1}, we can apply Corollary~\ref{covertrans} to find an $F$-factor of $H$.
\end{proof}
\section{Proof of Theorem~\ref{factor3-2}}
In this section, we shall prove Theorem~\ref{factor3-2} by Corollary~\ref{covertrans} and the regularity method. By the definitions of $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}$ and $\textbf{COVER}_3^{\ppoints{scale=0.11},2}$, we trivially have $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}\subseteq \textbf{COVER}_3^{\ppoints{scale=0.11},2}$.
In~\cite{DHSWZ}, we proved the following result using a random construction.
\begin{observation}{\rm ~\cite[Observation 6.4]{DHSWZ}.}\label{construction-trans}
For $2\le s\le k-1$, $\textbf{FACTOR}_k^{\ppoints{scale=0.11}, s} \subseteq {\bf TRANS}^{s}_k$.
\end{observation}
Therefore, we have $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}\subseteq {\bf TRANS}_{3}^2$. The following lemma shows that $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}\subseteq \{F:F\ {\rm is}\ {\rm a}\ {\text{3-graph}}\ {\rm and}\ \pi_{\ppoints{scale=0.11}}(F)=0\}$.
\begin{lemma}\label{turan-density}
For any $F\in \textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$, $\pi_{\ppoints{scale=0.11}}(F)=0$.
\end{lemma}
\begin{proof}
Suppose that
\[
0<1/n \ll \mu \ll \epsilon<1/{(f-1)},p<1.
\]
Let $F\in \textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$ with $v(F)=f$ and $H$ be an $(n,p,\mu,\ppoints{scale=0.11})$ 3-graph.
It suffices to show that $H$ contains a copy of $F$. We construct an auxiliary $3$-graph $H'$ as follows. Let $V(H')=V(H)\cup V_0$ where $V_0$ and $V(H)$ are disjoint and $|V_0|=\epsilon n$. Now we define $E(H')$. First, we add all $e\in E(H)$ into $E(H')$. Then, for any 3-set $e$ which contains at least one vertex in $V_0$, we add $e$ into $E(H')$. One can easily verify that $H'$ is $((1+\epsilon )n,p,\mu',\ppoints{scale=0.11})$-dense with $\delta_2(H')\geq \epsilon n$ and $\mu'=\mu/(1+\epsilon)^3$. As $F\in \textbf{FACTOR}_3^{\ppoints{scale=0.11},2}$, $H'$ has an $F$-factor. As $\epsilon <\frac{1}{f-1}$, there is at least one copy of $F$ in $H$, which implies $\pi_{\ppoints{scale=0.11}}(F)=0$.
\end{proof}
For the converse implication of Theorem~\ref{factor3-2}, we only need to prove the following lemma.
\begin{lemma}\label{transferral3-2}
Suppose that $1/n\ll \mu \ll \lambda \ll p,\zeta<1$ and $n\in \mathbb{N}$. Let $F\in {\bf TRANS}_3^2$ and $\pi_{\ppoints{scale=0.11}}(F)=0$.
If $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$
$3$-graph with vertex partition $\mathcal{P}=\{X,Y\}$ and $|X|,|Y|\geq \zeta n$, then $(-1,1)\in L_{\mathcal{P},F}^{\lambda }(H)$.
\end{lemma}
We shall use the following lemmas to prove Lemma~\ref{transferral3-2}.
Recall that for a $k$-graph $F$, $\partial F:=\{S\in [V(F)]^{k-1}: \exists\ e\in E(F), S\subseteq e\}$.
In our previous work~\cite{DHSWZ}, based on the work of Reiher-R\"odl-Schacht~\cite{Reiher2018Hypergraphs}, we show that a $3$-graph $F$ with $\pi_{\ppoints{scale=0.11}}(F)=0$ admits a structural $3$-coloring of $\partial F$ w.r.t. a 2-shadow disjoint partition $\mathcal{P}$.
\begin{lemma}{\rm (\cite[Lemma 4.2]{DHSWZ}).}
\label{alter} For a $3$-graph $F$ with $\pi_{\ppoints{scale=0.11}}(F)=0$ and a 2-shadow disjoint partition $\mathcal{P}=\{A,B\}$ of $V(F)$, there is an enumeration $\{v_1,v_2,\dots,v_f\}$ of $V(F)$ with $\{v_1,\cdots,v_{|A|}\}=A$ and $\{v_{|A|+1},\cdots,v_f\}=B$ and there is a $3$-coloring $\varphi:\partial F \longmapsto \{{\color {red}red},~{\color{blue}blue},~{\color{green}green}\}$ such that every hyperedge $\{v_i,v_j,v_k\}\in E(F)$ with $i<j<k$ satisfies:
\[
\varphi(v_i,v_j)={\color {red}red},\ \ \varphi(v_i,v_k)={\color{blue}blue},\ \ \varphi(v_j,v_k)={\color{green}green}.
\]
\end{lemma}
\begin{lemma}\label{Rembed}{\rm (\cite[Lemma 3.1]{Reiher2018Hypergraphs}).}
Given $\eta>0$ and $h\in \mathbb{N}$, there exists an integer $q$ such that the following holds. If $\mathcal{A}$ is an $\eta$-dense reduced hypergraph with index set $[q]$, vertex class $\mathcal{P}^{ij}$ and constituents $\mathcal{A}^{ijk}$, then
\begin{itemize}
\item [{\rm (i)}] there are indices $\lambda(1)<\cdots<\lambda(h)$ in $[q]$ and
\item [{\rm (ii)}] for each pair $1\leq r<s\leq h$ there are three vertices $P^{\lambda(r)\lambda(s)}_{{\rm{\color{red} red}}}$, $P^{\lambda(r)\lambda(s)}_{{\rm {\color{blue}blue}}}$ and $P^{\lambda(r)\lambda(s)}_{{\rm {\color{green}green}}}$ in $\mathcal{P}^{\lambda(r)\lambda(s)}$ such that for every triple indices $1\leq r<s<t\leq h$ the three vertices $P^{\lambda(r)\lambda(s)}_{{\rm{\color{red} red}}}$, $P^{\lambda(r)\lambda(t)}_{{\rm {\color{blue}blue}}}$ and $P^{\lambda(s)\lambda(t)}_{{\rm {\color{green}green}}}$ form a hyperedge in $\mathcal{A}^{\lambda(r)\lambda(s)\lambda(t)}$.
\end{itemize}
\end{lemma}
\begin{lemma}{\rm (\cite[Lemma 4.8]{DHSWZ}).}
\label{biembed}
Given $\eta>0$ and $h_1,h_2\in \mathbb{N}$, there exist integers $q_1$ and $q_2$ such that the following holds. If $\mathcal{A}$ is an $\eta$-dense reduced hypergraph with index set $[q_1+q_2]$, vertex class $\mathcal{P}^{ij}$ and constituents $\mathcal{A}^{ijk}$, then there are indices $\lambda(1)<\cdots<\lambda(h_1)$ in $[q_1]$ and $\sigma(1)<\cdots<\sigma(h_2)$ in $[q_1+q_2]\setminus [q_1]$ satisfying that
\begin{itemize}
\item [{\rm (i)}] for each pair $1\leq r<s\leq h_1$ there is a vertex $P^{\lambda(r)\lambda(s)}_{{\rm {\color{red}red}}}$;
\item [{\rm (ii)}] for any $1\leq r\leq h_1$ and any $1\leq t\leq h_2$ there are two vertices $P^{\lambda(r)\sigma(t)}_{{\rm {\color{blue}blue}}}$ and $P^{\lambda(r)\sigma(t)}_{{\rm {\color{green}green}}}$;
\item [{\rm (iii)}] for every triple indices $1\leq r<s\leq h_1$ and $1\leq t\leq h_2$ the three vertices $P^{\lambda(r)\lambda(s)}_{{\rm {\color{red}red}}}$, $P^{\lambda(r)\sigma(t)}_{{\rm {\color{blue}blue}}}$ and $P^{\lambda(s)\sigma(t)}_{{\rm {\color{green}green}}}$ form a hyperedge in $\mathcal{A}^{\lambda(r)\lambda(s)\sigma(t)}$.
\end{itemize}
\end{lemma}
\noindent{\textbf{Remark.}{\rm (\cite{DHSWZ}).}} In the following proof we also need another statement of Lemma~ \ref{biembed} as follows.
\emph{
Given $\eta>0$ and $h_1,h_2\in \mathbb{N}$, there exist integers $q_1$ and $q_2$ such that the following holds. If $\mathcal{A}$ is an $\eta$-dense reduced hypergraph with index set $[q_1+q_2]$, vertex class $\mathcal{P}^{ij}$ and constituents $\mathcal{A}^{ijk}$, then there are indices $\lambda(1)<\cdots<\lambda(h_1)$ in $[q_1]$ and $\sigma(1)<\cdots<\sigma(h_2)$ in $[q_1+q_2]\setminus [q_1]$ satisfying that
\begin{itemize}
\item [{\rm (i)}] for each pair $1\leq s<t\leq h_2$ there is a vertex $P^{\sigma (s)\sigma (t)}_{{\rm {\color{green}green}}}$;
\item [{\rm (ii)}] for any $1\leq r\leq h_1$ and any $1\leq t\leq h_2$ there are two vertices $P^{\lambda(r)\sigma(t)}_{{\rm {\color{red}red}}}$ and $P^{\lambda(r)\sigma(t)}_{{\rm {\color{blue}blue}}}$;
\item [{\rm (iii)}] for every triple indices $1\leq r\leq h_1$ and $1\leq s<t\leq h_2$ the three vertices $P^{\lambda(r)\sigma (s)}_{{\rm {\color{red}red}}}$, $P^{\lambda(r)\sigma(t)}_{{\rm {\color{blue}blue}}}$ and $P^{\sigma (s)\sigma(t)}_{{\rm {\color{green}green}}}$ form a hyperedge in $\mathcal{A}^{\lambda(r)\sigma (s)\sigma(t)}$.
\end{itemize}
}
\begin{proof}[Proof of Lemma~\ref{transferral3-2}]
Given $F\in {\bf TRANS}_3^2$ with $\pi_{\ppoints{scale=0.11}}(F)=0$ and $f:=v(F)$, let $\mathcal{F}_1=\{A_1,B_1\},\mathcal{F}_2=\{A_2,B_2\},\cdots,\mathcal{F}_m=\{A_{m},B_m\}$ be all 2-shadow disjoint bipartions of $V(F)$ with $|A_i|=a_i$ and $|B_i|=b_i$ for $i\in[m]$. Choose constants satisfying the following hierarchy:
\[
1/n,\lambda,\mu \ll 1/r,\delta\ll d_2, \delta_3,1/t \ll 1/M_1 \ll 1/M_2 \ll 1/m_2\ll 1/m_1, d_3\ll p,\zeta,1/f
\]
where $d_2\geq 1/t$ and $1/d_2\in \mathbb{N}$.
As in the proof of Lemma~\ref{transferral},
we apply Theorem~\ref{Reg-lem} to $H$ with the initial partition $\{X,Y\}$ and obtain a vertex partition $\mathcal{P}^{(1)}= \{V_1, V_2, \dots, V_{t_1+t_2}\} $ of $V(H)$ with $1/\eta \le t_1+t_2\le t$ and a $2$-edge partition $\mathcal{P}^{(2)}=\cup_{1\le i<j\le t_1+t_2} \mathcal{P}^{ij}$. In addition, $H$ is $(\delta_3,r)$-regular w.r.t. $\mathcal{P}(2)=\{\mathcal{P}^{(1)}, \mathcal{P}^{(2)}\}$. Without loss of generality, we may assume that $X=\bigcup _{i=1}^{t_1}V_i$, $Y=\bigcup _{i=t_1+1}^{t_2}V_i$ and $t!\mid n$ (by discarding up to $t!$ vertices if necessary).
Let $\mathcal{A}$ be the reduced hypergraph of $H$. Then $V(\mathcal{A})=\cup_{1\le i<j\le t_1+t_2} \mathcal{P}^{ij}$.
Similar to the proof of Lemma~\ref{transferral}, $\mathcal{A}$ is $p^*$-dense for some $p^*>0$.
For any $\mathcal{F}_i=\{A_i,B_i\}$ where $i\in [m]$, we first prove that $(a_i,b_i)\in I_{\mathcal{P},F}^{\lambda}(H)$. For any integer $0\leq j\leq 3$, let $F^{\mathcal{F}_i}_j\subseteq F$ be the 3-graph consisting of all 3-edges $e$ with $|e\cap A_i|=j$. An easy observation is that $\partial F^{\mathcal{F}_i}_0$, $\partial F^{\mathcal{F}_i}_1$, $\partial F^{\mathcal{F}_i}_2$, $\partial F^{\mathcal{F}_i}_3$ are pairwise disjoint as $\mathcal{F}_i$ is a 2-shadow disjoint bipartition.
In order to use Lemma~\ref{Counting-lem}, we need to find a regular $f$-partite graph.
We first apply Lemma~\ref{Rembed} with index set $[t_1]$ to find an index subset $\mathcal{I}_1$ of $[t_1]$ with $|\mathcal{I}_1|=M_1$ such that Property {\rm{(ii)}} from Lemma~\ref{Rembed} is satisfied. Similarly, we apply Lemma~\ref{Rembed} with index set $[t_1+t_2]\setminus [t_1]$ to find an index subset $\mathcal{I}_2$ of cardinality $M_2$ in $[t_1+t_2]\setminus [t_1]$. Next we apply Lemma \ref{biembed} with $h_1:=m_1$, $h_2:=m_2$ and the set $\mathcal{I}_1\cup\mathcal{I}_2$ to find a subset $\mathcal{J}_1$ of $\mathcal{I}_1$ with cardinality $m_1$ and a subset $\mathcal{J}_2$ of $\mathcal{I}_2$ with cardinality $m_2$ such that Properties {\rm(i)}--{\rm(iii)} from Lemma \ref{biembed} are satisfied.
Finally, we apply the Remark of Lemma \ref{biembed} with $h_1:=a_i$, $h_2:=b_i$ and the index set $\mathcal{J}_1\cup\mathcal{J}_2$ to find a subset $\mathcal{X}$ of $\mathcal{J}_1$ of cardinality $a_i$ and a subset $\mathcal{Y}$ of $\mathcal{J}_2$ of cardinality $b_i$ satisfying Properties {\rm(i)}--{\rm(iii)} from Lemma \ref{biembed}. Without loss of generality, we may assume that $\mathcal{X}=\{\lambda(1),\dots,\lambda(a_i)\}$ and $\mathcal{Y}=\{\lambda(a_i+1),\dots,\lambda(f)\}$ with $\lambda(1)<\dots<\lambda(f)$.
By Lemma~\ref{alter}, there is an enumeration of $V(F)$ and a $3$-coloring w.r.t. $\mathcal{F}_i$ satisfying the property in Lemma~\ref{alter}. For any $\{v_i,v_j,v_k\}\in \bigcup _{j=0}^3F^{\mathcal{F}_i}_j$, by our choices of $\mathcal{X}$ and $\mathcal{Y}$, we have $d(H|P^{\lambda(i)\lambda(j)\lambda(k)}_{{\rm {\color{red}red},{\color{blue}blue},{\color{green}green}}})\geq d_3$. Since $\partial F^{\mathcal{F}_i}_0$, $\partial F^{\mathcal{F}_i}_1$, $\partial F^{\mathcal{F}_i}_2$, $\partial F^{\mathcal{F}_i}_3$ are pairwise disjoint, by Lemma~\ref{Counting-lem}, there are at least $\frac{1}{2}(\frac{n}{t_1+t_2})^kd_3^{e(F)}d_2^{|\partial F|}\geq \lambda n^k$ copies of $F$ satisfying that $v_i$ is embedded into $V_{\lambda (i)}$ for $1\leq i\leq f$. Therefore $(a_i,b_i)\in I_{\mathcal{P},F}^{\lambda }(H)$.
As $F\in {\bf TRANS}_3^2$, i.e. $(1,-1)\in L_F^2$, we have $(1,-1)\in L_{\mathcal{P},F}^{\lambda }(H)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{factor3-2}]
By the definitions of $\textbf{COVER}_3^{\ppoints{scale=0.11},2}$ and $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}$, Observation~\ref{construction-trans} and Lemma~\ref{turan-density}, we have $\textbf{FACTOR} _{3}^{\ppoints{scale=0.11},2}\subseteq \textbf{COVER}_3^{\ppoints{scale=0.11},2}\cap {\bf TRANS}^2_3\cap \{F:F\ {\rm is}\ {\rm a}\ {\text{3-graph}}\ {\rm and}\ \pi_{\ppoints{scale=0.11}}(F)=0\}$. For the reverse direction, consider $F\in \textbf{COVER}_3^{\ppoints{scale=0.11},2}\cap {\bf TRANS}^2_3\cap \{F:F\ {\rm is}\ {\rm a}\ {\text{3-graph}}\ {\rm and}\ \pi_{\ppoints{scale=0.11}}(F)
\\
=0\}$ with $f:=v(F)$. Suppose that $1/n \ll\mu \ll \lambda' \ll \zeta,p,\alpha,1/f<1$ with $n\in f\mathbb{N}$.
Let $H$ be an $(n,p,\mu,\ppoints{scale=0.11})$ $3$-graph with $\delta_2(H)\ge \alpha n$.
For each induced subhypergraph $H'$ on $V'\subseteq V(H)$ with $|V'|\ge 2\zeta n$, $H'$ is a $(v(H'), p, \mu', \ppoints{scale=0.11})$ $3$-graph with $\mu'=\mu/(2\zeta)^3$. For $H'$ and a partition $\mathcal{P'}=\{X, Y\}$ of $V(H')$ with $|X|, |Y|\ge \zeta n$, we apply Lemma~\ref{transferral3-2} to get $\lambda'$, then the property ($\heartsuit$) in Corollary~\ref{covertrans} holds for $H$.
By Corollary~\ref{covertrans}, we can find an $F$-factor of $H$.
\end{proof}
\section{The proof of Theorem~\ref{gcd}}
In this section, we shall prove Theorem~\ref{gcd}. Similar to the proof of Theorem~\ref{1/8-dot-codegree}, we only need to prove a lemma similar to Lemma~\ref{transferral}.
First, we need the following proposition from~\cite{Han2017Minimum}.
\begin{prop}{\rm{(\cite[\rm Proposition 3.1]{Han2017Minimum})}.}
\label{supersaturation}
Given integers $k,r_0,a_1,\dots ,a_k\in \mathbb{N}$, suppose that $1/n\ll \lambda \ll \eta ,1/k,1/r_0,1/a_1,\dots ,1/a_k$. Let $H$ be a $k$-graph on $n$ vertices with a vertex partition $V_1\cup \cdots \cup V_r$ where $r\leq r_0$. Suppose $i_1,\dots ,i_k\in [r]$ and $H$ contains at least $\eta n^k$ edges $e=\{v_1,\dots ,v_k\}$ such that $v_1\in V_{i_1},\dots ,v_k\in V_{i_k}$. Then $H$ contains at least $\lambda n^{a_1+\cdots +a_k}$ copies of $K_{a_1,\dots ,a_k}$ whose jth part is contained in $V_{i_j}$ for all $j\in[k]$.
\end{prop}
\begin{lemma}\label{k-transferral}
Suppose that $1/n \ll \mu \ll \lambda \ll p,\zeta<1$ and $n\in \mathbb{N}$. Let $F$ be a $k$-partite $k$-graph with $\gcd(\mathcal S(F))=1$ and $f:=v(F)$.
If $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph with vertex partition $\mathcal{P}=\{X,Y\}$ of $V(H)$ and $|X|,|Y|\ge \zeta n$, then $(1,-1)\in L_{\mathcal{P},F}^{\lambda }(H)$.
\end{lemma}
\begin{proof}
Suppose that $F$ is a $k$-partite $k$-graph of order $f$ satisfying $\gcd(\mathcal S(F))=1$.
Let $K_1,\dots, K_t$ be all $k$-partite realisations of $F$ with $t\ge 1$. For any $i\in [t]$, let $V_{i1}, \dots, V_{ik}$ denote the vertex classes of $K_i$ with $|V_{ij}|= a_{ij}$
for $j\in [k]$. Choose constants satisfying the following hierarchy:
\[
1/n\ll \mu \ll \lambda \ll \eta \ll p,\zeta,1/k,1/f.
\]
As $H$ is $(p,\mu,\ppoints{scale=0.11})$-dense, we have $e(X,\dots ,X)\geq p|X|^k-\mu n^k\geq p(\zeta n)^k-\mu n^k$. Then there are at least $\frac{1}{k!}e(X,\dots ,X)\geq \eta n^k$ edges in $H[X]$. By Lemma~\ref{supersaturation}, there are at least $\lambda n^f$ copies of $K_i$ in $H[X]$ for each $i\in [t]$, which implies that $(f,0)\in L_{\mathcal{P},K_i}^{\lambda }(H)$. Since $e(Y,X,\dots ,X)\geq p|Y||X|^{k-1}-\mu n^k\geq p(\zeta n)^k-\mu n^k$, by Lemma~\ref{supersaturation} there are at least $\lambda n^f$ copies of $K_i$ where $V_{i1}$ is contained in $Y$ and $V_{ij}$ is contained in $X$ for $j\in [k]\setminus \{1\}$. This implies that $(\sum_{j\in [k]\setminus \{1\}}a_{ij}, a_{i1})\in L_{\mathcal{P},K_i}^{\lambda }(H)$. Similarly, $(\sum_{j\in [k]\setminus \{2\}}a_{ij}, a_{i2}),\dots ,(\sum_{j\in [k]\setminus \{k\}}a_{ij}, a_{ik})\in L_{\mathcal{P},K_i}^{\lambda }(H)$. Since $K_i$ is a $k$-partite realisation of $F$, $L_{\mathcal{P},K_i}^{\lambda }(H)=L_{\mathcal{P},F}^{\lambda }(H)$ for all $i\in [t]$. Therefore, $(a_{ij},-a_{ij})\in L_{\mathcal{P},F}^{\lambda }(H)$ for all $i\in [t]$ and $j\in [k]$. As $\gcd(\mathcal S(F))=1$, there exist integers $m_{ij}$ for $i\in [t]$ and $j\in [k]$ such that $\sum_{i,j}m_{ij}a_{ij}=1$.
Hence $(1,-1)=\sum_{i,j}m_{ij}(a_{ij},-a_{ij})\in L_{\mathcal{P},F}^{\lambda }(H)$.
\end{proof}
\begin{proof}[Proof of Theorem~\ref{gcd}]
Suppose that $1/n\ll \mu\ll \lambda' \ll \zeta, p,\alpha, 1/f <1$ and $f, n \in \mathbb{N}$ with $n\in f\mathbb{N}$.
Let $F$ be a $k$-partite $k$-graph $F$ with satisfying $\gcd(\mathcal S(F))=1$ and $f:=v(F)$. By the result of Erd\H{o}s~\cite{E-1964}, $\pi_{\ppoints{scale=0.11}}(F)=0$.
Suppose that $H$ is an $(n,p,\mu,\ppoints{scale=0.11})$ $k$-graph with $\delta_{k-1}(H)\ge \alpha n$.
For each induced subhypergraph $H'$ on $V'\subseteq V(H)$ with $|V'|\ge 2\zeta n$, $H'$ is a $(v(H'), p, \mu', \ppoints{scale=0.11})$ $k$-graph with $\mu'=\mu/(2\zeta)^k$. For $H'$ and a partition $\mathcal{P'}=\{X, Y\}$ of $V(H')$ with $|X|, |Y|\ge \zeta n$, we apply Lemma~\ref{k-transferral} to get $\lambda'$, then the property ($\heartsuit$) in Corollary~\ref{covertrans} holds for $H$.
By Proposition~\ref{cover1}, we can apply Corollary~\ref{covertrans} to find an $F$-factor of $H$.
In particular, if $F$ is a complete $k$-partite $k$-graph satisfying $\gcd(\mathcal S(F))>1$, then we claim that $F\notin {\bf TRANS}^{k-1}_k$.
This together with Observation~\ref{construction-trans} would imply $F\notin \textbf{FACTOR}_k^{\ppoints{scale=0.11}, k-1}$, which in turn implies the moreover part of the theorem.
Indeed, let $U_1, \dots, U_k$ be the vertex classes of $F$ and $\mathcal{P}=\{V_1, V_2\}$ be a $(k-1)$-shadow disjoint bipartition of $V(F)$. For any $i\in [k]$ and any $x,y \in U_i$ we may choose vertices $u_j\in U_j$ for each $j\neq i$, and since $F$ is complete both $\{x\}\cup \{u_j : j\neq i\}$ and $\{y\}\cup \{u_j : j\neq i\}$ are edges of $F$.
Both these sets therefore have the same index vector with respect to $\mathcal{P}$, so we must have $x, y\in V_1$ or $x, y\in V_2$. It follows that $U_i \subseteq V_1$ or $U_i \subseteq V_2$ for each $i\in [k]$, and therefore that $\gcd(\mathcal S(F))$ divides $|V_t|$ for $t\in [2]$. Since $L^{k-1}_{F}$ is the lattice generated by all $\textbf{i} _{\mathcal{P},F}$ such that $\mathcal{P}$ is $(k-1)$-shadow disjoint, $\gcd(\mathcal S(F))$ divides $a$ and $b$ for each $(a,b)\in L^{k-1}_{F}$. Due to $\gcd(\mathcal S(F))>1$, we conclude that $(1,-1)\notin L^{k-1}_{F}$, and so $F\notin {\bf TRANS}^{k-1}_k$.
\end{proof}
\section{Remarks}
In this paper, we studied the $F$-factor problems in quasi-random $k$-graphs.
We summarize our contributions as follows.
\begin{enumerate}[label=(\arabic*)]
\item For a 3-partite 3-graph $F$, $p>1/8$ and $\alpha>0$, every sufficiently large $n$-vertex $(p,\mu,\ppoints{scale=0.11})$-dense 3-graph $H$ with $\delta_2(H)\ge \alpha n$ contains an $F$-factor (Theorem~\ref{1/8-dot-codegree}).
The constant $1/8$ matches a construction of Lenz and Mubayi and thus is best possible. We also show that such a result does not hold if instead a minimum vertex degree is assumed and $p$ is close to 1 (Theorem~\ref{construction}).
\item For $k\geq 3$, using $\textbf{FACTOR}^{\pedge{scale=0.14},1}_k=\textbf{COVER}^{\pedge{scale=0.14},1}_k$ (Theorem~\ref{factor-cover2}), we show that all $k$-partite $k$-graphs are in $\textbf{FACTOR}^{\pedge{scale=0.14},1}_k$ (Theorem~\ref{dot-edge-partite}).
\item For $k=3$, towards a characterization of $\textbf{FACTOR}_3^{\ppoints{scale=0.11} ,2}$, we show that $\textbf{FACTOR}_3^{\ppoints{scale=0.11} ,2} = \textbf{COVER}_3^{\ppoints{scale=0.11} ,2}\cap {\bf TRANS}^2_3\cap \{F:F\ {\rm is}\ {\rm a}\ {\text{3-graph}}\ {\rm and}\ \pi_{\ppoints{scale=0.11}}(F)=0\}$ (Theorem~\ref{factor3-2}).
This reduces to the problem to the study of $\textbf{COVER}_3^{\ppoints{scale=0.11} ,2}$ as both ${\bf TRANS}^2_3$ and $\{F:\pi_{\ppoints{scale=0.11}}(F)=0\}$ are explicit families of 3-graphs $F$.
As a corollary, we obtain that every 3-partite 3-graph $F$ is in $\textbf{FACTOR}_3^{\ppoints{scale=0.11} ,2}$ if and only if it is in ${\bf TRANS}^2_3$ (Corollary~\ref{3-partite3-graph}).
\item For $k\ge 3$, we find that $k$-partite $k$-graphs $F$ with $\gcd(\mathcal{S}(F))=1$ belong to $\textbf{FACTOR}_k^{\ppoints{scale=0.11} , k-1}$ (Theorem~\ref{gcd}).
Moreover, the condition $\gcd(\mathcal{S}(F))=1$ is necessary for complete $k$-partite $k$-graphs $F$.
\end{enumerate}
Here we include a result for the following stronger quasi-randomness for $k$-graphs.
\begin{defi}
Given integers $n\geq k\ge 2$, let real numbers $p\in [0,1]$, $\mu >0$ and $H=(V,E)$ be a $k$-graph with $n$ vertices. We say that $H$ is ($p,\mu,\pcherry{scale=0.11}$)\emph{-dense} if for any two sets of ordered pairs $P,Q\subseteq V^{k-1}$,
\begin{equation}
e_{\pcherry{scale=0.11}}(P,Q)\geq p|\mathcal{K}_{\pcherry{scale=0.11}}(P,Q)|-\mu n^k,
\end{equation}
where $\mathcal{K}_{\pcherry{scale=0.11}}(P,Q)=\{(x_1,x_2,\dots,x_{k-1},x_k)\in V^k :(x_1,\dots,x_{k-1})\in P \text{\ and } (x_2,\dots,,x_k)\in Q \}$, and
$e_{\pcherry{scale=0.11}}(P,Q)$ is the size of the set $\{(x_1,x_2,\dots,x_{k-1},x_k)\in \mathcal{K}_{\pcherry{scale=0.11}}(P,Q): \{x_1,x_2,\dots,x_{k-1},x_k\}\in E \} $.
\end{defi}
Clearly, $\mathcal{C}(n,p,\mu, \pcherry{scale=0.11})\subseteq \mathcal{C}(n,p,\mu, \ppoints{scale=0.11})$ holds for all $p\in [0,1]$ and $\mu >0$. Therefore, the proof of Theorem~\ref{factor-cover1} in \cite{DHSWZ} also implies the following result.
\begin{thm}\label{cherry-equal}
For any $k \geq 2$, $\textbf{FACTOR}^{\pcherry{scale=0.11},1}_k=\textbf{COVER}^{\pcherry{scale=0.11},1}_k$.
\end{thm}
Let $K_4^{(3)-}$ be the (unique) $3$-graph consisting of three hyperedges on four vertices.
It is known that
\[
\pi_{\ppoints{scale=0.11}}(K_4^{(3)-})=\pi_{\pedge{scale=0.14}}(K_4^{(3)-})=1/4 \text{~~\cite{Glebov_2015, Reiher_2018} \ and \ } \pi_{\pcherry{scale=0.11}}(K_4^{(3)-})=0.
\]
Therefore, $K_4^{(3)-}\notin \textbf{FACTOR}_3^{\pedge{scale=0.14},1}$. However, applying Theorem~\ref{cherry-equal}, we show the following.
\begin{thm}\label{K-4-3}
$K_4^{(3)-}\in \textbf{FACTOR}_3^{\pcherry{scale=0.11},1}$.
\end{thm}
\begin{proof}
Suppose that $n_0$ is large enough and $\mu$ is sufficiently small. By Theorem~\ref{cherry-equal}, it suffices to show that $K_4^{(3)-}\in \textbf{COVER}^{\pcherry{scale=0.11},1}_3$. To see this, let $H$ be an $n$-vertex $(p,\mu, \pcherry{scale=0.11})$-dense $3$-graph with $\delta_1(H)\ge \alpha n^2$, $n\in 4\mathbb{N}$ and $n\geq n_0$.
Fixed $w\in V(H)$,
construct an auxiliary graph $G$ with $V(G)=V(H)\setminus \{w\}$ and $E(G)=\big\{\{v_1,v_2\}:\{v_1,v_2\}\in N(w)\big\}$. We shall count pairs $(v, T)$ where $T$ is a pair of vertices satisfying these two vertices are both adjacent to $v$ in $G$. By Jensen's inequality, the number of such pairs is
\[
\sum_{v\in V(G)}\binom{d_G(v)}{2}\ge (n-1)\binom {\frac{1}{n-1}\sum_{v\in V(G)}d_G(v)}{2}\ge (n-1)\binom {2\alpha n}{2}\ge \alpha n^3.
\]
Let $P=Q=N(w)$. Since $H$ is ($p,\mu,\pcherry{scale=0.11}$)\emph{-dense}, we have
\[
e_{\pcherry{scale=0.11}}(P,Q)\geq p|\mathcal{K}_{\pcherry{scale=0.11}}(P,Q)|-\mu n^3\ge (2\alpha p-\mu) n^3>0,
\]
which implies there are two pairs $\{x, y\}, \{x, z\}\in N(w)$ such that $\{x,y,z\}\in E(H)$. Therefore, $\{w,x,y\}$, $\{w,x,z\}$ and $\{x,y,z\}$ form a copy of $K_4^{(3)-}$ in $H$.
\end{proof}
\bibliographystyle{abbrv}
| 2024-02-18T23:39:46.472Z | 2021-11-30T02:19:31.000Z | algebraic_stack_train_0000 | 347 | 16,956 |
|
proofpile-arXiv_065-2178 | \section{Introduction}
Granular materials are actually very simple: they consist of a great number
of macroscopic particles interacting only with their neighbours. In case of dry
samples, the sole interactions are hard core repulsion and friction.
However, in spite of their apparent simplicity, their behaviour can be very
complex and often surprising \cite{degennes,jaeger,jaeger2}.
Stick slip motion has been observed in various frictional experiments from solid-on-solid friction of rough surfaces \cite{heslot} to delicate studies
of lubricated films between smooth mica surfaces \cite{gee}.
Since one of the dominating interactions in granular materials
is friction among grains, it is not surprising that dense granular materials
exhibit various stick slip phenomena.
There are many examples of stick slip behaviour in granular materials.
In the first type of experiments stick slip sliding between a granular medium
and a solid body is considered \cite{duran,nasuno,nasuno2,geminard,ovarlez,lubert,deryck,frye,lacombe}.
In the second class of experiments the granular sample is forced
to deform slowly; an intermittent motion is caused by successive
discrete reorganization events inside the material. Experiments in elastic
annular shear cells revealed both regular \cite{cain} and irregular \cite{miller,dalton} stick slip behaviour.
In the third class, it is friction between grains themselves which is studied
\cite{coste}. The formation and breakage of chain forces are an explanation to the stick slip \cite{geng}. Moreover, comparison between a sheared granular
medium and avalanches in the same type of media has been made \cite{manna}.
Humidity changes the interactions between grains: for example
chemical reactions \cite{olivi,gayvallet} or mechanical behaviour
\cite{bocquet,fraysse,restagno}.
Experiments have been made on the increase of the maximum angle of stability
of a humid granular media as a function of the quantity
of liquid added to the medium \cite{albert,barabasi} or as a function
of the humidity ratio of the surrounding atmosphere \cite{fraysse}.
In friction experiments, the effect of humidity has also been studied
\cite{ovarlez,frye,cain,dieterich}.
We will study here the effect of an under-saturated water vapor atmosphere
on the behaviour of a granular medium submitted to shear. The friction
studied here is a grain grain interaction obtained with an adapted apparatus.
The experimental set-up and experimental conditions will be detailed
in section II. Results will be discussed in section III. And finally
section IV contains the conclusion.
\section{Experimental Procedure}
\subsection{Set up}
We built an annular shear cell with an inner radius of 14 $mm$, an outer radius
of 31 $mm$ and of 40 $mm$ depth. The shear cell is made of two parts:
one lower and one upper. The lower part remains motionless and has a depth
of 20 $mm$, while upper part rotates freely; this last has a depth of 20 $mm$
and is divided by four large vanes (see fig.1).
A gap of about 150 $\mu m$ is kept between the lower and the upper part
to prevent any friction between the two parts; this gap is small enough
to prevent the grains from escaping the shear cell. Granular matter
fills the lower part and half of the upper part. Such a shear apparatus
allows us to measure the shear force between the granular matter
contained in the lower part and the granular matter contained in the upper
part. This shear force is measured with a viscometer HAAKE VT550 (measuring
a 0.01-100 Pa range in shear forces for rotation speeds 0.1-800 rpm); moreover
the data sampling frequency is 5 to 10 times per second. A cover plate
introduces a small constant normal load (about 150 Pa) on the granular matter;
this cover is free to move vertically, which allows the dilatancy
associated with the shear behavior.
The annular shear cell is kept inside a sealed heating unit, in which temperature (30$^oC \pm 1^oC$) is controlled by an adjustable water flow.
In order to control humidity, we used the method implemented
by Restagno et al. \cite{restagno} in which aqueous solutions saturated with
inorganic salts provide various levels of relative humidity (RH) in the atmosphere,
at a given temperature. For instance, to obtain a high humidity level
(95 \% RH), we used an ammonium dihydrogen phosphate ($NH_4H_2PO_4$)
solution and to obtain a low humidity level (about 5\% RH) we used silica gel.
A small air pump forces the humid atmosphere to flow through the shear cell. We have checked
that the relative humidity is homogeneous throughout the shear cell. During
the experiments, relative humidity is measured inside and outside the shear cell using
two humidity sensors, having an accuracy of $\pm 1 \%$ RH.
\subsection{Preparation of the samples}
One type of granular materials has been used in the experiments: glass beads
(sodo silicate glass, Potters Ballotini).
Following is the procedure used for the measurements:
(1) The beads are washed with distilled water and a cleaning solution
(RBS-25), then they are dried in an oven ( 12 hours at 130$^o$C). After that, the dried beads are sifted
using either two sieves of 315 and 400 $\mu$m pore size or
two sieves of 630 and 800 $\mu$m pore size. Note that the cleaning removes
the powder which is added to commercially available glass beads
in order to prevent the formation of clusters in presence of a humid atmosphere.
See for that fig. 3 which is a view taken with an optical
microscope using polarized light, of washed glass beads and non washed
glass beads: the non washed glass beads do not transmit polarized
light while washed beads do. This is the proof of the presence
of an additive on the surface of the glass beads.
(2) The beads are then poured into the sheared cell
and the cover plate is put. The granular matter is then
sheared in order to ensure the cover plate to be horizontal.
(3) The saline solution is then introduced in the experimental
set-up, and the set-up is kept aside for 7 hours in the temperature
controlled environment, to achieve a constant temperature and a homogeneous
distribution of humidity.
(4) Before starting the measurement, we rotate the shear cell for 40 s at an angular velocity
of 3.14 rad.s$^{-1}$. This defines the initial state of the granular sample.
(5) The shear stress is then measured for 600s at a small angular velocity.
Before each new measurement, the sample is changed.
\section{Results and Discussion}
\subsection{Results}
We studied the behaviour of the shear response of the granular medium,
at several levels of humidity ranging from 0 to 100 \%. Shear stress is plotted
as a function of the dimensionless linear displacement $\delta d$, defined as $\delta d= (\frac{\bar{R} \omega}{\bar{d}_{bead}})t$, where $\omega$ is the angular velocity
of the viscometer, $\bar{R}$ the mean cell radius, $t$ the time
and $\bar{d}_{bead}$ the mean diameter of the beads.
We studied the system for 2 values of the angular velocity: 1.7.10$^{-3}$ rad/s
and 6.9.10$^{-3}$ rad/s.
At such small angular velocities, the system is in a quasi static regime.
First of all, we have to make a difference between the mean amplitude MA of the stress
signal (difference between one local maximum and the very next local minimum of the signal) and the mean value MV (i.e. absolute value) of the same signal.
We observed a stick slip signal which corresponds
to the relaxation motion of the system (see fig.4).
Just after $t=0$ there is a transient regime where the stress increases
up to a maximum value and then decreases.
This transient regime is not reproducible between two experiments:
the maximum of the shear stress depends on the initial configuration
of the beads.
Hence it is not possible to analyze the total cohesion of the system by the way of the maximum value
of the shear stress.
After this transient regime,
the permanent regime is reached (see fig. 4 again).
The stick slip signal is quite irregular due to the small individual
displacements of the beads.
Note that there is an obvious difference between
the stick slip signal for a low humidity (13 \% RH) and for
a high humidity (93 \% RH): the mean value MV of the stress signal for
a low humidity is smaller than that for a high humidity and the amplitude MA
of the stress signal is larger for a higher value of the humidity.
Let us examine in details, in the following, these signals.
We computed the maximum and minimum values of each stick-slip event
during the permanent regime, i.e. after each stick event ( local increase
of the signal), the local maximum
was saved and after each slip event ( local decrease of the signal), the local minimum
was saved. This allowed us to compute the averages of the local maxima
and local minima during the permanent regime over which we saved the stress
signal during 600$s$.
From these mean maximum and minimum, a mean amplitude MA was deduced
over the same duration of the permanent regime.
Fig.5 corresponds to the mean maximum and minimum as a function of
relative humidity RH. The diameter of the beads used here was
enclosed between 630 and 800 $\mu$m, while the angular velocity
of the shear cell was equal to 1.7.10$^{-3}$ rad/s.
The curve corresponding to minima is first quasi constant then
increases abruptly from a value of 88 \% RH on. The curve corresponding
to maxima increases regularly until the largest value of the stress
is reached.
Fig.6 corresponds also to the mean maximum and minimum as a function
of relative humidity RH, but this time, the diameter of the beads
was enclosed between 310 and 400 $\mu$m and the angular
velocity was equal to 6.9.10$^-3$ rad/s.
Here, on the contrary to fig.5, the two curves corresponding
respectively to maxima and minima follow the same behavior
and increase first with a small slope then from value of 85\% RH on, with a large slope.
To understand the differences between the preceding behaviors,
let us examine fig.7 and fig.8. Fig.7 corresponds to the mean amplitude MA
of the stick slip signal as a function of relative humidity RH
for 630-800$\mu$m diameter large beads.
Two curves have been traced on fig.7: the mean amplitude MA
(for a angular velocity equal to 1.7.10$^{-3}$ rad/s) is first constant
then increases from RH=55\% on and finally decreases from RH=90 \% on.
The other curve (corresponding to a angular velocity equal
to 6.9.10$^{-3}$ rad/s) has a similar behaviour, with a maximum
which is lower.
Fig. 8 corresponds to the mean amplitude MA of the stick slip signal
as a function of relative humidity RH for 315-400 $\mu$m diameter large beads.
Here again the same velocities as for fig.7 have been imposed to the shear cell.
We observe a similar behavior of the two curves which have been traced on fig.8
than that on fig.7.
\subsection{Discussion of results in the light of the geometry of the assembly of beads}
The difference of amplitudes (MA) between two experiments comes from the number of contacts
inside the granulate: if this number is relatively large, the stick stage
will have a longer duration (because the granulate can resist
a larger shear stress) and hence, the observed stress will
be larger until the end of the stick stage where the granulate
begins to relax.
In the light of these preceding 2 figures (Fig. 7 and Fig. 8), we can say first,that there
is a velocity dependent behavior of our beads under shear, second,
that the size of the beads has also an influence on their behavior.
We can analyze the role of the angular velocity in terms of number of contacts
between the beads. For the largest angular velocity that we used,
the number of constant contacts (number of contacts which are not lost during one stick stage) between the beads
is smaller as the contacts have a larger probability to be lost during
the shear process due: i.e. the number of lost contacts per second is larger.
This explains why the maximum mean amplitude MA for RH=90\%, $\bar{d}_{bead}=630-800\mu$m
is lower for $\omega$=6.9.10$^{-3}$ rad/s than for $\omega$=1.7.10$^{-3}$ rad/s (Fig. 7).
Similarly, the size of the beads plays the same role in terms of number of constant contacts between the beads for one stick stage: if the glass beads are larger, the total number
of contacts between particles is smaller inducing a smaller resistance to shear and hence
smaller amplitudes MA of the stick-slip stress (Fig. 5 and 6). These two behaviors have an
effect on a very short time corresponding to a microscopic length.
Let us examine now the behavior of the mean value MV,
i.e. the action of shear on humid beads on the macroscopic level.
If we compare the mean values of the maxima of the MV (resp. minima)
in figures 5 and 6, the MV is larger for larger beads. So we can say that
the macroscopic resistance to shear is larger for larger beads.
Humidity has an effect on this macroscopic behavior
only for high degrees of humidities. We will analyze this last behavior
in section V-C.
\subsection{Physico chemistry of water on glass}
Let us recall some previous results about the interaction
between water and sodo silicate glass beads \cite{olivi,gayvallet,iler,ehrenstein}.
Under the influence of humidity or water a reaction of the glass bead
surface with water follows that forms a 'water skin'. The alkali ions of the glass surface
are taken out of the silicic acid structure and are replaced by protons. So grows a permanent water-skin that cannot be taken away, not even under a vacuum. With
longer influence of water there grows, in addition, a very strongly adhering
temporary water-skin, which thickness depends on the temperature and the degree
of humidity, which can be taken away by drying. Through the presence of water,
the strength of the glass bead surface can be reduced.
Note also that our glass beads have been washed with a commercial cleaning solution
(RBS-25). This cleaning solution is basic.
The attack of bases on glass increase with increasing temperature and time.
Our glass beads have been washed in a aqueous solution containing
90 \% distilled water and 10 \% cleaning solution (RBS-25) for $t=1h$
at ambient temperature.
The resulting pH of the cleaning solution was equal to 12.
Roughly, the following reactions occur under the action of alkali
at the glass surface:
\begin{equation}
-Si-O-R+H_2O \rightarrow -Si-OH+R^++OH^-
\end{equation}
where $R=Na,K, Li$ and others.
And:
\begin{equation}
-Si-O-Si-+OH^- \rightarrow -Si-OH+-Si-O-
\end{equation}
The rate of corrosion (resp. dissolution) is not determined
by a diffusion controlled process but by the active dissolution
of the $SiO_2$ network \cite{ehrenstein}. The loss of mass is proportional to time.
The smooth surface of the beads under the two water-skins (permanent and temporary) gets rough.
This is analogous to an aging process, but not during the shear experiment itself, but during the preparation of the granular sample.
The surface of the glass
beads was also attacked by the basic cleaning solution: the silicic structure
of the surface allows water to penetrate more profoundly
and hence leading to a thicker adsorbed water layer.
\subsection{Discussion: microscopic scale}
On the microscopic scale, there are three regimes
occurring during the shearing experiment and as a function
of humidity.Here we will follow
the theory of Halsey and Levine \cite{halsey}.
We can say, that for RH lower than 50\%, the quantity of water
on the surface of the beads is not sufficient to change their behavior
during shear compared to the shearing of 'dry' beads.The
friction process between the beads is dominated by the deformation
of the nano asperities on the surface of the beads and the resistance
to shear of the beads surface is negligible \cite{halsey}.
Between RH=50\% and RH=90\%, is the friction regime \cite{halsey}: we can say
that the water adsorbed layer on the surface of the beads induce a non negligible resistance to shear on the microscopic scale inducing an increase
in the amplitude MA of the stick slip signal.
And finally between RH=90\% and RH=100\%, the quantity of water
on the surface of the beads is sufficient to obtain a lubrication
of the surface of grains (i.e. always on the microscopic scale), and hence, the microscopic resistance
to shear decreases, and so does the amplitude MA. In that last case,
the quantity of water on the surface of the beads dissolves
the sodo-silicate glass.
This behavior of the shear stress as a function of the relative humidity
leads to the conclusion that the the quantity of water increases
first cohesion on the scale of one bead and then decreases it, for short time intervals.
\subsection{Discussion: macroscopic scale}
On the macroscopic scale, and for humidity ranging from 0 to 80 \%, the resistance to shear of humid beads
is similar to that of dry beads (increased resistance to shear
for large beads with respect to small beads, if comparison is made
between fig 5 and 6). This behavior is
the consequence of geometrical congestion for relatively
monodisperse spherical beads.
Always on the macroscopic scale, the action of water on the beads is only visible
when its quantity is sufficient (RH larger than 80 \%) to attack and lubricate
the surface of the beads. Then
the assembly of beads has its resistance to shear increased (value of the MV
of the shear signal, Fig. 5 and 6) when the quantity of water
is high: a comparison may be made with dense colloidal dispersions under shear.
It may seem contradictory that for high degrees of humidity, on the microscopic
scale water lubricates the surface of the beads leading to an increased
resistance to shear on the macroscopic scale. This may be a consequence
of the non Newtonian characteristics of very humid grains.
Let us compare the very humid glass beads with colloidal dispersions. Each colloidal particle
has a repulsive interaction with other colloidal particles (the attractive
part of the interaction can be neglected), but dense colloidal dispersions
observe a solid like behavior under shear even if there is no attractive
interactions between the particles \cite{botet}: when no shear is applied
the colloidal dispersion is fluid like becoming solid like as soon as
shear is applied on it.
Indeed, the behavior of dense colloidal dispersions is non Newtonian and
observes a transition from fluid like to solid like as a function of
the concentration of the colloidal dispersions and as a function of shearing rate \cite{botet}.When the colloidal dispersion is in the solid like state,
its resistance to shear is increased compared to the state where it is fluid like.
Similarly, the humid glass grains are lubricated and the microscopic
resistance to shear decreases when humidity is larger than 90\%: the beads
have a non attractive contact force like colloidal particles.
The explanation of the apparently contradictory behaviors on microscopic
and macroscopic scales may be that the very humid granulate is non Newtonian
leading to a higher macroscopic resistance to shear: the assembly
of grains observes a transition from a divided media to a solid like state
under shear.
So we join the conclusion of Tegzes \cite{tegzes}, that for high degrees
of humidity, the assembly of grains behaves as a non Newtonian fluid
(a viscoplastic fluid).
\section{Conclusion}
There are two scales which appear in grain grain friction
for a shearing experiment on glass beads in a water vapor atmosphere.
On the microscopic scale, the effect of humidity can be separated
into three regimes: the asperity regime where humidity has a negligible
role, the friction regime where the surface area of contacts between beads
increases the cohesion of the granular medium and a lubricated regime
where the quantity of water is sufficient reduce the viscosity
of the surface of the beads.
Moreover, there is a size effect in the cohesion of the granulate:
for large beads, the number of contacts per unit volume between particles is smaller
tan for small beads
leading to a smaller total cohesion. The effect of shear rate is similar:
for higher shear rates, the number of lost contacts per unit time increases
leading also to a smaller total cohesion.
On the macroscopic scale, the behavior of the resistance to shear
is similar to that of dry grains if RH is lower than 80\%.
For RH larger than 80\%, the assembly of beads behaves as
a non Newtonian fluid becoming solid like under shear.
| 2024-02-18T23:39:47.932Z | 2005-10-20T09:57:57.000Z | algebraic_stack_train_0000 | 441 | 3,503 |
|
proofpile-arXiv_065-2198 | \section{Introduction}
The nature and origin of dark energy stand out as two of the great
unsolved mysteries of cosmology. Two of the more popular explanations are
either a cosmological constant $\Lambda$, or a new, slowly rolling scalar
field (a quintessence field). If the solution of the dark energy problem
proved to be a cosmological constant, one would have to explain why it is not
120 orders of magnitude larger (as would be expected in a non-supersymmetric
field theory), nor exactly zero (as it would be if some hidden symmetry
were responsible for the solution of the cosmological constant problem), and
why it has become dominant only recently in the history of the universe. These
are the ``old'' and ``new'' cosmological constant problems in the parlance
of \cite{Weinberg:2000yb}. To date, this has not been accomplished
satisfactorily, despite intensive efforts. If, instead of $\Lambda$, the
solution rested on quintessence, one would need to justify the existence of
the new scalar fields with the finely tuned properties required of a
quintessence field (e.g. a tiny mass of about $10^{-33}$eV if the field
is a standard scalar field). Clearly, both of the above approaches
to explaining dark energy lead directly to serious, new cosmological problems.
In this paper, we will explore an approach to explaining dark energy which
does not require us to postulate any new matter fields.
There exist tight constraints on $\Lambda$ from various sources -
Big Bang Nucleosynthesis (BBN) \cite{Freese:1986dd}, cosmic microwave
background (CMB) anisotropies \cite{Bean:2001xy}, cosmological structure
formation \cite{Doran:2001rw} - which rule out models where the vacuum
energy density is comparable to the matter/radiation energy density at the
relevant cosmological times in the past. However, it could still be hoped
that a variable $\Lambda$ model might be compatible with observation since
the value of $\rho_{\Lambda}$ is constrained only for certain redshifts.
In fact, the above constraints taken together with the results from recent
supernovae observations \cite{Riess:1998cb},\cite{Perlmutter:1998np} leads
one to posit that the vacuum energy density might be evolving in time.
This leads directly to the proposal of tracking quintessence
\cite{Ratra:1987rm}. However, some of the drawbacks of quintessence were
mentioned above. A preferable solution would combine the better features of
both quintessence and a cosmological constant: a tracking cosmological
``constant''.
In this letter, we discuss the possibility that the energy-momentum
tensor of long wavelength cosmological perturbations might provide
an explanation of dark energy. The role of such perturbations in
terminating inflation and relaxing the bare cosmological constant
was investigated some time ago in \cite{Mukhanov:1996ak,Abramo:1997hu} (see
also \cite{WT}). However, this mechanism can only set in if the
number of e-foldings of inflation is many orders of magnitude larger
than the number required in order to solve the horizon
and flatness problems of Standard Big Bang cosmology. Here, we are
interested in inflationary models with a more modest number of
e-foldings. We discover that, in this context, the EMT of long wavelength
cosmological perturbations results in a tracking cosmological ``constant''
of purely gravitational origin and
can be used to solve the ``new'' cosmological constant problem.
We begin by reviewing the formalism of the effective EMT of cosmological
perturbations in Section 2. We recall how, in the context of
slow-roll inflation, it could solve the graceful exit problem of
certain inflationary models. We then extend these results beyond the
context of slow-roll inflation in Section 3. In Section 4, we investigate
the behaviour of the EMT during the radiation era and show that the
associated energy density is sub-dominant and tracks the cosmic fluid.
We examine the case of the matter era and show how the EMT
can solve the dark energy problem in section 5. In Section 6 we consider the
effects of back-reaction on the scalar field dynamics. We then
summarize our results and
comment on other attempts to use the gravitational back-reaction of
long wavelength fluctuations to explain dark energy.
\section{The EMT}
The study of effective energy-momentum tensors for gravitational
perturbations is not new \cite{Brill,Isaacson}. The interests of
these early authors revolved around the effects of high-frequency
gravitational waves. More recently, these methods were applied
\cite{Mukhanov:1996ak,Abramo:1997hu} to the study of the effects of
long-wavelength scalar metric perturbations and its application to
inflationary cosmology.
The starting point was the Einstein equations in a background defined by
\begin{eqnarray}
ds^{2} \, &=& \, a^{2}(\eta)((1+2\Phi(x,\eta))d{\eta}^{2}\nonumber \\
&-&(1-2\Phi(x,\eta))(\delta_{ij}dx^{i}dx^{j}))
\end{eqnarray}
where $\eta$ is conformal time, $a(\eta)$ is the cosmological
scale factor, and $\Phi(x, \eta)$ represents the scalar perturbations (in
a model without anisotropic stress). We are using longitudinal gauge
(see e.g. \cite{MFB} for a review of the theory of cosmological fluctuations,
and \cite{RHBrev03} for a pedagogical overview). Matter is, for simplicity,
treated as a scalar field $\varphi$.
The modus operandi of \cite{Mukhanov:1996ak} consisted of expanding both the
Einstein and energy-momentum tensor in metric ($\Phi$) and matter
($\delta\varphi$) perturbations up to second order. The linear equations
were assumed to be satisfied, and the remnants were spatially averaged,
providing the equation for a new background metric which takes into
account the back-reaction effect of linear fluctuations computed up to
quadratic order
\begin{equation}
G_{\mu\nu} \, = \, 8\pi G\,[T_{\mu\nu}+\tau_{\mu\nu}],
\end{equation}
where $\tau_{\mu\nu}$ (consisting of terms quadratic in metric and matter
fluctuations) is called the effective EMT.
The effective energy momentum tensor, $\tau_{\mu\nu}$, was found to be
\begin{eqnarray} \label{tzero}
\tau_{0 0} &=& \frac{1}{8 \pi G} \left[ + 12 H \langle \phi \dot{\phi} \rangle
- 3 \langle (\dot{\phi})^2 \rangle + 9 a^{-2} \langle (\nabla \phi)^2
\rangle \right] \nonumber \\
&+& \,\, \langle ({\delta\dot{\varphi}})^2 \rangle + a^{-2} \langle
(\nabla\delta\varphi)^2 \rangle \nonumber \\
&+& \,\,\frac{1}{2} V''(\varphi_0) \langle \delta\varphi^2 \rangle + 2
V'(\varphi_0) \langle \phi \delta\varphi \rangle \quad ,
\end{eqnarray}
and
\begin{eqnarray} \label{tij}
\tau_{i j} &=& a^2 \delta_{ij} \left\{ \frac{1}{8 \pi G} \left[ (24 H^2 + 16
\dot{H}) \langle \phi^2 \rangle + 24 H \langle \dot{\phi}\phi \rangle
\right. \right. \nonumber \\
&+& \left. \langle (\dot{\phi})^2 \rangle + 4 \langle \phi\ddot{\phi}\rangle
- \frac{4}{3} a^{-2}\langle (\nabla\phi)^2 \rangle \right] + 4 \dot{{%
\varphi_0}}^2 \langle \phi^2 \rangle \nonumber \\
&+& \,\, \langle ({\delta\dot{\varphi}})^2 \rangle - a^{-2} \langle
(\nabla\delta\varphi)^2 \rangle -
4 \dot{\varphi_0} \langle \delta \dot{\varphi}\phi \rangle \nonumber \\
&-& \left. \,\, \, \frac{1}{2}V''(\varphi_0) \langle \delta\varphi^2
\rangle + 2 V'( \varphi_0 ) \langle \phi \delta\varphi \rangle
\right\} \quad ,
\end{eqnarray}
where H is the Hubble expansion rate and the $\langle \rangle$ denote
spatial averaging.
Specializing to the case of slow-roll inflation (with $\varphi$ as the
inflaton) and focusing on the effects of long wavelength or IR modes
(modes with wavelength larger than the Hubble radius), the EMT simplifies to
\begin{equation}
\tau _0^0 \cong \left( 2\,{\frac{{V^{\prime \prime }V^2}}
{{V^{\prime }{}^2}}}-4V\right) <\phi ^2> \, \cong \, \frac 13\tau_i^i,
\end{equation}
and
\begin{equation}
p \, \equiv -\frac 13\tau _i^i\cong -\tau_{0}^{0}\,.
\end{equation}
so that $\rho_{eff}<0$ with the equation of state $\rho\,=\,-p$.
The factor $\langle \phi^{2} \rangle$ is proportional to the IR phase space
so that, given a sufficiently long period of inflation (in which the phase
space of super-Hubble modes grows continuously), $\tau_{0}^{0}$ can become
important and act to cancel any positive energy density (i.e. as associated
with the inflaton, or a cosmological constant) and bring inflation to an end
- a natural graceful exit, applicable to any model in which inflation
proceeds for a sufficiently long time.
Due to this behaviour during inflation, it was speculated
\cite{Brandenberger:1999su} that this could also be used as a mechanism
to relax the cosmological constant, post-reheating - a potential solution to
the old cosmological constant problem. However, this mechanism works (if
at all - see this discussion in the concluding section) only if inflation
lasts for a very long time (if the potential of $\varphi$ is quadratic,
the condition is that the initial value of $\varphi$ is larger than
$m^{-1/3}$ in Planck units).
\section{Beyond Slow-Roll}
Here, we will ask the question what role back-reaction of IR modes
can play in those models of inflation in which inflation ends naturally
(through the reheating dynamics of $\varphi$) before the phase space of
long wavelength modes has time to build up to a dominant value.
In order to answer this question, we require an expression for
$\tau_{\mu\nu}$ unfettered by the slow-roll approximation. Doing this provides
us with an expression for the EMT which is valid during preheating and,
more importantly, throughout the remaining course of cosmological evolution.
In the long wavelength limit, we have \footnote{We've ignored terms proportional to $\dot{\phi}$ on the
basis that such terms are only important during times when the equation of
state changes. Such changes could lead to large transient effects during
reheating but would be negligible during the subsequent history of the
universe.},
\begin{eqnarray} \label{one}
\tau_{0 0} &=& \frac{1}{2} V''(\varphi_0) \langle \delta\varphi^2 \rangle + 2
V'(\varphi_0) \langle \phi \delta\varphi \rangle, \quad
\end{eqnarray}
and
\begin{eqnarray} \label{two}
\tau_{i j} &=& a^2 \delta_{ij} \left\{ \frac{1}{8 \pi G} \left[ (24 H^2 + 16
\dot{H}) \langle \phi^2 \rangle] + 4 \dot{{\varphi_0}}^2 \langle \phi^2 \rangle\} \right. \right. \nonumber \\
&-& \left. \,\, \, \frac{1}{2}V''(\varphi_0) \langle \delta\varphi^2
\rangle + 2 V'( \varphi_0 ) \langle \phi \delta\varphi \rangle
\right\}.
\end{eqnarray}
As in the case of slow-roll, we can simplify these expressions by making use
of the constraint equations which relate metric and matter
fluctuations \cite{MFB}, namely
\begin{equation} \label{constr}
-(\dot{H} + 3H^2) \phi \, \simeq \, 4 \pi G V^{'} \delta \varphi \, .
\end{equation}
Then, (\ref{one}) and (\ref{two}) read
\begin{equation}{\tau_{0 0}\,=\, (2\kappa^2\frac{V''}{(V')^{2}}(\dot{H}+3H^{2})^{2}-4\kappa(\dot{H}+3H^{2}))\langle \phi^{2}\rangle,}\end{equation}\label{1}
\begin{eqnarray}
\tau_{i j}\,&=&\, a^{2}\delta_{i j}(12\kappa(\dot{H}+H^{2})+4\dot{\varphi_{0}(t)}^{2}\\ \nonumber &-&2\kappa^{2}\frac{V''}{(V')^{2}}(\dot{H}+3H^{2})^{2})\langle \phi^{2}\rangle,\label{2}
\end{eqnarray}
with $\kappa\,=\,\frac{M^{2}_{Pl}}{8\pi}$.
The above results are valid for all cosmological eras. With this in mind,
we now turn an eye to the post-inflation universe and see what the above
implies about its subsequent evolution.
In what follows, we take the scalar field potential to be
$\lambda \varphi^{4}$.
As was shown in \cite{Shtanov:1994ce}, the equation of state of the
inflaton after reheating is that of radiation, which implies
$\varphi(t)\varpropto 1/a(t)$.
\section{The Radiation Epoch}
The radiation epoch followed on the heels of inflation. The EMT in this
case reads
\begin{equation}
\tau_{00} \, = \,(\frac{1}{16}\kappa^{2}\frac{V''}{(V')^{2}}\frac{1}{t^{4}}-\frac{\kappa}{t^{2}}) \langle \phi^{2} \rangle,
\end{equation}
\begin{equation}
\tau_{ij} \, = \,a^{2}(t)\delta_{ij}(-3\frac{\kappa^{2}}{t^{2}}+4(\dot{\varphi})^{2}-\frac{1}{16}\kappa^{2}\frac{V''}{(V')^{2}}\frac{1}{t^{4}})\langle \phi^{2}\rangle .
\end{equation}
The first thing we notice is that, if the time dependence of
$\langle \phi^{2} \rangle$ is negligible, the EMT acts as a tracker with
every term scaling as $1/a^{4}(t)$ (except for the $\dot{\varphi}$ which
scales faster and which we ignore from now on).
We now determine the time dependence of $\langle \phi^{2} \rangle$, where
\begin{equation}
\langle \phi^{2} \rangle\,=\,\frac{\psi^{2}}{V}\int{\,d^{3} \vec{x}\,\,d^{3} \vec{k_{1}}\,d^{3} \vec{k_{2}}}\,f( \vec{k_{1}})f( \vec{k_{2}})e^{i( \vec{k_{1}}+ \vec{k_{2}})\cdot \vec{x}},
\end{equation}
with
\begin{equation}
f( \vec{k}) \, = \,
\sqrt{V}(\frac{k}{k_{n}})^{-3/2-\xi}k^{-3/2}_{n}e^{i\alpha( \vec{k})}.
\end{equation}\label{integral}
Here, $\psi$ represents the amplitude of the perturbations (which is
constant in time), $\xi$
represents the deviation from a Harrison-Zel'dovich spectrum,
$\alpha( \vec{k})$ is a random variable, and $k_{n}$ is a normalization scale.
Taking $\frac{\Lambda}{a(t)}$ as a time-dependent, infra-red cutoff and
the Hubble scale as our ultra-violet cutoff, and focusing on the case of a
nearly scale-invariant spectrum, the above simplifies to
\begin{eqnarray} \label{evalue}
\langle \phi^{2} \rangle\,&=&\, 4\pi\psi^{2}k^{-2\xi}_{n}\int^{H}_{\frac{\Lambda}{a(t)}}{}dk_{1}\frac{1}{k_{1}^{1-2\xi}}\\
\end{eqnarray}
In the limit of small $\xi$, the above reduces to
\begin{equation}
\langle \phi^{2} \rangle\,\cong\,4\pi\psi^{2}\ln(\frac{a(t)H}{\Lambda}).
\end{equation}
The time variation of the above quantity is only logarithmic in time
and hence not important for our purposes. As well, given the small
amplitude of the perturbations, $\langle \phi^{2} \rangle \ll 1$. Note that
this condition is opposite to what needs to happen in the scenario
when gravitational back-reaction ends inflation.
Now that we have established that the EMT acts as a tracker in this epoch,
we still have to determine the magnitude of $\tau_{00}$ and the
corresponding equation of state. In order to do this, as in
\cite{Shtanov:1994ce}, we assume that the preheating temperature is
$T=10^{12}$GeV, the quartic coupling $\lambda=10^{-12}$, and the
inflaton amplitude following preheating is $\varphi_{0}=10^{-4}M_{Pl}$.
Making use of
\begin{equation}
a(t)\,=\,(\frac{32\pi\rho_{0}}{3M^{2}_{Pl}})^{1/4}t^{1/2},
\end{equation}
where $\rho_{0}$ is the initial energy density of radiation, we find
\begin{equation}
\tau_{00}\,=-\kappa(\frac{32\pi\rho_{0}}{3M^{2}_{Pl}})\frac{1}{a^{4}(t)}[1-\frac{1}{8}]\langle \phi^{2} \rangle\,\cong\,-\frac{4}{3}\frac{\rho_{0}}{a^{4}(t)}\langle \phi^{2} \rangle,
\end{equation}
\begin{equation}
\tau_{ij}\,=\,-a^{2}(t)\delta_{ij}\kappa(\frac{32\pi\rho_{0}}{3M^{2}_{Pl}})\frac{1}{a^{4}(t)}[3+\frac{1}{8}]\langle \phi^{2} \rangle\,\cong\,-4\frac{\rho_{0}}{a^{4}(t)}\langle \phi^{2} \rangle.
\end{equation}
We find that, as in the case of an inflationary background, the energy
density is negative. However, unlike during inflation, the equation of state
is no longer that of a cosmological constant. Rather, $w\,\cong\,3$.
Clearly, due to the presence of $\langle \phi^{2}\rangle$, this energy
density is sub-dominant. Using the value of $\psi$ in (\ref{evalue})
determined by the normalization of the power spectrum
of linear fluctuations from CMB experiments \cite{COBE}, we can estimate
the magnitude to be approximately four orders of magnitude below that of
the cosmic fluid. Any observational constraints that could arise during
the radiation era (e.g. from primordial nucleosynthesis, or the CMB) will
hence be satisfied.
\section{Matter Domination}
During the period of matter domination, we find that the EMT reduces to
\begin{equation}
\tau_{00}\,=\,(\frac{2}{3}\frac{\kappa^{2}}{\lambda}\frac{a^{4}(t)}{\varphi^{4}}\frac{1}{t^{4}}-\frac{8}{3}\frac{1}{t^{2}})\langle \phi^{2} \rangle.
\end{equation}
\begin{equation}
\tau_{ij}\,=\,(-\frac{2}{3}\frac{\kappa^{2}}{\lambda}\frac{a^{4}(t)}{\varphi^{4}}\frac{1}{t^{4}}-\frac{8}{3}\frac{1}{t^{2}})\langle \phi^{2} \rangle.
\end{equation}
In arriving at these equations, we are assuming that the matter fluctuations
are carried by the same field $\varphi$ (possibly the inflaton)
as in the radiation epoch, a field
which scales in time as $a^{-1}(t)$
\footnote{Even if we were to add a second scalar
field to represent the dominant matter and add a corresponding second
matter term in the constraint equation (\ref{constr}), it can be seen that
the extra terms in the equations for the effective EMT decrease in time
faster than the dominant term discussed here.}.
This result is quite different from what was obtained in the radiation era
for the following reason: previously, we found that both terms in
$\tau_{00}$ scaled in time the same way. Now, we find (schematically)
\begin{equation}
\tau_{00} \propto \frac{\kappa^{2}}{a^{2}(t)}-\frac{\kappa}{a^{3}(t)}.
\end{equation}
The consequences of this are clear: the first term will rapidly come to
dominate over the second, which is of approximately the same magnitude at
matter-radiation equality. This will engender a change of sign for the
energy density and cause it to eventually overtake that of the cosmic fluid.
The same scaling behaviour is present in $\tau_{ij}$ and so the equation of
state of the EMT will rapidly converge to that of a cosmological constant,
but this time one corresponding to a positive energy density.
Matter-radiation equality occurred at a redshift of about
$z \approx 10^4$ and we find that
\begin{equation}
\tau_{00}(z=0)\,\simeq\,\rho_{m}(z=0),\quad w\,\simeq\,-1,
\end{equation}
and thus we are naturally led to a resolution of the both aspects of
the dark energy problem. We have an explanation for the presence of
a source of late-time acceleration, and a natural solution of the
``coincidence'' problem: the fact that dark energy is rearing its
head at the present time is directly tied to the observationally
determined normalization of the spectrum of cosmological perturbations.
\section{Dark Energy Domination and Inflaton Back-reaction}
Does this model predict that, after an initial stage of matter domination,
the universe becomes perpetually dominated by dark energy? To answer this
question, one needs to examine the effects of back-reaction on the late time
scalar field dynamics.
The EMT predicts an effective potential for $\varphi$ that differs from the
simple form we have been considering so far. During slow-roll, we have that
\begin{equation}
V_{eff} \, = \,V+\tau_{0}^{0}.\label{effective potential}
\end{equation}
One might expect that this would lead to a change in the spectral index of
the power spectrum or the amplitude of the fluctuations. To show that this
is not the case, we can explicitly calculate the form of $V_{eff}$ for the
case of an arbitrary polynomial potential and see that, neglecting any
$\varphi$ dependence of $\langle\phi^{2}\rangle$, (\ref{effective potential})
implies an (a priori small) renormalization of the scalar field coupling. We
find that the inclusion of back-reaction does not lead to any change in the
spectral index (in agreement with \cite{Martineau:2005aa}) or to any
significant change in the amplitude of the perturbations.
During radiation domination, we find that the ratio of
$\frac{\tau_{0}^{0}}{V}$ is fixed and small, so that scalar field
back-reaction does not play a significant role in this epoch.
In fact, back-reaction on the scalar field does not become important
until back-reaction begins to dominate the cosmic energy budget. In that case,
\begin{equation}
V_{eff} \sim \frac{1}{\varphi^{4}},
\end{equation}
causing the $\varphi$ to ``roll up'' it's potential. Once $\varphi$ comes
to dominate, the form of the effective potential changes to
\begin{equation}
V_{eff}\sim \varphi^{4},
\end{equation}
and $\varphi$ immediately rolls down it's potential, without the benefit
of a large damping term (given by the Hubble scale).
Thus, this model predicts alternating periods of dark energy/matter
domination, which recalls the ideas put forth in \cite{Brandenberger:1999su}.
From the point of view of perturbation theory, we see that in the regime where the higher-order terms begin to dominate and the series would be expected to diverge, these corrections are then suppressed and become sub-dominant again.
\section{Discussion and Conclusions}
To recap, we find that, in the context of inflationary cosmology, the EMT of
long wavelength cosmological perturbations can provide a candidate for
dark energy which resolves the ``new cosmological constant'' (or
``coincidence'' problem in a natural way. Key to the success of
the mechanism is the fact that the EMT acts as a tracker during the period of radiation domination, but redshifts
less rapidly than matter in the matter era. The fact that our dark energy
candidate is beginning to dominate today, at a redshift $10^4$ later than
at the time of equal matter and radiation is related to the observed
amplitude of the spectrum of cosmological perturbations.
We wish to conclude by putting our work in the context of other recent
work on the gravitational back-reaction of cosmological perturbations.
We are making use of non-gradient terms in the EMT (as was done in
\cite{Mukhanov:1996ak,Abramo:1997hu}). As was first realized by Unruh
\cite{Unruh} and then confirmed in more detail in \cite{Ghazal1,AW1},
in the absence of entropy fluctuations, the effects of these terms
are not locally measurable (they can be undone by a local time
reparametrization). It is important to calculate the effects of
back-reaction on local observables measuring the expansion history.
It was then shown \cite{Ghazal2} (see also \cite{AW2}) that
in the presence of entropy fluctuations, back-reaction of the non-gradient
terms is physically measurable, in contrast to the statements
recently made in \cite{Wald} \footnote{There are a number of problems present in the arguments of \cite{Wald}, in addition to this point. We are currently preparing a response that addresses the criticisms of these authors. See \cite{us}.}. In our case, we are making use of
fluctuations of the scalar field $\varphi$ at late times. As long as
this fluctuation is associated with an isocurvature mode, the
effects computed in this paper using the EMT approach should also
be seen by local observers.
Our approach of explaining dark energy in terms of back-reaction is different
from the proposal of \cite{Kolb:2005me}. In that approach, use is made of the
leading gradient terms in the EMT. However, it has subsequently been shown
\cite{Ghazal3} that these terms act as spatial curvature and that hence their
magnitude is tightly constrained by observations. Other criticism was raised
in \cite{Seljak} where it was claimed that, in the absence of a bare
cosmological constant, it is not possible to obtain a cosmology which changes
from deceleration to acceleration by means of back-reaction. This criticism is
also relevant for our work. However, as pointed out
in \cite{Buchert}, there are subtleties when dealing with spatially averaged
quantities, even if the spatial averaging is over a limited domain, and that
the conclusions of \cite{Seljak} may not apply to the quantities we are
interested in.
There have also been attempts to obtain dark energy from the back-reaction of
short wavelength modes \cite{Rasanen,Alessio,Kolb2}. In these approaches,
however, nonlinear effects are invoked to provide the required magnitude of the
back-reaction effects.
We now consider some general objections which have been raised regarding
the issue of whether super-Hubble-scale fluctuations can induce
locally measurable back-reaction effects. The first, and easiest to refute,
is the issue of causality. Our formalism is based entirely on the equations of
general relativity, which are generally covariant and thus have causality
built into them. We are studying the effects of super-Hubble but sub-horizon
fluctuations \footnote{We remind the reader that it is exactly because
inflation exponentially expands the horizon compared to the Hubble radius
that the inflationary paradigm can
create a causal mechanism for the origin of structure in the universe.
In our back-reaction work, we are using modes which, like those which we now
observe in the CMB, were created inside the Hubble radius during the early
stages of inflation, but have not yet re-entered the Hubble radius in the
post-inflationary period.}. Another issue is locality. As shown in \cite{Lam},
back-reaction effects such as those discussed here can be viewed in terms of
completely local cosmological equations. For a more extensive discussion, the reader is referred to \cite{us}.
In conclusion, we have presented a model which can solve the dark energy
problem without resorting to new scalar fields, making use only of
conventional gravitational physics. The effect of the back-reaction of the
super-Hubble modes is summarized in the form of an effective energy-momentum
tensor which displays distinct behaviour during different cosmological epochs.
\begin{acknowledgments}
This work is supported by funds from McGill University,
by an NSERC Discovery Grant and by the Canada Research Chair program. P.M. would like to thank Mark Alford and the Washington University physics department for their hospitality while part of this work was being completed.
\end{acknowledgments}
| 2024-02-18T23:39:48.020Z | 2005-10-18T17:59:26.000Z | algebraic_stack_train_0000 | 443 | 4,204 |
|
proofpile-arXiv_065-2247 | \section{Introduction}
\label{sect1} In recent years, the perovskite-related manganites
have attracted tremendous interest, mainly triggered by the
discovery of the colossal magnetoresistance (CMR) \cite{cmr}. But
these systems, being governed by many competing interactions
(superexchange, double exchange, charge ordering, and Jahn-Teller
effect) leading to very rich phase diagrams, are also renowned for
providing prototypical examples of temperature, doping, and
magnetic-field induced metal-to-insulator (MI) transitions. The
theoretical \cite{furukawa,millis1,horsch} and experimental
\cite{oki1,kaplan,jung1,jung2,quijada,machida,seeger,oki2,take,pao1,mayr1,sichel,tobe,hart1}
investigation of the dynamical conductivity played a significant
role in the clarification of the charge transport processes and
the types of charge carriers in these manganites. However, still
our understanding of the many puzzling properties of these
materials is far from complete and the theoretical understanding
of the importance of the electron-phonon coupling
\cite{millis2,roder} has to be verified in optical experiments..
In the present work, we report on the evolution of the dynamic
conductivity of the classical CMR manganite
La$_{1-x}$Sr$_{x}$MnO$_{3}$ ($0 \leq x \leq 0.2$) when approaching
the MI transition from the insulating side. Both, dielectric
measurements in the Hz-GHz frequency range and optic measurements
in the far- to near-infrared range were performed. They provide
complementary information giving insight into the mechanisms that
are active when the metallic state, hallmarked by a well
pronounced Drude contribution, develops out of the insulating one,
characterized by strongly localized charge carriers. Our results,
together with those obtained on various other transition metal
oxides, allow for the development of a general scenario for the
evolution of the broadband dynamic conductivity when approaching
the metallic state from the insulating side.
\begin{vchfigure}[htb]
\includegraphics[angle=270,width=.6\textwidth,clip]{PhaseDiaghatch.eps}
\vchcaption{The ($x$, $T$) phase diagram of
La$_{1-x}$Sr$_{x}$MnO$_{3}$
\cite{ParaJMMM,ParaJPhys,Griffiths,hemb}. The structural (O,
O$^{\prime}$, O$^{\prime\prime}$, R), magnetic (PM, CA, FM), and
electronic (M, I) phases are indicated. The diagonally hatched
areas indicate the metallic states. In the horizontally hatched
area, electronic phase-separation may exist \cite{ParaJPhys}. "GP"
denotes the Griffiths phase, proposed in \cite{Griffiths}. }
\label{PhaseDiag}
\end{vchfigure}
LaMnO$_{3}$ is a charge-transfer insulator. In the octahedral
environment the 3$d$-levels of the Mn$^{3+}$ ions are split into
a lower $t_{2g}$ triplet and an excited $e_{g}$ doublet. The
$e_{g}$ states are filled with one electron and hence are strongly
Jahn-Teller active. Indeed, below 800~K LaMnO$_{3}$ exhibits
orbital order with a half-filled lower and an empty excited
$e_{g}$ level. On substituting La$^{3+}$ by Sr$^{2+}$, holes are
introduced into the lower $e_{g}$ level, weakening the JT
interactions and driving the system into a metallic state
\cite{jonker}. Already in the early days it was speculated that
for the metal-to-insulator transition double exchange might play
an important role, driving the system from a paramagnetic
insulator into a ferromagnetic metal \cite{gennes}.
The rich phase diagram of the La$_{1-x}$Sr$_{x}$MnO$_{3}$ system
for low doping levels is given in Figure \ref{PhaseDiag}, based on
dc, magnetic, optic, and electron spin resonance measurements
\cite{ParaJMMM,ParaJPhys,Griffiths,hemb}. Depending on doping and
temperature, the system exhibits a rhombohedral (R) and three
different orthorhombic structures (O, O$^{\prime}$,
O$^{\prime\prime}$; see \cite{ParaJMMM,ParaJPhys,hemb} for
details). At low concentrations ($x<0.1$) there is a Jahn-Teller
(JT) distorted and insulating orthorhombic phase O$^{\prime}$,
which at low temperatures reveals canted antiferromagnetism (CA).
The ground state is an orbitally ordered and ferromagnetic
insulator (FM/O$^{\prime\prime}$/I) for $0.1 \leq x \leq 0.15$,
and a FM metal for $x>0.17$ in the non-JT distorted O and R phases
(diagonally hatched area). Finally, the triangular region, marked
"GP", denotes the Griffith phase, which was postulated in
\cite{Griffiths}, based on electron spin resonance and magnetic
measurements. The formation of this phase, which is characterized
by ferromagnetic clusters formed below a characteristic
temperature $T_{G}$ ($\approx 270$~K in the present case), was
ascribed to quenched disorder of the randomly distributed FM bonds
in this material \cite{Griffiths}. At the FM phase boundary, the
clusters achieve FM order and percolate.
\section{Experimental Details}
\label{sect2} Well characterized single crystals of
La$_{1-x}$Sr$_{x}$MnO$_{3}$, grown by the floating zone method,
were kindly provided by Profs. A.A. Mukhin and A.M. Balbashov
\cite{samples}. Measurements of the dc conductivity were performed
using a standard four-point technique. For the non-metallic
samples, the ac conductivity at frequencies below 1~GHz was
recorded using an autobalance bridge HP4284 covering frequencies
20~Hz~$\leq \nu \leq$ 1~MHz and an HP4291 impedance analyzer at
frequencies 1~MHz~$\leq \nu \leq$ 1.8~GHz \cite{ferro}. The
microwave conductivity was measured at 7.3 GHz utilizing a
microwave perturbation technique within a $^{4}$He-flow cryostat.
In the IR regime the temperature-dependent conductivity was
determined via reflectivity measurements using a Bruker IFS 113v
Fourier transform spectrometer. From the reflectivity $R$, the
conductivity $\sigma ^{\prime}$ was calculated via the
Kramers-Kronig transformation, requiring an extrapolation of the
data towards low and high frequencies. For $\nu \rightarrow 0$, a
constant extrapolation was used for the insulating states and a
Hagen-Rubens law for the metallic states. At high frequencies the
spectra were extrapolated by a power law $R\propto \nu^{-1.5}$ up
to $10^{6}$~cm$^{-1}$, leading to a smooth transition to the
experimental data, which was followed by a $\nu^{-4}$ power law at
higher frequencies \cite{mayrprom}. For some samples, additional
measurements at 7.3~GHz and around 100~GHz were performed using a
microwave perturbation technique and a quasi-optic Mach-Zehnder
spectrometer \cite{subm}, respectively.
\section{Results and Discussion}
In Fig. \ref{refl} the frequency dependent reflectivity of
La$_{1-x}$Sr$_{x}$MnO$_{3}$ is shown for four doping levels up to
$x=0.2$. The sharp resonance features showing up in the region
between 100 and 1000~cm$^{-1}$ (3~-~30~THz) are due to phonon
excitations. They are most prominent in the insulating regions of
the phase diagram, i.e. for the low doping levels shown in Figs.
\ref{refl}(a) and (b) and for the higher temperatures at the
higher doping levels given in Figs. \ref{refl}(c) and (d). For a
detailed discussion of these phonon modes the reader is referred
to references
\cite{mayr1,ParaJMMM,mayrprom,kim,fedorov,abrashev,pao2,hart2,hart3,mayr2}.
For the insulating states, immediately following the last phonon
mode the reflectivity increases sharply with increasing frequency.
This can be ascribed to the onset of electronic excitations across
a band gap, which at higher frequencies leads to a peak in the
reflectivity close to $10^{4}$~cm$^{-1}$ ($\approx 1.3$~eV). Above
about $2\times 10^{4}$~cm$^{-1}$, the onset of a further
electronic excitation is observed for all doping levels shown in
Fig. \ref{refl}. It is clear from the electronic structure of the
doped manganites that below $10^{4}$~cm$^{-1}$ excitations between
and within the JT-split $e_{g}$ levels dominate. The
charge-transfer gap, corresponding to a transition between the
oxygen 2$p$- and the manganese $e_{g}$ states, is expected close
to $4 ~\textrm{eV} \approx 3\times10^{4} ~\textrm{cm}^{-1}$ and is
just covered at the high-frequency end of the reflectivity spectra
for all compounds shown in Fig. 2. For the metallic sample with
$x=0.2$ [Fig. \ref{refl}(d)] and when the metallic state is
approached with decreasing temperature at for $x=0.175$ [Fig.
\ref{refl}(c)], the gaplike excitations below $10^{4}$~cm$^{-1}$
and the phonon modes are superimposed by a strong additional
contribution, smoothly increasing and finally tending to saturate
with decreasing frequency. These are the characteristic features
of a Drude contribution due to free charge carriers.
\begin{vchfigure}[htb]
\includegraphics[width=0.7\textwidth,clip]{refl.eps}
\vchcaption{Frequency dependent infrared reflectivity of
La$_{1-x}$Sr$_{x}$MnO$_{3}$ for four different doping levels and
various temperatures.} \label{refl}
\end{vchfigure}
To gain deeper insight into the various processes contributing to
the infrared reflectivity and to relate the results to those
obtained at sub-GHz frequencies, it is helpful to calculate the
frequency dependent conductivity from the reflectivity using the
Kramers-Kronig transformation. Figure \ref{x0} shows the frequency
dependence of $\sigma ^{\prime}$ of pure LaMnO$_{3}$ at selected
temperatures, combining results at low frequencies \cite{seeger}
and in the infrared region \cite{mayrprom}. Only those parts of
the spectra are shown that are not affected by contact
contributions \cite{contact}, which were shown in \cite{seeger} to
dominate at very low frequencies and high temperatures. As already
noted in \cite{seeger}, for $\nu<1$~GHz, three contributions can
be identified: For $\nu\rightarrow0$, $\sigma^{\prime}$ approaches
the frequency-independent dc conductivity. Its temperature
dependence follows the prediction of the Variable-Range-Hopping
model \cite{seeger}. With increasing frequency, $\sigma^{\prime}$
increases smoothly, which can be parameterized by a sublinear
power law $\sigma^{\prime}\propto\nu^{s}$ with $s<1$
\cite{seeger}, a behavior termed "Universal Dielectric Response"
(UDR) due to its universal occurrence in a variety of materials
\cite{udr}. Such a behavior is commonly regarded as a hallmark
feature of hopping conduction of charge carriers subjected to
disorder-induced localization \cite{hopping}. Superimposed to this
power law, a shoulder shows up at low temperatures. As revealed by
the upper inset of Fig. \ref{x0}, it corresponds to
well-pronounced peaks in the dielectric loss
$\varepsilon^{\prime\prime}\propto \sigma^{\prime}/\nu$ shifting
through the frequency window with temperature. This is typical for
a relaxational process, which was ascribed to localized hopping of
polarons in \cite{seeger}. However, it should be noted that the
energy barrier of 86~meV, determined from the temperature
dependence of the relaxation time \cite{seeger}, obviously has no
relation to the energy scales of the polaronic excitations
observed in the infrared region, which are discussed below.
\begin{vchfigure}[htb]
\includegraphics[width=0.7\textwidth,clip]{x0.eps}
\vchcaption{Frequency dependent conductivity of LaMnO$_{3}$ at
sub-GHz and infrared frequencies for various temperatures. The
upper inset shows the dielectric loss below GHz; the lines are
fits with the empirical Cole-Cole function \cite{seeger}. The
lower inset shows $\sigma ^{\prime}$ in the region of the
electronic excitation close to 1.5~eV for various temperatures.}
\label{x0}
\end{vchfigure}
It is not possible to extrapolate the sublinear power law found
below GHz towards the infrared results in the THz regime, even if
assuming $s=1$ as indicated by the dash-dotted line in Fig.
\ref{x0}. Clearly an additional contribution must be present in
the so-far uninvestigated intermediate region. We propose that
there is an additional superlinear power law (SLPL),
$\sigma^{\prime}\propto\nu^{n}$, with $n\approx1.2$ as indicated
by the dashed line. Such a SLPL was clearly detected, e.g., in the
CMR manganite Pr$_{0.65}$(Ca$_{0.8}$Sr$_{0.2}$)$_{0.35}$MnO$_{3}$
\cite{sichel} and evidenced to be a universal feature of
disordered matter \cite{univ}, e.g. supercooled liquids or doped
semiconductors. In the present case of a nominally undoped
material without any substitutional disorder, a slight
off-stoichiometry or impurities at the ppm level seem to be
sufficient to produce this typical response of disordered matter,
as it is also the case, e.g., for LaTiO$_{3}$ \cite{latio}.
In the infrared region, at frequencies beyond the phonon
resonances a peak shows up at about 380~THz, corresponding to
1.6~eV. As shown in the lower inset of Fig. \ref{x0}, its peak
frequency is nearly independent of temperature. This peak most
probably corresponds to an electronic transition between the
JT-split $e_{g}$ bands. The excitation from one $e_{g}$ orbital to
another at the same site is forbidden and should have negligible
oscillator strength. Hence, this peak structure must correspond to
a transition between Mn-ions on adjacent sites, which, due to
Hund's coupling, is favorable for parallel spin alignment. Thus,
at first glance at the magnetic phase transition an anomaly in the
intensity is expected, but experimentally only a moderate
temperature dependence is observed (lower inset of Fig. \ref{x0}).
This finding may be explained considering that in the paramagnetic
phase the spins are thermally disordered while in the type-A
antiferromagnetic state, order is established with ferromagnetic
spin alignment within and antiferromagnetic order between the
($a$,$b$)-planes. But the above interpretation is not corroborated
by polarized light-scattering experiments, which indicate a strong
charge-transfer character of this transition \cite{tobe}. Further
experiments will be necessary to unravel the nature of this
transition.
\begin{vchfigure}[htb]
\includegraphics[width=0.5\textwidth]{x125.eps}
\vchcaption{Frequency dependent conductivity of
La$_{0.875}$Sr$_{0.125}$MnO$_{3}$ at three temperatures. In
addition, the results of a four-point dc-measurement are
indicated. (reproduced from \cite{ParaJMMM})} \label{x125}
\end{vchfigure}
In Fig. \ref{x125}, reproduced from ref. \cite{ParaJMMM}, the
frequency-dependent conductivity of
La$_{0.875}$Sr$_{0.125}$MnO$_{3}$ is shown for three temperatures.
At room temperature, where the dc conductivity is rather high, the
low-frequency measurements provide information on the intrinsic
conductivity in a restricted frequency range only, because at low
frequencies the contact contributions and at high frequencies the
inductance of the sample dominates \cite{seeger}. Nevertheless,
from Fig. \ref{x125} it seems clear that in the sub-GHz range the
room-temperature conductivity is frequency independent. However,
interpolating from GHz to the lowest frequencies of the FIR
experiment, thereby taking into account the single point at
7.3~GHz from a microwave resonance measurement, it seems likely
that in this region a small increase of $\sigma^{\prime}(\nu)$ is
present. It can be suspected to be caused by hopping conduction of
localized charge carriers, the corresponding sublinear power law
emerging from the dc background with increasing frequency. The
presence of this contribution, pointing to a localization of the
charge carriers also in this compound, becomes even more obvious
at the lower temperatures, where a succession of a clearly
pronounced sublinear power law and a SLPL can be suspected. The
non-metallic character of this compound is further corroborated by
the complete absence of a Drude contribution in the infrared
region. Instead, above the phonon modes, the conductivity exhibits
a strong increase, which can be ascribed to superposed
contributions from polaron absorption and interband transitions,
as discussed in detail in \cite{mayr1,mayrprom}.
\begin{vchfigure}[htb]
\includegraphics[width=0.85\textwidth]{xhigh.eps}
\vchcaption{Frequency dependent conductivity of
La$_{1-x}$Sr$_{x}$MnO$_{3}$ for $x=0.175$ and 0.2 at various
temperatures. In addition, the results of a four-point
dc-measurement are indicated.} \label{xhigh}
\end{vchfigure}
For Sr contents $x=0.175$ and 0.2, having ferromagnetic metallic
ground states, the measured low-frequency conductivity is
dominated by inductance and skin effect contributions
\cite{seeger}. The intrinsic $\sigma^{\prime}(\nu)$ can be
expected to reveal a significant frequency dependence only at high
frequencies, in the infrared and optical region, as shown in Fig.
\ref{xhigh}. For $x=0.175$, at high temperatures, where the sample
is close to the MI transition (cf. Fig. \ref{PhaseDiag}),
$\sigma^{\prime}(\nu)$ shows a broad peak at about
$10^{4}$~cm$^{-1}$. With decreasing temperature, spectral weight
is shifted from this excitation towards lower frequencies, leading
to a peak or shoulder strongly shifting towards lower frequencies
when the sample becomes more metallic with decreasing temperature.
Finally, at 6~K $\sigma^{\prime}$ increases markedly towards low
frequencies, in good accord with the strongly enhanced dc
conductivity, thus exhibiting the clear signature of a Drude
contribution of free charge carriers. A detailed analysis of the
Drude response in La$_{0.825}$Sr$_{0.175}$MnO$_{3}$ was provided
in \cite{take}. It seems reasonable to assume that the excitation
at about 11000~cm$^{-1}$ or 1.4~eV has a similar origin as that at
1.6~eV for pure LaMnO$_{3}$. But now we have a fraction of
Mn$^{4+}$ sites with no $e_{g}$ electrons. Hence, in addition to
transitions between JT-split $e_{g}$ orbitals, optical transitions
between empty and occupied $e_{g}$ levels have to be taken into
account. We believe that the broad peak at room temperature at
1.4~eV is a mixture of both processes. With decreasing
temperatures a remainder of the JT-split derived peak still is
visible at 1.4~eV, indicating the significantly weaker JT energy
in the doped crystals compared to the 1.6~eV observed in the pure
compound (Fig. \ref{x0}). The second spectral feature branching
off from this excitation and being strongly temperature dependent
can be ascribed to a polaronic excitation. Similar behavior has
been reported in \cite{oki1} exemplifying the importance of strong
electron-phonon interactions in the manganites. The strong
softening of this polaronic excitation qualitatively resembles the
behaviour predicted by Millis \cite{millis1} for a scenario of
strong electron-phonon coupling. However, in the case of
half-filling, the polaronic excitation transforms into a Drude
type behavior at low temperatures. In La$_{1-x}$Sr$_{x}$MnO$_{3}$
($0 \leq x \leq 0.2$) a Drude peak evolves with doping
simultaneously with the polaronic excitation, with the Drude peak
being characterized by a low relaxation rate and a very low
optical weight. The rather symmetric shape of the polaron peak
with the absence of a clear cut-off frequency at low frequencies
indicates the dominance of small polarons \cite{hart1}. Finally,
at the high-frequency edge of the investigated spectral range, a
further strong increase of the conductivity shows up, indicating
the onset of the charge-transfer excitation as in pure
LaMnO$_{3}$. For $x=0.2$ [Fig. \ref{xhigh}(b)], the overall
behavior is qualitatively similar to that observed for $x=0.175$.
However, the excitation ascribed to the polaronic transition seems
to be shifted to lower energy, now located at about 0.8~eV at room
temperature and of significantly lower optical weight. At the
lowest temperatures a narrow Drude peak dominates the optical
conductivity and the polaronic excitation appears as a weak
shoulder close to 3000~cm$^{-1}$.
\begin{vchfigure}[htb]
\includegraphics[width=0.5\textwidth]{fits.eps}
\vchcaption{Frequency dependent conductivity of
La$_{0.825}$Sr$_{0.175}$MnO$_{3}$ at 6~K (solid line). The dashed
line is a fit of the experimental spectrum using the sum of a
Drude and a Lorentz term. These two contributions are indicated by
the dotted and dash-dotted lines, respectively.} \label{fits}
\end{vchfigure}
In Fig. \ref{fits}, a least-square fit of the conductivity of
La$_{0.825}$Sr$_{0.175}$MnO$_{3}$ measured at 6~K is shown (dashed
line). For the fit, the sum of a Drude term dominating at low
frequencies and a Lorentz term (peak frequency $\nu
_{0}=2040~\textrm{cm}^{-1}$, damping constant $\gamma =
7180~\textrm{cm}^{-1}$) taking into account the polaronic
excitation was used. The three phonon modes were neglected. A good
description of the experimental data could be achieved in this
way, revealing a Drude relaxation time of
$\tau=1.8\times10^{-14}$s. Using a Fermi velocity for the
orthorhombic structure of $2.2\times10^{7}$~cm/s \cite{singh}, we
obtain a mean free path of 4~nm, which characterizes this compound
as a bad metal. As discussed earlier, the symmetric form of the
polaron absorption points towards the existence of small polarons
\cite{hart1}. The small ratio of the optical weight of the Drude
conductivity to that of the polaron absorption (Fig. \ref{fits})
indicates that only a small fraction of free charge carriers
contributes to the Drude-like conductivity, while the majority
still is localized by strong electron-phonon coupling effects.
\section{Summary and Conclusions}
The ac conductivity of the CMR manganite
La$_{1-x}$Sr$_{x}$MnO$_{3}$ was reported for four doping levels
$x$. The combination of low-frequency and optical measurements
revealed the importance of hopping conduction of localized charge
carriers for the non-metallic parts of the phase diagram of this
system. In the infrared region beyond the phonon modes, evidence
for two electronic modes were obtained, one of them showing the
clear signature of a polaronic excitation. Deep in the metallic
state a strong Drude contribution is observed.
\begin{vchfigure}[htb]
\includegraphics[width=0.6\textwidth]{scheme.eps}
\vchcaption{Schematic plot of the conductivity of a system
approaching a MI transition from the insulating side as observed,
e.g., in transition metal oxides. The figure covers a broad range,
starting at frequencies of typical dielectric experiments up to
those covered by optical spectrometers. The lowest curves
correspond to the most insulating, the highest to the most
metallic states. The values of the tick labels are intended to
give a rough estimate only.} \label{scheme}
\end{vchfigure}
Taking together the results of the present work and those obtained
on a variety of further transition metal oxides and other systems
being close to a MI transition
\cite{seeger,sichel,univ,latio,broadband}, allows for the
development of a general scenario for the broadband ac
conductivity when approaching the MI transition from the
insulating side. As schematically indicated in Fig. \ref{scheme},
deep in the insulating phase most materials show a succession of a
dc plateau, a sublinear, and a superlinear power law \cite{univ}.
Such a behavior seems to be closely connected to hopping
conduction of charge carriers that are localized due to disorder.
One should note, however, that this notion is mainly based on
theories that were developed for amorphous or heavily doped
conventional semiconductors \cite{hopping,hoptheo} and whose
predictions can be approximated by a sublinear power law. It is
not clear, if these theories also should be applicable to
electronically correlated materials as, e.g., the CMR manganites.
Also so far there is no well-founded explanation for the SLPL,
which seems to occur concomitantly with the sublinear law.
Whatsoever, it seems clear now that in the insulating and
semiconducting states these power laws are also active in the only
rarely investigated transition region between the classical
dielectric frequencies ($\sim$ below GHz) and those of typical
infrared experiments. Often they even seem to represent the
background for the phonon modes, however, vanishing abruptly
beyond the highest phonon resonance. When the system further
approaches the metallic state, the dc conductivity, growing more
rapidly than the UDR and the nearly constant SLPL, dominates over
an increasingly broader frequency range, which finally extends
well up to the first phonon modes (second curve from above in Fig.
\ref{scheme}). This typically occurs for a dc conductivity of
about 100~-~1000~$\Omega^{-1}$cm$^{-1}$ and marks the transition
into the metallic state. Up to this point, the infrared response
is dominated by strong phonon modes, followed by electronic
excitations due to interband and polaronic contributions. Within
the metallic phase, the phonons and low-lying electronic modes are
successively superimposed by the typical free carrier Drude
behavior as indicated by the uppermost curve in Fig. \ref{scheme}.
It is interesting that the value of $\sigma^{\prime}$ marking the
transition from insulating to a Drude-like metallic behavior seems
to be rather universal and a connection of this finding to the
concept of the "minimum metallic conductivity" proposed by Mott
\cite{mott} may be suspected.
\begin{acknowledgement}
We thank A.A. Mukhin and A.M. Balbashov for providing the samples
and A. Pimenov, K. Pucher, and A. Seeger for performing part of
the measurements. Stimulating discussion with Th. Kopp and Ch.
Hartinger are gratefully acknowledged. This work was supported by
the Deutsche Forschungsgemeinschaft via the
Sonderforschungsbereich 484 and by the BMBF via VDI/EKM.
\end{acknowledgement}
| 2024-02-18T23:39:48.164Z | 2005-10-10T14:36:51.000Z | algebraic_stack_train_0000 | 453 | 4,060 |
|
proofpile-arXiv_065-2351 | \section{Introduction}
The proximity and the relative simplicity of the Local Group (LG)
Dwarf Spheroidal (dSph) Galaxies make these systems excellent
laboratories to test assumptions regarding the nucleosynthesis of
chemical elements and theories of galaxy evolution. Several
studies addressing the observation of red giants stars in local
dSph galaxies with high resolution spectroscopy allow one to
infer accurately the abundances of several elements including
$\alpha$-, iron-peak and very heavy elements,
such as barium and europium (Smecker-Hane $\&$ McWilliam
1999; Bonifacio et al. 2000; Shetrone, Cot\'e $\&$ Sargent
2001; Shetrone et al. 2003; Bonifacio et al. 2004; Sadakane
et al. 2004; Geisler et al. 2005). These abundances and abundance
ratios are not only central ingredients in galactic chemical
evolution studies but are also very important in the attempt
to clarify some aspects of the processes responsible for the
formation of chemical elements.
There is a certain consensus regarding the general aspects of the
production of iron peak and $\alpha$- elements, but for some other
elements several questions concerning their production are still
open for debate (Fran\c cois et al. 2004). It is largely
accepted that $\alpha$- elements are mainly
produced on short time-scales in explosions of type II
supernovae (SNe II, Woosley $\&$ Weaver 1995; Thielemann, Nomoto $\&$
Hashimoto 1996; Nomoto et al. 1997), whereas the iron-peak ones are
the main products of type Ia supernovae (SNe Ia, Nomoto et al. 1997),
which occur on longer time-scales. The different time-scales
involved in these explosions are the main reason for using the
ratio between these two sets of elements (the [$\alpha$/Fe] ratio)
as a cosmic clock, which can be used to impose constraints in
the star formation (SF) history of the
systems where this ratio is observed (Tinsley 1980; Matteucci 1996).
On the other hand, the sites of formation and
the processes which are responsible for the production of
elements heavier than Fe remain unclear. These elements are
synthesized mainly by neutron capture reactions and are normally
divided into r-process and s-process elements according to the
velocity of the neutron capture reaction, rapid and slow being relative
to the duration of the $\beta$-decay process, respectively.
Among these elements the production of Barium and Europium
is more often discussed, due to the characteristics of their
formation. They are believed to be produced by the two neutron
capture processes in different ways: while Eu is claimed to
be a pure r-process element it is believed that the process
which dominates the Ba production is the s-process, but with
a low fraction of Ba being produced also by the
r-process (Wheeler, Sneden $\&$ Truran 1989). The stellar mass
ranges where these elements are formed are not yet fully understood.
The r-process is generally accepted to take place in SNe II
explosions (Hill et al. 2002; Cowan et al. 2002).
The s-nuclei can be divided into the main component and
the weak component elements. The main component elements are synthesized
during the thermally pulsing asymptotic giant branch (AGB) phase of
low mass stars (Gallino et al. 1998; Busso et al. 2001). A different
site is required for the production of the weak
component of the s-process elements. These elements are believed to
be produced also in advanced evolutionary stages of massive stars
(Raiteri et al. 1993).
Models of stellar evolution predict
that the
main formation (by s-process) of Ba occurs during the
thermal pulses of AGB stars with initial masses between 1 to 3
$M_{\odot}$ (Gallino et al. 1998; Busso et al. 2001), with a negligible
contribution from intermediate mass stars of
($ 3 < M < 8 M_{\odot}$). As a consequence, the Ba enrichment
of the interstellar medium (ISM)
due to the s-process is delayed
relative to the Fe enrichment, which is produced also in massive stars,
and one should expect a low [Ba/Fe]
at low metallicities, which would later increase due to the injection
of s-processed Ba by the low mass stars (LMS) as [Fe/H] also increases.
It has also been
suggested that a low fraction of Ba is produced by
r-process in stars with masses at the low end of high
mass stars ($ 8 < M < 10 M_{\odot}$) (Travaglio et al. 2001), in order
to reproduce the [Ba/Fe] ratios observed at low metallicities in the
Milky Way. This production would contribute with Ba at early
stages of the evolution of the galaxy, prior to the peak of the iron
enrichment by SNe Ia.
Europium, on the other hand, is produced only by the r-process
which is believed to occur in massive stars ($M > 8 M_{\odot}$)
(Woosley et al. 1994). The details of the r-process nucleosynthesis
remain, however, unclear and several scenarios have been
proposed (Woosley et al. 1994;
Freiburghaus et al. 1999; Wanajo et al. 2003). Assuming
this site of production for Eu, one should observe a relatively
high [Eu/Fe] at low metallicities, which would then decrease
due to the injection of Fe in the ISM by SNe Ia explosions.
However, there is a large spread in the observational data
concerning both Eu and Ba in the stars of our Galaxy
(Ryan, Norris $\&$ Beers 1996; Sneden et al. 1998; Burris et al.
2000), which prevents one from drawing very firm conclusions
regarding the production of both Eu and Ba in the Milky Way.
In local dSph galaxies, red giant stars have been observed with
high resolution spectroscopy and the abundances of Eu and Ba,
among others elements, have been derived (Shetrone, Cot\'e $\&$
Sargent 2001; Shetrone et al. 2003; Sadakane et al. 2004;
Geisler et al. 2005).
Shetrone, Cot\'e $\&$
Sargent (2001) argued that Draco and Ursa Minor stars exhibit an
abundance pattern consistent with one dominated by the r-process, i.e.
[Ba/Eu] ranges from solar values at high metallicities to [Ba/Eu]
$\sim$ -0.5 at [Fe/H] $\leq$ -1 dex. The pattern of [Ba/Fe] and [Eu/Fe]
also resembles the one observed in the halo field stars according to
these authors. The same conclusion was reached by Shetrone et al. (2003),
who analysed these abundance ratios in Sculptor, Fornax and Carina.
Shetrone et al. (2003) claimed also that in Sculptor, Fornax and Leo
I the
pattern of [Eu/Fe] is consistent with the production of Eu in SNe II.
On the other hand, Venn et al. (2004), pointed out that, despite the
general similarity, the dSph stars span a larger range in [Ba/Fe]
and [Eu/Fe] ratios at intermediate metallicities than
the Galactic stars and, more important, that about half of the dSph
stars exhibit lower [Y/Eu] and 2/3 higher [Ba/Y] than the Galactic stars
at the same metallicity, thus suggesting a clear difference between the
chemical evolution of our Galaxy and the one of dSph galaxies.
The [$\alpha$/Fe] ratios observed in dSphs also are different from the same
ratios in the Milky Way showing in general lower [$\alpha$/Fe] ratios than
the Galactic stars with the same [Fe/H] (Smecker-Hane $\&$ McWilliam
1999; Bonifacio et al. 2000; Shetrone, Cot\'e $\&$ Sargent
2001; Shetrone et al. 2003; Bonifacio et al. 2004; Sadakane et al. 2004;
Geisler et al. 2005).
These observations not only shed some light into
the chemical evolution history of these galaxies but allowed also
the construction of chemical evolution models aimed at reproducing
important observational constraints, such as the elemental abundance
ratios, the present gas mass and total mass (Carraro et al. 2001;
Carigi, Hernandez $\&$ Gilmore 2002; Ikuta $\&$ Arimoto 2002;
Lanfranchi $\&$ Matteucci 2003 (LM03); Lanfranchi $\&$ Matteucci
2004 (LM04)).
Among these models the one proposed by LM03 and LM04
for 6 local dSph galaxies (namely Draco, Carina, Sculptor, Sextan,
Ursa Minor
and Sagittarius) succeeded in reproducing the observed [$\alpha$/Fe]
ratios, the present
gas mass and final total mass by adopting a very low star formation rate,
$\nu$ $\sim$ 0.01 to 0.5 Gyr$^{-1}$ (with lower values for Draco
and higher ones for Sagittarius) and a high wind efficiency
(6-13 times the star formation rate).
Besides that, LM04 predicted the stellar metallicity
distribution of these galaxies which were later on compared to
observational data for Carina with a reasonably good
agreement (Koch et al. 2004).
The success of LM03 and LM04 models in reproducing several
observational constraints allows us to use them as tools to test the
theories about the sites of production and the processes responsible for
the synthesis of Ba and Eu in dSph galaxies. By adopting the
nucleosynthesis prescriptions for these elements which are able to
reproduce the most recent observed data for our Galaxy (Cescutti
et al. 2005)
and comparing the predictions of the models with observational data,
it is possible to verify if the assumptions made regarding the
nucleosynthesis of Ba and Eu can also fit the data of local dSph
galaxies.
The paper is organized as follows: in Sect. 2 we present
the observational data concerning the dSph galaxies,
in Sect. 3 the adopted chemical evolution models,
the star formation and the nucleosynthesis prescriptions
are described, in Sect. 4 the predictions of our models are
compared to the observational data and the results discussed,
and finally in Sect. 5 we draw some conclusions. All elemental
abundances are normalized to the solar values
([X/H] = log(X/H) - log(X/H)$_{\odot}$) measured by
Grevesse $\&$ Sauval (1998).
\section{Data Sample}
Recently, red giant stars of dSph galaxies have been the subject
of several works with the aim of determining with high-resolution
spectroscopy the abundance of several chemical elements
including heavy elements such as barium and europium
(Bonifacio et al. 2000; Shetrone, Cot\'e $\&$ Sargent 2001;
Shetrone et al. 2003; Venn et al. 2004; Sadakane et al. 2004;
Fulbright, Rich $\&$ Castro, 2004; Geisler et al. 2005).
From these observations we gathered the data from the galaxies
that were analysed in LM03
and LM04 and for which there are abundance determinations for
both Ba and Eu. They are Carina, Draco, Sculptor, Ursa Minor and Sagittarius.
Despite of the relative small number of data points, it is
possible to compare the observed abundance ratios
with the model predictions. We choose to compare the observed
ratios [Ba/Fe], [Eu/Fe] and [Ba/Eu] with the ones predicted by
the models, since these ratios can provide some clues not only
to the nucleosynthesis of Ba and Eu, but also to all s-process
and r-process elements.
In order to properly compare different
data from different authors with the predictions of the models we adopted
the abundance values of Shetrone, Cot\'e $\&$ Sargent (2001),
Shetrone et al. (2003), and Sadakane et al. (2004) updated by
Venn et al. (2004). Venn et al. (2004) homogenized the atomica data
for spectral
lines of Ba and Eu providing data with improved quality which allow a
consistent comparison between data from different sources. Otherwise,
the effect
of combining these different data would be seen as a larger spread
in the abundances and possibly in the abundance ratios of 0.1 to 0.2 dex
(see Venn et al. 2004). In the case of Bonifacio et al. (2000) data, Eu
is obtained using hyper-fine splitting (HFS) (see their Table 5), but Ba
is not. The authors claimed that the Ba abundances obtained with HFS would
exhibit no significant difference since the line observed (Ba II 6496.9)
is a strong line which is not affected by this correction (Bonifacio
private communication, see also Shetrone et al. 2003).
Some of the observed stars, however, exhibit anomalous values of
[Ba/H] or [Eu/H], and for this reason were excluded from the sample.
Two stars in Ursa Minor, K and 199 (in Shetrone, Cot\'e $\&$
Sargent 2001), exhibit heavy-element abundance ratios enhanced
relative to those typical for other dSph stars: the Ursa Minor K star
has an abundance pattern dominated by the s-process and
was classified as a Carbon star while Ursa Minor 199 is
dominated by r-process (see also Sadakane et al. 2004). In Sculptor,
there are also two stars with enhanced heavy-element abundance:
Sc982 (Geisler et al. 2005) and Sculptor H-400 (Shetrone et al. 2003).
While Shetrone et al. (2003) claimed that the r-process dominated
abundance could be attributed to inhomogeneous mixing of the SNe II
yields, Geisler et al. (2005) classified Sc982 as a heavy element star
which could have been enriched by an other star, which is now dead.
Either way, all these stars do not exhibit an abundance pattern
characterized only by the nucleosynthesis process occurring inside
the star, but also one which was contaminated by external factors.
The maintenance of these stars in the sample could lead to
an erroneous comparison with the model predictions and, as
a consequence, to a misleading interpretation and to wrong conclusions
regarding the processes and the site of production of the heavy
elements analysed. Therefore, we excluded these stars from our sample,
whereas all the other stars were considered and included in
the comparisons with the models predictions.
\begin{table*}
\begin{center}\scriptsize
\caption[]{Models for dSph galaxies. $M_{tot}^{initial}$
is the baryonic initial mass of the galaxy, $\nu$ is the star-formation
efficiency, $w_i$ is the wind efficiency, and $n$, $t$ and $d$
are the number, time of occurrence and duration of the SF
episodes, respectively.}
\begin{tabular}{lccccccc}
\hline\hline\noalign{\smallskip}
galaxy &$M_{tot}^{initial} (M_{\odot})$ &$\nu(Gyr^{-1})$ &$w_i$
&n &t($Gyr$) &d($Gyr$) &$IMF$\\
\noalign{\smallskip}
\hline
Sculptor &$5*10^{8}$ &0.05-0.5 &11-15 &1 &0 &7 &Salpeter\\
Draco &$5*10^{8}$ &0.005-0.1 &6-10 &1 &6 &4 &Salpeter\\
Ursa Minor &$5*10^{8}$ &0.05-0.5 &8-12 &1 &0 &3 &Salpeter\\
Carina &$5*10^{8}$ &0.02-0.4 &7-11 &2 &6/10 &3/3 &Salpeter\\
Sagittarius &$5*10^{8}$ &1.0-5.0 &9-13 &1 &0 &13 &Salpeter\\
\hline\hline
\end{tabular}
\end{center}
\end{table*}
\section{Models}
We use in this work the same chemical evolution model for dSphs
galaxies as described in LM03 and LM04. The model is able to reproduce
the [$\alpha$/Fe] ratios, the present gas mass and the inferred total mass
of six dSph galaxies of the Local Group, namely Carina, Draco, Sculptor,
Sextan, Sagittarius and Ursa Minor, and also the stellar metallicity
distribution of Carina (Koch et al. 2004). The scenario representing
these galaxies is characterized by one long episode (two episodes
in the case of Carina) of star formation (SF) with very low efficiencies
(except in the case of Sagittarius) - $\nu$ = 0.001 to 0.5 Gyr $^{-1}$ -
and by the occurrence of very intense galactic winds - $w_i$ = 6-13.
The model allows one to follow in detail the evolution of the
abundances of several chemical elements, starting from
the matter reprocessed by the stars and restored into the
ISM by stellar winds and type II and Ia supernova explosions.
The main features of the model are:
\begin{itemize}
\item
one zone with instantaneous and complete mixing of gas inside
this zone;
\item
no instantaneous recycling approximation, i.e. the stellar
lifetimes are taken into account;
\item
the evolution of several chemical elements (H, D, He, C, N, O,
Mg, Si, S, Ca, Fe, Ba and Eu) is followed in detail;
\end{itemize}
In the scenario adopted in the previous works, the dSph galaxies
form through
a continuous and fast infall of pristine gas until a mass of
$\sim 10^8 M_{\odot}$ is accumulated. One crucial
feature in the evolution of these galaxies is the occurrence
of galactic winds, which develop when the thermal
energy of the gas equates its binding energy (Matteucci $\&$
Tornamb\'e 1987). This quantity is strongly influenced by
assumptions concerning the presence and distribution
of dark matter (Matteucci 1992). A diffuse ($R_e/R_d$=0.1,
where $R_e$ is the effective radius of the galaxy and $R_d$ is
the radius of the dark matter core) but massive
($M_{dark}/M_{Lum}=10$) dark halo has been assumed for each galaxy.
\subsection{Theoretical prescriptions}
The evolution in time of the fractional mass of the element $i$
in the gas within a galaxy, $G_{i}$, is described by the basic
equation:
\begin{equation}
\dot{G_{i}}=-\psi(t)X_{i}(t) + R_{i}(t) + (\dot{G_{i}})_{inf} -
(\dot{G_{i}})_{out}
\end{equation}
where $G_{i}(t)=M_{g}(t)X_{i}(t)/M_{tot}$ is the gas mass in
the form of an element $i$ normalized to a total fixed mass
$M_{tot}$ and $G(t)= M_{g}(t)/M_{tot}$ is the total fractional
mass of gas present in the galaxy at the time t. The quantity
$X_{i}(t)=G_{i}(t)/G(t)$ represents the abundance by mass
of an element $i$, with the summation over all elements
in the gas mixture being equal to unity. The star formation rate (SFR),
i.e. the fractional amount of gas turning into stars per unit
time, is given by $\psi(t)$, while the returned fraction
of matter in the form of an element $i$ that the stars
eject into the ISM through stellar winds and supernova
explosions is represented by $R_{i}(t)$. This term contains
all the prescriptions
concerning the stellar yields and the supernova progenitor
models. The infall of external gas and
the galactic winds are accounted for by the two terms
$(\dot{G_{i}})_{inf}$ and $(\dot{G_{i}})_{out}$, respectively.
The prescription adopted for the star formation history is the
main feature which characterizes the dSph galaxy models.
The SFR $\psi(t)$ has a simple form and is given by:
\begin{equation}
\psi(t) = \nu G(t)
\end{equation}
where $\nu$ is the inverse of the typical time-scale
for star formation, the SF efficiency, and is expressed
in $Gyr^{-1}$.
The star formation is not halted even after the onset of the
galactic wind but proceeds at a lower rate since a large
fraction of the gas ($\sim\, 10\%$) is carried out of the galaxy.
The details of the star formation, such as number of episodes,
time of occurrence and duration, are taken from the star
formation history of each individual galaxy as inferred by
CMDs taken from Dolphin (2002) and Hernandez, Gilmore
$\&$ Valls-Gabaud (2000). It is generally adopted 1 episode of
SF (2 in the case of Carina), with durations which
vary from 3 Gyr to 7 Gyr (see Table 1 for more details).
The rate of gas infall is defined as:
\begin{eqnarray}
(\dot G_{i})_{inf}\,=\,Ae^{-t/ \tau}
\end{eqnarray}
with A being a suitable constant and $\tau$ the infall
time-scale which is assumed to be 0.5 Gyr.
The rate of gas loss via galactic winds for each element
{\it i} is assumed to be proportional to the star formation
rate at the time {\it t}:
\begin{eqnarray}
(\dot{G_{i}})_{out}\,=\,w_{i} \, \psi(t)
\end{eqnarray}
where $w_{i}$ is a
free parameter that regulates the efficiency of the galactic
wind. The wind is assumed to be differential, i.e. some
elements, in particular the products of
SNe Ia, are lost from the galaxy more efficiently than others
(Recchi, Matteucci $\&$ D'Ercole 2001; Recchi et al. 2002).
This fact translates into slightly different values
for the $w_i$ corresponding to different elements. Here we will
always refer to the maximum value of $w_i$. It should be pointed out
that the differential aspect of the wind has only a small influence on
the abundance ratio patterns (including the [$\alpha$/Fe], [Ba/Fe] and
[Eu/Fe] ratios). If, instead, a normal wind (one where all elements are lost
with the same efficiency) is used, the results will not change significantly.
The efficiency of the wind, on the other hand, is crucial.
It is always high, but different for each dSph galaxy,
in order to account for the observational constraints. It is important
particularly to reproduce the [$\alpha$/Fe] ratio, the final total mass
and the present gas mass and to define the shape of the predicted
stellar metallicity distributions (for more details see LM03
and LM04).
The initial mass function (IMF) is usually assumed to be
constant in space and time in all the models and is
expressed by the formula:
\begin{equation}
\phi(m) = \phi_{0} m^{-(1+x)}
\end{equation}
where $\phi_{0}$ is a normalization constant.
Following LM03 we assume a Salpeter-like IMF (1955)
($x=1.35$) in the mass range $0.1-100 M_{\odot}$.
In table 1 we summarize the adopted parameters for the models
of dSph galaxies.
\subsection{Nucleosynthesis prescriptions}
The nucleosynthesis prescriptions are essentially the same
as those adopted in LM04, with few modifications. Here we use,
as in LM04, Nomoto et al.
(1997) for type Ia supernovae, but for massive stars
(M $> 10 M_{\odot}$) the yields of Woosley $\&$ Weaver (1995)
are used instead of those from Thielemann, Nomoto $\&$ Hashimoto
(1996) and Nomoto et al. (1997). This modification does not
change the results obtained in the previous papers, and it is based
on the best fit to the data of very metal-poor
stars of our Galaxy found by Fran\c cois et al. (2004). The only
difference is that the predicted [$\alpha$/Fe] ratios
exhibit values somewhat higher, but still in agreement
(actually a better one) with the data.
The type Ia SN progenitors are assumed to be white dwarfs in binary
systems according to the formalism originally developed by Greggio
\& Renzini (1983a) and Matteucci \& Greggio (1986).
The major difference in the nucleosynthesis prescriptions
is the inclusion of the yields of barium and
europium in the code, following the procedure adopted
by Cescutti et al. (2005), where details can be found.
{\bf Barium}
We assume two main different processes in the production of barium, taking
place in two different sites. The dominant s-process occurring in
low mass stars ($1 \le M/M_{\odot} \le 3$) and a low fraction of
Ba being produced in massive stars through the r-process. The yields from
LMS are taken from Busso et al. (2001). The Ba yields resulting from
the r-process are assumed to be produced in massive stars in the mass
range 10 - 30 $M_{\odot}$. Travaglio et al. (1999) already predicted
production of Ba in massive stars, but in the range
8 - 10 $M_{\odot}$. Cescutti et al. (2005), on the other hand,
showed that the production of Ba in massive stars must be extended
to higher masses in order to fully reproduce the most recent
observed trend of [Ba/Fe] in Galactic metal-poor stars (Fran\c cois et al.
2005). Following their procedure, we adopted the same yields
for the r-processed Ba, which are shown in Table 2.
\begin{table}
\begin{center}
\scriptsize
\caption[]{The stellar yields of massive stars
for barium and europium from Cescutti et al. (2005).}
\begin{tabular}{ccc}
\hline\hline
\noalign{\smallskip}
$M_{star}(M_{\odot}$) &$X_{Ba}$ &$X_{Eu}$\\
\noalign{\smallskip}
\hline
12. &$9.00*10^{-7}$ &$4.50*10^{-8}$\\
15. &$3.00*10^{-8}$ &$3.00*10^{-9}$\\
30. &$1.00*10^{-9}$ &$5.00*10^{-10}$\\
\hline\hline
\end{tabular}
\end{center}
\end{table}
{\bf Europium}
The yields of europium are adopted assuming that its production occurs
only through the r-process, which takes place in massive stars in a large
range of masses ($10 - 30 M_{\odot}$). Even though the production of
r-process elements is still matter of debate, in Cescutti et al. (2005) a
series of models adopting different nucleosynthesis prescriptions for Eu were
tested in order to reproduce the trend observed in Galactic metal
poor stars. Here we adopted the yields, shown in Table 2, of the model which
best reproduced the data, i.e. their model 1.
\section{Results}
In order to follow the evolution of Ba and Eu in local dSph galaxies and,
in this way, to test the adopted nucleosynthesis prescriptions
for these two elements we make use of the dSph chemical
evolution models from LM03 and LM04 following the procedure
and results described there.
In LM03 and LM04, we were able to fit the
observed [$\alpha$/Fe] ratios and the estimated final total mass and
present day gas mass of six local dSph galaxies by varying the most
important parameters, such as the SF efficiency and the
galactic wind efficiency. The observational constraints were very
well reproduced by models adopting very low SF efficiencies
($\nu$ = 0.005 - 0.5 Gyr $^{-1}$) and high wind efficiencies
(w$_i$ = 6-13). These two parameters together are the main responsibles
for the shape of the observed abundance ratios in dSphs. At low
metallicities, at the beginning of the evolution of the system,
the [$\alpha$/Fe] ratios are relatively high ($\sim
0.4$ dex) due to the almost instantaneous injection of $\alpha$-elements
in the ISM
by massive stars which die in the form of SNe II.
As the metallicity increases, the [$\alpha$/Fe] values start decreasing
slowly, and soon after the first explosions of the SNe Ia they
go down fast to sub-solar values. This intense decrease after the first
SNe Ia explosions is caused by the injection of Fe in the ISM by these
explosions and by the occurrence of the galactic wind triggered
by them. Since the galactic winds are very intense, with high
efficiencies, they remove a large fraction of the gas reservoir
which feeds the SF and, consequently, the SFR drops to very low values.
With an almost negligible SF, the injection of $\alpha$- elements
in the ISM is almost halted, whereas
the enrichment of Fe proceeds
for a very long time (a few Gyr) due to the long lifetime of
the stars responsible for its production and injection into the ISM.
Consequently, soon after the development of the wind, the
predicted [$\alpha$/Fe] ratios suffer an abrupt decrease,
as it is observed. The final mass and present day gas mass
are also controlled by the SF and wind efficiencies: the higher the
wind efficiency, the larger the mass of gas lost from the galaxy and the lower
the HI gas/total mass ratio. This scenario represents very well
the six Local Group dSph galaxies analysed and is able to reproduce the
observational data.
Here we adopt the same scenario as described above and the same range of
values for the main parameters of the models of LM04 (see
table 1 for more details). By means of these models we follow the
evolution of Ba and Eu
and test the adopted nucleosynthesis prescriptions for these two
elements. The predictions of the models are compared with [Ba/Fe],
[Eu/Fe] and [Ba/Eu] as functions of [Fe/H].
\subsection{Europium}
\begin{figure}
\centering
\epsfig{file=lan05_fig1.eps,height=8cm,width=8cm}
\caption[]{[Eu/Fe] vs. [Fe/H] observed in Carina dSph
galaxy compared to the predictions of the chemical evolution
model for Carina. The solid line represents the best model
($\nu = 0.1\;Gyr^{-1}$, w$_i$ = 7) and the dotted
lines the lower ($\nu = 0.02\;Gyr^{-1}$) and upper
($\nu = 0.4\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
The [Eu/Fe] ratio as a function of [Fe/H] observed in the four
Local Group dSph galaxies is compared with the model predictions
in the Figures 1 to 4 (Carina, Draco, Sculptor and Ursa Minor,
respectively). The predicted behaviour seen in the plots is the same for
all galaxies: [Eu/Fe] is almost constant with supra-solar values
($\sim 0.5$ dex) until [Fe/H] $\sim -1.7$ dex (depending on the
galaxy). Above this metallicity, the [Eu/Fe] values start decreasing
fast in Sculptor and Carina (there are no points at these metallicities
for Draco and Ursa Minor) similar to what is observed in the
case of the [$\alpha$/Fe] ratio. This behaviour is consistent with the production of
Eu by r-process taking place in massive stars with $M > 10 M_{\odot}$.
Stars in this mass range have short lifetimes
and enrich
the ISM at early stages of galactic evolution
giving rise to high values of [Eu/Fe], since the production of Fe
in these stars is lower than in type Ia SNe occurring later.
When the SNe Ia begin to occur,
the Fe abundance increases and, consequently, the [Eu/Fe] ratio
decreases, as one can see also in the data.
\begin{figure}
\centering
\epsfig{file=lan05_fig2.eps,height=8cm,width=8cm}
\caption[]{[Eu/Fe] vs. [Fe/H] observed in Draco dSph
galaxy compared to the predictions of the chemical evolution
model for Draco. The solid line represents the best model
($\nu = 0.03\;Gyr^{-1}$, w$_i$ = 6) and the dotted
lines the lower ($\nu = 0.005\;Gyr^{-1}$) and upper
($\nu = 0.1\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
\begin{figure}
\centering
\epsfig{file=lan05_fig3.eps,height=8cm,width=8cm}
\caption[]{[Eu/Fe] vs. [Fe/H] observed in Sculptor dSph
galaxy compared to the predictions of the chemical evolution
model for Sculptor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 13) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
The predicted [Eu/Fe] ratios in all four dSph galaxies well reproduce
the observed trend: an almost constant value
at low metallicities, and an abrupt decrease starting at [Fe/H]
$>$ -1.7 dex. In the model this decrease is caused not only by
the nucleosynthesis prescriptions and stellar lifetimes, but also
by the effect of a very intense galactic wind on the star
formation rate and, consequently,
on the production of the elements involved.
In fact, since the wind is very efficient, a large fraction of the gas reservoir
is swept from the galaxy. At this point, the SF is almost halted
and the production of Eu goes down to negligible values. The injection
of Fe in the ISM, on the other hand, continues due to the large
lifetimes of the stars responsible for its production. The main
result is an abrupt decrease in the [Eu/Fe] ratios, larger than the
one that one would expect only from the nucleosynthetic point of view if there
was no such intense wind. The abrupt decrease follows the
trend of the data very well, especially in the case of Sculptor and
Carina. For these two galaxies there are stars observed with
metallicities higher than the one corresponding to the time when the wind
develops ([Fe/H] $>$ -1.7 dex), and which are characterized by lower
values of
[Eu/Fe], in agreement with our predictions. The observed
stars of the other two dSph galaxies, Draco and Ursa Minor, exhibit
[Fe/H] values which place them before the occurrence of SNe Ia,
so it is not possible to verify if the abrupt decrease in the
[Eu/Fe] occurs also in these objects. Only observations of more
stars will confirm the trend. It should be said again that
the same phenomena explain very well the [$\alpha$/Fe] ratios
and the final total mass and present day gas mass observed in these
galaxies (LM03, LM04).
\begin{figure}
\centering
\epsfig{file=lan05_fig4.eps,height=8cm,width=8cm}
\caption[]{[Eu/Fe] vs. [Fe/H] observed in Ursa Minor dSph
galaxy compared to the predictions of the chemical evolution
model for Ursa Minor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 10) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
The small differences in the SF and wind efficiencies do not affect
strongly the
predictions of the models. As one can see in Table 1, the range of
values for the SF efficiency is practically the same for Carina, Sculptor
and Ursa Minor ($\nu$ = 0.02-0.4, 0.05-0.5, 0.05-0.5 $Gyr^{-1}$,
respectively), whereas Draco observational constraints are reproduced by
a model with lower values of $\nu$, $\nu$ = 0.005-0.1 $Gyr^{-1}$.
These values reflect in very similar curves for the first three
galaxies and a curve for Draco with only a small difference,
namely a [Eu/Fe] ratio which starts decreasing very slowly at metallicities
lower ([Fe/H] $\sim$ -2.0 dex) than in the other
three galaxies. However, the abrupt decrease starts at a similar
point. The same similarity can be seen in the values of the wind efficiency:
Carina - w$_i$ = 7-11, Draco - w$_i$ = 6-10, Sculptor - w$_i$ = 11-15 -
and Ursa Minor - w$_i$ = 8-12. Only Sculptor is characterized by
a wind efficiency a little bit higher, but this fact does not influence
the pattern of the abundances significantly. They all exhibit an
intense decrease in the [Eu/Fe] ratio after the wind develops. The small
differences in the ranges of values for w$_i$ are related more directly
to the gas mass and total mass observed.
What should be highlighted is that the nucleosynthesis prescriptions
adopted here allow the models to reproduce very well the data,
supporting the
assumption that Eu, also in dSph galaxies, is a pure r-process
element synthesized in massive stars in the range $M = 10-30
M_{\odot}$, as it is in the Milky Way (see Cescutti et al. 2005
for a more detailed discussion). Besides that, the low SF efficiencies
and the high wind efficiencies are required also to explain
the [Eu/Fe] observed pattern, especially the abrupt decrease of
the data in some dSph galaxies.
\subsection{Barium}
\begin{figure}
\centering
\epsfig{file=lan05_fig5.eps,height=8cm,width=8cm}
\caption[]{[Ba/Fe] vs. [Fe/H] observed in Carina dSph
galaxy compared to the predictions of the chemical evolution
model for Carina. The solid line represents the best model
($\nu = 0.1\;Gyr^{-1}$, w$_i$ = 7) and the dotted
lines the lower ($\nu = 0.02\;Gyr^{-1}$) and upper
($\nu = 0.4\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
The evolution of [Ba/Fe] as a function of [Fe/H] predicted by the models
and compared to the observed data in four
Local Group dSph galaxies is shown in the Figures 5 to 8
(Carina, Draco, Sculptor and Ursa Minor,
respectively). One can easily notice that the predicted curves
exhibit a similar behaviour in all four galaxies: the predicted [Ba/Fe]
ratio increases fast at very low metallicities ([Fe/H] $<$ -3.5 dex),
then remains almost constant, close to the solar value, at low-intermediate
metallicities (-3.5$<$ [Fe/H] $<$ -1.7 dex) and then starts decreasing soon
after the occurrence of the galactic wind at relatively high
metallicities ([Fe/H] $>$ -1.7 dex).
In this case, the decrease is not so intense as it is in the
case of [Eu/Fe], due to the differences in the nucleosynthesis
of Ba and Eu.
\begin{figure}
\centering
\epsfig{file=lan05_fig6.eps,height=8cm,width=8cm}
\caption[]{[Ba/Fe] vs. [Fe/H] observed in Draco dSph
galaxy compared to the predictions of the chemical evolution
model for Draco. The solid line represents the best model
($\nu = 0.03\;Gyr^{-1}$, w$_i$ = 6) and the dotted
lines the lower ($\nu = 0.005\;Gyr^{-1}$) and upper
($\nu = 0.1\;Gyr^{-1}$) limits for the SF efficiency. The thin line
represents the best model without Ba production in massive stars.}
\end{figure}
The predicted shape of the [Ba/Fe] vs. [Fe/H] relation in dSphs
can be associated
to the two different Ba
contributions, from stars in different mass ranges (high masses -
10 to 30 $M_{\odot}$ - and low masses - 1 to 3 $M_{\odot}$).
In the low metallicity
portion of the plot
the production of Ba is dominated by the r-process taking place
in massive stars which have lifetimes in the range from 6 to 25 Myr.
Therefore, the [Ba/Fe] ratio increases fast
reaching values above solar already at [Fe/H] $\sim$ -3.5 dex and
stays almost constant up to [Fe/H]=-1.7 dex.
It is worth noting that the massive star
contribution is more clearly seen when the SF efficiency is low.
In this regime, in fact, the stars are formed slowly and the
difference between the contribution of stars of different
masses is more evident, since the increase of the metallicity and
the evolution of the galaxy proceed at a low speed. On the other hand,
when the SF
efficiency is higher (like in the Milky Way), the early contribution of
massive stars is
more difficult to distinguish, because of the much
faster increase in metallicity.
At low-intermediate metallicities (-3.5 $<$ [Fe/H] $<$ -1.7 dex),
the production of Ba is still the one by r-process taking place
in massive stars, in particular in those with masses around the lower
limit for
the r-process Ba producers ($\sim 10M_{\odot}$).
The contribution to s-process Ba enrichment from LMS
(lifetimes from 3.8 x $10^{8}$ years
to 10 Gyrs)
affects significantly the predicted
[Ba/Fe] ratio only after the onset of the wind, consequently
only after the occurrence of the first SNe Ia. At this stage, the
[Ba/Fe] starts to decrease rapidly, since the first SNe Ia are
injecting large amounts of Fe into the ISM. Together with the
enrichment of Fe, the SNe Ia release also large quantities of energy
in the ISM which gives rise to a galactic wind. As the galactic
wind starts, the SFR goes down to very low values and the production
of Ba is limited only to the LMS, especially those at the low mass
end. The injection of Ba in the ISM at this stage is, however,
not so effective due to the galactic wind which removes
a large fraction of the material freshly released in the hot
medium (Ferrara $\&$ Tolstoy 1999, Recchi et al. 2001, 2004).
The effect of the Ba production in LMS is particularly
important to slow down the abrupt decrease
in [Ba/Fe] after the occurrence of the galactic wind. If this
production is not taken in account, the [Ba/Fe] values after the
onset of the wind would go down faster to very low values.
\begin{figure}
\centering
\epsfig{file=lan05_fig7.eps,height=8cm,width=8cm}
\caption[]{[Ba/Fe] vs. [Fe/H] observed in Sculptor dSph
galaxy compared to the predictions of the chemical evolution
model for Sculptor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 13) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
One can see in the Figures 5 to 8 that the observational
trends at high metallicities are very well reproduced by the model
predictions
supporting the assumptions made regarding the
nucleosynthesis of Ba. As already
mentioned the contribution from LMS to the enrichment
of Ba becomes important starting from intermediate to high
metallicities ([Fe/H] $>$ -1.9 dex), depending on the SF efficiency
adopted. In this
metallicity range, the data of all four galaxies are very well
reproduced, including the stars with low values of [Ba/Fe]
which should have formed soon after the onset of the
galactic wind.
On the other hand, at low metallicities ([Fe/H] $<$ -2.4 dex)
only the observational trend of Carina and Sculptor are well
fitted by the model predictions. In Ursa Minor and Draco there
are a few stars
which exhibits a very low [Ba/Fe]
($\sim$ -1.2 dex) at low [Fe/H] (Figures 6 and 8). These
points are well below the predicted curves and close to the values
of the Milky Way stars at similar metallicities, which are
reproduced by a chemical evolution model with the same
nucleosynthesis prescriptions adopted here (Cescutti et al. 2005) but with
a higher SF efficiency. In general, it seems like if the data for the solar
neighbourhood show values of [Ba/Fe]
lower than in the dSphs at the same metallicity, although this fact should
be confirmed by more data. For the $\alpha$-elements is the opposite, dSph
stars show lower [$\alpha$/Fe] ratios than Galactic stars at the same
metallicity (Shetrone \& al. 2001; Tolstoy et al. 2003).
LM03 and LM04 suggested that the difference in the behaviour of
$\alpha$-elements in the Milky Way and dSphs should be ascribed to their
different SF histories. In particular, the lower [$\alpha$/Fe]
ratios in dSphs
are due to their low star formation efficiency which produces a slow
increase of the [Fe/H] with the consequence of having the Fe restored
by type Ia SNe, and
therefore a decrease of the [$\alpha$/Fe] ratios,
at lower [Fe/H] values than in the Milky Way. This effect has been
described in Matteucci (2001) and is a consequence of the time-delay
model applied to systems with different star formation histories.
Therefore, in the light of what is said above,
can we explain also the differences between
the predicted [Ba/Fe] in
dSph galaxies and in the Milky Way? Again, the SF efficiency is the major
responsible parameter for this difference. In the Milky Way
model the SF efficiency is much larger (10 - 100 times) than
the ones adopted for the dSphs of the sample analysed here.
In the low efficiency regime, the contribution from
LMS appears at lower metallicities than in the high SF regime, exactly
for the same reason discussed for the [$\alpha$/Fe] ratios. As a
consequence, we predict a longer plateau for the [Ba/Fe] ratio in dSphs
than in the solar neighbourhood and starting at lower metallicities.
This prediction should in the future be confirmed or rejected by more data
at low metallicities in dSph galaxies.
\begin{figure}
\centering
\epsfig{file=lan05_fig8.eps,height=8cm,width=8cm}
\caption[]{[Ba/Fe] vs. [Fe/H] observed in Ursa Minor dSph
galaxy compared to the predictions of the chemical evolution
model for Ursa Minor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 10) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency. The thin line
represents the best model without Ba production in massive stars.}
\end{figure}
\begin{figure}
\centering
\epsfig{file=lan05_fig9.eps,height=8cm,width=8cm}
\caption[]{[Ba/Eu] vs. [Fe/H] observed in Carina dSph
galaxy compared to the predictions of the chemical evolution
model for Carina. The solid line represents the best model
($\nu = 0.1\;Gyr^{-1}$, w$_i$ = 7) and the dotted
lines the lower ($\nu = 0.02\;Gyr^{-1}$) and upper
($\nu = 0.4\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
Since there are no observed stars at
low [Fe/H] in Carina and Sculptor while there are three stars
(one with an upper limit) with very
low [Ba/Fe] in Draco and Ursa Minor, one could argue that the Ba production
from massive stars is not necessary.
To better see the effect of the r-process Ba production from massive stars,
we computed models suppressing this contribution.
In such a case, the
predictions of the models lie below all the observed data and are not
capable of fitting the stars with low [Ba/Fe]. If, on the other hand,
besides suppressing the contribution from r-processed Ba synthesised in
massive stars one expands the production of s-processed Ba to stars
with $M = 3 - 4 M_{\odot}$, with an yield $X_{Ba}$ = 0.5 x $10^{-6}$
in this mass range,
the models predict a trend similar to that observed
(see the thin lines in Figures 6 and
8). The assumption of Ba production
by s-process in stars with $M = 3 - 4 M_{\odot}$ is justified
by the fact that this production is predicted by models of stellar
evolution (Gallino et al. 1998, Busso et al. 2001), even
though there are no such yields available in the literature.
Besides that, Travaglio et al. (1999)
suggested that the dominat production of Ba comes from stars with
$2 - 4 M_{\odot}$.
As one can see from the thin lines in Figures 6 and 8,
the increase of the [Ba/Fe] ratio occurs at
metallicities similar to those of
the stars with low [Ba/Fe]. Besides that, this
model also reproduces the high values of [Ba/Fe] at high
metallicites and the [Ba/Eu] observed (see next section). In that sense,
the observed low values of [Ba/Fe], if confirmed by more observations,
could be explained by a model with Ba produced only by s-process in
stars with masses in the range $M = 1 - 4 M_{\odot}$. The problem
is that a model with such yields overpredicts the
[Ba/Fe] at high metallicities in our Galaxy. Consequently, either
the production of Ba is not the same in the dSph Galaxies and in
the Milky Way, as already suggested by Venn et al. (2004) or
these stars are characterised by anomalous metallicities, as others
observed in dSph galaxies (see Shetrone et al. 2001; Geisler et al. 2005).
Shetrone (2004) analysed the metal poor star Draco 119 (Fulbright, Rich
$\&$ Castro 2004) which exhibits an upper limit for [Ba/Fe] $\sim$
-2.6 (an arrow in the Figures 2, 6 and 10) and suggested that this
specific star might be contaminated by
inhomogeneus mixing, since other stars observed in Draco at similar
metallicities present higher values for [Ba/Fe] ($\sim$ -1.0 dex).
These other stars,
however, seem to indicate that these low values of [Ba/Fe]
are not uncommon in Draco and, as a possible consequence, that the
production of Ba in Draco (maybe also in other dSph galaxies) is
not the same as it is in our Galaxy.
Only more
observations of stars with similar metallicities (lower than
[Fe/H] $\sim$ -2.0 dex) in dSph galaxies could solve this problem.
\begin{figure}
\centering
\epsfig{file=lan05_fig10.eps,height=8cm,width=8cm}
\caption[]{[Ba/Eu] vs. [Fe/H] observed in Draco dSph
galaxy compared to the predictions of the chemical evolution
model for Draco. The solid line represents the best model
($\nu = 0.03\;Gyr^{-1}$, w$_i$ = 6) and the dotted
lines the lower ($\nu = 0.005\;Gyr^{-1}$) and upper
($\nu = 0.1\;Gyr^{-1}$) limits for the SF efficiency. The thin line
represents the best model without Ba production in massive stars.}
\end{figure}
\subsection{The ratio [Ba/Eu]}
The comparison between the observed [Ba/Eu] as a function of [Fe/H]
and the predicted curves for the four dSph galaxies is shown
in Figures 9 to 12. The models predict a similar pattern for all four
galaxies: an almost constant sub-solar value at low metallicities
([Fe/H] $<$ -1.7 dex) and, after that, a strong increase. This
pattern is explained again by the adopted nucleosynthesis
and by the effect of the galactic wind on the SFR and, consequently,
on the production of Ba and Eu. At the early stages of evolution, the
high mass stars provide the major contribution to the enrichment of the
ISM medium. Since Ba and Eu are both produced by the r-process taking place
in massive stars,
they both are injected in the ISM when the gas metallicity is still low.
The difference
is that Eu is considered to be a pure r-process element, while the
fraction of Ba that is produced by the r-process is low and its
bulk originates instead from LMS.
This fact translates into the sub-solar pattern observed in the predicted
curves: more Eu than Ba is injected in the ISM at low metallicities, at an
almost constant rate. When the LMS start to die and the first SNe Ia
start exploding the scenario changes significantly. The LMS
inject a considerable amount of Ba into the ISM causing an increase in the
[Ba/Eu] ratio. Besides that, the energy
released by the SNe Ia contributes to the onset of the galactic wind.
Since the wind is very intense, it removes from the galaxy a large
fraction of the gas reservoir which feeds the SF. Consequently,
the SFR drops down considerably and also the production of Eu
by massive stars, because the number of new formed stars is almost
negligible. Barium, on the other hand,
continues to be produced and injected in the ISM by the LMS
(s-process). This fact induces the increase of [Ba/Eu] to be even more
intense, as one can see in the predicted curves (Figures 9 to 12).
\begin{figure}
\centering
\epsfig{file=lan05_fig11.eps,height=8cm,width=8cm}
\caption[]{[Ba/Eu] vs. [Fe/H] observed in Sculptor dSph
galaxy compared to the predictions of the chemical evolution
model for Sculptor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 13) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency.}
\end{figure}
The observed trend is very well reproduced by the predicted curves
in all four galaxies, especially in the case of Carina and Sculptor
(Figures 9 and 11, respectively).
The abundance pattern of these two galaxies not only exhibits the
"plateau" at low metallicities, but also the sudden observed increase of
[Ba/Eu] after the onset of the wind, suggesting that the adopted
nucleosynthesis prescriptions for both Ba and Eu are appropriate
and that the scenario described by the chemical evolution models
is suitable to explain the evolution of these galaxies. In the case
of Draco and Ursa Minor (Figures 10 and 12, respectively), there are
no stars with metallicities larger than [Fe/H] $\sim$ -1.7 dex, the one
characteristic for the onset of the galactic wind. Therefore, one
cannot verify if this scenario (after the occurrence of the wind)
holds also for these systems. However, the "plateau" is very well
reproduced, even though there is some dispersion in the data,
especially in the case of Ursa Minor.
\begin{figure}
\centering
\epsfig{file=lan05_fig12.eps,height=8cm,width=8cm}
\caption[]{[Ba/Eu] vs. [Fe/H] observed in Ursa Minor dSph
galaxy compared to the predictions of the chemical evolution
model for Ursa Minor. The solid line represents the best model
($\nu = 0.2\;Gyr^{-1}$, w$_i$ = 10) and the dotted
lines the lower ($\nu = 0.05\;Gyr^{-1}$) and upper
($\nu = 0.5\;Gyr^{-1}$) limits for the SF efficiency. The thin line
represents the best model without Ba production in massive stars.}
\end{figure}
It is important to stress that the predicted [Ba/Eu] reproduces
all the observed trends, and that no star, given the uncertainties
(with the exception of Draco
119, which exhibits a very uncertain value due to the limits on
Ba and Eu abundances),
lies outside the predictions, as it was the case for the two stars
with very low [Ba/Fe]. This fact suggests strongly
that the outsider stars must be examined separately.
\subsection{The Sagittarius dSph galaxy}
In this section, we present the predictions for [Ba/Fe], [Eu/Fe] and
[Ba/Eu] as functions of [Fe/H] in Sagittarius dSph galaxy. Even
though there are only two stars (Bonifacio et al. 2000)
observed with Ba and Eu, it is
interesting to compare the predictions of the models to the data and to
analyse how these ratios would behave in this dSph galaxy. As
mentioned in LM04, the Sagittarius dSph galaxy exhibits chemical
properties which distinguish this galaxy from the other Local Group dSph
galaxies. In particular, the SF efficiency (required to reproduce the
observed [$\alpha$/Fe] ratios) and the predicted metallicity
distribution of this galaxy differ a lot from the other dSph
galaxies analysed - Draco, Carina, Ursa Minor, Sextan and Sculptor -
being more similar to the values assumed for the Milky Way disc. The
required SF
efficiency is much higher ($\nu$ = 1 - 5 Gyr$^{-1}$
compared to $\nu$ = 0.01 - 0.5 Gyr$^{-1}$) and the stellar metallicity
distribution exhibits a peak at higher metallicities ([Fe/H] $\sim$ -0.6
dex) than the other dSph galaxies ([Fe/H] $\sim$ -1.6 dex) and
close to the one from the solar neighborhood. As a consequence, one
would expect also [Ba/Fe], [Eu/Fe] in Sagittarius to be different
from the patterns observed in the other four dSph and more similar to
those observed in the metal-poor stars of the
Milky Way.
In order to predict the evolution of Ba and Eu as functions of Fe, we
made use of the Sagittarius dSph model as described in LM04, without
any changes in the most important parameters, such as SF efficiency
and wind efficiency, and with the same nucleosynthesis prescriptions
adopted for the other dSphs. This procedure is justified
by the fact that no modifications were required for the LM04 models
of the other galaxies (Carina, Draco, Sculptor and Ursa Minor) to
fit the observed [Ba/Fe], [Eu/Fe] and [Ba/Eu].
\begin{figure}
\centering
\epsfig{file=lan05_fig13.eps,height=8cm,width=8cm}
\caption[]{The predicted evolution of Ba and Eu as function of [Fe/H] for
Sagittarius dSph galaxy compared with the data. The solid line represents
the best model ($\nu =
3\;Gyr^{-1}$, w$_i$ = 9) and the dotted lines the lower ($\nu =
1\;Gyr^{-1}$) and upper ($\nu = 5\;Gyr^{-1}$) limits for the SF
efficiency. }
\end{figure}
In Figure 13, the predictions for Sagittarius dSph galaxy model for
[Ba/Fe], [Eu/Fe] and [Ba/Eu] are shown in comparison with the data.
As one can clearly see, all three predicted ratios reproduce very well the
data and exhibit significant differences (in particular [Ba/Fe])
when compared to the predictions (and observations) for
the other dSph galaxies. The decrease in the [Eu/Fe] at relatively
high metallicities ([Fe/H] $\sim$ -1.7 dex) observed in the four
dSph galaxies, and attributed to the effect of the galactic wind on
the SFR, is less intense in the case of Sagittarius. Moreover, one cannot
see the high values of [Ba/Fe] at low metallicities ([Fe/H]
$<$ -3.0 dex ), which were explained as an effect of the low SF
efficiency. Also, the predicted [Ba/Eu] ratios do not show the
almost constant "plateau" observed at low and intermediate metallicities
in the other dSph galaxies. All these differences can be also
found when one compares the pattern of these ratios in dSph galaxies
with those in the metal-poor stars of the Milky Way. The differences
between the predictions of Sagittarius and the other dSph and
the similarities with the Milky Way can be attributed to the high
values of the SF efficiency adopted for Sagittarius when compared
to the other dSph galaxies. These high values, in fact, are more
similar to the values generaly adopted for the solar neighborhoud
(Chiappini, Matteucci $\&$ Gratton, 1997).
Consequently, one could suggest that the chemical evolution of
Sagittarius follows roughly that of the Milky Way disc at the solar
neighborhoud in contrast to the other dSph Galaxies,
which exhibit a much slower chemical evolution.
\section{Summary}
By means of a chemical evolution model which is able to reproduce
several observational constraints of the Local Group dSph galaxies
(such as [$\alpha$/Fe], present day gas mass, estimated final total
mass, metallicity distribution) we followed the evolution of Ba and
Eu as a function of Fe in
these galaxies in order to verify some assumptions regarding
the production of these elements. The model galaxies are
specified by the SF prescriptions, such as the number, epoch
and duration of the episodes, the SF efficiency and also by the
wind efficiency. These two last parameters are the main responsible
in defining the shape of the abundance ratio patterns and for the
depletion of the gas content of the galaxy. The SF efficiency must
be characterised by low values ($\nu$ = 0.005 - 0.5 Gyr$^{-1}$)
in order to reproduce the observed data whereas the efficiency
of the galactic winds must be high ($w_i$ = 6 - 13). The effects
of a low SF efficiency and an intense wind efficiency on the production
of Ba and Eu coupled to the nucleosynthesis prescriptions suggested
by Cescutti et al. (2005) and adopted in this work enable us to
reproduce very well the observed trends of [Ba/Fe], [Eu/Fe] and
[Ba/Eu] as function of [Fe/H]. This agreement also allowed us
to suggest some constraints on the formation and production
of Ba and Eu.
The main conclusions can be summarized as follows:
\begin{itemize}
\item
the observed [Eu/Fe] ratio is very well reproduced by the models for all
four dSph galaxies with the assumption that this element is synthesized
by r-process in massive stars in a defined range of masses
(10-30 $M_{\odot}$). The pattern of [Eu/Fe] is explained by the
different sites of production of these
elements, Fe being mainly produced in SNe Ia on long time-scales and Eu in
SNe II on short time-scales, and by the effect of the galactic winds
on the SFR. At the early stages of evolution the major contributor
to the ISM enrichment are the massive stars,
giving rise to a high [Eu/Fe] at low [Fe/H] ($<$ -2.0 dex).
When the first SNe Ia start exploding and restoring the bulk of Fe
then the [Eu/Fe] ratio starts decreasing (time-delay
model, Tinsley 1980; Greggio $\&$ Renzini 1983b; Matteucci $\&$
Greggio 1986; Matteucci 1996). With the energy released by
these explosions a
galactic wind is triggered and, since it removes a large fraction
of the gas reservoir which fuels the
SF, the SFR drops down considerably. As a consequence, the enrichment
of Eu is almost halted but the one of Fe continues for a long
time giving rise to very low values of [Eu/Fe], as observed;
\item
the predictions of the models for [Ba/Fe] reproduce very well the
observed data in all four dSph galaxies with the exception of two
stars (one in Draco and another in Ursa Minor) which exhibit very low
values of [Ba/Fe] at low metallicities ([Fe/H] $<$ -2.4 dex) and could
be anomalous stars. The
[Ba/Fe] pattern is explained
by the different contributions to the production of Ba and by the
the effect of the galactic winds on the SFR. Ba is assumed to be produced
mainly by s-process in LMS (1-3 $M_{\odot}$), but also by r-process in
massive
stars in the range 10 - 30 $M_{\odot}$. This last production is
more important at early stages of galactic evolution, at low
and intermediate metallicities, and is responsible
for the high values observed in dSphs at low [Fe/H] and to maintain the
predicted [Ba/Fe] almost constant at supra-solar values at
intermediate metallicities. The s-processed Ba produced in
LMS becomes important later, at higher metallicities
([Fe/H] $>$ -1.9 dex), when the
wind develops.
\item
the low values of [Ba/Fe] at low [Fe/H] observed in Draco
and Ursa Minor,
if confirmed by more
observations, could be explained by a model in which Ba is produced
only by s-process occuring in stars with masses in the range
1 - 4 $M_{\odot}$.
However, this production of Ba is different from
the one adopted by a Milky Way model which successfully reproduces the
observations (Cescutti et al. 2005).
Consequently, if the low values of [Ba/Fe] in dSph galaxies
are real, then the production of Ba could be not the same in the dSph
galaxies
and in the Milky Way. On the other hand, if no other star is observed
with such low values, their abundance pattern could be anomalous.
This problem could be solved only with more observations of stars
in the same metallicity range of the star with reported low [Ba/Fe]
ratio;
\item
the observed [Ba/Eu] as a function of [Fe/H] is very well reproduced
by the models with the adopted assumptions regarding the
nucleosynthesis of Ba and Eu. The different sites for the production of
these elements and the effects of the galactic winds on the SFR are
again the main responsibles for the pattern observed in [Ba/Eu]. The
sub-solar "plateau"
at low metallicities is caused by the injection into the ISM of Ba and
Eu by massive stars, in different fractions: Eu is assumed to be a pure
r-process element, whereas the fraction of r-processed Ba is low. This
gives rise to a sub-solar [Ba/Eu]. When the first SNe Ia explode and the
wind develops, there is an abrupt change and the [Ba/Eu] suffers an
intense increase, due to the injection in the ISM of Ba by LMS
and to the decrease in the SFR. With a very low SFR, the production of
Eu is almost halted, but the one of Ba continues since the LMS
have a long lifetime (from several $10^{8}$ years to several Gyrs).
Therefore, [Ba/Fe] increases considerably, as observed in the dSph stars;
\item
the nucleosynthesis prescriptions adopted in this work are the same
adopted in a chemical evolution model for our Galaxy which reproduces
very well the [Ba/Fe], [Eu/Fe] and [Ba/Eu] trends observed
in Milky Way (Cescutti et al. 2005). This agreement,
coupled to the one achieved here (with red giant stars observed
in local dSph galaxies), strongly suggests that the assumptions
regarding the formation and production of Ba and Eu
are quite reasonable;
\item
we also compared the data of two observed stars with the predicted
evolution of Ba and Eu as functions of Fe in
Sagittarius using the same model as described in LM04 and with the same
nucleosynthesis prescriptions adopted in this work. The predictions
exhibit significant differences when compared to the predictions and
observations of the other four dSph galaxies, but similarities to the
metal-poor stars of the
Milky Way. Both facts can be attributed to the much higher values of SF
efficiencies adopted for the Sagittarius galaxy when compared to the other
galaxies. This galaxy is much larger and more massive than the
other dSphs and could be characterized by a chemical evolution more
similar to the one of the solar neighborhoud in the Milky Way disc
than to the one of the other dSph galaxies;
\item
finally, we are able to explain the different behaviour of the [Ba/Fe]
and [Eu/Fe] in the dSph galaxies and in the Milky Way (here one should
place Sagittarius together with the Milky Way instead of with the
dSph galaxies). The higher predicted and observed values
(although these should be confirmed by more data) of these ratios at low
metallicities in dSph galaxies are due to the much less efficient
star formation adopted for these galaxies. In this star formation
regime, in fact, the metallicity increases more slowly and the
different contributions for the Ba enrichment of the ISM appear
at lower metallicities than in the Milky Way.
For the same reason the [$\alpha$/Fe] ratios in dSphs are lower than the
same ratios in the Milky Way at the same metallicities, as suggested
already by LM03 and LM04.
\end{itemize}
\section*{Acknowledgments}
G.A.L. acknowledges financial support from the Brazilian agency
FAPESP (proj. 04/07282-2). F.M. acknowledges financial support
from INAF Project ``Blue Compact Galaxies: primordial helium and chemical
evolution'' and from COFIN2003 from the Italian Ministry for Scientific
Reasearch (MIUR) project ``Chemical Evolution of Galaxies: interpretation
of abundances in galaxies and in high-redshift objects''.
| 2024-02-18T23:39:48.463Z | 2005-10-06T02:03:44.000Z | algebraic_stack_train_0000 | 472 | 10,563 |
|
proofpile-arXiv_065-2392 | \section{Introduction and rationale}
The X-ray spectra of radio quiet Active Galactic Nuclei (AGN) are well represented as one or more broad continuum components with superimposed spectral features due to absorption and emission by the circumnuclear material. The key problem is that the decomposition of the observed spectrum is not unique.
The important role of both the emission and the absorption features in shaping the observed spectra is clear. Significant intrinsic absorption was already measured in Ariel 5 observation of NGC 4151 (Ives et al. 1976). The first X-ray narrow absorption feature was detected by Halpern (1984) in \textit{EINSTEIN} data. Now more than 50\% of well studied Seyfert galaxies and many quasars are known to possess absorbers at various ionization degrees (e.g. Seyfert 1: Nicastro et al. 2000, Behar et al. 2003, Steenbrugge et al. 2005, Young et al. 2005; Narrow Line Seyfert 1: Leighly et al. 1996, Gallo et al. 2004; PG QSO: Pounds et al. 2003a and 2003b, Porquet et al. 2004, Piconcelli et al. 2004, 2005; BAL QSO: Gallagher et al. 2002, Grupe et al. 2003). The first X-ray emission feature - the Fe fluorescent line - was already measured in 1978 (Mushotzky et al. 1978, Cen A). Later, the Compton reflection component was identified in a composite Ginga 12 spectrum (Pounds et al. 1990), and a broad Fe line was found (Tanaka et al. 1995) consistent with the expectations of a relativistically broadened feature due to X-ray reprocessing by an accretion disk. Nowadays many emission lines are identified in Seyfert 2 spectra (e.g. Sambruna et al. 2001, Kinkhabwala et al. 2002), iron lines are seen in most Seyfert 1 and 2 galaxies as well as in many quasars, and a few broad soft X-ray lines were possibly identified (Kaastra et al. 2002, R\' o\. za\' nska et al. 2004). Relatively narrow features come from a significantly ionized medium, called the warm absorber (hereafter WA), which is outflowing with velocities from hundreds to thousands km s$^{-1}$ from the nuclear region (cf. for instance Blustin et al. 2005 for a collation and an analysis of the results concerning 23 objects). This WA is located most probably somewhere between the Broad and the Narrow Line Region.
\begin{figure}
\centering
\includegraphics[width=9cm]{chevallier_fig1.eps}
\caption{Scheme of the absorption and reflection models showing the combination of spectra which constitutes the observed spectrum.}
\label{fig-scheme}
\end{figure}
However, determining whether a given feature is due to absorption or emission is not always simple. There are two main issues which are still under discussion:
\noindent - The first topic is the question whether objects like Mkn~766 possess strong and relativistically smeared soft X-ray emission lines or the observed feature is actually due to the (dusty?) warm absorber (Branduardi-Raymont et al. 2001, Turner et al. 2003, Sako et al. 2003).
\noindent - The second topic is the question of the nature of the apparent slope change in the overall X-ray spectrum at $\sim$ 1 keV in Seyfert 1 galaxies and radio quiet quasars. If the X-ray spectrum of an object is fitted with a power law plus absorption plus (eventually) the Compton reflection component plus (eventually) the iron line and (eventually) narrow spectral features, the model usually underpredicts the observed spectrum in the soft X-ray range. An additional component -- a soft X-ray excess -- is needed (Wilkes \& Elvis 1987). The spectral shape and the nature of this component is under discussion since many years (e.g. Czerny \& Zycki 1994). This component is usually modelled either as an additional continuum component (black body or multicolor blackbody; bremsstralhlung: Barvainis 1993; thermal Comptonization: Magdziarz et al. 1998), or as a strongly ionized reflection (i.e. scattering radiation with many atomic features like emission lines and recombination continua: Ross \& Fabian 1993, and subsequent works). However, as shown by Gierli\' nski \& Done (2004), this apparent change of slope is equally well modelled as being due to absorption of an originally rather soft power law intrinsic spectrum due to the warm absorber.
Resolving these issues is essential both for understanding the behaviour of the absorbing nuclear material and for the determination of the true intrinsic spectrum and, subsequently, for understanding the process of an accretion flow onto the central black hole.
It is natural that the presence of material surrounding the central source would affect the observed spectrum, and in particular lead to both emission and absorption features. The effect should depend on the location of the medium, its ionization, clumpiness, as well as the specific line of sight to the source. Some of the plasma parameters cannot be taken as arbitrary since the thermal state of the material is determined by the physical conditions.
The aim of this paper is to consider the advantages and drawbacks of the reflection versus absorption models, by modeling the transfer of radiation in the plasma surrounding the central black hole, taking into account its natural physical limitations.
In the next section, we recall some generalities about the models and our photoionization code. In Sect. 3 we consider pure absorption models, and we show that, unless a constraint of total pressure equilibrium is imposed, they should lead to a very large variation of the absorption spectrum. In Sect. 4 we consider intermediate models (partial coverage and absorption/emission), then reflection models, and we show that the observed spectrum is mainly a function of the coverage factor and of the column density. Finally, in the last section, we discuss some physical implications of these models.
\begin{figure}
\centering
\includegraphics[width=9cm]{chevallier_fig2a.eps}
\includegraphics[width=9cm]{chevallier_fig2b.eps}
\caption{Comparison of XSTAR and TITAN constant density calculations for physical conditions close to the Gierli\' nski \& Done (2004) best fitting model to PG 1211+143. Both panels represent the primary continuum and transmitted fluxes $\nu F_\nu$ (arbitrary units) vs. energy (in eV). The spectra are given at a spectral resolution 1000 (upper panel) -- differences for absorption lines, specially UTAs, are clearly visible -- and convolved with a velocity dispersion $v/c$=0.2 corresponding to a spectral resolution 2 (lowel panel). Differences on fluxes are less than 40 \%, and less than 10\% for the width of the trough taken at half maximum.}
\label{fig-comp-GD}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=18cm]{chevallier_fig3.eps}
\caption{Influence of the energy range of the primary continuum. The thin lines correspond to a primary continuum in the range 0.1 - 20 keV, and the thick lines show the corresponding results for a primary continuum in the range 0.01 - 100 keV. Both models have the same parameters as on Fig. \ref {fig-comp-GD} (note that the ionization parameter is integrated over the whole continuum, as explained in the text; this induces a slightly larger flux level for the ``truncated" continuum). For both models, we give the temperature profile (left panel), the fractional ionization abundances of oxygen (middle panel, including OVII to OIX for the ``extended'' continuum and OVIII and OIX -- which dominates -- for the ``truncated'' continuum) and the incident and absorption spectra (right panel). The spectral resolution is 2 and fluxes are in arbitrary units.}
\label{fig-comp-bornes}
\end{figure*}
\section{Some generalities}
\label{sect:generalities}
As already mentioned, AGN are characterized by an X-ray spectrum extending into the gamma ray range up to hundred keV, or several hundreds keV. This emission is assumed to be provided by a region located close to the central black hole. Any medium surrounding this primary source is radiatively heated and photoionized and will reprocess the primary photons as soft X-rays, UV, and possibly optical and IR photons. Whatever the structure of this medium - clumpy or continuous - the observed spectrum will thus be a combination of ``reflection'' by the illuminated side (which is actually not a real reflection, as it includes atomic and Compton reprocessing), ``outward emission'' (emission by the back side), transmission of the incident primary spectrum, and possibly the primary spectrum itself. The amount of each component depends on the coverage factor of the source by the medium and on its geometry (see Fig. \ref{fig-scheme}).
If the irradiated medium has a flat and continuous structure, like an accretion disk, one will see a combination of about equal proportions of primary and reflected radiation. This is the most simple case, because if the medium is distributed quasi-spherically, all components should be observed, though one or two of them can be negligible. If its coverage factor is very close to unity, only the outward emission plus transmission will be observed. If the coverage factor is small, the primary source plus the reflection and the outward emission will be observed.
So {\it only in some cases will a single component be observed}: a pure absorption spectrum requires the source to be completely covered and a small column density, in order for the outward emission to be negligible. It is indeed difficult to imagine that the line of sight is completely covered, and that the other directions are not covered; it would mean that the absorbing medium is confined in a very small cone or filament along the line of sight. This is certainly neither the case of the WA which is observed in a large fraction of Seyfert 1s, nor the case of the Gierli\' nski \& Done (2004) model, which is assumed to account for the soft X-ray excess in all bright quasars. A pure reflection spectrum requires a peculiar disposition of the medium, as in the inhomogeneous accretion disk model proposed by Fabian et al. (2002, see the discussion below).
All components are sensitive to several parameters. The thermal and ionization structure is determined mainly by the shape of the continuum in the soft X-ray range and by the ionization parameter. It is also not often realized that the upper and lower energies of the incident continuum play an important role in determining the free-free (at high density) and the Compton heating and cooling, as well as the ionization states of the elements. The absorption spectrum depends strongly on the column density, since the cool absorbing layers are located at the back side of our medium. The metal abundances influence not only directly the line spectrum but determine also the structure of the medium. On the contrary, micro-turbulence has a strong impact only on the line intensities in emission or in absorption, but a small influence on the thermal and ionization structure, at least for large column densities where the cooling is dominated by bound-free transitions. Also the density is not important, except for the relative intensities of the forbidden and permitted emission lines (for instance those of the helium-like ions like \ion{O}{vii}), and the overall spectrum is almost the same for a gas density varying from 10$^7$ to 10$^{12}$ cm$^{-3}$. It is why we do not specify the density used in the models, and why the computed fluxes in the figures are always given in arbitrary units (they are proportional to the density).
In this study, we used our photoionization code TITAN initially designed for ionized thick media (Thomson thickness up to several tens). The code is equally suited to model thinner media like the WAs. Its advantage over the other photoionisation codes like Cloudy (Ferland et al. 1998), XSTAR (Kallman \& Krolik 1995), or ION (Netzer 1993, 1996) is that it treats the transfer of both the lines and the continuum using the powerful ALI method (Accelerated Lambda Iteration), which permits to compute very precisely line and continuum fluxes. The other photoionization codes use, at least for the lines, an integral formalism called the ``escape probability approximation". In particular the computation of the absorption and emission lines is uncoupled when using this approximation, while with ALI both the lines and the continuum are treated in a consistent way. In the context of X-ray spectra of AGN, the escape approximation can lead to errors by a factor of several units on the line intensities. More important for the present study, in the case of thick media, it leads to large errors in the temperature and ionisation structure near the back side of the medium which gives rise to the absorption features (Dumont et al. 2003, and Collin et al. 2004).
TITAN has been described in several papers (for instance Dumont et al. 2000, 2003), so we will not insist on its properties, recalling only that it solves the transfer of about 1000 lines and of the continuum, and gives as output the ionization and temperature structures, and the reflected, emitted outward, and absorbed spectra. The sophisticated line transfer treatment precludes presently to take into account as many lines as in Cloudy, XSTAR, or ION, so in particular for the WA it misses some important features (like the Unresolved Transition Array - UTA - around 750 eV, and the inner shell transitions, except the iron K lines).
The previous versions of our code used a 2-stream approximation for the transfer, which is equivalent to the ALI method with a 1-point angular quadrature (cosine=$1/\sqrt{3}$). This approximation is consistent with a semi-isotropic illumination. For the study of pure absorption spectrum corresponding to a normal illumination, it was necessary to perform first a computation with the 2-stream code to obtain the opacities and optical thicknesses of the model. Hence normal absorption was computed by the extinction of a normal incident flux for the same ionization parameter using the optical thicknesses obtained earlier (R\'o\.za\'nska et al. 2005). A recent improvement was added to TITAN: to avoid this complex operation, we now use ALI in a multidirection version allowing to take into account a normal or inclined illumination. One can then determine the emitted or reflected intensity as a function of the direction. The advantage is to obtain the emission and absorption spectra in a completely consistent way for a given opening angle and coverage factor of the medium.
Our models consist in plane-parallel slabs illuminated on one side by an incident continuum. In the majority of the following models, the incident radiation field is concentrated in a small pencil normal to the slab. The value of the pencil opening angle is not important, provided it stays much smaller than a fraction of a radian. Otherwise the absorption spectrum is mixed with emission (cf. Sect.~4).
Finally, a basic ingredient of the models is the ionization parameter $\xi$. Its definition varies among authors. We adopt $\xi = L/n_H R^2$ (in the following $\xi$ will be given always in erg cm s$^{-1}$), where $n_H$ is the
hydrogen number density at the illuminated surface, $R$ the distance between the primary continuum source and the photoionized medium, and $L$ is the luminosity of the continuum. In the following we integrate $L$ over the whole primary continuum (10 eV to 100 keV in this study), but some authors prefer to integrate the luminosity only over the 0.1-10 keV range where most of the X-ray absorption takes place, or from 0.54 to 10 keV, the
energy range relevant for oxygen absorption, or from 1 to 1000 Rydberg as XSTAR does. The integration range should thus be taken into account when comparing the results of different authors.
\section{Pure absorption models}
\label{sect:absor}
\subsection{Constant density models}
In their paper, Gierli\' nski \& Done (2004) modelled the X-ray spectrum of the highly accreting source PG 1211+143 (a Narrow Line Seyfert 1, or NLS1) by a pure absorption spectrum produced by an ionized slab of constant density located on the line of sight of a source with a steep power law continuum between 0.1 and 20 keV. The photon index $\Gamma$ is equal to 2.7, quite a high value compared to typical observed indexes in the high energy range, but representative of radio quiet quasars and NLS1 in the soft X-ray band. The slab has a column density, $N$, equal to 3.3 $10^{23}$ cm$^{-2}$, a turbulent velocity of 100 km s$^{-1}$, cosmic abundances, and $\xi = 460$ (these latter informations were kindly provided to us by C. Done). In order to get a ``quasi-continuum''
with no narrow features, Gierli\' nski \& Done assumed that the lines and the photoelectric edges are smeared by a large velocity dispersion $v$, so they convolved the absorbed spectrum with a Gaussian velocity dispersion $v/c= 0.2$ (FWHM = $2 \sqrt{2\ln{2}} \ v/c \sim 2.35 \ v/c$) corresponding to a spectral resolution $R = 1/(2.35\ v/c) \sim 2$ (definition of $R$ used throughout this study). This high velocity can be due to an accelerated outflow, or to a disk wind dominated by Keplerian motion; in the latter case it should be produced very close to the black hole, at a distance around $(c/v)^2 = 25$ gravitational radii $R_{\rm G}$ ($R_{\rm G}=GM/c^2$). Such a model fits the data without any separate soft excess. Also Sobolewska \& Done (2004) have fitted the spectrum of another Narrow Line Seyfert 1 (1H 0707-495) with an absorption model.
We have built a set of models made of slabs of constant density, illuminated by power law continua, and we have computed the pure absorption spectra, assuming that there is no corresponding emission and that the slabs cover completely the primary source of radiation ($f_\mathrm{cov} \sim 1$). We illustrate the discussion first on Fig. \ref{fig-comp-GD} by a comparison between XSTAR and TITAN to get an idea of the influence of atomic data in our code, specially the missing UTAs. Using similar physical conditions as in Gierli\' nski \& Done (2004), with $\xi=2350$ for the incident flux from 10 eV to 100 keV, both codes provide similar results. The upper panel shows the incident and transmitted spectra for a high resolution ($\sim$ 1000), and the lower panel corresponds to those spectra being convolved with a dispersion velocity $v/c$ = 0.2 (i.e, spectral resolution 2). Since TITAN code take into account the electron scattering while this effect is neglected in XSTAR, the TITAN spectrum for the same incident flux is systematically below the XSTAR spectrum by a factor given by the Thomson optical depth of the medium, and the incident and transmitted fluxes at 200 eV are not equal in TITAN but equal in XSTAR. Therefore, for a better comparison of the effect of absorption on the spectra, we adjusted
the XSTAR spectrum to TITAN spectrum at 200 eV for the continuum. For these convolved spectra, the differences are weak (less than 40\%), and could be explained notably by the different approaches in the treatment of radiative transfer and energy balance. Also the influence of the atomic data, clearly visible at spectral resolution 1000 specially for the UTAs region around 900 eV, is weak for such a smearing. This comparison illustrates how much these models are sensitive to the computational methods, independently of the parameters (see also the comparison between several codes in Pequignot et al. 2001). We note however that for both models we see only one trough, located at around 2 keV, the absorption below 400 eV and above 8 keV are the same, and the difference for the width of this trough (taken at half maximum) is less than 10\%. UTAs included in XSTAR are the reason of a second trough around 800 eV, too weak to induce a difference in the absorption profile. \textit{Both codes give similar results for the constant density case, and our TITAN code can be used safely to explain WA features in this study in spite of its weaker number of lines treated}.
The influence of the physical parameters is illustrated on the following figures, which display a few examples of pure absorption spectra. In order to see easily the influence of the absorption on the soft X-ray excess, we have computed all models with simple power law incident continua ($F_{\nu} \propto \nu^{-\alpha}$).
\begin{figure*}
\centering
\includegraphics[width=18cm]{chevallier_fig4.eps}
\caption{Incident and absorption spectra for various constant density models, displayed with a spectral resolution of 2 for different models. The top panels correspond to a primary continuum with $\alpha=1$, and the panels on the bottom to continua with $\alpha$ larger than unity. The abundances are cosmic and the primary continua extend from 0.01 to 100 keV except when it is explicitely mentioned. Fluxes are in arbitrary units.}
\label{fig-divers-spe1}
\end{figure*}
Figure \ref{fig-comp-bornes} illustrates the influence of the primary continuum energy range. The thin lines correspond to a primary continuum from 0.1 to 20 keV (actually the Gierli\' nski-Done model), and the thick lines show the corresponding results for a primary continuum covering the 0.01 - 100 keV range. Both models have the same density, same column density, same metal abundances, same microturbulent velocity, and same ionization parameter (integrated over the whole continuum, which explains why the flux level is slightly larger for the ``truncated" continuum). The panel on the left shows the temperature profile: it is much higher for the ``truncated" continuum. Indeed with the ``extended" continuum the model includes more species able to cool the medium, as it is seen in the middle panel, which displays the fractional ionization abundances of oxygen for both models. For the ``truncated" continuum oxygen is mainly in the form of \ion{O}{IX}, while \ion{O}{VII} and \ion{O}{VIII} are predominant for the ``extended" continuum. As a result there is a strong imprint of the absorption in the case of the ``extended" continuum, as it can be seen on the panel on the right where the spectra are displayed with a spectral resolution of 2.
Figure \ref{fig-divers-spe1} shows a few examples of absorption spectra. They are not aimed at illustrating in detail the influence of the different physical parameters, but only at showing the variations from one spectrum to the other. All these models were calculated at a constant density. The top panels correspond to a primary continuum with $\alpha=1$, and the panels on the bottom to continua with $\alpha$ larger than unity. The abundances are cosmic except when it is mentioned. All primary continua extend from 0.01 to 100 keV except when it is mentioned (in the following we shall call ``standard" primary continuum the $\alpha=1$ power law within these limits). Among the top panels, the left one shows the influence of the ionization parameter, the middle one the influence of the column density, the right one the influence of the abundances. The bottom panels show the influence of the column density, the ionization parameter and the limits of the incident continuum. One can see that all absorption spectra have a trough around 1 keV. Its position is shifted towards slightly larger energies for higher values of the ionization parameter, as more ionized species become dominant.
\subsection{A fine tuning prescription for pure absorption models: total pressure equilibrium}
A conclusion which can be drawn from Fig. \ref{fig-divers-spe1} is that the intensity of the trough varies strongly with the values of the parameters. Indeed several spectra of this figure would be completely incompatible with any observed X-ray spectrum, because the absorption is much too large, though all parameters vary in only relatively small ranges. As a consequence, any small variation with time of the column density on the line of sight, or of the flux of the primary continuum, would induce a strong variation on the shape of the X-ray spectrum. It implies that there must be some kind of ``fine tuning" phenomenon which insures that the intensity of the trough cannot exceed some given value.
The previous models have been computed assuming, as it is generally done in this kind of problems, that the absorbing medium is one or several slabs of constant density. This is not necessarily appropriate, owing to the relatively short dynamical time scales of the absorbing gas. We will see in the last section that the pure absorption hypothesis implies a small distance between the primary source (or the black hole) and the absorber. The dynamics of the absorbing medium as a whole would thus be dominated by the gravitation of the black hole, and its dynamical time $t_\mathrm{dyn}$ would be of the order of the orbital time at the distance $R$ of the black hole:
\begin{equation}
t_{\rm dyn} \sim {R\over v_{\rm abs}} \sim {1\over \Omega} =5\times 10^4 r_{10}^{3/2} M_7 \ \ {\rm s},
\label{tq-tdyn}
\end{equation}
where $v_{\rm abs}$ is a typical velocity of the absorber (rotation, outflowing velocity, large scale turbulence...), $\Omega$ is the angular velocity, $r_{10}$ the distance to the black hole expressed in 10$R_{\rm G}$, and $M_7$ is the mass of the black hole in units of 10$^7$ M$_{\odot}$.
Note that the other important time scales (heating and cooling times, ionization and recombination times) are much smaller due to the high gas and radiation density in these central regions (for a discussion, see for instance Collin et al. 2003). So if any perturbation of the pressure equilibrium is produced, another equilibrium state will be established in less than a day for $r_{10} \sim M_7 \sim 1$. It might not correspond to a constant pressure, actually. If the dynamics of the absorbing medium is dominated by rotation, it could be close to the hydrostatic equilibrium of an accretion disk in the gravitation potential of the black hole. If it is dominated by outflowing motions it would be a dynamical equilibrium.
As a working approximation, let us assume that the medium reaches a static pressure equilibrium state determined by the total pressure, taken as the sum of the radiation pressure and the thermal gas pressure.
\begin{figure*}
\centering
\includegraphics[width=15cm]{chevallier_fig5.eps}
\caption{Temperature profiles for several constant total pressure models (thin lines) computed with the standard primary continuum and two values of the ionization parameter at the surface, $\xi=10^3$ and $\xi=10^4$. The solid lines correspond to the maximum thickness, while the dotted lines correspond to models with a smaller thickness. The figures also show the temperature profile for constant density models (thick lines) with the same values of the ionization parameter and of the surface density. }
\label{fig-pcst-ncst-compCD-T}
\end{figure*}
\begin{figure*}
\centering
\includegraphics[width=15cm]{chevallier_fig6.eps}
\caption{Incident and absorption spectra corresponding to the constant total pressure models of Fig. \ref{fig-pcst-ncst-compCD-T}, displayed with a spectral resolution of 2. The solid lines correspond to the maximum thickness and the dotted lines to a smaller thickness. The straight line is the primary continuum. Fluxes are in arbitrary units.}
\label{fig-pcst-compCD-spe1}
\end{figure*}
An important aspect of pressure - or hydrostatic - equilibrium models is that {\it the thickness of the illuminated slab cannot exceed a maximum value for a given ionization parameter} (R\' o\. za\' nska et al. 2005). In a constant density slab, the temperature and the ionization states of the various elements decrease slowly with increasing depth, owing to the absorption of the incident continuum and the reemission of a diffuse soft X-ray spectrum. When the column density is very large (say larger than 10$^{24}$ cm$^{-2}$), the back side of the slab becomes eventually cold and neutral, and the whole X-ray spectrum is absorbed, allowing us to see only optical and IR photons and, at the other extreme of the spectrum, gamma-ray photons. The object would thus be ``Compton thick", and would require a different study.
The behaviour of a slab in total pressure equilibrium is completely different, as the illumination by X-rays induces a thermal instability beyond a given layer in the slab. The phenomenon is due to the
S-shape of the temperature versus the radiation to gas pressure ratio, allowing the existence of two or even more stable phases for the same gas pressure. At a given gas pressure, which depends on the energy distribution of the specific intensity,
the gas can be in three states of thermal equilibrium, corresponding to a hot and a cold stable solution, and to an unstable intermediate solution. Sometimes there are even five states with three stable ones and two intermediate unstable ones. This was shown by Krolik et al. (1981) for an optically thin gas. In our case the phenomenon is slightly different, as the shape of the radiation spectrum and the radiation pressure itself depend on the location in the medium. In the deepest layers, close to the back surface of the slab, the radiation spectrum becomes harder, and contains only hard X-rays, and as a consequence the S-shape curve is more pronounced, inducing an important instability and producing a very strong jump in temperature when the gas adjusts to the cold solution.
The thermal instability problem was already discussed in the context of the TITAN code and of hydrostatic equilibrium by R\' o\. za\' nska et al. (2002), so we will just recall it here. It is difficult to know in which state (hot or cold) the gas can be when both are allowed. It probably depends on the previous history of the medium. Taking into account conductivity effects can help to solve the problem (R\' o\. za\' nska \& Czerny 2000), but it is a very difficult task when conductivity should be coupled with a complete transfer treatment like that of TITAN.
The present scheme of TITAN keeps the density constant in a given layer while searching for the equilibrium temperature. Such a numerical scheme produces a unique but approximate solution (intermediate between the hot and the cold solutions) in the unstable region. We are thus developing a new algorithm allowing to choose between the hot and the cold exact stable solutions. Even with the present scheme the approximate computation is possible only when the thermal instability is not too much pronounced. When the temperature drops suddenly to very low values (of the order of $10^4$ K), the radiation pressure becomes dominated by spectral lines and induces a numerical instability which corresponds to a real thermal instability. Thus the slab should necessarily be broken into cold dense clumps, possibly embedded into a warm dilute medium, and it does not exist any more as an entity. However, like in the constant density case, a cold region completely absorbing the primary source can exist beyond this instability.
A consequence of this maximum thickness of the slab is the existence of a ``maximum absorption trough", which cannot be exceeded. Smaller values of the trough can be obtained if the column density is smaller. In this case there is also a decrease of temperature at the back of the slab associated with an absorption, but the drop of temperature is smaller. The decrease occurs also in constant density models near the back surface, as the radiation can escape more easily and the medium cools. Simply it is more pronounced in constant total pressure models because as the radiation pressure decreases it induces an increase in the gas pressure and therefore in the density, so the medium cools even more rapidly. All these questions will be addressed in more detail in a future paper.
This discussion is illustrated on Fig. \ref{fig-pcst-ncst-compCD-T} which shows the temperature profile for several constant total pressure models (the thin lines). They have been computed with our standard primary continuum. The left panel corresponds to an ionization parameter at the surface $\xi=10^3$ and the right one to $\xi=10^4$. The solid lines correspond to the maximum thickness, and the dotted lines to models where the thickness has been imposed. In all cases the temperature decreases abruptly close to the back side, and has almost the same profile for the two ionization parameters. For comparison the two panels show also the temperature profiles for constant density models (the thick lines), with the same values of the ionization parameter and of the surface density. One sees that {\it the temperature is almost constant and always high when the density is constant}. We have added a constant density model with a very large column density (10$^{24}$ cm$^{-2}$), only to show that the temperature stays high for a larger thickness than the constant total pressure models. Note also that the thicker the medium, the higher the temperature, owing to the increased heating by radiation ``returning" from the back (cf. Dumont et al. 2000).
\begin{figure}
\centering
\includegraphics[width=7cm]{chevallier_fig7.eps}
\caption{ Absorption spectra for 3 constant total pressure models which have the maximum possible thickness for their given ionization parameter. The straight lines represent the primary continuum. The spectra are displayed with a spectral resolution of 2. Fluxes are in arbitrary units.}
\label{fig-pcst-comp-spe1}
\end{figure}
\begin{figure*}
\centering
\includegraphics[width=18cm]{chevallier_fig8.eps}
\caption{Fractional ionic abundances of oxygen near the back surface, as a function of the column density in the slab, for the three models of Fig. \ref{fig-pcst-comp-spe1}. The abscissae range corresponds to exactly the same total thickness of 10$^{22}$ cm$^{-2}$. One can see that the fractional abundances vary very similarly, though the range of $\xi$ is quite large, and consequently the OVII and OVIII layers are located at very different distances from the illuminated sides.}
\label{fig-pcst-deg0}
\end{figure*}
Figure \ref{fig-pcst-compCD-spe1} displays the incident and absorption spectra for the same constant total pressure models. The solid lines correspond to the maximum thickness and the dotted lines to a smaller thickness. Though the shape of the trough depends on the thickness of the slab, we see that {\it its strength cannot exceed a given value which does not seem to depend on the ionization parameter}. Considering only {\it constant pressure models with maximum thickness}, one can see that the shapes of the absorption spectra are very similar (cf. Fig. \ref{fig-pcst-comp-spe1}). In particular the absorption trough is located at the same energy, around 1 keV. Note however that the absorption {\it with respect to the primary continuum} is more important at all frequencies for a larger ionization parameter, and consequently a larger column density; this is because the overall spectrum goes down due to the loss of photons through electron scattering. Unfortunately this effect cannot be directly measurable since the primary continuum is not observed.
It is easy to understand the similarity of the absorption spectra. One can see on Fig. \ref{fig-pcst-ncst-compCD-T}
the temperature profiles of constant total pressure slabs with the maximum thickness, for $\xi$ = 10$^3$ and 10$^4$. It is really impressive how similar the temperature profiles are near the back surface. This is due to the fact that the gas enters in a multiple phase regime when the radiation spectrum has a given shape, whatever the physical state of the previous layers. This is well illustrated by Fig. \ref{fig-pcst-deg0} which shows the fractional ionic abundances of oxygen near the back surface, for the constant pressure models corresponding to the maximum thickness, for $\xi$ = 10$^2$, 10$^3$ and 10$^4$, as a function of the column density in the slab. The abscissae range corresponds exactly to the same total thickness of 2 10$^{22}$ cm$^{-2}$. One can see that the fractional abundances vary similarly, though the range of $\xi$ is quite large. This region is dominated by \ion{O}{vii} and \ion{O}{viii} ions which contribute to a large fraction of the absorption around 1 keV. The same result is obtained for the other elements, which explains why the absorption spectra are almost identical (cf. Fig. \ref{fig-pcst-compCD-spe1}). Therefore {\it the most important difference between constant density and constant total pressure models lies in the fact that the ``intermediate temperature layer" containing highly absorbing species, has the same thickness whatever its ionization parameter at the illuminated surface}.
\begin{figure}
\centering
\includegraphics[width=9cm]{chevallier_fig9.eps}
\caption{ Value of $\Gamma_{2-10 {\rm keV}}$ measured on the computed absorption spectra for constant total pressure models, as a function of the column density, for 3 values of the spectral index of the primary continuum. The curves are labelled with the value of the ionization parameter. Solid lines: $\alpha=0.8$, dashed lines: $\alpha=1$, dotted lines: $\alpha=1.3$. The squares mark the position of the maximum column density corresponding to a given ionization parameter. The small dashed lines are extrapolations which take into account that the upper value of $\Gamma$ for optically thin slabs is equal to $\alpha+1$. }
\label{fig-modeles-gammas}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=8cm]{chevallier_fig10.eps}
\caption{ Ratio of the 0.5 keV flux to the extrapolation of the $\Gamma_{2-10 {\rm keV}}$ power law measured on the computed absorption spectra versus $\Gamma_{2-10 {\rm keV}}$, for different values of the spectral index of the primary continuum $\alpha$ and different values of $\xi$. }
\label{fig-ratios-vs-gamma}
\end{figure}
\bigskip
With the computed absorption spectra, one can determine some parameters easily deduced from the observed spectra. We have chosen: 1) an ``average" photon spectral index $\Gamma_{2-10 {\rm keV}}$ measured from 2 to 10 keV and 2) the ratio of the 0.5 keV flux to the extrapolation of this power law ($Ratio$).
For different constant pressure models, Fig. \ref{fig-modeles-gammas} shows the variation of $\Gamma_{2-10 {\rm keV}}$ with the column density, for several values of the primary continuum spectral index $\alpha$ and of the ionization parameter at the illuminated surface $\xi$. Since the value of $\Gamma_{2-10 {\rm keV}}$ tends towards that of the primary continuum $\alpha\ ({\rm primary\; continuum})+1$ for very thin slabs (because the absorption spectrum is identical to the primary one), we have extrapolated the curves taking this constraint into account. Figure \ref{fig-ratios-vs-gamma} displays several curves giving $Ratio$ versus $\Gamma$: they correspond to different values of $\alpha$ and of $\xi$. It is interesting to see that the main variable is $\alpha$ and that the curves corresponding to different values of $\xi$ are almost aligned. It means that it would not be possible to find a unique fit for an observed spectrum, on the basis of these two parameters only.
Nevertheless, helped by the curves of {Fig. \ref{fig-modeles-gammas}, we have tried to fit some observed spectra with constant total pressure models.
Figure \ref {fig-comp-mod-obs} shows a comparison on PG 1307+085. The observed spectrum is the result of the ``best fit" of the EPIC data of {\it XMM} with phenomenological models including blackbodies and power laws, obtained by Piconcelli et al. (2005, see their Fig. 2). We measure $\Gamma_{2-10 {\rm keV}} = 1.5$ for this object. For the sake of clarity we divided the spectra by a power law with this value of $\Gamma=1.5$. The model is a constant total pressure slab illuminated by a power law continuum with $\alpha=0.9$, $\xi=10^4$, and $N$= 2 10$^{23}$ cm$^{-2}$. The absorption spectrum has been divided by the same power law as the observed spectrum and they are displayed with a spectral resolution of 2 (we recall that it corresponds to a dispersion velocity equal to $c/5$) and 100. This spectrum is well fitted, considering that the narrow emission feature around 0.5 keV - the OVII complex - must be provided by another emitting region. Note that this model corresponds to a thickness smaller than the maximum value allowed for its ionization parameter.
\begin{figure}
\centering
\begin{tabular}{cc}
\hspace{-0.5cm} & \includegraphics[width=8.5cm]{chevallier_fig11a.eps}
\end{tabular}
\includegraphics[width=8.3cm]{chevallier_fig11b.eps}
\caption{Comparison between the observed and computed X-ray spectra for PG 1307+085, from the sample of Piconcelli et al. (2005, see their Fig. 2). The model is a constant total pressure slab illuminated by a power law continuum with $\alpha=0.9$, $\xi=10^4$ and $N$= 2 10$^{23}$ cm$^{-2}$.
Top panel: the computed spectrum is displayed with a spectral resolution of 2. The observed and computed spectra have been both divided by a power law of photon index $\Gamma=1.5$.
Bottom panel: the computed absorption spectrum of the top panel is displayed with a spectral resolution of 2 (thick solid line), and 100 (thin solid line). The spectra have been divided by the same power law. It is extended up to 15 keV.}
\label{fig-comp-mod-obs}
\end{figure}
As we can see, it is thus difficult to obtain simultaneously a strong X-ray excess and a relatively flat 2-10~keV slope - i.e. not increasing too steeply. Moreover, a ``wiggle" appears always in the 1-12 keV spectrum, due to the decrease of absorbing species above 2 keV and to the presence of iron edges above 7 keV. Therefore, a pure power law cannot be obtained in spite of the considerable smearing. With a dispersion velocity smaller than $v/c=0.2$, several other ``wiggles" are apparent. To illustrate the effect of the large dispersion velocity, the spectrum is also displayed on Fig. \ref {fig-comp-mod-obs} with a spectral resolution of 100 (FWHM $\sim 1300$ km.s$^{-1}$), corresponding typically to the velocities in the Broad Line Region and of a classical WA. It allows to see the numerous features which are smeared by the assumed gas motions.\\
Those ``wiggles" become apparent with a resolution bigger than 5 (i.e. a dispersion velocity less than $0.08 c$)}. It is clear that {\it a smaller velocity dispersion would be unable to account for the smoothness of the observed spectra.}
\begin{figure*}[tbp]
\centering
\includegraphics[width=15cm]{chevallier_fig12.eps}
\caption{ Different combinations of the absorbed, emitted, and reflected spectra are displayed with a spectral resolution of 2, for a constant total pressure model which has the maximum possible thickness for its given ionization parameter, 2 10$^{23}$ cm$^{-2}$, and for a constant density model with a larger column density and the same ionization parameter. Thick solid line: pure absorption spectrum. Thin solid lines: absorption + outward emission spectra, assuming that the absorbing medium is a spherical envelope around the primary source. Dotted lines: reflection spectrum. Dashed lines: combination of equal proportions of primary and absorption spectra (i.e. partial coverage of the line of sight equal to 50$\%$ ). Large and small dashed lines: combination of equal proportions of primary, reflection, and absorption spectra. The straight lines represent the primary continuum. Fluxes are in arbitrary units.}
\label{fig-combinaisons-spe}
\end{figure*}
Note that a better fit could be obtained by summing up the contribution of two or more absorbing media, since one would then add two degrees of freedom for each new model ($\xi$ and the column density). The impact on the absorption spectrum depends also on the disposition of the absorbers: if they cover completely the primary source and are located at different distances from the source, the external absorber will be illuminated by an already absorbed spectrum; while if they are mixed together, they will receive the same incident spectrum. Therefore the degree of freedom would be increased, but the constant total pressure requirement would still limit the strength of the X-ray absorption.
\bigskip
Can we conclude that the pure absorption hypothesis can account for the soft X-ray excess in AGN?
We tried to fit other objects from Piconcelli et al. (2005) sample, without an agreement as good as with PG 1307+085. There are at least two reasons for that:
\begin{itemize}
\item our grid of models is restricted, and does not take into account all possible combinations of parameters;
\item two observational parameters are insufficient to determine the shape of the whole X-ray spectrum.
\end{itemize}
Our purpose in a further work is thus to extend the grid of models and to define several other observational parameters, in order to get (through $\chi^2$ tests) a unique and good solution for each object. Presently, though we can already state that - at the present uncertainty level of the observations, and in the range of energy from 0.3 to 10 keV - some spectra can be accounted for by pure absorption constant total pressure models, it is not clear at all if all spectra could be accommodated by such models.
\section {Emission, partial covering, and reflection}
\subsection{Low column density: emission and partial covering}
\label{sect:mixed}
Until now, we have considered pure absorption spectra, assuming that the emission produced by the medium surrounding the primary source is negligible. This is true only if its covering factor and/or its column density are small. A small covering factor would be in contradiction with the high occurence of soft X-ray excess, assumed here to be caused by the absorbing medium. It would thus be extremely unlikely that a large proportion of quasars and Seyfert nuclei display such an absorption, and that the absorbing medium would be confined to the line of sight of the primary source. Moreover, it is expected that the ``emission" coverage factor is about the same as that of the ``absorption" one, unless the medium has a very peculiar geometry. So the absorbing medium should surround the source with a coverage factor of the order of unity. In this case, it does not only absorb, but also emit in the UV and soft X-ray range, according to the temperature reached by the deep layers close to the back (non-illuminated) surface.
This emission can be important for relatively thick media illuminated by an intense radiation field. Even for a relatively thin model, the ``total" spectrum including the emission (computed in assuming that the absorbing medium is spherically distributed around the source with a coverage factor of unity) is slightly more intense than the pure absorption one. They would be more different in the absence of the strong smearing, since several narrow spectral features would be present in emission in the total spectrum, while they are absent in the pure absorption spectrum. With a resolution of 2, the difference between the two spectra would not be detectable, since the primary continuum itself is not observed.
Figure \ref{fig-combinaisons-spe} shows two examples of the influence of the emission, assuming that the absorbing medium is spherically distributed around the source with a coverage factor of unity. The left panel shows the spectrum of a constant total pressure model with its maximum thickness. Since this thickness is limited to a low value, the difference between the total and the pure absorption spectrum is small. The right panel shows the spectrum of a constant density medium with a large column density. Here the total spectrum differs appreciably from the pure absorption spectrum, and {\it the emission should be taken into account in the fitting procedure}. Fig. \ref{fig-speR30} displays the same spectra but with a velocity dispersion equal to $c/75$ (corresponding to a spectral resolution of 30). Now the differences between the emission and absorption spectra appear distinctly, as the emission spectrum contains several intense emission lines which are not present in the pure absorption spectrum.
Another effect not considered so far is a partial and not complete coverage factor of the absorbing medium. This possibility offers an enormous degree of freedom for the shape of the spectrum, and in particular it can erase efficiently the big absorption trough created in the pure absorption models. As an illustration Fig. \ref{fig-combinaisons-spe} shows (in dashed lines) the result of a partial coverage of 50$\%$ and one can see that the absorption is almost suppressed. It is clear that keeping the coverage factor as a free parameter would allow to fit all kinds of spectra, when added to the other important parameters, i.e. the spectral index of the primary continuum, the ionization parameter and the column density.
Finally, if the primary source is not completely obscured by the absorbing medium we have also to take into account another ingredient in the observed spectrum, namely the reflection spectrum. Since the whole covering factor is smaller than unity, one can possibly see also the side of the absorbing clouds facing the primary source. Of course, it implies that the primary source itself is not opaque, or that it is intimately mixed with the absorbing/emitting medium.
\begin{figure*}[tp]
\centering
\includegraphics[width=15cm]{chevallier_fig13.eps}
\caption{ The spectra corresponding to the same models as on Fig. \ref {fig-combinaisons-spe} but displayed with a spectral resolution of 30. Thick solid lines: pure absorption spectrum. Thin solid lines: absorption + emission spectrum, assuming that the absorbing medium is a spherical envelope around the primary source. Dotted lines: reflection spectrum. The straight lines represent the primary continuum. Fluxes are in arbitrary units.}
\label{fig-speR30}
\end{figure*}
\subsection {High column density: reflection models}
Up to now we have considered relatively thin media, whose Thomson thickness is smaller than unity.
Let us consider now thick media with a Thomson thickness of the order of or larger than unity (i.e., a column density larger than 10$^{24}$ cm$^{-2}$). If a thick medium covers completely the line of sight of the primary source, nothing except the extreme parts of this continuum (the UV and the Gamma bands) would be seen, as shown on Fig. \ref{fig-epais-ref-abs-em}. We enter here in the domain of the ``reflection" models, which were invoked to account for the presence of the soft X-ray band.
\begin{figure}
\centering
\includegraphics[width=9cm]{chevallier_fig14.eps}
\caption{ Spectrum for a constant density thick model, and the standard primary continuum displayed with a spectral resolution of 2. Thick solid lines: pure absorption spectrum. Thin solid lines: absorption + emission spectrum, assuming that the absorbing medium is a spherical envelope around the primary source. Thick dotted lines: reflection spectrum. The straight line represents the primary continuum. Fluxes are in arbitrary units.}
\label{fig-epais-ref-abs-em}
\end{figure}
\begin{figure}
\centering
\includegraphics[width=9cm]{chevallier_fig15.eps}
\caption{ A zoom on the reflection spectrum for the thick model given on Fig. \ref{fig-epais-ref-abs-em}. The solid and long dashed lines show the reflection spectra in the case of a normal and an isotropic illumination, and the dotted lines shows the ``observed spectrum", i.e. the half sum of the reflection and primary spectrum. The straight line represents the primary continuum. Fluxes are in arbitrary units.}
\label{fig-comp-iso-norm-spe}
\end{figure}
\begin{figure}[tb]
\centering
\includegraphics[width=9cm]{chevallier_fig16.eps}
\caption{ Computed spectra for thick reflection models for two values of the ionization parameter. The straight lines represent the primary continuum. Thin dotted lines: reflection spectrum. Thick solid lines: half sum of reflection plus primary spectrum. The figure illustrates the fact that the spectrum which reaches the observer (reflection plus primary) never displays a strong X-ray excess. The thick dashed line shows one of the ``observed" spectrum, after being absorbed by a constant total pressure slab of column density 10$^{22}$ cm$^{-2}$ and an ionization parameter $\xi=100$, with a dispersion velocity equal to 0.2$c$. Fluxes are in arbitrary units.}
\label{fig-x2x3-spe}
\end{figure}
In almost all the cases considered so far, the reflection spectrum was negligible, as it can be seen in the left panel on Fig. \ref{fig-combinaisons-spe}, where the reflection spectra are shown, as well as a combination of equal proportions of primary, absorption, and reflection spectra. For thick ionized media, reflection dominates completely the transmission and the outward emission. But it is not seen, unless the surrounding medium is located {\it behind or sideways} the primary source. The atmosphere of an accretion disk illuminated from above, either by a central source or by a patchy hot corona (Capriotti et al. 1981, Haardt \& Maraschi 1991, 1993, Ross \& Fabian 1993, Nayakshin et al. 2000, Nayakshin \& Kallman 2001), is considered as the best location for such a reflection. The disk atmosphere is then in hydrostatic equilibrium under the gravity of the central black hole, but it was often represented by a constant density slab (Ross \& Fabian 1993 and subsequent works), or by a slab in total pressure equilibrium (Collin et al. 2003), in both cases heated from below by the viscous release of gravitational energy.
We have run a set of constant density thick models to check whether it is possible to account for the soft X-ray excess in PG quasars by reflection. Instead of the normal illumination considered previously, we assumed an isotropic illumination, which is more realistic if the reflecting medium is located close to the primary source and has a comparable size (it was actually the type of illumination considered in all previous studies performed with the TITAN code). As an illustration, Fig. \ref{fig-comp-iso-norm-spe} shows a comparison between the reflection spectra in the case of normal and isotropic illuminations, and we see that, as expected (because the mean path of the photons in the normal direction is smaller in the case of an isotropic illumination), the reflection is more important in the isotropic case.
Figure \ref{fig-x2x3-spe} shows two examples of such models, with a small and with a large ionization parameter.
It is obvious that when the reflection spectrum is added to the primary continuum, the soft X-ray excess almost disappears, and one understands why it is so difficult, with such models, to account for the large soft excess observed in some PG quasars and in high accretors like NLS1s.
Actually we have tried without any success to fit some other observed spectra from the Piconcelli et al. (2005) sample, when the primary continuum was added to the reflection one. To summarize the problem:
\begin{itemize}
\item either the ionization parameter is small and the reflection spectrum displays a strong X-ray excess, but it is negligible compared to the primary one;
\item or the ionization parameter is large, and the reflection spectrum is comparable in flux to the primary one, but it has only a very small X-ray excess.
\end{itemize}
It is why a new disk model has been proposed by Fabian et al. (2002). They consider a disk made of a dilute hot gas creating a hard X-ray spectrum which is not seen directly, and whose radiation is reflected on sheets of dense material formed by disk instabilities. Thus the primary source is hidden and only the reflection spectrum is observed. This is presently the only available reflection model able to explain large X-ray excesses, but unfortunately it is valid only for objects accreting close to the Eddington rate, like NLS1s.
An alternative approach to hide the primary continuum could be due to the effect of ``light bending'' where a significant fraction of light emitted by the central source is bent onto the disc rather than escaping to the observer (Crummy et al. 2005).
\section{Discussion}
In summary, absorption models seem able to account for some of the soft X-ray excesses, and we do not know whether one could find good fits for the others with a better grid of models. A mixture of absorbing clouds spanning a range of optical thicknesses and ionization parameters, or covering only partially the primary source, would also help to accommodate all observed spectra. On the contrary, reflection models underpredict even relatively modest soft X-ray excesses, unless the primary continuum is hidden.
It is thus important to examine what are the implications of these models. Both absorption and reflection models require large ionization parameters and large values of the column density.
But {\it the most important condition is the smearing of the spectral features by a very large dispersion velocity}. We have indeed seen that a velocity dispersion smaller than $c/5$ cannot account for the smoothness of the observed spectrum. This large velocity implies that :
\begin{itemize}
\item either the absorbing/reflecting medium is located close to the black hole and is thus mixed with the primary source; this would be the case of a medium dominated by rotation and turbulent motions,
\item or it is located further out, but in this case it is not gravitationally bound to the black hole and it is necessarily in outflowing motion.
\end{itemize}
A likely solution for the overall X-ray emission of AGN is thus a hot medium emitting a primary hard X-ray continuum, surrounded by or embedded into a system of absorbing/reflecting/emitting photoionized clouds. In this hypothesis, the dominance of the absorption, or the emission, or the reflection spectrum, would be determined by the coverage factor of the primary source by the clouds, and by their thickness. This clumpy medium could be either a wind dominated by outflowing motions, or a thick inhomogeneous accretion flow.
\medskip
Let us first assume that the high velocity is due to an outflowing motion. Note that in this case the constant total pressure models considered so far would no more be valid, as the medium would be in a dynamical state. So possibly our constraint on the thickness due to the thermal instability would not hold, but it would be replaced by other constraints, such as those due to radiation pressure acceleration, which would also limit the column density of the absorbing matter. Second, the ``dispersion velocity" (actually a velocity gradient) would be accompanied by a blueshift of the lines, but it would not change appreciably the shape of the absorption spectrum.
The wind model is actually reminiscent of the recent discovery of several quasars showing extreme absorption properties (in particular blue-shifted highly ionized Fe lines and a sharp feature at 7 keV which is attributed to the K-shell edge), with outflowing velocities of a few tenths of the light speed, and column densities of the order of $10^{24}\ \mathrm{cm}^{-2}$ (Pounds et al. 2003a and 2003b, 2004). The ionization parameter at the surface of this medium is of the order of $10^3$, like in our models. The question is raised of whether these absorbers could be a general feature of AGN which is still undetected, as the highly ionised gas is hard to observe.
Assuming the stationary outflow to be a relatively thin shell as compared to its distance from the central black hole, one can estimate the mass outflowing rate $\dot{M}_{\rm out}$:
\begin {equation}
\dot{M}_{\rm out}\sim \Omega \ f_{\rm vol} \ m_{\rm H}\ V_{\rm out}\ n_\mathrm{H}\ R^2,
\label{eq-ouflow1}
\end{equation}
where $\Omega$ is the opening angle of the outflowing medium, $f_{\rm vol}$ is the volume filling factor (if the medium is made of small clumps), $V_{\rm out}$ the outflowing velocity, $m_{\rm H}$ the hydrogen mass. Dividing the outflowing rate by the Eddington accretion rate $\dot{M}_{\rm Edd}$, and using our definition of $\xi=L/n_\mathrm{H}R^2$, one gets:
\begin {equation}
{\dot{M}_{\rm out}\over \dot{M}_{\rm Edd}}\sim { \Omega f_{\rm vol} \ m_{\rm H}\ V_{\rm out}\ R_{\rm Edd}\ \eta\ c^2 \over \xi}
\label{eq-ouflow2}
\end{equation}
where $R_{\rm Edd}$ is the Eddington ratio $L/L_{\rm Edd}$ and $\eta$ is the mass-energy efficiency conversion factor. If we assume that the whole dispersion velocity is due to outflowing motions with an opening angle of the order of $0.5 \times 4\pi \sim 6.3$ if one admits that the number of Seyfert 1 = 50\% of Seyfert 2, we get:
\begin {equation}
{\dot{M}_{\rm out}\over \dot{M}_{\rm Edd}}\sim 2\ 10^3 \ f_{\rm vol}\ \left[{R_{\rm Edd}\over 0.3}\right] \left[{V_{\rm out}\over 0.2c}\right]\ \left[{\eta \over 0.1}\right]\ \left[{10^3\over \xi}\right].
\label{eq-ouflow3}
\end{equation}
Eq. \ref{eq-ouflow3} shows that $f_{\rm vol}$ should be smaller than $\sim 5\ 10^{-4}$, in order for the outflowing rate not to have an unrealistic value much larger than the Eddington rate.
It is interesting to deduce other physical properties of the absorbing medium. Since $n_\mathrm{H}\ f_{\rm vol}\ \Delta R$ is roughly equal to the column density $N$, where $\Delta R$ is the geometrical thickness of this outflow whose base is at distance $R$ from the central regions, and using the constraint $\Delta R / R < 1$, we obtain for the distance $R$:
\begin {equation}
R < {f_{\rm vol}\ R_{\rm Edd}\ L_{\rm Edd} \over \xi \ N},
\label{eq-ouflow4}
\end{equation}
or:
\begin {equation}
{R\over R_{\rm G}} < 10^6 \ f_{\rm vol}\ \left[{R_{\rm Edd}\over 0.3}\right] \left[{10^3\over \xi}\right] \left[{3\ 10^{23}\over N}\right].
\label{eq-ouflow5}
\end{equation}
As $f_{\rm vol}$ should be smaller than $\sim 5\ 10^{-4}$, $R/R_{\rm G} < 500$.
From our definition of $\xi$, we can also find the value of the density:
\begin {equation}
n_\mathrm{H} > 8\ 10^{11} \left[{R_{\rm Edd}\over 0.3}\right]
\left [ \frac{10^{3}}{\xi} \right ]
\left [ \frac{10^7 M_{\sun}}{M} \right ]
\left [ \frac{500 R_{\rm G}}{R} \right ] ^2\ \ {\rm cm}^{-3}.
\label{eq-ouflow10}
\end{equation}
Thus the density should be high, actually of the order of that of the atmosphere of a standard accretion disk.
On the other hand, the spread of velocity required by the absorption model implies the existence of many clouds on the line of sight, with a smooth velocity gradient up to $0.2c$ (FWHM = 60~000 km\ s$^{-1}$). The thermal velocity of hydrogen nuclei being of the order of 100 km\ s$^{-1}$ in the gas giving rise to the absorption spectrum, it means that at least 600 clouds on a line of sight are required to give a dispersion velocity equal to 0.2$c$. The ratio $ f_{\rm vol}/ f_{\rm cov}$ is of the order of $r_{\rm c} N_{\rm los}/\Delta R= N/(n_\mathrm{H}\Delta R)$, where $r_{\rm c}$ is the dimension of a cloud, and $N_{\rm los}$ the mean number of clouds on a line of sight. For $N_{\rm los}$= 600, one gets $r_{\rm c} < 6\ 10^8 \left[N\over {3\ 10^{23}}\right] \left[{8\ 10^{11}\over n_\mathrm{H}}\right]$ cm. Finally, since $f_{\rm cov} \sim 1$ (primary source completely covered), $ f_{\rm vol}/ f_{\rm cov}$ should be smaller than $5\ 10^{-4}$ and the following condition should be fullfilled:
\begin {equation}
\left [ \frac{8\ 10^{11}}{n_\mathrm{H}} \right ] \left[N\over {3\ 10^{23}}\right] \left [ \frac{10^7 M_{\sun}}{M} \right ] \left [ \frac{500 R_{\rm G}} {R}\right ] > 1.
\label{eq-ouflow11}
\end{equation}
We see that the inequality (\ref{eq-ouflow11}) is marginally realized.
Thus the model is consistent from the phenomenological point of view. It stays however to be justified on a physical ground, especially to seek for a confining mechanism of the small dense clouds (magnetic?). Note that they could be simply transient entities.
As the absorbing medium should be located inside 500 $R_{\rm G}$, its dynamics should comprise a fraction of rotational motion. It could perhaps be identified with the inner part of a radiatively driven wind launched by the disk, as proposed by Murray \& Chiang (1995, and subsequent works). However the wind does not reach
relativistic velocities in their model. The high velocities close to the black hole could also be triggered by the accretion disk releasing magnetically driven winds, like those proposed e.g. by K\"onigl \& Kartje (1994). In particular Czerny \& Goosmann (2004) showed that external X-ray heating by magnetic flares can account for vertical acceleration of disk material.
In this computation we have neglected the medium at a larger distance. The stationary assumption indeed implies the presence of matter at any distance from the center. Since the column density of this medium decreases as $R^{-2}$, the material located at, say, 1000 $R_{\rm G}$, would have a column density 100 times smaller than this value inside 100 $R_{\rm G}$ and would therefore be undetectable (or detectable as a ``classical" Warm Absorber).
Note also that we have assumed a stationary outflow, but of course it would have been equivalent to assume sporadic events, separated by the time (of the order of $R/V_{\rm out}$) that it would take for the absorbing medium to be replenished before its disappearence by dilution.
\medskip
An alternative solution to the outflow is a thick inhomogeneous accretion flow, whose dynamics is dominated by rotation and turbulent motions. But now the absorbing medium should be located at a distance of the order of 25~$R_{\rm G}$, as it should be gravitationally bound to the black hole and simultaneously have relativistic velocities.
This is actually close to the model proposed by Collin et al. (1996), consisting in a quasi-spherical (or a thick disk) distribution of clouds covering almost totally a primary source of X-rays. The difference with the present model lies in the very large column density of the Collin et al. accretion flow (Thomson opacity of the order of 10-100), which did not allows the leakage of any transmitted radiation, so partial covering was required and had to be fine tuned. The interest of this model was to account not only for the X-ray emission, but also for the UV spectrum, due to the outward emission of the optically thick cloud system. In the present model, another UV source is required, like a geometrically thin accretion disk.
\medskip
Finally, none of the models are very satisfactory from a physical point of view. Owing to its large column density, the wind implies too massive outflows. Both the wind and the accretion models require an additional UV emission, which has to be provided by a geometrically thin accretion disk. The coexistence of a spherical accretion flow and a thin disk seems quite artificial.
So we would prefer an ``hybrid model", including an accretion disk with a hot patchy corona emitting the whole X-ray spectrum, through the ``classical" Compton reflection on the disk and inverse Compton process in the corona initially proposed by Haardt \& Maraschi (1991 and 1993), the UV emission being produced by the viscous release in the disk. The emerging spectrum in the 0.1 - 10 keV range would be similar to the ``observed" spectra shown on Fig. \ref{fig-x2x3-spe} and would display a small soft X-ray excess. If such a spectrum is absorbed by a moderately thick wind, the excess would be increased and would become comparable to the observations. As an illustration, Fig. \ref{fig-x2x3-spe} shows one of the ``observed" spectra, after being absorbed by a constant total pressure slab of column density 10$^{22}$ cm$^{-2}$ with an ionization parameter $\xi=100$. Assuming that this slab has also a dispersion velocity equal to 0.2$c$, one gets a spectrum quite comparable to the observations. The big interest of this hybrid model is to require less extreme conditions for the outflowing mass, as the column density and the ionization parameter of the wind are smaller. According to Eqs. \ref{eq-ouflow3} and \ref{eq-ouflow5}, $f_{\rm vol}$, and thus $\dot{M}_{\rm out}/ \dot{M}_{\rm Edd}$, would be 300 and 30 times smaller than in the previous absorption model, respectively.
\section{Conclusion}
This study has shown that absorption models could account for some strong soft X-ray excesses, while reflection models are possibly able to account for weak soft X-ray excesses, but certainly not for the large ones (including those of typical PG quasars), unless the primary continuum is hidden from our view. An important conclusion is also that pure absorption models require a kind of ``fine tuning" of the absorber, in order to constraint the 1 keV trough, which otherwise could have any strength. We have suggested a medium in total pressure equilibrium, which leads to a maximum intensity of the trough, as well as a ``universal" shape of this maximum trough, due to the thermal instability mechanism. A complete grid of constant total pressure models, very demanding in computation time, is necessary to pursue this study.
In the absorption model, either a thick accretion flow, or a relativistic wind is required. None of them seem very realistic from a physical point of view, and moreover both models require an additional source of UV emission, like a geometrically thin accretion disk. Therefore we favor an ``hybrid" model, where the primary UV-X source could be produced by a disk-corona system, and then absorbed by a modest relativistic wind.
A clue to the problem could be obtained with observational data from Astro-E2, as this instrument should give good spectra above 10 keV. Looking at Fig. \ref{fig-comp-mod-obs}, where plotting the energy up to 15 keV reveals a well--defined slope, this might bring important limitations.
Variability is also a key point. It can help to disantangle the absorption and reflection models, as one would expect different behaviours of the light curves in the hard and soft X-ray bands: in the first case, the hard and the soft X-ray flux should vary simultaneously, while in the reflection model, there should be a time delay between the two light curves.
\begin{acknowledgements}
A.\,C. Gon\c{c}alves acknowledges support from the {\it {Funda\c{c}\~ao para a Ci\^encia e a Tecnologia}}, Portugal, under grant no. BPD/11641/2002. Part of this work was supported by the Laboratoire Europ\'een Associ\' e Astrophysique Pologne-France.
\end{acknowledgements}
| 2024-02-18T23:39:48.584Z | 2005-11-01T20:35:02.000Z | algebraic_stack_train_0000 | 485 | 11,680 |
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proofpile-arXiv_065-2515 | \section{Introduction}
In the last few years, the universality of the stellar initial mass function (IMF)
has been questioned by the results of theoretical studies, which consistently predict
that the first stars (hereafter Pop III stars) had characteristic masses of 100-600 ${M_\odot}$
(Omukai \& Nishi 1998; Abel, Bryan \& Norman 2002; Bromm, Coppi \& Larson 2002;
Omukai \& Palla 2003), about 100 times more massive than those observed today (Larson 2003).
Indeed, the physical conditions in primordial star-forming regions appear to systematically
favor the formation of very massive stars.
In particular, (i) the fragmentation scale of metal-free clouds is typically
$10^{3}\;{M_\odot}$ (Abel, Bryan \& Norman 2002; Bromm, Coppi \& Larson 2002)
(ii) because of the absence of dust grains the radiative feedback from
the forming star is not strong enough to halt further gas accretion
(Omukai \& Palla 2003).
(iii) since the accretion rate is as large as $10^{-3}$-$10^{-2}\;{M_\odot} \rm{yr}^{-1}$,
the star grows up to $\lower.5ex\hbox{$\; \buildrel > \over \sim \;$} 100\,{M_\odot}$ within its lifetime
(Stahler, Palla \& Salpeter 1986; Omukai \& Nishi 1998; Ripamonti et al. 2002).
In spite of the significant progresses made by numerical simulations
and semi-analytic models, many aspects of primordial star formation
remain to be fully understood, the more important of which are the radiative
feedback effects which operate during protostellar accretion
and which are likely to ultimately set the final stellar
mass. By modeling the structure of the accretion flow and the
evolution of the protostar, Tan \& McKee (2004) have recently shown
that radiative feedback becomes dynamically significant at
protostellar masses $\approx 30\,{M_\odot}$, and are likely to constrain the
mass of first stars in the range $100 - 300\,{M_\odot}$.
On the other hand, observations of present-day stellar populations show that stars form
according to a Salpeter IMF with a characteristic mass of 1 ${M_\odot}$, below which the IMF
flattens. Thus, unless the current picture of primordial star
formation is lacking in some fundamental physical process, a
transition in the modes of star formation must have
occurred during cosmic evolution.
The physical processes responsible for this transition,
as well as its cosmological consequences, are currently the subject of systematic
investigation. In particular, following the early study by Yoshii \& Sabano (1980), it
has been shown that a key element driving this transition is
the metallicity of the star-forming gas (Omukai 2000; Bromm et al. 2001;
Schneider et al. 2002, 2003). The fragmentation properties of the collapsing clouds change
as the mean metallicity of the gas increases above a critical threshold,
$Z_{\rm cr} = 10^{-5 \pm 1} \; {Z_\odot}$ (Schneider et al. 2002, 2003).
The fragmentation of clouds with $Z<Z_{\rm cr}$ proceeds only down to
relatively massive ($>100\;{M_\odot}$) cores,
whereas in clouds with $Z>Z_{\rm cr}$ lower mass fragments can be formed.
Within the critical metallicity range, low-mass cores can form if a sufficient amount of metals
are depleted onto solid dust grains, which provide an efficient cooling channel in the high
density regime (Schneider et al. 2003; Omukai et al. 2005).
On cosmic scales, the transition between an epoch dominated by the
formation of massive Pop III stars to an epoch dominated by the
formation of ordinary Pop II/I stars is
controlled by the strength of the so-called {\it chemical feedback}:
the explosions of the first massive supernovae starts
to metal-enrich the gas out of which subsequent stellar generations
form. In particular, in regions of the Universe where the metallicity
exceeds $Z_{\rm cr}$, low-mass stars can form. Chemical feedback
self-propagates through the metal enrichment process and therefore
depends on many poorly constrained parameters, such as the star
formation efficiency at high redshifts, the number of supernovae per
stellar mass formed, the efficiency of metal ejection and mixing. In
particular, the shape of the primordial
IMF governs the number of Pop III stars with masses
in the pair-instability supernova range (PISN, Heger \& Woosley 2002)
140 ${M_\odot}$ - 260 ${M_\odot}$ (which contribute to the metal
enrichment) relative to that of Pop III stars which end up as
massive black holes (which instead do not contribute to the metal
enrichment as all the heavy elements are captured by the black hole).
Using a semi-analytic approach, Scannapieco, Schneider \& Ferrara
(2003) have modeled the probability that a collapsing halo has been
impacted by metal-enriched outflows from neighboring
halos. Exploring different chemical feedback strengths, they concluded
that the transition between the Pop III and Pop II/I dominated
star formation epochs is essentially controlled by the spatial distribution of
metals. Because metal enrichment is generally inefficient (the filling
factor of metals is always found to be $< 1$), the transition epoch
is extended in time, with coeval Pop III and Pop II/I star formation
episodes, and Pop III stars still contributing to the global star
formation rate at redshifts $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5$. These findings have very
important implications for the detection of Pop III stars through
their strong Ly$\alpha$ emission (Scannapieco et al. 2003) as
well as the corresponding pair-instability supernova rate (Scannapieco
et al. 2005).
In this work, we investigate the cosmic transition between Pop III and
Pop II/I stars following a complementary approach. We use the code
PINOCCHIO\footnote{http://www.daut.univ.trieste.it/pinocchio.}
developed by Monaco, Theuns \& Taffoni (2002) and Taffoni,
Monaco \& Theuns (2002) to produce random catalogues of dark matter
halos at different redshifts. We then propagate metal enrichment
along the hierarchy of mergers of star-forming galaxies to predict
the number and mass distribution of metal-free objects, which can host
Pop III stars, at each redshift. We then couple these findings to the
star formation history (of Pop III and Pop II/I stars) and to the
history of cosmic reionization and quantify their interplay through
radiative feedback effects which can contribute to the extinction of Pop III stars.
A similar approach has been recently undertaken by Furlanetto \& Loeb (2005)
with the aim of assessing the plausibility of the double reionization
scenario suggested by Cen (2003) as a way to meet both the constraints
on the gas neutral fraction implied by observations of Gunn-Peterson
troughs in the spectra of distant quasars (Fan et al. 2002, 2003) and
on the gas ionized fraction derived by the high optical depth to
Thomson scattering measured by the WMAP satellite (Spergel et
al. 2003; Kogut et al. 2003).
Here we instead focus on the extinction epoch of Pop III stars which
we believe to be regulated by feedback both of radiative and chemical
types. We investigate the implications that complementary observations
on galaxy number counts at redshifts $z \approx 10$ (Bouwens et
al. 2005) and on the history of cosmic reionization (Kogut et al. 2003)
set on the transition epoch between Pop III and Pop II/I stars and on
the nature of Pop III stars as sources of UV photons contributing
to cosmic reionization.
In particular, the detection of galaxies at the highest redshifts is
challenging because of the redshifting of UV light into the infrared
and the well-known limitations of current IR instruments.
In spite of these difficulties, the search for high redshift galaxies,
based on the dropout technique, has provided the first tentative detection
of galaxies at redshifts beyond $z\approx 7$ (Bouwens et al.
2004), stimulating a re-analysis of the small amount of deep IR data.
Bouwens et al. (2005) looked at the prevalence of galaxies at $z\approx 8-12$
by applying the dropout technique to the wide variety of deep fields
that have been imaged with the Near Infrared Camera and Multi-Object
Spectrometer (NICMOS) on board of the Hubble Space Telescope.
Using an appropriate selection criterion for high-$z$
sources, Bouwens et al. (2005) concluded that the number
of $z\simeq 10$ sources must be three or fewer.
When combined with the high optical depth to Thomson scattering
inferred from the {\it Wilkinson Microwave Anisotropy Probe} (WMAP)
satellite (Kogut et al 2003), these data already provide a demanding
benchmark for theoretical models.
The paper is organized as follows. In Section~2 we describe the
different feedback effects that control the transition from a
Pop~III to a Pop~II dominated star formation epoch. We illustrate
the adopted chemical feedback model, predicting the number and
mass function of halos which can host Pop III stars at different
redshifts. In Section~3, we couple these predictions to a star formation
and reionization model.
In Section~4 we
summarize the observations of galaxy counts at redshifts $z=10$ and
discuss how these can be used to set important constraints on the model.
Finally, in Section~5 we present the main results and in Section~6 we
discuss their implications.
Throughout the paper, we adopt a $\Lambda$CDM cosmological model with
parameters $\Omega_M = 0.3$, $\Omega_{\Lambda} = 0.7$, $h=0.7$,
$\Omega_B=0.04$, $n=1$ and $\sigma_8=0.9$, which lie within the
experimental errorbars of WMAP experiment (Spergel et al. 2003).
We also use the AB magnitude
system\footnote{AB magnitudes are defined
as AB$=-2.5\log (F_{\nu_{0}}) -48.6$, where $F_{\nu_{0}}$ is the spectral energy density
within a given passband in units of erg s$^{-1}$ cm$^{-2}$ Hz$^{-1}$}.
\section{Feedback-regulated transition}
At least three types of feedback effects can be relevant to the extinction of
the first massive stars:
(i) radiative feedback caused by H$_2$ photo-dissociating radiation emitted
by the first stellar sources, which can inhibit gas cooling and star formation
inside small halos
(ii) chemical feedback due to the release of metals and dust in the first
Pop III supernova explosions, which pollute the gas out of which subsequent
generations of Pop II stars form
(iii) radiative feedback due to photo-heating filtering caused by
cosmic reionization, {\frenchspacing\it i.e. } the increased temperature of the cosmic gas
suppresses the formation of galaxies below the Jeans mass.
A thorough discussion of each of these processes is beyond the scope of this paper and
can be found in the recent extensive review by Ciardi \& Ferrara (2005).
As a first step, we build up the hierarchy of mergers of dark halos hosting
the star formation sites. We use PINOCCHIO code (Monaco, Theuns \& Taffoni 2002;
Taffoni, Monaco \& Theuns 2002) to generate random catalogues of dark matter halos
in a hierarchical universe.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig1.ps, height=8.0cm}}
\caption{\footnotesize Properties of the reference PINOCCHIO run. The
horizontal line shows the mass of the smallest resolved halo
and the line with points indicates the mass of the largest halo in
the box at each simulation output. The solid line shows the evolution of
the minimum halo mass allowed to form stars, corresponding to a virial temperature $T_{vir} = 10^4$~K.
The short- and long-dashed lines represent the
evolution of the filtering mass for two assumed reionization
histories where reionization is complete at redshifts $\approx 10$ and
$\approx 16$, respectively (see Section 2.2).}
\end{figure}
Our reference run has $256^3$ particles within a volume of comoving side 6 Mpc~$h^{-1}$
so that the smallest resolved dark matter halos have total mass $1.5 \times 10^7 {M_\odot}$
(in the merger trees only halos with at least 10 particles are considered).
Fig.~1 summarizes the details of our reference PINOCCHIO run and shows the redshift evolution of the
largest and smallest mass present in the simulation box and of the minimum halo mass that is allowed to form stars.
The latter corresponds, at each redshift, to halos with virial temperatures $T_{vir}=10^4~$K,
\begin{equation}
M_{min} = 10^8 {M_\odot} \left(\frac{1+z}{10}\right)^{-3/2}
\label{eq:min}
\end{equation}
\noindent
so that
stars do not form in halos which rely only on H$_2$ cooling
(radiative feedback).
The chosen resolution ensures that we can follow the formation
of stars in the lowest mass halos at the highest redshift.
Due to radiative feedback, the first star-forming halos have
a total mass $\approx 2 \times 10^7 {M_\odot}$ and start to form at
$z \approx 30$. This resolution is appropriate to follow
the transition from a Pop~III to a Pop~II~star formation mode around
redshift $z \approx 10$, which is the main scope of this analysis.
In Fig.~2 we compare the resulting mass function of existing halos at redshifts
30, 20, 10 and 5 in the simulation box, to the analytic predictions of
Press \& Schechter (1974) and Sheth \& Tormen (1999). The errorbars
are Poisson errors on the counts in each mass-bin.
The agreement is very good down to redshift $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5$ showing that
the chosen resolution ensures completeness in the simulation volume
in the redshift range relevant to our analysis.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig2.ps, height=8.0cm}}
\caption{\footnotesize Comparison between the predicted mass
function of halos within the simulation box (points with errorbars)
and the analytic predictions of Press \& Schechter (1974, solid lines) and Sheth \&
Tormen (1999 dashed) at redshifts 30, 20, 10 and 5 (from left to
right). Data at redshift 5 have been shifted upward by a factor 0.3
for clarity. The errorbars are Poisson errors on the counts in each
mass bin.}
\end{figure}
\subsection{Chemical feedback}
\label{sec:chem}
At each redshift, existing halos which are allowed to form stars
($T_{vir}\ge 10^4$~K) are classified as hosting Pop II (Pop III)
stars depending on whether the halo itself or any of its progenitors
have (have not) already experienced an episode of star formation, leading
to the formation of SN progenitors which then metal-enrich the gas.
Note that even a single 200 ${M_\odot}$ PISN can enrich the gas within a halo
with mass $M_{min}$ to a metallicity level well above the critical value as,
\begin{equation}
Z = \frac{0.45 M_*}{(\Omega_B/\Omega_M)M_{min}} \approx
0.02 Z_\odot \left (\frac{1+z}{10} \right)^{3/2}
\end{equation}
where, following Heger \& Woosley (2002), we have assumed that the mass of metals
released in a PISN explosion is 45\% of the initial progenitor mass.
This way of taking into account chemical feedback is very conservative as
(i) we are assuming that all metals synthesized during a star formation
episode remain
confined (or are re-accreted) within the host halo and/or
(ii) even if a fraction of metals is ejected out of the host halo,
the metals that are kept within the potential well are sufficient to pollute
the gas within the host system or within any unpolluted halo that
the system will later merge with, to a metallicity level larger than $Z_{\rm cr}$.
On the other hand, we are neglecting metal pollution from neighboring
galaxies through enriched-winds (Scannapieco et al. 2003; Sigward, Ferrara \& Scannapieco 2005).
Fig.~3 shows the fraction of halos which can host
Pop III stars (hereafter Pop III halos)
at different redshifts as a function of their mass.
The three different curves correspond to different assumptions on the
shape of the primordial IMF. In particular, we randomly classify
Pop~III halos as SN-forming or BH-forming depending on whether
they host stars with masses in the PISN range or outside this range.
SN-forming halos are metal-enriched and pollute their
descendants. BH-forming halos are not enriched in metals and therefore
do not propagate chemical feedback along their subsequent hierarchy of
mergers.
The fraction of SN-forming halos, $f_{\rm sn}$,
is therefore a free parameter that controls the strength of chemical
feedback.
The three curves shown in each panel of Fig.~3 correspond (from bottom
to top) to $f_{\rm sn}=1, 0.5, 0.1$.
Each curve represents an average over 50 realizations of the random
classification procedure and the errors are Poisson errors on the
counts in each mass bin.
When $f_{\rm sn} = 1$, all Pop III halos are SN-forming and
chemical feedback is strong. The total fraction of Pop III
halos decreases with time, being $\approx 0.25$ at $z=20$ and
$\approx 0.1$ at $z=5$; at all redshifts, Pop III stars are confined
to form in the smallest halo mass bins (close to the minimum mass to form stars shown
as the vertical dotted line in each panel). For $f_{\rm sn} = 0.5$
(50\% of Pop III halos host SN), the number of Pop III halos at each
redshift is larger than the previous case ($\approx 0.5$ at $z=20$ and
$\approx 0.2$ at $z \approx 5$) but Pop III halos are still
confined to the smallest mass bins ($ \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10^9 {M_\odot}$). Finally,
when $f_{\rm sn} = 0.1$ the strength of chemical feedback is
significantly reduced (as only 10\% of Pop III halos at each redshift
host SN) and therefore the fraction of Pop III halos is dominant
at all redshifts, being $\approx 0.8$ at $z=20$ and $\approx 0.65$ at $z=5$,
not negligible even in halos with the largest mass present
in the simulation box at each redshift. Hereafter we only consider
$f_{\rm sn} = 0.1$ and $=1$ as representative of weak and strong
chemical feedback models.
Despite the strength of chemical feedback, Pop III halos
continue to form stars even at redshifts as small as $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5$,
in agreement with the results of Scannapieco et al. (2003). Thus,
independently of whether it preferentially propagates through galaxy
mergers or through metal-enriched winds, chemical feedback always
leads to a smooth transition between a Pop III and a Pop II dominated
star formation epoch. Note that the above results depend only on the
parameter $f_{\rm sn}$ and on the hierarchy of galaxy mergers, and are
totally independent of the star formation efficiency or of the value
of critical metallicity for the transition in fragmentation scales
of collapsing pre-stellar clouds.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig3.ps, height=8.0cm}}
\caption{\footnotesize Fraction of star forming halos which,
at a given redshift, hosts Pop III stars as a function of
their total mass. The different panels correspond to
different output redshifts, $z =20, 15, 10$ and 5. In each panel,
the three curves correspond to $f_{\rm sn} = 0.1, 0.5, 1$ (from
top to bottom) and the vertical dotted line shows the mass that
at each considered redshift corresponds to the minimum halo
mass to form stars, $M = M_{min}$ (see text).}
\end{figure}
\begin{table*}
\begin{center}
\caption{\footnotesize
Properties of the three models considered: $f_{sn}$ is the
strength of chemical feedback (the fraction of Pop~III halos
which hosts SN and therefore contributes to metal enrichment),
columns 3-5 (6-8) show the initial stellar metallicity, IMF type and
Log of HI-ionizing photons per baryons in stars for Pop~III (Pop~II) stars.
By Top-Heavy IMF we mean that only stars with masses $\ge 100~{M_\odot}$
are considered.
Finally, the last two columns show the minimum and maximum values for
($f_{esc}, f_{\star}$) allowed by the observed NICMOS-UDFs counts at
redshift $z=10$ (see text).}
\begin{tabular}{ccc|c|c|ccc|c|c|ccc|c|cc|cc|}\hline
& & & Pop III & & & Pop II & & & \\ \hline
Model & $f_{sn}$ &$Z/Z_\odot$ & IMF & $N_{\gamma}$ &$Z/Z_\odot$ & IMF &
$N_{\gamma}$ & $[f_{esc},f_\star]_{min}$ & $[f_{esc},f_\star]_{max}$
\\ \hline
A & 0.1 & 0 & Top-Heavy & 4.862 & 0.2 & Salpeter & 3.602 & $[0,0.9 \times 10^{-3}]$ & $[0.5, 1.65 \times 10^{-3}]$ \\
B & 1 & 0 & Top-Heavy & 4.862 & 0.2 & Salpeter & 3.602 & $[0,1.15 \times 10^{-2}]$ & $[0.5, 2.15 \times 10^{-2}]$ \\
C & 1 & 0 & Salpeter & 4.235 & 0.2 & Salpeter & 3.602 & $[0,0.1]$ & $[0.5, 0.1]$ \\
\hline
\end{tabular}
\end{center}
\end{table*}
\subsection{Photoionization}
So far, we have taken into account radiative feedback only through the
choice of the minimum halo mass that can host stars. Namely, we have
assumed that mini-halos with virial temperatures less than $10^4~$K do
not significantly contribute to the cosmic star formation history.
However, radiative feedback does affect larger halos, particularly
when the universe is approaching the epoch of cosmic reionization.
Indeed, the increase in temperature of cosmic gas during reionization
will suppress the formation of small galaxies with mass below the Jeans mass.
To quantify this effect, we follow the method proposed by Gnedin (2000) and adopt the filtering mass as the
characteristic scale below which the gas fraction in star forming objects, hence the fraction
of stellar mass, is reduced with respect to the universal value.
In particular, before reionization, the average baryon fraction (the ratio of the mass in baryons over
the total halo mass) has the universal value $\Omega_B/\Omega_M$ while after reionization
it can be approximated by the fitting formula
(Gnedin 2000),
\begin{equation}
f_b = \frac{\Omega_B/\Omega_M}{[1+(2^{1/3}-1)M_C/M]^3},
\label{eq:fil}
\end{equation}
\noindent
where the characteristic mass, $M_C$, is the total mass of objects which, on average, retain 50\% of their
universal baryon mass, $0.5 (\Omega_B/\Omega_M) M_C$.
This characteristic mass is very well approximated by the linear-theory filtering mass, $M_C \approx M_F$,
with
\begin{equation}
M_F^{2/3} = \frac{3}{a} \int_0^a da^\prime M_J^{2/3}(T, a^\prime)[1-(\frac{a^\prime}{a})^{1/2}],
\end{equation}
\noindent
where the Jeans mass is given by
\begin{equation}
M_J(T) = 2.56 \times 10^5 {M_\odot} (\Omega_M h^2)^{-1/2}(T/\mu)^{3/2} (1+z)^{-3/2},
\end{equation}
\noindent
with $T$ and $\mu$ the temperature and molecular weight of the cosmic gas.
Here we assume the temperature of ionized regions to be $10^4$~K and
we modulate the increase in temperature of the cosmic gas due to reionization
with the volume filling factor of ionized regions at the corresponding redshift
$Q_{\rm HII}(z)$ (see Section~\ref{sec:reio}).
Thus, the characteristic mass depends on the thermal history of the gas
and it accounts for the finite time required for pressure to influence
the gas distribution. The redshift evolution of the filtering mass is
shown in Fig.~1 for a model where reionization occurs at redshift
$z_{\rm rei} \approx 10$ (short-dashed line) and $\approx 16$ (long-dashed line).
The filtering mass steeply increases with time and, as expected, its
value in the redshift range $5 \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 20$ depends strongly on
the reionization history. At redshift $z \approx 10$, the filtering mass
is $\approx 10^8 {M_\odot}$ for the late reionization case, and it is a
factor ten larger for the early reionization case.
As can be inferred from Fig.~1 and ~3, following reionization,
photoionization feedback can significantly
suppress the formation of Pop III stars since halos with masses
$10^8 {M_\odot} \le M \le 10^9 {M_\odot}$ have evaporated at least 50\% of
their mass.
\section{Reionization history}
\label{sec:reio}
Following Barkana \& Loeb (2001), we compute the redshift evolution of
the filling factor of HII regions as,
\begin{equation}
Q_{\rm HII}(z) = \int_{z}^{\infty} dz^\prime
\left\vert\frac{dt}{dz}\right\vert \frac{1}{n_H^0}
\frac{dn_{\gamma}}{dt} \mbox{e}^{F(z^\prime,z)},
\label{eq:reio}
\end{equation}
\noindent
where $n_H^0 = X_H n_B^0$ and $n_B^0$ are the present-day number densities of
hydrogen and baryons ($X_H=0.76$ is the hydrogen mass fraction), and
$dn_{\gamma}/dt$ is the production rate of ionizing photons that we
can express as,
\begin{equation}
\frac{dn_{\gamma}}{dt} = f_{esc} N_{\gamma} \dot{\rho}_\star,
\end{equation}
\noindent
where $f_{esc}$ is the escape fraction, {\frenchspacing\it i.e. } the fraction of
ionizing photons that is able to escape the host galaxy,
$N_{\gamma}$ is the time-averaged number of ionizing photons
emitted per unit stellar mass formed, and $\dot{\rho}_\star$
is the comoving star formation density,
\begin{equation}
\dot{\rho}_\star = f_\star f_b \frac{d}{dt} \int_{M_{min}}^{\infty} dM M \frac{dn}{dM}(M,z)
\end{equation}
where $f_\star$ is the star formation efficiency, $f_b$ is the baryon fraction given by
eq. (\ref{eq:fil}), $M_{min}$ is the minimum halo mass to form stars (see eq. \ref{eq:min})
and $n(M,z)$ is the number density of halos (Pop II and Pop III) at redshift $z$.
The parameter $f_{esc}$ also affects the relative strength of the stellar continuum and the
Ly$\alpha$ line and nebular luminosity of each galaxy. The latter
contributions are particularly important for massive Pop~III stars.
Finally, the function $F(z',z)$ takes into account the
effect of recombinations. Assuming a time-independent
volume-averaged clumping factor $C$, common to all HII regions,
we can write,
\begin{equation}
F(z^\prime,z) = -\frac{2}{3} \frac{\alpha_B n_H^0}{\sqrt{\Omega_M} H_0} C
[f(z^\prime)-f(z)],
\end{equation}
\noindent
and
\begin{equation}
f(z)=\sqrt{(1+z)^3 + \frac{1-\Omega_M}{\Omega_M}},
\end{equation}
\noindent
where $\alpha_B = 2.6 \times 10^{-13} \mbox{cm}^3 \mbox{s}^{-1}$ is
the hydrogen recombination rate.
For a given reionization history, it is straightforward to compute the optical
depth to electron scattering as,
\begin{equation}
\tau_e(z)=\int_0^z dz^\prime \left\vert c \frac{dt}{dz^\prime}\right\vert \sigma_T n_e(z^\prime),
\end{equation}
\noindent
where $\sigma_T = 6.65 \times 10^{-25} \mbox{cm}^{2}$ is the Thomson cross
Section and $n_e(z)$ is the number density of free electrons at redshift $z$.
In this paper we only consider the evolution of the filling factor of HII regions ($Q_{\rm HII}$).
However, for the purpose of computing the optical depth to
electron scattering, we assume that $Q_{\rm HeII} = Q_{\rm HII}$ and we neglect
the contribution of electrons coming from HeIII regions
(see also Haiman \& Holder 2003).
Under these assumptions we can write $n_e(z)=Q_{\rm HII}(z) n_B^0(1+z)^3$
where $Q_{\rm HII}(z)$ is given by eq.(6).
\section{Galaxy counts at redshift 10}
Bouwens et al. (2005) looked at the prevalence of galaxies at $z\approx 8-12$
by applying the dropout technique to the wide variety of deep
F110W- and F160W-band (hereafter $J_{110}$ and $H_{160}$, respectively) fields
that have been imaged with the Near Infrared Camera and Multi-Object
Spectrometer (NICMOS). The principal data set is constituted by two
1.3 arcmin$^2$ deep NICMOS parallels taken with the Advanced Camera
for Surveys (ACS) Hubble Ultra Deep Field (UDF; each has about $160$ orbits data).
Complementary shallower fields possessing similar $J_{110}+H_{160}$ imaging
have been analyzed as well. The 5$\sigma$ limiting magnitude for the UDFs
is $\approx 28.6$ in $J_{110}$ and $\approx 28.5$ in $H_{160}$
($0.6^{\prime\prime}$ aperture). Since these limiting magnitudes are
reached only in the deepest regions of the two NICMOS UDFs, we adopt the
conservative limiting magnitude $H_{160}=28$ (Bouwens, private communication).
The primary selection criterion for high-$z$
sources is $J_{110}-H_{160}>1.8$. Using this criterion, Bouwens et al. (2005) found
eleven sources. Eight of these are ruled out as credible $z\simeq 10$
sources, either as a result of detection ($>2\sigma$) blue-ward of $J_{110}$
or because of their colors red-ward of the break ($H_{160}-K \approx 1.5$). The nature
of the three remaining sources could not be assessed from the data, but
this number appears consistent with the expected contamination from
low-redshift interlopers. Hence, Bouwens et al. (2005) concluded that
the actual number of $z\simeq 10$ sources in the NICMOS parallel fields
must be three or fewer.
Adopting the same selection criterion as in Bouwens et
al. (2005), we compute the number of $z\approx
10$ galaxies detectable in the NICMOS UDFs predicted by our models.
Assuming that the three candidates selected by Bouwens et al. (2005)
are indeed $z\approx 10$ galaxies, we can set interesting constraints
on the models.
In particular, for a given chemical feedback strength $f_{\rm sn}$, we
compute the number density of Pop III and Pop II halos at each
redshift. We then assume a universal value for the star formation efficiency
$f_{\star}$ and escape fraction $f_{esc}$ of ionizing photons and we
characterize each stellar population (Pop III and Pop II/I stars) by
assuming a specific template emission spectrum.
In the following, we consider two models, A and B, whose details
are given in Table~1.
We assume two representative chemical feedback
strengths ($f_{\rm sn}$ = 0.1 for model A and $f_{\rm sn}$ = 1 for
model B). In both cases,
the emission properties of Pop~II stars have been computed
using a simple stellar population model taken from the GALAXEV library (Bruzual \& Charlot 2003).
Pop~II stars have an initial metallicity of $Z = 0.2\, Z_{\odot}$ and are assumed to form
according to a Salpeter IMF (with masses in the range 0.1 ${M_\odot}$ - 100 ${M_\odot}$).
Pop~III stars are assumed to be very massive
(with masses larger than 100 $M_{\odot}$) and their emission properties are computed using the
stellar spectra of Schaerer (2002), including the nebular continuum emission found to be very
important for stars with strong ionizing fluxes\footnote{Since the emission spectrum of very massive
Z=0 stars is found to be independent of the stellar mass for stars
with masses $\ge 100~{M_\odot}$ (Bromm, Kudritzki \& Loeb 2001; Schaerer
2002), we are not forced to make any assumption about the shape of the
IMF of Pop~III stars as long as these are assumed to be very
massive.}.
For each model, we then compute the number of sources per unit solid
angle with observed flux in the range $F_{\nu_0}$ and $F_{\nu_0}+dF_{\nu_0}$ as
\begin{equation}
\frac{dN}{d\Omega dF_{\nu_0}}(F_{\nu_0},z_{0})=\int^{\infty}_{z_{0}}
dz \left( \frac{dV_c}{dz d\Omega} \right) \, n(F_{\nu_0},z),
\label{eq:counts1}
\end{equation}
\noindent
where $dV/d\Omega dz$ is the comoving volume element per unit solid
angle and redshift, $n(F_{\nu_0},z)$ is the comoving number of objects at
redshift $z$ with observed flux in the range
$[F_{\nu_0}, F_{\nu_0} + dF_{\nu_0}]$, given by
\begin{equation}
n(F_{\nu_0},z)=\int^{\infty}_z dz^\prime \, \frac{dM}{dF_{\nu_0}}
(F_{\nu_0},z,z^\prime)
\, \frac{d^2 n}{dM dz^\prime}(M,z^\prime).
\label{eq:counts2}
\end{equation}
\noindent
Here $d^2 n/dMdz$ is the halo formation rate, and the factor $dM/dF_{\nu_0}$
converts the number density per unit mass in unit flux interval.
The average flux $F_{\nu_0}$ from a halo of mass $M$ at redshift $z$
that has formed at $z^\prime \ge z$ is,
\begin{equation}\label{eq:flux}
F_{\nu_0}=\frac{f_\star (\Omega_B/\Omega_M) M}{4 \pi
\Delta\nu_0\, d_L(z)^2} \int^{\nu_{max}}_{\nu_{min}}
d\nu \,\, l(\nu,t_{z,z^\prime}) \, \mbox{e}^{-\tau_{eff}(\nu_0,z_0,z)},
\label{eq:counts3}
\end{equation}
\noindent
where $l(\nu,t_{z,z^\prime})$
is the template luminosity per unit solar mass for a population of age
$t_{z,z^\prime} $ (the time elapsed between the redshift $z^\prime$ and
$z$), $d_L(z)$ is the luminosity distance, $\Delta\nu_0$ is the instrumental bandwidth,
$\nu_{min}$ and $\nu_{max}$ are the restframe frequencies corresponding to the observed ones.
Finally, $\tau_{eff}$ is the intergalactic medium (IGM) effective optical depth at $\nu_0$ between
redshift $z_0$ and $z$ (see Sect. 2.2 of Salvaterra \& Ferrara 2003)
\section{Results}
We compute the expected number of $z \approx 10$ galaxies detectable in the NICMOS UDFs
using eqs.~\ref{eq:counts1}-\ref{eq:counts3} and applying the same selection criterion of
Bouwens et al. (2005) ($J_{110}-H_{160}>1.8$, integrated up to the magnitude
limit of the survey, $H_{160}=28$). Assuming that only the three most
luminous sources should be above the detection threshold, we derive the
maximum allowed value for the star formation efficiency,
$f_\star$, corresponding to a given escape fraction of ionizing
photons.
The results obtained for models A and B are shown in Fig.~4 in terms of
comoving star formation density in Pop~III and Pop~II stars
as a function of redshift. Model C will be discussed in section \ref{sec:reio}.
In each panel, vertically (horizontally)
shaded regions represent the range of star formation
histories for Pop~III (Pop~II) stars allowed by the NICMOS-UDFs counts when
a value $0 \le f_{esc} \le 0.5$ in assumed (see Table~1 for the corresponding
values of $f_\star$).
For both models and all the explored values of
$f_{esc}$, the three most luminous objects in NICMOS UDFs are always
represented by galaxies hosting Pop~III stars, as their strong
Ly$\alpha$ and nebular emission (Schaerer 2002) is dominant. As
a consequence, $f_\star$ is restricted to relatively small values, which
increase with increasing $f_{esc}$.
Model B, being characterized by
the strongest chemical feedback, and therefore by the smallest fraction of
halos hosting Pop~III stars at redshift $\le 10$ (see also Fig.~3), has
larger allowed star formation efficiencies than model A.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig4.ps, height=8.0cm}}
\caption{\footnotesize Star formation densities as a function of
redshift in model A (upper panel), model B (middle panel) and
model C (lower panel, see
Table~1). The vertically (horizontally) shaded regions represent the
range of star formation densities in Pop~III (Pop~II) stars allowed
by the observed NICMOS-UDFs counts at redshift $z = 10$ (see text).
For all models, the IGM clumping factor is assumed to be $C=10$.}
\end{figure}
Fig.~4 shows that in model A chemical feedback is very inefficient
($f_{sn}=0.1$) and
Pop~III stars dominate the cosmic star formation history down to
redshift $z \le 5$. Conversely, model B shows that when chemical
feedback is maximally efficient, the star formation density in Pop~III
stars is sub-dominant at all but the highest redshifts, but Pop~III stars
still make a significant contribution to the cosmic star formation
history down to redshift 5.
Because of the moderate star formation efficiencies implied by
the observed NICMOS UDFs counts, radiative feedback is negligible
for the most massive galaxies. To evaluate its impact on the predicted
star formation densities shown in Fig.~4, we have assumed a clumping factor
$C=10$. For $0 \leq f_{esc} \leq 0.5$, the corresponding reionization redshifts
vary in the range $0 \leq z_{\rm rei} \leq 10$ for model A, and in the range
$0 \leq z_{\rm rei} \leq 14$ for model B.
In model A, radiative feedback effect on the overall Pop~III star formation density manifests
itself as a progressive decrement of the upper limit (which
corresponds to the largest $f_{esc}$ and $f_\star$) of the shaded
region for $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$. In model B, where the star
formation efficiencies implied by the observed counts are larger and,
at each redshift, Pop~III stars are confined to form only in the
smallest halos, radiative feedback is more significant; around redshift
10, the lines corresponding to the maximum and minimum Pop~III star formation
efficiencies cross each other (the highest the star formation
efficiency and escape fraction, the strongest the radiative feedback).
Overall, the observed galaxy counts at redshift $z \approx 10$ favor a
cosmic star formation history where the transition from Pop~III to
Pop~II stars is governed by strong chemical feedback, with all
Pop~III halos efficiently propagating metal enrichment to their
descendants along the hierarchy of galaxy mergers. In fact, in
less efficient chemical feedback scenarios ($f_{sn} = 0.1$), the predicted
cosmic star formation density is dominated by Pop III stars down to
redshift $z \approx 5$.
\subsection{Implications for reionization}
\label{sec:reio}
Stronger constraints on models A and B come from their corresponding
reionization histories. Figs.~5 and ~6 show isocontours of
redshifts of reionization as a function of the clumping factor and
the escape fraction (the star formation efficiency is set
by the observed counts at $z \approx 10$). We define the redshift of
reionization, $z_{rei}$, as the redshift when the filling factor of
ionized regions, given by eq.~(\ref{eq:reio}) is $Q_{\rm HII}(z_{\rm rei})=1$.
The shaded regions indicate the
parameter space which leads to an optical depth to electron scattering
within the range $\tau_e = 0.16 \pm 0.04$ observed by the WMAP
satellite (Spergel et al. 2003; Kogut et al. 2003), with the dashed
line corresponding to the central value.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig5.ps, height=8.0cm}}
\caption{\footnotesize Isocontours corresponding to reionization
redshifts 6, 8, 10, and 12 for model A in the plane ($C, f_{esc}$),
where $C$ is the clumping factor and $f_{esc}$ the escape fraction.
The dashed region indicates the parameter space which leads to
an optical depth to electron scattering within the range $\tau_e =
0.16 \pm 0.04$ observed by WMAP. The dashed line corresponds to the
central value.}
\end{figure}
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig6.ps, height=8.0cm}}
\caption{\footnotesize Same as Fig.~5 but for model B and reionization
redshifts 8, 10, 12, 14 and 16.}
\end{figure}
As it can be inferred from the figures, both models lead to reionization
histories which are in poor agreement with observations. Indeed,
large escape fractions and moderate clumping factors are required to
reproduce the observed range of optical depths.
Independent observations of the escape fraction give, with few exceptions, values
in the range $f_{esc} \le 0.15$ (Ciardi \& Ferrara 2005 and references
therein) whereas numerical simulations (Gnedin \& Ostriker 1997;
Iliev, Scannapieco \& Shapiro 2005)
and semi-analytic models (Madau, Haardt \& Rees 1999) predict clumping
factors in the range $1 < C \le 30$.
Fig.~5 shows that in model A, even assuming an homogeneous IGM
($C=1$), the escape fraction has to be $f_{esc}>0.15$ to be
marginally consistent with the observations. In model B,
the agreement improves but still requires extreme assumptions
on either the escape fraction or the clumping factor:
when $f_{esc} \le 0.15$, the predicted optical depth lies
within -1$\sigma$ error from the observed value only if $C < 8$.
Our analysis suggests that it is very hard to reconcile the observed
number counts at redshift $z \approx 10$ with the reionization history
inferred by WMAP observations if Pop~III stars are assumed to be all
very massive, with masses $\ge 100 \,{M_\odot}$. This result is somehow
counter-intuitive because metal-free very massive stars have always
believed to play an important role in an early epoch of cosmic
reionization (Cen 2003; Haiman \& Holder 2003; Sokasian et al. 2004,
2004; Venkatesan, Tumlinson \& Shull 2003; Wyithe \& Loeb 2003;
but see also Ciardi, Ferrara \& White 2003, and Ricotti \& Ostriker
2004 for alternative interpretations).
Still, due to their strong Ly$\alpha$ and nebular luminosity,
Pop~III stars with masses $\ge 100 \, {M_\odot}$ always dominate the
predicted galaxy number counts at $z \approx 10$, even assuming the
strongest chemical feedback model. The paucity of candidate objects
observed in NICMOS UDFs implies that star formation in halos with
$T_{vir} \ge 10^4$~K must have occurred very inefficiently.
Even if all three objects identified by Bouwens et al. (2005)
are truly $z \approx 10$ sources, the contribution of Pop~III stars
to cosmic reionization is strongly constrained.
The constraints are particularly severe when chemical
feedback is less efficient (model A), because Pop~III stars can
form at redshift $z \approx 10$ in relatively large, and therefore
luminous, galaxies. Even if chemical feedback is assumed to be maximally
efficient and Pop~III stars are confined to form in the smallest
halos, their contribution to cosmic reionization is
in agreement with observations only for moderate ($C<8$) clumping
factors.
Unless an additional type of feedback comes into play, the transition
between Pop~III to Pop~II halos, regulated by chemical and radiative
feedback, is very smooth. Thus, to relax the
severe upper limits placed on the star formation efficiencies
by NICMOS UDFs counts, the emission properties of Pop~III stars have
to be modified.
In particular, if Pop~III stars are assumed to form according to a
Salpeter IMF with masses in the range $1~{M_\odot} \le M_\star \le
100~{M_\odot}$, their Ly$\alpha$ and nebular emission is strongly
suppressed. In this limit, (see model C in Table~1), all Pop~III
halos contribute to chemical feedback, $f_{sn}=1$, because each
burst of star formation leads to the formation of SN progenitors
and NICMOS UDFs counts at $z \approx 10$ are dominated by Pop~II
galaxies.
As a consequence, the upper limit on the star formation efficiency
set by the observed counts is $f_\star = 0.1$, much higher than those
found for models A and B and independent of the escape
fraction.
The predicted evolution of the star formation density in Pop II and Pop III stars
is shown in the lowest panel of Fig.~4. Similar to model B, Pop II star
dominate the star formation history at all but the highest redshifts. Indeed, in our approach
the transition between Pop III and Pop II dominated star formation epochs is controlled
only by the strength of chemical feedback ($f_{sn}$) and it is independent of the assumed
star formation efficiencies (see section \ref{sec:chem}). However, contrary to model B,
the star formation density depends on the escape fraction only indirectly, through
the strength of radiative feedback. For this reason, the shaded regions
representing the range of allowed star formation densities are not degenerate only
after reionization which, for $C = 10$ and $f_{esc}=0.5$ occurs
at redshift $z_{\rm rei} = 15$ (for Pop II stars, which are preferentially
hosted in large halos, this effect is negligible).
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig7.ps, height=8.0cm}}
\caption{\footnotesize Same as Figs.~4 and ~5 but for model C and reionization
redshifts 6, 8, 10, 12, 14 and 16.}
\end{figure}
The implications for the history of reionization are
summarized in Fig.~7, which shows that model C is in much better agreement
with observations than model A and B. The optical depth observed by
WMAP can be reproduced with $f_{esc} \le 0.15$ even if the clumping
factor is as high as $C \le 25$.
Thus, when the observed $z \approx 10$ counts are used to set an
upper limit to the allowed star formation efficiency, Pop~III
stars forming with a conventional IMF are more efficient
sources of cosmic reionization than very massive stars.
Note that this conclusion is independent on issues related to
chemical and radiative feedback. Indeed, both models B and C assume
maximally efficient chemical feedback and radiative feedback plays
a minor role for model C because the predicted $z \approx 10$ counts are
dominated by massive Pop~II galaxies, which are largely unaffected by
photoevaporation.
\begin{figure}
\centerline{\epsfig{figure=popIIIextfig8.ps, height=8cm}}
\caption{\footnotesize Optical depths as a function of the
characteristic stellar mass of Pop~III stars, $M_c$, assumed to
form with a Larson IMF (see text). The different shaded regions
correspond to $C=1, 10,$ and 30. The emission properties of Pop~III
have been modified according to each assumed value of $M_c$ and
$f_{esc}=0.1$ has been adopted. For each $C$, the upper and lower
limits to the shaded region corresponds, respectively, to 3 and
0 $z \approx 10$ galaxies in NICMOS UDFs.}
\end{figure}
Fig.7 shows that the best-fit value of optical depth implied by WMAP
observations, $\tau_e = 0.16$, can only be reproduced by model C
with $f_{esc} \le 0.15$ if $C \le 8$.
If larger values of $C$ are to be favored or the observed optical
depth will be corrected up-ward in future WMAP data releases,
a revision of model C is to be considered.
However, unless the NICMOS UDFs counts have under-estimated
by a large factor the number of $z \approx 10$ sources, our
results suggest that the only plausible revision of model C is
to modify the primordial IMF by increasing the characteristic
mass but allowing Pop~III stars to form only in the
conventional mass range, $1~{M_\odot} \le M_\star \le 100~{M_\odot}$.
In Fig.~8 we show the predicted optical depth as a function of
the Pop~III characteristic stellar mass, $M_c$.
We assume that the characteristic stellar mass enters
in the definition of the stellar IMF as suggested by Larson (1998),
\begin{equation}
\frac{dN}{d\log M_\star} \propto \left ( 1 + \frac{M_\star}{M_c}\right)^{-1.35},
\end{equation}
\noindent
which, for $M_c=0$ leads to the standard Salpeter IMF.
For a given $M_c$, we compute $\tau_e$ considering
the proper emission rate of ionizing photons (taken from Table~1 of
Ciardi \& Ferrara 2005), assuming $f_{esc} = 0.1$, three
different clumping factors $C=1, 10,$ and 30, and that either all 3 or
none of the candidates identified by Bouwens et al. (2005) in NICMOS
UDFs represent $z \approx 10$ galaxies (upper and lower limits to each
shaded region).
Fig.~8 shows that the predicted optical depth depends more on the
clumping factor than on the assumed value of $M_c$.
If $C=10$, the predicted $\tau_e$ is in agreement with
the observed values even assuming $M_c=0$, consistent with the results
shown in Fig.~7. If $C=30$, WMAP and NICMOS UDFs data can
be reconciled only if all three candidates are confirmed to be
$z \approx 10$ galaxies and if the characteristic Pop~III stellar mass
is $\ge 20 \,{M_\odot}$. Finally, if the observed $\tau_e$ will be
corrected up-ward in future analyses of WMAP data, none of the models
considered, independently of the assumed $M_c$, will be able to
reconcile the observed reionization history with the counts at $z \approx
10$, unless $C<10$.
\section{Discussion}
The tentative detections of $z \approx 10$ galaxies in NICMOS UDFs by
Bouwens et al. (2005) provides tight constraints on the relative
contributions of Pop~III and Pop~II stars to the galaxy luminosity
function. When coupled to the reionization history of the universe
implied by the WMAP satellite, this collection of data sets represents
already a very demanding benchmark for galaxy evolution models.
Stimulated by these data, we have followed the
assembling history of high-redshift galaxies hosting the first stars
in a hierarchical model for structure formation.
Adopting simple prescriptions for chemical and radiative feedbacks along with
available data, we have constrained the evolution of the IMF of the first stars.
In particular we find that:
\begin{enumerate}
\item Two types of feedbacks need to be considered:
(ia) radiative feedback, which suppresses star formation
in H$_2$-cooling halos and the formation of low-mass galaxies
because of photoionization heating following reionization, and
(ib) chemical feedback due to the progressive
enrichment of (Pop~II) star forming gas by the metals released by the
first (Pop~III) stellar explosions.
\item If chemical feedback is weak ($f_{sn}=0.1$)
Pop~III stars can form in relatively large halos and
dominate the evolution of the cosmic star formation density down to
redshift $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5$. More realistic scenarios are found assuming
$f_{sn}=1$, where Pop~III stars are confined to form only in the
smallest halos at each redshift.
\item The combined effect of radiative and chemical feedbacks fails to
suppress the formation of massive Pop~III stars at $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 10$.
\item If Pop~III stars are assumed to form with a Top-Heavy IMF, that
is with masses $M_\star \ge 100 \, {M_\odot}$, Pop~III galaxy are always
found to dominate the predicted $z \approx 10$ NICMOS UDFs counts.
Because of their strong Ly$\alpha$ and nebular luminosities, very
tight upper limits on the star formation efficiency in $T_{vir} \ge
10^4$~K halos are placed by the paucity of observed counts. The
corresponding reionization histories fail to reproduce the observed
WMAP optical depths unless uncomfortably large values for the escape
fraction of ionizing photons and/or too small values for the IGM
clumping factor are assumed.
\item Reionization constraints from WMAP and observed $z \approx 10$
counts in NICMOS UDFs can be reconciled if Pop~III stars form
according to a Larson IMF with stellar masses in the
conventional range $1 \, {M_\odot} \le M_\star \le 100 \, {M_\odot}$ and
a characteristic mass $M_c \ge 0$ if the clumping factor is $\le
10$, or $M_c \ge 20 \, {M_\odot}$ if the clumping factor is as large as
30. In the latter case, at least 1 candidate identified by
Bouwens et al. (2005) has to be a galaxy at $z \approx 10$.
\end{enumerate}
Our analysis is based on the assumption that chemical feedback
propagates along the hierarchy of galaxy mergers from progenitors
to descendants rather than through metal-enriched galaxy outflows
(Scannapieco et al. 2003). By adopting this simplified scheme, we
are implicitly assuming that (i) even if a fraction of metals is ejected
out of the host halo, the metals that are kept within the potential
well are sufficient to pollute the gas within the host system or
within any unpolluted halo that the system will later merge with,
to a metallicity level larger than $Z_{\rm cr}$ and that (ii) metal
pollution from neighboring galaxies is negligible.
In spite of these simplifying assumptions, the resulting
Pop~III and Pop~II star formation histories are consistent with
the predictions of Scannapieco et al. (2003). In particular, even
assuming a maximally efficient chemical feedback, the transition is
always predicted to be very smooth, with Pop~III stars which continue
to form in the smallest halos at each redshift down to $z \lower.5ex\hbox{$\; \buildrel < \over \sim \;$} 5$.
Interestingly enough, our conclusion is independent of the primordial
IMF and/or star formation efficiency. We show that models where all
Pop~III halos are able to metal pollute their descendants ($f_{sn}=1$)
are favored by observations. The smoothness of the cosmic IMF
transition is therefore strictly linked to the inhomogeneous character
of the metal enrichment process, independently of how it is
implemented in the models.
Even if we have adopted a very approximate description of the thermal
evolution of the cosmic gas, we find that radiative feedback following
reionization can play an important role in the extinction of Pop~III
stars. In particular, if reionization occurs at redshifts $z \approx 15$
($z \approx 10$) the baryon fraction in $10^8\,{M_\odot}$ halos is reduced
to 70\% (99\%) of its universal value ($\Omega_B/\Omega_M$) at redshift
10 and to 10\% (30\%) of this value at redshift 5. Thus, the
baryon content in the smallest halos where Pop~III stars
can form at low redshift is severely limited by radiative feedback.
This is strongly related to the epoch of cosmic reionization which,
in turn, depends on the degree of inhomogeneities which characterize
the cosmic gas. We have parametrized this quantity through a constant
clumping factor $C$ and we have explored different values of $C$
ranging from 1 to 50. However, it is likely that the degree of
clumpiness of the cosmic gas varies with time, rising from $C \approx 1$
at high redshift to a few tens at $z \approx 10$ as structures go
increasingly non-linear.
Recently, Iliev et al. (2005) have studied the effect
of IGM clumping at redshifts $z \ge 10$, providing a fit to the
evolution of the clumping factor (see their Fig.~7) where $C \approx 2$
at redshift 30 and raises to $C \approx 8$ at redshift 10. Adopting this
evolution for the clumping factor and $f_{esc}=0.1$ we find that
models A and B are unable to reproduce the optical depths inferred
from WMAP data whereas in model C we find that reionization is
complete at redshift 13.5 and that the resulting optical depth is
$\tau_e = 0.1734$ in very good agreement with the observed value.
Thus, our analysis suggests that Pop~III stars forming with masses in
the same dynamical range as Pop~II/I stars, but possibly with higher
characteristic stellar mass, appear to be favored by observations.
Several scenarios that enable the formation of
low-mass stars in metal-free environments have been proposed,
among which the bi-modal IMF (Nakamura \& Umemura 1999, 2002),
enhanced HD-cooling in relic HII regions (Uehara \& Inutsuka 2000;
Nakamura \& Umemura 2002), the influence of
a strong UV radiation field due to the vicinity of a very massive star
(Omukai \& Yoshii 2003), fragmentation in shock-compressed shells
induced by the first very massive SN explosions (Mackey, Bromm \& Hernquist
2003; Salvaterra, Ferrara \& Schneider 2004).
However, all these processes either require very special conditions,
generally related to the existence of a previous generation of
very massive stars, or have failed to be realized in realistic
numerical simulations.
In spite of this, observations seem to indicate that either
radiative feedback effects during the accretion phase on protostellar
cores and/or a combination of the above mentioned processes
can lead to the formation of Pop~III stars with masses $\le 100 {M_\odot}$.
According to the results of our analysis, the latter contribution to
the Pop~III stellar IMF must be dominant at least at redshifts $z \approx 10$.
If this scenario is correct, relics of the low-mass Pop~III star
formation mode should be still present in our Galaxy,
eventually contributing to the lowermost end of the
metallicity distribution function (Salvadori, Schneider \& Ferrara in prep).
\section*{Acknowledgments}
We acknowledge the use of PINOCCHIO v 1.0 package written by Pierluigi Monaco, Tom Theuns \& Giuliano Taffoni
(http://www.daut.univ.trieste.it/pinocchio). This work has been partially supported by the Research and
Training Network ``The Physics of the Intergalactic Medium'' set up by
the European Community under the contract HPRN-CT-2000-00126.
| 2024-02-18T23:39:48.932Z | 2005-10-24T16:41:03.000Z | algebraic_stack_train_0000 | 500 | 9,088 |
|
proofpile-arXiv_065-2560 | \section{Introduction}
Small-world systems exhibit remarkable cooperation, not found in
complex systems with e.g.\@ an Erd\"os-Renyi
structure. The origins of the appellation `small-world' can be
traced to the now famous experiment by the Harvard social
psychologist Stanley Milgram \cite{milgram}. The outcome of this
experiment pointed to the fact that the structure of many real
networks is such that distant nodes can in fact be connected via
long-range short-cuts. This architectural property leads to small
path lengths between any pair of nodes and thus enhances
information processing and cooperation.
The importance and ubiquitous nature of small-world structures in
complex networks received further attention by the seminal paper
of Watts and Strogatz \cite{wattsstrogatz} in which the authors
proposed the small-world structure as a way to interpolate between
so-called `regular' and `random' networks. Surprisingly, the
simple small-world architecture can be found in many different
circumstances, ranging from linguistic, epidemic and social
networks to the world-wide-web. By now, a large body of work has
been devoted to the study of small-world networks, mainly
numerical, with emphasis e.g.\@ on biophysical
\cite{girvan,simingli,barabasi_nature} or social networks
\cite{newman_pnas} and to a lesser extent analytical
\cite{barrat,theodore}. For recent reviews in the area of
small-worlds see e.g.\@ the articles
\cite{albert-barabasi,newman_review}, or the books
\cite{bookwatts,bookdoro}.
From a statistical mechanical point of view such systems combine
two universality classes: a sparse `graph' structure, which is
superimposed upon a one-dimensional `ring'. Thus, every node on
the ring has a local neighborhood and a certain number of
long-range connections to distant parts of the chain. It was shown
in \cite{theodore} (and also in \cite{XYsmall} for the case of XY
spins) that this construction significantly enlarges the region in
parameter space where ferromagnetism occurs. In particular, it was
shown that the ferromagnetic-paramagnetic transition always occurs
at a finite temperature for any value of the average long-range
connectivity (however small). On a technical level, in evaluating
the relevant disorder-averaged free energy one is immediately
confronted with the problem of diagonalising a $2^n\times 2^n$
transfer matrix where $n$ represents the replica dimension.
Although for an infinite system size obtaining only the largest
eigenvalue suffices for the evaluation of the free energy, one can
in principle follow the systematic analysis of
\cite{diagonalisation} to derive the entire spectrum of
eigenvalues and thus evaluate e.g.\@ correlation functions.
Several important issues remain to be understood for small-world
systems. In this paper we evaluate the number of metastable
configurations, or more precisely, the number of equilibrium
configurations in which spins align to their local fields. With
this definition the energy of the system in a metastable state
cannot be decreased by flipping a single spin. Such
configurations can be e.g.\@ responsible for trapping the
microscopic update dynamics in locally stable states. Thus, from
an experimental point of view it is advantageous to know what is
the relevant size in phase space occupied by such states.
The computation of the number of metastable configurations in
small-world systems is generally an involved problem, both
analytically and numerically. Indicatively, on sparse random
graphs structures without the superimposed `ring' the evaluation of the
number of metastable configurations, or the so-called configurational
entropy, has only recently taken off
\cite{bergsellito,pagnani,lefevre} following the course of the
relevant analytic techniques (as e.g.\@ in
\cite{mezardparisi,mezardparisi0,leone}). In particular, using the
replica method, the solution of the ferromagnetic Poisson graph
has been studied in \cite{bergsellito} while with the cavity
method the authors of \cite{pagnani} examined the bond-disordered
Bethe lattice. These results agree well with results of numerical
enumerations \cite{boettcher1,boettcher2} and also serve as
good limiting tests of our findings.
This paper is organized as follows: in Section \ref{sec:definitions} we
first define the small-world model. In Section \ref{sec:saddle-points} we
express the generating function of the system as a saddle-point
problem which we then evaluate in Section \ref{sec:RTM} using
replicated transfer matrix techniques and within the replica symmetric
approximation. In Section \ref{sec:results} we present our
results and finally we present some concluding remarks in Section
\ref{sec:conclusions}.
\section{The model}
\label{sec:definitions}
Our model describes a system of $N$ Ising spins
$\bsigma=(\sigma_1,\ldots,\sigma_N)$, with $\sigma_i\in\{-1,1\}$,
arranged on a one-dimensional lattice. There are two different
couplings in this system: firstly, nearest-neighbor interactions of
uniform strength $J_0$ and secondly, sparse long-range ones. To
model the latter we will assign the random variable $c_{ij}$ for
every pair of sites $(i,j)$ representing whether a connection
exists ($c_{ij}=1$) or not ($c_{ij}=0$). This variable will be taken for
all $i<j$ from the distribution
\begin{equation}
Q_c(c_{ij})=\frac{c}{N}\delta_{c_{ij},1}+\left(1-\frac{c}{N}\right)\delta_{c_{ij},0},\label{eq.dist}
\end{equation}
so that, on average, every site has $c$ long-range connections. In
the small-world context one takes $c$ to be a small number of
order $\mathcal{O}(1)$ while $c/N\to 0$. The bond-strength $J_{ij}$ of the
long-range coupling between any pair of spins $(i,j)$ (with $i<j$)
will be taken from the distribution
\begin{equation}
Q_J(J_{ij})=p\,\delta_{J_{ij},J}+(1-p)\,\delta_{J_{ij},-J},
\label{eq:bond_disorder}
\end{equation}
for some $J>0$, so that $p=1$ corresponds to a model with strictly
ferromagnetic interactions. To allow for detailed balance we will
also consider absence of self-interactions and symmetry of the
connectivity matrix, namely $c_{ii}=0$, $c_{ij}=c_{ji}$ and
$J_{ij}=J_{ji}$. At thermal equilibrium the above system can be
described by the Hamiltonian:
\begin{equation}
\mathcal{H}(\bsigma)=-\frac12\sum_i \s_i\, h_i(\bsigma),
\label{eq:H}
\end{equation}
with the local fields defined as:
\begin{eqnarray*}
h_i(\bsigma)&=&
\sum_j\left[J_0(\delta_{j,i+1}+\delta_{j,i-1})+\frac{c_{ij}}{c}J_{ij}\right]\s_j.
\end{eqnarray*}
We now impose the condition for metastability: in a similar
spirit as e.g.\@ in \cite{braymoore,gardner} we call a
configuration $\bsigma$ metastable if all spins align to their
local fields, i.e.\@
\begin{eqnarray*}
\prod_i \Theta(\s_ih_i(\bsigma)) = 1,
\end{eqnarray*}
where $\Theta(x)=1$ for $x\geq 0$ and $\Theta(x)=0$ otherwise.
Notice that we have taken $\Theta(0)=1$ which is dictated by the underlying
physics: for spins which receive a zero local field, the
energetic cost of aligning to either of the two possible
directions is identical. A consequence of the above definition is
that any metastable configuration $\bsigma^{\text{MS}}$ is a local
minimum of the Hamiltonian:
\begin{equation*}
\forall \btau :
\frac{1}{2}\sum_i\vert\sigma_i^{\text{MS}}-\tau_i\vert=1
\Rightarrow \mathcal{H}(\btau)\geq\mathcal{H}(\bsigma^{\text{MS}}).
\end{equation*}
Since we are interested in evaluating the number of metastable
configurations we will define the following generating function:
\begin{eqnarray}
-\beta f&=& \lim_{N\to\infty}\frac1N\Bra \log
\sum_{\bsigma}e^{-\beta
\mathcal{H}(\bsigma)}\prod_{i=1}^N\Theta(\s_ih_i(\bsigma))\Ket_{\bc,\bJ}
\label{eq:generating_function}
\end{eqnarray}
where $\beta$ represents the inverse temperature and
$\bc=\{c_{ij}\}$, $\bJ=\{J_{ij}\}$. As in \cite{bergsellito,pagnani}, the entropy density $s_m$
of the metastable configurations
can be evaluated from (\ref{eq:generating_function}) via $s_m=\beta\partial_\beta(\beta f)-\beta f$.
\section{Saddle-point equations}
\label{sec:saddle-points}
To evaluate the disorder average in (\ref{eq:generating_function})
we begin by invoking the replica identity: $\bra \log
Z\ket=\lim_{n\to 0}\frac1n\log \bra Z^n\ket$. As the disorder
variables $\{c_{ij},J_{ij}\}$ lie within the \mbox{$\Theta$-function} we
insert the following unity into our expression
\begin{equation}
1 = \int\prod_{i,\alpha} dh_i^\alpha\
\delta[h_i^\alpha-h_i(\bsigma^\alpha)],
\label{eq:field_def}
\end{equation}
with $\bsigma^{\alpha}=(\sigma_1^\alpha, \cdots, \sigma_N^\alpha)$
and $\alpha=1,\ldots,n$. This allows us to conveniently re-locate
$c_{ij},J_{ij}$ into exponents where averages can be taken more
easily, namely
\begin{eqnarray}
\lefteqn{-\beta f=\lim_{N\to\infty}\lim_{n\to
0}\frac{1}{Nn}\log\int
\{dh_i^\alpha\,d\hat{h}_i^\alpha\}e^{-i\sum_{i,\alpha}h_i^\alpha\hat{h}_i^\alpha}\nonumber}
\\
& & \times\,\sum_{\{\sigma_i^\alpha\}}
\prod_{i,\alpha}\left[e^{iJ_0\hat{h}_i^\alpha(\s_{i+1}^\alpha+\s_{i-1}^\alpha)+
\frac12\beta\s_i^\alpha h_i^\alpha}\Theta(\s_i^\alpha
h_i^\alpha)\right]\nonumber
\\
& & \times\,\Bra
e^{i\sum_{i,\alpha}\hat{h}_i^\alpha\sum_j\frac{c_{ij}}{c}J_{ij}\s_j^\alpha}\Ket_{\bc,\bJ},
\label{eq:f_inter}
\end{eqnarray}
where
$\{dh_i^\alpha\,d\hat{h}_i^\alpha\}=\prod_{i,\alpha}(2\pi)^{-1}
dh_i^\alpha\,d\hat{h}_i^\alpha$. Let us concentrate on the last
line of the above which contains the disorder. After symmetrizing
with respect to the sites $i<j$ it leads for $N\to\infty$ to:
\begin{eqnarray}
\lefteqn{\Bra
e^{i\sum_{i,\alpha}\hat{h}_i^\alpha\sum_j\frac{c_{ij}}{c}J_{ij}\s_j^\alpha}\Ket_{\bc,\bJ}}
\nonumber
\\
&=& \exp\left[\frac{c}{2N}\sum_{ij}\left(\Bra
e^{i\frac{\mathcal{J}}{c}\sum_\alpha (\hat{h}_i^\alpha\sigma_j^\alpha+
\hat{h}_j^\alpha\sigma_i^\alpha)}\Ket_{\mathcal{J}}-1\right)\right]
\label{eq:dilav}
\end{eqnarray}
where $\bra \cdots\ket_{\mathcal{J}}$ denotes an average over the
binary random variable $\mathcal{J}$ taken from the distribution
$Q_J(\cdot)$, eq.\@ (\ref{eq:bond_disorder}). We have used the fact
that $c/N\to 0$ to recast the result of averaging over
$\{c_{ij}\}$ into an exponential form. Note now that upon
inserting the unities $1=\sum_{\bsigma}\delta_{\bsigma,\bsigma_i}$
and $1=\sum_{\btau}\delta_{\btau,\bsigma_j}$ where $\bsigma,\btau$
are auxiliary vectors in replica space (and we have denoted
$\bsigma_i=(\sigma_i^1,\ldots,\sigma_i^n)$), one has effectively
created an order function $P_{\mathcal{J}}(\bsigma,\btau)$. As usual it
can be inserted into
our generating function via:
\begin{eqnarray}
\lefteqn{1=\int\prod_{\bsigma\btau}\prod_{\mathcal{J}=\pm
J}dP_\mathcal{J}(\bsigma,\btau)\nonumber}
\\
&&
\times\,\delta\left[P_{\mathcal{J}}(\bsigma,\btau)-\frac{1}{N}\sum_i\delta_{\bsigma,\bsigma_i}\,
e^{i\frac{\mathcal{J}}{c}\hat{\bh}_i\cdot\btau}\right].
\label{eq:orderfunction_def}
\end{eqnarray}
As in \cite{topologies}, to understand the physical meaning of the
above order function one needs to add a generating term in the
replicated Hamiltonian
$\sum_{\alpha}\mathcal{H}(\bsigma^\alpha)\to
\sum_\alpha\mathcal{H}(\bsigma^\alpha)+\eta
P_\mathcal{J}(\bsigma,\btau)$ and take the derivative $\partial
f/\partial \eta|_{\eta=0}$ in (\ref{eq:generating_function}). One
then sees that upon introducing the identities (\ref{eq:field_def})
the order function becomes the distribution of replicated spins
with one connection removed (equivalently, it becomes the
distribution of replicated `cavity' spins).
We now aim to eliminate from our expressions the set of fields
$\{h_i^\alpha,\hat{h}_i^\alpha\}$.
This can be done by replacing the delta function
in (\ref{eq:orderfunction_def}) by its Fourier representation (for details see appendix \ref{sec:ap1}).
As an end result we obtain an extremisation problem over the density
$P_\mathcal{J}(\bsigma,\btau)$ expressed in terms of a trace over a transfer function, namely
\begin{widetext}
\begin{equation}
-\beta f= \lim_{n\to 0}\frac{1}{n}\underset{P}{\rm Extr}
\left[-\frac{c}{2}\sum_{\bsigma\btau}\Bra
P_\mathcal{J}(\bsigma,\btau)P_\mathcal{J}(\btau,\bsigma)\Ket_\mathcal{J}
+\lim_{N\to\infty}\frac1N\log\,{\rm tr}\,(T^N[P])\right],
\label{eq:extr}
\end{equation}
\end{widetext}
where $\mathcal{J}\in\{-J,J\}$ and with the abbreviation ${\rm
tr}(T^N[P])={\rm tr}(T^N[P_{J},P_{-J}])$. The order functions
$P_{\pm J}$ are to be evaluated self-consistently from
\begin{eqnarray}
\lefteqn{P_\mathcal{J}(\bsigma,\btau)\propto
\frac{1}{N}\sum_{j=1}^N\sum_{\bsigma_1\cdots\bsigma_N}\delta_{\bsigma_j,\bsigma}\nonumber}
\\
&\times& \
F^{(\mathcal{J})}_{\bsigma_{j-1},\bsigma_j,\bsigma_{j+1}}[P,\bsigma,\btau]\prod_{i\neq
j} T_{\bsigma_{i-1},\bsigma_i,\bsigma_{i+1}}[P] \label{eq:Psigmatau}
\end{eqnarray}
We have absorbed the normalisation constant
$\sum_{\{\sigma^\alpha_i\}} \prod_i
T_{\bsigma_{i-1},\bsigma_i,\bsigma_{i+1}}[P]$ in the
proportionality symbol. The traces in (\ref{eq:extr}) and
(\ref{eq:Psigmatau}) involve correlations between next-nearest
neighbors and can be evaluated in a spirit similar to the transfer
matrix technique. The relevant tensor is defined over a $2^n\times
2^n\times 2^n$ space with elements
\begin{eqnarray}
\lefteqn{T_{\bsigma_{i-1},\bsigma_i,\bsigma_{i+1}}[P]=\sum_{\mu\geq
0}\frac{e^{-c}c^\mu}{\mu!} \sum_{\btau_1\cdots\btau_\mu} }
\label{eq:matrix_Ta}
\\
& & \times\,\Bra[\prod_{\nu\leq \mu}
P_{\mathcal{J}_\nu}(\btau_\nu,\bsigma_i)]\prod_\alpha
R_i^\alpha(\sum_{\nu\leq \mu}\mathcal{J}_\nu\tau^\alpha_\nu)\Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu}, \nonumber
\end{eqnarray}
with the convention that the $\mu=0$ term of the above equals $e^{-c}\prod_\alpha R_i^\alpha(0)$. The quantity $F$ that specifies our order function is
\begin{eqnarray}
\lefteqn{F^{(\mathcal{J})}_{\bsigma_{i-1},\bsigma_i,\bsigma_{i+1}}[P,\bsigma,\btau]=\sum_{\mu\geq
0}\frac{e^{-c}c^\mu}{\mu!}
\label{eq:matrix_Fa}
\sum_{\btau_1\cdots\btau_\mu}
}
\\
& & \times \Bra[\prod_{\nu\leq \mu}
P_{\mathcal{J}_\nu}(\btau_\nu,\bsigma)]\prod_\alpha
R_i^\alpha(\sum_{\nu\leq
\mu}\mathcal{J}_\nu\tau^\alpha_\nu+\mathcal{J}\tau^\alpha)\Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu}.
\nonumber
\end{eqnarray}
We have used the abbreviation
\begin{eqnarray}
\lefteqn{R_i^\alpha(x) =} \label{eq:R}
\\
& &
e^{\frac{1}{2}\beta\sigma_i^{\alpha}(J_0(\s_{i+1}^\alpha+\s_{i-1}^\alpha)+\frac{x}{c}
)}
\Theta[\sigma_i^{\alpha}(J_0(\s_{i+1}^\alpha+\s_{i-1}^\alpha)+\frac{x}{c}
)].\nonumber
\end{eqnarray}
Note that due to our symmetrization with respect to site indices in
(\ref{eq:dilav}) we have ended up with a symmetric quantity, namely
$T_{L\bx_i}=T_{\bx_i}$ where $L$ is the 3$\times$3 matrix $L_{\ell k}=\delta_{\ell+k,4}$ and $\bx_i=
(\bsigma_{i-1},\bsigma_i,\bsigma_{i+1})$.
The structure of the function $R_i^\alpha(x)$ indicates that the
input $x$ is related to the long-range field received by a site
$i$. The tensors $T$ and $F$ differ only in their input to this
function. Since this is proportional to $\pm J/c$ we understand
that it is related to the `effective' (or, `cavity') and true
local-field, respectively.
\section{Replicated transfer matrix analysis}
\label{sec:RTM}
To interpret the spin summations as matrix multiplications in
equations (\ref{eq:extr}) and (\ref{eq:Psigmatau}) we need
to transform our variables such that the traces in these equations
involve nearest-neighbor correlations only. This can be done in several
ways. For instance, let us introduce the auxiliary spins:
\begin{equation}
\bs_i^\alpha\equiv(s_i^{\alpha,1},s_i^{\alpha,2})=(\s_i^\alpha,\s^\alpha_{i+1}).
\label{eq:transform}
\end{equation}
To suppress the replica index above, we will occasionally use
the more compact notation
$\bs_i=(\bs^{(1)}_i,\bs^{(2)}_i)=(\bsigma_{i},\bsigma_{i+1})$ with
vectors now defined in replica space. With the above, we can now transform
$T_{\bsigma_{i-1},\bsigma_i,\bsigma_{i+1}}$ into
$T_{\bs_{i-1},\bs_i}$ and in particular
\begin{eqnarray}
\lefteqn{\hspace{-4mm}T_{\bs_{i-1},\bs_i}[P]=
\delta_{\bs_{i-1}^{(2)},\bs_{i}^{(1)}}\sum_{\mu\geq
0}\frac{e^{-c}c^\mu}{\mu!}\sum_{\btau_1\cdots\btau_\mu} \nonumber}
\\
& & \times\Bra[\prod_{\nu\leq \mu}
P_{\mathcal{J}_\nu}(\btau_\nu,\be\cdot\bs_i)]\prod_\alpha
\hat{R}_i^\alpha(\sum_{\nu\leq \mu}\mathcal{J}_\nu\tau^\alpha_\nu)\Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu}
\label{eq:matrix_T}
\end{eqnarray}
with $\be=(1,0)$. Similarly for $F$:
\begin{eqnarray}
\lefteqn{F^{(\mathcal{J})}_{\bs_{i-1},\bs_i}[P,\bsigma,\btau]=
\delta_{\bs_{i-1}^{(2)},\bs_{i}^{(1)}}\sum_{\mu\geq
0}\frac{e^{-c}c^\mu}{\mu!} \sum_{\btau_1\cdots\btau_\mu}}
\nonumber \label{eq:matrix_F}
\\
& & \times\Bra[\prod_{\nu\leq \mu}
P_{\mathcal{J}_\nu}(\btau_\nu,\bsigma)]\prod_\alpha \hat{R}_i^\alpha(\sum_{\nu\leq
\mu}\mathcal{J}_\nu\tau^\alpha_\nu+\mathcal{J}\tau^\alpha)\Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu}. \nonumber
\end{eqnarray}
The Kronecker deltas on the right-hand side of the above
expressions impose the transformation (\ref{eq:transform}). Similarly
to equation (\ref{eq:R}) we now have
\begin{eqnarray}
\hat{R}_i^\alpha(x)= e^{\frac12\beta\bs_i^\alpha\cdot(
J_0\bs_{i-1}^\alpha+ \frac{x }{c}\be ) }\
\Theta\left[\bs_i^\alpha\cdot( J_0\bs_{i-1}^\alpha+ \frac{x
}{c}\be )\right]. \nonumber
\end{eqnarray}
This transfer matrix is also symmetric. In particular, it obeys $T_{\hat{L}\by_i}=T_{\by_i}$ where
$\hat{L}$ is the 4$\times$4 matrix $\hat{L}_{\ell k}=\delta_{\ell+k,5}$ (the so-called Dirac matrix $E_{11}$) and $\by_i=(\bs_i,\bs_{i+1})$.
With the above definitions we can now write the self-consistent
equation (\ref{eq:Psigmatau}) in a transparent way:
\begin{equation}
P_\mathcal{J}(\bsigma,\btau)=\sum_{\bsigma'}\frac{{\rm
tr}\left(Q^{(\mathcal{J})}[P,\bsigma',\bsigma,\btau]\,
T^{N-1}[P]\right)} {{\rm tr}\left(T^{N}[P]\right)}, \label{eq:P_tr}
\end{equation}
with the auxiliary matrix
\begin{equation}
Q^{(\mathcal{J})}_{\bs_a,\bs_b}[P,\bsigma',\bsigma,\btau]\equiv
F^{(\mathcal{J})}_{\bs_a,\bs_b}[P,\bsigma,\btau]\
\delta\left[{\bs_a^{(1)} \choose \bs_a^{(2)}}-{\bsigma'
\choose \bsigma}\right] \label{eq:Q}
\end{equation}
To proceed with the evaluation of
the traces involved in (\ref{eq:extr}) and (\ref{eq:P_tr}) we now
aim at diagonalizing the transfer matrix
$T$. Our analysis hereafter will follow closely
\cite{theodore}. To this end, let us consider the eigenvector equation
corresponding to the largest eigenvalue $\lambda_0(n)$, namely
\begin{eqnarray}
\sum_{\bs'}T_{\bs\bs'}[P]\ u(\bs')&=&\lambda_{0}(n)\ u(\bs).
\label{eq:right}
\end{eqnarray}
Note that we have only defined a `right' eigenvector. It is
sufficient due the symmetry of our transfer matrix. Next, using
(\ref{eq:Q}) and (\ref{eq:right}) we can rewrite the
self-consistent equation (\ref{eq:P_tr}) in terms of $u(\bs)$
\begin{eqnarray}
\lefteqn{P_\mathcal{J}(\bsigma,\btau)=\label{eq:P_decomp}}
\\
&&
\frac{\sum_{\bsigma'\bs\bs'}F_{\bs\bs'}^{(\mathcal{J})}\,[P,\bsigma,\btau]\, u(\bs')\,u(\bs)\,
\delta_{\bs^{(1)},\bsigma'}\,\delta_{\bs^{(2)},\bsigma}}
{\lambda_0(n)\, \sum_{\bs}u(\bs)\,u(\bs)}. \nonumber
\end{eqnarray}
Equations (\ref{eq:right}) and (\ref{eq:P_decomp}) are the basis
of our analysis in the subsequent sections.
\subsection{Replica symmetry and self-consistent equations}
\label{sec:RS}
Since the order function $P$ depends on $n$ via the
dimensionality of its arguments we must now make an ansatz that
will allow us eventually to take the limit $n\to 0$. The simplest choice
corresponds to considering permutation invariance of $P$ with
respect to its replica indices. This symmetry is guaranteed by
considering e.g.\@ the following form:
\begin{eqnarray}
P_\mathcal{J}(\bsigma,\btau)=
\int dW_\mathcal{J}(\bh)
\frac{e^{\beta \sum_\alpha (h_1\s^\alpha+h_2\tau^\alpha +h_3\s^\alpha\tau^\alpha)}}{[\mathcal{N}(\bh)]^n},
\label{eq:W_def}
\end{eqnarray}
with the short-hand notation $\bh=(h_1,h_2,h_3)$,
$dW_\mathcal{J}(\bh)=d\bh\,W_\mathcal{J}(\bh)$ and $\mathcal{N}(\bh)$
ensures that $\int d\bh\,W_\mathcal{J}(\bh)=1$. We also assume that the
eigenvector $u(\bs)$ takes the form
\begin{eqnarray}
u(\bs) = \int d\Phi(\bx)\ \label{eq:Phi_def}
e^{\beta \sum_\alpha(x_1 s^{\alpha,1} + x_2 s^{\alpha,2}+ x_3 s^{\alpha,1} s^{\alpha,2})}.
\end{eqnarray}
With these assumptions we can now proceed further and rewrite the
extremisation problem (\ref{eq:extr}) in terms of the pair of
densities $\Phi$ and $W_{\mathcal{J}}$. The starting point is
equations (\ref{eq:right}) and (\ref{eq:P_decomp}) respectively.
Inserting our assumptions (\ref{eq:W_def}) and (\ref{eq:Phi_def})
leads after some algebra to the following set of closed equations
for $n\to0$:
\begin{eqnarray}
\lefteqn{\lambda_0(0)\,\Phi(\bx')\nonumber}
\\
&& = \sum_{\mu\geq 0}\frac{e^{-c}c^\mu}{\mu!}\Bra\int
[\prod_{\nu=1}^\mu dW_{\mathcal{J}_{\nu}}(\bh_\nu)] \int
d\Phi(\bx)\right.\nonumber
\\
&&
\left.\times\prod_{i=1}^3 \delta\left[x_i'-\frac{1}{4\beta} \sum_{\sigma\tau}f_i(\sigma,\tau)G_{\sigma,\tau}(\bx,\bh_\nu)\right]\Ket_{\mathcal{J}_{1}\cdots\mathcal{J}_{\mu}}
\label{eq:self_Phi}
\end{eqnarray}
\begin{eqnarray}
\lefteqn{\lambda_0(0)\, W_\mathcal{J}(\bh')\nonumber}
\\
&& =\sum_{\mu\geq 0}\frac{e^{-c}c^\mu}{\mu!}\Bra\int
[\prod_{\nu=1}^\mu dW_{\mathcal{J}_\nu}(\bh_\nu)] \int d\Phi(\bx)
d\Phi(\bx')\right.
\nonumber\\
&\times&
\left.\prod_{i=1}^3\delta\left[h_i'-\frac{1}{4\beta}\sum_{\sigma\tau}f_i(\sigma,\tau)
H_{\sigma,\tau}^{(\mathcal{J})}(\bx,\bx',\bh_\nu)\right]
\Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu}\label{eq:self_W}
\end{eqnarray}
In the above we have used the function $f_i(\sigma,\tau)$ with
\[
f_1(\sigma,\tau) = \sigma,\quad f_2(\sigma,\tau) = \tau,\quad
f_3(\sigma,\tau) = \sigma\tau,
\]
while $G,H$ correspond to
\begin{eqnarray}
\lefteqn{G_{\sigma,\tau}(\bx,\bh)=\nonumber}
\\
&&\hspace{-3mm}
\log\left\{\sum_{\tau_1\cdots\tau_\mu}\sum_{\eta=\pm}
e^{\beta(x_1\tau+\eta x_2+\tau
x_3)}S_{\tau,\sigma,\eta}(\sum_{\nu\leq
\mu}\mathcal{J}_\nu\tau_\nu)\right\}\label{eq:update_G}
\end{eqnarray}
\begin{eqnarray}
H^{(\mathcal{J})}_{\sigma,\tau}(\bx,\bx',\bh)&=&\log\left\{\sum_{\tau_1\cdots\tau_\mu}
\sum_{\eta,\omega=\pm} e^{\beta(x_1\sigma+\omega(x_2+\sigma
x_3))}\right.\nonumber
\\
& &\hspace{-20mm}\times\ \left.e^{\beta(\sigma x_2'+\eta
(x_1'+\sigma x_3'))}S_{\sigma,\omega,\eta}(\sum_{\nu\leq
\mu}\mathcal{J}_\nu\tau_\nu+\mathcal{J}\tau)\right\}\label{eq:update_H}
\end{eqnarray}
and
\begin{eqnarray*}
\lefteqn{S_{\sigma_1,\sigma_2,\sigma_3}(x)=\prod_{\nu\leq
\mu}e^{\beta(h_{1\nu}\tau_\nu+h_{2\nu}\sigma_1+h_{3\nu}\sigma_1\tau_\nu+\frac{1}{2c}\sigma_1
\mathcal{J}_\nu\tau_\nu)}}
\\
& &\times\ e^{\frac12\beta J_0\sigma_1(\sigma_2+\sigma_3)}
\Theta\left(J_0\sigma_1(\sigma_2+\sigma_3)+\frac{\sigma_1}{c}x\right).
\end{eqnarray*}
Finally, to calculate (\ref{eq:extr}) we need to determine the
largest eigenvalue $\lambda_0(n)$ for $n\to 0$. The starting point
here is our eigenvector equation (\ref{eq:right}). Evaluating the
traces over the spin variables with the
definitions of the transfer matrix (\ref{eq:matrix_T}) and
eigenvectors (\ref{eq:Phi_def}) leads for $n\to 0$ to
\begin{eqnarray}
\lefteqn{\lambda_0(n)=1+n\left\{\sum_{\mu\geq
0}\frac{e^{-c}c^\mu}{\mu!}\int
[\prod_{\nu\leq\mu}dW_{\mathcal{J}_\nu}(\bh_{\nu})]\right.\label{eq:lambda}}
\\
&& \hspace{-4mm} \left. \frac{\Bra\int
d\Phi(\bx)\left[\log\left(\frac{K(\bx,\{\bh_\nu\})}{A(\bx)}\right)-\sum_{\nu\leq
\mu}\log\mathcal{N}(\bh_\nu)\right] \Ket_{\{\mathcal{J}_\nu\}}}{\int
d\bx'\Phi(\bx')}
\right\}
\nonumber
\\
& &
+\mathcal{O}(n^2), \nonumber
\end{eqnarray}
so that $\lambda_0(0)=1$. We have defined
\begin{eqnarray}
A(\bx) &=& \sum_{\sigma,\sigma'=\pm}e^{\beta( x_1\sigma+
x_2\sigma'+x_3\sigma\sigma')} \label{eq:A}
\\
K(\bx,\{\bh_\nu\}) & = & \sum_{\tau^1\cdots
\tau^\mu}\sum_{\sigma_1,\sigma_2,\sigma_3}e^{\beta (x_1 \s_1+
x_2\sigma_3+ x_3\sigma_1 \s_3)}\nonumber
\\
& & \times S_{\sigma_1,\sigma_2,\sigma_3}(\sum_{\nu\leq
\mu}\mathcal{J}_\nu\tau_\nu)\label{eq:K}.
\end{eqnarray}
With the expression (\ref{eq:lambda}) we can now evaluate
(\ref{eq:extr}):
\begin{eqnarray}
\lefteqn{-\beta f
= -\frac{c}{2}
\Bra\int dW_{\mathcal{J}}(\bh)\,dW_{\mathcal{J}}(\bh')\log D(\bh,\bh')\Ket_{\mathcal{J}}} \nonumber
\\
& & +\Bra\sum_{\mu\geq 0}\frac{e^{-c}c^\mu}{\mu!}\int
\prod_{\nu\leq \mu} dW_{\mathcal{J}_\nu}(\bh_\nu)\right.\nonumber
\\
& & \times\left.\frac{1} {\int d\bx'\ \Phi(\bx')}\int
d\bx\,\Phi(\bx) \,\log \left[\frac{K(\bx,\{\bh_\nu\})}{ A(\bx)}
\right] \Ket_{\mathcal{J}_1\cdots\mathcal{J}_\mu} \label{eq:f_final}
\end{eqnarray}
with
\[
D(\bh,\bh')=\sum_{\sigma,\sigma'=\pm} e^{\beta( h_1\sigma+
h_2\sigma' +h_3\sigma\sigma'+h_1'\sigma'+h_2'\sigma +
h_3'\sigma\sigma')} \nonumber,
\]
which is our final result. From this equation we can inspect the physical meaning
of the densities $W_{\mathcal{J}}(\bh)$ and $\Phi(\bx)$. The Poisson distribution $e^{-c}c^\mu/\mu!$ of mean $c$ can be clearly associated to the degree distribution of the graph. Once a degree has been sampled from this distribution one performs $\mu$ integrals over the densities $\{W_{\mathcal{J}_\nu}\}$ and one over $\Phi(\bx)$. Thus, we can think of the $W_{\mathcal{J}_\nu}(\bh)$ as the distribution of `effective' fields (or, so-called `messages') coming from the long-range connections and $\Phi(\bx)$ as those coming from the ring neighborhood.
\subsection{Benchmark tests of the theory}
\label{sec:tests}
Given the complicated structure of our equations, we now wish to
inspect the validity of the theory against simple benchmark tests.
Firstly, in the absence of the `ring' structure, and for
strictly ferromagnetic interactions, the equations must reduce to
those found in Ref.\@ \cite{bergsellito}. Indeed, setting into the
update functions (\ref{eq:update_G},\ref{eq:update_H}) $J_0=0$
and $p=1$ we find that after the first iteration of
(\ref{eq:self_Phi}) the density $\Phi$ collapses to
$\Phi(x_1,x_2,x_3)=\delta(x_1)\phi(x_2)\delta(x_3)$. Details of
the non-trivial function $\phi$ are not important for the purposes
of this section. Filling in this information in the right-hand
side of (\ref{eq:self_W}) leads to several simplifications as a
result of which the dependence of $W_{\mathcal{J}}(\bh)$ on $\phi(x)$ drops out
completely. The resulting closed equation is the one found in
\cite{bergsellito}. Thus, at the level of the self-consistent
equation the expressions reproduce the correct result. Next, we
consider the free energy. Clearly, the energetic part of
(\ref{eq:f_final}) depends only on $W_{\mathcal{J}}(\bh)$ and in the special
benchmark case takes the same form as the energetic term of
\cite{bergsellito}. The entropic term on the other hand, depends
explicitly on the reduced density of fields $\phi(x)$ which is
coupled to the functions $A(x)$ and $K(x,\bh)$, eq.\@ (\ref{eq:A})
and (\ref{eq:K}) respectively. Here, it turns out that one can
write $K(x,\bh)=A(x)\,\tilde{K}(\bh)$ which effectively removes
$A(x)$ and $\phi(x)$ completely from (\ref{eq:f_final}). The
resulting expression reproduces the free energy of
\cite{bergsellito}.
A second test of the theory is against the small-world
thermodynamic analysis of Ref.\@ \cite{theodore,guzai_rsb}. To map
the generating function (\ref{eq:generating_function}) to the free
energy of that system we set $\Theta(x)=1$ for all $x$. This
removes the stability condition from our definitions. After the
first iteration of (\ref{eq:self_Phi}) we now find that the
function $\Phi(\bx)$ collapses to
$\Phi(x_1,x_2,x_3)=\delta(x_1)\tilde{\phi}(x_2)\delta(x_3-\frac12J_0)$ and
using this to iterate (\ref{eq:self_W}) we obtain that
$W(h_1,h_2,h_3)=w(h_1)\delta(h_2)\delta(h_3-J/{2c})$. Thus, in
both cases only one of the three components is non-trivially
distributed. With these relations we recover at the second
iteration of (\ref{eq:self_Phi},\ref{eq:self_W}) the
self-consistent equations of \cite{guzai_rsb}. Equations (\ref{eq:self_Phi},\ref{eq:self_W})
also reduce to (29,30,32) of \cite{theodore} if the analysis of
\cite{theodore} would have been based on symmetric
transfer matrices. In this case the final result
(\ref{eq:f_final}) reduces to the correct free energy.
Finally, by inspection of the physical interpretation of equations
(\ref{eq:self_Phi},\ref{eq:self_W}) and (\ref{eq:f_final}) we can map
our model onto the one of \cite{pagnani} which evaluates the
number of metastable configurations on a Bethe lattice. This can be done by
appropriately converting the Poisson degree distribution to a Bethe-lattice one.
We have done this test numerically and within the limits of precision we find good agreement with
the results of \cite{pagnani}.
\section{Results}
\label{sec:results}
We are now interested in obtaining the energy $e$ and entropy densities $s$
of the metastable states. These can be generated from $f$ through
simple relations, i.e.\@ $e=
\partial_\beta (\beta f)$ and $s=\beta\big(e-f\big)$.
To obtain these we have solved equations (\ref{eq:self_Phi}) and
(\ref{eq:self_W}) through `population dynamics'
\cite{mezardparisi} and used simple Monte Carlo integration
recipes to evaluate (\ref{eq:f_final}) (typically, in executing
population dynamics the population size has been taken of the
order of $10^5$ and we assumed algorithmic equilibration after 100
steps). Since the profiles of $W_\mathcal{J}(\bh)$ and $\Phi(\bx)$
depend on the temperature, differentiation of $f$ with respect to
$\beta$ will involve derivatives of both of these densities. One
of these, namely $\partial_W f\cdot\partial_\beta W$, trivially
vanishes as we are at saddle-points of the order function
$P_\mathcal{J}(\bsigma,\btau)$ and, consequently, also of
$W_\mathcal{J}(\bh)$. The derivative $\partial_\Phi f
\cdot\partial_\beta \Phi$ however, may not necessarily vanish as
we have not extremized $f$ with respect to $\Phi$. Therefore, we
cannot assume that the energy $e=
\partial_\beta (\beta f)$ is given by a simple partial differentiation of
(\ref{eq:f_final}) (which indeed leads to incorrect results). To
proceed analytically, one is required to obtain further closed relations
for $\hat{\Phi}=\partial_\beta\Phi$ which, given the complexity of
the equations involved, appears to be a hard task. Here, we have
chosen to carry out the differentiations numerically.
In all cases we have taken the average connectivity to be $c=2$.
Let us now describe the results. First we take the simplest case
where long-range interactions are of uniform strength, i.e.\@
$p=1$. Thus, the only source of disorder in the system comes
through the connectivity variables $\{c_{ij}\}$. In figure
\ref{fig:energy-beta} we plot the energy against inverse
temperature for three different values of $J_0$ and with $J=1$. In
each case we compare the energy density of the system $e$ when we
allow all configurations to be visited in phase space
(i.e.\@ with $\Theta(x)=1$ for all $x$) against the energy $e_m$
of only the metastable subset of configurations (with
$\Theta(x)=1$ if $x\geq 0$ and 0 otherwise).
Since, by definition, metastable states are minima of the energy
landscape we expect that $e_m\leq e$ at any given temperature
which indeed is verified by the numerics. As we increase the
strength of the short-range couplings $J_0$ the system will
typically require a higher noise level to destroy the order. For
this reason we see that the location of the phase transition
towards low energy values decreases with $\beta$ as $J_0$ is
increased. For $\beta\to\infty$ one can find the ground state
energy of the system by simple inspection of the Hamiltonian
(\ref{eq:H}), namely $e_{\rm gr}=-J_0-J/2$. Furthemore, since in
the regime of low temperatures we expect the system to be in a
locally stable state we can anticipate that $e \approx e_m$. These
physical arguments are in agreement with the numerical results of
figure \ref{fig:energy-beta}. On the other hand, for $\beta \to
0$, noise dominates the microscopic spin dynamics and thus the
energy of the system $e$ typically averages to zero for all
values of $J_0$. We also observe that the transition to the
ordered phase is less smooth for $e_m$ than for $e$. This effect
which has also been reported in \cite{bergsellito} for the special
case of $J_0=0$ is due to non-linearities induced by the Heaviside
function.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{energy-beta-final.eps}
\caption{Comparison of the energy $e$ of a small-world system in
which spin configurations can visit all possible configurations against
the energy $e_m$ of the metastable configurations, plotted as
function of the inverse temperature $\beta$ and for
$J_0=\{0,1,2\}$. We see that $e_m<e$ and that for $\beta\to\infty$
$e_m\approx e$. Parameter values: $J=1$, $p=1$ and $c=2$.
\label{fig:energy-beta}}
\end{center}
\end{figure}
\begin{figure}[h]
\vspace{4mm}
\begin{center}
\includegraphics[width=.45\textwidth]{entropy-energy1.eps}
\caption{Comparison of the entropy $s$ of a small-world system in
which spin configurations can visit all possible configurations against
the entropy $s_m$ of the metastable configurations, plotted
against their energy $e,e_m$, for $J_0=\{0,1,2\}$.
For $J_0=0$ the ground state entropy has a finite value which
vanishes as soon as $J_0>0$. Parameters values:
$J=1$, $p=1$ and $c=2$.\label{entropy-energy1}}
\end{center}
\end{figure}
In figure \ref{entropy-energy1} we plot the entropy against the
energy for different values of $J_0$, and with $p=1$, $J=1$, $c=2$. The
low-energy part of the graph corresponds to regimes of low
temperatures. As before, we compare the entropy $s$ that would
follow from a thermodynamic calculation in the entire
configuration space against the entropy $s_m$ of the metastable
states. First, we see that one always has $s_m\leq s$, as one
would expect. The difference between the two entropies varies
significantly across the energy axis. For instance, for high
energy values (where the temperature is practically infinite) this
difference reaches its maximum value. On the other hand, for low
temperatures, both entropies reach their minimum value which for
any $J_0> 0$ is zero. In the special case where $J_0=0$ the graph
will typically consist of disconnected clusters which causes the
observed degeneracy. However, as soon as the `ring' connects all
spins together this degeneracy is lost and the ground state
entropy is zero.
Let us now examine the case of bond-disorder. In figure
\ref{fig:spinglass2} we present the entropies $s$ and $s_m$
against the energy for different values of $J_0$ and with $J=1$,
$c=2$, $p=1/2$. Firstly, we observe that the ground state energy
is significantly higher compared to the case of $p=1$. This is due
to the value of the local fields which will on average be smaller
for $p<1$ than for $p=1$. We also observe that the ground state
entropy can take a non-zero finite value even at $J_0>0$. This is
due to the presence of anti-ferromagnetic couplings in the system
which increases the fraction of sites with a zero local field.
However, as one increases the strength of the (ferromagnetic)
short-range couplings this fraction of sites becomes smaller and
the degeneracy of the ground state gradually disappears. To
illustrate this effect we plot in figure \ref{fig:minentropy} the
ground state entropy against the short-range coupling strength
$J_0$. We also see a `jump' precisely at $J_0=0$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=.45\textwidth]{spinglass2.eps}
\caption{We show the entropies $s$ and $s_m$ for small-worlds with
bond-disorder. In this case the degeneracy of the ground state
remains finite even for $J_0>0$. Parameter values: $J=1$, $p=1/2$ and $c=2$.}\label{fig:spinglass2}
\end{center}
\end{figure}
\begin{figure}[t]
\vspace{3mm}
\begin{center}
\includegraphics[width=.45\textwidth]{smin_final.eps}
\caption{The ground state entropy density $s_0$ for a small-world system with
bond-disorder as a function of $J_0$. There is a distinct jump
precisely at $J_0=0$ while the entropy remains finite even for
$J_0>0$. Parameter values: are $J=1$,
$p=1/2$ and $c=2$.} \label{fig:minentropy}
\end{center}
\end{figure}
\section{Discussion}
\label{sec:conclusions}
In recent years, the theory of complex networks has witnessed a
remarkable growth. Within the area of complex systems, the special
subset of `small-worlds', has aroused the curiosity of theorists
and experimentalists alike due to the striking cooperativity
phenomena that they allow. In particular, for any value of the
average long-range connectivity (however small), small-world
networks can have a phase transition to an ordered phase at a
finite temperature. Small world architectures have been observed
in a wide range of real complex systems.
For a theorist, several important questions arise regarding the
emergent collective properties on such systems. In this paper we
have evaluated the number of metastable configurations. In a
spirit similar to \cite{gardner,bergsellito,pagnani} the `metastability' condition
constrains the partition sum over configurations in which spins
align to their local fields. From an analytic point of view, two
are the main stumbling blocks: firstly, the non-linear nature of
the stability condition, and secondly, the diagonalisation of the
relevant transfer matrix. Our numerical results suggest that, for
low temperatures and in the case of bond-disorder, the metastable
configurations tend to dominate the space of equilibrium states.
We also see that superimposing the one-dimensional `backbone'
structure leads to a significantly smaller degeneracy of the
ground state (which, in fact, vanishes for strictly ferromagnetic
couplings). As function of the short-range coupling $J_0$, there
is a jump in the ground state entropy exactly at $J_0=0$
which is due to the formation of disconnected clusters within the
graph.
\begin{acknowledgements}
We would like to thank J.\@ Berg, T.\@ Coolen, J.\@ Hatchett, I.\@
P\'erez Castillo, A.\@ Pagnani and G.\@ Semerjian for insightful
discussions. This work is partially funded by the Fund for
Scientific Research Flanders-Belgium.
\end{acknowledgements}
| 2024-02-18T23:39:49.072Z | 2005-10-18T20:02:39.000Z | algebraic_stack_train_0000 | 509 | 6,376 |
|
proofpile-arXiv_065-2630 | \section{Introduction}\label{introduction}
Over the past few years there has been a heightened interest in
testing the statistical properties of Cosmic Microwave Background
(CMB) data. The process has been accelerated by the release of the
WMAP first year results. The WMAP data provide the first ever,
full-sky maps which are signal dominated up to scales of a few
degrees. Thus, for the first time we can test the Gaussianity and
isotropy assumptions of the cosmological signal over large scales in
the sample variance limit.
Ever since the release of the COBE-DMR results \citep{cobe} a
consensus has been hard to reach on tests of non-Gaussianity with some
studies reporting null results \citep{kog96a,contaldi2000,sandvik}
while others claimed detections of non-Gaussian features
\citep{fmg,joao,novikov, pando}. With the release of the WMAP first
year results a limit on the non-Gaussianity of primordial
perturbations in the form of an estimate of the non-linear factor
$f_{\rm NL}$ was obtained by \cite{wmapKomatsu}. However a number of
authors \citep{hbg,erik1,coles,park,copi,teglarge,kate,efstathiou,
football,npoint,jaffe} have also reported analysis of the maps that
suggest violations of the Gaussian or isotropic nature of the signal.
One of problems with testing Gaussianity is that one can devise a
plethora of tests to probe the infinite degrees of non-Gaussianity,
therefore different tests represent different perspectives on the
statistical patterns of the signal. For WMAP there are already a
number of detections of so called anomalies, most pointing to
different unexpected features in the microwave sky. The most
documented case \citep{peiris,efstathiou,slos,teglarge,erik1}
is the low amplitude of the quadrupole and octupole in comparison to
the inflationary prediction, something we can categorize as
{\sl amplitude anomalies}. Although it is simple to design inflationary
spectra with sharp features which reproduce, more or less closely, the
amplitude anomaly (see e.g. \cite{contaldi2003,bridle,salopek}) these
invariably suffer fine tuning problems. Another approach is to relate
the anomaly to the breakdown of statistical isotropy or Gaussianity.
Other reported features relate to the correlation of phases in the
multipole coefficients which are an indication of
non-Gaussianity. These can be dubbed {\sl phase anomalies}. One
example is the hemisphere asymmetries \citep{erik1}; the northern
ecliptic hemisphere is practically flat while the southern hemisphere
displays relatively high fluctuations in the power spectrum. Other
functions, such as the bispectrum \citep{kate} and n-point correlation
functions \citep{npoint} also show related asymmetries. Furthermore,
there is the anomalous morphology of several multipoles, in
particular, the striking planarity of the quadrupole and octupole and
the strong alignment between their preferred directions
\citep{teglarge}. Overall, there is a strong motivation to continue
probing the statistical properties of the data and find possible
sources for these signals, be it instrumental, astrophysical or
cosmological.
The first test to have provided indications of possible non-Gaussian
features in the CMB data was reported by \cite{fmg} and \cite{joao}
using a bispectrum estimator, the fourier analog of the three point
function. Both those detections were later found to be caused by
systematic effects rather than by cosmological source as reported by
\cite{banday}. For the case of the bispectrum signal detected by
\cite{joao}, which used an estimator tuned to detect correlations
between neighbouring angular scales, finding the source of the signal
had to wait for the release of the high precision WMAP data
\citep{wmap} which was able to provide a comparative test of the
cosmological signal. The WMAP data did not reproduce COBE's result and
systematic errors were found to be the cause \citep{joaoes}. The WMAP
data was later analysed with the bispectrum in more detail
by \cite{kate}. In that paper, the bispectrum of the clean, coadded maps
was analysed and a connection between the hemisphere asymmetries in
the 3-point correlation function and the bispectrum was established,
although the full sky as a whole was found to be consistent with
Gaussianity.
In this paper, we study the effect that foreground contaminations have
on bispectrum estimators. In section~\ref{sec:def} we define a
set of bispectrum estimators with set $\ell$ configurations. In
section~\ref{sec:foregrounds} we describe the template dust, free-free
and synchrotron maps used to characterize the effect on the
bispectrum. In section~\ref{sec:method} we determine the distribution
of the estimators in the presence of residual foregrounds with
different amplitudes and discuss the application of this method to
detect residuals in the data by introducing a number of statistical
and correlation measures. In section~\ref{sec:application} we discuss
the application of the the statistical tools developed in the previous
sections to the raw and cleaned WMAP first year maps. We conclude with
a discussion of our method and results in section~\ref{sec:disc}.
\section{The Angular Bispectrum}\label{sec:def}
We now introduce the angular bispectrum estimator \citep{fmg}. The
bispectrum is related to the third order moment of the spherical
harmonic coefficients $a_{\ell m}$ of a temperature fluctuation map
$\Delta T({\bf \hat n})/T$. The coefficients describe the usual
expansion of the map over the set of spherical harmonics $Y_{\ell
m}({\bf \hat n})$ as
\begin{eqnarray}
\frac{\Delta T}{T}({\bf \hat n})=
\sum_{\ell m}a_{\ell m}Y_{\ell m}({\bf \hat n}).
\end{eqnarray}
Given a map, either in pixel space or harmonic space, and assuming
statistical isotropy, one can construct a set hierarchy of
rotationally invariant statistical quantities characterizing the
pattern of fluctuations in the maps. These are the n-point correlation
functions in the temperature fluctuations $\langle\frac{\Delta
T}{T}({\bf \hat m})\frac{\Delta T}{T}({\bf \hat n})...\frac{\Delta
T}{T}({\bf \hat p})\rangle$ or in the spherical harmonic coefficients,
$\langle a_{\ell_1 m_1}a_{\ell_2m_2}...a_{\ell_nm_n}\rangle$.
The unique {\it quadratic} invariant is the angular power spectrum
defined as
$\langle a_{\ell_1 m_1} a^\star_{\ell_2 m_2}\rangle = \delta_{\ell_1\ell_2}\delta_{m_1m_2}C_\ell$, whose estimator can be written as ${\hat
C}_\ell=\frac{1}{2\ell+1}\sum_m|a_{\ell m}|^2$. This gives a measure
of the overall intensity for each multipole $\ell$. Following
\cite{fmg}, the most general {\it cubic} invariant defines the angle
averaged bispectrum,
\begin{equation}
\left<a_{\ell_1 m_1}a_{\ell_2 m_2}a_{\ell_3 m_3}\right>=
B_{\ell_1\ell_2\ell_3}\left (
\begin{array}{ccc} \ell_1 & \ell_2 & \ell_3 \\ m_1 & m_2 & m_3
\end{array} \right ),
\end{equation}
where the $(\ldots)$ is the Wigner $3J$ symbol. Parity invariance of
the spherical harmonic functions dictates that the bispectrum be
non-zero only for multipole combinations where the sum
$\ell_1+\ell_2+\ell_3$ is even. An unbiased estimator (for the full
sky) can be evaluated as
\begin{eqnarray}
{\hat B}_{\ell_1\ell_2\ell_3}&=&\frac{{\cal N}^{-1}_{\ell_1\ell_2\ell_3}}{\sqrt{4\pi}}\sum_{m_1m_2m_3}\left ( \begin{array}{ccc} \ell_1 & \ell_2 & \ell_3
\\ m_1 & m_2 & m_3
\end{array} \right )\times\\
&&a_{\ell_1 m_1}a_{\ell_2 m_2} a_{\ell_3 m_3},\nonumber
\end{eqnarray}
with the normalization factor defined as
\begin{eqnarray}
{\cal N}_{\ell_1\ell_2\ell_3}&=&{\left
(\begin{array}{ccc} \ell_1 & \ell_2 & \ell_3 \\ 0 & 0 & 0\end{array}
\right
)}\times\\&&\sqrt{\frac{(2\ell_1+1)(2\ell_2+1)(2\ell_3+1)}{4\pi}}.\nonumber
\end{eqnarray}
The bispectrum can be related to the three-point correlation functions
of the map just as the power spectrum $C_\ell$ can be related to the
correlation function $C(\theta)$ through the well known expression
\begin{equation}
C(\theta) = \frac{1}{4\pi}\sum_\ell (2\ell+1)C_\ell P_\ell(\cos\theta).
\end{equation}
For example, the pseudo-collapsed,
three-point correlation function, $C^{(3)}(\theta)=\langle
\frac{\Delta T}{T}({\bf \hat n})^2\frac{\Delta T}{T}({\bf \hat m})
\rangle$, is related to our definition of the bispectrum $
B_{\ell_1\ell_2\ell_3}$ as
\begin{equation}\label{3pt}
C^{(3)}(\theta)= \frac{1}{4\pi}\sum_{\ell_1\ell_2\ell_3}{\cal N}_{\ell_1\ell_2\ell_3}B_{\ell_1\ell_2\ell_3}P_{\ell_3}(\cos\theta),
\end{equation}
where ${\bf \hat n}\cdot{\bf \hat m}=\cos\theta$.
It is important to use both tools, the bispectrum and the three-point
correlation function, to probe the sky maps as they have the capacity
to highlight different features of the data. In principle, harmonic
space based methods are preferred for the study of primordial
fluctuations whereas real space methods are more sensitive to
systematics and foregrounds, which are strongly localized in real
space. In addition, the three-point correlation function is intrinsically very
sensitive to the low-$\ell$ modes, whereas the bispectrum can pick up
different degrees of freedom with respect to the different mode
correlations we want to probe
For the choice $\ell_1=\ell_2=\ell_3=\ell$ we can define the
single-$\ell$ bispectrum ${\hat B_\ell}=\hat B_{\ell\, \ell \,\ell}$
\citep{fmg}, which probes correlations between different $m$'s. Other
bispectrum components are sensitive to correlations between different
scales $\ell$. This can be extended to study correlations from
different angular scales. The simplest of these is the $\Delta\ell=1$
inter-$\ell$ bispectrum between neighbouring multipoles defined as
$\hat B_{\ell-1\, \ell \,\ell+1}$ \citep{joao}. It is convenient to
consider estimators normalized by their expected Gaussian variance
$\hat C_{\ell_1}\hat C_{\ell_2}\hat C_{\ell_3}$ which have been shown
to be more optimal and more Gaussian distributed than the unnormalized
estimators, and are not sensitive to the overall power in the
maps. Here we will introduce the $\hat I_\ell$,$\hat J_\ell$, and
$\hat K_\ell$ bispectra defined as
\begin{equation}\label{i3j3}
I^3_\ell = { {\hat B}_{\ell} \over ({\hat C}_{\ell})^{3/2}} , \ \
J^3_\ell = { \hat B_{\ell-1\, \ell \,\ell+1} \over ({\hat C}_{\ell-1}{\hat
C}_{\ell} {\hat C}_{\ell+1})^{1/2}},
\end{equation}
and
\begin{equation}\label{k3}
K^3_\ell = { \hat B_{\ell-2\, \ell \,\ell+2}
\over ({\hat C}_{\ell-2}{\hat C}_{\ell}
{\hat C}_{\ell+2})^{1/2}},
\end{equation}
where have extended the formalism to a separation $\Delta\ell=2$ to
probe signals with both odd and even parity in the inter-$\ell$
correlations.
\section{Foreground Templates}\label{sec:foregrounds}
The standard method of foreground removal used by cosmologists
makes use of a set of template maps for each of the dominant
sources of foreground contamination in the CMB frequency maps.
These are maps obtained from independent astronomical full-sky
observations at frequencies where the respective mechanisms of
emission are supposed to be dominant. These templates are the H
$\alpha$ map \citep{halpha}, for the free-free emission, the 408
MHz Haslam map \citep{haslam}, for the synchrotron emission, and
the FDS 94 GHz dust map \citep{FDS}. These are then subtracted
from the WMAP data with coupling coefficients determined by cross
correlating with the observed maps in the Q (41 GHz), V (61 GHz),
and W (94 GHz) bands. Nevertheless the templates are a poor
approximation of the of the real sky near the galactic plane, so a
Kp2 mask must still be used in the analysis. The method is
described in \cite{wmapfor} and \cite{komatsu};
\begin{eqnarray}\label{eq:amp}
\overline{T}_{Q} &=& T_{Q} - 1.044\,[1.036\, T^{\rm FDS} +
\frac{1.923}{\eta}\,T^{H\alpha} \nonumber\\ &&+1.006\,T^{\rm
Sync}],\nonumber\\ \overline{T}_{V} &=& T_{V} - 1.100\,[0.619\,
T^{\rm FDS} + \frac{1.923}{\eta} \,\left(\frac{\nu_{V}}{
\nu_{Q}}\right)^{-2.15}\,T^{H\alpha} \nonumber\\ && +1.006
\,\left(\frac{\nu_{V}}{ \nu_{Q}}\right)^{-2.7}\,T^{\rm Sync}],\\
\overline{T}_{W} &=& T_{W}- 1.251[0.873\, T^{\rm FDS} +
\frac{1.923}{\eta}\, \left(\frac{\nu_{W}}{
\nu_{Q}}\right)^{-2.15}\,T^{H\alpha} \nonumber \\
&&+1.006\,\left(\frac{\nu_{W}}{ \nu_{Q}}\right)^{-2.7}\,T^{\rm
Sync}],\nonumber
\end{eqnarray}
where $ \eta$ is a correction factor due to reddening in the free-free
template and $\nu_{Q}=40.7$ GHz, $\nu_{Q}=60.8$ GHz and $\nu_{W}=93.5$
GHz . The values in front of the left bracket convert the detector's
temperature to thermodynamic temperature. It is considered that this
is a sufficiently good method to remove the foregrounds outside the
Kp2 plane since it matches the correct amplitudes quite well, however
the usual doubts remain, especially in the light of the alignment/low
multipoles controversies. Another point one can make is that
whereas this may be a satisfactory technique to correct the
foregrounds at the power spectrum level, its effect on higher order
statistics is unknown and may actually induce unexpected correlations.
\section{The effect of foregrounds on the bispectrum}\label{sec:method}
\begin{figure*}
\centerline{\psfig{file=fig1.ps,angle=270,width=15cm}}
\caption{ The average functions for the angular spectra of the
simulations. Black (solid) is Gaussian, red (short-dashed) is for the
contaminated simulations with $\alpha=0.5$ and blue (long-dashed) is
for the contaminated simulations with $\alpha=1.0$. The top panel
shows the power spectrum $\hat C_\ell$, second panel shows the
single-$\ell$ bispectrum $I^3_\ell$, the third panel the
$\Delta\ell=1$ inter-$\ell$ bispectrum $J^3_\ell$ and the bottom
panel shows the $\Delta\ell=2$ inter-$\ell$ bispectrum
$K^3_\ell$. The shaded regions represent the Gaussian variance
measured directly from the ensemble of simulations. For the Gaussian
simulations, the average power spectrum is just the input $\Lambda$CDM
power spectrum and the average bispectra is effectively zero. On the
other hand, the average angular spectra of the contaminated
simulations have an emerging pattern of intermittency in both second-
and third-order statistics. This intermittent pattern comes about
due to the even parity of galactic foregrounds, ie, even modes
are enhanced relatively to the odd modes. This can be seen in the
significant increase of power in the even modes of the power
spectrum. In terms of the bispectrum, we see that the $\Delta\ell=0$
and the $\Delta\ell=2$ inter-$\ell$ components will be more
significantly enhanced than the $\Delta\ell=1$ inter-$\ell$
bispectrum because the latter includes correlations between even and
odd modes.}\label{fig:bisp}
\end{figure*}
We have generated a set of 3000 Gaussian, CMB simulations of the WMAP
first year Q, V, and W maps in
\textsc{HEALPix}\footnote{http://healpix.jpl.nasa.gov}\citep{healpix}
format with a resolution parameter $N_{\rm side}=512$. Each simulation
is smoothed with the Q, V and W frequency channel beams and channel
specific noise is added. We adopted the WMAP best-fit $\Lambda$CDM
with running index power spectrum\footnote{http://lambda.gsfc.nasa.gov} to generate the $a_{\ell m}$
coefficients of the maps. The Kp2 galactic mask is imposed on each
map. The masked maps are then decomposed into spherical harmonic
coefficients $a_{\ell m}$ using the \textsc{Anafast} routine. We then
calculate the four spectra; namely the the power spectrum $\hat
C_\ell$, single-$\ell$ bispectrum $I^3_\ell$, $\Delta\ell=1$
inter-$\ell$ bispectrum $J^3_\ell$ and $\Delta\ell=2$ inter-$\ell$
bispectrum $K^3_\ell$ as described in section~\ref{sec:def}.
We then add channel-specific foregrounds outside the Kp2 zone to the
same set of Gaussian simulations with amplitudes set as in
Eqn.~(\ref{eq:amp}). The addition of the foreground is scaled linearly
by a factor $\alpha$ as
\begin{equation}
T_{\rm \{Q,V,W\}} = T_{\rm CMB} + \alpha\overline T_{\rm \{Q,V,W\}},
\end{equation}
which we use to check the sensitivity of the bispectra to the
foregrounds (typically $\alpha=1.0 $ or $ \alpha=0.5$). The power
spectrum and bispectra are then calculated for the set of
contaminated maps.
In Fig.~\ref{fig:bisp} we show the mean angular spectra of the
simulations obtained by averaging over the ensembles. We show the mean
spectra for the Gaussian (solid, black) and the contaminated
simulations for $\alpha=0.5$ (short-dashed, red) and $\alpha=1.0$
(long-dashed, blue). The shaded area shows the variance of the three
bispectra obtained directly from the Gaussian simulations.
We see that even for the fully contaminated set of maps ($\alpha=1.0$)
the average signal is not significantly larger than the expected
Gaussian variance indicating that a detection would require averaging
over a large number of modes. However we see some important
distinguishing features in the signal in that it is sensitive to the
parity of the multipole, being suppressed for odd $\ell$. This is due
to the approximate symmetry of the foreground emission about the
galactic plane which means that most of the signal will be in even
$\ell$ modes since these have the same symmetry. This effect can be
seen in all the spectra but most significant is the suppression of the
odd inter-$\ell$ bispectrum $J^3_{\ell}$ with respect to the even
inter-$\ell$ bispectra $I^3_{\ell}$ and $K^3_{\ell}$.
Another obvious feature of the even parity nature of the signal
is the correlation between the spectra. In particular the absolute
values of the $I^3_\ell$ and the $K^3_\ell$ are correlated with
the structure visible in the fully contaminated power
spectrum.
Overall the $K^3_\ell$ is the most sensitive statistic with the
largest amplitude with respect to the Gaussian variance although
still quite small even at $50\%$ contamination. We now describe a
number of statistical estimators we use to test the detectability
of the template matched foregrounds in the Q, V, and W channel
maps.
\subsection{Chi-Squared Test}\label{sec:chisq}
Having seen how foregrounds affect the angular statistics of CMB
maps, we can now devise specific tests to probe these properties on
the bispectrum and test their sensitivity. The standard way to use
the bispectrum as a test of general non-Gaussianity is to use a
reduced $\chi ^2 $ statistic \citep{joao,joaoes,kate}. This is defined
as
\begin{equation}
\chi^2={1\over N_\ell}{\sum_{\ell = \ell_{\rm min}}^{\ell_{\rm max}}
\chi_\ell^2}= {1\over N_\ell} \sum_{\ell = \ell_{\rm min}}^{\ell_{\rm
max}} { {( X_\ell-\langle X_\ell\rangle)^2} \over
{\sigma_\ell^2} }.
\end{equation}
where $ X_{\ell} $ is a given bispectrum statistic, $\langle
X_\ell\rangle$ is its mean value computed over the Monte Carlo
ensembles, and $\sigma_\ell^2$ is the variance for each angular
scale. The $\chi^2 $ test is a measure of the deviation of the
observed data from the expected mean, weighted by the Gaussian
variance of the estimator.
Foregrounds increase the amplitude of the bispectra foregrounds, but
as shown in Fig.~\ref{fig:bisp}, we can see that only $K^3_\ell$ seems
to stand of chance of significant detections since the average
amplitude of the signal is comparable to the variance, unlike the
other components of the bispectrum.
The detectability of the template matched signals using any of the
bispectra can be tested by comparing the distribution of the $\chi
^2 $ values obtained from the contaminated simulations with that
obtained from Gaussian simulations. We compute the $\chi^2 $
values for the contaminated maps using the mean and the variance
obtained from the Gaussian simulations, ie, the expected Gaussian
functions.
We compare the distribution of the $\chi^2 $ values for the Gaussian
simulations against the distribution obtained for the simulations with
contamination ($\alpha=1.0$). We concentrate on the Q band since it is
the most contaminated frequency. The histograms of the $\chi^2$ are
shown in the left column of Fig.~\ref{fig:histo}. For the $I^3_\ell$
and $J^3_\ell$ spectra the histograms overlap completely. This means
that the probability of finding contaminated simulations with a high
$\chi^2$ is the same as for the Gaussian simulations indicating that
the $\chi ^2 $ test is insensitive to the presence of foreground
contaminations at this level. However the $K^3_\ell$ spectrum tells a
different story. There is a significant shift between the two
distributions which implies that this component of the bispectrum has
more sensitivity to foregrounds.
The sensitivity can be quantified in terms of the fraction of the
contaminated simulations (with $\alpha=1.$) with a $\chi^2$ larger
value than 95.45 $\%$ (i.e. 2 $\sigma$) of the Gaussian simulations
(with $\alpha =0$). The sensitivity for the $I^3_\ell$ and $J^3_\ell$
is low (< 0.05), whereas for $K^3_\ell$ the fraction increases to
0.355.
\subsection{Template Correlation Test}\label{sec:correlation}
A template matched statistic can be defined by correlating the
observed bispectra in the data with those of the foreground
templates. This is more sensitive to the structure in the template
signal as opposed to the $\chi^2$ test introduced above. We define
a cross correlation statistic $\rho$ as
\begin{equation}\label{corr_coef}
\rho=\frac{\sum_{\ell = \ell_{\rm min}}^{\ell_{\rm max}}{ X_{\ell} X^{F}_{\ell} }}{\left(\sum_{\ell = \ell_{\rm
min}}^{\ell_{\rm max}} X_{\ell}^2\sum_{\ell = \ell_{\rm
min}}^{\ell_{\rm max}} X^{F\,2}_{\ell}\right)^{1/2}}
\end{equation}
where $X_{\ell}$ are the bispectra obtained from the data and
the $X_{\ell}^{F}$ are those obtained from the foreground
templates.
In the middle column of Fig.~\ref{fig:histo} we display the histograms
for the $\rho $ values for the Gaussian simulations against the
distribution obtained for the contaminated ($\alpha=1$) simulations of
the Q band maps. The sensitivity has improved over the $\chi^2 $ test,
with the histograms of the input and output data sets being clearly
shifted, meaning that there is a higher probability of detection of
foregrounds using this method. Again the effect is stronger in the
$K^3_\ell$. This result simply quantifies the statement that a matched
template search for a contamination signal is more sensitive than a `blind'
statistic such as the $\chi^2$ test. The values for the sensitivity of the
test are given in table~\ref{tab:corr} for all three WMAP bands.
\subsection{Power Spectrum and Bispectra Cross-Correlation Test}\label{sec:rstat}
For a Gaussian field, the normalized bispectrum is statistically
uncorrelated with the power spectrum \citep{conf}. However,
foreground residuals in the map induce non-Gaussian correlations which
in turn will induce correlations between the normalized bispectra and
the power spectrum of the maps. This can provide another specific
signature that one can use to detect the presence of foreground
contamination.
For Gaussian simulations, the average power spectrum
is just the input $\Lambda$CDM power spectrum and the bispectrum is
effectively zero. On the other hand, the average angular spectra of
the contaminated simulations have an emerging pattern of intermittency
in both first- and second-order statistics. Correlations between the
power spectrum and the bispectra therefore come about due to the even
parity induced by the characteristic galactic foregrounds. This means
that the even modes of the power spectrum will be correlated
with the even modes of the bispectra, whereas odd modes will remain
uncorrelated. In order to test this effect on the maps, we introduce
the ${\rm R}$ correlation statistic defined as
\begin{equation}
{\rm R}^X = \sum^{\ell_{\rm max}}_{\ell=\ell_{\rm min}}(-1)^{{\rm
int}[\frac{\ell}{2}]+1} \hat C_\ell X_\ell
\end{equation}
where $\hat C_\ell$ is the observed power spectrum. We have chosen
$\ell_{\rm max}=30$ and $\ell_{\rm min}=4$ as we are interested in the
large angular scales where the effects of foreground contamination
will dominate. We use the absolute value of the bispectrum in order to
avoid the discrimination between negative and positive correlations
which would affect our sum. We are only interested in the
discrimination between the existence of absolute correlations against
null correlations between the $\hat C_\ell $ and the bispectra $X_\ell$.
Again we test the sensitivity of this method by computing a
distribution of ${\rm R}$ for Gaussian ensembles against the
contaminated ensembles. We make sure that for Gaussian ensembles we
use the correlation of $\hat C^{\rm S+F}_\ell$ with $X^{\rm S}_\ell$
and for the contaminated ensemble the correlation of $\hat C^{\rm
S+F}_\ell$ with $X^{\rm S+F}_\ell$ where $\rm S$ stands for the
Gaussian CMB signal and $\rm S+F$ indicates contaminated
ensembles. This allows us to cancel the effect of the increase of
power due to foregrounds in the correlation of the two statistics
between the two tests. The results for the contaminated ensemble,
$\alpha=1$, are plotted in the right column of Fig.~\ref{fig:histo}
and are summarized in table~\ref{tab:corr} for all three bands.
\begin{table}
\caption{ Sensitivity of the $\rho$ and ${\rm R}$ tests in terms of
the fraction of the contaminated simulations ($\alpha=1.0$) with a
larger value than 95.45 $\%$, ie 2 $\sigma$, of the Gaussian
simulations ($\alpha =1.0$). We present values for the Q,V and W
frequency channels and for the $I^3_\ell$, $J^3_\ell$ and
$K^3_\ell$. Note that $\rho(X_\ell)$, where $X_\ell$ is a given
bispectrum component stands for the correlation between $X_\ell(\rm
data)$ with$X_\ell(\rm template)$, whereas ${\rm R}(X_\ell)$,
represent the correlation of that specific bispectrum component with
the respective power spectrum of the map. The values in the table
quantify what can be seen in the histograms in
figure~\ref{fig:histo}. Applying the tests for the $K^3_\ell$
component provides better sensitivity to the foregrounds. Between
the two tests $\rho$ seems to provide a marginally better
sensitivity. Also, the Q channel, being the most foreground
contaminated yields the higher chances of detection.}
\begin{tabular}[b]{|c|c|c|c||c|c|c}
\hline & \multicolumn{1}{|c|}{ $\rho_{\rm Q}$ } &
\multicolumn{1}{|c|}{ $\rho_{\rm V}$ } & \multicolumn{1}{|c|}{
$\rho_{\rm W}$ } & \multicolumn{1}{|c|}{ ${\rm R}_{\rm Q}$ } &
\multicolumn{1}{|c|}{ ${\rm R}_{\rm V}$} & \multicolumn{1}{|c|}{
${\rm R}_{\rm W}$ } \\
\hline\hline
$I^3_\ell$ & 0.541& 0.085 & 0.139 & 0.280& 0.030 & 0.080 \\
$J^3_\ell$ & 0.225& 0.100 & 0.091 & 0.080& 0.060 & 0.060 \\
$K^3_\ell$ & 0.714& 0.072& 0.113& 0.690& 0.290 & 0.110 \\
\hline
\end{tabular}
\label{tab:corr}
\end{table}
\begin{figure*}
\centerline{\psfig{file=fig2.ps,width=15cm}}
\caption{ Distributions of values obtained for the three different
tests applied to the Q channel to detect the presence of foregrounds
($\chi^2$, $ \rho $ and ${\rm R}$), for Gaussian simulations (black)
and contaminated ($\alpha=1.0$) simulations (grey). The level of
sensitivity of a given method can be determined in terms of the shift
between the histograms for the Gaussian and the contaminated case. We
see that the $\rho^{K}$ and ${\rm R}^K$ statistics are the most
sensitive channels to probe the existence of foregrounds.}
\label{fig:histo}
\end{figure*}
\begin{table}
\caption{ Results for the WMAP data for $\rho$ and ${\rm R}$. The
results are shown as the fraction of Gaussian simulations below the
level obseverved in the data. We have highlighted values with
greater than 98\% in the foreground cleaned maps.}
\begin{tabular}[b]{|c|c|c|c|c|c|c}
\hline & \multicolumn{3}{|c|}{ RAW } & \multicolumn{3}{|c|}{
CLEANED}\\
\hline & \multicolumn{1}{|c|}{
${\rm Q}$ } & \multicolumn{1}{|c|}{ ${\rm V}$} &
\multicolumn{1}{|c|}{ ${\rm W}$ }& \multicolumn{1}{|c|}{ ${\rm Q}$ }
& \multicolumn{1}{|c|}{ ${\rm V}$} & \multicolumn{1}{|c|}{ ${\rm W}$
}\\
\hline\hline
${\rm R}^I$ & 0.726 & 0.178 & 0.328 & 0.475 & 0.421 & 0.775 \\
${\rm R}^J$ & 0.758 & 0.802 & 0.749 & 0.822 & 0.869 & $\fbox{0.983}$\\
${\rm R}^K$ & 0.983 & 0.450 & 0.486 & 0.362 & 0.364 & 0.188 \\
\hline
$\rho^{I}$ & 0.998 & 0.408 & 0.762 & 0.491& 0.550& 0.452\\
$\rho^{J}$ & 0.933 & 0.906 & 0.856 & $\fbox{0.998}$& $\fbox{0.985}$ & $\fbox{0.986}$\\
$\rho^{K}$ & 0.922 & 0.166 & 0.272 & 0.013 & 0.021 & 0.044 \\
\hline
\end{tabular}
\label{tab:data}
\end{table}
\section{Application to the WMAP data}\label{sec:application}
We have applied the statistical tools described above to the WMAP
first year data \citep{wmap}. We considered both the raw and
cleaned maps of the Q, V, and W channels
using the Kp2 exclusion mask. We summarise the results in
table~\ref{tab:data} showing the separate confidence limits from
each channel for both the raw and cleaned maps.
For the raw maps we find that only the Q channel ${\rm R}^K$
result is above the 95\% threshold while for the Q channel $\rho$
statistic, all confidence levels are above the 90\% level with the $\rho^I$
above the 95\% level. This is consistent with there being a component
most correlated to the foreground templates at the lowest frequencies
and with significant correlations between the $\Delta\ell= 2$ inter-$\ell$
bispectrum and the power spectrum. Since the raw maps do not have any
foreground subtracted from them this is not a surprise although the
confidence level suggests that the correlations are larger than what
was found for the expected amplitude ($\alpha=1$) of the foregrounds.
For all $I^3_\ell$ and $K^3_\ell$ statistics the cleaned map results
show confidence levels below the 95\% level and indeed show an overall
reduction in the significance of the correlations, indicating that the
cleaning has removed a component correlated to the foreground
templates, as one would expect. However for the $J^3_\ell$ statistics,
which should in principle be the least sensitive to the foregrounds
considered, we see that the confidence levels have all
increased. Indeed all three channels now have correlations significant
above the 95\% level in the $\rho$ statistic with the W channel also
having a~$>95\%$ confidence level. The cleaning algorithm appears to
have introduced significant correlations with the foreground templates
in the $\Delta\ell=1$ inter-$\ell$ bispectra and significant correlations
between the $\Delta\ell=1$ inter-$\ell$ bispectrum and power spectrum of
the W channel which is indicative of a non-Gaussian component.
In figure~\ref{fig:bisp_data} we show the bispectra for each cleaned
channel map and compare to the bispectra of the foreground
template ($\alpha=1$) for each channel. This shows the nature of
the result above. For both the $I^3_\ell$ and $K^3_\ell$ the
cleaned map bispectra are anti-correlated with the foreground
templates. In addition the the $K^3_\ell$ for all channels are
heavily suppressed in the cleaned maps for multipoles $\ell < 20$
compared to the expected Gaussian variance shown in
figure~\ref{fig:bisp}. The $J^3_\ell$ gives the only bispectra that
are correlated with the those of the templates.
Figure~\ref{fig:data} shows the break down of the $R^J$ result into
individual multipole contributions for each of the three bands. In
particular it is interesting to note how the W band ${\rm R}^J$ result
shown in table~\ref{tab:data} is dominated by an outlier at $\ell=26$.
\begin{figure*}
\centerline{\psfig{file=fig3.ps,angle=270,width=15cm}}
\caption{ The bispectra for each cleaned channel map (solid line)
against the bispectra of the foreground template ($\alpha=1.0$)
(dashed lines). The three frequency channels are shown as Q (red), V
(black) and W (blue). Both the cleaned map $I^3_\ell$ and
$K^3_\ell$ bispectra are anti-correlated with the foreground
templates. In addition the the $K^3_\ell$ for all channels are heavily
suppressed in the cleaned maps for multipoles $\ell < 20$ compared to
the expected Gaussian variance shown in figure~\ref{fig:bisp}. The
$J^3_\ell$ gives the only bispectra that are correlated with the those
of the templates.} \label{fig:bisp_data}
\end{figure*}
\begin{figure*}
\centerline{\psfig{file=fig4.ps,angle=270,width=15cm}}
\caption{ ${\rm R}^J$ as a function of angular scale, $\ell$. The
values displayed correspond to the foreground cleaned Q (red squares),
V (black triangles) and W (blue circles) frequency channels. The
values are offset by $\ell=0.25$ and $\ell=0.5$ for the Q and W bands
respectively. The $3 \sigma$ detection in the W foreground-cleaned
channel is dominated mainly by the $\ell=26$ mode. The error bars are
computed from 3000 Gaussian simulations assuming the specific channel
noise and beam.}
\label{fig:data}
\end{figure*}
\section{Discussion}\label{sec:disc}
At first sight our results appear contradictory. We have studied the
effect of foreground contamination on the maps and concluded that
foregrounds mainly affect the $I^3_\ell$ and $K^3_\ell$ components
of the bispectrum due to its parity. By comparing the results for
the raw and the foreground-cleaned maps, we are able to verify that
the amplitude of $I^3_\ell$ and $K^3_\ell$ reduces as expected after
foreground subtraction.
On the other hand, as shown in table~\ref{tab:data}, the correlations
induced in the $J^3_\ell$ appear to be close to inconsistent to a
Gaussian hypothesis with the correlation with the foreground templates
at a significance above the $3\sigma$ level for the Q-band, cleaned
map. It is also of interest to note that the cleaned maps do worse in
all bands for the $\rho$ measure.
This is not what we naively expected since the foregrounds considered
here have the wrong parity and their $J^3_\ell$ signal is heavily
suppressed. However the cleaning procedure used by the WMAP team {\sl
does} appear to increase the correlations $\rho$ of $J^3_\ell$
bispectrum to the input maps and its correlation $\rm R$ with the
power spectrum. Recall that we expect the normalized bispectra to be
independent of the power spectrum only in the Gaussian case.
The possibility of the foregrounds being more complex than accounted
for in this type of treatment is to be considered carefully as this
work has shown. The results shown here would suggest that the
procedure used to go from the {\sl raw} to {\sl cleaned} WMAP maps is
under or over subtracting a component with $\ell\pm 1$ parity in the
bispectrum. This is probably not an indication that the procedure is
faulty but rather that the templates used are not accurate enough to
subtract the foregrounds. One source of inaccuracy is the simple
scaling of the templates with respect to frequency. The {\sl cleaned}
maps are obtained assuming uniform spectral index and \cite{wmapfor}
acknowledge that this is a bad approximation particularly for the 408
MHz Haslam (synchrotron) template. This is seen when producing the
Internal Linear Combination (ILC) map which accounts for variation of
the spectral index of the various component. Unfortunately ILC maps
cannot be used in quantitative studies as their noise attributes are
complicated by the fitting procedure and one cannot simulate them
accurately.
Future WMAP ILC maps or equivalent ones obtained by
`blind' foreground subtraction \citep{tegclean,erik2} may be
better suited for this kind of analysis once their statistical
properties are well determined. It is expected that the impending
second release of WMAP data will allow more accurate foreground
analysis and the statistical tools outlined in this work will be useful in
determining the success of foreground subtraction.
It may be worthwile to include information of the higher order
statistics when carrying out the foreground subtraction itself, for
example by extending the ILC method to minimise higher order map
quantities such as the skewness and kurtosis of the maps.
\section*{Acknowledgments}
We thank H. K. Eriksen for advice and for making the
simulations available to us. We are also grateful to Jo\~ao
Magueijo, Kate Land and A.J. Banday for useful conversations
throughout the preparation of this work. Some of the results in
this paper have been derived using the HEALPix package. J.
Medeiros acknowledges the financial support of Fundacao para a
Ciencia e Tecnologia (Portugal).
| 2024-02-18T23:39:49.293Z | 2005-10-30T17:57:00.000Z | algebraic_stack_train_0000 | 523 | 5,995 |
|
proofpile-arXiv_065-2631 | \section{Introduction}
The origin of exponential disks has been difficult to observe
directly because the earliest examples of clear spiral galaxies
already have them (O'Neil, Bothun, \& Impey 2000; P\'erez 2004;
Elmegreen, Elmegreen, \& Sheets 2004; Toft et al. 2005).
Exponential disks have also been difficult to model in cosmology
simulations (van den Bosch 2001; Robertson et al. 2004), although
models with high numerical resolution (Governato et al. 2004),
star formation feedback (Westera et al. 2002; Abadi et al. 2003),
or nuclear black-hole feedback (Robertson et al. 2005) have made
exponential disks.
In two previous surveys, we distinguished between spiral galaxy
disks, which have bulges, spiral arms, and approximately
exponential profiles, and highly irregular disks, which we called
clump cluster galaxies (Elmegreen, Elmegreen \& Sheets 2004;
Elmegreen et al. 2005b). The association of clump clusters with
disks was made on the basis of models for the distribution of
axial ratios (Elmegreen, Elmegreen, \& Hirst 2004a; Elmegreen \&
Elmegreen 2005; Elmegreen et al. 2005b). Edge-on clump clusters
are apparently chain galaxies, which also have no central bulges
or exponential disks, and have similar clumps (Elmegreen,
Elmegreen, \& Hirst 2004a). In a study of 10 clump-cluster
galaxies, the average clump mass was found to be $\sim6\times10^8$
M$_\odot$ and the average clump diameter was 1.8 kpc (Elmegreen \&
Elmegreen 2005).
Clump clusters, by definition, do not have central light
concentrations resembling bulges, nor do they have exponential
disks along any radial profile. Some clump clusters, however, have
nearly exponential disks in their average radial light profile,
even though the major axis profile is irregular. Generally, the
elliptical contours for clump cluster galaxies cannot be
determined automatically by the IRAF routine {\it ellipse} because
the galaxy disks are too irregular. Still, we can use a constant
ellipticity and position angle to determine the average radial
profile, and obtain the values of the center, ellipticity and
position angle from the outer $2\sigma$ contour, where $\sigma$ is
the image noise level. Three examples of intensity profiles are
shown in Figure \ref{fig:profile}, along with the galaxy images.
On the left is a spiral with exponential profiles for both the
major axis (top panel) and the ellipse fits. In the central column
is a clump cluster with a typically irregular major axis profile
but a somewhat exponential elliptically-averaged profile. On the
right is another clump cluster with no exponential at all.
Exponential light profiles may be the result of cosmological
collapse during galaxy formation (Freeman 1970; and see references
above) or the result of radial flows in viscous, star-forming
disks (e.g., Lin \& Pringle 1987; Yoshii \& Sommer-Larsen 1989;
Zhang \& Wyse 2000; Ferguson \& Clarke 2001). In the collapse
simulations, the gas may be accreted smoothly from the halo
(Murali et al. 2002; Westera et al. 2002; Sommer-Larsen, G\"otz,
\& Portinari 2003; Keres et al. 2005) or erratically from
cannibalized dwarf companions (Walker, Mihos, \& Hernquist 1996;
Abadi et al. 2003) and intergalactic clouds (Brook et al. 2004).
In the viscous models, the inner regions tend to move in, making a
central concentration, while the outer regions tend to spread out,
making an exponential in the outer disk. The time scale for the
development of the outer exponential can be long, comparable to
the star formation time, if the initial disk is significantly
flatter than exponential or if it is highly irregular. The
presence of exponential disks in local dwarf irregular galaxies
(Hunter \& Elmegreen 2005), where there is little shear for
viscous effects, and the early appearance of exponentials in
high-redshift spirals, suggest that this profile is related to
galaxy assembly, not shear.
In order to understand star formation in the disks of young
galaxies, we have been studying the properties of the bright blue
clumps which dominate the appearance of clump clusters (Elmegreen
\& Elmegreen 2005). Clump masses can be as large as $10^9$
M$_\odot$. Most of these clumps have average densities only
slightly larger than the critical tidal densities in their disks,
so they should eventually disperse. Many also have tails or other
structures which make them look like they are in the process of
dispersing. As these galaxies also look very young (van den Bergh
et al. 1996; Conselice et al. 2004), clump dispersal in
clump-cluster galaxies could offer a clue to the origin of
exponential disks.
Here we show that even though the radial light profiles of clump
cluster galaxies are not exponential, the average radial profiles
of the clump positions are exponential, as are the average radial
profiles of the integrated clump fluxes. Thus, these galaxies may
evolve into exponential disks by the dissolution of the observed
clumps. In that case, it may be concluded that cosmological
accretion promotes exponential disks whether the gas comes in
smoothly or in the form of giant clumps. If the accretion is
smooth, then the disk has to be highly turbulent so that
gravitational instabilities form clouds and stellar clumps (Immeli
et al. 2004) with a Jeans mass as large as a clump mass.
\section{Small Exponential Scale Lengths in UDF Spirals}
The spiral or disk-like galaxies in our morphological catalog of
the Hubble Space Telescope Ultra Deep Field (UDF; Beckwith et al.
2005) were fit to elliptical isophotes using IRAF (Elmegreen et
al. 2005b). We considered only objects larger than 10 pixels in
diameter on the i$_{775}$ image. Those which had exponential light
profiles, spiral arms, and central bulges were designated as
spirals (269 galaxies) and those which did not were designated as
clump clusters (178 of them). All UDF spirals have an exponential
disk with a central bulge, although most of these bulges are
small.
For the spirals, we determined the exponential scale lengths in
pixels using the median value of the local slope on a log-linear
plot of ellipse-fit intensity versus radius in i$_{775}$ band.
This wavelength was chosen because it has the deepest UDF
exposures. The central pixel was not included in the ellipse fits.
The median technique avoids the small scale length of the bulge
region and possible scale length shifts in the far outer region.
The result is a good approximation to the scale length of the
main-disk exponential. Figure \ref{fig:samples} shows 4 examples
of the radial intensity profiles. The dotted lines are the fits
and the dashed horizontal lines are the 2$\sigma$ background noise
levels. We do not discuss here double exponential disks or disk
truncations, such as those observed in local galaxies by Kregel \&
van der Kruit (2004), Erwin, Beckman, \& Pohlen (2005), and
others. Such profiles are present in our UDF sample and they have
also been noted for other high-redshift galaxies by P\'erez
(2004).
Figure \ref{fig:rdisk} shows the distribution function of the
spiral galaxy scale lengths measured in pixels. Although the UDF
spirals probably span a wide range of redshifts, this distribution
function has physical meaning because the conversion between
angular size and physical size is relatively flat at redshifts
from 0.5 to 5. The conversion factor is shown in Figure
\ref{fig:chains6e} for the WMAP $\Lambda$CDM cosmology (Spergel et
al. 2003). The peak at $\sim7$ pixels in Figure \ref{fig:rdisk}
converts into 0.21 arc seconds for the ACS camera, in which case
Figure \ref{fig:chains6e} (dashed line) suggests that most spirals
have an exponential scale length of around 1.5 kpc, to within a
factor of $\sim1.5$.
This resultant average of $\sim1.5$ kpc for the UDF is smaller
than the average exponential scale length in local spiral
galaxies. van der Kruit (1987) obtained a local scale length
distribution with a peak at $3\pm2$ kpc for 42 galaxies having
known distances; the 31 spirals in his sample had an average scale
length of 4 kpc and the 11 dwarfs had an average scale length of 1
kpc. de Jong (1996a) observed B and K-band scale lengths somewhat
uniformly distributed between 1 and 7 kpc for 86 local galaxies,
but the volume-corrected distribution, which included substantial
corrections for missing small galaxies, peaked at 1.5 kpc and
extended from 0.8-11 kpc. Courteau (1996) found a scale length
distribution for 290 local Sb-Sc galaxies that peaked at $\sim3.5$
kpc with a range between 1.5 and 6 kpc. McGaugh \& Bothun (1994)
obtained a similar range of exponential scale lengths (1.2-6.3
kpc) for low surface brightness galaxies, as did de Blok, van der
Hulst, \& Bothun (1995), who got a median of 3.2 kpc. For local
dwarf galaxies, Hunter \& Elmegreen (2005) obtained an average
V-band scale length of 1 kpc for 94 galaxies of type Im, and 1.7
kpc for 18 galaxies of type Sm. Similar sizes were obtained in a
survey of 171 dwarf galaxies by Swaters \& Balcells (2002), who
found the distribution of scale lengths peaked at $\sim1$ kpc and
extended from approximately 0.2 to 4 kpc. Evidently, the UDF
spirals are systematically smaller than local spirals by a factor
of $\sim2$, and comparable in size to local spiral dwarfs (type
Sm). The local dwarf Irregulars (type Im) may be slightly smaller
than the UDF spirals.
The exponential scale length is a better measure of spiral galaxy
size than outer isophotal radius, which depends on the surface
brightness relative to the survey limit and varies with redshift
through cosmological dimming. The scale length may depend on
redshift through band shifting, as local disks have $\sim20$\%
larger B band scale lengths than K-band (de Jong 1996b; de Grijs
1998). Such color gradients typically come from star formation,
metallicity, and extinction gradients. The local-disk color
gradients are small, however, and if present at high z, would
increase the observed sizes, rather than decrease them, as the
i$_{775}$ band shifts to the rest frame blue. The scale length
also depends only slightly on bulge-disk decomposition, becoming
smaller if the fitted exponential includes the bulge (Courteau
1996). This is not a consideration here because the median local
slope of the profile avoids the steeper slope of the bulge region,
as mentioned above, and because the UDF spirals do not have such
prominent bulges as local early-type galaxies.
de Jong (1996a) and Beijersbergen, de Blok, \& van der Hulst
(1999) noted that local galaxies have a slight correlation between
extrapolated central disk surface brightness and exponential scale
length in the sense that higher surface brightness galaxies are
intrinsically smaller (see also McGaugh \& Bothun 1994). The
difference in central surface brightness between local galaxies
with a scale length of 3.5 kpc (the average for the local-galaxy
surveys quoted above), and local galaxies with a scale length of
1.5 kpc (the average for our UDF spirals) corresponds to between 1
and 1.5 mag arcsec$^{-2}$ in rest frame B band. Thus the small
sizes observed here for UDF spirals might be the result of an
observational bias toward high central surface brightness galaxies
in the UDF. Independent evidence for a $\sim25$\% loss of spirals
larger than 10 pixels below the UDF sensitivity limit was
presented elsewhere (Elmegreen, et al. 2005b), based partly on the
distribution of axial ratios. The magnitude of the selection
effect required for scale lengths seems too large to explain our
UDF distribution, however. To get the small scale lengths, we
would have to be selecting only UDF spirals that are intrinsically
brighter than the face-on average at high redshift by $1-1.5$ mag
arcsec$^{-2}$ in the center. This contrasts with our previous
study, where we estimated that the detection limit is only
$0.25-0.5$ mag arcsec$^{-2}$ brighter than the average surface
brightness. Larger galaxy losses at low surface brightness should
shift the peak in the axial ratio distribution to values lower
than $\sim0.5$ (minor to major axis ratio), where it is now. This
shift might be possible if internal dust removes such highly
inclined galaxies from the UDF survey. The galaxies that are in
our survey in fact have no correlation between central surface
brightness and scale length (the two quantities make a scatter
plot, which is not shown here). Thus the hypothesized large and
low-surface brightness galaxies have to be missing already from
our UDF survey. Nevertheless, it remains possible that UDF spirals
appear small because of surface brightness or other selection
effects.
A small size for high redshift galaxies has been noted before
(Lowenthal et al. 1997; Bouwens, Broadhurst, \& Silk 1998; Bouwens
\& Silk 2002; Ferguson, Dickinson, \& Giavalisco 2004; Papovich,
Dickinson, \& Giavalisco 2005; B\"ohm \& Ziegler 2005). Lyman
Break galaxies at higher redshift are even smaller (Baugh et al.
1998; Somerville, Primack \& Faber 2001). Small exponential scale
lengths ($\sim2$ kpc) were also found for $z\sim0.8$ barred
spirals in the Tadpole galaxy field (Elmegreen, Elmegreen \& Hirst
2004b). Disk scale lengths are expected to be smaller in galaxy
evolution models in proportion to the dark matter virial radius,
which varies as $H(z)^{-2/3}\sim\left(1+z\right)^{-1}$ for
constant mass, Hubble parameter $H(z)$, and large $z$ (Mo, Mau \&
White 1998; Bouwens, Broadhurst, \& Illingworth 2003; Ferguson,
Dickinson, \& Giavalisco 2004).
The scale length distribution function in Figure \ref{fig:rdisk}
is more revealing about galaxy size then the distribution of scale
length versus luminosity. Trujillo et al. (2005) found relatively
small scale lengths at high z for a given luminosity, but this may
be the case regardless of absolute length because high redshift
galaxies tend to have high surface brightnesses from high star
formation rates; then a fixed luminosity covers a smaller radius.
High intrinsic surface brightness is also expected from sampling
effects even without higher star formation rates because surface
brightness dimming hides the fainter populations.
The size evolution of a distribution of galaxies can differ from
the size evolution of any particular galaxy. If there is little
size evolution for particular galaxies, as suggested by Simard et
al. (1999) and Ravindranath et al. (2004), then most of our
spirals will become dwarfs or perhaps merge to form ellipticals.
Only the largest members of our sample could turn into spirals
today (e.g., Labb\'e et al. 2003).
One problem with this model of no size evolution is that the small
star-forming galaxies observed locally, which are typically types
Im, Sm, or BCD, do not have such clear spirals as the small UDF
spiral galaxies. The local dwarf disks are relatively thick (van
den Bergh 1988) and their stellar velocity dispersions relatively
large compared to their small rotation speeds, so they have little
ability to sustain spiral waves. Thus, distant spirals are
distinct from local dwarfs even at the same physical size.
To sustain a global spiral wave, the Toomre (1964) length, $2\pi
G\Sigma/\kappa^2$ (for epicyclic frequency $\kappa$ and total disk
surface density $\Sigma$) has to be comparable to the disk scale
length. The Toomre length is essentially the separation between
stellar spiral arms. To generate a strong wave while avoiding
catastrophic collapse, the Toomre (1964) instability parameter,
$Q=\kappa c/\left(3.36 G \Sigma\right)$ for stellar velocity
dispersion $c$ has to be between 1 and 3. If the disk scale
length is small for a galaxy that is morphologically spiral but
has a high surface brightness, then $\kappa$ has to be large. The
stability parameter then requires $c$ to be fairly small, which is
possible if the disk mass is dominated by gas. For the UDF
spirals, these constraints would be satisfied if the visible disks
are the inner gassy regions of fairly dense halo potentials, where
the density is about the same as in the inner regions of spiral
galaxies today. For a flat or solid body inner rotation curve,
$\kappa$ scales with the square root of the total enclosed
density. At high redshift, galaxies should be not only smaller
but also denser in some inverse proportion (Mo, Mau \& White
1998), allowing tiny spirals to appear somewhat normal.
The Toomre length may also be viewed in another way. The ratio of
the Toomre length to the galaxy size is comparable to the ratio of
the disk mass to the total mass in the disk region. Thus, the
appearance of global spirals implies that disk masses are
comparable to halo masses out to the same radius. This is known to
be true for local spiral galaxies. Our observation of high
redshift spirals implies it is true there too, regardless of their
size. The situation is different for local dwarf irregulars,
however. There the ratio of disk mass to halo mass is relatively
small, i.e., they have relatively massive dark halos (e.g.,
Carignan \& Beaulieu 1989; Persic \& Salucci 1995). It follows
that the Toomre length is small compared to the galaxy size in
dwarfs, and so a stellar disturbance makes relatively small
epicycles and no global density waves.
These considerations lead us to believe that small UDF spirals are
dense like the inner regions of modern spirals. This should be the
case even if we select only the small members of a larger sample
because of surface brightness bias or other effects (see above).
The small spirals should grow to become large spirals by adding
low-density halo material to the periphery over time and by
filling out their disks with accreted gas and radially increasing
star formation (if they are not cannibalized first). This is the
commonly discussed inside-out formation model of galaxy disks but
now with an increasing scale length in addition to an increasing
overall size. A similar conclusion was reached by Papovich et al.
(2005) on the basis of an observed increase in half-light radius
and color gradient for $z\sim1$ compared to $z\sim2.3$ galaxies in
the Hubble Deep Field North. The color gradient in the $z\sim1$
galaxies was in the sense of a red core and a blue envelope, as if
star formation moved outward. Spiral arms were not so readily
resolved in that survey compared to the UDF, but some of the
galaxies studied by Papovich et al. are clearly spirals. They
also point out that for such color gradients to exist, the major
merger activity has to be mostly finished by $z\sim1$, a
conclusion also reached by others for different reasons (e.g.,
Conselice, Blackburne \& Papovich 2005).
The higher density of high-redshift spirals has implications for
the formation and evolution of central bars. Bar formation is
faster at higher density, and in a gas-rich young galaxy, also
more dissipative than usually considered in N-body simulations of
modern stellar disks. Bar destruction with a central mass
concentration and gas torques should be faster at early times too,
for the same reasons. Moreover, if a fully-grown nuclear black
hole is present at this stage, then the disk mass fraction
represented by the black hole will be larger than it is for a
modern disk, thus promoting bar destruction by the degeneration of
x$_1$ orbits (e.g., Hasan, Pfenniger, \& Norman 1993). Modern
disks do not have the central mass concentration necessary for
such bar destruction (Shen \& Sellwood 2004; Athanassoula,
Lambert, \& Dehnen 2005), but very young disks could have. For
these reasons, it might be possible for young spiral disks that
are small, dense, and gas-rich to form and disperse relatively
small bars in their centers, thereby quickening the build up a
bulge from the disk.
\section{Clump Luminosity Fractions in UDF Disk Galaxies}
Galaxies at high redshift typically have giant blue clumps that
are associated with recent star formation. In spirals, these
clumps are often in the arms, as they are in modern galaxies,
while in clump clusters, they are dispersed throughout the disk.
Equally large clumps are in chain galaxies (Cowie, Hu, \& Songaila
1995), high redshift-elliptical galaxies (Elmegreen, Elmegreen, \&
Ferguson 2005), tadpole galaxies (van den Bergh et al. 1996), and
double-clump galaxies (van den Bergh 2002). A review of the
various morphologies seen in the UDF is in Elmegreen et al.
(2005b).
Clump clusters differ from spiral galaxies not only in the
regularity of the clump positions, but also in the fraction of the
total light that is in the form of clumps. Figure
\ref{fig:fraction} shows histograms of this fraction using
magnitudes at i$_{775}$. Magnitudes were determined in IRAF using
a central aperture of 3 pixels radius and a background subtraction
annulus of 5 pixels width with a separation of 2 pixels between
them. The lower two panels are for the 91 relatively face-on
spiral galaxies in the UDF, with the bottom panel including the
central clumps or bulges in these galaxies and the middle panel
excluding the bulges. The bulges are generally more luminous than
the other clumps in the spirals, so the clump fraction is much
higher when the bulges are included. The top panel is for all 178
clump clusters, which have a very high fraction of their total
flux in the form of clumps. The average flux fractions are
$0.27\pm0.14$ for clump clusters, $0.16\pm0.10$ for spirals
including their bulges, and $0.080\pm0.065$ for spirals not
including their bulges. Ten extreme cases of clump clusters were
studied in Elmegreen \& Elmegreen (2005) in which an average of
40\% of the i$_{775}$ flux was in the clumps.
The i$_{775}$ band corresponds to a different rest wavelength for
each galaxy, and the clump flux fraction is generally higher at
shorter wavelengths for local galaxies (i.e., star-forming regions
are blue). Thus one could interpret Figure \ref{fig:fraction} as
an indication that the clump clusters have systematically higher
redshifts than the spirals, with the shorter rest wavelengths more
clumpy. However, the large clump fraction at short wavelengths for
local galaxies is the result of a big difference between clump and
disk colors, the clumps being much bluer. This difference is not
as large at high redshift because the interclump stars are not as
old as they are locally. The average $B_{435}-V_{606}$ and
$V_{606}-i_{775}$ color differences between clump and interclump
emissions in the 10 clump cluster galaxies studied earlier are
$0.19\pm0.17$ mag and $0.14\pm 0.30$ mag, respectively, with the
clumps being bluer than the interclump regions. Similarly, the
V$_{606}$-I$_{814}$ difference between the clump and surrounding
emission for the chain and spiral galaxies in the Tadpole field is
$0.2-0.3$ mag (Elmegreen, Elmegreen, \& Sheets 2004). In contrast,
the difference between clump and average colors in local spiral
galaxies is much greater. Local spiral galaxies have total
$B-V\sim0.7\pm0.1$ depending on Hubble type, and star forming
regions with $B-V\sim0$, so the clumps are bluer than the average
by $\sim0.7$ mag. The clump-interclump color difference is even
larger for local galaxies because the interclump emission is
redder than the average.
We consider in this paper that the difference in clump flux
fraction between spiral and clump cluster galaxies is a reasonable
indication that there is a real difference in the intrinsic
clumpiness of the two types of galaxies. A more quantitative
measure of disk clumpiness will require redshift information.
\section{Clump Position Distribution in UDF Disk Galaxies}
To study the relation between clump positions and exponential
disks, the (x,y) positions and B, V, i, and z magnitudes of all
the bright clumps in the low-inclination spiral and clump cluster
galaxies of the UDF were determined individually using IRAF. This
represents 632 objects (including bulges) in the spirals and 904
objects in the clump clusters. The centers of all the galaxies
were measured also, using the middle of the central peak for the
spirals and the geometrically centered position of the outer
2$\sigma$ contour for the clump clusters. Deprojected
galactocentric radii were then determined from these positional
measurements, assuming a circular disk-like geometry with an
inclination from the axial ratio.
Because the galaxies span a wide range of redshifts, we normalized
the radii to the exponential scale length in the spirals, and to
the outer 2$\sigma$ isophotal radius for both the spirals and the
clump clusters. This normalization also makes the rest wavelength
of the measurement relatively unimportant (it differs for each
galaxy) because the radial distribution of intensity is
effectively determined only for each galaxy separately and then
added together for all galaxies. Different rest wavelengths do not
affect the relative distribution of intensity inside each disk as
long as there are no systematic correlations between clump age and
radius. In fact, clump B-V color has no measurable correlation
with deprojected radius for either galaxy type (not shown here).
Figure \ref{fig:magdist} plots the i$_{775}$ magnitudes of the
clumps versus the normalized galactocentric radii. The points at
zero radius for the spirals are the bulges. The clumps get fainter
further from the center. The dotted line in the lower panel has
unit slope, corresponding to the magnitude-radius relation of each
underlying exponential disk. The distribution of points lies
parallel to this dotted line, suggesting a selection effect: we
can measure only the clumps that stand out sufficiently above the
background disk. Such an effect seems likely because star-forming
clumps in local galaxies have a wide range of luminosities with a
power-law distribution function dominated in number by the
smallest members. Thus, there may be countless small clumps that
were not measured. However, this should not be a problem for our
analysis because the local power laws have slopes of $\sim-2$ or
shallower for linear intervals of luminosity, and this means that
the integrated luminosity in all of the clumps in a particular
radial range is some fixed factor (less than $\sim5$) times the
luminosity of the largest member. The fact that there are no high
luminosity clumps in the outer regions (the lower envelopes in
Fig. \ref{fig:fraction} also increase with radius) means that
there really is an overall gradient in clump properties, although
we may not be seeing the true gradient with our selection effect.
This conclusion is supported by the observation that there is also
a tendency for the clumps to get fainter with galactocentric
radius in clump cluster galaxies. There are no significant
underlying disks in these cases and no likely systematic confusion
about clump definition and brightness at any radius.
Figure \ref{fig:his} shows the radial distributions of clump
number density and clump flux density, converted to a magnitude
scale. The lower panels are for the clump number densities. These
were determined by summing the number of clumps in all galaxies of
each type that lie between radial intervals separated by 0.1 in
normalized units. On the left, the normalization of radius is with
the exponential scale length (for the spirals only) and on the
right, the normalization is with the radius of the outer 2$\sigma$
contour. The top panels are for the clump flux densities. These
were determined by summing the i$_{775}$ fluxes, $10^{-0.4m_I}$,
for all the clumps that lie between the same normalized radial
intervals. In both cases, the sums were divided by the areas of
the annuli corresponding to the radial interval and then converted
to a (negative) magnitude scale by the operation $2.5\log({\rm
sum/area})$. The dotted lines on the left have unit slope,
corresponding to the exponential disks in each spiral galaxy. The
vertical scale in the figure has an arbitrary zero point with
higher densities toward the top.
The clump distributions in the spiral and clump cluster galaxies
show a remarkable similarity to each other even though the radial
profiles of the two types differ significantly in individual
cases. This similarity is shown best by the agreement between
the dashed and solid lines in the right-hand panels. Evidently,
the clumps in clump cluster galaxies are mapping out an
exponential disk that is just like the exponential in spiral
galaxies. This implies that if the clumps in clump cluster
galaxies are blended together, by dispersal of the associated
giant star complexes, for example, then an exponential disk will
result that is essentially the same as the disk in a spiral
galaxy. This similarity does not extend to the bulges, however:
the average density of clump fluxes at the centers of spirals is
higher than it is at the centers of clump clusters by 2 magnitudes
(a factor of 6 in the top right panel). The bulges can be seen
again as the sharp rise at zero radius in the left-hand panels.
The correlation between clump flux and normalized galactocentric
radius (Fig. \ref{fig:magdist}) implies that the radial
distribution of average clump flux density (top of Fig.
\ref{fig:his}) is slightly steeper than the radial distribution of
clump number density (bottom of Fig. \ref{fig:his}). For radii
between $1.5R_{disk}$ and $5R_{disk}$, the slope of the
distribution on the left-hand plot of number density is $-1.3$ and
the slope of the distribution on the left-hand plot of flux
density is $-1.8$. Thus the integrated flux from clumps in the
spiral galaxies has an exponential scale length equal to
$1/1.8=0.55$ times the scale length of the total light. Dispersing
the clumps will shorten the overall scale length slightly (in
proportional to the relative luminosity of the clumps compared to
the total disk).
\section{Conclusions}
The exponential scale lengths for spiral galaxies in the UDF
appear to be a factor of $\sim2$ smaller than for local spiral
galaxies. Selection effects could give this result but the
required high central surface brightness seems more extreme than
permitted by the distribution of the ratio of axes. Because the
presence of global spiral waves implies that stellar disks are
relatively massive compared to the total mass inside the disk
radius, the small spiral galaxies in the UDF are probably only the
dense inner regions of today's disks. Subsequent accretion at
lower density should build up both the disk and the halo over
time, increasing the scale length and the overall galaxy size.
Bars that form early in dense, gas-rich disks would be more prone
to orbit degeneration and dispersal into a bulge than bars which
form later in fully developed disks. N-body simulations of bar
formation should be revised to include more gas and to use
smaller, denser initial disks.
Galaxies in the UDF, including the spirals, are highly clumped
into blue star-forming regions that contain up to $\sim10^9$
M$_\odot$ of stars $\sim1$ Gy old or younger. The most clumpy of
the disk galaxies are called clump-clusters. In the
low-inclination clump-cluster galaxies studied here, $0.27\pm0.14$
of the total i$_{775}$ flux comes from giant disk clumps, compared
to only $0.080\pm0.065$ for the spirals. In a previous paper, we
suggested that many of these clumps should be dispersing as a
result of tidal forces. Here we addressed the question of what may
happen to the clumps after they disperse.
The evidence suggests that the clumps in clump-cluster galaxies
disperse into a more uniform disk that has an exponential radial
light profile. We found that the clumps in both spirals and
clump-clusters were distributed in an exponential fashion, and
that their total flux was also distributed as an exponential. Even
though clump cluster galaxies do not presently have an exponential
light profile in a smooth disk, their average clump distribution
is as exponential as the profile in a spiral galaxy.
A similar conclusion was based on the near-exponential nature of
the average radial profiles of clump cluster galaxies made from
elliptical contours. While the major axis or single-cut profiles
are highly irregular, the average radial profiles, averaged over
projected ellipses in azimuth, are closer to exponential in many
cases (Fig. \ref{fig:profile}).
We also showed that spiral galaxy bulges stand out above the
exponential profile and are not part of it. Clump clusters
generally do not have such bulges. Thus it remains to be
determined whether clump clusters form primarily late Hubble-type
disks over time, or whether some of the clumps merge into a bulge,
as in models by Immeli et al. (2004).
Acknowledgments: We are grateful to the referee for comments that
improved the presentation of this paper. B.G.E. is supported by
the National Science Foundation through grant AST-0205097.
| 2024-02-18T23:39:49.295Z | 2005-10-09T04:34:04.000Z | algebraic_stack_train_0000 | 524 | 5,423 |
|
proofpile-arXiv_065-2952 | \section{Introduction}
The Vela Pulsar is the brightest object in the sky at 100\,MeV energies
\citep{3EGcatalog}, with emission extending beyond 10\,GeV
\citep{thompson}.
The emission at these energies is totally pulsed \citep{kanbach}.
The pulsar
undergoes large glitches \citep[e.g.,][]{dodson}, and
has one of the highest values of
$\dot{E}$/d$^2$ of all pulsars,
making it a prime target for southern
hemisphere searches for TeV gamma-rays \citep[e.g.,][]{nel}.
Early searches for TeV gamma-ray emission relied on the detection
of a pulsed signal, and produced a number of
suggestive but inconsistent results \citep[see][for a review]{pge}.
The first search sensitive to both pulsed and steady emission
and made using the proven imaging atmospheric Cherenkov technique
was undertaken with the CANGAROO-I 3.8\,m telescope.
A total of 119 hours of usable on-source data taken between 1993 and 1995
yielded a 5.8$\sigma$
excess, corresponding to a flux of
(2.9$\pm$0.5$\pm$0.4) $\times$ 10$^{-12}$ cm$^{-2}$s$^{-1}$ above
2.5$\pm$1.0\,TeV,
offset from the
Vela Pulsar by 0.13~degrees and with no significant modulation
at the pulsar period \citep{yoshikoshi}.
A further 29\,hours data from
CANGAROO-I observations in 1997 showed a 4.1$\sigma$
excess at a position consistent with peak from the
earlier observations \citep{yoshikoshi,yoshikoshi2}.
Observations by the Durham group in 1996 produced 3$\sigma$ upper limits above
300\,GeV of $5\times 10^{-11}$ and $1.3\times 10^{-11}$\,cm$^{-2}$s$^{-1}$
for steady and pulsed emission, respectively, at the
pulsar position \citep{chadwick}.
No significant excess was found in a 2$^\circ \times$2$^\circ$ area
centered at the pulsar position.
Based on independent Monte-Carlo simulations of the CANGAROO-I
telescope \citep{dazeley1}, \cite{dazeley2} searched for a
TeV gamma-ray signal in the 29\,hour 1997 dataset,
but found no evidence of emission. \cite{dazeley2} noted that increasing the
value of the {\it length} cut toward the value used in
the earlier analysis did result in a somewhat increased excess, though
not to a significant level, and acknowledged that the
simulations had not included several known characteristics of the
CANGAROO-I telescope hardware. An analysis of data from the
Crab Nebula taken between December 1996 and January 1998
with two different sets of cuts also
failed to yield a significant excess \citep{dazeley2}, although
the resulting upper limit was higher than the established flux
of the Crab Nebula. The TeV flux from the Crab Nebula from
earlier CANGAROO-I \citep{tanimori} observations is consistent with
the most recent measurements \citep{hegra2004}.
The first preliminary results from observations with the H.E.S.S.\
stereoscopic imaging Cherenkov telescopes showed no evidence
of emission from the Vela Pulsar nor from previous CANGAROO-I position
\citep{gamma2004}.
However, very recently the H.E.S.S.\ group reported the detection
of broad TeV emission
$\sim$0.5~degree south from the pulsar position
extended over a region $\lesssim$ 0.6~degree \citep{icrc2005}.
The preliminary non-detection of emission from the pulsar was
confirmed, yet the extended emission from the region of the Vela
pulsar wind nebula was at the level of 0.5~crab
\citep{icrc2005}.
Here we report the results from our observations of the Vela Pulsar region
made in early of 2004
with the new CANGAROO-III stereoscopic imaging atmospheric system.
The previous CANGAROO-I analyses used slightly different cut values,
especially in the image analyses,
due in part to differences in source declination and sky brightness.
Analysis using three or four image parameters requires
selection of six to eight cut parameters
(i.e., both upper- and lower-cut values for each parameter).
While the cuts were guided by simulations,
which endeavored to incorporate the characterstics of
all telescope hardware and electronics \citep[e.g.,][]{dazeley1},
any fine tuning introduces a number of degrees of freedom into the analysis,
which can be difficult to account for a posteriori.
The number of parameters can be reduced to one
with the use of mathematical methods such as
Likelihood \citep{enomoto_nature}
or the Fisher Discriminant \citep{fisher}.
In this report, we emphasize the application of such methods
in order to remove any potentially biased operation from the analysis
\citep[see. e.g.,][]{weekes}.
Previous CANGAROO-II analyses have been carried out
using the Likelihood ratio, a Likelihood analysis with only
one image cut \citep{enomoto_nature}.
The Fisher Discriminant analysis introduced here
now removes all cut uncertainties from the image analysis.
In \S2 we describe the CANGAROO-III system, and in \S3 we present the
various calibration checks made to confirm its performance,
introducing unbiased analysis methods based on
Likelihood \citep{enomoto_nature} and the Fisher Discriminant
\citep{fisher}. In \S\S4 and 5 the results of the CANGAROO-III
observations of the Vela Pulsar and the surrounding field are presented,
with conclusions given in \S6.
\section{The CANGAROO-III Stereoscopic System}
The imaging atmospheric Cherenkov technique (IACT) was
pioneered with the Whipple group's detection of TeV gamma-rays
from the Crab Nebula \citep{whipple}.
This technique enabled TeV gamma-rays to be distinguished
from the huge background of cosmic rays with the use of the
``Hillas moments" of the Cherenkov images \citep{hillas}.
Stereoscopic observations were successfully demonstrated by the
HEGRA group to significantly improve the rejection of the
cosmic ray background \citep{hegra}.
More recently, the H.E.S.S.\ group has reported the detection of
a number of faint gamma-ray sources
with angular resolutions of only a few-arc minutes \citep{HESS_science}.
The CANGAROO-III stereoscopic system consists of four imaging atmospheric
imaging telescopes located near Woomera, South Australia (31$^\circ$06'S,
136$^\circ$47'E, 160 m a.s.l.).
Each telescope has a 10\,m diameter reflector
consisting of 114 spherical mirror segments, each of 80\,cm
diameter and with an average radius of curvature of 16.4\,m.
The segments are made of Fiber Reinforced Plastic (FRP)
\citep{kawachi} and are aligned on a parabolic
frame with a focal length of 8\,m.
The total light collecting area is 57\,m$^2$.
The four telescopes are located on the corners of a diamond
with spacings of approximately 100\,m \citep{enomoto_app}.
The first telescope, designated T1, is the CANGAROO-II
telescope, and data from this telescope was not used in the analysis
presented here
as its smaller field of view (FOV) was less well suited for these
stereoscopic observations.
The second and third CANGAROO-III telescopes, T2 and T3, were
used for the observations described here, with the fourth telescope (T4)
becoming operational after these observations were completed.
The camera systems for T2, T3, and T4 are identical and are
described in detail in \cite{kabuki}.
The pixel timing resolution is 1\,ns at 20 photoelectron (p.e.) input.
The noise level is 0.1\,p.e.
The relative gains of pixels are adjusted to within 10\% at the
hardware level
by adjusting the high voltages applied to photomultipliers.
The offline calibration are carried out using LED flasher
data which were taken once per day. The gain uniformity
of pixels is believed to be less than 5\%.
The observations were made using the so-called ``wobble mode"
in which the pointing position of each telescope was
alternated between $\pm$0.5~degree in declination from the
center of the target every 20 minutes \citep{wobble}.
Two telescopes, T2 and T3, were operational at this time, with
T3 located 100\,m to the south-south-east of T2.
The data from each telescope were recorded with the trigger condition
requiring that more than four pixels exceeded 7.6~p.e.
The GPS time stamp was also recorded for each event.
In the offline analysis, we combine these datasets by comparing
the GPS timing.
The distribution of timing differences is shown in Fig.~\ref{f_gps}.
\begin{figure}
\epsscale{1.0}
\plotone{f1.eps}
\caption{
GPS time differences in units of $\mu$s for all events.
}
\label{f_gps}
\end{figure}
The broadness of the distribution was due to the time resolution
of the GPS clocks.
We required an off-line coincidence between the two telescopes'
trigger times within $\pm$100\,$\mu$s.
Events outside this time window were considered to be accidental
coincidences.
The trigger rates of the individual telescopes were at most 80\,Hz
and this was
reduced to 8\,Hz by requiring the above coincidence.
\section{Calibration}
In order to determine the sensitivity of the telescope, the efficiency of
the light collecting system is the most important parameter to be precisely
measured.
Electromagnetic showers can be accurately simulated by computer code
and cosmic ray showers can be observed in OFF-source observations
and generated using Monte-Carlo codes such as GEANT.
We measured the point spread function (PSF)
and reflectivity of each mirror segment
during their production and the quantum efficiency of
each photomultiplier (PMT) pixel was also measured.
Some deterioration in these values after the construction of the
telescope is inevitable, and was monitored
by regular PSF measurements of bright stars.
Light collection efficiencies were monitored via optical systems
such as photo-diodes and commercial CCD cameras observing stars
both directly, and after reflection by the mirrors.
These showed gradual degradation in the form of broadening spot
sizes and decreasing reflectivities.
However, these data only relate to the reflectors, and we need to know
the total performance of the whole system including the cameras.
In order to measure these, we need ``standard candles''; sources of
Cherenkov photons or non-variable gamma-ray sources such as the Crab Nebula.
Here, we introduce our calibration procedure using these methods.
\subsection{Muon-ring Analysis}
The ideal light source is one that produces the same Cherenkov radiation
that the telescope is designed to detect.
Sub-TeV cosmic rays produce hadronic showers in the upper atmosphere at a
depth of around three inelastic interaction lengths, corresponding
to an altitude of $\sim$10\,km.
Secondary hadrons are produced and interact further,
with only gamma-rays, electrons and muons surviving to near ground level.
Of these, relativistic muons radiate Cherenkov rings near the Earth's
surface, and these produce distinct complete or partial ring
images on the camera.
The contribution of these rings to the trigger rate of our telescopes
is critical,
because small Cherenkov images resemble
gamma-ray images.
Local muon ring images can, however, be removed by requiring the
coincidence of two or more telescopes separated by $\sim$100\,m.
Here, for calibration, we select images produced by
muons at altitudes of 100--200\,m.
This can be done by selecting a somewhat larger ``arc-length".
Arc-length is measured after fitting a circle to the images, and
is the fitted radius multiplied by the recorded opening
angle of the image.
If larger images are required, we can simply restrict the selection
to geometrically closer images such
as those with altitudes $<$100--200\,m.
The inverse of the radius (curvature) should have a
Gaussian distribution, so we therefore plot the
curvature as a solid histogram in Fig.~\ref{curvature_all_t3}.
\begin{figure}
\plotone{f2.eps}
\caption{Curvature distribution for the circular fitting results of all events
for T3. The solid histogram is for all events and the dashed one for
the selected events (with the vertical scale multiplied by 5).
}
\label{curvature_all_t3}
\end{figure}
A peak can be seen around 1/1.3 [1/degree] which corresponds to the
inverse of the Cherenkov angle initiated by relativistic muons at
normal temperature and pressure.
The distribution for T2 (which was constructed before T3), however,
was not as good, indicating some degradation in its performance,
as shown by the solid histogram in Fig.~\ref{curvature_all_t2}.
\begin{figure}
\plotone{f3.eps}
\caption{Curvature distribution for the circular fitting results on all events
for T2. The solid histogram is for all events and the dashed one for
the selected events (with the vertical scale multiplied by 5).
}
\label{curvature_all_t2}
\end{figure}
These data were taken in 2003 December. We need to know our light
collection efficiency in each observing period, i.e., we
need to improve the S/N ratio to select a pure muon-ring sample only.
The following are the cut criteria used to achieve this.
\begin{itemize}
\item Hit threshold more than 1 photo-electron (p.e.).
\item TDC range within $\pm$30\,ns from the mean arrival time of the event.
\item At least 15 triggered (or ``hit'') PMTs, with a clustering
requirement that there be at most two non-triggered PMTs between
nearest-neighbor hit PMTs.
\item A circle, or ring segment, can be fitted to the image.
\item The circle has an ``arc-length" of more than 2 degrees.
\item A $\chi^2$ per hit-pixel, normalized to the pixel size, of less than 1.5.
\end{itemize}
The curvature distributions of the samples meeting these criteria are shown
as dashed histograms
in Figs.~\ref{curvature_all_t3} and \ref{curvature_all_t2}.
Muon ring events are selected with a good S/N ratio.
Geometrically, the arc-length cut restricts selection to events
occurring at $<$100--200\,m.
The statistics of the accepted events are sufficient for
calibration to be undertaken with only several hours data.
There is some dependence of light yield on the atmospheric
temperature; however, this is monitored and recorded every second.
The night-time temperature near Woomera ranges from near 0\,C
in winter to over 30\,C in summer.
The systematic error,
which includes the reflectivity of
the present mirror, the PSF, and the quantum efficiency of the PMTs,
is thought to be at the 5\% level of the
total light collection efficiency.
The light collection efficiencies can be derived from the ``size/arc-length"
distribution
and the PSFs by the $\chi^2$ of the ring fits.
The observation periods for the Crab Nebula and Vela Pulsar data
considered here were
between the end of 2003 and the beginning of 2004.
The calibrated light collection efficiencies for T2 and T3
were both 70\%, with 5\% systematic errors.
These values are the ratios of the efficiencies derived during the observation
period and those measured at the production time,
i.e., the deterioration factors.
The PSFs were
approximated by Gaussian distributions with standard deviations of
0.14, and 0.12 degrees, respectively,
which are somewhat
larger than the results obtained via bright star measurements at the
initial installation time.
(The corresponding values for T4, which was just coming on-line,
were 85\% and 0.09 degrees, respectively.)
The $\chi^2$ distributions for the experiments (the solid
histogram) and the tuned Monte-Carlo simulations (the hatched area)
are shown in Fig.~\ref{f_mu_spot} for T2 (upper panel) and T3 (lower panel).
\begin{figure}
\epsscale{0.7}
\plotone{f4.eps}
\caption{
Distributions of $\chi^2$ for the experimental data and the tuned Monte-Carlo.
The solid histograms are the experimental data and the hatched the Monte-Carlo.
The upper panel is for T2 and the lower T3.
}
\label{f_mu_spot}
\end{figure}
The PSF of T2 is worse than that of T3.
The size/arc-length distributions for the experiments (the solid
histogram) and the tuned Monte-Carlo (the hatched area)
are shown in Fig.~\ref{f_mu_size} for T2 (upper panel) and T3 (lower panel).
\begin{figure}
\plotone{f5.eps}
\caption{
Distributions of size/arc-length
for the experimental data and the tuned Monte-Carlo.
The solid histograms are the experiment and the hatched the Monte-Carlo.
The upper panel is for T2 and the lower T3.
}
\label{f_mu_size}
\end{figure}
The PSFs obtained for the three telescopes are somewhat
larger than those of H.E.S.S.,
corresponding to a relatively lower
cosmic-ray rejection efficiency. The resulting S/N ratio for
gamma-rays is discussed in the next section;
however, it is sufficient to detect gamma-ray sources
with flux levels flux of $\sim$0.1~Crab in several tens of hours.
The measured efficiencies do not depend strongly on
climate or atmospheric conditions, as they are local (near-surface)
phenomena.
Thus uncertainty due to Mie scattering, for instance, still remains.
This is thought to be significant at the 10\% level; however,
the average effect can be gauged in the Crab Nebula data
described in the following section.
In fact, data taken during cloudy nights showed a better S/N ratio
in the curvature distribution.
As the muon rings are a local phenomenon,
there is naturally no dependence on the
elevation angle of the observation, whereas the
cosmic ray rate varies strongly with elevation.
Using these characteristics, we were able to measure the
degradation as a function of time for the
three telescopes (T2, T3, and T4), with the light collection efficiency
being found to decrease by 5\% per year.
Hereafter in the analyses of the individual sources, we used
muon-ring data from the
corresponding periods to tune our Monte-Carlo code.
\subsection{Crab Analysis}
The description on the Crab data used in the following section can be found
in Appendix \ref{a_crab_obs}.
At first, the Hillas moments were calculated for each telescope's image.
The intersection of the major axes of the two images is the incident direction
of the gamma-ray in the stereoscopic observation.
The $\theta^2$ distribution of the Monte-Carlo data for this observation
condition
calculated from the intersection points
is plotted in Fig.~\ref{f_fit} (the dashed line).
\begin{figure}
\plotone{f7.eps}
\caption{
$\theta^2$ distributions from Monte-Carlo simulations.
The dashed line is obtained without
the fitting procedure described in the text and the solid line is that
with it.
}
\label{f_fit}
\end{figure}
The angular resolution is, as expected,
several times worse than that of H.E.S.S.\ \citep{hess_gc},
due to the larger PSFs described above.
The Crab Nebula observations were carried out at relatively
low elevation angles
and the opening angle between the T2 and T3 images are typically small,
resulting in many parallel images.
This results in an increased uncertainty in the intersection point
of the two images.
To avoid this, we
reanalyzed the data
with a constraint
on the distances between images' center of gravity and
the intersection point.
(H.E.S.S.\ observations are made at elevation angles
typically 10 degrees higher and with smaller mirror PSF,
resulting a good S/N ratio without this constraint.
If the PSFs of mirrors were as small as specified
in the original design, we could also avoid this procedure.)
The $\chi^2$ was defined as
$$\sum_{T=2,3}\left[W_T^2+\frac{(D_T-\langle Distance\rangle)^2}
{\sigma_{Distance}^2}\right],$$
where $W_T$ is the {\em width} seen from the intersection point,
and $D_T$ is the {\em distance}
between the image center of gravity and the intersection for each telescope
(Fig.~\ref{ipfit}),
$\langle Distance\rangle$ is the mean {\em distance} obtained by
Monte-Carlo simulations for gamma-rays,
and $\sigma_{Distance}$ is its standard deviation.
\begin{figure}
\plotone{f6.eps}
\caption{
Definitions of parameter $W_T$ and $D_T$ which are used
in the fitting procedure described in the text.
}
\label{ipfit}
\end{figure}
The improved $\theta^2$ distribution is shown as the solid line
in Fig.~\ref{f_fit}.
The number of events with $\theta^2<$\,0.05\,degree$^2$ increases
by a factor of 1.8.
Note that the angular resolution was estimated to be 0.23 degree
and 0.23$^2$ is roughly 0.05\,degree$^2$.
We first used the conventional (``square cuts") method as follows;
we accepted events with
{\em width}\,$<$\,0.2 (for T2)
and 0.15 (for T3), and {\em length}\,$<$\,0.3 (for T2) and 0.25 (for T3),
respectively, where all values are in units of degrees. The
cut values differ for T2 and T3 due to the PSF
differences.
As the $\theta$ resolution is 0.23 degree, we
took six background points on the 0.5-degree-radius circle
from the pointing direction.
The resulting $\theta^2$ distributions for the ON-source points
(the points with error bars)
and the OFF-source points (the solid histogram)
are shown in Fig.~\ref{f_sq}.
\begin{figure}
\plotone{f8.eps}
\caption{
$\theta^2$ distribution for the standard analysis of Crab Nebula data.
The points with error bars are from the ON-source region and the
solid histogram is the background region.
}
\label{f_sq}
\end{figure}
The background normalization factor is 1/6, equal to the inverse
of the number
of background points in these wobble mode observations.
The number of excess events with $\theta^2<0.05$ degree$^2$ is
(2207$-$1990=) 216, where the numbers in parentheses are
the on-source and estimated background counts, respectively.
This corresponds to a \cite{lima} significance of 4.4$\sigma$.
The number of events predicted by our Monte-Carlo simulation,
using the \cite{hegra_crab} flux, an $E^{-2.5}$ gamma-ray energy spectrum,
and minimum and maximum
gamma-ray energies of 500\,GeV and 20\,TeV,
was 195 events.
(For comparison, we also undertook ``mono'', i.e., single telescope, analyses
using the regular CANGAROO-II procedures.
Excesses for both telescopes were found using
both by square cut and Likelihood cut analyses.
The statistical significances of these excesses were
at the 3$\sigma$ level,
confirming the power of the stereo technique.)
Using the events with $\theta^2 < 0.05$ degree$^2$, we can
derive the Hillas moment distributions after background
subtractions.
This provides a check of how well the Monte-Carlo simulations agree
with the real gamma-ray data.
These are shown in Fig.~\ref{f_shape}.
\begin{figure}
\plotone{f9.eps}
\caption {
Hillas moment distributions: upper-left (T2 {\em widths}), upper-right
(T2 {\em lengths}), lower-left (T3 {\em widths}),
and lower-right (T3 {\em lengths}).
The points with error bars were obtained after the subtraction
of background events. The solid histograms are the Monte-Carlo
predictions where the total number of entries were normalized to those
of the observations.
}
\label{f_shape}
\end{figure}
Our Monte-Carlo simulations are consistent with the data
within statistical uncertainty.
Note that the {\em width} distribution is sensitive to the PSF
of the mirror system and that that of T3 is better than T2.
After this standard analysis, we investigated two
bias-free analyses.
The first is the Likelihood method introduced by \cite{enomoto_nature}.
In the standard ``square cuts", there are four
parameters on which cuts can be made, and in the absence of
a strong gamma-ray source or detailed simulations accurately
incorporating the characteristics of the telescope hardware,
some freedom in choosing the exact cuts is available.
\citep[see, e.g., the discussion in][]{dazeley2}.
In the Likelihood method,
we proceed by making probability density functions (PDFs) from
the histograms of {\em width} and {\em length}, for the two telescopes,
and some other
parameters such as the opening angle and the distance between two images'
center of gravity (image separation) (Fig.~\ref{f_shape2}).
\begin{figure}
\plotone{f10.eps}
\caption{
Distributions of opening angle and separation angle of two images.
These were obtained also via the subtraction described above for
the Crab data.
}
\label{f_shape2}
\end{figure}
In order to treat the energy dependence of these
parameters, we used two-dimensional histograms:
the PDFs are therefore 2-D functions.
For the gamma-ray sample, we used the data from Monte-Carlo simulations,
and for the cosmic ray background sample, we used real observation
data, since the ``contamination'' by
gamma-rays is much less than 1\%.
Thus for each event, we can derive a probability ($L$) for the
event being initiated by a gamma-ray or a cosmic ray,
where normalization ambiguities still remain.
The Likelihood ratio $LR$ is defined as
$$LR=\frac{L(gamma-ray)}{L(background)+L(gamma-ray)}.$$
The distributions for the Crab Nebula data are shown in Fig.~\ref{f_l}.
\begin{figure}
\plotone{f11.eps}
\caption{
Likelihood Ratio ($LR$) distributions. The points with error bars
are from Crab Nebula gamma-rays, the solid histogram from the gamma-ray
Monte-Carlo, and the dashed histogram from all events.
The latter two distributions are
normalized to the total number of the observed excess.
}
\label{f_l}
\end{figure}
The observational data are in reasonable agreement with the Monte-Carlo
predictions.
Observationally, the optimal cut was found to be $LR>$0.9.
The number of excess events is (390$-$289=) 101
(a Li and Ma significance of 5.2\,$\sigma$)
where the Monte-Carlo expectation
is 121 events.
The determination of the cut position is, however, still somewhat subjective.
A simple figure of merit ---
the number of the accepted events in the Monte-Carlo simulation
divided by the square-root of those in the observation --- is
plotted as a function of $LR$ in Fig.~\ref{f_foml}.
\begin{figure}
\plotone{f12.eps}
\caption{
Statistical significance of the Crab excess versus $LR$ cut position
(the solid line). The dashed curve is the figure of merit
calculated as described in the text. The normalization
is arbitrary.
}
\label{f_foml}
\end{figure}
The Monte-Carlo prediction is a smoothly increasing function
of the cut position (the dashed line).
On the other hand, the statistical
significance of the observed excess is shown by the solid line.
The two show good agreement over a wide range; however, some discrepancy
is apparent in the extremely tight cut positions.
For example, a $LR$ cut at 0.5 gives an excess (Fig.~\ref{f_l05}),
of (1049$-$895=) 154 events (4.6$\sigma$) where the expectation is 182 events.
\begin{figure}
\plotone{f13.eps}
\caption{
$\theta^2$ distribution for Crab Nebula data
obtained using the $LR>$ 0.5 cut.
}
\label{f_l05}
\end{figure}
Further fine-tuning of the Monte-Carlo code is
necessary in the future and such tight cuts will for now
be avoided.
The optimal cut position of 0.9 is located on the edge of this tight cut
region.
We now investigate an alternative approach for the comparison of
observational and Monte-Carlo data: the Fisher Discriminant \citep{fisher}.
When we use multi-parameters:
$$\vec{P}=(Vector~of~Hillas~moments~for~two~telescopes)$$
and assume that a linear combination of
$$F=\vec{\alpha}\cdot\vec{P},$$
provides the
best separation between signal and background, then the set of
linear coefficients ($\vec{\alpha}$) should be uniquely determined as
$$\vec{\alpha}=\frac{\vec{\mu}_{sig}-\vec{\mu}_{BG}}{E_{sig}+E_{BG}},$$
where $\vec{\mu}$ is a vector of the mean value of $\vec{P}$ for each
sample, i.e., $=\langle\vec{P}\rangle$, and $E$ is their error matrix,
i.e., $=\langle\vec{P}\vec{P}^T\rangle-\langle\vec{P}\rangle\langle\vec{P}^T\rangle$.
The values of
$\vec{\mu}_{sig}$, $\vec{\mu}_{BG}$,
$E_{sig}$, and $E_{BG}$ can be calculated from the
Monte-Carlo gamma-ray events for signal and observational data for background.
Our purpose is to separate ``sharp images'' from ``smeared ones".
This method is regularly used in high-energy physics experiments, such
as the $B$-factory's ``Super Fox-Wolfram moment" \citep{two},
in order to separate spherical events from
jet-like events.
Here, the assumption strongly depends on which linear combination
is best. We must, therefore, select parameters which are
similar to each other. {\em Width} and {\em length} are
both second order cumulative moments of shower images
and thus a linear combination is a reasonable assumption.
We used four image parameters: {\em T2-width}, {\em T3-width},
{\em T2-length}, and {\em T3-length}.
Here, the energy dependence of {\em width} and {\em length} for each telescope
were corrected using the energy estimated from the summation of
ADC values of hit pixels for that telescope
using the Monte-Carlo expectations. This correction function was
a second-order polynomial and we carry out an offset correction
using it, i.e.,
$$\vec{P}=\vec{P}_{raw}-\vec{a}_0-\vec{a}_1log(\sum ADC)-
\vec{a}_2log(\sum ADC)^2,$$
where $\vec{a_i}$s were determined from the two-dimensional
plots of raw ``Hillas moments" versus $log(\sum ADC)$ using the
Monte-Carlo gamma-ray simulations and $\vec{P}_{raw}$ is a vector made
of the raw values of ``Hillas moments", respectively.
The corrected
{\em width} and {\em length} are distributed around zero and
the $\langle\vec{P}_{sig}\rangle$ is automatically located exactly
at zero,
$$\langle\vec{P}_{sig}\rangle=0$$
resulting in the Monte-Carlo expectation of $F$
location being zero:
$$\langle F_{sig} \rangle = 0.$$
This removes cut-selection bias: a cut at zero ensures 50\%
acceptance (for gamma-rays) automatically, a typical
value adopted in IACT analyses. From the $\theta^2$
Crab Nebula excess we obtain the following
distribution of Fisher Discriminant, $F$, as shown in Fig.~\ref{f_f}.
\begin{figure}
\plotone{f14.eps}
\caption{
Distributions of the Fisher Discriminant, $F$.
The points with error bars are obtained from the
observation, the solid histogram from the gamma-ray
Monte-Carlo, and the dashed from all events representing
background behavior.
}
\label{f_f}
\end{figure}
The behavior of the distributions for gamma-ray signals,
background events, and Monte Carlo expectations agree reasonably
well and are similar or a little bit better
than the result obtained for the Likelihood method.
This figure should be compared with Fig.~2 in the recently published
H.E.S.S.\ paper for the distribution of their
{\em mean reduced scaled width} \citep{hess_1303}.
The H.E.S.S.\ discrimination is better; however, improved results can be
expected for CANGAROO-III if the quality of mirrors were improved to the
levels of H.E.S.S.\ or MAGIC.
We consider the figure of merit for this method in Fig.~\ref{f_fomf}.
\begin{figure}
\plotone{f15.eps}
\caption{
Statistical significance of the Crab Nebula excess versus $F$ cut position
(the solid line). The dashed curve is the figure of merit
calculated as described in the text. The normalization
is arbitrary.
}
\label{f_fomf}
\end{figure}
This time, the agreement is very good in all regions.
Although the cut at $F>-$0.16 showed the best statistical
significance, 5.8$\sigma$, with an
excess of (1177$-$974=) 203 events,
where the Monte-Carlo expectation is 162 events.
A conservative choice of cut point is zero which,
as noted above,
is exactly the mean expected position for a gamma-ray
acceptance of 50\%.
In the following we use the Fisher Discriminant analysis as the default,
with a cut position at zero,
thus removing all subjective or potentially biased
elements from the analysis.
The excess is (744$-$604=) 140 events, where the expectation is 110 events.
The final $\theta^2$ distribution is shown in Fig.~\ref{f_zerof}.
\begin{figure}
\plotone{f16.eps}
\caption{
The final $\theta^2$ plot for the Crab Nebula, using the Fisher Discriminant
analysis with the cut at zero.
}
\label{f_zerof}
\end{figure}
In this analysis, we estimated the gamma-ray energy
using the average of the T2 and T3 summations of ADC values,
which probably provides the optimal energy resolution
for this analysis.
The usage of the intersection constraint,
due to finite point spread function of the individual mirror
segments, does not
allow a core-distance correction of energy.
The overall energy resolution was considered to be 30\%
from the simulation study.
We show the differential flux for the Crab Nebula
in Fig.~\ref{f_dfcrab}.
\begin{figure}
\plotone{f17.eps}
\caption{
Differential gamma-ray flux from the Crab Nebula as a function of energy.
The red line is the HEGRA result \citep{hegra_crab}
and the green is the Whipple result \citep{whipple_crab}.
}
\label{f_dfcrab}
\end{figure}
The points with error bars are our data where the errors
are statistical. The systematic error is considered
to be 15\% at this stage.
HEGRA \citep{hegra_crab} and Whipple \citep{whipple_crab}
results are also shown.
The results agree within the statistical and systematic errors.
The deviation in the CANGAROO flux at higher energies may be due to
saturation effects, which are not yet fully implemented in our
Monte-Carlo code; however, this in not an important
consideration for Vela observations, which are the main focus
of this paper.
We leave further consideration of this for the future, noting
that while the analysis template defined here
for CANGAROO-III is, at present, most suitable, it
may be able to be further optimized with refined Monte-Carlo simulations.
We show the 2D profile of gamma-ray images in Fig.~\ref{f_profile}.
\begin{figure}
\plotone{f18.eps}
\caption{
Excess number plot in the equatorial frame.
The unit is events per $0.1\times 0.1$ degree$^2$ in the FOV coordinates.
The cross indicates the center of the Crab Nebula.
The resolution is considered to be 0.23 degrees
in declination in this frame.
}
\label{f_profile}
\end{figure}
The angular resolution
is estimated to be 0.23 degrees (1$\sigma$).
Here, only high energy events ($E>$10.6\,TeV)
are plotted in order to improve the
S/N ratio and to remove uncertainties in the background subtraction.
The background level at each position was estimated using
the events with values of $F$ below $\langle F_{BG} \rangle$
which are considered to be mostly background protons.
Here, however, we need to note that the subtraction template for the background
was made using events with low $F$ sample.
The Fisher Discriminant uses a linear combination of
the energy-corrected {\em width} and {\em length}
for T2 and T3, respectively.
The subtraction sample for morphological studies may tend to be overwhelmed
by the larger sized events which have poorer angular resolutions.
The important thing is that the background sample tends to be flatter than
the gamma-ray events. Therefore, we only show a very restricted area in this
figure. An improved analysis, required for diffuse gamma-ray detection,
which will be introduced later.
As a result of the considerations above,
the analysis template is:
\begin{itemize}
\item the Fisher Discriminant is adopted.
\item the cut position on $F$ is exactly at zero.
\item others parameters,
such as elevation cut and shower-rate cut,
are determined as appropriate for the source under
investigation (depending on its declination and galactic coordinates).
\end{itemize}
\section{Observations of the Vela Pulsar}
The Vela Pulsar was observed between 2004 January 17 and February 25.
In total, the preselected data correspond to an analyzable
period of 1311 min.\ after the elevation and cloud cuts,
where the minimum elevation angle was set at 60~degrees
and the shower rate at 9\,Hz.
The mean elevation angle was 70.9~degrees, corresponding
to an energy threshold of 600\,GeV.
The observations were carried out using the same wobble mode
as for the Crab Nebula observations.
In this period, T2 and T3 were in operation and
we analyzed the stereo data from these two telescopes.
As the Vela Pulsar is at a declination of $-$45$^\circ$, the
relative orientation of the two telescopes does not present
any problems.
We used the optimized analysis procedure described in the
previous section and so there are no {\em a posteriori} trials
to consider in the interpretation of results --- it is
a ``blind analysis".
The resulting $\theta^2$ distribution is shown in Fig.~\ref{f_fzero}.
\begin{figure}
\plotone{f19.eps}
\caption{
The $\theta^2$ distribution for Vela Pulsar observations
resulting from application of the analysis template
described in \S3.3.
}
\label{f_fzero}
\end{figure}
After the background subtraction, we obtained an excess
of (677$-$722=) $-$45$\pm$29 events.
Monte Carlo simulations, with minimum and maximum gamma-ray energies
of 100\,GeV and 20\,TeV, respectively, and an energy spectrum
proportional to $E^{-2.5}$, predict 394 events would
be detected for a 100\%\,Crab level gamma-ray source.
The CANGAROO-I flux was 60\% of the Crab Nebula flux.
The $F$ distribution, obtained using subtraction,
is shown in Fig.~\ref{f_f_vela}.
\begin{figure}
\plotone{f20.eps}
\caption{
$F$ distributions.
The points with error bars are obtained from the
observations, the solid curve from the gamma-ray
Monte-Carlo simulations, and the dashed histogram
from all observed events to represent the cosmic ray
background behavior.
The Monte-Carlo histogram is normalized to a 100\%\,Crab Nebula flux level.
}
\label{f_f_vela}
\end{figure}
There is no excess of gamma-ray--like events around the predicted
region, offset by 0.13~degrees from the Vela Pulsar.
Thus observations with significantly improved instrumentation
and a robust analysis procedure do not support the previous
claim for TeV gamma-ray emission from this region.
The 2$\sigma$ upper limits are shown in Fig.~\ref{f_fvela},
together with the results of other observations.
\begin{figure}
\plotone{f21.eps}
\caption{
2$\sigma$ upper limits for the gamma-ray flux.
C-I represents the CANGAROO-I result \citep{yoshikoshi}
and C-III indicates the CANGAROO-III result reported here.
Also shown are Durham \citep{chadwick},
BIGRAT \citep{dazeley3} and H.E.S.S.\ preliminary
results \citep{gamma2004} \citep{icrc2005}.
}
\label{f_fvela}
\end{figure}
The upper limits in the figure are a factor of 5 below the
CANGAROO-I result. This analysis has used the point
offset by 0.13 degrees to the south-east of the Vela Pulsar, i.e.,
offset is ($\Delta$RA,$\Delta$dec)=(0.14$^\circ$,$-$0.1$^\circ$)
from the pulsar position.
This position
which was the maximum of the excess detected with the CANGAROO-I
telescope: an analysis at the position of the Vela Pulsar position
yields similar upper limits.
These are summarized in Table \ref{t_flux}.
\begin{table}
\begin{center}
\caption{
2-$\sigma$ upper limits to the integrated fluxes.
}
\label{t_flux}
\begin{tabular}{c c c}
\tableline\tableline
& $F$($>E$) at & $F$($>E$) at \\
Energy, $E$ & offset position & pulsar position \\
(GeV) & (cm$^{-2}$s$^{-1}$) & (cm$^{-2}$s$^{-1}$) \\
\tableline
~600 & ~ ~ $<5.8\times 10^{-12}$ ~ ~ & ~ ~ $<5.8\times 10^{-12}$ ~ ~ \\
~710 & ~ ~ $<4.5\times 10^{-12}$ ~ ~ & ~ ~ $<4.5\times 10^{-12}$ ~ ~ \\
~860 & ~ ~ $<3.4\times 10^{-12}$ ~ ~ & ~ ~ $<3.4\times 10^{-12}$ ~ ~ \\
1200 & ~ ~ $<2.0\times 10^{-12}$ ~ ~ & ~ ~ $<2.4\times 10^{-12}$ ~ ~ \\
2700 & ~ ~ $<4.7\times 10^{-13}$ ~ ~ & ~ ~ $<4.7\times 10^{-13}$ ~ ~ \\
\tableline\tableline
\end{tabular}
\end{center}
\end{table}
\section{Vela Pulsar Wind Nebula}
Recently, extended TeV gamma-ray emission coincident with the
Vela Pulsar Wind Nebula (PWN) was reported by the H.E.S.S.\ group
\citep{icrc2005}. Their preliminary report
claimed that the center of the emission is
(RA, dec) = (8$^h$35$^m$, $-$45$^\circ$36$'$)
and that the flux within a 0.6 degree
radius from this position is 50\% of the Crab Nebula at 1\,TeV.
They also noted that no gamma-rays were detected
from the Vela Pulsar, and placed a tight upper limit
on the pulsar flux at 0.26\,TeV
\citep{gamma2004,icrc2005}.
Thus far, we have focused on a point source analysis based on the
wobble mode observations.
The peak PWN source position coincides with the one of two wobble
pointing direction in the coordinate of the field of view of the camera.
We, therefore, can not carry out background estimation using the usual
wobble method.
In this section we undertake an optimized analysis for an extended source.
Another difficulty is that we don't have sufficient statistics
for OFF source regions, and so
the background subtractions should be carried out using the ON source
data runs. Gamma-ray--like events can be extracted by fitting
position-by-position $F$
distributions under the assumption that
gamma-rays obey the Monte-Carlo predictions, the proton background
follows the average $F$ distribution of all directions, and the total
distribution is a linear combination of those two.
The separation between those two distributions is likely to be worse
at lower energies due to the smaller image sizes.
These distributions are plotted in Figs~\ref{fvse} for
various energy ranges.
\begin{figure}
\plotone{f22.eps}
\caption{
$F$ distributions in various energy regions. The blank histograms are
for Monte Carlo simulations of gamma-rays and the hatched histograms
for background regions of real data. From the top panel to the bottom,
the central gamma-ray energies are
540, 780, 1200, 2700, 3300, and 8800\,GeV, respectively.
The histogram entries are normalized to one event in the shown regions.
}
\label{fvse}
\end{figure}
From the upper to lower panels,
the central energies of the energy bins are
540, 780, 1200, 2700, 3300, and 8800\,GeV, respectively.
The blank histograms are gamma-rays and the hatched, protons.
As shown in figure, the separation begins at 780\,GeV and
becomes significant at 1200\,GeV. Therefore, we first analyzed events with
energy greater than 1200\,GeV.
Then we checked the directional dependence of the $F$ distributions.
The field of view was segmented into 0.2$\times$0.2~degree$^2$ regions
and each $F$ distribution was compared with the average.
The reduced $\chi^2$ distribution is shown by the histogram in
Fig.~\ref{fchisq}.
\begin{figure}
\plotone{f23.eps}
\caption{
Reduced $\chi^2$ distribution for various arrival directions (the histogram).
The pixel size for each direction is 0.2\,$\times$\,0.2\,degree$^2$.
The curve is the expected distribution for the number of
degrees of freedom, 23.
}
\label{fchisq}
\end{figure}
The curve shown is the predicted distribution
for the 23 degrees of freedom. There is good agreement,
i.e., there is no significant directional dependence.
This method was then checked with Crab Nebula data. The $F$ distributions
at various $\theta^2$ slices were taken at energies greater
than 5.7\,TeV. The background $F$ distribution was obtained in the
higher $\theta^2$ region, 0.1--0.3~degree$^2$.
The only fitting parameter is the percentage of gamma-ray--like events
relative to the total events.
The result is shown in Fig.~\ref{crab}.
\begin{figure}
\plotone{f24.eps}
\caption{
The $\theta^2$ plot obtained by the fitting method
described in the text for Crab Nebula data.
The vertical scale is the percentage of gamma-ray--like events relative to
all events.
}
\label{crab}
\end{figure}
The statistical significance of the peak is 4.0$\sigma$, while the
ordinary wobble analysis gave a 3.6$\sigma$ excess.
Having demonstrated the validity of this method,
we carried out an analysis of the Vela PWN region.
The H.E.S.S.\ group detected a gamma-ray excess extended over
a 0.6~degree radius from
the center of the emission
[(RA, dec) = (8$^h$35$^m$, $-$45$^\circ$36$'$)].
In our case, the angular resolution (0.23~degree)
is significantly worse than
that of H.E.S.S. We, therefore, chose the background region
to be more than 0.8 degree from the center.
The result of fitting is shown in Fig.~\ref{f2_off}.
\begin{figure}
\plotone{f25.eps}
\caption{
The wide range $\theta^2$ plot for the Vela PWN region, where
$\theta$ was calculated from (RA, dec) = (8$^h$35$^m$, $-$45$^\circ$36'),
i.e., the peak of the emission detected by H.E.S.S.
The gamma-ray--like events were
extracted by the fitting procedure described in the text.
}
\label{f2_off}
\end{figure}
An excess was observed
at $\theta^2 < 0.6$ degree$^2$ around the center of Vela PWN region.
The excess radius is marginally consistent
with H.E.S.S.\ considering our angular resolution.
The total number of gamma-ray--like events is 561$\pm$114.
The differential flux was obtained and is shown in Fig.~\ref{fitdflux_off}.
\begin{figure}
\plotone{f26.eps}
\caption{
Gamma-ray flux in the Vela PWN region.
}
\label{fitdflux_off}
\end{figure}
Although the statistics are poor,
the spectrum looks hard, consistent with the preliminary H.E.S.S.\ results,
and the fluxes are in general agreement.
This excess is below the 5$\sigma$ level
generally required for a firm detection \citep{weekes} and is
spatially
extended over a significant
portion of our field of view,
and thus a high resolution (0.1 degree) morphology map
is not justified.
We are, however, able to offer supporting evidence of the
H.E.S.S.\ result and, in this light, are planning to observe
the PWN region in more detail next year.
\section{Conclusion}
We have observed the Vela Pulsar region from 2004 January 17 to February 25
with the CANGAROO-III stereoscopic imaging Cherenkov telescopes.
At that time, two telescopes (T2 and T3) were in operation
and events coincident to the two telescopes were analyzed.
Calibration was performed using muon rings and the performance
of the telescopes confirmed with observations of the Crab Nebula.
The estimated energy threshold for this analysis was 600\,GeV.
The use of the Fisher Discriminant has been introduced and
a template for CANGAROO-III analysis presented.
No significant excess of events was found from the
Vela Pulsar direction or from the peak of the emission detected
with the CANGAROO-I telescope.
The upper limits obtained are a factor of five less than
the CANGAROO-I fluxes (assuming an $E^{-2.5}$ spectrum).
In addition, we have confirmed, at the 4$\sigma$ level
the gamma-ray emission recently reported from the
Vela Pulsar Wind Nebula. The TeV emission from the PWN peaks
$\sim$0.5 degree south of the pulsar position
and has a $\sim$0.6 degree extension, consistent with the H.E.S.S.\ report.
A detailed morphological study of this source will
require more observations.
\acknowledgments
This work was supported by a Grant-in-Aid for Scientific Research by
the Japan Ministry of Education, Culture, Sports, Science and Technology,
the Australian Research Council, and by JSPS Research Fellowships.
We thank the Defense Support Center Woomera and BAE Systems.
| 2024-02-18T23:39:50.283Z | 2005-10-13T04:10:35.000Z | algebraic_stack_train_0000 | 590 | 7,560 |
|
proofpile-arXiv_065-2985 | \section{Introduction: what can we learn from the H{\,\small I}\ in nearby radio galaxies
}
There are a number of important aspects of radio galaxies that can be
investigated using observations of neutral hydrogen. Because of sensitivity
limitations of present day radio telescopes, some of these studies are restricted
to nearby radio galaxies. Nevertheless, they can give important insights on
phenomena that are likely to be very common in high redshift objects.
Here I summarize some recent results obtained for low-$z$ radio galaxies,
underlining their relevance for galaxies in the far away Universe.
\begin{figure*}
\centerline{\psfig{figure=Morganti_fig1.ps,width=9cm,angle=0}}
\caption{Numerical simulations of major mergers resulting in an
elliptical galaxy (from Mihos 1999).
}
\end{figure*}
Hierarchical (major) merging and accretion of small clumps appears to be a
good description of the formation of early-type galaxies, i.e. the typical
host of radio galaxies (see Fig. 1, Mihos 1999). If this is the case, the
presence and morphology of extended H{\,\small I}\ can be used as tracer to investigate
the origin and evolution of these galaxies. This is particularly interesting
in the case of radio galaxies as the origin of activity in galaxies is often
explained as triggered by merger and/or interaction processes. Torques and
shocks during the merger can remove angular momentum from the gas in the
merging galaxies and this provides injection of substantial amounts of
gas/dust into the central nuclear regions (see e.g.\ Mihos \& Hernquist
1996). Indeed, this appears to be the case for radio galaxies as suggested by
morphological and kinematical evidence (e.g.\ Heckman et al.\ 1986; Tadhunter
et al.\ 1989).
Another aspect that can be investigated using the neutral hydrogen is the
effect of the interaction between the radio plasma and the interstellar
medium (ISM) that surrounds the radio source. Indeed, the phase of nuclear activity
is now increasingly recognized to play an important role in the evolution of
the galaxy itself. Particularly important in this respect are gas outflows
that can be generated by this activity and the effect they can have on the
ISM. This feedback can be extremely important for the
evolution of the galaxy, up to the point that it could limit the growth of the
nuclear black-hole (e.g.\ Silk \& Rees 1998; Wyithe \& Loeb 2003). Thus, the
processes of assembly of the host galaxy, the supply of gas to the central
region, as well as the effects that the triggering of the (radio) activity has
on this gas, are tightly related and essential for our understanding of radio
galaxies. An other aspect related to the interaction is the possibility of
jet-induced star formation, a phenomenon considered to be particularly
relevant for high-$z$ radio galaxies.
\begin{figure*}
\hfill
\includegraphics[width=.4\textwidth]{Morganti_fig2a.ps}
\hfill
\includegraphics[width=.4\textwidth]{Morganti_fig2b.eps}
\hfill\null
\caption[]{Total intensity (left) and position-velocity plot along the
major axis (right) for the early-type galaxy IC~4200. The
H{\,\small I}\ disk is about 200 kpc in size. Contour levels are:
$2, 4, 8, 16 \times 10^{19}$ cm$^{-2}$ (left)
and -3, 3, 6, 8 mJy beam$^{-1}$ (right). }
\end{figure*}
\section{Large-scale H{\,\small I}\ (emission) in nearby early-type galaxies}
Large-scale H{\,\small I}\ (in emission) around galaxies hosting radio sources provides
an important signature of whether a (major) merger has occurred in the life of
these galaxies. More details about the results of this study are presented in
Emonts et al. (these Proceedings). However, to put these results in the more
general context of the formation and evolution of early-type galaxies, we
describe first the results obtained on the occurrence and morphology of the
neutral hydrogen in "normal" early-type galaxies. These results will be later
(Sec 2.3) compared with what found for radio galaxies.
\subsection{Shallow surveys}
It is already known since many years that HI-rich early-type galaxies do exist
(e.g.
Knapp et al. 1985; Morganti et al. 1997; van Gorkom etal. 1997; Oosterloo et
al. 2002). More recently, however, a systematic, albeit shallow, H{\,\small I}\ survey
of all early-type galaxies south of $\delta < -25^{\circ}$ with $V < 6000$ km
s$^{-1}$ (based on the Parkes All Sky H{\,\small I}\ Survey: HIPASS, Barnes et al. 2001
and followed up with the ATCA, Oosterloo et al. in preparation) has revealed that in
most of these H{\,\small I}-rich early-type galaxies the H{\,\small I}\ is distributed in very
large (up to 200 kpc in size), regular disks of low column density
H{\,\small I}. Fig. 2 shows the case of IC~4200. {\sl The amount of H{\,\small I}\ and
the size of the structures can be explained as result of major mergers between
gas-rich disk galaxies}. This is indeed how at least some early-type galaxies
are believed to be formed (see e.g. the case of NGC~7252, Hibbard \& van
Gorkom 1996).
As mentioned above, this survey is shallow, therefore able to detected only
galaxies with associated more than $10^9$ $M_{\odot}$ of neutral
hydrogen. Nevertheless, the results are telling us not only that such large
amount of neutral hydrogen (with H{\,\small I}\ masses up to $10^{10}$ $M_{\odot}$ and
log M$_{\rm H{\,\small I}}$/L$_B$ is between -1 and 0 for the H{\,\small I}\ detected galaxies)
appear to be present around 5-10\% of early-type galaxies but also that,
because of their regular and large appearance, these disks must be quite old,
in many cases well over $5 \times 10^9$ yr. Hence, they are not related to
recent accretions. The H{\,\small I}\ column density in these disks peaks at only
$10^{20}$ atoms cm$^{-2}$ (i.e. 0.5-1 M$\odot$/pc$^2$). Thus, despite the
large H{\,\small I}\ reservoir, no significant star formation is occurring, therefore
the HI is not used and the H{\,\small I}\ disks will evolve only very slowly. Finally,
these large, regular structures can be used to get information about the dark
matter content of early-type galaxies. So far it appears that this is similar
to what found in spirals galaxies (see Fig. 3).
\begin{figure}
\centerline{\psfig{figure=Morganti_fig3.ps,width=7cm,angle=0}}
\caption{ The log($M/L_B$) - log($R/R_{\rm e}$)
diagram (modified from Bertola et al.\ 1993). The dashed thick line
represents the cumulative $M/L_B$ as function of radius for spiral galaxies.
The symbols represent the data for the elliptical galaxies obtained from the optical
and the H{\,\small I}\ data. From Morganti et al. 1999.}
\end{figure}
\subsection{Deep H{\,\small I}-search}
The shallow surveys described above have shown that H{\,\small I}-rich early-type galaxies
do exist. However, it is also important to investigate how common is, around
these galaxies, the presence of (even a modest amount of) neutral hydrogen and
what are its characteristics. To investigate this, a representative sample of
12 early-type galaxies has been observed using the WSRT. These galaxies were
selected because they are part of the sample studied in the optical with the SAURON
panoramic integral-field spectrograph on the William Herschel Telescope, La
Palma. SAURON provides a very detailed view of the kinematics of the ionised
gas and of the stellar component in the inner regions of these objects (de
Zeeuw et al. 2002).
This deep, albeit so far small, survey can detect H{\,\small I}\ masses down to
$<10^7$ M$_\odot$. In nine of the 12 objects observed (75\%) H{\,\small I}\ associated
with the galaxy has been detected (Morganti, de Zeeuw, Oosterloo et
al. 2005a). This finding indicates that {\sl neutral hydrogen appears to be a
common characteristic of these galaxies, provided that deep enough
observations are available}. The neutral hydrogen shows a variety of
morphologies, from complex structures (tails, offset clouds) to regularly
rotating disks. Four of the detections belong to the latter group and they
will be the main test cases for probing the dark matter content in the inner
and outer part of the galaxies in a consistent way. The most extreme case in
our sample is a gas disk of only $2 \times 10^6$ M$_\odot$ and a column density
of $5 \times 10^{18}$ cm$^{-2}$ that has been found in the young SO galaxy
NGC~4150.
Figure 4 shows the results for the elliptical galaxy NGC 4278. This galaxy has
an H{\,\small I}\ disk that extends well beyond the optical image. The velocity maps of
the ionised gas and of the stars (as obtained with SAURON) together with the
velocity map of the H{\,\small I}\ illustrate one of the recurrently found
characteristics: the kinematics of the H{\,\small I}\ is very similar to that of the
ionised gas, indicating that they form one single structure. This is
underlined by the fact that galaxies with little or no ionised gas are less
likely to show H{\,\small I}. In NGC~4278, an offset exists between the position angle
of the rotating gas disk (H{\,\small I}\ and ionised) and that of the rotation of the
stars, something that is observed in several cases and may reflect the
non-axisymmetric nature of the galaxies. These offsets in kinematic alignment
can vary from co- to counter-rotating, sometimes showing dramatic twists
within a single object.
\begin{figure}
\centerline{\psfig{figure=Morganti_fig4.eps,width=7cm,angle=-90}}
\caption{WSRT total H{\,\small I}\ intensity (orange) of the galaxy NGC~4278 superimposed to
the optical image (grey).}
\end{figure}
\begin{figure*}
\centerline{\psfig{figure=Morganti_fig5.eps,angle=0,width=13cm}
}
\caption{{\sl Left} H{\,\small I}\ total intensity (contours) superimposed to an optical
image of B2~0258+35 (from Emonts et al. in preparation). {\sl Right} The
continuum image (from Fanti et al. 1986).}
\end{figure*}
\subsection{H{\,\small I}\ in nearby Radio Galaxies}
The results presented above have been obtained for samples of ``normal''
early-type galaxies, therefore not biased toward radio-loud objects. A study of
H{\,\small I}\ in radio galaxies has been instead carried out by Emonts et al. (in
prep). The sample includes more than 20 objects (both in the northern
and in the southern hemisphere). The selection and observations of the
northern sample are described in Emonts et al. (these proceedings).
The H{\,\small I}\ mass limit of this survey is somewhat in between the surveys
described above (few time $10^7$ up to $10^8$ M$_\odot$) and therefore the
comparison between them is difficult. Among the radio galaxies, 25\% of the
objects are detected in H{\,\small I}\ in emission. Interestingly, the most H{\,\small I}-rich
objects (again with H{\,\small I}\ masses up to $10^{10}$ $M_{\odot}$ and log M$_{\rm
H{\,\small I}}$/L$_B$ is between -1 and 0) have the neutral hydrogen distributed in very
large disks with regular kinematics as found in the shallow survey described
in Sec. 2.1. An example of an extended disk of H{\,\small I}\ around a radio galaxy is
shown in Fig.~5. Thus, as for those galaxies, the origin of the large amount
of H{\,\small I}\ is likely to be major mergers of gas-rich disk galaxies. The
formation of regular disk structures is explained in numerical simulations as
result of merger of similar size galaxies (from 1:1 to 1:4) with high angular
momentum (Barnes 2002; Burkert \& Naab 2003). The gas from the progenitors
falls back at a late stage of the merger (after the starburst phase) and
settles in a disk. The combination of H{\,\small I}\ and study of the stellar
population (in progress) allows to put the AGN activity in the evolutionary
sequence of early-type galaxies (see the case of B2 0648+27 in Emonts et
al. these Proceedings).
Other radio galaxies appear to have a much smaller amount of H{\,\small I}\ (with H{\,\small I}\
masses between $10^{8}$ and $10^{9}$ $M_{\odot}$ and log M$_{\rm H{\,\small I}}$/L$_B$
is between -2 and -1) and the neutral hydrogen is either distributed in disks
or in blobs or tails. These objects appears to be more similar to what found
in the deep SAURON survey (see Sec. 2.2). Thus, although the comparison is not
completely fair and based on small numbers statistic, we do not find so far
major differences in the H{\,\small I}\ characteristics (detection rate, morphologies,
masses, etc.) between ``normal'' early-type and radio galaxies. This may
indicate that indeed the radio-loud phase is just a short period in the life
of many (all?) early-type galaxies. However, it should be noticed that in our
sample of radio galaxies all the large H{\,\small I}\ disks have been detected so far
around compact radio sources, see Fig. 5 for an example. The reason for this
is not yet clear. Some possible explanations are given in Emonts et al. (these
Proceedings).
Finally, we would like to point out possible similarities between the large
quiescent Ly$\alpha$ structures detected in high-$z$ galaxies (Villar-Mart\'in
et al. 2002, 2005 in these proceedings) and the large H{\,\small I}\
disks detected in low-$z$ radio galaxies.
\begin{figure*}
\centerline{\psfig{figure=Morganti_fig6a.eps,width=8cm,angle=-90}
\psfig{figure=Morganti_fig6b.eps,width=8cm,angle=-90}}
\caption{{\sl Left} H{\,\small I}\ contours of the southern region of the H{\,\small I}\ cloud,
drawn on top of a broad band optical image (kindly provided by M.\
Rejkuba). The hatched area indicates the location where the anomalous H{\,\small I}\
velocities are detected while the arrow indicates the location and the flow
direction of the radio jet. The filament of ionised gas is visible in the top
left. The dashed lines roughly indicate the locations of young stars. {\sl
Right} Position-velocity plots taken through the location of the anomalous
H{\,\small I}\ (bottom) and, for comparison, taken just above that region (top). Taken
from Oosterloo \& Morganti (2005).}
\end{figure*}
\section{Jet-induced star formation}
As shown above, the host galaxies of radio-loud AGN can be gas rich. Thus, the
interaction between the non-thermal plasma ejected from the active nucleus and
the ISM of a galaxy can have important consequences and can be responsible for a
variety of phenomena in radio galaxies such as ionisation of the gas, AGN
driven outflows and jet-induced star formation. Such interactions are
considered to be particularly relevant in high redshift radio galaxies, as
they are typically living in a gas-rich environments (see e.g.\ van Breugel
2000 and references therein). As mentioned above, one aspect of jet-ISM
interaction is that it can trigger star formation and this is considered a
possible mechanism to explain the UV continuum emission observed in the host
galaxies of distant radio sources and the "alignment effect" between the radio
emission and this continuum (Rees 1989). Detecting and studying star
formation produced by this mechanism in high-$z$ radio galaxies is very
challenging. The only case where this has been done is 4C~41.17 (Dey et al.\
1997). Because of the observational problems for high-redshift sources, it is
important to find nearby examples of star formation triggered by the radio jet
that can be studied in more detail. The nearby, best examples are Centaurus A
(Oosterloo \& Morganti 2005) and the Minkowki's Object (van Breugel et al. 1985).
\subsection{Jet-induced star formation in Centaurus ~A}
In the case of Centaurus~A, new 21-cm \rm H{\hskip 0.05cm\scriptsize I}\ observations of the large \rm H{\hskip 0.05cm\scriptsize I}\
filament located about 15 kpc NE from the centre of this galaxy and discovered
by Schiminovich et al.\ (1994) have been carried out using the ATCA (Oosterloo
\& Morganti 2005). This \rm H{\hskip 0.05cm\scriptsize I}\ cloud is situated (in projection) near the
radio jet of Centaurus A (see Fig. 6), as well as near a large filament of
ionised gas of high excitation and turbulent velocities and near regions with
young stars. The higher velocity- and spatial-resolution of the new data
reveal that, apart from the smooth velocity gradient corresponding to the
overall rotation of the cloud around Centaurus A (Schiminovich et al. 1994),
\rm H{\hskip 0.05cm\scriptsize I}\ with anomalous velocities up to 130$\,$km$\,$s$^{-1}$\ is present at the southern tip
of this cloud (see Fig. 6 and Oosterloo \& Morganti 2005). This is
interpreted as evidence for an ongoing interaction between the radio jet and
the \rm H{\hskip 0.05cm\scriptsize I}\ cloud. Gas stripped from the \rm H{\hskip 0.05cm\scriptsize I}\ cloud gives rise to the large
filament of ionised gas and the cooling of the gas is then responsible for the
star formation regions that are found downstream from the location of the
interaction.
From the displacement of the
young stars from the location of the anomalous velocities we derive a flow
velocity of about 100$\,$km$\,$s$^{-1}$, very similar to the observed anomalous \rm H{\hskip 0.05cm\scriptsize I}\ velocities..
The jet induced star formation appears to be fairly
inefficient, of the order of few percent.
Recent numerical simulation have shown that radio jets can indeed drive
radiative shocks in interstellar clouds, causing them to compress and break up
into numerous dense, cloud fragments. These fragments survive for many
dynamical timescales and are presumably precursors to star formation (Mellema
et al. 2002; Fragile et al. 2004).
\begin{figure*}
\centerline{\psfig{figure=Morganti_fig7.eps,width=8cm,angle=-90}}
\caption{C-array H{\,\small I}\ cloud (thick, blue contours) at the end of the radio continuum
jet (thin, black contours) superimposed onto a Keck image of the Minkowski's
Object (van Breugel et al. in prep). }
\end{figure*}
\subsection{Jet-induced Star Formation in ``Minkowski's Object''}
Minkowski's Object is a peculiar starburst galaxy (at $z$ = 0.0187) at end of
the jet from NGC~541 (van Breugel et al 1985; van Breugel et al. 2004; Croft
et al. 2004). This object is an ideal candidate for a detailed study of the
effects of the interaction between the radio plasma and the ISM. Its
morphology is strongly suggestive of a collision of a low luminosity FR-I type
jet from NGC 541 with a gas rich cloud/object (see Fig.~7 and van Breugel et
al. 1985).
Recent C-array VLA observations have shown that a cloud of H{\,\small I}\ is detected
just down-stream from the main star formation region (see Fig.~7). This cloud
has a total H{\,\small I}\ mass of $\sim 3 \times 10^8$ M$_\odot$ (corresponding to a
M$_{\rm H{\,\small I}}$/ L$_B \sim 0.17$), it has the same transverse size as the radio
jet and consists of two main components separated at the centre of the radio
jet. A velocity gradient in the H{\,\small I}\ of $\sim 60$ $\,$km$\,$s$^{-1}$\ is observed.
Although, as in the case of Centaurus~A, H{\,\small I}\ is detected close to the
location of the star formation and radio jet, it is not clear whether the same
scenario can explain the case of the Minkowski's Object.
In this object the radio jet itself may be the cause
of the formation of the H{\,\small I}\ as predicted in the jet-triggered radiative
cooling model (Fragile et al. 2004). This model shows that star formation, and the
H{\,\small I}\ that preceeds this, can occur in relatively warm gas due to radiative
cooling triggered by the radio jet. However, until stronger constrains to this
model will be available from new, higher resolution H{\,\small I}\ observations the
possibility that the origin of the starburst is instead due to a
pre-existing peculiar galaxy or a pre-existing cold gas (like in the case of
Centaurus~ A) cannot be ruled out.
\subsection{Results from jet-induced star formation}
The kinematic of the H{\,\small I}\ supports the idea that at least in two nearby
objects (Centaurus ~A and Minkowski's Objects) radio jets can trigger star
formation. In both cases, H{\,\small I}\ is observed in regions where jet-induced star
formation has been claimed to be present. The kinematics of the neutral
hydrogen, its morphology and its relation to ionized gas give constraints on
the on-going process. The next step will be a more detailed comparison between
the results from the observations (in particular from high resolution H{\,\small I}\
data) and the results from numerical simulations. Because the gas densities
and the AGN activity are higher in early Universe, jet-induced star formation
is, therefore, likely to be a more common phenomenon in high-$z$ radio
galaxies.
\begin{figure}
\centerline{\psfig{figure=Morganti_fig8.eps,angle=0,width=7cm}}
\caption{ A zoom-in of the \ion{H}{i}\ absorption spectra of 3C~293 clearly
showing the broad \ion{H}{i}\ absorption. The spectra are plotted in flux
(mJy) against optical heliocentric velocity in km/s. From Morganti et al. 2003.}
\end{figure}
\section{Fast H{\,\small I}\ outflows}
Gaseous outflows appear to be a widespread phenomenon in galaxies, both in the
local as well as in the far-away Universe (e.g. Crenshaw et al. 2003; Veilleux
et al. 2005; Frye et al. 2002). They can be driven
by super-winds in the starburst phase or by the energy released in the active
phase of the nucleus. AGN-driven outflows of ionized gas have been detected in
many nearby galaxies. These outflows are a key ingredient in galaxy
evolution. The correlation found between the mass of the super-massive
black-hole and the mass of the central bulge of the galaxy are most
easily understood in terms of feedback models (Silk \& Rees 1998). Numerical
simulations suggest that the energy released by a quasar expels enough gas to
quench both star formation and further black-hole growth (e.g. di Matteo,
Springel, Hernquist 2005).
Recent results underline the importance of radio jets in producing such
outflows. These outflows have been found not only associated with ionized gas
(see Tadhunter these Proceedings) but also with neutral hydrogen. Recent
sensitive, broad-band 21-cm observations of the radio sources IC~5063
(Oosterloo et al. 2000) and 3C~293 (see Fig. 8; Morganti et al. 2003; Emonts
et al. 2005) have revealed that fast outflows of neutral hydrogen can occur in
galaxies with an AGN. In addition to these two initial objects, more cases of
broad and blueshifted H{\,\small I}\ absorption have been found using very sensitive and
broad-band 21-cm radio observations with the Westerbork Synthesis Radio
Telescope. Thus, fast and massive outflows may be common, in particular in
young powerful radio sources (Morganti et al. 2005b). Interestingly, the mass
outflow rates detected for the neutral gas are much larger than those
typically found for the ionised gas and are large enough to have a significant
impact on the evolution of the galaxies. The WSRT broad band covers $\pm
2000$ $\,$km$\,$s$^{-1}$\ around the central velocity (systemic velocity of the galaxy). The
observed borad H{\,\small I}\ absorption features have a width up to 2000 $\,$km$\,$s$^{-1}$\ and a
typical optical depth $<<1$\%, corresponding to a column density few times
$10^{21}$ cm$^{-2}$ (for T$_{spin}$ = 1000 K) The detected H{\,\small I}\ absorption
features are mostly blueshifted therefore corresponding to gas outflows. In
order to understand the origin of such fast outflows it is important to know
the location where the outflow is occurring.
\begin{figure}
\centerline{\psfig{figure=Morganti_fig9.eps,angle=0,width=7cm}}
\caption{Panel showing {\sl (top)} the radio continuum image
of 3C~305. {\sl (Bottom)}
The position-velocity plot from a slice passing through the two
lobes. The broad H{\,\small I}\ absorption is detected against the NE radio lobe,
about 1.6 kpc from the nucleus. The contour levels for the continuum image are 10
mJy beam$^{-1}$\ to 830 mJy beam$^{-1}$\ in steps of a factor 2. The
grey scale image represents the total intensity of the H{\,\small I}\
absorption. The contour levels of the H{\,\small I}\ are $-0.7$, ..., -7.7
mJy beam$^{-1}$\ in steps of 0.7 mJy beam$^{-1}$. The arrow represents the
systemic velocity. Taken from Morganti et al. (2005c)}
\end{figure}
\begin{figure}
\centerline{\psfig{figure=Morganti_fig10.eps,angle=0,width=7cm}}
\caption{{\sl Top} WHT spectrum of the [O{\,\small III}]\ region of 3C~305 (after the
subtraction of the continuum from the galaxy) taken in p.a.\ 60$^\circ$, i.e.\
along the galaxy's major axis (NE to the left, SW to the right). The two
arrows represent the approximate position of the peak of the radio lobes.
{\sl Bottom} WHT spectrum of the [O{\,\small III}]\ region of 3C~305 (black contours and
grey scale) taken in p.a.\ 42$^\circ$, i.e. along the radio axis. White
contours represent the H{\,\small I}\ position-velocity plot taken along the radio axis
(as in Fig.\ 9).}
\end{figure}
\subsection{The case of 3C~305}
High-spatial resolution 21-cm \rm H{\hskip 0.05cm\scriptsize I}\ VLA observations were obtained for the
radio galaxy 3C~305 (Morganti et al. 2005c). These new high-resolution data
show that the $\sim 1000$ $\,$km$\,$s$^{-1}$\ broad H{\,\small I}\ absorption, earlier detected in
low-resolution WSRT observations, is occurring against the bright, eastern
radio lobe, about 1.6 kpc from the nucleus (see Fig.~9).
We also used new optical spectra taken with the WHT to make a detailed
comparison of the kinematics of the neutral hydrogen with that of the ionised
gas (see Fig. 10). The striking similarity between the complex kinematics of
the two gas phases suggests that both the ionised gas and the neutral gas are
part of the same outflow. Earlier studies of the ionised gas (Heckman et
al. 1982; Jackson et al. 2003) had already found evidence for a strong
interaction between the radio jet and the ISM at the location of the eastern
radio lobe. These results show that the fast outflow produced by this
interaction also contains a component of neutral atomic hydrogen. The most
likely interpretation is that the radio jet ionises the ISM and accelerates it
to the high outflow velocities observed. These observations demonstrate that,
following this strong jet-cloud interaction, not all gas clouds are destroyed
and that part of the gas can cool and become neutral. The mass outflow rate
measured in 3C~305 (but also in other objects, see below) is comparable,
although at the lower end of the distribution, to that found in Ultra Luminous
IR galaxies. This suggests that AGN-driven outflows, and in particular
jet-driven outflows, can have a similar impact on the evolution of a galaxy as
starburst-driven superwinds.
\subsection{Results from fast H{\,\small I}\ outflows}
The results obtained so far show that fast outflows of neutral hydrogen can be
produced by the interaction between the radio jet and the surrounding dense
medium. The presence of neutral gas in these regions indicates that the gas
can cool very efficiently following a strong jet-cloud interaction.
Interestingly, the associated mass outflow rates range from a few tens to
about almost hundred M$_\odot$ yr$^{-1}$, comparable to (although at the lower
end of the distribution) the outflow rates found for starburst-driven
superwinds in Ultra Luminous IR Galaxies (Rupke et al., 2002; Heckman
2002). Thus, as these superwinds, the massive, jet-driven HI outflows in the
radio-loud AGN can have a major impact on the evolution of the host galaxy.
High-$z$ H{\,\small I}\ absorbers have been also found in Ly$\alpha$ profiles (although
with lower column density). As for the low-$z$ cases, a possible way to
explain these absorptions is via highly supersonic jet expanding into the
dense medium of a young radio galaxy that then will be surrounded by an
advancing quasi-spherical bow shock (Wilman et al. 2003; Krause 2002). Thus,
the detailed results obtained for the nearby objects can help in understanding
better the mechanism at work in their far away cousins.
\section{Conclusions}
This review aimed to illustrate the importance that studies of the neutral
hydrogen can have for the understanding of different phenomena observed in
radio galaxies. In nearby radio galaxies, the H{\,\small I}\ is beginning to tell us
about the origin of the host galaxy and about the presence and location of
fast gaseous outflows, that appear to have a significant impact on the
evolution of the galaxy. Nevertheless, despite the recent progress, a number
of questions remain still open. For example: what is the link (if any)
between large H{\,\small I}\ disks (low-$z$) and quiescent Ly$\alpha$ structures
(high-$z$)? what are the details of the physical process that produces fast
H{\,\small I}\ outflows? How common/important is the jet-induced starformation at
high-$z$ (and at low-$z$)?
Unfortunately at present deep studies of the neutral hydrogen as presented in
this review are only limited to nearby galaxies. The need for the new
generation of radio telescopes, and in particular for the Square Kilometer
Array is clear. A detailed overview of what this new instrument will do for us
can be found in the ``SKA Science Case'' (Carilli \& Rawlings 2004).
Hopefully we do not only have to wait for SKA but we may have some answers to
these questions before the next meeting!
\acknowledgements
I would like to thanks the organizing commetee and in particular Montse
Villar-Martin for organizing and inviting me at this very pleasant and
interesting workshop. The results presented in this review would not have
been obtained without the help of my collaborators. In particular I would like
to thanks T. Oosterloo, C.N. Tadhunter, B. Emonts, E. Sadler, T. de Zeeuw, W. van
Breugel, J. van Gorkom.
| 2024-02-18T23:39:50.364Z | 2005-10-10T10:48:54.000Z | algebraic_stack_train_0000 | 597 | 5,008 |
|
proofpile-arXiv_065-3072 | \section{\label{sec:intro}Introduction}
The presence of noise in physical systems is a well known
phenomenon. It is common opinion that noise affecting the dynamics
of a system introduces some degree of instability to the system
itself, but there is evidence that in some cases the noise can
increase the stability of a
system~\cite{NESChaos-TD,NESPiecewise,NESothers,NESreview}. Noise
modeling through the use of the stochastic processes formalism has
applications that involve many systems including physics, biology,
ecology~\cite{GenericNoise,MBE,Ecology} and even financial
markets~\cite{Mand,MSt,Bou}. The most basic model for financial
market is the geometric Brownian motion~\cite{Hull}. This model has
different drawbacks, it cannot reproduce in fact three important
stylized facts observed in financial time series: (i) the non
Gaussian distribution of returns, (ii) the fat tails~\cite{MSt,Bou},
and (iii) the stochastic character of volatility, which is
characterized by long range memory and
clustering~\cite{MSt,Bou,Bouchaud,Dacorogna}. More complex models
have been developed to reproduce the dynamics of the volatility. It
is worthwhile citing the ARCH~\cite{Arch} and GARCH~\cite{Garch}
models, where the actual volatility depends on the past values of
squared return (ARCH) and also on the past values of the volatility
(GARCH). Another class of models use a system of stochastic
equations writing the price as a geometric Brownian motion coupled
with a non-constant volatility described by a second stochastic
differential equation. The Heston model uses for the volatility a
multiplicative stochastic process characterized by mean
reversion~\cite{Heston,Hull-White}. Both the models presented so far
have exponential autocorrelation function so they are not able to
reproduce quantitatively the long range memory observed in real
markets. Nonetheless using values of the characteristic time that
are sufficiently high, they are able to give accurate statistic for
the stock prices, by tuning only few parameters. Another important
characteristic of financial markets is the presence of different
regimes. Markets indeed present days of normal activity and extreme
days where very high or very low price variations can be observed.
These are known as crash and rally days. A nonlinear Langevin market
model has already been proposed~\cite{BouchaudCont}, where different
regimes are modelled by means of an effective potential for price
returns. In some circumstances this potential has a cubic shape with
a metastable state and a potential barrier. The dynamics inside the
metastable state represents the days of normal evolution while the
escape after the potential barrier represents the beginning of a
crisis. Metastable states are ubiquitous in physics and the effect
of noise in such systems has been extensively
studied~\cite{NESChaos-TD,NESPiecewise,NESothers,NESreview}, but
considering the noise intensity as a parameter (physical models
described by additive stochastic differential equations for
example). Financial markets with their stochastic volatility are an
example of systems where the noise intensity is far from being a
constant parameter, but it is indeed a stochastic process itself.
Moreover there is evidence in nature that the noise intensity is not
a constant parameter and can be modelled as a multiplicative
noise~\cite{MBE,Ecology,OltreNES}. So it is interesting to release
the hypothesis of parametric noise intensity in financial market
models as well as in natural systems.
\section{\label{sec:nes_heston}The Heston model with a metastable state}
The Heston model introduced in the previous section is described
by the following system of coupled stochastic differential
equations~\cite{Heston}
\begin{eqnarray}
dx(t) & = & (\mu - v(t)/2) \cdot dt + \sqrt{v(t)} \cdot dZ(t) \nonumber \\
dv(t) & = & a(b-v(t)) \cdot dt + c \sqrt{v(t)} \cdot dW(t)
\label{Eqn:Heston}
\end{eqnarray}
The price $p(t)$ follows a geometric random walk whose standard
deviation is another stochastic process. Here $x(t)= \ln p(t)$ is
the $log$ of the price, $Z(t)$ and $W(t)$ are uncorrelated Wiener
processes with the usual statistical properties: (i) $\langle
dZ(t) \rangle =0$ and $\langle dZ(t) \cdot dZ(t') \rangle =
\delta(t-t') dt$; (ii) $\langle dW(t) \rangle =0$ and $\langle
dW(t) \cdot dW(t') \rangle = \delta(t-t') dt$. The $v(t)$ process
is characterized by mean reversion, i.e. its deterministic
solution has an exponential transient with characteristic time
equal to $a^{-1}$, after which the process tends to its asymptotic
value $b$. The process for $v(t)$ exhibits the phenomenon of
volatility clustering, alternating calm with burst periods of
volatility, and has an exponential autocorrelation function. The
smaller the value of $a$ the longer are the bursts in volatility.
Heston model has been subject of recent investigation by
econophysicists~\cite{BonannoPhysicaA,Yakovenko,BonannoFNL,Silva}.
The equations of the system $(1)$ are well known in finance, they
represent respectively the log-normal geometric Brownian motion
stock process used by Black and Scholes for option pricing~\cite{BS,
Merton}, and the Cox-Ingersoll-Ross (CIR) mean-reverting stochastic
differential equation first introduced for interest rate
models~\cite{Cox,Chalasani}.
Here we consider a generalization of the Heston model, by replacing
the geometric Brownian motion with a random walk in the presence of
a cubic nonlinearity. This generalization represents a
"\emph{Brownian particle}" moving in an \emph{effective} potential
with a metastable state, in order to model those systems with two
different dynamical regimes like financial markets in normal
activity and extreme days~\cite{BouchaudCont}. The equations of the
new model are
\begin{eqnarray}
dx(t) & = & - \left(\frac{\partial U}{\partial x} +
\frac{v(t)}{2}\right) \cdot dt + \sqrt{v(t)}
\cdot dZ(t) \nonumber \\
dv(t) & = & a(b-v(t)) \cdot dt + c \sqrt{v(t)} \cdot dW(t),
\label{Eqn:BS}
\end{eqnarray}
where $U$ is the \emph{effective} cubic potential $U(x)=px^3+qx^2$,
with $p=2$ and $q=3$ (see Fig.~\ref{Fig:Cubico}), $Z(t)$ and $W(t)$
are standard Wiener processes.
\begin{figure}[htbp]
\vspace{5mm}
\centering{\resizebox{9cm}{!}{\includegraphics{SpagnoloFig1.eps}}}
\caption{\label{Fig:Cubico} Cubic potential used in the dynamical
equation for the price x(t). The points in the figure indicate the
starting positions used in our simulations.}
\end{figure}
Let us call $x_M$ the abscissa of the potential maximum and $x_I$
the abscissa where the potential intersects the $x$ axes. The
intervals $x_o < x_I$ and $I=[x_I,x_M]$ are clearly regions of
instability for the system. In systems with a metastable state like
this, the noise can originate interesting effects that increase
instead of decrease the stability, by enhancing the lifetime of the
metastable
state~\cite{NESChaos-TD,NESPiecewise,NESothers,NESreview}. A good
example is the Noise Enhanced Stability (NES) phenomenon. The mean
escape time $\tau$ for a Brownian particle moving throughout a
barrier $\Delta U$, with a noise intensity $v$, is given by the well
known exponential Kramers law~\cite{Hanggi,Gardiner}
\begin{equation}
\tau = A exp\left[{\frac{\Delta U}{v}}\right],
\label{Eqn:Kramers}
\end{equation}
where $\tau$ is a monotonically decreasing function of the noise
intensity $v$, and $A$ is a prefactor which depends on the potential
profile. This is true only if the random walk starts from initial
positions inside the potential well. When the starting position is
chosen in the instability region $x_o < x_M$, $\tau$ exhibits an
enhancement behavior, with respect to the deterministic escape time,
as a function of $v$. Particularly for initial positions $x_0 <
x_I$, we have nonmonotonic behavior of $\tau$ as a function of
$v$~\cite{NESPiecewise,NES03}. This is the NES effect and can be
explained considering the barrier "\emph{seen}" by the Brownian
particle starting at the initial position $x_0 $, that is $\Delta
U_{in} = U(x_{max})-U(x_0)$. Moreover $\Delta U_{in}$ is less than
$\Delta U$ as long as the starting position $x_0$ lyes into the
interval $I=[x_I,x_M]$. Therefore for a Brownian particle, from a
probabilistic point of view, it is easier to enter into the well
than to escape from, when the particle is entered. So a small amount
of noise can increase the lifetime of the metastable
state~\cite{NESChaos-TD,NESPiecewise,NESreview,NES03}. For a
detailed discussion on this point and different dynamical regimes
see Refs.~\cite{NESPiecewise,NES03}. When $v$ is much greater than
$\Delta U$, the Kramers behavior is recovered. The NES effect has
been experimentally observed in a tunnel diode and theoretically
predicted in a wide variety of systems such as for example chaotic
map, Josephson junctions, chemical reaction kinetics, and neuronal
dynamics models~\cite{NESChaos-TD,NESPiecewise,NESothers,NESreview}.
Our modified Heston model, characterized by a stochastic volatility
and a nonlinear Langevin equation for the returns, has two limit
regimes, corresponding to the cases $a=0$, with only the noise term
in the equation for the volatility $v(t)$, and $c=0$ with only the
reverting term in the same equation. This last case corresponds to
the usual parametric constant volatility case. In fact, apart from
an exponential transient, the volatility reaches the asymptotic
value $b$. The NES effect should be observable in the latter case as
a function of $b$, which is the average volatility. In this case, in
fact, we have the motion of a Brownian particle in a fixed cubic
potential with a metastable state and an enhancement of its lifetime
for particular initial conditions (see
Refs.~\cite{NESPiecewise,NES03}).
\section{\label{sec:correlation}Enhancement of the escape time}
The two processes of Eqs.~(\ref{Eqn:Heston}) and (\ref{Eqn:BS}) are
actually uncorrelated. In financial markets the two processes can be
correlated, and a negative correlation between the processes is
known as \emph{leverage effect}~\cite{Fouque}. Heston model with
correlation has been recently discussed in the scientific
literature~\cite{Yakovenko,Silva}. Our modified Heston model becomes
therefore
\begin{eqnarray}
dx(t) & = & - \left(\frac{\partial U}{\partial x} +
\frac{v(t)}{2}\right) \cdot dt + \sqrt{v(t)} \cdot dZ(t) \nonumber \\
dv(t) & = & a(b-v(t)) \cdot dt + c \sqrt{v(t)} \cdot dW_c(t) \nonumber \\
dW_c(t) & = & \rho \cdot dZ(t) + \sqrt{1-\rho^2} \cdot dW(t),
\label{Eqn:BSCorr}
\end{eqnarray}
where $Z(t)$ and $W(t)$ are uncorrelated Wiener processes as in
Eqs.~(\ref{Eqn:Heston})~and~(\ref{Eqn:BS}), and $\rho$ is the cross
correlation coefficient between the noise sources. The investigation
is performed simulating the process of Eqs.~(\ref{Eqn:BSCorr}) with
time integration step $\Delta t=0.01$, and for the fixed starting
position $x_0=-1.25$ in the $I$ interval (this initial position is
shown as a black square point in Fig.~\ref{Fig:Cubico}). The
absorbing barrier is located at $x=-6.0$, and the results are
averaged over $10^5$ escape events. The algorithm used to simulate
the noise sources in Eqs.~(\ref{Eqn:BSCorr}) is based on the
\emph{Box-Muller} method for generating random processes with a
Gaussian distribution. The numerical integration of
Eqs.~(\ref{Eqn:BSCorr}) is done by using the \emph{forward Euler}
method~\cite{NumRec}.
Our first result shows that the curve $\tau$~vs.~$b$ is weakly
dependent on the value of the $\rho$ parameter. This is shown in
Fig.~\ref{Fig:t_b_corr}, where all the curves correspond to the
region of the parameters space where the effect is observable. There
is indeed a weak variation in the maximum value. The highest maximum
values correspond to the highest absolute $\rho$ values. It is worth
noting that the correlation affects directly the noise term in the
$v(t)$ equation, and it has negligible influence on the reverting
term of the same equation.
\begin{figure}[htbp]
\vspace{5mm}
\centering{\resizebox{9cm}{!}{\includegraphics{SpagnoloFig2.eps}}}
\caption{\label{Fig:t_b_corr} Mean escape time $\tau$ as a function
of the mean volatility $b$. The other parameters have the fixed
values $a=10^{-2}$ and $c=10^{-2}$. The six curves correspond to the
following values of the correlation: $\rho=-0.8, -0.5, -0.1, 0.1,
0.5, 0.8$. The fixed starting position is $x_0 = -1.25$. The values
of the potential parameters (see Eq.~(\ref{Eqn:BS})) are: $p = 2$,
$q = 3$. }
\end{figure}
The curves of $\tau~vs.~c$ however have an evident dependence on the
parameter $c$. This is shown in Fig.~\ref{Fig:t_c_corr}, for two
values of parameter $a$ for which the NES effect is observable. The
correlation affects the position as well as the value of the maximum
of $\tau$ (see Fig.~\ref{Fig:t_c_corr}), but in a different way. The
position of the maximum increases only for very high positive values
of the correlation coefficient. The maximum value of $\tau$
increases with positive correlation but decreases with negative
correlation. This effect is more evident for the higher value of
$a$, as shown in Fig.~\ref{Fig:t_c_corr}b.
\begin{figure}[htbp]
\vspace{13mm}
\centering{\resizebox{12cm}{!}{\includegraphics{SpagnoloFig3.eps}}}
\caption{\label{Fig:t_c_corr} Mean escape time $\tau$ as a function
of the model noise intensity $c$ for two different values of the
parameter $a$: (a) $a=2.0$ and (b) $a=20$. The value of the
parameter $b$ is fixed to $b=10^{-2}$. The fixed starting position
$x_0$ and the potential parameters $p$ and $q$ are the same of
Fig.~\ref{Fig:t_b_corr}. The different curves correspond to the
following values of $\rho$: -0.8 (circle), 0.0 (square), 0.8
(triangle).}
\end{figure}
To comment this result we first note that a negative correlation
between the logarithm of the price and the volatility means that a
decrease in $x(t)$ induces an increase in the volatility $v(t)$, and
this causes the Brownian particle to escape easily from the well. As
a consequence the mean lifetime of the metastable state decreases,
even if the nonmonotonic behavior is still observable. On the
contrary, when the correlation $\rho$ is positive, the Brownian
particle stays more inside the well, decrease in $x(t)$ indeed is
associated with decrease in the volatility. The escape process
becomes slow and this increases further the lifetime of the
metastable state, causing an increase in the value of the maximum of
the curve of Fig.~\ref{Fig:t_c_corr}. To illustrate better this
aspect and the behaviour of Figs.~\ref{Fig:t_b_corr}
and~\ref{Fig:t_c_corr} near the maximum, we plot the values
$\tau_{max}$ of the maximum as a function of the correlation
coefficient $\rho$ and we show these curves in
Fig.~\ref{Fig:max_rho}. Specifically in Fig.~\ref{Fig:t_b_corr}a we
report the values of $\tau_{max}$ related to the curves of $\tau$ of
Fig.~\ref{Fig:t_b_corr}, and in Fig.~\ref{Fig:t_b_corr}b those
related to Fig.~\ref{Fig:t_c_corr}.
\begin{figure}[htbp]
\vspace{5mm}
\centering{\resizebox{12cm}{!}{\includegraphics{SpagnoloFig4.eps}}}
\caption{\label{Fig:max_rho} The values of $\tau_{max}$ as a
function of the correlation coefficient of the curves reported in
Figs.~\ref{Fig:t_b_corr} and~\ref{Fig:t_c_corr}. Specifically: (a)
$\tau_{max}$ from Fig.~\ref{Fig:t_b_corr} ($\tau$ as a function of
parameter b) with $a = 10^{-2}$, $c = 10^{-2}$); (b) $\tau_{max}$
from Fig.~\ref{Fig:t_c_corr} with $a = 2.0$~(circles) and $a =
20$~(squares), $b = 10^{-2}$. The fixed starting position $x_0$ and
the potential parameters $p$ and $q$ are the same of
Fig.~\ref{Fig:t_b_corr}.}
\end{figure}
The increase of stability is evident in panel (b) as $\tau_{max}$
increases with $\rho$.
It is interesting to show the probability density function (PDF) of
stock price returns for the model described by the
Eqs.~(\ref{Eqn:BSCorr}). This is done in Fig.~\ref{Fig:PDF-returns}.
\begin{figure}[htbp]
\vspace{5mm}
\centering{\resizebox{9cm}{!}{\includegraphics{SpagnoloFig5.eps}}}
\caption{\label{Fig:PDF-returns} Probability density function of
stock price returns for the Heston model with a metastable state and
correlated noise sources (Eqs.~(\ref{Eqn:BSCorr})). The parameters
of the Cox-Ingersoll-Ross process of Eqs.~(\ref{Eqn:BSCorr}) are: $a
=2$, $b = 0.01$, $c = 26$. The cross correlation coefficient is
$\rho = -0.3$. The potential parameters are the same of
Fig.~\ref{Fig:t_b_corr}.}
\end{figure}
As one can see the qualitative behavior of a fat tail distribution,
typical of real financial market data, is recovered~\cite{MSt,Bou},
but with a peculiar asymmetry. To characterize quantitatively the
PDF of returns (Fig.~\ref{Fig:PDF-returns}) as regards the width,
the asymmetry and the fatness of the distribution, we calculate the
mean value $<\Delta x>$, the variance $\sigma_{\Delta x}$, the
skewness $\kappa_3$ and the kurtosis $\kappa_4$. We obtained the
following values: $<\Delta x> = - 0.434$, $\sigma_{\Delta x} =
0.903$, $\kappa_3 = - 2.086$, $\kappa_4 = 9.417$. These statistical
quantities clearly show the asymmetry of the distribution and its
leptokurtic nature observed in empirical investigations,
characterized by a narrow and larger maximum, and by fatter tails
than in the Gaussian distribution~\cite{MSt,Bou}.
The presence of the asymmetry is very interesting and it will be
subject of future investigations. It is worthwhile to note, however,
that the PDF of returns become asymmetric in crash and rally
days~\cite{Lil00,LiMa00}, that are just the time periods of
financial data where the related dynamical regimes could be
described by the models~(\ref{Eqn:BS}) and~(\ref{Eqn:BSCorr}). Of
course the quantitative agreement between the PDF of real data and
that obtained from these models requires further investigations on
the parameter value choice of the cubic potential, the parameters of
the simple Heston model and the correlation coefficient. This
analysis is outside the aim of the present work and will be further
investigated in a forthcoming paper.
Finally in the following Fig.~\ref{Fig:PDF-escape times} we report
the comparison between the probability density function of the
escape times of daily price returns from real market data and that
obtained from the model described by the Eqs.~(\ref{Eqn:BSCorr}).
The data set used here consists of daily closure prices for 1071
stocks traded at the NYSE and continuously present in the 12-year
period $1987-1998$ (3030 trading days). The same data were used in
previous investigations by one of the
authors~\cite{BonannoPhysicaA,Bon_PRE_03,Bon_EPJB04}. From this data
set we obtained the time series of the returns and we calculated the
time to hit a fixed threshold starting from a fixed initial
position. The two thresholds were chosen as a fraction of the
standard deviation $\sigma_n$ observed for each stock during the
above mentioned whole time period (n is the stock index, varying
between $1$ and $1071$). Specifically we chose: $(\Delta x_i)_n =
-0.1 \sigma_n$ and $(\Delta x_f)_n = -1.0 \sigma_n$. The parameters
of the CIR process are: $a = 20$, $b = 0.01$, $c = 2.4$. The cross
correlation coefficient is $\rho = -0.9$, and the potential
parameters are the same of Fig.~\ref{Fig:t_b_corr}. As one can see
the agreement between real and theoretical data is very good, except
at small escape times. The choice of this parameter data set is not
based on a fitting procedure as that used for example in
Ref.~\cite{Yakovenko}, where the minimization of the mean square
deviation between the PDF of the returns extracted from financial
data and that obtained theoretically is done. We chose the parameter
set in the range in which we observe the nonmonotonic behaviour of
the mean escape time as a function of the parameters $b$ and $c$.
Then by a trial and error procedure we selected the values of the
parameters $a$, $b$, and $c$ for which we obtain the best fitting
between the PDF of escape times of the price returns calculated from
the modified Heston model (Eqs.~(\ref{Eqn:BSCorr})) and that
obtained from time series of real market data. Of course a better
quantitative procedure could be done, by considering also the
potential parameters. This will be done, together with a detailed
analysis of PDF of returns and its asymmetry, in a forthcoming
paper.
\begin{figure}[htbp]
\vspace{5mm}
\centering{\resizebox{9cm}{!}{\includegraphics{SpagnoloFig6.eps}}}
\caption{\label{Fig:PDF-escape times} Comparison between the
probability density function of escape times of the price returns
for the modified Heston model with a metastable state
(Eqs.~(\ref{Eqn:BSCorr})) (solid line) and the PDF of escape times
of returns for real market data (cirles). The parameter values of
the model~(\ref{Eqn:BSCorr}) are: $p = 2$, $q = 3$, $a = 20$, $b =
10^{-2}$, and $c = 2.4$. The cross correlation coefficient between
the noise sources is: $\rho = -0.9$.}
\end{figure}
\section{Conclusions}
We have investigated the statistical properties of the escape time
in a generalized Heston market model, characterized by the presence
of a metastable state in the effective potential of the logarithm of
the price $p(t)$. We observe the NES effect in the system
investigated. The presence of correlation between the stochastic
volatility and the noise source which affects directly the dynamics
of the quantity $x(t) = \ln p(t)$ (as in usual market models), can
influence the stability of the market. Specifically a positive
correlation between $x(t)$ and volatility $v(t)$ slows down the
walker escape process, that is it delays the crash phenomenon by
increasing the stability of the market. A negative correlation on
the contrary accelerates the escape process, lowering the stability
of the system.
\\
\\
\\
\noindent This work was partially supported by MIUR.
| 2024-02-18T23:39:50.676Z | 2006-10-09T11:55:26.000Z | algebraic_stack_train_0000 | 617 | 3,538 |
|
proofpile-arXiv_065-3454 | \section{Introduction}
The semiclassical Peierls-Onsager-Lifshitz theory\cite{peierls_substitution,onsager,lifshitz_kosevich} connects experimentally-accessible quantities in magnetic phenomena to Fermi-surface parameters of the solid at zero field. For example, field-induced quantum oscillations of magnetization\cite{dHvA} and resistivity\cite{SdH} have become the leading method to map out the shape of the Fermi surface of normal metals\cite{shoenberg,ashcroft_mermin} and superconductors\cite{champel_mineev} -- this phenomenology has been coined `Fermiology'.\cite{shoenberg}
The semiclassical theory has been extended\cite{kohn_effham,rothI,blount_effham,wilkinson_semiclassical_harper} to incorporate two modern notions: a wavepacket orbiting in quasimomentum $(\boldsymbol{k})$ space acquires a geometric phase ($\phi_{\sma{B}}$),\cite{berry1984,mikitik_berryinmetal} as well as a second phase ($\phi_{\sma{R}}$) originating from the orbital magnetic moment of a wavepacket around its center of mass.\cite{chang_niu_hyperorbit} Further accounting for the well-known Zeeman coupling, $\lambda{:=}\phi_{\sma{B}}{+}\phi_{\sma{R}}{+}\phi_{\sma{Z}}$ is known\cite{rothI,rothII,fischbeck_review,Gao_Niu_zerofieldmagneticresponse} to be the complete, subleading (in powers of the field) correction to the Bohr-Sommerfeld quantization rule for nondegenerate bands.\cite{lifshitz_kosevich,onsager,zilberman_wkb} $\lambda$ is measurable as a phase offset in oscillations of the magnetization/resistivity in 3D solids, as well as in fixed-bias oscillations of the differential conductance in tunneling spectroscopy.\cite{Sangjun_Cd3As2,Ilija_SnTe}
While it is conventionally believed that $\phi_{\sma{B}}{=}0$ vs $\pi$ distinguishes between Schr\"odinger and Dirac systems,\cite{igor_kopelevich} we propose to view $\phi_{\sma{B}}/\pi$ as a continuous quantity that is sometimes fixed to an integer in certain space groups and for certain types of field-dependent orbits; moreover, while $\phi_{\sma{R}}$ vanishes for centrosymmetric metals without spin-orbit coupling (SOC), it plays an oft-ignored role in most other space groups. Our comprehensive symmetry analysis identifies the (magnetic) space groups in which $\lambda/\pi$ is robustly integer-valued -- we will formulate $\lambda$ as a topological invariant in magnetotransport, which is distinct from the traditional formulation of topological invariance in band insulators.\cite{schnyder2008A,kitaev2009} We also extend our symmetry analysis to the multi-band generalization of $\lambda$, with envisioned application to bands of arbitrary degeneracy ($D$); $D{=}2$ is exemplified by spin degeneracy.
Let us outline the organization of the main text. We begin in \s{sec:multibandquantrule} by introducing the multi-band quantization rule, and describing how $\lambda$ appears as the subleading phase correction. Experimental methods to extract $\lambda$ are discussed in \s{sec:extractlambda}; we present here generalized Lifshitz-Kosevich formulae for the oscillatory magnetization and density of states. These formulae extend previous works\cite{lifshitz_kosevich} in their applicability to orbits of any energy degeneracy and symmetry, including orbits in magnetic solids. In \s{sec:symmetrylambda}, we provide a general group-theoretic framework to identify solids for which $\lambda$ takes only discrete values. In addition, our symmetry analysis identifies the symmetry-enforced degeneracy of Landau levels; where degeneracy is not enforced, symmetry may nevertheless constrain the possible splittings of Landau levels. We exemplify our symmetry analysis with several case studies in \s{sec:casestudies}, including: graphene, transition metal dichalchogenides, surface states of topological insulators, and 3D Weyl and Dirac metals. In particular, we point out that a $\pi$ phase offset in the fundamental oscillation should \emph{not} be viewed as a smoking gun for a 3D Dirac metal. We recapitulate our main results in the concluding \s{sec:discussion}; a final remark broadens the applicability of our symmetry analysis to matrix representations of holonomy\cite{barrysimon_holonomy} in the Brillouin torus, also known as Wilson loops\cite{wilczek1984} of the Berry gauge field.\cite{berry1984}
\section{Multi-band Bohr-Sommerfeld quantization rule}\la{sec:multibandquantrule}
The quantization rule is derived from the effective Hamiltonian (${\cal H}$) that describes the low-energy dynamics of Bloch electrons in a field.\cite{kohn_effham,blount_effham,wannier_fredkin,rothI,nenciu_review} In a basis comprising $D$ field-modified Bloch functions at each wavevector, ${\cal H}(\boldsymbol{K})$ is a Weyl-symmetrized, matrix function of the kinetic quasimomentum operators, whose noncommutivity is manifest in $\boldsymbol{K}{\times}\boldsymbol{K}{=}{-i}e\boldsymbol{B}/c$. ${\cal H}$ is asympotically expandable in powers of the field: $H_0{+}H_1{+}\ldots$, where the leading-order term is the Peierls-Onsager Hamiltonian,\cite{peierls_substitution,onsager} while the subleading terms $H_1{:}{=}H_{{1}}^{\sma{B}}{+}H_1^{\sma{R}}{+}H_1^{\sma{Z}}$ respectively encode the geometric phase, the orbital moment, and the Zeeman effect.\cite{kohn_effham,blount_effham,rothI}
In the WKB approximation, the $D$-component vector wavefunction of ${\cal H}$ generalizes\cite{AALG_100} the known single-component solution developed by Zilberman\cite{zilberman_wkb} and Fischbeck\cite{fischbeck_review} for a nondegenerate band.
In the absence of breakdown,\cite{cohen_falicov_breakdown,blount_effham,azbel_quasiclassical} continuity of the vector wavefunction around a closed orbit ($\mathfrak{o}$) affords us the following quantization rule:
\e{ l^2 S(E,k_z)+ \lambda_a(E,k_z) = 2\pi j+ \phi_{\sma{M}}.\la{rule3b}}
We have assumed here that the field is oriented in $\vec{z}$, such that $\mathfrak{o}$ is a contour of the band dispersion at
fixed energy $E$ and $k_z$. By `closed orbit', we mean that $\mathfrak{o}$ does not wrap around the Brillouin torus. $\mathfrak{o}$ bounds a region in $\boldsymbol{k}^{{\sma{\perp}}}{:=}(k_x,k_y)$-space with positive-definite area $S$. $l{:}{=}(\hbar c/e|\boldsymbol{B}|)^{\sma{1/2}}$ above is the magnetic length, $j$ an integer, $a {\in} \mathbb{Z}_D{:}{=}\{0,1,\ldots,D-1\}$. The Maslov correction $(\phi_{\sma{M}})$ depends on the topology of the Fermi-surface orbit, e.g., it equals $\pi$ for orbits that are deformable to a circle,\cite{keller1958} but vanishes for a figure-of-eight orbit.\cite{AALG_breakdown} To leading order in $l^{\text{-}2}$, \q{rule3b} without $\lambda_a$ is a well-known result by Lifshitz and Onsager.\cite{lifshitz_kosevich,onsager,zilberman_wkb}
The remaining term $\lambda_a$ is defined through the spectrum $(\{e^{\sma{i\lambda}_a}\}_{\sma{a=1}}^{\sma{D}})$ of the propagator (${\cal A}$) that is generated by $H_1$ over the cyclotron period. ${\cal A}$ may be expressed as a path-ordered exponential (denoted $\overline{\exp}$):
\e{{\cal A}[\mathfrak{o}] = \overline{\exp}\left[i\textstyle{\oint}_{{\mathfrak{o}}} \left\{(\boldsymbol{\mathfrak{X}}+\boldsymbol{\mathfrak{A}}) \cdot d\boldsymbol{k} + \tf{g_0\hbar}{4mv^{\sma{\perp}}}\sigma^z{|d\boldsymbol{k}|} \right\} \right], \la{definenonabelianunitary}}
with $\mathfrak{o}$ carrying a clockwise orientation; the above three one-forms represent contributions by $H_{{1}}^{\sma{B}}$, $H_1^{\sma{R}}$ and $H_1^{\sma{Z}}$ respectively. The first one-form is the non-abelian Berry connection\cite{berry1984,wilczek1984} for the $D$-fold-degenerate subspace (henceforth denoted by $P$), and is defined by $\boldsymbol{\mathfrak{X}}(\boldsymbol{k})_{mn}{:=}i\braket{u_{m\boldsymbol{k}}}{\nabla_{\boldsymbol{k}} u_{n\boldsymbol{k}}}$, with $e^{i\boldsymbol{k} \cdot \boldsymbol{r}} u_{n\boldsymbol{k}}$ the Bloch function of a band labelled by $n{\in}\mathbb{Z}_D$. The multi-band orbital magnetic moment is encoded in the second one-form
\e{ \boldsymbol{\mathfrak{A}}_{mn} \cdot d\boldsymbol{k} = {\sum}_{l\notin \mathbb{Z}_D} {\mathfrak{X}^x_{ml}\Pi^y_{ln}} {dk_x}/{2{v}_y}+ (x \leftrightarrow y), \la{definerothoneform}}
with $m,n{\in} \mathbb{Z}_D$. $\boldsymbol{\Pi}(\boldsymbol{k})_{ln}{=}i\bra{u_{l\boldsymbol{k}}}e^{-i\boldsymbol{k} \cdot \hat{\boldsymbol{r}}}[\hat{H}_0,\hat{\boldsymbol{r}}]e^{i\boldsymbol{k} \cdot \hat{\boldsymbol{r}}}\ket{u_{n\boldsymbol{k}}}/\hbar$ are matrix elements of the velocity operator, with $\hat{H}_0$ the single-particle, translation-invariant Hamiltonian and $\hat{\boldsymbol{r}}$ the position operator. For $n{\in}\mathbb{Z}_D$, $\boldsymbol{\Pi}_{nn}{=}\boldsymbol{v}$ is the velocity of each band in $P$, and $v^{\sma{\perp}}{:}{=}(v^{\sma{2}}_x+v^{\sma{2}}_y)^{\sma{1/2}}$. $\boldsymbol{\mathfrak{X}}$ and $\boldsymbol{\Pi}$ in \q{definerothoneform} comprise only off-block-diagonal matrix elements between $P$ and its orthogonal complement. The third one-form in \q{definenonabelianunitary} is the well-known Zeeman coupling, with $\hbar\sigma^z_{mn}(\boldsymbol{k})/2$ the matrix elements of the spin operator $S_z$, $g_0{\approx}2$ the free-electron g-factor and $m$ the free-electron mass. While the definition of ${\cal A}$ presumes a basis choice $\{u_{n\boldsymbol{k}}\}_{\sma{n=1}}^{\sma{D}}$ within $P$, one may verify that the eigenvalues of ${\cal A}$ are independent of this choice. By setting $D$ to $1$ in the above equations, the line integral of the three one-forms in \q{definenonabelianunitary} give $\lambda_1{:}{=}\lambda{=}\phi_{\sma{B}}{+}\phi_{\sma{R}}{+}\phi_{\sma{Z}}$ respectively, as we have introduced in the second paragraph of the paper.
When \q{rule3b} is viewed at fixed field, the discrete energetic solutions $\{E_{a,j}\}$ correspond to $D$ sets of sub-Landau levels; within each set labelled by $a$, the difference between two adjacent levels ($|E_{a,j+1}{-}E_{a,j}|$) is approximately $\hbar \omega_c{:=}2\pi/(l^2|\partial S/\partial E|)$ evaluated at $E_{a,j}$. When \q{rule3b} is viewed at constant energy (e.g., the chemical potential $\mu$), the discrete solutions ($\{l^2_{a,j}\}$) correspond to values of the field where Landau levels successively become equal to $\mu$. In thermodynamic equilibrium, these are also the fields where Landau levels are suddenly depopulated with a periodicity: $l^2_{a,j+1}{-}l^2_{a,j}{=}2\pi/S(\mu)$, for each of $a{\in} \{1,\ldots,D\}$. This results in various oscillatory phenomena from which we may extract $\lambda_a$.
\section{Generalized Lifshitz-Kosevich formulae to extract $\lambda$} \la{sec:extractlambda}
In the de Haas-van Alphen effect,\cite{dHvA} each extremal orbit ($\mathfrak{o}$) on the Fermi surface of a 3D metal is associated to an oscillatory contribution to the longitudinal magnetization (parallel to the field in $\vec{z}$):
\e{ &\delta{{\cal M}} =- \f1{(2\pi)^{3/2}}\f{kT}{|\boldsymbol{B}|} \, \f{S}{l|S_{zz}|^{1/2}}\notag \\
&\times \sum_{a=1}^D\sum_{r=1}^{\infty}e^{-\tf{r\pi }{\omega_c\tau}}\f{\sin\left[ r\left(l^2S{+}\lambda_a{-}\phi_{\sma{M}}\right) {\pm} \pi/4\right]}{r^{1/2} \,\text{sinh}\,(2\pi^2 rkT/\hbar \omega_c)},\la{oscmag3D}}
which is a sum of $D$ sets of harmonics. Being valid in the degenerate ($\mu {\gg} kT$) and semiclassical ($\mu {\gg} \hbar \omega_c$) limits, \q{oscmag3D} is our generalization of the Lifshitz-Kosevich formula\cite{lifshitz_kosevich} to orbits of any energy degeneracy ($D$) and symmetry, including orbits in magnetic solids. In comparison, the commonly-employed Lifshitz-Kosevich formula with a `spin reduction factor'\cite{Dingle_I,shoenberg} is only applicable to two-fold degenerate orbits in solids with both time-reversal and spatial-inversion symmetries. All quantities on the right-hand side of \q{oscmag3D} are evaluated on $\mathfrak{o}$, which may be electron- or hole-like; the sign of $\pi/4$ (in the argument of the sine function) is negative (resp.\ positive) for a maximal (resp.\ minimal) orbit. $S_{zz}$ is the double derivative of $S$ with respect to $k_z$, and we have introduced Dingle's damping factor\cite{Dingle_collisions} that depends on the quasiparticle's mean free time ($\tau$).
For $D{=}1$, the field-independent phase in the argument of the fundamental harmonic is sometimes referred to as the Onsager phase:
\e{-2\pi \gamma := \lambda-\phi_{\sma{M}} \pm \pi/4.\la{onsagerphase}}
If a Fermi surface has multiple extremal orbits, each extremal orbit additively contributes a term with the same functional form as \q{oscmag3D}. If two extremal orbits ($\mathfrak{o}_i$ and $\mathfrak{o}_{i+1}$) are symmetry-related, they contribute oscillatory terms that are identical in the parameters $\{S,S_{zz},\omega_c,\phi_{\sma{M}}\}$, but not necessarily for the $\lambda$-phase corrections. Generally, $\{\lambda^i_a\}_{a=1}^D{=}\{\pm \lambda^{i+1}_a\}_{a=1}^D$ (defined modulo $2\pi$), with the sign depending on the symmetry class of the orbit, as we will elaborate in \q{IIBconst} of \s{sec:symmetrylambda}.
In 2D metals, the analogous oscillatory formula is
\e{\delta{{\cal M}} {=} {-}\f1{\pi} \f{kT}{|\boldsymbol{B}|} S \sum_{a=1}^D\sum_{r=1}^{\infty} e^{-\tf{r\pi }{\omega_c\tau}}\f{\sin[ r(l^2S{+}\lambda_a{-}\phi_{\sma{M}})]}{ \text{sinh}\,(2\pi^2 rkT/\hbar \omega_c)}\bigg|_{\mu}.\la{oscmag2D}}
The field dependence of $\mu$ is negligible in the semiclassical limit for 3D metals,\cite{shoenberg} as well as for 2D surface states of 3D solids.\cite{champel} In strictly-2D metals with a fixed particle density, field-induced oscillations in $\mu$ render the extraction of $\lambda$ implausible.
An alternative method to extract $\lambda_a$ is to measure the temperature-broadened, 3D density of states, defined as
\e{{\cal G}_{T}(E+\mu):= -\int_{-\infty}^{\infty}d\varepsilon\,f_{T}'(\varepsilon-\mu-E)\,g(\varepsilon).\la{defineTbroadenedDOS}}
Here, $g(E)$ is the density of states of 3D Landau levels (smoothened by the Dingle factor); $f_T'(x)$ is the derivative of the Fermi-Dirac distribution function, and approaches ${-}\delta(x)$ in the zero-temperature limit. The oscillatory component of ${\cal G}$ may be expressed as a harmonic expansion:
\e{&\delta {\cal G}(E)= \sqrt{2\pi} \f{kT}{(\hbar\omega_c)^2} \f1{l^3|S_{zz}|^{1/2}} \notag \\
&\times \sum_{a=1}^{D}\sum_{r=1}^{\infty}r^{\sma{1/2}}e^{-\tf{r\pi }{\omega_c\tau}}\f{\cos\left[ r(l^2S{+}\lambda_a{-}\phi_{\sma{M}}) {\pm} \pi/4\right]}{ \text{sinh}\,(2\pi^2 r kT/\hbar \omega_c)}\bigg|_{E,\bar{k}_z}.\la{oscillatorydIdV}}
This quantity may be accessed via the scanning tunneling microscope (STM)\cite{Sangjun_Cd3As2,Ilija_SnTe,satya_na3bi} or by planar tunneling junctions.\cite{ivar_planartunneljunction,sun_planartunnel_SmB6} The oscillatory component of the differential conductance ($dI/dV$), averaged over the surface of a 3D metal at fixed bias voltage ($V$) for the STM, is directly proportional to
$\delta {\cal G}(\mu+eV)$ in the absence of surface-localized states;\cite{stm_chen} for the planar tunneling junction, no spatial averaging is needed for a sufficiently large junction size. The just-mentioned proportionality presupposes that: (a) the tunneling matrix elements and the density of states of the STM tip are featureless in the energy range that is accessed by the bias voltage, and that (b) the tip and sample have thermally equilibrated.\cite{stm_chen,ivar_planartunneljunction} Landau-level spectroscopy via the scanning tunneling microscope has already been reported,\cite{Sangjun_Cd3As2,Ilija_SnTe,satya_na3bi} but we are not aware that the phase offset of the oscillations has ever been measured. Further details on the derivation of \qq{oscmag3D}{oscillatorydIdV} are provided in \app{app:dhva}.
Let us discuss how to extract $\lambda$ from dHvA data. For simplicity in presentation, we consider the magnetization oscillations contributed by a single orbit (extremal orbit in 3D metals). We assume that $l^2S$, $\hbar \omega_c$ and the Dingle lifetime $\tau$ have already been extracted by standard techniques.\cite{dhva_Dhillon} Let us first consider either $\omega_c\tau {\ll} 1$ or $kT {\gg} \hbar \omega_c$, such that the harmonic expansion is dominated by the fundamental ($r{=}1$) harmonic. If $D{=}1$, then the experimental data may directly be fitted to a single sine function offset by the Onsager phase [cf.\ \q{onsagerphase}]. If $D{=}2$ (e.g., spin degeneracy), the sum of two fundamental harmonics produces an equi-frequency harmonic proportional to\cite{rothII}
\e{ 2\bigg|\cos\bigg(\f{\lambda_1{-}\lambda_2}{2}\bigg)\bigg| \sin\left(l^2S{+}\Theta {-}\phi_{\sma{M}} {\pm} \f{\pi}{4}\right)\la{oscmag3DD2}}
with
\e{ \Theta:= \f{\lambda_1{+}\lambda_2}{2}+\pi \,\bigg(1-\text{sign}\bigg[\cos\bigg(\f{\lambda_1{-}\lambda_2}{2}\bigg) \bigg]\bigg) \la{defineTheta}}
defined to be invariant (modulo $2\pi$) under $\lambda_j {\rightarrow} \lambda_j{+}2\pi$; these formulae will be applied to a case study of Bi$_2$Se$_3$ in \s{sec:bi2se3}. If $|S_{zz}|$ is not otherwise measurable (for 3D metals), it is not possible to fully determine $\lambda_{1,2}$ owing to our ignorance of the amplitude of the fundamental harmonic; measuring the phase offset of the fundamental harmonic merely determines $\Theta$, a suitably-defined \emph{average} of $\lambda_{1,2}$. $\Theta$ alone does not completely characterize the non-abelian transport within the two-band subspace.
In the interest of measuring individual values of $\lambda_{1,2}$, we propose higher-field dHvA measurements of cleaner samples ($\omega_c\tau$ not ${\ll} 1$) and at lower temperatures ($kT$ not ${\gg} \hbar \omega_c$). In this regime, not just the fundamental but also higher ($r{>}1$) harmonics are needed to accurately represent the dHvA data.\cite{dhva_Dhillon} Since $\{\lambda_a\}_{a{=}1}^D$ is encoded in the interference of multiple harmonics, $\{\lambda_a\}_{a{=}1}^D$ may be extracted without knowledge of the absolute amplitude of a single harmonic.
In metals with multiple extremal orbits, there may exist field orientations where all orbits are related by symmetry, which simplifies the fitting to the Lifshitz-Kosevich formula.\cite{gfactorBismuth_cohen_blount} Independent of the field orientation, there exists one simplification for any non-magnetic metal: due to time-reversal ($T$) symmetry, the set of all $\{\lambda\}$ comprises only pairs that are invariant under inversion about zero ($\lambda{\rightarrow}{-}\lambda$), which effectively halves the independent parameters that require fitting. We quote this result here to exemplify the utility of a symmetry analysis of $\{\lambda\}$, which we explore in greater generality in the next section [\s{sec:symmetrylambda}]; the above mentioned constraint by $T$ symmetry is elaborated subsequently in \s{sec:globalsumrule}.
\section{Symmetry analysis of the $\lambda$ phase}\la{sec:symmetrylambda}
In certain (magnetic) space groups, $\lambda$ [or ${\sum}_{\sma{a=1}}^{\sma{D}}\lambda_a/\pi$ for $D{>}1$] is integer-valued owing to the symmetry of the extremal orbit. To identify these orbits and space groups, it is useful to distinguish ten symmetry classes for closed orbits -- to each class we associate certain constraints for the propagator ${\cal A}$ and its spectrum [$\{\lambda_a\}_{a=1}^D$], as summarized in \tab{tab:tenfold}. The goal of this section is identify the relevant symmetry class of closed orbit -- for any physical system one chooses to study. Once this identification is made, the resultant symmetry constraints on $\{\lambda_a\}_{a=1}^D$ may be read off from the last column of \tab{tab:tenfold}, and verified experimentally from any of the methods detailed in \s{sec:extractlambda}. This will be exemplified for several case studies in \s{sec:casestudies}.
Let us first restate the problem in simple terms: the dynamics of Bloch electrons immersed in $\vec{z}$ are restricted to Brillouin two-tori ($BT_{\sma{\perp}}$) of fixed $k_z$. For a $D$-fold degenerate band subspace with dispersion $\varepsilon(\boldsymbol{k})$, semiclassical motion occurs along (assumed) closed orbits defined by $\varepsilon(\boldsymbol{k}^{{\sma{\perp}}},k_z){=}E$, with $\boldsymbol{k}^{{\sma{\perp}}}$ parametrizing $BT_{\sma{\perp}}$$(k_z)$. If multiple disconnected orbits exist within the same $BT_{\sma{\perp}}$, we assume they are sufficiently separated in $\boldsymbol{k}^{{\sma{\perp}}}$-space that tunneling is negligible. Neglecting the subleading term ($H_1$) in the effective Hamiltonian, all Landau levels are at least $D$-fold degenerate owing to the Onsager-Lifshitz quantization rule; here and henceforth, the `degeneracy of a Landau level' is defined in units where the extensive degeneracy of a Landau level (associated to a single spinless orbit) equals $1$ [cf.\ \q{LLdeg}]. For a subset of Landau levels, this zeroth-order degeneracy is enhanced to $LD$ if a symmetry ($g$) constrains $L$ disconnected orbits to have identical shape. We may ask if (and how) $H_1$ splits this $LD$-fold (or $D$-fold) degeneracy;\footnote{Where $L{>}1$, we consider $L$ symmetry-related propagators ${\cal A}_i$ with an additional index $i{\in} \mathbb{Z}_L$; each of ${\cal A}_i$ is a matrix of dimension $D$, and its eigen-phases are denoted by $\lambda_a^i$, with $a\in \mathbb{Z}_D$.} if $L{=}D{=}1$, we ask if $H_1$ shifts the zeroth-order Landau-level spectrum at all. The answer to these questions depends on the class of symmetric orbit, which we proceed to analyze in full generality.
\subsection{Tenfold classification of symmetric orbits}\la{sec:tenfoldorbits}
Since lattice translations trivially constrain ${\cal A}$, we shall henceforth focus on symmetries ($g$) of the solid that correspond to nontrivial elements in the (magnetic) point group ($P$) of the solid; examples include (screw) rotations, (glide) reflections, spatial inversion and time reversal. Technically speaking, magnetic point groups differ from point groups in that the former includes a symmetry that reverses time; however this distinction is irrelevant to the following classification of symmetric orbits, hence we hereafter use `point group' democratically.
We are interested only in $g$ that maps $BT_{\sma{\perp}}$ to itself; such $g$ correspond to a subgroup ($P_{\perp}$) of $P$ that generally depends on the field orientation as well as $k_z$. Any configuration of closed orbits in $BT_{\sma{\perp}}$ may be divided into a disjoint set of elementary orbits $\{(g,{\cal O}_i)\}$, where ${\cal O}_i$ is defined to be the smallest, closed orbit configuration that is invariant under $g$. By `invariance', we mean that for every $\boldsymbol{k}^{{\sma{\perp}}} {\in} {\cal O}_i$, the map of $\boldsymbol{k}^{{\sma{\perp}}}$ under $g$ (denoted as $g {\circ} \boldsymbol{k}^{{\sma{\perp}}}$) belongs also in ${\cal O}_i$. Similarly, if a closed orbit $\mathfrak{o} {\in} {\cal O}_i$, so would $g{\circ}\mathfrak{o}{\in}{\cal O}_i$.
There are three topologically distinct mappings of $\boldsymbol{k}^{{\sma{\perp}}}{\in}\mathfrak{o}$. The simplest is the identity map, where each $\boldsymbol{k}^{{\sma{\perp}}}$ in $BT_{\sma{\perp}}$ (but not necessarily in the entire 3D torus) is individually invariant under $g$. Such mappings are labelled as class I, and all other mappings are of class II. We further distinguish between class-II mappings where $g\circ \mathfrak{o}$ is identical to $\mathfrak{o}$ up to orientation [class II-A], or they are disconnected orbits [class II-B].
There are two classes of class-I elementary orbits distinguished by whether $g$ is purely a spatial transformation, or otherwise includes a time reversal. We introduce a $\mathbb{Z}_2$ index $s(g)$ which equals $0$ in the former, and $1$ in the latter. Class $[I,s{=}0]$ is exemplified by $BT_{\sma{\perp}}$ being a mirror/glide-invariant plane, and $[I,s{=}1]$ by $g{=}T\mathfrak{i}$, which is the composition of time reversal ($T$) with spatial inversion ($\mathfrak{i}$); all class-I symmetries are order two. Class-II elementary orbits are likewise distinguished by whether $g$ inverts time; they are additionally distinguished by whether $g$ acts on $\boldsymbol{k}^{{\sma{\perp}}}$ as a two-dimensional rotation ($u{=}0$), or as a two-dimensional reflection ($u{=}1$). Equivalently, given that $\mathfrak{o}$ is clockwise-oriented, $u(g)$ distinguishes between symmetries that preserve ($u{=}0$) or invert ($u{=}1$) this orientation. In each of II-A and II-B, there are then four classes of elementary orbits distinguished by $s,u{\in}\mathbb{Z}_2$. This gives ten classes of elementary orbits in total, whose defining characteristics are summarized in the first three columns of \tab{tab:tenfold}.
\begin{table}[ht]
\centering
\scalebox{0.85}{
\begin{tabular} {|r|c|c|l|l|c|} \cline{2-6}
\multicolumn{1}{c}{} &\multicolumn{1}{|c}{$u$}& \multicolumn{1}{|c}{$s$} & \multicolumn{2}{|c}{Symmetry constraints} & \multicolumn{1}{|c|}{$\lambda$} \\ \hline \hline
(I)$\;\;\, \forall \, \boldsymbol{k}^{{\sma{\perp}}},$ & $0$& $0$ & ${\cal A}{=}\bar{g}{\cal A}\bar{g}^{\text{-}1}$ & $\bar{g}^{\sma{2}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $-$ \\ \cline{2-6}
$\boldsymbol{k}^{{\sma{\perp}}}{=}g{\sma{\circ}}\boldsymbol{k}^{{\sma{\perp}}}$ & $0$& $1$ & ${\cal A}{=}\bar{g}{\cal A}^*\bar{g}^{\text{-}1}$ & $(\bar{g}K)^{\sma{2}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $e^{\sma{ i{\sum}_a\lambda_a}}{\in} \mathbb{R}$ \\ \hline
\multicolumn{1}{|l|}{(II-A)} & $0$& $0$ & ${\cal A}{=}\bar{g}{\cal A}\bar{g}^{\text{-}1}$ & $\bar{g}^{\sma{N}}{=}{\cal A}^{\sma{{\pm} N{/}L}}e^{\sma{i\pi F\mu}}$ & \multicolumn{1}{c|}{$-$} \\ \cline{2-6}
$\boldsymbol{k}^{{\sma{\perp}}} \in \mathfrak{o},$ & $0$& $1$ & ${\cal A}{=}\bar{g}{\cal A}^*\bar{g}^{\text{-}1}$ & $(\bar{g}K)^{\sma{N}}{=}{\cal A}^{\sma{\pm N/L}}e^{\sma{i\pi F\mu}}$ & $e^{\sma{ i{\sum}_a\lambda_a}}{\in} \mathbb{R}$ \\ \cline{2-6}
$|\mathfrak{o}| {=} |g{\sma{\circ}}\mathfrak{o}|$ & $1$& $0$ & ${\cal A} {=}\bar{g}{\cal A}^{\text{-}1}\bar{g}^{\text{-}1}$ & $\bar{g}^{\sma{N}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $e^{\sma{ i{\sum}_a\lambda_a}}{\in} \mathbb{R}$ \\ \cline{2-6}
& $1$& $1$ & ${\cal A} {=}\bar{g}{\cal A}^{t}\bar{g}^{\text{-}1}$ & $(\bar{g}K)^{\sma{N}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$& \multicolumn{1}{c|}{$-$} \\ \hline
\multicolumn{1}{|l|}{(II-B)} & $0$& $0$ & ${\cal A}_{\sma{i+1}}{=}\bar{g}_{\sma{i}}{\cal A}_{\sma{i}}\bar{g}_{\sma{i}}^{\text{-}1}$ & $\bar{g}_{\sma{N}}\ldots \bar{g}_{\sma{1}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $\{\lambda^{\sma{i{+}1}}_{\sma{a}}\}{=}\{\lambda^{\sma{i}}_{\sma{a}}\}$ \\ \cline{2-6}
$\boldsymbol{k}^{{\sma{\perp}}} \in \mathfrak{o},$ & $0$& $1$ & ${\cal A}_{\sma{i+1}}{=}\bar{g}_{\sma{i}}{\cal A}_{\sma{i}}^*\bar{g}_{\sma{i}}^{\text{-}1}$ & $\bar{g}_{\sma{N}}K\ldots \bar{g}_{\sma{1}}K{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $\{\lambda^{\sma{i{+}1}}_{\sma{a}}\}{=}\{\text{-}\lambda^{\sma{i}}_{\sma{a}}\}$ \\ \cline{2-6}
$|\mathfrak{o}| {\neq} |g{\sma{\circ}}\mathfrak{o}|$ & $1$& $0$ & ${\cal A}_{\sma{i+1}}{=}\bar{g}_{\sma{i}}{\cal A}^{\text{-}1}_{\sma{i}}\bar{g}_{\sma{i}}^{\text{-}1}$ & $\bar{g}_{\sma{N}}\ldots \bar{g}_{\sma{1}}{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $\{\lambda^{\sma{i{+}1}}_{\sma{a}}\}{=}\{\text{-}\lambda^{\sma{i}}_{\sma{a}}\}$ \\ \cline{2-6}
& $1$& $1$ & ${\cal A}_{\sma{i+1}}{=}\bar{g}_{\sma{i}}{\cal A}^t_{\sma{i}}\bar{g}_{\sma{i}}^{\text{-}1}$ & $\bar{g}_{\sma{N}}K\ldots \bar{g}_{\sma{1}}K{=}e^{\sma{i\pi F\mu \text{-} i\boldsymbol{k} \cdot \boldsymbol{R}}}$ & $\{\lambda^{\sma{i{+}1}}_{\sma{a}}\}{=}\{\lambda^{\sma{i}}_{\sma{a}}\}$ \\ \hline
\end{tabular}
}
\caption{The first three columns distinguish between ten classes of elementary orbits. The map of $\boldsymbol{k}^{{\sma{\perp}}}$ under $g$ is $g{\circ}\boldsymbol{k}^{{\sma{\perp}}}{=}(-1)^{s(g)}\check{g}^{\sma{\perp}}\boldsymbol{k}^{{\sma{\perp}}}$, with $\check{g}^{\sma{\perp}}$ a two-by-two orthogonal matrix that represents the point-group component of $g$ in the plane orthogonal to the field; $s(g){=}0$ if $g$ is purely a spatial transformation, and ${=}1$ if $g$ inverts time. $|\mathfrak{o}| {=} |g{\sma{\circ}}\mathfrak{o}|$ indicates that $\mathfrak{o}$ is mapped to itself under $g$, modulo a change in orientation. $u(g)$ distinguishes between proper and improper transformations on $\boldsymbol{k}^{{\sma{\perp}}}$: $(-1)^{\sma{u}}{:=}$det$\,\check{g}^{\sma{\perp}}$. Fourth and fifth columns describe how unitary matrices $\bar{g}$ (that represent the symmetry $g$) constrain the propagator. Column six summarizes the constraints on $\lambda_a$; if there are none, we indicate this by $-$.
\label{tab:tenfold}}
\end{table}
In class I and II-A, ${\cal O}_i$ is composed of a single orbit $\mathfrak{o}$ which is self-constrained by $g$. In II-B, ${\cal O}_i$ is composed of $L$ disconnected orbits which are mutually constrained as $g{\circ}\mathfrak{o}_i{=}(-1)^{\sma{u}}\mathfrak{o}_{i+1}$ and $\mathfrak{o}_{i+L}{:=}\mathfrak{o}_i$. To clarify, $\mathfrak{o}$ and $\{\mathfrak{o}_j\}_{\sma{j{=}1}}^{\sma{L}}$ are all clockwise-oriented, and $-\mathfrak{o}_i$ denotes an anticlockwise-oriented orbit. $L$ was introduced in the second paragraph of \s{sec:symmetrylambda}, and is more precisely defined here as the smallest integer for which $g^L{\circ}\boldsymbol{k}^{{\sma{\perp}}}{=}\boldsymbol{k}^{{\sma{\perp}}}$ for all $\boldsymbol{k}^{{\sma{\perp}}}$; generally, $L$ divides the order ($N$) of $g$, e.g., $L{=}3$ and $N{=}6$ for the composition of $T$ and a six-fold rotation.
\subsection{Symmetry constraints on the propagator ${\cal A}$}
Column four summarizes how $g$ constrains ${\cal A}$ [I,II-A] and $\{{\cal A}_j\}$ [II-B], which are respectively the propagators for the self-constrained $\mathfrak{o}$ and mutually-constrained $\{\mathfrak{o}_j\}$. The corresponding spectra of the propagators are denoted as $\{e^{i\lambda_a}\}_{\sma{a=1}}^{\sma{D}}$ and $\{e^{i\lambda^j_a}\}_{\sma{a=1}}^{\sma{D}}$.
The unitary matrices $\bar{g}$ that constrain these propagators form a projective representation\cite{Cohomological,shiozaki_review} of the point group $P_{\perp}$, as summarized in column five. Any $g$ that inverts time $[s{=}1]$ has the antiunitary representation $\bar{g}K$, with $K$ implementing complex conjugation; otherwise $[s{=}0]$, $g$ has the unitary representation $\bar{g}$. The relations in column five are closely analogous to the space-group relations\cite{Lax} satisfied by $g$:
\e{ g^N{=}\mathfrak{e}^{\mu}\tra_{\sma{\boldsymbol{R}}}, \ins{with} \mu(g){\in}\{0,1\}. \la{spacegrouprel}}
Here, $N(g){\in}\mathbb{N}$ is the smallest integer such that $g^N$ is a translation ($\tra$) by the lattice vector $\boldsymbol{R}$, possibly composed with a $2\pi$ rotation (denoted $\mathfrak{e}$). Note the similarity in definition of $N$ with $L$, as was defined in \s{sec:tenfoldorbits}; in general, $L$ divides $N$. $\boldsymbol{R}$ is nonzero for nonsymmorphic symmetries such as screw rotations and glide reflections;\cite{Lax} the translation $\tra_{\sma{\boldsymbol{R}}}$ is represented on Bloch functions with wavevector $\boldsymbol{k}$ by the phase factor $e^{{-}i\boldsymbol{k} \cdot \boldsymbol{R}}$ [cf.\ column five]. $\mathfrak{e}$ in \q{spacegrouprel} is represented in column five by a phase factor [$(-1)^{\sma{F}}$] that a wavefunction accumulates upon a $2\pi$ rotation; $F{=}0$ (resp.\ ${=}1$) for integer-spin (resp.\ half-integer-spin) representations. The former case is useful in analyzing solids with negligible spin-orbit coupling, as we will exemplify with a case study of graphene in \s{sec:orbitsmutualT}.
The constraints in columns four and five are derived from the symmetry transformation of the Berry connection\cite{AA2014} and the one-form $\boldsymbol{\mathfrak{A}} {\cdot} d\boldsymbol{k}$ of \q{definerothoneform}. The latter may be expressed through Hamilton's equation [$\hbar \dot{\boldsymbol{k}}{=}{-}|e| \boldsymbol{v} {\times} \boldsymbol{B}/\hbar c$] as ${-}\boldsymbol{M}{\cdot}\boldsymbol{B} dt/\hbar$, with $\boldsymbol{M}(\boldsymbol{k})$ an orbital moment that transforms under $g$ like the spatial components of a (3+1)-dimensional pseudovector
\e{ \boldsymbol{M}\big|_{g\circ \boldsymbol{k}} = (-1)^{s(g)}\det[\check{g}] \,\bar{g}\, K^{s(g)} \, (\check{g}\boldsymbol{M})\, K^{s(g)} \,\bar{g}^{-1}\big|_{\boldsymbol{k}}. \la{gactsonMmultiband}}
$\check{g}$ here is a three-by-three orthogonal matrix representing the point-group component of $g$, and $K^s\boldsymbol{M} K^s{=}\boldsymbol{M}^*$ if $s{=}1$.
\subsection{Symmetry constraints on $\lambda$}\la{sec:constrainlambda}
From taking the determinant of each equation in column four, we derive constraints on $\lambda_a$ which are summarized in column six.
Three of six classes in [I,II-A] are characterized by the reality condition $e^{\sma{ i{\sum}_a\lambda_a}}{\in} \mathbb{R}$. This implies $\lambda{=}0$ or $\pi$ for a nondegenerate band, i.e., the orbit respectively encircles an even or odd number of Dirac points. $\lambda{=}\pi$ is exemplified by the Dirac surface state\footnote{While it is known that $\phi_{\sma{B}}{=}\pi$ for a time-reversal-invariant 2D Dirac fermion, it is not generally appreciated that $\phi_{\sma{R}}{=}\phi_{\sma{Z}}{=}0$.} of the $\mathbb{Z}_2$ topological insulator Bi$_2$Se$_3$;\cite{fu2007b,Inversion_Fu,moore2007,Rahul_3DTI} an orbit on the Dirac cone is self-constrained by time-reversal ($T$) symmetry [class II-A,$u{=}0,s{=}1$]. $\lambda{=}\pi$ may result solely from crystalline symmetry, as exemplified by a mirror-symmetric orbit [class II-A,$u{=}1,s{=}0$] that encircles a surface Dirac cone of the topological crystalline insulator SnTe\cite{SnTe} [further elaborated in \s{sec:mirror}]. Finally, $\lambda{=}\pi$ can be protected by a composition of $T$ and crystalline symmetry [class I, $s{=}1$], as exemplified by Weyl metals having the same space group as WTe$_2$; the robustness of $\lambda$ depends sensitively on the field orientation with respect to the crystallographic symmetry axis, as elaborated in \s{sec:fieldorientation}.
For degenerate bands, the reality condition fixes ${\sum}_{\sma{a=1}}^{\sma{D}}\lambda_a$ to $0$ or $\pi$, but not $\lambda_a$ individually. This will be exemplified by our case studies of 3D Dirac metals and $\mathbb{Z}_2$ topological insulators, in \s{sec:spindegSOC} and \s{sec:bi2se3} respectively. In \s{sec:orbitselfT}, we further demonstrate the reality constraint may be further strengthened for spin-degenerate orbits ($D{=}2$) that are self-constrained by $T$ symmetry -- into a zero-sum rule: $\lambda_1{+}\lambda_2{=}0$ mod $2\pi$. Moreover, if one considers the set of $\{\lambda\}$ contributed by all closed orbits in a $T$-symmetric solid, we propose that $\{\lambda\}$ comprise only pairs of $\lambda$, such that each pair individually satisfies a zero-sum rule, as elaborated in \s{sec:globalsumrule}. This global constraint on $\{\lambda\}$ might be viewed as an analog of the Nielsen-Ninomiya fermion-doubling theorem,\cite{NIELSEN1981} which states that there are always an equal number of left- and right-handed Weyl fermions on a lattice.
The four II-B classes are characterized by
\e{\{\lambda^{\sma{i{+}1}}_{\sma{a}}\}_{\sma{a=0}}^{\sma{D-1}}{=}\{(\text{-}1)^{\sma{u+s}} \lambda^{\sma{i}}_{\sma{a}}\}_{\sma{a=0}}^{\sma{D-1}}; \as i{\in} \mathbb{Z}_{\sma{L}}. \la{IIBconst}}
Let us discuss the implications of the above equation for the three following cases (i-iii):\\
\noi{i} For $s{\neq}u$ and even $L$ (which necessarily holds if $u{=}1$), the minimal Landau-level degeneracy is $L/2$. As illustration, the two valley-centered Fermi surfaces ($L{=}2$) in the transition-metal dichalchogenide WSe$_2$ are mutually constrained by $T$ symmetry [class II-B,$u{=}0,s{=}1$]; the Landau levels are nondegenerate but exhibit a symmetric splitting, as elaborated in \s{sec:orbitsmutualT}. \\
\noi{ii} For $s{\neq}u$ and odd $L$, the minimal Landau-level degeneracy is $L$; if $D$ is also odd, it is necessary that $\{\lambda\}$ contains either $0$ or $\pi$. This is exemplified by three of four disconnected Fermi pockets ($L{=}3$) per valley of bilayer graphene (with trigonal warping);\cite{mccann_bilayergraphene,varlet_bilayergraphene} we refer to the three Fermi pockets that are mutually constrained by the composition of $T$ and six-fold rotational symmetry $\rot_{6z}$ [class II-B, $u{=}0,s{=}1$]. For pedagogy, it is instructive to consider a model of bilayer graphene with spinless electrons, hence the orbit associated to each Fermi pocket is nondegenerate ($D{=}1$). The three-fold degenerate $\lambda{=}\pi$ reflects that each spinless pocket encircles a Dirac point.\cite{mccann_bilayergraphene,varlet_bilayergraphene}\\
\noi{iii} We would demonstrate that the same set of Fermi pockets, if equipped with multiple point-group symmetries, can belong to multiple symmetry classes in \tab{tab:tenfold}. For example, the above-mentioned three pockets are \emph{also} invariant under
the three-fold rotational symmetry $\rot_{3z}$ ($u{=}s{=}0$). We may therefore apply \q{IIBconst} with $u{=}s$, which generally implies that the the minimal Landau-level degeneracy is $L$. Again, $L{=}3$ in the present example. We remind the reader that this three-fold degeneracy constraint was consistently implied by $T\rot_{6z}$; in addition, $T\rot_{6z}$ implies a stronger constraint that $\{\lambda\}$ contains either $0$ or $\pi$.
\section{Case studies}\la{sec:casestudies}
The utility of \tab{tab:tenfold} is illustrated in the following case studies of existing conventional and topological metals, which were introduced in the previous section [\s{sec:constrainlambda}].
\subsection{Orbits mutually constrained by time-reversal symmetry: Application to graphene and transition metal dichalcogenides}\la{sec:orbitsmutualT}
Our first study encompasses materials with two time-reversal-related valleys in their band dispersion,\cite{xiaodong_tmd_review} as exemplified by graphene and monolayer WSe$_2$. We will demonstrate: (i) how orbits in time-reversal-invariant ($T$) solids can nevertheless develop a nonzero orbital magnetic moment, and (ii) the role of point-group symmetry in discretizing the Berry phase of valley-centered orbits.
To explain (i), we point out that $T$ symmetry relates the magnetic moment of wavepackets at $\boldsymbol{k}$ and $-\boldsymbol{k}$ [cf.\ \q{gactsonMmultiband}]; this mapping in $\boldsymbol{k}$-space distinguishes the symmetry transformation of magnetic moments in solids from that in atoms. This allows for valley-centered orbits that are separated in $\boldsymbol{k}$-space to \emph{individually} develop a magnetic moment -- since time reversal relates one valley to the other, the net magnetic moment must vanish.
In more detail, let us consider a finite chemical potential (as measured from the Dirac point) where valley-centered orbits ($\mathfrak{o}_i$) are disconnected; we introduce here a valley index $i{\in}\{1,2\}$. The two orbits are mutually constrained as $T{\circ}\mathfrak{o}_1{=}\mathfrak{o}_2$ [class II-B,$u{=}0,s{=}1$]; each of $\mathfrak{o}_i$ is also self-constrained as $T\mathfrak{i}{\circ}\mathfrak{o}_i{=}\mathfrak{o}_i$ [class I,s{=}1]. $T\mathfrak{i}$ imposes reality for the orbital component of wavefunctions at each $\boldsymbol{k}$, leading to $\boldsymbol{M}{=}0$ [cf.\ \q{gactsonMmultiband} with $D{=}s{=}1$ and $\check{g}{=}{-}I$]. In analyzing graphene (which has negligible spin-orbit coupling), it is useful to first neglect the spinor structure of its wavefunction and then account for the Zeeman effect after -- this was implicit in our previous assumption of $D{=}1$. Such `spinless' wavefunctions transform in an integer-spin representation ($F{=}0$ in \tab{tab:tenfold}); the corresponding ${\cal A}$ in Tab.\ \ref{tab:tenfold} should be interpreted as \q{definenonabelianunitary} without the Zeeman term.
The Berry phase of graphene is $\pi$, and therefore $\tilde{\lambda}^i{:=}\phi^i_{\sma{R}}{+}\phi^i_{\sma{B}}{=}0{+}\pi$ for each valley labelled by $i{=}1,2$, as is consistent with the reality constraint in [class I,$s{=}1$] of \tab{tab:tenfold}. We have added an accent to $\tilde{\lambda}^i$ to remind ourselves that it is the purely-orbital contribution to $\lambda^i$. Further accounting for the Zeeman contribution,
\e{\lambda_{\pm}^i = \tilde{\lambda}^i \pm \pi \,\f{g_0}{2}\,\f{m_c}{m},\la{noSOC}}
with $\tilde{\lambda}^i{=}\pi$, $\pm$ distinguishing two spin species, and $m_c$ the cyclotron mass. The symmetric splitting of $\lambda_{\pm}$ about $\pi$ implies an invariance under inversion: $\{\lambda_+^i,\lambda_-^i\}{=}\{{-}\lambda_+^i,{-}\lambda_-^i\}$ mod $2\pi$. $T$ symmetry imposes the mutual constraint:
\e{\{\lambda_+^1,\lambda_-^1\}=\{{-}\lambda_+^2,{-}\lambda_-^2\} \ins{mod} 2\pi,\la{Tmutual}}
which follows from [class II-B,$u{=}0,s{=}1$] of \tab{tab:tenfold}. \qq{noSOC}{Tmutual} together imply $\{\lambda_+^1,\lambda_-^1\}{=}\{\lambda_+^2,\lambda_-^2\}$. To recapitulate, we have reproduced the well-known fact that graphene's Landau levels are valley-degenerate but spin-split by the Zeeman effect.
The valley degeneracy of the Landau levels may be split by breaking spatial-inversion ($\mathfrak{i}$) symmetry, e.g., with a hexagonal BN\cite{hBN_gap} or SiC\cite{Zhou_SiCsubstrate} substrate. In zero field, the $\mathfrak{i}$ asymmetry is predicted to produce a band gap of $53$ and $260$ meV respectively -- this may be interpreted as a nonzero Semenoff mass for the Dirac fermion.\cite{semenoff_mass} Since each valley-centered orbit is no longer self-constrained by $T\mathfrak{i}$, it develops a nonzero orbital moment (as was first noted in Ref.\ \onlinecite{DiXiao_Magneticmoment_valley}), as well as a non-integer $\phi^i_{\sma{B}}/\pi$. Consequently, \q{noSOC} remains valid with $\tilde{\lambda}^i$ deviating from $\pi$, as we confirm with a first-principles calculation in \fig{fig:graphene_wse2}(a).\footnote{The approximate equality of $\phi_{\sma{B}}{+}\phi_{\sma{R}}$ to $\pi$ may be understood from a linearized, two-band model without next-nearest-neighbor hoppings\cite{fuchs_topologicalberryphase}} While $\{\lambda_+^i,\lambda_-^i\}$ (that is associated to one valley) is no longer invariant under inversion, we remark that the invariance persists for the full set:
$\{\lambda_+^1,\lambda_-^1,\lambda_+^2,\lambda_-^2\}$, owing to the mutual constraint by $T$ symmetry [cf.\ \q{Tmutual}].
\begin{figure}[ht]
\centering
\includegraphics[width=7 cm]{valley.png}
\caption{(a) Characterization of a single valley of non-centrosymmetric graphene on hBN substrate, assuming a translation-invariant, commensurate\cite{Woods_incommensuratehBN} phase. Left inset: plot of $\phi_{\sma{B}}$ (blue), $\phi_{\sma{R}}$ (orange), $\tilde{\lambda}{:=}\phi_{\sma{B}}{+}\phi_{\sma{R}}$ (green) and $\lambda_{\pm}$ [cf.\ \q{noSOC}] (red) against energy. Right inset: plot of the orbit area ($S$) vs $E$. (b) Characterization of a single valley of monolayer WSe$_2$. Left inset: plot of $\phi_{\sma{B}}$ (blue), $\phi_{\sma{R}}$ (orange), and $\lambda{:=}\phi_{\sma{B}}{+}\phi_{\sma{R}}{+}\phi_{\sma{Z}}$ (red) vs $E$. Right inset: plot of $S(E)$. (a-b) are obtained by \emph{ab initio} calculations detailed in \app{app:abinitio}.}\label{fig:graphene_wse2}
\end{figure}
The predicted Semenoff masses induced by substrates tend to be small -- it is instructive to compare graphene to WSe$_2$, a transition metal dichalcogenide with a large Semenoff mass due to the natural absence of inversion symmetry in its space group. WSe$_2$ is similar to graphene in that its low-energy bands (at zero-field) are also centered at two valleys, but differs from graphene in that its bands are spin-split by spin-orbit coupling. \fig{fig:graphene_wse2}(b) illustrates our calculated $\lambda^1$ as a function of energy, with $i{=}1,2$ again a valley index, and $\lambda^1{=}{-}\lambda^2$ owing to $T$ symmetry; at $0.2$ eV below the band maximum, $\lambda^1{-}\lambda^2{\approx}\pi/2$.
\subsection{Orbits self-constrained by mirror/glide symmetry: Application to topological crystalline insulators}\la{sec:mirror}
$\lambda/\pi{\in}\mathbb{Z}$ may originate from crystalline symmetry alone. This occurs for orbits that are self-constrained by a mirror/glide symmetry [IIA,$u{=}1,s{=}0$], which may arise in time-reversal-invariant and magnetic solids. Any circular orbit in [IIA,$1,0$] intersects a mirror/glide-invariant line at two points denoted by $\boldsymbol{k}_a$ and $\boldsymbol{k}_b$; at each point, the assumed-nondegenerate band transforms in either the even or odd representation of mirror/glide. The reality condition for $e^{i\lambda}$ in [IIA,$1,0$] of \tab{tab:tenfold} has a simple interpretation: $\lambda{=}0$ if the representations at $\boldsymbol{k}_a$ and $\boldsymbol{k}_b$ are identical, and $\pi$ if the two representations are distinct.\cite{AA2014,LMAA,JHAA} The former is exemplified by a band that is nondegenerate at all $\boldsymbol{k}^{{\sma{\perp}}}$ bounded by $\mathfrak{o}$, which implies that $\mathfrak{o}$ is contractible to a point -- due to continuity of the representation along the high-symmetry line, the representations at $\boldsymbol{k}_a$ and $\boldsymbol{k}_b$ must be identical. $\lambda{=}\pi$ occurs iff there is an odd number of linear band touchings along the segment of the mirror line within $\mathfrak{o}$ -- at each touching (a Dirac point), the mirror/glide representation flips discontinuously. $\lambda{=}\pi$ is exemplified by the mirror-symmetric surface states of the SnTe-class\cite{SnTe} of topological crystalline insulators, as well as by glide-symmetric Dirac cones in band-inverted, nonsymmorphic metals.\cite{LMAA}
\subsection{Effect of field orientation on the crystalline symmetry of extremal orbits: Application to 3D Weyl fermion}\la{sec:fieldorientation}
The previous examples demonstrate that deforming the crystal structure offers a way to tune ${\lambda}$. For 3D solids, we may continuously tune between integer and non-integer $\lambda/\pi$ without explicitly breaking any symmetry -- by modifying the orientation of the field with respect to the crystal structure, we effectively modify the symmetry of the extremal orbit. For specific orientations, the extremal orbit may be invariant under a point-group symmetry that stabilizes integer-valued $\lambda/\pi$. We shall illustrate this with a 3D topological metal whose type-I Weyl points lie on a high-symmetry plane that is invariant under $T\scr_{2z}$ (the composition of time reversal with a two-fold screw rotation), as exemplified by strained WTe$_2$.\cite{WTe2Weyl} When the field is aligned parallel to the screw axis, the maximal orbit of a Weyl point lies within the $T\scr_{2z}$-invariant plane [class-I,$s{=}1$], and is characterized robustly by $\lambda{=}\pi$. As illustrated in Fig.\ \ref{fig:WTe2-model}, $\lambda$ deviates from $\pi$ when the field is tilted away, owing to the absence of any symmetry for the tilted maximal orbit.
\begin{figure}[ht]
\centering
\includegraphics[width=1\columnwidth]{wte2anisotropic.png}
\caption{\label{fig:WTe2-model}
(a) Fermi surface of a Weyl fermion centered on a generic wavevector in a high-symmetry plane; red circle indicates an extremal orbit that has no symmetry. (b) For a model of the Weyl fermion, we plot $\lambda$ as a function of the field orientation, which we parametrize by spherical coordinates illustrated in (a).
}
\end{figure}
\subsection{Orbits self-constrained by time-reversal symmetry}\la{sec:orbitselfT}
Let us consider orbits ($\mathfrak{o}$) which are self-constrained as $T{\circ}\mathfrak{o}{=}\mathfrak{o}$ [II-A,$u{=}0,s{=}1$]; these are orbits that lie in $T$-invariant cross-sections of the Brillouin torus, and encircle Kramers-degenerate points. The contexts in which self-constrained orbits will be discussed include:
\noi{a} $T$-invariant solids with negligible SOC,
\noi{b} SO-coupled solids with both $T$ and spatial-inversion ($\mathfrak{i}$) symmetries, and
\noi{c} SO-coupled solids with $T$ but without $\mathfrak{i}$ symmetry.\\
\noindent Cases (a) and (b) correspond to spin-degenerate bands. In the above cases, we would show respectively that:
\noi{a} The lack of SOC allows us to constrain the purely-orbital component ($\tilde{\lambda}$) of $\lambda_{\pm}$, where $\pm$ distinguishes the two spin species; recall that $\tilde{\lambda}$ and $\lambda_{\pm}$ differ only by the Zeeman splitting, as shown in \q{noSOC}. The orbit-averaged orbital moment vanishes, and $\tilde{\lambda}{=}0$ reflects the trivial Berry phase of band extrema at $T$-invariant wavevectors. Combining $\tilde{\lambda}{=}0$ with \q{noSOC}, we obtain the following zero-sum rule: $\lambda_+{+}\lambda_-{=}0$ mod $2\pi$.
\noi{b} This zero-sum rule is also satisfied for spin-degenerate orbits in spin-orbit-coupled solids: $\lambda_1{+}\lambda_2{=}0$ mod $2\pi$.
\noi{c} Both the orbital moment and Zeeman effect average out, and $\lambda{=}\pi$ reflects the nontrivial Berry phase associated to the Kramers-degenerate Dirac point at $T$-invariant wavevectors. \\
\noindent \emph{Demonstration}
In all cases, the propagator ${\cal A}[\mathfrak{o}]$ satisfies
\e{\bar{T} {\cal A}^*\bar{T}^{\text{-}1}={\cal A}, \as \bar{T}\bar{T}^*=(-1)^{\sma{F}}{\cal A}^{\text{-}1}, \as \bar{T}^{\text{-}1} = \dg{\bar{T}}, \la{projectivealgebraAT}}
with $\bar{T}K$ an antiunitary representation of $T$ [cf.\ row 4 of \tab{tab:tenfold} with $N{=}L{=}2$]. The second equation may be contrasted with the usual antiunitary representation (denoted $\tilde{T}K$) of $T$, which satisfies $\tilde{T}\tilde{T}^*{=}(-1)^{\sma{F}}$; the additional factor of ${\cal A}^{\text{-}1}$ in \q{projectivealgebraAT} indicates that \q{projectivealgebraAT} represents an extension of the magnetic point group by contractible loop translations (represented by ${\cal A}$) in $\boldsymbol{k}$-space,\cite{AALG_100} which generalizes a previous work on noncontractible loop translations.\cite{Cohomological}
The first equation in \q{projectivealgebraAT} implies det$\,{\cal A}{=}e^{i{\sum}_{a=1}^{\sma{D}}\lambda_a}{=}{\pm}1$, with $D{=}2$ and $1$ for cases (b) and (c) respectively; in case (a), we should interpret det$\,{\cal A}{=}e^{i\tilde{\lambda}}{=}{\pm}1$. For cases (a) and (b), we may further restrict det$\,{\cal A}$ to ${+}1$ by the following argument: while preserving det$\,{\cal A}$, contract $\mathfrak{o}$ $T$-symmetrically to an infinitesimal loop that encircles the $T$-invariant point. In the classes of (a-b), the band dispersion is extremized at $T$-invariant points, hence
the band velocity $v(\boldsymbol{k})$ vanishes along the infinitesimal loop.\footnote{In principle, there may be a four-fold band touching (e.g., 3D Dirac point) at a $T$-invariant point that is accidental or imposed by crystalline symmetries. Note that such band touchings are not stable under a $T$-symmetric perturbation that breaks every other symmetry except, respectively, spin SU(2) symmetry in class (a), and $\mathfrak{i}$ symmetry in class (b). The value of det$\,{\cal A}$ cannot change due to this perturbation, since such value is quantized only by $T$ symmetry. The result of this argument is that we may always simplify the dispersion at the $T$-invariant point to a band extremum, and apply the proof in the main text. This perturbation argument is developed more carefully in \ocite{AALG_100}} det$\,{\cal A}$ is thus solely contributed by the non-geometric one forms which depend inversely on the velocity. Further applying the identity: det\,$\overline{\text{exp}}\int V(\boldsymbol{k})|d\boldsymbol{k}|{=}$exp$\int$Tr$V(\boldsymbol{k})|d\boldsymbol{k}|$ for any matrix $V(\boldsymbol{k})$, and the time-reversal constraints: Tr$\,\boldsymbol{M}(\boldsymbol{k}){=}{-}$Tr$\,\boldsymbol{M}({-}\boldsymbol{k})$ [cf.\ \q{gactsonMmultiband} with $g{=}T$] and Tr$\,\boldsymbol{\sigma}(\boldsymbol{k}){=}{-}$Tr$\,\boldsymbol{\sigma}({-}\boldsymbol{k})$, we derive the desired result. The above proof required path-ordering and matrix traces in case (b), where ${\cal A}, \boldsymbol{M}$ and $\boldsymbol{\sigma}$ are two-by-two matrices; such matrix operations are not needed in case (a).
In case (c), we contract $\mathfrak{o}$ $T$-symmetrically to an infinitesimal loop encircling the Kramers-degenerate Dirac point\footnote{By a $T$-symmetric perturbation, one may always simplify the dispersion to a Dirac point. This argument is analogous to that presented in the previous footnote.} -- since the velocity is finite in this limit, the non-geometric contribution to $\lambda$ vanishes; what remains is the $\pi$-Berry-phase contribution. This completes the demonstration.
One implication of (a-b) for spin-degenerate orbits that are self-constrained by $T$ symmetry: the net phase offset $\Theta$ [cf.\ \q{defineTheta}], of two time-reversal-related fundamental harmonics, can only be $0$ or $\pi$. The former occurs if the two values of $\{\lambda\}$ are closer to $0$ than to $\pi$, and vice versa.
\subsection{Global constraint on $\{\lambda\}$ for time-reversal-symmetric solids}\la{sec:globalsumrule}
Let us impose a global constraint on the set [denoted $\{\lambda\}$] of all $\lambda_a^i$ that are contributed by closed orbits of bulk states in a $T$-symmetric solid. To clarify, for a $d$-dimensional solid, a `bulk state' is spatially extended in $d$ directions. By combining our results for orbits that are self-constrained [cf.\ \s{sec:orbitselfT}] and mutually constrained [cf.\ \s{sec:orbitsmutualT}] by time-reversal ($T$) symmetry, we would show that $\{\lambda\}$ comprise only of inversion-invariant pairs, i.e., pairs that are symmetric about zero. In our counting of `pairs', each closed orbit with energy degeneracy $D$ contributes $D$ values of $\lambda$, independent of whether any of these $D$ values are mutually degenerate, or degenerate with $\lambda$ from a distinct orbit. A corollary of this result is the global sum rule: $\sum \lambda {=}0$ mod $2\pi$.
Let us first prove this claim for spin-degenerate orbits. All orbits in a $T$-symmetric solid are either self- or mutually constrained by $T$ symmetry. As proven in \s{sec:orbitselfT}, each self-constrained orbit contributes an inversion-invariant pair $\{\lambda,{-}\lambda\}$. Utilizing [class II-B,$u{=}0,s{=}1$] of \tab{tab:tenfold}, the net contribution of any $T$-related pair of spin-degenerate orbits is an inversion-invariant quartet $\{\lambda_1,\lambda_2,{-}\lambda_1,{-}\lambda_2\}$; this was exemplified by our case study of graphene in \s{sec:orbitsmutualT}.
We would next prove the global constraint for spin-orbit-coupled solids lacking spatial-inversion symmetry [case (c) in \s{sec:orbitselfT}]. By similar reasoning as in the previous paragraph, each $T$-related pair of spin-\emph{non}degenerate orbits contributes $\{\lambda,{-}\lambda\}$. A self-constrained orbit necessarily contributes $\lambda{=}\pi$, as proven in \s{sec:orbitselfT}. Furthermore, self-constrained orbits always come in pairs; this follows because each self-constrained orbit encircles a $T$-invariant wavevector, and there are always $2^d$ such wavevectors in a $d$-dimensional lattice. This completes the proof.
It is instructive to draw analogies between our result and the Nielsen-Ninomiya fermion-doubling theorem (as applied to $T$-invariant solids).\cite{NIELSEN1981} In full generality, the theorem states that there are always an equal number of left- and right-handed Weyl fermions on a lattice; a simple implication is that Weyl fermions come in pairs. If Weyl fermions exist at wavevectors which are \emph{not} $T$-invariant, they necessarily come in pairs due to $T$ symmetry mapping $\boldsymbol{k}$ to $-\boldsymbol{k}$. In the absence of point-group symmetry in the space group of the lattice, bands always come in pairs that touch at isolated $T$-invariant wavevectors -- such Kramers-degenerate points are also Weyl points. That there are even number of Weyl fermions on a lattice thus complements our previous observation that there are an even number of Kramers-degenerate points. If the Fermi surface encircles Kramers-degenerate Weyl points instead of intersecting them, we recover our previous claim that self-constrained orbits come in pairs.
To recapitulate, the global zero-sum rule for $\{\lambda\}$ is exemplified by all case studies of bulk orbits in this work; more generally it constrains the bulk oscillatory phenomena of all $T$-symmetric solids.
Let us extend our discussion to closed orbits contributed by surface states of a 3D $T$-symmetric solid; these orbits lie in a 2D surface Brillouin zone. By a `surface state', we mean a state localized spatially to a surface that is translation-invariant in two directions; this surface lies at the interface between vacuum and a bulk that is semi-infinite in the direction orthogonal to the surface. If the solid is spin-orbit-coupled, surface bands are generically nondegenerate, and we would use nearly the same argument (given above) for spin-nondegenerate, bulk orbits; the sole difference is that the fermion-doubling theorem does not apply, and it is possible to have an odd number of self-constrained orbits (associated to an odd number of surface Dirac fermions). One implication is that $\chi_s{:}{=}\sum \lambda$ (summed over surface orbits) may equal $0$ or $\pi$. If the bulk of the solid is insulating, $\chi_s$ may be viewed as a $\mathbb{Z}_2$ index that classifies insulators from the perspective of its surface magnetotransport; this is equivalent to the $\mathbb{Z}_2$ strong classification\cite{fu2007b,Inversion_Fu,moore2007,Rahul_3DTI} of 3D insulators in Wigner-Dyson class AII. If the solid has negligible spin-orbit coupling, then the surface orbits satisfy the same type of global constraint as for bulk orbits -- to demonstrate this, one may utilize the above argument for spin-degenerate bulk orbits.
\subsection{Spin-degenerate orbits in spin-orbit-coupled, centrosymmetric metals: Application to the 3D Dirac metal Na$_3$Bi}\la{sec:spindegSOC}
A zero-sum rule also applies individually to each $T\mathfrak{i}$-invariant orbit ($\mathfrak{o}$ in class [I,$s{=}1$]); this rule applies whether or not the orbit is self-constrained by $T$ symmetry. We have previously discussed such orbits in the context of graphene; here, we extend our discussion to $T\mathfrak{i}$-invariant, spin-degenerate orbits in SO-coupled solids. The reality of det$\,{\cal A}$ and contractibility of $\mathfrak{o}$ together imply $\lambda_1{+}\lambda_2{=}0$ mod $2\pi$.
Let us apply this result to a 3D massless Dirac fermion -- a four-band touching point between conically-dispersing bands which are spin-degenerate at generic wavevectors (owing to $T\mathfrak{i}$ symmetry). We shall assume that the 3D Dirac point is \emph{not} centered at a $T$-invariant point, hence an orbit at finite chemical potential (as measured from the Dirac point) is $T\mathfrak{i}$-invariant but not $T$-invariant, e.g., such Dirac points in the topological metal Na$_3$Bi are stabilized by three-fold rotational symmetry.\cite{Na3BiZhijun} For a field aligned along the rotational (trigonal) axis, the equidistant splitting of $\lambda_a$ is illustrated in \fig{fig:Na3Bi}(a) for constant-$k_z$ orbits on a surface of constant energy ($0.08$eV below the Dirac point); $\lambda_{1,2}{\approx} {\pm}\pi/4$ for the maximal orbit (occurring at wavevector $\bar{k}_z$, as indicated by a dashed line on the plot). In \fig{fig:Na3Bi}(b), we further plot $\lambda_a(\mu,\bar{k}_z)$ for a range of $\mu$ below the Dirac point. The ab-initio calculation of $\lambda_a$ is detailed in \app{app:abinitio}.
\begin{figure}[ht]
\centering
\includegraphics[width=8cm]{Na3Bi.png}
\caption{\label{fig:Na3Bi}
Characterization of the 3D Dirac metal Na$_3$Bi. (a) Left: surface of fixed energy ($\bar{E}{=}{-}0.08$ eV measured from the 3D Dirac point); contours of fixed energy and $k_z$ are illustrated by red circles. Right: plot of $\{\lambda_a(\bar{E},k_z)\}_{a=1,2}$ for a field parallel to the trigonal axis ($k_z$). $k_z$ is measured in units where the reciprocal period ($0.64{\text{\AA}}^{\text{-}1}$) is unity; the minimum (resp.\ maximum) $k_z$ on the plot corresponds to the south (resp.\ north) poles of the constant-energy surface. The extremal orbit occurs at $k_z=\bar{k}_z(\bar{E})$ indicated by a dashed line. (b) $\lambda_{1,2}(E,\bar{k}_z(E))$ vs $E$, for a range of $E$ below the Dirac point.}
\end{figure}
Let us discuss the experimental implications for Na$_3$Bi. A single Dirac point contributes to the oscillatory magnetization a term of the form \q{oscmag3D}, with $D{=}2$ and $\lambda_{1,2}(\mu,\bar{k}_z)$ plotted in \fig{fig:Na3Bi}(b). Na$_3$Bi actually has two $T$-related Dirac points, which both lie on the rotation-invariant line through $\Gamma$.\cite{Na3BiZhijun} Assuming the Fermi surfaces of the two Dirac points are disconnected, the effect of the second Dirac point is simply to double the magnitude of the oscillations. This follows because the maximal orbits ($\mathfrak{o}_1,\mathfrak{o}_2$) on disconnected Fermi surfaces are mutually constrained by $T$ symmetry, hence from class [II-B,$u{=}0,s{=}1$] of \tab{tab:tenfold}, $\{\lambda_1^1,\lambda_2^1\}{=}\{-\lambda_1^2,-\lambda_2^2\}$ mod $2\pi$. We remind the reader that $\lambda_1^i{=}{-}\lambda_2^i$ owing to $T\mathfrak{i}$ symmetry, hence in combination $\{\lambda_1^1,\lambda_2^1\}{=}\{\lambda_1^2,\lambda_2^2\}$.
To reiterate, the two Dirac points contribute identically to the oscillatory magnetization, which we plot in \fig{fig:Na3Bi-harmonics} for a temperature of 0.37 K and three representative Fermi energies. These plots are obtained from \q{oscmag3D}, which receive as inputs: (i) the ab-initio calculated band parameters $\{S, S_{zz},m_c,\lambda_{1,2}\}$ for the maximal orbit, as well as the Dingle lifetime $\tau{=}10^{-12}$s. For the range of $B$ plotted, $kT/\hbar\omega_c{\sim} 10^{-2}$ and $\omega_c\tau{\sim} 1$, which implies that the Dingle factor (rather than temperature smearing) is the limiting factor in observing higher harmonics; for the parameters considered, the third harmonic does not appreciably modify the plots. The second harmonic is especially transparent at $\mu_c{:}{=}{-}69$meV [\fig{fig:Na3Bi-harmonics}(b)], where $\lambda_1{=}\pi/2{=}{-}\lambda_2$ [cf.\ \fig{fig:Na3Bi}(b)] leads to the complete destructive interference of all time-reversed pairs of odd harmonics (including the fundamental harmonic) [cf.\ \q{oscmag3DD2}], i.e., the dHvA period is effectively halved. The point of destructive interference intermediates two energy intervals where the phase offset $\Theta$ [cf.\ \q{defineTheta}] in the fundamental harmonic differs by $\pi$: $\Theta{=}\pi$ for $\mu{>}\mu_c$ and closer to the Dirac point [cf.\ green shaded region in \fig{fig:Na3Bi}(b)]; $\Theta{=}0$ for $\mu{\leq} \mu_c$ [blue shaded region]. Our case study demonstrates that the experimental tunability\cite{satya_na3bi,Junxiong_Na3Bi,xiong2015signature} of $\mu$ in Na$_3$Bi should expose a wide range of $T$-symmetric interference patterns.
\begin{figure}[ht]
\centering
\includegraphics[width=8.5cm]{Na3Bi-harmonics.png}
\caption{\label{fig:Na3Bi-harmonics}
Magnetization oscillations calculated for Na$_3$Bi using \q{oscmag3D} at T=0.37K, $\tau{=}10^{-12}$s and Fermi energies (a) -72meV, (b) -69meV, and (c) -66meV. The corresponding $\lambda_{1,2}$ are indicated by dashed lines in \fig{fig:Na3Bi}(b). The sum of the two lowest harmonics [$r{=}1,2$ in \q{oscmag3D}], as illustrated by orange solid lines, faithfully represents the oscillations; however, accounting only for the fundamental harmonic ($r{=}1$, dashed blue line) is inadequate. The unit of magnetization is chosen as $10^{-6}\mu_{\sma{B}}/\text{\AA}^{\sma{3}}$, with $\mu_{\sma{B}}$ the Bohr magneton.}
\end{figure}
For the parameter ranges in \fig{fig:Na3Bi} and \fig{fig:Na3Bi-harmonics}, the Fermi surfaces enclosing each Dirac point are indeed disconnected. As $\mu$ is increased beyond ${\sim} 10$ meV, the two Fermi surfaces intersect at a saddlepoint and merge into a dumbbell Fermi surface. For a field parallel to the trigonal axis, this merged Fermi surface has a single minimal orbit $\mathfrak{o}_3$ (encircling the saddlepoint), in addition to the aforementioned two maximal orbits. At slightly higher energy (${\sim} 20$ meV), the dumbbell transforms into an ellipsoid, and correspondingly the three extremal orbits merge into a single maximal orbit (denoted $\mathfrak{o}_4$). $\mathfrak{o}_3$ (and also $\mathfrak{o}_4$) is $T$-invariant [under class II-A,$u{=}0,s{=}1$] and $T\mathfrak{i}$-invariant [class I,$s{=}1$]; this exemplifies how a single orbit having multiple symmetries may fall under different classes in \tab{tab:tenfold}. Both classes impose (consistently) a zero-sum-rule constraint on $\{\lambda\}$.
\subsubsection{Comment on magnetotransport experiments of 3D Dirac metals}
We conclude this section by commenting on the interpretation of several magnetotransport experiments on 3D Dirac metals.\cite{he_transportCd3As2,xiang_sdHcd3as2,pariari_dhva_cd3as2,kumar_bi2se3,narayanan_cd3as2,zhaoyanfei_transportcd3as2} The experimental data in these works have only been fitted to the fundamental harmonic ($r{=}1$) in the traditional Lifshitz-Kosevich formula.\cite{shoenberg} The phase offset in this fitted fundamental harmonic is interpreted as $\phi_{\sma{B}}{-}\phi_{\sma{M}}{\pm} \pi/4$; $\phi_{\sma{B}}{=}\pi$ that is inferred from such a fitting is viewed as a smoking gun for a 3D Dirac metal. These interpretations have been justified\cite{he_transportCd3As2,xiang_sdHcd3as2,pariari_dhva_cd3as2,kumar_bi2se3} by appealing to the \emph{one-band} Onsager-Lifshitz quantization rule [\q{rule3b} with $D{=}1$].
We propose an alternative interpretation that nevertheless makes sense of the robustness of their measurements, across different experimental groups and slightly-varying samples -- this robustness is solely a consequence of $T$ symmetry, rather than anything intrinsic to a 3D-Dirac Fermi surface. In other words, $\Theta{=}\pi$ should not be viewed as a smoking gun for a 3D Dirac metal.
From our perspective, spin-degenerate orbits in spin-orbit-coupled metals are semiclassically described by the multi-band quantization rule [cf.\ \q{rule3b}] and a generalized Lifshitz-Kosevich formula [cf.\ \qq{oscmag3D}{oscillatorydIdV}]. The phase corrections in these expressions encode the \emph{non-abelian} Berry, orbital-moment and Zeeman one-forms [cf.\ \q{definenonabelianunitary}]. Owing to the generic non-commutivity of the three one-forms in the path-ordered integral [cf.\ \q{definenonabelianunitary}], $\lambda_{\pm}$ cannot be decomposed into a sum of Berry, orbital-moment and Zeeman phases.\footnote{ If $D{=}1$, the three one-forms commute, and therefore $\lambda$ may by decomposed. We remark that the $\pi$ Berry phase of a Weyl point originates from an integral of the \emph{abelian} Berry connection in a single-band subspace.}
As described in \qq{oscmag3D}{defineTheta}, the phase offset ($\Theta$) of the summed fundamental harmonic, after substracting the Maslov and Lifshitz-Kosevich corrections, is a suitably-defined \emph{average} of $\lambda_{1,2}$. As proven in \s{sec:orbitselfT}, $\Theta$ is restricted to $0$ or $\pi$ for orbits that are self-constrained by $T$ symmetry. We believe that previous measurements\cite{he_transportCd3As2,xiang_sdHcd3as2,pariari_dhva_cd3as2,kumar_bi2se3,narayanan_cd3as2,zhaoyanfei_transportcd3as2} of $\Theta$ have been misinterpreted as a measurement of the Berry phase. The convergence in measurements of several groups indicate the quantization of $\Theta$ due to $T$ symmetry. However, $\Theta{=}\pi$ is not an intrinsic property of 3D Dirac metals, as we have exemplified by our case study of Na$_3$Bi. Moreover, we point out that $\Theta{=}\pi$ has been measured in conventional (i.e., non-topological) metals, even in the first metal (3D, single-crystal Bismuth) for which quantum oscillations were reported.\cite{SdH,dHvA} As far back as 1954, it was remarked by Dhillon and Shoenberg that: `For Bismuth good agreement was found by the theoretical formula except that the signs of the fundamental and odd harmonics had to be reversed.'\cite{dhva_Dhillon} Here, the `theoretical formula' refers to the traditional Lifshitz-Kosevich formula\cite{lifshitz_kosevich} without the $\Theta$ correction to the phase. In a later work, the quantization of $\Theta$ is implicit in the form of Shoenberg's spin `reduction factor',\cite{Dingle_I,shoenberg} which involves a \emph{phenomenological} g-factor in the presence of spin-orbit coupling.
One contribution of this work is to present an \emph{analytic} formula for the effective g-factor through the spectrum of \q{definenonabelianunitary}.\footnote{For small orbits encircling a high-symmetry wavevector, this effective g-factor may alternatively be calculated in the effective-mass approximation.\cite{gfactorBismuth_cohen_blount} For larger orbits, the effective-mass approximation is inappropriate, but \q{definenonabelianunitary} remains valid. Our method is essentially identical to previous works by Roth\cite{rothII}, Mikitik and Sharlai;\cite{Mikitik_quantizationrule} the only difference is that a basis choice (gauge) has been chosen in previous works such that their one-forms are traceless.\cite{AALG_100} In comparison, no gauge-fixing is assumed in \q{definenonabelianunitary}, and therefore \q{definenonabelianunitary} is the most natural object to calculate numerically.} Our proposed formula is calculable from \emph{ab initio} wavefunctions [as elaborated in \app{app:abinitio}], which allows for a quantitative comparison between theory and experiments. Another contribution of this work is the recognition that the quantization of $\Theta$ originates from $T$ symmetry; hence in magnetic materials, Shoenberg's spin `reduction factor' is not generally applicable, but our generalized Lifshitz-Kosevich formulae in \s{sec:extractlambda} remain valid.
We further emphasize that $\Theta$ is an incomplete characterization of the Fermi-surface orbit; a more complete characterization involves measuring the individual values of $\lambda_{1,2}$, which carry the complete non-abelian information about the Fermi-surface wavefunction. Two methods are available to measure $\lambda_{1,2}$: (i) where either of $\omega_c\tau {\ll} 1$ or $kT {\gg} \hbar \omega_c$ applies, one needs to measure both the amplitude and the phase offset of the fundamental harmonic; $\lambda_{1,2}$ is then extracted from the $r{=}1$ terms in \q{oscmag3D} and \qq{oscmag3DD2}{defineTheta}.\footnote{As mentioned in \s{sec:extractlambda}, this procedure requires an independent determination of $|S_{zz}|$.} (ii) Where neither of the above inequalities apply, the experimental data should be fitted to the full harmonic expansion of \q{oscmag3D}, and $\lambda_{1,2}$ extracted accordingly. Note that the zero-sum rule for $\lambda_{1,2}$ implies that there is only one parameter to be fitted.
Going beyond general symmetry constraints, one may ask if $\Theta$ and $\lambda_{1,2}$ may be analytically calculated in a case-by-case basis. Given material-specific information about the magnetic point group of the 3D Dirac point, as well as the relevant symmetry representations that span the low-energy Hilbert space at the Dirac point, one may indeed determine $\Theta$ from a $\boldsymbol{k} \cdot \boldsymbol{p}$ analysis, as exemplified in the next section.
\subsection{Calculating $\lambda_{1,2}$ from $\boldsymbol{k} \cdot \boldsymbol{p}$ analysis: Application to 3D massive Dirac fermions in $\mathbb{Z}_2$ topological insulator Bi$_2$Se$_3$} \la{sec:bi2se3}
In certain symmetry classes, and for sufficiently small (but nonzero) chemical potentials ($\mu$) relative to the Dirac point, 3D massless Dirac fermions may exhibit $\lambda_1{\approx}\lambda_2{\approx}\pi$; this would imply from \q{defineTheta} that $\Theta{=}\pi$. Precisely, we restrict $|\mu|$ to be small enough that a linearized $\boldsymbol{k}\cdot \boldsymbol{p}$ Hamiltonian is a good approximation -- how small $|\mu|$ needs to be to fulfil the above criterion depends on material-specific band parameters. As far as the linear approximation is valid, $\lambda_{1,2}$ that follows from the subsequent $\boldsymbol{k}\cdot \boldsymbol{p}$ analysis is approximately $\pi$, hence our use of $\approx$ throughout this section. $\lambda_{1,2}$ should manifest in the dHvA oscillations for sufficiently weak field, such that the semiclassical approximation remains valid, i.e., $l^2S(\mu) {\gg}1$.\footnote{Note this implies a double constraint on $|\mu|$.}
Let us demonstrate how $\lambda_{1,2}{\approx}\pi$ arises for 3D massless Dirac fermions centered at a wavevector ($\Gamma$) with a magnetic point group ($P$) that combines $T$ symmetry with the $D_{3d}$ point group.\cite{tinkhambook} $\lambda_{1,2}{\approx}\pi$ relies not just on $P$, but also on symmetries that are \emph{absent} in $P$; these additional symmetries emerge only at long wavelength ($\boldsymbol{k} {\rightarrow} \boldsymbol{0}$), where the averaged-out crystal field has \emph{higher} symmetry than the magnetic point group.
Our study of $D_{3d}$ is motivated by the identical symmetry class of Bi$_2$Se$_3$, a well-known $\mathbb{Z}_2$ topological insulator.\cite{fu2007b,Inversion_Fu,moore2007,Rahul_3DTI} For the symmetry representations of Bi$_2$Se$_3$, the $\boldsymbol{k} \cdot \boldsymbol{p}$ Hamiltonian around $\Gamma$ (of the 3D Brillouin torus) assumes the form:\cite{zhang2009,chaoxingliu_modelhamiltonian}
\begin{align}
H_0 {=} \hbar\big(v(k_x\sigma_{\sma{1}}{+}k_y \sigma_{\sma{2}}){+}wk_z\sigma_{\sma{3}}\big) {\otimes} \tau_{\sma{1}} {+}M(\boldsymbol{k}) I_{\sma{2\times 2}}{\otimes} \tau_{\sma{3}}.\la{kdotpD3d}
\end{align}
to quadratic accuracy in $\boldsymbol{k}$. Here, $\sigma_j$ and $\tau_j$ are Pauli matrices spanning spin and orbital (labelled $P1_z^+$ and $P2_z^-$)\cite{zhang2009} subspaces respectively, and $M(\boldsymbol{k}){=}\Delta{+}b_1k_z^2{+}b_2(k_x^2{+}k_y^2)$ with $\Delta$, the Dirac mass, vanishing at the critical point between trivial and topological insulator.
Assuming $\Delta{=}0$ for now, $[H_0(\boldsymbol{k}),I_{\sma{2\times 2}}{\otimes}\tau_{\sma{1}}]{=}O(\boldsymbol{k}^2)$ is an emergent conservation law of the linearized $H_0$. Block-diagonalizing $H_0$ with respect to $I_{\sma{2\times 2}}{\otimes}\tau_{\sma{1}}{=}{\pm}1$, we derive two decoupled 3D Weyl Hamiltonians $H_{\pm}$ with opposite chirality, each satisfying the time-reversal constraint $TH_{\pm}(\boldsymbol{k})T^{\text{-}1}{=}H_{\pm}({-}\boldsymbol{k})$ with $T{=}i\sigma_{\sma{2}} K$ squaring to ${-}I$ in a half-integer-spin representation ($F{=}1$). Independent of the field orientation, the spin-degenerate extremal orbit effectively decouples to two nondegenerate $(D{=}1)$ extremal orbits which are individually $T$-invariant -- this implies from a previous demonstration [case (c) of \s{sec:orbitselfT}] that $\lambda_{1,2}{\approx}\pi$, which may be viewed as the Berry phase ($\phi_{\sma{B}}$) for each decoupled orbit. This would effectively imply a single set of dHvA harmonics (indexed by $r$ in \q{oscmag3D}) with twice the usual amplitude, and a Lifshitz-Kosevich phase correction of ${-}\pi/4$ for a maximal orbit.
Away from the critical point, we prove in \app{sec:cancellation} that the degeneracy is Zeeman-split as $\lambda_{1,2}{\approx}\pi{\pm} \pi \Delta/mv^2$ with $m$ the free-electron mass,
for a field parallel to the trigonal axis ($k_z$).
It is worth remarking that $\phi_{\sma{B}}$ reduces from $\pi$, and the phase ($\phi_{\sma{R}}$) associated to the orbital moment increases from $0$, however their sum ($\phi_{\sma{B}}{+}\phi_{\sma{R}}$) remains fixed to $\pi$.\footnote{ This robustness is an artifact of a restricted class of $\boldsymbol{k}\cdot \boldsymbol{p}$ models described in \app{sec:cancellation}. The robustness is not generally valid at larger chemical potentials where the effect of other bands become significant. A similar observation was first found in \ocite{fuchs_topologicalberryphase} in the context of graphene with a Semenoff mass.}
In fact, naturally-occurring Bi$_2$Se$_3$ does not lie at the critical point, i.e., the corresponding 3D Dirac fermion is massive: we estimate $\Delta/mv^2{\approx}0.13$ utilizing fitted parameters from an ab-initio calculation in Ref.\ \onlinecite{zhang2009}; utilizing \q{defineTheta} with $|\Delta/mv^2|{<}0.5$, we further derive $\Theta{=}\pi$. This splitting of $\lambda$ manifests as two sets of harmonics in the dHvA oscillation, which should be measurable utilizing techniques outlined in \s{sec:spindegSOC}. The tunability of $\Delta$ by doping\cite{Bi2se3_dope} or strain\cite{Bi2se3_strain,Young_Bi2se3_strain,Bi2se3_strain_expt} suggests that the dHvA oscillations are correspondingly tunable -- in particular, $|\Delta/mv^2|{=}0.5$ is the point of complete destructive interference for the fundamental harmonic; this point separates two regimes where $\Theta{=}0$ vs $\pi$.
There exist galvano-magnetic evidence that supports the quantization of $\Theta$. The fundamental Shubnikov-de Haas harmonic\cite{SdH,adams_holstein,magneticquantumeffects_Roth_argyres} in the transverse resistivity of Bi$_2$Se$_3$ has been fitted\cite{kazuma_bi2se3oscillations} as
\e{\Delta \rho_{xx} {\propto} \cos (l^2S{+}\Theta{+}\phi_{\sma{M}}{-}\pi/4) {\approx} \cos[l^2S{+}2\pi(0.4)],\la{fit}}
with the higher harmonics suppressed by temperature. The above proportionality may be derived from: (a) $\rho_{xx}{\approx}\sigma_{xx}/\sigma_{xy}^2$, which is valid for a large Hall angle $\sigma_{xy}{\gg}\sigma_{xx}$, and (b) the proportionality between the oscillatory components of the transverse conductivity and the longitudinal magnetic susceptibility: $\Delta \sigma_{xx}\propto m_c \Delta \chi$;\cite{quantumcollision_kosevich_andreev,physicalkinetics_lifshitzpitaev} the cyclotron mass ($m_c$) is positive for the lone electron-like orbit in Bi$_2$Se$_3$,\cite{kazuma_bi2se3oscillations} and the oscillatory susceptibility ($\Delta \chi$) is obtained from differentiating \q{oscmag3D} (with $D{=}2$) with respect to $B$ and keeping only the fastest oscillatory terms. The proportionality stated in (b) is valid for metals having only a single extremal orbit, and for sufficiently weak fields such that $\mu {\gg}\hbar \omega_c$.\footnote{In addition, there are auxiliary conditions such as $\hbar \omega_c{\ll}(kT \mu)^{\sma{1/2}}$ if $kT$ and $\hbar/\tau {\ll}\hbar \omega_c$.\cite{physicalkinetics_lifshitzpitaev} It has been assumed in \ocite{physicalkinetics_lifshitzpitaev} that impurity-induced scattering between spin-split Landau levels is absent; we are not certain that this assumption is generally valid in spin-orbit-coupled metals.} The fit in \q{fit} implies $\Theta{=}0.4{-}1/2{+}1/8{\approx}0$ to the accuracy of their fit. It was further remarked in \ocite{kazuma_bi2se3oscillations}, without explanation, that the measured $\Theta{\approx}0$ was independent of the field orientation. To elaborate, four measurements, carried out at angles $0$, $\pi/4$, $\pi/3$ and $\pi/2$ relative to the trigonal axis in Bi$_2$Se$_3$, produced the same value for $\Theta$. Our explanation for this robustness: independent of the field orientation, the extremal orbit is invariant under time-reversal symmetry, which quantizes $\Theta$ to $0$ or $\pi$. In contrast, crystalline symmetries of an extremal orbit depend sensitively on the field orientation, as explained in \s{sec:fieldorientation}.
\section{Discussion}\la{sec:discussion}
In fermiology, the shape of the Fermi surface is deducible from the period of dHvA oscillations; here, we propose that the topology of the Fermi-surface wavefunctions are deducible from the phase offset ($\lambda_a$) of dHvA oscillations in 3D solids, as well as in fixed-bias oscillations of the differential conductance in scanning-tunneling microscopy. $\lambda_a$ manifests as the \emph{complete}, subleading [$O(1)$] corrections to the Bohr-Sommerfeld quantization rules for closed orbits without breakdown [cf.\ \q{rule3b}]; we have formulated $\lambda_a$ as eigen-phases of a propagator ${\cal A}$ defined in \q{definenonabelianunitary}.
In certain solids and for certain field orientations with respect to a crystal axis, ${\sum}_{\sma{a{=}1}}^{\sma{D}}\lambda_{\sma{a}}$ (with $D$ the degeneracy of the low-energy band subspace) are topologically invariant under deformations of the zero-field Hamiltonian that respect the symmetry of the orbit, as well as preserves the global shape of the orbit. Precisely, globally-equivalent orbits correspond to a graph with a homotopy equivalence defined in Ref.\ \onlinecite{AALG_100}. To identify orbits with robustly integer-valued ${\sum}_{\sma{a{=}1}}^{\sma{D}}\lambda_{\sma{a}}/\pi$, as well as identify the degeneracy of Landau levels, we classify symmetric orbits into ten (and only ten) classes summarized in the first three columns of \tab{tab:tenfold}; the rest of the table describes the corresponding constraints on the propagator ${\cal A}$.
The results of this table remain valid if we substitute ${\cal A} {\rightarrow}{\cal W}$ and $\lambda_a{\rightarrow}\phi_{\sma{B},a}$; here, ${\cal W}$ is defined as the unitary generated by the Berry connection, and may be viewed as the purely-geometric component of ${\cal A}$; $\phi_{\sma{B},a}$ are defined as the eigen-phases of ${\cal W}$, and may be viewed as the non-abelian generalization\cite{wilczek1984} of the Berry phase. ${\cal W}$ provides a purely-geometric characterization of bands that is intimately related to the topology of wavefunctions over the Brillouin torus.\cite{zak1989,Cohomological,z2pack} While ${\cal W}$ (resp.\ $\phi_{\sma{B},a}$) generically differs from ${\cal A}$ (resp.\ $\lambda_a$) due to the orbital moment and the Zeeman effect, ${\cal W}$ and ${\cal A}$ transform identically under symmetry, and therefore satisfy the same constraints. In particular, if we define $\{\phi_{\sma{B}}\}$ as the set of $\phi_{\sma{B}}$ contributed by all bulk orbits in a $T$-symmetric solid, then $\{\phi_{\sma{B}}\}$ comprises only pairs of $\phi_{\sma{B}}$ which individually satisfy a zero-sum rule, in close analogy with the global constraint on $\{\lambda\}$ that is described in \s{sec:globalsumrule}.
Higher-order (in $|\boldsymbol{B}|$) corrections to the Bohr-Sommerfeld quantization rule become relevant in higher-field experiments that intermediate the semiclassical and quantum limits;\cite{rothII,fischbeck_review} these corrections may be interpreted as zero-field magnetic response functions.\cite{Gao_Niu_zerofieldmagneticresponse} These corrections are accounted for in the generalized Lifshitz-Kosevich formulae [cf.\ \qq{oscmag3D}{oscillatorydIdV}] by the simple replacement $l^2S(E) {+} \lambda_a(E) {\rightarrow}l^2S(E) {+}\lambda_a(E,|\boldsymbol{B}|),$ with $\lambda_a$ expandable asymptotically in powers of $|\boldsymbol{B}|$; $\lambda_a(E,|\boldsymbol{B}|{=}0){=}\lambda_a(E)$ is obtained from diagonalizing the propagator ${\cal A}$ [cf.\ \q{definenonabelianunitary}]. It would be interesting to extend our symmetry analysis to the field-dependent component of the phase offset.
\begin{acknowledgments}
The authors thank Zhijun Wang and Ilya K. Drozdov for their expert opinions on 3D Dirac metals and tunneling spectroscopy. We also thank Yang Gao and Qian Niu for communicating their work on higher-order corrections in the quantization rule. We acknowledge support by the Yale Postdoctoral Prize Fellowship (AA), NSF DMR Grant No.\ 1603243 (LG), the Ministry of Science and Technology of China, Grant No.\ 2016YFA0301001 (CW and WD), and the National Natural Science Foundation of China, Grants No.\ 11674188 and 11334006 (CW and WD).\\
\end{acknowledgments}
| 2024-02-18T23:39:52.127Z | 2018-01-10T02:04:37.000Z | algebraic_stack_train_0000 | 704 | 13,316 |
|
proofpile-arXiv_065-3551 | \section{Introduction}
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\caption{Example commands for accented characters, to be used in, e.g., \BibTeX\ names.}\label{tab:accents}
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\section{Introduction}
Producing true causal explanations requires deep understanding of the domain.
This is beyond the capabilities of modern AI.
However, it is possible to collect large amounts of causally related events, and, given powerful enough representational variability,
to construct cause-effect chains by selecting individual pairs appropriately and linking them together.
Our hypothesis is that chains composed of locally coherent pairs can suggest overall causation.
In this paper, we view \textit{causality} as (commonsense) cause-effect expressions that occur frequently in online text such as news articles or tweets. For example, ``\textit{greenhouse gases causes global warming}" is a sentence that provides an `atomic' link that can be used in a larger chain.
By connecting such causal facts in a sequence, the result can be regarded as a \textit{causal explanation} between the two ends of the sequence
(see Table~\ref{tab:exchains} for examples).
\noindent This paper makes the following contributions:
\begin{itemize}[leftmargin=*,noitemsep,topsep=0pt]
\item we define the problem of causal explanation generation,
\item we detect causal features of a time series event (\textsc{CSpikes}) using Granger~\cite{granger1988some} method with features extracted from text such as N-grams, topics, sentiments, and their composition,
\item we produce a large graph called \textsc{CGraph}~of local cause-effect units derived from text and develop a method to produce causal explanations by selecting and linking appropriate units, using neural representations to enable unit matching and chaining.
\end{itemize}
\begin{figure}[t]
\small
\centerin
{
{\includegraphics[width=.89\linewidth]{figs/{words_FB_False_3_2013-01-01_2013-12-31}.png}}
\caption{\label{fig:example}
Example of causal features for Facebook's stock change in 2013.
The causal features (e.g., \textit{martino}, \textit{k-rod}) rise before the Facebook's rapid stock rise in August.
\end{figure}
The problem of causal explanation generation arises for systems that seek to determine causal factors for events of interest automatically.
For given time series events such as companies' stock market prices, our system called \textsc{CSpikes}~detects events that are deemed causally related by time series analysis using Granger Causality regression~\cite{granger1988some}.
We consider a large amount of text and tweets related to each company, and produces for each company time series of values for hundreds of thousands of word n-grams, topic labels, sentiment values, etc.
Figure~\ref{fig:example} shows an example of causal features that temporally causes Facebook's stock rise in August.
\begin{table}[t]
\centering
\small
\caption{\label{tab:exchains}Examples of generated causal explanation between some temporal causes and target companies' stock prices.
}
\begin{tabularx}{\columnwidth}{@{}X@{}}
\toprule
\textbf{\color{Sepia}party} $\xmapsto[]{cut}$ budget\_cuts $\xmapsto[]{lower}$ budget\_bill $\xmapsto[]{decreas}$ republicans $\xmapsto[]{caus}$ obama $\xmapsto[]{lead to}$ facebook\_polls $\xmapsto[]{caus}$ \textbf{\color{BlueViolet}facebook's stock} $\downarrow$\\\hline
\bottomrule
\end{tabularx}
\end{table}
However, it is difficult to understand how the statistically verified factors actually cause the changes, and whether there is a latent causal structure relating the two.
This paper addresses the challenge of finding such latent causal structures, in the form of \textit{causal explanations} that connect the given cause-effect pair.
Table~\ref{tab:exchains} shows example causal explanation that our system found between \textit{party} and \textit{Facebook's stock fall ($\downarrow$)}.
To construct a general causal graph, we extract all potential causal expressions from a large corpus of text. We refer to this graph as \textsc{CGraph}. We use FrameNet~\cite{baker1998berkeley} semantics to provide various causative expressions (verbs, relations, and patterns),
which we apply to a resource of $183,253,995$ sentences of text and tweets.
These expressions are considerably richer than previous rule-based patterns~\cite{riaz2013toward,kozareva2012cause}.
\textsc{CGraph}~ contains 5,025,636 causal edges.
Our experiment demonstrates that our causality detection algorithm outperforms other baseline methods for forecasting future time series values. Also, we tested the neural reasoner on the inference generation task using the BLEU score.
Additionally, our human evaluation shows the relative effectiveness of neural reasoners in generating appropriate lexicons in explanations.
\section{\textsc{CSpikes}: Temporal Causality Detection from Textual Features}
\label{sec:method}
The objective of our model is, given a target time series $y$, to find the best set of textual features $F = \{f_1, ..., f_k\} \subseteq X$, that maximizes sum of causality over the features on $y$, where $X$ is the set of all features. Note that each feature is itself a time series:
\begin{equation}\label{objective}
\argmax_{F} \mathbf{C}{ (y, \Phi(X, y)) }
\end{equation}
where $\mathbf{C}(y,x)$ is a causality value function between $y$ and $x$, and $\Phi$ is a linear composition function of features $f$.
$\Phi$ needs target time series $y$ as well because of our graph based feature selection algorithm described in the next sections.
We first introduce the basic principles of Granger causality in Section~\ref{subsec:granger}. Section~\ref{subsec:feature} describes how to extract good source features $F = \{f_1, ..., f_k\}$ from text. Section~\ref{subsec:causality} describes the causality function $\mathbf{C}$ and the feature composition function $\Phi$.
\subsection{Granger Causality}\label{subsec:granger}
The essential assumption behind Granger causality is that a cause must occur before its effect, and can be used to predict the effect.
Granger showed that given a target time series $y$ (effect) and a source time series $x$ (cause), \textit{forecasting} future target value $y_t$ with both past target and past source time series $E(y_t | y_{<t}, x_{<t})$ is significantly powerful than with only past target time series $E(y_t | y_{<t})$ (plain auto-regression), if $x$ and $y$ are indeed a cause-effect pair.
First, we learn the parameters $\alpha$ and $\beta$ to maximize the prediction expectation:
\begin{align}\label{granger}
&E(y_t | y_{<t}, x_{t-l}) = \sum_{j=1}^{m} \alpha_j y_{t-j} + \sum_{i=1}^{n} \beta_i x_{t-i}
\end{align}
where $i$ and $j$ are size of lags in the past observation. Given a pair of causes $x$ and a target $y$, if $\beta$ has magnitude significantly higher than zero (according to a confidence threshold), we can say that $x$ causes $y$.
\subsection{Feature Extraction from Text}\label{subsec:feature}
Extracting meaningful features is a key component to detect causality.
For example, to predict future trend of presidential election poll of \textit{Donald Trump}, we need to consider his past poll data as well as people's reaction about his pledges such as \textit{Immigration}, \textit{Syria} etc.
To extract such ``good'' features crawled from on-line media data, we propose three different types of features: $F_{words}$, $F_{topic}$, and $F_{senti}$.
$F_{words}$ is time series of N-gram words that reflect popularity of the word over time in on-line media.
For each word, the number of items (e.g., tweets, blogs and news) that contains the N-gram word is counted to get the day-by-day time series.
For example, $x^{\small Michael\_Jordan} = [ 12,51,..]$ is a time series for a bi-gram word \textit{Michael Jordan}.
We filter out stationary words by using simple measures to estimate how dynamically the time series of each word changes over time. Some of the simple measures include Shannon entropy, mean, standard deviation, maximum slope, and number of rise and fall peaks.
$F_{topic}$ is time series of latent topics with respect to the target time series.
The latent topic is a group of semantically similar words as identified by a standard topic clustering method such as LDA~\cite{blei2003latent}.
To obtain temporal trend of the latent topics, we choose the top ten frequent words in each topic and count their occurrence in the text to get the day-by-day time series.
For example, $x^{healthcare}$ means how popular the topic \textit{healthcare} that consists of \textit{insurance}, \textit{obamacare} etc, is through time.
$F_{senti}$ is time series of sentiments (positive or negative) for each topic.
The top ten frequent words in each topic are used as the keywords, and tweets, blogs and news that contain at least one of these keywords are chosen to calculate the sentiment score.
The day-by-day sentiment series are then obtained by counting positive and negative words using OpinionFinder~\cite{wilson2005recognizing}, and normalized by the total number of the items that day.
\subsection{Temporal Causality Detection}\label{subsec:causality}
We define a causality function $\mathbf{C}$ for calculating causality score between target time series $y$ and source time series $x$.
The causality function $\mathbf{C}$ uses Granger causality~\cite{granger1988some} by fitting the two time series with a Vector AutoRegressive model with exogenous variables (VARX)~\cite{hamilton1994time}: $y_t = \alpha y_{t-l} + \beta x_{t-l} + \epsilon_t$
where $\epsilon_t$ is a white Gaussian random vector at time $t$ and $l$ is a lag term.
In our problem, the number of source time series $x$ is not single so the prediction happens in the $k$ multi-variate features $X=(f_1, ... f_k)$ so:
\begin{align}
y_t & = \alpha y_{t-l} + \bm{\beta} (f_{1,t-l} + ... + f_{k,t-l}) + \epsilon_t
\end{align}
where $\bm{\alpha}$ and $\bm{\beta}$ is the coefficient matrix of the target $y$ and source $X$ time series respectively, and $\epsilon$ is a residual (prediction error) for each time series.
$\bm{\beta}$ means contributions of each lagged feature $f_{k,t-l}$ to the predicted value $y_t$.
If the variance of $\bm{\beta_k}$ is reduced by the inclusion of the feature terms $f_{k,t-l} \in X $, then it is said that $f_{k,t-l}$ Granger-causes $y$.
Our causality function $\mathbf{C}$ is then $\mathbf{C}(y, f, l) =\Delta(\beta_{y,f,l})$ where $\Delta$ is change of variance by the feature $f$ with lag $l$.
The total Granger causality of target $y$ is computed by summing the change of variance over all lags and all features:
\begin{align}\label{eq:causality}
\mathbf{C}(y, X) =\sum_{k,l} \mathbf{C}(y, f_k, l)
\end{align}
We compose best set of features $\Phi$ by choosing top $k$ features with highest causality scores for each target $y$.
In practice, due to large amount of computation for pairwise Granger calculation, we make a bipartite graph between features and targets, and address two practical problems: \textit{noisiness} and \textit{hidden edges}.
We filter out noisy edges based on TFIDF and fill out missing values using non-negative matrix factorization (NMF)~\cite{hoyer2004non}.
\begin{table*}[h!]
\centering
\smal
\caption{\label{tab:causalgraph} Example (relation, cause, effect) tuples in different categories (manually labeled): \textit{general}, \textit{company}, \textit{country}, and \textit{people}. FrameNet labels related to causation are listed inside parentheses. The number of distinct relation types are 892.
\resizebox{\textwidth}{!}{%
\begin{tabular}{@{}r|r||r|l@{}}
\toprule
&\textbf{Relation} & \multicolumn{2}{c}{\textbf{Cause $\mapsto$ Effect $\qquad$}} \\\hline
\midrule
\parbox[t]{1mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\tiny{General}}}} &
causes (Causation) & the virus (Cause) & aids (Effect) \\
&cause (Causation) & greenhouse gases (Cause) & global warming (Effect)\\
&forced (Causation) & the reality of world war ii (Cause) & the cancellation of the olympics (Effect)\\
\midrule
\parbox[t]{1mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\tiny{Company}}}} &heats (Cause\_temperature\_change) & microsoft vague on windows (Item) & legislation battle (Agent) \\
&promotes (Cause\_change\_of\_position\_on\_a\_scale) & chrome (Item) & google (Agent)\\
&makes (Causation) & twitter (Cause) & love people you 've never met facebook (Effect)\\
\midrule
\parbox[t]{1mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\tiny{Country}}}}
&developing (Cause\_to\_make\_progress) & north korea (Agent) & nuclear weapons (Project)\\
&improve (Cause\_to\_make\_progress) & china (Agent) & its human rights record (Project)\\
&forced (Causation) & war with china (Cause) & the japanese to admit , in july 1938 (Effect)\\
\midrule
\parbox[t]{1mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\tiny{People}}}}
&attracts (Cause\_motion) & obama (Agent) & more educated voters (Theme)\\
&draws (Cause\_motion) & on america 's economic brains (Goal) & barack obama (Theme) \\
&made (Causation) & michael jordan (Cause) & about \$ 33 million (Effect)\\
\bottomrule
\end{tabular}
}
\end{table*}
\section{\textsc{CGraph}~Construction}\label{sec:graph}
Formally, given source $x$ and target $y$ events that are causally related in time series, if we could find a sequence of cause-effect pairs $(x \mapsto e_1)$, $(e_1 \mapsto e_2)$, ... $(e_t \mapsto y)$, then $e_1 \mapsto e_2, ... \mapsto e_t$ might be a good causal explanation between $x$ and $y$.
Section~\ref{sec:graph} and \ref{sec:reasoning} describe how to bridge the causal gap between given events ($x$, $y$) by (1) constructing a large general cause-effect graph (\textsc{CGraph}) from text, (2) linking the given events to their equivalent entities in the causal graph by finding the internal paths ($x \mapsto e_1, ... e_t \mapsto y$) as causal explanations, using neural algorithms.
\textsc{CGraph}~is a knowledge base graph where edges are directed and causally related between entities.
To address less representational variability of rule based methods~\cite{girju2003automatic,blanco2008causal,sharp2016creating} in the causal graph construction, we used FrameNet~\cite{baker1998berkeley} semantics.
Using a semantic parser such as SEMAFOR~\cite{chen2010semafor} that produces a FrameNet style analysis of semantic predicate-argument structures, we could obtain lexical tuples of causation in the sentence.
Since our goal is to collect only causal relations, we extract total 36 causation related frames\footnote{Causation, Cause\_change, Causation\_scenario, Cause\_ benefit\_or\_detriment, Cause\_bodily\_experience, etc.} from the parsed sentences.
\begin{table}[h]
\centerin
\small
\caption{\label{tab:graphstats} Number of sentences parsed, number of entities and tuples, and number of edges (\textit{KB-KB}, \textit{KBcross}) expanded by Freebase in \textsc{CGraph}.
\resizebox{\columnwidth}{!}{%
\begin{tabular}{c|c|c|c|c}
\toprule
\# Sentences & \# Entities & \# Tuples & \# \textit{KB-KB} & \# \textit{KBcross}\\
\midrule
183,253,995 & 5,623,924 & 5,025,636 & 470,250 & 151,752\\
\bottomrule
\end{tabular}
}
\end{table}
To generate meaningful explanations, high coverage of the knowledge is necessary.
We collect six years of tweets and NYT news articles from 1989 to 2007 (See Experiment section for details).
In total, our corpus has 1.5 billion tweets and 11 million sentences from news articles.
The Table~\ref{tab:graphstats} has the number of sentences processed and number of entities, relations, and tuples in the final \textsc{CGraph}.
Since the tuples extracted from text are very noisy~\footnote{SEMAFOR has around $62\%$ of accuracy on held-out set.}, we constructed a large causal graph by linking the tuples with string match and filter out the noisy nodes and edges based on some graph statistics.
We filter out nodes with very high degree that are mostly stop-words or auto-generated sentences.
Too long or short sentences are also filtered out.
Table~\ref{tab:causalgraph} shows the (case, relation, effect) tuples with manually annotated categories such as \textit{General}, \textit{Company}, \textit{Country}, and \textit{People}.
\section{Causal Reasoning}\label{sec:reasoning}
To generate a causal explanation using \textsc{CGraph}, we need traversing the graph for finding the path between given source and target events.
This section describes how to efficiently traverse the graph by expanding entities with external knowledge base and how to find (or generate) appropriate causal paths to suggest an explanation using symbolic and neural reasoning algorithms.
\subsection{Entity Expansion with Knowledge Base}
A simple choice for traversing a graph are the traditional graph searching algorithms such as Breadth-First Search (BFS). However, the graph searching procedure is likely to be incomplete (\textit{low recall}), because simple string match is insufficient to match an effect to all its related entities, as it misses out in the case where an entity is semantically related but has a lexically different name.
To address the \textit{low recall} problem and generate better explanations, we propose the use of knowledge base to augment our text-based causal graph with real-world semantic knowledge.
We use Freebase~\cite{freebase} as the external knowledge base for this purpose.
Among $1.9$ billion edges in original Freebase dump, we collect its first and second hop neighbours for each target events.
While our \textsc{CGraph}~is lexical in nature, Freebase entities appear as identifiers (MIDs).
For entity linking between two knowledge graphs, we need to annotate Freebase entities with their lexical names by looking at the wiki URLs.
We refer to the edges with freebase expansion as \textit{KB-KB} edges, and link the \textit{KB-KB} with our \textsc{CGraph}~using lexical matching, referring as \textit{KBcross} edges (See Table~\ref{tab:graphstats} for the number of the edges).
\subsection{Symbolic Reasoning}
Simple traversal algorithms such as BFS are infeasible for traversing the \textsc{CGraph}~due to the large number of nodes and edges.
To reduce the search space $k$ in $e_{t} \mapsto \{e_{t+1}^1, ...e_{t+1}^k\}$, we restricted our search by depth of paths, length of words in entity's name, and edge weight.
\begin{algorithm}[h]
\caption{ Backward Causal Inference. $y$ is target event, $d$ is depth of BFS, $l$ is lag size, $BFS_{back}$ is Breadth-First search for one depth in backward direction, and $\sum_l\mathbf{C}$ is sum of Granger causality over the lags. \label{alg:inference}}
\begin{algorithmic}[1]
\State $\mathbb{S} \gets {\textit{y}}$, $d = 0 $
\While {($\mathbb{S} = \emptyset$) or ($d > D_{max} $)}
\State $\{e_{-d}^1, ...e_{-d}^k\} \gets BFS_{back}(\mathbb{S})$
\State $d = d + 1$, $\mathbb{S} \gets \emptyset $
\For{\texttt{$j$ in $\{1,...,k\}$}}
\If {\textit{$\sum_l\mathbf{C}(y,e_{-d}^j,l) < \epsilon$}} $\mathbb{S} \gets e_{-d}^j $
\EndIf
\EndFor
\EndWhile
\end{algorithmic}
\end{algorithm}
For more efficient inference, we propose a backward algorithm that searches potential causes (instead of effects) $\{e_{t}^1, ...e_{t}^k\} \mapsfrom e_{t+1}$ starting from the target node $y = e_{t+1}$ using Breadth-first search (BFS).
It keeps searching backward until the node $e_{i}^j$ has less Granger confident causality with the target node $y$ (See Algorithm~\ref{eq:causality} for causality calculation).
This is only possible because our system has temporal causality measure between two time series events.
See Algorithm~\ref{alg:inference} for detail.
\subsection{Neural Reasoning}
While symbolic inference is fast and straightforward, the sparsity of edges may make our inference semantically poor.
To address the \textit{lexical sparseness}, we propose a lexically relaxed reasoning using a neural network.
Inspired by recent success on alignment task such as machine translation~\cite{bahdanau2014neural}, our model learns the causal alignment between cause phrase and effect phrase for each type of relation between them.
Rather than traversing the \textsc{CGraph}, our neural reasoner uses \textsc{CGraph}~ as a training resource.
The encoder, a recurrent neural network such as LSTM~\cite{hochreiter1997long}, takes the causal phrase while the decoder, another LSTM, takes the effectual phrase with their relation specific attention.
In original attention model~\cite{bahdanau2014neural}, the contextual vector $c$ is computed by $c_i = a_{ij} * h_j$ where $h_j$ is hidden state of causal sequence at time $j$ and $a_{ij}$ is soft attention weight, trained by feed forward network $a_{ij} = FF (h_j, s_{i-1})$ between input hidden state $h_j$ and output hidden state $s_{i-1}$.
The global attention matrix $a$, however, is easy to mix up all local alignment patterns of each relation.
For example, a tuple,
\textit{\small{(north korea (Agent)
$\xmapsto[(Cause\_to\_make\_progress)]{developing}$ nuclear weapons (Project))}},
is different with another tuple,
\textit{\small{(chrome (Item) $\xmapsto[(Cause \_change\_of\_position)]{promotes}$ google (Agent))}} in terms of local type of causality.
To deal with the \textit{local attention}, we decomposed the attention weight $a_{ij}$ by relation specific transformation in feed forward network:
\begin{align*}
&a_{ij} = FF (h_j, s_{i-1}, r)
\end{align*}
where $FF$ has relation specific hidden layer and $r \in R$ is a type of relation in the distinct set of relations $R$ in training corpus (See Figure~\ref{fig:proposedmodel}).
\begin{figure}[t]
\centering
\includegraphics[trim=2.2cm 5.5cm 2.1cm 5cm,clip,width=.96\linewidth]{figs/srs.pdf}
\caption{\label{fig:proposedmodel} Our neural reasoner. The encoder takes causal phrases and decoder takes effect phrases by learning the causal alignment between them. The MLP layer in the middle takes different types of FrameNet relation and locally attend the cause to the effect w.r.t the relation (e.g., ``because of'', ``led to'', etc).
\end{figure}
Since training only with our causal graph may not be rich enough for dealing various lexical variation in text, we use pre-trained word embedding such as word2vec~\cite{mikolov2013distributed} trained on GoogleNews corpus\footnote{https://code.google.com/archive/p/word2vec/} for initialization.
For example, given a cause phrase \textit{weapon equipped}, our model could generate multiple effect phrases with their likelihood: \textit{($\xmapsto[0.54]{result}$war)}, \textit{($\xmapsto[0.12]{force}$army reorganized)}, etc, even though there are no tuples exactly matched in \textsc{CGraph}.
We trained our neural reasoner in either forward or backward direction.
In prediction, decoder inferences by predicting effect (or cause) phrase in forward (or backward) direction.
As described in the Algorithm~\ref{alg:inference}, the backward inference continue predicting the previous causal phrases until it has high enough Granger confidence with the target event.
\section{Experiment}\label{sec:experiment}
\textbf{Data}. We collect on-line social media from tweets, news articles, and blogs.
Our Twitter data has one million tweets per day from 2008 to 2013 that are crawled using Twitter's Garden Hose API.
News and Blog dataset have been crawled from 2010 to 2013 using Google's news API.
For target time series, we collect companies' stock prices in NASDAQ and NYSE from 2001 until present for 6,200 companies.
For presidential election polls, we collect polling data of the 2012 presidential election from 6 different websites, including USA Today , Huffington Post, Reuters, etc.
\begin{table}[t]
\centering
\setlength\tabcolsep{2pt}
\caption{\label{tab:dynamics} Examples of $F_{words}$ with their temporal dynamics: Shannon entropy, mean, standard deviation, slope of peak, and number of peaks.}
\resizebox{\columnwidth}{!}{%
\begin{tabular}{@{}r|cccccc@{}}
\toprule
& \textbf{entropy} & \textbf{mean} & \textbf{STD} & \textbf{max\_slope} & \textbf{\#-peaks} \\
\midrule
\#lukewilliamss & 0.72 & 22.01 & 18.12 & 6.12 & 31 \\
happy\_thanksgiving & 0.40 & 61.24& 945.95 &3423.75 &414 \\
michael\_jackson & 0.46 &141.93 &701.97 &389.19 &585 \\
\bottomrule
\end{tabular}
\end{table}
\textbf{Features}. For N-gram word features $F_{word}$,we choose the spiking words based on their temporal dynamics (See Table~\ref{tab:dynamics}).
For example, if a word is too frequent or the time series is too burst, the word should be filtered out because the trend is too general to be an event.
We choose five types of temporal dynamics: Shannon entropy, mean, standard deviation, maximum slope of peak, and number of peaks;
and delete words that have too low or high entropy, too low mean and deviation, or the number of peaks and its slope is less than a certain threshold.
Also, we filter out words whose frequency is less than five.
From the $1,677,583$ original words, we retain $21,120$ words as final candidates for $F_{words}$ including uni-gram and bi-gram words.
For sentiment $F_{senti}$ and topic $F_{topic}$ features, we choose 50 topics generated for both politicians and companies separately using LDA, and then use top 10 words for each topic to calculate sentiment score for this topic.
Then we can analyze the causality between sentiment series of a specific topic and collected time series.
\textbf{Tasks}. To show validity of causality detector, first we conduct random analysis between target time series and randomly generated time series.
Then, we tested forecasting stock prices and election poll values with or without the detected textual features to check effectiveness of our causal features.
We evaluate our reasoning algorithm for generation ability compared to held-out cause-effect tuples using BLEU metric.
Then, for some companies' time series, we describe some qualitative result of some interesting causal text features found with Granger causation and explanations generated by our reasoners between the target and the causal features.
We also conducted human evaluation on the explanations.
\subsection{Random Causality Analysis}
\begin{figure}[t]
\centerin
{
\subfloat[\footnotesize{$y \xleftarrow[]{lag=3} rf_1, ..., rf_k$ }]{
\fbox{\includegraphics[clip,trim=2.4cm 0.3cm 0.5cm 1.5cm,width=.84\linewidth, height=60px]{figs/{random_GOOGL_False_3_2013-01-01_2013-12-31}.jpg}
}}\\
\subfloat[\footnotesize{$y \xrightarrow[]{lag=3} rf_1, ..., rf_k$ }]{
\fbox{\includegraphics[clip,trim=2.4cm 0.3cm 0.5cm 0.7cm,width=.84\linewidth, height=60px]{figs/{random_GOOGL_True_3_2013-01-01_2013-12-31}.jpg}
}}\\
\caption{\label{fig:random} Random causality analysis on \textbf{Googles}'s stock price change ($y$) and randomly generated features ($rf$) during 2013-01-01 to 2013-12-31.
(a) shows how the random features $rf$ cause the target $y$, while (b) shows how the target $y$ causes the random features $rf$ with lag size of 3 days.
The color changes according to causality confidence to the target (blue is the strongest, and yellow is the weakest).
The target time series has y scale of prices, while random features have y scale of causality degree $\mathbf{C}(y,rf) \subset [ 0,1 ]$.
\end{figure}
To check whether our causality scoring function $\mathbf{C}$ detects the temporal causality well, we conduct a random analysis between target time series and randomly generated time series (See Figure~\ref{fig:random}).
For Google's stock time series, we regularly move window size of 30 over the time and generate five days of time series with a random peak strength using a SpikeM model~\cite{DBLP:conf/kdd/MatsubaraSPLF12a}\footnote{SpikeM has specific parameters for modeling a time series such as peak strength, length, etc.}.
The color of random time series $rf$ changes from blue to yellow according to causality degree with the target $\mathbf{C}(y,rf)$.
For example, blue is the strongest causality with target time series, while yellow is the weakest.
We observe that the strong causal (blue) features are detected just before (or after) the rapid rise of Google' stock price on middle October in (a) (or in (b)).
With the lag size of three days, we observe that the strength of the random time series gradually decreases as it grows apart from the peak of target event.
The random analysis shows that our causality function $\mathbf{C}$ appropriately finds cause or effect relation between two time series in regard of their strength and distance.
\subsection{Forecasting with Textual Features}\label{sec:forecasting}
\begin{table}[h]
\footnotesize
\caption{\label{tab:forecasting} Forecasting errors (RMSE) on \textbf{Stock} and \textbf{Poll} data with time series only (\textit{SpikeM} and \textit{LSTM}) and with time series plus text feature (\textit{random}, \textit{words}, \textit{topics}, \textit{sentiment}, and \textit{composition}).}
\centering
\setlength\tabcolsep{2pt}
\begin{tabular}{r|r|cc|ccccc}
\toprule
\multicolumn{2}{r}{\textit{}} & \multicolumn{2}{c}{\textbf{Time Series}} & \multicolumn{5}{c}{\textbf{Time Series + Text}} \\\hline
\multicolumn{2}{r}{\textit{Step}} & SpikeM & LSTM & $\mathbf{C}_{rand}$ & $\mathbf{C}_{words}$ & $\mathbf{C}_{topics}$ & $\mathbf{C}_{senti}$ & $\mathbf{C}_{comp}$\\
\midrule
\multirow{3}{2mm}{\rotatebox[origin=c]{90}{\textbf{Stock}}}
&1 & 102.13 & 6.80 & 3.63 & 2.97 & 3.01 & 3.34 & \underline{1.96} \\
&3 & 99.8 & 7.51 & 4.47 & 4.22 & 4.65 & 4.87 & \underline{3.78} \\
&5& 97.99 & 7.79 & 5.32 & \underline{5.25} & 5.44 & 5.95 & 5.28 \\
\hline
\multirow{3}{2mm}{\rotatebox[origin=c]{90}{\textbf{Poll}}}
&1 &10.13 & 1.46 &1.52 & 1.27 & 1.59 & 2.09 & \underline{1.11} \\
&3 & 10.63 & 1.89 & 1.84 & 1.56 & 1.88 & 1.94 & \underline{1.49}\\
&5 & 11.13 & 2.04 & 2.15 & 1.84 & 1.88 & 1.96 &\underline{1.82}\\
\bottomrule
\end{tabular}
\end{table}
We use time series forecasting task as an evaluation metric of whether our textual features are appropriately causing the target time series or not.
Our feature composition function $\Phi$ is used to extract good causal features for forecasting.
We test forecasting on stock price of companies (\textbf{Stock}) and predicting poll value for presidential election (\textbf{Poll}).
For stock data, We collect daily closing stock prices during 2013 for ten IT companies\footnote{Company symbols used: TSLA, MSFT, GOOGL, YHOO, FB, IBM, ORCL, AMZN, AAPL and HPO}.
For poll data, we choose ten candidate politicians~\footnote{Name of politicians used: Santorum, Romney, Pual, Perry, Obama, Huntsman, Gingrich, Cain, Bachmann} in the period of presidential election in 2012.
\begin{table}[t]
\centering
\small
\caption{\label{tab:beam} Beam search results in neural reasoning. These examples could be filtered out by graph heuristics before generating final explanation though.
\begin{tabular}{@{}l@{}|l@{}}
\toprule
Cause$\mapsto$Effect in \textsc{CGraph} & Beam Predictions\\\hline
\midrule
\specialcell{the dollar's \\$\xmapsto[]{caus}$ against the yen} & \specialcell{$[1]$$\xmapsto[]{caus}$ against the yen\\ $[2]$$\xmapsto[]{caus}$ against the dollar \\ $[3]$$\xmapsto[]{caus}$ against other currencies} \\\hline
\specialcell{without any exercise \\$\xmapsto[]{caus}$ news article} & \specialcell{$[1]$$\xmapsto[]{lead to}$ a difference \\ $[2]$$\xmapsto[]{caus}$ the risk \\ $[3]$$\xmapsto[]{make}$ their weight} \\
\bottomrule
\end{tabular
\end{table}
For each of stock and poll data, the future trend of target is predicted only with target's past time series or with target's past time series and past time series of textual features found by our system.
Forecasting only with target's past time series uses \textit{SpikeM}~\cite{DBLP:conf/kdd/MatsubaraSPLF12a} that models a time series with small number of parameters and simple \textit{LSTM}~\cite{hochreiter1997long,nnet} based time series model.
Forecasting with target and textual features' time series use Vector AutoRegressive model with exogenous variables (VARX)~\cite{hamilton1994time} from different composition function such as $\mathbf{C}_{random}$, $\mathbf{C}_{words}$, $\mathbf{C}_{topics}$, $\mathbf{C}_{senti}$, and $\mathbf{C}_{composition}$.
Each composition function except $\mathbf{C}_{random}$ uses top ten textual features that causes each target time series.
We also tested LSTM with past time series and textual features but VARX outperforms LSTM.
Table~\ref{tab:forecasting} shows root mean square error (RMSE) for forecasting with different step size (time steps to predict), different set of features, and different regression algorithms on stock and poll data.
The forecasting error is summation of errors over moving a window (30 days) by 10 days over the period.
Our $\mathbf{C}_{composition}$ method outperforms other time series only models and time series plus text models in both stock and poll data.
\subsection{Generating Causality with Neural Reasoner}
The reasoner needs to predict the next effect phrase (or previous cause phrase) so the model should be evaluated in terms of generation task.
We used the BLEU ~\cite{papineni2002bleu} metric to evaluate the predicted phrases on held out phrases in our \textsc{CGraph}~.
Since our \textsc{CGraph}~ has many edges, there may be many good paths (explanations), possibly making our prediction diverse. To evaluate such diversity in prediction, we used ranking-based BLEU method on the $k$ set of predicted phrases by beam search.
For example, $B@k$ means BLEU scores for generating $k$ number of sentences and $B@kA$ means the average of them.
Table~\ref{tab:beam} shows some examples of our beam search results when $k=3$.
Given a cause phrase, the neural reasoner sometime predicts semantically similar phrases (e.g., \textit{against the yen}, \textit{against the dollar}), while it sometimes predicts very diverse phrases (e.g., \textit{a different}, \textit{the risk}).
Table~\ref{tab:bleu} shows BLEU ranking results with different reasoning algorithms: \textbf{S2S} is a sequence to sequence learning trained on \textsc{CGraph}~by default, \textbf{S2S+WE} adds word embedding initialization, and \textbf{S2S+REL+WE} adds relation specific attention.
Initializing with pre-trained word embeddings (\textbf{+WE}) helps us improve on prediction.
Our relation specific attention model outperforms the others, indicating that different type of relations have different alignment patterns.
\begin{table}[t]
\centering
\caption{\label{tab:bleu} BLEU ranking.
Additional word representation \textbf{+WE} and relation specific alignment \textbf{+REL} help the model learn the cause and effect generation task especially for diverse patterns.
\begin{tabular}{@{}l|c|c|c}
\toprule
&B@1 & B@3A & B@5A\\
\midrule
\textbf{S2S} & 10.15 & 8.80 & 8.69 \\
\textbf{S2S + WE} & 11.86 & 10.78 & 10.04 \\
\textbf{S2S + WE + REL} & 12.42 & 12.28 & 11.53 \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Generating Explanation by Connecting}
\begin{table*}[h]
\centering
\caption{\label{tab:explanation} Example causal chains for explaining the rise ($\uparrow$) and fall ($\downarrow$) of companies' stock price. The temporally causal {\color{Sepia}$feature$} and {\color{BlueViolet}$target$} are linked through a sequence of predicted cause-effect tuples by different reasoning algorithms: a symbolic graph traverse algorithm \textit{SYMB} and a neural causality reasoning model \textit{NEUR}.
\resizebox{\linewidth}{!}{%
\begin{tabular}{@{}c|l@{}}
\midrule
\parbox[t]{1.0mm}{\multirow{3}{*}{\rotatebox[origin=c]{90}{\textit{SYMB}}}}
& \textbf{\color{Sepia}medals} $\xmapsto[]{match}$ gold\_and\_silver\_medals $\xmapsto[]{swept}$ korea $\xmapsto[]{improving}$ relations $\xmapsto[]{widened}$ gap $\xmapsto[]{widens}$ \textbf{\color{BlueViolet}facebook} $\uparrow$ \\
& \textbf{\color{Sepia}excess}$\xmapsto[]{match}$excess\_materialism$\xmapsto[]{cause}$people\_make\_films$\xmapsto[]{make}$money $\xmapsto[]{changed}$ twitter $\xmapsto[]{turned} $\textbf{\color{BlueViolet}facebook} $\downarrow$\\
& \textbf{\color{Sepia}clinton} $\xmapsto[]{match}$president\_clinton $\xmapsto[]{raised}$antitrust\_case $\xmapsto[]{match}$government's\_antitrust\_case\_against\_microsoft $\xmapsto[]{match}$microsoft $\xmapsto[]{beats}$\textbf{\color{BlueViolet}apple} $\downarrow$\\
\hline
\parbox[t]{1.0mm}{\multirow{4}{*}{\rotatebox[origin=c]{90}{\textit{NEUR}}}}
& \textbf{\color{Sepia}google} $\xmapsto[]{forc}$ microsoft\_to\_buy\_computer\_company\_dell\_announces\_recall\_of\_batteries $\xmapsto[]{cause}$ \textbf{\color{BlueViolet}microsoft} $\uparrow$\\
& \textbf{\color{Sepia}the\_deal} $\xmapsto[]{make}$ money $\xmapsto[]{rais}$ at\_warner\_music\_and\_google\_with\_protest\_videos\_things $\xmapsto[]{caus}$ \textbf{\color{BlueViolet}google} $\downarrow$\\
& \textbf{\color{Sepia}party} $\xmapsto[]{cut}$ budget\_cuts$\xmapsto[]{lower}$ budget\_bill$\xmapsto[]{decreas}$ republicans$\xmapsto[]{caus}$ obama$\xmapsto[]{lead to}$ facebook\_polls$\xmapsto[]{caus}$ \textbf{\color{BlueViolet}facebook} $\downarrow$\\
& \textbf{\color{Sepia}company} $\xmapsto[]{forc}$ to\_stock\_price $\xmapsto[]{lead to}$ investors $\xmapsto[]{increas}$ oracle\_s\_stock $\xmapsto[]{increas}$ \textbf{\color{BlueViolet}oracle} $\uparrow$
\\
\bottomrule
\end{tabular}
\end{table*}
Evaluating whether a sequence of phrases is reasonable as an explanation is very challenging task.
Unfortunately, due to lack of quantitative evaluation measures for the task, we conduct a human annotation experiment.
Table~\ref{tab:explanation} shows example causal chains for the rise ($\uparrow$) and fall ($\downarrow$) of companies' stock price, continuously produced by two reasoners: \textit{SYBM} is symbolic reasoner and \textit{NEUR} is neural reasoner.
\begin{table}[h]
\centering
\caption{\label{tab:eval} Human evaluation on explanation chains generated by symbolic and neural reasoners.
\begin{tabular}{r|c|c}
\toprule
\textbf{Reasoners} &SYMB & NEUR\\
\midrule
\textbf{Accuracy (\%)}& 42.5 & 57.5 \\
\bottomrule
\end{tabular}
\end{table}
We also conduct a human assessment on the explanation chains produced by the two reasoners, asking people to choose more convincing explanation chains for each feature-target pair.
Table~\ref{tab:eval} shows their relative preferences
\section{Related Work}\label{sec:related}
Prior works on causality detection~\cite{acharya2014causal,websummary,qiu2012granger} in time series data (e.g., gene sequence, stock prices, temperature) mainly use Granger~\cite{granger1988some} ability for predicting future values of a time series using past values of its own and another time series.
\cite{hlavavckova2007causality} studies more theoretical investigation for measuring causal influence in multivariate time series based on the entropy and mutual information estimation.
However, none of them attempts generating explanation on the temporal causality.
Previous works on text causality detection use syntactic patterns such as $X \xmapsto[]{verb} Y$, where the $verb$ is causative~\cite{girju2003automatic,riaz2013toward,kozareva2012cause,do2011minimally} with additional features~\cite{blanco2008causal}.
\cite{kozareva2012cause} extracted cause-effect relations, where the pattern for bootstrapping has a form of $X^* \xmapsto[Z^*]{verb} Y$ from which terms $X^*$ and $Z^*$ was learned.
The syntax based approaches, however, are not robust to semantic variation.
As a part of SemEval~\cite{girju2007semeval}, \cite{mirza2016catena} also uses syntactic causative patterns~\cite{mirza2014analysis} and supervised classifier to achieve the state-of-the-art performance.
Extracting the cause-effect tuples with such syntactic features or temporality~\cite{bethard2008building} would be our next step for better causal graph construction.
\cite{grivaz2010human} conducts very insightful annotation study of what features are used in human reasoning on causation.
Beyond the linguistic tests and causal chains for explaining causality in our work, other features such as counterfactuality, temporal order, and ontological asymmetry remain as our future direction to study.
Textual entailment also seeks a directional relation between two given text fragments~\cite{dagan2006pascal}.
Recently, \cite{rocktaschel2015reasoning} developed an attention-based neural network method, trained on large annotated pairs of textual entailment, for classifying the types of relations with decomposable attention~\cite{parikh2016decomposable} or sequential tree structure~\cite{chen2016enhancing}.
However, the dataset~\cite{bowman2015large} used for training entailment deals with just three categories, \textit{contradiction}, \textit{neutral}, and \textit{entailment}, and focuses on relatively simple lexical and syntactic transformations~\cite{kolesnyk2016generating}. Our causal explanation generation task is also similar to \textit{future scenario generation}~\cite{hashimoto2014toward,hashimoto2015generating}.
Their scoring function uses heuristic filters and is not robust to lexical variation.
\section{Conclusion}\label{sec:conclusion}
This paper defines the novel task of detecting and explaining causes from text for a time series.
First, we detect causal features from online text. Then, we construct a large cause-effect graph using FrameNet semantics. By training our relation specific neural network on paths from this graph, our model generates causality with richer lexical variation. We could produce a chain of cause and effect pairs as an explanation which shows some appropriateness. Incorporating aspects such as time, location and other event properties remains a point for future work.
In our following work, we collect a sequence of causal chains verified by domain experts for more solid evaluation of generating explanations.
| 2024-02-18T23:39:52.498Z | 2017-07-28T02:06:54.000Z | algebraic_stack_train_0000 | 729 | 8,202 |
|
proofpile-arXiv_065-3743 | \section{Asymptotic analysis of GS-SVM}\label{sec:gs-svm_proof}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.4\textwidth,height=0.3\textwidth]{gssvm_delta_2_edited_noaxis.pdf}
\end{center
\caption{Visualizing the Gaussian mixture model of Section \ref{sec:generalization} with $K=2$ imbalanced groups in the two-dimensional space ($d=2$). Different colors (resp., markers) correspond to different class (resp., group) membership. Examples in the minority group correspond to cross markers ($\times$). The means of the majority / minority groups are depicted in white / green markers. The purple line illustrates the group-sensitive SVM (GS-SVM) classifier that forces larger margin to the minority group examples in relation to standard SVM in green.}
\label{fig:gssvm_2d}
\end{figure}
In Theorem \ref{thm:main_imbalance} we derived the asymptotic generalization performance of CS-SVM under the Gaussian mixture data model. Here, we state the counterpart result for GS-SVM with an appropriate Gaussian mixture data model with group imbalances, which we repeat here for convenience.
\noindent\textbf{Data model.}~We study a binary Gaussian-mixture generative model (GMM) for the data distribution $\mathcal{D}$. For the label $y\in\{\pm1\}$ let $\pi:=\mathbb{P}\{y=+1\}.$ Group membership is decided conditionally on the label such that $\forall j\in[K]:$ $\mathbb{P}\{g=j|y=\pm1\}=p_{\pm,j}$, with $\sum_{j\in[K]}p_{+,j}=\sum_{j\in[K]}p_{-,j}=1$. Finally, the feature conditional given label $y$ and group $g$ is a multivariate Gaussian of mean $\boldsymbol\mu_{y,g}\in\mathbb{R}^d$ and covariance $\boldsymbol\Sigma_{y,g}$, that is,
$\mathbf{x}\big|(y,g)\,\widesim{} \mathcal{N}(\boldsymbol\mu_{y,g},\boldsymbol\Sigma_{y,g}).
$
We focus on two groups $K=2$ with $p_{+,1}=p_{-,1}=p<1-p=p_{+,2}=p_{-,2},\,j=1,2$ and $\mathbf{x}\,|\,(y,g) \sim \mathcal{N}(y\boldsymbol\mu_{g},\sigma_g\mathbf{I}_d)$, for $\sigma_1^2,\sigma_2^2$ the noise variances of the minority and the majority groups, respectively. As before, let $\mathbf{M}$ denote the matrix of means (that is $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{+} & \boldsymbol\mu_{-}\end{bmatrix}$ and $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{1} & \boldsymbol\mu_{2}\end{bmatrix}$, respectively) and consider the eigen-decomposition of its Gramian:
$
\mathbf{M}^T\mathbf{M} = \mathbf{V}\mathbf{S}^2\mathbf{V}^T,~~ \mathbf{S}\succ \mathbf{0}_{r\times r},\mathbf{V}\in\mathbb{R}^{2\times r}, r\in\{1,2\},
$
with $\mathbf{S}$ an $r\times r$ diagonal positive-definite matrix and $\mathbf{V}$ an orthonormal matrix obeying $\mathbf{V}^T\mathbf{V}=\mathbf{I}_r$. We study linear classifiers with $h(\mathbf{x})=\mathbf{x}$.
\noindent\textbf{Learning regime.}
Again, as in Theorem \ref{thm:main_imbalance}, we focus on a regime where training data are linearly separable. Specifically, there exists threshold $\widetilde\gamma_{\star}:=\widetilde\gamma_{\star}(\mathbf{V},\mathbf{S},\pi,p)\leq 1/2$, such
that GMM data with groups are linearly separable with probability approaching one provided that $\gamma>\widetilde\gamma_{\star}$ (see Section \ref{sec:PT_GS-SVM}).
We assume $\gamma>\widetilde\gamma_{\star}$, so that GS-SVM is feasible with probability approaching 1.
Although similar in nature, the result below differs to Theorem \ref{thm:main_imbalance} since now each class itself is a Gaussian mixture.
\begin{theorem}[Sharp asymptotics of GS-SVM]\label{thm:GS_app}\label{thm:main_group_fairness}
Consider the GMM with feature distribution and priors as specified in the `Data model' above. Fix $\delta>0$ (corresponding to group VS-loss with $\Delta_{y,g}=\Delta_{g}, g=1,2$ such that $\delta=\Delta_2/\Delta_1$). Define $G, Y, S, {\widetilde{\Delta}}_S,\Sigma_S\in\mathbb{R},$ and $\widetilde{E}_S\in\mathbb{R}^{2\times 1}$ as follows:
$G\sim \mathcal{N}(0,1)$; $Y$ is a symmetric Bernoulli with $\mathbb{P}\{Y=+1\}=\pi$; $S$ takes values $1$ or $2$ with probabilities $p$ and $1-p$, respectively;
$\widetilde{E}_S=\mathbf{e}_1\ind{S=1}+\mathbf{e}_2\ind{S=2}$; ${\widetilde{\Delta}}_S=\delta\cdot\ind{S=1} +1\cdot\ind{S=2}$ and $\Sigma_S=\sigma_1\ind{S=1}+\sigma_2\ind{S=2}$.
With these define function $\widetilde\eta_\delta:\mathbb{R}_{\geq0}\times{\mathcal{S}}^{r}\times\mathbb{R}\rightarrow\mathbb{R}$ as
$$
\widetilde\eta_\delta(q,\boldsymbol\rho,b):=
\mathbb{E}\big(G+\Sigma_S^{-1}\widetilde{E}_S^T\mathbf{V}\mathbf{S}\boldsymbol\rho +\frac{b \Sigma_S^{-1} Y-\Sigma_S^{-1}{\widetilde{\Delta}}_S}{q}\big)_{-}^2 \notag- (1-\|\boldsymbol\rho\|_2^2)\gamma.
$$
Let $(\widetilde q_\delta,\widetilde \boldsymbol\rho_\delta, \widetilde b_\delta)$ be the unique triplet satisfying \eqref{eq:eta_eq_main} but with $\eta_\delta$ replaced with the function $\widetilde \eta_\delta$ above.
Then, in the limit of $n,d\rightarrow\infty$ with $d/n=\gamma>\widetilde\gamma_{\star}$ it holds for $i=1,2$ that
$
\mathcal{R}_{\pm,i} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_i^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta\pm \widetilde b_\delta/{\widetilde q_\delta}}\big)$. In particular,
$
\mathcal{R}_{\rm deo} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_1^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big) - Q\big({\mathbf{e}_2^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big).$
\end{theorem}
\subsection{Proof of Theorem \ref{thm:GS_app}}
The proof of Theorem \ref{thm:GS_app} also relies on the CGMT framework and is very similar to the proof of Theorem \ref{thm:main_imbalance}. To avoid repetitions, we only present the part that is different. As we will show the PO is slightly different as now we are dealing with a classification between mixtures of mixtures of Gaussians. We will derive the new AO and will simplify it to a point from where the same steps as in Section \ref{sec:cs_proof_app} can be followed mutatis mutandis.
Let $(\hat{\w},\hat{b})$ be solution pair to the GS-SVM for some fixed parameter $\delta>0$, which we rewrite here expressing the constraints in matrix form:
\begin{align}\label{eq:GS-SVM_app}
\min_{\mathbf{w},b}\|\mathbf{w}\|_2~~\text{sub. to}~\begin{cases} y_i(\mathbf{w}^T\mathbf{x}_i+b) \geq\delta ,~g_i=1 \\ y_i(\mathbf{w}^T\mathbf{x}_i+b)\geq 1,~g_i=2 \end{cases}\hspace{-10pt},~i\in[n]~~=~~ \min_{\mathbf{w},b}\|\mathbf{w}\|_2~~\text{sub. to}~\mathbf{D}_\mathbf{y}(\mathbf{X}\mathbf{w}+b\mathbf{1}_n)\geq \boldsymbol\delta_\mathbf{g},
\end{align}
where we have used the notation
\begin{align*}
\mathbf{X}^T&=\begin{bmatrix} \mathbf{x}_1 & \cdots & \mathbf{x}_n \end{bmatrix},~\mathbf{y}=\begin{bmatrix} y_1 & \cdots & y_n \end{bmatrix}^T,
\\\mathbf{D}_\mathbf{y}&=\rm{diag}(\mathbf{y})~\text{and}~\boldsymbol\delta_\mathbf{g}=\begin{bmatrix}
\delta\ind{g_1=1}+\ind{g_1=2}& \cdots &\delta\ind{g_n=1}+\ind{g_n=2}
\end{bmatrix}^T.
\end{align*}
We further need to define the following one-hot-encoding for group membership:
$$
\mathbf{g}_i = \mathbf{e}_1 \ind{g_i=1} + \mathbf{e}_2 \ind{g_i=2},\quad\text{and}\quad {G}_{n\times 2}^T=\begin{bmatrix}\mathbf{g}_1 & \cdots & \mathbf{g}_n \end{bmatrix}.
$$
where recall that $\mathbf{e}_1,\mathbf{e}_2$ are standard basis vectors in $\mathbb{R}^2$. Finally, let
$$
\mathbf{D}_\sigma = {\rm diag}\big( \begin{bmatrix} \sigma_{g_1} & \cdots & \sigma_{g_n} \end{bmatrix} \big).
$$
With these,
notice for later use that under our model, $\mathbf{x}_i=y_i\boldsymbol\mu_{g_i}+\sigma_{g_i}\mathbf{z}_i = y_i\mathbf{M}\mathbf{g}_i+\sigma_{g_i}\mathbf{z}_i,~\mathbf{z}_i\sim\mathcal{N}(0,1)$. Thus, in matrix form with $\mathbf{Z}$ having entries $\mathcal{N}(0,1)$:
\begin{align}\label{eq:model_gs}
\mathbf{X} = \mathbf{D}_\mathbf{y}{G}\mathbf{M}^T+\mathbf{D}_\sigma\mathbf{Z}.
\end{align}
As usual, we express the GS-SVM program in a min-max form to bring it in the form of the PO as follows:
\begin{align}
&\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{X}\mathbf{w}+b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{g}\notag\\
=&\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{D}_\sigma\mathbf{Z}\mathbf{w} + \mathbf{u}^T{G}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{g}.\label{eq:gs_PO}
\end{align}
where in the last line we used \eqref{eq:model_gs} and $\mathbf{D}_\mathbf{y}\mathbf{D}_\mathbf{y}=\mathbf{I}_n$.
We immediately recognize that the last optimization is in the form of a PO and the corresponding AO is as follows:
\begin{align}\label{eq:gs_AO}
\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~ \frac{1}{2}\|\mathbf{w}\|_2^2+\|\mathbf{w}\|_2\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{D}_\sigma\mathbf{h}_n + \|\mathbf{D}_\mathbf{y}\mathbf{D}_\sigma\mathbf{u}\|_2\mathbf{h}_d^T\mathbf{w} + \mathbf{u}^T{G}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{g}.
\end{align}
where $\mathbf{h}_n\sim\mathcal{N}(0,\mathbf{I}_n)$ and $\mathbf{h}_d\sim\mathcal{N}(0,\mathbf{I}_d)$.
As in Section \ref{sec:cs_proof_app} we consider the one-sided constrained AO in \eqref{eq:gs_AO}. Towards simplifying this auxiliary optimization, note that $\mathbf{D}_y\mathbf{h}_n\sim\mathbf{h}_n$ by rotational invariance of the Gaussian measure. Also, $\|\mathbf{D}_\mathbf{y}\mathbf{D}_\sigma\mathbf{u}\|_2=\|\mathbf{D}_\sigma\mathbf{u}\|_2$. Thus, we can express the AO in the following more convenient form:
\begin{align*
&\min_{\|\mathbf{w}\|_2^2+b^2\leq R}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \|\mathbf{w}\|_2\mathbf{u}^T\mathbf{D}_\sigma\mathbf{h}_n + \|\mathbf{D}_\sigma\mathbf{u}\|_2\mathbf{h}_d^T\mathbf{w} + \mathbf{u}^T{G}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{g}\\
=&\min_{\|\mathbf{w}\|_2^2+b^2\leq R}\max_{\mathbf{v}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \|\mathbf{w}\|_2\mathbf{v}^T\mathbf{h}_n + \|\mathbf{v}\|_2\mathbf{h}_d^T\mathbf{w} + \mathbf{v}^T\mathbf{D}_{\sigma}^{-1}{G}\mathbf{M}^T\mathbf{w} +b(\mathbf{v}^T\mathbf{D}_\sigma^{-1}\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{v}^T\mathbf{D}_\sigma^{-1}\boldsymbol\delta_\mathbf{g},
\end{align*}
where in the second line we performed the change of variables $\mathbf{v}\leftrightarrow\mathbf{D}_\sigma\mathbf{u}$ and used positivity of the diagonal entries of $\mathbf{D}_\sigma$ to find that $\mathbf{u}\leq0 \iff \mathbf{v}\leq 0 $.
Notice that the optimization in the last line above is very similar to the AO \eqref{eq:cs_AO_11} in Section \ref{sec:cs_proof_app}. Following analogous steps, omitted here for brevity, we obtain the following scalarized AO:
\begin{align}\label{eq:gs_AO_scal_cvx}
&\min_{ \substack{ q\geq \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \\ q^2+b^2\leq R} }~\frac{1}{2}q^2\quad
\\
&\text{sub. to}\quad\frac{1}{\sqrt{n}}\Big\| \big( \, q \mathbf{h}_n + \mathbf{D}_\sigma^{-1}{G}\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2 \end{bmatrix} +b\,\mathbf{D}_\sigma^{-1}\mathbf{D}_y\mathbf{1}_n-\mathbf{D}_\sigma^{-1}\boldsymbol\delta_{\mathbf{g}} \, \big)_- \Big\|_2 \nonumber\\&-\mu_1\frac{\mathbf{h}_d^T\mathbf{u}_1}{\sqrt{n}} - \mu_2\frac{\mathbf{h}_d^T\mathbf{u}_2}{\sqrt{n}} - \alpha\frac{\|\mathbf{h}_d^T\mathbf{U}^\perp\|_2}{\sqrt{n}}\leq 0.\notag
\end{align}
where as in Section \ref{sec:cs_proof_app} we have decomposed the matrix of means $\mathbf{M}=\mathbf{U}\mathbf{S}\mathbf{V}^T$ and $\mu_1,\mu_2,\alpha$ above represent $\mathbf{u}_1^T\mathbf{w}$, $\mathbf{u}_1^T\mathbf{w}$ and $\|\mathbf{w}^\perp\|_2$.
Now, by law of large numbers, notice that for fixed $(q,\mu_1,\mu_2,\alpha,b)$, the functional in the constraint above converges in probability to
\begin{align}\label{eq:Ln2L}
\bar L(q,\mu_1,\mu_2,\alpha,b):=\sqrt{\mathbb{E}\big(qG+\Sigma_S^{-1}\widetilde{E}_S^T\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2\end{bmatrix} + {b\,\Sigma_S^{-1}Y-\Sigma_S^{-1}{\widetilde{\Delta}}_S}\big)_{-}^2} - \alpha\sqrt{\gamma},
\end{align}
where the random variables $G,\widetilde{E}_S,Y,{\widetilde{\Delta}}_S$ and $\Sigma_S$ are as in the statement of the theorem.
Thus, the deterministic equivalent (high-dimensional limit) of the AO expressed in variables
$
\boldsymbol\rho = \begin{bmatrix}\boldsymbol\rho_1\\\boldsymbol\rho_2 \end{bmatrix}:=\begin{bmatrix}\mu_1/q \\ \mu_2/q\end{bmatrix}
$
becomes (cf. Eqn. \eqref{eq:cs_AO_scal_det2}):
\begin{align}\label{eq:gs_AO_scal_det2}
&\min_{ q^2+b^2 \leq R, q>0, \|\boldsymbol\rho\|_2\leq 1 }~\frac{1}{2}q^2
\\
&\text{sub. to}\quad {\mathbb{E}\big(G+\Sigma_S^{-1}\widetilde{E}_S^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\,\Sigma_S^{-1}Y-\Sigma_S^{-1}\widetilde\Delta_S}{q}\big)_{-}^2} \notag \leq \left({1-\|\boldsymbol\rho\|_2^2}\right){\gamma}.
\end{align}
Now, recall the definition of the function $\widetilde\eta_\delta$ in the statement of the theorem and observe that the constraint above is nothing but
$$
\widetilde\eta_\delta(q,\boldsymbol\rho,b)\leq 0.
$$
Thus, \eqref{eq:gs_AO_scal_det2} becomes
\begin{align}\label{eq:gs_AO_scal_det3}
\min\left\{q^2~\Big|~ 0\leq q\leq \sqrt{R} \quad\text{ and }\quad \min_{b^2 \leq R-q^2, \|\boldsymbol\rho\|_2\leq 1} \widetilde\eta_\delta(q,\boldsymbol\rho,b)\leq 0 \right\}.
\end{align}
The remaining steps of the proof are very similar to those in Section \ref{sec:cs_proof_app} and are omitted.
\subsection{Phase transition of GS-SVM}\label{sec:PT_GS-SVM}
The phase-transition threshold $\widetilde \gamma_\star$ of feasibility of the GS-SVM is the same as the threshold of feasibility of the standard SVM for the same model (see Section \ref{sec:proof_gd}). But, the feasibility threshold of SVM under the group GMM with $K=2$ groups is different from that of Section \ref{sec:PT_CS-SVM} for $K=1$, since now each class is itself a mixture of Gaussians. We derive the desired result from \cite{gkct_multiclass}, who recently studied the separability question for the more general case of a multiclass mixture of mixtures of Gaussians.
\begin{propo} Consider the same data model and notation as in Theorem \ref{thm:main_group_fairness} and consider the event
$$
\mathcal{E}_{{\rm sep}, n}:=\left\{\exists (\mathbf{w},b)\in\mathbb{R}^d\times\mathbb{R}~~\text{s.t.}~~y_i(\mathbf{w}^T\mathbf{x}_i+b)\geq 1,\,~\forall i\in[n]\right\}.
$$
Define threshold $\gamma_\star:=\gamma_\star(\mathbf{V},\mathbf{S},\pi)$ as follows:
\begin{align}
\widetilde\gamma_\star:=\min_{\mathbf{t}\in\mathbb{R}^r, b\in\mathbb{R}}\mathbb{E}\left[\Big(\sqrt{1+\|\mathbf{t}\|_2^2} \,G + \widetilde{E}_S^T\mathbf{V}\mathbf{S}\mathbf{t} - bY\Big)_-^2\right].
\end{align}
Then, the following hold:
\begin{align}
\gamma>\widetilde\gamma_\star \Rightarrow \lim_{n\rightarrow\infty}\mathbb{P}(\mathcal{E}_{{\rm sep}, n}) = 1\qquad\text{ and }\qquad \gamma<\widetilde\gamma_\star \Rightarrow \lim_{n\rightarrow\infty}\mathbb{P}(\mathcal{E}_{{\rm sep}, n}) = 0.\notag
\end{align}
In words, the data are linearly separable with overwhelming probability if and only if $\gamma>\widetilde\gamma_\star$. Furthermore, if this condition holds, then GS-SVM is feasible with overwhelming probability for any value of $\delta>0$.
\end{propo}
\subsection{Algorithms}
\vspace{-0.1in}
\noindent\textbf{Cross-entropy adjustments.}~We introduce the \textbf{vector-scaling (VS) loss}, which combines \emph{both} additive and {{multiplicative}} logit adjustments, previously suggested in the literature in isolation. The following is the \textbf{binary VS-loss} for labels $y\in\{\pm1\}$, weight parameters $\omega_\pm>0$, \blue{additive} logit parameters $\blue{\iota_\pm}\in\mathbb{R}$, and \red{multiplicative} logit parameters $\red{\Delta_\pm}>0$:
\begin{align}\label{eq:loss_ours_bin}
\ell_{\rm VS}(y,f_\mathbf{w}(\mathbf{x})) = \omega_y\cdot\log\left(1+e^{\blue{\iota_y}}\cdot e^{-{\red{\Delta_{y}}} yf_\mathbf{w}(\mathbf{x})}\right).
\end{align}
For imbalanced datasets with $C>2$ classes, the \textbf{VS-loss} takes the following form:
\begin{align}\label{eq:loss_ours}
\ell_{\rm VS}(y,\mathbf{f}_\mathbf{w}(\mathbf{x})) &= -\omega_y \,\log\Big({e^{{\color{red}\Delta_y}\mathbf{f}_y(\mathbf{x})+\blue{\iota_y}}}\big/{\sum_{c\in[C]}e^{{\color{red}\Delta_{c}}\mathbf{f}_{c}(\mathbf{x})+\blue{\iota_{c}}}}\Big).
\end{align}
Here $\mathbf{f}_\mathbf{w}:\mathbb{R}^d\rightarrow\mathbb{R}^C$ and $\mathbf{f}_\mathbf{w}(\mathbf{x})=[\mathbf{f}_1(\mathbf{x}),\ldots,\mathbf{f}_C(\mathbf{x})]$ is the vector of logits. The VS-loss (Eqns. \eqref{eq:loss_ours_bin},\eqref{eq:loss_ours}) captures existing techniques as special cases by tuning accordingly the \blue{additive}/\red{multiplicative} hyperparameters.
Specifically, we recover: (i) \textbf{weighted CE (wCE) loss} by ${\Delta_y}=1,{\iota_y}=0,\omega_y=\pi_y^{-1}$; (ii) \textbf{LA-loss} by ${\Delta_y}=1$; (iii) \textbf{CDT-loss} by ${\iota_y}=0$.
With the goal of (additionally) ensuring fairness with respect to \emph{sensitive groups}, we extend the VS-loss by introducing parameters $(\red{\Delta_{y,g}},\blue{\iota_{y,g}},\omega_{y,g})$ that depend \emph{both} on class and group membership (specified by $y$ and $g$, respectively). Our proposed \textbf{group-sensitive VS-loss} is as follows (multiclass version can be defined accordingly):
\begin{align}\label{eq:loss_ours_bin_group}
\ell_{\rm Group-VS}(y,g,f_\mathbf{w}(\mathbf{x})) = \omega_{y,g}\cdot\log\big(1+e^{\blue{\iota_{y,g}}}\cdot e^{-\red{\Delta_{y,g}} yf_\mathbf{w}(\mathbf{x})}\big).
\end{align}
\noindent\textbf{CS-SVM.}
For linear classifiers $f_\mathbf{w}(\mathbf{x}) = \inp{\mathbf{w}}{h(\mathbf{x})}$ with $h:\mathcal{X}\rightarrow\mathbb{R}^p$, CS-SVM \cite{masnadi2010risk} solve
\begin{align}\label{eq:CS-SVM}
\hspace{-0.1in}\min_{\mathbf{w}}~\|\mathbf{w}\|_2~\text{sub. to} \begin{cases} \inp{\mathbf{w}}{h(\mathbf{x}_i)}\geq\delta &\hspace{-0.05in}, y_i=+1 \\ \inp{\mathbf{w}}{h(\mathbf{x}_i)}\leq -1 &\hspace{-0.05in}, y_i=-1 \end{cases}, i\in[n],
\end{align}
for hyper-parameter $\delta\in\mathbb{R}_+$ representing the ratio of margins between classes. $\delta=1$ corresponds to (standard) SVM, while tuning $\delta > 1$ (resp. $\delta<1$) favors a larger margin $\delta/\|\hat{\w}_\delta\|_2$ for the minority vs $1/\|\hat{\w}_\delta\|_2$ for the majority classes.
Thus, $\delta\rightarrow+\infty$ (resp. $\delta\rightarrow 0$) corresponds to the decision boundary starting right at the boundary of class $y=-1$ (resp. $y=+1$).
\noindent\textbf{Group-sensitive SVM.}~The \emph{group-sensitive} version of CS-SVM (GS-SVM), for $K=2$ protected groups adjusts the constraints in \eqref{eq:CS-SVM} so that $y_i\inp{\mathbf{w}}{h(\mathbf{x}_i)} \geq\delta $ (or $\geq1$), {if} $g_i=1$ (or $g_i=2.$)
$\delta>1$, GS-SVM favors larger margin for the sensitive group $g=1$. Refined versions when classes are also imbalanced modify the constraints to $y_i\inp{\mathbf{w}}{\mathbf{h}(\mathbf{x}_i)}\geq \delta_{y_i,g_i}$.
{Both CS-SVM and GS-SVM are feasible iff data are linearly separable (see SM). However, we caution that the GS-SVM hyper-parameters are in general harder to interpret as ``margin-ratios".
\section{Asymptotic analysis of CS-SVM}\label{sec:cs-svm_proof}
\subsection{Preliminaries}
The main goal of this appendix is proving Theorem \ref{thm:main_imbalance}. For fixed $\delta>0$, let $(\hat{\w},\hat b)$ be the solution to the CS-SVM in \eqref{eq:CS-SVM}. (See also \eqref{eq:CS-SVM_con} below.) In the following sections, we will prove the following convergence properties for the solution of the CS-SVM:
\begin{align}\label{eq:2prove_app}
(\|\hat\mathbf{w}\|_2, \frac{\hat{\w}^T\boldsymbol\mu_+}{\|\hat{\w}\|_2}, \frac{\hat{\w}^T\boldsymbol\mu_-}{\|\hat{\w}\|_2} , \hat b) \stackrel{{P}}{\longrightarrow} (q_\delta,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta,\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta, b_\delta).
\end{align}
where the triplet $(q_\delta,\boldsymbol\rho_\delta,b_\delta)$ is as defined in the theorem's statement, that is, the unique triplet satisfying
\begin{align}\label{eq:eta_eq_main}
\eta_\delta(q_\delta,\boldsymbol\rho_\delta,b_\delta)= 0\quad\text{and}\quad
(\boldsymbol\rho_\delta,b_\delta):=\arg\min_{\substack{\|\boldsymbol\rho\|_2\leq 1, b\in\mathbb{R}}}\eta_\delta(q_\delta,\boldsymbol\rho,b).
\end{align}
In this section, we show how to use \eqref{eq:2prove_app} to derive the asymptotic limit of the conditional class probabilities.
Consider the class conditional
$\mathcal{R}_+=\mathbb{P}\left\{ (\mathbf{x}^T\hat{\w}+b)<0 \,|\, y=+1 \right\}$. Recall that conditioned on $y=+1$, we have $\mathbf{x}=\boldsymbol\mu_++\mathbf{z}$ for $\mathbf{z}\sim\mathcal{N}(\mathbf{0},\mathbf{I}).$ Thus, the class conditional can be expressed explicitly in terms of the three summary quantities on the left hand side of \eqref{eq:2prove_app} as follows:
\begin{align*}
\mathcal{R}_+ &=\mathbb{P}\left\{ (\mathbf{x}^T\hat{\w}+\hat b)<0 \,|\, y=+1 \right\} = \mathbb{P}\left\{ \mathbf{z}^T\hat{\w}+ \boldsymbol\mu_+^T\hat{\w}+\hat b<0 \,|\, y=+1 \right\}\notag
\\
&=\mathbb{P}\left\{ \mathbf{z}^T\hat{\w} > \boldsymbol\mu_+^T\hat{\w}+\hat b \right\}
\\
&= \mathbb{P}_{G\sim\mathcal{N}(0,1)}\left\{ G\|\hat{\w}\|_2 > \boldsymbol\mu_+^T\hat{\w}+\hat b \right\} = \mathbb{P}_{G\sim\mathcal{N}(0,1)}\left\{ G > \frac{\boldsymbol\mu_+^T\hat{\w}}{\|\hat{\w}\|_2}+\frac{\hat b}{\|\hat{\w}\|_2} \right\}\\
&= Q\left(\frac{\boldsymbol\mu_+^T\hat{\w}}{\|\hat{\w}\|_2}+\frac{\hat b}{\|\hat{\w}\|_2}\right).
\end{align*}
Then, the theorem's statement follows directly by applying \eqref{eq:2prove_app} in the expression above.
In order to prove the key convergence result in \eqref{eq:2prove_app} we rely on the convex Gaussian min-max theorem (CGMT) framework. We give some necessary background before we proceed with the proof.
\subsection{Background and related literature}\label{sec:HD_lit}
\noindent\textbf{Related works:}~Our asymptotic analysis of the CS-SVM fits in the growing recent literature on sharp statistical performance asymptotics of convex-based estimators, e.g. \cite{bean2013optimal,montanari13,thrampoulidis2018precise,thrampoulidis2018symbol} and references therin. The origins of these works trace back to the study of sharp phase transitions in compressed sensing, e.g. see \cite{thrampoulidis2018precise} for historical remarks
and performance analysis of the LASSO estimator for sparse signal recovery. That line of work led to the development of two analysis frameworks: (a) the approximate message-passing (AMP) framework \cite{BayMon,AMPmain}, and, (b) the convex Gaussian min-max theorem (CGMT) framework \cite{StojLAS,thrampoulidis2015regularized}. More recently, these powerful tools have proved very useful for the analysis of linear classifiers \cite{salehi2019impact,montanari2019generalization,deng2019model,gkct_dd2020,mignacco2020role,liang2020precise,taheri2020sharp,candes2020phase,aubin2020generalization,taheri2020fundamental}. Theorems \ref{thm:main_imbalance} and \ref{thm:main_group_fairness} rely on the CGMT and contribute to this line of work. Specifically, our results are most closely related to \cite{deng2019model} who first studied max-margin type classifiers together with \cite{montanari2019generalization}.
\vspace{5pt}
\noindent\textbf{CGMT framework:}~Specifically, we rely on the CGMT framework. Here, we only
summarize the framework's essential ideas and refer the reader to \cite{thrampoulidis2015regularized,thrampoulidis2018precise} for more details and precise statements. Consider the following two Gaussian processes:
\begin{subequations}\label{eq:POAO}
\begin{align}
X_{\mathbf{w},\mathbf{u}} &:= \mathbf{u}^T \mathbf{A} \mathbf{w} + \psi(\mathbf{w},\mathbf{u}),\\
Y_{\mathbf{w},\mathbf{u}} &:= \norm{\mathbf{w}}_2 \mathbf{h}_n^T \mathbf{u} + \norm{\mathbf{u}}_2 \mathbf{h}_d^T \mathbf{w} + \psi(\mathbf{w},\mathbf{u}),\label{eq:AO_obj}
\end{align}
\end{subequations}
where: $\mathbf{A}\in\mathbb{R}^{n\times d}$, $\mathbf{h}_n \in \mathbb{R}^n$, $\mathbf{h}_d\in\mathbb{R}^d$, they all have entries iid Gaussian; the sets $\mathcal{S}_{\mathbf{w}}\subset\mathbb{R}^d$ and $\mathcal{S}_{\mathbf{u}}\subset\mathbb{R}^n$ are compact; and, $\psi: \mathbb{R}^d\times \mathbb{R}^n \to \mathbb{R}$. For these two processes, define the following (random) min-max optimization programs, which are refered to as the \emph{primary optimization} (PO) and the \emph{auxiliary optimization} (AO) problems:
\begin{subequations}
\begin{align}\label{eq:PO_loc}
\Phi(\mathbf{A})&=\min\limits_{\mathbf{w} \in \mathcal{S}_{\mathbf{w}}} \max\limits_{\mathbf{u}\in\mathcal{S}_{\mathbf{u}}} X_{\mathbf{w},\mathbf{u}},\\
\label{eq:AO_loc}
\phi(\mathbf{h}_n,\mathbf{h}_d)&=\min\limits_{\mathbf{w} \in \mathcal{S}_{\mathbf{w}}} \max\limits_{\mathbf{u}\in\mathcal{S}_{\mathbf{u}}} Y_{\mathbf{w},\mathbf{u}}.
\end{align}
\end{subequations}
According to the first statement of the CGMT Theorem 3 in \cite{thrampoulidis2015regularized} (this is only a slight reformulation of Gordon's original comparison inequality \cite{Gor2}), for any $c\in\mathbb{R}$, it holds:
\begin{equation}\label{eq:gmt}
\mathbb{P}\left\{ \Phi(\mathbf{A}) < c\right\} \leq 2\, \mathbb{P}\left\{ \phi(\mathbf{h}_n,\mathbf{h}_d) < c \right\}.
\end{equation}
In other words, a high-probability lower bound on the AO is a high-probability lower bound on the PO. The premise is that it is often much simpler to lower bound the AO rather than the PO.
However, the real power of the CGMT comes in its second statement, which asserts that if the PO is \emph{convex} then the AO in can be used to tightly infer properties of the original PO, including the optimal cost and the optimal solution.
More precisely, if the sets $\mathcal{S}_{\mathbf{w}}$ and $\mathcal{S}_{\mathbf{u}}$ are convex and \emph{bounded}, and $\psi$ is continuous \emph{convex-concave} on $\mathcal{S}_{\mathbf{w}}\times \mathcal{S}_{\mathbf{u}}$, then, for any $\nu \in \mathbb{R}$ and $t>0$, it holds \cite{thrampoulidis2015regularized}:
\begin{equation}\label{eq:cgmt}
\mathbb{P}\left\{ \abs{\Phi(\mathbf{A})-\nu} > t\right\} \leq 2\, \mathbb{P}\left\{ \abs{\phi(\mathbf{h}_n,\mathbf{h}_d)-\nu} > t \right\}.
\end{equation}
In words, concentration of the optimal cost of the AO problem around $q^\ast$ implies concentration of the optimal cost of the corresponding PO problem around the same value $q^\ast$. Asymptotically, if we can show that $\phi(\mathbf{h}_n,\mathbf{h}_d)\stackrel{{P}}{\longrightarrow} q^\ast$, then we can conclude that $\Phi(\mathbf{A})\stackrel{{P}}{\longrightarrow} q^\ast$.
In the next section, we will show that we can indeed express the CS-SVM in \eqref{eq:CS-SVM} as a PO in the form of \eqref{eq:PO_loc}. Thus, the argument above will directly allow us to determine the asymptotic limit of the optimal cost of the CS-SVM. In our case, the optimal cost equals $\|\hat{\w}\|_2$; thus, this shows the first part of \eqref{eq:2prove_app}. For the other parts, we will employ the following ``deviation argument" of the CGMT framework \cite{thrampoulidis2015regularized}. For arbitrary $\epsilon>0$, consider the desired set
\begin{align}\label{eq:Sc}
{\mathcal{S}}:=\left\{ (\mathbf{v},c)~\Big|~ \max\Big\{\abs{\|\mathbf{v}\|_2- q_\delta}\,,\, \Big|\frac{\mathbf{v}^T\boldsymbol\mu_+}{\|\mathbf{v}\|_2} - \mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta\Big|,,\,\Big|\frac{\mathbf{v}^T\boldsymbol\mu_-}{\|\mathbf{v}\|_2} - \mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta\Big|
\,,\, |c- b_\delta| \Big\}\leq\epsilon\, \right\}.
\end{align}
Our goal towards \eqref{eq:2prove_app} is to show that with overwhelming probability $(\mathbf{w},b)\in{\mathcal{S}}$. For this, consider the following constrained CS-SVM that further constraints the feasible set to the complement ${\mathcal{S}}^c$ of ${\mathcal{S}}$:
\begin{align}\label{eq:CS-SVM_con}
\hspace{-0.1in}\Phi_{{\mathcal{S}}^c}(\mathbf{A}):=\min_{(\mathbf{w},b)\in{\mathcal{S}}^c}~\|\mathbf{w}\|_2~\text{sub. to} \begin{cases} \mathbf{w}^T\mathbf{x}_i+b \geq\delta &\hspace{-0.05in}, y_i=+1 \\ \mathbf{w}^T\mathbf{x}_i+b\leq -1 &\hspace{-0.05in}, y_i=-1 \end{cases}, i\in[n],
\end{align}
As per Theorem 6.1(iii) in \cite{thrampoulidis2018precise} it will suffice to find costants $\bar\phi, \bar\phi_S$ and $\eta>0$ such that the following three conditions hold:
\begin{align}\label{eq:cgmt_conditions}
\begin{cases}
\text{(i)~~\,\, $\bar\phi_S \geq \bar\phi + 3\eta$}\\
\text{(ii)~~ \,$\phi(\mathbf{h}_n,\mathbf{h}_d) \leq \bar\phi + \eta$~~ with overwhelming probability}\\
\text{(iii)~~ $\phi_{{\mathcal{S}}^c}(\mathbf{h}_n,\mathbf{h}_d) \geq \bar\phi_S - \eta$~~ with overwhelming probability,}
\end{cases}
\end{align}
where $\phi_{{\mathcal{S}}^c}(\mathbf{h}_n,\mathbf{h}_d)$ is the optimal cost of the constrained AO corresponding to the constrained PO in \eqref{eq:CS-SVM_con}.
To prove these conditions for the AO of the CS-SVM, in the next section we follow the principled machinery of \cite{thrampoulidis2018precise} that allows simplifying the AO from a (random) optimization over vector variables to an easier optimization over only few scalar variables, termed the ``scalarized AO".
\subsection{Proof of Theorem \ref{thm:main_imbalance}}\label{sec:cs_proof_app}
Let $(\hat{\w},\hat{b})$ be solution pair to the CS-SVM in \eqref{eq:CS-SVM} for some fixed margin-ratio parameter $\delta>0$, which we rewrite here expressing the constraints in matrix form:
\begin{align}\label{eq:CS-SVM_app0}
\min_{\mathbf{w},b}\|\mathbf{w}\|_2~~\text{sub. to}~\begin{cases} \mathbf{w}^T\mathbf{x}_i+b \geq\delta ,~y_i=+1 \\ -(\mathbf{w}^T\mathbf{x}_i+b)\geq 1,~y_i=-1 \end{cases}\hspace{-10pt},~i\in[n]~~=~~ \min_{\mathbf{w},b}\|\mathbf{w}\|_2~~\text{sub. to}~\mathbf{D}_\mathbf{y}(\mathbf{X}\mathbf{w}+b\mathbf{1}_n)\geq \boldsymbol\delta_\mathbf{y},
\end{align}
where we have used the notation
\begin{align*}
\mathbf{X}^T&=\begin{bmatrix} \mathbf{x}_1 & \cdots & \mathbf{x}_n \end{bmatrix},~\mathbf{y}=\begin{bmatrix} y_1 & \cdots & y_n \end{bmatrix}^T,
\\\mathbf{D}_\mathbf{y}&=\rm{diag}(\mathbf{y})~\text{and}~\boldsymbol\delta_\mathbf{y}=\begin{bmatrix}
\delta\ind{y_1=+1}+\ind{y_1=-1}& \cdots &\delta\ind{y_n=+1}+\ind{y_n=-1}
\end{bmatrix}^T.
\end{align*}
We further need to define the following one-hot-encoding of the labels:
$$
\mathbf{y}_i = \mathbf{e}_1 \ind{y_i=1} + \mathbf{e}_2 \ind{y_i=-1},\quad\text{and}\quad \mathbf{Y}_{n\times 2}^T=\begin{bmatrix}\mathbf{y}_1 & \cdots & \mathbf{y}_n \end{bmatrix}.
$$
where recall that $\mathbf{e}_1,\mathbf{e}_2$ are standard basis vectors in $\mathbb{R}^2$.
With these,
notice for later use that under our model, $\mathbf{x}_i=\boldsymbol\mu_{y_i}+\mathbf{z}_i = \mathbf{M}\mathbf{y}_i+\mathbf{z}_i,~\mathbf{z}_i\sim\mathcal{N}(0,1)$. Thus, in matrix form with $\mathbf{Z}$ having entries $\mathcal{N}(0,1)$:
\begin{align}\label{eq:model_cs}
\mathbf{X} = \mathbf{Y}\mathbf{M}^T+\mathbf{Z}.
\end{align}
Following the CGMT strategy \cite{thrampoulidis2015regularized}, we express \eqref{eq:CS-SVM_app0} in a min-max form to bring it in the form of the PO as follows:
\begin{align}
&\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{X}\mathbf{w}+b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{y}\notag\\
=&\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{Z}\mathbf{w} + \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{y}.\label{eq:cs_PO}
\end{align}
where in the last line we used \eqref{eq:model_cs} and $\mathbf{D}_\mathbf{y}\mathbf{D}_\mathbf{y}=\mathbf{I}_n$.
We immediately recognize that the last optimization is in the form of a PO (cf. \eqref{eq:PO_loc}) and the corresponding AO (cf. \eqref{eq:AO_loc}) is as follows:
\begin{align}\label{eq:cs_AO}
\min_{\mathbf{w},b}\max_{\mathbf{u}\leq0}~ \frac{1}{2}\|\mathbf{w}\|_2^2+\|\mathbf{w}\|_2\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{h}_n + \|\mathbf{D}_\mathbf{y}\mathbf{u}\|_2\mathbf{h}_d^T\mathbf{w} + \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{y}.
\end{align}
where $\mathbf{h}_n\sim\mathcal{N}(0,\mathbf{I}_n)$ and $\mathbf{h}_d\sim\mathcal{N}(0,\mathbf{I}_d)$.
In order to apply the CGMT in \cite{thrampoulidis2015regularized}, we need boundedness of the constraint sets. Thus, we restrict the minimization in \eqref{eq:cs_AO} and \eqref{eq:cs_PO} to a bounded set $\|\mathbf{w}\|_2^2+b^2\leq R$ for (say) $R:=2\left( q_\delta^2+ b_\delta^2\right)$. This will allow us to show that the solutions $\hat{\w}_R,\hat b_R$ of this constrained PO satisfy $\hat{\w}_R\stackrel{{P}}{\longrightarrow} q_\delta$ and $\hat b_R\stackrel{{P}}{\longrightarrow} b_\delta$. Thus, with overwhelming probability, $\|\hat{\w}_R\|_2^2+{\hat b}_R^2 < R$. From this and convexity of the PO, we can argue that the minimizers $\hat{\w},\hat b$ of the original unconstrained problem satisfy the same convergence properties. Please see also Remark 4 in App. A of \cite{deng2019model}.
For the maximization, we follow the recipe in App. A of \cite{deng2019model} who analyzed the standard SVM. Specifically, combining Remark 3 of \cite{deng2019model} together with (we show this next) the property that the AO is reduced to a convex program, it suffices to consider the unconstrained maximization.
Thus, in what follows we consider the one-sided constrained AO in \eqref{eq:cs_AO}. Towards simplifying this auxiliary optimization, note that $\mathbf{D}_\mathbf{y}\mathbf{h}_n\sim\mathbf{h}_n$ by rotational invariance of the Gaussian measure. Also, $\|\mathbf{D}_\mathbf{y}\mathbf{u}\|_2=\|\mathbf{u}\|_2$. Thus, we can express the AO in the following more convenient form:
\begin{align}\label{eq:cs_AO_11}
&\min_{\|\mathbf{w}\|_2^2+b^2\leq R}\max_{\mathbf{u}\leq0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \|\mathbf{w}\|_2\mathbf{u}^T\mathbf{h}_n + \|\mathbf{u}\|_2\mathbf{h}_d^T\mathbf{w} + \mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{M}^T\mathbf{w} +b(\mathbf{u}^T\mathbf{D}_\mathbf{y}\mathbf{1}_n)-\mathbf{u}^T\boldsymbol\delta_\mathbf{y}.
\end{align}
We are now ready to proceed with simplification of the AO. First we optimize over the direction of $\mathbf{u}$ and rewrite the AO as
\begin{align*}
&\min_{\|\mathbf{w}\|_2^2+b^2\leq R}\max_{\beta\geq 0}~\frac{1}{2}\|\mathbf{w}\|_2^2+ \beta\left( \Big\| \big( \, \|\mathbf{w}\|_2\mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{M}^T\mathbf{w}+b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2 -\mathbf{h}_d^T\mathbf{w} \right)
\\
=&\min_{\|\mathbf{w}\|_2^2+b^2\leq R}~\frac{1}{2}\|\mathbf{w}\|_2^2\quad\text{sub. to}~\Big\| \big( \, \|\mathbf{w}\|_2\mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{M}^T\mathbf{w}+b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2 \leq \mathbf{h}_d^T\mathbf{w}.
\end{align*}
Above, $(\cdot)_-$ acts elementwise to the entries of its argument.
Now, we wish to further simplify the above by minimizing over the direction of $\mathbf{w}$ in the space orthogonal to $\mathbf{M}$. To see how this is possible consider the SVD $\mathbf{M}^T=\mathbf{V}\mathbf{S}\mathbf{U}^T$ and project $\mathbf{w}$ on the columns of $\mathbf{U}=\begin{bmatrix}
\mathbf{u}_1 & \mathbf{u}_2\end{bmatrix}\in\mathbb{R}^{d\times 2}$ as follows:
$$
\mathbf{w} = \mathbf{u}_1(\mathbf{u}_1^T\mathbf{w}) + \mathbf{u}_2(\mathbf{u}_2^T\mathbf{w}) + \mathbf{w}^\perp,
$$
where $\mathbf{w}^\perp=\mathbf{U}^\perp\mathbf{w}$, $\mathbf{U}^\perp$ is the orthogonal complement of $\mathbf{U}$. For simplicity we will assume here that $\mathbf{M}$ is full column rank, i.e. $\mathbf{S}\succ\mathbf{0}_{2\times 2}$. The argument for the case where $\mathbf{M}$ is rank 1 is very similar.
Let us denote $\mathbf{u}_i^T\mathbf{w}:=\mu_i, i=1,2$ and $\|\mathbf{w}^\perp\|_2:=\alpha$. In this notation, the AO becomes
\begin{align*}
&\min_{\mu_1^2+\mu_2^2+\|\mathbf{w}^\perp\|_2^2+b^2\leq R}~\frac{1}{2}(\mu_1^2+\mu_2^2+\alpha^2)\quad\\
&\text{sub. to}\quad\Big\| \big( \, \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2 \end{bmatrix} +b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2\\ &\quad\quad\quad\leq \mu_1(\mathbf{h}_d^T\mathbf{u}_1) + \mu_2(\mathbf{h}_d^T\mathbf{u}_2) + \mathbf{h}_d^T\mathbf{U}^\perp\mathbf{w}^\perp.
\end{align*}
At this point, we can optimize over the direction of $\mathbf{w}^\perp$ which leads to
\begin{align*}
&\min_{ \mu_1^2+\mu_2^2+\alpha^2+b^2\leq R }~\frac{1}{2}(\mu_1^2+\mu_2^2+\alpha^2)\quad
\\
&\text{sub. to}\quad\Big\| \big( \, \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2 \end{bmatrix} +b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2\\ &\quad\quad\quad \leq \mu_1(\mathbf{h}_d^T\mathbf{u}_1) + \mu_2(\mathbf{h}_d^T\mathbf{u}_2) + \alpha\|\mathbf{h}_d^T\mathbf{U}^\perp\|_2.
\end{align*}
As a last step in the simplification of the AO, it is convenient to introduce an additional variable
$
q=\sqrt{\mu_1^2+\mu_2^2+\alpha^2}.
$
It then follows that the minimization above is equivalent to the following
\begin{align}\label{eq:cs_AO_scal_cvx}
&\min_{ \substack{ q\geq \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \\ q^2+b^2\leq R} }~\frac{1}{2}q^2\quad
\\
&\text{sub. to}\quad\Big\| \big( \, q \mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2 \end{bmatrix} +b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2 \leq \mu_1(\mathbf{h}_d^T\mathbf{u}_1) + \mu_2(\mathbf{h}_d^T\mathbf{u}_2) + \alpha\|\mathbf{h}_d^T\mathbf{U}^\perp\|_2.\notag
\end{align}
In this formulation it is not hard to check that the optimization is jointly convex in its variables $(\mu_1,\mu_2,\alpha,b,q)$. To see this note that: (i) the constraint $q\geq \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \iff q\geq \|\begin{bmatrix} \mu_1 & \mu_2 & \alpha \end{bmatrix}\|_2$ is a second-order cone constraint, and, (ii) the function
\begin{align}\label{eq:Ln}
\mathcal{L}_n(q,\mu_1,\mu_2,\alpha,b)&:=\frac{1}{\sqrt{n}}\Big\| \big( \, q \mathbf{h}_n + \mathbf{D}_\mathbf{y}\mathbf{Y}\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2\end{bmatrix} +b\, \mathbf{D}_\mathbf{y}\mathbf{1}_n- \boldsymbol\delta_{\mathbf{y}} \, \big)_- \Big\|_2\notag\\ &\quad- \mu_1\frac{\mathbf{h}_d^T\mathbf{u}_1}{\sqrt{n}} - \mu_2\frac{\mathbf{h}_d^T\mathbf{u}_2}{\sqrt{n}} - {\alpha}\frac{\|\mathbf{h}_d^T\mathbf{U}^\perp\|_2}{\sqrt{n}}
\end{align}
is also convex since $\|(\cdot)_-\|_2:\mathbb{R}^n\rightarrow\mathbb{R}$ is itslef convex and is composed here with an affine function.
Now, by law of large numbers, notice that for fixed $(q,\mu_1,\mu_2,\alpha,b)$, $\mathcal{L}_n$ converges in probability to
\begin{align}\label{eq:Ln2L_cs}
\mathcal{L}_n(q,\mu_1,\mu_2,\alpha,b)\stackrel{{P}}{\longrightarrow} L(q,\mu_1,\mu_2,\alpha,b):=\sqrt{\mathbb{E}\big(qG+E_{Y}^T\mathbf{V}\mathbf{S}\begin{bmatrix}\mu_1 \\ \mu_2\end{bmatrix} + {b\,Y-\Delta_Y}\big)_{-}^2} - \alpha\sqrt{\gamma},
\end{align}
where the random variables $G,E_Y,Y,\Delta_Y$ are as in the statement of the theorem. But convergence of convex functions is uniform over compact sets as per Cor.~II.I in \cite{andersen1982cox}. Therefore, the convergence in \eqref{eq:Ln2L_cs} is in fact uniform in the compact feasible set of \eqref{eq:cs_AO_scal_cvx}.
Consider then the deterministic high-probability equivalent of \eqref{eq:cs_AO_scal_cvx} which is the following convex program:
\begin{align}
&\min_{ \substack{ q\geq \sqrt{\mu_1^2+\mu_2^2+\alpha^2} \\ q^2+b^2\leq R \\ L(q,\mu_1,\mu_2,\alpha,b)\leq 0} }~\frac{1}{2}q^2\notag.
\end{align}
Since $q$ is positive and the constraint $q\geq\sqrt{\mu_1^2+\mu_2^2+\alpha^2}$ must be active at the optimum, it is convenient to rewrite this in terms of new variables
$
\boldsymbol\rho = \begin{bmatrix}\boldsymbol\rho_1\\\boldsymbol\rho_2 \end{bmatrix}:=\begin{bmatrix}\mu_1/q \\ \mu_2/q\end{bmatrix}
$
as follows:
\begin{align}\label{eq:cs_AO_scal_det2}
&\min_{ q^2+b^2 \leq R, q>0, \|\boldsymbol\rho\|_2\leq 1 }~\frac{1}{2}q^2
\\
&\text{sub. to}\quad {\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\,Y-\Delta_Y}{q}\big)_{-}^2\Big]} \notag \leq \left({1-\|\boldsymbol\rho\|_2^2}\right){\gamma}.
\end{align}
Now, recall the definition of the function $ \eta_\delta$ in the statement of the theorem and observe that the constraint above is nothing but
$$
\eta_\delta(q,\boldsymbol\rho,b)\leq 0.
$$
Thus, \eqref{eq:cs_AO_scal_det2} becomes
\begin{align}\label{eq:cs_AO_scal_det3}
\min\left\{q^2~\Big|~ 0\leq q\leq \sqrt{R} \quad\text{ and }\quad \min_{b^2 \leq R-q^2, \|\boldsymbol\rho\|_2\leq 1} \eta_\delta(q,\boldsymbol\rho,b)\leq 0 \right\}.
\end{align}
We will prove that
\begin{align}\label{eq:eta_dec_cs}
\text{the function $f(q):=\min_{b,\|\boldsymbol\rho\|_2\leq1} \eta_\delta(q,\boldsymbol\rho,b)$ is strictly decreasing}.
\end{align}
Before that, let us see how this completes the proof of the theorem. Let $ q_\delta$ be as in the statement of the theorem, that is such that $f( q_\delta)=0$. Then, we have the following relations
$$
f(q)\leq 0 ~\Rightarrow~ f(q) \leq f( q_\delta) ~\Rightarrow~ q \geq q_\delta.
$$
Thus, the minimizers in \eqref{eq:cs_AO_scal_det3} are $( q_\delta, \boldsymbol\rho_\delta, b_\delta)$, where we also recall that we have set $R> q_\delta^2 + b_\delta^2$.
With all these, we have shown that the AO converges in probability to $ q_\delta^2$ (cf. condition (ii) in \eqref{eq:cgmt_conditions}). From the CGMT, the same is true for the PO. Now, we want to use the same machinery to prove that the minimizers $(\hat{\w},\hat b)$ of the PO satisfy \eqref{eq:2prove_app}.
To do this, as explained in the previous section, we use the standard strategy of the CGMT framework , i.e., to show that the PO with the additional constraint
$
(\mathbf{w},b)\in{\mathcal{S}}^c
$
for the set ${\mathcal{S}}$ in \eqref{eq:Sc} has a cost that is strictly larger than $q_\delta^2$ (i.e. the cost of the unconstrained PO). As per the CGMT this can be done again by showing that the statement is true for the correspondingly constrained AO (i.e. show condition (iii) in \eqref{eq:cgmt_conditions}). With the exact same simplifications as above, the latter program simplifies to \eqref{eq:cs_AO_scal_cvx} with the additional constraints:
$$
|q- q_\delta|>\epsilon\,,\, \big|\mu_i/q - \boldsymbol\rho_{\delta,i}\big|>\epsilon,\, i=1,2\,,\, |b- b_\delta|>\epsilon.
$$
Also, using the uniform convergence in \eqref{eq:Ln2L_cs}, it suffices to study the deterministic equivalent \eqref{eq:cs_AO_scal_det3} with the additional constraints above. Now, we can show the desired (cf. condition (i) in \eqref{eq:cgmt_conditions}) again by exploiting \eqref{eq:eta_dec_cs}. This part of the argument is similar to Section C.3.5 in \cite{deng2019model} and we omit the details.
\noindent\underline{Proof of \eqref{eq:eta_dec_cs}:}~To complete the proof, it remains to show \eqref{eq:eta_dec_cs}. Specifically, we show that $\frac{\mathrm{d}f}{\mathrm{d}q}<0$ by combining the following three observations.
First,
\begin{align}\notag
\frac{\mathrm{\partial} \eta_\delta}{\mathrm{\partial}q} &= \frac{2}{q^2}\mathbb{E}\Big[ (G+ E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\, Y- \Delta_Y}{q}\big)_{-}\cdot \Delta_Y \Big] - \frac{2b}{q^2}\mathbb{E}\Big[ (G+ E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\, Y- \Delta_Y}{q}\big)_{-}\cdot Y \Big]
\\
&<
- \frac{2b}{q^2}\mathbb{E}\Big[ (G+ E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\, Y- \Delta_Y}{q}\big)_{-}\cdot Y \Big]\label{eq:gs_arg1}
\end{align}
where for the inequality we observed that $(\cdot)_-$ is always non-positive, its argument has non-zero probability measure on the negative real axis, and $ \Delta_Y$ are positive random variables.
Second, letting $\boldsymbol\rho^\star:=\boldsymbol\rho^\star(q)$ and $b^\star:=b^\star(q)$ the minimizers of $ \eta_\delta(q,\boldsymbol\rho,b)$, it follows from first-order optimality conditions that
\begin{align}\label{eq:gs_arg2}
\frac{\mathrm{\partial} \eta_\delta}{\mathrm{\partial}b} = 0 \iff \mathbb{E}\Big[ (G+ E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho^* + \frac{b^*\, Y- \Delta_Y}{q}\big)_{-}\cdot Y \Big] = 0.
\end{align}
Third, by the envelope theorem
\begin{align}\label{eq:gs_arg3}
\frac{\mathrm{d}f}{\mathrm{d}q} = \frac{\mathrm{\partial} \eta_\delta}{\mathrm{\partial}q}\big|_{\boldsymbol\rho^\star,b^\star}.
\end{align}
The desired inequality $\frac{\mathrm{d}f}{\mathrm{d}q}<0$ follows directly by successively applying \eqref{eq:gs_arg3}, \eqref{eq:gs_arg1} and \eqref{eq:gs_arg2}.
\noindent\textbf{Uniqueness of triplet $(q_\delta,\boldsymbol\rho_\delta,b_\delta)$.}~ First, we prove that the minimizers $\boldsymbol\rho_\delta,b_\delta$ are unique. This follows because $\eta_\delta(q,\boldsymbol\rho,b)$ is jointly strictly convex in $(\boldsymbol\rho,b)$ for fixed $q$. To see this note that the function $x\mapsto (x)_-^2$ is strictly convex for $x<0$ and that the random variable $G+E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho+(bY-\Delta_Y)/q$ has strictly positive measure on the real line (thus, also in the negative axis). Next, consider $q_\delta$, which was defined such that $f(q_\delta)=0$ for the function $f(\cdot)$ in \eqref{eq:eta_dec_cs}. From \eqref{eq:eta_dec_cs} we know that $f(\cdot)$ is strictly decreasing. Thus, it suffices to prove that the function has a zero crossing in $(0,\infty)$, which we do by proving $\lim_{q\rightarrow0} f(q)=\infty$ and $\lim_{q\rightarrow\infty} f(q)<0.$ Specifically, we have
\begin{align*}
\lim_{q\rightarrow0} f(q) &\geq \lim_{q\rightarrow0} \,\min_{b\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\,Y-\Delta_Y}{q}\big)_{-}^2\Big]}-\gamma\notag\\
&\geq \lim_{q\rightarrow0}\, \min_{b\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\,Y-\Delta_Y}{q}\big)_{-}^2\,\ind{G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho +(b/q)Y\leq 0}\Big]}-\gamma\\
&\geq \lim_{q\rightarrow0}\, \min_{b\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{1/q^2}-\gamma = \infty,
\end{align*}
where in the last inequality we used the facts that $x\mapsto (x)_-^2$ is decreasing and the event $\{G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho +(b/q)Y\leq 0\leq 0\}$ has non-zero measure for all $\|\boldsymbol\rho\|_2\leq 1, b\in \mathbb{R}$, as well as, $\Delta_Y\geq 1$ (because $\delta>1$). Moreover,
\begin{align*}
\lim_{q\rightarrow\infty} f(q) &= \lim_{1/q\rightarrow 0^+} f(q) = \lim_{1/q\rightarrow 0^+} \,\min_{b\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b\,Y-\Delta_Y}{q}\big)_{-}^2\Big]}-(1-\|\boldsymbol\rho\|_2^2)\gamma\\
&=\lim_{1/q\rightarrow 0^+} \,\min_{\tilde{b}\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \tilde{b} Y - \frac{\Delta_Y}{q}\big)_{-}^2\Big]}-(1-\|\boldsymbol\rho\|_2^2)\gamma\\
&\leq\min_{\tilde{b}\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}{\mathbb{E}\Big[\big(G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \tilde{b} Y\big)_{-}^2\Big]}-(1-\|\boldsymbol\rho\|_2^2)\gamma\\
&\leq\min_{\tilde{b}\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}(1-\|\boldsymbol\rho\|_2^2)\cdot\Big( {\mathbb{E}\Big[\Big((G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \tilde{b} Y)/\sqrt{1-\|\boldsymbol\rho\|_2^2}\Big)_{-}^2\Big]}-\gamma\Big) \\
&\leq\min_{\tilde{b}\in\mathbb{R}, \|\boldsymbol\rho\|_2\leq1}\, {\mathbb{E}\Big[\Big((G+E_{Y}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \tilde{b} Y)/\sqrt{1-\|\boldsymbol\rho\|_2^2}\Big)_{-}^2\Big]}-\gamma \\
&\leq\min_{\breve{b}\in\mathbb{R}, \mathbf{t}\in\mathbb{R}^r}\, {\mathbb{E}\Big[\big(\sqrt{1+\|\mathbf{t}\|_2^2}\,G+E_{Y}^T\mathbf{V}\mathbf{S}\mathbf{t} + \breve{b} Y\big)_{-}^2\Big]}-\gamma=\gamma_\star-\gamma < 0,
\end{align*}
where to get the penultimate inequality we used the change of variables $\mathbf{t}=\boldsymbol\rho/\sqrt{1-\|\boldsymbol\rho\|_2^2}$ and $\breve{b}=\tilde{b}/\sqrt{1-\|\boldsymbol\rho\|_2^2}$. Also, in the last line above, we used the definition of the phase-transition threshold $\gamma_\star$ in Equation \eqref{eq:gamma_star_CS} and the theorem's assumption that $\gamma>\gamma_\star$ (aka separable regime).
We note that similar uniqueness argument was presented in \cite{deng2019model} for the special case of antipodal means, \emph{no} intercept and $\delta=1.$
\subsection{Antipodal means and non-isotropic data}
\noindent\textbf{Antipodal means.}~In the special case of antipodal means of equal energy $\boldsymbol\mu_+=-\boldsymbol\mu_-=\boldsymbol\mu$ with $s:=\|\boldsymbol\mu\|_2$, the formulas of Theorem \ref{thm:main_imbalance} simplify as we have $r=1$ with $\mathbf{S}=s\sqrt{2}$ and $\mathbf{V}=[1/\sqrt{2}\,,\,-1\sqrt{2}]^T$. Now, the function $\eta_\delta$ can be written as
$
\mathbb{E}\big[(G+\tilde \rho s + \frac{\tilde b}{\tilde q}Y-\frac{1}{\tilde q}\Delta_Y)_{-}^2\big] \notag- \left(1-{\tilde \rho}^2\right)\gamma.
$
The asymptotic performance of SVM for this special geometry of the means has been recently studied in \cite{deng2019model,mignacco2020role}. We extend this to the CS-SVM classifier, to general means for the two classes and to $\boldsymbol\Sigma\neq\mathbf{I}$.
\noindent\textbf{Non-isotropic data.}~We show how Theorem \ref{thm:main_imbalance} for the isotropic case can still be applied in the general case $\boldsymbol\Sigma\neq \mathbf{I}$. Assume $\boldsymbol\Sigma\succ0$. Write $\mathbf{x}_i=y_i\boldsymbol\mu_{y_i}+\boldsymbol\Sigma^{1/2}\mathbf{h}_i$ for $\mathbf{h}_i\sim\mathcal{N}(0,\mathbf{I}_d).$ Consider whitened features $\mathbf{z}_i:=\boldsymbol\Sigma^{-1/2}\mathbf{x}_i = y_i\boldsymbol\Sigma^{-1/2}\boldsymbol\mu_{y_i}+\mathbf{h}_i$ and let
\begin{align}
(\hat{\w},\hat b) &= \arg\min_{\mathbf{w},b}\frac{1}{n}\sum_{i\in[n]}\ell(y_i(\mathbf{x}_i^T\mathbf{w}+b)), \notag
\\
(\hat\vb,\hat c) &= \arg\min_{\mathbf{v},c}\frac{1}{n}\sum_{i\in[n]}\ell(y_i(\mathbf{z}_i^T\mathbf{v}+c)).\notag
\end{align}
Clearly, $\hat{\w}=\boldsymbol\Sigma^{-1/2}\hat\vb$ and $\hat b = \hat c$.
Thus,
\begin{align}
\mathcal{R}_+\left((\hat{\w},\hat b) \right) &= \mathbb{P}\{ (\mathbf{x}^T\hat{\w}+\hat b) <0 \,|\,y=+1\} = \mathbb{P}\{ \boldsymbol\mu_+^T\hat{\w}+\hat{\w}^T\boldsymbol\Sigma^{1/2}\mathbf{h}+\hat b <0 \} = Q\left( \frac{\boldsymbol\mu_+^T\hat{\w}+\hat b}{\|\boldsymbol\Sigma^{1/2}\hat{\w}\|_2} \right) \notag\\
&=Q\Big( \frac{\boldsymbol\mu_+^T\boldsymbol\Sigma^{-1/2}\hat\vb+\hat c}{\|\hat\vb\|_2} \Big)=\mathbb{P}\{ (\mathbf{z}^T\hat\vb+\hat c) <0 \,|\,y=+1\}\notag\\
&=\mathcal{R}_+\left((\hat\vb,\hat c) \right)\notag
\end{align}
Similar derivation holds for $\mathcal{R}_-$.
Thus, we can just apply Theorem \ref{thm:main_imbalance} for $\mathbf{S},\mathbf{V}$ given by the eigendecomposition of the new Grammian $\mathbf{M}^T\boldsymbol\Sigma^{-1}\mathbf{M}$.
\subsection{Phase transition of CS-SVM}\label{sec:PT_CS-SVM}
Here, we present a formula for the threshold $\gamma_\star$ such that the CS-SVM of \eqref{eq:CS-SVM} is feasible (resp., infeasible) with overwhelming probability provided that $\gamma>\gamma_\star$ (resp., $\gamma<\gamma_\star$). The first observation is that the phase-transition threshold $ \gamma_\star$ of feasibility of the CS-SVM is the same as the threshold of feasibility of the standard SVM for the same model; see Section \ref{sec:CS_svm_easy_claim}. Then, the desired result follows \cite{gkct_multiclass} who very recently established separability phase-transitions for the more general multiclass Gaussian mixture model
\begin{propo}[\cite{gkct_multiclass}]
Consider the same data model and notation as in Theorem \ref{thm:main_imbalance} and define the event
$$
\mathcal{E}_{{\rm sep}, n}:=\left\{\exists (\mathbf{w},b)\in\mathbb{R}^d\times\mathbb{R}~~\text{s.t.}~~y_i(\mathbf{w}^T\mathbf{x}_i+b)\geq 1,\,~\forall i\in[n]\right\}.
$$
Define threshold $\gamma_\star:=\gamma_\star(\mathbf{V},\mathbf{S},\pi)$ as follows:
\begin{align}\label{eq:gamma_star_CS}
\gamma_\star:=\min_{\mathbf{t}\in\mathbb{R}^r, b\in\mathbb{R}}\mathbb{E}\left[\Big(\sqrt{1+\|\mathbf{t}\|_2^2} \,G + E_Y^T\mathbf{V}\mathbf{S}\mathbf{t} - bY\Big)_-^2\right].
\end{align}
Then, the following hold:
\begin{align}
\gamma>\gamma_\star \Rightarrow \lim_{n\rightarrow\infty}\mathbb{P}(\mathcal{E}_{{\rm sep}, n}) = 1\quad\text{ and }\quad
\gamma<\gamma_\star \Rightarrow \lim_{n\rightarrow\infty}\mathbb{P}(\mathcal{E}_{{\rm sep}, n}) = 0.\notag
\end{align}
In words, the data are linearly separable (with overwhelming probability) if and only if $\gamma>\gamma_\star$. Furthermore, if this condition holds, then CS-SVM is feasible (with overwhelming probability) for any value of $\delta>0$.
\end{propo}
\section{Concluding remarks}\label{sec:future_work
\vspace{-0.1in}
We presented a theoretically-grounded study of recently introduced cost-sensitive CE modifications for imbalanced data. To optimize key fairness metrics, we formulated a new such modification subsuming previous techniques as special cases and provided theoretical justification, as well as, empirical evidence on its superior performance against existing methods.
We suspect the VS-loss and our better understanding on the individual roles of different hyperparameters can benefit NLP and computer vision applications; we expect future work to undertake this opportunity with additional experiments.
When it comes to group-sensitive learning, it is of interest to extend our theory to other fairness metrics of interest. Ideally, our precise asymptotic theory could help contrast different fairness definitions and assess their pros/cons. Our results are the first to theoretically justify the benefits/pitfalls of specific logit adjustments used in \cite{cosen,TengyuMa,Menon,CDT}. The current theory is limited to settings with fixed features. While this assumption is prevailing in most related theoretical works \cite{ji2018risk,nacson2019stochastic,hastie2019surprises,bartlett2020benign,muthukumar2020classification},
it is still far from deep-net practice where (last-layer) features are learnt jointly with the classifier. We expect recent theoretical developments on that front \cite{NC,mixon2020neural,lu2020neural} to be relevant in our setting when combined with our ideas.
\subsection{VS-loss vs LA-loss for a group-sensitive GMM}
In Figure \ref{fig:gs_deo_gd_convergence} we test the performance of our theory-inspired VS-loss against the logit-adjusted (LA)-loss in a group-sensitive classification setting with data from a Gaussian mixture model with a minority and and a majority group. Specifically, we generated synthetic data from the model with class prior $\pi=1-\pi=1/2$, minority group membership prior $p=0.05$ (for group $g=1$) and $\boldsymbol\mu_{1}=3\mathbf{e}_1, \boldsymbol\mu_{2}=3\mathbf{e}_2 \in\mathbb{R}^{500}$. We trained homogeneous linear classifiers based on a varying number of training sample $n=d/\gamma$. For each value of $n$ (eqv. $\gamma$) we ran normalized gradient descent (see Sec. \ref{sec:gd_num_app}) on
\begin{itemize}
\item CDT-loss $\ell(y,\mathbf{w}^T\mathbf{x},g):=\log(1+e^{-\Delta_g y(\mathbf{w}^T\mathbf{x})})$ with $\Delta_g=\delta_0\ind{g=1}+\ind{g=2}$.
\item the LA-loss modified for group-sensitive classification $\ell(y,\mathbf{w}^T\mathbf{x},g):=\log(1+e^{\iota_g}e^{ y(\mathbf{w}^T\mathbf{x})})$ with $\iota_g=p^{-1/4}\ind{g=1}+(1-p)^{-1/4}\ind{g=2}$. This value for $\iota$ is inspired by \cite{TengyuMa}, but that paper only considered applying the LA-loss in label-imbalanced settings.
\end{itemize}
For $\gamma>0.5$ where data are necessarily separable, we also ran the standard SVM and the GS-SVM with $\delta=\delta_0$.
Here, we chose the parameter $\delta_0$ such that the GS-SVM achieves zero DEO. To do this, we used the theoretical predictions of Theorem \ref{thm:main_group_fairness} for the DEO of GS-SVM for any value of $\delta$ and performed a grid-search giving us the desired $\delta_0$; see Figure \ref{fig:gs_deo_gd_convergence} for the values of $\delta_0$ for different values of $\gamma$.
Figure \ref{fig:gs_deo_gd_convergence}(a) verifies that the GS-SVM achieves DEO (very close to) zero on the generated data despite the finite dimensions in the simulations. On the other hand, SVM has worse DEO performance. In fact, the DEO of SVM increases with $\gamma$, while that of GS-SVM stays zero by appropriately tuning $\delta_0$.
The figure further confirms the message of Theorem \ref{thm:implicit_loss_gen}:
In the separable regime, GD on logit-adjusted loss converges to the standard SVM performance, whereas GD on our VS-loss converges to the corresponding GS-SVM solution, thus allowing to tune a suitable $\delta$ that can trade-off misclassification error to smaller DEO magnitudes. The stopping criterion of GD was a tolerance value on the norm of the gradient. The match between empirical values and the theoretical predictions improves with increase in the dimension, more Monte-Carlo averaging and a stricter stopping criterion for GD.
\section{Experiments}\label{sec:experiments}
\vspace{-0.1in}
We show experimental results further justifying theoretical findings. {(Code available in \cite{code}).}
\vspace{-0.1in}
\subsection{Label-imbalanced data}\label{sec:exp_label}
\vspace{-0.1in}
Our first experiment (Table \ref{table:CIFAR}) shows that \emph{non-trivial combinations} of additive/multiplicative adjustments
can improve balanced accuracy over \emph{individual} ones. Our second experiment (Fig. \ref{fig:tau_gamma_effect_on_training_main}) validates the theory of Sec. \ref{sec:insights_gen} by examining how these adjustments affect training.
\noindent\textbf{Datasets.}
Table \ref{table:CIFAR} evaluates LA/CDT/VS-losses on imbalanced instances of CIFAR-10/100.
Following \cite{TengyuMa}, we consider: (1) \textit{STEP} imbalance,
reducing the sample size of half of the classes to a fixed number. (2) Long-tailed (\textit{LT}) imbalance,
which exponentially decreases the number of training images across different classes. We set an imbalance ratio $N_{\max}/ N_{\min}=100$, where $N_{\max}=\max_y{N_y}, N_{\min}=\min_y{N_y}$ and $N_y$ are sample sizes of class $y$. For consistency with \cite{he2016deep, TengyuMa, Menon, CDT} we keep a balanced test set and in addition to evaluating our models on it, we treat it as our validation set and use it to tune our hyperparameters. More sophisticated tuning strategies (perhaps using bi-level optimization) are deferred to future work.
We use data-augmentation exactly as in \cite{he2016deep, TengyuMa, Menon, CDT}.
See SM for more implementation details.
\noindent\textbf{Model and Baselines.}
We compare the following:
\textit{(1) CE-loss}. \textit{(2) Re-Sampling} that includes each data point in the batch with probability ${\pi_y}^{-1}$. \textit{(3) wCE} with weights $\omega_y = {\pi_y}^{-1}$. \textit{(4) LDAM-loss} \cite{TengyuMa}, special case of LA-loss where $\iota_y = \frac{1}{2} ({N_{\min}}/{N_y})^{1/4}$ is \emph{subtracted} from the logits.
\begin{wraptable}{r}{0.7\textwidth}
\caption{Top-1 accuracy results on balanced validation set ($\%$).
}
\label{table:CIFAR}
\begin{center}\resizebox{0.7\textwidth}{!}{
\begin{tabular}{lllllllll} \hline
\textbf{Dataset} & \multicolumn{2}{c}{\textbf{CIFAR 10}} & \multicolumn{2}{c}{\textbf{CIFAR 100}} \\ \hline
\textbf{Imbalance Profile} & \multicolumn{1}{c}{\textbf{LT-100}} & \multicolumn{1}{c}{\textbf{STEP-100}} & \multicolumn{1}{c}{\textbf{LT-100}} & \multicolumn{1}{c}{\textbf{STEP-100}} \\ \hline \hline
CE & $71.94 \pm 0.38$ & $62.69 \pm 0.50$ & $38.82 \pm 0.69$ & $39.49 \pm 0.16$ \\
Re-Sampling & 71.2 & 65.0 & 34.7 & 38.4 \\
wCE & 72.6 & 67.3 & 40.5 & 40.1 \\ \hline
LDAM \cite{TengyuMa}. & 73.35 & 66.58 & 39.60 & 39.58 \\
LDAM-DRW \cite{TengyuMa} & 77.03 & 76.92 & 42.04 & 45.36 \\
LA ($\tau=\tau^{*}$) \cite{Menon} & $80.81 \pm 0.30$ & $78.23 \pm 0.52$ & $42.87 \pm 0.32$ & $45.69 \pm 0.27$ \\
CDT ($\gamma=\gamma^{*}$) \cite{CDT} & $79.55 \pm 0.35$ & $73.26 \pm 0.29$ & $42.57 \pm 0.32$ & $44.12 \pm 0.17$ \\ \hline \hline
VS ($\tau=\tau^{*}, \gamma=\gamma^{*}$) & $\textbf{80.82} \pm 0.37 $ & $\textbf{79.10} \pm 0.66$ & $\textbf{43.52} \pm 0.46$ & $\textbf{46.53} \pm 0.17$ \\
\end{tabular}}
\end{center}
\end{wraptable}
\textit{(5) LDAM-DRW} \cite{TengyuMa}, combining LDAM with deferred re-weighting. \textit{(6) LA-loss} \cite{Menon}, with the Fisher-consistent parametric choice $\iota_y = \tau \log(\pi_y)$. \textit{(7) CDT-loss} \cite{CDT}, with $\Delta_y = ({N_{y}}/{N_{max}})^{\gamma}$. \textit{(8) VS-loss}, with combined hyperparameters $\iota_y = \tau \log(\pi_y)$ and $\Delta_y = ({N_{y}}/{N_{\max}})^{\gamma}$, parameterized by $\tau, \gamma>0$ respectively \footnote{{Here, the hyperparameter $\gamma$ is used with some abuse of notation and is important to not be confused with the parameterization ratio in the linear models in Sec. \ref{sec:insights_gen} and \ref{sec:generalization}. We have opted to use the same notation as in \cite{CDT} to ease direct comparisons of experimental findings.}}.
The works introducing (5)-(7) above, all trained for a different number of epochs, with dissimilar regularization and learning rate schedules. For consistency, we follow the training setting in \cite{TengyuMa}. Thus, for LDAM we adapt results reported by \cite{TengyuMa}, but for LA and CDT, we reproduce our own in that setting. Finally, for a fair comparison we ran LA-loss for optimized $\tau=\tau^*$ (rather than $\tau=1$ in \cite{Menon}).
\noindent\textbf{VS-loss balanced accuracy.} Table \ref{table:CIFAR} shows Top-1 accuracy on balanced validation set (averaged over 5 runs).
We use a grid to pick the best $\tau$ / $\gamma$ / ($\tau,\gamma$)-pair for the LA / CDT / VS losses
on the validation set.
Since VS includes LA and CDT as special cases (corresponding to $\gamma=0$ and $\tau=0$ respectively), we expect that it is at least as good as the latter over our hyper-parameter grid search.
We find that the optimal $(\tau^*,\gamma^*)$-pairs correspond to non-trivial combinations of each individual parameter.
Thus, VS-loss has better balanced accucy as shown in the table.
See SM for optimal hyperparameters choices.
\begin{figure}[h]
\begin{center}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{cifar100_exp100_CDT_training_error_vs_gamma.png}
\caption{$\Delta_y$'s (parameterized by $\gamma$) can hurt training.}
\end{subfigure}
\begin{subfigure}[b]{0.49\textwidth}
\centering
\includegraphics[width=0.5\textwidth]{cifar100_exp100_LA_training_error_vs_tau.png}
\caption{LA trains easier than CDT.}
\end{subfigure}
\\
\begin{subfigure}[b]{\textwidth}
\includegraphics[width=\textwidth]{cifar100_exp100_VS_error_vs_tau_1x4_edited.png}
\caption{$\iota_y$'s mitigate the effect of $\Delta_y$'s (c1,c2), but $\Delta_y's$ dominate TPT performance (c3,c4).}
\end{subfigure}
\end{center}
\caption{ Experiments on CIFAR10 with Long-tailed LT-100 imbalance demonstrating the effects of additive/multiplicative parameters at different phases of training. All results are averaged over 5 runs and shaded regions indicate the $95\%$ confidence intervals. See text for details and interpretations.}
\label{fig:tau_gamma_effect_on_training_main}
\end{figure}
\vspace*{-0.5cm}
\noindent\textbf{How hyperparameters affect training?} We perform three experiments. \textbf{(a)} Figure \ref{fig:tau_gamma_effect_on_training_main}(a) shows that larger values of hyperparameter $\gamma$ (corresponding to more dispersed $\Delta_y$'s between classes) hurt training performance and delay entering to TPT. Complementary Figures \ref{fig:tau_gamma_effect_on_training_main}(c1,c2) show that eventually, if we train longer, then, train accuracy approaches 100\%. These findings are in line with Observation \ref{obs:CDT_bad} in Sec. \ref{sec:best}. \textbf{(b)} Figure \ref{fig:tau_gamma_effect_on_training_main}(b) shows training accuracy of LA-loss for changing hyperparameter $\tau$ controlling additive adjustments. On the one hand, increasing values of $\tau$ delay training accuracy to reach 100\%. On the other hand, when compared to the effect of $\Delta_y$'s in Fig. \ref{fig:tau_gamma_effect_on_training_main}(a), we observe that the impact of additive adjustments on training is significantly milder than that of multiplicative adjustments. Thus, LA trains easier than CDT. \textbf{(c)} Figure \ref{fig:tau_gamma_effect_on_training_main}(c) shows train and balanced accuracies for (i) CDT-loss in blue: $\tau=0$, $\gamma=0.15$, (ii) VS-loss in orange: $\tau=-0.5$, $\gamma=0.15$. In Fig. \ref{fig:tau_gamma_effect_on_training_main}(c1,c3) we trained for $200$ epochs, while in Fig. \ref{fig:tau_gamma_effect_on_training_main}(c2,c4) we trained for $300$ epochs. For $\gamma=0.15$, CDT-loss does \emph{not} reach good training accuracy within 200 epochs ($\sim93$\% at epoch 200 in Fig. \ref{fig:tau_gamma_effect_on_training_main}(c1)), but the addition of $\iota_y$'s with $\tau=-0.5$ mitigates this effect achieving improved $\sim97$\% accuracy at 200 epochs. This also translates to balanced test accuracy: VS-loss has better accuracy at the end of training in Fig. \ref{fig:tau_gamma_effect_on_training_main}(c3). Yet, CDT-loss has not yet entered the interpolating regime in this case. So, we ask: What changes if we train longer so that both CDT and VS loss get (closer) to interpolation. In Fig. \ref{fig:tau_gamma_effect_on_training_main}(c2) train accuracy of both algorithms increases when training continues to 300 epochs. Again, thanks to the $\iota_y$'s VS-loss trains faster. However, note in Figure \ref{fig:tau_gamma_effect_on_training_main}(c4) that the balanced accuracies of the two methods are now very close to each other. Thus, in the interpolating regime what dominates the performance are the multiplicative adjustments which are same for both losses. This is in agreement with the finding of Theorem \ref{propo:gd} and the synthetic experiment in Fig. \ref{fig:mismatch_intro}(b,c).
\input{group_exp}
\subsection{Label-imbalanced data}
\noindent\textbf{Datasets:}
We evaluate the VS-loss on CIFAR-10 and CIFAR-100 \cite{krizhevsky2009learning}; specifcally, on imbalanced instances generated by \cite{TengyuMa}.
We study two types of imbalance. (1) \textit{STEP} imbalance \cite{Buda_2018}, reduces the number of training instances of half of the classes to a fixed size. (2) Long-tailed (\textit{LT}) imbalance \cite{cui2019classbalanced} exponentially decreases the number of training images across different classes. In both cases we define the imbalance ratio as $\rho = N_{\max}/ N_{\min} \in \{10, 100\}$, where $N_{\max}=\max_y{N_y}, N_{\min}=\min_y{N_y}$ and $N_y$ are the sample sizes of each class $y$. For consistency with \cite{he2016deep, TengyuMa, Menon, CDT} we keet the test set balanced and in addition to evaluating our models on it, we treat it as our validation set and use it to tune our hyperparameters. \red{In the SM we discuss a more practical hyperparameter tuning strategy using a validation set held out of the training set and we demonstrate that this approach achieves similar performance on the test set.} {Same as in \cite{he2016deep, TengyuMa, Menon, CDT} we augment the data by padding the images to size $40 \times 40$, flipping them horizontally at random and then random cropping them to their original size. More results on other datasets are in the SM.}
\\
\noindent\textbf{Model and Baselines:}
We compare the VS-loss against several different state-of-the-art baselines:
\textbf{(1) CE-loss} (standard cross-entropy). \textbf{(2) Re-Sampling}, where every training instance is assigned a ${\pi_y}^{-1}$ probability to be included in the batch. \textbf{(3) wCE}, weighted CE with weights $\omega_y = {\pi_y}^{-1}$. \textbf{(4) LDAM-loss} \cite{TengyuMa}, a special case of LA-loss where $\iota_y = \frac{1}{2} ({N_{\min}}/{N_y})^{1/4}$ is \emph{subtracted} from the logits. \textbf{(5) LDAM-DRW} \cite{TengyuMa}, combining LDAM with deferred re-weighting (i.e. $\omega_y \neq 1$ only as we enter TPT). \textbf{(6) LA-loss} \cite{Menon}, with the Fisher-consistent parametric choice $\iota_y = \tau \log(\pi_y)$ added to the logits. \textbf{(7) CDT-loss} \cite{CDT}, where the logits are multiplied by $\Delta_y = ({N_{y}}/{N_{max}})^{\gamma}$. \textbf{(8) VS-loss}, with combined hyperparameters $\iota_y = \tau \log(\pi_y)$ and $\Delta_y = ({N_{y}}/{N_{\max}})^{\gamma}$, parameterized by $\tau, \gamma>0$ respectively. We note that the three papers that first introduced (5), (6) and (7) above, all trained CIFAR for a different number of epochs and with dissimilar regularization and learning rate schedules. Here, for consistency, we follow the training setting by \cite{TengyuMa} (see below). Thus, for LDAM we show the results reported by \cite{TengyuMa}, but for the LA/CDT, we reproduce the results of \cite{Menon,CDT} in the above setting.
\\
\noindent\textbf{Technical details:} \red{We follow \cite{TengyuMa}.} We train a ResNet-32 model \cite{he2016deep}, using batch size $128$ and SGD with momentum $0.9$ and weight decay $2\times10^{-4}$. For the first $5$ epochs we use a linear warm up schedule until baseline learning rate of $0.1$. We train for a total of 200 epochs, while decaying our learning rate by $0.1$ at epochs $160$ and $180.$ We develop our code using PyTorch \cite{paszke2017automatic} building on codes provided by \cite{TengyuMa,CDT}. Training is performed on 2 NVIDIA RTX-3080 GPUs. \red{We remark that LDAM failed to converge in this setting, thus we used learning rate decay 0.01. For that same loss we also normalized the features norms after the last layer before the classifiers and the weights of the classifiers as done by \cite{TengyuMa}. We avoid that for the rest of the losses to isolate the impact of loss-modification which is our focus.}
\\
\noindent\textbf{Results:} Table \ref{table:CIFAR} shows Top-1 accuracy results on the balanced validation set (averages over 5 runs) on CIFAR 10/100, for STEP/LT imbalances with imbalance-ratios 10 and 100. We tune $\gamma$ for the CDT-loss and both $\tau$ and $\gamma$ for the VS-loss on the validation set. \red{The best value found is denoted $\tau^*, \gamma^*$ repsectively (see SM).} We also evaluated the performance of VS-loss with DRW. We see that the VS loss frequently outperforms other methods. In theory, as VS includes LA and CDT as special cases, it should always perform at least as good as them given that the optimal hyperparameters were chosen. However, we observe that in some cases VS performs slightly worse than these losses, that is because in practice the distribution of the validation set on which the hyperparameters were chosen differs from that of the test set. \op{Here the validation and the test set are the same, so the real reason here is that I don't use seeds, thus there is some randomness and over a small amount of runs LA and CDT can perform better than VS} Additional results, including histograms of the per class error rates and plots of the dynamics of training are included in the SM.
\section{Generalization analysis and fairness tradeoffs}\label{sec:generalization}
\vspace{-0.1in}
Our results in the previous section regarding VS-loss/CS-SVM hold for arbitrary linearly-separable training datasets. Here, under additional distributional assumptions, we establish a sharp asymptotic theory for VS-loss/CS-SVM and their group-sensitive counterparts.
\noindent\textbf{Data model.}~We study binary Gaussian-mixture generative models (GMM) for the data distribution $\mathcal{D}$. For the label $y\in\{\pm1\}$, let $\pi:=\mathbb{P}\{y=+1\}.$ Group membership is decided conditionally on the label such that $\forall j\in[K]:$ $\mathbb{P}\{g=j|y=\pm1\}=p_{\pm,j}$, with $\sum_{j\in[K]}p_{+,j}=\sum_{j\in[K]}p_{-,j}=1$. Finally, the feature conditional given label $y$ and group $g$ is a multivariate Gaussian of mean $\boldsymbol\mu_{y,g}\in\mathbb{R}^d$ and covariance $\boldsymbol\Sigma$, i.e.
$\mathbf{x}\big|(y,g)\,\widesim{} \mathcal{N}(\boldsymbol\mu_{y,g},\boldsymbol\Sigma).
$
Specifically for \emph{label-imbalances}, we let $K=1$ and $
\mathbf{x}\big|y\,\widesim{} \mathcal{N}(\boldsymbol\mu_{y},\mathbf{I}_d)$ (see SM for $\boldsymbol\Sigma\neq \mathbf{I}_d$). For \emph{group-imbalances}, we focus on two groups with $p_{+,1}=p_{-,1}=p<1-p=p_{+,2}=p_{-,2},\,j=1,2$ and $\mathbf{x}\,|\,(y,g) \sim \mathcal{N}(y\boldsymbol\mu_{g},\mathbf{I}_d)$. In both cases, $\mathbf{M}$ denotes the matrix of means, i.e. $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{+} & \boldsymbol\mu_{-}\end{bmatrix}$ and $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{1} & \boldsymbol\mu_{2}\end{bmatrix}$, respectively. Also, consider the eigen-decomposition:
$
\mathbf{M}^T\mathbf{M} = \mathbf{V}\mathbf{S}^2\mathbf{V}^T,~~ \mathbf{S}\succ \mathbf{0}_{r\times r},\mathbf{V}\in\mathbb{R}^{2\times r}, r\in\{1,2\},
$
with $\mathbf{S}$ an $r\times r$ diagonal positive-definite matrix and $\mathbf{V}$ an orthonormal matrix obeying $\mathbf{V}^T\mathbf{V}=\mathbf{I}_r$. We study linear classifiers with $h(\mathbf{x})=\mathbf{x}$.
\noindent\textbf{Learning regime.}
We focus on the separable regime. For the models above, linear separability undergoes a sharp phase-transition as $d,n\rightarrow\infty$ at a proportional rate $\gamma=\frac{d}{n}$. That is, there exists threshold $\gamma_\star:=\gamma_\star(\mathbf{V},\mathbf{S},\pi)\leq1/2$
for the label-case, such
that data are linearly separable with probability approaching one provided that $\gamma>\gamma_\star$ (accordingly for the group-case) \cite{candes2020phase,montanari2019generalization,deng2019model,kammoun2020precise,gkct_icassp}. See SM for formal statements and explicit definitions.
\noindent\textbf{Analysis of CS/GS-SVM.}~We use $\stackrel{{P}}{\longrightarrow}$ to denote convergence in probability and $Q(\cdot)$ the standard normal tail. We let $(x)_-:=\min\{x,0\}$; $\ind{\mathcal{E}}$ the indicator function of event $\mathcal{E}$; ${\mathcal{B}}_2^r$ the unit ball in $\mathbb{R}^r$; and, $\mathbf{e}_1=[1,0]^T, \mathbf{e}_2= [0,1]^T$ standard basis vectors in $\mathbb{R}^2$.
We further need the following definitions. Let random variables as follows:
$
G\sim \mathcal{N}(0,1)$, $Y$ symmetric Bernoulli with $\mathbb{P}\{Y=+1\}=\pi$,
$E_Y=\mathbf{e}_1\ind{Y=1} - \mathbf{e}_2\ind{Y=-1}$ and
$\Delta_Y=\delta\cdot\ind{Y=+1} +\ind{Y=-1}$, for $\delta>0$.
With these define key function $\eta_\delta:\mathbb{R}_{\geq0}\times{\mathcal{B}}_2^{r}\times\mathbb{R}\rightarrow\mathbb{R}$ as
$\eta_\delta(q,\boldsymbol\rho,b):=\mathbb{E}\Big[\big(G+E_Y^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{b Y-\Delta_Y}{q}\big)_{-}^2\big] \notag- (1-\|\boldsymbol\rho\|_2^2)\gamma.
$ Finally, define $(q_\delta,\boldsymbol\rho_\delta,b_\delta)$ as the \emph{unique} triplet (see SM for proof) satisfying $\eta_\delta(q_\delta,\boldsymbol\rho_\delta,b_\delta)= 0$ and
$(\boldsymbol\rho_\delta,b_\delta):=\arg\min_{\substack{\|\boldsymbol\rho\|_2\leq 1, b\in\mathbb{R}}}\eta_\delta(q_\delta,\rho,b).$ Note that these triplets can be easily computed numerically for given values of $\gamma, \delta, \pi, p$ and means' Gramian $\mathbf{M}^T\mathbf{M} = \mathbf{V}\mathbf{S}^2\mathbf{V}^T.$
\begin{theorem}[Balanced error of CS-SVM]\label{thm:main_imbalance}
Let GMM data with label imbalances and learning regime as described above.
Consider the CS-SVM classifier in \eqref{eq:CS-SVM} with $h(\mathbf{x})=\mathbf{x}$, intercept $b$ (i.e. constraints $\inp{\mathbf{x}}{\mathbf{w}}+b\geq\{\delta \text{ or } 1\}$ in \eqref{eq:CS-SVM}) and fixed margin-ratio $\delta>0.$
Define
$\overline\Rc_+:=Q\left(\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta+b_\delta/q_\delta\right)$ and $\overline\Rc_-:=Q\left(-\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta-b_\delta/q_\delta\right).$
Then, as $n,d\rightarrow\infty$ with $d/n=\gamma>\gamma_\star$, it holds that
$\mathcal{R}_+\stackrel{{P}}{\longrightarrow}\overline\Rc_+$ and $\mathcal{R}_-\stackrel{{P}}{\longrightarrow}\overline\Rc_-$.
In particular,
$\Rc_{\text{bal}}\stackrel{{P}}{\longrightarrow}\overline{\Rc}_{\text{bal}}:=(\overline\Rc_++\overline\Rc_-)\big/2.$
\end{theorem}
The theorem further shows
$(\|\hat\mathbf{w}_\delta\|_2, \frac{\hat{\w}_\delta^T\boldsymbol\mu_+}{\|\hat{\w}_\delta\|_2}, \frac{\hat{\w}_\delta^T\boldsymbol\mu_-}{\|\hat{\w}_\delta\|_2} , \hat b_\delta) \stackrel{{P}}{\longrightarrow} (q_\delta,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta,\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta, b_\delta).
$
Thus, $b_\delta$ is the asymptotic the intercept, $q_\delta^{-1}$ is the asymptotic classifier's margin ${1}/{\|\hat{\w}_\delta\|_2}$ to the majority, and $\boldsymbol\rho_\delta$ determines the asymptotic alignment of the classifier with the class mean. The proof uses the convex Gaussian min-max theorem (CGMT) framework \cite{StojLAS,thrampoulidis2015regularized}; see SM for background, the proof, as well as, (a) simpler expressions when the means are antipodal ($\pm \boldsymbol\mu$) and (b) extensions to general covariance model ($\boldsymbol\Sigma\neq\mathbf{I}$). The experiment (solid lines) in Figure \ref{fig:mismatch_intro}(a) validates the theorem's predictions.
Also, in the SM, we characterize the DEO of GS-SVM for GMM data.
Although similar in nature, that characterization differs to Thm. \ref{thm:main_imbalance} since each class is now itself a Gaussian mixture as described in the model above.
\begin{figure}[t]
\begin{subfigure}[b]{0.55\textwidth}
\centering
\includegraphics[width=0.8\textwidth,height=0.42\textwidth]{fig_bal_misclas_tradeoff_s=3_p_min=0p05.PNG
\caption{}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=0.8\textwidth,height=0.48\textwidth]{fig_deo_vs_misclas_theory_fixed_s_priors_varying_gamma_alpha_s_min=3_s_maj=3_p_minor_class=0p5_p_minor_group=0p05.PNG
\caption{}
\end{subfigure}
\caption{Fairness tradeoffs between classification error and error-imbalance/balanced-error/DEO on GMM data achieved by \textbf{(a)} CS-SVM for class prior $\pi=0.05$ and \textbf{(b)} GS-SVM for group prior $p=0.05$, as a function of the margin-ratio hyperparameter $\delta\geq 1$ and for various values of overparameterization $\gamma$. Plots in (a) are generated using our sharp predictions in Theorem \ref{thm:main_imbalance}. Plots in (b) use corresponding result for GS-SVM given in the SM.
See text for interpretations.
}
\label{fig:tradeoffs}
\end{figure}
\noindent\textbf{Fairness tradeoffs.}~The theory above allow us to study tradeoffs between misclassification / balanced error / DEO in Fig. \ref{fig:tradeoffs}. Fig. \ref{fig:tradeoffs}(a) focuses on label imbalances. We make the following observations. (1) The optimal value $\delta_\star$ minimizing $\Rc_{\text{bal}}$ also achieves perfect balancing between the conditional errors of the two classes, that is $\mathcal{R}_+=\mathcal{R}_-=Q(\frac{\ell_-+\ell_+}{2}).$ We prove this interesting property in the SM
by deriving an explicit formula for $\delta_\star$ that only requires computing the triplet $(q_1,\boldsymbol\rho_1,b_1)$ for $\delta=1$ corresponding to the standard SVM. Such {closed-form} formula is rather unexpected given the seemingly involved nonlinear dependency of $\Rc_{\text{bal}}$ on $\delta$ in Thm. \ref{thm:main_imbalance}. In the SM, we also use this formula to formulate a theory-inspired heuristic for hyperparameter tuning, which shows good empirical performance on simple datasets such as imbalanced MNIST. (2) The value of $\delta$ minimizing standard error $\mathcal{R}$ (shown in magenta) is not equal to $1$, hence CS-SVM also improves $\mathcal{R}$ (not only $\Rc_{\text{bal}}$). In Fig. \ref{fig:tradeoffs}(b), we investigate the effect of $\delta$ and the improvement of GS-SVM over SVM. The largest DEO and smallest misclassification error are achieved by the SVM ($\delta=1$). But, with increasing $\delta$, misclassification error is traded-off for reduction in absolute value of DEO. Interestingly, for some $\delta_0=\delta_0(\gamma)$ (with value increasing with $\gamma$) GS-SVM guarantees Equal Opportunity (EO) $\mathcal{R}_{\rm deo}=0$ (without explicitly imposing such constraints as in \cite{olfat2018spectral,donini2018empirical}).
\subsection{Implicit bias of Gradient flow with respect to VS-loss}\label{sec:gradient_flow}
Theorem \ref{thm:implicit_loss_gen} does not consider the effect of the optimization algorithm. Instead here, we study gradient flow (the limit of gradient descent for infinitesimal step-size) and characterize its implicit bias when applied to the VS-loss. Similar, to Theorem \ref{thm:implicit_loss_gen}, we find that the iterations of gradient flow converge to the solution of a corresponding CS-SVM. For simplicity, we consider a VS-type adjusted exponential loss $\ell(t)=e^{-t}$, rather than logistic loss $\ell(t)=\log(1+e^{-t})$. Recent work makes it clear that both loss functions have similar implicit biases and similar lines of arguments are used to analyze the convergence properties \cite{ji2021characterizing,ji2018risk}. Thus, one would expect that insights also apply to logistic loss.
\begin{theorem}[Implicit bias of the gradient flow]\label{thm:gradient_flow}
Consider the gradient flow iteration
$
\dot{\mathbf{w}}_t=-\nabla\mathcal{L}(\mathbf{w}_t),
$
on the exponential VS-loss $\mathcal{L}(\mathbf{w})=\sum_{i\in[n]}\omega_i\exp(-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}+\iota_i).$ Recall that $\hat\mathbf{w}$ is the solution to the CS-SVM in \eqref{eq:CS-SVM_app}.
For almost every dataset which is linearly separable and any starting point $\mathbf{w}_0$ the gradient flow iterates will behave as $\mathbf{w}(t)=\hat\mathbf{w}\log(t)+\boldsymbol\rho_t$ with a bounded residual $\boldsymbol\rho_t$ so that $\lim_{t\rightarrow\infty}\frac{\mathbf{w}_t}{\|\mathbf{w}_t\|_2}=\frac{\hat\mathbf{w}}{\|\hat\mathbf{w}\|_2}.$
\end{theorem}
Note that \cite{soudry2018implicit} previously studied the implicit bias of the gradient flow on standard CE or exponential loss. The theorem above studies the gradient flow applied to the VS-loss and its proof is similar to \cite{soudry2018implicit}.
\begin{proof} Let ${\mathcal{S}}\subset[n]$ be the set of indices such that $\forall i\in{\mathcal{S}}: \Delta_iy_i\mathbf{x}_i^T\hat\mathbf{w}=1$, i.e. the set of support vectors of the CS-SVM. By KKT conditions (eg. see Equation \eqref{eq:KKT_2}), there exist $\epsilon_i>0$ such that $\hat\mathbf{w}=\sum_{i\in{\mathcal{S}}}\epsilon_iy_i\mathbf{x}_i.$ Moreover, by \cite[Lemma 12]{soudry2018implicit}, for almost all datasets it is true that $|{\mathcal{S}}|\leq d$ and $i\in{\mathcal{S}}\implies\epsilon_i>0$. Thus, for almost all datasets we can define vector $\widetilde\w$ satisfying the following equation
$\omega_i\Delta_i\exp(-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i) = \epsilon_i, \forall i\in[S].$
Note then that
\begin{align}\label{eq:kkt_GF}
\hat{\w}=\sum_{i\in[S]}\omega_i\Delta_ie^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}y_i\mathbf{x}_i
\end{align}
Let us define
$\mathbf{r}_t=\boldsymbol\rho_t-\widetilde\w=\mathbf{w}_t-\log(t)\hat\mathbf{w}-\widetilde\w.$ It suffices to show that $\|\mathbf{r}(t)\|_2$ is bounded, since that would automatically give $\boldsymbol\rho_t$ is bounded. By the gradient flow equation, we have that $$
\dot\mathbf{r}_t=-\nabla\mathcal{L}(\mathbf{w}_t)-\frac{\hat\mathbf{w}}{t}=\sum_{i\in[n]}\omega_i\Delta_iy_ie^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}_t+\iota_i}\mathbf{x}_i-\frac{\hat\mathbf{w}}{t}.
$$ Therefore,
\begin{align}
\frac{1}{2}\frac{\mathrm d}{\mathrm{d} t}\|\mathbf{r}_t\|_2^2&=\dot\mathbf{r}_t^T\mathbf{r}_t
=\sum_{i\in[n]}\omega_i\Delta_iy_ie^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}_t+\iota_i}\mathbf{x}_i^T\mathbf{r}_t-\frac{1}{t}\hat\mathbf{w}^T\mathbf{r}_t\notag\\
&=\underbrace{\sum_{i\in{\mathcal{S}}}\omega_i\Delta_iy_ie^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}_t+\iota_i}\mathbf{x}_i^T\mathbf{r}_t-\frac{1}{t}\hat\mathbf{w}^T\mathbf{r}_t}_{:=A} + \underbrace{\sum_{i\not\in{\mathcal{S}}}\omega_i\Delta_iy_ie^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}_t+\iota_i}\mathbf{x}_i^T\mathbf{r}_t}_{:=B}\label{eq:implicit_step1}
\end{align}
We now study the two terms $A$ and $B$ separately. In doing so, recall that $\mathbf{w}_t=\mathbf{r}_t+\log(t)\hat\mathbf{w}+\widetilde\w.$ Hence, using the fact that ${\Delta_iy_i\mathbf{x}_i^T\hat{\w}}={\begin{cases}=1 &i\in{\mathcal{S}} \\ \geq m>1 &i\not\in{\mathcal{S}}\end{cases}}$, it holds that
\begin{align*}
\exp\big({-\Delta_iy_i\mathbf{x}_i^T\mathbf{w}_t+\iota_i}\big)
&\begin{cases}
=\frac{1}{t}\cdot\exp\big({-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t}\big)\cdot\exp\big({-\Delta_iy_i\mathbf{x}_i^T\widetilde\w}+\iota_i\big) &i\in{\mathcal{S}} \\
\leq\frac{1}{t^m}\cdot\exp\big({-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t}\big)\cdot\exp\big({-\Delta_iy_i\mathbf{x}_i^T\widetilde\w}+\iota_i\big) &i\not\in{\mathcal{S}}
\end{cases}
\end{align*}
Using this and \eqref{eq:kkt_GF}, the term $A$ becomes
\begin{align*}
A&=\frac{1}{t}\sum_{i\in[S]}\omega_ie^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}\cdot e^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t} \Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t-\frac{1}{t}\sum_{i\in[S]}\omega_i\Delta_ie^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}y_i\mathbf{x}_i^T\mathbf{r}_t\\
&=\frac{1}{t}\sum_{i\in[S]}\omega_ie^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}\cdot\Big(e^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t} \Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t\Big)\leq 0,
\end{align*}
since $\forall x, x\geq xe^{-x}.$
Similarly, for term $B$:
\begin{align}
B\leq\frac{1}{t^m}\sum_{i\not\in{\mathcal{S}}}\omega_i e^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}\cdot e^{-\Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t}\cdot \Delta_iy_i\mathbf{x}_i^T\mathbf{r}_t \leq
\frac{1}{t^m}\sum_{i\not\in{\mathcal{S}}}\omega_i e^{-\Delta_iy_i\mathbf{x}_i^T\widetilde\w+\iota_i}\,,\label{eq:m>1}
\end{align}
since $\forall x, xe^{-x}\leq 1$.
To finish the proof it only takes now using the above bounds on $A, B$ and integrating both sides of Equation \eqref{eq:implicit_step1}. This gives that for all $t_0, t>t_0$, there exists finite constant $C$ such that $\|\mathbf{r}_t\|^2\leq \|\mathbf{r}_{t_0}\|^2+C$ where it was critical that $m>1$ in \eqref{eq:m>1} for the corresponding integral to be finite. This proves that $\|\mathbf{r}_t\|_2$ is bounded as desired.
\end{proof}
We note that the above proof is a straightforward extension of \cite{soudry2018implicit} for analysis of CDT, with a simple rescaling of the features of the training set according to the labels, however the analysis for VS-loss with additive logit-adjustments (although similar) cannot be obtained as a special case of \cite{soudry2018implicit}.
\subsection{Numerical illustrations of Theorems \ref{propo:gd} and \ref{thm:gradient_flow}}\label{sec:gd_num_app}
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{gd_normalized_VSloss.pdf}
\caption{CDT-loss vs CS-SVM}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=\textwidth]{gd_normalized_LAloss.pdf}
\caption{LA-loss vs SVM}
\end{subfigure}
\end{center}
\caption{
Convergence properties of GD (blue) and normalized GD (red) iterates $\mathbf{w}_t, t\geq 1$ on VS-loss with $f_\mathbf{w}(x)=\mathbf{w}^T\mathbf{x}$ for two set of parameter choices: (a) $\omega_y=1,\iota_y=0,\Delta_y=\delta\ind{y=1}+\ind{y=-1}$ (aka CDT-loss) with $\delta=20$; (b) $\omega_y=1,\iota_y=\pi^{-1/4}\ind{y=1}+(1-\pi)^{-1/4}\ind{y=-1},\Delta_y=1$ (aka LA-loss).
We plotted the angle gap $1-\frac{\hat{\w}^T\mathbf{w}_t}{\|\mathbf{w}_t\|_2\|\hat{\w}\|_2}$
and norm gap $\|\frac{\mathbf{w}_t}{\|\mathbf{w}_t\|_2}-\frac{\hat{\w}}{\|\hat{\w}\|_2}\|_2$
of $\mathbf{w}_t$ to $\hat{\w}$, for two values of $\hat{\w}$ for the two subfigures as follows: (a) $\hat{\w}$ is the CS-SVM solution in \eqref{eq:CS-SVM} with parameter $\delta$; (b) $\hat{\w}$ is the standard SVM solution. Data were generated from a Gaussian mixture model with $\boldsymbol\mu_1=2\mathbf{e}_1, \boldsymbol\mu_2=-3\mathbf{e}_1 \in\mathbb{R}^{220}$, $n=100$ and $\pi=0.1$. For (standard) GD we used a constant rate $\eta_t=0.1$. For normalized GD, we used $\eta_t=\frac{1}{\sqrt{t}\|\nabla\mathcal{L}(\mathbf{w}_t)\|_2}$ as suggested in \cite{nacson2019convergence}.}
\label{fig:normalized_gd_experiments}
\end{figure}
Figure \ref{fig:normalized_gd_experiments} numerically demonstrate the validity of Theorems \ref{propo:gd} and \ref{thm:gradient_flow}. Here, we solved the VS-loss in Equation \eqref{eq:loss_ours_bin} using gradient descent (GD) for GMM data with class imbalance $\pi=0.1$. We ran two experiments for two choices of parameters in \eqref{eq:loss_ours_bin} corresponding to CDT-loss (with non-trivial multiplicative weights) and the LA-loss (with non-trivial additive weights); see the figure's caption for details. For each iterate outcome $\mathbf{w}_t$ of GD, we report the (i) angle and (ii) vector-norm gap to CS-SVM and SVM for the VS-loss and LA-loss, respectively, as well as, the (iii) value of the loss $\mathcal{L}(\mathbf{w}_t)$ and the (iv) norm of the weights $\|\mathbf{w}_t\|_2$ at current iteration. Observe that the loss $\mathcal{L}(\mathbf{w}_t)$ is driven to zero and the norm of the weights $\|\mathbf{w}_t\|_2$ increases to infinity with increasing $t$.
The experiment confirms that the VS-loss converges (aka angle/norm gap vanishes) to the CS-SVM solution, while the LA-loss converges to the SVM.
In Figure \ref{fig:normalized_gd_experiments}, we also study (curves in red) the convergence properties of \emph{normalized GD}. Following \cite{nacson2019convergence}, we implemented a version of normalized GD that uses a variable learning rate $\eta_t$ at iteration $t$ normalized by the gradient of the loss as follows: $\eta_t=\frac{1}{\|\nabla\mathcal{L}(\widetilde\w)\|_2\sqrt{t+1}}$. \cite{nacson2019convergence} (see also \cite{ji2021characterizing}) demonstrated that this normalization speeds up the convergence of standard logistic loss to SVM. Figure \ref{fig:normalized_gd_experiments} suggests that the same is true for convergence of the VS-loss to the CS-SVM.
\subsection{Group-sensitive data} \label{sec:GS_exp_main}
\vspace{-0.2cm}
The message of our experiments on group-imbalanced datasets is three-fold. (1) We demonstrate the practical relevance of logit-adjusted CE modifications to settings with imbalances at the level of (sub)groups. (2) We show that such methods are competitive to alternative state-of-the-art; specifically, distributionally robust optimization (DRO) algorithms. (3) We propose combining logit-adjustments with DRO methods for even superior performance.
\noindent\textbf{Dataset.}
We study a setting with spurious correlations ---strong associations between label and background in image classification--- which can be cast as a subgroup-sensitive classification problem
\cite{sagawa2019distributionally}.
We consider the Waterbirds dataset
\cite{sagawa2019distributionally}.
The goal is to classify images as either `waterbirds' or `landbirds', while their background ---either `water' or `land'--- can be spuriously correlated with the type of birds.
Formally, each example has label $y\in\mathcal{Y}=\{\pm1\}\equiv\{\text{waterbird},\text{landbird}\}$ and belongs to a group
$g\in{\mathcal{G}}=\{\pm1\}\equiv\{\text{water},\text{land}\}$. Let then $s=(y,g) \in \{\pm1\}\times\{\pm1\}$ be the four sub-groups with $(+1,-1)$, $(-1,+1)$ being minorities (specifically, $\hat p_{+1,+1}=0.22,\hat p_{+1,-1}=0.012,\hat p_{-1,+1}=0.038$ and $\hat p_{-1,-1}=0.73.$). Denote $N_{s}$ the number of training examples belonging to sub-group $s$ and $N_{\text{max}}:=\max_s N_s$.
For notational consistency with Sec. \ref{sec:problem_setup}, we note that the imbalance here is in subgroups; thus, Group-VS-loss in \eqref{eq:loss_ours_bin_group} consists of logit adjustments that depend on the subgroup $s=(y,g)$.
\noindent\textbf{Model and Baselines.}
As in \cite{sagawa2019distributionally},
we train a ResNet50 starting with pretrained weights on Imagenet. Let $\beta_{s=(y,g)}=(N_{(y,g)}/N_{\text{max}})$. We propose training with the group-sensitive VS-loss in \eqref{eq:loss_ours_bin_group} with $\Delta_{y,g}=\Delta_{s}=\beta_{s}^\gamma$ and $\iota_{s}=-\beta_{s}^{-\gamma}$ with $\gamma=0.3$. We compare against CE and the DRO method of \cite{sagawa2019distributionally}. We also implement a new training scheme that combines Group-VS$+$DRO. We show additional results for Group-LA/CDT (not previously used in group contexts).
For fair comparison, we reran the baseline experiments with CE and report our reproduced numbers. Since class $+1$ has no special meaning here, we use
$\text{Symm-DEO}=(|\mathcal{R}_{(+1,+1)}-\mathcal{R}_{(+1,-1)}|+|\mathcal{R}_{(-1,+1)}-\mathcal{R}_{(-1,-1)}|)/2$ and also report balanced and worst sub-group accuracies.
We did not fine-tune $\gamma$ as the heuristic choice already shows the benefit of Group-VS-loss. We expect further improvements tuning over validation set.
\noindent\textbf{Results.}
Table \ref{table:waterbirds} reports test values obtained at last epoch ($300$ in total).
\begin{wraptable}{r}{0.55\textwidth}
\caption{Symmetric DEO, balanced and worst-case subgroup accuracies on Waterbirds dataset; averages over 10 runs, along with standard deviations.}
\label{table:waterbirds}
\begin{center}\resizebox{0.55\textwidth}{!}{
\begin{tabular}{lccc}
\toprule
\textbf{Loss} & \textbf{Symm. DEO} & \textbf{Bal. acc.} & \textbf{Worst acc.}\\
\toprule
CE & 25.3$\pm$0.66 & 84.9$\pm$0.29 & 68.1$\pm$2.2\\
Group LA & 24.0$\pm$2.4 & 84.2$\pm$3.0 & 70.1$\pm$2.6 \\
Group CDT & 18.5$\pm$0.46 & 87.2$\pm$1.2 & 75.4$\pm$2.2 \\
Group VS & \textbf{18.1$\pm$0.65} & \textbf{88.1$\pm$0.38} & \textbf{76.7$\pm$2.3}\\ \midrule
CE + DRO & 16.3$\pm$0.37 & 88.7$\pm$0.31 & 75.2$\pm$2.1\\
Group LA + DRO & 16.3$\pm$0.82 & 88.7$\pm$0.40 & 74.3$\pm$2.5\\
Group CDT + DRO & \textbf{11.7$\pm$0.15} & \textbf{90.3$\pm$0.2} & \textbf{79.9$\pm$1.5} \\
Group VS + DRO & {11.8$\pm$0.70} & {90.2$\pm$0.22} & {78.9$\pm$1.0} \\
\bottomrule
\end{tabular}}
\end{center}
\end{wraptable}
Our Group-VS loss significantly improves performance (measured with all three fairness metrics)
over CE, providing a cure for the poor CE performance under overparameterization reported in \cite{sagawa2020investigation}.
Group-CDT/VS have comparable performances, with or without DRO. Also, both outperform Group-LA that only uses additive adjustments. While these conclusions hold for the specific heuristic tuning of $\iota_y$'s, $\Delta_y$'s described above, they are in alignment with our Theorem \ref{propo:gd}.
Interestingly, Group-VS improves by a small margin the worst accuracy over CE+DRO, despite the latter being specifically designed to minimize that objective.
Our proposed Group-VS + DRO outperforms the CE+DRO algorithm used in \cite{sagawa2019distributionally} when training continues in TPT.
Finally, Symm. DEO appears correlated with balanced accuracy, in alignment with our discussion in Sec. \ref{sec:generalization} (see Fig. \ref{fig:tradeoffs}(a)).
\section{Further details and additional experiments on group-imbalances}\label{sec:exp_group_app}
\subsection{Deep-net experiments}
In this section, we elaborate on our proposed method of combining our group logit-adjusted losses with the DRO method. In all experiments, we chose $\Delta_{s}=(N_{s}/N_{\max})^\gamma$, $\iota_{s}=-(N_{s}/N_{\max})^{-\gamma}$ with $\gamma=0.3$. For example, Group-LA has $\iota_s=-(N_{s}/N_{\max})^{-0.3}$ and $\Delta_s=0$.
\noindent\textbf{Group-VS+DRO algorithm.}~For completeness, we elaborate on our proposed method of combining DRO with our Group VS-loss (see bottom half of Table \ref{table:waterbirds}). We recall from \cite{sagawa2019distributionally} that their proposed CE+DRO algorithm seeks a model that minimizes the worst subgroup empirical risk by instead minimizing the worst subgroup CE-loss:
$
\max_{s\in\mathcal{S}}\mathbb{E}_{(\mathbf{x},y)\sim\hat P_s}[\ell_{\rm CE}(y,f_\mathbf{w}(\mathbf{x}))],
$
where $\hat P_s$ is the empirical distribution on training samples from subgroup $s$. Instead, our Group-VS+DRO method attempts to solve the following distributionally robust optimization problem:
\begin{align*}
\min_\mathbf{w}\max_{s\in\mathcal{S}}\,\mathbb{E}_{(\mathbf{x},y)\sim\hat P_s}[\ell_{\rm Group-VS}(y,s,f_\mathbf{w}(\mathbf{x}))],
\end{align*}
with $\ell_{\rm Group-VS}(y,s,f_\mathbf{w}(\mathbf{x}))=\omega_{s}\cdot\log\big(1+e^{{\iota_{s}}}\cdot e^{-{\Delta_{s}} yf_\mathbf{w}(\mathbf{x})}\big)$ (see Equation \eqref{eq:loss_ours_bin_group}). To solve the above non-convex non-differentiable minimization, we employ the same online optimization algorithm given in \cite[Algorithm 1]{sagawa2019distributionally}, but changing the CE loss to the Group-VS.
\subsection{GS-SVM experiments}
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=0.65\textwidth]{waterbirds_svm_worst_group_error_new.pdf}
\caption{Worst case sub-group error}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=0.65\textwidth]{waterbirds_svm_misclas_error_new.pdf}
\caption{Misclassification error}
\end{subfigure}
\end{center}
\caption{The benefit of GS-SVM (corr. Group-VS loss) compared to SVM (corr. wCE) in achieving smaller \emph{worst case sub-group error} without significant loss on the misclassification error in the Waterbirds dataset. Training a linear model with $N$-dimensional Random-feature map over pretrained ResNet-18 features as in \cite{sagawa2020investigation}. }
\label{fig:waterbirds_expt}
\end{figure}
Section \ref{sec:GS_exp_main} demonstrated, for a deep-net model trained on the Waterbird dataset, the efficacy of the Group-VS loss compared to the CE and DRO algorithms used in \cite{sagawa2019distributionally}. Here, we follow \cite{sagawa2020investigation} who, similar to us, focused in overparameterized training in the TPT. Specifically, \cite{sagawa2020investigation} showed that wCE trained on a Random-feature model applied on top of a pretrained ResNet results in large \emph{worst-group error} when trained in TPT. In their analysis, they observed that this is because weighted logistic loss in the separable regime behaves like SVM, which is insensitive to groups. Here, we repeat their experiment only this time we use the Group-VS loss. In line with our results thus far, Group-VS loss shows improved performance in this setting as well.
\noindent\textbf{Algorithm.}~Concretely, since we are training linear models (on random feature maps), we know from Theorem \ref{propo:gd} that Group-VS loss converges to GS-SVM. Thus, for simplicity, we directly trained the following instance of GS-SVM and compared it against SVM:
\begin{align}\label{eq:gs_svm_waterbirds}
~~\min_{\mathbf{w}}~~\|\mathbf{w}\|_2 \qquad\text{sub. to}~~y_i(h(\mathbf{x}_i)^T\mathbf{w}+b)\geq \delta_{s_i}, ~i\in[n].
\end{align}
Above, $\delta_{s_i}=\delta_{(y_i,g_i)}=(\frac{1}{\hat p_{(y,g)}})^4$, $h:\mathcal{X}\rightarrow\mathbb{R}^N$ is the random-feature map (see Section \ref{sec:MNIST}), and
$\mathbf{x}_i,i \in[n]$ are $d$-dimensional pretrained ResNet18 features (same as those used in \cite{sagawa2020investigation}). Here, $n=4795$, $N$ took a range of values from $500$ to $10000$ and $d=512$. For those values of $N$ the data are separable, thus SVM/GS-SVM are feasible.
\noindent\textbf{Experiment \#1: GS-SVM vs SVM (or, Group-VS vs wCE).}~Figure \ref{fig:waterbirds_expt} shows worst-group and missclassification errors of GS-SVM and SVM as a function of the feature dimension $N$. The curves show averages over $10$ realizations of the random projection matrix along with standard deviations depicted using shaded error-bars. We confirm that:
\begin{itemize}
\item GS-SVM consistently outperforms standard SVM in the overparameterized regime in terms of worst-group error
\item This gain comes without significant losses on the misclassification error.
\end{itemize}
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[width=0.44\textwidth]{waterbirds_svm_gs_ss_misclas_error.pdf}
\caption{misclassification error}
\end{subfigure}
\\
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{waterbirds_svm_gs_ss_group0_error.pdf}
\caption{Sub-group-0 error}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{waterbirds_svm_gs_ss_group1_error.pdf}
\caption{Sub-group-1 error}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{waterbirds_svm_gs_ss_group2_error.pdf}
\caption{Sub-group-2 error}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{waterbirds_svm_gs_ss_group3_error.pdf}
\caption{Sub-group-3 error}
\end{subfigure}
\end{center}
\caption{Misclassification and conditional sub-group errors of SVM (blue), GS-SVM with heuristic tuning $\delta_{(y,g)}={p^{-4}_{(y,g)}}$ (red), and, SVM with subsampling (green) for the Waterbirds dataset. GS-SVM has lower worst-case error (Sub-group-2) compared to the SVM without significant increase on the misclassification error. SVM with subsampling has the best worst-group error performance, but also worst misclassification error in subfigure (a).}
\label{fig:waterbirds_group_errors_svm_gs_ss}
\end{figure}
\noindent\textbf{Experiment \#2: GS-SVM vs Sub-sampling.}~
As a means of improving over wCE, \cite{sagawa2020investigation} proposed instead the use of \emph{CE with subsampling,} for better \emph{worst case sub-group error}. In Figure \ref{fig:waterbirds_group_errors_svm_gs_ss} we compare the performance of three algorithms: (i) SVM, (ii) GS-SVM, and (iii) SVM with subsampling (corresponding to CE with subsampling). For the latter,
we chose $56$ examples from every sub-group (this is the size of the smallest sub-group) and ran SVM on the resulting (smaller), now balanced, dataset. Figure \ref{fig:waterbirds_group_errors_svm_gs_ss} reports missclassification error, as well as, \emph{conditional sub-group errors}. Recall that in the original dataset, sub-groups- 0 and 3 were the \emph{majority} with 3498 and 1057 examples respectively, while sub-groups 1 and 2 were the \emph{minorities} with 184 and 56 examples, respectively. We find the following:
\begin{itemize}
\item Consistent with \cite{sagawa2020investigation} SVM with subsampling achieves low \emph{worst case sub-group error}, lower than both SVM and GS-SVM (at least, when tuned with $\delta_{(y,a)}=\big(\frac{1}{p_{(y,a)}}\big)^4$).
\item Specifically, note the very low errors achieved by SVM with subsampling for minority sub-groups 2 and 3.
\item However, the gain comes at a significant cost paid for the majority sub-groups- 1 and 3 resulting in an increase of the misclassification error by more than $3-$ times compared to standard SVM and GS-SVM.
\end{itemize}
We expect that, with more careful tuning of the hyper-parameters $\delta_{(y,g)}$, GS-SVM can eventually achieve even lower \emph{sub-group errors} for the minority sub-groups without hurting the \emph{majority sub-group errors} significantly. We leave this to future work.
\subsection{Analysis of GS-SVM}\label{sec:sharp_GS}
Now, we focus on the group-sensitive case to characterize the DEO of GS-SVM.
\textbf{Sharp asymptotics:}~First, we need the following definitions. Let $G,Y$ as defined in Sec. \ref{sec:CS-SVM_sharp}. Fix $\delta>0$. Define additional random variables $GS, \Delta_S\in\mathbb{R},$ and $E_{S}\in\mathbb{R}^{2\times 1}$ as follows:
Fix $\delta>0$. Define random variables $G, Y, S, \Delta_S\in\mathbb{R},$ and $E_{S}\in\mathbb{R}^{2\times 1}$ as follows:
\begin{equation*}
\begin{cases}
~\text{$S$ takes values $1$ or $2$ with probabilities $p$ and $1-p$, respectively}\\
~E_{S}=\mathbf{e}_1\ind{S=1}+\mathbf{e}_2\ind{S=2}\quad\text{and}\quad\Delta_S=\delta\cdot\ind{S=1} +1\cdot\ind{S=2},
\end{cases}
\end{equation*}
and function $\widetilde\eta_\delta:\mathbb{R}_{\geq0}\times{\mathcal{B}}_2^{r}\times\mathbb{R}\rightarrow\mathbb{R}$ as
$
\widetilde\eta_\delta(q_\delta,\boldsymbol\rho_\delta,b_\delta):=\mathbb{E}\big(G+E_{S}^T\mathbf{V}\mathbf{S}\tilde \rhob + \frac{\tilde b Y-\Delta_S}{\tilde q}\big)_{-}^2 \notag- (1-\|\tilde \rhob\|_2^2)\gamma.
$
\begin{theorem}[Equal Opportunity of GS-SVM]\label{thm:main_group_fairness}
Let the data model and learning regime as described above.
Consider the GS-SVM classifier with $h(\mathbf{x})=\mathbf{x}$, intercept $b$ and fixed margin-ratio $\delta>0$ (corresponding to group VS-loss with $\Delta_{y,g}=\Delta_{g}, g=1,2$ such that $\delta=\Delta_2/\Delta_1$).
Let $(\widetilde q_\delta,\widetilde \boldsymbol\rho_\delta, \widetilde b_\delta)$ be the unique triplet satisfying same equations as in Theorem \ref{thm:main_imbalance}, but with $\eta_\delta$ replaced with the function $\widetilde \eta_\delta$ above.
Then, in the limit of $n,d\rightarrow\infty, d/n=\gamma>\widetilde\gamma_{\star}$ it holds
$
\mathcal{R}_{\pm,i} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_i^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta\pm \widetilde b_\delta/{\widetilde q_\delta}}\big), i=1,2.$ In particular, the difference of equal opportunity (DEO) satisfies
$
\mathcal{R}_{\rm deo} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_1^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big) - Q\big({\mathbf{e}_2^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big).$
\end{theorem}
The proof (given in the SM) differs to that of Theorem \ref{thm:main_imbalance} since now each class itself is a mixture of Gaussians.
\noindent\textbf{Tradeoffs:}~Our theorem allows us to study tradeoffs between DEO and accuracy in Fig. \ref{fig:fairness_deo_misclas_tradeoff}(b). Specifically, we investigate the effect of $\delta$ and the improvement of GS-SVM over standard SVM (corresponding to $\delta=1$). The largest DEO and smallest misclassification error are achieved by the SVM ($\delta=1$). But, with increasing $\delta$, misclassification error is traded-off for reduction in absolute value of DEO. Interestingly, for some $\delta_0=\delta_0(\gamma)$ (with value increasing as function of $\gamma$) GS-SVM guarantees Equal Opportunity (EO) $\mathcal{R}_{\rm deo}=0$ (without explicitly imposing such constraints as in \cite{olfat2018spectral,donini2018empirical}).
\section{Additional Experiments on Label-Imbalanced Datasets}\label{sec:exp_label_app}
In this section, we provide omitted information on the results of Section \ref{sec:exp_label}, as well, as additional experiments.
\subsection{Deep-net experiments}
Here we provide additional implementation details and a more extensive discussion on the results presented in Table \ref{table:CIFAR} in Section \ref{sec:exp_label} of the main text.
\noindent\textbf{Technical details:} Following \cite{TengyuMa}, we train a ResNet-32 \cite{he2016deep}, using batch size $128$ and SGD with momentum $0.9$ and weight decay $2\times10^{-4}$. For the first $5$ epochs we use a linear warm up schedule until baseline learning rate of $0.1$. We train for a total of 200 epochs, while decaying our learning rate by $0.1$ at epochs $160$ and $180.$ For STEP-100 imbalance we trained for $300$ rather than $200$ epochs and adjusted the learning rate accordingly as we found this type of imbalance more difficult to learn. We remark that the values for LDAM (adapted from \cite{TengyuMa}) used learning rate decay 0.01 and last-layer feature/classifier normalization. We have found convergence difficult otherwise. For other losses, we do \emph{not} use the above normalization of weights to isolate the impact of loss modifications.
\noindent\textbf{Implementation details.}~A seed is used for each of the $5$ runs and the weights of the network are initialized with the same values for all the losses that we train. We only show $95\%$ confidence intervals for CE, LA, CDT and VS losses which we implemented. For the remaining algorithms (e.g., LDAM), we report averages over $5$ realizations as given in \cite{TengyuMa}. For LA, CDT and VS losses, we have tuned the hyper-parameters $(\tau,\gamma)$ over the validation set as described in Section \ref{sec:exp_label} (see Remark \ref{rem:taugamma} and Table \ref{table:hyperparameters}). More sophisticated tuning strategies over the validation set (e.g., based on bilevel optimization or Hyperband \cite{lorraine2020optimizing,li2017hyperband}) and the corresponding performance assessment on test set are left to future work. Same as in \cite{he2016deep, TengyuMa, Menon, CDT} before training we augment the data by padding the images to size $40 \times 40$, flipping them horizontally at random and then random cropping them to their original size. We use PyTorch \cite{paszke2017automatic} building on codes provided by \cite{TengyuMa,CDT}. Training is performed on 2 NVIDIA RTX-3080 GPUs.
\begin{remark}[On the $(\tau,\gamma)$ parameterization of $\iota_y$'s \& $\Delta_y$'s]\label{rem:taugamma} As mentioned in Section \ref{sec:exp_label}, our deep-net experiments with VS-loss for label-imbalances, use the following parameterization for the additive and multiplicative logit factors in terms of two hyperparameters $\tau$ and $\gamma$:
\begin{align}\label{eq:tau_gamma}
\iota_y = \tau\log(N_y/N_{\rm tot})\qquad\text{and}\qquad\Delta_y=(N_y/N_{\max})^\gamma,
\end{align}
where $N_y$ is the train-sample size of class $y$, $N_{\max}=\max_y N_y$ and $N_{\rm tot}=\sum_y N_y$. This parameterizations follow \cite{Menon} and \cite{CDT}, respectively. A convenient feature is that setting $\tau=0$ recovers the CDT-loss, and setting $\gamma=0$ recovers the LA-loss.
\end{remark}
\noindent\textbf{Results and discussion.}~Table \ref{table:CIFAR} shows that our VS-loss performs favorably over the other methods across all experiments. The margins of improvement depend on the dataset / imbalance-type. Also, observe that in most cases LA-loss performs better than CDT-loss. This is likely because the CDT loss enters the TPT slower for the shown amount of training. Interestingly, VS-loss, even though it resembles the CDT-loss in the fact that it also adjusts the logits multiplicatively, does \emph{not} seem to suffer from the same problem. In Section \ref{sec:insights}, we presented experiments showing that: (i) If given enough time to train, CDT-loss can achieve similar or better results than LA-loss. (ii) The addition of the $\iota_y$'s in the VS-loss can mitigate the effect of $\Delta_y$ on the speed of convergence. In that sense, VS-loss fulfills the theoretical intuition in Section \ref{sec:insights}, as the method that combines additive and multiplicative adjustments for high accuracy and fast convergence.
\noindent\textbf{Tuning results.}~To promote reproducibility of our results and to give some insight on the range of $\tau$ and $\gamma$, in Table \ref{table:hyperparameters} we present the values of the hyperparameters that we determined through tuning and used to generate Table \ref{table:CIFAR}. As we discussed in Sec. \ref{sec:insights}, large values of $\tau$ and $\gamma$ can hinder training. Thus, when training with the VS loss, which adjusts the logits both in an additive and in a multiplicative way, it seems beneficial to use smaller values of these parameters, than when training with the LA or the CDT losses.
Additionaly, note that if searching over a grid, it is possible that the best values found for the VS-loss, will be the same as those of the LA or CDT losses (but never worse than them). Searching over a fine enough grid though should yield parameter values for which VS-loss outperforms both of them. Finally, note that the $(\tau,\gamma)$-parameterization of the $\iota_y, \Delta_y$'s is itself restrictive and other alternatives might yield further improvements when combining both types of adjustments as observed in the other cases.
\begin{table}[t]
\caption{Hyperparameter tuning results for each dataset, imbalance profile and loss function.}
\label{table:hyperparameters}
\begin{center}{
\begin{tabular}{lllllllll} \hline
\textbf{Dataset} & \multicolumn{2}{c}{\textbf{CIFAR 10}} & \multicolumn{2}{c}{\textbf{CIFAR 100}} \\ \hline
\textbf{Imbalance Profile} & \multicolumn{1}{c}{\textbf{LT-100}} & \multicolumn{1}{c}{\textbf{STEP-100}} & \multicolumn{1}{c}{\textbf{LT-100}} & \multicolumn{1}{c}{\textbf{STEP-100}} \\ \hline \hline
LA ($\tau=\tau^{*}$) \cite{Menon} & $2.25$ & $2.25$ & $1.375$ & $0.875$ \\
CDT ($\gamma=\gamma^{*}$) \cite{CDT} & $0.4$ & $0.3$ & $0.1$ & $0.1$ \\ \hline \hline
VS ($\tau=\tau^{*}, \gamma=\gamma^{*}$) & $(1.25, 0.15)$ & $(1.5, 0.2)$ & $(0.75, 0.05)$ & $(0.5, 0.05)$ \\
\end{tabular}}
\end{center}
\end{table}
\input{mnist_Orestis}
\section{Insights on the VS-loss}\label{sec:insights_gen}
\vspace{-0.1in}
Here, we shed light on the distinct roles of the VS-loss hyper-parameters $\omega_y, \iota_y$ and $\Delta_y$.
\vspace{-0.1in}
\subsection{CDT-loss vs LA-loss: Why multiplicative weights?}\label{sec:insights}
\vspace{-0.1in}
We first demonstrate the unique role played by the multiplicative weights $\Delta_y$ through a motivating experiment on synthetic data in Fig. \ref{fig:mismatch_intro}.
We generated a binary Gaussian-mixture dataset of $n=100$ examples in $\mathbb{R}^{300}$ with data means sampled independently from the Gaussian distribution and normalized such that $\|\boldsymbol\mu_{+1}\|_2=2\|\boldsymbol\mu_{-1}\|_2=4$. We set prior $\pi_+=0.1$ for the minority class $+1$. For varying model size values $p\in[5:5:50\,,\,75:25:300]$ we trained linear classifier $f_\mathbf{w}(x)=\inp{\mathbf{w}}{h(\mathbf{x})}$ using only the first $p$ features, i.e. $h(\mathbf{x})=\mathbf{x}(1:p)\in\mathbb{R}^p$. This allows us to investigate performance versus the parameterization ratio $\gamma=p/n.$ \footnote{Such simple models have been used in e.g. \cite{hastie2019surprises,deng2019model,chang2020provable,dhifallah2020precise,Sur14516} for analytic studies of double descent \cite{belkin2018understand,nakkiran2019deep} in terms ofclassification error. Fig. \ref{fig:mismatch_intro}(a) reveals a double descent for the balanced error.} We train the model $\mathbf{w}$ using the following special cases of the VS-loss (Eqn.~\eqref{eq:loss_ours_bin}): (i) \emph{CDT-loss} with $\Delta_+=\delta_\star^{-1}, \Delta_-=1$ ($\delta_\star>0$ is set to the value shown in the inset plot; {see SM for details}). (ii) \emph{LDAM-loss:} $\iota_+=\pi^{-1/4}, \iota_-=(1-\pi)^{-1/4}$ (special case of \emph{LA-loss} \cite{TengyuMa}). (iii) \emph{LA loss:} $\iota_+=\log\big(\frac{1-\pi}{\pi}\big), \iota_-=\log\big(\frac{\pi}{1-\pi}\big)$ (Fisher-consistent values \cite{Menon}). We ran gradient descent and averaged over $25$ independent experiments.
The \emph{balanced error} was computed on a test set of size $10^4$
and reported values are shown in red/blue/black markers.
We also plot the training errors, which are zero for $\gamma\gtrsim0.45$. The shaded region highlights the transition to the overparameterized / separable regime.
In this regime, we continued training in the TPT.
The plots reveal the following clear message:
\emph{The CDT-loss has better balanced-error performance compared to the LA-loss when both trained in TPT.} Moreover, they offer an intuitive explanation by uncovering a connection to max-margin classifiers: \emph{In the TPT,
(a) LA-loss performs the same as SVM, and, (b) CDT-loss performs the same as CS-SVM.}
We formalize those empirical observations in the theorem below, which holds for arbitrary linearly separable datasets (beyond Gaussian mixtures of the experiment). Specifically, for a sequence of norm-constrained minimizations of the VS-loss, we show that: As the norm constraint $R$ increases (thus, the problem approaches the original unconstrained loss), the direction of the constrained minimizer $\mathbf{w}_R$ converges to that of the CS-SVM solution $\hat{\w}_{\Delta_-/\Delta_+}$.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.85\textwidth,height=0.3\textwidth]{Story_withCE6.png}
\end{center}
\caption{Insights on various cost-sensitive modifications of the CE-loss. \textbf{(a)} CDT has superior balanced-error performance over LA in the separable regime.
Also, its performance matches that of CS-SVM, unlike LA matching SVM; Sec. \ref{sec:insights} for more details. Solid lines follow theory of Sec. \ref{sec:generalization}.
\textbf{(b)} Although critical in TPT, multiplicative weights (aka CDT) can harm minority classes in initial phase of training by guiding the classifier in the wrong direction. Properly tuned additive weights (aka LA) can mitigate this effect and speed up convergence. This explains why VS can be superior compared to CDT (see Observation \ref{obs:CDT_bad}).
Dashed lines show where TPT starts for each loss. \textbf{(c)} CDT and VS converge to CS-SVM, unlike LA and wCE. We prove this in Theorem \ref{propo:gd}.
}
\label{fig:mismatch_intro}
\end{figure}
\begin{theorem}[VS-loss=CS-SVM]\label{propo:gd} Fix a binary training set $\{\mathbf{x}_i,y_i\}_{i=1}^n$ with at least one example from each of the two classes. Assume feature map $h(\cdot)$ such that the data are linearly separable, that is $\exists \mathbf{w}: y_i\mathbf{w}^Th(\mathbf{x}_i)\geq1, \forall i\in[n].$ Consider training a linear model $f_\mathbf{w}(\mathbf{x})=\inp{\mathbf{w}}{h(\mathbf{x})}$ by minimizing the VS-loss $\mathcal{L}_n(\mathbf{w}):=\sum_{i\in[n]}\ell_{\rm VS}(y_i,f_\mathbf{w}(\mathbf{x}_i))$ with $\ell_{\rm VS}$ defined in \eqref{eq:loss_ours_bin} for positive parameters $\Delta_\pm, \omega_\pm \geq 0$ and arbitrary $\iota_{\pm}$. Define the norm-constrained optimal classifier
$
\mathbf{w}_R=\arg\min_{\|{\mathbf{w}}\|_2\leq R}\mathcal{L}_n(\mathbf{w}).
$
Let $\hat{\w}_\delta$ be the CS-SVM solution of \eqref{eq:CS-SVM} with $\delta=\Delta_-/\Delta_+$. Then,
$
\lim_{R\rightarrow\infty}{\mathbf{w}_R}\big/{\|\mathbf{w}_R\|_2} = {\hat\mathbf{w}_\delta}\big/{\|\hat\mathbf{w}_\delta\|_2}
$
\end{theorem}
\vspace{-0.1in}
On the one hand, the theorem makes clear that $\omega_\pm$ and $\iota_\pm$ become ineffective in the TPT as they all result in the same SVM solutions.
On the other hand, the multiplicative parameters $\Delta_\pm$ lead to the same classifier as that of CS-SVM, thus favoring solutions that move the classifier towards the majority class provided that $\Delta_->\Delta_+\Leftrightarrow \delta>1.$
The proof is given in the SM together with extensions for multiclass datasets.
In the SM, we also strengthen Theorem \ref{propo:gd} by characterizing the \emph{implicit bias} of gradient-flow on VS-loss. Finally, we show that group-sensitive VS-loss with $\Delta_{y,g}=\Delta_g$ converges to the corresponding GS-SVM.
\begin{remark}
Thm \ref{propo:gd} is reminiscent of Thm. 2.1 in \cite{rosset2003margin} who
showed for a regularized ERM with CE-loss that when the regularization parameter vanishes, the normalized solution converges to the SVM classifier. Our result connects nicely to \cite{rosset2003margin} extending their theory to VS-loss / CS-SVM, as well as, to the group-case.
In a similar way, our result on the implicit bias of gradient-flow on the VS-loss connects to more recent works \cite{soudry2018implicit,ji2018risk} that pioneered corresponding results for CE-loss. Although related, our results on the properties of the VS-loss are not obtained as special cases of these existing works. As a final remark, in Fig. \ref{fig:mismatch_intro}(b,c) we kept constant learning rate $0.1$. Significantly faster convergence is observed with normalized GD schemes
\cite{nacson2019convergence,ji2021characterizing}; see the SM for a detailed numerical study. We also note that Thm. \ref{propo:gd} gives a modern interpretation to the CS-SVM via the lens of implicit bias theory.
\end{remark}
\vspace{-0.1in}
\subsection{VS-loss: Best of two worlds}\label{sec:best}
\vspace{-0.1in}
We have shown that multiplicative weights $\Delta_\pm$ are responsible for good balanced accuracy in the TPT. Here, we show that, at the initial phase of training, the same multiplicative weights can actually harm the minority classes. The following observation supports this claim.
\begin{observation}\label{obs:CDT_bad}
Assume $f_\mathbf{w}(x)=0$ at initialization. Then, the gradients of CDT-loss with multiplicative logit factors $\Delta_y$ are \underline{identical} to the gradients of wCE-loss with weights $\omega_y=\Delta_y.$ Thus, we conclude the following where say $y=+1$ is minority. On the one hand, wCE, which typically sets $\omega_{\rm +}>\omega_{\rm -}$ (e.g., $\omega_y=1/\pi_y$), {helps minority examples by weighing down the loss over majority}. On the other hand, the CDT-loss requires {the reverse direction} ${\Delta_{+}<\Delta_{-}}$ as per Theorem \ref{propo:gd}, thus initially it {guides} the classifier in the wrong direction {to penalize} minorities.
\end{observation}
To see why the above is true note that for $f_\mathbf{w}(x)=\inp{\mathbf{w}}{h(\mathbf{x})}$ the gradient of VS-loss is $\nabla_\mathbf{w}\ell_{\rm VS}(y,f_\mathbf{w}(\mathbf{x}))=-\omega_y \Delta_y\,\sigma\big(-\Delta_y y f_\mathbf{w}(x)+\iota_y\big)\cdot {y h(\mathbf{x})}$ where $\sigma(t)=(1+\exp(-t))^{-1}$ is the sigmoid function. It is then clear that at $f_\mathbf{w}(\mathbf{x})=0$, the logit factor $\Delta_y$ plays the same role as the weight $\omega_y$. From Theorem \ref{propo:gd}, we know that pushing the margin towards majorities (which favors balancing the conditional errors) requires $\Delta_+<\Delta_-$. Thus, gradient of minorities becomes smaller, initially pushing the optimization in the wrong direction. Now, we turn our focus at the impact of $\iota_y$'s at the start of training. Noting that $\sigma(\cdot)$ is increasing function, we see that setting $\iota_+>\iota_-$ increases the gradient norm for minorities. This leads us to a second observation:
\emph{By properly tuning the additive logit adjustments $\iota_y$ we can counter the initial negative effect of the multiplicative adjustment, thus speeding up training.}
The observations above naturally motivated us to formulate the VS-loss in Eqn. \eqref{eq:loss_ours} bringing together the best of two worlds: the $\Delta_y$'s that play a critical role in the TPT and the $\iota_y$'s that compensate for the harmful effect of the $\Delta_y$'s in the beginning of training.
Figure \ref{fig:mismatch_intro}(b,c) illustrate the discussion above. In the binary linear classification setting of Fig. \ref{fig:mismatch_intro}(a), we investigate the effect of the additive adjustments on the training dynamics. Specifically, we trained using gradient descent: (i) \emph{CE}; (ii) \emph{wCE} with $\omega_y = 1/\pi_y$; (iii) \emph{LA-loss} with $\iota_y=\log(1/\pi_{y})$; (iv) \emph{CDT-loss} with $\Delta_+=\delta_\star^{-1}, \Delta_-=1$; (v) \emph{VS-loss} with $\Delta_+=\delta_\star^{-1}, \Delta_-=1$, $\iota_y=\log(1/\pi_{y})$ and $\omega_y=1$; (vi) \emph{VS-loss} with same $\Delta$'s, $\iota_y=0$ and $\omega_y=1/\pi_y.$
Figures \ref{fig:mismatch_intro}(b) and (c) plot balanced test error $\Rc_{\text{bal}}$ and angle-gap to CS-SVM solution as a function of iteration number for each algorithm. The vertical dashed lines mark the iteration after which training error stays zero and we enter the TPT. Observe in Fig. \ref{fig:mismatch_intro}(c) that CDT/VS-losses, both converge to the CS-SVM solution as TPT progresses verifying Theorem \ref{propo:gd}. This also results in lowest test error in the TPT in Fig. \ref{fig:mismatch_intro}(b). However, compared to CDT-loss, the VS-loss enters faster in the TPT and converges orders of magnitude faster to small values of $\Rc_{\text{bal}}$. Note in Fig. \ref{fig:mismatch_intro}(c) that this behavior is correlated with the speed at which the two losses converge to CS-SVM. Following the discussion above, we attribute this favorable behavior during the initial phase of training to the inclusion of the $\iota_y$'s. This is also supported by Fig. \ref{fig:mismatch_intro}(c) as we see that LA-loss (but also wCE) achieves significantly better values of $\Rc_{\text{bal}}$ at the first stage of training compared to CDT-loss. In Sec. \ref{sec:exp_label} we provide deep-net experiments on an imbalanced CIFAR-10 dataset that further support these findings.
\section{Margin properties and implicit bias of VS-loss}\label{sec:propo_proof}
\subsection{A more general version and proof of Theorem \ref{propo:gd}}\label{sec:gen_gd}
We will state and prove a more general theorem to which Theorem \ref{propo:gd} is a corollary. The new theorem also shows that the group-sensitive adjusted VS-loss in \eqref{eq:loss_ours_bin_group} converges to the GS-SVM
\begin{remark} Theorem \ref{propo:gd} and the content of this section are true for \emph{arbitrary} linear models $\mathbf{f}_\mathbf{w}(\mathbf{x})=\inp{h(\mathbf{x})}{\mathbf{w}}$ and feature maps $h:\mathcal{X}\rightarrow\mathbb{R}^p.$ To lighten notation in the proofs, we assume for simplicity that $h$ is the identity map, that is $\mathbf{h}(\mathbf{x})=\mathbf{x}.$ For the general case, just substitute the raw features $\mathbf{x}_i\in\mathcal{X}$ below with their feature representation $h(\mathbf{x}_i)\in\mathbb{R}^p$.
\end{remark}
Consider the VS-loss empirical risk minimization (cf. \eqref{eq:loss_ours_bin} with $f(\mathbf{x})=\mathbf{w}^T\mathbf{x}$):
\begin{align}\label{eq:loss_app}
\mathcal{L}(\mathbf{w}):=\sum_{i\in[n]}\ell(y_i,\mathbf{w}^T\mathbf{x}_i,g_i):=\omega_i\log\left(1+e^{\iota_i}\cdot e^{- \Delta_i y_i(\mathbf{w}^T\mathbf{x}_i)}\right).
\end{align}
for strictly positive (but otherwise arbitrary) parameters $\Delta_i, \omega_i > 0$ and arbitrary $\iota_{i}$. For example, setting $\omega_i=\omega_{y_i,g_i}, \Delta_i = \Delta_{y_i,g_i}$ and $\iota_i=\iota_{y_i,g_i}$ recovers the general form of our binary VS-loss in \eqref{eq:loss_ours_bin_group}.
Also, consider the following general cost-sensitive SVM (to which both the CS-SVM and the GS-SVM are special instances)
\begin{align}\label{eq:CS-SVM_app}
\hat{\w}:= \arg\min_\mathbf{w} \|\mathbf{w}\|_2\quad\text{subject to}~~y_i(\mathbf{w}^T\mathbf{x}_i)\geq {1}\big/{\Delta_i}, \forall i\in[n].
\end{align}
First, we state the following simple facts about the cost-sensitive max-margin classifier in \eqref{eq:CS-SVM_app}. The proof of this claim is rather standard and is included in Section \ref{sec:CS_svm_easy_claim} for completeness.
\begin{lemma}\label{lem:CS_svm_easy_claim}
Assume that the training dataset is linearly separable, i.e. $\exists \mathbf{w}$ such that $y_i(\mathbf{w}^T\mathbf{x}_i)\geq 1$ for all $i\in[n]$. Then, \eqref{eq:CS-SVM_app} is feasible. Moreover, letting $\hat{\w}$ be the solution of \eqref{eq:CS-SVM_app}, it holds that
\begin{align}\label{eq:CS_svm_easy_claim}
\frac{\hat\mathbf{w}}{\|\hat\mathbf{w}\|_2} = \arg\max_{\|\mathbf{w}\|_2=1}\,\min_{i\in[n]}\, \Delta_{i}y_i\mathbf{x}_i^T\mathbf{w}.
\end{align}
\end{lemma}
Next, we state the main result of this section connecting the VS-loss in \eqref{eq:loss_app} to the max-margin classifier in \eqref{eq:CS-SVM_app}. After its statement, we show how it leads to Theorem \ref{propo:gd}; its proof is given later in Section \ref{sec:proof_gd}.
{
\begin{theorem}[Margin properties of VS-loss: General result]\label{thm:implicit_loss_gen}
Define the norm-constrained optimal classifier
\begin{align}\label{eq:con_ERM}
\mathbf{w}_R:=\arg\min_{\|{\mathbf{w}}\|_2\leq R}\mathcal{L}(\mathbf{w}),
\end{align}
with the loss $\mathcal{L}$ as defined in \eqref{eq:loss_app} for positive (but otherwise arbitrary) parameters $\Delta_i, \omega_i>0$ and arbitrary $\iota_i$.
Assume that the training dataset is linearly separable
and let $\hat{\w}$ be the solution of \eqref{eq:CS-SVM_app}.
Then, it holds that
\begin{align}\label{eq:implicit_2_prove}
\lim_{R\rightarrow\infty}\frac{\mathbf{w}_R}{\|\mathbf{w}_R\|_2} = \frac{\hat\mathbf{w}}{\|\hat\mathbf{w}\|_2} \,.
\end{align}
\end{theorem}
}
\subsubsection{Proof of Theorem \ref{propo:gd}}
Theorem \ref{propo:gd} is a corollary of Theorem \ref{thm:implicit_loss_gen} by setting $\omega_i=\omega_{y_i}$, $\iota_i=\iota_{y_i}$ and $\Delta_i=\Delta_{y_i}$. Indeed for this choice the loss in Equation \eqref{eq:loss_app} reduces to that in Equation \eqref{eq:loss_ours_bin}. Also, \eqref{eq:CS-SVM_app} reduces to \eqref{eq:CS-SVM}. The latter follows from the equivalence of the following two optimization problems:
\begin{align*}
&\big\{\,\,\arg\min_\mathbf{w} \|\mathbf{w}\|_2\quad\text{subject to}~~\mathbf{w}^T\mathbf{x}_i \begin{cases}\geq 1/\Delta_+ & y_i=+1 \\ \leq -1/\Delta_- & y_i=-1\end{cases}\,\, \big\}~~
\\
&\qquad\qquad\qquad=~~\big\{\,\,\arg\min_\mathbf{v} \|\mathbf{v}\|_2\quad\text{subject to}~~\mathbf{v}^T\mathbf{x}_i \begin{cases} \geq\Delta_-/\Delta_+ & y_i=+1 \\ \leq-1 & y_i=-1\end{cases} \,\,\big\},
\end{align*}
which can be verified simply by a change of variables $\mathbf{v}/\Delta_-\leftrightarrow\mathbf{w}$ and $\Delta_->0$.
\vspace{5pt}
\noindent{\textbf{The case of group-sensitive VS-loss.}~}
As another immediate corollary of Theorem \ref{thm:implicit_loss_gen} we get an analogue of Theorem \ref{propo:gd} for a group-imbalance data setting with $K=2$ and balanced classes. Then, we may use the VS-loss in \eqref{eq:loss_app} with margin parameters $\Delta_{i}=\Delta_g, g=1,2$.
From Theorem \ref{thm:implicit_loss_gen}, we know that in the separable regime and in the limit of increasing weights, the classifier $\mathbf{w}_R$ (normalized) will converge to the solution of the GS-SVM with $\delta=\Delta_2/\Delta_1.$
\subsubsection{Proof of Theorem \ref{thm:implicit_loss_gen}}\label{sec:proof_gd}
{First, we will argue that for any $R>0$ the solution to the constrained VS-loss minimization is on the boundary, i.e.
\begin{align}\label{eq:boundary}
\|\mathbf{w}_R\|_2=R.
\end{align}
We will prove this by contradiction. Assume to the contrary that $\mathbf{w}_R$ is a point in the strict interior of the feasible set. It must then be by convexity that $\nabla \mathcal{L}(\mathbf{w}_R)=0$. Let $\widetilde\w$ be any solution feasible in \eqref{eq:CS-SVM_app} (which exists as shown above) such that $y_i(\mathbf{x}_i^T\widetilde\w)\geq 1/\Delta_i$. On one hand, we have $\widetilde\w^T\nabla \mathcal{L}(\mathbf{w}_R)=0$. On the other hand, by positivity of $\omega_i,\Delta_i, \forall i\in[n]$:
\begin{align}\label{eq:no_finite_cp}
\widetilde\w^T\nabla \mathcal{L}(\mathbf{w}_R) = \sum_{i\in[n]}\underbrace{\frac{-\omega_{i} \Delta_{i}e^{-\Delta_i y_i\mathbf{x}_i^T\mathbf{w}_R+\iota_i}}{ 1+ e^{\iota_{i}}e^{-\Delta_i y_i\mathbf{x}_i^T\mathbf{w}_R}}}_{<0}\, \underbrace{y_i\widetilde\w^T\mathbf{x}_i}_{>0} <0,
\end{align}
which leads to a contradiction.}
Now, suppose that \eqref{eq:implicit_2_prove} is not true. This means that there is some $\epsilon_0>0$ such that there is always an arbitrarily large $R>0$ such that $\frac{\mathbf{w}_R^T\hat{\w}}{\|\mathbf{w}_R\|_2\|\hat{\w}\|_2}\leq 1-\epsilon_0$. Equivalently, (in view of \eqref{eq:boundary}):
\begin{align}\label{eq:corr_gd}
\frac{\mathbf{w}_R^T\hat{\w}}{R\|\hat{\w}\|_2}\leq 1-\epsilon_0.
\end{align}
Towards proving a contradiction, we will show that, in this scenario using $\hat{\w}_R=R\frac{\hat{\w}}{\tn{\hat{\w}}}$ yields a strictly smaller VS-loss (for sufficiently large $R>0$), i.e.
\begin{align}
\mathcal{L}(\hat{\w}_R) < \mathcal{L}(\mathbf{w}_R),\qquad\text{for sufficiently large $R$}.
\end{align}
We start by upper bounding $\mathcal{L}(\hat{\w}_R)$. To do this, we first note from definition of $\hat{\w}_R$ the following margin property:
\begin{align}\label{eq:key_margin}
y_i\hat{\w}_R^T\mathbf{x}_i = \frac{R}{\|\hat\mathbf{w}\|_2} y_i\hat{\w}^T\mathbf{x}_i \geq \frac{R}{\|\hat\mathbf{w}\|_2} (1/\Delta_i)=:\frac{\bar{R}}{\Delta_i},
\end{align}
where the inequality follows from feasibility of $\hat{\w}$ in \eqref{eq:CS-SVM_app} and we set $\bar{R}:=R/\|\hat{\w}\|_2$. Then, using \eqref{eq:key_margin} it follows immediately that
\begin{align}
\mathcal{L}(\hat{\w}_R)&=\sum_{i=1}^n \omega_{i}\log\left(1+ e^{\iota_i}e^{-\Delta_{i}y_i\hat{\w}_R^T\mathbf{x}_i}\right) \notag\\
&\leq \sum_{i=1}^n \omega_{i}\log\left(1+ e^{\iota_i}e^{-\frac{\bar{R}}{\Delta_i}\Delta_{i}}\right)\notag\\
&=\sum_{i=1}^n \omega_{i}\log\left(1+ e^{\iota_i}e^{-\bar{R}}\right) \notag\\
&\leq \omega_{\max}n e^{\iota_{\max}-\bar{R}}.\label{eq:ub}
\end{align}
In the first inequality above we used \eqref{eq:key_margin} and non-negativity of $\omega_i, \Delta_i \geq 0$. In the last line, we have called $\omega_{\max} := \max_{i\in[n]}\omega_{i}>0$ and $\iota_{\max} := \max_{i\in[n]}\iota_{i}>0$\,.
Next, we lower bound $\mathcal{L}(\mathbf{w}_R)$. To do this, consider the vector $$\bar{\mathbf{w}}=\frac{\tn{\hat{\w}}}{R}\mathbf{w}_R=\mathbf{w}_R/\bar{R}.$$ By feasibility of $\mathbf{w}_R$ (i.e. $\|\mathbf{w}_R\|_2\leq R$), note that $\|\bar\mathbf{w}\|_2 \leq \|\hat{\w}\|_2$. Also, from \eqref{eq:corr_gd}, {we know that $\bar\mathbf{w}\neq\hat{\w}$}. {Indeed, if it were $\bar\mathbf{w}=\hat{\w} \iff \hat{\w}/\|\hat{\w}\|_2 = \mathbf{w}_R/R$, then
$$
\frac{\hat{\w}^T\mathbf{w}_R}{R\|\hat{\w}\|_2} = 1,
$$
which would contradict \eqref{eq:corr_gd}. Thus, it must be that $\bar\mathbf{w}\neq\hat{\w}$.
From these and strong convexity of the objective function in \eqref{eq:CS-SVM_app}, it follows that $\bar\mathbf{w}$ must be \emph{infeasible} for \eqref{eq:CS-SVM}. Thus, there exists at least one example $\mathbf{x}_j,~j\in[n]$ and $\epsilon>0$ such that
\[
y_j\bar{\mathbf{w}}^T\mathbf{x}_j \leq(1-\epsilon)(1/\Delta_j).
\]
}
But then
\begin{align}
y_j\mathbf{w}_R^T\mathbf{x}_j \leq \bar{R}(1-\epsilon)(1/\Delta_j),\label{eq:point2}
\end{align}
which we can use to lower bound $\mathcal{L}(\mathbf{w}_R)$ as follows:
\begin{align}
\mathcal{L}(\mathbf{w}_R)&\geq \omega_{j} \log\left(1+ e^{\iota_{j}-\Delta_{j}y_j\mathbf{w}_R^T\mathbf{x}_j}\right)\notag\\
&\geq \omega_{j} \log\left(1+ e^{\iota_{y_j}-\bar{R}\Delta_{j}\frac{(1-\epsilon)}{\Delta_j}}\right)\notag\\
&\geq \omega_{\min}\log\left(1+ e^{\iota_{\min}-\bar{R}(1-\epsilon)}\right).\label{eq:lb}
\end{align}
The second inequality follows fron \eqref{eq:point2} and non-negativity of $\Delta_{\pm},\omega_{\pm}.$
To finish the proof we compare \eqref{eq:lb} against \eqref{eq:ub}. If $\epsilon\geq 1$, clearly $\mathcal{L}(\hat{\w}_R)<\mathcal{L}(\mathbf{w}_R)$ for sufficiently large $R$. Otherwise $e^{-\bar{R}(1-\epsilon)}\rightarrow 0$ with $R\rightarrow\infty$. Hence,
\[
\mathcal{L}(\mathbf{w}_R)\geq \omega_{\min}\log\left(1+ e^{\iota_{\min}-\bar{R}(1-\epsilon)}\right)\geq 0.5 \omega_{\min}e^{\iota_{\min}-\bar{R}(1-\epsilon)}.
\]
Thus, again
\[
\mathcal{L}(\hat{\w}_R)< \mathcal{L}(\mathbf{w}_R)\impliedby \omega_{\max}n e^{\iota_{\max}-\bar{R}}<0.5 \omega_{\min}e^{\iota_{\min}-\bar{R}(1-\epsilon)}\iff e^{\bar{R}\epsilon}>\frac{2n\omega_{\max}}{\omega_{\min}}e^{\iota_{\max}-\iota_{\min}},
\]
because the right side is true by picking $R$ arbitrarily large.
\subsubsection{Proof of Lemma \ref{lem:CS_svm_easy_claim}}\label{sec:CS_svm_easy_claim}
The proof of Lemma \ref{lem:CS_svm_easy_claim} is standard, but included here for completeness. The lemma has two statements and we prove them in the order in which they appear.
\vspace{5pt}
\noindent\textbf{Linear separability $\implies$ feasibility of \eqref{eq:CS-SVM_app}.} Assume $\mathbf{w}$ such that $y_i(\mathbf{w}^T\mathbf{x}_i)\geq 1$ for all $i\in[n]$, which exists by assumption. Define $M:=\max_{i\in[n]}\frac{1}{\Delta_i}>0$ and consider $\widetilde\w=M\mathbf{w}$. Then, we claim that $\widetilde\w$ is feasible for \eqref{eq:CS-SVM_app}. To check this, note that
\begin{align*}
y_i = +1 &~\implies~ \mathbf{x}_i^T\widetilde\w=M(\mathbf{x}_i^T\mathbf{w})\geq M\geq 1/\Delta_i\quad\text{since}~\mathbf{x}_i^T\mathbf{w}\geq 1,\\
y_i = -1 &~\implies~ \mathbf{x}_i^T\widetilde\w=M(\mathbf{x}_i^T\mathbf{w})\leq -M\leq -1/\Delta_i\quad\text{since}~\mathbf{x}_i^T\mathbf{w}\leq -1.
\end{align*}
Thus, $y_i(\mathbf{x}_i^T\widetilde\w)\geq 1/\Delta_i$ for all $i\in[n]$, as desired.
\vspace{5pt}
\noindent\textbf{Proof of \eqref{eq:CS_svm_easy_claim}.} For the sake of contradiction let $\widetilde\w\neq \frac{\hat\mathbf{w}}{\|\hat\mathbf{w}\|_2}$ be the solution to the max-min optimization in the RHS of \eqref{eq:CS_svm_easy_claim}. Specifically, this means that $\|\widetilde\w\|_2=1$ and
$$
\tilde m:= \min_{i\in[n]} \Delta_iy_i\mathbf{x}_i^T\widetilde\w > \min_{i\in[n]} \Delta_iy_i\mathbf{x}_i^T \frac{\hat\mathbf{w}}{\|\hat\mathbf{w}\|_2}=: m.
$$
We will prove that the vector $\mathbf{w}^\prime:=\widetilde\w/\tilde m$ is feasible in \eqref{eq:CS-SVM_app} and has smaller $\ell_2$-norm than $\hat\mathbf{w}$ contradicting the optimality of the latter. First, we check feasibility. Note that, by definition of $\tilde m$, for any $i\in[n]$:
$$
\Delta_iy_i\mathbf{x}_i^T\mathbf{w}^\prime = \frac{\Delta_iy_i\mathbf{x}_i^T\widetilde\w}{\tilde m} \geq 1,
$$
Second, we show that $\|\mathbf{w}^\prime\|_2< \|\hat \mathbf{w}\|_2$:
$$
\|\mathbf{w}^\prime\|_2 = \frac{\|\widetilde\w\|_2}{\tilde m} = \frac{1}{\tilde m} < \frac{1}{m} = \frac{\|\hat\mathbf{w}\|_2}{\min_{i\in[n]} \Delta_iy_i\mathbf{x}_i^T\hat\mathbf{w}} \leq \|\hat\mathbf{w}\|_2,
$$
where the last inequality follows by feasibility of $\hat\mathbf{w}$ in \eqref{eq:CS-SVM_app}. This completes the proof of the lemma.
\subsection{Multiclass extension}
In this section, we present a natural extension of Theorem \ref{thm:implicit_loss_gen} to the multiclass VS-loss in \eqref{eq:loss_ours}. Here, let we let the label set $\mathcal{Y}=\{1,2,\ldots,C\}$ for a $C$-class classification setting and consider the cross-entropy VS-loss:
\begin{align}\label{eq:loss_app_multi}
\mathcal{L}(\mathbf{W}):=\sum_{i\in[n]}\ell(y_i,\mathbf{w}_1^T\mathbf{x}_i,\ldots,\mathbf{w}_K^T\mathbf{x}_i) = \sum_{i\in[n]}\omega_{y_i}\log\Big(1+\sum_{\substack{y'\in[C]\\mathbf{y}'\neq y_i}}e^{\iota_{y'}-\iota_{y_i}}e^{- (\Delta_{y_i}\mathbf{w}_{y_i}^T\mathbf{x}_i-\Delta_{y'}\mathbf{w}_{y'}^T\mathbf{x}_i)}\Big),
\end{align}
where $\mathbf{W}=[\mathbf{w}_1,\ldots,\mathbf{w}_C]\in\mathbb{R}^{C\times d}$ and $\mathbf{w}_y$ is the classifier corresponding to class $y\in[C]$. We will also consider the following multiclass version of the CS-SVM in \eqref{eq:CS-SVM_app}:
\begin{align}\label{eq:CS-SVM_app_multi}
\hat\mathbf{W} = \arg\min_{\mathbf{W}} \|\mathbf{W}\|_F\quad\text{subject to }\mathbf{x}_i^T(\Delta_{y_i}\mathbf{w}_{y_i}-\Delta_{y'}\mathbf{w}_{y'})\geq 1, ~\forall y'\neq y_i\in[C]~\text{ and }~\forall i\in[n].
\end{align}
Similar to Lemma \ref{lem:CS_svm_easy_claim}, it can be easily checked that \eqref{eq:CS-SVM_app_multi} is feasible provided that the training data are separable, in the sense that
\begin{align}\label{eq:sep_multi}
\exists \mathbf{W}=[\mathbf{w}_1,\ldots,\mathbf{w}_K] \text{ suc that } \mathbf{x}_i^T(\mathbf{w}_{y_i}-\mathbf{w}_{y'})\geq1, \forall y'\in[C], y'\neq y_i~\text{ and }~\forall i\in[n].
\end{align}
Moreover, it holds that
$$
\hat\mathbf{W}/\|\hat\mathbf{W}\|_F = \arg\max_{\|\mathbf{W}\|_F=1}\min_{i\in[n]}\min_{y'\neq y_i} \,\mathbf{x}_i^T(\Delta_{y_i}\mathbf{w}_{y_i}-\Delta_{y'}\mathbf{w}_{y'}).
$$
The theorem below is an extension of Theorem \ref{thm:implicit_loss_gen} to multiclass classification.
{
\begin{theorem}[Margin properties of VS-loss: Multiclass]\label{thm:implicit_loss_multi}
Consider a $C$-class classification problem and define the norm-constrained optimal classifier
\begin{align}\label{eq:con_ERM_multi}
\mathbf{W}_R=\arg\min_{\|{\mathbf{W}}\|_F\leq R}\mathcal{L}(\mathbf{W}),
\end{align}
with the loss $\mathcal{L}$ as defined in \eqref{eq:loss_app_multi} for positive (but otherwise arbitrary) parameters $\Delta_y, \omega_y>0, y\in[C]$ and arbitrary $\iota_{y}, y\in[C]$.
Assume that the training dataset is linearly separable as in \eqref{eq:sep_multi}
and let $\hat\mathbf{W}$ be the solution of \eqref{eq:CS-SVM_app_multi}.
Then, it holds that
\begin{align}\label{eq:implicit_2_prove_multi}
\lim_{R\rightarrow\infty}\frac{\mathbf{W}_R}{\|\mathbf{W}_R\|_F} = \frac{\hat\mathbf{W}}{\|\hat\mathbf{W}\|_2} \,.
\end{align}
\end{theorem}
}
\begin{proof}
The proof follows the same steps as in the proof of Theorem \ref{thm:implicit_loss_gen}. Thus, we skip some details and outline only the basic calculations needed.
It is convenient to introduce the following notation, for $\ell\in[C]$:
$$
p(\ell | \mathbf{x}, y, \mathbf{W}):= \frac{e^{\iota_y}e^{\Delta_{y} \mathbf{x}^T\mathbf{w}_\ell}}{\sum_{y'\in[C]}e^{\iota_{y'}}e^{\Delta_{y'} \mathbf{x}^T\mathbf{w}_{y'}}}.
$$
In this notation,
$\mathcal{L}(\mathbf{W})=-\sum_{i\in[n]}\log\big(p(y_i|\mathbf{x}_i,y_i,\mathbf{W})\big)$ and
for all $\ell\in[C]$ it holds that
$$
\nabla_{\mathbf{w}_\ell}\mathcal{L}(\mathbf{W}) = \sum_{i\in[n]} \omega_{y_i}\Delta_{y_i} \left(p(\ell | \mathbf{x}_i, y_i, \mathbf{W}) - \ind{y_i=\ell} \right) \mathbf{x}_i.
$$
Thus, for any $\widetilde\mathbf{W}$ that is feasible in \eqref{eq:CS-SVM_app_multi}
\begin{align*}
\sum_{\ell\in[C]}\widetilde\mathbf{w}_\ell^T\nabla_{\mathbf{w}_\ell}\mathcal{L}(\mathbf{W}) &= \sum_{i\in[n]}\sum_{\ell\in[C]}\omega_{y_i}\Delta_{y_i} \left(p(\ell | \mathbf{x}_i, y_i, \mathbf{W}) - \ind{y_i=\ell} \right) \mathbf{x}_i^T\widetilde\mathbf{w}_\ell\\
&= \sum_{i\in[n]}\sum_{\ell\neq y_i}\omega_{y_i}\Delta_{y_i} p(\ell | \mathbf{x}_i, y_i, \mathbf{W}) \,\mathbf{x}_i^T\widetilde\mathbf{w}_\ell - \omega_{y_i}\Delta_{y_i}\left(1 - p(y_i | \mathbf{x}_i, y_i, \mathbf{W}) \right) \mathbf{x}_i^T\widetilde\mathbf{w}_{y_i}\\
&= \sum_{i\in[n]}\underbrace{-\omega_{y_i} \big(\sum_{\ell\neq y_i}p(\ell | \mathbf{x}_i, y_i, \mathbf{W})\big)}_{<0}\,\underbrace{\Delta_{y_i}\mathbf{x}_i^T\left(\widetilde\w_\ell-\widetilde\w_{y_i}\right)}_{>0}~<~0,
\end{align*}
where in the third line we used that $\sum_{\ell\in[C]}p(\ell | \mathbf{x}, y, \mathbf{W})=1$. With the above it can be shown following the exact same argument as in the proof of \eqref{eq:boundary} for the binary case that $\|\mathbf{W}_R\|_F=R$, the minimizer of \eqref{eq:con_ERM_multi} satisfies the constraint with equality.
The proof continues with a contradiction argument similar to the binary case. Assume the desired \eqref{eq:implicit_2_prove_multi} does not hold. We will then show that for $\hat\mathbf{W}_R=\frac{R}{\|\hat\mathbf{W}\|_F}\hat\mathbf{W}$ and sufficiently large $R>0$:
$\mathcal{L}(\hat\mathbf{W}_R)<\mathcal{L}(\mathbf{W}_R)$.
Using feasibility of $\hat\mathbf{W}$ in \eqref{eq:CS-SVM_app_multi} and defining $\omega_{\max}:=\max_{y\in[C]}\omega_y$ and $\iota_{\max}=\max_{y\neq y'\in[C]}\iota_{y'}-\iota_y$, it can be shown similar to \eqref{eq:ub} that
\begin{align}\label{eq:multi_proof_1}
\mathcal{L}(\hat\mathbf{W}_R)&=\sum_{i\in[n]}\omega_{y_i}\log\Big(1+\sum_{\substack{y'\in[C]\\mathbf{y}'\neq y_i}}e^{\iota_{y'}-\iota_{y_i}}e^{-(R/\|\mathbf{W}\|_F) (\Delta_{y_i}\hat{\w}_{y_i}^T\mathbf{x}_i-\Delta_{y'}\hat{\w}_{y'}^T\mathbf{x}_i)}\Big),\notag
\\
& \leq n \omega_{\max} \log\Big(1+(K-1)e^{\iota_{\max}}e^{-R/\|\hat\mathbf{W}\|_F}\Big)\leq n(K-1) e^{\iota_{\max}}e^{-R/\|\hat\mathbf{W}\|_F}.
\end{align}
Next, by contradiction assumption and strong convexity of \eqref{eq:CS-SVM_app_multi}, for $\bar\mathbf{W}=\frac{\|\hat\mathbf{W}\|_2}{R}\mathbf{W}_R$, there exist $\epsilon>0$ and at least one $j\in[n]$ and $y'\neq y_j$ such that $\mathbf{x}_j^T(\Delta_{y_j}\bar\mathbf{w}_{j} -\Delta_{y'} \bar\mathbf{w}_{y'})\leq (1-\epsilon)$. With this, we can show similar to \eqref{eq:lb} that
\begin{align}\label{eq:multi_proof_2}
\mathcal{L}(\mathbf{W}_R) \geq \log\Big(1+e^{\iota_{y'}-\iota_{y_j}}e^{R/\|\hat\mathbf{W}\|_F(1-\epsilon)}\Big).
\end{align}
The proof is complete by showing that for sufficiently large $R$ the RHS of \eqref{eq:multi_proof_2} is larger than the LHS of \eqref{eq:multi_proof_1} leading to a contradiction. We omit the details for brevity.\end{proof}
\section{Introduction}\label{sec:introduction}
\vspace{-0.1in}
\subsection{Motivation and contributions}
\vspace{-0.1in}
Equitable learning in the presence of data imbalances is a classical machine learning (ML) problem, but one with increasing importance as ML decisions are adapted in increasingly more complex applications directly involving people \cite{barocas2016big}.
Two common types of imbalances are those appearing in \emph{label-imbalanced} and \emph{group-sensitive} classification. In the first type, examples from a target class are heavily outnumbered by examples from the rest of the classes. The standard metric of average misclassification error is insensitive to such imbalances and among several classical alternatives
the \emph{balanced error} is a widely used metric. In the second type, the broad goal is to ensure fairness with respect to a protected underrepresented group (e.g. gender, race). While acknowledging that there is no universal fairness metric \cite{kleinberg2016inherent,friedler2016possibility}, several suggestions have been made in the literature including
\emph{Equal Opportunity} favoring same true positive rates across groups \cite{hardt2016equality}.
Methods for imbalanced data are broadly categorized into data- and algorithm- level ones. In the latter category, belong \emph{cost-sensitive methods} and, specifically, those that modify the {training loss} to account for varying class/group penalties. Corresponding state-of-the-art (SOTA) research is motivated by observations that classical methods, such as weighted cross-entropy (wCE) fail when training overparameterized deep nets without regularization and with train-loss minimization continuing well beyond zero train-error, in the so-called \emph{terminal phase of training (TPT)} (\cite{NC} and references therein). Intuitively, failure of wCE when trained in TPT is attributed to the failure to appropriately adjust the relative margins between different classes/groups in a way that favors minorities. To overcome this challenge, recent works have proposed a so-called {logit-adjusted (LA) loss} that modifies the cross-entropy (CE) loss by including extra \emph{additive} hyper-parameters acting on the logits \cite{cosen,TengyuMa,Menon}. Even more recently, \cite{CDT} suggested yet another modification that introduces \emph{multiplicative} hyper-parameters on the logits leading to a {class-dependent temperature (CDT) loss.} Empirically, both adjustments show performance improvements over wCE. However, it remains unclear: \emph{Do both additive and multiplicative hyper-parameters lead to margin-adjustments favoring minority classes? If so, what are the individual mechanisms that lead to this behavior? How effective are different adjustments at each stage of training?}
This paper answers the above questions. Specifically, we argue that multiplicative hyper-parameters are most effective for margin adjustments in TPT, while additive parameters can be useful in the initial phase of training. Importantly, this intuition justifies our algorithmic contribution: we introduce the \emph{vector-scaling (VS) loss} that combines both types of adjustments and attains improved performance on SOTA imbalanced datasets. Finally, using the same set of tools, we extend the VS-loss to instances of group-sensitive classification. We make multiple contributions as summarized below; see also Figure \ref{fig:intro_table}.
\noindent $\bullet$~\textbf{Explaining the distinct roles of additive/multiplicative adjustments.}~We show that when optimizing in TPT \emph{multiplicative} logit adjustments are critical. Specifically, we prove for linear models that multiplicative adjustments find classifiers that are solutions to cost-sensitive support-vector-machines (CS-SVM), which by design create larger margins for minority classes. While effective in TPT, we also find that, at the start of training, the same adjustments can actually harm minorities. Instead, \emph{additive} adjustments can speed up convergence by countering the initial negative effect of the multiplicative ones. The analytical findings are consistent with our experiments.
\noindent $\bullet$~\textbf{An improved algorithm: VS-loss.}~Motivated by the unique roles of the two different types of adjustments, we propose the vector-scaling (VS) loss that combines the best of both worlds and outperforms existing techniques on benchmark datasets.
\begin{wrapfigure}{h}{0.41\textwidth}
\centering
\vspace{-0.05in}
\includegraphics[width=0.38\textwidth,height=0.23\textwidth]{Contribution4.pdf}
\caption{Summary of contributions.}
\label{fig:intro_table}
\end{wrapfigure}
\noindent $\bullet$~\textbf{Introducing logit-adjustments for group-imbalanced data.}~We introduce a version of VS-loss tailored to {group-imbalanced datasets}, thus treating, for the first time, loss-adjustments for label and group imbalances in a unifying way. For the latter, we propose a new algorithm combining our VS-loss with the previously proposed DRO-method to achieve state-of-the-art performance
in terms of both Equal Opportunity and worst-subgroup error.
\noindent $\bullet$~\textbf{Generalization analysis / fairness tradeoffs.}~We present a sharp generalization analysis of the VS-loss on binary overparameterized Gaussian mixtures.
Our formulae are explicit in terms of data geometry, priors, parameterization ratio and hyperparameters; thus, leading to tradeoffs between standard error and fairness measures. We find that VS-loss can improve both balanced and standard error over CE. Interestingly, the optimal hyperparameters that minimize balanced error also optimize Equal Opportunity.
\section{What is going on with the loss?!}
Let the label set $\mathcal{Y}=\{1,2,\ldots,K\}$ for a K-class classification setting and consider the cross-entropy VS-loss:
\begin{align}\label{eq:loss_app_multi_A}
\mathcal{L}(\mathbf{W})&:= -\sum_{i\in[n]}\omega_{y_i}\log\left(\frac{e^{\Delta_{y_i}\mathbf{w}_{y_i}^T\mathbf{x}+\iota_{y_i}}}{\sum_{c\in[K]}e^{\Delta_{y_i}\mathbf{w}_c^T\mathbf{x}+\iota_c}}\right)
\\
&=\sum_{i\in[n]}\omega_{y_i}\log\Big(1+\sum_{\substack{y'\in[K]\\mathbf{y}'\neq y_i}}e^{\iota_{y'}-\iota_{y_i}}e^{- \Delta_{y_i} (\mathbf{w}_{y_i}^T\mathbf{x}_i-\mathbf{w}_{y'}^T\mathbf{x}_i)}\Big),\notag
\end{align}
where $\mathbf{W}=[\mathbf{w}_1,\ldots,\mathbf{w}_K]\in\mathbb{R}^{K\times d}$ and $\mathbf{w}_y$ is the classifier corresponding to class $y\in[K]$. The VS-loss converges to the following CS $K$-class SVM:
\begin{align}\label{eq:CS-SVM_app_multi_A}
\boxed{\hat\mathbf{W} = \arg\min_{\mathbf{W}} \|\mathbf{W}\|_F\quad\text{subject to }\Delta_{y_i}\mathbf{x}_i^T(\mathbf{w}_{y_i}-\mathbf{w}_{y'})\geq 1, ~\forall y'\neq y_i\in[K]~\text{ and }~\forall i\in[n].}
\end{align}
Now consider the other loss
\begin{align}\label{eq:loss_app_multi_A_2}
\mathcal{L}(\mathbf{W})&:= -\sum_{i\in[n]}\omega_{y_i}\log\left(\frac{e^{\Delta_{y_i}\mathbf{w}_{y_i}^T\mathbf{x}+\iota_{y_i}}}{\sum_{c\in[K]}e^{\Delta_c\mathbf{w}_c^T\mathbf{x}+\iota_c}}\right)
\\
&=\sum_{i\in[n]}\omega_{y_i}\log\Big(1+\sum_{\substack{y'\in[K]\\mathbf{y}'\neq y_i}}e^{\iota_{y'}-\iota_{y_i}}e^{- (\Delta_{y_i}\mathbf{w}_{y_i}-\Delta_{y'}\mathbf{w}_{y'})^T\mathbf{x}_i)}\Big)
\end{align}
This loss converges to the following
\begin{align}\label{eq:CS-SVM_app_multi_A_2}
\boxed{\hat\mathbf{W} = \arg\min_{\mathbf{W}} \|\mathbf{W}\|_F\quad\text{subject to }(\Delta_{y_i}\mathbf{w}_{y_i}-\Delta_{y'}\mathbf{w}_{y'})^T\mathbf{x}_i\geq 1, ~\forall y'\neq y_i\in[K]~\text{ and }~\forall i\in[n].}
\end{align}
\subsection{Binary case}
Consider the binary case: $y_i=\pm1$.
\vspace{5pt}
\noindent$\bullet$~~\eqref{eq:CS-SVM_app_multi_A} becomes
\begin{align}
\min \|\mathbf{w}_+\|_2^2 + \|\mathbf{w}_-\|_2^2\quad\text{subject to } \begin{cases}
\Delta_+(\mathbf{w}_+-\mathbf{w}_-)^T\mathbf{x}_i\geq 1 & y_i=+1 \\
\Delta_-(\mathbf{w}_--\mathbf{w}_+)^T\mathbf{x}_i\geq 1 & y_i=-1
\end{cases}
\end{align}
Change variables to $\mathbf{v}=\mathbf{w}_+-\mathbf{w}_-$ and $\mathbf{u}=\mathbf{w}_++\mathbf{w}_-$. Note that
$$
\|\mathbf{w}_+\|_2^2 + \|\mathbf{w}_-\|_2^2 = \frac{1}{4}\|\mathbf{v}+\mathbf{u}\|_2^2 + \frac{1}{4}\|\mathbf{v}-\mathbf{u}\|_2^2 = \frac{1}{2}\|\mathbf{v}\|_2^2+\frac{1}{2}\|\mathbf{u}\|_2^2.
$$
Thus, the above reduces to
\begin{align}
\min \|\mathbf{v}\|_2^2 + \|\mathbf{u}\|_2^2\quad\text{subject to } \begin{cases}
\Delta_+\mathbf{v}^T\mathbf{x}_i\geq 1 & y_i=+1 \\
-\Delta_-\mathbf{v}^T\mathbf{x}_i\geq 1 & y_i=-1
\end{cases} \qquad = \qquad
\min \|\mathbf{v}\|_2^2 \quad\text{subject to }
y_i\Delta_{y_i}\mathbf{v}^T\mathbf{x}_i\geq 1
\end{align}
Recognize that the latter is exactly the binary CS-SVM.
To conclude, letting $\mathbf{v}_{\rm CS-SVM}$ be the corresponding CS-SVM solution, we have shown that
$$
\mathbf{w}_+=-\mathbf{w}_-={\mathbf{v}_{\rm CS-SVM}}/{2},
$$
So, decision rule for a new sample $\mathbf{x}$ is $\hat y=\arg\max_{y=\pm} \mathbf{w}_y^T\mathbf{x}=\rm{sign}((\mathbf{w}_+-\mathbf{w}_-)^T\mathbf{x})=\rm{sign}(\mathbf{v}_{\rm CS-SVM}^T\mathbf{x})$.
\vspace{5pt}
\noindent$\bullet$~~\eqref{eq:CS-SVM_app_multi_A_2} becomes
\begin{align}
\min \|\mathbf{w}_+\|_2^2 + \|\mathbf{w}_-\|_2^2\quad\text{subject to } \begin{cases}
(\Delta_+\mathbf{w}_+-\Delta_-\mathbf{w}_-)^T\mathbf{x}_i\geq 1 & y_i=+1 \\
(\Delta_-\mathbf{w}_--\Delta_+\mathbf{w}_+)^T\mathbf{x}_i\geq 1 & y_i=-1
\end{cases}
\end{align}
Change variables to $\mathbf{v}=\Delta_+\mathbf{w}_+-\Delta_-\mathbf{w}_-$ and $\mathbf{u}=\Delta_+\mathbf{w}_++\Delta_-\mathbf{w}_-$.
Compute
$$
\|\mathbf{w}_+\|_2^2 + \|\mathbf{w}_-\|_2^2 = \frac{1}{4\Delta_+^2}\|\mathbf{v}+\mathbf{u}\|_2^2 + \frac{1}{4\Delta_-^2}\|\mathbf{v}-\mathbf{u}\|_2^2 = \frac{\alpha}{2}\|\mathbf{v}\|_2^2 + \frac{\alpha}{2}\|\mathbf{u}\|_2^2 + \beta\mathbf{v}^T\mathbf{u}
$$
with $\alpha=\frac{1}{\Delta_+^2}+\frac{1}{\Delta_-^2}$ and $\beta=\frac{1}{2\Delta_+^2}-\frac{1}{2\Delta_-^2}$.
Note then that the minimization over $\mathbf{u}$ is unconstrained leading to the objective:
$$
\frac{\alpha}{2}\|\mathbf{v}\|_2^2 - \frac{\beta^2}{2\alpha}\|\mathbf{v}\|_2^2 = \frac{1}{2\alpha}(\alpha^2-\beta^2)\|\mathbf{v}\|_2^2
$$
Thus, the optimization is equivalent to
\begin{align}
\min~\|\mathbf{v}\|_2^2\quad\text{subject to } y_i\mathbf{x}_i^T\mathbf{v}\geq 1
\end{align}
which is of course nothing but standard SVM. To conclude, letting $\mathbf{v}_{\rm SVM}$ be the corresponding CS-SVM solution, we have shown that
$$
\mathbf{w}_+=\frac{1}{2\Delta_+}(\mathbf{u}+\mathbf{v})= \frac{1}{2\Delta_+}(1-\beta/\alpha)\mathbf{v}_{\rm SVM}\quad\text{and}\quad\mathbf{w}_-=\frac{1}{2\Delta_-}(\mathbf{u}-\mathbf{v})= -\frac{1}{2\Delta_-}(1+\beta/\alpha)\mathbf{v}_{\rm SVM}.
$$
So, decision rule for a new sample $\mathbf{x}$ is $\hat y=\arg\max_{y=\pm} \mathbf{w}_y^T\mathbf{x}=\rm{sign}((\mathbf{w}_+-\mathbf{w}_-)^T\mathbf{x})=\rm{sign}(\mathbf{v}_{\rm SVM}^T\mathbf{x})$.
\section*{Acknowledgments}
This work is supported by the National Science Foundation under grant Numbers CCF-2009030, by
HDR-193464, by a CRG8 award from KAUST and by an NSERC Discovery Grant. C. Thrampoulidis would also like to acknowledge his affiliation with University of California, Santa Barbara. S. Oymak is partially supported by the NSF award
CNS-1932254 and by the NSF CAREER award CCF-2046816.
\newpage
\bibliographystyle{alpha}
\subsection{Experiments on the MNIST dataset}\label{sec:MNIST}
Here, we present additional results on imbalanced MNIST data trained with linear and random-feature models. These results complement the synthetic experiment of Figure \ref{fig:mismatch_intro}(a).
Specifically, we designed an experiment where we perform binary one-vs-rest classification on the MNIST dataset to classify digit $7$ from the rest. Specifically, we split the dataset in two classes, the minority class containing images of the digit $7$ and the majority class containing images of all other digits. To be consistent with our notation we assign the label $+1$ to the minority class and the label $-1$ to the majority class. Here, $d=784$ and $\pi=0.1$ is the prior for the minority class. All test-error evaluations were performed on a test set of $1000$ samples. The results of the experiments were averaged over 200 realizations and the $90\%$ confidence intervals for the mean are shown in Figure \ref{fig:mnist_delta_tuning} as shaded regions.
\begin{figure}[t]
\begin{center}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{MNIST_digit_7_p_01_reps_200_shaded_conf.png}
\caption{Linear classifier}
\end{subfigure}
\begin{subfigure}[b]{0.45\textwidth}
\centering
\includegraphics[width=0.8\textwidth]{MNIST_digit_7_p_01_reps_200_shaded_conf_rf.png}
\caption{Random-features classifier}
\end{subfigure}
\end{center}
\caption{A comparison of CS-SVM balanced error against the overparameterization ratio $\gamma$, for the standard hard margin SVM ($\delta=1$), for a heuristic $\delta = (\frac{1-\pi}{\pi})^\frac{1}{4}$ and for our \emph{approximation} of the optimal $\delta$ ($\delta=\tilde\delta_\star$) obtained by the data-dependent heuristic in Section \ref{sec:delta_star_estimate}. The experiment is performed on the MNIST dataset in a one-vs-rest classification task where the goal is to separate the minority class containing images of the digit $7$ from the majority class containing images of all other digits. See text for details.}
\label{fig:mnist_delta_tuning}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.5\textwidth]{LA_VS_loss_binary_MNIST_digit_7_p_01_reps_200_shaded_conf.png}
\caption{In the overparameterized regime, our VS loss converges to the CS-SVM classifier, while the LA-loss converges to the inferior ---in terms of balanced-error performance--- SVM. The experiment was performed on the MNIST dataset in a one-vs-rest classification task where the goal is to separate the minority class containing images of the digit $7$ from the majority class containing images of all other digits. See text for details.}
\label{fig:la_vs_mnist_f}
\end{figure}
We ran two experiments. In the first one depicted in Figure \ref{fig:mnist_delta_tuning}(a), we trained linear classifiers using the standard SVM (blue), the CS-SVM with a heuristic value $\delta=(\frac{1-\pi}{\pi})^\frac{1}{4}$ (orange), and the CS-SVM with our heuristic data-dependent estimate of the optimal $\tilde\delta_\star$ (green). We compute such an estimate based on a recipe inspired by our exact expression in \eqref{eq:delta_star} for the GMM; see Section \ref{sec:delta_star_estimate} for details. We compute the three classifiers on training sets of varying sizes $n=d/\gamma$ for a range of values of $\gamma$ and report their balanced error. We observe that CS-SVM always outperforms SVM (aka $\delta=1$) and the heuristic optimal tuning of CS-SVM consistently outperforms the choice $\delta=(\frac{1-\pi}{\pi})^\frac{1}{4}$.
Next, in Figure \ref{fig:mnist_delta_tuning}(b) for the same dataset we trained a Random-features classifier. Specifically, for each one of the $n=300$ training samples $\mathbf{x}_i\in\mathbb{R}^{d=784}$ we generate random features $\widetilde\mathbf{x}_i={\rm ReLU}(\mathbf{A}\mathbf{x}_i)$ for a matrix $\mathbf{A}\in\mathbb{R}^{N\times d}$ which we sample once such that it has entries IID standard normal and is then standardized such that each column becomes unit norm.
In this case we control $\gamma$ by varying the number $N=\gamma n$ of rows of that matrix $\mathbf{A}$. Observe here that the balanced error decreases as $\gamma$ increases (an instance of benign overfitting, e.g. \cite{hastie2019surprises,bartlett2020benign,mei2019generalization} and that again the estimated optimal $\delta_\star$ results in tuning of CS-SVM that outperforms the other depicted choices.
In Figure \ref{fig:la_vs_mnist_f} we repeat the experiment of Figure \ref{fig:mnist_delta_tuning}(a) only this time additionally to training CS-SVM for $\delta=1$ and for $\delta=\tilde\delta_\star$ we also train using the LA-loss and our VS-loss.
For the VS loss we use \eqref{eq:loss_ours_bin} with the following choice of parameters: $\omega_\pm=1$, $\iota_\pm=0$ and $\Delta_y=\tilde\delta_\star^{-1}\,\ind{y=+1}+\ind{y=-1}$ (see Section \ref{sec:delta_star_estimate} for $\tilde\delta_\star$). In a similar manner, LA-loss is defined using the same formula \eqref{eq:loss_ours_bin}, but with parameters $\Delta_\pm=1$, $\omega_\pm=1$ and $\iota_+=\pi^{-1/4}, \iota_-=(1-\pi)^{-1/4}$ (as suggested in \cite{TengyuMa}).
The figure confirms our theoretical expectations: training with gradient descent on the LA and VS losses asymptotically (in the number of iterations) converge to the SVM and CS-SVM solutions respectively.
The training is performed over 200 epochs and for computing the gradient we iterate through the dataset in batches of size 64. The results are averaged over 200 realizations and the $90\%$ confidence intervals are plotted as shaded regions for the CS-SVM model and as errorbars for the VS loss.
\section{Additional numerical results}\label{sec:more_num}
\subsection{Multiplicative vs Additive adjustments for label-imbalanced GMM
}
In Figure \ref{fig:figure_intro_more} we show a more complete version of Figure \ref{fig:mismatch_intro}(a), where we additionally report standard and per-class accuracies. We minimized the CDT/LA losses in the separable regime with normalized gradient descent (GD), which uses increasing learning rate appropriately normalized by the loss-gradient norm for faster convergence; refer to Figure \ref{fig:normalized_gd_experiments} and Section \ref{sec:gd_num_app} for the advantages over constant learning rate. Here, normalized GD was ran until the norm of the gradient of the loss becomes less than $10^{-8}$. We observed empirically that the GD on the LA-loss reaches the stopping criteria faster compared to the CDT-loss. This is in full agreement with the CIFAR-10 experiments in Section \ref{sec:insights} and theoretical findings in Section \ref{sec:insights}.
In all cases, we reported both the results of Monte Carlo simulations, as well as, the theoretical formulas predicted by Theorem \ref{thm:main_imbalance}. As promised, the theorem sharply predicts the conditional error probabilities of minority/majority class despite the moderate dimension of $d=300$.
As noted in Section \ref{sec:insights}, CDT-loss results in better balanced error (see `Top Left') in the separable regime (where $\mathcal{R}_{\rm train}=0$) compared to LA-loss. This naturally comes at a cost, as the role of the two losses is reversed in terms of the misclassification error (see `Top Right'). The two bottom figures better explain these, by showing that VS sacrifices the error of majority class for a significant drop in the error of the minority class. All types of errors decrease with increasing overparameterization ratio $\gamma$ due to the mismatch feature model
Finally, while balanced-error performance of CDT-loss is clearly better compared to the LA-loss in the separable regime, the additive offsets $\iota_y$'s improve performance in the non-separable regime. Specifically, the figure confirms experimentally the superiority of the tuning of the LA-loss in \cite{Menon} compared to that in \cite{TengyuMa} (but only in the underparameterized regime). Also, it confirms our message: VS-loss that combines the best of two worlds by using both additive and multiplicative adjustments.
\subsection{Multiplicative vs Additive adjustments with $\ell_2$-regularized GD }
In this section we shed more light on the experiments presented in Figure \ref{fig:mismatch_intro}(b,c), by studying the effect of $\ell_2$-regularization.
Specifically, we repeat here the experiment of Fig. \ref{fig:mismatch_intro}(b) with $p=d=50,n=30$. We train with CE, CDT, and LA-losses in TPT with a weight-decay implementation of $\ell_2$-regularization, that is GD with update step:
$
\mathbf{w}_{t+1} = (1-\beta)\mathbf{w}_t-\eta\nabla_{\mathbf{w}}\mathcal{L}(\mathbf{w}_t),
$
where $\beta$ is the weight-decay factor and we used $\beta\in\{0,10^{-3},10^{-2}\}$.
For our discussion, recall our findings in Section \ref{sec:insights}: (i) CDT-loss trained without regularization in TPT converges to CS-SVM, thus achieving better balanced error than LA-loss converging to SVM; (ii) however, at the beginning of training, multiplicative adjustment of CDT-loss can hurt the balanced error; (iii) Additive adjustments on the other hand helped in the beginning of GD iterations but were not useful deep in TPT.
We now turn our focus to the behavior of training in presence of $\ell_2$-regularization. The weight-decay factor was kept small enough to still achieve zero training error. A few interesting observations are summarized below:
\begin{figure}[h]
\begin{center}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{cdt_gen.pdf}
\caption{CDT: bal. error}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{cdt_norms.pdf}
\caption{CDT:classifier norm}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{la_gen.pdf}
\caption{LA: bal. error}
\end{subfigure}
\begin{subfigure}[b]{0.24\textwidth}
\centering
\includegraphics[width=\textwidth]{la_norms.pdf}
\caption{LA: classifier norm}
\end{subfigure}
\end{center}
\caption{Training dynamics of a linear classifier trained with gradient descent on LA and CDT losses, with and without weight decay (parameter $\beta$).}
\label{fig:gd_weight_decay}
\end{figure}
\begin{itemize}
\item The classifier norm plateaus when trained with regularization (while it increases logarithmically without regularization; see Theorem \ref{thm:gradient_flow}). The larger the weight decay factor, the earlier the norm saturates; see Fig. \ref{fig:gd_weight_decay}(b) and (d).
\item Suppose a classifier is trained with a small, but non-zero, weight decay factor in TPT, and the resulting classifier has a norm saturating at some value $\zeta>0$.
The final balanced error performance of such a classifier closely matches the balanced error produced by a classifier trained without regularization but with training stopped early at that iteration for which the classifier-norm is equal to $\zeta$; compare for example, the value of yellow curve (CDT, $\beta=10^{-3}$) at $t=10^6$ with the value of the red curve (CDT, $\beta=0$) at around $t=300$ in Fig. \ref{fig:gd_weight_decay}(c) and (d). \footnote{See also \cite{raskutti2014early,ali2019continuous} for the connection between gradient-descent and regularization solution paths.}
\item If early-stopped (appropriately) before entering TPT, LA-loss can give better balanced performance than CDT-loss. In view of the above mentioned mapping between weight-decay and training epoch, the use of weight decay results in same behavior.
Overall, this supports that VS-loss, combining both additive and multiplicative adjustments is a better choice for a wide range of $\ell_2-$ regularization parameters.
\end{itemize}
\subsection{Additional information on Figures \ref{fig:mismatch_intro}(b),(c) and \ref{fig:tradeoffs}(a),(b)} \label{sec:details_intro_fig}
\noindent\textbf{Figures \ref{fig:mismatch_intro}(b,c).}~We generate data from a binary GMM with $d=50, n=30$ and $\pi=0.1$. We generate mean vectors as random iid Gaussian vector and scale their norms to $5$ and $1$, respectively. For training, we use gradient descent with constant learning rate $0.1$ and fixed number of $10^6$ iterations. The balanced test error in Figure \ref{fig:mismatch_intro}(b) is computed by Monte Carlo on a balanced test set of $10^5$ samples. Figure \ref{fig:mismatch_intro}(c) measures the angle gap of GD outputs $\mathbf{w}^t$ to the solution $\hat\mathbf{w}_\delta$ of CS-SVM in \eqref{eq:CS-SVM} with $\delta=\delta_\star$ and $\mathbf{h}(\mathbf{x}_i)=\mathbf{x}_i$.
\noindent\textbf{Figures \ref{fig:tradeoffs}(a,b).}~In (a), we generated GMM data with $\|\boldsymbol\mu_+\|=3, \boldsymbol\mu_-=-\boldsymbol\mu_+$ and $\pi=0.05.$ In (b), we considered the GMM of Section \ref{sec:generalization} with $\|\boldsymbol\mu_{y,g}\|=3, y\in\{\pm1\}, g\in\{1,2\}$ and $\boldsymbol\mu_{+,1}\perp\boldsymbol\mu_{+,2} \in \mathbb{R}^d$, sensitive group prior $p=0.05$ and equal class priors $\pi=1/2$.
\input{convergence_group}
\input{waterbirds_experiments}
\subsection{Validity of theoretical performance analysis}
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.38\textwidth,height=0.25\textwidth]{fig_semilogy_alg_comparison_arbmeans_sim_theory_bal_error_fixed_snr_p_min_varying_gamma_antipodal=1s_min=4s_maj=8_p_min=0p05_1.pdf}
\includegraphics[width=0.38\textwidth,height=0.25\textwidth]{fig_semilogy_alg_comparison_arbmeans_sim_theory_misclas_error_fixed_snr_p_min_varying_gamma_antipodal=1s_min=4s_maj=8_p_min=0p05_edited.pdf}
\end{center}
\caption{Balanced (Left) and misclassification (Right) errors as a function of the parameterization ratio $\gamma=d/n$ for the following algorithms: SVM with and without majority class resampling, CS-SVM with different choices of $\delta=\big(\frac{1-\pi}{\pi}\big)^\alpha, \pi=0.05$ and $\delta=\delta_\star$ (cf. Eqn. \eqref{eq:delta_star}) plotted for different values of $\gamma=d/n$. {Solid lines show the theoretical values thanks to Theorem \ref{thm:main_imbalance} and the discrete markers represent empirical errors over 100 realizations of the dataset.} Data were generated from a GMM with $\boldsymbol\mu_{+}=4\mathbf{e}_1, \boldsymbol\mu_{-}=-\boldsymbol\mu_{+} \in\mathbb{R}^{500}$, and $\pi=0.05$. SVM with resampling outperforms SVM without resampling in terms of balanced error, but the optimally tuned CS-SVM is superior to both in terms of both balanced and misclassification errors for all values of $\gamma$.
}
\label{fig:cls_imbalance_antipodal_misclass_errors}
\end{figure}
Figures \ref{fig:cls_imbalance_antipodal_misclass_errors} and \ref{fig:group_fairness_deo_misclas} demonstrate that our Theorems \ref{thm:main_imbalance} and \ref{thm:main_group_fairness} provide remarkably precise prediction of the GMM performance even when dimensions are in the order of hundreds. Moreover, both figures show the clear advantage of CS/GS-SVM over regular SVM and naive resampling strategies in terms of balanced error and equal opportunity, respectively.
\begin{figure}[t!]
\begin{center}
\includegraphics[width=0.5\textwidth]{fig_alg_comparison_sim_theory_DEO_misclas_fixed_s_priors_varying_gamma_s_min=3_s_maj=3_p_minor_class=0p5p_minor_group=0p05_rlz=100.PNG}
\end{center}
\caption{DEO and misclassification error of SVM and GS-SVM with different choices of $\delta=\big(\frac{1-p}{p}\big)^\alpha$ for minority group prior $p=0.05$ plotted against $\gamma=d/n$. {Solid lines show the theoretical values and the discrete markers represent empirical errors over 100 realizations of the dataset.} Data generated from a GMM with $\boldsymbol\mu_{+,1}=3\mathbf{e}_1, \boldsymbol\mu_{+,2}=3\mathbf{e}_2 \in\mathbb{R}^{500}$. While SVM has the least misclassification error, it suffers from a high DEO. By trading off misclassification error, it is possible to tune GS-SVM (specifically, $\alpha=0.75$) so that it achieves DEO close to 0 for all the values of $\gamma$ considered here. }
\label{fig:group_fairness_deo_misclas}
\end{figure}
The reported values for the misclassification error and the balanced error / DEO were computed over $10^5$ test samples drawn from the same distribution as the training examples.
Additionally, Figure \ref{fig:cls_imbalance_antipodal_misclass_errors} validates the explicit formula that we derive in Equation \eqref{eq:delta_star} for $\delta_\star$ minimizing the balanced error.
Specifically, observe that CS-SVM with $\delta=\delta_\star$ (`$\times$' markers) not only minimizes balanced error (as predicted in Section \ref{sec:delta_star_proof}), but also leads to better misclassification error compared to SVM for all depicted values of $\gamma$. The figure also shows the performance of our data-dependent heuristic of computing $\delta_\star$ introduced in Section \ref{sec:delta_star_estimate}. The heuristic appears to be accurate for small values of $\gamma$ and is still better in terms of balanced error compared to the other two heuristic choices of $\delta=(\frac{1-\pi}{\pi})^{\alpha}, \alpha=1/4,1$. Finally, we also evaluated the SVM+subsampling algorithm; see Section \ref{sec:undersampling} below for the algorithm's description and performance analysis. Observe that SVM+resampling outperforms SVM without resampling in terms of balanced error, but the optimally tuned CS-SVM is superior to both.
\subsubsection{Max-margin SVM with random majority class undersampling}\label{sec:undersampling}
For completeness, we briefly discuss here SVM combined with undersampling, a popular technique that first randomly undersamples majority examples and only then trains max-margin SVM. The asymptotic performance of this scheme under GMM can be analyzed using Theorem \ref{thm:main_imbalance} as explained below.
Suppose the majority class is randomly undersampled to ensure equal size of the two classes. This increases the effective overparameterization ratio by a factor of $\frac{1}{2\pi}$ (in the asymptotic limits). In particular, the conditional risks converge as follows:
\begin{align}
\mathcal{R}_{+,\text{undersampling}}(\gamma,\pi)&\stackrel{{P}}{\longrightarrow} \overline\Rc_{+,\text{undersampling}}(\gamma,\pi)=\overline\Rc_+\big(\frac{\gamma}{2\pi},0.5\big)\nonumber\\\mathcal{R}_{-,\text{undersampling}}(\gamma,\pi)&\stackrel{{P}}{\longrightarrow}\overline\Rc_{-,\text{undersampling}}(\gamma,\pi)=\overline\Rc_{+,\text{undersampling}}(\gamma,\pi).
\end{align}
Above, $\mathcal{R}_{+,\text{undersampling}}$ and $\mathcal{R}_{-,\text{undersampling}}$ are the class-conditional risks of max-margin SVM after random undersampling of the majority class to ensure equal number of training examples from the two classes. The risk $\overline\Rc_+\big(\frac{\gamma}{2\pi},0.5\big)$ is the asymptotic conditional risk of a \emph{balanced} dataset with overparameterization ratio $\frac{\gamma}{2\pi}$. This is computed as instructed in Theorem \ref{thm:main_imbalance} for the assignments $\gamma\leftarrow \frac{\gamma}{2\pi}$ and $\pi\leftarrow 1/2$ in the formulas therein.
Our numerical simulations in Figure \ref{fig:cls_imbalance_antipodal_misclass_errors} verify the above formulas.
\section{Optimal tuning of CS-SVM}\label{sec:opt_tune_app}
\subsection{An explicit formula for optimal tuning}
The parameter $\delta$ in the CS-SVM constraints in \eqref{eq:CS-SVM} aims to shift the decision space towards the majority class so that it better balances the conditional errors of the two classes. But, how to best choose $\delta$ to achieve that? That is, how to find $\arg\min_\delta \mathcal{R}_+(\delta)+\mathcal{R}_-(\delta)$ where $\mathcal{R}_\pm(\delta):=\mathcal{R}_\pm\big((\hat\mathbf{w}_\delta,\hat b_\delta)\big)$? Thanks to Theorem \ref{thm:main_imbalance}, we can substitute this hard, data-dependent parameter optimization problem with an analytic form that only depends on the problem parameters $\pi, \gamma$ and $\mathbf{M}$. Specifically, we seek to solve the following optimization problem
\begin{align}\notag
&\arg\min_{\delta>0}~ Q(\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta+b_\delta/q_\delta) + Q(-\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta-b_\delta/q_\delta)\\
&~~\text{sub. to~~~ $(q_\delta,\boldsymbol\rho_\delta,b_\delta)$ defined as \eqref{eq:eta_eq_main}.}\label{eq:delta_tune_1}
\end{align}
Compared to the original data-dependent problem, the optimization above has the advantage that it is explicit in terms of the problem parameters. However, as written, the optimization is still cumbersome as even a grid search over possible values of $\delta$ requires solving the non-linear equation \eqref{eq:eta_eq_main} for each candidate value of $\delta$. Instead, we can exploit a structural property of CS-SVM (see Lemma \ref{lem:1_to_delta} in Section \ref{sec:lemma1_proof}) to rewrite \eqref{eq:delta_tune_1} in a more convenient form. Specifically, we will show in Section \ref{sec:delta_star_proof} that \eqref{eq:delta_tune_1} is equivalent to the following \emph{explicit minimization}:
\begin{align}\label{eq:delta_tune_2}
\hspace{-0.08in}\arg\min_{\delta>0}~ Q\Big(\ell_{+} +\big(\frac{\delta-1}{\delta+1}\big){q_1^{-1}}\Big) + Q\Big(\ell_{-} -\big(\frac{\delta-1}{\delta+1}\big){q_1^{-1}}\Big),
\end{align}
where we defined $\ell_{+}:=\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1+b_1/q_1,~\ell_-:=-\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1-b_1/q_1$, and, $(q_1,\boldsymbol\rho_1,b_1)$ are as defined in Theorem \ref{thm:main_imbalance} for $\delta=1$. In other words, $(q_1,\boldsymbol\rho_1,b_1)$ are the parameters related to the standard hard-margin SVM, for which the balanced error is then given by $\left(Q(\ell_+)+Q(\ell_-)\right)\big/2$. To summarize, we have shown that one can optimally tune $\delta$ to minimize the \emph{asymptotic} balanced error by minimizing the objective in \eqref{eq:delta_tune_2} that only depends on the parameters $(q_1,\boldsymbol\rho_1,b_1)$ characterizing the asymptotic performance of SVM. In fact, we obtain explicit formulas for the optimal value $\delta_\star$ in \eqref{eq:delta_tune_2} as follow
\begin{align}\label{eq:delta_star}
\delta_\star := ({\ell_- -\ell_++2q_1^{-1}})\big/{\left(\ell_+-\ell_-+2q_1^{-1}\right)_+},
\end{align}
where it is understood that when the denominator is zero (i.e. $\ell_+-\ell_-+2q_1^{-1}\leq0$) then $\delta_\star\rightarrow\infty$. When $\ell_+-\ell_-+2q_1^{-1}>0$,
setting $\delta=\delta_\star$ in \eqref{eq:CS-SVM} not only achieves minimum balanced error among all other choices of $\delta$, but also it achieves perfect balancing between the conditional errors of the two classes, i.e. $\mathcal{R}_+=\mathcal{R}_-=Q(\frac{\ell_-+\ell_+}{2}).$
Formally, we have the following result.
\begin{theorem}[Optimal tuning of CS-SVM]\label{propo:delta_star}
Fix $\gamma>\gamma_\star$. Let $\overline{\Rc}_{\text{bal}}(\delta)$ denote the asymptotic balanced error of the CS-SVM with margin-ratio parameter $\delta>0$ as specified in Theorem \ref{thm:main_imbalance}. Further let $(q_1,\boldsymbol\rho_1,b_1)$ the solution to \eqref{eq:eta_eq_main} for $\delta=1$. Finally, define
$$
\ell_{+}:=\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1+b_1/q_1,\quad\ell_-:=-\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1-b_1/q_1,$$
Then, for all $\delta> 0$ it holds that
$$
\overline{\Rc}_{\text{bal}}(\delta) \geq \overline{\Rc}_{\text{bal}}(\delta_\star)
$$
where $\delta_\star$ is defined as
\begin{align}\label{eq:delta_star_app}
\delta_\star = \begin{cases}
\frac{\ell_- -\ell_++2q_1^{-1}}{\ell_+-\ell_-+2q_1^{-1}} & \text{if } \ell_++\ell_-\geq0 \text{ and } \ell_+-\ell_-+2q_1^{-1}> 0,\\
\rightarrow\infty & \text{if } \ell_++\ell_-\geq0 \text{ and } \ell_+-\ell_-+2q_1^{-1}\leq 0, \\
\rightarrow 0 & \text{if } \ell_++\ell_-<0.
\end{cases}
\end{align}
Specifically, if $\ell_+ + \ell_- \geq 0$ and $\ell_+-\ell_-+2q_1^{-1}>0$ hold, then the following two hold: (i)
$\overline{\Rc}_{\text{bal}}(\delta_\star)=Q\left(\left({\ell_-+\ell_+}\right)\big/{2}\right)$, and, (ii) the asymptotic conditional errors are equal, i.e.
$
\mathcal{R}_+(\delta_\star)=\mathcal{R}_-(\delta_\star).
$
\end{theorem}
See Figures \ref{fig:five over x} and \ref{fig:delta_star_theory_formula_2} for numerical illustrations of the formula in Theorem \ref{propo:delta_star}, specifically how $\delta_\star$ depends on $\pi$ and $\gamma$.
\subsubsection{Data-dependent heuristic to estimate $\delta_\star$}\label{sec:delta_star_estimate}
It is natural to ask if formula \eqref{eq:delta_star_app} can be used for tuning in practice. To answer this, observe that evaluating the formula requires knowledge of the true means, which are typically unknown. In this section, we propose a \emph{data-dependent heuristic} to estimate $\delta_\star$. More generally, tuning $\delta$ (or $\Delta_y$ in VS-loss) requires a train-validation split by creating a balanced validation set from the original training data which would help assess balanced risk. Since there is only a single hyperparameter we expect this approach to work well with fairly small validation data (without hurting the minority class sample size).
Recall from Equation \eqref{eq:delta_star} that
$
\delta_\star := ({\ell_- -\ell_++2q_1^{-1}})\big/{\left(\ell_+-\ell_-+2q_1^{-1}\right)_+},
$
where $\ell_{+}:=\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1+b_1/q_1$ and $\ell_-:=-\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1-b_1/q_1$. Also, according to Theorem \ref{thm:main_imbalance} and for $\delta=1$ it holds that
\begin{align}\label{eq:remember_mnist}
(\|\hat\mathbf{w}_1\|_2, {\hat{\w}_1^T\boldsymbol\mu_+}/{\|\hat{\w}_1\|_2}, {\hat{\w}_1^T\boldsymbol\mu_-}/{\|\hat{\w}_1\|_2} , \hat b_1) \stackrel{{P}}{\longrightarrow} (q_1,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1,\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1, b_1).
\end{align}
The first key observation here is that $\hat{\w}_1,\hat b_1$ are the solutions to SVM, thus they are data-dependent quantities to which we have access to. Hence, we can simply run SVM and estimate $q_1$ and $b_1$ using Equation \eqref{eq:remember_mnist}. Unfortunately, to further estimate $\boldsymbol\rho_1$ we need knowledge of the data means. When this is not available, we propose approximating the data means by a simple average of the features, essentially pretending that the data follow a GMM.
Concretely, our recipe for approximating the optimal $\delta$ is as follows. First, using the training set we calculate the empirical means for the two classes, $\tilde{\boldsymbol\mu}_+$ and $\tilde{\boldsymbol\mu}_-$. (Ideally, this can be done on a balanced validation set.) Then, we train standard SVM on the same set of data and keep track of the coefficients $\hat{\w}_1$ and the intercept $\hat b_1$. Then, we can reasonably approximate the optimal $\delta$ as:
\begin{align}
\tilde\delta_\star := \frac{\tilde{\ell_-} -\tilde{\ell}_++2\|\hat\mathbf{w}_1\|_2^{-1}}{\left(\tilde{\ell_+}-\tilde{\ell_-}+2\|\hat\mathbf{w}_1\|_2^{-1}\right)_+}, \text{ with }
\tilde{\ell_{+}}:=\frac{\hat{\w}_1^T\tilde{\boldsymbol\mu}_+ + \hat b_1}{\|\hat{\w}_1\|_2}, \quad \tilde{\ell_{-}}:=-\frac{\hat{\w}_1^T\tilde{\boldsymbol\mu}_- + \hat b_1}{\|\hat{\w}_1\|_2}.
\end{align}
We expect this data-dependent theory-driven heuristic to perform reasonably well on data that resemble the GMM. For example, this is confirmed by our experiments in Figures \ref{fig:mnist_delta_tuning} and \ref{fig:cls_imbalance_antipodal_misclass_errors}. More generally, we propose tuning $\delta$ with a train-validation split by creating a balanced validation set from the original training data which would help assess balanced risk. Since there is only a single hyperparameter we expect this approach to work well with a fairly small validation data (without hurting the minority class sample size).
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig_semilogy_delta_tuning_theory_fixed_snr_p_min_varying_gamma_delta_s=4_p_min=0p1.pdf}
\caption{$\pi=0.1$}
\label{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig_semilogy_delta_tuning_theory_fixed_snr_p_min_varying_gamma_delta_s=4_p_min=0p0025.pdf}
\caption{$\pi=0.0025$}
\label{}
\end{subfigure}
\hfill
\begin{subfigure}[b]{0.3\textwidth}
\centering
\includegraphics[width=\textwidth]{fig_semilogy_delta_tuning_theory_fixed_snr_p_min_varying_gamma_delta_s=4_p_min=0p001.pdf}
\caption{$\pi=0.0001$}
\label{fig:five over x}
\end{subfigure}
\caption{Graphical illustration of the result of Theorem \ref{propo:delta_star}: Balanced errors of CS-SVM against the margin-ratio parameter $\delta$ for a GMM of antipodal means with $\|\mu_+\|=\|\mu_-\|=4$ and different minority class probabilities $\pi$. The balanced error is computed using the formulae of Theorem \ref{thm:main_imbalance}. For each case, we studied three different values of $\gamma$. The value $\delta_\star$ at which the curves attain (or approach) their minimum are predicted by Theorem \ref{propo:delta_star}. Specifically, note the following for the three different priors. (a) For all values of $\gamma$, the minimum is attained (cf. first branch of \eqref{eq:delta_star_app}). (b) For $\gamma=2,5$ the minimum is approached in the limit $\delta\rightarrow\infty$ (cf. second branch of \eqref{eq:delta_star_app}), but it is attained for $\gamma=0.5$ (c) The minimum is always approached as $\delta_\star\rightarrow\infty$. }
\label{fig:delta_star_theory_formula}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[width=0.3\textwidth]{fig_semilogy_delta_tuning_theory_fixed_snr_p_min_varying_gamma_delta_s=1_p_min=0p1.pdf}
\caption{An example showing the dependence of $\delta_\star$ on the data geometry. The above figure is similar to Fig \ref{fig:delta_star_theory_formula} but with a smaller $\|\mu_+\|=\|\mu_-\|=1$, and for $\pi=0.1$. While in Fig \ref{fig:delta_star_theory_formula}, the value of $\delta_\star$, whenever finite, can be seen to increase with increase in $\gamma$, for the current setting, it is observed to decrease. Note also that $\delta_\star\rightarrow\infty$ for $\gamma=0.5$, but finite for $\gamma=2,5$.}
\label{fig:delta_star_theory_formula_2}
\end{figure}
\subsection{CS-SVM as post-hoc weight normalization}\label{sec:lemma1_proof}
We need the lemma below to prove Theorem \ref{propo:delta_star}. But the results is interesting on its own right as it allows us to view CS-SVM as an appropriate ``post-hoc weight normalization"-approach.
\begin{lemma}\label{lem:1_to_delta}
Let $(\hat\mathbf{w}_1,\hat b_1)$ be the hard-margin SVM solution. Fix any $\delta>0$ in \eqref{eq:CS-SVM} and define:
$
\hat\mathbf{w}_\delta:=\big(\frac{\delta+1}{2}\big)\,{\hat\mathbf{w}_1}$
and $\hat b_\delta:=\big(\frac{\delta+1}{2}\big)\,\hat b_1 + \big(\frac{\delta-1}{2}\big).
$
Then, $(\hat\mathbf{w}_\delta,\hat b_\delta)$ is optimal in \eqref{eq:CS-SVM}.
\end{lemma}
Thus, classification using \eqref{eq:CS-SVM} is equivalent to the following. First learn $(\hat\mathbf{w}_1,\hat b_1)$ via standard hard-margin SVM, and then simply predict:
$
\hat y = {\rm sign}\big( (\hat\mathbf{w}_1^T\mathbf{x} + \hat b_1) + \frac{\delta-1}{\delta+1} \big).
$
The term $\frac{\delta-1}{\delta+1}$ can be seen as an additive form of post-hoc weight normalization to account for class imbalances. In the literature this post-hoc adjustment of the threshold $b$ of standard SVM is often referred to as boundary-movement SVM (BM-SVM) \cite{shawe1999optimizing,wu2003class}. Here, we have shown the equivalence of CS-SVM to BM-SVM for a specific choice of the boundary shift. The proof of Lemma \ref{lem:1_to_delta} presented in Appendix \ref{sec:lemma1_proof} shows the desired using the KKT conditions of \eqref{eq:CS-SVM}.
\begin{proof}
From optimality of $(\hat\mathbf{w}_1,\hat b_1)$, convexity of \eqref{eq:CS-SVM} and the KKT-conditions, there exist dual variables $\beta_i, i\in[n]$ such that:
\begin{align}\label{eq:KKT_1}
&\hat \mathbf{w}_1 = \sum_{i\in[n]}y_i\beta_i \mathbf{x}_i,\quad\sum_{i\in[n]}y_i\beta_i = 0,\\
&\forall i\in[n]~:~\beta_i\big(\mathbf{x}_i^T\hat \mathbf{w}_1+\hat b_1\big)=\beta_iy_i\notag,\quad \beta_i\geq 0.
\end{align}
Let $(\hat\mathbf{w}_\delta,\hat b_\delta)$ defined as in the statement of the lemma and further define $\epsilon_i:=\big(\frac{\delta+1}{2}\big)\beta_i,~i\in[n]$. Then, it only takes a few algebra steps using \eqref{eq:KKT_1} to check that the following conditions hold:
\begin{align}\label{eq:KKT_2}
&\hat \mathbf{w}_\delta = \sum_{i\in[n]}y_i\epsilon_i \mathbf{x}_i,\quad\sum_{i\in[n]}y_i\epsilon_i = 0,\\
&\forall i\in[n]~:~\epsilon_i\big(\mathbf{x}_i^T\hat \mathbf{w}_\delta+\hat b_\delta\big)=\epsilon_i\cdot\begin{cases}\delta &,\text{if } y_i=+1\\ -1 &,\text{if } y_i=-1\end{cases}\notag,\quad \epsilon_i\geq 0.
\end{align}
It can also be verified that \eqref{eq:KKT_2} are the KKT conditions of the CS-SVM with parameter $\delta$. This proves that $(\hat\mathbf{w}_\delta,\hat b_\delta)$ is optimal in \eqref{eq:CS-SVM} as desired.
\end{proof}
\subsection{Proof of Theorem \ref{propo:delta_star}}\label{sec:delta_star_proof}
As discussed in the section above the proof proceeds in two steps:
(i) First, starting from \eqref{eq:delta_tune_1}, we prove \eqref{eq:delta_tune_2}.
(ii) Second, we analytically solve \eqref{eq:delta_tune_2} to derive the explicit expression for $\delta_\star$ in \eqref{eq:delta_star_app}.
\noindent\textbf{Proof of \eqref{eq:delta_tune_2}.}~
Fix any $\delta>0.$ From Lemma \ref{lem:1_to_delta},
\begin{align}\label{eq:tune_proof_0}
\hat{\w}_\delta= \big(\frac{\delta+1}{2}\big) {\hat{\w}_1}\quad\text{and}\quad \hat{b}_\delta=\big(\frac{\delta+1}{2}\big)\hat{b}_1+\big(\frac{\delta-1}{2}\big).
\end{align}
Recall from Theorem \ref{thm:main_imbalance} that $\|\hat{\w}_\delta\|_2\stackrel{{P}}{\longrightarrow} q_\delta$, $\|\hat{\w}_1\|_2\stackrel{{P}}{\longrightarrow} q_1$, $\hat{b}_\delta\stackrel{{P}}{\longrightarrow} b_\delta$, $\hat{b}_1\stackrel{{P}}{\longrightarrow} b_1$, and, for $i=1,2$: $\frac{\hat{\w}_\delta^T\boldsymbol\mu_i}{\|\hat{\w}_\delta\|_2}\stackrel{{P}}{\longrightarrow}\mathbf{e}_i^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta$ and $\frac{\hat{\w}_1^T\boldsymbol\mu_i}{\|\hat{\w}_1\|_2}\stackrel{{P}}{\longrightarrow}\mathbf{e}_i^T\mathbf{V}\mathbf{S}\boldsymbol\rho_1$. Here, $q_\delta,\rho_\delta,b_\delta$ and $q_1,\rho_1,b_1$ are as defined in Theorem \ref{thm:main_imbalance}. Thus, from \eqref{eq:tune_proof_0} we find that
\begin{align}
\boldsymbol\rho_\delta = \boldsymbol\rho_1,\qquad q_\delta=\big(\frac{\delta+1}{2}\big) q_1\qquad \text{and} \qquad b_\delta=\big(\frac{\delta+1}{2}\big)b_1 + \big(\frac{\delta-1}{2}\big).
\end{align}
Hence, it holds:
$$
Q\left(\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta+b_\delta/q_\delta\right) = Q\Big(\underbrace{\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho_\delta+b_1/q_1}_{=\ell_+}+\frac{\delta-1}{\delta+1}q_1^{-1}\Big).
$$
A similar expression can be written for the conditional error of class $-1$. Putting these together shows \eqref{eq:delta_tune_2}, as desired.
\noindent\textbf{Proof of \eqref{eq:delta_star_app}.}~Recall from \eqref{eq:delta_tune_2} that we now need to solve the following constrained minimization where for convenience we call $a=\ell_+$, $b=\ell_-$ and $c=q_1^{-1}$:
$$
\min_{\delta>0}~Q\left(a+\frac{\delta-1}{\delta+1}c\right) + Q\left(b-\frac{\delta-1}{\delta+1}c\right).
$$
We define a new variable $x=\frac{\delta-1}{\delta+1}c$. The constraint $\delta>0$ then writes $x\leq c$. This is because the function $\delta\in(0,\infty)\mapsto\frac{\delta-1}{\delta+1}$ is onto the interval $(-1,1).$
Thus, we equivalently need to solve
$$
\min_{-c<x< c}~f(x):=Q(a+x) + Q(b-x).
$$
Define function $f(x) = Q(a+x)+Q(b-x)$ for some $a,b\in\mathbb{R}$.
Direct differentiation gives
$
\frac{{\rm d}f}{{\rm d}x} = \frac{1}{\sqrt{2\pi}}\left(e^{-(b-x)^2/2}-e^{-(a+x)^2/2}\right).
$
Furthermore, note that $\lim_{x\rightarrow\pm\infty}f(x)=1$.
With thes and some algebra it can be checked that $f(\cdot)$ behaves as follows depending on the sign of $a+b.$ Denote $x_\star = (b-a)/2$.
\begin{itemize}
\item If $a+b\geq 0$, then $1>f(x)\geq f(x_\star)$ and $x_\star$ is the unique minimum.
\item If $a+b< 0$, then $1<f(x)\leq f(x_\star)$ and $x_\star$ is the unique maximum.
\end{itemize}
Thus, we conclude with the following:
\begin{align*}
\arg\inf_{-c<x< c}~f(x) = \begin{cases}
x_\star & \text{if } a+b\geq0 \text{ and } b-a< 2c,\\
c & \text{if } a+b\geq0 \text{ and } b-a\geq 2c, \\
-c & \text{if } a+b<0.
\end{cases}
\end{align*}
Equivalently,
\begin{align*}
\arg\inf_{\delta>-1}~Q\left(\ell_++\frac{\delta-1}{\delta+1}q_1^{-1}\right) + Q\left(\ell_--\frac{\delta-1}{\delta+1}q_1^{-1}\right) = \begin{cases}
\frac{\ell_- -\ell_++2q_1^{-1}}{\ell_+-\ell_-+2q_1^{-1}} & \text{if } \ell_++\ell_-\geq0 \text{ and } \ell_+-\ell_-+2q_1^{-1}> 0,\\
\infty & \text{if } \ell_++\ell_-\geq0 \text{ and } \ell_+-\ell_-+2q_1^{-1}\leq 0, \\
0 & \text{if } \ell_++\ell_-<0.
\end{cases}
\end{align*}
This shows \eqref{eq:delta_star_app}. The remaining statement of the theorem is easy to prove requiring simple algebra manipulations.
\section*{Organization of the supplementary material}
The supplementary material (SM) is organized as follows.
\begin{enumerate}
\item In Section \ref{sec:exp_label_app} we provide additional technical information on the label-imbalanced experiments of Sec. \ref{sec:exp_label}. We also show experiments of imbalanced MNIST dataset.
\item In Section \ref{sec:exp_group_app} we provide missing details and additional results on the group-imbalanced experiments of Sec. \ref{sec:GS_exp_main}.
\item In Section \ref{sec:more_num} we present synthetic experiments on both label-imbalanced and group-sensitive datasets further supporting our theoretical findings in Sections \ref{sec:insights_gen} and \ref{sec:generalization}.
\item In Section \ref{sec:propo_proof} we present and prove a more general version of Theorem \ref{propo:gd} (specifically, see Theorem \ref{thm:implicit_loss_gen}) on the connection of overparameterized VS-loss and to CS-SVM. We also discuss multiclass extensions (see Theorem \ref{thm:implicit_loss_multi}) and implicit bias of gradient flow (see Theorem \ref{thm:gradient_flow}).
\item In Section \ref{sec:opt_tune_app}, we present theoretical results on optimal tuning of CS-SVM. First, we state and prove Lemma \ref{lem:1_to_delta} which establishes a structural connection between the solution of CS-SVM to the solution of the standard SVM, allowing to view the former as a post-hoc adjustment to the latter. Then, we use this property together with the sharp characterizations of Theorem \ref{thm:main_imbalance} to derive an explicit formula for the optimal margin ratio under Gaussian mixture data.
\item In Section \ref{sec:cs-svm_proof} we prove Theorem \ref{thm:main_imbalance} on generalization of CS-SVM. We also discuss related works on sharp high-dimensional asymptotics and provide necessary background on the convex Gaussian min-max theorem. Finally, we include formulas for the phase-transition threshold of CS-SVM.
\item Finally, in Section \ref{sec:gs-svm_proof} we {state and prove Theorem \ref{thm:main_group_fairness} characterizing the DEO of GS-SVM} as mentioned in Section \ref{sec:generalization}.
\end{enumerate}
\subsection{Connections to related literature}\label{sec:lit}
\vspace{-0.1in}
\noindent\textbf{CE adjustments.}~The use of wCE for imbalanced data is rather old \cite{xie1989logit}, but it becomes ineffective under overparameterization, e.g. \cite{byrd2019effect}. This deficiency has led to the idea of \emph{additive} label-based parameters $\iota_{y}$ on the logits \cite{cosen,TengyuMa,tan2020equalization,Menon,Wang_2018}. Specifically, \cite{Menon} proved that setting $\iota_y=\log(\pi_y)$ ($\pi_y$ denotes the prior of class $y$) leads to a \emph{Fisher consistent} loss, termed LA-loss, which outperformed other heuristics (e.g., focal loss \cite{lin2018focal}) on SOTA datasets. However, Fisher consistency is only relevant in the large sample size limit. Instead, we focus on overparameterized models. In a recent work, \cite{CDT} proposed the CDT-loss, which instead uses \emph{multiplicative} label-based parameters $\Delta_y$ on the logits. The authors arrive at the CDT-loss as a heuristic means of compensating for the empirically observed phenomenon of that the last-layer minority features deviate between training and test instances \cite{KimKim}. {Instead, we arrive at the CDT-loss via a different viewpoint: we show that the multiplicative weights are necessary to move decision boundaries towards majorities when training overparameterized linear models in TPT. Moreover, we argue that while additive weights are not so effective in the TPT, they can help in the initial phase of training. Our analysis sheds light on the individual roles of the two different modifications proposed in the literature and naturally motivates the VS-loss in \eqref{eq:loss_ours}. Compared to the above works we also demonstrate the successful use of VS-loss in group-imbalanced setting and show its competitive performance over alternatives in \cite{sagawa2019distributionally,hu_does,oren2019distributionally}.} Beyond CE adjustments there is active research on alternative methods to improve fairness metrics, e.g. \cite{kang2020decoupling,zhou2020bbn,liu2019largescale,ouyang2016factors}. These are orthogonal to CE adjustments and can potentially be used in conjunction.
\\
\noindent\textbf{Relation to vector-scaling calibration.} Our naming of the VS-loss
is inspired by the vector scaling (VS) calibration \cite{guo2017calibration}, a \emph{post-hoc procedure} that modifies the logits $\mathbf{v}$ \emph{after training} via $\mathbf{v}\rightarrow \boldsymbol{\Delta}\odot\mathbf{v}+\boldsymbol{\iota}$, where $\odot$ is the Hadamard product. \cite{zhao2020role} shows that VS can improve calibration for imbalanced classes, but, in contrast to VS calibration, the multiplicative/additive scalings in our VS-loss are part of the loss and directly affect training.
\\
\noindent\textbf{Blessings/curses of overparameterization.}
Overparameterization acts as a catalyst for deep neural networks \cite{nakkiran2019deep}. In terms of optimization, \cite{soudry2018implicit,oymak2019overparameterized,ji2018risk,azizan2018stochastic} show that gradient-based algorithms are \emph{implicitly biased}
towards favorable min-norm solutions. Such solutions, are then analyzed in terms of generalization showing that they can in fact lead to benign overfitting e.g. \cite{bartlett2020benign,hastie2019surprises}. While implicit bias is key to benign overfitting it may come with certain downsides. As a matter of fact, we show here that certain hyper-parameters (e.g. additive ones) can be ineffective in the interpolating regime in promoting fairness. Our argument essentially builds on characterizing the implicit bias of wCE/LA/CDT-losses. Related to this, \cite{sagawa2020investigation} demonstrated the ineffectiveness of $\omega_y$ in learning with groups.
\\
\section{Problem setup}\label{sec:problem_setup}
\vspace{-0.15in}
\textbf{Data.}~Let training set $\{(\mathbf{x}_i,g_i,y_i)\}_{i=1}^n$ consisting of $n$ i.i.d. samples from a distribution $\mathcal{D}$ over $\mathcal{X}\times{\mathcal{G}}\times\mathcal{Y}$; $\mathcal{X}\subseteq\mathbb{R}^d$ is the input space, $\mathcal{Y}=[C]:=\{1,\ldots,C\}$ the set of $C$ labels, and, ${\mathcal{G}}=[K]$ refers to group membership among $K\geq 1$ groups. Group-assignments are known for training data, but unknown at test time. For concreteness, we focus here on the binary setting, i.e. $C=2$ and $\mathcal{Y}=\{-1,+1\}$; we present multiclass extensions in the Experiments and in the Supplementary Material (SM).
We assume throughout that $y=+1$ is minority class.
\textbf{Fairness metrics.}~Given a training set we learn $f_\mathbf{w}:\mathcal{X}\mapsto\mathcal{Y}$ parameterized by $\mathbf{w}\in\mathbb{R}^p$. For instance, linear models take the form $f_\mathbf{w}=\inp{\mathbf{w}}{h(\mathbf{x})}$ for some feature representation $h:\mathcal{X}\mapsto\mathbb{R}^p$.
Given a new sample $\mathbf{x}$, we decide class membership $\hat y = {\rm sign}(f_\mathbf{w}(\mathbf{x})).$
The (standard) \emph{risk} or \emph{misclassification error} is
$
\mathcal{R}:=\mathbb{P}\left\{\hat y \neq y\right\}.
$
Let $s=(y,g)$ define a subgroup for given values of $y$ and $g$. We also define the \emph{class-conditional risks}
$
\mathcal{R}_{\pm}=\mathbb{P}\left\{\hat y \neq y\,| y=\pm1 \right\},
$
and,
the \emph{sub-group-conditional risks}
$
\mathcal{R}_{\pm,j}=\mathbb{P}\left\{\hat y \neq y\,| y=\pm1, g=j \right\},~~j\in[K].
$
The \emph{balanced error} averages the conditional risks of the two classes: $
\Rc_{\text{bal}} := \left({\mathcal{R}_+ + \mathcal{R}_-\,}\right)\big/{2}.
$
Assuming $K=2$ groups, Equal Opportunity requires $\mathcal{R}_{+,1}=\mathcal{R}_{+,2}$ \cite{hardt2016equality}. More generally, we consider the ({signed}) \emph{difference of equal opportunity (DEO)}
$
\Rc_{\text{deo}}:=\mathcal{R}_{+,1}-\mathcal{R}_{+,2}.
$
In our experiments, we also measure the worst-case subgroup error $\max_{(y\in{\pm1},g\in[K])} \mathcal{R}_{y,g}.$
\textbf{Terminal phase of training (TPT).}~Motivated by modern training practice,
we assume overparameterized $f_\mathbf{w}$ so that $\Rc_{\text{train}}=\frac{1}{n}\sum_{i\in[n]}\ind{{\rm{sign}}(f_\mathbf{w}(\mathbf{x}_i))\neq y_i}$ can be driven to zero.
Typically, training such large models continues well-beyond zero training error as the training loss is being pushed toward zero.
As in \cite{NC}, we call this the \emph{terminal phase of training}.
\section{Story}
\subsection{Contributions}
\subsubsection{Insights on state-of-the-art methods}
Tons of recent papers on how to mitigate imbalances. Mostly methods based on interesting, but heuristic/empirical, observations. Here, we give theoretical intuitions on some of the key findings. Later, we use the insights towards algorithmic contributions.
\vspace{5pt}
\noindent$\bullet$~\textbf{Multiplicative vs additive scalings (or LA vs CDT):}
Ye at al. introduce the CDT and report significant improvements over LA \ct{@Orestis,Ganesh: can we please check here how/if their reported numbers compare to those in the Google paper? Since Google paper never compares to them.}. Here, we offer a theoretical explanation on what multiplicative logit corrections can offer over additive ones.
We prove that for binary linear classifiers multiplicative is advantageous over additive. We show this via a connection to cost-sensitive SVMs. In short, our key finding is that for overparameterized binary linear classifiers LA=CE=SVM and CDT=CS-SVM. Intuitively, CS-SVM is better than SVM. We prove this to be the case for a stylized Gaussian-mixtures dataset.
\vspace{5pt}
\noindent$\bullet$~\textbf{CE fails in group-case} Sagawa et al. argue and demonstrate empirically that CE loss fails in the presence of group-imbalances. We propose a natural extension of the CDT, called GDT, and we prove that GDT=GS-SVM where GS-SVM is the natural counterpart of CS-SVM for the group case that pushes decision boundary towards dominant groups. We analytically showcase the efficacy of GS-SVM (and thus GDT) in a stylized group-GMM.
\vspace{5pt}
\noindent$\bullet$~\textbf{Cost-sensitive methods can improve standard accuracy:} Cost-sensitive methods are typically designed to improve balanced accuracy. Empirically it has been reported (Khan et al. \cite{cosen} \ct{@Ganesh: can you please add the new citations})\gk{added} that they can also improve standard accuracy as a byproduct. Motivated by this, for the GMM we compute the fundamental tradeoff achieved by binary CDT loss between standard and balanced accuracy identifying scenarios where the empirical observations are confirmed. Perhaps surprisingly, we also prove for the same model that the tuning of CDT minimizing balanced error automatically balances the minority/majority errors. Talk about DEO here. That you can achieve zero.
\subsubsection{Algorithmic contributions}
\noindent$\bullet$~\textbf{VS loss (aka LA + CDT)}
\begin{itemize}
\item \ct{Hopefully Oresti's Credit Fraud experiment is a first example here}
\item \ct{Can we do a CIFAR 10 experiment where we simply combine best $\Delta$'s of CDT paper with best $\iota$'s of Menon/Google paper and hopefully show better balanced accuracy. If we show one such \textbf{benchmark} example we are golden.}
\end{itemize}
\vspace{5pt}
\noindent$\bullet$~\textbf{VS loss for group imbalances}
\begin{itemize}
\item \ct{Here any improvement over ERM is new, so we don't have to fight to show VS>LA. Just VS (with any combination of $\Delta$'s/$\iota$'s) is better than ERM. We need again one \textbf{DNN} example.}
\end{itemize}
\subsection{Discussion}
Our analysis offers useful insights to existing methods. We hope that our work motivates further similar investigations as there admittedly remain many open questions towards navigating the zoo of recently proposed methods for learning under imbalances. Below we discuss some of the most fascinating to us and how they relate to our work.
\vspace{5pt}
\noindent$\bullet$~\textbf{Learned representations vs hand-crafted features:} To the best of our knowledge, our result is the first to theoretically justify the benefit of multiplicative weights. We do this in a linear setting, but most importantly assuming \emph{fixed} feature representations (perhaps through some kernel method). In particular this means that the distribution of the training samples is same as that of test samples. On the other hand, \cite{ye2020identifying} (but also see \cite{kimkim_adjusting}) introduce CDT motivated by an empirical finding that the distributions of the \emph{learned feature representations} differ between training and test. Our implicit-bias theory does not extend to this setting and it would be fascinating extending to this direction (perhaps following \cite{mixon2020neural}).
\vspace{5pt}
\noindent$\bullet$~\textbf{LA vs CDT redux:} Why is still LA better than CE? Understand training dynamics. Empirically, we find that VS-loss is more unstable. Why?
\vspace{5pt}
\noindent$\bullet$~\textbf{Multiclass vs binary:}
\section{Introduction}
\vspace{5pt}
\noindent $\bullet$~\textbf{Connection to cost-sensitive SVM:}
For linear predictors and separable datasets, we argue that optimizing the loss in \eqref{eq:loss_Menon} with gradient descent leads to a classifier whose direction converges to the max-margin SVM solution \emph{irrespective} of the choice of the parameters $\omega_y$ and $\iota_y$. Instead, the VS loss leads to the solution of another old friend: the \emph{cost-sensitive SVM (CS-SVM)}. For binary classification (i.e. $y\in\{\pm1\}$), in particular, the resulting CS-SVM is parameterized by a \emph{margin-ratio parameter} $\delta=\Delta_{-1}/\Delta_{+1}$; see \eqref{eq:CS-SVM}.
\vspace{5pt}
\noindent $\bullet$~\textbf{CS-SVM through a modern lens:} For an insightful Gaussian mixtures model (GMM), we present formulae that sharply predict the classification and balanced errors of CS-SVM (and thus, of the VS-loss as well) in a \emph{high-dimensional} separable regime. Our formulae are explicit in terms of data geometry, class priors, parameterization ratio and tuning parameter $\delta$.
Additionally, we identify a key structural property of CS-SVM that together with our asymptotic theory lead to an explicit formula for the optimal margin ratio $\delta_\star$ that minimizes the balanced error. For example, we show that $\delta_\star$ not only depends on the class priors, but is also sensitive to the parameterization ratio.
\vspace{5pt}
\noindent $\bullet$~\textbf{Group-sensitive SVM:} We propose natural modifications to \eqref{eq:loss_ours} and to CS-SVM for classification in the presence of sensitive groups. We then extend the GMM to capture the presence of imbalanced groups, and for binary labels we develop a sharp analysis of our algorithms under sufficient overparameterization.
\vspace{5pt}
\noindent $\bullet$ \textbf{Numerical experiments:} We present numerical experiments that corroborate our findings above. Also, using our sharp analysis we study key tradeoffs between balanced error / equal opportunity and misclassification error.
\subsection{VS-loss vs LA-loss for a synthetic GMM (Additional results on the experiment of Figure \ref{fig:mismatch_intro})}
In Figure \ref{fig:figure_intro_more} we have implemented the same experiment as in Figure \ref{fig:mismatch_intro} detailed in Section \ref{sec:example_imbalance}. Here, additionally to the balanced error in Figure \ref{fig:mismatch_intro}, we present results for the misclassification error and the two conditional errors. In all cases, we report both the results of Monte Carlo simulations, as well as, the theoretical formulas predicted by Theorem \ref{thm:main_imbalance}. As promised, the theorem sharply predicts the conditional error probabilities of both the minority and the majority class. Note the almost perfect match with the numerical averages despite the relatively small problem dimension ($d=300$).
As noted in Section \ref{sec:example_imbalance}, we observe in the `Top Left' of the figure that the VS-loss results in a better balanced error in the separable regime (where $\mathcal{R}_{\rm train}=0$) compared to the LA-loss. This naturally comes at a cost, as the role of the two losses is reversed in terms of the misclassification error (see `Top Right'). The two bottom figures explain these observations showing that VS-loss sacrifices the error of majority class for a significant drop in the error of the minority class. All types of errors decrease with increasing overparameterization ratio $\gamma$ due to the mispecified model; see also \cite{hastie2019surprises,deng2020model}.
For the numerical experiments in Figure \ref{fig:figure_intro_more} we minimized the VS-loss and the LA-loss in the separable regime using normalized gradient descent. Specifically, we use an in increasing learning rate that is appropriately normalized by the norm of the loss gradient for faster convergence. Please refer to Figure \ref{fig:normalized_gd_experiments} and Section \ref{sec:gd_num_app} for a discussion on the advantage of this over a constant learning rate.
\begin{figure}[t]
\centering
\includegraphics[width=0.6\textwidth,height=0.4\textwidth]{Figures/Appendix/details_synthetic_Christos.eps}
\caption{Performance of the VS-loss vs the Logt-adjusted loss for a GMM with missing features. The experimental setting is identical to that of Figure \ref{fig:mismatch_intro}. \emph{Top Left:} balanced error $\Rc_{\text{bal}}$ and training error $\mathcal{R}_{\rm train}$ (in dashed lines). \emph{Top Right:} misclassification error $\mathcal{R}$. \emph{Bottom Left:} majority class error $\mathcal{R}_-$. \emph{Bottom Right:} minority class error $\mathcal{R}_+$. Solid lines correspond to theoretical formulas obtained thanks to Theorem \ref{thm:main_imbalance}.}
\label{fig:figure_intro_more}
\end{figure}
\section{Generalization Analysis for Group-VS-loss / GS-SVM}\label{sec:thm_gssvm_app}
In Theorem \ref{thm:main_imbalance}, we stated a sharp asymptotic theoretical result for VS-loss/CS-SVM under the Gaussian mixture data model. Here, we will state the counterpart result for GS-SVM with an appropriate Gaussian mixture data model with group imbalances.
\\
\noindent\textbf{Data model:}~We study a binary Gaussian-mixture generative model (GMM) for the data distribution $\mathcal{D}$. For the label $y\in\{\pm1\}$ let $\pi:=\mathbb{P}\{y=+1\}.$ Group membership is decided conditionally on the label such that $\forall j\in[K]:$ $\mathbb{P}\{g=j|y=\pm1\}=p_{\pm,j}$, with $\sum_{j\in[K]}p_{+,j}=\sum_{j\in[K]}p_{-,j}=1$. Finally, the feature conditional given label $y$ and group $g$ is a multivariate Gaussian of mean $\boldsymbol\mu_{y,g}\in\mathbb{R}^d$ and covariance $\boldsymbol\Sigma$, that is,
$\mathbf{x}\big|(y,g)\,\widesim{} \mathcal{N}(\boldsymbol\mu_{y,g},\boldsymbol\Sigma).
$
We focus on two groups $K=2$ with $p_{+,1}=p_{-,1}=p<1-p=p_{+,2}=p_{-,2},\,j=1,2$ and $\mathbf{x}\,|\,(y,g) \sim \mathcal{N}(y\boldsymbol\mu_{g},\mathbf{I}_d)$. As before, let $\mathbf{M}$ denote the matrix of means (that is $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{+} & \boldsymbol\mu_{-}\end{bmatrix}$ and $\mathbf{M}=\begin{bmatrix} \boldsymbol\mu_{1} & \boldsymbol\mu_{2}\end{bmatrix}$, respectively) and consider the eigen-decomposition of its Gramian:
$
\mathbf{M}^T\mathbf{M} = \mathbf{V}\mathbf{S}^2\mathbf{V}^T,~~ \mathbf{S}\succ \mathbf{0}_{r\times r},\mathbf{V}\in\mathbb{R}^{2\times r}, r\in\{1,2\},
$
with $\mathbf{S}$ an $r\times r$ diagonal positive-definite matrix and $\mathbf{V}$ an orthonormal matrix obeying $\mathbf{V}^T\mathbf{V}=\mathbf{I}_r$. We study linear classifiers with $h(\mathbf{x})=\mathbf{x}$.
\\
\noindent\textbf{Learning regime:}
Again, as in Theorem \ref{thm:main_imbalance}, we focus on a regime where training data are linearly separable. It turns out once again that linear separability undergoes a sharp phase-transition as $d,n\rightarrow\infty$ at a proportional rate $\gamma=\frac{d}{n}$. There exists threshold $\widetilde\gamma_{\star}:=\widetilde\gamma_{\star}(\mathbf{V},\mathbf{S},\pi,p)\leq 1/2$, such
that data are linearly separable with probability approaching one provided that $\gamma>\widetilde\gamma_{\star}$ \cite{candes2020phase,montanari2019generalization,deng2019model,kammoun2020precise,gkct_icassp}.
We assume $\gamma>\widetilde\gamma_{\star}$, so that GS-SVM is feasible with probability approaching 1.
\noindent\textbf{Definitions:}~Following the notation introduced before Theorem \ref{thm:main_imbalance},
we let random variables as follows:
\begin{equation*}
\begin{cases}
~G\sim \mathcal{N}(0,1),\quad \text{$Y$ symmetric Bernoulli with $\mathbb{P}\{Y=+1\}=\pi$,}\\
~\text{$S$ takes values $1$ or $2$ with $\mathbb{P}\{S=1\}=p$,}
\end{cases}
\end{equation*}
\begin{equation*}
~\widetilde{E}_{S}=\mathbf{e}_1\ind{S=1}+\mathbf{e}_2\ind{S=2}\quad\text{and}\quad \widetilde\Delta_S=\delta\cdot\ind{S=1} +1\cdot\ind{S=2},\quad\text{for}~\delta>0.
\end{equation*}
With these define the following functions $ \widetilde\eta_\delta:\mathbb{R}_{\geq0}\times{\mathcal{B}}_2^{r}\times\mathbb{R}\rightarrow\mathbb{R}$ as \\
\begin{align*}
\widetilde\eta_\delta(q,\boldsymbol\rho,b)&:=\mathbb{E}\big[\big(G+\widetilde E_{S}^T\mathbf{V}\mathbf{S}\boldsymbol\rho + \frac{bY-\widetilde\Delta_S}{q}\big)_{-}^2\big] \notag- (1-\|\boldsymbol\rho\|_2^2)\gamma.
\end{align*}
Finally, define $(q_\delta,\boldsymbol\rho_\delta,b_\delta)$ as the \emph{unique} triplet satisfying $\widetilde \eta_\delta(\widetilde q_\delta,\widetilde \boldsymbol\rho_\delta,\widetilde b_\delta)= 0$ and
$(\widetilde \boldsymbol\rho_\delta,\widetilde b_\delta):=\arg\min_{\substack{\|\boldsymbol\rho\|_2\leq 1, b\in\mathbb{R}}}\eta_\delta(q_\delta,\rho,b).$ Again, these triplets can be easily computed numerically for given values of $\gamma, \delta, \pi, p$ and means' Gramian $\mathbf{M}^T\mathbf{M} = \mathbf{V}\mathbf{S}^2\mathbf{V}^T.$
Although similar in nature, the result differs to Theorem \ref{thm:main_imbalance} since now each class itself is a Gaussian mixture.
\begin{theorem}[DEO of GS-SVM]\label{thm:main_group_fairness}
Let group-sensitive GMM data and learning regime as described above.
Consider the GS-SVM classifier with $h(\mathbf{x})=\mathbf{x}$, intercept $b$ and fixed margin-ratio $\delta>0$ (corresponding to group VS-loss with $\Delta_{y,g}=\Delta_{g}, g=1,2$ such that $\delta=\Delta_2/\Delta_1$).
Then, in the limit of $n,d\rightarrow\infty, d/n=\gamma>\widetilde\gamma_{\star}$ it holds
$
\mathcal{R}_{\pm,i} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_i^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta\pm \widetilde b_\delta/{\widetilde q_\delta}}\big), i=1,2.$ In particular,
$
\mathcal{R}_{\rm deo} \stackrel{{P}}{\longrightarrow} Q\big({\mathbf{e}_1^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big) - Q\big({\mathbf{e}_2^T\mathbf{V}\mathbf{S}\widetilde \boldsymbol\rho_\delta+ \widetilde b_\delta/{\widetilde q_\delta}}\big).$
\end{theorem}
\section{Underparameterized Regime}
\ct{I did some guessing of the formulas here. Need to verify at some point.}
\begin{theorem}\label{thm:main_imbalance_underP}
Consider the Gaussian mixture data model in Section \ref{sec:data} with $K=1$, priors $(\pi, 1-\pi)$ for the two classes and eigen-decomposition of Grammian matrix of means as in \eqref{eq:SVD}. Let $(\hat{\w},\hat{b})$ be the solution to the following ERM with logit-adjusted cost:
\begin{align}
\min_{\mathbf{w},b}\frac{1}{n}\sum_{i\in[n]}\log\left(1+e^{\Delta_{y_i}} e^{-y_i(\mathbf{x}_i^T\mathbf{w}+b)}\right),
\end{align}
where $\Delta_{y_i}=\begin{cases}\Delta_{+1}&, y_i=1\\ \Delta_{-1}&, y_i=-1\end{cases}$.
Define random variables $G, Y, \Delta_Y\in\mathbb{R}$ and $E_Y\in\mathbb{R}^{2\times 1}$ as follows: $G\sim \mathcal{N}(0,1)$, $Y$ is a symmetric Bernoulli with $\mathbb{P}\{Y=+1\}=\pi$, $E_Y=\mathbf{e}_1^T\ind{Y=1} - \mathbf{e}_2^T\ind{Y=-1}$, and $\Delta_Y=\Delta_{+1}\cdot\ind{Y=+1} + \Delta_{-1}\ind{Y=-1}$. Finally, let $\mathcal{M}_{\ell}(\cdot;\la)$ be the Moreau envelope function at scale $\la$ of the sigmoid function $\ell(t)=\log(1+e^{-t}).$ Denote $\mathcal{M}_{\ell}^\prime(\cdot;\la)$ its derivative. With these, let $(q,\boldsymbol\rho,\la,b)\in\mathbb{R}_+\times{\mathcal{S}}^{r-1}\times\mathbb{R}_{+}\times\mathbb{R}$ be the unique solution to the following system of four equations in four unknowns:
\begin{align*}
\mathbb{E}\left[(G+q\,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho)\cdot\mathcal{M}_{\ell,1}^\prime\left(qG+q\,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho+bY-\Delta_Y;\la\right)\right] &= 0\\
\la^2\mathbb{E}\left[\mathcal{M}_{\ell}^\prime\left(qG+q\,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho+bY-\Delta_Y;\la\right)\right] &= \gamma q^2 (1-\|\boldsymbol\rho\|_2^2)\\
\la\mathbb{E}\left[G\cdot\mathcal{M}_{\ell}^\prime\left(qG+q\,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho+bY-\Delta_Y;\la\right)\right] &= \gamma q \sqrt{1-\|\boldsymbol\rho\|_2^2}\\
\mathbb{E}\left[Y\cdot\mathcal{M}_{\ell}^\prime\left(qG+q\,\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho+bY-\Delta_Y;\la\right)\right] &= 0
\end{align*}
Then, in the limit of $n,d\rightarrow\infty$ with $d/n=\gamma<\gamma_\star$ it holds that
\begin{align*}
\mathcal{R}_{+} \stackrel{{P}}{\longrightarrow} Q\left(\mathbf{e}_1^T\mathbf{V}\mathbf{S}\boldsymbol\rho+b/q\right)\text{ and }
\mathcal{R}_{-} \stackrel{{P}}{\longrightarrow} Q\left(\mathbf{e}_2^T\mathbf{V}\mathbf{S}\boldsymbol\rho-b/q\right).
\end{align*}
\end{theorem}
\subsection{Experiments on ``Waterbirds" dataset}
| 2024-02-18T23:39:53.755Z | 2021-11-09T02:32:18.000Z | algebraic_stack_train_0000 | 762 | 36,394 |
|
proofpile-arXiv_065-3824 |
\section{Introduction} \label{sec:intro}
\input{sections/intro.tex}
\section{Preliminaries} \label{sec:preliminaries}
\input{sections/preliminaries.tex}
\section{Multi Copy Atomic model} \label{sec:mca}
\input{sections/mca_intro.tex}
\section{\textsl{C11}\xspace Memory Model Semantics} \label{sec:c11 hb}
\input{sections/c11_memory_order.tex}
\section{\textsl{MCA}\xspace restriction for \textsl{C11}\xspace } \label{sec:our hb}
\input{sections/our_hb.tex}
\section {\textsl{C11}\xspace-\textsl{MCA}\xspace aware source-DPOR\xspace} \label{sec:algo}
\input{sections/algo_working.tex}
\section{Experimental Validation} \label{sec:results}
\input{sections/results.tex}
\section{Related Work} \label{sec:related-work}
\input{sections/related_work.tex}
\section{Concluding Remarks} \label{sec:conclusion}
\input{sections/conclusion.tex}
\begin{footnotesize}
\bibliographystyle{IEEEtran}
| 2024-02-18T23:39:54.074Z | 2021-05-28T02:26:47.000Z | algebraic_stack_train_0000 | 774 | 117 |
|
proofpile-arXiv_065-3928 |
\section{Background}\label{sec:background}
\input{steadystate}
\input{jmh}
\input{dyn-reconfig}
\newpage
\section{Conclusion} \label{sec:conclusion}
This paper presents a comprehensive investigation on Java steady state performance assessment.
Through a rigorous assessment, we showed that Java microbenchmarks do not always reach a steady state of performance, thus confirming the finding of \cite{Barrett2017} in the more defined scope of ``testing-oriented'' Java microbenchmarks.
Even when microbenchmarks consistently reach a steady state of performance, a reliable assessment remains far from trivial.
According to our results, the current state-of-practice, which mostly relies on developer static configurations, show poor effectiveness for steady state performance assessment.
Developers often fail to accurately estimate the end of the warmup phase, thereby causing either large time wastes or poor results quality.
Dynamic reconfiguration provides a significant leap forward over developer static configurations by providing more accurate warmup estimates and less pronounced side effects. Still, the achieved results highlight non-trivial estimation errors, large time wastes, and distorted performance measurements.
The findings of our work have implications for both practitioners and researchers.
For the former, it is important to be aware that benchmark forks may not always reach a steady state of performance. The recommendation here is to perform an adequate number of forks (\emph{e.g.,}\xspace 10) to mitigate the noise introduced by ``non-steady'' forks.
Another important lesson for practitioners is to favor dynamic reconfiguration over static configuration when possible.
Indeed, when compared to developer configurations, dynamic reconfiguration techniques provide more accurate estimates of warmup time, though this improvement may (sometimes) come at the expense of a more time-consuming performance assessment process.
Further empirical studies are needed to assess whether this ``cost'' is acceptable for practitioners.
Nonetheless, the achieved results are also helpful for suggesting which technique to use depending on the practitioner's need.
On the researchers' side, given the promising results of dynamic reconfiguration and the large room for improvement suggested by our investigation, we envision research aimed at designing novel and more effective dynamic reconfiguration techniques to (i) reduce the time effort devoted to performance assessment and (ii) strengthen the reliability of performance measurements.
This is a direction we aim to investigate in future work.
We have made the code and the data used in our study publicly available to encourage further research on this topic.
\section{Experimental Design} \label{sec:design}
In order to answer our research questions, we first collect performance measurements from the execution of 586\xspace microbenchmarks across 30\xspace systems.
Then, we analyze collected measurements to determine whether and when each fork reaches a steady state of performance.
Finally, we perform post-hoc analysis on the collected measurements to assess developers and dynamic configurations.
In this section, we first describe the microbenchmarking setup we use to collect performance measurements and the benchmark subjects. Then, we describe in detail the steady state detection technique used in our empirical study. Finally, we present the process we use to extract both developer and dynamic configurations.
\subsection{Microbenchmarking setup}\label{sec:microbench}
Following the methodology used in the study of \cite{Barrett2017}, we execute each benchmark for a substantially longer time than ``usual'' JMH configurations (on average 171~times longer than developer configurations).
We perform 10 JMH forks for each benchmark (as suggested by \cite{Barrett2017}), where each fork involves an overall execution time of at least 300 seconds and 3000 benchmark invocations.
To do so, we configure the execution of each benchmark via JMH CLI arguments.
Specifically, we configure 3000 measurement iterations (\verb|-i 3000|) and 0 warmup iterations (\verb|-wi 0|) to collect all the measurements along the fork.
Each iteration continuously executes the benchmark method for 100ms (\verb|-r 100ms|)\footnote{We chose 100ms as iteration time because this value enables us to ``replicate'' every possible configuration considered in our study (see Section 4.4 for more details on how we obtain the set performance measurements for a particular JMH configuration).}.
The number of fork is configured to 10 (\verb|-f 10|). As benchmarking mode, we use sample (\verb|-bm sample|), which returns nominal execution times for a sample of benchmark invocations within the measurement iteration.
The execution environment and external events occurring during the benchmark runs have a remarkable influence on the accuracy of results.
This is especially true when executing microbenchmarks, as they tend to measure small portions of code that may last less than a microsecond and are, therefore, more prone to be affected by even small changes in the environment.
Hence, we tried to control as many sources of variability as possible in order to obtain more reliable measurements.
We disabled Intel Turbo Boost, \emph{i.e.,}\xspace a feature that automatically raises the CPU operating frequency when demanding tasks are running~\citep{RHEL}.
We also disabled hyper-threading, \emph{i.e.,}\xspace a feature in modern processors that executes two threads simultaneously on the same physical core~\citep{RHEL}.
This is achieved by replicating the architectural state but sharing execution resources such as ALUs and caches.
For this reason, hyper-threading may lead to contention patterns that continuously vary during the execution.
Another potential cause of variability among repeated runs is represented by Address Space Layout Randomization (ASLR), which is a security technique to randomly arrange the address space positions at each execution.
We disabled ASLR as it may cause variability in the measurements from one fork to another.
The amount of available memory can also affect execution times.
We fixed (through the \texttt{-Xmx} flag) the total amount of heap memory available to the JVM to 8GB, because this is the most important factor affecting garbage collection performance.
In fact, the throughput of garbage collections is inversely proportional to the amount of memory available, since collections occur when memory fills up~\citep{JavaPerformance}.
A large variety of operating system events may have a noticeable impact on execution times because they increase context switching in most cases.
For this reason, we tried to keep the events that are not related to the benchmarks to a minimum.
We disabled any Unix daemon that is not strictly necessary.
We also disabled SSH logins for the entire duration of the experiments.
To further reduce context switching, we used priority scheduling and increased the \emph{niceness} of the JMH process running the benchmarks and all its children.
Finally, we ensured that the state of the system was consistent at each run by monitoring the \emph{dmesg} log and the \emph{systemd} journal for anomalies, as well as the shell environment of the process for changes in size~\citep{MytkowiczDHS09}.
The benchmarks were executed on a bare metal server running Linux Ubuntu 18.04.2 LTS on a dual Intel Xeon E5-2650v3 CPU at 2.30GHz, with a total of 40 cores and 80GiB of RAM.
\subsection{Subject Benchmarks}
\input{tables_systems}
\tabref{tab:systems} reports the list of the 30\xspace Java open source systems we use in this study.
We selected such systems because they are relatively popular (\emph{i.e.,}\xspace they have more than 100 Github stars), have non-trivial JMH suites (\emph{i.e.,}\xspace have at least 20 benchmarks), and span different domains (\emph{e.g.,}\xspace application servers, logging libraries, databases).
Given the large size of the benchmark suites, we randomly sample 20 benchmarks for each system.
14 out of the 600 sampled benchmarks failed in our experimental setup\footnote{The 14 failed benchmarks are distributed as follows: 3 \texttt{cantaloupe}, 1~\texttt{jetty.project}, 4~\texttt{vert.x}, 3~\texttt{hazelcast} and 5~\texttt{jbdi}}.
Overall, in our empirical study, we assess the behavior of 586\xspace randomly sampled benchmarks across 30\xspace Java systems.
In order to investigate the correctness of the benchmarks selected for this study, we ran the SpotJMHBugs tool by \cite{Costa2019} on the systems.
The tool was able to detect only one potential bad practice, of type LOOP, in the \texttt{netty} project.
Therefore, we consider our selection of benchmarks suitable for the study.
\subsection{Steady State Detection}\label{sec:exp_steadystate}
Detecting the end of the warmup phase, and consequently the start of the steady state, is no trivial task, as the notion of ``\emph{steady}'' resides on how much stability of results one wants to achieve during performance testing or, to put it in another way, how much variability one is willing to tolerate.
Nonetheless, any analyst who wants to study steady state performance must establish the length of the warmup phase.
In our study, we are interested in automatically detecting the length of such phase to determine whether and when a benchmark reached a steady state.
For this task, we build on the steady state detection approach proposed by \cite{Barrett2017}, which we adapted to the purposes of our study.
The approach is fully automated, and it is based on changepoint analysis~\citep{Eckley2011}, which is a statistical technique to detect shifts in timeseries data. In our experimental setup, each datapoint of the timeseries represents the average execution time within a JMH iteration, and the whole timeseries represents a fork.
\begin{figure}[htbp]
\centering
\includegraphics[width=0.98\linewidth]{img_steady_state_detection}
\caption{Steady state detection process.}%
\label{fig:steady_state_detection}
\end{figure}
An overview of the approach can be found in \figref{fig:steady_state_detection}.
As a first step of the approach, we identify and remove potential outliers as we are only interested in detecting shifts in execution time that appear to stay consistent for a period of time. As done by \cite{Barrett2017}, we use the method by \cite{tukey1977} to identify as outliers the datapoints that lie outside the median $\pm3 \times (99\%ile - 1\%ile)$ in a 200 datapoints window \circled{1}.
Out of the $1.8 \times 10^7$ datapoints in the study, $0.27\%$ are classified as outliers, with the most of any fork being $1.3\%$.
After filtering outliers, we apply a changepoint algorithm to detect shifts in execution time.
Changepoint algorithms are designed to divide the entire timeseries into segments, within which the behavior of the timeseries is considered to remain unchanged.
On the basis of the segmentation of the timeseries we can detect if and when a benchmark execution reached a steady state.
The specific changepoint algorithm we used is called \emph{PELT}~\citep{Killick2012}.
We applied the algorithm to timeseries data gathered from individual forks in order to detect changes in both the mean and the variance of execution time.
An important parameter of the PELT algorithm is the \emph{penalty}, namely an argument designed to avoid under/over-fitting and, therefore, directly impacting the number of changepoints the algorithm will detect.
The higher the penalty value, the more difficult will be for the algorithm to detect changepoints.
Conversely, lower penalty values will result in more changepoints.
Barrett~\emph{et~al.}\xspace set this parameter to $15\log(n)$, where $n$ is the number of datapoints in the timeseries after discarding the outliers.
We concluded that a single penalty value for all the timeseries (\emph{i.e.,}\xspace all the forks in the experiment) was not suitable for tuning the algorithm to the differences we found among benchmark execution data.
As a consequence, we decided to employ a method to derive an appropriate penalty value for each timeseries: we used the \emph{CROPS} algorithm~\citep{Haynes2014} to efficiently generate, for each timeseries, optimal changepoint segmentations for all penalty values in a continuous range ($[4, 10^5]$, in our case) \circled{2}.
The number of changepoints in the alternative segmentations and the corresponding penalty values can be used to derive an optimization curve (sometimes referred to as \emph{elbow diagram}).
\figref{fig:crops_example} shows an example of such a curve computed on the timeseries in \figref{fig:non_steady_example}.
As suggested by \cite{Lavielle05}, this diagram can be visually inspected to find suitably parsimonious penalty values in the area of the \emph{elbow}.
A penalty point in the elbow area can be automatically selected using the \emph{Kneedle} algorithm~\citep{SatopaaAIR11}, which is a method to find the point of maximum curvature in the continuous approximation of an optimization curve \circled{3}.
A red \emph{x} in \figref{fig:crops_example} marks the point in the curve that was chosen by the \emph{Kneedle} algorithm in that case.
Using this procedure, we were able to automatically derive a different penalty value to guide the segmentation of each fork.
\begin{figure}
\centering
\includegraphics[width=10cm]{img_crops_example}
\caption{Example of the selection procedure of a penalty value from an elbow diagram.}
\label{fig:crops_example}
\end{figure}
Once we obtain a segmentation of the timeseries \circled{4}, we can proceed to detect a possible steady state.
A steady state should be detected when the execution time is reasonably stable after the end of the warmup phase.
It is a matter of interpretation how much the execution time is allowed to vary but, following the approach from Barrett~\emph{et~al.}\xspace, we consider a fork to have reached a steady state if the last 500 measurements are contained in a single segment (\emph{i.e.,}\xspace no changepoint was detected within the last 500 datapoints).
In general, to find when the steady state was first reached we could just take the start of the last segment.
However, this would not consider practical cases in which the smallest variation in mean or variance would generate different segments, even if, from a performance evaluation point of view, the benchmark has completed its warmup phase.
Therefore, we need to establish a suitable tolerance to allow the steady state period to span multiple segments whose variation is not meaningful to determine the execution time of the benchmark.
In Barrett~\emph{et~al.}\xspace the tolerance was provided by combining the maximum number of consecutive segments from the last one, such that a segment $s_i$ is equivalent to the final segment $s_f$ if mean($s_i$) is within (mean($s_f$) $\pm$ max(variance($s_f$), 0.001s)).
We did not intend to apply a fixed threshold in units of time (like 0.001 seconds) because our benchmarks vary from tens of nanoseconds to few seconds, nor we wanted to use the variance of the last segment as a threshold because it leads to an extremely low tolerance for most of the benchmarks\footnote{In 60\% of benchmarks forks, the ratio between the variance and mean of last segment is smaller than 0.0000002. Using variance as a threshold in these forks, it would imply that any negligible performance shift larger than 0.00002\% would be considered as a meaningful performance change.}.
Hence, we preferred to compare the segments by applying a 5\% tolerance on the confidence interval for the mean relative performance change, by using the approach of \cite{Kalibera2013} \circled{5} (see \secref{sec:rq2} for details).
On the basis of this information, we can classify individual forks as \emph{steady state} if we were able to detect a steady state, or \emph{no steady state} when the opposite occurred.
Moreover, we classify a benchmark as \emph{steady state} if all its forks reached a steady state, \emph{no steady state} if all its forks did not reach a steady state, and \emph{inconsistent} if at least one fork was classified as \emph{steady state} while at least another one was classified as \emph{no steady state}.
The same information is used to derive the \emph{steady state starting time} (\emph{st}), which is the beginning of the first segment $s_i$ that is considered to be steady.
Consequently, for a given fork, the set of measurements in the range between \emph{st} and the end of the timeseries is the set of the steady state performance measurements ($M^{stable}$) of our experiment.
Based on that, given a specific benchmark, we can further define the entire set of steady measurements $\mathcal{M}^{stable}$ as the union of the steady measurements sets $M^{stable}$ collected across all the steady forks of the benchmark.
\subsection{Benchmark configurations}\label{sec:bench_conf}
Each benchmark configuration determines for each fork two relevant pieces of information: the \emph{estimated warmup time} (\emph{wt}) and a set of performance measurements $M^{conf}$.
We use this information to (i) compute warmup estimation error ($WEE$), (ii) determine whether a configuration underestimates or overestimates the \emph{steady state starting time} (\emph{st}), and (iii) assess potential side effects due to wrong estimation (\emph{i.e.,}\xspace \emph{time waste} or \emph{relative performance deviation}).
In the following, we describe the process we use to derive \emph{wt} and $M^{conf}$ for each benchmark fork, for both developer and dynamic configurations.
\input{algo_estimate_warmup}
\input{algo_measurements}
\subsubsection{Software developer configurations}
Software developers define JMH configurations in benchmark code through Java annotations.
When a benchmark is launched, JMH executes the benchmark according to developer configurations (\emph{e.g.,}\xspace no. measurements and warmup iterations).
A trivial approach to obtain both \emph{wt} and the set of performance measurements $M^{conf}$ for each fork would be to simply run the benchmark, and extract this information from the execution logs and JSON result files produced by JMH.
Unfortunately, this approach would be extremely expensive in terms of time, and would not fit our own needs.
Instead, we use post-hoc analysis: We first obtain the JMH configurations as defined by developers, then we use this information to compute both \emph{wt} and $M^{conf}$ based on the performance measurements collected in our microbenchmarking setup (see \secref{sec:microbench}).
In order to obtain developer configurations, we leverage a JMH feature that allows to overwrite configurations on-the-fly via CLI arguments\footnote{ We use dynamic analysis (\emph{i.e.,}\xspace running benchmarks by overwriting CLI arguments) instead of static analysis because this methodology ensures a better coverage and lower margin of errors. Indeed, developers may rely on other mechanisms than JMH annotations to configure benchmarks, (\emph{e.g.,}\xspace see \texttt{OptionsBuilder} at \url{https://bit.ly/3OkxzJS}). Our approach allows to safely retrieve configurations also in these cases, while this would have been impractical through static analysis.}.
We exploit this capability to reduce benchmark execution time and speed-up developer configurations retrieval.
We first execute each benchmark twice while reducing execution time through JMH CLI arguments.
Then, we retrieve developer configurations in the JSON result files of each individual execution.
Specifically, we first obtain the number of measurement and warmup iterations (\emph{i.e.,}\xspace $wi$ and $i$), and the number of forks $f$ by executing each benchmark while setting the measurement and warmup time of each iteration to 1 nanoseconds (\verb|-w 1ns -r 1ns|).
Then, we retrieve the measurement and warmup time (\emph{i.e.,}\xspace $w$ and $r$) of each iteration by running each benchmark while setting the number of forks, warmup and measurement iterations to 1 (\verb|-f 1 -wi 1 -i 1|).
At the end of this process, we obtain a tuple $(w, wi, r, i, f)$, where $w$ denotes the time of a warmup iteration, $wi$ denotes the number of warmup iterations, $r$ denotes the time of a measurement iteration, $i$ the number of measurement iterations and $f$ the number of forks.
We exploit $w$, $wi$, $r$ and $i$ along with the fork measurements $M$ (as collected in our microbenchmarking setup) to compute, for each fork, the \emph{estimated warmup time} (\emph{wt}) and the set of performance measurements $M^{conf}$.
Specifically, we estimate the time spent in each warmup/measurement iteration using the average execution time of each iteration as observed in our microbenchmarking setup; then, we derive $wt$ and $M^{conf}$ based on the JMH configuration.
We report the detailed process to obtain both \emph{wt} and $M^{conf}$ in \algoref{alg:estimate_wt} and \algoref{alg:select_measurements}, respectively.
Based on the above, we can exploit the number of configured forks $f$ defined by software developers to obtain the whole set of performance measurements gathered from the entire benchmark execution, namely $\mathcal{M}^{conf}$. In other words, we derive $\mathcal{M}^{conf}$ by joining all the sets of measurements $M^{conf}$ gathered from the first $f$ forks of the benchmark.
\subsubsection{Dynamic configurations}
In order to obtain $(w, wi, r, i, f)$ for each dynamic reconfiguration variant, we leverage the replication package provided by \cite{Laaber2020}.
Specifically, we use the scripts provided for post-hoc analysis, which take as input JMH result JSON files, and return, for each fork, the number of warmup iterations ($wi$) according to stability criteria.
The warmup iteration time $w$, the measurement time $r$, and the number of measurement iterations $i$ are fixed to $w=1s$, $r=1s$ and $i=10$, respectively, according to the experimental setup defined in \citep{Laaber2020}.
We then obtain $wt$ and $M^{conf}$ using the same approach adopted for developer configurations.
\section{Discussion} \label{sec:discussion}
Overall, we can observe that Java microbenchmarking is still subject to some flaws.
The results for RQ$_1$ provide evidence that benchmarks do not always reach a steady state of performance.
About 11\% of benchmark forks never reach a steady state of performance, and 43\% of benchmark executions involve at least one fork that doesn't hit the steady state.
These results are consistent with the seminal study on VM microbenchmarking of \cite{Barrett2017}, thus showing, on a larger corpus of benchmarks and in the more defined scope of ``testing-oriented'' Java microbenchmarks, that the ``\emph{two-phase assumption}'' does not always hold.
With this finding, we aim to raise awareness among developers (and researchers) that deal with Java microbenchmarking. An important lesson here is that some benchmark forks (mean $\sim$10\%) may not be representative of ``actual'' steady state performance, since their performance may continuously fluctuate over time, with a non-negligible deviation from steady state performance. Unfortunately, the only way to avoid this issue is to execute each fork for a large number of iterations, and then run the \cite{Barrett2017} technique to determine if the steady state of performance is reached or not. While this methodology may be appropriate in a research context (like ours), it may be impractical in real-word performance assurance processes, where benchmarks are repeatedly executed against software evolution, and time/resources are subject to constraints~\citep{traini2021}. Nonetheless, there are certain measures that can be put in place to (partially) mitigate this problem. For example, the analyses performed for RQ$_2$ showed that performance deviations of non-steady forks can be significantly reduced by using a minimum of 50 warmup iterations that, based on our microbenchmarking setup, correspond to 5 seconds of continuous benchmark execution and no less than 50 invocations. The performance deviations can be further mitigated by increasing the number of warmup iterations up to 300 (\emph{i.e.,}\xspace 30 seconds of continuous benchmark execution and no less than 300 invocations). It is worth to notice that these values are considerably different than those provided by JMH defaults, which define 50 seconds of continuous benchmark execution and no less than 5 invocations for warmup. On the basis of our results, our practical suggestion is to never execute a benchmark for less than 5 seconds (and less than 50 invocations) before starting to collect measurements. When time does not represent a major concern, warmup should last for at least 30 seconds of continuous benchmark execution, and no less than 300 invocations.
Microbenchmarking is far from trivial even when benchmarks consistently reach a steady state of performance. The results of RQ$_2$ sheds a light on the potential pitfalls of using non-steady measurements. Performance measurements gathered in non-steady phases of benchmark execution substantially deviate from those collected during steady phases ($\sim$124k\% on average). Hence, relying on them can significantly mislead performance assessment.
To deal with this problem, the current practice mostly rely on developers' \emph{guesses} to estimate the end of the warmup phase and discard measurements subject to performance fluctuations.
Based on the results for RQ$_3$, this approach seems to drastically mitigate these large deviations, \emph{i.e.,}\xspace developer configurations lead to an average deviation from steady measurements of 8\%.
Nonetheless, warmup estimation remains challenging and subject to (large) errors.
The results for RQ$_3$ show that developer static configurations fail to accurately estimate the end of the warmup phase, often with a non-trivial estimation error (median: 28 seconds). Developers tend to overestimate warmup time more frequently than underestimating it (48\% \emph{vs} 32\%). Nonetheless, both of these kinds of estimation errors produce relevant (though diverse) side effects.
For example, we showed that overestimation produces severe time wastes (median: 33 seconds), thereby hampering the adoption of benchmarks for continuous performance assessment.
On the other hand, underestimation often leads to performance measurements that significantly deviate from those collected in the steady state (median 7\%), thus leading to poor results quality and potentially wrong judgements.
The latter side effect can be partially mitigated by running an adequate number of forks (\emph{e.g.,}\xspace 5). Indeed, as we have shown in RQ$_2$, forks play a significant role in reducing performance deviations of non-steady measurements. Unfortunately, they also largely increase benchmark execution time, and this may be impractical in real-word contexts.
Another option is to leverage automated techniques that can effectively estimate the end of the warmup time at run-time.
Prior work tried to address this challenge through dynamic reconfiguration \citep{Laaber2020}.
Based on the results for RQ$_5$, dynamic reconfiguration techniques significantly improve the effectiveness of the state-of-practice. The achieved results show that dynamic reconfiguration techniques outperform developer static configurations in terms of \emph{warmup estimation error} with statical significance (p $\leq$ 0.05) and large/medium effect sizes (see \secref{sec:dyn_vs_dev_wee}). Nevertheless, this improvement may come at the expense of an increased microbenchmark execution time. For example, $RCIW$ produces higher estimates of the warmup time with non-negligible effect size in 20 out of the 30\xspace systems (see \secref{sec:dyn_vs_dev_wt}).
On the other hand, $CV$ and $KLD$ have more heterogeneous behaviors depending on the system, but they still report higher warmup estimates than those of developers in 10 systems.
Further empirical studies are needed to assess whether such time increase is acceptable for practitioners.
The results for RQ$_4$ also highlight a substantial diversity among different dynamic reconfiguration techniques. One peculiar example is $RCIW$ that, on one hand, induces the highest increase in microbenchmark execution time, but on the other hand, it provides the most reliable set of performance measurements.
Microbenchmark practitioners that do not have specific concerns on time (\emph{e.g.,}\xspace small benchmarks suites) should adopt $RCIW$ for a reliable steady state performance assessment.
In the other cases, $KLD$ and $CV$ represent the best alternatives.
Despite the promising results highlighted in RQ$_5$, our findings suggest room for improvement for dynamic reconfiguration.
As shown for RQ$_4$, all dynamic reconfiguration techniques lead to a substantial estimation error, with $RCIW$ providing by far the largest error (median: 48 seconds), and, $CV$ and $KLD$ producing smaller, but still relevant, estimation errors (median of 19 and 17 seconds, respectively).
These errors induce significant, though diverse, side effects depending on the technique.
For example, $RCIW$ induces substantial side effects in terms of time waste (median: 46 seconds), while $KLD$ and $CV$ induce more frequent and impactful side effects on the reliability of performance measurements (median performance deviation of 10\% and 9\%, respectively).
Nonetheless, half of the warmup estimates of dynamic reconfiguration techniques can be reduced by at least 96\%, when only considering overestimated forks.
These results highlight a large space for improvement in dynamic reconfiguration techniques, and call for further research on designing and developing more effective dynamic reconfiguration techniques.
For example, future research may explore the use of other stability metrics (\emph{e.g.,}\xspace autocorrelation metrics, other confidence interval metrics~\citep{Fieller1954}), or combinations of them to more effectively determine the end of the warmup phase.
Another suggestion is to focus more on improving warmup estimation accuracy, rather than finding stability criteria that are suitable to both stop forks and warmup iterations.
Indeed, based on our results, dynamically stopping forks does not produce any tangible improvement over developer static configurations. In this regard, our suggestion is to allow practitioners manually configuring forks based on their own needs and time constraints. Nonetheless, we always recommend to run at least 5 forks (\emph{i.e.,}\xspace the default in JMH) to mitigate the impact of non-steady measurements. Practitioners may decide to run less forks when time represents a major concern, however they should be aware of potential implications on results quality.
\subsection{Dynamic Reconfiguration} \label{sec:dyn-reconfig}
JMH allows to statically define the expected length of the warmup phase using configuration parameters, such as warmup iteration time $w$ and warmup iterations $wi$.
Such estimation is typically performed on the basis of developer expertise and/or benchmark nature.
Previous studies have shown that a static definition of the warmup time can be quite detrimental for steady state performance assessment~\citep{Georges2007, Kalibera2013, Barrett2017}.
An alternative to JMH static configuration can be found in a recent approach called dynamic reconfiguration~\citep{Laaber2020}.
This approach is able to determine, during a JMH benchmark execution, whether the measurements appear to be stable, and more executions are unlikely to improve their accuracy.
The rationale behind dynamic reconfiguration is that, by using automated stability criteria to halt the benchmark execution, developers can save some of the time dedicated to performance testing while keeping an acceptable level of accuracy.
To achieve this, the dynamic reconfiguration approach uses a sliding window to compare the last iterations to a stability criterion.
\cite{Laaber2020} propose and evaluate three stability criteria to dynamically estimate the end of the warmup phase:
\begin{itemize}
\item \emph{Coefficient of variation (CV)\footnote{It is worth to notice that, although both the approaches of \cite{Laaber2020} and \cite{Georges2007} are based on coefficient of variation (CV), they have some relevant differences. Indeed, Georges~\emph{et~al.}\xspace's heuristic uses a fixed threshold on CV, while Laaber~\emph{et~al.}\xspace's approach computes the difference between the minimum and the maximum CV in a sliding window of iterations, and it determines whether this difference exceeds a predefined threshold. Interestingly, \cite{Laaber2020} reported that the usage of the Georges~\emph{et~al.}\xspace's heuristic to dynamically estimate the end of the warmup is unrealistic for JMH microbenchmarks. (Indeed, in our experimental setup, it stops warmup iterations in only 46.7\% of forks, when using a measurements window of 30, and a CV threshold of 0.02).}}: CV is the ratio of the standard deviation to the mean, and it can be used to compare normally distributed data. Even when data is not normally distributed, as it is often the case for benchmark data, CV can still provide an estimate of measurement variability. A fork is considered stable when the difference between the largest and the smallest value of CV computed on the sliding window is within a fixed threshold.
\item \emph{Relative confidence interval width (RCIW)}: in this case, the variability in measurement data is estimated using a technique by \cite{Kalibera2013,kalibera2020} that employs hierarchical bootstrapping to compute the RCIW for the mean. The hierarchical levels are invocations, iterations, and forks.
\item \emph{Kullback-Leibler divergence (KLD)}: a technique described by \cite{He2019} to compute the probability that two distributions are similar based on the Kullback-Leibler divergence (KLD)~\citep{kullback1951}. In this case, the first distribution contains all the measurements in the sliding window excluding the last one, while the second distribution includes also the last measurement. As a consequence, stability is reached when the mean of the computed similarity probabilities is above some threshold.
\end{itemize}
In order to further reduce benchmark execution time, dynamic reconfiguration leverages the same stability criteria also to dynamically determine whether to execute the next fork or not.
\section{Introduction}
Microbenchmarking is a form of lightweight performance testing, widely used to assess the execution time of Java software \citep{Leitner2017}.
Although less demanding than other performance testing techniques (\emph{e.g.,}\xspace load tests \citep{Jiang2015}), Java microbenchmarking requires careful design \citep{Costa2019, Kalibera2013, Georges2007} to enable a reliable performance assessment.
A key challenge is the inherent non-linearity of Java performance: The Java Virtual Machine (JVM) uses just-in-time compilation to translate ``hot'' parts of the Java code into efficient machine code at run-time \citep{Barrett2017}, leading to (often severe) performance fluctuations and potentially unstable results.
To tackle this problem, practitioners rely on the \emph{assumption} that microbenchmarking is characterized by \emph{two distinct phases}.
During an initial \emph{warmup phase}, the JVM determines which parts of the software under test would most benefit from dynamic compilation, then, in a subsequent phase, the benchmark reaches a \emph{steady state} of performance.
Based on that, benchmarks are typically designed to discard measurements of the \emph{warmup phase} and focus on \emph{steady state} performance~\citep{Georges2007, Kalibera2013, Barrett2017}.
Java Microbenchmark Harness (JMH), \emph{i.e.,}\xspace the most popular Java microbenchmarking framework \citep{Leitner2017}, leverages this concept and enables developers to manually \emph{configure} the expected \emph{warmup time} of a benchmark.
Once launched, a JMH benchmark continuously executes the software under test for the configured warmup time, and, only after that, starts to collect steady state performance measurements.
Although considered the cornerstone of most of the current Java microbenchmarking practice, the \emph{two-phase assumption} is not yet confirmed by empirical studies.
Quite the opposite, there are indications that such an assumption oversimplifies the actual microbenchmarking behavior.
In a recent study, \cite{Barrett2017} studied a set of small and deterministic benchmarks \citep{Bagley2004, Bolz2015} across different types of VMs (including the JVM), and found that a relevant portion of benchmarks never hit the steady state.
It is worth to notice that these benchmarks significantly differ from ``software testing oriented'' benchmarks (\emph{e.g.,}\xspace JMH). Indeed, they are not aimed at assessing specific software, rather, they are typically used as optimization targets by VM authors. Even more, they are generally more effectively optimized by VMs than average software \citep{Ratanaworabhan2009}.
Despite the peculiarity of these benchmarks, the finding of \cite{Barrett2017} raises concerns on the current Java microbenchmarking practice, and calls for further empirical investigation.
Nevertheless, even when benchmarks consistently reach a steady state, performance assessment remains far from trivial.
A key challenge is to effectively estimate \emph{warmup time}.
An overestimated warmup time may waste too much time, thereby potentially hampering the adoption of benchmarks in the Continuous Integration (CI) pipeline \citep{Laaber2020, traini2021}.
On the other hand, an underestimated warmup time may easily mislead steady state performance assessment \citep{Georges2007, Kalibera2013}.
The current state-of-practice mostly relies on software developers' knowledge to determine warmup time.
Software developers estimate warmup time based on their expertise, and statically \emph{configure} benchmark execution according to this estimation.
Unfortunately, no empirical studies so far have directly investigated the effectiveness of this practice for steady state performance assessment.
Recently, an alternative approach to static benchmark configuration has been proposed by \cite{Laaber2020}.
This approach, called \emph{dynamic reconfiguration}, leverages stability criteria \citep{Kalibera2013, He2019} to automatically determine the end of the warmup phase at run-time.
According to their results, when compared to JMH default configurations, dynamic reconfiguration can significantly reduce execution time with low impact on results quality.
Despite these promising results, there is still little knowledge on the effectiveness of dynamic reconfiguration for steady state performance assessment.
And even more, it is yet unclear whether such techniques can improve the effectiveness of the current practice, \emph{i.e.,}\xspace developer static configurations.
In the past years, Java microbenchmarks have been widely studied in the literature.
\cite{Leitner2017} found that JMH was one of the predominant microbenchmarking framework in the Java community. \cite{Costa2019} empirically studied five JMH bad practices. \cite{Laaber2019} performed an exploratory study on software microbenchmarking in the cloud. \cite{Samoaa2021} studied the impact of parameterization in JMH microbenchmarks.
Despite these efforts, there is still a lack of knowledge on the effectiveness of modern Java microbenchmarking for steady state performance assessment.
In this paper, we aim to fill this gap by presenting the first comprehensive study that investigates steady state performance assessment in Java microbenchmarking.
After an extensive experimentation of 586\xspace JMH benchmarks from 30\xspace Java systems, involving $\sim$9.056 billion benchmark invocations for an overall execution time of $\sim$93~days\xspace, we determined whether and when each benchmark reaches a steady state using an automated statistical approach by \cite{Barrett2017} based on changepoint analysis \citep{Killick2012}.
At the time of writing, \citeauthor{Barrett2017}'s approach represents one of the most advanced automated technique to determine steady state execution in Java benchmarks.
Besides investigating whether benchmarks ever reach a steady state or not, we also comprehensively evaluated the effectiveness of the current state-of-practice and state-of-the-art.
In particular, we investigated to what extent statically-defined developer configurations (\emph{i.e.,}\xspace state-of-practice), and dynamic reconfiguration techniques (\emph{i.e.,}\xspace state-of-the-art) are effective in ensuring a reliable and time-efficient assessment of steady state performance in JMH benchmarks. Even more, we quantified the potential side effects due to inaccurate warmup estimation both in terms of execution time waste and misleading performance measurements.
\cite{Laaber2020} have already investigated the effectiveness of dynamic reconfiguration. However, their study was mainly concerned with a particular aspect of Java microbenchmarking, \emph{i.e.,}\xspace reducing execution time.
In this paper, instead, we aim to provide a comprehensive investigation on the effectiveness of dynamic reconfiguration (and developer static configurations) for steady state performance assessment.
Due to our goal, we do not use JMH defaults as baselines, as done by Laaber~\emph{et~al.}\xspace, since there is no guarantee that they can effectively capture steady state performance.
Indeed, although JMH defaults are a reasonable baseline for comparison\footnote{JMH defaults are configurations defined by JMH developers, which are undoubtedly experts in the field of Java microbenchmarking.}, their use may have downsides when studying steady state performance.
Several studies have shown that benchmarks often reach their steady states in different numbers of iterations \citep{Georges2007, Kalibera2013}, thus it may be misleading to use a unique (though reasonable) configuration as baseline, \emph{i.e.,}\xspace JMH defaults may be effective for some benchmarks and suboptimal for other ones.
In order to avoid this problem, we leverage a different (and more rigorous) approach to assess dynamic reconfiguration effectiveness.
We first use a state-of-the-art steady state detection technique to determine if/when a benchmark reaches a steady state of performance, then we base on its outcome to assess dynamic reconfiguration effectiveness.
Besides this evaluation of dynamic reconfiguration techniques, this paper presents the following novel investigations.
We present the first study that investigates if/when JMH benchmarks reach a steady state of performance. Second, we perform the first evaluation on the effectiveness of statically defined developer configurations. Third, we introduce the first comparison between the effectiveness of developer configurations (\emph{i.e.,}\xspace the current state-of-practice) and dynamic reconfiguration techniques (\emph{i.e.,}\xspace the current state-of-the-art).
Our results show that JMH benchmarks do not always reach a steady state of performance, thereby demystifying the current cornerstone of Java microbenchmarking, \emph{i.e.,}\xspace the \emph{two-phase assumption}. This finding implies that practitioners may rely on measurements that are not representative of ``actual'' steady state performance.
In addition, our results suggest that developer static configurations are often ineffective for warmup estimation, and may cause either improperly long execution times or misleading performance assessment.
On the other hand, dynamic reconfiguration techniques show significant improvement over the current state of practice, but they still produce inaccurate estimates of the warmup time, hence causing time-consuming benchmark executions and distorted results.
This finding highlights room for improvement for dynamic reconfiguration, and it calls for further research on this topic.\\
The main contributions of this paper are:
\begin{itemize}
\item a statistically rigorous investigation of steady state performance in JMH microbenchmarks.
\item an empirical evaluation of developer static configurations in JMH microbenchmarks.
\item a comprehensive comparison among the effectiveness of developer static configurations and state-of-the-art dynamic reconfiguration techniques.
\item a large dataset of labeled benchmark executions to facilitate future research on Java steady state performance assessment, and foster further innovations in dynamic reconfiguration.\\
\end{itemize}
The remainder of this paper is organized as follows. \secref{sec:background} introduces steady state performance assessment and JMH microbenchmarks. \secref{sec:rqs} describes our research questions and \secref{sec:design} explains the experimental design. \secref{sec:results} reports the results. \secref{sec:discussion} discusses some implications of our findings. \secref{sec:threats} describes threats to validity. \secref{sec:related} presents related work, and \secref{sec:conclusion} concludes this paper.
\subsection{Java Microbenchmark Harness (JMH)}\label{sec:jmh}
JMH is the de-facto standard framework for writing and executing microbenchmarks for Java software.
It enables software developers to easily develop and execute microbenchmarks that measure fine-grained performance of specific units of Java code (\emph{e.g.,}\xspace methods).
JMH supports steady state performance assessment by providing facilities that enable developers to statically \emph{configure} the number of times each benchmark execution will be repeated (without compromising the reliability of results\footnote{JMH code samples. Pitfalls of using loops in Java microbenchmarking. \url{https://bit.ly/3lCqCZ4}}).
\figref{fig:jmh} depicts a typical JMH benchmark execution.
JMH supports three different levels of repetitions: \emph{forks}, \emph{iterations} and \emph{invocations}.
\emph{Invocations} (\emph{i.e.,}\xspace the lower level of repetition) are nominal benchmark executions that are continuously performed within a predefined amount of time, namely an \emph{iteration}.
In turn, a \emph{fork} is constituted by a sequence of \emph{iterations} performed on a fully clear instantiation of new JVM.
Indeed, as suggested by best practices \citep{Georges2007, Kalibera2013, Barrett2017, Costa2019}, iterations should be repeated multiple times on fresh JVM instantiations (\emph{i.e.,}\xspace forks) to mitigate the contextual effects of confounding factors.
Each fork is usually composed by two distinct types of iterations: \emph{warmup} and \emph{measurement iterations}.
\emph{Warmup iterations} are intended to bring the fork (\emph{i.e.,}\xspace the fresh JVM) into a steady state of performance, while \emph{measurement iterations} are the ones where performance measurements are actually collected.
Each measurement iteration typically returns a set of performance measurements (\emph{e.g.,}\xspace a sample of benchmark invocation execution times) or a performance statistic (\emph{e.g.,}\xspace average execution time or throughput).
JMH provides a set of configuration parameters to define the different levels of repetitions involved during microbenchmarking. These parameters include: warmup iteration time $w$, measurement iteration time $r$, warmup iterations $wi$, measurement iterations $i$, and forks $f$.
Iteration time parameters ($w$ and $r$) define the minimum time spent within an iteration.
Given an iteration time $w$ (resp. $r$), a warmup (resp. measurement) iteration will continuously perform benchmark \emph{invocations} until the iteration time will expire.
Warmup and measurement iterations (\emph{i.e.,}\xspace $wi$ and $i$), instead, define the number of iterations performed within each fork. Finally, the fork parameter $f$ defines the number of fresh JVM instantiation, \emph{i.e.,}\xspace the higher level of benchmark repetition.
\begin{figure}
\centering
\includegraphics[width=8cm]{img_jmh}
\caption{\label{fig:jmh} The JMH microbenchmark life cycle.}
\end{figure}
Typically, Java developers directly set JMH configuration parameters on benchmark code through Java annotations.
Nonetheless, when launching the benchmark, JMH allows to override developer configurations via Command Line Interface (CLI) arguments.
\section{Related work} \label{sec:related}
\begin{comment}
\LUCA{Alcuni related work da aggiungere:
\begin{itemize}
\item Steady state: \citep{Georges2007} and \citep{Kalibera2013}.
\item Microbencmarking/JMH studies: \citep{Leitner2017}, \citep{Costa2019}, \citep{Samoaa2021}, \citep{Laaber2019} and \citep{Laaber2018}.
\end{itemize}}
\DDP{\citep{Laaber2019} qui si parla di benchmark in cloud, non mi sembra un related. \citep{Samoaa2021} qui invece si parla dell'impatto delle configurazioni di JMH sulle performance dei test. Anche questo non mi sembra un related del nostro lavoro. \citep{Leitner2017} questo invece piu' che related mi sembra motivazionale. Anzi dice che i test di performance in OSS non sono mantenuti, quindi metterebbe anche a rischio un po il nostro approccio, visto che abbiamo assunto che i microbenchmark eseguiti siano utilizzati dagli sviluppatori.}
\LUCA{Tutti gli studi empirici che riguardano benchmark JMH, a mio parere, dovrebbero essere considerati related (non ce ne sono molti), visto che noi analizziamo proprio questi tipi di benchmark. I seguenti lavori sono tutti studi empirici che investigano diversi aspetti dei benchmarks JMH: \citep{Samoaa2021}, \citep{Laaber2019}, \citep{Laaber2018} e \citep{Costa2019}. Effettivamente \citep{Leitner2017} è un pò bordeline visto che riguarda il performance testing in generale. \citep{Laaber2019} invece analizza la stabilità dei benchmark nel cloud (un concetto molto vicino allo steady state), non capisco perché consideri questo lavoro meno related di (ad esempio) \citep{He2019}, che invece studia load tests (quindi un'altro tipo di test) sempre in contesti cloud. Ovviamente, non sto dicendo di rimuovere \citep{He2019}, anzi penso che ci stia molto bene, e ho capito che l'hai inserito per quella componente dell'approccio che stoppa automaticamente il load test, però non capisco perché \citep{Laaber2019} debba essere escluso visto la presenza di questo lavoro.}
\DDP{\citep{Laaber2018} questo lo devo guardare un po meglio per capire se realmente un related. Anche se abbiamo gia' considerato un lavoro di Laaber mi sembra un po brutto aggiungerne un altro dello stesso autore. Se dovesse essere un related riscrivo un attimo il pezzo di laaber.}
\LUCA{A mio parere la citazione di un articolo non dovrebbe essere preclusa solo perché di un autore già citato. }
\DDP{\citep{Georges2007} descrivere metodologie di performance analysis e spiega il perche' queste tecniche possono portare a risultati non veritieri. Anche qui non intravedo il related con il nostro lavoro. Ma mi sembra stato correttamente citato quando e' stato descritto lo study design motivando le scelte prese.}
\LUCA{ È il paper seminale sulle analisi di performance a steady state nei benchmark Java (``We advocate approaches to quantify startup as well as steady state performance...''). Il nostro è uno studio sull'analisi di performance a steady state attraverso microbenchmark Java. Lo ritengo un related importante.}
\DDP{\citep{Kalibera2013} e' un paper che fornisce guidelines per ottenere risultati validi in termini statistici. Perche' dovrebbe essere related al nostro? non non forniamo queste info, anzi le usiamo ed e' stato citato nella parte di study design.}
\LUCA{È tra gli studi più importanti sull'analisi di performance a steady state nei benchmark. È il primo studio a dire che i benchmark potrebbe non raggiungere mai uno steady state. Mi sembra un related importante (stessa motivazione del precedente punto). }
\end{comment}
There are different perspectives of tackling performance analysis of software systems, through models at runtime \citep{GieseL020,CORTELLESSA2022111084}, or by means of benchmarking.
Recently, benchmarking technique has played a key role to discover potential performance flaws \citep{DBLP:conf/wosp/StefanHBT17}.
One of the main problems with benchmarking results is the reliability of the data.
Recently, different approaches have defined rigorous processes to interpret those data.
For example, some approaches rely on statistical inference for identifying and measuring the reliability of benchmarking results~\citep{Kalibera2013, kalibera2020}.
Other approaches, instead, have presented performance analysis methodologies to extract data in a more reliable way \citep{Georges2007}.
\cite{Barrett2017} introduced a fully automated statistical approach based on changepoint analysis.
In their work, \cite{Barrett2017} studied a set of small and deterministic VM benchmarks across different types of VMs, including the JVM.
They found that VM microbenchmarks may not always reach a steady state of performance.
On the other hand, performance benchmarking is a time-demanding process.
Recently, some approaches investigated solutions to reduce the time for performance analysis while preserving reliable results~\citep{DBLP:conf/issta/MostafaWX17, He2019, alghamdi_2020}.
\cite{He2019} have studied the reduction of performance testing in the cloud. They introduced a statistical tool, namely PT4Cloud, that provides stop conditions in order to obtain reliable performance indices. Another way to reduce testing time is, for example, by reusing the ``functional'' unit tests, which are likely available and maintained. For example, \cite{DBLP:journals/ase/BulejBHKMTT17} extended ``functional'' unit tests with performance knowledge by equipping them with stochastic performance logic.
Java Microbenchmark Harness (JMH) is a popular benchmarking framework for Java software. JMH allows defining performance testing to reduce variability in the measurements as well as external factor contributions during the microbenchmark testing phase. Hence, different studies spanned over different JMH aspects \citep{Costa2019, Samoaa2021, Laaber2019, Laaber2018}.
A JMH microbenchmark might be affected by bad practices that could degrade performance results.
\citep{Costa2019} studied those bad practices by analyzing a corpus of 123 OSS, and they extracted those bad practices that more likely lead to bad performance indicators.
\cite{Laaber2020} focused their study on reducing the required execution time of microbenchmarking tests through dynamic reconfiguration. They have defined three stability criteria to dynamically estimate the end of the warmup phase and halt warmup iterations accordingly. \cite{Samoaa2021} studied, instead, the impact of benchmark parameters and how they affect performance results.
In this work, we studied the effectiveness of modern Java microbenchmarking for steady state performance assessment.
Similarly to \cite{Barrett2017}, we investigated whether microbenchmarks reach a steady state of performance. However, unlike them, we studied this aspect in the more defined scope of ``testing-oriented'' Java benchmarks, \emph{i.e.,}\xspace JMH benchmarks specifically designed to assess performance of a particular software.
Our results are consistent with those gathered by \cite{Barrett2017}, thus confirming that, even in a different context, Java benchmarks may not always reach a steady state of performance.
\cite{Costa2019} broadly studied bad practices in JMH benchmarks, instead we specifically investigated the effectiveness of developer configurations for steady state performance assessment. \cite{Laaber2020} presented dynamic reconfiguration as a viable alternative to developer static configurations. In their study, they compared dynamic reconfiguration to JMH default configurations, and they observed a significant reduction in execution time with a negligible loss of result quality. In our study, instead, we evaluated the effectiveness of dynamic reconfiguration for steady state performance assessment. Furthermore, we showed, through a rigorous comparison, that dynamic reconfiguration is significantly more effective than developer configurations and, as such, it produces less pronounced side effects.
\section*{Data Availability} \label{sec:replication}
To aid reproducibility we provide the data and scripts needed to replicate our findings. The complete replication package is available at
\center{\href{https://doi.org/10.5281/zenodo.7058361}{\texttt{DOI:10.5281/zenodo.7058361}}}
\section{Results} \label{sec:results}
This section presents the results of the experiments and provides answers to the RQs formulated in \secref{sec:rqs}.
\input{rq1}
\input{rqnew}
\input{rq2}
\input{rq3}
\input{rq4}
\subsection{RQ$_1$ - Steady state assessment}\label{sec:results:rq1}
As described in Section~\ref{sec:exp_steadystate}, we classify the benchmarks in our study on the basis of their ability to eventually reach a steady state.
Such classification is first performed at fork level, and then at benchmark level by combining results from the steady state detection on forks.
\begin{figure}[h]
\center
\begin{subfigure}[b]{5.8cm}
\center
\includegraphics[width=\linewidth]{img_rq1_forks_heatmap}
\caption{Forks classification.}
\label{fig:rq1_fork_class}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.8cm}
\center
\includegraphics[width=\linewidth]{img_rq1_benchmarks_heatmap}
\caption{Benchmarks classification.}
\label{fig:rq1_bench_perf}
\end{subfigure}
\caption{RQ$_1$. Steady state classification.}
\end{figure}
\begin{figure}
\center
\includegraphics[width=12cm]{img_perc_no_steady_state}
\caption{RQ$_1$. Percentages of no steady state forks within each benchmark, grouped by subject system.}
\label{fig:perc_no_steady_state}
\end{figure}
In order to provide an overview of how many forks reached a steady state, Figure~\ref{fig:rq1_fork_class} reports the percentage of forks classified as \emph{steady state} or otherwise, as grouped by systems.
The percentage of forks that reached steady state varies between 69\% (\texttt{JCTools}) and 98.9\% (\texttt{cantaloupe}).
Even if there is some variability among systems, 28 of them, out of the 30 in our study, show a percentage of steady state forks above 80\%, with 18 of them above 90\%.
Globally, in most cases (89.1\% in the last row of Figure~\ref{fig:rq1_fork_class}), individual forks were able to reach a steady state according to our detection technique.
When we examine how the classified forks are distributed among the benchmarks, we get a less obvious outlook.
In Figure~\ref{fig:rq1_bench_perf}, we report the percentage of benchmarks classified as \emph{steady state}, or \emph{inconsistent} in the respective systems (note we do not report percentages for ``no steady state'' classification, since we didn't find any benchmark classified as such).
The first clear result is that there are no cases in which all the forks of a benchmark did not reach a steady state, since the totality of benchmarks is always distributed among the \emph{steady state} and \emph{inconsistent} columns.
On the one hand, this might encourage the assumption that, in the vast majority of cases, benchmarks reach and measure steady state performance.
On the other hand, we can assess that the percentage of benchmarks in which all the forks reached a steady state is subject to large variability depending on the specific system.
In fact, the percentage of \emph{steady state} benchmarks varies between 20\% (\texttt{JCTools}) and 95\%(\texttt{rdf4j}).
Only 5 systems overcome a 80\% percentage and, as opposed to what one would expect, \emph{no steady state} forks are unevenly distributed among benchmarks, thus causing most systems to have a low percentage of \emph{steady state} benchmarks even though the percentage of \emph{steady state} forks was higher.
Since there are no benchmarks in which all the forks did not reach a steady state, all the remaining benchmarks are classified as \emph{inconsistent} (43.5\%), which means that their forks showed mixed behavior.
Another viewpoint on the classification of benchmarks is provided in Figure~\ref{fig:perc_no_steady_state}, where we report the distribution of the percentages of no steady state forks in benchmarks, as grouped by systems, which further clarifies what contributes to the percentages of inconsistent benchmarks.
We can notice that, in most cases (with very few exceptions like \texttt{JCTools}), the systems tend to exhibit inconsistent benchmarks with small percentages of no steady state forks.
It is worth recalling that a single fork (\emph{i.e.,}\xspace 10\% in a benchmark) is enough to flip the classification from \emph{steady state} to \emph{inconsistent}.
This is, in fact, the most common case, as we can see in the distribution computed over all the systems (\emph{i.e.,}\xspace \emph{Total} in Figure~\ref{fig:perc_no_steady_state}) that shows a mean around 10\%.
\begin{figure}[]
\centering
\includegraphics[width=.98\linewidth]{img_steady_states_by_project_boxplot}
\caption{RQ$_1$. Time to reach the steady states, in seconds. The y-axis uses a logarithmic scale.}%
\label{fig:steady_states_time_by_project_boxplot}
\end{figure}
From a practical perspective, it is also important to estimate how long it takes to reach a steady state, in the cases in which it is reached.
This provides a better view on how the time budget could be spent when executing the benchmarks.
\figref{fig:steady_states_time_by_project_boxplot} shows the distributions of the time to reach a steady state, as grouped by system and in total.
We can observe that the time spent considerably varies, even within a single system, therefore it can hardly be generalized.
This result is not surprising, because the attainment of a steady state inherently depends on the nature of the benchmark.
As most of the instability during the warmup is due to the JIT activity, we can imagine that, beside the size of the benchmark method itself, also the number of loaded classes plays a crucial role, since it will induce a different amount of compilation.
\figref{fig:steady_states_time_by_project_boxplot} also shows how the time spent to reach a steady state compares to the JMH default setting of 50 seconds for the warmup phase (dashed horizontal line in the figure).
We can observe that the difference between the detected end of the warmup phase and the JMH defaults largely differs from one system to the other, by extreme values for \emph{eclipse-collections} and \emph{jgrapht}.
The clear picture that emerges from this is that, in most cases, by using the JMH defaults we would overestimate the time needed to warm the benchmark up, therefore wasting a considerable amount of time dedicated to performance testing.
While the amount of time that would be wasted considerably varies from one system to another, the percentage of overestimated forks is quite consistent across all the systems.
This leads to the conclusion that, more often than not, the JMH defaults for the warmup time should not be used, rather one should rely on techniques to assess the actual amount of time a specific benchmark requires to reach a steady state.
\\
\begin{tcolorbox}[colback=black!3!white,colframe=black!33!white]
\textbf{RQ$_1$ summary} - When we only look at individual forks, measurements appear to reach a steady state in the majority of cases.
However, when combining forks at the benchmark level, we obtain mixed results.
These results provide evidence that benchmarks do not always reach a steady state of performance, thus showing, on a large corpus of JMH benchmarks, that the ``\emph{two-phase assumption}'' does not always hold.
Moreover, in most cases, the JMH defaults for the warmup time tend to overestimate the time needed to reach a steady state.
\end{tcolorbox}
\subsection{RQ$_3$ - Developer configuration assessment}
In this subsection, we first present results of the assessment of developer static configurations, thereafter we provide answer to RQ$_3$.
\begin{figure}[hbtp]
\centering
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\linewidth]{img_new_warmup_estimation_dev_barplot}
\caption{}\label{fig:developer_configurations_barplot}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\linewidth]{img_new_warmup_estimation_dev_boxplot}
\caption{}\label{fig:developer_configurations_boxplot}
\end{subfigure}
\caption{RQ$_3$. Developer configurations - Warmup estimation accuracy.
The left plot reports the percentages of overestimated, underestimated, and correctly estimated forks.
The right plot depicts the distribution of $WEE$ across all benchmark forks.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure.}
\label{fig:developer_configurations}
\end{figure}
\subsubsection{Warmup Estimation Accuracy}
In the first part of this subsection, we investigate to what extent developers accurately estimate \emph{steady state starting time} ($st$), \emph{i.e.,}\xspace the end of the warmup phase.
\figref{fig:developer_configurations_boxplot} depicts the distribution of \emph{warmup estimation error} ($WEE$) across all benchmark forks.
Developers configurations lead to a $WEE$ that ranges between 9 and 50 seconds in half of the cases (\emph{i.e.,}\xspace the interquartile range (IQR)), and they lead to a median and mean $WEE$ of 28 and 90 seconds, respectively.
Interestingly, we found that the estimation error is approximately as large as the steady state starting time (or more) in half of the forks, \emph{i.e.,}\xspace the median of the ratios between $WEE$ and $st$ is 0.997 (IQR: 0.79-43.5).
It might be easier for some systems to assess the time required to reach a steady state than for other ones.
With respect to the steady state we detected, most systems (27 out of 30) show an estimation error of less than 100 seconds.
However, the developer configurations in \texttt{eclipse-collections}, \texttt{presto}, and \texttt{RxJava} seem to be less effective in estimating the warmup phase.
The most evident case is represented by \texttt{eclipse-collections} with an error, on average, of around 5 minutes.
Overall, these results suggest that, in most of the cases, \emph{software developers fail to accurately estimate the end of the warmup phase}, and often \emph{with a non-trivial estimation error}.
Besides investigating the estimation accuracy of developers, we also check whether they provide more accurate estimates than JMH defaults\footnote{In our evaluation, we use the default configuration defined for JMH versions $\geq 1.21$, \emph{i.e.,}\xspace 5 warmup and measurements iterations ($wi=5$ and $i=5$), and iteration time of 10 seconds for both measurement and warmup ($w=10s$ and $r=10s$).} (\emph{i.e.,}\xspace the default configuration provided by JMH developers). To do so, we compare the $WEE$s provided by developer configurations against those provided by JMH defaults using the Wilcoxon Rank-Sum test~\citep{cohen2013}, and the Vargha Delaney's effect size measure~\citep{Vargha2000}. We found that developer configurations outperform JMH defaults with statistical significance ($p<0.001$) and small effect size ($\hat{A}_{12}$\xspace=0.64). The median and the mean $WEE$ provided by JMH defaults are larger than those provided by developer configurations, \emph{i.e.,}\xspace mean of 97 seconds and median of 50 seconds (IQR: 36-50 seconds). These results suggest that the estimates provided by developers are more accurate than those provided by JMH defaults.
\figref{fig:developer_configurations_barplot} reports the percentages of overestimated, underestimated, and correctly estimated forks across all forks (\emph{i.e.,}\xspace the estimated warmup time $wt$ is respectively smaller or larger than $st$ by at least 5s).
The bar chart shows that overestimation is more common than underestimation.
Developers overestimate the end of the warmup phase in 48\% of the forks (median WEE: 33 seconds, IQR: 19-50 seconds), whereas underestimation is reported in 32\% of the cases (median WEE: 150 seconds , IQR: 36-240 seconds).
Developers accurately estimate it in only 19\% of the cases.
In the following subsections, we investigate side effects in both overestimated and underestimated forks.
\subsubsection{Overestimation side effects}
\figref{fig:overestimation_dev} shows the \emph{time waste} due to overestimation.
87\% of the overestimated forks waste more than 10 seconds (\emph{i.e.,}\xspace 37\% of all the forks).
As can be observed by \figref{fig:overestimation_dev_barplot}, overestimation leads to a time waste between 10 and 25 seconds in 30\% of the cases, between 25 and 50 seconds in 38\% of the cases, and it leads to a time waste higher than 50 seconds in 19\% of the cases.
\figref{fig:overestimation_dev_boxplot} depicts the distribution of \emph{time waste} across overestimated forks. The box plot shows that the average \emph{time waste} is 37 seconds, and the median is 33 seconds (IQR: 19-49 seconds).
\begin{figure}[hbtp]
\centering
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\textwidth]{img_new_overestimation_effect_excluded_0-10_dev_barplot}
\caption{}\label{fig:overestimation_dev_barplot}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\linewidth]{img_new_overestimation_effect_excluded_0-10_dev_boxplot}
\caption{}\label{fig:overestimation_dev_boxplot}
\end{subfigure}
\caption{RQ$_3$. Developer configurations - Overestimation side effects (\emph{Time waste}).
The left plot reports the percentages of overestimated forks where $\emph{Time waste}\, (sec) \in \{[10,25), [25,50), [50,100), [100, inf)\}$.
The right plot depicts the distribution of \emph{Time waste} across all overestimated forks.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure.}
\label{fig:overestimation_dev}
\end{figure}
The amount of time wasted on warmup, after reaching a steady state, considerably varies from one system to another.
In most systems, by using the developers configurations, each fork wastes, on average, more than 20 seconds.
More extremes behaviors can be found, for instance, in \texttt{jdbi}, \texttt{netty}, and \texttt{rdf4j}, with an average wasted time of more than 50 seconds per fork.
On the contrary, \texttt{cantaloupe}, \texttt{JCTools}, \texttt{r2dbc-h2}, and \texttt{zipkin} might waste just a few seconds, and probably their configurations do not need any adjustment.
Such absolute \emph{time wastes}, when contextualized within concrete performance assurance processes (that involve multiple benchmarks), can have substantial effects on the overall execution time and, as consequence, can hamper microbenchmarks adoption for continuous performance assessment.
For example, a time waste of 33 seconds in a relatively small performance testing suite (\emph{e.g.,}\xspace \texttt{r2dbc-h2}), which involves a typical number of 5 forks\footnote{The default number of forks in JMH is 5 (see \url{https://bit.ly/3mOBvHy})}, could lead to an overall time waste of approximately one hour\footnote{\label{fn:timewaste}To compute the overall time waste, we multiplied the time waste (33 seconds) by the number of forks (5) and the number of benchmarks in the testing suite (20 for \texttt{r2dbc-h2} and 1,302 for \texttt{RxJava}).}.
In larger testing suites, such as \texttt{RxJava}, the same \emph{time waste} could lead to an overall waste of about 2 days and a half.
Besides this, $wt$ is mostly composed by time waste in a large number of cases. In fact, we have measured that the median of the ratios between the time waste and the estimated warmup time $wt$ is approximately 0.97 (IQR: 0.77-0.99), \emph{i.e.,}\xspace in half of overestimated forks, at least 97\% of the estimated warmup time $wt$ consists of time waste.
The reported results highlight a substantial portion of time wasted during microbenchmarking, and stress the need for better microbenchmark configuration approaches that reduce their execution time. These findings further motivate prior efforts in reducing execution time through dynamic reconfiguration \citep{Laaber2020}, and highlight huge opportunities for execution time reduction in microbenchmarks.
\subsubsection{Underestimation side effects}
Although less frequent than overestimation, underestimation can have relevant side effects on microbenchmarking.
Indeed, it can lead to consider performance measurements that significantly differ from steady state performance, as in practice they fall within the warmup phase.
\figref{fig:underestimation_dev} reports the distribution of the relative performance deviation ($RPD$) across underestimated forks.
Underestimation leads to an $RPD$ of at least 5\% in 57\% of cases (\emph{i.e.,}\xspace 22\% of all the forks).
\figref{fig:underestimation_dev_barplot} depicts that underestimation induces an $RPD$ between 5\% and 10\% in 15\% of the cases, between 10\% and 25\% in 20\% of the cases, and it induces an $RPD$ greater than 50\% in 9\% of the cases.
The box plot (\figref{fig:underestimation_dev_boxplot}) shows a mean $RPD$ of 17\%, and a median of 7\%~(IQR: 1-21\%).
\begin{figure}[hbtp]
\centering
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\textwidth]{img_new_underestimation_effect_excluded_5-10_dev_barplot}
\caption{}\label{fig:underestimation_dev_barplot}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.5cm}
\includegraphics[width=\linewidth]{img_new_underestimation_effect_excluded_5-10_dev_boxplot}
\caption{}\label{fig:underestimation_dev_boxplot}
\end{subfigure}
\caption{RQ$_3$. Developer configurations - Underestimation side effects (\emph{performance change}).
The left plot reports the percentages of underestimated forks where $\emph{performance change}\, (\%) \in \{[5,10), [10,25), [25,50),[50,inf) \}$.
The right plot depicts the distribution of \emph{performance change} across all underestimated forks.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure.}
\label{fig:underestimation_dev}
\end{figure}
Some systems show a large performance deviation when underestimating the warmup time.
In nine systems, developers will observe measurements that deviate at least by 20\% from the performance reached in the steady state.
Such large deviations prevent the benchmarks to spot smaller performance changes, in fact defeating their purpose in practical performance testing scenarios.
These results highlight significant performance deviations due to underestimation.
In Java systems, even relatively small performance regressions (\emph{e.g.,}\xspace 5\%) may lead to rejections of code revisions.
In fact, a microbenchmark regression, for example due to software refactoring \citep{traini}, can have huge impact at system level, as microbenchmarks measure performance at fine-grained level \citep{Laaber2018, Leitner2017}.
For this reason, the reported $RPD$s can have severe consequences on steady state performance assessment, as they can easily lead to faulty judgments of code revisions.\\
According to our results, developers may rely on measurements that significantly differ from those collected during steady state execution.
This finding sheds a light on the perils of underestimation, and on how such inaccuracy can disrupt performance assessment.
Indeed, the reported results show that underestimation is not rare in the current developer practice~(32\% of forks), and it often leads to potentially misleading results~(57\% of underestimated forks lead to a performance deviation $\geq$5\%).
\subsubsection{Benchmark level assessment}
\begin{figure}[htpb]
\centering
\includegraphics[width=9cm]{img_wholeconfig_dev}
\caption{RQ$_3$. Developer configurations: Benchmark level assessment. The left plot depicts the RPD distribution across all benchmarks. The right plot depicts the execution time distribution. }
\label{fig:dev_perfdev}
\end{figure}
Here, we discuss the results of the analysis of developer configurations at benchmark level. The left plot of \figref{fig:dev_perfdev} reports the distribution of performance deviations of measurements gathered by software developers when compared to steady state measurements. The first aspect that we can notice by observing the plot is that the impact of underestimation seems to be considerably mitigated when considering aggregated measurements. This finding is in line with our previous analysis on forks (see RQ$_2$), in which we have shown that the number of forks can substantially mitigate deviations of non-steady measurements. Performance deviations are not statistically significant in about half of the benchmarks (\emph{i.e.,}\xspace 53\% of the cases), the median RPD is 0\% and the IQR is 0-6\%. Nonetheless, 47\% of benchmarks report statistically significant performance deviations (\emph{i.e.,}\xspace the confidence interval does not contain zero), the mean RPD is 8\%, and a quarter of benchmarks report a deviation higher 6\%. As we have already discussed in our prior analysis on underestimation, these magnitudes of deviation can be harmful in Java microbenchmarking, as they can mislead performance assessment and lead to wrong judgements of software revisions.
The right plot of \figref{fig:dev_perfdev} reports the distribution of benchmark execution times based on developer configurations.
Developer configurations lead to extremely different execution times, with durations ranging from a minimum of 1 second to a maximum of 893 seconds.
The average execution time is 210 seconds, and the median is 100 seconds (IQR 30-403 seconds).
Our previous analysis on overestimation has already highlighted large opportunities for execution time reduction.
In the light of the above results, these opportunities appear even more significant.
For example, our prior analysis has reported a median time waste of 33 seconds in overestimated forks.
If we compare this result with the median execution time of a benchmark, \emph{i.e.,}\xspace 100 seconds, the time waste appears extremely relevant, \emph{i.e.,}\xspace approximately one third of the entire benchmark execution.
In that, it is worth to remark that time wastes are measured on individual forks, while the execution times (reported in \figref{fig:dev_perfdev}) measure the entire duration of a benchmark, which typically involves multiple forks (3 on average in the case of developer configurations).
This finding further remarks on the need for better techniques to reduce benchmark execution time without affecting result quality.
\\
\begin{tcolorbox}[colback=black!3!white,colframe=black!33!white]
\textbf{RQ$_3$ summary} -
Developer configurations have limited effectiveness for steady state performance assessment. Developers fail to accurately estimate the end of the warmup phase, often with a non-trivial estimation error. In a large number of cases, this error leads to a substantial increase in the execution time (\emph{i.e.,}\xspace overestimation).
Nevertheless, underestimation is not rare in the current developer practice, and when this happens it significantly distorts performance assessment.
\end{tcolorbox}
\subsection{RQ$_4$ - Dynamic reconfiguration assessment}\label{sec:results:rq3}
In this subsection, we first present results of the assessment of dynamic reconfiguration techniques, then we answer to RQ$_4$ .
\subsubsection{Warmup Estimation Accuracy}
\begin{figure}
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_warmup_estimation_stacked}
\caption{}
\label{fig:warmup_estimation_DyCo_barplot}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_warmup_estimation_boxplot}
\caption{}
\label{fig:warmup_estimation_DyCo_boxplot}
\end{subfigure}
\caption{RQ$_4$. Dynamic reconfiguration - Warmup estimation accuracy.
The left plot reports the percentages of overestimated, underestimated, and correctly estimated forks per dynamic reconfiguration technique.
The right plot depicts the distribution of $WEE$ across all benchmark forks per dynamic reconfiguration technique.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure.}
\label{fig:warmup_estimation_DyCo}
\end{figure}
\figref{fig:warmup_estimation_DyCo_boxplot} depicts the distributions of warmup estimation errors ($WEE$) per dynamic reconfiguration technique across all benchmark forks.
As it can be seen from the figure, \emph{RCIW} leads to higher $WEE$ (median: 48 seconds, IQR: 30-50 seconds) when compared to other techniques, whereas $CV$ and $KLD$ report similar distributions.
\emph{CV} leads to a median $WEE$ of 19 seconds (IQR: 10–41 seconds), while \emph{KLD} reports a median of 17 seconds (IQR: 9–47 seconds).
Also, we have measured that, in half of the forks, all dynamic techniques report a $WEE$ approximately as large as the steady state starting time $st$ (or more), \emph{i.e.,}\xspace the medians of the ratios between $WEE$ and $st$ for \emph{CV}, \emph{RCIW} and \emph{KLD} are respectively 0.97 (IQR: 0.79-45.7), 2.04 (IQR: 0.78-75) and 0.97 (IQR: 0.79-36.1).
These results indicate that all dynamic reconfiguration techniques lead to a substantial error in the estimate of steady state starting time.
Nevertheless, \emph{CV} and \emph{KLD} clearly provide more accurate estimates than \emph{RCIW}.
The bar chart in \figref{fig:warmup_estimation_DyCo_barplot} confirms the specificity of \emph{RCIW}.
As it can be observed from the bar chart, \emph{RCIW} reports a strong tendency toward overestimation, while \emph{CV} and \emph{KLD} show similar frequencies both in terms of underestimation and overestimation. \emph{RCIW} overestimates 72\% of forks (median WEE: 46 seconds, IQR: 31-50 seconds), and it underestimates only 25\% forks (median WEE: 181 seconds, IQR: 86-236 seconds).
On the other hand, \emph{CV} reports 50\% of overestimations (median WEE: 17 seconds, IQR: 12-24 seconds) and 39\% of underestimations (median WEE: 131 seconds, IQR: 29-249 seconds). Similarly, \emph{KLD} reports 55\% overestimated forks (median WEE: 13 seconds, IQR: 9-21 seconds) and 37\% underestimated forks (median WEE: 145 seconds, IQR: 29-255 seconds).
\subsubsection{Overestimation side effects}
\begin{figure}
\centering
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_overestimation_effect_excluded_0-10_stacked}
\caption{}
\label{fig:overestimation_stacked}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_overestimation_effect_excluded_0-10_boxplot}
\caption{}
\label{fig:overestimation_boxplot}
\end{subfigure}
\caption{RQ$_4$. Dynamic reconfiguration - Overestimation side effects (\emph{time waste}).
The left plot reports the percentages of overestimated forks where $\emph{time waste}\, (sec) \in \{[10,25), [25,50), [50,100), [100, inf)\}$.
The right plot depicts the distribution of \emph{time waste} across all overestimated forks per dynamic reconfiguration technique.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure).}
\label{fig:overestimation_DyCo}
\end{figure}
Besides the large number of overestimated forks, \emph{RCIW} also causes higher \emph{time wastes} when compared to other techniques.
As it can be observed in \figref{fig:overestimation_DyCo}, \emph{RCIW} reports a mean \emph{time waste} of 45 seconds and a median of 46 seconds (IQR: 31-55 seconds).
95\% of overestimated forks lead to a \emph{time waste} of at least 10 seconds, 81\% lead to a time waste of at least 25 seconds, and 13\% to a time waste of at least 50 seconds.
On the other hand, \emph{CV} and \emph{KLD} report a median time waste of 17 seconds (IQR: 12-24 seconds) and 14 seconds (IQR: 9-21 seconds), respectively (see \figref{fig:overestimation_boxplot}).
Nonetheless, we have measured that in half of the overestimated forks, all techniques lead to an estimated warmup time that mostly consists of time waste, \emph{i.e.,}\xspace the medians of the ratios between the \emph{time waste} and \emph{wt} for \emph{CV}, \emph{RCIW}, and \emph{KLD} are respectively 0.98 (IQR: 0.88-0.99), 0.96 (IQR: 0.7-0.99), and 0.97 (IQR: 0.88-0.99).
Overall, these results indicate that different techniques lead to diverse outcomes in terms of overestimation.
For example, RCIW induces more frequent overestimations and higher time wastes when compared to other techniques (see \figref{fig:overestimation_stacked}).
Despite this diversity, overestimation is frequent across all dynamic reconfiguration techniques, and it often leads to a non-trivial time waste, which can hamper continuous performance assessment.
\subsubsection{Underestimation side effects}
\begin{figure}
\centering
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_underestimation_effect_excluded_5-10_stacked}
\caption{}
\label{fig:underestimation_stacked}
\end{subfigure}
\hfill
\begin{subfigure}[b]{5.7cm}
\centering
\includegraphics[width=\linewidth]{img_new_underestimation_effect_excluded_5-10_boxplot}
\caption{}
\label{fig:underestimation_boxplot}
\end{subfigure}
\caption{RQ$_4$. Dynamic reconfiguration - Underestimation side effects (\emph{performance change}).
The left plot reports the percentages of underestimated forks where $\emph{performance change}\, (\%) \in \{[5,10), [10,25), [25,50),[50,inf) \}$.
The right plot depicts the distribution of \emph{performance change} across all underestimated forks per dynamic reconfiguration technique.
$n$ is the total amount of data points, and the number on the top of the plot is the amount of outliers not drawn in the figure).}
\label{fig:underestimation_DyCo}
\end{figure}
\emph{RCIW} is less prone to underestimation than other techniques, and it also has less marked side effects due to underestimation.
As it can be observed in \figref{fig:underestimation_DyCo}, \emph{CV} and \emph{KLD} lead to higher performance changes when compared to \emph{RCIW}.
\emph{RCIW} reports a median $RPD$ of 3\% (IQR: 0-11\%), whereas \emph{CV} and \emph{KLD} report medians of 10\% (IQR: 5-20\%) and 9\% (IQR: 3\%-17\%), respectively (see \figref{fig:underestimation_boxplot}).
Additionally, \emph{CV} and \emph{KLD} cause performance changes of at least 5\%, respectively, in 76\% and 69\% of the cases, while \emph{RCIW} achieves a similar deviation in only 43\% of forks (see \figref{fig:underestimation_stacked}).
These results suggest that underestimation side effect varies depending on the dynamic reconfiguration technique.
Some dynamic reconfiguration techniques (\emph{i.e.,}\xspace CV and KLD) are more prone to induce underestimation, and they often lead to a non-trivial performance deviation which can potentially mislead steady state performance assessment.
Other techniques, such as \emph{RCIW}, are instead ``safer'' in terms of underestimation, and ensure higher results quality in terms of performance assessment.
\subsubsection{Benchmark level assessment}
\begin{figure}[htpb]
\centering
\includegraphics[width=12cm]{img_wholeconfig_dyn}
\caption{RQ$_4$. Dynamic reconfiguration: Benchmark level assessment. The left plot depicts the RPD distribution across all benchmarks. The right plot depicts the execution time distribution. }
\label{fig:dyn_perfdev}
\end{figure}
The analysis at benchmark level confirms the trend observed in our prior analysis.
\figref{fig:dyn_perfdev} reports the deviations ($RPD$) of measurements gathered through dynamic reconfiguration techniques when compared to steady measurements (left plot), along with the distributions of benchmark execution times (right plot).
As it can be observed, \emph{RCIW} is by far the most reliable technique in terms of performance deviation. It reports an average $RPD$ of 2\%, a median of 0\% and an IQR of 0-0\%. About 91\% of the benchmarks report an $RPD$ smaller than 5\%.
On the other hand, \emph{RCIW} provides also the most time consuming process with a mean execution time of 283 seconds and a median of 300 seconds (IQR 265-300 seconds).
This is not surprising, as our prior analysis has highlighted a large overestimation rate (72\%) and relevant time wastes (45 seconds on average) in \emph{RCIW}.
Contrariwise, \emph{CV} and \emph{KLD} are less reliable in terms of results quality, by reporting a mean $RPD$ of respectively 8\% (median: 1\%, IQR: 0-10\%) and 7\% (median: 0.5\%, IQR 0-8\%), but they are also less demanding in terms of execution time. \emph{CV} reports a mean execution time of 93 seconds (median: 75 seconds, IQR: 51-113 seconds), while \emph{KLD} reports a mean of 154 seconds (median: 121 seconds, IQR: 94-156 seconds).
\emph{CV} and \emph{KLD} show similar behaviors in terms of execution times and performance deviations, however, as it emerges by the box plots in \figref{fig:dyn_perfdev}, they show two opposite tendencies. In particular, \emph{CV} shows a slight tendency towards faster execution times and less reliable results, while \emph{KLD} shows the opposite behavior, \emph{i.e.,}\xspace better results quality and more time-consuming executions.
\\
\begin{tcolorbox}[colback=black!3!white,colframe=black!33!white]
\textbf{RQ$_4$ summary} -
Dynamic reconfiguration techniques provide far from optimal estimates of the warmup phase, often with a non-trivial error.
The side effects vary depending on the technique.
\emph{RCIW} is more prone to overestimation than other techniques, and it induces more time-consuming benchmark executions (\emph{i.e.,}\xspace higher time waste).
On the other hand, \emph{CV} and \emph{KLD} often lead to performance measurements that differ from those collected during the steady state, while \emph{RCIW} provides a ``safer'' assessment of steady state performance.
\end{tcolorbox}
\subsection{RQ$_5$ - Dynamic reconfiguration \emph{vs} Developer configuration }
In this subsection, we first present results of the comparison between dynamic reconfiguration techniques and developer static configurations for each considered metric: (i) \emph{warmup estimation error} ($WEE$), (ii) \emph{estimated warmup time} ($wt$), and (iii) \emph{relative performance deviation} ($RPD$). Then, we provide answer to RQ$_5$.
\subsubsection{Warmup Estimation Error}\label{sec:dyn_vs_dev_wee}
\input{tables_rq4_time}
\begin{figure}[p]
\centering
\includegraphics[width=7.5cm]{img_rq4_timescore}
\caption{RQ$_5$. $WEE$ comparison summary. Each cell reports the number of systems whose comparison leads to a statistically significant change ($p\leq~0.05$) within a specific $\hat{A}_{12}$\xspace effect size range: negligible (N), small (S), medium (M) and large (L). $\hat{A}_{12}$\xspace~$>0.5$ indicates that dynamic configurations perform better than developer configurations.}
\label{fig:rq4_timescore}
\end{figure}
We performed the Wilcoxon test to check the significance of the difference between $WEE$ of dynamic reconfiguration techniques and developer static configurations. The detailed results of the comparison within and across systems are reported in \tabref{tab:rq4_time}. As it can be observed by the last row of the table, the differences are statistically significant for all dynamic reconfiguration techniques (p$\leq$0.05), with two techniques reporting extremely small p-values (p$<$0.001). The comparison leads to a medium effect size in $RCIW$ ($\hat{A}_{12}$\xspace $\geq$ 0.64), and to a large effect size in $CV$ and $KLD$ ($\hat{A}_{12}$\xspace $\geq$ 0.71). These results indicate that, when compared to developer static configurations, dynamic reconfigurations techniques overall provide more accurate estimates of the end of the warmup phase.
Nevertheless, \tabref{tab:rq4_time} also shows that the difference between $WEE$ of dynamic reconfiguration techniques and developer configurations varies across systems.
\figref{fig:rq4_timescore} shows a summary of these results, where each cell reports the number of projects whose comparison leads to a statically significant difference (p$\leq$0.05) within a specific $\hat{A}_{12}$\xspace effect size range.
As it can be observed by the figure, $CV$ provides better warmup estimates than developers in 15 of the 30\xspace systems (13 with large effect sizes, and 2 with medium effect sizes). Conversely, developer configurations provide lowers estimation errors than $CV$ in only 2 systems (respectively, negligible and small effect size).
$KLD$ shows a similar trend to that observed for $CV$. $KLD$ provides better warmup estimates than developers in 13 out of the 30\xspace systems (11 with large effect sizes, 1 with medium effect size, and 1 with negligible effect size), whereas developers outperform $KLD$ in only 2 systems (respectively, small and medium effect size).
$RCIW$ also shows improvement over static configurations in a considerable number of systems (10 out of 30\xspace), though with lower effect sizes (4 large, 2 medium, and 4 small). Again, developer configurations provide better estimations in only 2 systems \footnote{Interestingly, \texttt{cantaloupe} and \texttt{hive} are the same two systems where developers provide better warmup estimations than whatever dynamic reconfiguration technique.}.
Overall, we can observe that dynamic reconfiguration techniques provide more accurate warmup estimates than software developer ones.
In particular, $CV$ and $KLD$ outperform developer configurations in terms of $WEE$ on a considerable number of projects with high effect sizes.
\subsubsection{Estimated Warmup Time}\label{sec:dyn_vs_dev_wt}
\input{tables_rq4_wt}
\begin{figure}[p]
\centering
\includegraphics[width=7.5cm]{img_rq4_wt}
\caption{RQ$_5$. Summary $wt$ comparison. Each cell reports the number of systems whose comparison leads to a statistically significant change ($p\leq~0.05$) within a specific $\hat{A}_{12}$\xspace effect size range: negligible (N), small (S), medium (M) and large (L). $\hat{A}_{12}$\xspace~$>0.5$ indicates that dynamic configurations lead to shorter $wt$ than developer configurations.}
\label{fig:rq4_wt}
\end{figure}
In this subsection, we investigate the difference between the estimated warmup time ($wt$) provided by dynamic reconfiguration techniques and developer static configurations.
\tabref{tab:rq4_wt} reports results of Wilcoxon tests for each system, and across all systems.
As it can be observed by the last row of the table, $CV$ and $KLD$ report a statistically significant difference (p$<$0.001) with tendency toward improvement (\emph{i.e.,}\xspace smaller $wt$ values) but with negligible effect size.
$RCIW$ also reports statistically significant difference (p$<$0.001), but with the opposite tendency, \emph{i.e.,}\xspace larger $wt$ ($\hat{A}_{12}$\xspace $<$ 0.5), and a small effect size.
If we look at project-level results, we can observe a remarkable diversity among projects for $CV$ and $KLD$.
As it can be seen in \figref{fig:rq4_wt}, $CV$ leads to higher $wt$ than those defined by developers ($\hat{A}_{12}$\xspace $<$ 0.5) in 10 of the 30\xspace systems, and it reports lower $wt$ ($\hat{A}_{12}$\xspace $>$ 0.5) in 17 systems, thus not showing a clear trend.
A similar behavior can be observed for $KLD$: 10 of the 28 systems that report a statically significant difference (p $\leq$ 0.05) have $\hat{A}_{12}$\xspace$<$0.5, while 17 of them have $\hat{A}_{12}$\xspace$>$0.5.
On the other hand, $RCIW$ performs worse than developer configurations in most of the systems. $RCIW$ leads to higher $wt$ than developers in 22 of the 30\xspace systems: 12 with large effect sizes, 4 with medium effect sizes, and 4 with small effect sizes, 2 with negligible effect sizes. It leads to shorter $wt$ in only 5 projects (1 negligible, 3 small, and 1 medium effect sizes).
Furthermore, if we compare overestimation frequency/side effect of $RCIW$ (see Figs.~\ref{fig:warmup_estimation_DyCo_barplot} and \ref{fig:overestimation_DyCo}) and developer configurations (see Figs. \ref{fig:developer_configurations} and \ref{fig:overestimation_dev}), we can observe that $RCIW$ reports more frequent overestimations, namely 72\% \emph{vs} 48\%, and higher \emph{time wastes}, namely median of 46 seconds (IQR: 31-55 seconds) \emph{vs} median of 33 seconds (IQR: 19-49 seconds).
Overall, our results indicate that $CV$ and $KLD$ can lead to different behaviors based on the context, that is they can provide either higher or lower $wt$ than developers depending on the system.
On the other hand, $RCIW$ induces higher estimates of the warmup time $wt$ when compared to developer configurations, thus increasing microbenchmark execution time.
\subsubsection{Relative Performance Deviation}
\input{tables_rq4_perf}
\begin{figure}[p]
\center
\includegraphics[width=7.5cm]{img_rq4_perfscore}
\caption{RQ$_5$. Summary $RPD$ fork level comparison. Each cell reports the number of systems whose comparison leads to a statistically significant change ($p\leq~0.05$) within a specific $\hat{A}_{12}$\xspace effect size range: negligible (N), small (S), medium (M) and large (L). $\hat{A}_{12}$\xspace~$>0.5$ indicates that dynamic configurations perform better than developer configurations.}
\label{fig:rq4_perfscore}
\end{figure}
In this subsection, we assess the difference between $RPD$ of dynamic reconfiguration techniques and developer static configurations.
Overall, we can observe that dynamic reconfiguration techniques slightly improve developer configurations, as shown in the last row of \tabref{tab:rq4_perf}. The comparisons report statistically significant differences for all techniques (p$<$0.001), respectively with small ($CV$ and $RCIW$) and negligible ($KLD$) effect sizes.
If we look at project-level results (see \tabref{tab:rq4_perf} and \figref{fig:rq4_perfscore}), we can observe that $RCIW$ leads to statistically significant improvements over developer configurations in a large number of projects. In particular, $RCIW$ improves developer configurations in 18 of the 30\xspace systems (5 with large, 3 with medium, 5 with small, and 5 with negligible effect sizes), while it degrades $RPD$ in only 2 systems (both with negligible effect sizes).
By comparing the $RPD$ distributions of $RCIW$ (\figref{fig:underestimation_DyCo}) and developer configurations (\figref{fig:underestimation_dev}), we can observe that the former produces lower deviations with respect to steady state measurements. For example, using developer configurations, about 57\% of the forks lead to an $RPD$ of at least 5\%, while the same performance deviation is achieved in 43\% of forks when using $RCIW$.
Even more, $RCIW$ provides a median $RPD$ of 3\% (IQR: 0-11\%), whereas developer configurations lead to a deviation of 7\% (IQR: 1-21\%).
These results demonstrate that performance measurements gathered through $RCIW$ deviate less from steady state measurements than those collected through developer static configurations, thereby ensuring better results quality.
$CV$ and $KLD$ also report statistically significant improvements over static configurations in 16 and 17 systems, respectively. Nonetheless, developer configurations perform better than $CV$ and $KLD$ in, respectively, 10 and 12 systems.
Despite this, by looking at \figref{fig:rq4_perfscore}, we can observe that systems where developer configurations perform better than $CV$ and $KLD$, tend to have lower effect sizes than those where they provide improvement.
For example, if we exclude negligible effect sizes, we can observe that $CV$ still leads to improvement ($\hat{A}_{12}$\xspace $\geq$ 0.56) in 7 systems (5 large and 2 medium), while it performs worse than developer configurations ($\hat{A}_{12}$\xspace $\leq$ 0.44) in only one case with small effect size.
Similarly, $KLD$ leads to an effect size $\hat{A}_{12}$\xspace $\geq$ 0.56 in 10 systems (5 large, 2 medium, 3 small), while it reports a small effect size $\hat{A}_{12}$\xspace $\leq$ 0.44 in 5 systems.
These results suggest that the comparison between $RPD$ of $CV$/$KLD$ and developer configurations lead to different outcomes depending on the system, though with a slight overall tendency towards improvement.
The results across benchmarks of all systems confirm this tendency. As shown in the last row of \tabref{tab:rq4_perf}, $CV$ and $KLD$ report statistically significant improvements (p$<$0.001), respectively with small and negligible effect sizes.
Given these results, we can safely state that $CV$ and $KLD$ only provide marginal improvements over developer static configurations in terms of performance deviation.
\subsubsection{Benchmark level assessment}
\input{tables_rq4_benchlevel}
\begin{figure}[p]
\centering
\includegraphics[width=7.5cm]{img_rq4_benchlevel}
\caption{RQ$_5$. Summary $RPD$ benchmark level comparison. Each cell reports the number of systems whose comparison leads to a statistically significant change ($p\leq~0.05$) within a specific $\hat{A}_{12}$\xspace effect size range: negligible (N), small (S), medium (M) and large (L). $\hat{A}_{12}$\xspace~$>0.5$ indicates that dynamic configurations perform better than developer configurations.}
\label{fig:rq4_benchlevel}
\end{figure}
Interestingly, when we look at benchmark level results, we can observe significant differences
(see \figref{fig:rq4_benchlevel} and \tabref{tab:rq4_benchlevel}). \emph{CV} notably shifts from a tendency towards improvement to a tendency towards regression ($p$$<$$0.001$ and $\hat{A}_{12}$\xspace=0.45).
\emph{KLD}, which reported statistically significant differences and tendency towards improvement at fork level, reports neither improvement nor regression ($p>0.05$) at benchmark level. By analyzing project level results, we can further appreciate this shift.
At fork level, \emph{CV} reports worse RPDs than developers in only one project (with non-negligible effect size). At benchmark level instead, if we exclude negligible effect sizes, it reports worse performance deviations than developer configurations in 11 projects (3 with small effect sizes, 2 with medium, and 6 with large).
Likewise, \emph{KLD} reports worse RPDs in 9 projects at benchmark level (2 small, 2 medium and 5 large effect sizes), while, at fork level, it reports worse RPDs in only 5 project with small effect sizes.
The only technique that seems to provide improvement over developer configurations both at benchmark and fork level is \emph{RCIW}, which reports statistically significant improvement ($p<0.001$) and small effect size ($\hat{A}_{12}$\xspace=0.64).
\emph{RCIW} outperforms developer configurations in 17 projects (10 with large effect sizes, 5 with medium and 2 with small), and the effect sizes provided at benchmark level are even better than those provided at fork level (\emph{i.e.,}\xspace higher $\hat{A}_{12}$\xspace) in 15 out of the 30 projects.
These results may suggest that the capability of \emph{RCIW} to dynamically stop forks at run-time may further improve performance deviations when compared to those of developers.
To further investigate this aspect, we analyzed the results of these 15 projects, and we found that, in 80\% of the benchmarks, \emph{RCIW} involves 5 fork executions, \emph{i.e.,}\xspace the maximum number of forks for dynamic reconfiguration techniques (based on the original parameterization provided in \citep{Laaber2020}). That is, forks are not halted by stability criteria, rather they are stopped because the technique has reached the maximum number of allowed forks. This is somehow equivalent to statically fix the number of forks to 5.
Our analysis at fork level has shown that dynamic reconfiguration outperforms developer configurations in terms of performance deviation due to its capability to dynamically stop warmup iterations.
The same cannot be said when stability criteria are applied to halt forks.
Indeed, the analysis at benchmark level has shown that this capability of dynamic reconfiguration has only neutral or negative effects on performance deviations, and it is hard to perceive any improvement brought by this specific feature.
\\
\begin{tcolorbox}[colback=black!3!white,colframe=black!33!white]
\textbf{RQ$_5$ summary} - When compared to developer configurations, dynamic reconfiguration techniques provide more accurate estimates of the warmup time and better results quality (in most of the cases). $CV$ and $KLD$ show the largest improvement in terms of estimation accuracy, but only provide slight improvement in terms of results quality.
On the other hand, $RCIW$ outperforms developer configurations both in terms of estimation accuracy and results quality, but this improvement often comes at the expense of an increased warmup time. When time doesn't not represent a key concern, $RCIW$ should be the primary choice.
\end{tcolorbox}
\subsection{RQ$_2$ - Steady state impact}
In this subsection, we present results of our analysis on the impact of steady state on performance.
\begin{figure}[h]
\centering
\includegraphics[width=4cm]{img_steady-impact-within-fork}
\caption{RQ$_2$. Steady state impact within forks. The box plot reports the distribution of RPD between steady and non-steady phases of the execution across all the steady forks. The y-axis is in logarithmic scale.}%
\label{fig:impact-within-fork}
\end{figure}
\input{tables_steadyimpact-withinfork}
\figref{fig:impact-within-fork} and \tabref{tab:impact-within-fork} report results of the analysis that investigates how performance changes (within each fork) when the steady state is reached.
In particular, \figref{fig:impact-within-fork} depicts the distribution of the \emph{relative performance deviation} (RPD) across all the steady forks of our study. The figure highlights strong performance deviations when the steady state is reached, with an average RPD of 123,937\% and a median of 41\% (IQR 14-195\%).
By looking at the detailed results reported in \tabref{tab:impact-within-fork}, we can observe that the large mean is highly influenced by some specific projects (\emph{e.g.,}\xspace \texttt{camel}, \texttt{crate}, \texttt{h2o-3} and \texttt{presto}), which report extremely high RPD (up to 3.5 billion \%). Nonetheless, even when considering projects with smaller RPD (\emph{e.g.,}\xspace \texttt{jgrapht}, \texttt{RoaringBitmap} and \texttt{SquidLib}), we can observe considerable performance deviations between steady and non-steady measurements (respectively, 36\%, 40\% and 57\% on average).
Besides the diversity across projects, \tabref{tab:impact-within-fork} also highlights a substantial diversity within each project. Indeed, performance deviations substantially differ across benchmarks of the same projects, as they report extremely high standard deviations (maximum standard deviation of $\sim$16 billion (\texttt{presto}), and minimum of 56 (\texttt{RoaringBitmap})).
The above results suggest that performance substantially changes when forks reach a steady state of performance, and provide empirical evidence on the danger of using non-steady measurements during performance assessment.
Indeed, given these large magnitudes, even a tiny portion of non-steady measurements can substantially distort performance indices, with significant implications on performance assessment.
\begin{figure}[h]
\centering
\includegraphics[width=4cm]{img_steady-impact-across-fork}
\caption{RQ$_2$. Steady state impact across forks. The box plot reports the distribution of RPDs between steady and non-steady forks across all the inconsistent benchmarks.}%
\label{fig:impact-across-fork}
\end{figure}
\input{tables_steadyimpact-acrossfork}
\figref{fig:impact-across-fork} and \tabref{tab:impact-across-fork} report the results of our second analysis, which investigates performance deviations between steady and non-steady forks. By observing \figref{fig:impact-across-fork}, we can notice that the reported performance deviations are significantly smaller than those within forks (see \figref{fig:impact-within-fork}).
The average RPD between steady and non-steady forks is 5\%, while the median RPD is 2\% (IQR 0-5\%).
Although these deviations may appear negligible at a first glance, they are still significant if placed in the context of Java microbenchmarking.
Indeed, in these contexts, even relatively small performance regressions (\emph{e.g.,}\xspace 5\%) may lead to rejections of code revisions~\footnote{As an example, see \texttt{netty} pull request \texttt{8614} at \url{https://bit.ly/33MqlMZ}.}, as they can have significant impact at system level.
Moreover, if we look at some specific projects, such as \texttt{byte-buddy}, \texttt{JCTools} and \texttt{RxJava}, the reported deviations are even more conspicuous (average RPDs of respectively 14\%, 19\% and 13\%, see \tabref{tab:impact-across-fork}).
Still, the deviations between steady and non-steady forks are substantially smaller than those within forks.
In that, it is worth to remark that we compare steady and non-steady forks using on purpose the same measurement window (\emph{i.e.,}\xspace we discard all measurements collected before $st$, where $st$ denotes the steady starting time of the paired steady fork).
Indeed, this methodology may potentially discard measurements that are related to the most unstable phases of the (non-steady) fork execution, and it can consequently smooth deviations from steady measurements.
Nonetheless, this fact may also suggest that performance deviations tend to improve over time during benchmark execution, even in forks that do not reach a steady state of performance.
\begin{figure}[htpb]
\centering
\includegraphics[width=12cm]{img_iteration-impact}
\caption{RQ$_2$. The impact of warmup iterations on RPD. In each plot the x-axis denotes the number of warmup iterations, while the y-axis report RPD statistics. The dark shadow around the line represents the 95\% confidence interval. The plots in the first column report the mean RPD, while those in the second column report the median RPD.
Plots in the first row present the overall results, while those in the second row zoom on the first 500 iterations.
}%
\label{fig:iteration-impact}
\end{figure}
\begin{figure}[htpb]
\centering
\includegraphics[width=12cm]{img_fork-impact}
\caption{RQ$_2$. The impact of forks on RPD. In each plot the x-axis denotes the number of non-steady forks considered in the comparison, while the y-axis report RPD statistics. The dark shadow around the line represents the 95\% confidence interval. The left plot reports the mean RPD, while the right plot reports the median RPD.}%
\label{fig:fork-impact}
\end{figure}
To further investigate this aspect, we assess how warmup iterations impact performance deviation. In particular, we investigate to what extent warmup iterations mitigate performance deviations (RPD) of non-steady forks. \figref{fig:iteration-impact} reports the results of our analysis. As it can be seen from the figure, RPDs seem to be considerably influenced by the number of warmup iterations, especially at the early stages of benchmark execution. Indeed, we can see a substantial drop in the first 50 iterations, where the mean RPD decreases from 578\% to 17\% and the median from 30\% to 12\%. Moreover, we found a clear trend toward RPD reduction in the first 300 iterations. After this point, the decreasing trend seems to disappear, and RPDs begin to show fluctuations with means ranging from a minimum of 6\% to a maximum of 9\%, and medians ranging from 1\% to 3\%.
The above results suggest that performance deviations of non-steady forks may be substantially mitigated through a reasonable amount of warmup iterations. In particular, a significant reduction of performance deviations can be obtained (\emph{i.e.,}\xspace -561\% on average and -18\% on median) by using at least 50 warmup iterations, which correspond to 5 seconds of continuous benchmark execution and no less than 50 invocations. These deviations can be further reduced by using a higher number of warmup iterations up to 300 (\emph{i.e.,}\xspace 30 seconds of continuous execution and at least 300 invocations). After this point, increasing the number of warmup iterations barely affect RPDs.
Besides investigating the impact of warmup iterations, we also analyze the impact of forks. Specifically, we study whether using a higher number of forks reduces performance deviations of non-steady measurements. As it can be observed in \figref{fig:fork-impact}, the analysis shows an overall trend toward RPD reduction when increasing the number of forks. The average RPD is reduced by each additional fork, while the median shows a swinging trend only in the first two forks, and then it constantly decreases. If we compare RPDs of individual forks to those of 5 forks, we observe a significant RPD reduction, \emph{i.e.,}\xspace -244,237\% on the mean and -21\% on the median. This trend leads to an RPD reduction of -307,555\% (mean) and -31\% (median) when using 10 forks.
Still, the reported RPDs remain extremely high, \emph{i.e.,}\xspace mean of 35,797\% and median of 40\%.
The latter results are not surprising if we consider that our analysis deliberately targets non-steady measurements, which are typically subject to severe performance deviations (as we have shown in the first analysis of this RQ, see \figref{fig:impact-within-fork}).
In fact, in this analysis we were not particularly interested on the absolute RPD, rather we wanted to assess how RPDs change with respect to the number of forks.
In this regard, our results suggest that increasing the number of forks can effectively mitigate the impact of non-steady measurements.
\\
\begin{tcolorbox}[colback=black!3!white,colframe=black!33!white]
\textbf{RQ$_2$ summary} - The attainment of steady state has relevant effects on software performance.
Performance substantially changes within forks when their execution reaches steady state.
This difference in performance is less pronounced when comparing forks that never reach steady state against those that consistently reach it. Nonetheless, the reported performance deviations are still considerable and potentially harmful for performance assessment.
The use of an appropriate number of warmup iterations can significantly mitigate performance deviations induced by non-steady forks.
In addition, the use of an adequate number of forks can alleviate deviations that are induced by unstable measurements, which are collected before steady state execution occurs.
\end{tcolorbox}
\newpage
\section{Research Questions} \label{sec:rqs}
In this work we aim to answer the following Research Questions (RQs):
\begin{enumerate}
\item[RQ$_1$] \emph{Do Java microbenchmarks reach a steady state of performance?}
\item[RQ$_2$] \emph{How does steady state impact microbenchmark performance?}
\item[RQ$_3$] \emph{How effective are developer configurations in assessing Java steady state performance?}
\item[RQ$_4$] \emph{How effective is dynamic reconfiguration in assessing Java steady state performance?}
\item[RQ$_5$] \emph{Does dynamic reconfiguration provide more effective warmup estimates than developers do?}
\end{enumerate}
In the following subsections, we discuss in detail the motivation for each of the RQs and the methodology used to gather the answers. In \secref{sec:design}, we describe the experimental setup along with the benchmarks used in our empirical study.
\subsection{RQ$_1$ - Steady state assessment}
With this research question, we aim to evaluate whether Java microbenchmarks reach a steady state of performance.
We first use a state-of-the-art steady state detection technique~\citep{Barrett2017} to determine whether and when each fork reaches a steady state of performance (see \secref{sec:exp_steadystate} for details).
Based on these results, we then classify each benchmark as either (i) \emph{steady state}, if all the forks reach a steady state of performance, (ii) \emph{no steady state}, if none of the forks reach a steady state or (iii) \emph{inconsistent}, if the execution involves both \emph{steady state} and \emph{no steady state} forks.
We report the classification shares for both benchmarks and forks.
Additionally, we report the percentages of \emph{no steady state} forks for each benchmark.
\subsection{RQ$_2$ - Steady state impact}\label{sec:rqnew}
With this research question, we want to investigate to what extent the attainment of a steady state impacts benchmark performance. To do so, we compare the measurements collected during steady state phases against those collected in non-steady phases of benchmark execution. We perform two different analyses: the first one investigates the difference between steady and non-steady performance within the same fork, the second one assesses the same aspect across different forks.
Besides these two analyses, we also investigate two potential countermeasures to mitigate performance deviations in non-steady phases of benchmark execution.
All the aforementioned analyses involve a comparison between a set of steady measurements $M^{stable}$ and a set of non-steady measurements $M^{unstable}$. In order to assess to what extent non-steady measurements differ from steady measurements, we use \emph{relative performance deviation} ($RPD$).
In the following, we first explain the process we use to compute $RPD$. Then, we describe in the detail the four analyses we use to gather the answer.
\subsubsection*{Relative Performance Deviation}
In order to quantify the relative performance deviation of $M^{unstable}$ compared to $M^{stable}$, we use the technique proposed by \cite{Kalibera2013}.
The main benefit of this technique is that it provides a clear and rigorous account of the relative performance change and the uncertainty involved. For example, it can indicate that a set of execution time measurements is higher than another by $X\% \pm Y\%$ with 95\% confidence. Following the guidelines of \cite{Kalibera2013,kalibera2020}, we build each confidence interval using bootstrapping with random re-sampling and replacement \citep{davison1997}, with a confidence level of 95\%.
We run 10,000 bootstrap iterations. At each iteration, new realizations $\hat{M}^{unstable}$ and $\hat{M}^{stable}$ (respectively, of $M^{unstable}$ and $M^{stable}$ measurements) are simulated and the relative performance change is computed.
The simulation of the $\hat{M}^{unstable}$ new realizations randomly selects a subset of real data from $M^{unstable}$ with replacement. Similarly, $\hat{M}^{stable}$ is simulated by randomly sampling $M^{stable}$. The two means ($\mu^{unstable}$ and $\mu^{stable}$) and the relative performance change ($\rho$) for simulated measurements are computed as follows:
{
\begin{equation*}
\mu^{unstable}=\frac{\sum_{i=1}^{n}{\hat{M}^{unstable}_{i}}}{n}
\end{equation*}
\begin{equation*}
\mu^{stable}=\frac{\sum_{i=1}^{m}{\hat{M}^{stable}_{i}}}{m}
\end{equation*}
\begin{equation*}
\rho=\frac{\mu^{unstable}-\mu^{stable}}{\mu^{stable}}
\end{equation*}
where $n$ is the number of measurements in $\hat{M}^{unstable}$, $m$ the number of measurements in $\hat{M}^{stable}$.
After the termination of all iterations, we collect a set of simulated realizations of the relative performance change $P=\{\rho_i \mid 1\leq i \leq 10,000\}$ and estimate the 0.025 and 0.975 quantiles on it, for a 95\% confidence interval.
We consider a relative performance change as statistically significant if the confidence interval does not contain 0.
For example, given a confidence interval of $(0.05, 0.07)$, we can say that the mean execution time in $M^{unstable}$ is higher than the one in $M^{stable}$ with a relative performance change that ranges between 5\% and 7\% with 95\% confidence.
We leverage the confidence interval of the mean relative performance change $(lb, ub)$ to compute the \emph{relative performance deviation}:
\begin{equation*}
RPD=\begin{cases}
\text{~~~\small$0$} & \text{\small if \xspace $lb \leq 0 \leq ub$} \\
\text{\large$|\frac{lb + ub}{2}|$} & \text{\small otherwise}
\end{cases}
\end{equation*}
In other words, we define $RPD$ as the center of the 95\% confidence interval of the mean relative performance change, if the interval doesn't contain zero. On the other hand, $RPD$ evaluates to 0 if the confidence interval does not report a statistically significant performance change, \emph{i.e.,}\xspace the interval does contain 0.
Higher values of $RPD$ indicate that $M^{unstable}$ strongly deviates from $M^{stable}$.
\subsubsection*{Analyses}
\begin{figure}
\centering
\begin{subfigure}{12cm}
\includegraphics[width=12cm]{img_rq2_1}
\caption{\label{fig:rq2_1} Performance deviation within fork.}
\end{subfigure}
\hfill
\begin{subfigure}{12cm}
\includegraphics[width=12cm]{img_rq2_2.pdf}
\caption{\label{fig:rq2_2} Performance deviation across forks.}
\end{subfigure}
\hfill
\begin{subfigure}{12cm}
\includegraphics[width=12cm]{img_rq2_3.pdf}
\caption{\label{fig:rq2_3} Influence of warmup iterations on performance deviation.}
\end{subfigure}
\hfill
\begin{subfigure}{12cm}
\includegraphics[width=12cm]{img_rq2_4}
\caption{\label{fig:rq2_4} Influence of forks on performance deviation.}
\end{subfigure}
\caption{\label{fig:rq2_analyses} RQ$_2$ analyses.}
\end{figure}
In order to analyze performance deviation within forks, we consider only forks that have reached a steady state of performance, and we analyze how performance changes when the steady state is reached. In particular, we partition the set of measurements of each fork in two distinct sets, namely $M^{stable}$ and $M^{unstable}$, and we compare them to quantify the relative performance deviation ($RPD$). $M^{stable}$ contains the measurements collected during steady state execution, \emph{i.e.,}\xspace those gathered after the \emph{steady state starting time} $st$, while $M^{unstable}$ contains measurements collected before $st$.
\figref{fig:rq2_1} provides a graphical representation of this process for a given steady fork.
To investigate performance deviation across forks, instead, we assess how performance differs between steady and non-steady forks. To do that, we exclusively consider inconsistent benchmarks (\emph{i.e.,}\xspace the ones that contain both steady and non-steady forks), and we randomly pick from each benchmark a pair of forks ($f$, $\hat{f}$): one that has reached a steady state of performance ($f$) and one that does not ($\hat{f}$). We then use each pair to compare the measurements collected in the steady fork ($M^{stable}$) against those collected in the non-steady fork ($M^{unstable}$), and we quantify $RPD$.
In order to enable a fair comparison, we use the same measurement window for both steady and non-steady forks. Namely, given a pair of forks ($f$, $\hat{f}$), we define $M^{unstable}$ (resp., $M^{stable}$) as the set of measurements collected in $\hat{f}$ (resp., $f$) after $st$, where $st$ denotes the steady starting time of the steady fork $f$.
\figref{fig:rq2_2} depicts this process for one inconsistent benchmark.
Besides the aforementioned analyses, we investigate potential countermeasures to mitigate performance deviations of non-steady measurements.
In particular, we analyze two specific aspects: number of warmup iterations and number of forks.
With our analysis on warmup iterations, we aim to tackle performance deviations of non-steady forks, \emph{i.e.,}\xspace forks that never reach a steady state of performance. In particular, we want to assess to what extent an increase in the number of warmup iterations can mitigate performance deviations of non-steady forks.
With our analysis on forks, instead, we aim to tackle deviations of non-steady measurements in forks that consistently reach a steady state of performance. That is, we investigate to what extent an increase in the number of forks mitigates performance deviations of measurements gathered during non-steady phases of benchmark execution.
In order to investigate the impact of warmup iterations, we first randomly sample one steady fork $f$ and one non-steady fork $\hat{f}$ from each inconsistent benchmark. Then, for each pair ($f$, $\hat{f}$), we use a sliding window of 50 consecutive measurements ($M^{unstable}$) in $\hat{f}$, and we assess the deviation from the set of steady state measurements collected in $f$, namely $M^{stable}$. In particular, we partition the sequence of measurements gathered from $\hat{f}$ in 60 segments of equal size (\emph{i.e.,}\xspace 50 measurements), and we compute the relative performance deviation of each segment from $M^{stable}$. By doing so, we can assess whether increasing the number of warmup iterations can mitigate performance deviations of non-steady forks.
\figref{fig:rq2_3} shows the process used to compute the $RPD$ for one specific segment ($M^{unstable}$) of $\hat{f}$, \emph{i.e.,}\xspace for a specific number of warmup iterations.
In order to investigate the influence of forks, instead, we assess how performance deviation of non-steady measurements changes when using different numbers of forks. Given the goal of this analysis, we consider only steady benchmarks, \emph{i.e.,}\xspace benchmarks that exclusively involve steady forks. For each benchmark, we progressively increment the number of considered forks and, at each increment $i$, we form a set $\mathcal{M}^{unstable}_{i}$ composed by all the non-steady measurements gathered from these forks. Then, we compute, for each set $\mathcal{M}^{unstable}_{i}$, the deviation from the entire set of steady measurements gathered from all the forks ($\mathcal{M}^{stable}$), as showed in \figref{fig:rq2_4}.
In particular, we start by comparing the set of non-steady measurements collected from the first fork ($\mathcal{M}^{unstable}_{1}$) to the whole set of steady measurements $\mathcal{M}^{stable}$. Subsequently, we consider the set of non-steady measurements from the first two forks together ($\mathcal{M}^{unstable}_{2}$) and, again, we compare it to $\mathcal{M}^{stable}$. We proceed in this way for each set of non-steady measurements $\mathcal{M}^{unstable}_{i}$, with $i$ ranging from 1 to 10.
At the end of this process, we obtain one result (\emph{i.e.,}\xspace relative performance deviation) for each pair ($b$, $i$), where $b$ denotes a steady benchmark and $i$ denotes the number of considered forks.
Through this analysis, we can assess if/how the increases in the number of forks mitigate the inherent deviations of non-steady measurements. It is worth to notice that we perform this analysis in an extreme situation, \emph{i.e.,}\xspace by considering exclusively non-steady measurements.
We decided to do so because, if can we demonstrate that the increases in the number of forks can mitigate performance deviations in such an extreme case, then there are strong indications that they can effectively mitigate the impact of non-steady measurements.
\subsection{RQ$_3$ - Developer configuration assessment}\label{sec:rq2}
With this research question, we aim to evaluate the effectiveness of developer configurations for steady state performance assessment.
To do so, we assess how well software developers capture \emph{steady state starting time} (\emph{st}), \emph{i.e.,}\xspace the execution time required to reach a steady state of performance in a fork.
Specifically, in each fork, we compare the \emph{estimated warmup time} (\emph{wt}) defined by software developers with the \emph{steady state starting time} \emph{st} detected by the technique of \cite{Barrett2017}.
We measure the error of $wt$ using \emph{warmup estimation error} ($WEE$), \emph{i.e.,}\xspace how far $wt$ is to the steady state starting time $st$:
\begin{equation*}
WEE = |wt - st|
\end{equation*}
Additionally, we report the proportion of underestimated and overestimated forks (\emph{i.e.,}\xspace \emph{wt} is respectively smaller or larger than \emph{st} by at least 5 seconds), and we investigate potential side effects.
An overestimated warmup phase wastes execution time, since it leads to a surplus of warmup iterations, which, in turn, delay the beginning of measurement iterations.
This inevitably increases benchmark execution time and causes potentially harmful practical implications.
For example, the ``time effort'' of a performance test is often considered a critical factor when selecting the tests to execute before a software release \citep{traini2021,DBLP:conf/icsm/ChenS17}.
In that, an overestimated warmup phase can hamper the adoption of microbenchmarks for continuous performance assurance, especially when software releases happen frequently \citep{Rubin2016}, and tests are executed as part of a Continuous Integration (CI) pipeline \citep{Fowler2006}.
To investigate side effects of overestimated warmup time, we use \emph{time waste}, \emph{i.e.,}\xspace the overestimation error induced by developer configurations.
We quantify \emph{time waste} through the difference (in terms of time) between \emph{wt} and \emph{st}.
In other words, \emph{time waste} represents the execution time that can be potentially saved in a fork.
On the other hand, an underestimated warmup phase may easily mislead steady state performance assessment.
Indeed, unstable measurements (\emph{i.e.,}\xspace measurements gathered before $st$) may distort performance results, thereby leading to potentially wrong conclusions \citep{Georges2007, Kalibera2013}.
We assess the impact of underestimated warmup time using \emph{ relative performance deviation} ($RPD$), \emph{i.e.,}\xspace the magnitude of performance deviation compared to steady state measurements.
That is, for each underestimated fork, we compare performance measurements in the steady state ($M^{stable}$) with those in the measurement time window defined by software developers ($M^{conf}$) using $RPD$.
A high $RPD$ indicates that performance measurements used by software developers strongly deviate from those collected in the steady state.
Besides investigating the effectiveness of developer configurations at fork level, we also analyze their effectiveness at benchmark level, \emph{i.e.,}\xspace across multiple forks. Through this analysis, we aim to evaluate developer configurations not only in terms of warmup and measurement iterations, but also based on the number of configured forks.
To do that, we consider the entire set of measurements gathered through developer configuration (\emph{i.e.,}\xspace across the configured forks), and we compare it to the whole set of steady measurements gathered across all the steady forks of the benchmark.
In particular, for each benchmark, we assess the extent to which the set $\mathcal{M}^{conf}$ of developer measurements deviate from the set $\mathcal{M}^{stable}$ of steady measurements.
In addition, we report the time effort of benchmarks based on developer configurations by computing their overall execution time.
\subsection{RQ$_4$ - Dynamic reconfiguration assessment}
With this research question, we aim to evaluate the effectiveness of dynamic reconfiguration approaches~\citep{Laaber2020} for steady state performance assessment.
These approaches dynamically determine the warmup time during benchmark execution using stability criteria.
In our evaluation, we consider three stability criteria proposed by \cite{Laaber2020}: (i) Coefficient of variation (CV), (ii) Relative confidence interval width (RCIW), and (iii) Kullback-Leibler divergence (KLD).
In order to assess the effectiveness of each dynamic reconfiguration variant (\emph{i.e.,}\xspace stability criteria), we use the same methodology adopted in RQ$_3$.
We first report the \emph{warmup estimation error} ($WEE$) of forks, along with the proportion of underestimated and overestimated warmup time.
Then, we consider the potential side effects of wrong warmup estimation using \emph{time waste} and \emph{relative performance deviation} ($RPD$).
Finally, we investigate the effectiveness of dynamic reconfiguration at benchmark level (\emph{i.e.,}\xspace across forks) by assessing their RPDs and the overall execution time.
\subsection{RQ$_5$ - Dynamic vs Developer configurations}
To answer RQ$_5$, we compare dynamic reconfiguration techniques against developer configurations using three evaluation metrics: (i) \emph{warmup estimation error}, (ii) \emph{estimated warmup time}, and (iii) \emph{relative performance deviation}.
\emph{Warmup estimation error} ($WEE$) measures the accuracy of $wt$, \emph{i.e.,}\xspace how close $wt$ is to the steady state starting time $st$.
Lower values of $WEE$ indicate better estimates of the warmup time.
\emph{Estimated warmup time} ($wt$) measures the time spent to warmup a fork with respect to a specific configuration.
Higher values of $wt$ increase the time effort devoted to performance testing, and, therefore, can potentially hamper the adoption of benchmarks for continuous performance assessment.
\emph{Relative performance deviation} ($RPD$), instead, measures in each fork the magnitude of performance deviation of $M^{conf}$ compared to steady state performance measurements ($M^{stable}$).
Higher values of $RPD$ indicate that $M^{conf}$ strongly deviates from $M^{stable}$, thus implying that performance measurements determined by the configuration may potentially mislead steady state performance assessment.
As a complementary analysis, we further use $RPD$ to measure performance deviations at benchmark level. That is, we quantify (for each benchmark) the magnitude of deviation of the entire set of measurements gathered through developers/dynamic configurations ($\mathcal{M}^{conf}$ ) when compared to the whole set of steady measurements collected through the entire benchmark execution ($\mathcal{M}^{stable}$).
The results of $WEE$, $wt$, and $RPD$ are compared using the Wilcoxon Rank-Sum test~\citep{cohen2013}, which is a non-parametric test that makes no assumption about underlying data distribution, hence, raises the bar for significance for both normally and non-normally distributed data.
Additionally, a standardized non-parametric effect size measure, namely the Vargha Delaney's $\hat{A}_{12}$\xspace statistic~\citep{Vargha2000}, is used to assess the effect size.
Given a dynamic reconfiguration technique $D$, $\hat{A}_{12}$\xspace measures the probability of $D$ performing better than developer configurations with reference to a specific evaluation metric. $\hat{A}_{12}$\xspace is computed using \eqqref{eq:vda}, where $R_1$ is the rank sum of the first data group we are comparing, and $m$ and $n$ are the numbers of observations in the first and second data sample, respectively.
\begin{equation}\label{eq:vda}
\hat{A}_{12} = \frac{(\frac{R_1}{m} - \frac{m+1}{2})}{n}
\end{equation}
We interpret $\hat{A}_{12}$\xspace using the thresholds provided by \cite{Vargha2000}.
Based on \eqqref{eq:vda}, if dynamic configurations and developer configurations are equally good, $\hat{A}_{12}$\xspace$=0.5$. Respectively, $\hat{A}_{12}$\xspace higher than 0.5 means that dynamic reconfiguration is more likely to produce better results. The effect size is considered small for $0.56 \leq \hat{A}_{12} < 0.64$, medium for $0.64 \leq \hat{A}_{12} < 0.71$, and large for $\hat{A}_{12} \geq 0.71$.
On the other hand, $\hat{A}_{12}$\xspace smaller than 0.5 means that developer configurations provide better results.
In this case, the effect size is considered small for $0.34 > \hat{A}_{12} \leq 0.44$, medium for $0.29 > \hat{A}_{12} \leq 0.34$, and large for $\hat{A}_{12} \leq 0.29$.
In order to avoid misleading interpretations, we perform transformation on the $\hat{A}_{12}$\xspace effect size~\citep{Neumann2015}, since we consider values of $WEE$ smaller than 5 seconds and $RPD$ smaller than 5\%, as negligible~\citep{222561}, and, therefore, ``equally good''.
In particular, we apply \emph{Pre-Transforming Data}~\citep{Neumann2015} by replacing each value of $WEE<5sec$ and $RPD<0.05$ with zero.
When it comes to \emph{estimated warmup time} ($wt$), no transformation is performed on the $\hat{A}_{12}$\xspace effect size, since we are interested in any improvement~\citep{Neumann2015, Sarro2016}.
\subsection{Steady State Performance} \label{sec:steadystate}
\begin{figure*}[p]
\centering
\begin{subfigure}[b]{10cm}
\centering
\includegraphics[width=\linewidth]{img_steady_example}
\caption{Example of benchmark execution that consistently reaches a steady state of performance.
}
\label{fig:steady_example}
\end{subfigure}
\hfill
\begin{subfigure}[b]{10cm}
\centering
\includegraphics[width=\linewidth]{img_non_steady_example}
\caption{Example of benchmark execution that doesn't reach a steady state of performance.}
\label{fig:non_steady_example}
\end{subfigure}
\caption{Two examples of benchmarks executions from our results. The grey line represents the execution times of each benchmark iteration visualized as a time-series. The x-axis represents the benchmark iteration number, while the y-axis represents the mean execution time within the iteration. Shifts in performance behaviour (\emph{i.e.,}\xspace changepoints) are indicated by dashed vertical red lines. The height of each dashed horizontal red line denotes the mean execution time within the changepoint segment. The plot titles show: the system that the benchmark belongs to (\emph{e.g.,}\xspace \texttt{h2o-3}), the benchmark name (\emph{e.g.,}\xspace~\texttt{water.util.IcedHashMapBench.writeMap}), and the JMH fork number.}
\end{figure*}
At the beginning of its execution, typically, a Java microbenchmark is slowly executed by the JVM.
In a subsequent phase, the JVM detects ``hots'' (\emph{i.e.,}\xspace frequently executed) loops or methods, and it dynamically compiles them into optimized machine code.
As a consequence, subsequent executions of those loops or methods (usually) become faster.
Once dynamic compilation is completed, the JVM is said to have finished warming up, and the benchmark is said to be executing at a \emph{steady state of performance}.
Java microbenchmarking aims at assessing steady state performance (\emph{e.g.,}\xspace execution time) of Java software.
The typical approach to collect steady state measurements is straightforward: A microbenchmark is executed for a certain number of times, and the first $n$ benchmark executions are discarded (\emph{i.e.,}\xspace those related to the \emph{warmup phase}) to prevent potentially misleading results.
Unfortunately, the fixed number of $n$ executions does not guarantee that warmup has ended.
In order to investigate this issue, researchers started to develop data-driven methodologies to identify the end of the warmup phase.
Prominent works in this regard are the methodologies proposed by \cite{Georges2007}, and \cite{Kalibera2013}.
The former uses preset thresholds on the coefficient of variation to determine whether the warmup phase is finished or not, while the latter leverages data visualization techniques (\emph{i.e.,}\xspace auto-correlation function plots, lag plots and run-sequence plots).
Unfortunately, each one of these methodologies has its own drawback.
Kalibera and Jones showed that the Georges \emph{et~al.}\xspace's heuristic~(\citeyear{Georges2007}) often fails to accurately determine the end of the warmup phase \citep{Kalibera2013}.
On the other hand, the methodology proposed by \cite{Kalibera2013} is mostly based on a manual process, which typically implies some major limitations: (i) humans are prone to error/disagreement, (ii) manual analysis doesn't enable automation, and therefore it is not scalable.
To overcome these limitations, \cite{Barrett2017} recently proposed a novel automated technique based on change point detection~\citep{Eckley2011}.
The main advantage of this technique is that it provides a more rigorous approach compared to the Georges~\emph{et~al.}\xspace's simple heuristic, while still enabling a fully automated process (unlike Kalibera and Jones' approach).
The Barrett~\emph{et~al.}\xspace's technique leverages a standard change point detection algorithm, namely PELT~\citep{Killick2012}, to determine shifts in benchmark execution time. The identified shifts (\emph{i.e.,}\xspace change points) are then post-processed (\emph{e.g.,}\xspace by removing negligible performance shifts) to determine \emph{if}/\emph{when} a benchmark reaches a steady state of performance.
To the best of our knowledge, this technique currently represents the state-of-the-art for steady state detection.
\figref{fig:steady_example} and \figref{fig:non_steady_example} show two examples of benchmark executions along with the performance shifts identified by the PELT algorithm. The former consistently reaches a steady state of performance, while the latter doesn't.
\section{Threats to validity} \label{sec:threats}
Our study may be affected by different threats that span from how we collected performance data to the subject project domains, and we describe them in the following.
\paragraph{Construct validity}
We assume that the benchmarks that reach a steady state will be able to do it within the execution time we defined.
There may be benchmarks that need more time to show some stability.
However, we chose the execution time to be considerably longer than the time we found in developer configurations (171 times longer on average).
Also, as done in other studies~\citep{Laaber2020}, measurement time, the number of iterations, and the number of forks are fixed for every benchmark.
The consequence is that the number of invocations varies from one benchmark to another.
Nonetheless, no benchmark is invoked less than 3000 times per fork, that is 1000 times more than in \citep{Barrett2017}.
The number of forks is fixed at 10, as recommended in \citep{Barrett2017}.
Our experiment was performed in an environment where we tried our best to reduce the measurement noise and external influencing factors~\citep{DBLP:journals/tse/PapadopoulosVBH21,DBLP:conf/asplos/MytkowiczDHS09}.
Such settings are effective in improving the accuracy of results, but may not represent the more common environment in which developers execute the benchmarks.
However, general reference environments can hardly exist for benchmarks.
In fact, different developers can potentially execute the same microbenchmark on a wide range of different machines/environments, given the inherently distributed nature of open-source software development.
On top of that, there is an increasing interest in promoting the adoption of microbenchmarks in CI \citep{Laaber2021, Laaber2020,Laaber2019, Laaber2018}, which are most likely uncontrolled/noisy environments. All these aspects make it impractical to identify a reference environment for each benchmark/system.
For this reason, we have deliberately chosen to execute benchmarks on the same bare-metal server, using precautionary measures to mitigate measurement noise and external influencing factors (see \secref{sec:microbench} for details).
In this respect, we preferred to control the confounding variables rather than losing accuracy in more noisy and unreproducible settings.
This consideration was especially motivated by the first goal of our study (RQ$_1$), \emph{i.e.,}\xspace checking whether benchmarks reach a steady state of performance.
Indeed, we prefer to run benchmarks on an environment that is considerably less subject to noise than those of developers, rather than possibly causing non-steady executions due to the noise that is specific to our environment.
Nonetheless, we do believe that future research should further strengthen our investigation beyond controlled environments, \emph{i.e.,}\xspace by studying steady state performance even in uncontrolled and noisy environments. For example, future studies may replicate our experiments in cloud contexts to assess how these virtualized environments affect steady state performance. We envision that this latter step is crucial to foster the adoption of Java microbenchmarking in CI.
Developer configurations for warmup iterations $wi$ and measurement iterations $i$ are derived from the JSON reports generated by JMH (details in \secref{sec:bench_conf}).
We do not consider other ways in which benchmarks might be executed by developers, by using, for example, different command line arguments at launch time or by configuring additional parameters in build automation pipelines.
Nonetheless, to the best of our knowledge, no study so far showed evidence of microbenchmarks being used as part of build automation pipelines~\citep{Rausch2017, Vassallo2017, Beller2017}.
Following the methodology of \cite{Laaber2020}, we performed post-hoc analysis to derive measurements for developers and dynamic configurations.
Indeed, a comprehensive execution of all benchmarks across all developer static configurations and dynamic reconfiguration alternatives would have made our experimental evaluation impractical due to extremely long execution times (the execution of all 586\xspace benchmarks using our single setup took about 93 days).
In order to mitigate the impact of post-hoc analysis on the validity of our study, we employed a carefully defined process to derive measurements.
We first estimated the time spent in each warmup/measurement iteration using the average execution time of each iteration as observed in our microbenchmarking setup. Then, we derived the \emph{estimated warmup time} $wt$ and the set of measurements $M^{conf}$ based on the JMH configurations provided by software developers or dynamic reconfiguration techniques.
The detailed process to estimate $wt$ and derive $M^{conf}$ can be found in \secref{sec:bench_conf}.
Nevertheless, our results may be marginally affected by approximations in converting measurement samples from our configuration to the others.
\paragraph{Internal validity}
The steady state is detected on the basis of execution time measurements.
We do not consider other event-based stability criteria, such as JIT activity, as these are not part of any JMH report and, therefore, are not something we can compare against when examining developer configurations.
However, considering such additional criteria may lead to a different steady state classification.
Before performing the change point detection, we filtered the outliers using Tukey's fences on a sliding window, as described in \secref{sec:exp_steadystate}. The parametrization of this procedure might affect the detection of the steady state. However, we tried to select parameters that would result in a very conservative outliers filtering. In fact, we only filter $0.27\%$ of the datapoints. Nonetheless, the outliers filtering is a necessary procedure when employing change point detection algorithms, as the vast majority of such algorithms cannot distinguish between actual changes and outliers~\citep{fearnhead2019outliers}, thus leading to an overestimation of changes.
\paragraph{External validity}
We only focused on GitHub repositories.
Therefore, it is unclear if the findings are valid for other open-source hosting platforms or industrial software.
Nonetheless, we conducted our experiment on 30 systems, a number larger than most recent empirical studies on performance (\emph{e.g.,}\xspace see \citeauthor{Laaber2018,Laaber2020,DBLP:conf/icse/DingCS20,DBLP:conf/kbse/ReicheltKH19}).
We chose to limit the scope of the experiment to JMH microbenchmarks, because JMH is a mature and widely adopted Java microbenchmark harness.
We ran the benchmarks on JVM 8 (JDK 1.8.0 update 241) or 11 (JDK 11.0.6), depending on the requirements of the specific system.
Using other JVM versions or JVMs from other vendors may change the results.
All the benchmarks were executed on the same bare-metal server running Linux.
Nothing can be said about other hardware characteristics or other operating systems.
\paragraph{Conclusion validity}
The changepoint detection method we used assumes independence in the time series.
Even when the data contain some dependence, changepoint methods can still be used, provided that larger penalty values are used~\citep{ANTOCH1997291,Barrett2017}.
The penalty values we dynamically compute for each fork, as explained in \secref{sec:exp_steadystate}, tend to be larger (the average penalty is 504.54) than the value used, for example, in \cite{Barrett2017}.
Also, we manually inspected some of the time series to ensure that the segmentation was reasonable given the goal of the experiment.
Wherever possible, we used appropriate statistical procedures with p-value and effect size measures to test the significance of the differences and their magnitude.
| 2024-02-18T23:39:54.598Z | 2022-10-03T02:11:37.000Z | algebraic_stack_train_0000 | 792 | 22,419 |
|
proofpile-arXiv_065-3969 |
\section{Default Notation}
\section{Introduction}
\looseness=-1Neural Networks (NN) learn to transform high dimensional data into meaningful representations that are helpful for solving downstream tasks. Typically, these representations are seen as elements of a vector space, denoted as latent space, which corresponds to the constrained output (explicitly or implicitly) of a key component of the NN, e.g., the bottleneck in an Autoencoder (AE), or the word embedding space in an NLP task.
The underlying assumption is that the learned latent spaces should be the best encoding given the data distribution, the downstream task, and the network constraints.
In practice, however, the learned latent spaces are subject to changes even when the above assumptions remain fixed. We illustrate this phenomenon in \Cref{fig:latent-rotation}, where we show the latent spaces produced by an AE with a two-dimensional bottleneck, trained on the \mnist{} dataset several times from scratch. These spaces differ from one another, breaking the fundamental assumptions made above. The distribution of the latent embeddings is affected by several factors, such as the random initialization of the network weights, the data shuffling, hyperparameters, and other stochastic processes in the training phase.
Although different, the learned representations in \Cref{fig:latent-rotation} are {\em intrinsically} similar: the distances between the embedded representations are approximately the same across all spaces, even if their absolute coordinates differ. Indeed, the learned latent spaces are the same up to a nearly isometric transformation.\footnote{To the best of our knowledge, the first to acknowledge this behavior was \cite{Colah2015} in a blogpost.}
This symmetry is a consequence of the implicit biases underlying the optimization process \citep{Soudry2017Oct} forcing the model to generalize and, therefore, to give similar representations to similar samples with respect to the task.
There exist infinitely many spatial arrangements complying with these similarity constraints, each associated with a different isometry. But while the resulting models will be equally good in terms of the task, one still encounters
several practical problems. For example,
it is notoriously challenging to compare latent spaces across different trainings or across different NNs;
perhaps more importantly, re-using neural components trained on different embeddings of the same data becomes impossible, since they are incompatible.
To overcome this, we propose adopting a local coordinate system defined by the data itself. Each data point becomes a set of coefficients that encode the point as a function of other data samples, instead of an independent point in $\mathbb{R}^d$.
The proposed \emph{relative} representation directly encodes the intrinsic information underlying the data, and can be made fully invariant to isometries by construction.
Remarkably, this enables a form of compositionality between learning models; it allows, for instance, to stitch together an encoder trained on ImageNet with a decoder trained on CIFAR, as we showcase in our experiments.
Our main contributions can be summarized as follows:
\begin{itemize}
\itemsep0em
\item We show that the representations learned by NNs are subject to change due to several factors in the training process, and model these changes via latent space isometries.
\item We introduce a novel {relative representation} for latent embeddings, that is invariant by construction to the isometries induced by stochastic factors in the training process.
\item For the first time, we successfully demonstrate \emph{zero-shot stitching} of neural components produced by distinct training regimens, e.g., due to different seeds or different neural architectures; we validate our findings on different data modalities (e.g. images, text).
\item Our framework also provides a \emph{quantitative} measure of performance while training neural models, which is differentiable, does not need any labeled data, and is correlated with standard performance measures such as accuracy.
\end{itemize}
\begin{figure}[t]
\centering
\begin{overpic}[trim=0cm .3cm 0cm -.1cm,clip,width=.95\linewidth]{figures/latent_rotation.pdf}
%
\put(12, 24.25){\small Train 1}
\put(37, 24.25){\small Train 2}
\put(61, 24.25){\small Train 3}
\put(85, 24.25){\small Train 4}
\end{overpic}
\caption{Latent spaces learned by distinct trainings of the same AE on the \mnist{} dataset.
The bottleneck has size $2$, thus there is no dimensionality reduction involved in the visualization of the latent space.
The stochasticity in the training phase induces the same representations up to isometry. As we show in \Cref{fig:latent-rotation-pca-proof}, this property holds even for high-dimensional latent spaces.}
\label{fig:latent-rotation}
\end{figure}
\section{Related Work}
\vspace{1ex}\noindent\textbf{Representation similarity.}
Recently, there has been growing agreement that good networks learn similar representations across a variety of architectures, tasks and domains \citep{Morcos2018-ra,Li2015-jo,Kornblith2019-sz,Bonheme2022-tk,Tsitsulin2019-gg,Barannikov2021-eb,Vulic2020-zb,Conneau2017-vv,Lenc2014-gy}, although this is still debated \citep{Wang2018-ho} and missing strong theoretical justifications.
Supported by the empirical evidence widely reported in these works, our method assumes that well-performing neural networks trained on similar tasks and data produce similar latent spaces, which allows us to define a \enc{}{rep} that unifies all these spaces.
\vspace{1ex}\noindent\textbf{Model stitching.}
\cite{Lenc2014-gy} introduced \emph{trainable} stitching layers that allow swapping parts of different networks,
while \cite{Bansal2021-oj,Csiszarik2021-yi} employed stitching to quantitatively verify statements such as ``good networks learn similar representations'' and ``more data, width or time is better''.
Other works, such as \cite{Gygli2021-qb}, tried to directly produce compatible and reusable network components without stitching layers; more in general, stitching has been adopted in the literature to analyze neural networks. In our work, we sidestep the need for trainable stitching layers and propose \emph{zero-shot} model stitching to effectively reuse models.
\vspace{1ex}\noindent\textbf{Relative information.}
\looseness=-1The attention mechanism \citep{Vaswani2017-qw} and its variants \citep{Kossen2021-mj} exploit the relationship between features to extract meaningful representations.
Prototypical Networks \citep{Snell2017-fe} learn a metric space where the classification can be performed by measuring the distances to prototype representations.
\cite{Shalam2022-hj} proposed the Self Optimal Transport feature transform to enrich the sample representations with higher order relations between the instance features, while
\cite{Alvarez-Melis2018-kr} proposed a general formulation of the optimal transport that accounts for global invariances in the underlying feature spaces.
Mathematically, our method bears resemblance to a kernel method~\citep{hofmann2008kernel} as it employs inner products of embedded features as a core ingredient. However, differently from kernel methods, we do not introduce learnable parameters and, crucially, we compute the representations explicitly without resorting to a kernel trick.
\section{Method}
Given a training set ${\mathbb{X}}$, standard NNs learn an embedding function $E_\theta: {\mathbb{X}} \rightarrow \mathbb{R}^d$, parametrized by $\theta$, which maps each sample ${\bm{x}}^{(i)} \in {\mathbb{X}}$ to its latent representation, or \newterm{\enc{abs}{rep}}, ${\bm{e}}_{{\bm{x}}^{(i)}} = E_\theta({\bm{x}}^{(i)})$. This representation is then exploited to solve downstream tasks, such as classification, reconstruction or generation, optimizing over some objective function of the general form:
\begin{equation} \label{eq:task_obj}
\min_\theta \mathbb{E}_{x \in {\mathbb{X}}} [\mathcal{L}(E_\theta(x)) + Reg(\theta)] \,.
\end{equation}
Here, $\mathbb{E}_{\mathbb{X}} $ denotes the expectation over the training distribution, and $Reg(\theta)$ encodes additional constraints on the weights $\theta$. As previously discussed, we argue that the learned weights $\theta^*$ are not only a function of ${\mathbb{X}}$ and of the specific loss appearing in Equation~\ref{eq:task_obj}, but in practice they are also affected by the optimization process used to train the network due to weight initialization, data shuffling, hyperparameters, and other stochastic factors.
We denote these factors collectively by $\phi$. In particular, as shown in Figure \ref{fig:latent-rotation}, changing these factors induces a transformation $T$ over the latent space, i.e., $\phi \rightarrow \phi' $ implies $E_\theta( {\bm{x}}^{(i)}) \rightarrow T E_\theta( {\bm{x}}^{(i)}) $. We make the core assumption that $T$ is an \emph{isometric} transformation, preserving the distances between elements of the latent space, namely $d({\bm{e}}_{{\bm{x}}^{(i)}},{\bm{e}}_{{\bm{x}}^{(j)}}) = d(T {\bm{e}}_{{\bm{x}}^{(i)}},T {\bm{e}}_{{\bm{x}}^{(j)}})$ for every $({\bm{x}}^{(i)},{\bm{x}}^{(j)}) \in {\mathbb{X}}$.
While the isometry assumption might seem too restrictive, in practice it arises in several real scenarios as we show in the sequel.
\subsection{Relative Representations}\label{sec:relative-representations}
\looseness=-1To build our representation, we start by selecting a subset ${\mathbb{A}}$ of the training data ${\mathbb{X}}$, which we denote as \newterm{anchor} samples.
Every sample in the training distribution will be represented with respect to the embedded anchors
${\bm{e}}_{{\bm{a}}^{(j)}} = E({\bm{a}}^{(j)}) \text{ with } {\bm{a}}^{(j)} \in {\mathbb{A}}$.
As a measure capturing the relation between the anchors and the other samples, we consider a generic similarity function $sim:\mathbb{R}^d \times \mathbb{R}^d \rightarrow \mathbb{R}$, yielding a scalar score $r$ between two \enc{abs}{reps} $r = sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{x}}^{(j)}})$.
Given the anchors ${\mathbb{A}}$ in an arbitrary ordering $a^{(1)}, \dots, a^{(|{\mathbb{A}}|)}$, we define the \newterm{\enc{rel}{rep}} of ${\bm{x}}^{(i)} \in {\mathbb{X}}$ as:
\begin{align}
{\bm{r}}_{{\bm{x}}^{(i)}} =
(
sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(1)}}),
sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(2)}}),
\dots,
sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(|{\mathbb{A}}|)}})
) \,.
\end{align}
\Cref{fig:relative-distances} illustrates the key differences between \enc{abs}{} and \enc{rel}{} representations.
\begin{figure}[h]
\centering
\begin{overpic}[trim=0cm 1.7cm 0cm 1.8cm,clip,width=0.9\linewidth, ]{figures/relative-distances/teaser.pdf}
\put(35, 8){\huge $\Rightarrow$}
\put(53, 8){\huge $\Rightarrow$}
\end{overpic}
\caption{{\em Left}: Three anchors (colored stars) are selected on the data manifold; given a point on the manifold (blue dot), we compute its similarity w.r.t. the three anchors, yielding a vector of dimensionality 3 (middle). {\em Right}: Each dimension is treated as coefficients in a coordinate system defined by the anchors. Anchors are orthogonal in this example only for visualization purposes.}
\label{fig:relative-distances}
\end{figure}
\vspace{1ex}\noindent\textbf{Choice of the anchors.}
Anchors directly affect the expressivity of the relative representation space, and are related to the task at hand. For example, in a classification task, we should sample anchors from each class in the training set, in order to well represent each data sample in ${\mathbb{X}}$.
One case of interest arises when the data comes from different domains ${\mathbb{X}}$, ${\mathbb{Y}}$, and we are given a partial correspondence $\Gamma: P_{\mathbb{X}} \mapsto P_{\mathbb{Y}} $ mapping from a subset of ${\mathbb{X}}$ to a subset of ${\mathbb{Y}}$. In this case, we can sample anchors $\mathbb{A}_{\mathbb{X}} \subseteq P_{\mathbb{X}}$ and obtain corresponding anchors on the other domain directly as $\Gamma (\mathbb{A})$. We refer to these as \emph{parallel anchors}. We show an example of parallel anchors in Section~\ref{sec:nlp-app}, where ${\mathbb{X}}$ and ${\mathbb{Y}}$ represent Amazon reviews written in two different languages.
The choice of the anchors is not restricted to elements in the training distribution. Given an encoder pre-trained on a fixed training distribution, we can pick elements from a set $ \tilde {\mathbb{A}} $ that is out-of-domain w.r.t. ${\mathbb{X}}$, and build the relative representations on top of $\tilde{\mathbb{A}}$. We refer to these as \emph{OOD anchors} and exploit them, e.g., to solve domain adaptation tasks where we do not have access to a correspondence, and have scarce data labels. We refer again to the \Cref{sec:nlp-app,sec:cv-app} for real-world examples.
\vspace{1ex}\noindent\textbf{Isometry invariance.}
In this work, we choose the cosine similarity as the similarity function due to the properties it induces on the relative representation.
The cosine similarity $S_C$ is the dot product of unit vectors, corresponding to the cosine of the angle $\theta$ between the two:
\begin{align}
S_C({\bm{a}}, {\bm{b}}) = \frac{{\bm{a}} {\bm{b}}}{|| {\bm{a}} || || {\bm{b}} || } = \cos\theta \,.
\end{align}
Importantly, $\cos\theta$ does not change if we apply the same orthogonal transformation ${\bm{T}}$ to two vectors ${\bm{a}}$ and ${\bm{b}}$,
i.e., the cosine similarity is invariant to rotations, reflections, and rescaling.
While this is not true for translations, NNs commonly employ normalization techniques (e.g., InstanceNorm \citep{instance-norm}) to center the latent spaces around zero. Under this assumption, cosine similarity guarantees a \enc{rel}{rep} ${\bm{r}}_{{\bm{x}}^{(i)}}$ invariant to isometric transformations.
This means we have the freedom to change the embedding function $E_\theta$ with any other function $\tilde{E}$ that produces different but isometric representations, i.e.:
\begin{align}
[
S_C({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(1)}}),
\dots,
S_C({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(|{\mathbb{A}}|)}}]=[
S_C( \tilde {\bm{e}}_{{\bm{x}}^{(i)}}, \tilde {\bm{e}}_{{\bm{a}}^{(1)}}),
\dots,
S_C( \tilde {\bm{e}}_{{\bm{x}}^{(i)}}, \tilde {\bm{e}}_{{\bm{a}}^{(|{\mathbb{A}}|)}})
] \,,
\end{align}
where $\tilde{{\bm{e}}}_{{\bm{x}}^{(i)}} = \tilde{E}({\bm{x}}^{(i)}) = {\bm{T}} E({\bm{x}}^{(i)})$ and ${\bm{T}}$ is an arbitrary isometric transformation.
A practical application of this invariance is the possibility of comparing latent spaces across multiple trainings, and re-using models as demonstrated in Sections~\ref{sec:latent-communication} and ~\ref{sec:model-reusability}.
\vspace{1ex}\noindent\textbf{Quasi-isometry invariance.}\label{sec:quantization}
We remark that other choices of similarity function can be made to enforce different invariances into the representation. For example, to relax the isometry assumption, one may impose invariance to non-isometric deformations with bounded distortion.
We did not find this to be a necessity in our experiments, as typically NNs that generalize sufficiently well can handle small perturbations of the input. Nevertheless, this invariance can be enforced by design by using vector quantization algorithms. We preliminarily explore this approach in Figure~\ref{fig:latent-rotation-comparison-quantization}, leaving further exploration to future work.
\section{Latent Space Communication}\label{sec:latent-communication}
In this section, we demonstrate how our relative representations can effectively be used to produce latent spaces that are stable under a variety of factors. Our main question is the following: Given two different learning models that are trained separately on different data, can we compare their latent embeddings? In asking this, we assume that the two models are trained on a similar phenomenon, e.g., on two different samplings of the English language or on two different modalities.
\looseness=-1We answer in the positive, showing the gained invariance enables effective communication between different, but semantically equivalent latent spaces. %
In particular, we analyze how different word embedding spaces, once projected onto \enc{rel}{reps}, are intrinsically the same (\Cref{sec:word-embeddings}); we then show how the similarity between the relative counterparts of two or more embedding spaces is a surprisingly good predictor of model performance (\Cref{sec:manifold-performance}); finally, we confirm that relative representations in the training phase are not detrimental to performance (\Cref{sec:abs-vs-rel}).
\subsection{Word Embeddings}\label{sec:word-embeddings}
\vspace{1ex}\noindent\textbf{Experimental setting.}
We select two different word embeddings on the English language, namely \texttt{FastText} \citep{bojanowski2016enriching} and \texttt{Word2Vec} \citep{word2vec}. Both models are pre-trained on different data, but partly share a vocabulary from which we extract $\approx20$K words. For each embedding space, we convert their absolute embeddings to corresponding relative representations, using $300$ randomly drawn parallel anchors.
In Table~\ref{fig:latent-rotation-comparison} (left) we show the original embeddings and the resulting relative ones. We measure the degree of similarity in two ways:
(i) {\em Jaccard}: for each word vector in the source space, we compute the Jaccard similarity between its top-k source neighbors and its top-k target neighbors; (ii) {\em Mean Reciprocal Rank}: for each embedded word in the source space, we compute its (reciprocal) ranking among the top-k neighbors in the target.
\begin{table}[h]
\noindent\makebox[\textwidth][c]{%
\begin{minipage}{.30\textwidth}
\begin{overpic}[trim=-1cm 1cm -0.5cm 0cm,width=1\linewidth]{figures/latent-rotation-comparison/word-embeddings-spaces-no-quant.pdf}
%
\put(11, 93){\texttt{FastText}}
\put(55, 93){\texttt{Word2Vec}}
\put(-1, 53){\rotatebox{90}{\small Absolute}}
\put(-1, 8){\rotatebox{90}{\small Relative}}
\end{overpic}
\end{minipage}
\hspace{0.5cm}
\begin{minipage}{.55\textwidth}
\begin{tabular}{lcccc}
& Source & Target & \multicolumn{1}{c}{Jaccard ↑} & \multicolumn{1}{c}{MRR ↑}
\\ \midrule
\multirow{4}{*}{{\STAB{\rotatebox[origin=c]{90}{\enc{Abs}{}}}}} & \texttt{{FastText}} & \texttt{{FastText}} & $1.00$ & $1.00$ \\%& $0.38$ \\
& \texttt{{FastText}} & \texttt{{Word2Vec}} & $0.00$ & $0.00$ \\%& $0.00$ \\
& \texttt{{Word2Vec}} & \texttt{{FastText}} & $0.00$ & $0.00$ \\%& $0.00$ \\
& \texttt{{Word2Vec}} & \texttt{{Word2Vec}} & $1.00$ & $1.00$ \\%& $0.37$ \\
\cmidrule(lr){1-5}
\multirow{4}{*}{{\STAB{\rotatebox[origin=c]{90}{\enc{Rel}{}}}}} & \texttt{{FastText}} & \texttt{{FastText}} & $1.00$ & $1.00$ \\%& $0.33$ \\
& \texttt{{FastText}} & \texttt{{Word2Vec}} & $\mathbf{0.34}$ & $\mathbf{0.94}$ \\%& $0.28$ \\
& \texttt{{Word2Vec}} & \texttt{{FastText}} & $\mathbf{0.39}$ & $\mathbf{0.98}$ \\%& $0.30$ \\
& \texttt{{Word2Vec}} & \texttt{{Word2Vec}} & $1.00$ & $1.00$ \\%& $0.34$ \\
\bottomrule
\end{tabular}
\vspace{0.4cm}
\end{minipage}
}
\caption{Qualitative \emph{(left)} and quantitative \emph{(right)} comparison of English word embeddings using \protect\enc{abs}{} and \protect\enc{rel}{reps}. All metrics are calculated with $K=10$.
}
\label{fig:latent-rotation-comparison}
\end{table}
\vspace{1ex}\noindent\textbf{Result analysis.}
\Cref{fig:latent-rotation-comparison} \emph{(left)} highlights clusters of semantically similar words and shows that the absolute representations are incoherent across the two latent spaces, while the relative embeddings are highly similar.
The average Jaccard distance reported in \Cref{fig:latent-rotation-comparison} \emph{(right)}, says that the neighborhoods of the relative representations are matched exactly 34\% of the time in one direction, and 39\% of the time in the other one (the missing 61\% is due to semantic differences, that are not taken into account by the discrete nature of the Jaccard metric). By contrast, the absolute embeddings are never matched exactly (Jaccard score equal to zero); for a match to happen, it would mean that the \texttt{FastText} and \texttt{Word2Vec} embeddings of a given English word are almost the same, which is highly unlikely. The performance gap is even more evident in terms of MRR, which is close to a perfect score for the relative representations.
Overall, these results show that relative representations are preserved across different word embedding models, validating our isometry assumptions.
\subsection{Latent distance as a performance proxy}\label{sec:manifold-performance}
\looseness=-1\vspace{1ex}\noindent\textbf{Experimental setting.}
In this experiment, we consider a node classification task on the \cora{} graph dataset~\citep{cora}. We first train a \emph{reference} model that achieves good accuracy on a validation set. Then, we train $\approx2000$ models with various combinations of seed, number of epochs, number of layers, dropout probability, activation functions, optimizer type, learning rate or type of graph embedder (\Cref{tab:manifold-performance-parameters}).
All the models are classically trained using absolute representations, which are converted to relative post-training
by projecting the embeddings onto $300$ randomly drawn but fixed anchors. For each model, we measure its classification accuracy and compute the similarity of its space with the reference one. This similarity is computed as the average cosine similarity between the node embeddings produced by a given model and the corresponding embeddings in the reference.
\begin{figure}[h]
\centering
\begin{overpic}[trim=-.5cm -.4cm -.5cm 0cm,clip,width=.73\linewidth]{figures/data-manifold/correlation_subfigure.pdf}
\put(0, 14){\rotatebox{90}{Performance}}
\put(19, 0){Similarity}
\put(49, 14){\rotatebox{90}{\color{pltblue}Performance}}
\put(71, 0){Epochs}
\put(97.5, 16){\rotatebox{90}{\color{pltorange}Similarity}}
\end{overpic}
\caption{Graph node classification task on \cora. \emph{Left:} Correlation between the performance of $\approx2000$ models and the similarity of their latent spaces with respect to a well-performing reference model.
\emph{Right:} The same correlation plotted over time. The mean Pearson correlation over all models is $0.955$, after filtering out the models having best validation accuracy below $0.5$.}
\label{fig:data-manifold}
\end{figure}
\vspace{1ex}\noindent\textbf{Result analysis.}
The scatter plot in \Cref{fig:data-manifold} \emph{(left)} shows that better-performing models tend to be the ones with the latent spaces most similar to the reference. The performance-similarity correlation also holds over time, as shown in \Cref{fig:data-manifold} \emph{(right)}.
Additional correlation examples are in \Cref{fig:correlation-grid}.
Interestingly, this metric is differentiable, enabling an explicit supervision signal on the latent space, which does not require labeled data and could be readily exploited in a teacher-student framework.
Overall, these results suggest that the similarity between the relative representations of latent spaces is a remarkably good proxy to evaluate model performance.
\subsection{Training with Absolute vs. Relative representations}\label{sec:abs-vs-rel}
\vspace{1ex}\noindent\textbf{Experimental setting.}
Finally, we compare architectures that do or do not employ the \enc{rel}{rep} while training. In these experiments, the models vary slightly according to the dataset; however, the relative and absolute versions are always comparable in terms of architecture, number of learnable parameters and hyperparameters. We refer to the supplementary material and the open-source code for further details on their implementation.
In this section we consider classification tasks on several datasets, spanning the image domain \citep{Lecun1998-we,Xiao2017-er,Krizhevsky2009-hv} and the graph domain \citep{Planetoid}.
\begin{table}[h]
\caption{Performance comparison between \protect\enc{rel}{} and \protect\enc{abs}{reps} on several image and graph datasets. The metric is the classification weighted F1 score ($\pm$ std), over 6 seeds.}
\label{tab:abs-rel-performance-comparison}
\tiny
\begin{center}
\begin{tabular}{ccccccccccccc}
\toprule
&\multicolumn{4}{c}{\bf Image Classification} & \multicolumn{3}{c}{\bf Graph Node Classification} \\
\cmidrule(lr){2-5}\cmidrule(lr){6-8}
\multicolumn{1}{c}{}
& \multicolumn{1}{c}{\bf \mnist{}}
& \multicolumn{1}{c}{\bf \fmnistshort{}}
& \multicolumn{1}{c}{\bf \cifart{}}
& \multicolumn{1}{c}{\bf \cifarh{}}
& \multicolumn{1}{c}{\bf \cora{}}
& \multicolumn{1}{c}{\bf \citeseer{}}
& \multicolumn{1}{c}{\bf \pubmed{}}
\\ \midrule
\textbf{\protect\enc{Rel}{}} & $97.91 \pm 0.07$ & $90.19 \pm 0.27$ & $87.70 \pm 0.09$ & $66.72 \pm 0.35$ & $0.89 \pm 0.02$ & $0.77 \pm 0.03$ & $0.91 \pm 0.01$ \\
\textbf{\protect\enc{Abs}{}} & $97.95 \pm 0.10$ & $90.32 \pm 0.21$ & $87.85 \pm 0.06$ & $68.88 \pm 0.14$ & $0.90 \pm 0.01$ & $0.78 \pm 0.03$ & $0.91 \pm 0.01$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\vspace{1ex}\noindent\textbf{Result analysis.}
The results, reported in \Cref{tab:abs-rel-performance-comparison}, show that \enc{rel}{reps}, when used at training time, are not detrimental to performance in general. This is further shown in
\Cref{%
tab:cross-train-performance-comparison,%
tab:cross-model-monolingual,%
tab:cifar100-imagenet,%
tab:multilingual-full-coarse-grained,%
tab:multilingual-en-coarse-grained,%
tab:multilingual-full-fine-grained,%
tab:xmlr-multilingual-full-fine-grained,%
tab:cifar100-fine},
where a subset of the results compares the \enc{abs}{} and \enc{rel}{} representations on a variety of domains, datasets and tasks.
Overall, these results show that relative representations are effective when involved in end-to-end training, without significant performance drops.
\section{Zero-Shot Model Stitching}\label{sec:model-reusability}
In this section, we illustrate how the latent space communication demonstrated in \Cref{sec:latent-communication} enables zero-shot interoperability of pre-trained neural components. In previous works, such as \cite{Lenc2014-gy,Bansal2021-oj}, stitching layers are \emph{trainable} linear projections that allow swapping parts of different networks.
Instead, relative representations unlock the possibility of \emph{zero-shot stitching} different neural components, treating them as frozen black-box modules.
We define a generic \emph{stitched model} as the composition of an encoder, that embeds data, plus a decoder specialized in a downstream task (classification, reconstruction).
The stitching operation is always performed without training or fine-tuning, in a zero-shot fashion.
Hereafter,
we showcase stitching capabilities across combinations of
different stochasticity sources (\Cref{fig:encoder-decoder-swap,tab:cross-train-performance-comparison}),
neural architectures (\Cref{tab:cross-model-monolingual,tab:multilingual-en-coarse-grained})
or datasets (\Cref{tab:cifar100-imagenet}).
Finally, we present strong real-world applications in NLP (\Cref{sec:nlp-app}) and CV (\Cref{sec:cv-app}), e.g. zero-shot predictions on novel languages.
Additional implementation details are given in the supplementary materials.
\begin{figure}[h]
\centering
\begin{overpic}[trim=-0.27cm -0.15cm -.3cm 0cm,clip,width=1\linewidth]{figures/encoder-decoder-swap/source.pdf}
%
\put(0, 2.75){\rotatebox{90}{S}}
\end{overpic}
\begin{overpic}[trim=-0.27cm 0cm -.3cm 0cm,clip,width=1\linewidth]{figures/encoder-decoder-swap/stitching.pdf}
%
\put(0, 31){\rotatebox{90}{\small \textit{Abs.}}}
\put(0, 21.75){\rotatebox{90}{\small \textit{Rel.}}}
\put(0, 12.25){\rotatebox{90}{\small \textit{Abs.}}}
\put(0, 3){\rotatebox{90}{\small \textit{Rel.}}}
%
\put(98.5, 26.5){\rotatebox{90}{\small \textbf{AE}}}
\put(98.5, 7.25){\rotatebox{90}{\small \textbf{VAE}}}
\put(98, 19.25){\color{black}\line(0,1){17.75}}
\put(98, .75){\color{black}\line(0,1){17.5}}
\end{overpic}
\caption{Reconstruction examples. Each column is a different image, row pairs are different architectures. In each pair, we first report the non-stitched reconstructions, then the stitched ones.}
\label{fig:encoder-decoder-swap}
\end{figure}
\subsection{Image Reconstruction}\label{sec:model-reusability-ae}
\looseness=-1\vspace{1ex}\noindent\textbf{Experimental setting.}
We perform zero-shot stitching with AEs and VAEs trained end-to-end on several image datasets.
For each combination of model and dataset, we perform five trainings, identical but with different seeds. Thus, we zero-shot stitch together the resulting encoders and decoders. The stitching operation simply consists of composing an encoder with a decoder that is trained on a different seed.
\vspace{1ex}\noindent\textbf{Result analysis.}
In \Cref{fig:encoder-decoder-swap} the stitched models that employ absolute representations (\emph{Abs.}) produce erroneous predictions, since the latent spaces obtained from distinct trainings are incompatible.
Interestingly, although the absolute VAE does not produce compatible latent spaces, it is regularized, thus all embeddings produced by the encoders correspond to wrong but semantically meaningful reconstructions.
Relative representations (\emph{Rel.}) exhibit almost indistinguishable reconstructions between the models trained end-to-end and the stitched ones. Quantitative results are in \Cref{tab:cross-train-performance-comparison}.
Overall, these results prove that relative representations are invariant to training stochasticity.
\begin{table}[h]
\caption{Stitching performance.
The MSE ($\pm$ std) between the ground truth ${\mathbb{X}}$ and the reconstructions is computed over 5 different seeds. Stitching with our relative representations yields an error up to two orders of magnitude less than the absolute counterpart.}
\label{tab:cross-train-performance-comparison}
\scriptsize
\begin{center}
\begin{tabular}{cccrrrr|r}
\toprule
& \multicolumn{1}{c}{\bf }
& \multicolumn{1}{c}{\bf }
& \multicolumn{1}{c}{\bf \mnist{}}
& \multicolumn{1}{c}{\bf \fmnistshort{}}
& \multicolumn{1}{c}{\bf \cifart{}}
& \multicolumn{1}{c}{\bf \cifarh{}}
& \multicolumn{1}{c}{\textbf{MSE ↓}}\\
\midrule
\multirow{4}{*}{\STAB{\rotatebox[origin=c]{90}{\textbf{AE}}}} & \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{\textit{Abs.}}}} & Non-Stitch. &
$0.66 \pm 0.02$ & $1.57 \pm 0.03$ & $1.94 \pm 0.08$ & $2.13 \pm 0.08$ & $1.58 \pm 0.05$ \\
& & Stitch. &
$97.79 \pm 2.48$ & $120.54 \pm 6.81$ & $86.74 \pm 4.37$ & $97.17 \pm 3.50$ & $100.56 \pm 4.29$ \\[1ex]
& \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{\textit{Rel.}}}} & Non-Stitch. &
$1.18 \pm 0.02$ & $3.59 \pm 0.04$ & $2.83 \pm 0.13$ & $3.50 \pm 0.08$ & $2.78 \pm 0.07$ \\
& & Stitch. &
$2.83 \pm 0.20$ & $6.37 \pm 0.29$ & $5.39 \pm 1.18$ & $18.03 \pm 12.46$ & $8.16 \pm 3.53$ \\
\midrule
\multirow{4}{*}{\STAB{\rotatebox[origin=c]{90}{\textbf{VAE}}}} & \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{\textit{Abs.}}}} & Non-Stitch. &
$1.31 \pm 0.04$ & $4.38 \pm 0.03$ & $2.68 \pm 0.06$ & $3.00 \pm 0.03$ & $2.84 \pm 0.04$ \\
& & Stitch. &
$98.51 \pm 1.49$ & $118.96 \pm 2.96$ & $69.02 \pm 1.54$ & $78.57 \pm 1.88$ & $91.27 \pm 1.97$\\[1ex]
& \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{\textit{Rel.}}}} & Non-Stitch. &
$2.97 \pm 0.14$ & $6.81 \pm 0.06$ & $5.18 \pm 0.22$ & $5.93 \pm 0.14$ & $5.22 \pm 0.14$ \\
& & Stitch. &
$13.43 \pm 6.79$ & $24.03 \pm 13.15$ & $11.20 \pm 3.15$ & $11.23 \pm 2.38$ & $14.97 \pm 6.37$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\subsection{Text Classification}\label{sec:nlp-app}
In this Section, we show practical examples of the use of parallel anchors (Sec ~\ref{sec:relative-representations}).
\begin{table}[h]
\scriptsize
\centering
\caption{
Cross-lingual stitching performance comparison.
The table reports the mean weighted F1 ($\pm$ std) and MAE on \amazon{} coarse-grained, across 5 seeds.}
\label{tab:multilingual-en-coarse-grained}
\begin{tabular}{clcccccc}
\toprule
& & \multicolumn{2}{c}{\enc{Abs}{}} & \multicolumn{4}{c}{\enc{Rel}{}} \\
\cmidrule(lr){3-4}
\cmidrule(l){5-8}
& & \multicolumn{2}{c}{} & \multicolumn{2}{c}{Translated} & \multicolumn{2}{c}{Wikipedia} \\
\cmidrule(lr){5-6}
\cmidrule(l){7-8}
\stdecoder & \stencoder & FScore & MAE & FScore & MAE & FScore & MAE \\
\midrule
\multirow{4}{*}{en} & en & $91.54 \pm 0.58$ & $0.08 \pm 0.01$ & $90.06 \pm 0.60$ & $0.10 \pm 0.01$ & $90.45 \pm 0.52$ & $0.10 \pm 0.01$ \\
& es & $43.67 \pm 1.09$ & $0.56 \pm 0.01$ & $82.78 \pm 0.81$ & $0.17 \pm 0.01$ & $78.53 \pm 0.30$ & $0.21 \pm 0.00$ \\
& fr & $54.41 \pm 1.61$ & $0.45 \pm 0.02$ & $78.49 \pm 0.66$ & $0.21 \pm 0.01$ & $70.41 \pm 0.57$ & $0.29 \pm 0.01$ \\
& ja & $48.72 \pm 0.90$ & $0.51 \pm 0.01$ & $65.72 \pm 0.55$ & $0.34 \pm 0.01$ & $66.31 \pm 0.80$ & $0.34 \pm 0.01$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[h]
\small
\caption{Cross-architecture stitching performance comparison. The table reports the mean weighted F1 ($\pm$ std) for each dataset, across 5 different seeds.}
\label{tab:cross-model-monolingual}
\begin{center}
\begin{tabular}{llcccc}
\toprule
& & \multicolumn{1}{c}{\trec{}} & \multicolumn{1}{c}{\dbpedia{}} & \multicolumn{2}{c}{\amazon} \\ \cmidrule{5-6}
& & & & \multicolumn{1}{c}{\textit{Coarse}} & {\textit{Fine}} \\
\midrule
\multirow{2}{*}{{\STAB{\rotatebox[origin=c]{90}{Abs.}}}} & Non-Stitch & $91.70 \pm 1.39$ & $98.62 \pm 0.58$ & $87.81 \pm 1.58$ & $55.35 \pm 3.19$ \\
& Stitch & $21.49 \pm 3.64$ & $6.96 \pm 1.46$ & $49.58 \pm 2.95$ & $19.01 \pm 2.04$ \\
\midrule
\multirow{2}{*}{{\STAB{\rotatebox[origin=c]{90}{Rel.}}}} & Non-Stitch & $88.08 \pm 1.37$ & $97.42 \pm 2.05$ & $85.08 \pm 1.93$ & $48.92 \pm 3.57$ \\
& Stitch & $75.89 \pm 5.38$ & $80.47 \pm 21.14$ & $72.37 \pm 7.32$ & $33.24 \pm 7.21$ \\
\bottomrule
\end{tabular}
\end{center}
\end{table}
\vspace{1ex}\noindent\textbf{Experimental setting.}
We consider two different text classification settings.
\emph{Cross-lingual}: given a review predict the associated star rating, done on multi-lingual data from the \amazon{} dataset \citep{amazon-reviews}.
Following the original paper, we work on a binarized version of the task, with FScore and MAE as metrics. In the supplementary material, we report results on the fine-grained formulation.
We adopt four different pre-trained language-specific RoBERTa transformers \citep{roberta} and evaluate their zero-shot stitching performance on languages never seen by the classifier.
We use parallel anchors in two modalities: i) \emph{Translated}: consider English reviews translated\footnote{We used the \texttt{=GOOGLETRANSLATE} function available in Google Sheets.} into the other languages; ii) \emph{Wikipedia}: adopt an external corpus, WikiMatrix \citep{wikimatrix}, providing parallel sentences extracted from Wikipedia.
\emph{Cross-architecture}: assessed on three different datasets: \trec{} (coarse) \citep{trec}, \dbpedia{} \citep{dbpedia}, \amazon{} (English split). We adopt two different pre-trained BERT \citep{bert} transformers (cased and uncased version), ELECTRA \citep{electra} and RoBERTa.
\vspace{1ex}\noindent\textbf{Result analysis.}
With these results, we show for the first time that it is possible to learn to solve a downstream task on a specific language or transformer, and perform predictions on another.
Tables~\ref{tab:multilingual-en-coarse-grained} and~\ref{tab:cross-model-monolingual} show how the \enc{rel}{reps} allow reusing a trained classification head across different languages or encoder architectures.
Models stitched with absolute representations show performance that is comparable to random guessing across the board, proving that relative representations are a key element for the success of this kind of zero-shot stitching. Indeed, one can only stitch together models with absolute representations if they happen to have the same latent dimensionality.
Moreover, the anchor selection results in \Cref{tab:multilingual-en-coarse-grained} highlight the robustness that relative representations have on the choice of anchors, even when they are noisy (\emph{Translated} case), or their distribution differs from the one of the downstream task (\emph{Wikipedia} case), as long as their encoding can be handled correctly by the encoder. In our case, the encoder is pre-trained to represent a variety of texts in a specific language, thus, even if WikiMatrix has a completely different domain from \amazon{}, the transformer still computes a meaningful and comparable representation with those of the reviews.
\looseness=-1We report in \Cref{tab:multilingual-full-coarse-grained,tab:multilingual-full-fine-grained} complete results on all languages combination, and in \Cref{tab:xmlr-multilingual-full-fine-grained} the performance obtained by a multi-lingual transformer.
To the best of our knowledge, it is the only alternative to obtaining compatible representations across languages.
According to these results, relative representations show invariance to different architectures and data distribution shifts (e.g., different train languages).
\subsection{Image Classification}\label{sec:cv-app}
In this Section, we show practical examples of the use of OOD anchors (Sec ~\ref{sec:relative-representations}).
\begin{table}[h]
\scriptsize
\centering
\caption{
Stitching performance comparison with different encoding techniques. The table reports the mean weighted F1 ($\pm$ std) on \cifarh{} coarse-grained and \imagenet{}, across 5 seeds.}
\label{tab:cifar100-imagenet}
\begin{tabular}{llllll}
\toprule
& & \multicolumn{2}{c}{\cifarh{}} & \multicolumn{2}{c}{\imagenet{}} \\\cmidrule(lr){3-4}\cmidrule(lr){5-6}
\stdecoder & \stencoder & \multicolumn{1}{c}{\enc{Abs}{}} & \multicolumn{1}{c}{\enc{Rel}{}} & \multicolumn{1}{c}{\enc{Abs}{}} & \multicolumn{1}{c}{\enc{Rel}{}} \\
\midrule
\multirow{4}{*}{rexnet-100} & rexnet-100 & $82.06 \pm 0.15$ & $80.22 \pm 0.28$ & $73.78 \pm 0.29$ & $72.61 \pm 0.16$ \\
& vit-base-patch16-224 & \multicolumn{1}{c}{-} & $54.98 \pm 0.44$ & \multicolumn{1}{c}{-} & $37.39 \pm 0.36$ \\
& vit-base-resnet50-384 & \multicolumn{1}{c}{-} & $53.33 \pm 0.37$ & \multicolumn{1}{c}{-} & $42.36 \pm 0.36$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $59.82 \pm 0.32$ & \multicolumn{1}{c}{-} & $43.75 \pm 0.27$ \\
\cmidrule{1-6}
\multirow{4}{*}{vit-base-patch16-224} & rexnet-100 & \multicolumn{1}{c}{-} & $76.81 \pm 0.49$ & \multicolumn{1}{c}{-} & $30.78 \pm 0.81$ \\
& vit-base-patch16-224 & $93.15 \pm 0.05$ & $91.94 \pm 0.10$ & $80.91 \pm 0.29$ & $78.86 \pm 0.33$ \\
& vit-base-resnet50-384 & $6.21 \pm 0.33$ & $81.42 \pm 0.38$ & $0.07 \pm 0.05$ & $44.72 \pm 0.57$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $84.29 \pm 0.86$ & \multicolumn{1}{c}{-} & $48.31 \pm 0.72$ \\
\cmidrule{1-6}
\multirow{4}{*}{vit-base-resnet50-384} & rexnet-100 & \multicolumn{1}{c}{-} & $79.79 \pm 0.43$ & \multicolumn{1}{c}{-} & $53.46 \pm 0.68$ \\
& vit-base-patch16-224 & $4.69 \pm 0.07$ & $84.46 \pm 0.19$ & $0.08 \pm 0.04$ & $62.21 \pm 0.54$ \\
& vit-base-resnet50-384 & $91.41 \pm 0.09$ & $90.77 \pm 0.16$ & $82.55 \pm 0.30$ & $81.88 \pm 0.16$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $84.66 \pm 0.16$ & \multicolumn{1}{c}{-} & $61.32 \pm 0.36$ \\
\cmidrule{1-6}
\multirow{4}{*}{vit-small-patch16-224} & rexnet-100 & \multicolumn{1}{c}{-} & $75.35 \pm 0.41$ & \multicolumn{1}{c}{-} & $37.58 \pm 0.44$ \\
& vit-base-patch16-224 & \multicolumn{1}{c}{-} & $81.23 \pm 0.31$ & \multicolumn{1}{c}{-} & $50.08 \pm 0.63$ \\
& vit-base-resnet50-384 & \multicolumn{1}{c}{-} & $78.35 \pm 0.69$ & \multicolumn{1}{c}{-} & $45.45 \pm 1.41$ \\
& vit-small-patch16-224 & $90.07 \pm 0.19$ & $88.85 \pm 0.44$ & $77.73 \pm 0.41$ & $76.36 \pm 0.40$ \\
\bottomrule
\end{tabular}
\end{table}
\vspace{1ex}\noindent\textbf{Experimental setting.}
We consider a classification task on \imagenet{} and \cifarh{} with coarse labels (20), and 4 different pre-trained image encoders:
three variants of the ViT transformer \citep{vit} and RexNet \citep{rexnet}.
\vspace{1ex}\noindent\textbf{Result analysis.}
The results in \Cref{tab:cifar100-imagenet} highlight how the relative representations allow to stitch modules with different encoding dimensionality, since the encodings are converted to a representation with size equal to the number of anchors. Further, the results demonstrate the ability to generalize and perform zero-shot stitching on \cifarh{}, although that data was never seen by the encoder since it is a frozen transformer trained on \imagenet{}.
Interestingly, \texttt{rexnet-100} is the only transformer whose latent dimensionality is higher than the number of anchors, and the biggest drop in stitching performance happens when the decoder is trained on it. This suggests the number of anchors is an important hyperparameter; we refer to \Cref{fig:anchor-num} for a deeper analysis.
Overall, these results prove that relative representations can bridge general-purpose encoders and pre-trained task-specific decoders.
\section{Conclusion}
\looseness=-1In this work, we introduced the concept of relative representations to enable zero-shot latent space communication, with several practical consequences as showcased in our discussion and experiments.
Our work proves that a latent semantic correspondence between data domains, when present, can be exploited through a simple shift of representation, without resorting to sophisticated processing or heavy training.
\looseness=-1\vspace{1ex}\noindent\textbf{Limitations and future work.}
Our work is open to several follow-up directions.
While in this paper we considered the cosine similarity, different functions can enforce additional invariances in the relative representation. The study of {invariant latent spaces} as a general direction has the potential to lead to further impact; in \Cref{fig:latent-rotation-comparison-quantization} we showed preliminary results of this possibility, obtaining representations that are invariant to near-isometries with {\em guaranteed} bounded distortion via vector quantization.
Another interesting line of research to improve the representation expressivity would be to estimate \emph{geodesic} distances over the data manifold instead of adopting Euclidean approximations.
Similarly, we believe that the connections between the composition of the anchors set ${\mathbb{A}}$ and the expressivity of relative representations demands additional research.
For example, the training cost is directly affected by the number and update frequency of the anchors.
Finally, the stitching procedure may be extended to multiple layers, promoting {reusable network components}.
\subsubsection*{Reproducibility Statement}
We describe in detail the relative representation computation in \Cref{sec:relative-representations}.
We describe the experimental settings for the various scenarios, and refer to the supplementary material for further implementation details (\Cref{sec:implementatin-details}).
Moreover, we release a well-documented and modular codebase, with the relative representation layer being implemented as a stand-alone PyTorch module. All the checkpoints used in the experiments are versioned with DVC \citep{dvc} to easily reproduce all the figures and tables. The stand-alone module allows the integration of the relative representations in any existing neural network effortlessly.
\subsubsection*{Acknowledgments}
This work is supported by the ERC Starting Grant No. 802554 (SPECGEO).
\subsection{Latent Space Isometries}\label{sec:high-dim-rotation}
In \Cref{fig:latent-rotation}, multiple trainings of the same two-dimensional AE produce isometric latent spaces; in \Cref{fig:latent-rotation-pca-proof} we show this property also holds on AEs with a high-dimensional bottleneck. In the first row, PCA is fitted indipendently in each column, and since the PCA transformation produces the same output everywhere the latent spaces are isometric. In the second row, PCA is fitted only on the first latent space; since in this case the PCA transformations produces different outputs, the latent spaces although isometric are extrinsically different.
\begin{figure}[h]
\centering
\begin{overpic}[trim=0cm 0cm 0cm 0cm,clip,width=\linewidth]{figures/pca-proof/pca-proof-row1.pdf}
%
\end{overpic}
\begin{overpic}[trim=0cm 0cm 0cm 0cm,clip,width=\linewidth]{figures/pca-proof/pca-proof-row2.pdf}
%
\end{overpic}
\caption{Latent spaces learned by distinct trainings of the same high-dimensional AE on the \mnist{} dataset. Each column is the latent space obtained by the AE with a different seed.
On the first row, the dimensionality reduction is performed through PCAs fitted independently on each latent space, meanwhile, on the second row PCA is fitted on the leftmost latent space and then applied to all of them.
}
\label{fig:latent-rotation-pca-proof}
\end{figure}
\subsection{Anchors analysis}\label{sec:anchor-num}
The cardinality of the anchors set ${\mathbb{A}}$ and the choice of specific anchors is crucial to the quality of the relative representations. At the extreme, selecting one single anchor or the same repeated data points for all anchors, will produce collapsed relative representations. We believe that additional research is required to obtain a better understanding on the optimal choice for ${\mathbb{A}}$. Questions like ``Are anchors set composed only by stopwords worse than the ones composed by meaningful and diverse words?'' require empirical evidence and could help revealing the semantics of the latent space. Indeed, each anchor is associated with a dimension in a relative representation; one could inspect the anchor data point to get a sense of the meaning of that latent dimension.
Below, we report a preliminary study on the performance sensitivity against the cardinality of the anchors set. In \Cref{fig:anchor-num} we report the performance on the node classification task on \cora{}, with a model trained end-to-end adopting the relative representations while training, and on image classification tasks on \cifarh{}, with a frozen encoder.
The performance improves monotonically as the number of anchors increase when the absolute representations are frozen \emph{(right)}. Differently, training models end-to-end proves to be more susceptible to model collapse and instabilities, as increasing the number of anchors does not always improve the performance \emph{(left)}. Further research on the relation between the absolute latent space dimensionality and the relative representation dimensionality (i.e., the number of anchors) is needed to clarify how the two quantities impact the performance, when training end-to-end or not.
\begin{figure}[h]
\centering
\begin{overpic}[trim=-.5cm -.4cm -.5cm 0cm,clip,width=0.9\linewidth]{figures/anchor-analysis/anchors_num_analysis.pdf}
%
\put(0, 15){\rotatebox{90}{Performance}}
\put(17, 0){Number of anchors}
\put(70, 0){Number of anchors}
\end{overpic}
\caption{Accuracy vs Number of anchors. Each point is a trained model. \textit{Left}: Trained embedder on Cora, node classification. \textit{Right}: Frozen transformer on Cifar100 coarse-grained, image classification. Left is less stable because the absolute embeddings are trained, and we are working on a domain that is less stable (graphs). Some collapsed examples are not visualized.}
\label{fig:anchor-num}
\end{figure}
\subsection{Quasi-Isometry Invariance with guaranteed bounds}\label{sec:anchor-quant}
In this section, we explore a slightly modified version of the similarity function adopted in the main paper.
The experimental setting is the same as in \Cref{sec:word-embeddings}. We want to measure the similarity between pairs of absolute embeddings and their relative counterparts. To get some kind of quantitative measure, we add a similarity score calculated as the pairwise cosine distance between the two embedding types, averaged. Therefore, a lower score indicates the spaces are more similar. On top of the standard relative representations, the ones computed with $sim=S_C$, here we try to improve the similarity measure with guaranteed robustness to quasi-isometries up to a desired distortion bound. In \Cref{fig:latent-rotation-comparison-quantization} we report preliminary results that adopt this technique: a vector-quantized similarity function produces relative representations which are more similar (they have a lower score). The vector-quantization is done through agglomerative clustering on the absolute embeddings at various thresholds $t$. We leave to future works the study of the trade-off between a guaranteed bound on the near-isometries invariance and the expressiveness of the resulting representations.
\begin{figure}[h]
\centering
\begin{overpic}[trim=-0.25cm 0cm -0.5cm 0cm,clip,width=1\linewidth]{figures/latent-rotation-comparison/word-embeddings-spaces.pdf}
%
\put(0, 24){\rotatebox{90}{\texttt{FastText}}}
\put(0, 5.5){\rotatebox{90}{\texttt{Word2Vec}}}
\put(12, 39){\small Abs.}
\put(31, 39){\small Rel.}
\put(44, 39){\small Rel. Qnt. $t=1$}
\put(61.5, 39){\small Rel. Qnt. $t=1.5$}
\put(81, 39){\small Rel. Qnt. $t=2$}
\put(0, -1){{\small Score ↓}}
\put(11.5, -1){{\small $3.30$}}
\put(30, -1){{\small $1.24$}}
\put(48, -1){{\small $1.19$}}
\put(66.5, -1){{\small $1.09$}}
\put(85, -1){{\small $1.02$}}
\end{overpic}
\caption{The \texttt{FastText} and \texttt{Word2Vec} embeddings of a subset of the English dictionary.
The score is the pairwise distance average between the two embedding types, thus a lower score indicates the spaces are more similar.
The \protect\enc{abs}{reps} appear very dissimilar meanwhile the \protect\enc{rel}{reps} yield almost identical spaces.
Quantizing the \protect\enc{abs}{reps} by performing agglomerative clustering with distance threshold $t$ produces even more similar spaces.
}
\label{fig:latent-rotation-comparison-quantization}
\end{figure}
\subsection{Dataset Information}
In \Cref{tab:dataset-info} we summarize the datasets utilized in our work, and for each one, we specify the number of classes, to give an idea about the classification difficulty.
\begin{table}[h]
\centering
\caption{All the datasets utilized in our work with their number of classes.}\label{tab:dataset-info}
\begin{tabular}{lll}
\toprule
& \textbf{Dataset} & \textbf{Number of Classes} \\
\midrule
\multirow{5}{*}{\rotatebox{90}{\textbf{Image}}} & \mnist{} & 10 \\
& \fmnist{} & 10 \\
& \cifart{} & 10 \\
& \cifarh{} & 20 (coarse) | 100 (fine) \\
& \imagenet{} & 1000 \\
\midrule
\multirow{3}{*}{\rotatebox{90}{\textbf{Graph}}} & \cora{} & 7 \\
& \citeseer{} & 6 \\
& \pubmed{} & 3 \\
\midrule
\multirow{3}{*}{\rotatebox{90}{\textbf{Text}}} & \trec{} & 6 (coarse) | 50 (fine) \\
& \dbpedia{} & 14 \\
& \amazon{} & 2 (coarse) | 5 (fine) \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Implementation Details}\label{sec:implementatin-details}
In this Section, following the corresponing sections in the main paper, we report implementation details for all the experimental settings considered.
\paragraph{Tools \& Technologies}
In all the experiments presented in this work, the following tools were used:
\begin{itemize}
\item \textit{PyTorch Lightning}, to ensure reproducible results while also getting a clean and modular codebase;
\item \textit{Weights and Biases}, to log experiments and compare runs across huge sweeps;
\item \textit{Transformers by HuggingFace}, to get ready-to-use transformers for both text and images;
\item \textit{Datasets by HuggingFace}, to access most of the NLP datasets and ImageNet for CV;
\item \textit{DVC}, for data versioning;
\item \textit{PyTorch Geometric}, to handle graph datasets and get ready-to-use GNN architectures.
\end{itemize}
\subsubsection{Word Embeddings}
For both the Figure and the Table in \Cref{sec:word-embeddings}, the number of anchors is set to 300 for a fair comparison with the dimensionality of the original spaces.
For visualization purposes, we needed the figure to both show an easy clusterable and restricted set of word embeddings. They are obtained by subsampling the shared vocabulary with the following procedure: we select 4 random pivot words, and for each of them we consider the top-200 words in their neighborhood. This results in a total of 800 points divided in 4 clusters, the ones used only for the visualization part.
For the quantitative part (table results), we select 20K random words from the shared vocabulary with a fixed seed for reproducibility purposes.
\subsubsection{Latent distance as a performance proxy}
The hyperperameters used in \Cref{sec:manifold-performance} are summarized in \Cref{tab:manifold-performance-parameters}.
\begin{table}[h]
\centering
\caption{The {reference} model and exhaustive hyperparameter combinations pertaining \Cref{sec:manifold-performance}.}
\label{tab:manifold-performance-parameters}
\begin{tabular}{lll}
\toprule
Hyperparameter & Reference Model & Sweep \\
\midrule
Seed & \texttt{1} & \texttt{0}, \texttt{1}, \texttt{2}, \texttt{3}, \texttt{4} \\
Epochs & \texttt{500} & \texttt{10}, \texttt{30}, \texttt{50} \\
Number of layers & \texttt{32} & \texttt{32}, \texttt{64} \\
Dropout Probability & \texttt{0.5} & \texttt{0.1}, \texttt{0.5} \\
Hidden Activations & \texttt{ReLU} & \texttt{ReLU}, \texttt{Tanh} \\
Convolution Activation & \texttt{ReLU} & \texttt{ReLU}, \texttt{Tanh} \\
Optimizer & \texttt{Adam} & \texttt{Adam}, \texttt{SGD} \\
Learning Rate & \texttt{0.02} & \texttt{0.01}, \texttt{0.02} \\
Graph Embedder & \texttt{GCNConv} & \texttt{GCNConv}, \texttt{GINConv} \\
\bottomrule
\end{tabular}
\end{table}
\subsubsection{Training with Absolute vs. Relative Representations}
\paragraph{Image Classification} The architecture is a standard deep CNN. We run a sweep for each dataset where we vary only the random seed (over 10 possible in total). We then aggregate by dataset and encoding type to obtain the final results with their standard deviation.
\paragraph{Graph Classification} We run a sweep identical to the one in \Cref{tab:manifold-performance-parameters} for the reference model, except that we sweep on the ``Number of layers'' with two values: 32 and 64. Each configuration is repeated with 10 different seeds, then we aggregate by dataset and encoding type to obtain the final results with their standard deviation.
\subsubsection{Image Reconstruction}
The relative and absolute models appearing in Figure~\ref{fig:encoder-decoder-swap} are vanilla AEs and VAEs, the same for all the datasets, and have a comparable number of trainable parameters. Their architecture is composed by simple convolutions, deconvolutions and mean squared error as reconstruction loss. The number of anchors is $500$ and the latent dimensionality of the absolute representations is $500$.
\subsubsection{Text Classification}
We report in \Cref{tab:transformers-nlp,tab:transformers-nlp-mono,tab:wikimatrix} details on the transformers and anchors adopted in \Cref{sec:nlp-app}.
\begin{table}[h]
\centering
\caption{The HuggingFace transformers employed in \Cref{sec:nlp-app} to tackle the \textit{Cross-lingual} setting.}
\label{tab:transformers-nlp}
\begin{tabular}{lll}
\toprule
Language & HuggingFace transformers name & Encoding Dim \\
\midrule
English & roberta-base & 768 \\
Spanish & PlanTL-GOB-ES/roberta-base-bne & 768 \\
French & ClassCat/roberta-base-french & 768 \\
Japanese & nlp-waseda/roberta-base-japanese & 768 \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[h]
\centering
\caption{The HuggingFace transformers employed in \Cref{sec:nlp-app} to tackle the \textit{Cross-architecture} setting.}
\label{tab:transformers-nlp-mono}
\begin{tabular}{ll}
\toprule
HuggingFace transformers name & Encoding Dim \\
\midrule
bert-base-cased & 768 \\
bert-base-uncased & 768 \\
google/electra-base-discriminator & 768 \\
roberta-base & 768 \\
\bottomrule
\end{tabular}
\end{table}
\paragraph{Preprocessing} Following the original work in which the \amazon{} dataset was proposed \citep{amazon-reviews}, we utilize both the \textit{title} and \textit{body} of each review. We differ in not using the category and in how we merge them; namely, we add the title as prefix for the body and add a full stop as separator when needed (avoiding duplicates). To obtain a single latent encoding for each sample, with fixed shape, we take the last hidden state and select the representation corresponding to the \emph{[CLS]} token.
\paragraph{Wikipedia anchors}
We use WikiMatrix, a corpus of sentences extracted from Wikipedia. The sentences are parallel between pairs of languages (i.e., same sentences translated in two languages), and since we are looking for a collection of parallel anchors between all 4 languages, we decided to use the English language as a pivot to compute the intersection. To get the final results, we considered only the sentences with margin score $\ge 1.06$, getting high-quality sentence alignments.
In \Cref{tab:wikimatrix} we show the total number of parallel sentences when computing the intersections. We randomly selected 768 samples to use as anchors.
\begin{table}
\centering
\caption{
WikiMatrix analysis. Each row shows the number of parallel sentences having a translation available in all the languages of that row. Since we consider all four languages, we have $3338$ parallel sentences available.}
\label{tab:wikimatrix}
\begin{tabular}{ll}
\toprule
Languages & Number of Sentences \\ \midrule
en, es & 2302527 \\
en, ja & 264259 \\
en, fr & 1682477 \\
en, es, fr & 23200 \\
en, es, ja & 147665 \\
en, fr, ja & 20990 \\
en, es, fr, ja & \textbf{3338} \\ \bottomrule
\end{tabular}
\end{table}
\subsubsection{Image Classification}
The details of the transformers used in \Cref{sec:cv-app} are summarized in \Cref{tab:transformers-cv}.
\begin{table}[h]
\centering
\caption{Timm transformers used in \Cref{sec:cv-app}.}
\label{tab:transformers-cv}
\begin{tabular}{llll}
\toprule
Version & Timm model name & Encoding Dim & Training data \\
\midrule
ViT & vit\_base\_patch16\_224 & 768 & JFT-300M, ImageNet \\
ViT & vit\_small\_patch16\_224 & 384 & ImageNet \\
ViT & vit\_base\_resnet50\_384 & 768 & ImageNet \\
RexNet & rexnet\_100 & 1280 & ImageNet \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Additional results}
In this section we report additional results on the correlation between latent similarity and performance in \Cref{fig:correlation-grid}, results on the multilingual stitching both with Amazon coarse-grained in \Cref{tab:multilingual-full-coarse-grained} and fine-grained in \Cref{tab:multilingual-full-fine-grained}, results on the image classification stitching on \cifarh{} fine-grained in \Cref{tab:cifar100-fine}. Moreover, we evaluate the stitching performance of a multilingual transformer in \Cref{tab:xmlr-multilingual-full-fine-grained}.
\begin{figure}[!h]
\centering
\begin{overpic}[trim=-0.27cm -0.15cm 0cm 0cm,clip,width=1\linewidth]{figures/data-manifold/correlation_grid.pdf}
%
\end{overpic}
\caption{Correlation plot between performance and latent similarity with the reference model for multiple different models, over time.}
\label{fig:correlation-grid}
\end{figure}
\begin{table}[!h]
\scriptsize
\centering
\caption{Stitching performance comparison with different encodings techniques. The table reports the mean weighted F1 (± std) and MAE classification performance on \amazon{} coarse-grained, across 5 different seeds. All the language pairs are shown.}
\label{tab:multilingual-full-coarse-grained}
\begin{tabular}{clcccccc}
\toprule
& & \multicolumn{2}{c}{\enc{Abs}{}} & \multicolumn{4}{c}{\enc{Rel}{}} \\
\cmidrule(lr){3-4}
\cmidrule(l){5-8}
& & \multicolumn{2}{c}{} & \multicolumn{2}{c}{Translated} & \multicolumn{2}{c}{Wikipedia} \\
\cmidrule(lr){5-6}
\cmidrule(l){7-8}
\stdecoder & \stencoder & FScore & MAE & FScore & MAE & FScore & MAE \\
\midrule
\multirow{4}{*}{en} & en & $91.54 \pm 0.58$ & $0.08 \pm 0.01$ & $90.06 \pm 0.60$ & $0.10 \pm 0.01$ & $90.45 \pm 0.52$ & $0.10 \pm 0.01$ \\
& es & $43.67 \pm 1.09$ & $0.56 \pm 0.01$ & $82.78 \pm 0.81$ & $0.17 \pm 0.01$ & $78.53 \pm 0.30$ & $0.21 \pm 0.00$ \\
& fr & $54.41 \pm 1.61$ & $0.45 \pm 0.02$ & $78.49 \pm 0.66$ & $0.21 \pm 0.01$ & $70.41 \pm 0.57$ & $0.29 \pm 0.01$ \\
& ja & $48.72 \pm 0.90$ & $0.51 \pm 0.01$ & $65.72 \pm 0.55$ & $0.34 \pm 0.01$ & $66.31 \pm 0.80$ & $0.34 \pm 0.01$
\\[2ex]
\multirow{4}{*}{es} & en & $33.23 \pm 1.00$ & $0.66 \pm 0.01$ & $78.68 \pm 2.74$ & $0.21 \pm 0.03$ & $76.65 \pm 3.23$ & $0.23 \pm 0.03$ \\
& es & $91.64 \pm 1.02$ & $0.08 \pm 0.01$ & $89.96 \pm 0.77$ & $0.10 \pm 0.01$ & $89.62 \pm 0.94$ & $0.10 \pm 0.01$ \\
& fr & $47.66 \pm 0.70$ & $0.52 \pm 0.01$ & $78.57 \pm 1.80$ & $0.21 \pm 0.02$ & $75.25 \pm 0.76$ & $0.25 \pm 0.01$ \\
& ja & $53.10 \pm 2.27$ & $0.46 \pm 0.02$ & $67.69 \pm 0.24$ & $0.32 \pm 0.00$ & $61.84 \pm 0.61$ & $0.38 \pm 0.01$
\\[2ex]
\multirow{4}{*}{fr} & en & $51.00 \pm 2.63$ & $0.49 \pm 0.03$ & $83.32 \pm 1.80$ & $0.17 \pm 0.02$ & $75.55 \pm 0.37$ & $0.24 \pm 0.00$ \\
& es & $51.96 \pm 2.81$ & $0.48 \pm 0.03$ & $82.50 \pm 0.83$ & $0.17 \pm 0.01$ & $77.12 \pm 0.88$ & $0.23 \pm 0.01$ \\
& fr & $88.22 \pm 0.75$ & $0.12 \pm 0.01$ & $85.68 \pm 1.37$ & $0.14 \pm 0.01$ & $86.45 \pm 0.96$ & $0.13 \pm 0.01$ \\
& ja & $50.32 \pm 4.16$ & $0.50 \pm 0.04$ & $69.38 \pm 0.73$ & $0.31 \pm 0.01$ & $62.79 \pm 0.27$ & $0.37 \pm 0.00$
\\[2ex]
\multirow{4}{*}{ja} & en & $53.82 \pm 2.62$ & $0.46 \pm 0.03$ & $68.66 \pm 3.62$ & $0.31 \pm 0.04$ & $70.26 \pm 3.16$ & $0.29 \pm 0.03$ \\
& es & $44.91 \pm 2.21$ & $0.55 \pm 0.02$ & $70.37 \pm 6.94$ & $0.29 \pm 0.06$ & $58.54 \pm 1.21$ & $0.41 \pm 0.01$ \\
& fr & $66.46 \pm 1.30$ & $0.34 \pm 0.01$ & $76.49 \pm 1.13$ & $0.23 \pm 0.01$ & $63.94 \pm 2.70$ & $0.36 \pm 0.02$ \\
& ja & $83.30 \pm 0.67$ & $0.17 \pm 0.01$ & $81.04 \pm 0.82$ & $0.19 \pm 0.01$ & $80.80 \pm 1.25$ & $0.19 \pm 0.01$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}[!h]
\scriptsize
\centering
\caption{Stitching performance comparison with different encodings techniques. The table reports the mean weighted F1 (± std) and MAE classification performance on \amazon{} fine-grained, across 5 different seeds. All the language pairs are shown. }
\label{tab:multilingual-full-fine-grained}
\begin{tabular}{clcccccc}
\toprule
& & \multicolumn{2}{c}{\enc{Abs}{}} & \multicolumn{4}{c}{\enc{Rel}{}} \\
\cmidrule(lr){3-4}
\cmidrule(l){5-8}
& & \multicolumn{2}{c}{} & \multicolumn{2}{c}{Translated} & \multicolumn{2}{c}{Wikipedia} \\
\cmidrule(lr){5-6}
\cmidrule(l){7-8}
\stdecoder & \stencoder & FScore & MAE & FScore & MAE & FScore & MAE \\
\midrule
\multirow{4}{*}{en} & en & $65.46 \pm 2.89$ & $0.38 \pm 0.02$ & $61.18 \pm 1.92$ & $0.44 \pm 0.02$ & $62.36 \pm 2.23$ & $0.43 \pm 0.02$ \\
& es & $22.70 \pm 0.41$ & $1.39 \pm 0.03$ & $51.67 \pm 1.20$ & $0.62 \pm 0.01$ & $45.40 \pm 0.68$ & $0.76 \pm 0.01$ \\
& fr & $30.75 \pm 0.67$ & $1.19 \pm 0.02$ & $49.18 \pm 0.83$ & $0.69 \pm 0.02$ & $40.29 \pm 0.90$ & $0.91 \pm 0.02$ \\
& ja & $24.85 \pm 0.91$ & $1.37 \pm 0.07$ & $37.34 \pm 1.49$ & $0.99 \pm 0.02$ & $37.73 \pm 0.70$ & $1.01 \pm 0.02$
\\[2ex]
\multirow{4}{*}{es} & en & $21.24 \pm 0.81$ & $1.43 \pm 0.07$ & $51.02 \pm 2.54$ & $0.68 \pm 0.05$ & $47.70 \pm 5.08$ & $0.73 \pm 0.10$ \\
& es & $61.29 \pm 3.04$ & $0.43 \pm 0.02$ & $57.89 \pm 3.80$ & $0.48 \pm 0.03$ & $57.96 \pm 4.40$ & $0.48 \pm 0.03$ \\
& fr & $29.02 \pm 0.85$ & $1.26 \pm 0.05$ & $48.40 \pm 1.02$ & $0.71 \pm 0.02$ & $44.92 \pm 1.83$ & $0.77 \pm 0.01$ \\
& ja & $29.23 \pm 1.32$ & $1.22 \pm 0.02$ & $37.22 \pm 1.56$ & $1.03 \pm 0.04$ & $34.56 \pm 0.87$ & $1.08 \pm 0.04$
\\[2ex]
\multirow{4}{*}{fr} & en & $27.39 \pm 1.22$ & $1.23 \pm 0.06$ & $45.55 \pm 3.55$ & $0.76 \pm 0.09$ & $39.01 \pm 1.25$ & $0.88 \pm 0.06$ \\
& es & $29.47 \pm 3.68$ & $1.18 \pm 0.07$ & $40.29 \pm 1.72$ & $0.90 \pm 0.04$ & $41.29 \pm 2.01$ & $0.83 \pm 0.04$ \\
& fr & $56.40 \pm 1.89$ & $0.51 \pm 0.01$ & $53.58 \pm 0.70$ & $0.57 \pm 0.01$ & $54.23 \pm 0.95$ & $0.56 \pm 0.01$ \\
& ja & $25.92 \pm 1.31$ & $1.25 \pm 0.05$ & $38.60 \pm 1.03$ & $0.96 \pm 0.02$ & $35.22 \pm 0.56$ & $1.08 \pm 0.02$
\\[2ex]
\multirow{4}{*}{ja} & en & $29.36 \pm 0.59$ & $1.17 \pm 0.04$ & $38.19 \pm 2.28$ & $0.88 \pm 0.03$ & $36.57 \pm 1.72$ & $0.98 \pm 0.02$ \\
& es & $25.64 \pm 1.77$ & $1.28 \pm 0.04$ & $34.23 \pm 2.62$ & $1.00 \pm 0.05$ & $33.16 \pm 2.28$ & $1.06 \pm 0.03$ \\
& fr & $31.79 \pm 1.91$ & $1.06 \pm 0.02$ & $38.50 \pm 2.46$ & $0.89 \pm 0.02$ & $36.68 \pm 3.14$ & $1.00 \pm 0.05$ \\
& ja & $54.09 \pm 1.35$ & $0.60 \pm 0.02$ & $50.89 \pm 1.70$ & $0.65 \pm 0.02$ & $51.64 \pm 1.47$ & $0.65 \pm 0.02$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\small
\centering
\caption{
Stitching performance comparison on XLM-R, a multilingual model by design.
The table reports the mean weighted F1 ($\pm$ std) and MAE classification performance on \amazon{} fine-grained, across 5 different seeds.}
\label{tab:xmlr-multilingual-full-fine-grained}
\begin{tabular}{llcccc}
\toprule
& & \multicolumn{2}{c}{Absolute} & \multicolumn{2}{c}{Relative} \\
\stdecoder & \stencoder & FScore & MAE & FScore & MAE \\
\midrule
\multirow{4}{*}{en} & en & $65.27 \pm 0.94$ & $0.41 \pm 0.01$ & $58.24 \pm 1.92$ & $0.51 \pm 0.03$ \\
& es & $59.55 \pm 0.76$ & $0.48 \pm 0.01$ & $52.81 \pm 1.57$ & $0.62 \pm 0.02$ \\
& fr & $58.58 \pm 1.04$ & $0.49 \pm 0.01$ & $54.01 \pm 1.34$ & $0.59 \pm 0.02$ \\
& ja & $57.98 \pm 0.77$ & $0.52 \pm 0.01$ & $48.47 \pm 2.67$ & $0.71 \pm 0.04$ \\
\cmidrule{1-6}
\multirow{4}{*}{es} & en & $60.32 \pm 1.50$ & $0.47 \pm 0.01$ & $45.69 \pm 2.19$ & $0.87 \pm 0.07$ \\
& es & $61.25 \pm 1.74$ & $0.44 \pm 0.01$ & $57.61 \pm 0.73$ & $0.51 \pm 0.01$ \\
& fr & $59.50 \pm 1.41$ & $0.47 \pm 0.01$ & $45.16 \pm 3.30$ & $0.83 \pm 0.09$ \\
& ja & $58.24 \pm 1.31$ & $0.51 \pm 0.02$ & $41.14 \pm 1.76$ & $0.99 \pm 0.05$ \\
\cmidrule{1-6}
\multirow{4}{*}{fr} & en & $58.00 \pm 4.21$ & $0.49 \pm 0.03$ & $52.37 \pm 1.66$ & $0.66 \pm 0.03$ \\
& es & $56.87 \pm 3.79$ & $0.49 \pm 0.03$ & $54.99 \pm 0.46$ & $0.57 \pm 0.01$ \\
& fr & $57.99 \pm 3.88$ & $0.47 \pm 0.02$ & $57.00 \pm 0.90$ & $0.52 \pm 0.01$ \\
& ja & $55.83 \pm 3.32$ & $0.53 \pm 0.03$ & $39.15 \pm 1.21$ & $1.02 \pm 0.03$ \\
\cmidrule{1-6}
\multirow{4}{*}{ja} & en & $59.53 \pm 1.73$ & $0.48 \pm 0.01$ & $39.46 \pm 2.34$ & $1.04 \pm 0.07$ \\
& es & $57.02 \pm 1.36$ & $0.51 \pm 0.00$ & $40.74 \pm 2.75$ & $0.97 \pm 0.09$ \\
& fr & $57.48 \pm 1.06$ & $0.51 \pm 0.01$ & $43.36 \pm 3.70$ & $0.89 \pm 0.11$ \\
& ja & $61.43 \pm 0.97$ & $0.45 \pm 0.01$ & $57.67 \pm 1.17$ & $0.51 \pm 0.01$ \\
\bottomrule
\end{tabular}
\end{table}
\begin{table}
\small
\centering
\caption{
Stitching performance comparison with different encodings techniques.
The table reports the mean weighted F1 ($\pm$ std) classification performance on \cifarh{} fine-grained, across 5 different seeds.}
\label{tab:cifar100-fine}
\begin{tabular}{llll}
\toprule
\stdecoder & \stencoder & \multicolumn{1}{c}{\enc{Abs}{}} & \multicolumn{1}{c}{\enc{Rel}{}} \\
\midrule
\multirow{4}{*}{rexnet-100} & rexnet-100 & $72.77 \pm 0.19$ & $71.39 \pm 0.18$ \\
& vit-base-patch16-224 & \multicolumn{1}{c}{-} & $40.68 \pm 0.50$ \\
& vit-base-resnet50-384 & \multicolumn{1}{c}{-} & $38.18 \pm 0.24$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $44.11 \pm 0.84$ \\
\cmidrule{1-4}
\multirow{4}{*}{vit-base-patch16-224} & rexnet-100 & \multicolumn{1}{c}{-} & $57.81 \pm 0.39$ \\
& vit-base-patch16-224 & $88.69 \pm 0.14$ & $87.05 \pm 0.34$ \\
& vit-base-resnet50-384 & $1.08 \pm 0.19$ & $66.65 \pm 1.79$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $73.73 \pm 0.60$ \\
\cmidrule{1-4}
\multirow{4}{*}{vit-base-resnet50-384} & rexnet-100 & \multicolumn{1}{c}{-} & $66.91 \pm 0.79$ \\
& vit-base-patch16-224 & $1.10 \pm 0.09$ & $75.70 \pm 0.68$ \\
& vit-base-resnet50-384 & $85.85 \pm 0.18$ & $85.04 \pm 0.38$ \\
& vit-small-patch16-224 & \multicolumn{1}{c}{-} & $75.52 \pm 0.36$ \\
\cmidrule{1-4}
\multirow{4}{*}{vit-small-patch16-224} & rexnet-100 & \multicolumn{1}{c}{-} & $56.60 \pm 0.39$ \\
& vit-base-patch16-224 & \multicolumn{1}{c}{-} & $70.14 \pm 0.46$ \\
& vit-base-resnet50-384 & \multicolumn{1}{c}{-} & $62.85 \pm 1.22$ \\
& vit-small-patch16-224 & $84.11 \pm 0.14$ & $83.24 \pm 0.13$ \\
\bottomrule
\end{tabular}
\end{table}
\subsection{Subspaces}\label{sec:subspaces}
This work does not assume the training data type, domain, or complexity, the embedding function $E$, or the downstream task to solve.
Nevertheless, the data Neural Networks deal with is often complex and presents intricate interactions.
Even when the function $E$ is learned end-to-end, in some cases a single scalar $r = sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(j)}})$ is not enough to meaningfully express the relationship between the sample ${\bm{x}}^{(i)}$ and the anchor ${\bm{a}}^{(j)}$.
To address these complex settings, we propose a multidimensional representation for the similarity measure $r = sim({\bm{e}}_{{\bm{x}}^{(i)}}, {\bm{e}}_{{\bm{a}}^{(j)}})$: we subdivide the absolute representations ${\bm{e}}$ of the samples and the anchors in ${\mathbb{S}} = \{1, \dots, s\}$ subspaces with size $l = \left\lfloor\dfrac{d}{s}\right\rfloor$.
The absolute representation of the $j$ subspace is a contiguous portion of the original embedding ${\bm{e}}^j = {\bm{e}}_{(j-1) l : jl} \text{ with } j \in {\mathbb{S}}$.
After that, we compute the subspaces relative representation independently:
$$
{\bm{r}}^j_{{\bm{x}}^{(i)}} =
(
sim({\bm{e}}^j_{{\bm{x}}^{(i)}}, {\bm{e}}^j_{{\bm{a}}^{(1)}}),
sim({\bm{e}}^j_{{\bm{x}}^{(i)}}, {\bm{e}}^j_{{\bm{a}}^{(2)}}),
\dots,
sim({\bm{e}}^j_{{\bm{x}}^{(i)}}, {\bm{e}}^j_{{\bm{a}}^{(|{\mathbb{A}}|)}})
)
\text{ with } j \in {\mathbb{S}}
$$
Projecting the data on independent subspaces, enable us to obtain more expressive relative representations and to distinguish more cases than considering the whole space. We show this behavior in the toy example in \cref{fig:subspace}
We define the relative representation of ${\bm{x}}^{(i)}$ as the concatenation of the representation of each subspace:
$$
{\bm{r}}^{{\mathbb{S}}}_{{\bm{x}}^{(i)}} = ({\bm{r}}^1_{{\bm{x}}^{(i)}}, \dots, {\bm{r}}^s_{{\bm{x}}^{(i)}})
$$
This formulation describes the similarity measure between a sample and each anchor with $s$ scalar numbers,
enabling the modeling of complex data relationships. We demonstrate the effectiveness of this strategy in \Cref{sec:subspace-comparison}.
\begin{figure}
\centering
\includegraphics[trim={0cm 8cm 0 0},width=\linewidth]{figures/p2.png}
\caption{Consider two sets of points (in red and green respectively) lying on two different great circles on a sphere (top left). Taking the inner product between these sets of points (bottom left) give us limited information on their spatial distribution on the sphere. Instead if we project the points on the three subspaces, represented as 1 dimensional spheres (top right) and measure the inner products on each of them independently (bottom right), we can efficiently detect the symmetries underlying the point distribution on the sphere and distinguish that the points lie on two different great circles.}
\label{fig:subspace}
\end{figure}
Moreover, suppose the embedding function $E$ is trained end-to-end. In that case, the independent subspaces allow the embedder to specialize each subspace on different data characteristics, enabling the measurement of similarities on distinct data features.
\marco{Sarebbe bello mettere un toy example che fa vedere come questa divisione in sottospazi sia utile a distinguere casi che vogliamo distinguere, o ad essere invarianti. In maniera simile rispetto agli esempi dell'isometria e della quasi isometria. Forse ne ho in mente uno } \luca{Yes sembra perfetto! Dici aggiungiamo direttamente qui l'esempio?}
\subsection{Subspace comparison}\label{sec:subspace-comparison}
In this section, we demonstrate the effectivness of the subspace strategy described \Cref{sec:subspaces}.
\paragraph{Experimental setting.} We consider the classification and reconstruction tasks on the \mnist{} and \cifart{}. The models compared a classifier and autoencoder that employs the relative representation with an increasing number of subspaces.
The results reported in \Cref{tab:subspace-performance-comparison} show that subdiving the latent space in multiple subspaces improves the expressivness of the architecture and the performance on classification and reconstruction tasks.
\begin{table}[ht]
\caption{\draft{TBA} Performance comparison between different number of subspaces subdivision.}
\label{tab:subspace-performance-comparison}
\begin{center}
\begin{tabular}{ccccccc|cccccc}
& \multicolumn{6}{c}{\bf Classification } & \multicolumn{6}{c}{\bf Reconstruction} \\
Subspaces & $1$ & $2$ & $4$ & $8$ & $16$ & $32$ & $1$ & $2$ & $4$ & $8$ & $16$ & $32$ \\
\\ \hline \\
\mnist{} & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? \\
\cifart{} & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? & ? \\
\end{tabular}
\end{center}
\end{table}
\subsection{Interpretability}\label{sec:interpretability}
The similarity function infuses specific properties in the latent representation. In this work, our focus has been on the cosine similarity that induces an isometry invariant representation.
This section contains an example of an alternative similarity function, that besides the isometry invariance, infuses other properties in the latent representation.
\textbf{Experimental setting.}
We define two \glspl{ae} that utilize the relative representations in the bottleneck. The first \gls{ae} adopts the cosine similarity as described in \Cref{sec:isometry-invariance}; the second \gls{ae} performs a change of basis solving the following linear system ${\bm{e}}_{{\mathbb{A}}} {\bm{r}}_{{\bm{x}}^{(i)}} = {\bm{e}}_{{\bm{x}}^{(i)}}$. That is, we represent the absolute embedding ${\bm{e}}_{{\bm{x}}^{(i)}}$ of a sample ${\bm{x}}^{(i)}$ as the coefficients of the basis formed by the embeddings ${\bm{e}}_{{\mathbb{A}}}$ of all the anchors ${\mathbb{A}}$. We train both \glspl{ae} end-to-end on the \mnist{} dataset.
The basis-change method forces the anchors to be a basis for the relative latent space, thus a higher similarity value on the $i$ dimension indicates a higher coefficient for the $i$-th anchor. In \Cref{fig:interpretability} we compare the reconstructions performed by decoders trained with relative representations obtained with the cosine similarity (bottom row) and the basis-change method (top row). The relative representations provided are zero everywhere except that in one dimension. When adopting the basis-change strategy, the reconstructed images are similar to the anchor associated with the non-zero dimension.
Since the anchors ${\bm{a}}^{(j)}$ have intelligible also the corresponding dimension in the relative representation ${\bm{r}}_{j}$ becomes interpretable.
\begin{figure}[ht]
\centering
\begin{overpic}[trim=0cm 0cm 0cm 0cm,clip,width=1\linewidth]{figures/mockup/interpretability.png}
%
\end{overpic}
\vspace{-.5cm}
\caption{\draft{TBA} Relative representation properties comparison. Each image is a reconstruction given the relative representation shown on its left. The relative representation shows the anchors associated with each dimension, it is zero in all dimensions except one. The images in the top row are obtained with a decoder trained with the basis change as similarity measure; the bottom row images are produced by a decoder trained with the cosine similarity. }
\label{fig:interpretability}
\end{figure}
\subsection{Word Analogies}\label{sec:analogies}
Possibile esperimento 1 per validare questa cosa: testiamo il benchmark base di word analogies (e simili, molto semplice/immediato) sia con encoding assoluto che relativo (stessa lingua). Se il relativo va meglio: siamo piu' astratti. Se l'assoluto va meglio: non si mette l'esperimento (tanto c'e' la citazione a https://www.frontiersin.org/articles/10.3389/frai.2021.698809/full) .
Consideriamo il task di calcolare la distanza fra coppie di parole
Il gold standard contro cui testarsi è annotazionie umana
We evaluate the correlations between the gold standard distances and the distances predicted by the model
Prendiamo gli embedding fasttext
In this setting we do not train any neural network, but we evaluate the goodness of the word representation
\begin{table}[ht]
\caption{\luca{WIP}
}
\label{tab:word-analogies}
\begin{center}
\begin{tabular}{lccc}
\multicolumn{1}{c}{ \bf Embedding Type} &
\multicolumn{1}{c}{ \bf Score } &
\\[.5ex] \hline \\
Relative FastText & ? & ? & ? \\
Absolute FastText & ? & ? & ? \\
\end{tabular}
\end{center}
\end{table}
\subsubsection{Cross-domain stitching}\label{sec:model-reusability-domain}
In this section we demonstrate relative representations generate compatible latent spaces even \emph{across different data domains}, as long as the data represents the same information and there are parallel anchors in correspondence between the domains.
\paragraph{Experimental setting.} We consider a classification task on the datasets:
\thingi{} \cite{Zhou_undated-jc},
\surreal{} \cite{Varol2017-et},
\deformingthings{} \cite{Li2021-yv}, \shapenet{}. Note that these datasets contains the 3D model of each object associated with the corresponding 2D rendering.
We define two encoders:
one to embed images $E^{RGB}$
and one to embed points clouds $E^{PC}$.
Thus, we choose image-3D parallel anchors that allow to define relative representation between the embeddings of images and between the embeddings of point clouds.
Furthermore, we define a neural component $D$ that takes a representation and predicts the object category.
We performs experiments to showcase the resulting alignment of the relative representations in the two domains and the zero-shot cross-domain capabilities, switching encoder without re-training or fine-tuning the classifier component.
The results in \Cref{tab:cross-domain} prove that the absolute representation completely fail when performing zero-shot cross-domain classification. Meanwhile, the relative representation demonstrate a certain degree of compatibility between the latent spaces of images and point clouds.
\begin{table}[ht]
\caption{\draft{TBA} Cross-domain performance comparison. Each column is trained on the indicated dataset with the reported encoder, thus the resulting classifier component $D$ is stitched together with the encoder reported on each row to perform zero-shot classification. The row also indicates whether the relative representations are employed or not. The numbers reported are the accuracy on the test set of each dataset. }
\label{tab:cross-domain}
\begin{center}
\begin{tabular}{cllllllllll}
& & & \multicolumn{8}{c}{\bf Training } \\
& & & \multicolumn{2}{c}{\thingi{} }
& \multicolumn{2}{c}{\surreal{} }
& \multicolumn{2}{c}{\deformingthings{} }
& \multicolumn{2}{c}{\shapenet{} } \\[.5ex]
& & & $E^{RGB}$ & $E^{PC}$
& $E^{RGB}$ & $E^{PC}$
& $E^{RGB}$ & $E^{PC}$
& $E^{RGB}$ & $E^{PC}$
\\ \hline \\
\multirow{4}{*}{\STAB{\rotatebox[origin=c]{90}{\bf Stitching}}} & \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{Rel.}}} & $E^{RGB}$ & ? & ? & ? & ? & ? & ? & ? & ? \\
& & $E^{PC}$ & ? & ? & ? & ? & ? & ? & ? & ? \\[1ex]
& \multirow{2}{*}{\STAB{\rotatebox[origin=c]{90}{Abs.}}} & $E^{RGB}$ & ? & ? & ? & ? & ? & ? & ? & ? \\
& & $E^{PC}$ & ? & ? & ? & ? & ? & ? & ? & ? \\
\end{tabular}
\end{center}
\end{table}
\subsection{Self-representation computation}
\begin{figure}[!h]
\centering
\begin{overpic}[trim=0cm 0cm 0cm 0cm,clip,width=.5\linewidth]{figures/mockup/method.pdf}
\end{overpic}
\caption{\draft{TBA} \luca{Change image symbol} Relative representation computation:
(1) the function $E$ embeds the sample $x^{(i)}$;
(2) the function $E$ embeds all the anchors $a^{(1)}, \dots, a^{(|{\mathbb{A}}|)}$;
(3) the similarities between the sample embedding and anchors embeddings are the relative representation of the sample.}
\label{fig:method}
\end{figure}
\subsection{Robustness Analysis}\label{sec:quantization-performance}
\begin{figure}[ht]
\centering
\begin{overpic}[trim=0cm 0cm 0cm 0cm,clip,width=.7\linewidth]{figures/mockup/marco-quantization.png}
\end{overpic}
\caption{\draft{TBA} Reconstruction errors on the \cifart{} dataset at different quantization bin sizes. The models shown are: $RAE^b_{E_1D_1}$ quantized and trained end-to-end, $RAE_{E_1D_1}$ non-quantized and trained end-to-end, $RAE^b_{E_1D_2}$ quantized and stitched together, $RAE_{E_1D_2}$ non-quantized and stitched together. The quantization improves the robustness of the models stitched together since enhances the latent space isometry invariance.}
\label{fig:quantization-performance}
\end{figure}
| 2024-02-18T23:39:54.811Z | 2022-10-03T02:13:19.000Z | algebraic_stack_train_0000 | 801 | 12,811 |
|
proofpile-arXiv_065-4132 | \section{Introduction}
The electronic property in a stack of two-dimensional (2D) materials sensitively depends
on the relative twist angle $\theta$ between the adjacent layers,
and we often have dramatic angle-dependent phenomena which are
never observed in an isolated layer.
The best known example is the twisted bilayer graphene (twisted BLG), or
a rotationally stacked pair of monolayer graphenes,
where a long-period moir\'{e} interference pattern significantly modifies the Dirac dispersion
\cite{lopes2007graphene,mele2010commensuration,trambly2010localization,shallcross2010electronic,morell2010flat,bistritzer2011moirepnas,moon2012energy,de2012numerical,moon2013opticalabsorption,weckbecker2016lowenergy}.
Recently, the superconductivity and correlated insulating
states are discovered in the magic-angle twisted BLG with extremely flat bands,
\cite{cao2018unconventional,cao2018mott, yankowitz2019tuning}
and it is followed by a number of theoretical studies on the detailed properties of the flat bands
and the possible mechanism of the superconductivity.
\cite{yuan2018model,po2018origin,xu2018topological,kang2018symmetry,koshino2018maximally,ochi2018possible,isobe2018unconventional,dodaro2018phases,padhi2018doped,wu2018theory,tarnopolsky2019origin,zou2018band}
Graphene on hexagonal boron nitride (hBN) also exhibits
the moir\'{e}-induced physics such as the formation of the secondary Dirac bands
and the miniband structure. \cite{kindermann2012zero, wallbank2013generic, mucha2013heterostructures, jung2014ab, moon2014electronic,dean2013hofstadter,ponomarenko2013cloning,hunt2013massive,yu2014hierarchy}
A recent experiment reported the correlated insulating states in ABC-trilayer graphene on hBN,
which is tunable by the external gate electric field. \cite{chen2019evidence}
Controlling the twist angle in a stack of 2D materials provides powerful means to manipulate quantum properties of the electronic systems.
\begin{figure}
\begin{center}
\leavevmode\includegraphics[width=0.81\hsize]{fig_lattice_BZ.eps}
\end{center}
\caption{
(a) Atomic structure of the twisted AB-AB double BLG
and (b) that of the twisted AB-BA double BLG.
(c) Brillouin zone folding in the double BLG.
Two large hexagons represent the first Brillouin zones of the first bilayer graphene,
and the small hexagon is the moir\'{e} Brillouin zone.
}
\label{fig_lattice_BZ}
\end{figure}
In this paper, we study a different type of moir\'{e} system, the twisted double bilayer graphene,
which is composed of a pair of AB-stacked BLGs rotationally stacked on top of each other.
The AB-stacked BLG is the most stable form of bilayer graphene which has the stacking structure of graphite.
\cite{mccann2013electronic}
Here we consider two different arrangements, AB-AB and AB-BA, as illustrated
in Figs.\ \ref{fig_lattice_BZ}(a) and 1(b), respectively,
where the AB-BA is obtained just by 180$^\circ$ rotation of the second BLG in the AB-AB.
For each case, we derive the continuum Hamiltonian by extending the approach for
the twisted BLG\cite{lopes2007graphene,bistritzer2011moirepnas,kindermann2011local,PhysRevB.86.155449,moon2013opticalabsorption,koshino2015interlayer,koshino2015electronic,weckbecker2016lowenergy},
and calculate the energy bands as well as the valley Chern numbers.
Here we include the interlayer asymmetric potential $\Delta$ induced by the gate electric field.
The energy band structures are found to be similar between the AB-AB and the AB-BA,
but the topological nature is different.
In the absence of $\Delta$, the lowest electron and hole bands of the AB-AB
are entangled by the symmetry protected band touching points,
while they are separated in the AB-BA due to the different space symmetry.
In both cases, the asymmetric potential $\Delta$ immediately opens an energy gap
at the charge neutral point.
We find that the graphite band parameters such as $\gamma_3$ and $\gamma_4$
play an important role in the electron-hole asymmetry, where the electron band
becomes much narrower than the hole band as increasing $\Delta$.
The crucial difference between AB-AB and AB-BA is found in the Chern number.
In the absence of $\Delta$, in particular, the AB-AB double bilayer becomes a trivial insulator because
the symmetry requires all the Chern numbers to vanish,
while the AB-BA is a valley Hall insulator with finite Chern number.
We demonstrate the evolution of the Chern numbers as a function of $\Delta$,
where we see that the energy bands of AB-AB and AB-BA carry completely different topological numbers,
even though the band structures are similar.
The difference in the Chern number would be observed by the
measurement of the valley Hall conductivity \cite{mak2014valley,shimazaki2015generation},
and also by the Landau level structure in the magnetic field.
This paper is organized as follows: In Sec.\ \ref{sec_atomic},
we define the lattice structures of AB-AB and AB-BA double bilayers,
and then introduce a continuum Hamiltonian for each system in Sec.\ \ref{sec_continuum}.
In Sec.\ \ref{sec_band}, we study the band structures and the evolution of Chern numbers
as a function of the twist angle and the asymmetric potential,
where we discuss in detail about similarity and difference between the two systems.
A brief conclusion is presented in Sec.\ \ref{sec_conclusion}.
\section{Atomic structure}
\label{sec_atomic}
The AB-stacked BLG is composed of a pair of monolayer graphenes,
with four atoms in the unit cell, labeled $A_1$, $B_1$ on the layer 1 (upper layer)
and $A_2$, $B_2$ on the layer 2 (lower layer). \cite{mccann2013electronic}
The two graphene layers are arranged so that $B_1$ and $A_2$ are vertically located.
We refer to these two atomic sites as dimer sites because the electronic orbitals on them are strongly coupled.
The other two atoms, $A_1$ and $B_2$ are directly above or below the hexagon center of the other layer,
and are referred to as non-dimer sites.
We compose the twisted AB-AB double bilayer graphene by stacking the first AB-stacked BLG
(layers 1 and 2) on top of the second AB-stacked BLG (layers 3 and 4), as Fig.\ \ref{fig_lattice_BZ}(a).
We start from the non-rotated geometry where $B_1$, $A_2$, $B_3$ and $A_4$ are
vertically aligned at the origin $O$, and then rotate the first and the second BLGs
around $O$ by $-\theta/2$ and $+\theta/2$, respectively.
The system has a three-fold in-plane rotation $C_{3z}$ symmetry along the $z$-axis (perpendicular to the layer),
and a two-fold rotation $C_{2x}$ along the $x$-axis.
The twisted AB-BA double bilayer can be defined just by rotating the second BLG (layer 3 and 4)
of the AB-AB by 180$^\circ$ as in Fig.\ \ref{fig_lattice_BZ}(b),
where we flip the definition of $A$ site and $B$ site for layer 3 and 4.
The system is symmetric under a three-fold in-plane rotation $C_{3z}$
and a two-fold rotation $C_{2y}$ along the $y$-axis.
We define $\mathbf{a}_1 = a(1,0)$ and $\mathbf{a}_2 = a(1/2,\sqrt{3}/2)$
as the lattice vectors of the initial BLGs before the rotation, where $a \approx 0.246\,\mathrm{nm}$
is the lattice constant of graphene. The corresponding reciprocal lattice vectors
are $\mathbf{a}^*_1 = (2\pi/a)(1,-1/\sqrt{3})$ and $\mathbf{a}^*_2=(2\pi/a)(0,2/\sqrt{3})$.
After the rotation, the lattice vector of the $l$-th BLG is given by $\mathbf{a}_i^{(l)} =R(\mp \theta/2)\mathbf{a}_i$
with $\mp$ for $l=1,2$, respectively, where $R(\theta)$ represents the rotation matrix by $\theta$.
The reciprocal lattice vectors become $\mathbf{a}_i^{*(l)} =R(\mp \theta/2)\mathbf{a}^*_i$.
In a small $\theta$, the reciprocal lattice vectors for the moir\'{e} pattern is given by
$ \mathbf{G}^{\rm M}_{i} = \textbf{a}^{*(1)}_i - \textbf{a}^{*(2)}_i \, (i=1,2)$,
and the real-space lattice vectors $\mathbf{L}^{\rm M}_{j}$ can then be obtained from
$\mathbf{G}^{\rm M}_i\cdot\mathbf{L}^{\rm M}_{j} = 2\pi\delta_{ij}$.
A moir\'{e} unit cell is spanned by $\mathbf{L}^{\rm M}_{1}$ and $\mathbf{L}^{\rm M}_2$.
The lattice constant $L_{\rm M} = | \mathbf{L}^{\rm M}_{1}|=| \mathbf{L}^{\rm M}_2|$
is $L_{\rm M} = a/[2\sin (\theta/2)]$.
Figure \ref{fig_lattice_BZ}(c) illustrates the Brillouin zone folding,
where two large hexagons represent the first Brillouin zones of the first and the second BLGs,
and the small hexagon is the moir\'{e} Brillouin zone of the twisted double BLG.
The graphene's Dirac points (the band touching points)
are located at $\mathbf{K}^{(l)}_\xi = -\xi [2\mathbf{a}^{(l)*}_1+\mathbf{a}^{(l)*}_2]/3$
for the $l$-th BLG, where $\xi=\pm 1$ is the valley index.
We label the symmetric points of the moir\'{e} Brillouin zone as
$\bar{\Gamma}$, $\bar{M}$, $\bar{K}$ and $\bar{K'}$ as in Fig.\ \ref{fig_lattice_BZ}(c).
\section{Continuum Hamiltonian}
\label{sec_continuum}
To describe the electronic band structure of the twisted double bilayers,
we adopt the continuum method based on the Dirac Hamiltonian.\cite{lopes2007graphene,bistritzer2011moirepnas,kindermann2011local,PhysRevB.86.155449,moon2013opticalabsorption,koshino2015interlayer,koshino2015electronic,weckbecker2016lowenergy}
The validity of the continuum model was verified for twisted BLG
by the direct comparison to the tight-binding model. \cite{moon2013opticalabsorption,weckbecker2016lowenergy}
We define the Bloch bases of $p_z$ orbitals at sublattice
$X= A_1,B_1, \cdots ,A_4,B_4$ as
$ |\mathbf{k},X\rangle = N^{-1/2} \sum_{\mathbf{R}_{X}} e^{i\mathbf{k}\cdot\mathbf{R}_{X}}$,
where $|\mathbf{R}_{X} \rangle$ is the atomic $p_z$ orbital at the site $\mathbf{R}_{X}$,
$\mathbf{k}$ is the two-dimensional Bloch wave vector
and $N$ is the number of same sublattices in the system.
The continuum Hamiltonian for
twisted AB-AB double bilayer graphene at small twist angle $\theta (\ll 1)$
is written in 8 $\times $ 8 matrix for the Bloch bases of
$(A_1,B_1,A_2,B_2,A_3,B_3,A_4,B_4)$ as
\begin{align}
&
{H}_{\textrm{AB-AB}} =
\begin{pmatrix}
H_0(\mathbf{k}_1) & g^\dagger(\mathbf{k}_1) & & \\
g(\mathbf{k}_1) & H'_0(\mathbf{k}_1) & U^\dagger &\\
& U & H_0(\mathbf{k}_2) & g^\dagger(\mathbf{k}_2) \\
& & g(\mathbf{k}_2) & H'_0(\mathbf{k}_2) \\
\end{pmatrix} + V,
\label{eq_AB-AB}
\end{align}
where $ \mathbf{k}_l = R(\pm \theta/2)({\mathbf{k}}-\mathbf{K}^{(l)}_\xi)$
with $\pm$ for $l=1$ and 2, respectively, and
\begin{align}
& H_0(\mathbf{k})
=
\begin{pmatrix}
0 & -\hbar v k_- \\
-\hbar v k_+ & \Delta'
\end{pmatrix},
\,
H'_0(\mathbf{k})
=
\begin{pmatrix}
\Delta' & -\hbar v k_- \\
-\hbar v k_+ & 0
\end{pmatrix}
\\
& g(\mathbf{k})
=
\begin{pmatrix}
\hbar v_4 k_+ & \gamma_1 \\
\hbar v_3 k_- & \hbar v_4 k_+
\end{pmatrix},
\end{align}
with $k_\pm = \xi k_x \pm i k_y$.
$H_0$ and $H'_0$ are the Hamiltonian of monolayer graphene
where $\Delta' = 0.050 $eV\cite{mccann2013electronic}
represents the on-site potential of dimer sites with respect to non-dimer sites.
The parameter $v$ is the band velocity of monolayer graphene,
and it is taken as $\hbar v /a = 2.1354$ eV.\cite{moon2013opticalabsorption,koshino2018maximally}
The matrix $g$ describes the interlayer coupling of the AB-stacked BLG,
where $\gamma_1 = 0.4$eV is the coupling between dimer sites, and
$v_3$ and $v_4$ are related to diagonal hoppings $\gamma_3 = 0.32$eV and $\gamma_4=0.044$ eV
with the relation $v_i = (\sqrt{3}/2) \gamma_i a /\hbar \, (i=3,4)$. \cite{mccann2013electronic}
In the AB-stacked BLG, $v_3$ is responsible for the trigonal warping of the energy band
and $v_4$ is for the electron-hole asymmetry.
The matrix $U$ is the moir\'{e} interlayer coupling between twisted layers given by
\cite{bistritzer2011moirepnas,moon2013opticalabsorption,koshino2018maximally}
\begin{align}
U &=
\begin{pmatrix}
u & u'
\\
u' & u
\end{pmatrix}
+
\begin{pmatrix}
u & u'\omega^{-\xi}
\\
u'\omega^\xi & u
\end{pmatrix}
e^{i\xi \mathbf{G}^{\rm M}_1\cdot\mathbf{r}}
\nonumber\\
&
\qquad \qquad \qquad +
\begin{pmatrix}
u & u'\omega^\xi
\\
u'\omega^{-\xi} & u
\end{pmatrix}
e^{i\xi(\mathbf{G}^{\rm M}_1+\mathbf{G}^{\rm M}_2)\cdot\mathbf{r}},
\label{eq_interlayer_matrix}
\end{align}
where $\omega =e^{2\pi i/3}$,
$u = 0.0797$eV and $u' = 0.0975$eV \cite{koshino2018maximally}
are the amplitudes of diagonal and off-diagonal terms, respectively.
The difference between $u$ and $u'$ effectively describe the out-of-plane corrugation effect,
which enhances the energy gaps between the lowest energy bands
and the excited bands.\cite{koshino2018maximally,nam2017lattice,tarnopolsky2019origin}
Lastly, $V$ is the interlayer asymmetric potential,
\begin{align}
& V =
\begin{pmatrix}
\frac{3}{2}\Delta \hat{1} &&& \\
& \frac{1}{2}\Delta \hat{1} && \\
&& -\frac{1}{2}\Delta \hat{1} & \\
&&& -\frac{3}{2}\Delta \hat{1}
\end{pmatrix},
\end{align}
where $\hat{1}$ is $2\times 2$ unit matrix, and $\Delta$ represents
the difference in the electrostatic energy between the adjacent layers.
Here we simply assumed the perpendicular electric field is constant.
Noting that the lattice structure of the AB-AB double bilayer has $C_{2x}$ symmetry
and also the valley degree of freedom $\xi=\pm$ is unchanged under $C_{2x}$,
the Hamiltonian $H_\textrm{AB-AB}$ of each single valley commutes with $C_{2x}$,
given that the asymmetric potential $\Delta$ is absent.
Similarly, the Hamiltonian of the twisted AB-BA double bilayer graphene is given by
\begin{align}
&
{H}_{\textrm{AB-BA}} =
\begin{pmatrix}
H_0(\mathbf{k}_1) & g^\dagger(\mathbf{k}_1) & & \\
g(\mathbf{k}_1) & H'_0(\mathbf{k}_1) & U^\dagger &\\
& U & H'_0(\mathbf{k}_2) & g(\mathbf{k}_2) \\
& & g^\dagger(\mathbf{k}_2) & H_0(\mathbf{k}_2) \\
\end{pmatrix}
+ V.
\label{eq_AB-BA}
\end{align}
where $H_0(\mathbf{k}_2)$ and $H'_0(\mathbf{k}_2)$ are interchanged and
also $g(\mathbf{k}_2)$ and $g^\dagger(\mathbf{k}_2)$ are swapped
in ${H}_{\textrm{AB-AB}}$.
The lattice structure of the AB-BA double bilayer has $C_{2y}$ symmetry,
and $C_{2y}$ interchanges the valleys $\xi=\pm$.
As a result, the Hamiltonian $H_\textrm{AB-BA}$ with $\Delta = 0$
commutes with $C_{2y} T$, where $T$ is the time reversal operator.
The calculation of the energy bands and the eigenstates
is performed in the $k$-space picture.
For a single Bloch vector $\mathbf{k}$ in the moir\'{e} Brillouin zone,
the interlayer coupling $U$ hybridizes the graphene's eigenstates at
$\mathbf{q} = \mathbf{k} + \mathbf{G}$, where $\mathbf{G} = m_1 \mathbf{G}^{\rm M}_1 + m_2 \mathbf{G}^{\rm M}_2$
and $m_1$ and $m_2$ are integers.
The low-energy eigenstates can be obtained by numerically diagonalizing the Hamiltonian
within the limited number of $\mathbf{q}$'s inside the cut-off circle $|\mathbf{q}-\mathbf{q}_0| < q_c$.
Here $\mathbf{q}_0$ is taken as the midpoint between $\mathbf{K}^{(1)}_\xi$ and $\mathbf{K}^{(2)}_\xi$,
and $q_c$ is set to $4 |\mathbf{G}^{\rm M}_1|$.
The calculation is done independently for each of $\xi=\pm$ as the intervalley coupling can be neglected
in small twist angles.
We calculate the Chern number of moir\'{e} subbands by the standard definition,
\begin{align}
& C_n = \frac{1}{2\pi}\int_{\rm MBZ} {\cal F}_{n,\mathbf{k}} \, d^2k,
\end{align}
where $n$ is the band index,
MBZ represents the moir\'{e} Brillouin zone, and $F_{n,\mathbf{k}}$ is the Berry curvature defined by
\begin{align}
{\cal F}_{n,\mathbf{k}} = \frac{\partial a^{(y)}_{n,\mathbf{k}}}{\partial k_x} - \frac{\partial a^{(x)}_{n,\mathbf{k}}}{\partial k_y}, \quad
a^{(\mu)}_{n,\mathbf{k}} = \frac{1}{i} \langle u_{n,\mathbf{k}} | \frac{\partial}{\partial k_\mu} | u_{n,\mathbf{k}} \rangle,
\end{align}
where $u_{n,\mathbf{k}}$ is the Bloch wave function of $n$-th subband.
We numerically calculate the Chern numbers using the discretizing method. \cite{fukui2005chern}
The symmetry imposes constraints on the Chern number.
For the AB-AB double bilayer at $\Delta =0$, the $C_{2x}$
symmetry requires ${\cal F}_{n,(k_x,-k_y)} = - {\cal F}_{n,(k_x,k_y)}$,
so that the Chern number of each single band must vanish.
In the AB-BA double bilayer at $\Delta =0$, the $C_{2y} T$
symmetry requires ${\cal F}_{n,(k_x,-k_y)} = {\cal F}_{n,(k_x,k_y)}$, and the Chern number can be finite.
\begin{figure}
\begin{center}
\leavevmode\includegraphics[width=0.95\hsize]{fig_band_1.33minimal.eps}
\end{center}
\caption{(a) Band structure of the twisted AB-AB double bilayer at the twist angle $\theta= 1.33^\circ$
with $\Delta = 0$ and 20 meV, calculated by the minimal model.
(b) Corresponding plots for the twisted AB-BA double bilayer.
Black numbers indicate the Chern numbers for the energy bands in $\xi =+$,
and the blue numbers between the bands are the integrated Chern numbers
summed over all the energy bands of $\xi =+$ below.
The Chern numbers for $\xi =-$ bands are opposite in sign.
}
\label{fig_band_1.33minimal}
\end{figure}
\section{Band structures and topological properties}
\label{sec_band}
\subsection{Minimal model}
\label{sec_minimal}
Before calculating the band structure with all the band parameters fully included,
it is intuitive to consider the minimal model which neglects the relatively small parameters, $v_3$, $v_4$, $\Delta'$
and the rotation matrix $R(\pm \theta/2)$ in the definitions of $\mathbf{k}^{(1)}$ and $\mathbf{k}^{(2)}$.
Then the AB-AB Hamiltonian Eq.\ (\ref{eq_AB-AB}) has a fictitious particle-hole symmetry
similar to TBG\cite{moon2013opticalabsorption},
\begin{align}
\Sigma^{-1} H_{\textrm{AB-AB}} \Sigma = - H^*_{\textrm{AB-AB}}, \nonumber\\
\Sigma =
\begin{pmatrix}
&&& \sigma_x \\
&& -\sigma_x &\\
& \sigma_x &&\\
- \sigma_x &&&
\end{pmatrix}.
\label{eq_e-h_sym_AB-AB}
\end{align}
This immediately leads to the electron-hole symmetry in the energy bands,
$E_{n,\mathbf{k}} = -E_{-n,-\mathbf{k}}$, and also the anti-symmetric relation in the Chern number,
$C_{-n} = - C_n$, where $n$ and $-n$ stand for the band indexes of
the corresponding electron and hole bands, respectively.
Note that Eq.\ (\ref{eq_e-h_sym_AB-AB}) holds even
in the presence of the interlayer asymmetric potential $\Delta$.
The AB-BA Hamiltonian Eq.\ (\ref{eq_AB-BA}) has a different type of symmetry
between the electron and the hole bands,
\begin{align}(
\Sigma'^{-1} \tilde{P}) H_{\textrm{AB-BA}} (\tilde{P} \Sigma') = - H_{\textrm{AB-BA}}, \nonumber\\
\Sigma' =
\begin{pmatrix}
&&& \hat{1} \\
&& - \hat{1} &\\
& \hat{1} &&\\
- \hat{1} &&&
\end{pmatrix},
\label{eq_e-h_sym_AB-BA}
\end{align}
where $\tilde{P}$ is a space inversion operator which works on envelop function as
$\tilde{P} F_X(\mathbf{r}) = F_X(-\mathbf{r})$,
while it does not change the sublattice degree of freedom ($X=A_1,B_1,\cdots$).
This again forces the electron-hole symmetry $E_{n,\mathbf{k}} = -E_{-n,-\mathbf{k}}$,
but the Chern number becomes electron-hole symmetric, $C_{-n} = C_n$,
because the operation lacks the complex conjugate.
Figures \ref{fig_band_1.33minimal}(a) and (b) show the minimal-model band structure,
calculated for the AB-AB double bilayer and the AB-BA double bilayer, respectively,
at the twist angle $\theta= 1.33^\circ$ with $\Delta = 0$ and 20 meV.
The band structures of the two systems closely resemble each other.
At $\Delta =0$, we have a pair of energy bands touching at Dirac point,
which are isolated from the excited bands by energy gaps as in the twisted BLG. \cite{koshino2018maximally}
A finite $\Delta$ immediately opens an energy gap at the charge neutral point.
This is in a sharp contrast to the twisted BLG, where the perpendicular electric field never opens a gap
at the charge neutral point, because the band touching is protected by $C_2 T$ symmetry.
Now the twisted double bilayer lacks $C_2$ symmetry.
Although the band structures are pretty much similar between the AB-AB and the AB-BA cases,
the properties of the Chern number are completely different.
In Fig.\ \ref{fig_band_1.33minimal}, the black numbers indicate the Chern numbers
of the central two bands in $\xi =+$ valley,
and the blue numbers between the bands are the integrated Chern numbers
summed over all the energy bands of $\xi =+$ below.
Because of the time reversal symmetry, the Chern number of $\xi =-$ valley is opposite in sign to $\xi =+$.
We actually see the expected relation $C_{-n} = - C_n$ for the AB-AB, and $C_{-n} = C_n$ for the AB-BA.
In the absence of the asymmetric potential $\Delta$, the Chern numbers all vanish in the AB-AB because of the
rigorous symmetry $C_{2x}$ mentioned in the previous section,
while it is finite in the AB-BA.
When the Fermi energy is inside one of those gaps, therefore,
the AB-BA double bilayer is a valley Hall insulator, while AB-AB is a trivial insulator.
The Chern numbers can be finite in the AB-AB once the asymmetric potential $\Delta$ is switched on,
because it breaks $C_{2x}$.
The integrated Chern numbers inside the central gap is $-2$ in the AB-AB, while 0 in the AB-BA.
This is just equal to the sum of Chern numbers of two independent gapped BLGs,
which is $-1$ for the AB stack while $+1$ for the BA stack. \cite{martin2008topological,koshino2008electron,zhang2013valley,vaezi2013topological}
\begin{figure*}
\begin{center}
\leavevmode\includegraphics[width=0.75\hsize]{fig_band_1.33full.eps}
\end{center}
\caption{(a) Band structure of the AB-AB double bilayer at the twist angle $\theta= 1.33^\circ$
with $\Delta = 0, 5$ and 20 meV, calculated by the full parameter model,
(b) Corresponding plots for the AB-BA double bilayer.
Thin green lines indicate the energy bands of the minimal model [Fig.\ \ref{fig_band_1.33minimal}].
}
\label{fig_band_1.33full}
\end{figure*}
\begin{figure*}
\begin{center}
\leavevmode\includegraphics[width=1.\hsize]{fig_band_angle_dep.eps}
\end{center}
\caption{
(a) Band structure of the AB-AB double bilayer at various twist angles with $\Delta = 0$,
calculated by the full parameter model.
(b) Corresponding plots for the AB-BA double bilayer.
}
\label{fig_band_angle_dep}
\end{figure*}
\subsection{Full parameter model}
Inclusion of the additional band parameters neglected in the minimal model
causes a significant change particularly in the low-energy sector.
Figures \ref{fig_band_1.33full}(a) and (b) show the full-parameter band structure
of the AB-AB and the AB-BA double bilayers, respectively, at the twist angle $\theta= 1.33^\circ$
with $\Delta = 0, 5$ and 20 meV.
The thin green lines indicate the energy bands of the minimal model [Fig.\ \ref{fig_band_1.33minimal}]
for quantitative comparison.
We see that the energy bands are now electron-hole asymmetric
because the fictitious symmetry of Eq.\ (\ref{eq_e-h_sym_AB-AB}) or Eq.\ (\ref{eq_e-h_sym_AB-BA})
is broken.
The band structures of AB-AB and AB-BA are still similar,
but there are several important differences.
At $\Delta =0$, in particular,
the central energy bands of the AB-AB are touching at two points on the $\bar{\Gamma}-\bar{M}$ line,
while they are anti-crossing in the AB-BA.
The band touching of the former is protected by the $C_{2x}$ symmetry.
Since the $k$-points on $\bar{\Gamma}-\bar{M}$ are invariant under $C_{2x}$ operation,
the Bloch states on the line can be characterized by the eigenvalues of $C_{2x}$.
The energy bands crossing at the center actually have
the opposite eignevalues $C_{2x} = \pm 1$, so that they are never hybridized.
The energy bands form a two-dimensional Dirac cone around each band touching point.
Because of $C_{3}$ symmetry, we have six touching points in each single valley.
Note that the energy bands of $\xi=-$ is just 180$^\circ$ rotation of $\xi=+$ band,
so the band touching of $\xi=-$ are not seen in the figure.
In increasing $\Delta$, we see that
the upper central band (the first conduction band) becomes much narrower than
the lower central band (the first valence band), in both of the AB-AB and the AB-BA.
As a result, the energy gap just above the upper band survives
in relatively large $\Delta$,
while the gap below the lower band is easily masked by the wide dispersion.
The properties of the Chern number are mostly carried over from the minimal model.
A difference is seen in $\Delta = 5$ meV in the AB-AB case, where the central bands have the
Chern number $\pm 3$, unlike $\pm 2$ in the minimal model.
This is attributed to the six Dirac points at $\Delta =0$,
each of which contributes to the Berry curvature $\pi$ when gapped out.
In even increasing $\Delta$, we have a band touching at $\bar{K}$ around $\Delta \sim 9$ meV,
where the Chern number $+1$ is transferred from the lower central band to the higher central band.
As a result, the Chern number of the central bands becomes $\pm 2$ as in the minimal model.
A similar topological change is also observed in the AB-BA double bilayer,
where the Chern numbers of the central two bands change
from $(2,0)$ to $(1,1)$.
Finally, we present in Fig.\ \ref{fig_band_angle_dep} the twist angle dependence of the band structure in
(a) the AB-AB and (b) the AB-BA double bilayers with $\Delta = 0$.
For the AB-AB, the central electron and hole bands are separated by an energy gap at $\theta = 2^\circ$,
and they get closer in decreasing $\theta$.
At $\theta \sim 1.44^\circ$, there is a quadratic band touching on $\bar{\Gamma}-\bar{M}$ line,
and a pair of the Dirac points are formed below that angle.
Those band touching points are protected by $C_{2x}$ symmetry as already argued.
In even lower angles, the central energy bands become narrower and narrower,
and at the same time the higher excited bands collides with the central bands.
At $\theta = 1.1^\circ$ and $0.9^\circ$, we have an insulating gap between
the third and the fourth valence bands.
Because of $C_{2x}$ symmetry, the Chern number is zero everywhere as long as $\Delta=0$.
In the AB-BA case, we have a similar evolution of the band structure,
while the Chern number is generally non-zero.
At $\theta = 2^\circ$, the charge neutral point is a valley Hall insulator with the Chern number 2.
In decreasing $\theta$, we have a topological change at $\theta \sim 1.45^\circ$,
where the Chern number 3 is transferred from the lower band to the higher band
through the three touching points arranged in 120$^\circ$ symmetry.
Unlike the AB-AB, the band touching occurs only at the topological transition,
and the bands are separated again after the transition.
In smaller angles less than 1$^\circ$, the central bands touch with the excited bands,
and there are complex topological changes between them.
We have a new insulating gap between the third and the fourth valence bands, where the Chern number is $-2$.
In this work, we assumed different parameters $u$ and $u'$
to describe possible corrugation effect, where the adopted values, $u = 0.0797$eV and $u' = 0.0975$eV,
are taken from the twisted BLG (monolayer-monolayer) \cite{koshino2018maximally}.
In the twisted double BLG, however, the corrugation would be reduced to some extent
considering that bilayer graphene is stiffer than monolayer graphene,
and then the difference between $u$ and $u'$ should also decrease accordingly.
To see this effect, we present the band structure assuming $u=u' = 0.0975$eV
in Fig.\ \ref{fig_band_1.33full_no_corr}, where thin blue lines represent the original calculation
from Fig.\ \ref{fig_band_1.33full}.
We see that the qualitative features are similar, while
the energy gaps between the lowest bands and the excited bands are smaller
in the $u = u'$ model
than in the $u \neq u'$ model, as in the twisted BLG. \cite{koshino2018maximally, nam2017lattice,tarnopolsky2019origin}.
The real situation should be somewhere between the two cases.
\section{Conclusion}
\label{sec_conclusion}
We have studied the electronic band structure and the Chern numbers of
AB-AB and AB-BA twisted double bilayer graphenes,
and found that the two systems have similar band structures, but with completely different
topological properties. In the absence of the asymmetric potential $\Delta$ (perpendicular electric field),
in particular,
the AB-BA double bilayer is a valley Hall insulator when the Fermi energy is in a gap,
while the AB-AB is a trivial insulator due to the symmetry constraint.
Also, the energy bands of the AB-AB in $\Delta= 0$
are entangled by the symmetry protected band touching points, while they are all separated in the AB-BA.
The common features shared by the two systems
is that a pair of narrow bands at the charge neutral point
are immediately gapped by applying the perpendicular electric field, unlike the twisted BLG (monolayer-monolayer).
There the graphite band parameters such as $\gamma_3$, $\gamma_4$ play an important role
in the electron-hole asymmetry, where the electron branch
becomes much narrower than the hole branch in increasing the perpendicular electric field.
\section*{Acknowledgments}
MK thanks the fruitful discussions with Jeil Jung, Pablo Jarillo-Herrero,
Philip Kim, Eslam Khalaf, Jong Yeon Lee and Ashvin Vishwanath.
MK acknowledges the financial support of JSPS KAKENHI Grant Number JP17K05496.
{\it Note added:}
The recent preprints reported experimental observations of
superconductivity and correlated insulating states in the twisted double bilayer graphenes.
\cite{shen2019observation,liu2019spin,cao2019electric}
During the completion of this work,
we became aware of recent theoretical works on the electronic properties of twisted AB-AB double BLG \cite{chebrolu2019flatbands,choi2019intrinsic,lee2019theory}.
Just after the submission of this manuscript, we have come to notice a recent theoretical study on the
electronic and topological properties on twisted multilayer graphene systems
with various stacking configurations. \cite{liu2019quantum}
\begin{figure*}
\begin{center}
\leavevmode\includegraphics[width=0.75\hsize]{fig_band_1.33full_no_corr.eps}
\end{center}
\caption{(a) Band structures of the AB-AB double bilayer at $\theta= 1.33^\circ$
calculated by the full parameter model with $u=u' = 0.0975$eV.
Thin blue lines are the original results with $u = 0.0797$eV and $u' = 0.0975$eV [Fig.\ \ref{fig_band_1.33full}].
(b) Corresponding plots for the AB-BA double bilayer.
}
\label{fig_band_1.33full_no_corr}
\end{figure*}
| 2024-02-18T23:39:55.510Z | 2019-06-10T02:05:11.000Z | algebraic_stack_train_0000 | 836 | 4,894 |
|
proofpile-arXiv_065-4153 | \section{Introduction}
There is rising interest in employing robots such as unmanned aerial vehicles (UAVs) for search and rescue operations. Locating injured victims early is crucial to reduce suffering and mortality rates, and UAVs have the potential to search large areas quickly. Emergency services have begun experimenting with incorporating remote controlled drones or aerial robots with some level of autonomy. However, how to do this efficiently remains an open problem
Rescue robots is an active research area in AI and robotics, tackling a range of problems from the logistics of aid delivery and communications deployment, down to sensor and motion planning of individual robots.
Here we focus exclusively on how to improve the search part of the problem, envisioning an automatic and near-optimal solution for aerial search that can be used both as a component of a larger robotic system, or decision support for first-responders.
Most work on directed exploration in robotics stems from mapping or monitoring of the environment, e.g. \cite{singh2009efficient}, which ignores the full structure of the SAR problem. Even works explicitly on SAR tend to take a narrow view of the problem, such as controlling a sensor to maximize information gain. Heuristics \citep{waharte2010supporting} and exhaustive search via terrain coverage maximization \citep{huang2001optimal,xu2011optimal} seem common in practice. However, many disasters such as earthquakes, flooding, and even terrorist attacks have a multitude of victims and can cover large areas, where approaches needing human guidance take valuable time away from rescuers. We instead seek a more hands-off probabilistic solution, where humans encode their domain knowledge via priors, and are then free to focus on rescue efforts while the aerial robots do what they do best - scout large areas.\looseness=-1000
Ideally, the system should be autonomous, robust to the complexities and uncertainty of disasters, and flexible to the requirements of disaster management. Here we leverage probabilistic generative models that reflect the structure of the problem, and solve it as accurately as possible within the stipulated real-time requirements by using approximate probabilistic inference and planning. We need a model that reflects the desiderata that goes into human decision making, and update it in real-time. We propose to separately model the population density, injury probability and detection probability. In conjunction with a terrain-based exploration time, inferring these factors allows us to optimize the sequence of exploratory moves to minimize discovery time. Further, we both allow strong priors on these variables, and can revert back to a spatial process when the priors are uninformative or inaccurate. For example, we can automatically pull estimates of population density from geographic information systems (GIS). A priori, a densely populated area is likely to contain more victims than a sparsely populated one, and a field or road is both faster and easier to explore than a forest. In the case of an earthquake, areas near buildings are likely to contain more victims than roads. During a terrorist attack, whatever early information exists can be encoded and explored first. Without further information, the system should sample high population areas, and if it stumbles on victims, \textit{learn} a local adjustment in the injury model and focus on that area. Although simpler intensity maps and Bayesian methods have previously been used for prioritized search (c.f. \cite{waharte2010supporting,morere2017sequential}), to the best of our knowledge this is the first attempt at capturing the full structure of the problem in probabilistic model that can be updated in real-time.
However, real-time search in such spatial problems is very challenging for three main reasons: i) only a small part of the point pattern is observed early in the search, ii) the parameter space is high-dimensional and real-time sequential inference is therefore computationally challenging, and iii) optimal search requires solving a computationally hard planning problem under uncertainty, and we need to solve it in real-time.
The main contributions of this paper are:
\vspace{-1em}
\begin{itemize}
\setlength\itemsep{-0.2em}
\item A powerful structured probabilistic model for the search and rescue domain with the possibility to make effective use of prior information
\item Real-time inference via deterministic Integrated Nested Laplace Approximation (INLA), so the model can be updated online
\item A real-time approximation to the search planning problem using a novel variant of Monte-Carlo tree search with options.
\end{itemize}
The remainder of this paper is organized as follows. We first define the proposed structured probabilistic model and the hierarchical spatial point processes underpinning it. Then we describe how INLA was used for real-time approximate inference in section~\ref{sec:inla}. The search planning problem and the Monte-Carlo tree search approximation is introduced in section~\ref{sec:mcts}, and finally we present results on a range of search and rescue scenarios in section~\ref{sec:experiments}.
\section{A Hierarchical Spatial Point Process for Search-and-Rescue Applications}\label{sec:spatial_processes}
\subsection{Model overview}
Learning a spatial process for real-time search is a difficult inference since only a small part of the point pattern is observed at any given point in time. We therefore develop a structured hierarchical spatial point process model that allows us to complement the observed point pattern with prior knowledge, for example about the terrain or cell phone traffic in the search area.
Let $\mathbf{Y}=(\mathbf{y}_{1},\ldots,\mathbf{y}_{n})$
denote a spatial \emph{point pattern} over a region of interest $S \subset\mathbb{R}^{2}$, for example the observed spatial locations of $n$ individuals. The simplest example of a \emph{point process} model for such data is the homogeneous Poisson process for which points are uniformly distributed over $S$ with constant intensity $\lambda$.
Search-and-rescue scenarios have a more complex \emph{marked point pattern} where a given person i) may or may not be detected by a searching UAV, and ii) may or may not be injured. We propose a model built up by three interacting spatial fields:
\begin{itemize}
\setlength\itemsep{-0.2em}
\item the population intensity $\lambda(\mathbf{s})$,
\item the detection probability $r(\mathbf{s})$,
\item the probability of being injured $q(\mathbf{s})$,
\end{itemize}
where $\mathbf{s} \in S$. The following subsection gives the details for each of the fields, and how specific prior information can be used in each of the three parts of the process.
\begin{figure*}[ht]
\vspace{1in}
\includegraphics[width=81mm]{figs/b_r_f_fo_w_layers.png}\hspace{0.5cm}
\includegraphics[trim=0mm 3mm 0mm 0mm,width=77mm]{figs/dataRealization.png}
\caption{Left: Map of the town of Gamleby, Sweden with buildings (orange), forest (dark green), fields (light green), roads (light grey) and water (dark grey) marked out. Right: a sample realization from the model showing the population intensity overlayed by persons in the area (filled dots), detected persons (green circles) and injured persons (blue crosses). There is increased population intensity at buildings and roads, decreased population density in water, lower detection probability in the forest and increased probability of injury due to an explosion in the southwest part of the town. The displayed points have been thinned out by a factor 10 for visualization purposes.}\label{fig:modelRealization}
\end{figure*}
\subsection{Population model}
Let $\mathbf{Y^\star}=(\mathbf{y}_{1}^\star,\ldots,\mathbf{y}_{n}^\star)$ denote the point pattern of persons over a region $S$, and let $N_{y^\star}(\tilde{S})$ denote the number of persons in the subset $\tilde{S} \subset S$. We model the point pattern $\mathbf{Y^\star}$ by a Log Gaussian Cox Process (LGCP, \cite{moller1998log})
\[
N_{y^\star}(\tilde{S})|\lambda\sim\mathsf{Poisson}\left(\int_{\mathbf{s}\in\tilde{S}}\lambda(\mathbf{s})d\mathbf{s}\right),
\]
with log intensity surface given by
\[
\log\lambda(\mathbf{s})=\alpha_{\lambda}+\mathbf{x}_{\lambda}^{\top}(\mathbf{s})\boldsymbol{\beta}_{\lambda}+\xi_{\lambda}(\mathbf{s}),
\]
where $\alpha_{\lambda}$ is an intercept, $\mathbf{x}_{\lambda}(\mathbf{s})$
are spatial covariates, $\boldsymbol{\beta}_{\lambda}$ are regression
coefficients and $\xi_{\lambda}(\mathbf{s})$ is a zero mean Gaussian
Process (GP) over $S$. The spatial covariates $\mathbf{x}_{\lambda}(\mathbf{s})$
could contain any spatial prior information that helps explain the population intensity $\lambda$, for example the location of buildings and water; such information is readily available from GIS systems. The remaining part
of $\lambda$ is modeled as a GP $\xi_{\lambda}(\mathbf{s})$ with a smooth kernel, as people tend to cluster together. We will throughout this
paper focus on GP kernels from the Mat\'ern family.
\subsection{Detection model}
The interpretation of $\mathbf{Y^\star}$ are all persons that could be possibly observed by a UAV if flown over at low height and at good sighting conditions. In practice, conditions may not be perfect and we model the persons actually observed $\mathbf{Y}$ through the detection probability $r(\mathbf{s})$ of observing a person at point $\mathbf{s}$, generating a thinned Poisson process
\[
N_{y}(\tilde{S})|r,\lambda\sim\mathsf{Poisson}\left(\int_{\mathbf{s}\in\tilde{S}}r(\mathbf{s})\lambda(\mathbf{s})d\mathbf{s}\right),
\]
where $N_{y}(\tilde{S})$ denotes the number of observed
persons in $\tilde{S}$. The detection probability $r(\mathbf{s})$ is modeled with
\[
\log r(\mathbf{s})=\mathbf{x}_{r}^{\top}(\mathbf{s})\boldsymbol{\beta}_{r},
\]
where $\mathbf{x}_{r}(\mathbf{s})$ contain prior information about for example terrain type that might affect visibility. $r(\mathbf{s})$ is technically a detection \emph{rate}, since positive values of $\mathbf{x}_{r}^{\top}(\mathbf{s})\boldsymbol{\beta}_{r}$ leads to values for $r(\mathbf{s})$ greater than $1$. However, it can be interpreted as a detection probability when $\mathbf{x}_{r}^{\top}(\mathbf{s})\boldsymbol{\beta}_{r}$ is non-negative for all $\mathbf{s}$, which, if not already the case, could always be achieved by re-balancing the model and increasing the base population level $\alpha_\lambda$ and modifying $\mathbf{x}_{r}^{\top}(\mathbf{s})$ and $\boldsymbol{\beta}_{r}$.
\subsection{Injury model}
We assume that when the disaster strikes all persons have a spatially varying probability $q(\mathbf{s})$ of being injured. Let $\mathbf{w}$ be a binary vector of length $n$ with $w_{i}=1$ iff the $i$th detected person is injured, and $w_{i}=0$ otherwise. This is an example of marked point pattern where each observed point is marked by a binary variable. We assume a geostatistical marking process where the marking process (injured) is independent of the point pattern process (pattern of detected persons) and assumed to follow
\[
w_{i}|q \sim \mathsf{Bernoulli}\left(q(\mathbf{y}_{i})\right),
\]
where
\[
\log\left(\frac{q(\mathbf{s})}{1-q(\mathbf{s})}\right)=\alpha_{q}+\mathbf{x}_{q}^{\top}(\mathbf{s})\boldsymbol{\beta}_{q}+\xi_{q}(\mathbf{s}),
\]
where $\alpha_{q}$ is an intercept, $\mathbf{x}_{q}(\mathbf{s})$
are spatial covariates, $\boldsymbol{\beta}_{q}$ are regression coefficients
and $\xi_{q}(\mathbf{s})$ is a GP. The covariates $\mathbf{x}_{q}(\mathbf{s})$ could for example be large near buildings in an earthquake scenario.
Figure \ref{fig:modelRealization} displays a sample realization from the model over a map of the town of Gamleby in Sweden. A graphical representation of the model is given later in Figure \ref{fig:graphicalModel} when the inference of the model parameters is discussed.
\section{Real-time online learning using INLA}\label{sec:inla}
The Integrated Nested Laplace Approximation (INLA, \cite{rue2009approximate}) is a fast and memory efficient approximate Bayesian learning algorithm which we show can be successfully applied to our model for sequential learning under real-time constraints.
\subsection{Lattice formulation}\label{sec:inla_lattice}
When learning the model, we assume that the domain $\left(0,x_{1,max}\right)\times\left(0,x_{2,max}\right)$
is rectangular and has been split up into $a_{1}\times a_{2}$ equally
sized rectangles, each with area $\Delta=\frac{x_{1,max}x_{2,max}}{a_{1}a_{2}}$. This approach was previously used for spatial point process models with INLA by \cite{illian2012toolbox}.
Define $n_{ij}$ as the number of detectable persons in cell $s_{ij}$
and $m_{ij}$ as the number of detectable injured persons in $s_{ij}$.
Define $E_{ij}=\Delta$ if $s_{ij}$ has not been visited by the UAV and
$E_{ij}=0$ otherwise. Now, conditional on the latent fields, the joint distribution of the number of persons and injured in cell $s_{ij}$ is given by
\[
n_{ij}|z_{\lambda},z_{r}\sim\mathsf{Poisson}\left(E_{ij}\exp\left(z_{\lambda,ij}+z_{r,ij}\right)\right)
\]
\[
m_{ij}|n_{ij},z_{q}\sim\mathsf{Binomial}\left(n_{ij},\frac{\exp\left(z_{q,ij}\right)}{1+\exp\left(z_{q,ij}\right)}\right)
\]
where $z_{\lambda,ij}$, $z_{r,ij}$, and $z_{q,ij}$ are representative
values of $Z_{\lambda}(s)\equiv \log \lambda(\mathbf{s})$, $Z_{r}(s) \equiv \log r(\mathbf{s}) $ and $Z_{q}(s)\equiv \log(q(\mathbf{s})/(1-q(\mathbf{s})))$ in $s_{ij}$.
Inference is simplified by observing since $\alpha_{q,ij}$ and $\boldsymbol{\beta}_{q}$ only enter the binomial injury model and the log link in both the population and the detection models imply that
\[
z_{\lambda,ij}+z_{r,ij}=\alpha_{\lambda,ij}+\mathbf{x}^\top_{\lambda,ij}\boldsymbol{\beta}_{\lambda}+\mathbf{x}^\top_{r,ij}\boldsymbol{\beta}_{r}+\xi_{\lambda,ij}.
\]
To avoid identification problems between $\boldsymbol{\beta}_{\lambda}$ and $\boldsymbol{\beta}_{r}$ we need to use different covariates in the population process $\mathbf{x}_{\lambda}$ and in the detection process $\mathbf{x}_{r}$.
A graphical representation of the model is given in Figure \ref{fig:graphicalModel}.
\begin{figure}[ht]
\tikz{
\node[latent, minimum size=1.9cm, xshift=+0.0cm, yshift = -0.2cm] (nUnObs) {};
\node[obs, minimum size=1.9cm, xshift=+0.4cm, yshift = +0.2cm] (n) {${\scriptstyle n_{ij}\sim \mathrm{Pois}}$};
\node[rectangle, draw, inner sep=0, xshift=+0.2cm] (poisLattice)
{
\tikz{\draw[step=5mm, thin, dotted] (0,0) grid (2.4,2.4);}
};
\node[cloud, scale=0.8, draw, above=of poisLattice, xshift=1.5cm, yshift=-0.05cm] (xiLambda) {$\xi_\lambda$};
\node[latent, right=of xiLambda, xshift=-0.9cm, yshift=-1.2cm] (thetaLambda) {$\boldsymbol{\theta}_\lambda$};
\edge[thick] {thetaLambda} {xiLambda} ;
\node[const, above=of poisLattice, xshift=-1cm, yshift=0.3cm] (Xlambda) {$\mathbf{x}_\lambda$} ;
\node[latent, above=of poisLattice, xshift=0.0cm, yshift=0.1cm] (betaLambda) {$\boldsymbol{\beta}_\lambda$} ;
\edge[thick] {betaLambda,xiLambda} {poisLattice} ;
\edge[thick, shorten <= 2pt] {Xlambda,xiLambda} {poisLattice} ;
\node[const, right=of poisLattice, xshift=0.2cm, yshift=-0.2cm] (Xr) {$\mathbf{x}_r$} ;
\node[latent, right=of poisLattice, xshift=0.0cm, yshift=-0.8cm] (betar) {$\boldsymbol{\beta}_r$} ;
\edge[thick] {betar} {poisLattice}
\edge[thick, shorten <= 2pt] {Xr} {poisLattice} ;
\node[latent, minimum size=1.9cm, xshift=+5.3cm, yshift = -0.2cm] (mUnObs) {};
\node[obs, right=of n, minimum size=1.9cm, xshift=+2.4cm] (m) {${\scriptstyle m_{ij}\vert n_{ij} \sim \mathrm{Bin}}$};
\edge[thick] {n} {m} ;
\node[rectangle, draw, inner sep=0, xshift=+5.5cm] (binLattice)
{
\tikz{\draw[step=5mm, thin, dotted] (0,0) grid (2.4,2.4);}
};
\node[cloud, scale=0.8, draw, above=of binLattice, xshift=-1.5cm, yshift=-0.05cm] (xiq) {$\xi_q$};
\node[latent, left=of xiq, xshift=+0.8cm, yshift=-1.2cm] (thetaq) {$\boldsymbol{\theta}_q$};
\edge[thick] {thetaq} {xiq} ;
\node[const, above=of binLattice, xshift=1cm, yshift=0.3cm] (Xq) {$\mathbf{x}_q$} ;
\node[latent, above=of binLattice, xshift=0.0cm, yshift=0.1cm] (betaq) {$\boldsymbol{\beta}_q$} ;
\edge[thick] {betaq,xiq} {binLattice} ;
\edge[thick, shorten <= 1pt] {Xq} {binLattice} ;
}
\caption{Graphical model representation of the lattice formulation of the model at a given search iteration. The chequered squares represent lattices and the nodes inscribed in them represent the variables $n_{ij}$ and $m_{ij}\vert n_{ij}$ over the lattice (grey = observed, white = unobserved). Cloud nodes represent Gaussian Processes over $\mathbb{R}^2$.}\label{fig:graphicalModel}
\end{figure}
\subsection{Integrated Nested Laplace Approximation}
Inference in models with high-dimensional spatial random fields is a challenging problem, and exact methods such as Markov Chain Monte Carlo (MCMC) are much too slow for real-time learning. Variational inference is the go-to method in machine learning for fast Bayesian inference, but is well known to underestimate posterior uncertainty, particularly in high-dimensional spatial problems, see e.g. \cite{rue2009approximate}.
INLA \citep{rue2009approximate} is a framework for fast accurate approximation of Bayesian posterior distributions in the class of latent Gaussian models. INLA is by now a standard method for spatial problems in the statistical literature, but is rarely used in Robotics. To describe the class of latent Gaussian models, let $\mathbf{z} \in \mathbb{R}^d$ be a (high-dimensional) vector of Gaussian variables with prior $\pi(\mathbf{z} \vert \boldsymbol{\theta})=N(\mathbf{0},Q(\boldsymbol{\theta})^{-1})$, where $Q(\boldsymbol{\theta})$ is a sparse precision (inverse covariance) matrix. In our case $\mathbf{z}$ contains the Gaussian random fields $\xi_{\lambda}(\mathbf{s})$, $\xi_q(\mathbf{s})$ as well as the fixed effects $\beta_\lambda$, $\beta_r$ and $\beta_q$. The (low-dimensional) vector of hyperparameters in the prior $\boldsymbol{\theta}$ contains the variances and length scales of the Mat\'ern kernel functions for $\xi_{\lambda}(\mathbf{s})$ and $\xi_q(\mathbf{s})$, and unknown hyperparameters in the priors for the fixed effects. INLA further assumes the $n$ observations in $\mathbf{y}$ are independent conditional on the latent variables $\mathbf{z}$, with likelihood function
\begin{equation}\label{eq:INLAlikelihood}
\pi(\mathbf{y} \vert \mathbf{z}, \boldsymbol{\theta}) = \prod_{i=1}^n \pi(y_i \vert z_i, \boldsymbol{\theta}).
\end{equation}
The vector $\boldsymbol{\theta}$ can include additional hyperparameters needed to describe $\pi(y_i \vert z_i, \boldsymbol{\theta})$. Our likelihood for the Poisson model for detected persons, $n_{ij}$, and the likelihood of the binomial model for number of injured persons, $m_{i,j}$, are clearly both of the form \eqref{eq:INLAlikelihood}.
INLA uses an intricate mix of several Laplace approximations for the high-dimensional $\mathbf{z}$ combined with numerical integration of the low-dimensional hyperparameters $\boldsymbol{\theta}$ to approximate the marginal posteriors of the latent variables $z_i$ and the joint posterior of the hyperparameters $\pi (\boldsymbol{\theta}\vert \mathbf{y})$. The basic INLA approximation is of the form
\[
\pi(z_i \vert \mathbf{y}) \approx \int \tilde \pi(z_i \vert \boldsymbol{\theta},\mathbf{y}) \tilde \pi(\boldsymbol{\theta}\vert\mathbf{y})d\boldsymbol{\theta}.\label{eq:INLAmargApprox}
\]
where $\tilde \pi(z_i \vert \boldsymbol{\theta},\mathbf{y})$ is obtained by marginalizing a Laplace approximation \citep{tierney1986accurate} of $\pi(\mathbf{z} \vert \boldsymbol{\theta},\mathbf{y})$ and
\[
\tilde \pi(\boldsymbol{\theta}\vert\mathbf{y}) = \frac{\pi( \mathbf{z},\boldsymbol{\theta}, \mathbf{y})}{\tilde \pi_G (\mathbf{z} \vert \boldsymbol{\theta}, \mathbf{y} )} \bigg\vert_{\mathbf{z}=\mathbf{z}^\star(\boldsymbol{\theta})},
\]
where $\tilde \pi_G (\mathbf{z} \vert \boldsymbol{\theta}, \mathbf{y} )$ is a Gaussian approximation to the full conditional posterior of $\mathbf{z}$ and $\mathbf{z}^\star$ is the mode of $\mathbf{z}$ for a given $\boldsymbol{\theta}$. The integral in \eqref{eq:INLAmargApprox} is performed numerically by summing over a set of carefully selected support points in $\boldsymbol{\theta}$, see \cite{rue2009approximate} for details. Note that INLA does not approximate $\pi(\boldsymbol{\theta}\vert\mathbf{y})$ by a Gaussian, which is important since $\pi(\boldsymbol{\theta}\vert\mathbf{y})$ is often highly non-Gaussian.
The use of \emph{nested} Laplace approximations makes INLA extremely accurate for latent Gaussian models, see for example \cite{rue2009approximate} and \cite{teng2017bayesian} for some evidence for the Log Gaussian Cox Process. In particular, INLA has been shown to be much more accurate than variational approximations.
INLA is many orders of magnitude faster than Markov Chain Monte Carlo (MCMC) and Hamiltonian Monte Carlo (HMC), and can therefore be successfully applied in a real-time context. By exploiting the sparsity of the precision matrix $Q(\boldsymbol{\theta})$ that results from conditional independencies that appear naturally in spatial and temporal problems and efficient reordering schemes \citep{rue2005gaussian}, INLA scales favorably as $O(d^{3/2})$ in 2D, where $d$ is the total number of cells where the fields are evaluated. Moreover, since our focus here is on $\pi(x_i \vert \mathbf{y})$ rather than hyperparameter inference per se, we will use the Empirical Bayes (EB) to optimize wrt $\boldsymbol{\theta}$. This gives additional speed-ups since we can also benefit from a warm start with excellent initial values from the previous search iteration, followed by an fast update of the posterior $\pi(x_i \vert \mathbf{y}, \boldsymbol{\hat\theta}_{EB})$.
The core of INLA is implemented in C++ with the convenient \textsf{r-inla} interface to the statistical programming language R, see \cite{rue2017bayesian} for details.
\section{Planning Exploratory Moves via MCTS}\label{sec:mcts}
The search part of search and rescue has traditionally been solved by a variety of exploration methods, as e.g. a coverage maximization problem \citep{huang2001optimal}, travelling salesman problem to minimize a path cost, or as information maximization in POMDPs \citep{waharte2010supporting}. However, attempting to solve these sequential decision problems optimally usually result in computationally hard problems. Greedy heuristics or simple coverage algorithms, such as lawnmower patterns, are therefore often employed in search and rescue applications.
As we employ a sophisticated probabilistic model, it is natural to view the decision problem as a partially observable Markov decision process, or POMDP \citep{ASTROM1965174}. Only the visited regions are observed, which we use to infer a belief over the remaining spatial point process.
Formally, we define the search state of our problem to be $\mathbf{x} = (\mathrm{Bel}\(\mathbf{m}), \mathbf{p})$. The matrix $\mathbf{m}$ consists of all $m_{ij}$, random variables for the number of injured in the $\Delta$-sized discretization of the spatial point process from section~\ref{sec:inla_lattice}. We use the notation $\mathrm{Bel}\(\mathbf{m})$ for the distribution of $\mathbf{m}$. We also define $\mathbf{p} \in \{s_{ij}:\forall\,(i,j)\}$ as the position of the UAV in the cell grid, which can be considered known by GPS at the lengthscales of our cell size. As the $\mathrm{Bel}\(\mathbf{m})$ are probability distributions, this is technically a belief-augmented MDP formulation (c.f. \cite{thrun2005probabilistic}).
The UAV has to sequentially decide on which cell to explore, in the form of actions $a \in \mathcal{A} = \{s_{ij} : \forall\,(i,j)\}$ that map uniquely to exploration of a cell $s_{ij}$. Cell exploration takes time $T_{ij}= \mathbf{x}_r^{\top}(\mathbf{s})\boldsymbol{\beta}_{T}$, where $\mathbf{x}_r(\mathbf{s})$ are the aforementioned spatial covariates for terrain type from section~$\ref{sec:inla}$, and $\boldsymbol{\beta}_T$ are user-supplied estimates of their respective exploration time. However, the UAV cannot teleport between cells, it additionally takes time $T_f \cdot \mathrm{dist}(\mathbf{p}, s_{ij})$ to reach a non-adjacent cell $s_{ij}$. Clearly, a sequential exploration of adjacent cells takes less time but may not discover the most injured, resulting in a difficult trade-off.
Finally, we want to solve the optimal exploration problem
\[
\argmin_{\pi(\mathbf{x})} \; \mathbb{E}_{\boldsymbol{\tau}|\pi(\mathbf{x})}[c(\boldsymbol{\tau})],
\]
where $c(\boldsymbol{\tau})$ is a cost function and $\boldsymbol{\tau}=\{\mathbf{x}_{t_i}...\mathbf{x}_{t_N}\}$ is the trajectory through the belief-augmented state space $\mathbf{x}_t$, from current time $t_i$ until all cells have been explored at time $t_N$. The trajectory is uncertain as the injuries $\mathbf{m}$ are partially unknown at decision time. By taking action $a_t = \pi(\mathbf{x}_t)$, where $a_t$ maps to a cell $s_{ij}$, data $o_t=(n_{ij},m_{ij})$ will be observed and beliefs will be updated by $\mathrm{Bel}\_{t+1}(\mathbf{m}) = f(\mathrm{Bel}\_{t}(\mathbf{m}),a_t,o_t)$, where the transition function $f$ updates the spatial point process as described in section \ref{sec:inla}.
While maximizing information is popular in POMDP formulations of search (c.f. \cite{morere2017sequential, waharte2010supporting}), not all information is equally useful in more complex models, e.g. one may have large uncertainty but low expectation in sparsely populated areas. Contrary to earlier work, we take a more direct approach that at each time $t_i$ attempts to minimize the total remaining harm to victims,
\[
c(\boldsymbol{\tau}) = \int_{t_i}^{t_N} \sum_{ij} m^*_{ij}(t)h(t)\,dt,
\]
where $m^*_{ij}(t)$ is the number of \textit{unexplored} injured in cell $s_{ij}$ at time $t$, and $h(t)$ is the rate of harm. This is integrated with the trapezoid rule over the varying durations of the discrete sequence of actions. For convenience, in the following we assume the rate of harm (e.g. mortality) is proportional to the time spent undiscovered, $h(t) \propto 1$.
Unfortunately, the complexity of solving POMDPs is doubly-exponential \citep{thrun2005probabilistic}. Even with a discretized state lattice and four directional actions, this problem will be infeasible for real-world sizes.
Monte-Carlo tree search (MCTS) is an approximate solution to discrete sequential decision making problems. In its purest form, MCTS is a tree search algorithm that treats the problem of finding good branches (actions) as a sequence of bandit problems solved by the UCB (Upper Confidence Bounds) algorithm. The resulting UCT algorithm \citep{kocsis2006bandit} effectively treats finding good plans as an exploration problem in itself. By guiding the search, the effective branching factor can be significantly reduced.
While most well-known for its successes in the game of Go \citep{browne2012survey,silver2017go}, MCTS has recently found uses in motion planning problems \citep{hennes2015interplanetary}. In \cite{morere2017sequential} MCTS was also used to plan belief trajectories in a POMDP for environment monitoring. However, POMDP planning is still very expensive, and they could only afford to plan three steps forward. The reason is that for each tested action in the search tree, uncertain outcomes have to be sampled, and new beliefs inferred as in section~\ref{sec:inla}.
To create an effective MCTS algorithm for our search domain we propose three modifications, i) incorporating action-abstractions, a small set of long-range moves $\mathcal{A}_{\mathrm{max}}$ directly to the cells $s_{ij}$ with the maximum expected number of unexplored injured $m^*_{ij}(t_i)$, ii) a certainty equivalence assumption where random variables are replaced with their expected values, and iii) using receding-horizon planning with warm-starts and a fast domain-specific cost-to-go approximation.
We test two variants. The first, simply called \texttt{MCTS}, can only move through adjacent squares $\mathcal{A}_\mathtt{MCTS} = \{s_{ij} : \operatorname{adj}(\mathbf{p}, s_{ij})\}$, to explore, or if explored, fly through. The second, called \texttt{MCTSJump}, additionally includes the long-range moves from i) in the first time step $t_i$ of each plan, i.e. $\mathcal{A}_{\mathtt{MCTSJump},t_i} = \mathcal{A}_{\mathrm{max}}\,\cup\,\mathcal{A}_\mathtt{MCTS}$. An example search pattern from \texttt{MCTSJump} can be seen in Figure~\ref{fig:MCTS_example}.
These action-abstractions can be seen as a type of option policies \citep{stolle2002learning,subramanian2016efficient} in reinforcement learning. Adding more actions increases the branching factor, which results in a difficult trade-off. However, we found that instead of selecting target cells equally spaced or random, using just the top ten best cells resulted in a noticeable performance increase. This also has the benefit of remedying the biggest drawback of iii), the finite planning horizon often made it leave some cells unexplored in the end.
\begin{figure}[ht]
\includegraphics[clip,width=\linewidth]{figs/mcts_plan.png}
\caption{\label{fig:MCTS_example}Search scenario using \texttt{MCTSJump}, overlaid on heatmap for expected number of detectable injured.}
\end{figure}
The certainty equivalence assumption $\mathrm{Bel}\_{t+1}(\mathbf{m}) \approx f(\delta_{\mathbb{E}_t(\mathbf{m})}(\mathbf{m}),a_t,o_t)$, where $\delta_a(\cdot)$ is the Dirac spike at $a$, allows us to forego sampling from the outcomes to update the belief model, which is the main bottleneck of planning in POMDPs. This alone let us effectively plan $20$ steps ahead. However, it can be a strong assumption, because plans are \textit{evaluated} on the premise that the future is predictable, which means it will not value recourse, the possibility to later change the plan if it turns out worse than expected. In practice it still replans at each step however, and due to the spatial correlation in our problem, we also argue that the effect is small. Significant recourse is costly - it often takes several moves to see large changes.
Finally, the receding-horizon formulation from iii) is standard in control, where it is sometimes known as model-predictive control. By cutting the planning horizon from $t_N$ to $t_H$ and adding a domain-specific approximation of the remaining cost $c(\boldsymbol{\tau})=c(\boldsymbol{\tau}_{t_i..t_H})+\hat{c}_{t_H..t_N}(\mathbf{x}_{t_H})$, computation cost can be greatly decreased. Warm-starts at from the best plan at $t_{i-1}$ allows computation to be amortized over several iterations. Here we just assume the cost-to-go decreases linearly to zero as for an ideal lawnmower pattern. This allows us to compare plans of different duration, such as action-abstractions moves from i).
As MCTS is an any-time algorithm we give it a fixed 3 second compute budget. It is implemented in C++ and evaluates about 100 000 plans. As it does not assume any fixed search pattern or observation order, it also allows human operators the flexibility to take control if needed.
\section{Experiments}\label{sec:experiments}
Here we test the proposed real-time probabilistic SAR framework and MCTS exploration algorithm. To the best of our knowledge, this is the first principled attempt at probabilistic modelling of the full complexity of the SAR problem, and explicit minimization of harm, in real-time or otherwise. As noted in the introduction, related probabilistic approaches appear to have a narrower focus in what they model. On the planning side, existing principled optimization-based approaches typically scale poorly \citep{waharte2010supporting,morere2017sequential} to the large scenarios we envison. Maximum coverage algorithms \citep{xu2011optimal,huang2001optimal}, or some heuristic combination of locally and globally greedy behavior, are often used in practice. Unfortunately, heuristics tend to be sensitive to scenario-specific tuning. We therefore use a maximum-coverage algorithm as baseline. Fortunately, tree-search algorithms like \texttt{MCTS} subsume locally-greedy algorithms if planning horizon is taken to zero, and should in expectation be a dominating strategy as the planning horizon is increased. Further, our proposed \texttt{MCTSJump} algorithm will also always include the globally-greedy choice in its list of action abstractions, and can therefore be seen as a principled way of solving this trade-off as an optimization problem, rather than as a heuristic needing tuning to a scenario.
We use real-world GIS data from the Swedish government, which offered easy access to a wealth of data on favorable terms\cite{lantmateriet}. We selected a 4.0x2.7km area around the town Gamleby seen in Figure~\ref{fig:modelRealization}. It contains a variety of terrain and is within a proposed UAV test zone, where we may be permitted to test the algorithm with real UAVs.
Data was simulated using the the hierarchical spatial point process model in section~\ref{sec:spatial_processes}, then discretized to 50x33 lattice cells for search and model inference, each about the size of a soccer field. We assume the UAV can fly at a speed of $10$m/s for fly-to moves. The cell explore times are set to $\boldsymbol{\beta_T}=\{1,2,0.5,0.5,0.75\}$ minutes for terrain covariates "buildings", "forest", "road", "field", and "water" respectively. This reflects a fast overhead search with some compensation for difficult terrain such as forests needing multiple angles. The detection covariate for forest was similarly set lower. This rapid search pace also highlights the importance of real-time performance, and why we capped MCTS to 3 seconds. For reference, the entire inference and planning loop in our prototype implementation takes about 5 seconds on a Core i7 CPU, imposing minimal overhead on the search.
In the following we test four scenarios reflecting different types of real-world disasters and the level of prior information available. A summary of the scenarios, the covariates used in the data generating process, as well as those used for the inference models, can be seen in Table~\ref{table:Scenarios}. Each scenario is replicated 15 times from different seeds, except for the first one, which used 30. As our mission area is rectangular, a simple baseline coverage algorithm is a zick-zack, or "lawnmower" pattern. In terms of covering the largest area in the shortest time, this is optimal. In all cases, we attempted to find reasonably wide distributions for parameters in the data generating processes, $\boldsymbol\beta_\lambda$, $\theta_\lambda$, $\boldsymbol\beta_\lambda$ and $\theta_\lambda$, by sampling realizations of the spatial point process and comparing to real-world expectations. We also experimented with perturbing the inference priors and found remarkable small changes in the results, see the supplementary material.
\begin{table*}
\caption{Scenario Settings. Covariates and spatial fields in data generating
process and inferred model. \protect \\
$B$=buildings, $R$=roads, $W$=water, $F$=forest, $G_{i}$=Gaussian no $i$, $S$=spatial field. Deviations from truth in red.}
\label{table:Scenarios}
\centering{}%
\begin{tabular}{llllllll}
\hline
& & & & Scenario A & Scenario B & Scenario C & Scenario D \tabularnewline
& & Population & & $B$ & $B+R+W+S$ & $B+R+W+S$ & $B+R+W+S$\tabularnewline
\textbf{Truth} & & Detection & & $-$ & $F$ & $F$ & $F$\tabularnewline
& & Injury & & $-$ & $B+S$ & $G_{1}$ & $G_{2}+G_{3}$\tabularnewline
& & & & & & & \tabularnewline
& & Population & & ${\color{red}S}$ & $B+R+W+S$ & $B+R+W+S$ & $B+R+W+S$\tabularnewline
\textbf{Model} & & Detection & & $-$ & $F$ & $F$ & $F$\tabularnewline
& & Injury & & $-$ & $B+S$ & ${\color{red}G_{1}+S}$ & ${\color{red}G_{2}+S}$\tabularnewline
\hline
\end{tabular}
\end{table*}
\begin{table*}
\caption{Time until half of injured have been found}
\label{table:Halftime}
\centering{}%
\begin{tabular}{lrrrrr}
\hline
& & Scenario A & Scenario B & Scenario C & Scenario D \tabularnewline
Lawnmower & & 835 & 1113 & 164 & 271 \tabularnewline
MCTS & & 273 & 118 & 88 & 121 \tabularnewline
MCTSjump & & 241 & 98 & 69 & 100 \tabularnewline
\hline
\end{tabular}
\end{table*}
\subsection{Scenario A: No prior information}
In this scenario we assume the model does not have access to any useful spatial covariates. While in practice some information tends to be available, this was designed to test the capability of the model to fall back to the spatial fields, to cover for unexpected situations.
The ground truth is a population distribution drawn from GIS building covariates not available to the agent, see Table~\ref{table:Scenarios}. For simplicity we ignore the injury part of the model and focus only on maximizing the number of people found in this scenario. As can be seen from the results in the top row of Figure~\ref{fig:scenario2prop} and Table~\ref{table:Halftime}, our model with \texttt{MCTS} and \texttt{MCTSJump} significantly outperforms the lawnmower coverage strategy. Just relying on the spatial field was sufficient to capture the natural clustering in population data. In this case however, the improvements offered by long-range moves ("jumps") was not statistically significant, which is not surprising considering the spatial correlation captured by the field only gives local information.
\subsection{Scenario B: Earthquake}
\vspace{-0.7em}
Here we simulate a classical earthquake scenario. We generate population using all five GIS covariates, as well as a spatial field. As this is an earthquake, both building covariates and a field was used to draw realizations of injured people. We use the same structure of the model for inference, and reasonably uninformative priors.
Figure~\ref{fig:scenario2prop} and Table~\ref{table:Halftime} show that by drawing on the GIS covariates, even corrupted by a spatial field, our algorithm is significantly faster than in A, and also increases its lead to lawnmower by a wide margin. This also showcases the advantage of \texttt{MCTSJump}, which outperformed regular \texttt{MCTS} by first drawing on the injury covariate to make informed jumps directly to the urban areas, then using jumps at the end to complete the map. Finite-horizon MCTS can leave some areas unvisited.
\subsection{Scenario C: Terrorist attack - known site}
\vspace{-0.7em}
In this scenario there has been a localized terrorist attack, represented by a Gaussian in the injury field southwest of town. We show that using the proposed model, this can easily be encoded on the fly by first-responders, via e.g. a Gaussian spatial covariate prior centered on the reported site. Figure~\ref{fig:scenario2prop} shows similar performance to Scenario B.
\subsection{Scenario D: Terrorist attack - one site unknown}
\vspace{-0.7em}
Finally, we showcase all the capabilities of the structured model by extending Scenario C. In this case, there is a terrorist attack with one site encoded by a Gaussian spatial prior. However, early information during catastrophes is often incomplete. In this case there is also a second attack site unknown to us. The results show the model quickly picks up on this. In particular, \texttt{MCTSJump} flies to and explores the a priori known site, then without further information will jump around and explore high population areas. At some point it stumbles on injured near the second site, the spatial field quickly learns the local anomaly in injury probability, and the planner focuses on that area. A simulation run of Scenario D is shown in the supplementary video material\footnote{\url{https://youtu.be/wyD0O5hF5tE}} and Figure~\ref{fig:scenario6maps}.\looseness=-1000
\section{Conclusions}
We present a new framework for search-and-rescue based on real-time learning and decision making with a hierarchically structured spatial point process. The model is built from spatially referenced components on which there is usually ample prior information in search-and-rescue problems: i) the distribution of persons, ii) the probability of detecting a person, and iii) the probability of injury. Learning spatial processes and acting on them in real-time is a hard problem. We propose a novel combination of approximate Bayesian learning using INLA combined with a MCTS strategy adapted to the search problem
We assess the empirical performance of the method on several simulated scenarios on a real map with publicly available GIS data, and show that prior information can be very efficiently used in our model to clearly outperform a conventional search strategy. We also demonstrate that the spatial fields can fill in for missing prior information in a very adaptable manner.
The framework proposed here can be extended in many interesting directions, for example to dynamic problems where the intrinsic state variables evolve over time, such as disasters involving gas leakage or a rescue operation at sea. It would be interesting to generalize the model and the approximate inference method to other data distributions and other link functions than the exponential and logistic. While this work indicates that using just one UAV very cleverly can make a large difference, in future work we also intend to extend it to search with a team of real UAV
\vfill
\begin{figure*}[h!]
\vspace{1in}
\includegraphics[width=40mm]{figs/injuredFoundScenario2lawnmower.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario2mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario2mcts_jump.pdf}
\includegraphics[width=40mm]{figs/diffInjuredFoundScenario2mcts_jump_vs_mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario3lawnmower.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario3mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario3mcts_jump.pdf}
\includegraphics[width=40mm]{figs/diffInjuredFoundScenario3mcts_jump_vs_mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario4lawnmower.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario4mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario4mcts_jump.pdf}
\includegraphics[width=40mm]{figs/diffInjuredFoundScenario4mcts_jump_vs_mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario6lawnmower.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario6mcts.pdf}
\includegraphics[width=40mm]{figs/injuredFoundScenario6mcts_jump.pdf}
\includegraphics[width=40mm]{figs/diffInjuredFoundScenario6mcts_jump_vs_mcts.pdf}
\caption{Comparing the proportion of injured found as a function of search time (minutes) for the different strategies. The rows correspond to each of the four scenarios A to D (top down). The graphs show the mean proportion of injured found as a function of search time (solid line) for the three strategies over 30 replicates, as well as the 95\% confidence bands for the mean (darker regions) and 95\% predictive bands for individual proportions in individual replicated datasets (lighter regions). The final column shows that same properties, but for the differences in proportions between MCTS and MCTSjump.}\label{fig:scenario2prop}
\end{figure*}
\bibliographystyle{apalike}
\section{Tables}
All tables must be centered, neat, clean, and legible. Table number
and title always appear above the table. See Table~\ref{table:Scenarios}.
One line space before the table title, one line space after the table
title, and one line space after the table. The table title must be
initial caps and each table numbered consecutively.
\begin{table*}
\caption{Scenario Settings. Covariates and spatial fields in the data generating
process (truth) in the inferred model (model). \protect \\
$B=$buildings, $R=$roads, $W=$water, $F=$forest, $G_{1}=$Gaussian
no 1, $G_{2}=$Gaussian no 2. Differences in bold red font.}
\label{table:Scenarios}
\centering{}%
\begin{tabular}{llll>{\raggedleft}p{2.2cm}>{\raggedleft}p{2.2cm}>{\raggedleft}p{2.2cm}>{\raggedleft}p{2.2cm}>{\raggedleft}p{2.2cm}}
\hline
& & & & Scenario 2 & Scenario 3 & Scenario 4 & Scenario 5 & Scenario 6\tabularnewline
\cline{5-9} \cline{6-9} \cline{7-9} \cline{8-9} \cline{9-9}
& & Population & & $B$ & $B+R+W+S$ & $B+R+W+S$ & & $B+R+W+S$\tabularnewline
\textbf{Truth} & & Detection & & $-$ & $F$ & $F$ & & $F$\tabularnewline
& & Injury & & $-$ & $B+S$ & $G_{1}$ & & $G_{1}+G_{2}$\tabularnewline
& & & & & & & & \tabularnewline
& & Population & & $\mathbf{{\color{red}S}}$ & $B+R+W+S$ & $B+R+W+S$ & & $B+R+W+S$\tabularnewline
\textbf{Model} & & Detection & & $-$ & $F$ & $F$ & & $F$\tabularnewline
& & Injury & & $-$ & $B+S$ & $\mathbf{{\color{red}G_{1}+S}}$ & & $\mathbf{{\color{red}G_{1}+S}}$\tabularnewline
\hline
\end{tabular}
\end{table*}
\newpage
\end{document}
| 2024-02-18T23:39:55.603Z | 2019-03-26T01:41:27.000Z | algebraic_stack_train_0000 | 840 | 6,967 |
|
proofpile-arXiv_065-4295 | \section{Background and introduction}
The aim of this paper is to solve the following tensor problem
\begin{equation}\label{eq1}
{\mathcal{M}} (\mathscr{X}) = \mathscr{C},
\end{equation}
where ${\mathcal M}$ is a linear operator that could be described as
\begin{equation}\label{eq2}
{\mathcal M} (\mathscr{X}) = \mathscr{A} \ast\mathscr{X},
\end{equation}
or as
\begin{equation}\label{eql}
{\mathcal M} (\mathscr{X}) = \mathscr{A} \ast\mathscr{X}\ast\mathscr{B},
\end{equation}
where $\mathscr{A}$, $\mathscr{X}$, $\mathscr{B}$ and $\mathscr{C}$ are three-way tensors, leaving the specific dimensions to be defined later, and $\ast$ is the T-product to be also defined later. To mention but a few applications, problems of these types arise
in engineering \cite{qllz}, signal processing \cite{lb}, data mining \cite{lxnm}, tensor complementarity problems\cite{lzqlxn}, computer vision\cite{vt1, vt2} and graph analysis \cite{kolda2}. For those applications, and so many more, one have to take advantage of this multidimensional structure to build rapid and robust iterative methods for solving large-scale problems. We will then, be interested
in developing robust and fast iterative tensor Krylov subspace methods under tensor-tensor product framework between third-order
tensors, to solve regularized problems originating from color image and video processing applications. Standard and global Krylov subspace methods are suitable when dealing with grayscale images, e.g, \cite{belguide, belguide2, reichel1, reichel2}. However, these methods might be time consuming to numerically solve problems related to multi channel images (e.g. color images, hyper-spectral images and videos). \\
For the Einstein product, both the Einstein tensor global Arnoldi and Einstein
tensor global Gloub-Kahan bidiagonalization algorithms have been established \cite{Elguide}, which makes so natural to generalize these methods using the T-product. In this paper, we will show that the
tensor-tensor product between third-order
tensors allows the application of the global iterative methods, such as the global Arnoldi and global Golub-Kahan algorithms. The tensor form of the proposed Krylov methods, together with using the fast
Fourier transform (FFT) to compute the T-product between third-order
tensors can be efficiently implemented on many modern computers and allows to significantly reduce the overall computational complexity. It is also worth mentioning that our approaches can be naturally generalized to higher-order tensors in a recursive manner.
Our paper is organized as follows. We shall first present in Section 2
some symbols and notations used throughout paper. We also recall the concept T-product between two tensors. In Section 3, we define tensor global Arnoldi and tensor global Golub-Kahan algorithms that allow the use of the T-product. Section 4 reviews the adaptation of Tikhonov regularization for tensor equation (\ref{eq1}) and then proposing a restarting strategy of the so-called tensor global GMRES and tensor global Golub-Kahan approach in connection with Gauss-type quadrature rules to inexpensively compute solution of the regularization of (\ref{eq1}). In Section 5, we give a tensor formulation in the form of (\ref{eq1}) that describes the cross-blurring of color image and then we present a few numerical examples on restoring blurred and noisy color images and videos. Concluding remarks can be found in Section 6.
\section{Definitions and Notations} A tensor is a multidimensional array of data. The number of indices of a tensor is called modes or ways.
Notice that a scalar can be regarded as a zero mode tensor, first mode tensors are vectors and matrices are second mode tensor. The order of a tensor is the dimensionality of the array needed to represent it, also known as
ways or modes.
For a given N-mode (or order-N) tensor $ \mathscr {X}\in \mathbb{R}^{n_{1}\times n_{2}\times n_{3}\ldots \times n_{N}}$, the notation $x_{i_{1},\ldots,i_{N}}$ (with $1\leq i_{j}\leq n_{j}$ and $ j=1,\ldots N $) stands for the element $\left(i_{1},\ldots,i_{N} \right) $ of the tensor $\mathscr {X}$. The norm of a tensor $\mathscr{A}\in \mathbb{R}^{n_1\times n_2\times \cdots \times n_\ell}$ is specified by
\[
\left\| \mathscr{A} \right\|_F^2 = {\sum\limits_{i_1 = 1}^{n_1 } {\sum\limits_{i_2 = 1}^{n_2 } {\cdots\sum\limits_{i_\ell = 1}^{n_\ell} {a_{i_1 i_2 \cdots i_\ell }^2 } } } }^{}.
\] Corresponding to a given tensor $ \mathscr {A}\in \mathbb{R}^{n_{1}\times n_{2}\times n_{3}\ldots \times n_{N}}$, the notation $$ \mathscr {A}_{\underbrace{::\ldots:}_{(N-1)-\text{ times}}k}\; \; {\rm for } \quad k=1,2,\ldots,n_{N}$$ denotes a tensor in $\mathbb{R}^{n_{1}\times n_{2}\times n_{3}\ldots \times n_{N-1}}$ which is obtained by fixing the last index and is called frontal slice. Fibers are the higher-order analogue of matrix rows and columns. A fiber is
defined by fixing all the indexes except one. A matrix column is a mode-1 fiber and a matrix row is a mode-2 fiber. Third-order tensors have column, row and tube fibers. An element $c\in \mathbb{R}^{1\times 1 \times n}$ is called a tubal-scalar of length $n$. More details are found in \cite{kimler1,kolda1}. \\
\begin{figure}[!h]
\begin{center}
\includegraphics[scale=0.4]{fibers1.png}
\caption{ (a) Frontal, (b) horizontal, and (c) lateral slices of a third order tensor. (d) A mode-3 tube fibers. }
\label{fig:fibre111}
\end{center}
\end{figure}
\subsection{Discrete Fourier Transformation}
In this subsection we recall some definitions and properties of the discrete Fourier transformation and the T-product. The Discrete Fourier Transformation (DFT) plays a very important role in the definition of the T-product of tensors. The DFT on
a vector $v \in {\mathcal{R}}^n$ is defined by
\begin{equation}
\label{dft1}
\tilde v= F_n(v) \in {\mathbb{C}}^n,
\end{equation}
where $F_n$ is the matrix defined as
\begin{equation}
\label{dft2}
F_n(v) = \left (
\begin{array}{ccccc}
1 & 1 & 1 &\ldots&1\\
1 & \omega & \omega^2&\ldots&\omega^{n-1}\\
\vdots & \vdots & \vdots &\ldots&\vdots\\
1 & \omega^{n-1}& \omega^{2(n-1)} &\ldots&\omega^{(n-1)(n-1)}\\
\end{array}
\right) \in {\mathbb{C}}^{n \times n},
\end{equation}
where $\omega=e^{\frac{-2 \pi i}{n}}$ with $i^2=-1$. It is not difficult to show that (see \cite{golub1})
\begin{equation}
\label{dft3}
F_n^{*} =\overline F_n,\; {\rm and}\; F_n^{*} F_n=F_n F_n^{*} =nI_n.
\end{equation}
Then $F_n^{-1}= \displaystyle \frac{1}{n} \overline F_n$ which show that $\displaystyle \frac{1}{\sqrt {n}}F_n$ is a unitary matrix.\\ The cost of computing the vector $\tilde v$ directly from \eqref{dft1} is $O(n^2)$. Using the Fast Fourier Transform ({\tt fft}), it will costs $O(nlog(n))$. It is known that
\begin{equation}\label{dft5}
F_n \, {\rm circ}(v)\, F_n^{-1} = {\rm Diag}(\tilde v),\\
\end{equation}
which is equivalent to
\begin{equation}\label{dft6}
F_n \, {\rm circ}(v)\, F_n^{*} = n\,{\rm Diag}(\tilde v),\\
\end{equation}
where
$$
{\rm circ}(v)= \left (
\begin{array}{cccc}
v_1 & v_2 & \ldots & v_n\\
v_2 & v_1 & \ldots & v_3\\
\vdots & \vdots & \ldots & \vdots \\
v_n & v_{n-1} & \ldots & v_1\\
\end{array}
\right ),
$$
and $ {\rm Diag}(\tilde v),$ is the diagonal matrix whose $i$-th diagonal element is $ {\rm Diag}(\tilde v)_i$.
The decomposition \eqref{dft5} shows that the columns of $F_n$ are the eigenvectors of ${\rm circ}(v))^T$.
\subsection{Definitions and properties of the T-product}
In this part, we briefly review some concepts and notations, which play a central role for the elaboration of the global iterative methods based on T-product; see \cite{Braman, Hao,kilmer0,kimler1} for more details. Let $\mathscr {A} \in \mathbb{R}^{n_{1}\times n_{2}\times n_{3}} $ be a third-order tensor, then the operations {\tt bcirc}, {\tt unfold} and {\tt fold} are defined by
$$ {\tt bcirc}(\mathscr {A})=\left( {\begin{array}{*{20}{c}}
{{A_1}}&{{A_{{n_3}}}}&{{A_{{n_{3 - 1}}}}}& \ldots &{{A_2}}\\
{{A_2}}&{{A_1}}&{{A_{{n_3}}}}& \ldots &{{A_3}}\\
\vdots & \ddots & \ddots & \ddots & \vdots \\
{{A_{{n_3}}}}&{{A_{{n_{3 - 1}}}}}& \ddots &{{A_2}}&{{A_1}}
\end{array}} \right) \in {\mathcal{R}}^{ n_1n_2 \times n_2n_3},$$
$${\tt unfold}(\mathscr {A} ) = \begin{pmatrix}
A_{1} \\
A_{2} \\
\vdots \\
A_{n_{3}}\end{pmatrix} \in \mathbb{R}^{n_{2}n_{3}\times m_{2}} , \qquad {\tt fold}({\tt unfold}(\mathscr {A}) ) = \mathscr {A}.$$ Let $\widetilde {\mathscr{A}}$ be the tensor obtained by applying the DFT on all the tubes of the tensor $\mathscr {A}$. With the Matlab command ${\tt fft}$, we have
$$\widetilde {\mathscr{A}}= {\tt fft}(\mathscr {A},[ ],3), \; {\rm and }\;\; {\tt ifft} (\widetilde {\mathscr{A}}, [ ],3)= \mathscr {A},$$
where ${\tt ifft}$ denotes the Inverse Fast Fourier Transform.\\
Let ${\bf A}$ be the matrix
\begin{equation}\label{dft9}
{\bf A}= \left (
\begin{array}{cccc}
{A}^{(1)}& &&\\
& {A}^{(2)}&&\\
&&\ddots&\\
&&&{A}^{(n_3)}\\
\end{array}
\right),
\end{equation}
and the matrices ${A}^{(i)}$'s are the frontal slices of the tensor ${\widetilde {\mathscr{A}}}$.\\
The block circulant matrix ${\tt bcirc}(\mathscr {A})$ can also be block diagonalized by using the DFT and this gives
\begin{equation}\label{dft8}
(F_{n_3} \otimes I_{n_1})\, {\tt bcirc}(\mathscr {A})\, (F_{n_3}^{-1} \otimes I_{n_2})={\bf A},
\end{equation}
As noticed in \cite{kimler1,lu}, the diagonal blocks of the matrix ${\bf A}$ satisfy the following property
\begin{equation}
\label{f1}
\left \{
\begin{array}{ll}
{A}^{(1)} \in {\mathcal{R}}^{n_1 \times n_2}\\
conj({A}^{(i)})= A^{(n_3-i+2)},\\
\end{array}
\right.
\end{equation}
where $conj ({A}^{(i)})$ is the complex conjugate of the matrix ${A}^{(i)}$. Next we recall the definition of the T-product.
\medskip
\begin{definition}
The \textbf{T-product} ($\star $) between two tensors
$\mathscr {A} \in \mathbb{R}^{n_{1}\times n_{2}\times n_{3}} $ and $\mathscr {B} \in \mathbb{R}^{n_{2}\times m\times n_{3}} $ is an ${n_{1}\times m\times n_{3}}$ tensor given by:
$$\mathscr {A} \star \mathscr {B}={\rm fold}({\rm bcirc}(\mathscr {A}){\rm unfold}(\mathscr {B}) ).$$
\end{definition}
Notice that from the relation \eqref{dft9}, we can show that the the product $\mathscr {C}= \mathscr {A} \star \mathscr {B}$ is equivalent to $ {\bf C}= {\bf A}\,{\bf B}$. So, the efficient way to compute the T-product is to use Fast Fourier Transform (FFT).
Using the relation \eqref{f1}, the following algorithm allows us to compute in an efficient way the T-product of the tensors $\mathscr {A}$ and
$\mathscr {B}$, see \cite{lu}.\\
\begin{algorithm}[!ht]
\caption{Computing the T-product via FFT}\label{algo1}
Inputs: $\mathscr {A} \in \mathbb{R}^{n_{1}\times n_{2}\times n_{3}} $ and $\mathscr {B} \in \mathbb{R}^{n_{2}\times m\times n_{3}} $\\
Output: $\mathscr {C}= \mathscr {A} \star \mathscr {B} \in \mathbb{R}^{n_{1}\times m \times n_{3}} $
\begin{enumerate}
\item Compute $\mathscr {\widetilde A}={\tt fft}(\mathscr {A},[ ],3)$ and $\mathscr {\widetilde B}={\tt fft}(\mathscr {B},[ ],3)$.
\item Compute each frontal slices of $\mathscr {\widetilde C}$ by\\
$$C^{(i)}= \left \{
\begin{array}{ll}
A^{(i)} B^{(i)} , \quad \quad\quad i=1,\ldots,\lfloor \displaystyle \frac{{n_3}+1}{2} \rfloor\\
conj({C}^{(n_3+i-2)}),\quad \quad i=\lfloor \displaystyle \frac{{n_3}+1}{2} \rfloor+1,\ldots,n_3 .\\
\end{array}
\right.$$
\item Compute $\mathscr {C}={\tt ifft}(\widetilde {C},[],3)$.
\end{enumerate}
\end{algorithm}
\noindent For the T-product, we have the following definitions
\begin{definition}
The identity tensor $\mathscr{I}_{n_{1}n_{1}n_{3}} $ is the tensor whose first frontal slice is the identity matrix $I_{n_1,n_1}$ and the other frontal slices are all zeros.
\end{definition}
\medskip
\begin{definition}
\begin{enumerate}
\item An $n_{1}\times n_{1} \times n_{3}$ tensor $\mathscr{A}$ is invertible, if there exists a tensor $\mathscr{B}$ of order $n_{1}\times n_{1} \times n_{3}$ such that
$$\mathscr{A} \star \mathscr{B}=\mathscr{I}_{ n_{1} n_{1} n_{3}} \qquad \text{and}\qquad \mathscr{B} \star \mathscr{A}=\mathscr{I}_{ n_{1} n_{1} n_{3}}.$$
In that case, we set $\mathscr{B}=\mathscr{A}^{-1}$. It is clear that $\mathscr{A}$ is invertible if and only if ${\rm bcirc}(\mathscr{A})$ is invertible.
\item The transpose of $\mathscr{A}$ is obtained by transposing each of the frontal slices and then reversing the order of transposed frontal slices 2 through $n_3$.
\item If $\mathscr {A}$, $\mathscr {B}$ and $\mathscr {A}$ are tensors of appropriate order, then
$$(\mathscr {A} \star \mathscr {B}) \star \mathscr {C}= \mathscr {A} \star (\mathscr {B} \star \mathscr {C}).$$
\item Suppose $\mathscr {A}$ and $\mathscr {B}$ are two tensors such $\mathscr {A} \star \mathscr {B}$ and $ \mathscr {B}^T \star \mathscr {A}^T$ are defined. Then $$(\mathscr {A} \star \mathscr {B})^T= \mathscr {B}^T \star \mathscr {A}^T.$$
\end{enumerate}
\end{definition}
\noindent \begin{exe}
If $\mathscr{A} \in \mathbb{R}^{n_{1}\times n_{2}\times 5}$ and its frontal slices are given by the $n_{1}\times n_{2}$
matrices $A_{1}, A_{2}, A_{3}, A_{4}, A_{5}$, then
$$\mathscr{A}^{T} ={\tt fold} \begin{pmatrix}
A_{1}^{T} \\
A_{5}^{T} \\
A_{4}^{T} \\
A_{3}^{T}\\
A_{2}^{T} \\
\end{pmatrix}.$$
\end{exe}
\begin{definition} Let $\mathscr{A}$ and $\mathscr{B}$ two tensors in $\mathbb{R}^{n_1 \times n_2 \times n_3}$. Then
\begin{enumerate}
\item The scalar inner product is defined by
$$\langle \mathscr{A}, \mathscr{B} \rangle = \displaystyle \sum_{i_1=1}^{n_1} \sum_{i_2=1}^{n_2} \sum_{i_3=1}^{n_3} a_{i_1 i_2 i_3}b_{i_1 i_2 i_3}.$$
\item The norm of $\mathscr{A}$ is defined by
$$ \Vert \mathscr{A} \Vert_F=\displaystyle \sqrt{\langle \mathscr{A} , \mathscr{A} \rangle}.$$
\end{enumerate}
\end{definition}
\medskip
\begin{remark}
Another interesting way for computing the scalar product and the associated norm is as follows:
$$\langle \mathscr{A}, \mathscr{B} \rangle = \displaystyle \frac{1}{n_3} \langle {\bf A}, {\bf B} \rangle;\; \; \Vert \mathscr{A} \Vert_F= \displaystyle \frac{1}{\sqrt{n_3}} \Vert {{\bf A}} \Vert_F,$$
where the block diagonal matrix ${\bf A}$ is defined by \eqref{dft9}.
\end{remark}
\begin{definition}
An $n_{1}\times n_{1} \times n_{3}$ tensor $\mathscr{Q}$ is orthogonal if
$$\mathscr{Q}^{T} \star \mathscr{Q}=\mathscr{Q} \star \mathscr{Q}^{T}=\mathscr{I}_{ n_{1} n_{1} n_{3}}.$$
\end{definition}
\begin{lemma}
If $\mathscr{Q}$ is an orthogonal tensor, then
$$ \left\|\mathscr{Q} \star \mathscr{A}\right\|_F= \left\| \mathscr{A}\right\|_F.$$
\end{lemma}
\begin{definition}\cite{kimler1}
A tensor is called f-diagonal if its frontal slices are orthogonal matrices. It is called upper triangular if all its frontal slices are upper triangular.
\end{definition}
\begin{definition}\label{bloctens0}\cite{miaoTfunction}{(Block tensor based on T-product)}
Suppose $\mathscr{A} \in {\mathbb R}^{n_{1}\times m_{1} \times n_{3}} $, $\mathscr{B}\in {\mathbb R}^{n_{1}\times m_{2} \times n_{3}}$, $\mathscr{C} \in {\mathbb R}^{n_{2}\times m_{1} \times n_{3}} $ and $\mathscr{D}\in {\mathbb R}^{n_{2}\times m_{2} \times n_{3}}$ are four tensors. The block tensor
$$\left[ {\begin{array}{*{20}{c}}
{\mathscr{A}}&{\mathscr{B}} \\
{\mathscr{C}}&{\mathscr{D}} \\
\end{array}} \right]\in {\mathbb R}^{(n_{1}+n_2)\times (m_1+m_{2}) \times n_{3}} $$
is defined by compositing the frontal slices of the four tensors.
\end{definition}
\medskip
\noindent Now we introduce the T-diamond tensor product. \\
\begin{definition
Let
$\mathscr{A} =[\mathscr{A}_{1},\ldots,\mathscr{A}_{p}]\in {\mathbb R}^{n_{1}\times ps \times n_3},$ where $ \mathscr{A}_{i} \; n_{1}\times s\times n_{3} , \, i =1,...,p$
and let $ \mathscr{B} =[\mathscr{B}_{1},\ldots,\mathscr{B}_{l}]\in {\mathbb R}^{n_{1}\times \ell s \times n_3}$ with $ \mathscr{B}_{j}; n_{1}\times s\times n_{3}, \, j =1,...\ell$. Then, the product $\mathscr{A}^{T} \diamondsuit \mathcal{B} $ is the size matrix $p \times \ell $ given by :
$$ (\mathscr{A}^{T} \diamondsuit \mathcal{B})_{i,j} = \langle \mathscr {A}_i ,\mathscr{B}_{j} \rangle \;\;.
$$
\end{definition}
\section{Global tensor T-Arnoldi and global tensor T-Golub-Kahan}
\subsection{The tensor T-global GMRES }
Consider now the following tensor linear system of equations
\begin{equation}\label{syslintens}
\mathscr{A}\star \mathscr{X}=\mathscr{C},
\end{equation}
where $\mathscr{A}\in \mathbb{R}^{n\times n \times p}$, $\mathscr{C}$ and $ \mathscr{X}\in \mathbb{R}^{n\times s \times p}$. \\
\noindent We introduce the tensor Krylov subspace $\mathcal{\mathscr{TK}}_m(\mathscr{A},\mathscr{V} )$ associated to the T-product, defined for the pair $(\mathscr{A},\mathscr{V})$ as follows
\begin{equation}
\label{tr3}
\mathcal{\mathscr{TK}}_m(\mathscr{A},\mathscr{V} )= {\rm Tspan}\{ \mathscr{V}, \mathscr{A} \star\mathscr{V},\ldots,\mathscr{A}^{m-1}\star\mathscr{V} \}\\
=\left\lbrace \mathscr{Z} \in \mathbb{R}^{n\times s \times n_3}, \mathscr{Z}= \sum_{i=1}^m \alpha_{i} \left(
\mathscr{A}^{i-1}\star\mathscr{V}\right) \right\rbrace
\end{equation}
where $\alpha_{i}\in \mathbb{R} $, $\mathscr{A}^{i-1}\star\mathscr{V}=\mathscr{A}^{i-2}\star\mathscr{A}\star\mathscr{V}$, for $i=2,\ldots,m$ and $\mathscr{A}^{0}$ is the identity tensor. We can now give a new version of the Tensor T-global Arnoldi algorithm.
\begin{algorithm}[H]
\caption{Tensor T-global Arnoldi} \label{TGA}
\begin{enumerate}
\item {\bf Input.} $\mathscr{A}\in \mathbb{R}^{n\times n \times p}$, $\mathscr{V}\in \mathbb{R}^{n\times s \times p}$ and and the positive integer $m$.
\item Set $\beta=\|\mathscr{V}\|_F$, $\mathscr{V}_{1} = \dfrac{\mathscr{V}}{ \beta}$
\item For $j=1,\ldots,m$
\begin{enumerate}
\item $\mathscr{W}= \mathscr{A}\star \mathscr{V}_j$
\item for $i=1,\ldots,j$
\begin{enumerate}
\item $h_{i,j}=\langle \mathscr{V}_i, \mathscr{W} \rangle$
\item $\mathscr{W}=\mathscr{W}-h_{i,j}\;\mathscr{V}_i$
\end{enumerate}
\item End for
\item $h_{j+1,j}=\Vert \mathscr {W} \Vert_F$. If $h_{j+1,j}=0$, stop; else
\item $\mathcal {V}_{j+1}=\mathscr {W}/h_{j+1,j}$.
\end{enumerate}
\item End
\end{enumerate}
\end{algorithm}
\begin{proposition}
Assume that m steps of Algorithm (\ref{TGA}) have been run. Then, the tensors $\mathscr{V}_{1},\ldots,\mathscr{V}_{m}$, form an orthonormal basis of the tensor global Krylov subspace $\mathscr{TK}^{g}_{m}(\mathscr{A},\mathscr{V})$.
\end{proposition}
\medskip
\begin{proof}
\noindent This can be shown easily by induction on $m$.
\end{proof}
\medskip
\noindent Let $\mathbb{V}_{m} $ be the $(n\times (sm)\times p)$ tensor with frontal slices $\mathscr{V}_{1},\ldots,\mathscr{V}_{m}$ and let $ {\widetilde{H}}_{m}$ be the $(m+1)\times m $ upper Hesenberg matrix whose elements are the $h_{i,j}$'s defined by Algorithm \ref{TGA}. Let $ {H}_{m}$ be the matrix obtained from $\widetilde{ { H}}_{m}$ by deleting its last row; $H_{.,j}$ will denote the $j$-th column of the matrix $H_m$ and $\mathscr{A}\star\mathbb{V}_{m} $ is the $(n\times (sm)\times p)$ tensor with frontal slices $\mathscr{A}\star\mathscr{V}_{1},\ldots,\mathscr{A}\star\mathscr{V}_{m}$ respectively given by
\begin{equation}
\label{ev1}
\mathbb{V}_{m}:=\left[ \mathscr{V}_{1},\ldots,\mathscr{V}_{m}\right], \;\;\; {\rm and}\;\;\; \mathscr{A}\star\mathbb{V}_{m}:=[\mathscr{A}\star\mathscr{V}_{1},\ldots,\mathscr{A}\star\mathscr{V}_{m}].
\end{equation}
We introduce the product $\circledast$ defined by: $$\mathbb{V}_{m}\circledast y=\sum_{j=1}^{m} {y}_{j}\mathscr{ {V}}_{j},\; y= (y_1,\ldots,y_m)^T\in \mathbb{R}^m.$$
We set the following notation:
\begin{equation*}
\mathbb{V}_{m}\circledast {{ {H}}_{m}}=\left[ \mathbb{V}_m\circledast H_{.,1} ,\ldots,\mathscr{V}_{m}\circledast H_{.,m} \right].
\end{equation*}
Then, it is easy to see that $\forall$ $u \,\text{and}\, v$ $\in \mathbb{R}^{m}$, we have
\begin{equation}
\mathbb{V}_{m}\circledast (u+v)=\mathbb{V}_{m}\circledast u + \mathbb{V}_{m}\circledast v \quad \text{and}\quad (\mathbb{V}_{m}\circledast H_m)\circledast u=\mathbb{V}_{m}\circledast(H_m\;u).
\end{equation}
With these notations, we can show the following result (proposition) that will be useful later on.
\medskip
\begin{proposition}\label{normfrobnorm2}
Let $\mathbb{V}_{m}$ be the tensor defined by $\left[ \mathscr{V}_{1},\ldots,\mathscr{V}_{m}\right]$ where $\mathscr{V}_{i}\in \mathbb{R}^{n\times s\times p} $ are defined by the Tensor T-global Arnoldi algorithm. Then, we have
\begin{equation}
\|\mathbb{V}_{m}\circledast y\|_F=\|y\|_2, \; \forall y= (y_1,\ldots,y_m)^T\in \mathbb{R}^m .
\end{equation}
\end{proposition}
\begin{proof}
From the definition of the product $\circledast$, we have $\sum_{j=1}^{m} {y}_{j}\mathscr{ {V}}_{j}=\mathbb{V}_{m}\circledast y$. Therefore,
$$\|\mathbb{V}_{m}\circledast y\|_F^2= \left< \sum_{j=1}^{m} {y}_{j}\mathscr{ {V}}_{j},\sum_{j=1}^{m} {y}_{j}\mathscr{ {V}}_{j} \right>_F.$$
But, since the tensors $ \mathscr{ {V}}_{i}$'s are orthonormal, it follows that
$$\|\mathbb{V}_{m}\circledast y\|_F^2= \sum_{j=1}^{m} {y}_{j}^2={\|y\|_2^2},$$
which shows the result.
\end{proof}
\noindent With the above notations, we can easily prove the results of the following proposition :
\begin{proposition}\label{T-GlobalArnolproposit}
Suppose that m steps of Algorithm \ref{TGA} have been run. Then, the following statements hold:
\begin{eqnarray}
\mathscr{A}\star\mathbb{V}_{m}&=&\mathbb{V}_{m}\circledast {{ {H}}_{m}} + h_{m+1,m}\left[ \mathscr{O}_{n\times s\times p},\ldots,\mathscr{O}_{n\times s\times p},\mathscr{V}_{m+1}\right],\\
\mathscr{A}\star\mathbb{V}_{m}&=&\mathbb{V}_{m+1} \circledast \widetilde{ { H}}_{m}, \\
\mathbb{V}_{m}^{T}\diamondsuit\mathscr{A}\star\mathbb{V}_{m}&=& {H}_{m}, \\
\mathbb{V}_{m+1}^{T}\diamondsuit \mathscr{A}\star\mathbb{V}_{m}&=&\widetilde{ { H}}_{m},\\
\mathbb{V}_{m}^{T} \diamondsuit\mathbb{V}_m&=& {I}_{ m },
\end{eqnarray}
where ${I}_{ m }$ the identity matrix and $\mathscr{O}$ is the tensor having all its entries equal to zero.
\end{proposition}
\medskip
\begin{proof}
From Algorithm \ref{TGA}, we have $ \mathscr{A}\star\mathscr{V}_{j}=\sum_{i =1}^{j+1}h_{i,j} \mathscr{V}_{i}$.
Using the fact that $\mathscr{A}\star\mathbb{V}_{m}=\left[\mathscr{A}\star\mathscr{V}_{1},\ldots,\mathscr{A}\star\mathscr{V}_{m}\right]$,
the $j$-th frontal slice of $\mathscr{A}\star\mathbb{V}_{m}$ is given by
\begin{align*}
(\mathscr{A}\star\mathbb{V}_{m})_j=\mathscr{A}\star\mathscr{V}_{j}&=\sum_{i =1}^{j+1}h_{i,j} \mathscr{V}_{i}.
\end{align*}
Furthermore, from the definition of the $\circledast$ product, we have
\begin{align*}
(\mathbb{V}_{m+1} \circledast \widetilde{ { H}}_{m})_j&=\mathbb{V}_{m+1}\circledast H_{.,j},\\
&= \sum_{i =1}^{j+1}h_{i,j} \mathscr{V}_{i},
\end{align*}
which proves the first two relations. The other relations follow from the definition of T-diamond product
\end{proof}
\medskip
\noindent In the sequel, we develop the tensor T-global GMRES algorithm for solving the problem \eqref{syslintens}. It could be considered as generalization of the well known global GMERS algorithm \cite{jbilou1}.
Let $\mathscr{ {X}}_{0}\in \mathbb{R}^{n\times s\times p}$ be an arbitrary initial guess with the corresponding residual
$\mathscr{R}_0=\mathscr{C}-\mathscr{A}\star \mathscr{X}_0$. The aim of tensor T-global GMRES method is to find and approximate solution $\mathscr{X}_{m}$ approximating the exact solution $\mathscr{X}^*$ of \eqref{syslintens} such that
\begin{equation}
\label{gmres1}
\mathscr{X}_{m}-\mathscr{X}_{0}\in \mathscr{TK}^{g}_{m}(\mathscr{A},\mathscr{R}_0),
\end{equation}
with the classical minimization property
\begin{equation}
\label{gmres2}
\Vert \mathscr{R}_{m}\Vert_F = \displaystyle \min_{ \mathscr{X} \in \mathscr{X}_{0} + \mathscr{TK}^{g}_{m}(\mathscr{A},\mathscr{R}_0)}\left\lbrace \|\mathscr{C}-\mathscr{A}\star \mathscr{X}\|_F
\right\rbrace.
\end{equation}
\noindent Let $\mathscr{X}_{m}=\mathscr{X}_{0}+\mathbb{V}_{m}\circledast y $ with $ {y} \in \mathbb{R}^m $, be the approximate solution satisfying \eqref{gmres1}. Then,
\begin{align*}
\mathscr{R}_m=&\mathscr{C}-\mathscr{A}\star\mathscr{X}_{m},\\
=& \mathscr{C}-\mathscr{A}\star\left(\mathscr{X}_{0}+ \mathbb{V}_{m}\circledast y\right), \\
=& \mathscr{C}-\mathscr{A}\star\mathscr{X}_{0}-\mathscr{A} \star(\mathbb{V}_{m}\circledast y),\\
=&\mathscr{R}_{0}-\left(\mathscr{A} \star\mathbb{V}_{m}\right)\circledast y.
\end{align*}
It follows then that
\begin{align*}
\|\mathscr{R}_m \|_{F}&= \displaystyle \min_{ y\in \mathbb{R}^{m }}
\|\mathscr{R}_{0}-(\mathscr{A}\star\mathbb{V}_{m})\circledast y\|_F,
\end{align*}
where $\mathscr{A}\star\mathbb{V}_{m}:=[\mathscr{A}\star\mathscr{V}_{1},\ldots,\mathscr{A}\star\mathscr{V}_{m}]
$ is the $(n\times sm\times p)$ tensor defined earlier.\\
\noindent Using Propositions \ref{normfrobnorm2} and the fact that $\mathscr{R}_{0}=\|\mathscr{R}_{0}\|_F \mathscr{V}_1 $ with $\mathscr{V}_1 = \mathscr{V}_{m+1}\circledast e_{1}$, where $e_{1}$ the first canonical basis vector in $\mathbb{R}^{m+1}$, we get
\begin{align*}
\|\mathscr{R}_{0}-(\mathscr{A}\star\mathbb{V}_{m})\circledast y\|_F&=\| \mathscr{R}_{0}- (\mathbb{V}_{m+1} \circledast \widetilde{ { H}}_{m}) \circledast y \|_F,\\
&=\|\|\mathscr{R}_{0}\|_F (\mathbb{V}_{m+1}\circledast e_1)- (\mathbb{V}_{m+1} \circledast \widetilde{ { H}}_{m}) \circledast y \|_F,\\
&=\| \mathbb{V}_{m+1}\circledast (||\mathscr{R}_{0}\|_F e_1-\widetilde{ { H}}_{m} y) \|_F ,\\
&=\|\; \|\mathscr{R}_{0}\|_F\; e_1-\widetilde{ { H}}_{m} y \|_2.\\
\end{align*}
Finally, we obtain
\begin{equation}\label{solutdegmresxm}
\mathscr{X}_{m}=\mathscr{X}_{0}+ \mathbb{V}_{m} \circledast y,
\end{equation}
where,
\begin{equation}\label{Gmressol}
y= \text{arg } \min_{{{y}}\in \mathbb{R}^{m }}||\; ||\mathscr{R}_{0}||_F\; e_1-\widetilde{ { H}}_{m} y) ||_2.
\end{equation}
\subsection{Tensor T-global Golub Kahan algorithm}
\noindent Instead of using the tensor T-global Arnoldi to generate a basis for the projection subspace, we can define T-version of the tensor global Lanczos process. Here, we will use the tensor Golub Kahan algorithm related to the T-product. We notice here that we already defined in \cite{Elguide} another version of the tensor Golub Kahan algorithm by using the $m$-mode or the Einstein products with applications to color image restoration.\\
Let $\mathscr{A} \in \mathbb{R}^{n\times \ell\times p}$ be a tensor and let $\mathscr{U} \in \mathbb{R}^{\ell\times s \times p}$ and $\mathscr{V} \in \mathbb{R}^{n\times s \times p}$ two other tensors. Then, the Tensor T-global Golub Kahan bidiagonalization algorithm (associated to the T-product) is defined as follows
\begin{algorithm}[h!]
\caption{The Tensor T-global Golub Kahan algorithm}\label{TG-GK}
\begin{enumerate}
\item {\bf Input.} The tensors $\mathscr {A}$, $\mathscr{V}$, and $\mathscr {U}$ and an integer $m$.
\item Set $\beta_1= \Vert \mathscr{V}\Vert_F$, $\alpha_1= \Vert \mathscr {U} \Vert_F$, $\mathscr {V}_1=\mathscr {V}/\beta_1$ and
$\mathscr {U}_1=\mathscr {U}/\alpha_1$.
\item for $j=2,\ldots,m$
\begin{enumerate}
\item $\widetilde {\mathscr {V}}= \mathscr {A} \star \mathscr {U}_{j-1} -\alpha_{j-1}\mathscr {V}_{j-1}$
\item $\beta_j=\Vert \widetilde {\mathscr {V}}\Vert_F$ if $\beta_j=0$ stop, else
\item $\mathscr {V}_j=\widetilde {\mathscr {V}}/\beta_j$
\item $\widetilde {\mathscr {U}}=\mathscr {A}^T \star \mathscr {V}_j-\beta_j \mathscr{U} _{j-1}$
\item $\alpha_j=\Vert \widetilde {\mathscr {U}} \Vert_F$
\item if $\alpha_j=0$ stop, else
\item $\mathscr {U}_j=\widetilde {\mathscr {U}}/\alpha_j$
\end{enumerate}
\end{enumerate}
\end{algorithm}
\medskip
\noindent Let $\widetilde{C}_m$ be the upper bidiagonal $((m+1) \times m )$ matrix
$$ \widetilde{ { {C}}}_m=\left[ \begin{array}{*{20}{c}}
{{{\alpha}_1 }}&{{ }}& & \\
{\beta}_{2}&{{{\alpha}_2}}&\ddots& \\
&\ddots&\ddots& \\
& & {\beta}_{m} & {\alpha}_{m}\\
& & & {\beta}_{m+1}
\end{array} \right]
$$
and let $ {{{C}}}_m$ be the $(m \times m )$ matrix obtain by deleting the last row of $\widetilde{{ {C}}}_m$. We denote by $C_{.,j}$ will denote the $j$-th column of the matrix $C_m$. Let $\mathbb{U}_{m} $ and $\mathscr{A}\star\mathbb{U}_{m} $ be the $(\ell\times (sm)\times p)$ and $(n\times (sm)\times p)$ tensors with frontal slices $\mathscr{U}_{1},\ldots,\mathscr{U}_{m}$ and $\mathscr{A}\star\mathscr{U}_{1},\ldots,\mathscr{A}\star\mathscr{U}_{m}$, respectively, and let $\mathbb{V}_{m} $ and $\mathscr{A}^T\star\mathbb{V}_{m} $ be the $(n\times (sm)\times p)$ and $(\ell\times (sm)\times p)$ tensors with frontal slices $\mathscr{V}_{1},\ldots,\mathscr{V}_{m}$ and $\mathscr{A}^T\star\mathscr{V}_{1},\ldots,\mathscr{A}^T\star\mathscr{V}_{m}$, respectively. We set
\begin{align}
\label{ev12}
\mathbb{U}_{m}:&=\left[ \mathscr{U}_{1},\ldots,\mathscr{U}_{m}\right], \;\;\; {\rm and}\;\;\; \mathscr{A}\star\mathbb{U}_{m}:=[\mathscr{A}\star\mathscr{U}_{1},\ldots,\mathscr{A}\star\mathscr{U}_{m}],\\
\mathbb{V}_{m}:&=\left[ \mathscr{V}_{1},\ldots,\mathscr{V}_{m}\right], \;\;\; {\rm and} \;\;\; \mathscr{A}^T\star\mathbb{V}_{m}:=[\mathscr{A}^T\star\mathscr{V}_{1},\ldots,\mathscr{A}^T\star\mathscr{V}_{m}].
\end{align}
\noindent Then, the following proposition can be established\\
\begin{proposition}\label{proptggkb}
The tensors produced by the tensor T-global Golub-Kahan algorithm satisfy the following relations
\begin{eqnarray} \label{equa20}
\mathcal {A} \star \mathbb{U}_m& = &\mathbb{V}_{m+1} \circledast {\widetilde { { {C}}}}_m , \\
& = &\mathbb{V}_m\circledast{ { { {C}}}}_m + {\beta}_{m+1} \left[ \mathscr{O}_{n\times s\times p},\ldots,\mathscr{O}_{n\times s\times p},\mathscr{V}_{m+1}\right], \\
\mathscr{A}^{T}\star\mathbb{V}_{m}& = &\mathbb{U}_m \circledast {\widetilde { { {C}}}}_m^T .
\end{eqnarray}
\end{proposition}
\begin{proof}
Using $\mathscr{A}\star\mathbb{U}_{m}=[\mathscr{A}\star\mathscr{U}_{1},\ldots,\mathscr{A}\star\mathscr{U}_{m}] \in \mathbb{R}^{n \times (sm)\times n_{3}} $ , the ($j-1$)-th lateral slice of $(\mathscr{A}\star\mathbb{U}_{m})$ is given by $$ (\mathcal {A} \star \mathbb{U}_m)_{j-1} =\mathscr {A} \star \mathscr {U}_{j-1}= {\alpha }_{j-1} \mathscr {V}_{j-1}+{\beta }_j \mathcal {V}_{j}.$$
Furthermore, from the definition of the $\circledast$ product, we have
\begin{align*}
(\mathbb{V}_{m+1} \circledast \widetilde{ { C}}_{m})_{j-1}&=\mathbb{V}_{m+1}\circledast C_{.,j-1},\\
&= \sum_{i =1}^{j+1}c_{i,j-1} \mathscr{V}_{i},\\
&={\alpha }_{j-1} \mathscr {V}_{j-1}+{\beta }_j \mathcal {V}_{j}
\end{align*}
and for $j=m$, $\mathbb{U}_{m} \circledast \mathscr{C}_{.,m}= \mathscr{A}\star \mathscr{U}_{m}+{\beta}_{m+1} \mathscr{V}_{m+1} $ and the result follows.\\
To derive (4.5) , one may first
notice that from Algorithm \ref{TG-GK}, we have $$(\mathcal {A}^T \star \mathbb{V}_m)_{j} =\mathscr {A} ^T\star \mathscr {V}_{j}= {\alpha }_{j} \mathscr {U}_{j}+{\beta }_j \mathcal {U}_{j-1}.$$
Considering now the $j$-th frontal slice of the right-hand side
of (4.5), the assertion can be easily deduced .
\end{proof}
\medskip
\begin{proposition}
Let $\mathscr{ {X}}_{m}= \mathscr{ {X}}_{0}+\mathbb{U}_{m}\circledast y \in \mathbb{R}^{\ell\times s\times p}$ with $ { {y}}\in \mathbb{R}^{m },$ where $\mathbb{U}_{m}$ is obtained from Algorithm \ref{TG-GK}, be an approximation of (\ref{syslintens}). Then, we have
\begin{equation}
\|\mathscr{C}-\mathscr{A}\star \mathscr{X}_m\|_F= \| \beta_{1}e_1- { \widetilde{C}}_m y \|_2,
\end{equation}
where $\beta_{1}=\|\mathscr{C}\|_F$.
\end{proposition}
\medskip
\begin{proof}
Using representation (\ref{equa20}) and the fact that $\mathscr{C}=\mathbb{V}_{m+1}\circledast (\beta_{1}e_1)$ with $\beta_{1}=\|\mathscr{C}\|_F$, we get
\begin{align*}
\|\mathscr{C}-\mathscr{A}\star \mathscr{X}_m\|_F&=\|\mathbb{V}_{m+1}\circledast (\beta_{1}e_1) - (\mathbb{V}_{m+1} \circledast { \widetilde{C}}_m)\circledast y |\|_{F},\\
&=|||\mathbb{V}_{m+1}\circledast (\beta_{1}e_1-{\widetilde{C}}_m y )||_F,\\
&=||\beta_{1}e_1- {\widetilde{C}}_m y ||_2.
\end{align*}
\end{proof}
\section{Application to discrete-ill posed tensor problems}
We consider the following discrete ill-posed tensor equation
\begin{equation}\label{tr1}
\mathscr{A} \star \mathscr{X}= \mathscr{C},\quad \mathscr{C}=\widehat{ \mathscr C}+ \mathscr{N},
\end{equation}
where $\mathscr{A} \in {\mathcal{R}}^{n \times n \times s}$, $\mathscr{X}$, $ \mathscr{N}$ (additive noise) and $\mathscr{C}$ are tensors in ${\mathcal{R}}^{n \times s \times p}$. \\
In color image processing, $p=3$, $\mathscr{A}$ represents the blurring tensor, $\mathscr{C}$ the blurry and noisy observed image, $\mathscr{X}$ is the image that we would like to restore and $\mathscr{N}$ is an unknown additive noise. Therefore, to stabilize the recovered image, regularization techniques are needed. There are several techniques to regularize the linear inverse problem given by
equation (\ref{tr1}); for the matrix case, see for example, \cite{belguide,reichel2,golubwahba,hansen1}. All of these techniques stabilize the restoration process by adding a regularization term, depending on some priori knowledge of the unknown image. One of the most regularization method is due to Tikhonov and is given as follows
\begin{equation}\label{tr2}
\underset{\mathscr{X}}{\text{min}}\{\|\mathscr{A} \star \mathscr{X} - \mathscr{C} \|_F^2+\mu \|\mathscr{X}\|_F^2\}.
\end{equation}
As problem \eqref{tr1} is large, Tikhonov regularization \eqref{tr2} may be very expensive to solve. One possibility is instead of regularizing the original problem, we apply the Tikhonov technique to the projected problem \eqref{Gmressol} which leads to the following problem
\begin{eqnarray}
y_{m,\mu}&=& \arg \min_{ y \in \mathbb{R}^{m }}\left( \|\mathscr{R}_0\| e_1-\widetilde{ { H}}_{m} y \|_ 2 +\mu \| y \|_2
\right),\\
&=&\arg \min_{ y \in \mathbb{R}^{m }}\left \Vert \left ( \begin{array}{ll}
\widetilde H_m\\
\mu I_m
\end{array}\right ) y - \left ( \begin{array}{ll}
\beta e_1\label{min}\\
0
\end{array}\right )
\right \Vert_2.
\end{eqnarray}
The minimizer $y_{m,\mu}$ can also be computed as the solution of the following normal equations associated with \eqref{min}
\begin{equation}{\label{tikho2}}
\widetilde H_{m,\mu} y=\widetilde H_m^T, \quad \widetilde H_{m,\mu}= (\widetilde H_m^T \widetilde H_m+ \mu^2 I_m).
\end{equation}
Note that since the Tikhonov problem \eqref{tikho2} is now a matrix one with small dimension as $m$ is generally small, the vector $y_{m,\mu}$, can thereby be inexpensively computed by some techniques such as the GCV method \cite{golubwahba} or the L-curve criterion \cite{hansen1,hansen2,reichel1,reichel2}. To choose the regularization parameter, we can use the
generalized cross-validation (GCV) method
\cite{golubwahba,wahbagolub}. Now for the GCV method, the regularization
parameter is chosen by minimizing the following function
\begin{equation}\label{gcv}
GCV(\mu)=\frac{\|\widetilde H_m y_{m,\mu}-{\bf
\beta e_1}\|_2^2}{[tr(I_m-\widetilde H_m \widetilde H_{m,\mu}^{-1}\widetilde H_m^T)]^2}=\frac{\|(I_m-\widetilde H_m \widetilde H_{m,\mu}^{-1} \widetilde H_m^T){\beta e_1}\|_2^2}{[tr(I_m-H_m H_{m,\mu}^{-1} \widetilde H_m^T)]^2}.
\end{equation}
To minimize (\ref{gcv}), we take advantage of the the SVD decomposition of the low dimensional matrix $\widetilde H_m$ to obtain a more simple and computable expression of $GCV(\mu)$. Consider the SVD decomposition of $\widetilde H_m=U\Sigma V^T$. Then, the GCV is now expressed as (see \cite{wahbagolub})
\begin{equation}
\label{gcv2}
GCV(\mu)=\frac{\displaystyle
\sum_{i=1}^m\left(\frac{\widetilde
g_i}{\sigma_i^2+\mu^2}\right)^2}{\displaystyle\left(\sum_{i=1}^m
\frac{1}{\sigma_i^2+\mu^2}\right)^2},
\end{equation}
where $\sigma_i$ is the $i$th singular value of the matrix
$\widetilde H_m$ and $\widetilde g= \beta_1 U^T e_1$.
In terms of practical implementations, it's more convenient to introduce a restarted version of the tensor Global GMRES. This strategy is essentially based on restarting the tensor T-global Arnoldi algorithm. Therefore, at each
restart, the initial guess $\mathscr{X}_0$ and the regularization parameter $\mu$
are updated employing the last values computed when the the number of inner iterations required is fulfilled. We note that as the number outer iterations increases it is possible to compute the $m$th residual without having to compute extra T-products. This is described in the following proposition.
\begin{proposition}
At step $m$, the residual $\mathscr{R}_{m}=\mathscr{C}-\mathscr{A}\ast \mathscr{X}_{m}$ produced by the tensor Global GMRES method for tensor equation (\ref{eq1}) has the following expression
\begin{equation}\label{resex}
\mathscr{R}_m=\mathbb{V}_{m+1} \circledast\left(\gamma_{m+1}Q_me_{m+1}\right),
\end{equation}
where $Q_m$ is the unitary matrix obtained from the QR decomposition of the upper Hessenberg matrix $\widetilde{H}_{m}$ and $\gamma_{m+1}$ is the last component of the vector $\left\|\mathscr{R}_{0}\right\|_{F} Q_{m}^{\mathrm{T}} e_{1}$ and $e_{m+1}=(0,0, \ldots, 1)^{\mathrm{T}} \in \mathbb{R}^{m+1}.\\$
Furthermore,
\begin{equation}\label{resnrm}
\left\|\mathscr{R}_{m}\right\|_{F}=\left|\gamma_{m+1}\right|.
\end{equation}
\end{proposition}
\begin{proof}
At step $m$, the residual $\mathscr{R}_m$ can be expressed as
$$\mathscr{R}_m=\mathbb{V}_{m+1} \circledast\left(\beta e_{1}-\widetilde{H}_{m} y_{m}\right),$$
by considering the QR decomposition $\widetilde{H}_{m}=Q_{m}\widetilde{U}_m$ of the $(m + 1) \times m$ matrix $\widetilde{H}_{m}$, we get
$$\mathscr{R}_m=\left(\mathbb{V}_{m+1} \circledast Q_m\right)\circledast\left(\beta Q_m^T e_{1}-\widetilde{U}_{m} y_{m}\right).$$
Since $y$ solves problem (\ref{Gmressol}), it follows that
$$\mathscr{R}_m=\mathbb{V}_{m+1} \circledast\left(\gamma_{m+1}Q_me_{m+1}\right),$$
where $\gamma_{m+1}$ is the last component of the vector $\beta Q_{m}^T e_{1}.$ Therefore,
\begin{eqnarray*}
\left\|\mathscr{R}_{m}\right\|_{F}&=&\left\|\mathbb{V}_{m+1} \circledast\left(\gamma_{m+1}Q_me_{m+1}\right)\right\|_F,\\
&=&\left\|\gamma_{m+1}Q_me_{m+1}\right\|_2,\\
&=&\left|\gamma_{m+1}\right|,
\end{eqnarray*}
which shows the results.
\end{proof}
\noindent The tensor T-global GMRES method is summarized in the following algorithm
\begin{algorithm}[h!]
\caption{Implementation of Tensor T-global GMRES(m)}\label{TG-GMRES(m)}
\begin{enumerate}
\item {\bf Input.} $\mathscr{A}\in \mathbb{R}^{n\times n \times n_3}$, $\mathscr{V},\mathscr{B},\mathscr{X}_{0}\in \mathbb{R}^{n\times s \times n_3}$, the maximum number of iteration $\text{Iter}_{\text{max}} $ and a tolerance $tol>0$ .
\item {\bf Output.} $ \mathscr{X}_{m}\in \mathbb{R}^{n\times s\times n_3}$ approximate solution of the system (\ref{syslintens}).
\item $k=1,\ldots,\text{Iter}_{\text{max}}$
\begin{enumerate}
\item Compute $\mathscr{R}_{0}=\mathscr{C}-\mathscr{A}\star\mathscr{X}_{0} $.
\item Apply Algorithm \ref{TGA} to compute $\mathbb{V}_{m}$ and ${ \widetilde{H}}_m$ .
\item Determine $\mu_{k}$ as the parameter minimizing the GCV function given by (\ref{gcv2})
\item Compute the regularized solution $y_{m_k,\mu}$ of the problem \eqref{min}.
\item Compute the approximate solution $\mathscr{X}_{m}=\mathscr{X}_{0}+ \mathbb{V}_{m} \circledast y_{m,\mu_k}
$
\end{enumerate}
\item If $\|\mathscr{R}_{m}\|_F<tol$, stop, else
\item Set $\mathscr{X}_{0}=\mathscr{X}_{m}$ and go to 3-a.
\item End
\end{enumerate}
\end{algorithm}
\noindent We turn now to the tensor T-global Golub Kahan approach for the solving the Tikhonov regularization of the problem (\ref{eq1}). Here, we apply the following Tikhonov regularization approach and solve the new problem
\begin{equation}\label{tikho3}
\underset{\mathscr{X}}{\text{min}}\{\|\mathscr{A} \star \mathscr{X} - \mathscr{C} \|_F^2+\mu^{-1} \|\mathscr{X}\|_F^2\}.
\end{equation}
The use of $\mu^{-1} $
in (\ref{tikho3}) instead of $\mu$ will be justified below. In the what follows, we briefly review the discrepancy principle approach to determine a suitable regularization parameter, given an approximation of the norm of the additive error. We then assume that a bound $\varepsilon$ for $\|\mathscr{N}\|_F$ is available. This priori information suggests that $\mu$ has to be determined as soon as
\begin{equation}\label{discrepancy}
\phi(\mu)\leq\eta\epsilon,
\end{equation}
where $\phi(\mu)=\|\mathscr{A} \star \mathscr{X} - \mathscr{C} \|_F^2$ and $ \eta\gtrapprox 1$ is refereed to as the safety factor for the discrepancy principle. A zero-finding method can be used to solve (\ref{discrepancy}) in order to find a suitable regularization parameter which also implies that $\phi(\mu)$ has to be evaluated for several $\mu$-values. When the tensor $\mathscr{A}$ is of moderate size, the quantity $\phi(\mu)$ can be easily evaluated. This evaluation becomes expensive when the matrix $\mathscr{A}$ is large, which means that its evaluation by a zero-finding method can be very difficult and computationally expensive. We will approximate $\phi$ to be able to determine an estimate of $\|\mathscr{A} \star \mathscr{X} - \mathscr{C} \|_F^2$. Our approximation is obtained by using T-global Golub-Kahan bidiagonalization (T-GGKB) and Gauss-type quadrature rules. This connection provides approximations of moderate sizes to the quantity $\phi$, and therefore gives a solution method to inexpensively solve (\ref{discrepancy}) by evaluating these small quantities that can successfully and inexpensively be employed to compute $\mu$ as well as defining a stopping criterion for the T-GGKB iterations; see \cite{belguide, belguide2} for discussion on this method.\\
Introduce the functions (of $\mu$)
\begin{eqnarray}\label{Gkfmu}
\mathcal{G}_m f_\mu&=&\|\mathscr{C}\|_F^2 e_1^T(\mu C_m C_m^T+I_m)^{-2}e_1,\\
{\mathcal R}_{m+1}f_\mu&=&\|\mathcal{C}\|_F^2 e_1^T(\mu \widetilde{C}_m\widetilde{C}_m^T+I_{m+1})^{-2}e_1;
\end{eqnarray}
The quantities $\mathcal{G}_m f$ and ${\mathcal R}_{m+1}f_\mu$ are refereed to as Gauss and Gauss-Radau quadrature rules, respectively, and can be obtained after $m$ steps of T-GGKB (Algorithm \ref{TG-GK}) applied to tensor $\mathscr{A}$ with initial tensor $\mathscr{C}$. These quantities approximate $\phi(\mu)$ as follows
\begin{equation}
\mathcal{G}_m f_\mu\leq\phi(\mu)\leq{\mathcal R}_{m+1}f_\mu.
\end{equation}
Similarly to the approaches proposed in \cite{belguide, belguide2}, we therefore instead solve for $\mu$ the low dimensional nonlinear equation
\begin{equation}\label{lin22}
{\mathcal G}_m f_\mu=\epsilon^2.
\end{equation}
We apply the Newton's method to solve (\ref{lin22}) that requires repeated evaluation of the function ${\mathcal G}_m f_\mu$ and its derivative, which are inexpensive computations for small $m$.\\
We now comment on the use of $\mu$ in (\ref{tikho3}) instead of $1 / \mu,$ implies that the left-hand side of (\ref{discrepancy}) is a decreasing convex function of $\mu .$ Therefore, there is a unique solution, denoted by $\mu_{\varepsilon},$ of
$$
\phi(\mu)=\varepsilon^{2}
$$
for almost all values of $\varepsilon>0$ of practical interest and therefore also of (\ref{lin22}) for $m$ sufficiently large; see \cite{belguide, belguide2} for analyses. We accept $\mu_m$ that solve (\ref{discrepancy}) as an approximation of $\mu$, whenever we have
\begin{equation}\label{upperbd}
{\mathcal R}_{m+1}f_{\mu}\leq\eta^2\epsilon^2.
\end{equation}
If (\ref{upperbd}) does not hold for $\mu_m$, we carry out one more GGKB steps, replacing $m$ by $m+1$ and solve the nonlinear equation
\begin{equation}\label{}
{\mathcal G}_{m+1}f_\mu=\epsilon^2;
\end{equation}
see \cite{belguide, belguide2} for more details. Assume now that (\ref{upperbd}) holds for some $\mu_m$. The corresponding regularized solution is then computed by
\begin{equation}\label{Xkmu}
\mathscr {X}_{m,\mu_m}=\mathbb{U}_m \circledast y_{m,\mu_m},
\end{equation}
where $y_{m,\mu_m}$ solves
\begin{equation}\label{normeq2}
(\widetilde{C}_m^T\widetilde{C}_m+\mu_m^{-1} I_m)y=\beta_1\widetilde{C}_m^Te_1,\qquad\beta_1=\|\mathscr{C}\|_F.
\end{equation}
It is also computed by solving the least-squares problem
\begin{equation}\label{leastsq}
\min_{y\in\mathbb{R}^m} \begin{Vmatrix}
\begin{bmatrix}
\mu_m^{1/2}\widetilde{C}_m\\
I_m
\end{bmatrix}
y-\beta_1\mu_m^{1/2}e_1 \end{Vmatrix}_2.
\end{equation}The following result shows an important property of the approximate solution (\ref{Xkmu}). We include a proof for completeness.
\begin{proposition}
Let $\mu_{m}$ solve (\ref{lin22}) and let $y_{m,\mu_m}$ solve (\ref{leastsq}). Then the associated approximate solution (\ref{Xkmu}) of (\ref{tikho3}) satisfies
$$
\left\|\mathscr{A}\ast \mathscr{X}_{m,\mu_m}-\mathscr{C}\right\|_{F}^{2}=R_{m+1} f_{\mu_{m}}.
$$
\end{proposition}
\begin{proof} The representation of Proposition \ref{proptggkb} show that
$$
\mathscr{A}\ast \mathscr{X}_{m,\mu_m}=(\mathscr{A}\ast \mathbb{U}_{m})\circledast y_{m,\mu_m}= \mathbb{V}_{m+1}\circledast(\widetilde{C}_m y_{m,\mu_m}).
$$
Using the above expression gives
$$
\begin{aligned}
\left\|\mathscr{A}\star \mathscr{X}_{m,\mu_m}-\mathscr{C}\right\|_{F}^{2}&=\left\|\mathbb{V}_{m+1}\circledast(\widetilde{C}_m y_{m,\mu_m})-\beta_1\mathscr{V}_1\right\|_{F}^{2}, \\
&=\left\|\mathbb{V}_{m+1}\circledast(\widetilde{C}_m y_{m,\mu_m})-\mathbb{V}_{m+1}\circledast(\beta_1e_1)\right\|_{F}^{2}, \\
&=\left\|\mathbb{V}_{m+1}\circledast\left(\widetilde{C}_m y_{m,\mu_m}-\beta_1e_1\right)\right\|_{F}^{2}, \\
&=\left\|\widetilde{C}_\ell y_{m,\mu_m}-\beta_1e_1\right\|_{2}^{2}.
\end{aligned}$$
where we recall that $\beta_1=\|\mathcal{C}\|_{F}$. We now express $y_{m,\mu_m}$ with the aid of (\ref{normeq2}) and apply the following identity
$$I-A\left(A^{T} A+\mu^{-1} I\right)^{-1} A^{T}=\left(\mu A A^{T}+I\right)^{-1}$$
with $A$ replaced by $\widetilde{C}_{m},$ to obtain
$$
\begin{aligned}
\left\|\mathscr{A}\ast \mathscr{X}_{m,\mu_m}-\mathscr{C}\right\|_{F}^{2} &=\beta_1^{2}\left\|e_{1}-\widetilde{C}_{m}\left(\widetilde{C}_{m}^{T} \widetilde{C}_{m}+\mu_{m}^{-1} I_{m}\right)^{-1} \widetilde{C}_{m}^{T} e_{1}\right\|_{F}^{2}, \\
&=\beta_1^{2} e_{1}^{T}\left(\mu_{m} \widetilde{C}_{m} \widetilde{C}_{m}^{T}+I_{m+1}\right)^{-2} e_{1}, \\
&=R_{m+1} f_{\mu_{m}}.
\end{aligned}
$$
\end{proof}
\noindent The following algorithm summarizes the main steps to compute a regularization parameter and a corresponding regularized solution of (\ref{eq1}), using Tensor T-GGKB and quadrature rules method for Tikhonov regularization.
\begin{algorithm}[!h]
\caption{ Tensor T-GGKB and quadrature rules method for Tikhonov regularization}\label{TG-GKB}
\begin{enumerate}
\item {\bf Input.} $\mathscr{A}\in \mathbb{R}^{n\times n \times n_3}$, $\mathscr {C}$, $\eta\gtrapprox 1$ and $\varepsilon$.
\item {\bf Output.} T-GGKB steps $m$, $\mu_{m}$ and $X_{m,\mu_m}$.
\item Determine the orthonormal bases $\mathbb{U}_{m+1}$ and $\mathbb{V}_{m}$ of tensors, and the bidiagonal $C_m$ and $\widetilde{C}_m$
matrices with Algorithm \ref{TG-GK}.
\item Determine $\mu_{m}$ that satisfies (\ref{lin22}) with Newton's method.
\item Determine $y_{m,\mu_m}$ by solving (\ref{leastsq}) and then compute $X_{m,\mu_m}$ by (\ref{Xkmu}).
\end{enumerate}
\end{algorithm}
\section{Numerical results}
This section performs some numerical tests on the methods of Tensor T-Global GMRES(m) and Tensor T-Global Golub Kahan algorithm given by Algorithm \ref{TG-GMRES(m)} and Algorithm \ref{TG-GKB}, rspectively,
when applied to the restoration of blurred and noisy color images and videos. For clarity, we only focus on the formulation of a tensor model (\ref{tr1}), describing the blurring that is taking place in the process of going from the exact to the blurred RGB image. We recall that an RGB image is just multidimensional array of dimension $m\times n\times 3$ whose entries are the light intensity. Throughout this section, we assume that the the three channels
of the RGB image has the same dimensions, and we refer to it as $n\times n\times 3$ tensor. Let $\widehat{X}^{(1)}$, $\widehat{X}^{(2)}$, and $\widehat{X}^{(3)}$ be the $n\times n$ matrices that constitute the three channels of the original error-free color image $\widehat{\mathscr{X}}$, and $\widehat{C}^{(1)}$, $\widehat{C}^{(2)}$, and $\widehat{C}^{(3)}$ the $n\times n$ matrices associated with error-free blurred color image $\widehat{\mathscr{C}}$. Because of some
unique features in images, we seek an image restoration model that utilizes blur information,
exploiting the spatially invariant properties. Let us also consider that both cross-channel and within-channel blurring take place in the blurring process of the original image. Let $\tt{vec}$ be the operator that
transforms a matrix to a vector by stacking the columns of the matrix from
left to right. Then, the full blurring model is described by the following
form
\begin{equation}\label{linmodel}
\left(\mathbf{A}_{\text {color }} \otimes \mathbf{A^{(1)}}\otimes\mathbf{A^{(2)}}\right) \widehat{\mathbf{x}}=\widehat{\mathbf{c}},
\end{equation}
where,
$$ \widehat{\mathbf{c}}=\left[\begin{array}{c}
\tt{vec}\left(\widehat{\mathbf{C}}^{(1)}\right) \\
\tt{vec}\left(\widehat{\mathbf{C}}^{(2)}\right) \\
\tt{vec}\left(\widehat{\mathbf{C}}^{(3)}\right)
\end{array}\right], \quad \widehat{\mathbf{x}}=\left[\begin{array}{c}
\tt{vec}\left(\widehat{\mathbf{X}}^{(1)}\right) \\
\tt{vec}\left(\widehat{\mathbf{X}}^{(2)}\right) \\
\tt{vec}\left(\widehat{\mathbf{X}}^{(3)}\right)
\end{array}\right], $$
and
$$\mathbf{A}_{\mathrm{color}}=\left[\begin{array}{ccc}
\alpha & \gamma& \beta \\
\beta & \alpha & \gamma \\
\gamma & \beta & \alpha
\end{array}\right]$$
$\mathbf{A}_{\text {color }}$ is the $3\times3$ matrix that models the cross-channel blurring, where each row sums to one. $\mathbf{A^{(1)}}\in\mathbb{R}^{n\times n}$ and $\mathbf{A^{(2)}}\in\mathbb{R}^{n\times n}$ define within-channel blurring and they model
the horizontal within blurring and the vertical within blurring matrices, respectively; for more details see \cite{HNO}. The notation $\otimes$ denotes the Kronecker product of matrices; i.e. the Kronecker product of a $n\times p$ matrix $A=(a_{ij})$ and a $(s\times q)$ matrix
$B=(b_{ij})$, is defined as the $(ns)\times(pq)$ matrix $A \otimes B = (a_{ij}B)$. By exploiting the circulant structure of the cross-channel blurring matrix $\mathbf{A}_{\text {color }}$ and the operators unfold and fold, it can be easily shown that (\ref{linmodel}) can be written in the following tensor form
\begin{equation}
\mathscr {A}\star \widehat{\mathscr{X}}\star \mathscr{B}= \widehat{\mathscr{C}},
\end{equation}
where $\mathscr {A}$ is a 3-way tensor such that $\mathscr {A}(:,:,1)=\alpha \mathbf{A^{(2)}}$, $\mathscr {A}(:,:,2)=\beta \mathbf{A^{(2)}}$ and $\mathscr {A}(:,:,3)=\gamma \mathbf{A^{(2)}}$ and $\mathscr {B}$ is a 3-way tensor with $\mathscr {B}(:,:,1)=(\mathbf{A^{(1)}})^T$, $\mathscr {B}(:,:,2)=0$ and $\mathscr {B}(:,:,3)=0$. To test the performance of algorithms, the within blurring matrices $A^{(i)}$ have the following
entries
$$a_{k \ell}=\left\{\begin{array}{ll}
\frac{1}{\sigma \sqrt{2 \pi}} \exp \left(-\frac{(k-\ell)^{2}}{2 \sigma^{2}}\right), & |k-\ell| \leq r \\
0, & \text { otherwise }
\end{array}\right.$$
Note that $\sigma$ controls the amount of smoothing, i.e. the larger the $\sigma$, the more ill posed the problem. We generated a blurred and noisy tensor image $\mathscr{C}=\widehat{\mathscr{C}}+\mathscr{N},$ where $\mathscr{N}$ is a noise tensor with normally distributed random entries with zero mean and with variance chosen to correspond to a specific noise level $\nu:=\|\mathscr{N}\|_F /\|\widehat{\mathscr{C}}\|_F.$
To determine the effectiveness of our solution methods, we evaluate
$$\text{Relative error}=\frac{\left\|\hat{ \mathscr X}-{\mathscr X}_{\text{restored}}\right\|_{F}}{\|\widehat{ \mathscr X}\|_{F}}$$
and the Signal-to-Noise
Ratio (SNR) defined by
\[\text{SNR}({ \mathscr X}_{\text{restored}})=10\text{log}_{10}\frac{\|\widehat{\mathscr X}-E(\widehat{\mathscr X})\|_F^2}{\|{\mathscr X}_{\text{restored}}-\widehat{\mathscr X}\|_F^2},\]
where $E(\widehat{\mathscr X})$ denotes the mean gray-level of the uncontaminated image $\widehat{\mathscr{X}}$.
All computations were carried out using the MATLAB environment on an Intel(R) Core(TM) i7-8550U CPU @ 1.80GHz (8 CPUs) computer with 12 GB of
RAM. The computations were done with approximately 15 decimal digits of relative
accuracy.
\subsection{Example 1}
In this example we present the experimental results recovered by Algorithm \ref{TG-GMRES(m)} and Algorithm \ref{TG-GKB} for the reconstruction of a cross-channel blurred color images that have been contaminated by both within and cross blur, and additive noise. The cross-channel blurring is determined by the matrix
$$\mathbf{A}_{\mathrm{color}}=\left[\begin{array}{ccc}
0.8 & 0.10& 0.10 \\
0.10 & 0.80 & 0.10 \\
0.10& 0.10 & 0.80
\end{array}\right].$$
We consider two $\mathrm{RGB}$ images from \textbf{MATLAB}, $\tt papav256$ ($\widehat{\mathscr X}\in\mathbb{R}^{256\times256\times3}$) and $\tt peppers$ ($\widehat{\mathscr X}\in\mathbb{R}^{512\times512\times3}$). They are shown on Figure \ref{fig1}. For the within-channel blurring, we let $\sigma=4$ and $r=6$. The considered noise levels are $\nu=10^{-3}$ and $\nu=10^{-2}$. The associated blurred and noisy RGB images $\mathscr{C}=\mathscr{A}\ast\widehat{\mathscr{X}}\ast\mathscr{B}+\mathscr{N}$ for noise level $\nu=10^{-3}$ are shown on Figure \ref{fig2}. Given the contaminated RGB image $\mathscr{C}$, we would like to recover an approximation of the original RGB image $\widehat{\mathscr X}$. The restorations for noise level $\nu=10^{-3}$ are shown on Figure \ref{fig3} and they are obtained by applying Algorithm \ref{TG-GMRES(m)} implementing the Tensor T-Global GMRES method, with $\mathscr{X}_0=\mathscr{O}$, $tol=10^{-6}$, $m=10$ and $\text{Iter}_\text{max}=10$. Using GCV, the computed optimal value for the projected problem was $\mu_{10}=3.82\times 10^{-5}.$ Table \ref{tab1} compares, the computing time (in seconds), the relative errors and the SNR of the computed restorations. Note that in this table, the allowed maximum number of outer iterations for Algorithm \ref{TG-GMRES(m)} with noise level $\nu=10^{-2}$ was $\text{Iter}_\text{max}=4$ and the maximum number of inner iterations was $m=4$. The restorations obtained with Algorithm \ref{TG-GKB} are shown on Figure \ref{fig4}. For the $\tt papav256$ color image, the discrepancy principle with $\eta=1.1$ is satisfied when $m=64$ steps of the Tensor T-GGKB method (Algorithm \ref{TG-GK}) have been carried out, producing a regularization parameter given by $\mu_m=5.57\times 10^{-5}$. For comparison with existing approaches in the literature, we report in Table \ref{tab1} the results obtained with the method
proposed in \cite{CR}. This method utilizes the connection between (standard)
Golub--Kahan bidiagonalization and Gauss quadrature rules for solving large
ill-conditioned linear systems of equations (\ref{linmodel}). We refer to this method as
GKB. It determines the regularization parameter analogously to Algorithm \ref{TG-GKB}, and uses a similar stopping criterion. We can see that the
methods yield restorations of the same quality, but the new proposed methods perform significantly better in terms of cpu-time.
\begin{table}[htbp]
\begin{center}\footnotesize
\renewcommand{\arraystretch}{1.3}
\begin{tabular}{cccccc}\hline
\multicolumn{1}{c}{{RGB images}} &\multicolumn{1}{c}{{Noise level}} & \multicolumn{1}{c}{{Method}}&
\multicolumn{1}{c}{{SNR}} & Relative error& CPU-time (sec) \\
\hline
\multirow{6}{*}{$\tt papav256$}&\multirow{4}{3em}{$10^{-3}$}&Algorithm \ref{TG-GMRES(m)} &$21.01$&$6.64\times10^{-2}$&$\phantom{1}6.62$\\
&&Algorithm \ref{TG-GKB}&$20.41$& $7.12\times10^{-2}$&$\phantom{1}5.87$\\
&&GKB&$20.99$&$7.12\times10^{-2}$&$18.61$\\
\cline{2-6}
&\multirow{4}{3em}{$10^{-2}$}&Algorithm \ref{TG-GMRES(m)}&$18.00$&$9.40\times10^{-2}$&$\phantom{1}1.18$\\
& &Algorithm \ref{TG-GKB}&$17.78$&$9.64\times10^{-2}$&$\phantom{1}1.11$\\
&&GKB&$17.78$&$9.64\times10^{-2}$&$5.79$\\
\hline
\multirow{6}{*}{$\tt peppers$}&\multirow{4}{3em}{$10^{-3}$}&Algorithm \ref{TG-GMRES(m)} &$19.39$&$5.50\times10^{-2}$&$\phantom{1}24.32$\\
&&Algorithm \ref{TG-GKB}&$19.11$& $5.68\times10^{-2}$&$\phantom{1}25.63$\\
&&GKB&$19.11$&$5.68\times10^{-2}$&$78.13$\\
\cline{2-6}
&\multirow{4}{3em}{$10^{-2}$}&Algorithm \ref{TG-GMRES(m)}&$16.23$&$7.92\times10^{-2}$&$\phantom{1}4.59$\\
& &Algorithm \ref{TG-GKB}&$15.61$&$8.50\times10^{-2}$&$\phantom{1}3.39$\\
&&GKB&$15.61$&$8.50\times10^{-2}$&$15.16$\\
\hline
\end{tabular}
\caption{Results for Example 1.}\label{tab1}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{rgb_orig.pdf}
\caption{Example 1: Original RGB images: $\tt peppers$ (left), $\tt papav256$ (right).}\label{fig1}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{rgb_blur.pdf}
\caption{Example 1: Blurred and noisy images, $\tt peppers$ (left), $\tt papav256$ (right).}\label{fig2}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{rgb_gmres.pdf}
\caption{Example 1: Restored images by Algorithm \ref{TG-GMRES(m)}, $\tt peppers$ (left), $\tt papav256$ (right).}\label{fig3}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{rgb_ggkb.pdf}
\caption{Example 1: Restored images by Algorithm \ref{TG-GKB}, $\tt peppers$ (left), $\tt papav256$ (right).}\label{fig4}
\end{center}
\end{figure}
\subsection{Example 2}
In this example, we evaluate the effectiveness of Algorithm \ref{TG-GMRES(m)} and Algorithm \ref{TG-GKB} when applied to the restoration of a color video defined by a sequence of RGB images. Video restoration is the problem of restoring a sequence of $k$ color images (frames). Each frame is represented by a tensor of $n \times n\times3$ pixels. In the present example, we are interested in restoring 10 consecutive frames of a contaminated video. Note that the processing of such given frames, one at a time, is extremely time consuming. We consider the xylophone video from MATLAB. The video clip is in MP4 format with each frame having $240 \times 240$ pixels. The (unknown) blur- and noise-free frames are stored in the tensor $\widehat{ \mathscr X} \in \mathbb{R}^{240 \times 240\times30}$, obtained by stacking the grayscale images that constitute the three channels of
each blurred color frame. These frames are blurred by $\mathscr {A}\star \widehat{\mathscr{X}}\star \mathscr{B}= \widehat{\mathscr{C}}$, where $\mathscr {A}$ and $\mathscr {B}$ are a 3-way tensors such that $\mathscr {A}(:,:,1)= \mathbf{A^{(2)}}$, $\mathscr {B}(:,:,1)= (\mathbf{A^{(1)}})^T$ and $\mathscr {A}(:,:,i)=\mathscr {B}(:,:,i)=0$, for $i=2,...,30$, using $\sigma=2$ and $r=4$ to build the blurring matrices. We consider white Gaussian noise of levels $\nu=10^{-3}$ or $\nu=10^{-2}$. Figure \ref{frame5ob} shows the 5th exact (original) frame and the contaminated version with noise level $\nu=10^{-3}$, which is to be restored. Table \ref{tab2} displays the performance of Algorithm \ref{TG-GMRES(m)} and Algorithm \ref{TG-GKB}. In Algorithm \ref{TG-GMRES(m)}, we have used as an input for noise level $\nu=10^{-3}$, $\mathscr{C}$, $\mathscr{X}_0=\mathscr{O}$, $tol=10^{-6}$, $m=10$ and $\text{Iter}_{\text{max}}=10$.The chosen inner and outer iterations for noise level $\nu=10^{-2}$ were $m=4$ and $\text{Iter}_{\text{max}}=4$, respectively. For the ten outer iterations, minimizing the GCV function produces $\mu_{10}=1.15 \times 10^{-5}$. Using Algorithm \ref{TG-GKB}, the discrepancy principle with $\eta=1.1$ have been satisfied after $m=59$ steps of T-GGKB method (Algorithm \ref{TG-GK}), producing a regularization parameter given by $\mu_m=1.06\times10^{-4}$. For completeness, the restorations obtained with Algorithm \ref{TG-GMRES(m)} and Algorithm \ref{TG-GKB} are shown on the left-hand and the right-hand side of Figure \ref{frame5r}, respectively.
\begin{table}[htbp]
\caption{Results for Example 2.}\label{tab2}
\begin{center}
\begin{tabular}{lcccc}
\hline Noise level & Method & SNR & Relative error & CPU-time (second) \\
\hline {$10^{-3}$}& Algorithm \ref{TG-GMRES(m)} & 19.97 & $4.07 \times 10^{-2}$ & 35.68 \\ & Algorithm \ref{TG-GKB} & 19.24 & $4.43 \times 10^{-2}$ & 25.52 \\
\hline {$10^{-2}$} & Algorithm \ref{TG-GMRES(m)} & 15.17 & $7.08 \times 10^{-2}$ & 6.12 \\
&Algorithm \ref{TG-GKB} & 15.13 & $7.11 \times 10^{-2}$ & 4.40 \\
\hline
\end{tabular}
\end{center}
\end{table}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{frame5ob.pdf}
\caption{Example 2: Original frame no. 5 (left), blurred and noisy frame no. 5 (right).
}\label{frame5ob}
\end{center}
\end{figure}
\begin{figure}
\begin{center}
\includegraphics[width=5in]{frame5resto.pdf}
\caption{Example 2: restored frame no. 5 by Algorithm \ref{TG-GMRES(m)} (left), and restored frame no. 5 by Algorithm \ref{TG-GKB}
(right).}\label{frame5r}
\end{center}
\end{figure}
\section{Conclusion}
In this paper we have proposed tensor version of GMRES and Golub–Kahan bidiagonalization
algorithms using the T-product, with applications to solving large-scale linear tensor equations arising in the reconstructions of blurred and noisy multichannel images and videos. The numerical experiments that we have performed show the
effectiveness of the proposed schemes to inexpensively computing regularized solutions of high quality.
\bibliographystyle{siam}
| 2024-02-18T23:39:56.196Z | 2020-06-15T02:15:39.000Z | algebraic_stack_train_0000 | 873 | 10,442 |
|
proofpile-arXiv_065-4434 | \section{Introduction and summary}
One of the most exciting developments the past few years, is the discovery of exactly solvable models of quantum gravity, starting with Kitaev's SYK models \cite{Kitaevtalks,*Sachdev:1992fk,*Polchinski:2016xgd,*Maldacena:2016hyu}, going through bulk Jackiw-Teitelboim (JT) gravity \cite{Jackiw:1984je,*Teitelboim:1983ux,Almheiri:2014cka,*Jensen:2016pah,*Maldacena:2016upp,*Engelsoy:2016xyb} and its correlation functions \cite{Bagrets:2016cdf,Stanford:2017thb, Mertens:2017mtv,Lam:2018pvp,Mertens:2018fds,Blommaert:2018oro,Kitaev:2018wpr, *Yang:2018gdb,Iliesiu:2019xuh}, and leading to the inclusion of higher genus and random matrix descriptions \cite{Saad:2019lba}, making contact with the black hole information paradox in its various incarnations \cite{Saad:2019pqd, Almheiri:2019qdq, *Penington:2019kki, Marolf:2020xie}.
It goes without saying that finding other models that are solvable to the same extent would be highly valuable, in particular to test the robustness of the ideas. For example, it is important to have a similar non-perturbative definition of theories of gravity as in \cite{Saad:2019lba} that are also coupled to matter.
In the same work \cite{Saad:2019lba}, it was proposed that JT gravity can be viewed as a parametric limit of the older minimal string model.
The latter can be viewed as a double-scaled matrix integral \cite{Brezin:1990rb, *Douglas:1989ve, *Gross:1989vs} that in the continuum description becomes a non-critical string theory described by Liouville CFT, coupled to a minimal model and the $bc$ ghost sector. We will call this combination \emph{Liouville gravity} in what follows. Since there is a substantial amount of evidence in favor of a random matrix description of these models, finding JT gravity within a limiting situation illustrates that it is in hindsight not a surprise at all that JT gravity is a matrix integral.
\\~\\
In this work, we will develop these UV ancestors of JT gravity in more detail. We will enlarge our scope slightly: instead of restricting to only minimal models to complete the Liouville CFT, we will consider a generic matter CFT for the first few sections. In that case, we do not have a (known) matrix description to guide us. At times, we will restrict to the minimal string and find perfect agreement between continuum and matrix descriptions.
A particular emphasis is placed on correlation functions within these theories and how precisely they approach the JT correlation functions in a certain limit.
We also highlight how the Riemann surface description of JT gravity at higher topology also generalizes (in fact, quantum deforms) to these models leading to generalizations of the Weil-Petersson (WP) volumes to glue surfaces together.
\\~\\
Let us sketch the set-up in more detail.
Consider a disk-shaped worldsheet with coordinates $(z,\bar{z})$ and boundary coordinate $x$. Within Liouville gravity, we are allowed to insert closed string tachyon vertex operators $\mathcal{T}_{i}$ and open string tachyon vertex operators $\mathcal{B}_{i}$. Denoting these operator insertions collectively by $\mathcal{O}$, we will define the disk amplitudes $\mathcal{A}_\mathcal{O}(\ell_1,\ldots, \ell_n)$ with fixed length boundaries $\ell_1 \hdots \ell_n$ (see discussion around \eqref{introbdycond} for more details on the boundary conditions) as
\begin{figure}[h]
\centering
\raisebox{18mm}{$\mathcal{A}_\mathcal{O}(\ell_1,\ldots, \ell_n) \quad = \quad$}
\begin{tikzpicture}[scale=0.8]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[fill] (-0.5,0) circle (0.06);
\node at (0,-1.985) {};
\node at (-0.1,0) {\small $\mathcal{T}_1$};
\draw[fill] (0.5,0) circle (0.06);
\node at (0.9,0) {\small $\mathcal{T}_2$};
\node at (0.75,-0.5) {\small $...$};
\draw[fill] (-1.3,-0.75) circle (0.06);
\node at (-1.65,-0.8) {\small $\mathcal{B}_1$};
\node at (-1.85,0) {\small $\ell_1$};
\node at (-1,1.6) {\small $\ell_2$};
\node at (1,1.6) {\small $\ell_3$};
\draw[fill] (-1.3,0.75) circle (0.06);
\node at (-1.65,0.8) {\small $\mathcal{B}_2$};
\draw[fill] (0,1.5) circle (0.06);
\node at (0,1.8) {\small $...$};
\end{tikzpicture}
\end{figure}
\noindent Since the string worldsheet theory is treated as 2d gravity (by imposing the Virasoro constraints), the operator insertions of interest $\mathcal{B}_i$ and $\mathcal{T}_i$ have to be worldsheet coordinate-invariant.
The familiar strategy from string theory is to restrict these to conformal weight one (in both holomorphic and anti-holomorphic sectors), and then integrate them over the entire worldsheet:
\begin{equation}\label{defopintro}
\mathcal{B}= \oint_{\partial \Sigma} dx \, \Phi_{\rm M}(x)e^{\beta \phi(x)} ,\qquad \mathcal{T}= \int_\Sigma d^2z \,\mathcal{O}_{\rm M}(z,\bar{z}) e^{2 \alpha \phi(z,\bar{z})}.
\end{equation}
Here $\Phi_{\rm M}$ and $\mathcal{O}_{\rm M}$ denote boundary and bulk matter operators, $\phi$ is the Liouville field (scale factor in physical metric) and the parameters $\beta$ and $\alpha$ are tuned to the matter operator to make the integrand marginal in both cases. These operators will be labeled by the Liouville parameters corresponding to the matter operators $\alpha_M$ and $\beta_M$ (see \eqref{bulklioupar2} and \eqref{bdylioupar} for the definition). The conventional interpretation of these formulas is that the bare matter operators $\Phi_{\rm M}$ and $\mathcal{O}_{\rm M}$ (as objects in only the matter CFT), are gravitationally dressed by the Liouville vertex operators $e^{\beta \phi(x)}$ and $e^{2 \alpha \phi(z,\bar{z})}$ to produce observable worldsheet diff-invariant operators. From this perspective, the matter fields are the more fundamental objects and we will indeed reach this conclusion throughout our work as well.
As well-known in string theory, we can use the SL$(2,\mathbb{R})$ isometries of the disk to gauge-fix the worldsheet location of three degrees of freedom (where a bulk operator counts as two, and a boundary operator as one). If one has more operator insertions, there are non-trivial integrations left over the moduli space of the punctured disk. Throughout this work, we only focus on the case without moduli integration. This leaves only four disk configurations which we explicitly investigate.
In the final section of this work, we investigate higher topology, and in particular the annulus diagram which has a single worldsheet modulus.
\\~\\
It should be emphasized that the worldsheet boundary coordinates $x_i$ (and their moduli) and the physical distances $\ell_i$ are distinct. They are only related by the non-local (and not so restrictive) constraints:
\begin{equation}\label{introbdycond}
\ell_i = \int_{x_i}^{x_{i+1}}dx\hspace{0.1cm} e^{b\phi(x)}
\end{equation}
in terms of the Liouville field $\phi$ appearing in the Liouville gravity models we will consider.
For all disk cases we study, the worldsheet coordinate $x$-dependence drops out due to gauge-fixing, but the final result depends explicitly on the physical distances $\ell$. In this sense, even though boundary operators are integrated over the worldsheet as in \eqref{defopintro}, they behave as local insertions in the physical space and their gravitational dressing has the effect of fixing geodesic distances between them. Moreover, even though the worldsheet theory is a CFT, the boundary amplitudes as a function of physical lengths do not respect conformal symmetry (see for example \eqref{twoa} below).
For the annulus amplitude, there is a single worldsheet modulus $\tau$ that needs to be integrated over. Doing so leads in the end to an amplitude that depends on the physical lengths of both boundaries of the annulus.
\\~\\
Next we present a summary of the main results regarding fixed length amplitudes, some known some new, that are computed in this paper. We introduce the quantities:
\begin{equation}
\mu_B(s) = \kappa \cosh 2\pi b s,~~~\kappa \equiv \frac{\sqrt{\mu}}{\sqrt{\sin \pi b^2}},
\end{equation}
where $\mu$ is the bulk cosmological constant, $\mu_B(s)$ is the boundary cosmological constant for FZZT boundaries labeled by $s$, and $b$ is defined through the central charge of the Liouville field $c_{\rm L}=1+6Q^2$, with $Q=b+1/b$.
\paragraph{Partition Function:} We compute the marked partition function
\begin{eqnarray}
Z(\ell) = N \mu^{\frac{Q}{2b}} \int_{0}^{\infty} ds ~e^{-\ell \mu_B(s)}\rho(s) ,
\end{eqnarray}
where we define the spectral weight
\begin{equation}
\rho(s)\equiv \sinh 2\pi b s \sinh\frac{2\pi s}{b},
\end{equation}
which coincides with the Virasoro modular S-matrix $S_0{}^s=\rho(s)$, and $N$ is a length independent normalization. After performing the integral, the partition function can be put in the more familiar form $Z(\ell) \sim \frac{1}{\ell} \mu^{\frac{1}{2b^2}}K_{1/b^2}(\kappa \ell)$. This quantity was previously obtained by \cite{Fateev:2000ik} (and from the dual matrix integral by \cite{Moore:1991ir}). We present a more systematic derivation which we found to be more useful in order to generalize this to correlation functions. \\
Following \cite{Saad:2019lba} we interpret $\mu_B(s)$ as the energy of the boundary theory dual to the bulk gravity, $\rho(s)$ as a density of states, and $\ell$ as an inverse temperature.
\paragraph{Bulk one-point function:} We compute the fixed length partition function with a bulk insertion $\mathcal{T}_{\alpha_M}$, and $P$ is the Liouville momentum associated to $\alpha_M$. This can be depicted as
\begin{equation}
\left\langle \mathcal{T}_{\alpha_M}\right\rangle_\ell =~ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}, scale=0.7]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[fill] (0,0) circle (0.06);
\node at (-1.8,0.2) {\small $\ell$};
\node at (0.5,0) {\small $\mathcal{T}$};
\end{tikzpicture}
\end{equation}
Repeating the previous procedure we obtain
\begin{equation}
\label{eq:b1}
\left\langle \mathcal{T}_{\alpha_M}\right\rangle_\ell = \frac{2}{b} \int_{0}^{\infty} ds\hspace{0.1cm} e^{-\ell \mu_B(s)} \cos 4 \pi P s.
\end{equation}
The integrand coincides with the Virasoro modular S-matrix $S_P{}^s=\cos 4 \pi P s$. We interpret the bulk operator as creating a defect (for $P$ imaginary) or a hole (for $P$ real) on the physical space. This interpretation is consistent with classical solutions of the Liouville equation, and also becomes clear in the JT gravity limit \cite{Mertens:2019tcm}.
\paragraph{Boundary two-point function:} The two point function between boundary operators, labeled by $\beta_M$, inserted between segments of fixed physical length is defined from the following diagram
\begin{equation}
\hspace{-0.2cm}\mathcal{A}_{\beta_M}(\ell_1,\ell_2) =~ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}, scale=0.7]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[fill] (-1.5,0) circle (0.06);
\draw[fill] (1.5,0) circle (0.06);
\node at (-1.9,0) {\small $\mathcal{B}$};
\node at (1.9,0) {\small $\mathcal{B}$};
\node at (0,-1.8) {\small $\ell_1$};
\node at (0,1.8) {\small $\ell_2$};
\end{tikzpicture}
\end{equation}
We obtain
\begin{equation}
\label{twoa}
\mathcal{A}_{\beta_M}(\ell_1,\ell_2)= N_{\beta_M} \int ds_1 ds_2 \rho(s_1) \rho(s_2)\hspace{0.05cm} e^{-\mu_B(s_1)\ell_1} e^{-\mu_B(s_2)\ell_2}\hspace{0.05cm}\mathcal{M}_{\beta_M}(s_1,s_2)^2,
\end{equation}
where $N_{\beta_M}$ is a length independent constant and we define the amplitude
\begin{equation}
\label{twoam}
\mathcal{M}_{\beta_M}(s_1,s_2) \equiv \frac{\prod_{\pm\pm}S_b\left(\beta_M \pm i s_1 \pm i s_2\right)^{1/2}}{S_b(2\beta_M)^{1/2}},
\end{equation}
where $S_b(x)$ is the double sine function. Its definition and properties that will be relevant in this paper can be found in Appendix B.1 of \cite{Mertens:2017mtv}. The appearance of this structure was derived somewhat cavalier in \cite{Mertens:2019tcm}, and we substantiate it here.
Following \cite{Saad:2019lba}, the amplitude $\mathcal{M}_{\beta_M}(s_1,s_2)$ can be interpreted as a matrix element of operators in the dual boundary theory between energy eigenstates. We interpret this result as an exact expression for the gravitational dressing by Liouville gravity of boundary correlators (notice that the boundary lengths are not necessarily large and therefore this corresponds to gravity in a finite spacetime region).
Another motivation for studying these correlators is the resemblance with exact results in double-scaled SYK derived in \cite{Berkooz:2018jqr,*Berkooz:2018qkz,*Berkooz:2020xne}, which we hope to come back to in future work.
\paragraph{Boundary three-point function:} The fixed length boundary three-point function is defined as
\begin{equation}
\hspace{-0.2cm}\mathcal{A}_{123}(\ell_1,\ell_2,\ell_3) =~ \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}, scale=0.7]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[fill] (0,1.5) circle (0.06);
\draw[fill] (-1.3,-0.75) circle (0.06);
\draw[fill] (1.3,-0.75) circle (0.06);
\node at (-1.7,-0.8) {\small $\mathcal{B}_1$};
\node at (1.75,-0.8) {\small $\mathcal{B}_3$};
\node at (0,-1.85) {\small $\ell_1$};
\node at (-1.6,1) {\small $\ell_2$};
\node at (1.6,1) {\small $\ell_3$};
\node at (0,1.85) {\small $\mathcal{B}_2$};
\end{tikzpicture}
\vspace{-0.4cm}
\end{equation}
and we get
\begin{eqnarray}
\label{threea}
\mathcal{A}_{123}(\ell_1,\ell_2,\ell_3) &=& N_{\beta_1\beta_2\beta_3} \int ds_1 ds_2 ds_3 \rho(s_1) \rho(s_2)\rho(s_3) e^{- \mu_B(s_1)\ell_1}e^{- \mu_B(s_2)\ell_2}e^{- \mu_B(s_3)\ell_3} \nonumber\\
&&\times \mathcal{M}_{\beta_{M2}}(s_2,s_3)\mathcal{M}_{\beta_{M1}}(s_1,s_2)\mathcal{M}_{\beta_{M3}}(s_1,s_3) \sj{\beta_{M1}}{\beta_{M2}}{\beta_{M3}}{s_3}{s_1}{s_2},
\end{eqnarray}
where $N_{\beta_1\beta_2\beta_3}$ is a length independent constant. The quantity appearing in the second line is the quantum deformed $6j$ symbols computed by Teschner and Vartanov \cite{Teschner:2012em, *Vartanov:2013ima} (this quantity is proportional to a Virasoro fusion kernel). This expression gives the universal Liouville gravitational dressing of boundary three-point functions.
\paragraph{Bulk-boundary correlator:}
The fixed length bulk-boundary two-point function is defined by
\begin{equation}
\hspace{0cm}\mathcal{A}_{\alpha_M, \beta_M}(\ell) =\hspace{0.5cm} \begin{tikzpicture}[baseline={([yshift=-.5ex]current bounding box.center)}, scale=0.7]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[fill] (1.5,0) circle (0.06);
\node at (0,-1.985) {};
\node at (0,1.8) {\small $\ell$};
\draw[fill] (0,0) circle (0.06);
\node at (-0.5,0) {\small $\mathcal{T}$};
\node at (1.9,0) {\small $\mathcal{B}$};
\end{tikzpicture}
\vspace{-0.4cm}
\end{equation}
where $\alpha_M$ (with momentum $P$) and $\beta_M$ label the bulk and boundary insertions. We obtain
\begin{align}
\label{twoabb}
\mathcal{A}_{\alpha_M, \beta_M}(\ell) = N_{\beta_M,P}\int_{0}^{+\infty} ds_1 ds_2 \rho(s_1) \rho(s_2) e^{-\mu_B(s_1) \ell} \, \frac{S_P{}^{s_2}}{S_0{}^{s_2}} \, \mathcal{M}_{\beta_M/2}(s_1,s_2)^2,
\end{align}
in terms of the Virasoro modular S-matrices defined above.
\\~\\
We will also define the JT classical limits of these equations, where we will reproduce known expressions found in \cite{Mertens:2019tcm,Mertens:2017mtv,Iliesiu:2019xuh}.
\\~\\
If we take the specific case of the minimal string (where the matter sector is a minimal model), we have the power of the matrix model at our disposal to aid our investigation. In particular, the set of minimal string boundary primaries correspond to setting $\beta_M = -bj$, for $j\in \mathbb{N}/2$. The two-point amplitude \eqref{twoam} becomes degenerate (due to a singularity in the denominator) and using the matrix description we will derive the answer:
\begin{equation}
\mathcal{M}_{\beta_M}(s_1,s_2)^2 = (2j)! \sum_{n=-j}^{j}\frac{\delta(s_1-s_2-in b)}{\prod_{\stackrel{m=-j}{m\neq n}}^{j} (\cosh 2\pi b (s+i nb) - \cosh 2\pi b (s+imb))}.
\end{equation}
These delta-functions have to be interpreted as causing a contour shift within the double integral \eqref{twoa}. One can also take the degenerate limit directly in \eqref{twoam} using quantum group methods, and we will find agreement.
Taking the JT classical limit for these correlators, we find the degenerate Schwarzian bilocal correlators, for which the first case $j=1/2$ was studied in appendix D of \cite{Mertens:2019tcm}, and the generic case is studied in \cite{Mertens:2020pfe}.
Next to these amplitudes, we also analyze multi-boundary amplitudes for the minimal string. A four-boundary example is drawn in Figure \ref{multiboundaryQ}.
\begin{figure}[h]
\centering
\includegraphics[width=0.3\textwidth]{multiboundaryQ.pdf}
\caption{Genus zero $n$-boundary loop amplitude (here $n=4$).}
\label{multiboundaryQ}
\end{figure}
For $n$ circular boundaries, we find the genus $g$ amplitude is of the form:
\begin{eqnarray}\label{eq:nloopcorr}
\Big\langle \prod_{i=1}^{n} Z(\ell_i) \Big\rangle_{g, \, {\rm conn.}} \sim \prod_{i=1}^{n}\int_0^\infty \lambda_i d\lambda_i \tanh \pi \lambda_i \,V_{g,n}(\bm{\lambda}) \, \left\langle \mathcal{T}_{\alpha_{Mi}}\right\rangle_{\ell_i},
\end{eqnarray}
where $\left\langle \mathcal{T}_{\alpha_{Mi}}\right\rangle$ is the bulk one-point function \eqref{eq:b1} with $P_i = b\lambda_i /2$ (which we interpret as a Liouville gravity trumpet partition function), the quantity $V_{g,n}(\bm{\lambda})$ is a symmetric polynomial of order $n+3g-3$ in the $\lambda_i^2$ and a quantum deformation of the WP volumes. The measure factor $\lambda_i d\lambda_i \tanh \pi \lambda_i$ generalizes the classical gluing formula for Riemann surfaces $b_i db_i$, where $b_i$ is the circumference of the gluing geodesic. Indeed, for large values of $\lambda_i$ (the classical JT limit), these formulas reduce to these classical WP gluing formulas. \\
In particular, we focus on the genus zero contributions, for which we give a general formula for the deformed volumes (and therefore by taking the appropriate limit, an explicit formula for the classical WP volumes). For higher genus, we argue they also take the form \eqref{eq:nloopcorr}. It would be interesting to develop a more geometrical interpretation of this quantum deformation of the WP volumes. Such derivation would confirm the choice of normalization of the one-point function and the integration measure in \eqref{eq:nloopcorr} \footnote{The ambiguity arises since, for example, the final answer (except for the special case of two boundaries and no handles) is unchanged under $d\mu(\lambda) \to f(\lambda) d\mu(\lambda)$ and $\langle \mathcal{T} \rangle \to f(\lambda)^{-1} \langle \mathcal{T} \rangle$, for an arbitrary $f(\lambda)$ that goes to one in the JT gravity limit. We argue below the choice in \eqref{eq:nloopcorr} is the most natural one.}.
\\~\\
The organization of the paper and summary of some more results is as follows.
In \textbf{section \ref{sec:review}} we give a quick review on the non-critical string, Liouville gravity and the minimal string. The knowledgeable reader can skip this section, although we do fix conventions and write down previous results that will be essential later on.
In \textbf{section \ref{sec:diskZ}} we describe a systematic way to compute fixed length amplitudes and illustrate it by reproducing known formulas for the fixed length partition function.
In \textbf{section \ref{sec:diskcorr}} we compute explicitly fixed length boundary correlation functions with and without bulk insertions. We also define and take the JT gravity limit of these observables.
\textbf{Section \ref{s:qg}} explains the structure of these equations as coming from a constrained version of the $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ quantum group. In particular, the vertex function is reproduced from a 3j-symbol computation with Whittaker function insertions. In \textbf{section \ref{sec:MM}} we show for the case of the minimal string how to produce the correlators directly from the matrix model. We check that the quantum group formulas from the previous section lead to the same structure.
Finally in \textbf{section \ref{sec:othertopo}} we study other topologies. We give a streamlined derivation of the cylinder amplitude. We also review the exact result presented in \cite{Ambjorn:1990ji, Moore:1991ir} for the $n$ boundary-loop correlator at genus zero for the minimal string theory and discuss its decomposition in terms of gluing measures, bulk one-point functions and quantum deformed WP volume factors. By taking the JT gravity limit we give a very simple generating function of WP volumes for $n$ geodesic boundaries at genus zero.
In \textbf{section \ref{sec:conclusions}} we end with a discussion and open problems for future work. In particular, we argue that the bulk gravity can be rewritten in terms of a 2d dilaton gravity model with a sinh dilaton potential. In the appendices, we include some related topics that would otherwise distract from the story. In particular, we discuss the role of poles in the complex $\mu_B$ plane as one transforms to fixed length amplitudes, we discuss degenerate bulk one-point functions, and degenerate (ZZ) branes as boundary segments. For the multi-boundary story for unoriented surfaces, we compute the crosscap spacetime contribution, which we show matches with a GOE/GSE matrix model calculation.
\section{Non-critical strings and 2d gravity} \label{sec:review}
This section contains review material on Liouville gravity and minimal string theory. We first discuss the bulk stories in \ref{s:qlg} and \ref{s:mst}, and then the boundary versions in \ref{s:bdy}.
\subsection{Quantum Liouville gravity}
\label{s:qlg}
We study two dimensional theories on Riemann surfaces $\Sigma$ with dynamical gravity, by summing over all metrics $g_{\mu\nu}(x)$ (in Euclidean signature) modulo diffeomorphisms. We also add a matter theory with fields $\chi(x)$ living on the Riemann surfaces with action $S_M[\chi;g]$. The starting point is the path integral
\begin{equation}\label{eq:deftheory}
Z=\sum_{\rm topologies} \int \frac{\mathcal{D}g \mathcal{D}\chi}{{\rm Vol}({\rm Diff})} e^{ - S_M[\chi;g] - \mu_0 \int_\Sigma d^2x \sqrt{g}},
\end{equation}
where $\mu_0$ is the bare cosmological constant. We will focus only on the case where the matter sector is a CFT with central charge $c_M$. We will also consider minimal models as matter CFT which might not have a path integral representation.
Following \cite{Polyakov:1981rd, *Distler:1988jt, *David:1988hj} we can gauge fix conformal gauge $g_{\mu\nu}=e^{2 b\phi(x)} \hat{g}_{\mu\nu}(x)$ with $\phi$ a dynamical scale factor, $b$ a normalization to be fixed later, and $\hat{g}$ a fiducial metric. This has the effect of adding the usual $bc$-ghosts with central charge $c_{\rm gh} = -26$ and a Liouville mode coming in part from the conformal anomaly in the path integral measure and also from the bare cosmological constant. One ends up with an action consisting of the matter on the fixed fiducial metric $S_M[\chi; \hat{g}]$, the ghost action, and a Liouville field theory with action \cite{Polyakov:1981rd}
\begin{equation}
S_L[\phi] = \frac{1}{4\pi} \int_{\Sigma} \left[ (\hat{\nabla} \phi)^2 + Q \hat{R} \phi + 4 \pi \mu e^{2 b \phi} \right] .
\end{equation}
This can be interpreted as CFTs living on the fiducial metric. It is important the matter sector is a CFT so that no explicit interactions appear between matter and the Liouville field. The renormalized bulk cosmological constant is $\mu$ and scale invariance fixes the background charge $Q = b + b^{-1}$. The central charge of the Liouville mode is $c_L = 1+6Q^2$. The three sectors are coupled through the conformal anomaly cancellation
\begin{equation}
c_M + c_L + c_{\rm gh} =0.
\end{equation}
The results in this paper are mostly independent of the details of the matter CFT but we will refer to two cases for concreteness. We will analyze timelike Liouville CFT as matter, with action
\begin{equation}\label{eq:timeLioaction}
S_M[\chi] = \frac{1}{4\pi} \int_{\Sigma} \left[ -(\hat{\nabla} \chi)^2 - q \hat{R} \chi + 4 \pi \mu_M e^{2 b \chi} \right].
\end{equation}
For simplicity we can also set its cosmological constant term $\mu_M$ to zero, in which case the theory becomes the usual Coulomb gas. The central charge for this theory is $c_M = 1- 6 q^2$. The matter and Liouville background charges are related from the anomaly cancellation
\begin{equation}
c_M + c_L =26,~~~\Rightarrow~~~q = 1/b - b,
\end{equation}
which for $\mu_M\neq 0$ is consistent with the choice of the exponential interaction in \eqref{eq:timeLioaction}. This theory is equivalent to a Liouville CFT with $\tilde{b} = i b$, $\tilde{Q} = i q$ and $\tilde{\mu}=\mu_M$. The case with non-vanishing matter cosmological constant was analyzed in detail in \cite{Zamolodchikov:2005fy}. In the next section we will also consider the case of a $(p,q)$ minimal model.
Now we will go through the construction of physical operators in these theories. First, generic bulk operators of the Liouville CFT and matter CFT, seen as two independent field theories, can be written as
\begin{eqnarray}\label{bulklioupar1}
{\rm Liouville:}&&~~~\hspace{0.1cm}~V_\alpha = \exp{(2 \alpha \phi)}~~~~~~~~\hspace{0.1cm}\Delta_\alpha = \alpha(Q-\alpha),\\
\label{bulklioupar2} \hspace{-0.3cm}{\rm Matter:}&&~~\mathcal{O}_{\alpha_M} = \exp{(2 \alpha_M \chi)}~~~~\Delta_{\alpha_M} = \alpha_M(q+\alpha_M),
\end{eqnarray}
where we also wrote their scaling dimension under worldsheet conformal transformations. When we consider other matter CFT we will still label their operators by the parameter $\alpha_M$. It is customary to also introduce the Liouville momentum and energy $\alpha= Q/2 + i P$ and $ \alpha_M= -q/2 + i E$. These can be interpreted as target space energy and momentum $(E,P)$ in a Minkowski 2D target space $(X^0,X^1)=(\chi, \phi)$ with a linear dilaton background.
If gravity was not dynamical, the only operators of the theory would be the matter $\mathcal{O}_{\alpha_M}$. When gravity is turned on diffeomorphism invariant observables are made out of physical operators that are marginal. The gravitational dressing necessary for this is achieved by combining matter and Liouville operators into the bulk vertex operator
\begin{equation}\label{tachyondef}
\mathcal{T}_{\alpha_M} \sim \int_{\Sigma} \hspace{0.1cm}\mathcal{O}_{\alpha_M}(x) \hspace{0.01cm}V_\alpha(x),
\end{equation}
with a normalization that will be fixed later. After gauge fixing, we can replace the integral by a local insertion with the ghosts $\mathcal{T}_{\alpha_M} \sim c \bar{c} \hspace{0.1cm}\mathcal{O}_{\alpha_M} \hspace{0.01cm}V_\alpha$. In the context of non-critical string theory, these insertions create bulk tachyons which will be labeled by its matter content. The parameter $\alpha$ controlling the gravitational dressing is fixed through the relation \cite{Knizhnik:1988ak}
\begin{equation}
\Delta_{\alpha_M} + \Delta_\alpha = 1 ,~~~\Rightarrow~~~\alpha_+=b-\alpha_M,~~\alpha_-=\frac{1}{b}+\alpha_M.
\end{equation}
For fixed $\mathcal{O}_{\alpha_M}$ these two choices are related through $\alpha_+ = Q-\alpha_-$, which up to reflection coefficients creates the same operator. For a given $\Delta_{\alpha_M}$ there are also two possible choices of $\alpha_M$ (related by $\alpha_M\to -q -\alpha_M$) giving four choices of pairs $(\alpha_M,\alpha)$ all related through Liouville reflection relations. In terms of momenta the dressing condition can be nicely summarized as $P^2 =E^2$ which is the on-shell condition of a massless field moving in the target space with 2-momentum $(E,P)$. Up to this identification between $\alpha_M$ and $\alpha$, when computing correlators of $\mathcal{T}_{\alpha_M}$ the answer factorizes into a matter, Liouville and ghost contributions, before the integration over the moduli.
A simple operator that we will use later is the area operator which can be defined as $\hat{A} = \int_\Sigma V_b$. This can also be written after gauge fixing in the form of a tachyon vertex operator as above, which corresponds to picking the identity in the matter sector $\mathcal{T}_{\rm id} \sim c\bar{c}\hspace{0.1cm} V_b$. This operator measures the total area of the surface in terms of the physical metric.
Before we moving on, we will enumerate some special set of operators in both the matter and Liouville sectors that will be useful to distinguish later on:
\paragraph{Degenerate Liouville operators:} These operators, labeled by two positive integers $m\geq1$ and $n\geq1$, are defined through the parameter
\begin{equation}\label{eq:liouvdeg}
\alpha_{(m,n)}= - \frac{(n-1)b}{2} - \frac{(m-1)b^{-1}}{2},~~~{\rm and}~~\alpha_{(m,n)} \to Q-\alpha_{(m,n)}.
\end{equation}
\paragraph{Degenerate matter operators:} We can analogously define operators which are degenerate in the matter sector also labeled by positive integers $m\geq1$ and $n\geq1$
\begin{equation}\label{eq:mattdeg}
\alpha_{M(m,n)} =- \frac{(n-1) b}{2} + \frac{(m-1)b^{-1}}{2},~~~{\rm and}~~\alpha_{M(m,n)} \to - q-\alpha_{M(m,n)}.
\end{equation}
Its important to notice that these operators never appear together in a tachyon vertex operator. We can easily see from the expressions above that if the matter content corresponds to a degenerate operator, then the Liouville dressing will be generic. One the other hand, if the Liouville dressing is degenerate, the matter operator will be generic instead. We can easily see this in the semiclassical (also related to JT gravity) limit:
\paragraph{Semiclassical limit:} Following \cite{Saad:2019lba} we will be interested in the limit $b\to0$ for which $c_M \to -\infty$ and $c_L \to \infty$. In this limit we will parametrize light matter operators as $\alpha_M = b h$, where $h$ is a continuous parameter which is fixed in the $b\to0$ limit. They are dressed by Liouville operators with $\alpha = b(1-h)$. In this limit, $h$ corresponds to the dimension of the matter operator $\Delta_{\alpha_M} \to h$, while the Liouville field has $\Delta_\alpha \to 1-h$. Degenerate matter operators have $h_{Mn} =\frac{1-n}{2}=0,-\frac{1}{2},-1,-\frac{3}{2},\ldots$, while Liouville degenerate operators have $h_{Ln} = \frac{1+n}{2}=1,\frac{3}{2}, 2, \ldots$. These carry a single index since the other set from \eqref{eq:liouvdeg} or \eqref{eq:mattdeg} become infinitely heavy.
\subsection{Minimal string theory}
\label{s:mst}
In this section we review the definition of the minimal string theory. This corresponds to the same theory of 2D gravity as before, but the matter CFT now consists on the $M_{p,p'}$ minimal model, for any $p'>p\geq 2$ coprime. The Liouville-like parametrization of the physical quantities that characterize this theory will be very useful later. For example, the central charge can still be written as $c_M = 1 - 6q^2$, where $q=1/b-b$ and $b=\sqrt{p/p'}$, which also matches the parameter $b$ of the gravitational Liouville mode, canceling the conformal anomaly.
The matter CFT for the $(p,p')$ minimal string has a discrete and finite set of operators $\mathcal{O}_{n,m}$. These can still be parametrized through the Liouville-like parameter $\alpha_M$. The spectrum of the minimal model consists of the matter degenerate states with label $\alpha_{M(n,m)}$ and dimension $\Delta_{n,m}$ given by
\begin{equation}
\mathcal{O}_{n,m}:~~~\alpha_{M(n,m)} =- \frac{(n-1)b}{2} + \frac{(m-1)b^{-1}}{2},~~~\Delta_{n,m} = \frac{(nb^{-1} -m b)^2-(b^{-1}-b)^2}{4}.
\end{equation}
where $n=1,\ldots, p'-1$ and $m=1,\ldots p-1$. Due to the reflection property the operators $\mathcal{O}_{n,m} \equiv \mathcal{O}_{p'-n,p-m}$ are identified this gives a total of $(p'-1)(p-1)/2$ operators. For some purposes, it is useful to define a fundamental domain $(n,m)\in E_{p'p}$ defined by $1\leq n \leq p'-1$ and $1\leq m \leq p-1$ and $p' m < p n$. We can construct physical string theory vertex operators $\mathcal{T}_{n,m}$ for each primary $\mathcal{O}_{n,m}$ by adding the gravitational dressing and integrating over the worldsheet as in equation \eqref{tachyondef}.
Since we will need them later, we will quote results for the torus characters for these degenerate representations
\begin{equation}\label{degcharacters}
\chi_{n,m}(q) = \frac{1}{\eta(q)} \sum_{k\in\mathbb{Z}} (q^{a_{n,m}(k)}-q^{a_{n,-m}(k)}),~~~~a_{n,m}(k) =\frac{( 2 p' p k + p n - p' m)^2}{4 p'p},
\end{equation}
where $q=e^{2\pi i \tau}$ and $\tau$ is the torus moduli. We will also need the modular S-matrix describing their transformation under $\tau \to - 1/\tau$, which is given by
\begin{equation}
S_{n,m}^{n',m'} = 2 \sqrt{\frac{2}{p'p}}(-1)^{1+mn'+n m'} \sin\Big( \pi \frac{p}{p'} n' n \Big)\sin\Big( \pi \frac{p'}{p}m'm \Big).
\end{equation}
More results regarding these representations such as their fusion coefficients $\mathcal{N}_{n_1,m_1;n_2,m_2}^{n_3,m_3}$ can be found in \cite{DiFrancesco:1997nk}.
We will be mostly interested in the $(2,2\mathfrak{m}-1)$ minimal string which is known to be dual to a single-matrix model \cite{Moore:1991ir}. This theory has $\mathfrak{m}-1$ bulk tachyons labeled by a single integer
\begin{equation}
\mathcal{T}_{n} \equiv \mathcal{T}_{n,1}\sim \int_{\Sigma} \hspace{0.1cm} \mathcal{O}_{n,1} \hspace{0.1cm} e^{2(b-\alpha_{M(n,1)})\phi},
\end{equation}
where $n=1,\ldots, \mathfrak{m}-1$. The matter sector for these operators has parameter $\alpha_{M(n,1)} = \frac{1-n}{2}b$ and its Liouville dressing insertion has $\alpha_{n,1} = (1+n)b/2$. We have chosen these parameters in order to have a smooth semiclassical limit.
We will be interested in the $\mathfrak{m}\to \infty$ limit of the $(2,2\mathfrak{m}-1)$ minimal string, which is equivalent to JT gravity \cite{Saad:2019lba}. This limit, since $b=\sqrt{2/(2\mathfrak{m}-1)}$, corresponds to $c_M \to -\infty$ and $c_L \to \infty$. We will focus on `light' operators $\mathcal{T}_n$ with fixed $n$ in the $k\to\infty$ limit. These are the semiclassical operators defined in the previous section with parameter $h=n/2$. Another interesting limit is given by heavy operators with $n/\mathfrak{m}$ fixed in the large $\mathfrak{m}$ limit.
\subsection{2D gravity on the disk}
\label{s:bdy}
We will be mostly interested in observables on the disk. We quickly review here results for Liouville theory with boundaries, focusing mostly on the gravitational part. The simplest boundary condition for the Liouville mode corresponds to the FZZT brane \cite{Fateev:2000ik}. This is labeled by a single parameter $\mu_B$ referred to as the boundary cosmological constant. A path integral representation is given by the Liouville Lagrangian plus the following boundary term
\begin{equation}
S_L^{\rm bdy} [\phi] = \frac{1}{2\pi} \oint_{\partial \Sigma} \left[ Q \hat{K} \phi + 2\pi \mu_B e^{b \phi} \right].
\end{equation}
It is convenient to parametrize the boundary cosmological constant in terms of the FZZT parameter $s$ as
\begin{equation}
\mu_B = \kappa \cosh 2\pi b s,~~~~\kappa\equiv\frac{\sqrt{\mu}}{\sqrt{\sin \pi b^2}}.
\end{equation}
It will also be useful to keep the parameter $\kappa=\mu_B(s=0)$, with an implicit dependence on the bulk cosmological constant $\mu$ and $b$. In the case of timelike Liouville matter we can introduce analogous branes labeled by another continuous parameter we will call $\tilde{s}$.
This boundary condition can be understood from the point of view of the boundary conformal bootstrap \cite{Fateev:2000ik}. Each boundary condition is related to a Liouville primary field with parameter $\alpha =\frac{Q}{2} + i s(\mu_B)$, analogously to the rational case \cite{Cardy:1989ir}. A different set of boundary conditions is given by the ZZ brane, which are labeled by degenerate representations \cite{Zamolodchikov:2001ah}. The FZZT boundary conditions can be represented through Cardy boundary states \cite{Cardy:1989ir}
\begin{eqnarray}
|{\rm FZZT}(s) \rangle &=& \int_0^{\infty} dP \hspace{0.1cm}\Psi_s(P) |P\rangle\hspace{-0.1cm}\rangle,\\
\Psi_s(P) &=&(\pi \mu \gamma(b^2))^{-iP/b} \frac{\Gamma(1+2i Pb)\Gamma(1+2iP/b)}{2^{1/4}(-2 i \pi P)} \cos 4 \pi s P
\end{eqnarray}
where $|P\rangle\hspace{-0.1cm}\rangle$ denotes the Ishibashi state \cite{Ishibashi:1988kg} corresponding to the primary $P$ and the wavefunction $\Psi_s(P)$ was found in \cite{Fateev:2000ik}.
A similar set of branes can be defined for the matter sector when written as a time-like Liouville theory. In the case of the minimal string we can also write boundary conditions in terms of boundary states. Their form for the minimal model sector is
\begin{equation}
|n,m \rangle = \sum_{n',m'} \frac{S_{n,m}^{n',m'}}{(S_{1,1}^{n',m'})^{1/2}} |n',m' \rangle\hspace{-0.1cm}\rangle,
\end{equation}
written in terms of the modular S-matrix. They are also labeled by primary operators \cite{Cardy:1989ir}.
We will be interested in the case of bulk and boundary correlators on the disk (following for example \cite{Kostov:2003uh}). The Liouville parametrization of boundary changing operators is
\begin{eqnarray}
{\rm Liouville:}&&~~~\hspace{0.1cm}~B_\beta^{s_1s_2} = \exp{( \beta \phi)}~~~~~~~~\hspace{0.1cm}\Delta_\beta = \beta(Q-\beta),\\
\label{bdylioupar}\hspace{-0.3cm}{\rm Matter:}&&~~~~~\Phi_{\beta_M}^{\tilde{s}_1\tilde{s}_2} = \exp{(\beta_M \chi)}~~~~~\hspace{-0.06cm}\Delta_{\beta_M} = \beta_M(q+\beta_M),
\end{eqnarray}
where we indicated explicitly the boundary conditions $s_i$/$\tilde{s}_i$ between which these operators interpolate. With this normalization, degenerate operators for both theories can be written in terms of the same expression as bulk operators so $\beta_{(n,m)}$ and $\beta_{M(n,m)}$ are equivalent to \eqref{eq:liouvdeg} and \eqref{eq:mattdeg}. Since it will be important later, we quote here the parameter for matter degenerates
\begin{equation}
\beta_{{\rm M}(m,n)} =- \frac{(n-1) b}{2} + \frac{(m-1)b^{-1}}{2},
\end{equation}
with $(n,m)$ a pair of positive integers. Similar operators can be written for the minimal string $\Phi_{(n,m)}^{n_1,m_1;n_2,m_2}$ which now generate a finite discrete set of dimension $\Delta_{(n,m)}$ interpolating between $(n_1,m_1)$ and $(n_2,m_2)$ branes.
We construct physical open tachyon vertex operators by gravitational dressing
\begin{equation}
\mathcal{B}_{\beta_M} \sim \oint_{\partial\Sigma} \hspace{0.1cm}\Phi_{\beta_M} \hspace{0.1cm}B_\beta,
\end{equation}
where from now on we omit the boundary conditions labels on each side of the insertion. After gauge fixing this is $\mathcal{B}_{\beta_M} \sim c \hspace{0.1cm}\Phi_{\beta_M}\hspace{0.1cm}B_\beta$. The relation between $\beta_M$ and the dressing parameter $\beta$ is the same as for the bulk operators, and we will pick $\beta_M=b-\beta$. Physical correlators factorize into the ghost, matter and Liouville contribution up to a possible integral over moduli. For the minimal string we have a discrete set $\mathcal{B}_{n,m}$ and for the $(2,2\mathfrak{m}-1)$ case we have $\mathcal{B}_{n}\equiv \mathcal{B}_{n,1}$.
A special operator that we will make use of analogous to the area operator in the bulk is $\mathcal{B}_{\rm id} \sim c B_b^{s_1,s_2} = c e^{b\phi}$, which we will refer to as the boundary \emph{marking operator}. It is the gravitationally dressed version of the matter identity operator $\mathbf{1}_M$. Before gauge fixing, this operator can also be written as $\hat{\ell} = \oint B_b$ which measures the physical length of the boundary.
Finally, we will need the boundary correlators of Liouville CFT for an FZZT boundary \cite{Fateev:2000ik, Ponsot:2001ng}. This is simplified if we choose the fiducial metric space to be the upper half plane $(z,\bar{z})$ with ${\rm Im}(z)\geq0$ and a boundary labeled by $z=\bar{z} = x$. The bulk one point function is
\begin{equation}
\langle V_\alpha(z,\bar{z}) \rangle_s = \frac{U_s(\alpha)}{|z-\bar{z}|^{2\Delta_\alpha}},
\end{equation}
with
\begin{equation}
\label{Lonep}
U_s(\alpha) = \frac{2}{b}(\pi \mu \gamma(b^2))^{(Q-2\alpha)/2b} \Gamma(2b \alpha - b^2) \Gamma\Big(\frac{2\alpha}{b}-\frac{1}{b^2}-1\Big)\cosh 2\pi (2\alpha-Q)s,
\end{equation}
The boundary two point function is
\begin{equation}
\langle B_{\beta_1}^{s_1s_2}(x)B_{\beta_2}^{s_2s_1}(0)\rangle = \frac{\delta(\beta_2 + \beta_1-Q)+ d(\beta|s_1,s_2)\delta(\beta_2-\beta_1)}{|x|^{2\Delta_{\beta_1}}}.
\end{equation}
where we define the quantity\footnote{There is an implicit product over all four sign combinations of the $S_b$ in this and in subsequent similar equations.}
\begin{equation} \label{liouvillebdy2pt}
d(\beta|s_1,s_2) = (\pi \mu \gamma(b^2)b^{2-2b^2})^{\frac{Q-2\beta}{2b}} \frac{\Gamma_b(2\beta-Q)\Gamma_b^{-1}(Q-2\beta)}{S_b(\beta \pm i s_1 \pm i s_2)}.
\end{equation}
The bulk-boundary two point function is of the form
\begin{equation}
\langle V_\alpha(z,\bar{z})B_\beta^{ss}(x)\rangle_s = \frac{R_s(\alpha,\beta)}{|z-\bar{z}|^{2\Delta_\alpha-\Delta_\beta}|z-x|^{2\Delta_\beta}}
\end{equation}
with
\begin{eqnarray}\label{bbdy}
R_s(\alpha,\beta)&=&2\pi (\pi \mu \gamma(b^2)b^{2-2b^2})^{\frac{Q-2\alpha-\beta}{2b}} \frac{\Gamma_b^3(Q-\beta)}{\Gamma_b(Q)\Gamma_b(Q-2\beta)\Gamma_b(\beta)} \frac{\Gamma_b(2\alpha-\beta)\Gamma_b(2Q-2\alpha-\beta)}{\Gamma_b(2\alpha) \Gamma_b(Q-2\alpha)}\nonumber\\
&&\times \int_{\mathbb{R}} dt \hspace{0.1cm} e^{4\pi i t s} \frac{S_b(\frac{1}{2}(2\alpha+\beta-Q)+i t)S_b(\frac{1}{2}(2\alpha+\beta-Q)-it)}{S_b(\frac{1}{2}(2\alpha-\beta+Q)+it) S_b(\frac{1}{2}(2\alpha-\beta+Q)-it)}
\end{eqnarray}
We will look at the boundary two-point function with $\beta_1=\beta_2$. A naive application of the formula given above would predict a divergent factor of $\delta(\beta_2-\beta_1)\to\delta(0)$. This zero-mode divergence is canceled when one divides by the full group of diffeomorphisms (an analogous thing was observed recently in \cite{Erbin:2019uiz} for the case of the bosonic critical string). The correct answer is given by
\begin{equation}
\langle \mathcal{B}_{\beta_M} \mathcal{B}_{\beta_M} \rangle = 2(Q-2\beta) d(\beta| s_1,s_2) \times ({\rm matter}),
\end{equation}
as explained for example in \cite{Aharony:2003vk, *Alexandrov:2005gf}. This result can be obtained by taking a derivative of the two point function with respect to the cosmological constant, producing a three point function with all symmetries fixed, which can then be integrated obtaining the relation above.
The on-shell condition relating $\beta$ with $\beta_M$ produces a cancellation of the worldsheet coordinate dependence $x$, after including the ghost two-point function. The last factor in the equation above comes from the matter normalization.
\section{Disk partition function}\label{sec:diskZ}
In this section we will analyze the disk partition function for the minimal string and Liouville gravity for fixed length boundary conditions.
\subsection{Fixed length boundary conditions} \label{s:bosdisk}
We will start by defining the fixed length boundary condition in the disk. We will mostly focus on the Liouville sector and therefore the answer will be valid for both the time-like Liouville string and the minimal string.
The starting point is the disk with FZZT brane boundary conditions, specified by the boundary cosmological constant $\mu_B$. It will be useful to distinguish two different notions of partition function of the disk. The first is the unmarked partition function $Z(\mu_B)^{\scriptscriptstyle \text{U}}$. We will refer to the second type as the mark partition function $Z(\mu_B)^{\scriptscriptstyle \text{M}}$ defined by
\begin{equation}
Z(\mu_B)^{\scriptscriptstyle \text{M}} \equiv \partial_{\mu_B}Z(\mu_B)^{\scriptscriptstyle \text{U}}= \left\langle c \hspace{0.1cm}e^{b\phi}\right\rangle_{\mu_B}.
\end{equation}
This is equivalent to the partition function on a marked disk, where a boundary base point has been chosen, and we do not consider translations of the boundary coordinate as a gauge symmetry \cite{Moore:1991ir}. We will refer to insertions of $e^{b\phi}$ as marking operators. This corresponds to inserting a factor of $\ell$ in the length basis. The fixed length partition function is then defined by the inverse Laplace transform
\begin{equation}\label{eq:deffixlength}
Z(\ell) \equiv -i \int_{-i\infty}^{i \infty} d\mu_B e^{\mu_B \ell} Z(\mu_B)^{\scriptscriptstyle \text{M}}.
\end{equation}
This is explained, for example, by Kostov in \cite{Kostov:2002uq}. One can check from the path integral definition of Liouville theory that this integral when combined with the boundary term produces a fixed length delta function, justifying this formula.
The first step is then to compute the FZZT partition function $Z(\mu_B)^{\scriptscriptstyle \text{U}}$. Following the calculation of Seiberg and Shih done in \cite{Seiberg:2003nm}, its useful to differentiate with respect to the bulk cosmological constant in order to fix all the symmetries in the problem
\begin{eqnarray}\label{eq:unmarkedZ}
\partial_\mu Z(\mu_B)^{\scriptscriptstyle \text{U}} &=& \langle c \bar{c}\hspace{0.1cm} e^{2b\phi} \rangle_{\mu_B} \\
&=& \frac{2}{b} (\pi \mu \gamma(b^2))^{\frac{1}{2b^2}-\frac{1}{2}} \Gamma(b^2) \Gamma(1-b^{-2}) \cosh 2 \Big( b-\frac{1}{b} \Big) \pi s
\end{eqnarray}
where in the second line we pick a normalization such that the result is precisely the bulk cosmological constant one-point function derived in \cite{Fateev:2000ik} (Seiberg and Shih make a different normalization choice). Integrating this with respect to the cosmological constant $\mu$ we obtain the unmarked disk partition function
\begin{equation}
\label{ses}
Z(\mu_B)^{\scriptscriptstyle \text{U}} = (\pi \mu \gamma(b^2))^{\frac{1-b^2}{2b^2}} \frac{4 \Gamma(b^2) \Gamma(1-b^{-2}) \mu b^2}{b(1+b^2)} \left(b^2 \cosh{2\pi b s} \cosh \frac{2\pi s}{b} - \sinh{2\pi b s} \sinh \frac{2\pi s}{b}\right),
\end{equation}
where the FZZT parameter should be understood as implicitly depending on $\mu_B$ and $\mu$. We compute now the marked partition function differentiating with respect to $\mu_B$ which simplifies the $\mu_B$ dependence considerably
\begin{equation}
\label{Zmarked}
Z(\mu_B)^{\scriptscriptstyle \text{M}} \sim \mu^{\frac{1}{2b^2}} \cosh \frac{2 \pi s}{b},
\end{equation}
where we omit the $s$ independent prefactor that we will put back later. The next step is to compute the integral defined in \eqref{eq:deffixlength}. This can be done by deforming the contour around the negative real axis, as shown in figure \ref{contourDeformfirst}.
\begin{figure}[t!]
\centering
\begin{tikzpicture}[scale=1.2]
\draw[->,thick, -latex] (0,-2) -- (0,2);
\draw[->,thick, -latex] (-4,0) -- (2,0);
\draw[red!80!black, thick,decoration = {zigzag,segment length = 2mm, amplitude = 0.5mm},decorate] (-4,0) -- (-1,0);
\filldraw[red!80!black] (-1,0) circle (0.07);
\draw[blue!70!black,thick] (-4,0.2) -- (-2,0.2);
\draw[blue!70!black,thick,latex-] (-2.2,0.2) -- (-1,0.2);
\draw[blue!70!black,thick] (-1,0.2) arc (90:-90:0.2);
\draw[blue!70!black,thick] (-1,-0.2) -- (-2,-0.2);
\draw[blue!70!black,thick, latex-] (-1.8,-0.2) -- (-4,-0.2);
\draw[green!60!black,thick, -latex] (0,-1.9) -- (0,0.5);
\draw[green!60!black,thick] (0,0.2) -- (0,1.7);
\draw[green!60!black,thick] (0,0.2) -- (0,1.7);
\node at (-3,0.5) {$\mathcal{C}$};
\node at (-1,-0.4) {\scriptsize $-\kappa$};
\node at (3,2) {\large $\mu_B$};
\draw[thick] (2.65,1.75+0.4) -- (2.65,1.75) -- (2.65+0.5,1.75);
\end{tikzpicture}
\caption{Contour deformation from the original one (in green) to a deformed one that wraps the negative real axis (blue line). The segment $(-\kappa,0)$ has no branch cut and the contour can be further deformed to the semi-infinite interval $(-\infty,-\kappa)$.}
\label{contourDeformfirst}
\end{figure}
This allows us to write the integral as
\begin{equation}
\label{toplug}
Z(\ell) =-i\int_{-\kappa}^{-\infty}d\mu_B e^{\mu_B \ell}~ \text{Disc}\left[Z(\mu_B)^{\scriptscriptstyle \text{M}}\right]
\end{equation}
in terms of the discontinuity $\text{Disc}\left[Z(\mu_B)^{\scriptscriptstyle \text{M}}\right]$ along the negative real axis of the marked partition function.
A first observation, as shown in figure \ref{contourDeformfirst}, is that the branch cut along the negative real axis starts at $\mu_B =-\kappa$, where $\kappa \equiv \sqrt{\mu/\sin \pi b^2} = \mu_B(s=0)$. The value of $ s \sim \cosh^{-1} (\mu_B/\kappa)$ on the negative real axis for $\mu_B \in \left(-\kappa,\kappa\right)$ is purely imaginary and conjugate above and below the real axis. Since any \emph{even} function of $s$ has no discontinuity, $\text{Disc}\left[Z(|\mu_B|<\kappa)^{\scriptscriptstyle \text{M}}\right]=0$.
In what follows we will be mostly interested in the $\ell$ dependence of the final answer. On the interval $(-\infty,-\kappa)$, we can use the fact that $\text{arccosh}(\frac{\mu_B}{\kappa} + i\varepsilon) = \text{arccosh}\frac{\left|\mu_B\right|}{\kappa} \pm i\pi$. Then the discontinuity of an arbitrary function $F(s)$ on this interval is given by ${\rm Disc}[F(s)]=F(s+i/2b) -F(s-i/2b)$. Using this fact we can compute explicitly the discontinuity as
\begin{equation}
{\rm Disc} \left[\cosh \Big( \frac{1}{b^2} \text{arccosh} \frac{\mu_B}{\kappa} \Big) \right]= 2i \sin \frac{\pi}{b^2} \sinh \Big( \frac{1}{b^2}\text{arccosh}\frac{ \left|\mu_B\right|}{\kappa} \Big).
\end{equation}
We can use this to compute $\text{Disc}\left[Z(\mu_B)^{\scriptscriptstyle \text{M}} \right]$ and inserting the answer into \eqref{toplug} we find the fixed-length marked disk amplitude
\begin{equation}
\label{fixedldisk}
Z(\ell) = N \mu^{\frac{1}{2b^2}} \int_{\kappa}^{\infty} d\mu_B e^{-\ell \mu_B} \sinh \Big( \frac{1}{b^2} \text{arccosh} \frac{\mu_B}{\kappa} \Big).
\end{equation}
This answer is consistent with the result of \cite{Fateev:2000ik}. Keeping track of the prefactor appearing in \eqref{eq:unmarkedZ}, the normalization is given by $N = (\pi \gamma(b^2))^{\frac{1}{2b^2}} \frac{8 \pi (1-b^2)}{b \Gamma(b^{-2})}$. Written in terms of the FZZT $s$ variable the partition function is
\begin{eqnarray}\label{partfuncs}
Z(\ell) \sim \mu^{\frac{1}{2b^2}+\frac{1}{2}} \int_{0}^{\infty} ds ~e^{-\ell \kappa \cosh(2\pi b s)}\rho(s),~~~~\rho(s)\equiv \sinh 2\pi b s \sinh\frac{2\pi s}{b}.
\end{eqnarray}
In the language of \cite{Saad:2019lba} where the boundary is identified as Euclidean time of a dual theory, we see $\ell$ can be interpreted as an inverse temperature $\beta \to \ell $, while $\mu_B$ is identified with the eigenvalue of the boundary Hamiltonian $E \to \mu_B = \kappa \cosh 2\pi b s$ \footnote{Interestingly, the density of states is equal to the Plancherel measure on the principal series irreps of the quantum group $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ \cite{Ponsot:1999uf} as a function of representations labeled by $s$. It is also equal to the vacuum modular S-matrix $S_0{}^s$. We expand on this in section \ref{s:qg}.}. In terms of the energy $E$, we write:
\begin{eqnarray}\label{partfuncs2}
Z(\beta) \sim \mu^{\frac{1}{2b^2}} \int_{\kappa}^{\infty} dE ~e^{-\beta E}\rho_0(E),~~~~\rho_0(E) = \sinh\Big(\frac{1}{b^2} \text{arccosh}\frac{E}{\kappa}\Big).
\end{eqnarray}
We will review some interesting properties of this expression in section \ref{sec:proprho}. The integral can be done explicitly using the identity
\begin{equation}
\int_{0}^{+\infty}ds e^{-\ell \kappa \cosh 2 \pi b s} \sinh 2 \pi b s \sinh \frac{2\pi s}{b} = \frac{1}{2\pi b^3} \frac{1}{\kappa \ell} K_{\frac{1}{b^2}}(\kappa \ell),
\end{equation}
where the right hand side involves a modified Bessel function of the second kind.
More generally, if we consider the $M$-marked fixed length partition function, then we would write:
\begin{equation}
\label{genmark}
Z(\ell) \sim \frac{1}{\ell^{2-M}} K_{i\lambda}(\kappa \ell), \qquad i\lambda= 1/b^2.
\end{equation}
This formula holds since taking $\mu_B$-derivatives to bring down $\oint e^{b\phi}$ corresponds in the fixed length basis to just including factors of $\ell$. In our case we set $M=1$. The unmarked Seiberg-Shih partition function \eqref{ses}, when transformed to the fixed length basis, corresponds to setting $M=0$ in \eqref{genmark}.
\subsection{Marking operators}\label{s:bosmark}
In this section, we demonstrate that inserting more marking operators $c e^{b\phi}$ between generic FZZT brane segments does not affect the final answer for the fixed length partition function. More precisely, the boundary $n$-point function of $n$ marking operators, in the fixed length basis, is precisely given by the fixed-length disk partition function itself \eqref{fixedldisk}, see figure \ref{markingthree}. Notice that this is different than marking by differentiating with respect to $\mu_B$ as in \eqref{genmark}. As explained before, these operators are physical by themselves and correspond to the dressed identity operator in the matter sector $\mathbf{1}_M$. The resulting equality we mention here is then indeed expected.
We illustrate this fact first with the simplest case of two operator insertions, after gauge fixing $\langle [c e^{b\phi}] [c e^{b\phi}] \rangle$. The Liouville CFT boundary two-point function is given in \eqref{liouvillebdy2pt} specialized to $\beta=b$, and its contribution to the full 2D quantum gravity two-point function is given by $2(Q-2b)d(b|s_1,s_2)$. We can simplify this expression considerably using
\begin{equation}
\frac{1}{S_b(b \pm i s_1 \pm i s_2)} =\frac{\sinh \frac{\pi}{b} (s_1-s_2) \sinh \frac{\pi}{b} (s_1+s_2)}{\sinh \pi b (s_1-s_2) \sinh \pi b(s_1+s_2)}= \kappa \frac{\cosh \frac{2\pi}{b} s_1 - \cosh \frac{2\pi}{b} s_2}{\mu_{B}(s_1)-\mu_{B}(s_2)},
\end{equation}
giving
\begin{equation}\label{eq:lft2ptmbo}
d(b|s_1,s_2) = \left[(\pi \gamma(b^2))^{\frac{1}{2b^2}-\frac{1}{2}} \Gamma(b^2)\Gamma(1-b^{-2}) \frac{\sqrt{\sin(\pi b^2)}}{\pi} \right] \mu^{\frac{1}{2b^2}} \frac{\cosh \frac{2\pi}{b} s_1 - \cosh \frac{2\pi}{b} s_2}{\mu_{B1}-\mu_{B2}}.
\end{equation}
\begin{figure}[t!]
\centering
\begin{tikzpicture}[scale=0.7]
\draw[fill=blue!50!white,opacity=0.7] (0,0) ellipse (2 and 1);
\draw[fill] (-2,0) circle (0.08);
\node at (-2.6,0) {$e^{b\phi}$};
\node at (-1.4,-1.1) {$\mu_{B3}$};
\draw[fill] (1,0.86) circle (0.08);
\node at (1.4,1.3) {$e^{b\phi}$};
\draw[fill] (1,-0.86) circle (0.08);
\node at (1.3,-1.2) {$e^{b\phi}$};
\node at (-1,1.2) {$\mu_{B1}$};
\node at (2.6,0) {$\mu_{B2}$};
\draw[-latex] (3.5,0) -- (4.5,0);
\draw[fill=blue!50!white,opacity=0.7] (7.5,0) ellipse (2 and 1);
\node at (10.5,1) {\small $\ell_1 + \ell_2 + \ell_3$};
\end{tikzpicture}
\caption{FZZT brane segments between $n$ marking operators leads upon transforming to the fixed length basis with length $\ell \equiv \sum_j \ell_j$. In the figure we show an example with $n=3$.}
\label{markingthree}
\end{figure}
The definition of the fixed length amplitude for two marking operator insertions between two intervals of length $\ell_1$ and $\ell_2$ is given by
\begin{equation}
\label{twolength}
\mathcal{A}_b(\ell_1,\ell_2) = i^{-2} \prod_{i=1,2}\int_{-i\infty}^{+i \infty}d\mu_{Bi} e^{\mu_{Bi} \ell_i} 2(Q-2b)d(b|s_1,s_2)
\end{equation}
Repeating the procedure outlined in the previous section and taking the double discontinuity, we find
\begin{align}
{\rm Disc}\left[\frac{\cosh \frac{2\pi}{b} s_1 - \cosh \frac{2\pi}{b} s_2}{\mu_{B1}-\mu_{B2}} \right] = - 2i \sin \frac{\pi}{b^2} \sinh \Big( \frac{1}{b^2}\text{arccosh} \frac{|\mu_B|}{\kappa} \Big)2\pi i \delta(\mu_{B1}-\mu_{B2}),
\end{align}
which is non vanishing only for $\mu_{B1} = \mu_{B2} < - \kappa$. Plugging this into the expression \eqref{twolength} after deforming the contour and using the delta function to do one of the integrals, we get the fixed-length amplitude with two marking operator insertions:
\begin{eqnarray}
\label{markedsame}
\mathcal{A}_b(\ell_1,\ell_2) &=& N \mu^{\frac{1}{2b^2}} \int_{\kappa}^{\infty} d\mu_B e^{-(\ell_1+\ell_2) \mu_B} \sinh \Big( \frac{1}{b^2} \text{arccosh} \frac{\mu_B}{\kappa} \Big) \nonumber\\
&=&Z(\ell_1 + \ell_2),
\end{eqnarray}
where we also checked that the final $b$ dependent prefactor in the equation above, derived from \eqref{eq:lft2ptmbo}, coincides with the one in the partition function derived from \eqref{Zmarked}.
This result can be generalized to an arbitrary number of marking operators. Hosomichi wrote down a generalization to an arbitrary $n$-point correlator of such $\beta=b$ insertions \cite{Hosomichi:2008th} interpolating between FZZT boundaries of parameter $\mu_{Bi}=\mu_i$,
\begin{eqnarray}
\left\langle {}^{\mu_1}[e^{b\phi_1}]{}^{\mu_2} \hdots {}^{\mu_n}[e^{b\phi_n}]{}^{\mu_1}\right\rangle = \frac{(-)^{\frac{n(n-1)}{2}}}{\Delta(\mu_i)}\text{det}\left(\begin{array}{ccccc}
1 & \mu_1 & \hdots & \mu_1^{n-2} & Z^{\scriptscriptstyle \text{M}}(s_1) \\
\vdots & \vdots & \vdots &\vdots & \vdots \\
1 & \mu_n & \hdots & \mu_n^{n-2} & Z^{\scriptscriptstyle \text{M}}(s_n)
\end{array}\right),
\end{eqnarray}
where we indicated by the indices the parameters that each operator interpolates between. The transformation to fixed length generalize immediately and yields the same outcome \eqref{markedsame}, which means that all of them are equal to the (singly-marked) partition function. The main result of this section is the check that
\begin{equation}\label{eq:markedtrivial}
\mathcal{A}_b(\ell_1,\ldots, \ell_n) = Z(\ell_1 + \ldots + \ell_n).
\end{equation}
This result has a simple explanation from the perspective of the matrix integral when applied to the minimal string that we mention in section \ref{sec:MMpf}.
\subsection{Properties of the density of states}\label{sec:proprho}
In this section we will present some properties regarding the density of states. We will first work out the JT gravity limit of these expressions, as defined by Saad Shenker and Stanford \cite{Saad:2019lba}. To begin, we will rescale the energy and boundary length in the following way
\begin{equation}\label{JTparam}
E = \kappa (1+ 2\pi^2b^4 E_{\rm JT}),~~~~\ell =\frac{\ell_{\rm JT}}{2 \pi^2 \kappa b^4}.
\end{equation}
In terms of these variables the partition function can be written as
\begin{equation}
\label{defrho0}
Z(\beta) \sim e^{- \ell_{JT} E_0} \int_0^\infty dE_{\rm JT}\hspace{0.1cm} e^{-\ell_{\rm JT} E_{\rm JT}} \sinh\Big(\frac{1}{b^2} \text{arccosh}\big(1 + 2 \pi^2 b^4 E_{\rm JT} \big)\Big),
\end{equation}
where the edge of the energy spectrum normalized to be conjugate to the rescaled length $\ell_{\rm JT}$ is given by $E_0 = 1/2\pi^2 b^4$. So far this is an exact rewriting. Now we can take the JT limit defined by $b\to0$ with $\ell_{\rm JT}$ fixed, which implies the integral is dominated by $E_{\rm JT}$ fixed in the limit. The density of states is approximately
\begin{equation}\label{jtdosap}
\rho_0 (E) \approx \sinh 2 \pi \sqrt{E_{\rm JT}},
\end{equation}
which precisely coincides with the JT gravity answer, as first pointed out in \cite{Saad:2019lba}. We will take this as a definition of JT gravity limit in the case of more general observables, where all boundary length go to infinity as $b$ goes to zero, following equation \eqref{JTparam}.
We can easily reproduce this result from the partition function written in terms of the parameter $s$ as in equation \eqref{partfuncs}. In this case the density of states is $\rho(s) = \sinh 2 \pi b s \sinh \frac{2\pi s}{b}$ and the energy $ E(s) = \kappa \cosh(2 \pi b s)$. When we pick the boundary length such that $\ell_{\rm JT}$ is fixed, the integral is dominated by $s = b k$, where we keep $k$ fixed as $b\to0$. In this limit we get $\rho(s) \sim k \sinh(2 \pi k)$ and $\ell (E(s)-\kappa)\sim \ell_{\rm JT} k^2$, reproducing the previous result after the $E_{\rm JT} = k^2$ identification. This representation will be more useful when applied to more general observables.
This derivation was done for a general Liouville gravity in the small $b$ limit. When applied beyond the minimal string theory its interpretation is not clear since the theory is not dual to a single matrix integral anymore. The minimal string corresponds to $b^2=2/(2\mathfrak{m}-1)$. In this case the density of states is a polynomial in $\sqrt{E}$ of order $2\mathfrak{m}-1$, since it can be rewritten as
\begin{equation}
\rho_\mathfrak{m}(E) = \frac{1}{\sqrt{2E}}(T_\mathfrak{m}(1+E)-T_{\mathfrak{m}-1}(1+E)),
\end{equation}
where $T_p(\cos \theta) = \cos( p\theta)$ is the Chebyshev polynomial of the first kind. In the JT gravity limit $\mathfrak{m}$ is large and the series becomes approximately infinite reproducing \eqref{jtdosap}.
Having presented the JT limit we will now give a more global picture of the density of states for general $b$. The energy density of states is sketched in Figure \ref{rho}.
\begin{figure}[h]
\centering
\includegraphics[width=0.45\textwidth]{rho.pdf}
\caption{(Blue) Energy density of states $\rho_0(E_{JT})$ defined in \eqref{defrho0} with $b=1/2$. (Red) JT limit which focusses on the middle region. (Green) spectral edge limit.}
\label{rho}
\end{figure}
This quantity has three regimes, the small $E$ regime close to the spectral edge where $\rho_0 \sim 2\pi \sqrt{E_{JT}}$, the intermediate JT range where effectively $E_{JT} \ll 1/b^2$, and the UV regime where a different power-law behavior is present $\rho_0(E) \sim E^{1/b^2}$ (this is evident for the minimal string but still true for arbitrary $b$).
An interesting feature is that the UV rise of the spectral density in this theory is slower than that of JT gravity, which has Cardy scaling $\sim e^{2\pi \sqrt{E}}$ at high energies. Since by the UV/IR connection in holography, the high energy states probe the asymptotic region, we propose that the bulk asymptotic region becomes strongly coupled and the geometry deviates from AdS. We will discuss further how this happens in the conclusion.
The saddle of the above Laplace integral \eqref{partfuncs2} gives the energy-temperature relation:
\begin{equation}
\label{firstlaw}
\sqrt{E^2-\kappa^2} = \frac{1}{b^2 \beta},
\end{equation}
where $\beta = \ell_{\rm JT}$. As above, this law changes qualitatively from $\sqrt{E_{JT}} \sim \beta^{-1}$, the AdS$_2$ JT black hole first law, into $E_{JT} \sim \beta^{-1}$ at high energies. This suggests the possibility that the UV region close to the boundary of the space is strongly coupled, even in the JT gravity limit. It is important to explain this entire thermodynamic relation as a black hole first law of the bulk gravity system. We comment on how this works in the conclusion.
\section{Disk correlators}\label{sec:diskcorr}
In this section, we extend the discussion to a larger class of correlators. We discuss the fixed length amplitudes of the bulk one-point function in \ref{sec:bulk1pt}, the boundary two-point function in \ref{s:bostwo}, the boundary three-point function in \ref{s:three} and the bulk-boundary two-point function \ref{s:bbtwo}. \\
Since the fixed length amplitudes are found by Fourier transforming the FZZT branes, one can also wonder whether the degenerate ZZ-branes have any relation to the fixed length branes directly. This question is only tangentially related to our main story, and we defer some of the details to appendix \ref{s:degbrane}.
\subsection{Bulk one-point function} \label{sec:bulk1pt}
In this section we will compute the fixed length partition function in the presence of a bulk tachyon insertion $\mathcal{T}_{\alpha_M}$ with dimension $\Delta_{\alpha_M}$. In general now we will get a contribution from the matter sector given by the matter one-point function.
First we will compute the bulk Liouville one-point function for an FZZT boundary.
We will normalize the tachyon vertex, after gauge fixing, in the following way
\begin{equation}
\label{defbulk}
\mathcal{T}_{\alpha_M} = N_{\alpha_M} c\bar{c}\hspace{0.1cm}\mathcal{O}_{\alpha_M=-\frac{q}{2}+iP}\hspace{0.1cm}V_{\alpha=\frac{Q}{2}+iP},
\end{equation}
where
\begin{equation}\label{onepointins}
N_{\alpha_M} = \frac{(\pi \mu \gamma(b^2))^{\frac{i P}{b}}}{4\pi^2 b}\frac{\Gamma(-2iP/b)}{\Gamma(1+2iP b)} \frac{1}{\text{(matter)}}.
\end{equation}
We divided out by the factor from the matter one-point function. In the case of the minimal string calculation of $\left\langle \mathcal{T}_{(n,m)}\right\rangle_\ell$ the matter contribution is given by the Cardy wavefunction $S_{\text{\tiny$(n',\hspace{-0.04cm}m')$}}{}^{\text{\tiny$(n,\hspace{-0.04cm}m)$}}/(S_{\text{\tiny$(1,\hspace{-0.04cm}1)$}}{}^{\text{\tiny$(n,\hspace{-0.04cm}m)$}})^{1/2}$ where the matter boundary state is a Cardy state associated to the primary $(n',m')$.
The fixed length amplitude with the bulk insertion is given by the same inverse Laplace transform as the partition function with respect to the boundary cosmological constant
\begin{eqnarray}
\left\langle \mathcal{T}_{\alpha_M}\right\rangle_\ell &=&-i \int_{-i\infty}^{+i \infty}d\mu_B e^{\mu_B \ell} \partial_{\mu_B}\left[\left\langle \mathcal{T}_{\alpha_M}\right\rangle_{\mu_B}\right].
\end{eqnarray}
Inserting the Liouville contribution \eqref{Lonep}, the marked partition function with the bulk insertion is proportional to
\begin{equation}
\label{fion}
\partial_{\mu_B} \left[\cos 4\pi P s\right] = - \frac{2P}{b\kappa} \frac{\sin 4\pi P s}{ \sinh 2\pi b s}.
\end{equation}
Notice that this amplitude is actually marked twice now; we will explicitly see it in the final formula below.
We can again deform the contour as we did for the partition function. The integrand is meromorphic (and actually analytic) in the complex $\mu_B$ plane except for a branch cut at negative values. The discontinuity is given by
\begin{equation}\label{eq:disccos}
\text{Disc }\partial_{\mu_B} \left[\cos 4\pi P s\right]= \frac{2P}{b\kappa} 2i \sinh \frac{2\pi P}{b} \frac{\cos 4\pi P s }{ \sinh 2\pi b s},
\end{equation}
valid for $\mu_B<-\kappa$. For $\mu_B \in (-\kappa,0)$, the function \eqref{fion} has no discontinuity as is readily checked, and seen immediately since \eqref{fion} is even in $s$. Finally the bulk one-point function at fixed length is given by
\begin{equation}
\label{bulkone}
\left\langle \mathcal{T}_{\alpha_M}\right\rangle_\ell = \frac{2}{b} \int_{0}^{\infty} ds\hspace{0.1cm} e^{-\ell \kappa \cosh(2\pi b s)} \cos 4 \pi P s.
\end{equation}
This integral can be done explicitly:\footnote{Using the identity
\begin{equation}
\label{idthree}
\int_{0}^{+\infty}ds e^{-\ell \kappa\cosh 2 \pi b s} \cos 2\pi b \lambda s = \frac{1}{2\pi b} K_{i\lambda}(\kappa \ell).
\end{equation}
}
\begin{equation}
\left\langle \mathcal{T}_{\alpha_M}\right\rangle_\ell = \frac{1}{\pi b^2} K_{\frac{2i P}{b}}(\kappa \ell).
\end{equation}
Notice that no prefactors of $1/\ell$ appear, comparing to \eqref{genmark}, making this amplitude interpretable as a twice-marked amplitude.
Intuitively, one marking is just as the partition function, the second marking happens because of the non-trivial bulk insertion that creates a branch cut in the chiral sector of the geometry that has to intersect the boundary somewhere, marking it a second time. We develop this intuition in appendix \ref{app:mark}.
It was mentioned below equation \eqref{partfuncs} that the integrand of the disk partition function in terms of $s$ is the vacuum modular S-matrix. Here, in the presence of a bulk state of momentum $P$, we find a similar structure with the non-vacuum modular S-matrix $S_P{}^{s}$ appearing.
One can parametrize microscopic bulk operators by setting $P = i \frac{\theta}{2b}$, in terms of a new parameter $\theta$. For the particular case of $\theta \in \mathbb{N}$, the Liouville one-point amplitude $U_s(\alpha)$ is divergent.
We argue in Appendix \ref{app:unif} that one should not additionally mark the boundary in this case. We do this by arguing that this case is embedded in the degenerate Virasoro Liouville insertions. We complement this argument by a bulk Liouville geometry discussion. The analogous expressions are written in \eqref{degvir} and \eqref{bulkexc}.
\subsection{Boundary two-point function}
\label{s:bostwo}
In this section we will compute the boundary two-point function between generic operators, for a fixed length boundary. We will consider a general matter operator labeled by the parameter $\beta_M$ and include its gravitational dressing Liouville operator with parameter $\beta$
\begin{equation}
\label{tocu}
\mathcal{A}_{\beta_M} ( \ell_1, \ell_2) = \left\langle \mathcal{B}^{+}_{\beta_M} \hspace{0.1cm}\mathcal{B}^{-}_{\beta_M}\right\rangle_{\ell_1,\ell_2},
\end{equation}
where we defined the boundary tachyon operators
\begin{eqnarray}
\mathcal{B}^{+}_{\beta_M} &=&(\pi \mu \gamma(b^2))^{\frac{2\beta-Q}{4b}} \frac{\Gamma(b(Q-2\beta))}{\pi} c \hspace{0.1cm}e^{\beta\phi} \hspace{0.1cm}e^{\beta_M\chi}, \\
\mathcal{B}^{-}_{\beta_M} &=&(\pi \mu \gamma(b^2))^{\frac{2\beta-Q}{4b}} \frac{\Gamma(b^{-1}(Q-2\beta))}{\pi} c \hspace{0.1cm}e^{\beta\phi} \hspace{0.1cm}e^{(-q-\beta_M)\chi},
\end{eqnarray}
where we included the leg-pole factor in the definition of the insertion. Since we will eventually consider light matter operators we will pick the Liouville dressing with $\beta=b-\beta_M$. We will omit the labels $+/-$ on the operators when its clear by context.
It is easy to account for the matter contribution since its independent of the boundary and bulk cosmological constant. In fact we can choose the matter operator to be normalized such that the boundary two point function has unit prefactor
\begin{equation}
\left\langle e^{\beta_M \chi} e^{ (-q - \beta_M) \chi}\right\rangle_M = \frac{1}{x^{\Delta_{\beta_M}}}.
\end{equation}
This correlator corresponds to the vacuum brane changing to the state $\beta_M$ brane and then back according to the fusion $\mathbf{1} \times \beta_M \to \beta_M$ and $\beta_M \times (-q-\beta_M) \to \mathbf{1}$ (see figure \ref{mattertwo}).
\begin{figure}[t!]
\centering
\begin{tikzpicture}[scale=0.9]
\draw[fill=blue!50!white,opacity=0.7] (0,0) ellipse (2 and 1);
\draw[fill] (-2,0) circle (0.08);
\node at (-2.7,0) {$e^{\beta_M \chi}$};
\draw[fill] (1,0.86) circle (0.08);
\node at (1.5,1.3) {$e^{(-q-\beta_M)\chi}$};
\node[red!70!black] at (1.3,-1.2) {$1$};
\node[red!70!black] at (-1,1.2) {$\beta_M$};
\draw[-latex] (3.5,0) -- (4.5,0);
\draw[thick, black] (6,-0.5)--(9,-0.5);
\draw[thick, black] (7.5,-0.5)--(7.5,-0.5+1.5);
\node[red!70!black] at (6.3,-0.18) {\small $1$};
\node[red!70!black] at (8.7,-0.18) {\small $\beta_M$};
\node[red!70!black] at (7.5+0.5,0.8) {\small $\beta_M$};
\node at (7.5,1.4) {$e^{\beta_M \chi}$};
\end{tikzpicture}
\caption{Matter Coulomb gas two-point function with a vacuum brane $\mathbf{1}$ injected with charge $\beta_M$ to form the state $\beta_M$-brane and then back.}
\label{mattertwo}
\end{figure}
Likewise for the ghost sector. This leaves again only the Liouville sector as the source of non-trivial dependence on the boundary lengths.
For these reasons we will focus again only on the Liouville sector. Starting with the boundary two-point function
\begin{equation}
\label{lou2}
d(\beta|s_1,s_2) = (\pi \mu \gamma(b^2)b^{2-2b^2})^{\frac{Q-2\beta}{2b}} \frac{\Gamma_b(2\beta-Q)\Gamma_b^{-1}(Q-2\beta)}{S_b(\beta \pm i s_1 \pm i s_2)},
\end{equation}
and denoting
\begin{equation}
D_{s_1,s_2} \equiv \frac{1}{S_b(\beta \pm i s_1 \pm i s_2)} = S_b(Q - \beta \pm i s_1 \pm i s_2),
\end{equation}
we can compute the fixed length amplitude with boundary segments $\ell_1$ and $\ell_2$ by computing the Fourier transform:
\begin{eqnarray}
\mathcal{A}_{\beta_M} ( \ell_1, \ell_2) &=&(\pi \mu \gamma(b^2))^{\frac{2\beta-Q}{2b}}2(Q-2\beta)\frac{\Gamma(b(Q-2\beta))}{\pi}\frac{\Gamma(b^{-1}(Q-2\beta))}{\pi} \nonumber\\
&&\times i^{-2}\prod_{i=1,2}\int_{-i\infty}^{+i \infty}d\mu_{Bi} e^{\mu_{Bi} \ell_i} d(\beta|s_1,s_2),
\end{eqnarray}
where we included all the prefactors coming from the Liouville mode. We can again deform the contour to wrap the negative real axis. The main quantity to compute (up to prefactors) is the following discontinuity of the product of double sine functions
\begin{equation}
\prod_{i=1,2}\int_{0}^{+\infty}d\mu_B e^{-\mu_{Bi} \ell_i} \text{Disc } D_{s_1,s_2}.
\end{equation}
The discontinuity of the object $D_{s_1,s_2}$ can be found by subtracting the terms with $s_i \pm \frac{i}{2b}$,namely
\begin{equation}
\text{Disc } D_{s_1,s_2} \equiv D_{s_1 + \frac{i}{2b}, s_2 + \frac{i}{2b}} - D_{s_1 +\frac{i}{2b}, s_2 -\frac{i}{2b}} - D_{s_1 - \frac{i}{2b}, s_2 + \frac{i}{2b}} + D_{s_1 - \frac{i}{2b}, s_2 - \frac{i}{2b}}.
\end{equation}
Using the shift formulas that this double sine function satisfies
\begin{align}
\label{Sshift}
S_b(b+x) = 2\sin \pi b x S_b(x), \qquad S_b\Big(\frac{1}{b}+x\Big) = 2 \sin \frac{\pi x}{b} S_b\left(x \right),
\end{align}
the discontinuity can be tremendously simplified into\footnote{This kind of relation is actually much more general. For example replace $b\to1/b$ from equations (3.18), (3.20) and (3.24) of \cite{Kostov:2003uh}.}
\begin{equation}\label{discmain}
\text{Disc} \Big[D_{s_1,s_2}\Big] =\Big[16 \sin \frac{2 \pi \beta}{b} \sin \frac{\pi}{b}(b^{-1}-2\beta)\Big] \sinh \frac{2\pi s_1}{b} \sinh \frac{2\pi s_2}{b}~S_b\left(b -\beta\pm i s_1 \pm i s_2\right),
\end{equation}
where the factors in brackets depend only on $\beta$ and $b$ and the rest include all the $\mu_B$ dependent terms that will affect the length dependence of the final answer. Note the first term in the argument of the double sine functions was shifted from $Q-\beta \to b-\beta=\beta_M$ which is precisely the Liouville parameter associated to the matter operator. This will be important when taking the JT gravity limit.
It is straightforward to check that in the range $\mu_{Bi} \in \left(-\kappa,0\right)$, one has instead a pure imaginary value of $s_i $ and its conjugate below the real axis. Since $D_{s_1,s_2}=D_{-s_1,s_2}$, $D_{s_1,s_2}=D_{s_1,-s_2}$, etc, there is again no discontinuity along this interval. Even though there are no more branch cuts, in this case there are now poles coming from the double sine functions. We can define the original $\mu_B$ contour in a way that does not pick them and the matrix model calculation of section \ref{sec:MM} supports this definition. Alternatively we will also show in Appendix \ref{app:poles} that they are negligible in the JT gravity limit.
The final answer for the two-point amplitude is
\begin{equation}\label{eq:2pt}
\boxed{\mathcal{A}_{\beta_M}(\ell_1,\ell_2)= N_{\beta_M}\kappa^2 \int ds_1 ds_2 \rho(s_1) \rho(s_2)\hspace{0.05cm} e^{-\mu_B(s_1)\ell_1} e^{-\mu_B(s_2)\ell_2}\hspace{0.05cm}\frac{S_b\left(\beta_M \pm i s_1 \pm i s_2\right)}{S_b(2\beta_M)}}.
\end{equation}
The prefactor in the right hand side can be obtained keeping track of it at each step of the calculation. Surprisingly all terms conspire to simplify drastically into the $\beta$ independent prefactor $N_{\beta_M}=16 \pi b^2$. In the case of the minimal string this factor should be multiplied by the matter contribution to the two point function.
When viewed as a holographic theory, the result \eqref{eq:2pt} can be interpreted (read from left to right) as a sum over two intermediate channels with their respective densities of states, their propagators over lengths $\ell_i$ weighted by energies $\mu_B(s_i)$, and a matrix element squared of the matter operator between energy eigenstates given by the product of double sine functions.
Finally we can analyze the UV behavior. We will pick $\ell_1<\ell_2$ and call $\tau\equiv \ell_1$ and $\beta=\ell_1+\ell_2$. The UV behavior without gravity is given by $G_0(\tau) \sim 1/\tau^{2h}$ for very small $\tau\to 0$. This arise from a combination of the fact that even though the density of states grows exponentially $\rho(E) \sim e^{\sqrt{E}}$ the matrix elements decay too, up to a power law $\rho(E)|\langle E | \mathcal{O} | E \rangle|^2 \sim E^{2h-1} $ at high energies. The situation when quantum gravity is turned on is surprisingly not too different. Now the density of states grows as a power law at large energies $\rho(E) \sim E^{p/2}$. We can use the asymptotics of the double sine function $S_b(x) = e^{i \delta(b)} e^{\mp i x (x-Q)}$ when ${\rm Im}(x)\to \pm \infty$, where $\delta(b)$ is a phase that is independent of $x$. We find that the amplitude goes as $S_b(\ldots)\sim E^{-p/2} E^{\frac{2}{b}\beta_M-1}$. The slower growth of the density of states is exactly compensated by a slower decay of matrix elements. This gives an asymptotics that is very similar to the case without gravity $G(\tau) \sim 1/ \tau^{2 h_{\rm eff}}$, with an effective gravitational dress scaling dimension $h_{\rm eff} \equiv \beta_M/b$. This is given, as a function of the bare scaling dimension $\Delta = \beta_M(q+\beta_M)$ as
\begin{equation}
h_{\rm eff} =\frac{\sqrt{2b^2(2\Delta-1)+b^4+1}-1+b^2}{2b^2},
\end{equation}
where we picked the root that has a smooth $b\to0$ limit. When gravity is weakly coupled $b\to0$ and $h_{\rm eff} (b\to0) \sim \Delta$. On the other hand when gravity is strong we get $h_{\rm eff}(b\sim 1) \sim \sqrt{\Delta}$ but the qualitative behavior in the UV is the same. In any case, including quantum gravity does not seem to smooth out the UV divergence.
\subsection{Boundary three-point function}
\label{s:three}
In this subsection we will compute the boundary three point function between three operators with matter parameters $\beta_{M1}$, $\beta_{M2}$ and $\beta_{M3}$, which we will denote as
\begin{equation}
\mathcal{A}_{123}(\ell_1,\ell_2,\ell_3) \equiv \langle \mathcal{B}_{\beta_{M1}} \mathcal{B}_{\beta_{M2}} \mathcal{B}_{\beta_{M3}} \rangle,
\end{equation}
and can be obtained as an inverse Laplace transform of FZZT boundary conditions as before.
The expressions required in this calculation are very involved so we will focus only on the length dependence to simplify the presentation. The first object we need is the Liouville three-point function between operators of parameter $\beta_1$, $\beta_2$ and $\beta_3$ which should be thought of as a function of the matter parameter $\beta_i = b - \beta_{Mi}$. The Ponsot-Teschner \cite{Ponsot:2001ng} boundary three-point function is
\begin{equation}\label{bdy3pt}
C_{\beta_3\beta_2\beta_1}^{s_3s_2s_1} = \frac{g_{Q-\beta_3}^{s_3s_1}}{g_{\beta_2}^{s_3s_2}g_{\beta_1}^{s_2s_1}} F_{s_2 \beta_3}\left[{}^{\beta_2}_{s_3} \hspace{0.1cm}{}^{\beta_1}_{s_1}\right],
\end{equation}
where following \cite{Ponsot:2001ng} we define
\begin{equation}
g_{\beta}^{s_2s_1}\equiv (\pi \mu \gamma(b^2)b^{2-2b^2})^{\beta/2b} \frac{\Gamma_b(Q) \Gamma_b(Q-2\beta) \Gamma_b(Q+ 2i s_1) \Gamma_b(Q-2is_2)}{\Gamma_b(Q-\beta\pm i s_1 \pm i s_2)}.
\end{equation}
The fusion matrix appearing in the right hand side of \eqref{bdy3pt} was also computed by Ponsot and Teschner previously in \cite{Ponsot:1999uf}. We can rewrite this boundary three point function in the following suggestive way
\begin{eqnarray}
C_{\beta_3\beta_2\beta_1}^{s_3s_2s_1} &=&\frac{S_b(2\beta_1)^{\frac{1}{2}}S_b(2\beta_2)^{\frac{1}{2}}S_b(2\beta_3)^{\frac{1}{2}}}{\sqrt{2\pi}} C_{\beta_1,\beta_2,\beta_3}{}^{\frac{1}{2}}\nonumber\\
&&\times \Big[S_b(\bar{\beta}_2\pm is_2\pm is_3)S_b(\bar{\beta}_1\pm is_1\pm is_2)S_b(\bar{\beta}_3\pm is_1\pm is_3) \Big]^{\frac{1}{2}} \sj{\bar{\beta}_1}{\bar{\beta}_2}{\bar{\beta}_3}{s_3}{s_1}{s_2},\nonumber
\end{eqnarray}
where we defined $\bar{\beta}=Q-\beta$ and used that $\Gamma_b(Q)^2= 2 \pi/\Upsilon'(0)$. The factor appearing in the first line is the DOZZ structure constant
\begin{equation}
C_{\beta_1,\beta_2,\beta_3} = \frac{(\pi \mu \gamma(b^2) b^{2-2b^2})^{(Q-\beta_{123})/b}\Upsilon'(0)\Upsilon(2\beta_1)\Upsilon(2\beta_2)\Upsilon(2\beta_3)}{\Upsilon(\beta_{1+2-3})\Upsilon(\beta_{3+2-1}) \Upsilon(\beta_{3+1-2})\Upsilon(\beta_{123}-Q)}.
\end{equation}
The final term is the b-deformed $6j$-symbol of SL$(2,\mathbb{R})$ computed by Teschner and Vartanov \cite{Teschner:2012em}.
Now we can compute the discontinuity of the boundary OPE along the negative $\mu_B$ axis. We can do this by applying three times equation (3.24) of \cite{Kostov:2003uh} and the result, up to a $s$ independent prefactor, is
\begin{equation}
{\rm Disc}[C_{\beta_3\beta_2\beta_1}^{s_3s_2s_1}] \sim \sinh \frac{2\pi s_1}{b} \sinh \frac{2\pi s_2}{b} \sinh \frac{2\pi s_3}{b} C_{\beta_3+\frac{1}{b}\beta_2+\frac{1}{b}\beta_1+\frac{1}{b}}^{s_3s_2s_1}.
\end{equation}
Putting everything together and using the relation $\beta=b-\beta_M$ we can write a final answer for the boundary three point function
\begin{eqnarray}\label{eq:3ptbdyfinal}
\mathcal{A}_{123}(\ell_1,\ell_2,\ell_3) &=& N_{\beta_1\beta_2\beta_3} \int \prod_{i=1}^3 \Big[ds_i\rho(s_i) e^{- \mu_B(s_i)\ell_i}\Big] \nonumber\\
&&\hspace{-4cm}\times \big[S_b(\beta_{M2}\pm is_2\pm is_3)S_b(\beta_{M1}\pm is_1\pm is_2)S_b(\beta_{M3}\pm is_1\pm is_3) \big]^{\frac{1}{2}} \sj{\beta_{M1}}{\beta_{M2}}{\beta_{M3}}{s_3}{s_1}{s_2} .
\end{eqnarray}
where the prefactor $N_{\beta_1\beta_2\beta_3}$ includes contributions from both the Liouville and the matter sectors. Interestingly it is proportional to the square root of the DOZZ structure constant. This prefactor is important since it quantifies the bulk coupling between the three particles created by the boundary operators, but to estimate its size it's important to include properly the matter contribution, which depends on the theory.
\subsection{Bulk boundary two-point function}
\label{s:bbtwo}
The bulk-boundary two-point function we will consider is of the form
\begin{equation}
\left\langle \mathcal{T}_{\alpha_M} \, \mathcal{B}_{\beta_M}^{+}\right\rangle_{\ell}.
\end{equation}
We will take the bulk operator with $\alpha = Q/2 + i P$, $\beta_1 = Q/2 + is$ as the FZZT boundary label and $\beta = b- \beta_M$ for the boundary operator. \\
The Liouville amplitude was listed in \eqref{bbdy}. We transform this to fixed length by evaluating the discontinuity across the branch cut on the negative real $\mu_B$-axis. To do this, the following functional discontinuity relation can be used:\footnote{The analogous relation for a shift in $b$ was written in eq (3.20) in \cite{Kostov:2003uh}, in turn extracted from the Teschner trick computation of \cite{Hosomichi:2001xc}. We corrected a typo in that equation in the middle Gamma-function in the denominator.}
\begin{align}
R_{s+\frac{i}{2b}} - R_{s-\frac{i}{2b}} &= \sinh \frac{2\pi}{b}s \,\, R_{s}(\alpha,\beta+1/b) \\
&\times 2\pi \left( \frac{\mu}{\pi \gamma(-b^2)}\right)^{1/2} \frac{\Gamma(1-\frac{2}{b}\beta)\Gamma(1-\frac{1}{b^2} - \frac{2}{b}\beta)}{\Gamma^2(1-\frac{1}{b}\beta)\Gamma(1 - \frac{1}{b}\beta - \frac{2}{b}\alpha + \frac{1}{b}Q) \Gamma(1- \frac{1}{b}\beta + \frac{2}{b}\alpha - \frac{1}{b}Q)}. \nonumber
\end{align}
The resulting bulk-boundary two-point function has the following complicated form:
\begin{align}
\label{oneone}
&\int_{0}^{+\infty} ds \rho(s) e^{-\mu_B(s) \ell} \, \Gamma(b(Q-2\beta)) \frac{1}{4\pi^2 b} \frac{\Gamma(-2iP/b)}{\Gamma(1+2iPb)}\\
&\times\frac{\Gamma(1-2b^{-1}\beta)\Gamma(1-b^{-2}-2b^{-1}\beta)}{\Gamma^2(1-b^{-1}\beta)\Gamma(1-b^{-1}\beta-2b^{-1}\alpha+b^{-1}Q)\Gamma(1-b^{-1}\beta+2b^{-1}\alpha-b^{-1}Q)} \nonumber \\
&\times \frac{\Gamma_b^3(\beta_M)}{\Gamma_b(Q) \Gamma_b(-Q+2\beta_M)\Gamma_b(Q-\beta_M)} \frac{\Gamma_b(2\alpha-Q + \beta_M)\Gamma_b(Q-2\alpha+\beta_M)}{\Gamma_b(2\alpha) \Gamma_b(Q-2\alpha)} \,\, I_{\beta_1\alpha}(\beta + 1/b). \nonumber
\end{align}
The first line contains the legpole factors of the boundary operator, and the normalization of the bulk operator \eqref{onepointins}. The second line contains the prefactors coming from deforming the contour. The final line is the Liouville bulk-boundary two-point function in terms of the modular $S$-matrix, defined by Teschner and Vartanov as \cite{Teschner:2012em}:
\begin{align}
S^{\scriptscriptstyle \text{PT}}_{\beta_1 \beta_2}&(\alpha_0) \equiv \frac{S_0{}^{\beta_2} e^{\frac{\pi i}{2} \Delta \alpha_0}}{S_b(\alpha_0)} I_{\beta_1\beta_2}(\alpha_0) \\
&= \frac{S_0{}^{\beta_2} e^{\frac{\pi i}{2} \Delta \alpha_0}}{S_b(\alpha_0)} \int_{\mathbb{R}}dt e^{2\pi t (2\beta_1-Q)}\frac{S_b(\frac{1}{2}(2\beta_2+\alpha_0-Q)+it)}{S_b(\frac{1}{2}(2\beta_2-\alpha_0+Q)+it)}\frac{S_b(\frac{1}{2}(2\beta_2+\alpha_0-Q)-it)}{S_b(\frac{1}{2}(2\beta_2-\alpha_0+Q)-it)}.
\end{align}
The integral $I_{\beta_1\alpha}(\beta + 1/b) $ can be evaluated as:
\begin{align}
I_{\beta_1\alpha}(\beta + 1/b) &= \int_{\mathbb{R}}dt e^{2\pi t (2\beta_1-Q)}\frac{S_b(\frac{1}{2}(2\alpha+(\beta+1/b)-Q)+it)}{S_b(\frac{1}{2}(2\alpha-(\beta+1/b)+Q)+it)}\frac{S_b(\frac{1}{2}(2\alpha+(\beta+1/b)-Q)-it)}{S_b(\frac{1}{2}(2\alpha-(\beta+1/b)+Q)-it)} \nonumber \\
&= \frac{1}{S_b(\beta_M)^2}\int_{\mathbb{R}}dt e^{4\pi t i P} S_b( \beta_M/2 \pm i s \pm it),
\end{align}
where we used the property $I_{\beta_1\beta_2}(\alpha_0) = S_b(\alpha_0)^2 I_{\beta_2\beta_1}(Q-\alpha_0)$ to swap the roles of $\alpha$ and $\beta_1$. This allows for a well-defined JT limit below.
Using the shift identities and the gamma-function reflection identity, the integrand of \eqref{oneone} can be simplified into:
\begin{align}
N_{\beta_M,P} \int_{\mathbb{R}}dt\hspace{0.1cm} e^{4\pi t i P} \frac{S_b( \beta_M/2 \pm i s \pm it)}{S_b(\beta_M)},
\end{align}
which contains (in order) the prefactor $2/b$ for the bulk operator, the boundary operator, and a coupling between these in the third prefactor, given by
\begin{equation}
\label{prebb}
N_{\beta_M, P}= \frac{2}{b} \frac{\Gamma_b(1/b +\beta_M)}{\Gamma_b(1/b + 2 \beta_M)}\frac{\Gamma_b(\frac{1}{b} + \beta_M \pm 2iP)}{\Gamma_b(\frac{1}{b} \pm 2iP)}.
\end{equation}
Upon using $t\to -t$ to write the $t$-integral over $\mathbb{R}^+$, we can write:
\begin{equation}
e^{4\pi t i P} \,\, \to \,\,\cos 4\pi P t = S_P{}^t = \frac{S_P{}^t}{S_0{}^t} S_0{}^t,
\end{equation}
in terms of the Virasoro modular $S$-matrix, where $S_0{}^t = \rho(t) = \sinh 2 \pi b s \sinh \frac{2\pi s}{b}$. We then obtain for the full result \eqref{oneone}:
\begin{equation}\label{eq:bulkbdyfinal}
\boxed{
\left\langle \mathcal{T}_{\alpha_M} \, \mathcal{B}_{\beta_M}^{+}\right\rangle = N_{\beta_M, P} \int_{0}^{+\infty} ds dt \rho(s) \rho(t) e^{-\mu_B(s) \ell} \, \frac{S_P{}^t}{S_0{}^t} \, \frac{S_b(\beta_M/2 \pm i s \pm it)}{S_b(\beta_M)}}.
\end{equation}
As a check on this formula, taking $\beta_M \to 0$, we can use the identity
\begin{equation}
\lim_{\beta_M \to 0}\frac{S_b(\beta_M/2 \pm i s \pm it)}{S_b(\beta_M)} = \frac{\delta(s-t)}{S_0{}^{t}},
\end{equation}
to obtain
\begin{align}
\frac{2}{b} \int_{0}^{+\infty} ds e^{-\mu_B(s) \ell} S_P{}^t,
\end{align}
which is indeed the bulk one-point function we derived in section \ref{sec:bulk1pt}.
\subsection{JT gravity limit}
In this section we take the semiclassical limit of the formulas derived above, for which the central charge of the Liouville mode becomes large. We will see in each case a match with the analogous calculation done previously in JT gravity.
\begin{center}
\textbf{Bulk one-point function}
\end{center}
We will begin with the bulk one-point function
\begin{equation}
\left\langle \mathcal{T}\right\rangle_\ell = \frac{2}{b} \int_{0}^{+\infty} ds e^{-\ell \kappa \cosh(2\pi b s)} \cos 4 \pi P s,
\end{equation}
We take the $b\to0$ limit and write it in terms of $\ell_{\rm JT}$ (see section \ref{sec:proprho} for its definition in terms of $\ell$). In order to have a non-trivial limit, we consider heavy matter operators such that the Liouville momenta scales as $P=\lambda/2b$, with finite $\lambda$. Then the one-point function becomes
\begin{equation}
\left\langle \mathcal{T}\right\rangle_\ell = 2 \int_{0}^{+\infty} dk e^{-\ell_{\rm JT} k^2 } \cos 2 \pi \lambda k.
\end{equation}
This expression coincides with the JT gravity partition function on a single trumpet of geodesic length $2\pi \lambda$. Therefore in this limit the bulk operator has the effect of creating a macroscopic hole of a given length.
These single defect partition functions in JT gravity are known to be related to functional integrals within the different Virasoro coadjoint orbits \cite{Mertens:2019tcm},\footnote{See also \cite{Nayak:2019evx}.} where the choice of defect selects a particular orbit. For $\lambda \in \mathbb{R}$, these can be identified with the hyperbolic orbits of the Virasoro group.
On the other hand, for imaginary $\lambda \equiv i\theta$ this partition function is equivalent to the JT gravity calculation with a conical defect inside the disk, with angular identification $\varphi \sim \varphi + 2 \pi \theta$. These are identified with functional integrals along the elliptic coadjoint orbits of the Virasoro group. For $\theta\in \mathbb{N}$, these become replicated geometries. Taking the JT limit of \eqref{bulkexc}, we get:
\begin{equation}
\left\langle \mathcal{T^{\scriptscriptstyle \text{U}}}\right\rangle_\ell = 4 \int_{0}^{+\infty} dk e^{-\ell_{\rm JT} k^2 } k \sinh 2 \pi n k,
\end{equation}
matching the JT exceptional elliptic defect amplitudes discussed in \cite{Mertens:2019tcm}.
Starting instead with \eqref{degvir}, and setting $n=\lambda/b^2$ with $\lambda$ a new continuous quantity, one gets the limit:
\begin{equation}
\left\langle \mathcal{T^{\text{deg}}}\right\rangle_\ell = 4 \int_{0}^{+\infty} dk e^{-\ell_{\rm JT} k^2 } \sinh 2 \pi \lambda k \sinh 2 \pi n k,
\end{equation}
which we proposed in \cite{Mertens:2019tcm} to be related to the exceptional hyperbolic Virasoro coadjoint orbits.
In conclusion, the insertion of a bulk operator has the effect of creating a hole (for real $P$) or a localized conical defect (for imaginary $P$). We checked this in the semiclassical JT limit but this is consistent with the classical solution of the Liouville equation, see for example the discussion in \cite{Moore:1991ir}.
\begin{center}
\textbf{Boundary two-point function}
\end{center}
Now we will take the JT gravity limit of the two point function computed in \eqref{eq:2pt}. We will take the matter operator with parameter $\beta_M = b h$ and keep $h$ fixed in the $b\to0$ limit. We will also take the boundary length to be large with $\ell_{{\rm JT}1}$ and $\ell_{{\rm JT}2}$ fixed. Then up to only $b$ dependent terms, we can write the two point function as
\begin{equation}
\label{JTtwo}
\mathcal{A}_{\beta_M}(\ell_1,\ell_2) \sim \mu^{\frac{1}{2b^2}} \int dk_1 dk_2 \rho_{\rm JT}(k_1)\rho_{\rm JT}(k_2)e^{-k_1^2 \ell_{{\rm JT}1}}e^{-k_2^2 \ell_{{\rm JT}2}}\frac{\Gamma(h \pm i k_1 \pm i k_2)}{\Gamma(2h)},
\end{equation}
where $\rho_{\rm JT}(k) = k \sinh 2\pi k$ and we used the small $b$ asymptotic of the double sine function $S_b(bx) \propto \Gamma(x)$. This expression coincides with the JT gravity two point function computed in \cite{Mertens:2017mtv,Lam:2018pvp}.
In the limit of large $\ell_{{\rm JT}1}$ and $\ell_{{\rm JT}2}$ this formula simplifies further since the Schwarzian mode becomes weakly coupled. Renaming $\tau = {\rm min} ( \ell_{{\rm JT}1}, \ell_{{\rm JT}2})$ and $\beta=\ell_{{\rm JT}1}+\ell_{{\rm JT}2}$, for large $\beta,\tau$ we get $\mathcal{A} \sim ( \sin \frac{\pi}{\beta}\tau)^{-2h}$. This is precisely the boundary correlator one would get if the gravitational mode would be turned off. In order to obtain this limit we need $b$ to be small. Therefore in general theories there is no regime where the gravitational dressing becomes weakly coupled \footnote{Similar drastic effects of gravitational dressings can happen also in higher dimensions \cite{Lewkowycz:2016ukf}.}.
\begin{center}
\textbf{Boundary three-point function}
\end{center}
Following the previous calculation we take the limit of the three-point function \eqref{eq:3ptbdyfinal} when the three boundary length to be large with fixed $\ell_{{\rm JT}i}$ and $\beta_{{\rm M}i}= b h_i$ for $i=1,2,3$. The integrals are then dominated by $s_i = b k_i$. Ignoring length independent prefactors, using the asymptotics of the double sine functions we can get
\begin{eqnarray}
\langle \mathcal{B}_1 \mathcal{B}_2 \mathcal{B}_3 \rangle&\sim& \int \prod_{i=1}^3 \Big[dk_i\rho_{\rm JT}(k_i) e^{- \ell_{{\rm JT}i}k_i^2}\Big] \nonumber\\
&&\hspace{-4cm}\times \big[\Gamma(h_2\pm ik_2\pm ik_3)\Gamma(h_1\pm ik_1\pm ik_2)\Gamma(h_3\pm ik_1\pm ik_3) \big]^{\frac{1}{2}} \sj{h_1}{h_2}{h_3}{k_3}{k_1}{k_2}_{\text{SL}(2,\mathbb{R})},
\end{eqnarray}
where the expression involves now the $6j$-symbol of the classical group SL$(2,\mathbb{R})$ between three principal series representations labeled by $k_i$ and three discrete representations labeled by $h_i$. This is precisely the same structure as the JT gravity three-point function computed in equation (4.35) of \cite{Iliesiu:2019xuh}.
\begin{center}
\textbf{Bulk-boundary two-point function}
\end{center}
Finally we will take the JT limit of the bulk boundary correlator given in equation \eqref{eq:bulkbdyfinal}. We set $\beta_M = bh$, and $P = \lambda/2b$. It is instructive to work this out for $h \in \mathbb{N}$. In this particular case, the last factor of the prefactor \eqref{prebb} simplifies to:
\begin{equation}
\frac{\Gamma_b(\frac{1}{b} + \beta_M \pm i \frac{\lambda}{b})}{\Gamma_b(\frac{1}{b} \pm i \frac{\lambda}{b})} \to 2\pi b \left(\frac{\sinh \pi \lambda}{\pi \lambda} \right)^{h},
\end{equation}
for a hyperbolic (macroscopic) defect with geodesic circumference $2\pi \lambda$.
\\~\\
For an elliptic (microscopic) insertion, we set $\lambda = i \theta$, and obtain instead $2\pi b \left(\frac{\sin \pi \theta}{\pi \theta} \right)^{h}$. Notice that this factor vanishes for $\theta \in \mathbb{N}_0$, which are precisely the values of the Virasoro exceptional elliptic coadjoint orbits.
The other prefactors scale in uninteresting ways and can be absorbed in normalization of the bulk and boundary operators separately.
To find a finite result, we rescale $t\to bt$ and use the small $b$-asymptotics of the $S_b$-function to get:
\begin{align}
\label{11lim}
\left\langle \mathcal{T}_{\alpha_M} \, \mathcal{B}_{\beta_M}^{+} \right\rangle &= 2\pi b \left(\frac{\sinh \pi \lambda}{\pi \lambda} \right)^{h} \int_{0}^{+\infty} dk dt \rho_{\rm JT}(k) \rho_{\rm JT}(t) e^{- \ell k^2} \chi_t(\lambda) \frac{\Gamma(h/2 \pm i k \pm it)}{\Gamma(h)},
\end{align}
in terms of the character insertion $\chi_t(\lambda)$ for $\lambda$ a hyperbolic conjugacy class element \cite{Mertens:2019tcm}:
\begin{equation}
\chi_t(\lambda) = \frac{\cos 2 \pi \lambda t}{t \sinh 2\pi t}.
\end{equation}
The $t$-momentum variable has no exponential factor, and hence no boundary segment.
The Schwarzian diagram is sketched in Figure \ref{modularS} with a bilocal line lasso-ing around the defect.
\begin{figure}[h!]
\centering
\begin{tikzpicture}[scale=1]
\draw[fill=blue!40!white,opacity=0.7] (0,0) ellipse (1.5 and 1.5);
\draw[red] (0,0.5) -- (1.5,0) -- (0,-0.5);
\draw[red] (0,-0.5) arc (180+94:180-94:0.5);
\draw[fill] (1.5,0) circle (0.06);
\node at (0,-1.985) {};
\node at (0,1.8) {\small $\ell$};
\node at (0.6,0) {\small $t$};
\node at (-0.8,0.8) {\small $k$};
\draw[fill,green!40!black] (0,0) circle (0.06);
\node[green!40!black] at (-0.23,0) {\small $\lambda$};
\node at (1.9,0) {\small $\mathcal{B}$};
\end{tikzpicture}
\caption{Schwarzian limit of the modular S-matrix, and hence the bulk-boundary propagator. The answer is given by the expectation value of a boundary-anchored bilocal line (red line) encircling the defect (green dot). This line separates two regions with energy parameters $k$ (region without defect) and $t$ (region with defect).}
\label{modularS}
\end{figure}
Notice that the bilocal line has \emph{half} the value of $h$ of the boundary operator. This can be appreciated by viewing this single boundary operator as the renormalized point-split version of two boundary operators with half the value of $h$ as:
\begin{equation}
:\lim_{x_2\to x_1} e^{\frac{\beta_M}{2}\chi_1}e^{\frac{\beta_M}{2}\chi_2}:\,\, \equiv \,\, e^{\beta_M \chi}.
\end{equation}
leading indeed to the vertex functions present in \eqref{11lim}. We also remark that this renormalization removes the coincident UV divergence of the two constituent boundary operators which would correspond in the JT limit to a contractible bilocal line (i.e. \emph{not} encircling the defect).
\section{A quantum group perspective}
\label{s:qg}
We have seen that the propagation factors in the amplitudes $e^{-\mu_B(s) \ell}$ (as in e.g. \eqref{twoa}) contain in the exponent the factor $\cosh 2 \pi b s$, and the measure is $\rho(s) = \sinh 2\pi b s \sinh \frac{2 \pi s}{b}$. In this section we highlight the quantum group structure that underlies these expressions. \\
The quantity $C_s \equiv \cosh 2 \pi b s$ is identified with the Casimir $C_s$ of the (continuous) self-dual irreps $\mathcal{P}_s$ labeled by $s$ of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ with $q=e^{\pi i b^2}$. The associated Plancherel measure on this set of representations is
\begin{equation}
d\mu(s) = ds \sinh 2\pi b s \sinh \frac{2 \pi s}{b}.
\end{equation}
This class of representations is characterized by the following \cite{Ponsot:1999uf,Ponsot:2000mt,*Bytsko:2002br,*Bytsko:2006ut,*Ip}:
\begin{itemize}
\item
It is a \emph{positive} representation, in the sense that all generators are represented by positive self-adjoint operators.
\item
They are closed under tensor product in the sense:
\begin{equation}
\mathcal{P}_{s_1} \otimes \mathcal{P}_{s_2} \simeq \int^{\oplus} d\mu(s) \mathcal{P}_s.
\end{equation}
\item
They are simultaneously representations of the dual quantum group $\mathcal{U}_{\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$ where $\tilde{q} = e^{\pi i b^{-2}}$. Hence they can be viewed naturally as representations of the modular double $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb{R})) \otimes \mathcal{U}_{\tilde{q}}(\mathfrak{sl}(2,\mathbb{R}))$.
\end{itemize}
This means the expressions \eqref{eq:b1}, \eqref{twoa}, \eqref{threea} and \eqref{twoabb} have the same group theoretic structure as those of 2d Yang-Mills or 2d BF theory, but based on the modular double of $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb{R}))$ as underlying quantum group structure. Notice that the restriction to only these self-dual representations is a strong constraint on the group-theoretic structure. But it is one that is necessary to make contact with geometric notions, as can be seen through the link with Teichm\"uller theory \cite{Nidaiev:2013bda}. Roughly speaking, the positivity constraint ensures one only has eigenstates of positive geodesic distance.
\\~\\
JT gravity can be realized in a similar group theoretical language, based on the subsemigroup SL${}^+(2,\mathbb{R})$ structure \cite{Blommaert:2018iqz}, where the defining representation of the subsemigroup consists of all positive $2\times 2$ matrices. This positivity is directly related to having only hyperbolic monodromies and hence only smooth (i.e. not punctured) geometries.
Additionally, one has to impose gravitational boundary conditions at all holographic boundaries. These boundary conditions enforce a coset structure of the underlying group and reduce the complete set of intermediate states from the full space of irrep matrix elements $R_{ab}(g)$ (by the Peter-Weyl theorem), to the double coset matrix elements $R_{00}(x)$ where both indices are fixed by the gravitational constraints. \\
From a SL${}^+(2,\mathbb{R})$ perspective, the generators $J^+$ and $J^-$ are constrained as $J^+=1$, $J^-=1$ for resp. the ket and the bra of the matrix element. This corresponds to imposing constraints on the parabolic generators, and we call the resulting matrix element a mixed parabolic matrix element. In the mathematics literature, such matrix elements are called Whittaker functions.
\\~\\
The vertex function in JT gravity $\frac{\Gamma(h \pm i k_1 \pm i k_2)^{1/2}}{\Gamma(2h)^{1/2}}$ is known to correspond to the integral definition of a 3j-symbol. For a compact group, one writes the expression as:
\begin{equation}
\int d g R_{1,m_1n_1}(g) R_{2,m_2n_2}(g) R_{3,m_3n_3}(g) = \tj{R_1}{R_2}{R_3}{m_1}{m_2}{m_3}\tj{R_1}{R_2}{R_3}{n_1}{n_2}{n_3}.\label{3R}
\end{equation}
In the JT gravity case, we have insertions of two principal series representation mixed parabolic matrix elements, and one insertion of a discrete representation (corresponding to the operator insertion):
\begin{equation}
\label{JT3j}
\int d x R_{k_1,00}(x) R_{h,00}(x) R_{k_2,00}(x) = \int_{-\infty}^{+\infty}dx\, K_{2ik_1}(e^{x}) e^{2 h x} K_{2ik_2}(e^{x}) =
2^{2h-3}\frac{\Gamma(h \pm i k_1 \pm i k_2)}{\Gamma(2h)} .
\end{equation}
\\~\\
\noindent We here illustrate that this structure persists in the $q$-deformed case and in particular to the vertex functions we wrote down \eqref{twoam} in this work. The Whittaker function of the principal series representation of $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ was derived in \cite{Kharchev:2001rs}:
\begin{equation}
\label{qmelb}
\psi^{\epsilon}_s(x) = e^{\pi i 2 s x} \int_{-\infty}^{+\infty} \frac{d\zeta}{(2\pi b)^{-2i\zeta/b-2is/b}} S_b(-i\zeta) S_b(-i 2 s -i \zeta ) e^{-\pi i \epsilon (\zeta^2 + 2s \zeta)} e^{2\pi i \zeta x} ,
\end{equation}
where $\epsilon = \pm 1$. In the notation of \cite{Kharchev:2001rs}, this corresponds to choosing $g = (2\pi b)^{1/b}$. It satisfies the following finite difference equation:
\begin{align}
\label{wdw}
\left(1+(2\pi b)^2 e^{2\pi b x - i \pi b^2} \right) \psi_s^{-}(x-ib) + \psi_s^{-}(x+ib) &= 2\cosh 2\pi b s \psi^{-}_s(x), \\
\psi_s^{+}(x-ib) + \left(1+(2\pi b)^2 e^{2\pi b x + i \pi b^2} \right) \psi_s^{+}(x+ib) &= 2\cosh 2\pi b s \psi^{+}_s(x),
\end{align}
which boils down from the Casimir equation on $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$ by constraining a parabolic generator in both the left- and right-regular representation. The rhs contains the Casimir eigenvalue in the irrep $s$. In the classical $b\to 0$ limit, this structure is precisely the same as how one constrains the $\mathfrak{sl}(2,\mathbb{R})$ Casimir equation to produce the 1d Liouville equation. Indeed, the classical $b\to 0$ limit transforms the finite difference equations both into the 1d Liouville differential equation. The options $\epsilon =\pm 1$ can be viewed as different discretizations (quantum versions) of the same classical problem. At the level of the eigenfunctions, one has the limiting behavior:
\begin{equation}
\lim_{b\to 0}\psi^{\epsilon}_s \left(\frac{x}{\pi b}\right) = \frac{1}{\pi b} K_{2 i s/b}\left(\frac{2}{b}e^{x}\right).
\end{equation}
Setting $s=bk$ and shifting $x$, the function $K_{2 i k}\left(e^{x}\right)$ is known as the Whittaker function of SL${}^+(2,\mathbb{R})$ and was inserted in \eqref{JT3j}. It is equally the 1d Liouville Schr\"odinger eigenfunction.\footnote{Crucially, in the same notation, the Whittaker function of SL$(2,\mathbb{R})$ is $\cosh \pi k \, K_{2i k}(e^{x})$ and this difference in prefactor in the end produces the SL$(2,\mathbb{R})$ Plancherel measure $d\mu(k) = dk \frac{k \sinh 2\pi k}{\cosh^2 \pi k} = 2 dk k \tanh \pi k$, in stark contrast to the SL${}^+(2,\mathbb{R})$ Plancherel measure $d\mu(k) = dk k \sinh 2\pi k$, relevant for gravity. One may encounter this Whittaker function with an additional factor of $e^x$ present: this compensates for the Haar measure on the group (coset) manifold, and one can choose to remove it and simultaneously take a flat measure in the $x$-integral as we have done.}
The modified Bessel function has a Mellin-Barnes integral representation as:
\begin{equation}
\label{mbk}
K_\nu(z) = \frac{1}{4\pi i }\left(\frac{z}{2}\right)^{\nu} \int_{-i\infty}^{+i\infty} dt \Gamma(t) \Gamma(t-\nu) \left(\frac{z}{2}\right)^{-2t},
\end{equation}
and the above formula \eqref{qmelb} is its $q$-deformed version. We need to scale $s \to b k$ in order to obtain a finite classical limit.
By analogy with the lhs of \eqref{JT3j}, we hence compute the integral of two Whittaker functions, and one discrete insertion, of the type ($\beta_M = b h$):
\begin{equation}
\int_{-\infty}^{+\infty} dx\, \psi^{\epsilon}_{s_1} (x) \psi^{\epsilon * }_{s_2} (x) e^{2 \beta_M \pi x}.
\end{equation}
Inserting the explicit expression \eqref{qmelb}, one can evaluate the $x$-integral as:
\begin{equation}
\int_{-\infty}^{+\infty} dx e^{\pi i (2s_1 - 2s_2 + 2 \zeta_1 - 2 \zeta_2) + 2 \beta_M \pi x} = \delta(\zeta_1-\zeta_2 + s_1 - s_2 - i \beta_M).
\end{equation}
We get:
\begin{align}
&\int_{-\infty}^{+\infty} dx \psi^{\epsilon}_{s_1} (x) \psi^{\epsilon * }_{s_2} (x) e^{2 \beta_M \pi x} = e^{-\pi i \epsilon(\beta_M^2-s_1^2+s_2^2+2i s_1 \beta_M)} \\
&\times \int_{-\infty}^{+\infty} \frac{d\zeta_1}{(2\pi b)^{2\beta_{M}/b}} e^{\pi 2 \epsilon \beta_M \zeta_1} S_b(-i\zeta_1) S_b(-i \zeta_1-2is_1) S_b(i\zeta_1+is_1 -is_2 + \beta_M)S_b(i\zeta_1+is_1 + is_2 + \beta_M). \nonumber
\end{align}
The $q$-deformed first Barnes lemma is:
\begin{align}
\label{qbarne}
\int d\tau e^{\pi \tau (\alpha+\beta+\gamma+\delta)}&S_b(\alpha+i\tau)S_b(\beta+i\tau)S_b(\gamma-i\tau)S_b(\delta-i\tau) \\
&= e^{\pi i (\alpha\beta-\gamma\delta)} \frac{S_b(\alpha+\gamma)S_b(\alpha+\delta)S_b(\beta+\gamma)S_b(\beta+\delta)}{S_b(\alpha+\beta+\gamma+\delta)}. \nonumber
\end{align}
Using \eqref{qbarne}, we can do the remaining integral and obtain finally:\footnote{The prefactor is immaterial and can be absorbed into the normalization of the boundary operator. Reinstating the parameter $g$ of \cite{Kharchev:2001rs}, the prefactor would be $\frac{1}{g^{2\beta_M}}$ instead.}
\begin{equation}\label{idwhitsb}
\boxed{
\int_{-\infty}^{+\infty} dx \hspace{0.1cm} \psi^{\epsilon}_{s_1} (x) \psi^{\epsilon * }_{s_2} (x) e^{2 \beta_M \pi x} = \frac{1}{(2\pi b)^{2\beta_{M}/b}} \frac{S_b(\beta_M \pm is_1 \pm is_2)}{S_b(2\beta_M) }}\, .
\end{equation}
Following the structure of \eqref{JT3j}, we interpret this as the square of the 3j-symbol with two mixed parabolic entries, and one discrete parabolic entry, of the quantum group $\mathcal{U}_q(\mathfrak{sl}(2,\mathbb{R}))$. As a check, taking the $b\to 0$ limit of both sides, we get the equality:
\begin{equation}
\left(\frac{b}{2}\right)^{2h}\frac{1}{(\pi b)^3} \int_{-\infty}^{+\infty} dx K_{2ik_1}(e^{x}) K_{2ik_2}(e^{x}) e^{2hx} = b^{2h} \frac{1}{(2\pi b)^3} \frac{\Gamma(h\pm i k_1 \pm ik_2)}{\Gamma(2h)},
\end{equation}
matching back onto \eqref{JT3j}
\subsection{Wheeler-DeWitt wavefunction}
We have computed above the partition function on a hyperbolic Euclidean disk with a fixed length boundary. We can cut this disk along a bulk geodesic with length function $L$, that joins two boundary points separated by a distance $\beta/2$. This can be interpreted as a Euclidean preparation of the Wheeler-DeWitt (WdW) wavefunction $\Psi_\beta(L)$ corresponding to the two-sided black hole, see figure \ref{fig:wdw}. This wavefunction has been studied in the context of JT gravity in \cite{Harlow:2018tqv}.\footnote{This is different than the radial-quantization WdW wavefunction studied for example in \cite{Maldacena:2019cbz,Iliesiu:2020zld}.} Based on the properties of the Whittaker function $\psi_s(x)$ above, we propose the following identification
\begin{equation}
\Psi_\beta (L) = \int ds\hspace{0.1cm} e^{- \frac{1}{2}\beta \mu_B(s)} \rho(s) \psi^+_s(L),
\end{equation}
where we take $\epsilon=+1$ for concreteness. When we take the JT gravity limit, the density of states becomes the Schwarzian density of states, while the Whittaker function becomes a Bessel function derived directly from JT gravity in \cite{Harlow:2018tqv}. We have identified the group (coset) parameter $x$ of the Whittaker function with the argument of the wavefunction $L$. In the classical $b\to 0$ limit, this quantity is related to the boundary-to-boundary geodesic length $d$ as $x \to e^{d/2}$. The wavefunction can also be interpreted as the Euclidean partition function in the disk with an end-of-the-world brane.
\begin{figure}[h!]
\begin{center}
\begin{tikzpicture}[scale=0.75]
\pgftext{\includegraphics[scale=0.3]{WdW.pdf}} at (0,0);
\draw (2,-1) node {\small $\beta/2$};
\draw (0,1) node {\small $L$};
\end{tikzpicture}
\caption{\label{fig:wdw} Depiction of the geometry creating the Hartle-Hawking state $\Psi_\beta(L)$. The state is labeled by a parameter $\beta$ that gives the proper length of the boundary segment preparing the state. The constant time slice is labeled by $L$, which is related to the geodesic distance along the slice.}
\end{center}
\end{figure}
To verify this identification we can rewrite the exact two point function \eqref{twoa} in the following form
\begin{equation}
\langle \mathcal{B}\hspace{0.05cm} \mathcal{B} \rangle = \int dL \hspace{0.1cm} e^{2 \beta_M \pi L} \hspace{0.1cm}\Psi_{\ell_1}(L)^\dagger \hspace{0.1cm} \Psi_{\ell_2}(L),
\end{equation}
where we used the relation \eqref{idwhitsb}. This expression can be interpreted as gluing two portions of the disk along their bulk geodesic with the inclusion of the matter propagator $e^{2 \beta_M \pi L}$. This is structurally identical to the JT gravity expressions, and it would be interesting to give a more rigorous derivation from Liouville gravity.
Finally, the wavefunction $\Psi_\beta(L)$ proposed here satisfies an interesting equation. We can rewrite the wavefunction for the same Hartle-Hawking state in an energy basis, which becomes the Whittaker function $\Psi_{E=\mu_B(s)}(L) = \psi^+_s(L)$ and satisfies the difference equation \eqref{wdw}. In terms of the fixed length basis this equation is
\begin{equation}
\Psi_\beta(L-ib) + \big(1+(2\pi b)^2e^{2\pi b L + i \pi b^2} \big) \Psi_\beta(L+ib) = 4 \frac{\partial}{\partial \beta} \Psi_\beta(L),
\end{equation}
which can be viewed as a discretized (due to the $q$-deformation) ancestor of the Wheeler-DeWitt equation. This suggests that Liouville quantum gravity effectively discretizes the spacetime in a way we do not understand sufficiently, and this discreteness might be related to the quantum group structure present in the theory.
\subsection{Degenerate fusion algebra}
Modified Bessel functions satisfy the following identity:
\begin{equation}
\label{besselprop}
K_{\alpha+1}(x) - K_{\alpha-1}(x) = \frac{2\alpha}{x}K_{\alpha}(x),
\end{equation}
which can be proved directly from the Mellin-Barnes representation \eqref{mbk}. This identity is important since they act as the degenerate fusion rules that directly lead to the degenerate $h\in -\mathbb{N}/2$ vertex functions for JT gravity \cite{Mertens:2020pfe}, where the vertex function in e.g. \eqref{JTtwo} is singular. Following a similar strategy with \eqref{qmelb}, one can prove the following fusion property for $\epsilon = \pm 1$:
\begin{equation}
\label{qfus}
\psi^{\epsilon}_{s+ib/2}(x) - \psi^{\epsilon}_{s-ib/2}(x) = \frac{\sinh 2 \pi b s}{ \pi i b\, e^{\pi b x}} \psi^{\epsilon}_s(x).
\end{equation}
This relation is the basis to derive the minimal string correlators where $\beta_M \in - b\, \mathbb{N}/2$ from the continuum approach directly. The trick is to successively apply it to compute ($j \in \mathbb{N}/2$):
\begin{equation}
\int_{-\infty}^{+\infty} dx \psi^{\epsilon}_{s_1}(x) \psi^{\epsilon * }_{s_2}(x) e^{- 2 \pi b j x},
\end{equation}
until we reach
\begin{equation}
\int_{-\infty}^{+\infty} dx \psi^{\epsilon}_{s_1}(x) \psi^{\epsilon * }_{s_2}(x) = \frac{\delta(s_1-s_2)}{S_0^{s_1}}.
\end{equation}
After providing a matrix model computation of these minimal string correlators, we will come back to this approach using \eqref{qfus} and check explicitly that they match indeed.
\section{Dual matrix models}\label{sec:MM}
In this section we will give a matrix model interpretation of some of the results in the previous sections for the case of the $(2,p)$ minimal string. This case is special since the dual is a single matrix model. The discrete calculation of disk boundary correlators was proposed in \cite{Kostov:2002uq} (see also \cite{Hosomichi:2008th, Ishiki:2010wb, Bourgine:2010ja}). Besides the explicit checks, the new ingredient is to interpret the dual matrix as a boundary Hamiltonian in the sense of holography, as suggested by \cite{Saad:2019lba}. Then we will see boundary correlators of the bulk theory are equal to boundary correlators of random operators.
\subsection{Partition function} \label{sec:MMpf}
Motivated by \cite{Saad:2019lba} we will denote the random matrix as $H$ since we will interpret it as a boundary random Hamiltonian. The matrix model dual of a marked disk partition function is
\begin{equation}
Z(\mu_B) = \left\langle \text{Tr}\hspace{0.1cm}\frac{1}{H-\mu_B}\right\rangle.
\end{equation}
After inverse Laplace transforming the fixed length partition function is instead
\begin{equation}
Z(\ell) = \left\langle \text{Tr}\hspace{0.1cm}e^{-\ell H}\right\rangle.
\end{equation}
By choosing an appropriate potential for the matrix model ensemble we can make this match with the continuum answer in the double scaling limit.
Before moving on, we want to show that the result \eqref{eq:markedtrivial} can actually be easily deduced using the matrix model language. According to this formulation of the theory, the $n$ marking operator correlator is given by the expectation value of the following product of matrices
\begin{equation}\label{eq:markcorrmamo}
\left\langle {}^{\mu_1}e^{b\phi_1}{}^{\mu_2} \hdots {}^{\mu_n}e^{b\phi_n}{}^{\mu_1}\right\rangle = \left\langle \text{Tr}\hspace{0.1cm}\frac{1}{(H-\mu_1)\hdots (H-\mu_n)}\right\rangle.
\end{equation}
Instead of finding the expectation value first, we can inverse Laplace transform directly the matrix model observable
\begin{equation}
\left\langle \text{Tr}\hspace{0.1cm}e^{-(\ell_1+\hdots + \ell_n)H}\right\rangle,
\end{equation}
which makes manifest that depends only on the total boundary length and is consistent with \eqref{eq:markedtrivial}, since the operator $\text{Tr}\hspace{0.1cm}e^{-\ell H}$ is dual to inserting a fixed length $\ell$ boundary.
\subsection{Amplitudes}
The matrix model dual to the minimal string with boundary insertions can be written by introducing vector degrees of freedom
\begin{equation}\label{eq:genfuncmm}
e^Z = \int DH D\bar{v}Dv \hspace{0.1cm} e^{- L {\rm Tr} V(H) - \bar{v}_a C^{ab}(H) v_b},
\end{equation}
where $v_a$ are $N$ dimensional vectors and $a=1,\ldots, N_f$. For example, the FZZT unmarked boundary partition function can be obtained by taking a single vector $N_f=1$ and an interaction $C(H)=\mu_B-H$. Similarly the boundary correlator of $n$ marking operators in the previous section can be obtained still by a single vector and a higher order polynomial interaction $C(H) = (\mu_{B1}-H)(\mu_{B2}-H)\ldots (\mu_{Bn}-H)$ which should be compared to \eqref{eq:markcorrmamo}. We will follow the presentation in \cite{Ishiki:2010wb}.
For the insertion of the two point function corresponding to $\mathcal{B}_{2,1}$ we need two vectors and the following interaction
\begin{equation}\label{matrixvectorinter}
C(H) = \begin{pmatrix}
\mu_B(s_1)-H&c^{12}\\
c^{21} & F_2(H)\\
\end{pmatrix},~~~~F_2(H)=\prod_{\pm} (\mu_B(s_2 \pm i b)-H).
\end{equation}
For this choice \eqref{eq:genfuncmm} is a generating function of $\mathcal{B}_{2,1}$ correlators for which $c^{12}$ and $c^{21}$ are sources and boundary conditions shift from $\mu_B(s_1) \to \mu_B(s_2)$. For the minimal string matrix model this produces the same answer as the star polymer operators in the context of the loop gas formalism \cite{Kostov:2002uq}. For example, the two point function is
\begin{equation}
\langle\mathcal{B}_{2,1}\mathcal{B}_{2,1} \rangle = \left\langle \text{Tr}\hspace{0.1cm}\frac{1}{(H-\mu_B(s_1))}\frac{1}{(H-\mu_B(s_2-ib))}\frac{1}{(H-\mu_B(s_2+ib))}\right\rangle.
\end{equation}
This can be compared directly in the fixed cosmological constant basis to the results from the continuum Liouville approach. Instead we will transform the observable directly into fixed length basis. For this we need to perform the inverse Laplace transform of the previous formula for the operator inside the trace
\begin{align}
\int_{-i\infty}^{+i\infty}dy \frac{e^{-y \ell_1}}{(y-H)}\int_{-i\infty}^{+i\infty}dx \frac{1}{(\cosh(2\pi b (s_2+ib/2)) - H) (\cosh( 2\pi b (s_2-ib/2) - H)} e^{-x\ell_2},
\end{align}
where for simplicity we set $\kappa=1$ and define $x=\cosh 2\pi b s_2$, $y=\cosh 2\pi b s_1$. The $y$-integral directly gives the marked length $\ell_1$ operator $e^{-\ell_1 H}$. The denominator can be written as $x^2 - 2H \cos \pi b^2 x + H^2 - \sin^2 \pi b^2$ and the integral can then be directly evaluated by residues, picking up two pole contributions, yielding
\begin{equation}
\langle\mathcal{B}_{2,1}\mathcal{B}_{2,1} \rangle = \left\langle \text{Tr}\hspace{0.1cm} e^{-\ell_1 H} e^{-\ell_2 H \cos \pi b^2} \frac{\sin \left( \ell_2 \sin \pi b^2 \sqrt{H^2-1}\right)}{\sin \pi b^2 \sqrt{H^2-1}} \right\rangle.
\end{equation}
This is for the matrix $H$ underlying the minimal string matrix integral. If we now identify
\begin{equation}
H \leftrightarrow \cosh 2\pi b s = \mu_B, \qquad \sqrt{H^2-1} \leftrightarrow \sinh 2\pi b s,
\end{equation}
we get for the full result at leading order in the genus expansion, using the leading density of states
\begin{align}
\langle\mathcal{B}_{2,1}\mathcal{B}_{2,1} \rangle &= \int_0^\infty ds \rho(s) \hspace{0.1cm} e^{-\ell_1 \cosh 2\pi b s } e^{-\ell_2 \cosh 2 \pi b s \cos \pi b^2} \frac{\sin \left( \ell_2 \sin \pi b^2 \sinh 2\pi b s \right)}{\sin \pi b^2 \sinh 2\pi b s} \nonumber \\
\label{tocmp}
&= \int_0^\infty ds \rho(s) \hspace{0.1cm} e^{-\ell_1 \cosh 2\pi b s }\left[ \frac{e^{-\ell_2 \cosh 2\pi b (s+ib/2)}}{\sin \pi b^2 \sinh 2\pi b s} - \frac{e^{-\ell_2 \cosh 2\pi b (s-ib/2)}}{\sin \pi b^2 \sinh 2\pi b s}\right].
\end{align}
Following the interpretation of \cite{Saad:2019lba} of the random matrix as a random Hamiltonian we can interpret the boundary correlator as inserting an operator. Since they match for fixed FZZT boundaries this correlator matches with the fixed length two-point function when $\beta_M = - b/2 $ corresponding to $\mathcal{B}_{2,1}$.
This correlator has a very simple JT gravity limit. Following the previous discussion, we set $s= bk$, with fixed $k$ as $b\to 0$ and define the renormalized length $\ell_{{\rm JT}i} \equiv 2\pi^2 b^4\ell_i$. This gives the simple answer
\begin{eqnarray}
\langle\mathcal{B}_{2,1}\mathcal{B}_{2,1} \rangle_{(2,p\to\infty)} &=& \int_0^\infty kdk \hspace{0.1cm} \sinh 2 \pi k e^{-(\ell_{{\rm JT}1}+\ell_{{\rm JT}2}) k^2}\, e^{ \frac{1}{4} \ell_{{\rm JT}2}} \frac{\sin \left( \ell_{{\rm JT}2} k \right)}{k}\\
&\sim& \Big( \frac{\beta}{\pi} \sin \frac{\pi \tau}{\beta} \Big) e^{\frac{\tau(\beta-\tau)}{4\beta}},
\end{eqnarray}
where in the second line we defined $\tau =\ell_{{\rm JT}2}$ and $\beta= \ell_{{\rm JT}1} + \ell_{{\rm JT}2}$. This is precisely equal to the exact Schwarzian two point function for operators of dimension $\Delta=-1/2$. This is equivalent to equation (D.7) of \cite{Mertens:2019tcm}, for $C_{\rm there}=1/2$. As explained there, only operators with negative half integer dimension have such a simpler form, and these correspond to the minimal model CFT dimensions.
This discussion can be extended to higher degenerate insertions $\mathcal{B}_{2j+1,1}$ where $\beta_M = -bj$ and $j\in \mathbb{N}/2$. This can be achieved still with two vectors interacting through the same two by two matrix in \eqref{matrixvectorinter}, but with $F_j(H) =\prod_{n=-j}^{j}(\cosh(2\pi b (s+inb)) - H)$. The two-point function of $\mathcal{B}_{2j+1,1}$ corresponds then to the matrix integral insertion
\begin{align}
\langle \mathcal{B}_{2j+1,1}\mathcal{B}_{2j+1,1}\rangle = \left\langle \text{Tr}\hspace{0.1cm}\frac{1}{\cosh 2\pi b s_1-H}(2j)! \prod_{n=-j}^{j} \frac{1}{\cosh(2\pi b (s_2+inb)) - H}\right\rangle .
\end{align}
Transferring to the fixed length basis, one has to perform the integral
\begin{align}
(2j)!\int_{-i\infty}^{+i\infty}dy \frac{e^{-y \ell_1}}{(y-H)}\int_{-i\infty}^{+i\infty}dx \frac{1}{\prod_{n=-j}^{j}(\cosh(2\pi b (s+inb)) - H)} e^{-x\ell_2}.
\end{align}
Combining the factors $\pm n$ together, we can play the same game, and combine the denominators into:
\begin{align}
x^2 - 2H \cos 2\pi n b^2 x + H^2 - \sin^2 2\pi n b^2
= (x - \cosh 2\pi b (s\pm i n b)).
\end{align}
If $2j+1$ is even, then these are all of the factors. If $2j+1$ is odd, then we have one additional factor $(x-H)$ in the denominator. What is left is just a sum of $2j+1$ residues, where the denominator is a polynomial in $H$ of order $2j$.
The previous procedure can be done for any $j \in \mathbb{N}/2$ and we get the complicated general expression:\footnote{We have conventionally divided by the partition function $Z$ in this equation.}
\begin{align}
\label{gendeg}
&\langle \mathcal{B}_{2j+1,1}\mathcal{B}_{2j+1,1} \rangle \nonumber \\
&= \frac{1}{Z}\int_0^{+\infty} ds \rho(s) \, e^{-\ell_1 \cosh 2\pi b s}\sum_{n=-j}^{+j} \frac{(2j)!e^{-\ell_2 \cosh 2\pi b (s+i nb)}}{\prod_{\stackrel{m=-j}{m\neq n}}^{j} (\cosh 2\pi b (s+i nb) - \cosh 2\pi b (s+imb))}.
\end{align}
One can check that in the UV limit $\ell_2 \to 0$, the entire sum becomes $\ell_2^{2j} + \mathcal{O}(\ell_2^{2j+1})$, and the expression reduces to
\begin{equation}
\label{UVdeg}
\langle \mathcal{B}_{2j+1,1}\mathcal{B}_{2j+1,1} \rangle \to \ell_2^{2j},
\end{equation}
matching the general analysis in section \ref{s:bostwo}. \\
In the JT limit, the pole contributions and exponentials are expanded as:
\begin{align}
\label{reslim}
\cosh 2\pi b (s+inb) - \cosh 2\pi b (s+imb)\quad &\to \quad b^4 2\pi^2 (m-n)\left(n+m -2 ik \right) + \mathcal{O}(b^6), \\
e^{- \ell \cosh 2\pi b (s+inb)} \quad &\to \quad e^{- \ell_{\rm JT} k^2} e^{\ell_{\rm JT} n^2 }e^{- 2 i \ell_{\rm JT} n k}.
\end{align}
This is precisely the structure expected for a degenerate Schwarzian insertion \cite{Mertens:2020pfe}: the denominators \eqref{reslim} produce a polynomial in $k$, while the $m-n$ factors conspire to give a binomial coefficient. In the end, we can identify this matrix insertion with the degenerate Schwarzian bilocal as:
\begin{equation}
\mathcal{B}_{2j+1,1}\mathcal{B}_{2j+1,1} \quad \to \frac{1}{b^{8j}(2\pi^2)^{2j}} \, \mathcal{I}^{j}(0)\mathcal{I}^{j}(\tau),
\end{equation}
where the prefactor is also readily determined from \eqref{UVdeg} combined with the relation between $\ell_{\rm JT}$ and $\ell$. Here $\mathcal{I}^j$ indicates an operator in the Schwarzian theory of dimension $\Delta=-j/2$.
\\~\\
This structure of the minimal string correlators \eqref{gendeg} matches with the continuum approach by using the fusion property \eqref{qfus}.
As an example, for the first minimal string $j=1/2$ insertion, a single application of \eqref{qfus} leads to the identity:
\begin{align}
\int_{-\infty}^{+\infty} dx \psi^{\epsilon}_{s_1}(x) \psi^{\epsilon * }_{s_2}(x) e^{- \pi b x} = \frac{\pi b}{i S_0{}^{s_1}}\left[\frac{\delta(s_1-s_2-ib/2)}{\sinh 2 \pi b s_2 }- \frac{\delta(s_1-s_2 + ib/2)}{\sinh 2 \pi b s_2 }.
\right]
\end{align}
The delta-function enforces the correct dependence in the exponential factor in \eqref{tocmp}. We also see the $1/\sinh 2 \pi b s$ factor in the denominator of \eqref{tocmp} appearing. \\
It is clear that for generic $j\in \mathbb{N}/2$, we will find a similar result. As an example, in Appendix \ref{app:one} we work out the formulas for $j=1$, and check indeed that the methods match.
\section{Other topologies}\label{sec:othertopo}
In this section we will extend previous calculations to situations with more general topologies and multiple boundaries. We will focus here on the minimal string theory since it has a direct interpretation as a one-matrix integral.
\subsection{Cylinder}
We will first study minimal string theory on a cylinder between fixed length boundaries. This was computed from a continuum approach by Martinec \cite{Martinec:2003ka} and from a discrete approach by Moore, Seiberg and Staudacher \cite{Moore:1991ir}. We will present a technically simplified derivation from the continuum limit and make a connection with JT gravity for the $(2,p)$ string with large $p$. As another example, we will apply our method to the crosscap spacetime in Appendix \ref{app:crosscap}, also reproducing the matrix model result.
Using the boundary state formalism \cite{Ishibashi:1988kg, Cardy:1989ir} we can described a boundary labeled by an FZZT parameter $s$ and matter labels $(n,m)$ by the following combination of Ishibashi states
\begin{equation}
|s; n,m \rangle = \sum_{n',m'} \int_0^{\infty} dP \hspace{0.1cm}\Psi_s(P) \frac{S_{n,m}^{n',m'}}{(S_{1,1}^{n',m'})^{1/2}} |P\rangle\hspace{-0.1cm}\rangle_L |n',m' \rangle\hspace{-0.1cm}\rangle_M.
\end{equation}
As pointed out by Seiberg and Shih this state can be simplified as a sum of matter identity branes over shifted FZZT parameters, modulo BRST exact terms that cancel when computing physical observables, see equation (3.8) in \cite{Seiberg:2003nm}. Therefore in the end of the calculation we will focus on the matter sector identity brane.
\begin{figure}[t!]
\centering
\begin{tikzpicture}[scale=0.9]
\node at (-3.5,0) {\small $s_1{}_{(n_1,m_1)}$};
\node at (3.5,0) {\small $s_2{}_{(n_2,m_2)}$};
\draw[thick] (-2,0) ellipse (0.4 and 1.5);
\draw[thick] (2,0) ellipse (0.4 and 1.5);
\draw[thick] (-1.95,1.49) to [bend right=50] (1.95,1.49);
\draw[thick] (-1.95,-1.49) to [bend left=50] (1.95,-1.49);
\node at (0,-1.985) {};
\end{tikzpicture}
\caption{The figure shows the cylinder amplitude we are computing between two FZZT boundaries with boundary cosmological constants $\mu_B(s_1)$ and $\mu_B(s_2)$ and matter boundary conditions labeled by $(n_1,m_1)$ and $(n_2,m_2)$.}
\label{fig:cylinder}
\end{figure}
As explained in \cite{Martinec:2003ka} the annulus partition function between unmarked $s_1; n_1,m_1$ and $s_2;n_2,m_2$ branes is computed as the overlap of the boundary states, integrated over the moduli. For the annulus, there is a single real modulus $\tau$ parametrizing the length along the cylinder. Notice that this is a coordinate on the worldsheet and it is integrated over. In the end we will find dependence on \emph{physical} lengths instead as emphasized in the Introduction.
Before integration the answer factorizes into the Liouville (L), matter (M) and ghost (G) contributions
\begin{equation}\nonumber
\langle Z(s_1;n_1,m_1)^{\scriptscriptstyle \text{U}} Z(s_2;n_2,m_2)^{\scriptscriptstyle \text{U}}\rangle = \int d\tau Z_{L} Z_{M} Z_{G},~~\begin{cases} Z_{L} = \int_0^\infty \frac{dP}{\pi} \frac{\cos 4 \pi s_1 P \cos 4 \pi s_2 P}{\sqrt{2}\sinh 2 \pi P b \sinh 2 \pi \frac{P}{b}}\chi_P(q), \\
~~\vspace{-0.5cm}\\
Z_{M} = \sum_{n,m}\mathcal{N}_{n,m}^{(n_1,m_1)(n_2,m_2)} \chi_{n,m}(q'),\\
~~\vspace{-0.5cm}\\
Z_{G} =\eta(q)^2, \end{cases}
\end{equation}
where $q'=e^{-2 \pi i/\tau}$ and $\mathcal{N}_{n,m}^{(n_1,m_1)(n_2,m_2)} $ denote the fusion coefficient of the matter theory. In the matter sector, we used the Verlinde formula \cite{Verlinde:1988sn} to simplify the boundary state inner product as a sum over the dual channel characters weighted by the fusion numbers. This simplifies the calculation compared to \cite{Martinec:2003ka}.
We will write $\tau = i t$ where $t$ is integrated over the positive real line. Then using the modular property of the Dedekind eta function the contribution from descendants cancel up to a factor of $t^{-1/2}$ and we can write
\begin{eqnarray}
\langle Z(s_1;n_1,m_1)^{\scriptscriptstyle \text{U}} Z(s_2;n_2,m_2)^{\scriptscriptstyle \text{U}}\rangle&=& \int_0^\infty \frac{dP}{\pi} \frac{\cos 4 \pi s_1 P \cos 4 \pi s_2 P}{\sqrt{2}\sinh 2 \pi P b \sinh 2 \pi \frac{P}{b}} \nonumber\\
&&\hspace{-3.5cm}\times \sum_{n,m}\mathcal{N}_{n,m}^{(n_1,m_1)(n_2,m_2)} \sum_k \int_0^\infty \frac{dt}{\sqrt{t}} e^{-2\pi t P^2} \big(e^{-\frac{2\pi}{t}a_{n,m}(k)}-e^{-\frac{2\pi}{t} a_{n,-m}(k)}\big),
\end{eqnarray}
where $a_{n,m}(k)$ was defined in equation \eqref{degcharacters}. We first integrate over $t$. The answer depends on whether $k>0$, $k<0$ or $k=0$ so each case has to be considered separately. We then sum over $k$ taking this into account. The final answer is very simple
\begin{equation}\nonumber
\sum_k \int \frac{dt}{\sqrt{2t}} e^{-2\pi t P^2} (e^{-\frac{2\pi}{t}a_{n,m}(k)}-e^{-\frac{2\pi}{t} a_{n,-m}(k)}) = \begin{cases} \frac{\sinh 2 \pi b P (p'-n) \sinh 2 \pi \frac{P}{b} m }{P \sinh 2 \pi p \frac{P}{b}} ~~~{\rm if}~~np>mp'\\
~~\vspace{-0.2cm}\\
\frac{\sinh 2 \pi b P n \sinh 2 \pi \frac{P}{b} (p-m) }{P \sinh 2 \pi p \frac{P}{b}} ~~~{\rm if}~~np<mp'\end{cases}
\end{equation}
When we sum over the primaries we can use the fundamental domain $E_{p'p}$ which corresponds precisely to the first case in the result above. Then the final answer for the annulus partition function becomes
\begin{eqnarray}
\langle Z(s_1;n_1,m_1)^{\scriptscriptstyle \text{U}} Z(s_2;n_2,m_2)^{\scriptscriptstyle \text{U}}\rangle &=& \sum_{(n,m)\in E_{p'p}}\mathcal{N}_{n,m}^{(n_1,m_1)(n_2,m_2)} \int_0^\infty \frac{dP}{\pi} \nonumber\\
&&\hspace{-3cm}\times \frac{\cos(4 \pi s_1 P) \cos(4 \pi s_2 P) \sinh(2 \pi b P (p'-n)) \sinh(2 \pi \frac{P}{b} m) }{P \sinh(2 \pi p \frac{P}{b}) \sinh(2 \pi b P) \sinh(2 \pi \frac{P}{b})},
\end{eqnarray}
where we make explicit that this expression is valid when $(n,m)$ are in the fundamental domain $E_{p'p}$. This generalizes the formula derived by Martinec, which only includes boundary states of the form $(n,1)$ to an arbitrary boundary state and also matches in this case with the expression derived in reference \cite{Kutasov:2004fg}. We can use the Seiberg-Shih relation between boundary states to justify focusing on the matter identity branes, and the partition function simplifies to
\begin{equation}
\langle Z(s_1)^{\scriptscriptstyle \text{U}} Z(s_2)^{\scriptscriptstyle \text{U}}\rangle_{(p,p')} = \int_0^\infty \frac{dP}{\pi} \frac{\cos(4 \pi s_1 P) \cos(4 \pi s_2 P) \sinh(2 \pi \frac{P}{b} (p-1)) }{P \sinh(2 \pi p \frac{P}{b}) \sinh(2 \pi \frac{P}{b})}.
\end{equation}
This is valid for the $(p,p')$ minimal string. Since we will be interested mostly in theories dual to single matrix models we can further take the $(2,p)$ minimal model and get
\begin{equation}
\langle Z(s_1)^{\scriptscriptstyle \text{U}} Z(s_2)^{\scriptscriptstyle \text{U}}\rangle_{(2,p)} = \int_0^\infty dP\frac{\cos(4 \pi s_1 P) \cos(4 \pi s_2 P)}{2\pi P \sinh 2 \pi \frac{P}{b} \cosh 2 \pi \frac{P}{b}}.
\end{equation}
In the rest of this section we will analyze this expression.
As indicated, these are unmarked FZZT boundaries. Using the methods described above we can first compute the marked boundary amplitude which is more directly related to the matrix integral. Taking derivatives with respect to the boundary cosmological constant using \eqref{fion} we get
\begin{eqnarray}\label{annulusunmarked}
\langle Z(s_1)^{\scriptscriptstyle \text{M}} Z(s_2)^{\scriptscriptstyle \text{M}}\rangle_{(2,p)} &=&\frac{2}{b^2} \int_0^\infty \frac{dP}{\pi} \frac{\sin(4 \pi s_1 P) \sin(4 \pi s_2 P)}{\kappa \sinh \pi b s_1 \kappa \sinh \pi b s_2} \frac{2P }{\sinh 4 \pi \frac{P}{b}} ,\nonumber\\
&=& \frac{1}{8\pi} \frac{1}{\sqrt{-\mu_1 + \kappa}\sqrt{-\mu_2 + \kappa}}\frac{1}{(\sqrt{-\mu_1+\kappa} + \sqrt{-\mu_2 + \kappa})^2},
\end{eqnarray}
where $\mu_i=\mu_B(s_i)$. The expression in the second line is precisely the connected component to the resolvent two point function (see for example equation (47) of \cite{Saad:2019lba}). When written in the appropriate variables this result is completely independent of $p$ and therefore independent of the precise density of states. This is evident in the matrix integral approach but unexpected from the continuum approach.
\begin{figure}[t!]
\centering
\begin{tikzpicture}[scale=0.9]
\node at (-2.6,0) {\small $\ell_1$};
\node at (2.6,0) {\small $\ell_2$};
\draw[thick] (-2,0) ellipse (0.3 and 1.5);
\draw[thick] (2,0) ellipse (0.3 and 1.5);
\draw[thick] (-1.937,1.47) to [bend right=50] (1.937,1.47);
\draw[thick] (-1.937,-1.47) to [bend left=50] (1.937,-1.47);
\node at (0,-1.985) {};
\node at (4.5,0) {\large $=\int $ {\small $d\mu(\lambda)$}};
\draw[thick] (6.5,0) ellipse (0.3 and 1.5);
\draw[thick] (8.5,0) ellipse (0.1 and 0.7);
\draw[thick] (6.54,1.49) to [bend right=20] (8.5,0.7);
\draw[thick] (6.54,-1.49) to [bend left=20] (8.5,-0.7);
\draw[thick] (11.5,0) ellipse (0.3 and 1.5);
\draw[thick] (9.5,0) ellipse (0.1 and 0.7);
\draw[thick] (11.46,1.49) to [bend left=20] (9.5,0.7);
\draw[thick] (11.46,-1.49) to [bend right=20] (9.5,-0.7);
\node at (5.9,0) {\small $\ell_1$};
\node at (12.1,0) {\small $\ell_2$};
\node at (8.5,-1) {\small $\lambda$};
\node at (9.5,-1) {\small $\lambda$};
\end{tikzpicture}
\caption{We depict the cylinder amplitude in physical space between fixed length boundaries. The final answer can be interpreted as gluing minimal string trumpets generalizing the procedure of JT gravity}
\label{fig:trumpetgluing}
\end{figure}
We can compute the fixed length amplitude in two ways. Firstly, we can apply the method above to compute the inverse Laplace transform through the discontinuity before integrating over $P$. In order to do this we can use the expression \eqref{eq:disccos} for the discontinuity. Secondly, we can apply this directly to the second line of \eqref{annulusunmarked}. Either way the result is the same, given after relabeling $\lambda = 2P/b$ by the formula
\begin{eqnarray}\label{eq:2loopcorr}
\langle Z(\ell_1) Z(\ell_2) \rangle &=&\frac{2}{\pi} \int_0^\infty \lambda d\lambda \tanh \pi \lambda ~K_{i\lambda}(\kappa \ell_1) K_{i \lambda}(\kappa \ell_2),\\
&=& \frac{\sqrt{\ell_1 \ell_2}}{\ell_1+\ell_2}e^{-\kappa (\ell_1+ \ell_2)}.
\end{eqnarray}
The first line of the previous equation has a very familiar form when comparing with JT gravity. As we explained before, inserting a bulk operator in the disk can be interpreted as creating a hole in the physical space which in the JT gravity limit becomes a geodesic boundary of length $\sim \lambda$. Therefore we can compare the integral above after replacing $ \lambda = b_{\rm JT}/(2\pi b^2)$ and $ \ell = \ell_{\rm JT}/(2\kappa \pi^2 b^4)$ as gluing two minimal string trumpets with a deformed measure\footnote{The prefactors of this equation can be tracked by using the integral representation \eqref{idthree}.}
\begin{eqnarray}
e^{\kappa \ell} K_{i\lambda}(\kappa \ell) &\to& \pi b^2 \sqrt{\frac{\pi}{\ell_{\rm JT}}}e^{- \frac{b_{\rm JT}^2}{4\ell_{\rm JT}}},\\
\lambda d\lambda \tanh \pi \lambda &\to& \frac{1}{4\pi^2b^4}b_{\rm JT}db_{\rm JT} ,~~~{\rm for}~\lambda \to \infty.
\end{eqnarray}
Liouville CFT is deeply intertwined with Teichm\"uller theory (the universal cover of the moduli space of Riemann surfaces), see e.g. \cite{Verlinde:1989ua,Teschner:2002vx}.\footnote{A related observation is the following. The partition function of group $G$ Chern-Simons theory on an annulus times $\mathbb{R}$ is known to be describable through the diagonal modular invariant of the $\hat{G}$ (non-chiral) WZW model, where the chiral sectors of the WZW model are each associated to one of the boundary cylindrical walls \cite{Elitzur:1989nr}. Something similar was observed in \cite{Blommaert:2018iqz} for Liouville CFT: the Liouville diagonal torus partition function yields the two-boundary sector of 3d gravity, but glued within Teichm\"uller space.} Here we see that when Liouville is combined with the minimal model into a full gravitational theory the integral becomes the WP measure over the moduli space instead, in accordance with the matrix model expectation. This is clear in the JT gravity limit, and it would be interesting to understand the origin of this tanh measure for finite $p$ minimal string, and to confirm this is its correct normalization.
\subsection{Multiple boundaries}
It will be useful to rephrase the minimal string as a matrix integral in the double scaling limit using the formalism of \cite{Brezin:1990rb, Banks:1989df}. A central object from this approach is the heat capacity $u(x)$ appearing in the string equation. This is related to the density of states as
\begin{equation}\label{eqdeff}
\rho_0(E) = \frac{1}{2\pi} \int_{E_0}^E \frac{du}{\sqrt{E-u}} f(u),
\end{equation}
where $\partial_x u = - f(u)^{-1}$ (see \cite{Johnson:2019eik, *Johnson:2020heh,Okuyama:2019xbv,*Okuyama:2020ncd} for recent discussions). It will be convenient for us to define shifted and rescaled quantities that will have a finite large $p$ limit, as:
\begin{equation}
\label{ujt}
E = \kappa \Big(1 + \frac{8\pi^2}{p^2} E_{\rm JT} \Big), \qquad u = \kappa\Big(1 + \frac{8\pi^2}{p^2} u_{\rm JT} \Big).
\end{equation}
For ease of notation, we set $\kappa=1$ in the following. With these conventions the undeformed minimal string will correspond to $x\to0$. The minimal string density of states according to the Liouville calculation is
\begin{eqnarray}
\rho_0(E_{\rm JT}) &=&\frac{1}{4\pi^2} \sinh \Big( \frac{p}{2} {\rm arccosh} \Big(1 + \frac{8\pi^2}{p^2} E_{\rm JT} \Big) \Big),\\
&=& \sum_{j=0}^\mathfrak{m} \frac{(2\pi)^{2j-3}}{(2j-1)!} \frac{4^{j-1}(\mathfrak{m}+j-2)!}{(2\mathfrak{m}-1)^{2j-2}(\mathfrak{m}-j)!} (\sqrt{E_{\rm JT}})^{2j-1} ~~~~{\rm with}~~p=2\mathfrak{m}-1,
\end{eqnarray}
where $\mathfrak{m}\in \mathbb{Z}$. For large $\mathfrak{m}$ we get the JT gravity density of states.
We can find the function $f(u)$ by solving \eqref{eqdeff} and get
\begin{equation}
f(u_{\rm JT}) = \frac{1}{2} {}_2F_1\left(\frac{1-p}{2},\frac{1+p}{2},1, - \frac{4\pi^2}{p^2}u_{\rm JT}\right),
\end{equation} where ${}_2F_1(a,b,c,x)$ is the hypergeometric function. Integrating this relation we can get an implicit formula for the minimal string heat capacity
\begin{equation}
\frac{u_{\rm JT}}{2} \hspace{0.0cm} {}_2F_1 \Big(\frac{1-p}{2},\frac{1+p}{2},2, - \frac{4\pi^2}{p^2}u_{\rm JT}\Big) = - x.
\end{equation}
This can be written in a more familiar form recognizing that for these values of parameters the hypergeometric function becomes a Legendre polynomial, e.g.:
\begin{equation}
f(u_{\rm JT}) = \frac{1}{2} P_{\mathfrak{m}-1}\Big(1+\frac{8\pi^2}{p^2}u_{\rm JT} \Big)
\end{equation}
The relation above becomes the string equation \cite{Brezin:1990rb} (to leading order in genus expansion) written in the usual form, given by
\begin{equation}
\sum_{j=0}^\mathfrak{m} t_j u_{\rm JT}^j = 0 ,~~~~~~t_j\equiv \frac{1}{2} \frac{\pi^{2j-2}}{ j! (j-1)!} \frac{4^{j-1}(\mathfrak{m}+j-2)!}{(\mathfrak{m}-j)! (2\mathfrak{m}-1)^{2j-2}}
\end{equation}
where $p=2\mathfrak{m}-1$, we introduced the couplings $t_j$ and defined $t_0=x$. As explained in \cite{Moore:1991ir}\cite{Belavin:2008kv} this is an analytic redefinition of coupling constants of the $\mathfrak{m}$'th multicritical point and for large $x$ behaves as $u \sim x^{1/\mathfrak{m}}$, as expected. Knowing the couplings $t_j$, it is possible to also compute higher genus corrections by replacing the power law in the equation above by the KdV hierarchy operators $u^j \to R_j[u]$ derived in \cite{Gelfand1975,*Gelfand2}.
Knowing the heat capacity for the minimal string $u(x)$ derived from the density of states a surprising formula can be written for the $n$th loop correlator first proposed by \cite{Ambjorn:1990ji, Moore:1991ir}. The relation can be written in different ways but we found a useful version to be
\begin{equation} \label{nbdyformula}
\Big\langle \prod_i Z(\ell_i)^{\scriptscriptstyle \text{M}} \Big\rangle =- \frac{\sqrt{\ell_1\ldots \ell_n}}{2 \pi^{n/2}} \Big( \frac{\partial}{\partial x} \Big)^{n-3} u'(x) e^{-u(x)(\ell_1+\ldots+\ell_n)} \Big|_{u\to 1}.
\end{equation}
From now one we will only work with marked boundaries and omit the $\scriptscriptstyle \text{M}$ suffix. \eqref{nbdyformula} is based on the discrete approach and its surprising such a simple answer exists from the continuum approach. Shifting to our variable $u_{\rm JT}$ as $u = 1 + \frac{8\pi^2}{p^2} u_{\rm JT}$, the final answer is evaluated at $u_{\rm JT}(x\to0)=0$.
To apply this formula we need the derivatives $\partial_x u_{\rm JT}$ but the relation $u_{\rm JT}(x)$ is given only implicitly. To find the necessary derivatives, we can apply the Lagrange inversion theorem to write
\begin{equation}\label{eq:inversederums}
\left.\partial_x^n u_{\rm JT}\right|_{u_{\rm JT} =0} = \lim_{u_{\rm JT}\to 0} \frac{d^{n-1}}{du_{\rm JT}^{n-1}} \left(- \frac{2}{{}_2F_1(\frac{1-p}{2},\frac{1+p}{2},2, - \frac{4\pi^2}{p^2}u_{\rm JT})} \right)^n.
\end{equation}
This can be used order by order to find all terms appearing in the loop correlators.
We will now use this to generate some $n$ loop correlators for fixed boundary length. The case $n=1$ is special and actually is used to fix $u(x)$. The case $n=2$ is also special and gives $\langle Z(\ell_1) Z(\ell_2) \rangle = \frac{1}{2\pi} \frac{\sqrt{\ell_1 \ell_2}}{\ell_1+\ell_2}$, which coincides with \eqref{eq:2loopcorr} after appropriate shifts and redefinitions mentioned above. The cases $n=3,4,5$ give\footnote{We have defined the length parameter $\ell^{\rm JT} = \kappa \frac{8\pi^2}{p^2} \ell = 2\pi^2b^4\kappa \ell$ as in \eqref{JTparam}, and have redefined the overall normalization by dropping a factor of $\big(\frac{8\pi^2}{p^2}\big)^{1-\frac{n}{2}} e^{-\kappa \sum_i \ell_i}$.}
\begin{align}
\Big\langle \prod_{i=1}^3 \frac{Z(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &=-\frac{1}{2\pi^{3/2}} \left.\frac{\partial u_{\rm JT} }{ \partial x}\right|_{u_{\rm JT}=0} =\frac{1}{2\pi^{3/2}} 2 , \\
\Big\langle \prod_{i=1}^4 \frac{Z(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &= \frac{1}{2\pi^{2}} \Big( 4 \Big(\sum_{i=1}^4\ell^{\rm JT}_{i}\Big) +4\pi^2\Big(1-\frac{1}{p^2} \Big) \Big), \\
\Big\langle \prod_{i=1}^5 \frac{Z(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &= \frac{1}{2\pi^{5/2}} \Big(8 \Big(\sum_{i=1}^5\ell^{\rm JT}_{i}\Big)^2 +24 \pi^2 (1-\frac{1}{p^{2}})\Big(\sum_{i=1}^5\ell^{\rm JT}_{i}\Big) + 4\pi^4 (5-\frac{2}{p^{2}} -\frac{3}{p^{4}}) \Big).
\end{align}
At this point it should be clear how to generalize it to arbitrary boundaries.
As a further check of these expressions we will take the JT gravity limit $p \to \infty$. First we will take the JT limit of the string equation. Using the following identity (Abramowitz and Stegun eq (9.1.71)):
\begin{equation}
\label{AS}
\lim_{\nu \to+\infty} P_\nu \left(\cos \frac{x}{\nu}\right) = J_0(x), \qquad \text{with }\cos \frac{x}{\mathfrak{m}-1} = 1 + \frac{8\pi^2}{p^2}u_{\rm JT},
\end{equation}
one shows that $f(u) \to \frac{1}{2} I_0(2\pi \sqrt{u})$. For large $p$ the couplings become $t_j \to \frac{1}{2} \frac{\pi^{2j-2}}{j!(j-1)!}+\mathcal{O}(1/p)$. The sum can be done explicitly and the JT gravity string equation becomes
\begin{equation}
\sum_{j=1}^\infty \frac{1}{2} \frac{\pi^{2j-2}}{ j! (j-1)!} u_{\rm JT}^j = \frac{\sqrt{ u_{\rm JT} }}{2\pi} I_1(2\pi \sqrt{ u_{\rm JT}}) = -x.
\end{equation}
The $n$-boundary JT gravity partition function to leading order in the genus expansion is then
\begin{equation} \label{nbdyformulaJT}
\Big\langle \prod_i Z_{\rm JT}(\ell^{\rm JT}_{i}) \Big\rangle =- \frac{\sqrt{\ell^{\rm JT}_{1}\ldots \ell^{\rm JT}_{n}}}{2\pi^{n/2}} \Big( \frac{\partial}{\partial x} \Big)^{n-3} u'_{\rm JT}(x) e^{-u_{\rm JT}(x)(\ell^{\rm JT}_{1}+\ldots+\ell^{\rm JT}_{n})} \Big|_{u_{\rm JT}\to 0}.
\end{equation}
This can be seen as a generating function for the genus $0$ WP volumes with $n$ geodesic boundaries $V_{g=0,n}(\mathbf{b})$ with length $\mathbf{b}=(b_1,\ldots, b_n)$, after an appropriate Laplace transform we will do explicitly in the next section.
We can check this formula computing some simple cases with $n=3,4,5,\ldots$. This can be obtained either from the $p\to\infty$ limit of the minimal string or directly using the JT string equation $u_{\rm JT}(x)$. The result is
\begin{eqnarray}
\Big\langle \prod_{i=1}^3 \frac{Z_{\rm JT}(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &=&\frac{1}{2\pi^{3/2}} 2 ,\\
\Big\langle \prod_{i=1}^4 \frac{Z_{\rm JT}(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &=& \frac{1}{2\pi^{2}} \Big( 4 \Big(\sum_{i=1}^4\ell^{\rm JT}_{i}\Big) +4\pi^2 \Big), \\
\Big\langle \prod_{i=1}^5 \frac{Z_{\rm JT}(\ell^{\rm JT}_{i})}{\sqrt{\ell^{\rm JT}_{i}} } \Big\rangle &=& \frac{1}{2\pi^{5/2}} \Big(8 \Big(\sum_{i=1}^5\ell^{\rm JT}_{i}\Big)^2 +24 \pi^2\Big(\sum_{i=1}^5\ell^{\rm JT}_{i}\Big) + 20\pi^4 \Big).
\end{eqnarray}
It is surprising that these correlators match with the direct JT gravity calculation \cite{Saad:2019lba} where
\begin{equation}
\Big\langle \prod_{i=1}^n Z_{\rm JT}(\ell^{\rm JT}_{i}) \Big\rangle = \int \prod_{i=1}^n b_i db_i Z_{\rm trumpet}(\ell^{\rm JT}_{i}, b_i) V_{0,n}(\mathbf{b}),
\end{equation}
where the trumpet partition function is given by $Z_{\rm trumpet}(\ell^{\rm JT}_{i}, b_i) =e^{-b_i^2/4\ell^{\rm JT}_{i}}/2\sqrt{\pi \ell^{\rm JT}_{i}} $. For the WP volumes we used the expressions in \cite{do2011moduli} and we also check this works for $n=6$ and $7$, although we did not write it here. Therefore we see that \eqref{nbdyformulaJT} gives a simple generating function for (a simple integral transform) of WP volumes.
\subsection{$p$-deformed Weil-Petersson volumes}
In this section we will point out some interesting structure in the minimal string multi-loop correlator. One can write the amplitude in two ways:
\begin{align}
\label{Wppdef}
\left\langle \prod_{i=1}^n Z(\ell_i)\right\rangle &= -\left. \frac{\sqrt{\ell_1 \hdots \ell_n}}{2\pi^{n/2}}\left(\frac{\partial}{\partial x}\right)^{n-3} u ' (x) e^{-u(x) (\ell_1 + \hdots \ell_n)}\right|_{u=1} \\
\label{defwp}
&= \frac{2^{n-1}}{\pi^{n/2}} \prod_i \int_{0}^{+\infty} d\lambda_i \lambda_i \sinh \pi \lambda_i K_{i\lambda_i}(\ell_i) \frac{V_{0,n}(\bm{\lambda})}{\cosh \pi \lambda_i}
\end{align}
where in the second line we have written the integral in terms of multiple gluing cycles $\lambda_i$, with the gluing measure $d\mu(\lambda) = d\lambda \lambda \tanh \pi \lambda$ which we have written suggestively. The quantity $V_{0,n}(\bm{\lambda}) \equiv V_{0,n}(\lambda_1,...\lambda_n)$ will turn out to be a polynomial in $\lambda_i^2$ and can be viewed as a generalization of the WP volumes to the $p$-deformed setup. The numerical prefactors were chosen such that the $p\to+\infty$ limit directly yields back the WP volumes. \\
We can find explicit expressions for the $V_{0,n}(\bm{\lambda})$ by applying the Kontorovich-Lebedev (KL) transform:
\begin{align}
\label{KL}
g(y) = \int_0^{+\infty} \frac{dx}{x} f(x) K_{iy}(x), \qquad f(x) = \frac{2}{\pi^2} \int_{0}^{+\infty} dy g(y) K_{iy}(x) y \sinh \pi y
\end{align}
leading to
\begin{align}
\label{wpfi}
V_{0,n}(\bm{\lambda}) = \left.\left(\frac{\pi}{2}\right)^{n/2} \left(\frac{\partial}{\partial x}\right)^{n-3} u ' \prod_i \mathcal{L}_u\left(\frac{K_{i\lambda_i}(\ell_i)}{\sqrt{\ell_i}}\right) \cosh \pi \lambda_i\right|_{u=1}
\end{align}
where we need the following Laplace transform:\footnote{The integral is convergent at $x=0$ since $\left|K_{i\lambda}\right|$ is bounded close to zero.}
\begin{align}
\mathcal{L}_u\left(\frac{K_{i\lambda_i}(\ell_i)}{\sqrt{\ell_i}}\right) = \int_0^{+\infty} \frac{dx}{\sqrt{x}}K_{i\lambda}(x) e^{-u x} &= \frac{(2\pi)^{3/2}}{4\cosh \pi \lambda} {}_2F_{1}\left(\frac{1}{4} + \frac{i \lambda}{2}, \frac{1}{4} - \frac{i \lambda}{2}, 1, 1-u^2\right),\\
\label{legp}
&= \frac{(2\pi)^{3/2}}{4\cosh \pi \lambda} P_{-\frac{1}{2}-i\lambda}(u)
\end{align}
where $ u\geq 1$. Notice that the KL transform is invertible, and hence the WP volumes are unambigously defined by the above relation \eqref{Wppdef}. It is convenient as before to work with the shifted and rescaled variable $u_{\rm JT}$ defined by the relation $u=1+\frac{8\pi^2}{p^2}u_{\rm JT}$. We hence write:
\begin{equation}
\label{WPp}
\boxed{
V_{0,n}(\bm{\lambda}) =\lim_{u_{\rm JT}\to 0}- \frac{1}{2} \Big( \frac{\partial}{\partial x} \Big)^{n-3} u'_{\rm JT}(x) \prod_{i=1}^n P_{-\frac{1}{2}-i\lambda}\left(1+\frac{8\pi^2}{p^2}u_{\rm JT}(x)\right)},
\end{equation}
where an explicit formula for $\partial_x^n u_{ \rm JT}(x=0) $ was given above in equation \eqref{eq:inversederums}.
Zograf proved a theorem about a generating function for WP volumes in the sphere with n punctures $V_{0,n}(\mathbf{0})$ \cite{zograf1998weilpetersson}. This formula \eqref{WPp} gives a minimal string version of it and extends it to finite size boundary lengths. \\
To find explicit formulas from \eqref{WPp}, we can have to differentiate and evaluate at $u_{\rm JT} = 0$ in the end. To that effect, we can use the result:
\begin{align}
\partial_{u_{\rm JT}}^m \left. P_{-\frac{1}{2}-i\lambda}(u)\right|_{u_{\rm JT} = 0} &= (-)^m \left(\frac{8\pi^2}{p^2}\right)^m\frac{4\cosh \pi \lambda}{(2\pi)^{3/2}}\int_0^{+\infty} dx x^{m-1/2}K_{i\lambda}(x) e^{-x} \\
\label{toins}
&= \left(\frac{8\pi^2}{p^2}\right)^m (-)^m \frac{1}{2^m m!}\prod_{j=1}^{m}(\lambda^2+(2j-1)^2/4)
\end{align}
The equality in the last line is the KL transform of equations written in Appendix C of \cite{Moore:1991ag}. Importantly, this produces a polynomial in $\lambda_i^2$, mirroring the analogous situation for the WP volumes. \\
Finally, in order to make contact with the Weil-Petersson volumes at $p\to + \infty$, we define
\begin{equation}
\label{geolim}
\lambda_i = \frac{p}{4\pi}b_i
\end{equation}
in terms of the geodesic length $b_i$ that stays finite as we take the limit.
As explicit examples, for $n=4,5,6$ we obtain by inserting \eqref{eq:inversederums} and \eqref{toins} into \eqref{WPp} :
\begin{align}
V_{0,4}(\bm{\lambda}) &= \Big( 2\pi^2 + \frac{6 \pi^2}{p^2}\Big)+\frac{1}{2} \sum_i b_i^2 \\
V_{0,5}(\bm{\lambda}) &=\Big( 10\pi^4 + \frac{56\pi^4}{p^2} + \frac{104 \pi^4}{p^4} \Big) +\left(3\pi^2+ \frac{10\pi^2}{p^2}\right) \sum_i b_i^2 + \frac{1}{2}\sum_{i<j}b_i^2b_j^2 + \frac{1}{8} \sum_i b_i^4\\
V_{0,6}(\bm{\lambda}) &= \Big(\frac{244}{3}\pi^6 + \frac{1972\pi^6}{3p^2} + \frac{6604\pi^6}{3p^4} + \frac{3060 \pi^6}{p^6}\Big) + \left(26\pi^4+ \frac{160\pi^4}{p^2} + \frac{916\pi^4}{3p^4}\right) \sum_i b_i^2 \\
&\hspace{-1cm} + \left(6\pi^2+ \frac{21\pi^2}{p^2}\right) \sum_{i<j} b_i^2 b_j^2 + \left(\frac{3\pi^2}{2}+ \frac{31\pi^2}{6p^2}\right) \sum_i b_i^4 + \frac{3}{4}\sum_{i<j<k}b_i^2b_j^2b_k^2 + \frac{3}{16}\sum_{i,j, i \neq j}b_i^4b_j^2 + \frac{1}{48}\sum_i b_i^6 \nonumber
\end{align}
All of these satisfy the correct $p\to+\infty$ WP limit, as can be seen by comparing to Appendix B of \cite{do2011moduli}, see also \cite{Mirzakhani:2006fta, *Mirzakhani:2006eta}.
\begin{center}
\textbf{Adding handles}
\end{center}
We will show some more evidence of the structure identified here. We will derive the simplest correction for a single boundary and higher genus $g=1$, and then discuss some properties of the generic higher genus result. This is very hard to do from the continuous approach but we can assume the duality is true and obtain the leading handle correction to the partition function using the matrix model.
To find higher genus amplitudes, we can use Eynard's topological recursion relations as follows \cite{Eynard:2004mh,Eynard:2007kz}. Provided the two quantities:
\begin{equation}
W_{0,1}(z) = 2 zy(z), \qquad W_{0,2}(z_1,z_2) = \frac{1}{(z_1-z_2)^2},
\end{equation}
the generic amplitude for a double-scaled matrix integral can be found recursively by computing the residue
\begin{align}
\label{eynard}
&W_{g,n}(z_1,J) = \\
&\text{Res}_{z\to 0}\left\{\frac{1}{(z_1^2-z^2)} \frac{1}{4y(z)}\left[W_{g-1,n-1}(z,-z,J) + \sum_{h,I,h',I'} W_{h,1+I}(z,I)W_{h',1+I'}(-z,I')\right]\right\}, \nonumber
\end{align}
where $h+h'=g$ and $I \cup I' =J$ denoting a subset of the labels $z_2, \hdots z_n$, and the sum excludes the cases $(h=g,I=J)$ and $(h'=g,I'=J)$.
Using the minimal string spectral curve as seed, and applying it to genus one with one boundary, we get the following correction to the partition function
\begin{equation}\label{eq:Z11}
Z(\ell_{\rm JT})_{g=1,n=1} = \frac{\sqrt{\ell_{\rm JT}}}{12\sqrt{\pi}} (\ell_{\rm JT} + \pi^2(1-p^{-4})),
\end{equation}
which we wrote in term of the normalized length $\kappa\ell=\ell_{\rm JT} \frac{p^2}{ 8\pi^2}$. Using the Kontorovich-Lebedev transform this correction can be written
\begin{equation}
Z(\ell)_{g=1} \sim \int \lambda d\lambda \tanh \pi \lambda K_{i \lambda} (\ell) V_{1,1}(\lambda),
\end{equation}
where we will not worry about the overall normalization. The $p$-deformed WP volume appearing from \eqref{eq:Z11} is given by
\begin{equation}
\label{pgen1}
V_{1,1}(\lambda)= \Big(\frac{\pi^2}{12} + \frac{\pi^2}{12 p^2} - \frac{\pi^2}{12 p^4}\Big)+ \frac{\pi^2}{3p^2}\lambda^2 .
\end{equation}
It is easy to see that after calling $\lambda = \frac{p}{4\pi} b_{\rm JT}$, the large $p$ limit of this expression reproduces the WP volume for torus with one geodesic boundary of length $b_{\rm JT}$, namely $V_{1,1}(\lambda)\approx (b_{\rm JT}^2 + 4\pi^2)/48$.
\\~\\
This $p$-deformed WP volume \eqref{pgen1} is again a polynomial in $\lambda_i^2$ as before. Using the recursion relation \eqref{eynard}, we can give an argument why this is so for arbitrary genus $g$ and boundaries $n$.
The resolvents $W_{g,n}(z_1, \hdots z_n)$ for a one-cut matrix model with edges at $z=a,b$ are symmetric rational functions of the $z_i$ with poles only at $z_i = a,b$, see e.g. section 4.2.3 in \cite{Eynard:2015aea}.\footnote{Except of course $W_{0,2}$.} In addition, they decay to zero as $z_i \to \infty$.
For a double-scaled matrix integral, for which we shift the edge to $z_i=0$, these properties fix the $W_{g,n}(z_1, \hdots z_n)$ to be multivariate polynomials of $1/z_i$.
If the spectral curve $y(z)$ is in addition an odd function of $z$, then the $W_{g,n}(z_1, \hdots z_n)$ are polynomials with only even powers of $1/z_i$, making it a polynomial in $1/z_i^2$.\footnote{The reason for this constraint is that $W_{0,2}$ is not an even function of the $z_i$, but for $y(z)$ odd, when computing the residue in \eqref{eynard}, the Taylor series of $W_{0,2}(z,z_1)$ around $z=0$ needs to select an even power of $z$ (and hence of $z_1$) in order to contribute to the residue.}
For the minimal string case at hand, the spectral curve is odd and hence this is true.
The resolvent $W_{g,n}(z_1, \hdots z_n)$ is related to the multi-loop amplitude $Z_{g,n}(\ell_1 \hdots \ell_n)$ through
\begin{equation}
W_{g,n}(z_1, \hdots z_n) = 2^n z_1\hdots z_n \int_{0}^{+\infty} \prod_i d\ell_i e^{- \ell_i z_i^2} Z_{g,n}(\ell_1 \hdots \ell_n),
\end{equation}
which is in turn related to the WP volume $V_{g,n}(\bm{\lambda})$ by \eqref{defwp}. Each such $1/z_i^{2(m+1)}$ term in $W_{g,n}(z_1, \hdots z_n)$, where $m=0,1,\hdots$, then gets inverse Laplace transformed and Kontorovich-Lebedev transformed to the WP volumes using consecutively:\footnote{The $e^{-\ell}$ factor is explained by our choice to shift the spectral edge to $z=0$.}
\begin{align}
\frac{2 (m+1)!}{z^{2(m+1)}} &= 2z \int_{0}^{+\infty} d\ell e^{- \ell z^2} \ell^{m+1/2}, \qquad m=0,1,\hdots, \\
\ell^{m+1/2} e^{-\ell} &= \sqrt{\frac{2}{\pi}} \frac{1}{2^m m!}\int_{0}^{+\infty} d\lambda \lambda \tanh \pi \lambda \prod_{j=1}^{m} (\lambda^2 + (2j-1)^2/4) K_{iE}(\ell).
\end{align}
Hence if $W_{g,n}$ is a multivariate polynomial in the $1/z_i^{2}$, as happens for the minimal string, then the $p$-deformed WP volumes are polynomials in the $\lambda_i^2$:
\begin{equation}
W_{g,n}(z_1,\hdots z_n) = \sum_{i_1\hdots i_n} \frac{c_{i_1\hdots i_n}}{z_1^{2i_1}z_2^{2i_2} \hdots z_n^{2i_n}} \qquad \to \qquad V_{g,n} = \sum_{i_1\hdots i_n \hdots =0}^{n+3g-3}\tilde{c}_{i_1\hdots i_n} \lambda_1^{2i_1}\lambda_2^{2i_2} \hdots \lambda_n^{2i_n},
\end{equation}
as was to be shown.
\begin{center}
\textbf{Classical WP volumes}
\end{center}
As a final application of these results we will write an explicit formula for WP volumes in the sphere. One can take the JT limit directly at the level of the generating functions. Considering the description in terms of a Legendre function \eqref{legp}, inserting \eqref{geolim} and \eqref{ujt}, and using \eqref{AS}, we get:
\begin{equation}
P_{-\frac{1}{2}-\frac{ip}{4\pi}b_i}\Big(1+\frac{8\pi^2}{p^2}u_{\rm JT}\Big) \,\, \to \,\, J_0(b_i\sqrt{u_{\rm JT}}),
\end{equation}
leading to the closed formula for the (undeformed) WP volumes:
\begin{equation}
\boxed{
V_{0,n}(\mathbf{b}) =\lim_{x\to0} -\frac{1}{2} \Big( \frac{\partial}{\partial x} \Big)^{n-3} u'_{\rm JT}(x) \prod_{i=1}^n J_0(b_i \sqrt{u_{\rm JT}(x)})},
\end{equation}
where the derivatives of $u_{\rm JT}(x)$ are equal to
\begin{equation}
\partial_x^{n} u_{\rm JT}(x=0) = \lim_{u\to0} \frac{d^{n-1}}{du^{n-1}} \Big( -\frac{2\pi \sqrt{u}}{I_1(2\pi\sqrt{u})}\Big)^{n}.
\end{equation}
For each value of $n$ it is easy to take the appropriate derivatives and obtain a formula for WP volumes with $n$ holes. We computed these explicitly for $n=1,\ldots,7$ matching previous results that use the loop equations presented, for example, in Appendix B of \cite{do2011moduli}. This is surprising since even though we derived this formula from the matrix model we did not use the loop equations explicitly.
As a special case, we can take the WP volume on the sphere with $n$ punctures which is equivalent to taking the limit $\mathbf{b}\to0$. It is easy to see that this gives $V_{0,n}(\mathbf{0}) =-\frac{1}{2}\partial_x^{n-2} u_{\rm JT}(0)$. Using the expression above for these derivatives using the Lagrange inversion theorem gives a somewhat more explicit formula
\begin{equation}
V_{0,n}(\mathbf{0}) =\lim_{u\to0} \frac{1}{2} \frac{d^{n-3}}{du^{n-3}} \left( \frac{2\pi \sqrt{u}}{J_1(2\pi\sqrt{u})}\right)^{n-2},
\end{equation}
where we used that the minus signs can be absorbed in a shift of Bessel functions $I_1\to J_1$. This result is equivalent to the WP volume extracted from the generating function derived by Zograf \cite{zograf1998weilpetersson}, which is precisely the string equation of JT gravity.
\begin{center}
\textbf{Summary}
\end{center}
With these polynomials, we can now explicitly decompose the $n$-loop amplitude as:
\begin{align}
\left\langle \prod_{i=1}^n Z(\ell_i)\right\rangle_g = 2^n (2\pi)^{n-3}(\pi b^2)^{n} \prod_{i=1}^{n} \int_0^\infty \lambda_i d\lambda_i \tanh \pi \lambda_i \,V_{g,n}(\bm{\lambda}) \left\langle \mathcal{T}_{\alpha_{Mi}}\right\rangle_{\ell_i},
\end{align}
in terms of the $p$-deformed gluing measure $d\mu(\lambda) \sim d\lambda_i \lambda_i \tanh \pi \lambda_i$, the $p$-deformed WP-volume polynomial $V_{g,n}(\bm{\lambda})$, and the bulk one-point functions \eqref{bulkone} with $\lambda = 2P/b$. Graphically, we have the situation:
\begin{align}
\left\langle \prod_{i=1}^n Z(\ell_i)\right\rangle \quad = \quad \prod_{i=1}^{n}\int d\mu(\lambda_i) \qquad \raisebox{-25mm}{\includegraphics[width=0.3\textwidth]{multiboundaryglue.pdf}}
\end{align}
Notice that one only integrates over the macroscopic labels where $\alpha_M = -q/2 + iP$ with $P \in \mathbb{R}$, in analogy with the JT limit. For finite $p$, one can deform the contour of integration and replace the integral by a discrete sum over minimal string physical operators \cite{Moore:1991ir}.
We studied this mainly for $g=0$, where we found explicit expressions \eqref{WPp}, but proposed a very similar structure for higher genus contributions, which we checked explicitly by computing $V_{1,1}(\lambda)$ and utilizing general arguments based on the topological recursion relations of the matrix model.
\section{Conclusions} \label{sec:conclusions}
Throughout this work, we have presented fixed length amplitudes of Liouville gravities, and in particular of the minimal string. We have developed both the continuum approach and the discrete matrix model approach. A particular emphasis was placed on the interpretation in terms of Euclidean gravity amplitudes at fixed temperature $\beta^{-1}$, and in their JT parametric limit. \\
We here present some open problems and preliminary results that will be left to future work.
\begin{center}
\textbf{Heavy boundary operators and cusps}
\end{center}
We have seen that taking $\beta_M = bh$ in \eqref{eq:2pt} and letting $b\to 0$, one finds the JT boundary two-point function. However, the expression \eqref{eq:2pt} is more general. In particular, if we set $\beta_M = Q-bh$, we would find a finite $b\to 0$ limit as well:
\begin{equation}
\mathcal{A}_{\beta_M}(\ell_1,\ell_2) \sim \int dk_1 dk_2 \rho_{\rm JT}(k_1)\rho_{\rm JT}(k_2)e^{-k_1^2 \ell_{{\rm JT}1}}e^{-k_2^2 \ell_{{\rm JT}2}}\frac{\Gamma(2h)}{\Gamma(h \pm i k_1 \pm i k_2)},
\end{equation}
with \emph{inverted} vertex functions. This corresponds to taking a heavy boundary insertion. Since we know heavy bulk insertions correspond geometrically to conical singularities in the Euclidean JT geometry, it is natural to suspect that the situation here corresponds geometrically to having cusps in the boundary at the location of the operators. Such expressions are ill-defined when $h \in -\mathbb{N}/2$.
\begin{center}
\textbf{Quantum groups}
\end{center}
In section \ref{s:qg} we have developed the quantum group perspective on these amplitudes, mirroring the structure of JT gravity based on SL$(2,\mathbb{R})$. An interesting question is to understand precisely how this structure persists for four- and higher-point functions. This is dependent on understanding how the moduli summation for multiple ($>3$) boundary insertions works when combining the Liouville and the matter sectors. \\
The group theoretic structure $\mathcal{U}_{q}(\mathfrak{sl}(2,\mathbb{R}))$ is present in 3d gravity as well \cite{Jackson:2014nla}.\footnote{Another connection with 3d (and higher dimensional) gravity was developed for example in \cite{Ghosh:2019rcj, Iliesiu:2020qvm}, but only works in the Schwarzian limit.} In that case however, one has angular dependence on all correlators, requiring a more complicated combination of these group theoretic building blocks. Our setup is based on the same (quantum) group structure, but does not require additional features. As such, it is one of the simplest quantum extensions of the SL$(2,\mathbb{R})$ case. \\
Another setting that generalizes JT gravity through $q$-deformation is the double-scaled SYK model, explicitly solved in \cite{Berkooz:2018jqr}. In that case the vertex functions were found to be of the form:
\begin{equation}
\frac{\Gamma_b(h\pm is_1 \pm is_2)}{\Gamma_b(2h)},
\end{equation}
which is not quite the same as the structure we have. This can be explained since that work argues that double-scaled SYK is governed by the $q$-deformation into SU$_q(1,1)$, which is a different quantum group theoretical structure than ours. In the classical regime $q\to 1$, both groups coincide since we have the classical isomorphism SL$(2,\mathbb{R}) \simeq$ SU$(1,1)$.
\begin{center}
\textbf{Multi-boundary and higher genus amplitudes}
\end{center}
In the last section \ref{sec:othertopo}, we have investigated the structure of multi-loop amplitudes, both in the continuum approach and through matrix model techniques. This leads to several unanswered questions. \\
We found the gluing measure for the minimal string for genus zero multi-loop amplitudes to be $d\mu(\lambda) = \lambda\, d\lambda\, \tanh\pi \lambda$, limiting to the Weil-Petersson measure $d\mu_{\rm WP}(b)=b\, db$ in the semi-classical limit where $\lambda \to \infty$. The quantity $b$ has a geometric interpretation as circumference of the gluing tube, and the factor of $b$ in $b\, db$ represents the sum over all possible twists, ranging from 0 to $b$, happening before gluing two tubes together. It would be interesting to find a similar geometric interpretation for the measure $d\lambda \lambda \tanh\pi \lambda$, perhaps as a gluing formula on quantum Riemann surfaces. \\
In the same vein, we can observe that for generic $c_M<1$ matter, the two-loop amplitude for fixed matter momentum $p$, can be written suggestively as \cite{Moore:1991ag,Martinec:2003ka}
\begin{equation}
\left\langle Z(\ell,p)Z(\ell',-p)\right\rangle \sim \int_{0}^{+\infty}dE \rho_{\widetilde{SL(2)}}\left( \frac{E}{2}, \frac{p}{2} \right) K_{iE}(\ell) K_{iE}(\ell'),
\end{equation}
with gluing measure the Plancherel measure of the universal cover of SL$(2,\mathbb{R})$:
\begin{equation}
\rho_{\widetilde{SL(2)}}(s,\mu) = \frac{s\sinh 2\pi s}{\cosh 2\pi s - \cos 2\pi \mu}, \quad 0\leq \mu \leq 1.
\end{equation}
For the $(2,2\mathfrak{m}-1)$ minimal string, the matter momentum takes on values $p=\pm 1/2$, and hence $\cos 2\pi \mu=0$. We do not understand the significance of this. \\
When summing over higher genus, it remains to be seen whether a simplification occurs. For the case of $c=1$ ($b=1$) several expressions for the all-genus result are known in a very concise form, see e.g. \cite{Moore:1991sf} for early work and \cite{Betzios:2020nry} for a recent account. \\
Finally, the expression \eqref{Wppdef} has some interesting implications. A different way to write it is the following
\begin{equation}
\Big\langle \prod_{i=1}^n Z(\ell_i)\Big\rangle =\lim_{x\to 0}\sqrt{\frac{\ell_1\ldots \ell_n}{\ell_1+\ldots+\ell_n}} \Big(\frac{\partial}{\partial x}\Big)^{n-1} \langle Z(x;\ell_1+ \ldots+ \ell_n)\rangle.
\end{equation}
Each derivative can be interpreted as an insertion for each boundary of the KdV operator associated to the parameter $x$ (corresponding to $t_0$ in the usual nomenclature). The undeformed ($x=0$) version of $Z(\ell_1+ \ldots+ \ell_n)$ is, in the JT limit, the answer one would obtain from a multi-loop amplitude in BF theory associated to (the universal cover of) $SL(2,\mathbb{R})$, as derived in \cite{Verlinde:2020upt}. It would be interesting to understand the BF nature of this KdV operator, since it allows to go from the moduli space of flat connections to the WP one, up to the simple length dependent prefactor in the equation above.
This formula also predicts a very simple behavior for the higher order spectral form factor correlator $ \langle |Z(\beta+ i T)|^{2n} \rangle_{\rm conn} \sim (\beta^2+T^2)^{n/2}$, which (to leading order in genus expansion) is valid for all times.
A possible application of the multi-loop amplitudes computed here is to study the structure of the baby universe Hilbert space introduced in \cite{Coleman:1988cy,*Giddings:1988cx,*Giddings:1988wv} (and recently further developed in \cite{Saad:2019pqd} and \cite{Marolf:2020xie}), which we leave for future work. These euclidean wormholes were recently found to be relevant towards understanding unitarity of black hole evaporation \cite{Saad:2019pqd, Almheiri:2019qdq, *Penington:2019kki, Marolf:2020xie}.\footnote{Although their Lorenzian interpretation is not clear \cite{Giddings:2020yes}.} Also, adding brane boundaries can be interpreted as fixing eigenvalues of the random matrix integral \cite{Maldacena:2004sn}, which allows one to simulate an underlying discrete system \cite{Blommaert:2019wfy,*Blommaert:2020seb}.
\begin{center}
\textbf{Supersymmetric versions}
\end{center}
Our construction of fixed length amplitudes can be generalized to the $\mathcal{N}=1$ minimal superstring, composed of $\mathcal{N}=1$ super-Liouville with a superminimal model, mimicking most of the steps in this work. The comparison to JT gravity can be made since both the disk partition functions, the bulk one-point function and the boundary two-point functions are all known \cite{Stanford:2017thb,Mertens:2017mtv,Stanford:2019vob}. The resulting structure of the amplitudes is quite analogous and is presented in \cite{Mertens:2020pfe}.
\begin{center}
\textbf{Dilaton gravity interpretation}
\end{center}
It would be of high interest to get a better understanding of the bulk gravitational interpretation of the Liouville gravities, with the holographic interpretations made in this work. We point out a connection of Liouville gravity to dilaton gravity in Appendix \ref{app:connliouville}, derived in \cite{StanfordSeiberg}, where we combine the Liouville $\phi$ and matter field $\chi$ into the conformal factor of the metric $\rho$ and the dilaton field $\Phi$. In particular, the dilaton potential is $V(\Phi) \sim \sinh 2b^2 \Phi$.
Assuming such a connection to dilaton gravity exists, we can substantiate the precise form of the potential purely from bulk gravity considerations as follows. It is known that for a generic model with dilaton potential $V(\Phi)$
\begin{equation}
\label{act}
S = - \frac{1}{2} \int d^2 \sqrt{g}(\Phi R + V(\Phi)),
\end{equation}
every classical solution to this system can be written in the form \cite{Gegenberg:1994pv,Witten:2020ert}:
\begin{equation}
\label{bhgen}
ds^2 = A(r) dt^2 + \frac{dr^2}{A(r)}, \qquad \Phi(r) = r,
\end{equation}
where the asymptotic region $r\to +\infty$, has a linearly diverging dilaton field, like in JT gravity. The classical solution is determined by the equations of motion in terms of the potential $V$ as:
\begin{equation}
A(r) = \int_{r_h}^{r}dr' V(r'),
\end{equation}
where $r_h$ is the horizon location. Moreover, the energy-temperature relation of the black hole is determined by
\begin{equation}
E = \frac{1}{2} \int^{V^{-1}(4\pi T)} V(\Phi) d\Phi,
\end{equation}
in terms of the dilaton potential $V(\Phi)$, where $V^{-1}$ denotes the inverse function. Given an $E(T)$ relation, one can solve this functional equation to find the dilaton potential $V(\Phi)$.\footnote{In fact, there is an explicit solution for the inverse function $\Phi(V)$. First computing the canonical entropy $S(T)$ as a function of temperature $T$, one finds:
\begin{equation}
\Phi = \frac{1}{2\pi} S\left(\frac{V}{4\pi}\right),
\end{equation}
which is uniquely invertible into $V(\Phi)$ given an assumption of monotonicity of $S(T)$ as a function of $T$.
}
Taking
\begin{equation}
V(\Phi) = 4\pi b^2 \kappa \, \sinh 2 \pi b^2\Phi,
\end{equation}
we indeed find
\begin{equation}
E = \sqrt{ T^2/b^4+ \kappa^2} ,
\end{equation}
reproducing the first law \eqref{firstlaw} we found for the fixed-length disk partition function, but now coming from a (thermodynamically stable) bulk black hole solution. This provides substantial evidence to our claim that the bulk gravity is a 2d dilaton gravity model with a $\sinh$ dilaton potential.\footnote{The precise coefficients in the sinh potential can be changed by rescalings and are not important for our purposes here.}
The (real-time) classical black hole solution \eqref{bhgen} is then:
\begin{equation}
\label{geom}
ds^2 = - 2\kappa \left[\cosh 2\pi b^2 r - \cosh 2 \pi b^2 r_h \right] dt^2 + \frac{dr^2}{2\kappa \left[\cosh 2\pi b^2 r - \cosh 2\pi b^2 r_h \right]}, \qquad \Phi(r) = r,
\end{equation}
where the horizon radius $r_h$ is related to the temperature $T$ as
\begin{equation}
r_h = \Phi_h = \frac{1}{2\pi b^2} \text{arcsinh} \frac{T}{\kappa b^2}.
\end{equation}
The thermal entropy of the system can be found as the Bekenstein-Hawking entropy, or directly by using the first law, and we get:
\begin{equation}
S = 2 \pi \Phi_h +S_0 = \frac{1}{b^2} \text{arcsinh} \frac{T}{\kappa b^2} +S_0.
\end{equation}
One checks that the Ricci scalar of this solution is indeed
\begin{equation}
R = - 8 \pi^2 b^4 \kappa\, \cosh 2\pi b^2 r = - V'(\Phi),
\end{equation}
as required by the $\Phi$ equation of motion of \eqref{act}. The geometry \eqref{geom} interpolates between the JT black hole for $r,r_h \ll 1/b^2$ with constant negative Ricci scalar, and an exponentially rising Ricci-scalar closer to the boundary.
This black hole solution has been written before in \cite{Kyono:2017jtc,*Kyono:2017pxs,*Okumura:2018xbh} in the context of a Yang-Baxter deformation of JT gravity.\footnote{The thermodynamical relations are not the same as there due to a coordinate transformation in the time coordinate.} It would be interesting to understand this connection and the dilaton gravity description better, which we postpone to future work.
To further probe the bulk gravitational dynamics, we can mention the following.
Heavy operator insertions serve as interesting probes of backreaction effects, which are expected to have a gravitational interpretation in terms of classical energy injections. For JT gravity, this setup was analyzed in \cite{Lam:2018pvp,Goel:2018ubv}.
In \cite{Blommaert:2019hjr,*Mertens:2019bvy,*Blommaert:2020yeo} JT bulk observables and their correlators were introduced, exploiting a radar definition to anchor bulk points to the holographic boundary. This relied strongly on the specifics of JT gravity as a theory of boundary frames (the Schwarzian description). While the bulk here would not be so easily treated, it would be very interesting to understand whether a similar construction in the bulk would be viable, and in particular whether bulk physics behaves similarly. Since the IR of the Liouville gravities studied here matches that of JT gravity, we do not expect strong deviations from conclusions made there.
Finally, it would be interesting to apply these methods to understanding closed universes. This can be done by considering fixed length boundaries with imaginary length \cite{Maldacena:2019cbz, Cotler:2019nbi}. In particular, the CFT perspective on Liouville gravity might help finding the correct inner product between no-boundary states.
\paragraph{Acknowledgements}
We thank V. Gorbenko, K. Jensen, S. Okumura, S. Shenker, D. Stanford, M. Usatyuk, H. Verlinde and W. Weng for useful discussions. We also thank A. Blommaert for initial collaboration. TM gratefully acknowledges financial support from Research Foundation Flanders (FWO Vlaanderen). GJT is supported by a Fundamental Physics Fellowship.
| 2024-02-18T23:39:56.748Z | 2021-02-08T02:17:45.000Z | algebraic_stack_train_0000 | 904 | 30,281 |
|
proofpile-arXiv_065-4440 | \section{Introduction}
\label{intro}
The last decade has witnessed the success of convolutional neural networks (CNNs) in various artificial intelligence applications, such as computer vision and natural language processing.
In CNNs, convolutional layers perform efficient linear transformation from their input data through a linear or circular convolutional operation. Inspecting such a linear transformation lends itself to theoretical understanding of the behaviors of convolutional layers in CNNs with respect to, e.g., stability, generalization performance, and gradient explosion or vanishing effects.
Such a linear transformation of convolutional layers plays the same role as the weight matrix of the fully-connected layers. As a powerful means to matrix analysis, spectral methods have been applied to understand the properties of the weight matrices of deep neural networks through inspecting the behavior of their singular values, such as \citet{bartlett2017spectrally,sedghi2018the,Singla2019,miyato2018spectral,yoshida2017spectral,farnia2018generalizable,neyshabur2017pac,roth2019adversarial,Long2020Generalization}, to name just a few.
The singular values play a key role in spectral analysis of deep neural networks, where spectral norm is the largest singular value and Frobenius norm involves all singular values of the weight matrices.
It has been shown in \citet{neyshabur2017pac} that generalization error is upper bounded by spectral and Frobenius norms of the weight matrices of the layers, for which suppressing singular values can reduce the gap and therefore enhance the generalization performance.
In addition, for CNNs, spectral regularization has been also applied to convolutional layers so as to guide the training process by e.g., clipping singular values within an interval to avoid explosion or vanishing of gradients \citet{sedghi2018the}, and bounding spectral norms to enhance generalization performance and robustness against adversarial examples \citet{yoshida2017spectral,Singla2019,miyato2018spectral}.
However, as the size of such linear transformation matrices grows with the input size of the layers, it is computationally challenging to find singular values. The straightforward singular value decomposition (SVD) incurs huge computational burden, which is even worse when singular values are required to be computed during the training process to guide spectral regularization and normalization \citet{miyato2018spectral,yoshida2017spectral}.
Fortunately, the structures of the linear transformation matrices can be exploited to reduce the computational complexity of SVDs. Of particular relevance is the work by \citet{sedghi2018the,bibi2018deep}, in which the linear convolutional layer is treated as a circular one by a ``wrapping round'' operation. In doing so, the linear transformation matrices are endowed with a circulant structure, by which efficient methods were proposed to compute a circular approximation of the convolutional layers with substantially reduced complexity. To further reduce computational complexity, upper bounds of spectral norm of the circular convolutional layers were derived in \citet{Singla2019} at the expense of degraded accuracy.
As a matter of fact, such a ``wrapping round'' operation is not always endowed in many convolutional layers, for which a linear, rather than circular, convolutional operation is applied. With such a linear convolution, the linear transformation matrix has a Toeplitz structure, which includes the circulant one as a special case. This has been pointed out by a number of previous works, e.g., \citet{goodfellow2016deep,wang2019orthogonal,appuswamy2016structured}, that the two-dimensional single-channel convolutional layer results in a doubly block Toeplitz matrix. A question then arises as to how close is the circular approximation to the exact linear Toeplitz case.\footnote{Although some theoretical analysis bounded the gap between large Toeplitz and circulant matrices \citet{ZhuTIT2017}, it seems only applied to Hermitian matrices (or symmetric for real matrices). The linear transformation matrices of linear convolution are {\em asymmetric} real matrices, which are {\em non-Hermitian} matrices.} This motivates the current work.
\paragraph{Our Contributions}
In this paper, we consider the linear convolutional layers, with main focus on the multi-channel two-dimensional linear convolution with stride size of 1, so that the linear transformation matrix is a block matrix with each block being a doubly Toeplitz matrix. By rows and columns permutation, we construct an alternative representation as a doubly block Toeplitz matrix with each element being a matrix, for which the singular values of both representations are identical.
As such, we propose a spectral representation of the linear transformation matrix
by a spectral density matrix,
by which the spectral analysis of the former can be alternatively done on the latter. Specifically, the main contributions are three-fold:
\begin{itemize}
\vspace{-2pt}
\item The singular value distribution of linear transformation matrix of CNNs is cast to that of its spectral density matrix, thanks to an extension of the celebrated Szeg\"o Theorem for Hermitian Toeplitz matrices to non-Hermitian block doubly Toeplitz matrices. In doing so, the asymptotic spectral analysis of the linear convolutional layers can be alternatively done by inspecting the corresponding spectral density matrix. The circular convolution by ``wrapping around'' is a special case of such a spectral representation, by which the singular values can also be produced by uniformly sampling the spectral density matrix.
%
\item By treating singular values of the spectral density matrix as random variables, the individual singular value distribution can be quantified by a quantile function. As such, we propose a simple yet effective algorithm to compute singular values of linear convolutional layers by subtly adjusting the singular value distribution obtained from the circular approximation.
\item To upper-bound the spectral norm of the linear transformation matrix, we instead upper-bound that of its corresponding spectral density matrix. As a consequence, we come up with three spectral norm bounds that can be used for spectral regularization.
\vspace{-2pt}
\end{itemize}
Experimental results demonstrate the superior accuracy of our singular value approximation method and the effectiveness of spectral norm bounds for regularization with respect to generalization in practical CNN models, e.g., ResNets. Notations and preliminaries can be found in Section \ref{sec:prelim}.
\section{Convolutional Neural Networks}
\subsection{Linear Convolutional Layer}
We consider multiple-channel two-dimensional {\em linear} convolutional layers with {\em arbitrary padding} schemes in CNNs before applying activation functions and pooling. For ease of presentation, we first consider the stride size 1, and the extension to larger stride size will be discussed in Section \ref{sec:discussion}.
Let the input be $\mat{x} \in \mathbb{R}^{c_{in}\times n \times n}$ and the linear convolutional filter be $\mat{k} \in \mathbb{R}^{c_{out} \times c_{in} \times h \times w}$ with $h,w \le n$, where $n,h,w,c_{in},c_{out}$ are input size, filter height, filter width, the numbers of input and output channels, respectively. For convenience, we let the output $\mat{y}$ have the same size as the input $\mat{x}$ by arbitrary padding strategies,
and abuse $\mat{x}$ as the input with padding.
By applying linear convolution of the filter $\mat{k}$ to the input $\mat{x}$, the output $\mat{y} \in \mathbb{R}^{c_{out}\times n \times n}$ can be given by
\begin{align}
\mat{y}_{c,r,s}=\sum_{d=1}^{c_{in}} \sum_{p=1}^n \sum_{q=1}^n \mat{x}_{d,r+p,s+q} \mat{k}_{c,d,p,q}
\end{align}
for $r,s \in [n]$ and $c \in [c_{out}]$ where $\mat{k}_{c,d,p,q}=0$ if $p,q$ exceed the ranges of $h,w$.
A compact form of the above input-output relation can be rewritten as
\begin{align}
\vec(\mat{y}) = \mat{a} \vec(\mat{x}),
\end{align}
where $\mat{a} \in \mathbb{R}^{c_{out}n^2 \times c_{in} n^2}$ is the linear transformation matrix of the convolutional layer.
For the general case with multiple-input and multiple-output channels, the linear transformation can be represented as a $c_{out}\times c_{in}$ block matrix, i.e.,
\begin{align} \label{eq:M-1}
\mat{a} = \begin{bmatrix}
\mat{a}_{1,1} & \mat{a}_{1,2} & \dots & \mat{a}_{1,c_{in}}\\
\mat{a}_{2,1} & \mat{a}_{2,2} & \dots & \mat{a}_{2,c_{in}}\\
\vdots & \vdots & & \vdots\\
\mat{a}_{c_{out},1} & \mat{a}_{c_{out},2} & \dots & \mat{a}_{c_{out},c_{in}}
\end{bmatrix},
\end{align}
where each block is a doubly Toeplitz matrix, i.e., $[\mat{a}_{c,d}]_{i_1,j_1}=\mat{a}_{i_1-j_1}^{c,d}$ with $[\mat{a}_{k}^{c,d}]_{i_2,j_2}=a_{k,i_2-j_2}^{c,d}$ (See a concrete representation in Section \ref{sec:prelim-toe}). In matrix analysis, $\mat{a}$ is usually referred to as multi-block multi-level (doubly) Toeplitz matrix.
For $k\in [-h_1:h_2]$ and $l \in [-w_1:w_2]$,
we have
\begin{align} \label{eq:M4}
a_{k,l}^{c,d} = \mat{k}_{c,d,h_1+k+1,w_1+l+1},
\end{align}
for all $c \in [c_{out}]$ and $d \in [c_{in}]$.
\subsection{Alternative Representation}
\label{sec:section2.2}
For ease of spectral analysis, we transform $\mat{a}$ into a multi-level block Toeplitz matrix (whose entries of the last level are matrices) via vec-permutation operation \citet{henderson1981vec}, for which the matrix spectrum keeps unchanged.
Denote by $\mat{t} \in \mbox{\bb R}^{c_{out}n^2 \times c_{in} n^2}$ the alternative representation as a block Toeplitz matrix with
$[\mat{t}]_{i_1,j_1}=\mat{t}_{i_1-j_1}$ where $[\mat{t}_k]_{i_2,j_2}=\mat{t}_{k,i_2-j_2}$ (See a concrete representation in Section \ref{sec:prelim-toe}).
For $k \in [-h_1:h_2]$ and $l\in [-w_1:w_2]$, each block $\mat{t}_{k,l} \in \mbox{\bb R}^{c_{out} \times c_{in}}$ is given by %
\begin{align} \label{eq:toe-blocks}
\mat{t}_{k,l}=\begin{bmatrix}
t_{1,1}^{k,l} & t_{1,2}^{k,l} & \cdots & t_{1,c_{in}}^{k,l}\\
t_{2,1}^{k,l} & t_{2,2}^{k,l} & \cdots & t_{2,c_{in}}^{k,l}\\
\vdots & \ddots & \ddots & \vdots \\
t_{c_{out},1}^{k,l} & t_{c_{out},2}^{k,l} & \cdots & t_{c_{out},c_{in}}^{k,l}
\end{bmatrix}.
\end{align}
By such an alternative representation, we have
\begin{align} \label{eq:T_k_l}
t_{c,d}^{k,l}=\mat{k}_{c,d,h_1+k+1,w_1+l+1} = a_{k,l}^{c,d},
\end{align}
for all $c \in [c_{out}]$ and $d \in [c_{in}]$.
In what follows, we show that the alternative representation $\mat{t}$ of the linear convolutional layers has the identical spectrum structure as the original form $\mat{a}$.
\begin{lemma} \label{lemma:toe-permutation-invariant}
$\{\sigma_j(\mat{t}), \; \forall j\}=\{\sigma_j(\mat{a}), \; \forall j\}$.
\end{lemma}
Lemma \ref{lemma:toe-permutation-invariant} says the block matrix with doubly Toeplitz matrix blocks (i.e., $\mat{a}$) has the same set of singular values as the block doubly Toeplitz matrix (i.e., $\mat{t}$). This holds for any Toeplitz matrices which are not necessarily banded, and for any multi-level case but not limited to doubly (i.e., 2-level) Toeplitz case.
Equipped with this lemma, we hereafter treat $\mat{t}$ as the linear transformation matrix of linear convolutional layers for spectral analysis.
\subsection{Circular Approximation}
The ``wrapping around'' operation makes linear transformation a circular convolution, which is deemed as a circular approximation of linear convolution.
As $h,w \le n$, we can construct a circulant matrix by ``wrapping around'' to assist the spectral analysis.
Given the doubly block Toeplitz matrix $\mat{t}=[\mat{t}_{i-j}]_{i,j=1}^n$ with $\mat{t}_k=0$ if $k>h_2$ or $k<-h_1$ and $\mat{t}_k=[\mat{t}_{k,p-q}]_{p,q=1}^n$ with $\mat{t}_{k,l}=0$ if $l>w_2$ or $l<-w_1$, the doubly block circulant matrix $\mat{c}=\circ(\mat{c}_0,\mat{c}_1,\dots,\mat{c}_{n-1})$ is as follows
\begin{align} \label{eq:Cm_k}
\mat{c}_k=\left\{
\Pmatrix{\mat{t}_{-k}, & k\in \{0\}\cup[h_1]\\
\mat{t}_{n-k}, & k \in n-[h_2]\\
0,& \text{otherwise}}
\right.
\end{align}
where $\mat{c}_k=\circ(\mat{c}_{k,0},\mat{c}_{k,1},\dots,\mat{c}_{k,n-1})$ with
\begin{align} \label{eq:C_k_l}
\mat{c}_{k,l}=\left\{
\Pmatrix{\mat{t}_{-k,-l}, & k\in \{0\}\cup[h_1], \; l\in\{0\}\cup[w_1]\\
\mat{t}_{-k,n-l}, & k\in \{0\}\cup[h_1], \; l \in n-[w_2]\\
\mat{t}_{n-k,-l}, &k \in n-[h_2], \; l\in\{0\}\cup[w_1]\\
\mat{t}_{n-k,n-l}, & k \in n-[h_2], \; l \in n-[w_2]\\
0,& \text{otherwise}}
\right.
\end{align}
where $\mat{t}_{k,l}$ is defined in \eqref{eq:toe-blocks}.
In a similar way, the original block doubly Toeplitz matrix $\mat{a}$ can also have a corresponding block doubly circulant matrix $\mat{c}(\mat{a})=[\mat{c}(\mat{a}_{c,d})]_{c,d=1}^{c_{out},c_{in}}$ where
\begin{align} \label{eq:CM-1}
\mat{c}(\mat{a}_{c,d})=\circ(&\mat{c}(\mat{a}_0^{c,d}),\mat{c}(\mat{a}_{-1}^{c,d}), \dots, \mat{c}(\mat{a}_{-h_1}^{c,d}), 0,
\dots,0, \mat{c}(\mat{a}_{h_2}^{c,d}),\dots,\mat{c}(\mat{a}_1^{c,d}))
\end{align}
with $\mat{c}(\mat{a}_{c,d}) \in \mbox{\bb R}^{n^2 \times n^2}$ where
\begin{align} \label{eq:CM-2}
\mat{c}(\mat{a}_k^{c,d})=\circ(a_{k,0}^{c,d}, &a_{k,-1}^{c,d}, \dots, a_{k,-w_1}^{c,d}, 0,
\dots,0, a_{k,w_2}^{c,d},\dots,a_{k,1}^{c,d})
\end{align}
with $\mat{c}(\mat{a}_k^{c,d}) \in \mbox{\bb R}^{n \times n}$.
Similarly to Lemma \ref{lemma:toe-permutation-invariant}, we have the following lemma.
\begin{lemma} \label{lemma:cir-permutation-invariant}
$\{\sigma_j(\mat{c}), \; \forall j\}=\{\sigma_j(\mat{c}(\mat{a})), \; \forall j\}$.
\end{lemma}
It can be easily verified that $\mat{c}(\mat{a})$ is essentially the linear transformation matrix of circular convolutional layers considered in \citet{sedghi2018the}.
As a byproduct of Lemma \ref{lemma:cir-permutation-invariant}, we present an alternative calculation of the singular values for the circular convolutional layers that were characterized in \citet{sedghi2018the}.
\begin{lemma} \label{lemma:toe-equal-cir}
The linear transformation matrix $\mat{c}(\mat{a})$ can be block-diagonalized as
\begin{align} \label{eq:block-diag}
\mat{c} = (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}}) &\mathrm{blkdiag} (\mat{b}_{1,1},\mat{b}_{1,2},\dots, \mat{b}_{1,n}, \mat{b}_{2,1}, \dots \mat{b}_{n,n}) (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})^{\H}
\end{align}
where both $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}})$ and $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})$ are unitary matrices.
Thus, the singular values of $\mat{c}(\mat{a})$ are the collection of singular values of
$
\{\mat{b}_{i,k}\}_{i,k=1}^n
$
where
\begin{align}
\mat{b}_{i,k}= \sum_{p=0}^{n-1} \sum_{q=0}^{n-1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(i-1)+q(k-1)}{n}}
\end{align}
with $\mat{c}_{p,q}$ defined in \eqref{eq:C_k_l}.
\end{lemma}
The computation of $\mat{b}_{i,k}$ can be seen as a two-dim DFT of $\mat{c}_{p,q}$. With $hw$ non-zero submatrices $\{\mat{c}_{p,q}\}$, the computational complexity consists in $hw$ FFTs and $n^2$ SVDs, which is identical to that in \citet{sedghi2018the}. We also point out that this alternative approach essentially has the same flavor as that in \citet{bibi2018deep}.
Given Lemmas 1-3, we hereafter take $\mat{t}$ as the linear transformation matrix of the linear convolutional layer and $\mat{c}$ as its circular approximation, for asymptotic spectral analysis.
\section{Asymptotic Spectral Analysis}
\label{sec:spectral}
In what follows, we present asymptotic spectral analysis for the linear transformation matrix $\mat{t}$ of convolutional layers in CNNs, taking advantage of its Toeplitz structure \citet{Gray1972,avram1988bilinear,parter1986distribution,voois1996theorem,Tilli1998,miranda2000asymptotic,Tyrtyshnikov1996,Zizler2002,Bogoya2015}. The proofs and insights are relegated to Section \ref{sec:proofs}. %
\subsection{Spectral Representation}
\begin{theorem} \label{thm:szg-limit}
Given the linear transformation matrix $\mat{t} \in \mbox{\bb C}^{rn^2\times sn^2}$, let a complex matrix-valued Lebesgue-measurable function $F:[-\pi,\pi]^2 \mapsto \mathbb{C}^{r \times s}$ be the generating function such that
\begin{align*}
\mat{t}_{k,l} = \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} F(\omega_1,\omega_2) e^{-\jmath (k \omega_1 + l \omega_2)} d\omega_1 d\omega_2.
\end{align*}
It follows that, for any continuous function $\Phi$ with compact support in $\mathbb{R}$, we have
\begin{align}
\MoveEqLeft \lim_{n \to \infty} \frac{1}{n^2} \sum_{j=1}^{\min\{r,s\}n^2} \Phi(\sigma_j(\mat{t}))
= \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \sum_{j=1}^{\min\{r,s\}} \Phi(\sigma_j(F(\omega_1,\omega_2))) d\omega_1 d\omega_2, \notag
\end{align}
for which $\mat{t}$ is said to be equally distributed as $F(\omega_1,\omega_2)$ with respect to singular values, i.e., $\mat{t} \sim_{\sigma} F$. Specifically, for linear convolutional layers, the linear transformation matrix $\mat{t}$ has doubly banded structures, so that the generating function can be explicitly written as
\begin{align} \label{eq:theorem-1-F}
F(\omega_1,\omega_2)=\sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \mat{t}_{k,l} e^{\jmath (k \omega_1 + l \omega_2)},
\end{align}
which is also referred to as the spectral density matrix of $\mat{t}$.
\end{theorem}
Theorem 1 endows the linear transformation matrix $\mat{t}$ of linear convolutional layers with an asymptotic spectral representation - the spectral density matrix $F(\omega_1,\omega_2)$ - by establishing the collective equivalence of their asymptotic singular value distributions. As such, the spectral analysis of linear convolutional layers of CNNs can be alternatively done on its spectral representation $F(\omega_1,\omega_2)$.
The singular values of $\mat{t}$ can be clustered into $\min\{r,s\}$ non-overlapping subsets. When $n$ is sufficiently large, the singular values in the $j$-th subset concentrate on $\sigma_j(F)$, where $\sigma_j(F)$ is the $j$-th singular value function of $F(\omega_1,\omega_2)$.
As such, the singular values of $\mat{t}$ can be approximately obtained by sampling $\sigma_j(F)$ over a uniform gird in $ [-\pi,\pi]^2$,
for all $j \in [\min\{r,s\}]$. It turns out that such approximation is equivalent to the circular approximation, which will be detailed in Theorem \ref{thm:circulant}.
\begin{theorem} \label{thm:circulant}
Given $\mat{t}$ and $\mat{c}$ as in \eqref{eq:Cm_k}-\eqref{eq:C_k_l},
there exists a constant $c_1>0$ such that
\begin{align}
\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{\min\{r,s\}n^2} \abs{\sigma_j(\mat{t})-\sigma_j(\mat{c})} \le c_1,
\end{align}
where the singular values of $\mat{c}$ are the collection of singular values of $\{\sigma_j(F(\omega_1,\omega_2))\}_j$ with
\begin{align}
\label{eq:uniform-samples}
(\omega_1,\omega_2) = (-\pi+ \frac{2 \pi j_1}{n}, -\pi &+ \frac{2 \pi j_2}{n}), \quad \forall j_1,j_2 \in [n]-1.
\end{align}
\end{theorem}
Theorem 2 shows that the singular values of the circular approximation of the linear convolution can be alternatively obtained by uniformly sampling the spectral density matrix $F(\omega_1,\omega_2)$ over $(\omega_1,\omega_2) \in [-\pi,\pi]^2$, where the {\em average} difference of the overall singular values from the exact ones is bounded by $O(\frac{1}{n})$,\footnote{The big O notation $O(n)$ follows the standard Bachmann–Landau notation, meaning that there exists a positive constant $c>0$ such that the term is upper-bounded by $cn$.} and tends to zero as $n$ increases.
\begin{remark}
The block diagonal matrices $\mat{b}_{i,k}$ of $\mat{c}$ in \eqref{eq:block-diag} is essentially the matrix-valued function $F(\omega_1,\omega_2)$ with uniform sampling on grids as in \eqref{eq:uniform-samples}, i.e.,
\begin{align}
\mat{b}_{j_1,j_2} = F\Big(\frac{ 2 \pi (j_1-1)}{n},&\frac{ 2 \pi (j_2-1)}{n}\Big), \quad
\forall j_1,j_2 \in [n].
\end{align}
\end{remark}
Collecting all singular values $\{\sigma_j(F)\}_j$ according to the uniform sampling grids as in \eqref{eq:uniform-samples}, we sort them in non-decreasing order as $(\kappa_1,\kappa_2,\dots,\kappa_{N})$.
Let $\psi: [0,1] \mapsto \mbox{\bb R}$ be a piece-wise linear non-decreasing function that interpolates the samples $(\kappa_1,\kappa_2,\dots,\kappa_{N})$ over the nodes $(0,\frac{1}{N},\frac{2}{N},\dots,1)$ such that $\psi(\frac{i}{N})=\kappa_i$ for all $i \in \{0\} \cup [N]$ and $\psi(\cdot)$ is linear between any two consecutive nodes. Then we have
\begin{align}
\frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \sum_{j=1}^{\min\{r,s\}} &\Phi(\sigma_j(F(\omega_1,\omega_2))) d\omega_1 d\omega_2
= \int_{0}^1 \Phi(\psi(t)) dt.
\end{align}
It means the singular values of $\mat{t}$ can be approximately obtained by sampling the density function $\psi(t)$ in $[0,1]$.
This motivates a singular value approximation method in Theorem \ref{thm:quantile}.
\subsection{Singular Value Approximation}
From a probabilistic perspective, Theorem \ref{thm:szg-limit} implies that the statistical average of the singular values of $\mat{t}$ converges to that of the singular values of the corresponding spectral density matrix $F$ in distribution with any continuous functions $\Phi$. Inspired by this, we propose a method to approximate $\sigma_j(\mat{t})$ through the singular value distribution of $\sigma_j(F)$ with bounded approximation error.
\begin{theorem} \label{thm:quantile}
Let $\phi_j:[-\pi,\pi]^2 \mapsto \mbox{\bb R}_+$ be the $j$-th singular value function of $F(\hbox{\boldmath$\omega$})$ and $\sigma_k^{(j)}(\mat{t})$ be $k$-th singular value of $j$-th cluster. It follows that
\begin{align}
\sup_{u \in (\frac{k-1}{n^2},\frac{k}{n^2}]} \abs{\sigma_k^{(j)}(\mat{t})-Q_{\phi_j}(u)} \le \frac{c_2}{n}, \quad
\forall 1\le k \le n^2, \;& 1 \le j \le \min\{r,s\}
\end{align}
where $c_2>0$ is a constant that only depends on $F(\hbox{\boldmath$\omega$})$, and
\begin{align}
Q_{\phi_j}(u)&=\inf\{v \in \mbox{\bb R}: u \le G_{\phi_j}(v)\}\\
G_{\phi_j}(v)&=\frac{1}{(2\pi)^2}\mu\{\hbox{\boldmath$\omega$} \in [-\pi,\pi]^2: \phi_j(\hbox{\boldmath$\omega$})\le v\}
\end{align}
are quantile and cumulative distribution functions for $\phi_j(\hbox{\boldmath$\omega$})$, respectively, and $\mu$ is Lebesgue measure.
\end{theorem}
\begin{wrapfigure}{R}{0.58\textwidth}
\begin{minipage}{0.58\textwidth}
\vspace{-18pt}
\begin{algorithm}[H]
\caption{Singular Values via Quantile Interpolation}
\label{alg:quantile}
\begin{algorithmic}[1]
\State {\bfseries Input:} Convolutional filter $\mat{k} \in \mbox{\bb R}^{c_{out} \times c_{in} \times h \times w}$
\State Initialize $h_1,h_2,w_1,w_2$
\State Construct $\mat{t}_{k,l}$ from $\mat{k}$ according to \eqref{eq:toe-blocks}
\For{$j_1=1$ {\bfseries to} $n$}
\For{$j_2=1$ {\bfseries to} $n$}
\State Set $(\omega_1,\omega_2)=(-\pi+\frac{2\pi j_1}{n},-\pi+\frac{2\pi j_1}{n})$
\State Compute $F(\omega_1,\omega_2)$ by \eqref{eq:theorem-1-F}
\State Compute SVD of $F(\omega_1,\omega_2)$
\EndFor
\EndFor
\For{$j=1$ {\bfseries to} $\min\{r,s\}$}
\State Collect singular values $\{\sigma_j(F(\omega_1,\omega_2))\}_{\omega_1,\omega_2}$
\State Arrange $\sigma_j(F(\omega_1,\omega_2))$ in descending order
\State{Estimate quantile $\hat{Q}_{\phi_j}$ by $\{\sigma_j(F(\omega_1,\omega_2))\}_{\omega_1,\omega_2}$}
\State Interpolate quantile using e.g., kernel smoothing
\State Select proper $u=\{\frac{j-\gamma_j}{n^2}\}_{j=1}^{n^2}$ with $\gamma_j \in (0,1)$
\State Compute $\{\hat{Q}_{\phi_j}(u)\}_u$ as singular value estimates
\EndFor
\State {\bfseries Output:} Singular values $\{\{\hat{Q}_{\phi_j}(u)\}_u\}_j$
\end{algorithmic}
\end{algorithm}
\end{minipage}
\end{wrapfigure}
Theorem \ref{thm:quantile} reveals that the individual singular value of $\sigma_j(\mat{t})$ can be approximated by sampling the quantile function of $\phi_j(\hbox{\boldmath$\omega$})$ within each interval $ (\frac{k-1}{n^2},\frac{k}{n^2}]$. If the estimation of the quantile function is perfect, this approach approximates each {\em individual} singular value with gap to the exact one within $O(\frac{1}{n})$.
\begin{remark}
It is challenging to compute the closed-form expression of the singular value function\footnote{As $F(\omega_1,\omega_2)$ is a Laurent polynomial matrix with respect to $e^{\jmath \omega_1}$ and $e^{\jmath \omega_2}$, the singular value functions $\phi_j(\hbox{\boldmath$\omega$})$ can be computed efficiently by, e.g., \citet{foster2009algorithm}.} $\phi_j(\hbox{\boldmath$\omega$})$ from $F(\hbox{\boldmath$\omega$})$, so is its quantile function.
Alternatively, $Q_{\phi_j}(u)$ can be estimated from some easily attainable samples, e.g., $\{\sigma_j(\mat{c})\}_j$, which are the uniform sampling of $\sigma_j(F)$ on $[-\pi,\pi]^2$, followed by quantile interpolation/extrapolation with e.g., kernel smoothing tricks. As such, the singular value approximation can be done by properly sampling the interpolated quantile function.
In this way, the approximation accuracy of $\{\sigma_j(\mat{t})\}_j$ depends on (1) the accuracy of quantile estimation from the samples, (2) the smoothing factors of quantile interpolation, and (3) the sampling grid in $ (\frac{k-1}{n^2},\frac{k}{n^2}]$. Alg.~\ref{alg:quantile} presents a simple approach to approximate $\{\sigma_j(\mat{t})\}_j$ through quantile estimation and interpolation.
\end{remark}
For quantile interpolation, a simple way is linear interpolation, which uses linear polynomials to interpolate new values between two consecutive data points. Kernel density estimation can be used to smooth interpolation. Some other interpolation methods, such as t-Digests \citet{dunning2019computing}, are also available in Python and MATLAB from 2019b onward.
\subsection{Spectral Norm Bounding}
Thanks to the spectral representation, spectral analysis on the linear transformation matrix $\mat{t}$ can be alternatively done on the spectral density matrix $F(\hbox{\boldmath$\omega$})$ with $\hbox{\boldmath$\omega$} \in [-\pi,\pi]^2$. For instance, to upper-bound spectral norm of $\mat{t}$, we can do it on $F$ due to the following lemma.
\begin{lemma}
\label{lemma:bounding-Tm}
$
\norm{\mat{t}}_2 \le \norm{F}_2.
$
\end{lemma}
Built upon Lemma \ref{lemma:bounding-Tm}, the spectral norm of $\mat{t}$ can be further upper-bounded in different ways.
\begin{theorem}
\label{theorem:bounding-norm}
The spectral norm $\norm{F}_2$ can be bounded by
\begin{align}
\label{eq:upper-bound-F-2norm}
\norm{F}_2 &\le \min \Big\{\sqrt{hw}\norm{\mat{r}}_2, \sqrt{hw}\norm{\mat{l}}_2 \Big\},\\
\label{eq:upperibound-oneinfnorm}
\norm{F}_2 &\le \max_{\hbox{\boldmath$\omega$}} \sqrt{\norm{F(\hbox{\boldmath$\omega$})}_1 \norm{F(\hbox{\boldmath$\omega$})}_{\infty}},\\
\label{eq:upperibound-2norm}
\norm{F}_2 &\le \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \norm{\mat{t}_{k,l}}_2,
\end{align}
where $\mat{r} \in \mbox{\bb R}^{hc_{out} \times wc_{in}}$ is a $c_{out} \times c_{in}$ block matrix with $(c,d)$-th block being $\mat{k}_{c,d,:,:} \in \mbox{\bb R}^{h \times w}$ and $\mat{l} \in \mbox{\bb R}^{wc_{out} \times hc_{in}}$ is a $c_{out} \times c_{in}$ block matrix with $(c,d)$-th block being $\mat{k}_{c,d,:,:}^{\scriptscriptstyle\mathsf{T}} \in \mbox{\bb R}^{w \times h}$.
\end{theorem}
In Theorem \ref{theorem:bounding-norm}, the first upper bound \eqref{eq:upper-bound-F-2norm} is identical to that in \citet{Singla2019}, however the derivation here is different as we directly work on $F$, while the bounds in \citet{Singla2019} is for the circulant approximation. This reveals that, with respect to spectral norm upper bounds, it may be not necessary to distinguish circular from linear convolutional layers.
With respect to computational complexity, the first bound \eqref{eq:upper-bound-F-2norm} requires to compute two spectral norms with sizes $hc_{out} \times wc_{in}$ and $wc_{out} \times hc_{in}$ respectively. The complexity of the second bound \eqref{eq:upperibound-oneinfnorm} depends on the sampling complexity of $\hbox{\boldmath$\omega$}$, which usually takes as $n^2$. As such, it requires to compute $n^2$ times of $\ell_1$ and $\ell_{\infty}$ norms with size $c_{out} \times c_{in}$. The third bound \eqref{eq:upperibound-2norm} requires to compute $hw$ spectral norms with size $c_{out} \times c_{in}$.
\section{Experiments}
\label{sec:experiments-main}
\subsection{Singular Value Approximation}
To verify the singular value approximation in Section \ref{sec:spectral}, we conduct experiments with respect to four different methods on singular values calculation. The weights of filters are extracted from either the pre-trained networks, e.g., GoogLeNet \citet{szegedy2015going}, with ImageNet dataset or from the training process of ResNet-20 \citet{ResNet} on CIFAR-10 dataset. More experimental results using randomly generated weights and weights from pre-trained networks are given in Section \ref{sec:experiments-sva}.
\begin{itemize}
\vspace{-3pt}
\item Exact Method: A block doubly Toeplitz matrix $\mat{t}$ is generated from the convolutional filter $\mat{k}$ according to \eqref{eq:T_k_l}. The exact singular values of linear convolutional layers are computed by applying SVD to $\mat{t}$ directly. %
\item Circular Approximation: A block doubly circulant matrix $\mat{c}$ is constructed according to \eqref{eq:Cm_k}-\eqref{eq:C_k_l}. The singular values are computed by applying SVD on $\mat{c}$ directly.
\item Uniform Sampling: The block diagonal matrices $\mat{b}_{j_1,j_2}$ is produced by uniformly sampling the spectral density matrix $F(\omega_1,\omega_2)$ with sampling grids $(\omega_1,\omega_2)=(-\pi+\frac{2\pi j_1}{n},-\pi+\frac{2\pi j_1}{n})$ for all $j_1, j_2 \in [n]$. The singular values are obtained by collecting all singular values of $\{\mat{b}_{j_1,j_2}\}_{j_1,j_2=1}^n$. This corresponds to lines 1-10 in Algorithm \ref{alg:quantile}.
\item Quantile Interpolation: The singular values obtained from uniform sampling are arranged for each $1\le j \le \min\{c_{in},c_{out}\}$ in descending order. By quantile estimation using linear interpolation methods, the singular values are recomputed by selecting properly shifted sampling grids as outlined in Algorithm \ref{alg:quantile}.
\vspace{-3pt}
\end{itemize}
The experiments are conducted on MATLAB 2020a, which is more friendly to matrix computation. For simplicity, we set $h_1=h_2$ and $w_1=w_2$, and the input size per channel is set to $10 \times 10$. Fig.~\ref{fig:Fig-1} presents the $(i-1)n+1$-th largest singular values ($i \in [n]$) of four methods with four different filter sizes.
The first two filters are from the pre-trained GoogLeNet, and the last two are from the training process of ResNet-20.
It can be observed that (1) both circular approximation and uniform sampling have identical singular values for different filter sizes, (2) quantile interpolation improves accuracy of the singular values over the circular approximation with negligible extra running time (see Section \ref{sec:experiments-sva}), and (3) during the training process the improvement of the largest singular value approximation is dominant, while for the well-trained networks, the improvement is mainly due to that on smaller singular values. This might be attributed to implicit regularization during training.
\begin{figure}[t]
\vskip 0.1in
\hspace{-12pt}
{\includegraphics[width=0.27\columnwidth]{googlenet-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.27\columnwidth]{googlenet-inception-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.27\columnwidth]{resnet-20-conv1-10epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.27\columnwidth]{resnet-20-conv1-100epoch-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\vspace{-0.1in}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. Input size per channel is set to $10 \times 10$. For illustration, only 10 singular values are plotted. Four types of convolutional filters are considered from left to right with sizes $64 \times 3 \times 7 \times 7$ (pre-trained GoogLeNet conv1), $32 \times 16 \times 5 \times 5$ (pre-trained GoogLeNet inception), $16 \times 3 \times 3 \times 3$ (ResNet-20 conv1 after 10 training epochs), and $16 \times 3 \times 3 \times 3$ (ResNet-20 conv1 after 100 training epochs), respectively. }
\label{fig:Fig-1}
\end{center}
\vskip -0.35in
\end{figure}
\subsection{Spectral Norm Bounding}
To verify the accuracy and running time of different spectral norm bounds, we conduct experiments on the pre-trained ResNet-18 model with ImageNet dataset on MATLAB 2020a on HP EliteBook. For the accuracy, we use the circular approximation as the reference and present the ratios to it. Table~\ref{sample-table} summarizes the results for different filters, where the numbers ``$a/b$'' read as $a$ times of the circular approximation in accuracy and $b$ milliseconds (ms) in running time.
We observe that (1) the first bound \eqref{eq:upper-bound-F-2norm} usually has the best accuracy except for the larger filter size, e.g., $7 \times 7$, while the second bound \eqref{eq:upperibound-oneinfnorm} works better for large filter size; (2) the third bound \eqref{eq:upperibound-2norm} has comparable accuracy as the first one \eqref{eq:upper-bound-F-2norm}, yet accounting for less than 10\% running time of the latter.
\begin{table}[t]
\caption{Comparison of spectral norm bounds ($a/b$: accuracy ratio/running time).}
\label{sample-table}
\vskip -0.1in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccr}
\toprule
Filter size & \eqref{eq:upper-bound-F-2norm} & \eqref{eq:upperibound-oneinfnorm} & \eqref{eq:upperibound-2norm}\\
\midrule
$64 \times 3 \times 7 \times 7$ & 3.00/12.84 & 2.14/51.51 & 4.33/1.146 \\
$64 \times 64 \times 3 \times 3$ & 1.63/77.68 & 3.21/54.27 & 2.20/5.427 \\
$128 \times 64 \times 3 \times 3$ & 1.48/155.3 & 3.52/102.3 & 2.10/8.981 \\
$256 \times 256 \times 3 \times 3$ & 1.27/1285 & 4.66/671.7 & 1.56/68.74 \\
$512 \times 256 \times 3 \times 3$ & 1.10/2516 & 4.72/2010 & 1.27/124.6 \\
$512 \times 512 \times 3 \times 3$ & 1.13/7232 & 4.51/3215 & 1.26/288.5 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.25in
\end{table}
Following the same setting as \citet{Singla2019}, we conduct experiments for generalization using spectral norm bounds \eqref{eq:upper-bound-F-2norm} and \eqref{eq:upperibound-2norm} as regularizers. The sum of spectral norm bounds of all convolutional and fully-connected layers are used during training. The bound \eqref{eq:upper-bound-F-2norm} has been already evaluated in \citet{Singla2019}, so our focus will be placed on the evaluation of \eqref{eq:upperibound-2norm} by replacing the matrices of interest in the forward and backward propagation. We test the accuracy of CIFAR-10 dataset on ResNet-20 model with no weight decay and $\beta=0.0014$ as in \citet{Singla2019}. We observe an improvement of 0.8\% over the non-regularization case (i.e., $\beta=0$) using \eqref{eq:upperibound-2norm}, which is slightly worse (0.3\%) than that of \eqref{eq:upper-bound-F-2norm} after 150 training epochs. The learning rate is initialized as 0.1 and changed to 0.01 after 100 epochs. %
Although test accuracy does matter in generalization, we argue that the regularizer \eqref{eq:upperibound-2norm} would be more preferable as it substantially reduces the computational complexity (with more than 30\% running time saving) at the expense of slight performance degradation.
The detailed experimental setup and more results are given in Section \ref{sec:experiments-snr}.
\section{Conclusion}
In this paper, we proposed to use spectral density matrices to represent the linear convolutional layers in CNNs, for which the linear transformation matrices are block doubly Toeplitz matrices constructed from the convolutional filters.
By doing so, spectral analysis of linear convolutional layers can be alternatively done on the corresponding spectral density matrices. Such a spectral representation has been demonstrated to be useful in singular value approximation and spectral norm bounding.
In particular, spectral norm bounds derived from the spectral density matrix can be used as regularizers to enhance generalization performance with substantially reduced computational complexity.
This spectral representation is expected to offer a different approach to understand linear convolutional layers, through analyzing the spectral density matrices associated to linear transformation.
\newpage
\section*{\Large Appendix}
\setcounter{lemma}{0}
\setcounter{theorem}{0}
The Appendix is organized as follows. In Section \ref{sec:prelim}, we present the notations used in this paper and the definitions of Toeplitz and circulant matrices. The detailed proofs of the key lemmas and theorems in the main text are detailed in Section \ref{sec:proofs}, together with some comments. For convenience, we restate these lemmas and theorems here. In Section \ref{sec:discussion}, the extensions are discussed with respect to larger stride size, higher dimensional linear convolution, and multiple convolutional layers in linear networks. We also present in Section \ref{sec:experiments} the detailed experimental setups in the main text as well as more results to demonstrate the applicability and practical usefulness of our methods in practical CNN models, e.g., VGG, ResNets.
\section{Notations and Preliminaries}
\label{sec:prelim}
\subsection{Notations and Definitions}
For two integers $m$ and $n$ satisfying $m<n$, define $[m] \triangleq \{1,2,\dots,m\}$, $n-[m] \triangleq \{n-1,n-2,\dots,n-m\}$, and $[m:n] \triangleq \{m,m+1,\dots,n\}$. $x \in [a,b]$ is such that $a\le x \le b$. $\jmath$ is the imaginary unit.
Denote by $a$, $\vect{a}$, $\mat{a}$ scalars, vectors, and matrices/tensors, respectively.
$\mat{a}^{{\scriptscriptstyle\mathsf{T}}}$ and $\mat{a}^{\H}$ represent matrix transpose and Hermitian transpose of $\mat{a}$, respectively. A complex-valued matrix $\mat{a}$ is Hermitian if $\mat{a}=\mat{a}^{\H}$. If $\mat{a}$ is real-valued, $\mat{a}$ is Hermitian is equivalent to $\mat{a}$ is symmetric, i.e., $\mat{a}=\mat{a}^{{\scriptscriptstyle\mathsf{T}}}$.
We denote by $\mathrm{blkdiag}(\mat{a}, \mat{b}, \dots)$ a block diagonal matrix with diagonal blocks being $\mat{a}, \mat{b}, \dots$, and by $\circ(a,b,\dots)$ a circulant matrix with elements in the first row being $a,b,\dots$. Likewise, $\circ(\mat{a},\mat{b},\dots)$ is the block circulant matrix with first row blocks being $\mat{a}, \mat{b}, \dots$. An $n \times n$ matrix $\mat{f}_n$ is called Discrete Fourier Transform (DFT) matrix, where $[\mat{f}_n]_{ik}=\frac{1}{\sqrt{n}} e^{-\jmath 2 \pi (i-1)(k-1)/n}$ for $i,k \in [n]$. $\mat{{I}}_n$ is the $n \times n$ identity matrix. For a tensor $\mat{a}$, $\vec(\mat{a})$ denotes the vectorized version of $\mat{a}$, and for a 4-order tensor $\mat{a}$, $\mat{a}_{i,j,k,l}$ is used to index its elements.
Denote by $\otimes$ the Kronecker product between two matrices. For a scalar $k$, it holds $\mat{a} \otimes (k\mat{b})=k(\mat{a} \otimes \mat{b})$ and $\mat{a} \otimes (\sum_i \mat{b}_i)=\sum_i \mat{a} \otimes \mat{b}_i$. For two matrices $\mat{a}$ and $\mat{b}$, $\mat{a} \otimes \mat{b}$ is permutation equivalent to $\mat{b} \otimes \mat{a}$, i.e., there exist permutation matrices $\hbox{\boldmath$\Pi$}_1$ and $\hbox{\boldmath$\Pi$}_2$ such that $\mat{b} \otimes \mat{a}= \hbox{\boldmath$\Pi$}_1 (\mat{a} \otimes \mat{b}) \hbox{\boldmath$\Pi$}_2$.
A matrix-valued function $F:[a,b]^k \mapsto \mbox{\bb C}^{m \times n}$ is such that $F(\vect{x}) \in \mbox{\bb C}^{m \times n}$ for $\vect{x} \in [a,b]^k$. $F$ is Lebesgue measurable (resp. bounded, continuous) in $[a,b]^k$ if each of its element $F_{ij}$ is Lebesgue measurable (resp. bounded, continuous) in $[a,b]^k$. $F \in {\mathcal L}^2([-\pi,\pi]^2)$ means $\Norm{F} \triangleq \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \Norm{F} d\omega_1 d \omega_2 < +\infty$.
For a matrix $\mat{a}=(a_{ij})_{i,j=1}^{m,n}$ with $\rank(\mat{a})=r$, we denote by $\{\sigma_j(\mat{a})\}_{j}$ the collection of singular values of $\mat{a}$ arranged in descending order, i.e., $\sigma_1(\mat{a}) \ge \sigma_2(\mat{a})\ge \dots \ge \sigma_r(\mat{a})$.
The norm $\norm{\mat{a}}_2 \triangleq \sigma_1(\mat{a})$ is called spectral norm.
The Schatten $p$-norm is defined as $\norm{\mat{a}}_p \triangleq (\sum_{j=1}^r \sigma_j^p(\mat{a}))^{\frac{1}{p}}$. When $p=2$, it coincides with Frobenius norm $\normf{\mat{a}}\triangleq \sqrt{\sum_{i=1}^m \sum_{j=1}^n \Abs{a_{ij}}}= \sqrt{\sum_{j=1}^r \sigma_j^2(\mat{a})}$. %
The matrix $\ell_1$ and $\ell_{\infty}$ norms are defined as $\norm{\mat{a}}_{1}\triangleq \max_{j} \sum_{i=1}^m \abs{a_{ij}}$ and $\norm{\mat{a}}_{\infty}\triangleq \max_{i} \sum_{j=1}^n \abs{a_{ij}}$, respectively.
$\abs{a}$ is the absolute value or modulus of a scalar $a$.
\subsection{Toeplitz and Circulant Matrices}
\label{sec:prelim-toe}
A Toeplitz matrix $\mat{t} = [t_{i-j}]_{i,j=1}^n$ is an $n \times n$ matrix for which the entries come from a sequence $\{t_k, k=0, \pm1, \pm2, \dots,\pm(n-1)\}$.
A circulant matrix is a special Toeplitz matrix, where $\mat{c}=[t_{(i-j) \mod n}]_{i,j=1}^n$. That is, $t_{-k}=t_{n-k}$ for $k=1,2,\dots,n-1$. We denote the circulant matrix by $\mat{c}=\circ(t_0,t_{-1},\dots,t_{-(n-1)})$ using its first row, where the rest rows are cyclic shift of the first row with $n$ times.
An $m \times m$ block Toeplitz matrix $\mat{b}=[\mat{a}_{i-j}]_{i,j=1}^m \in \mbox{\bb C}^{mp \times mq}$ is a Toeplitz matrix with each element being a $p \times q$ matrix.
Similarly, the block circulant matrix $\mat{c}$ is such that $\mat{c}=[\mat{a}_{(i-j)\!\!\mod m}]_{i,j=1}^m$ with $0\!\!\mod m = m\!\!\mod m = 0$. That is, $\mat{a}_{-k}=\mat{a}_{m-k}$ for $k=1,2,\dots,m-1$, such that $\mat{c}=\circ(\mat{a}_0,\mat{a}_{-1},\dots,\mat{a}_{-(m-1)})$ and the rest row blocks are block-wise cyclic shift of the first row block.
When $\{\mat{a}_k, k=0,\pm1,\dots,\pm(m-1)\}$ are also $n \times n$ Toeplitz/circulant matrices, $\mat{b}$ is a block Toeplitz/circulant matrix with Toeplitz/circulant blocks, which is also known as doubly Toeplitz/circulant matrix.
A banded (block) Toeplitz matrix is a special Toeplitz matrix $\mat{t}$ [resp.~$\mat{b}$] such that $t_k=0$ [resp. $\mat{a}_k=\mat{\mathrm{0}}$] when $k>r$ or $k<-s$ for some $1<r,s<n$ [resp.~$1<r,s<m$].
For the general case with multiple-input and multiple-output channels, the linear transformation of convolutional layers in CNNs can be represented as a $c_{out}\times c_{in}$ block matrix, i.e.,
\begin{align} \label{eq:M-1}
\mat{a} = \begin{bmatrix}
\mat{a}_{1,1} & \mat{a}_{1,2} & \dots & \mat{a}_{1,c_{in}}\\
\mat{a}_{2,1} & \mat{a}_{2,2} & \dots & \mat{a}_{2,c_{in}}\\
\vdots & \vdots & & \vdots\\
\mat{a}_{c_{out},1} & \mat{a}_{c_{out},2} & \dots & \mat{a}_{c_{out},c_{in}}
\end{bmatrix}.
\end{align}
Each block $\mat{a}_{c,d}$ is a banded block Toeplitz matrix with
\begin{align} \label{eq:M-2}
\mat{a}_{c,d}=\begin{bmatrix}
\mat{a}_0^{c,d} & \cdots & \mat{a}_{-h_1}^{c,d} & 0 & \dots & 0\\
\vdots & \mat{a}_0^{c,d} & \ddots & \ddots & \ddots & \vdots\\
\mat{a}_{h_2}^{c,d} & \ddots & \ddots & \ddots & \ddots & 0\\
0 & \ddots & \ddots & \ddots & \ddots & \mat{a}_{-h_1}^{c,d}\\
\vdots & \ddots & \ddots & \ddots & \mat{a}_0^{c,d} & \vdots\\
0 & \cdots & 0 & \mat{a}_{h_2}^{c,d} & \cdots & \mat{a}_{0}^{c,d}
\end{bmatrix}
\end{align}
where $h_1,h_2$ depend on the size of padding in height subject to $h=h_1+h_2+1$. Each block $\mat{a}_{k}^{c,d}$ is still a banded Toeplitz matrix with
\begin{align} \label{eq:M-3}
\mat{a}_{k}^{c,d}=\begin{bmatrix}
a_{k,0}^{c,d} & \cdots & a_{k,-w_1}^{c,d} & 0 & \cdots & 0\\
\vdots & a_{k,0}^{c,d} & \ddots & \ddots & \ddots & \vdots \\
a_{k,w_2}^{c,d} & \ddots & \ddots & \ddots & \ddots & 0\\
0 & \ddots & \ddots & \ddots & \ddots & a_{k,-w_1}^{c,d}\\
\vdots & \ddots & \ddots & \ddots & a_{k,0}^{c,d} & \vdots\\
0 & \cdots & 0 & a_{k,w_2}^{c,d} & \cdots & a_{k,0}^{c,d}
\end{bmatrix}
\end{align}
where $w_1,w_2$ subject to $w_1+w_2+1=w$ that are determined by the size of padding in width. The elements in $\mat{a}_{k}^{c,d}$ are weights in the filter [cf. \eqref{eq:M4}].
As stated in Section \ref{sec:section2.2} in the main text, the linear transformation matrix $\mat{a}$ can be alternatively represented by doubly block Toeplitz matrix $\mat{t}$ without change of spectrum.
The alternative representation $\mat{t} \in \mbox{\bb R}^{c_{out}n^2 \times c_{in} n^2}$ is a doubly block Toeplitz matrix
\begin{align} \label{eq:alter-repre}
\mat{t}=\begin{bmatrix}
\mat{t}_0 & \cdots & \mat{t}_{-h_1} & 0 & \dots & 0\\
\vdots & \mat{t}_0 & \ddots & \ddots & \ddots & \vdots\\
\mat{t}_{h_2} & \ddots & \ddots & \ddots & \ddots & 0\\
0 & \ddots & \ddots & \ddots & \ddots & \mat{t}_{-h_1}\\
\vdots & \ddots & \ddots & \ddots & \mat{t}_0 & \vdots\\
0 & \cdots & 0 & \mat{t}_{h_2} & \cdots & \mat{t}_{0}
\end{bmatrix}
\end{align}
with each block $\mat{t}_k$ for all $k\in [-h_1:h_2]$ being still a block Toeplitz matrix
\begin{align}
\mat{t}_k=\begin{bmatrix}
\mat{t}_{k,0} & \cdots & \mat{t}_{k,-w_1} & 0 & \dots & 0\\
\vdots & \mat{t}_{k,0} & \ddots & \ddots & \ddots & \vdots\\
\mat{t}_{k,w_2} & \ddots & \ddots & \ddots & \ddots & 0\\
0 & \ddots & \ddots & \ddots & \ddots & \mat{t}_{k,-w_1}\\
\vdots & \ddots & \ddots & \ddots & \mat{t}_{k,0} & \vdots\\
0 & \cdots & 0 & \mat{t}_{k,w_2} & \cdots & \mat{t}_{k,0}
\end{bmatrix}
\end{align}
where each block $\mat{t}_{k,l} \in \mbox{\bb R}^{c_{out} \times c_{in}}$ with $l\in [-w_1:w_2]$ is
\begin{align}
\mat{t}_{k,l}=\begin{bmatrix}
t_{1,1}^{k,l} & t_{1,2}^{k,l} & \cdots & t_{1,c_{in}}^{k,l}\\
t_{2,1}^{k,l} & t_{2,2}^{k,l} & \cdots & t_{2,c_{in}}^{k,l}\\
\vdots & \ddots & \ddots & \vdots \\
t_{c_{out},1}^{k,l} & t_{c_{out},2}^{k,l} & \cdots & t_{c_{out},c_{in}}^{k,l}
\end{bmatrix}.
\end{align}
Each element of $\mat{t}_{k,l}$ comes from the weight of the filter $\mat{k}$ [cf. \eqref{eq:T_k_l}].
\section{Proofs of Main Theorems}
\label{sec:proofs}
\subsection{Proof of Lemmas}
\begin{lemma}
$\{\sigma_j(\mat{t}), \ \forall j\}=\{\sigma_j(\mat{a}), \ \forall j\}$.
\end{lemma}
\begin{proof}
While the following proof is dedicated to the banded Toeplitz matrices, it can be straightforwardly extended to any Toeplitz matrix without loss of generality.
Let $\vect{e}_i$ be $i$-th column of identity matrix and $\mat{e}_{i,j}=\vect{e}_i \vect{e}_j^{{\scriptscriptstyle\mathsf{T}}}$ be a $c_{out} \times c_{in}$ matrix with only the $(i,j)$-th element being 1 and 0 elsewhere. Define $\mat{p}_k$ as an $n \times n$ matrix with $[\mat{p}_k]_{i,j}=1$ if $i-j=k$ and 0 otherwise. Thus, the original linear transformation matrix $\mat{a}$ can be represented as
\begin{align}
\mat{a} &= \sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} \mat{e}_{c,d} \otimes \mat{a}_{c,d} \\
&= \sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} \mat{e}_{c,d} \otimes (\sum_{k=-h_1}^{h_2} \mat{p}_k \otimes \mat{a}_k^{c,d}) \\
&= \sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} \mat{e}_{c,d} \otimes (\sum_{k=-h_1}^{h_2} \mat{p}_k \otimes (\sum_{l=-w_1}^{w_2} \mat{p}_l \otimes a_{k,l}^{c,d})) \\
&= \sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} a_{k,l}^{c,d} \mat{e}_{c,d} \otimes \mat{p}_k \otimes \mat{p}_l
\end{align}
where the last equality is because $a_{k,l}^{c,d}$ is a scalar.
The alternative one $\mat{t}$ can be represented as
\begin{align}
\mat{t}&=\sum_{k=-h_1}^{h_2}\mat{p}_k \otimes \mat{t}_{k} \\
&= \sum_{k=-h_1}^{h_2} \mat{p}_k \otimes (\sum_{l=-w_1}^{w_2} \mat{p}_l \otimes \mat{t}_{k,l})\\
&= \sum_{k=-h_1}^{h_2}\mat{p}_k \otimes (\sum_{l=-w_1}^{w_2} \mat{p}_l \otimes (\sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} t_{c,d}^{k,l} \mat{e}_{c,d} ))\\
&= \sum_{c=1}^{c_{out}} \sum_{d=1}^{c_{in}} \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} t_{c,d}^{k,l} \mat{p}_k \otimes \mat{p}_l \otimes \mat{e}_{c,d}
\end{align}
where the last equality is because $t_{c,d}^{k,l}$ is a scalar.
According to \cite{henderson1981vec}, $\mat{p}_k \otimes \mat{p}_l \otimes \mat{e}_{c,d}$ is permutation equivalent to $\mat{e}_{c,d} \otimes\mat{p}_k \otimes \mat{p}_l$, for which there exist two permutation matrices $\hbox{\boldmath$\Pi$}_1$ and $\hbox{\boldmath$\Pi$}_2$, such that $\mat{p}_k \otimes \mat{p}_l \otimes \mat{e}_{c,d} = \hbox{\boldmath$\Pi$}_1 (\mat{e}_{c,d} \otimes\mat{p}_k \otimes \mat{p}_l) \hbox{\boldmath$\Pi$}_2$. Given the fact that $a_{k,l}^{c,d}=t_{c,d}^{k,l}$, it follows that
\begin{align}
\mat{t} = \hbox{\boldmath$\Pi$}_1 \mat{a} \hbox{\boldmath$\Pi$}_2.
\end{align}
Because permutation matrices are also orthogonal matrices, and thus unitary, $\mat{t}$ and $\mat{a}$ have an identical set of singular values. This completes the proof.
\end{proof}
\begin{lemma}
$\{\sigma_j(\mat{c}), \ \forall j\}=\{\sigma_j(\mat{c}(\mat{a})), \ \forall j\}$.
\end{lemma}
\begin{proof}
The proof is similar to that of Lemma \ref{lemma:toe-permutation-invariant} and thus omitted. The only difference is that, for the representation of $n \times n$ circulant matrices, we have $[\mat{p}_k]_{i,j}=1$ if $(i-j) \!\! \mod n = k$ and 0 otherwise.
\end{proof}
\begin{lemma}
The linear transformation matrix $\mat{c}$ can be block-diagonalized as
\begin{align}
\mat{c} = (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}}) \mathrm{blkdiag} (\mat{b}_{1,1},\mat{b}_{1,2},\dots, \mat{b}_{1,n}, \mat{b}_{2,1}, \dots \mat{b}_{n,n}) (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})^{\H}
\end{align}
where $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}})$ and $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})$ are unitary matrices.
Thus, the singular values of $\mat{c}$ are the collection of singular values of
$
\{\mat{b}_{i,k}\}_{i,k=1}^n
$
where
\begin{align}
\mat{b}_{i,k}= \sum_{p=0}^{n-1} \sum_{q=0}^{n-1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(i-1)+q(k-1)}{n}}
\end{align}
with $\mat{c}_{p,q}$ defined in \eqref{eq:C_k_l}.
\end{lemma}
\begin{proof}
By extending Lemma 5.1 in \cite{book_block_toe} from block circulant matrices to doubly block circulant matrices, we conclude that
the linear transformation matrix $\mat{c}$ can be block-diagonalized as
\begin{align}
\mat{c} = (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}}) \mathrm{blkdiag} (\mat{b}_{1,1},\mat{b}_{1,2},\dots, \mat{b}_{1,n}, \mat{b}_{2,1}, \dots \mat{b}_{n,n}) (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})^{\H}
\end{align}
where both $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}})$ and $(\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})$ are unitary matrices. As such, the singular values of $\mat{c}$ are the collection of singular values of $n^2$ matrices $\{\mat{b}_{i,k}\}_{i,k=1}^{n}$.
By Lemma 5.1 in \cite{book_block_toe}, for each $i,k \in [n]$, we compute $\mat{b}_{i,k} \in \mbox{\bb C}^{c_{out} \times c_{in}}$ by
\begin{align}
\mat{b}_{i,k}= \sum_{p=0}^{n-1} \sum_{q=0}^{n-1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(i-1)+q(k-1)}{n}}.
\end{align}
The singular values of $\mat{b}_{i,k}$ can be therefore obtained by applying off-the-shelf singular-value decomposition algorithms.
\end{proof}
\subsection{Proof of Theorem 1}
\begin{theorem}
Given a block doubly Toeplitz matrix $\mat{t} \in \mbox{\bb C}^{rn^2\times sn^2}$, let a complex matrix-valued Lebesgue-measurable function $F:[-\pi,\pi]^2 \mapsto \mathbb{C}^{r \times s}$ be the generating function such that
\begin{align} \label{eq:App-T-F}
\mat{t}_{k,l} = \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} F(\omega_1,\omega_2) e^{-\jmath (k \omega_1 + l \omega_2)} d\omega_1 d\omega_2.
\end{align}
It follows that, for any continuous function $\Phi$ with compact support in $\mathbb{R}$, we have
\begin{align}
\MoveEqLeft \lim_{n \to \infty} \frac{1}{n^2} \sum_{j=1}^{\min\{r,s\}n^2} \Phi(\sigma_j(\mat{t}))
= \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \sum_{j=1}^{\min\{r,s\}} \Phi(\sigma_j(F(\omega_1,\omega_2))) d\omega_1 d\omega_2,
\end{align}
for which $\mat{t}$ is said to be equally distributed as $F(\omega_1,\omega_2)$ with respect to singular values, i.e., $\mat{t} \sim_{\sigma} F$. Specifically, for linear convolutional layers, the linear transformation matrix $\mat{t}$ has doubly banded structures, so that the generating function can be explicitly written as
\begin{align}
F(\omega_1,\omega_2)=\sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \mat{t}_{k,l} e^{\jmath (k \omega_1 + l \omega_2)},
\end{align}
which is also referred to as the spectral density matrix of $\mat{t}$.
\end{theorem}
\begin{remark}
Theorem \ref{thm:szg-limit} is a generalization of the celebrated Szeg\"o Theorem \cite{Gray1972}, which deals with real scalar-valued generating functions $F: [-\pi,\pi] \mapsto \mbox{\bb R}$ that correspond to Hermitian Toeplitz matrices. It was extended to non-Hermitian matrices \cite{avram1988bilinear,parter1986distribution}, block Toeplitz matrices \cite{Tilli1998}, and multi-level Toeplitz matrices \cite{Tyrtyshnikov1996,voois1996theorem}.
The linear transformation matrix $\mat{t}$ is an asymmetric real matrix and hence non-Hermitian, with doubly block Toeplitz structure, which corresponds to a complex matrix-valued generating function $F: [-\pi,\pi]^2 \mapsto \mbox{\bb C}^{s \times r}$.
In particular, when $s=r=1$, Theorem \ref{thm:szg-limit} reduces to single-channel 2D convolutional layers, for which
$\mat{t} \sim_{\sigma} \abs{F(\omega_1,\omega_2)}$.
When it comes to signal-channel 1D convolutional layer, Theorem \ref{thm:szg-limit} indicates $\mat{t} \sim_{\sigma} \abs{F(\omega)}$.
\end{remark}
\begin{proof}
The proof is an extension of
those in \cite{voois1996theorem,Tilli1998,miranda2000asymptotic,Tyrtyshnikov1996} that consider block Toeplitz matrices or doubly Toeplitz matrices.
The main proof technique is to relate Toeplitz matrices to their circulant counterpart, which has been shown efficient in many similar settings. This technique is also applied here.
In particular, we follow the footsteps of \cite{Tilli1998,miranda2000asymptotic} to extend the proofs to non-Hermitian block doubly Toeplitz matrices $\mat{t}$, by relating to the block doubly circulant matrices $\mat{c}$.
First, we show that both $\mat{t}$ and $\mat{c}$ have the same asymptotic singular values distribution as $n \to \infty$.
As $\mat{c}$ is constructed from $\mat{t}$ and both of them are banded matrices, by Lemma \ref{lemma:proof-thm1} below, it can be easily verified that
\begin{align}
\lim_{n \to \infty} \frac{1}{n^2} \Normf{\mat{t} -\mat{c}}=0
\end{align}
as the values of the elements in $\mat{c}$ and $\mat{t}$ are upper-bounded, and the total number of different elements between $\mat{c}$ and $\mat{t}$ does not scale as $n^2$.
According to Chapter 2 in \cite{gray2006toeplitz} , it follows that $\mat{c}$ and $\mat{t}$ are asymptotically equivalent.
Let us introduce two Hermitian matrices
\begin{align}
\tilde{\mat{c}} = \begin{bmatrix} \mat{\mathrm{0}} & \mat{c} \\ \mat{c}^H & \mat{\mathrm{0}} \end{bmatrix}, \qquad \tilde{\mat{t}} = \begin{bmatrix} \mat{\mathrm{0}} & \mat{t} \\ \mat{t}^H & \mat{\mathrm{0}} \end{bmatrix}.
\end{align}
It follows that $\tilde{\mat{c}}$ and $\tilde{\mat{t}}$ are asymptotically equivalent as well. It is worth noting that the sets of eigenvalues of $\tilde{\mat{c}}$ and $\tilde{\mat{t}}$ are exactly the respective sets of singular values of ${\mat{c}}$ and ${\mat{t}}$, according to Theorem 7.3.3 in \cite{horn2012matrix}. Thus, according to Theorem 2.1 in \cite{Gray1972}, we have
\begin{align}
\lim_{n \to \infty} \frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}n^2} (\sigma_j(\mat{t}))^p = \lim_{n \to \infty} \frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}n^2} (\sigma_j(\mat{c}))^p
\end{align}
for any positive integer $p$. By Stone-Weierstrass theorem \cite{Gray1972}, it follows that, any continuous function $\Phi(\cdot)$ with compact support, there exists a set of polynomials that uniformly converges to it. Thus, we have
\begin{align} \label{eq:equally-distributed}
\lim_{n \to \infty} \frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}n^2} (\Phi(\sigma_j(\mat{t})) -\Phi(\sigma_j(\mat{c})))=0.
\end{align}
Second, we show that the singular value distribution of the block doubly circulant matrix $\mat{c}$ converges to that of the generating function $F$.
The doubly circulant matrix $\mat{c}$ can be block-diagonalized as
\begin{align}
\mat{c} = (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{out}}) &\mathrm{blkdiag} (\mat{b}_{1,1},\mat{b}_{1,2},\dots, \mat{b}_{1,n}, \mat{b}_{2,1}, \dots \mat{b}_{n,n}) (\mat{f}_n \otimes \mat{f}_n \otimes \mat{{I}}_{c_{in}})^{\H}
\end{align}
where by Lemma \ref{lemma:toe-equal-cir} and \eqref{eq:C_k_l} in the main text, we have
\begin{align}
\mat{b}_{j_1,j_2}&= \sum_{p=0}^{n-1} \sum_{q=0}^{n-1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(j_1-1)+q(j_2-1)}{n}}\\
&=\sum_{p=0}^{h_1} \sum_{q=0}^{w_1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(j_1-1)+q(j_2-1)}{n}}
+ \sum_{p=0}^{h_1} \sum_{q=n-w_2}^{n-1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(j_1-1)+q(j_2-1)}{n}} \\
& \qquad + \sum_{p=n-h_1}^{n-1} \sum_{q=0}^{w_1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(j_1-1)+q(j_2-1)}{n}}
+ \sum_{p=n-w_2}^{n-1} \sum_{q=0}^{w_1} \mat{c}_{p,q} e^{-\jmath 2 \pi \frac{ p(j_1-1)+q(j_2-1)}{n}}\\
&=\sum_{k=-h_1}^{0} \sum_{l=-w_1}^{0} \mat{t}_{k,l} e^{\jmath 2 \pi \frac{ k(j_1-1)+l(j_2-1)}{n}}
+ \sum_{k=-h_1}^0 \sum_{l=1}^{w_2} \mat{t}_{k,l} e^{\jmath 2 \pi \frac{ k(j_1-1)+(l-n)(j_2-1)}{n}} \\
& \qquad + \sum_{k=1}^{h_2} \sum_{l=-w_1}^{0} \mat{t}_{k,l} e^{\jmath 2 \pi \frac{ (k-n)(j_1-1)+l(j_2-1)}{n}}
+ \sum_{k=1}^{h_2} \sum_{l=1}^{w_2} \mat{t}_{k,l} e^{\jmath 2 \pi \frac{ (k-n)(j_1-1)+(l-n)(j_2-1)}{n}}\\
&= \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \mat{t}_{k,l} e^{\jmath 2 \pi \frac{ k(j_1-1)+l(j_2-1)}{n}}\\
&= F(\frac{2\pi (j_1-1)}{n},\frac{2\pi (j_2-1)}{n}), \label{eq:B-to-F}
\end{align}
for $j_1,j_2 \in [n]$.
Consequently, the collection of singular values of block doubly circulant matrix $\mat{c}$ is the collection of singular values of $F$ over the uniform grids
\begin{align}
{\mathcal M} \triangleq \left\{ (\omega_1,\omega_2) = \left(-\pi+ \frac{2 \pi j_1}{n}, -\pi + \frac{2 \pi j_2}{n} \right), \forall\;j_1,j_2 \in [n]-1 \right\}.
\end{align}
As such, for any integer $p \ge 0$, we have
\begin{align}
\frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}n^2} (\sigma_j(\mat{c}))^p
&= \frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}} \sum_{(\omega_1,\omega_2) \in {\mathcal M}} (\sigma_j(F(\omega_1,\omega_2)))^p\\
&= \frac{1}{\min\{r,s\}} \sum_{j=1}^{\min\{r,s\}} \frac{1}{n^2} \sum_{(\omega_1,\omega_2) \in {\mathcal M}} (\sigma_j(F(\omega_1,\omega_2)))^p\\
& \stackrel{n \to \infty}{=} \frac{1}{\min\{r,s\}} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} (\sigma_j(F(\omega_1,\omega_2)))^p d \omega_1 d\omega_2
\end{align}
where the last equation is due to the fact that the Riemann sum converges to the integral of the function $(\sigma_j(F(\omega_1,\omega_2)))^p$ over $[-\pi, \pi]^2$, as $n \to \infty$.
Further, by Stone-Weierstrass theorem \cite{Gray1972}, we have
\begin{align}
\lim_{n \to \infty} \frac{1}{\min\{r,s\}n^2} \sum_{j=1}^{\min\{r,s\}n^2} \Phi(\sigma_j(\mat{c})) = \frac{1}{\min\{r,s\}} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \Phi(\sigma_j(F(\omega_1,\omega_2))) d \omega_1 d\omega_2
\end{align}
Finally, together with \eqref{eq:equally-distributed}, the proof of Theorem 1 is completed.
\end{proof}
\begin{lemma}
\label{lemma:proof-thm1}
Given the banded block doubly Toeplitz and circulant matrices $\mat{t}$ and $\mat{c}$, it follows that
\begin{align}
\norm{\mat{c}-\mat{t}}_p^p \le O(n).
\end{align}
where $\norm{\mat{a}}_p\triangleq (\sum_{i=1}^n \sum_{j=1}^n \abs{\mat{a}_{i,j}}^p)^{\frac{1}{p}}$ for $1 \le p < \infty$. When $p=2$, $\norm{\mat{a}}_p$ boils down to the Frobenius norm $\normf{\mat{a}}$.
\end{lemma}
\begin{proof}
Given $\mat{t}$ and $\mat{c}$, we define the difference of the $(k,l)$-th block $\hbox{\boldmath$\Delta$}_{k,l} \in \mbox{\bb C}^{r \times s}$, where $k \in [-(n-1),(n-1)]$ and $l \in [-(n-1),(n-1)]$ are indices of two levels of Toeplitz and circulant matrices but not the indices of rows and columns, in the following way
\begin{align}
\hbox{\boldmath$\Delta$}_{k,l} &\triangleq [\mat{c}-\mat{t}]_{k,l} \\
&\stackrel{(a)}{=} \sum_{m_1 \in \{-1,0,1\}} \sum_{m_2 \in \{-1,0,1\}} \mat{t}_{k+nm_1,l+nm_2} (1 - \delta(m_1,m_2))
\end{align}
where $\delta(m_1,m_2)=1$ if and only if $m_1=m_2=0$, and $(a)$ is due to the banded structure of circulant matrix as in \eqref{eq:C_k_l}.
It can be easily verified that $\mat{c}-\mat{t}$ is still a block doubly Toeplitz matrix with blocks $\{\hbox{\boldmath$\Delta$}_{k,l}\}_{k,l}$.
Thus, we have
\begin{align}
\MoveEqLeft \norm{\mat{c}-\mat{t}}_p^p \notag \\ &\stackrel{(a)}{=} \sum_{k=-(n-1)}^{n-1} \sum_{l=-(n-1)}^{n-1} (n-\abs{k})(n-\abs{l}) \norm{\hbox{\boldmath$\Delta$}_{k.l}}_p^p\\
&\stackrel{(b)}{\le} \sum_{k=-(n-1)}^{n-1} \sum_{l=-(n-1)}^{n-1} \sum_{m_1 = -1}^1 \sum_{m_2=-1}^1 (n-\abs{k})(n-\abs{l}) (1 - \delta(m_1,m_2)) \norm{\mat{t}_{k+nm_1,l+nm_2}}_p^p\\
%
%
&\stackrel{(c)}{=} \sum_{(k,l) \in {\mathcal B}_{12}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k,l+n}}_p^p + \sum_{(k,l) \in {\mathcal B}_{13}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k,l-n}}_p^p\\
& \qquad + \sum_{(k,l) \in {\mathcal B}_{21}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k+n,l}}_p^p + \sum_{(k,l) \in {\mathcal B}_{22}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k+n,l+n}}_p^p \\
&\qquad + \sum_{(k,l) \in {\mathcal B}_{23}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k+n,l-n}}_p^p
+ \sum_{(k,l) \in {\mathcal B}_{31}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k-n,l}}_p^p \\
&\qquad + \sum_{(k,l) \in {\mathcal B}_{32}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k-n,l+n}}_p^p + \sum_{(k,l) \in {\mathcal B}_{33}} (n-\abs{k})(n-\abs{l}) \norm{\mat{t}_{k-n,l-n}}_p^p\\
%
&\stackrel{(d)}{\le} hw_1^2C_pn + hw_2^2C_pn + h_1^2wC_pn + h_1^2 w_1^2 C_p + h_1^2w_2^2C_p + h_2^2 wC_pn + h_2^2w_1^2C_p + h_2^2w_2^2C_p \\
%
&\stackrel{(e)}{=} an+b
\end{align}
where $(a)$ is due the definition of the element-wise $p$-norm, $(b)$ is due to the sub-additivity of matrix norms, in $(c)$ we define
\begin{align}
{\mathcal B}_{11}&=\{(k,l):k \in [-h_1,h_2] \text{ and } l \in [-w_1,w_2]\}\\
{\mathcal B}_{12}&=\{(k,l):k \in [-h_1,h_2] \text{ and } l \in [-(n-1),-(n-w_1)]\}\\
{\mathcal B}_{13}&=\{(k,l):k \in [-h_1,h_2] \text{ and } l \in [(n-w_2),(n-1)]\}\\
{\mathcal B}_{21}&=\{(k,l):k \in [-(n-1),-(n-h_1)] \text{ and } l \in [-w_1,w_2]\}\\
{\mathcal B}_{22}&=\{(k,l):k \in [-(n-1),-(n-h_1)] \text{ and } l \in [-(n-1),-(n-w_1)]\}\\
{\mathcal B}_{23}&=\{(k,l):k \in [-(n-1),-(n-h_1)] \text{ and } l \in [(n-w_2),(n-1)]\}\\
{\mathcal B}_{31}&=\{(k,l):k \in [(n-h_2),(n-1)] \text{ and } l \in [-w_1,w_2]\}\\
{\mathcal B}_{32}&=\{(k,l):k \in [(n-h_2),(n-1)] \text{ and } l \in [-(n-1),-(n-w_1)]\}\\
{\mathcal B}_{33}&=\{(k,l):k \in [(n-h_2),(n-1)] \text{ and } l \in [(n-w_2),(n-1)]\}
\end{align}
for which $\mat{t}_{k+nm_1,l+nm_2} \ne \mat{\mathrm{0}}$ in ${\mathcal B}_{11}$ if and only if $m_1=m_2=0$ which invokes $\delta(m_1,m_2)=1$, $(d)$ is due to $\norm{\mat{t}_{k,l}}_p^p$ is upper-bounded by a constant, say $C_p$ for all $k,l$, and in $(e)$, $a=C_p(h(w_1^2+w_2^2)+(h_1^2+h_2^2)w)$ and $b=C_p(h_1^2+h_2^2)(w_1^2+w_2^2)$. This completes the proof.
\end{proof}
\subsection{Proof of Theorem 2}
\begin{theorem}
Given $\mat{t}$ and $\mat{c}$ as in \eqref{eq:Cm_k}-\eqref{eq:C_k_l},
there exists a constant $c_1>0$ such that
\begin{align}
\lim_{n \to \infty} \frac{1}{n} \sum_{j=1}^{\min\{r,s\}n^2} \abs{\sigma_j(\mat{t})-\sigma_j(\mat{c})} \le c_1,
\end{align}
where the singular values of $\mat{c}$ are the collection of singular values of $\{\sigma_j(F(\omega_1,\omega_2))\}_j$ with
\begin{align}
(\omega_1,\omega_2) = (-\pi+ \frac{2 \pi j_1}{n}, -\pi &+ \frac{2 \pi j_2}{n}), \quad \forall j_1,j_2 \in [n]-1.
\end{align}
\end{theorem}
\begin{remark}
It is worth noting that \cite{ZhuTIT2017} dealt with eigenvalues of Hermitian Toeplitz matrices that correspond to real scalar-valued generating functions, for which the Theorem 2 in \cite{ZhuTIT2017} regarding the bounded circular approximation error can not be taken as granted e.g., \cite{sedghi2018the,Singla2019}, to justify the circular approximation of linear convolutional layers whose transformation matrix $\mat{t}$ is non-Hermitian block doubly Toeplitz matrices. As the proof of \cite{ZhuTIT2017}[Theorem 2] relies on Sturmian Separation Theorem that deals with eigenvalues for Hermitian matrices, %
it is not guaranteed that the difference of individual singular values between non-Hermitian Toeplitz and circulant matrices can be bounded in the same way. %
\end{remark}
\begin{proof}
Given the generating function $F(\omega_1,\omega_2)$ defined in \eqref{eq:theorem-1-F}, we introduce an auxiliary matrix $\mat{c}(F)$ generated by $F$ in the following form
\begin{align}
\mat{c}(F) = (\mat{f}_n \times \mat{f}_n \times \mat{{I}}_{r}) \mathrm{blkdiag} \Big(\{F(\omega_1,\omega_2), (\omega_1,\omega_2) \in {\mathcal M}\}\Big) (\mat{f}_n \times \mat{f}_n \times \mat{{I}}_s)^{\H}
\end{align}
where ${\mathcal M}$ is the uniform sampling over $[-\pi,\pi]^2$ defined as \eqref{eq:uniform-samples}.
It can be readily verified that $\mat{c}(F)$ is also a block doubly circulant matrix, similar to $\mat{c}$.
First, we show $\mat{c}(F)$ and $\mat{c}$ are identical, and thus uniform sampling $F$ yields singular values of $\mat{c}$.
Denote by $[\mat{c}(F)]_{p,q} \in \mbox{\bb C}^{r \times s}$ the $(p,q)$-th block of $\mat{c}(F)$, where $p$ and $q$ indicate the indices of the first and second levels of circulant blocks, similar to the definition of $\mat{c}_{p,q}$ in \eqref{eq:C_k_l}. Therefore, we have
\begin{align}
[\mat{c}(F)]_{p,q}
&\stackrel{}{=} \frac{1}{n^2}\sum_{j_1=0}^{n-1} \sum_{j_2=0}^{n-1} F(\frac{2\pi j_1}{n},\frac{2\pi j_2}{n})e^{-\jmath 2 \pi \frac{ pj_1+qj_2}{n}}\\
&\stackrel{}{=} \frac{1}{n^2}\sum_{j_1=0}^{n-1} \sum_{j_2=0}^{n-1} \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \mat{t}_{k,l}e^{\jmath \frac{2 \pi}{n} ( (k-p)j_1+(l-q)j_2)}\\
&\stackrel{}{=} \frac{1}{n^2} \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \mat{t}_{k,l} \sum_{j_1=0}^{n-1} e^{\jmath \frac{2 \pi j_1}{n} (k-p)} \sum_{j_2=0}^{n-1} e^{\jmath \frac{2 \pi j_2}{n} (l-q)} \\
&\stackrel{(a)}{=} \sum_{m_1=-\infty}^{\infty} \sum_{m_2=-\infty}^{\infty} \mat{t}_{-p+nm_1,-q+nm_2}\\
&\stackrel{(b)}{=} \left\{
\Pmatrix{\mat{t}_{-p,-q}, & p\in \{0\}\cup[h_1], \; q\in\{0\}\cup[w_1]\\
\mat{t}_{-p,n-q}, & p\in \{0\}\cup[h_1], \; q \in n-[w_2]\\
\mat{t}_{n-p,-q}, &q \in n-[h_2], \; q\in\{0\}\cup[w_1]\\
\mat{t}_{n-p,n-q}, & p \in n-[h_2], \; q \in n-[w_2]\\
\mat{\mathrm{0}},& \text{otherwise}}
\right.\\
&\stackrel{(c)}{=}\mat{c}_{p,q}
\end{align}
for $p, q \in [n]-1$, where $(a)$ is due to
\begin{align}
\sum_{j=0}^{n-1} e^{\jmath \frac{2 \pi j}{n} (k-p)} = \left\{ \Pmatrix{n,& (k-p) \!\!\! \mod n = 0 \\ 0, & \text{otherwise}} \right.,
\end{align}
$(b)$ is due to $\mat{t}_{p,q}=\mat{\mathrm{0}}$ if $p \notin [-h_1,h_2]$ or $q \notin [-w_1,w_2]$, and $(c)$ is from \eqref{eq:C_k_l}.
For each $p,q \in [n]-1$, the $(p,q)$-th blocks of $\mat{c}(F)$ and $\mat{c}$ are identical. Thus, we have
\begin{align}
\mat{c}(F) = \mat{c}.
\end{align}
Therefore, by Lemma \ref{lemma:toe-equal-cir}, we conclude that the singular values of $\mat{c}$ can be given by those of $F(\omega_1,\omega_2)$ with uniform sampling on $[-\pi,\pi]^2$, i.e.,
\begin{align}
\left\{\sigma_{j}(F(\omega_1,\omega_2)): (\omega_1,\omega_2) \in {\mathcal M} \right\},
\end{align}
where ${\mathcal M}$ is the uniform sampling grids defined in \eqref{eq:uniform-samples}.
Second, we show that the accumulated difference of the singular values between $\mat{c}$ and $\mat{t}$ is upper-bounded.
By inspecting $\mat{t}$ and $\mat{c}$, we find from Lemma \ref{lemma:proof-thm1} that $\hbox{\boldmath$\Delta$}_{k,l}=0$ if and only if $(k,l) \in {\mathcal B}_{11}$. The number of rows and columns with indices outside ${\mathcal B}_{11}$ scales as $n$.
As such, invoking Theorem 3.1 in \cite{Zizler2002}, we conclude that
\begin{align}
\sum_{j=1}^{\min\{r,s\}n^2} \abs{\sigma_j(\mat{t})-\sigma_j(\mat{c})} \le O(n).
\end{align}
Thus, we have
\begin{align}
\lim_{n \to \infty} \frac{1}{n}\sum_{j=1}^{\min\{r,s\}n^2} \abs{\sigma_j(\mat{t})-\sigma_j(\mat{c})}=O(1)
\end{align}
This completes the proof.
\end{proof}
\begin{remark}
The intuition behind is that the number of different elements between two matrices scales as $n$ but not $n^2$ because of the banded structure of $\mat{c}$ and $\mat{t}$. Although not rigorously proved, it looks the equality holds with the term $O(1)$ strictly larger than 0, meaning that the circular approximation can be arbitrarily loose as $n$ tends to infinity.
\end{remark}
\subsection{Proof of Theorem 3}
\begin{theorem}
Let $\phi_j:[-\pi,\pi]^2 \mapsto \mbox{\bb R}_+$ be the $j$-th singular value function of $F(\hbox{\boldmath$\omega$})$ and $\sigma_k^{(j)}(\mat{t})$ be $k$-th singular value of $j$-th cluster. There exists a constant $c_2>0$ which only depends on $F(\hbox{\boldmath$\omega$})$ such that
\begin{align}
\sup_{u \in (\frac{k-1}{n^2},\frac{k}{n^2}]} \abs{\sigma_k^{(j)}(\mat{t})-Q_{\phi_j}(u)} &\le \frac{c_2}{n}, \quad
\forall 1\le k \le n^2, \; 1 \le j \le \min\{r,s\}
\end{align}
where
\begin{align}
Q_{\phi_j}(u)&=\inf\{v \in \mbox{\bb R}: u \le G_{\phi_j}(v)\}\\
G_{\phi_j}(v)&=\frac{1}{(2\pi)^2}\mu\{\hbox{\boldmath$\omega$} \in [-\pi,\pi]^2: \phi_j(\hbox{\boldmath$\omega$})\le v\}
\end{align}
are quantile and cumulative distribution functions for $\phi_j(\hbox{\boldmath$\omega$})$, respectively, and $\mu$ is Lebesgue measure.
\end{theorem}
\begin{proof}
Without loss of generality, we let $r \le s$, i.e., $r=\min\{r,s\}$. We divide all $\{\sigma_j(\mat{t})\}_{j=1}^{rn^2}$ into $r$ clusters $\{\sigma_k^{(j)}(\mat{t}),k\in[n^2]\}_{j=1}^{r}$ according to their localization, each of which is arranged in ascending order, i.e.,
\begin{align}
\sigma_1^{(j)}(\mat{t}) \le \sigma_2^{(j)}(\mat{t}) \le \dots \le \sigma_{n^2}^{(j)}(\mat{t}), \quad \forall j \in [r].
\end{align}
From Theorem 1, we have
\begin{align} \label{eq:theorem-1-proof}
\frac{1}{r} \sum_{j=1}^{r} \lim_{n \to \infty} \frac{1}{n^2} \sum_{k=1}^{n^2} \Phi(\sigma_k^{(j)}(\mat{t})) = \frac{1}{r}\sum_{j=1}^{r} \frac{1}{(2\pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \Phi(\sigma_j(F(\omega_1,\omega_2))) d\omega_1 d\omega_2.
\end{align}
Let $\phi_j:[-\pi,\pi]^2 \mapsto \mbox{\bb R}_+$ be the $j$-th singular value function of $F(\hbox{\boldmath$\omega$})$, i.e., $\phi_j(\hbox{\boldmath$\omega$})=\sigma_j(F(\omega_1,\omega_2))$. When taking $\hbox{\boldmath$\omega$}$ as a multivariate random variable with uniform distribution on $[-\pi,\pi]^2$, we can treat $\phi_j(\hbox{\boldmath$\omega$})$ as a continuous random variable, such that the right-hand side of \eqref{eq:theorem-1-proof} can be interpreted as
\[
\frac{1}{r} \sum_{j=1}^{r} \E_{\hbox{\boldmath$\omega$}} [\Phi(\phi_j(\hbox{\boldmath$\omega$}))]
\]
Similarly, we can treat $\{\sigma_k^{(j)}(\mat{t})\}_{k=1}^{n^2}$ as realizations of discrete random variable $X_n^{(j)}$ with equal probability $\Pr(X_n^{(j)}=\sigma_k^{(j)}(\mat{t}))=\frac{1}{n^2}$, and interpret the left-hand side of \eqref{eq:theorem-1-proof} as
\[
\frac{1}{r} \sum_{j=1}^{r} \lim_{n \to \infty} \E_{X_n^{(j)}} [\Phi(X_n^{(j)})]
\]
Thus, from a probabilistic perspective, Theorem 1 says, for the sequence of random variables $\{X_1^{(j)},X_2^{(j)},\dots,X_n^{(j)},\dots\}$, $\E_{X_n^{(j)}} [\Phi(X_n^{(j)})]$ converges to $\E_{\hbox{\boldmath$\omega$}} [\Phi(\phi_j(\hbox{\boldmath$\omega$}))]$ in distribution for any continuous function $\Phi$.
For both random variables $X_n^{(j)}$ and $\phi_j(\hbox{\boldmath$\omega$})$, let us define the cumulative distribution and quantile functions as
\begin{align}
G_{X_n^{(j)}}(v)&=\frac{1}{n^2} \max \{k \in [n^2]: \sigma_k^{(j)}(\mat{t}) \le v\}\\
Q_{X_n^{(j)}}(u)&=\inf \{v \in \mbox{\bb R}: u \le G_{X_n^{(j)}}(v)\}\\
G_{\phi_j}(v)&=\frac{1}{(2\pi)^2}\mu\{\hbox{\boldmath$\omega$} \in [-\pi,\pi]^2: \phi_j(\hbox{\boldmath$\omega$})\le v\}\\
Q_{\phi_j}(u)&=\inf\{v \in \mbox{\bb R}: u \le G_{\phi_j}(v)\}
\end{align}
where $\mu$ is the Lebesgue measure of $\hbox{\boldmath$\omega$}$ on $[-\pi,\pi]^2$. As $\{\sigma_k^{(j)}(\mat{t})\}_{k=1}^{n^2}$ is ordered and $G_{X_n^{(j)}}(v)$ is right continuous and non-decreasing over $v$, it follows from \cite{Bogoya2015}[Proposition 2.5] that
\begin{align} \label{eq:thm3-proof-Q-sigma}
Q_{X_n^{(j)}}(\frac{k}{n^2}) = \sigma_k^{(j)}(\mat{t}).
\end{align}
By Portmanteau Lemma \cite{Bogoya2015}[Lemma 3.1], the fact that $\E_{X_n^{(j)}} [\Phi(X_n^{(j)})]$ converges to $\E_{\hbox{\boldmath$\omega$}} [\Phi(\phi_j(\hbox{\boldmath$\omega$}))]$ in distribution for any continuous function $\Phi$ leads to (1) $G_{X_n^{(j)}}(v)$ converges to $G_{\phi_j}(v)$ for every $v \in \mbox{\bb R}$ at which $G_{\phi_j}$ is continuous, and (2) $Q_{X_n^{(j)}}(u)$ converges to $Q_{\phi_j}(u)$ for every $u \in (0,1]$ at which $Q_{\phi_j}$ is continuous.
Inspired by \cite{Zizler2002}[Theorem 3.2, Corollary 3.3], we can further bound the gap between $G_{X_n^{(j)}}(v)$ and $G_{\phi_j}(v)$.
\begin{lemma} \label{lemma:thm3-gap}
There exists a constant $c_1$ such that
\begin{align} \label{eq:apdx-G-gap}
\max_{j \in [r]} \;\; \abs{G_{X_n^{(j)}}(v)-G_{\phi_j}(v)} \le \frac{c_1}{n}
\end{align}
for every $n>1$.
\end{lemma}
\begin{proof}
Due to Theorem 2, the singular values of $\mat{c}$ can be given by those of $F(\hbox{\boldmath$\omega$})$ with uniform sampling on $[-\pi,\pi]^2$, i.e.,
\begin{align}
\{\sigma_k^{(j)}(\mat{c})\}_{k=1}^{n^2} = \left\{\sigma_{j}(F(\omega_1,\omega_2)): (\omega_1,\omega_2) = (-\pi+ \frac{2 \pi j_1}{n}, -\pi + \frac{2 \pi j_2}{n}), \forall j_1,j_2 \in [n]-1 \right\}
\end{align}
for $j \in [r]$.
Following the same footsteps of \cite{Zizler2002}[Theorem 2.2], we have
\begin{align}
\abs{\sum_{k=1}^{n^2} \sigma_k^{(j)}(\mat{c}) - \frac{n^2}{(2\pi)^2}\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \sigma_j(F) d \omega_1 d \omega_2 } \le c'_0 n
\end{align}
where $c'_0>0$ is a constant that does not depend on $n$.
Due to Theorem 2, there must exist a constant $c_0>0$ such that
\begin{align}
\sum_{k=1}^{n^2} \abs{\sigma_k^{(j)}(\mat{t}) - \sigma_k^{(j)}(\mat{c})} \le c_0 n.
\end{align}
It follows that, there exists a constant $c_1>0$ that does not depend on $n$ such that
\begin{align}
\abs{\sum_{k=1}^{n^2} \sigma_k^{(j)}(\mat{t}) - \frac{n^2}{(2\pi)^2}\int_{-\pi}^{\pi} \int_{-\pi}^{\pi} \sigma_j(F) d \omega_1 d \omega_2 } \le c_1n
\end{align}
By \cite{Zizler2002}[Corollary 3.3], for a real value $v$, we have
\begin{align}
\abs{G_{X_n^{(j)}}(v)-G_{\phi_j}(v)} \le \frac{c_1}{n}
\end{align}
for all $j$, where $\phi_j(\hbox{\boldmath$\omega$})$ takes values of $\sigma_j(F(\hbox{\boldmath$\omega$}))$ that are upper bounded given the fact that $F(\hbox{\boldmath$\omega$})$ is a Laurent polynomial matrix with respect to $e^{\jmath \hbox{\boldmath$\omega$}}$, each element of which is a Laurent polynomial. This completes the proof.
\end{proof}
Let $\epsilon=\frac{c_1}{n}$ and $\frac{k-1}{n^2} < u \le \frac{k}{n^2}$. By \cite{Bogoya2015}[Proposition 2.2], we have $u \le G_{\phi_j}(Q_{\phi_j}(u))$. Together with Lemma \ref{lemma:thm3-gap}, we have
\begin{align}
u &= u+\epsilon - \epsilon \le G_{\phi_j}(Q_{\phi_j}(u+\epsilon)) - \epsilon \le G_{X_n^{(j)}}(Q_{\phi_j}(u+\epsilon))
\end{align}
Let $\delta=c\epsilon$ with $c>0$ being a constant. Given the fact that $Q_{\phi_j}(u-\epsilon) \ge Q_{\phi_j}(u-\epsilon) - \delta$, we have
\begin{align}
u-\epsilon \ge G_{\phi_j}(Q_{\phi_j}(u-\epsilon)-\delta) \ge G_{X_n^{(j)}}(Q_{\phi_j}(u-\epsilon)-\delta) - \epsilon
\end{align}
Thus, due to the fact that $u \le G_{X_n^{(j)}}(v)$ if and only if $Q_{X_n^{(j)}}(u) \le v$, we have
\begin{align} \label{eq:thm3-proof-Q-bounds}
Q_{X_n^{(j)}}(u) &\le Q_{\phi_j}(u+\epsilon) \\
Q_{X_n^{(j)}}(u) &\ge Q_{\phi_j}(u-\epsilon) - \delta.
\end{align}
Before proceeding further, we investigate the Lipschitz continuity of $\phi_j$.
\begin{lemma} \label{lemma:thm3-lip}
The singular value function $\phi_j(\hbox{\boldmath$\omega$})=\sigma_j(F(\hbox{\boldmath$\omega$}))$ is Lipschitz continuous for every $j$.
\end{lemma}
\begin{proof}
According to the generalized Hoffman-Wielandt theorem for singular values \cite{mirsky1960symmetric}[Theorem 5] and \cite{sun1983perturbation}[Theorem 5.1], we have
\begin{align}
\sqrt{\sum_{j=1}^{r} \Abs{\sigma_j(F(\hbox{\boldmath$\omega$}))-\sigma_j(F(\hbox{\boldmath$\omega$}'))}} & \stackrel{}{\le} \normf{F(\hbox{\boldmath$\omega$})-F(\hbox{\boldmath$\omega$}')}\\
&= \normf{\sum_{k_1=-h_1}^{h_2} \sum_{k_2=-w_1}^{w_2} \mat{t}_{k_1,k_2} (e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}} - e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}'})}\\
&\stackrel{(a)}{\le} \sum_{k_1=-h_1}^{h_2} \sum_{k_2=-w_1}^{w_2} \normf{ \mat{t}_{k_1,k_2} }
\abs{e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}} - e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}'}}\\
&\stackrel{(b)}{\le} \sum_{k_1=-h_1}^{h_2} \sum_{k_2=-w_1}^{w_2} \normf{ \mat{t}_{k_1,k_2} }
\abs{ \vect{k}^{\scriptscriptstyle\mathsf{T}} (\hbox{\boldmath$\omega$}-\hbox{\boldmath$\omega$}') }\\
&\stackrel{(c)}{\le} \sum_{k_1=-h_1}^{h_2} \sum_{k_2=-w_1}^{w_2} \norm{\vect{k}} \normf{ \mat{t}_{k_1,k_2} }
\norm{\hbox{\boldmath$\omega$}-\hbox{\boldmath$\omega$}'}
\end{align}
where $(a)$ is due to the triangle inequality of matrix norm, $(b)$ is due to the non-negativity of matrix norms and the following inequality
\begin{align}
\abs{e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}} - e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\hbox{\boldmath$\omega$}'}} &= \abs{\int_{\hbox{\boldmath$\omega$}'}^{\hbox{\boldmath$\omega$}} \jmath e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\vect{t}} \vect{k}^{\scriptscriptstyle\mathsf{T}} d \vect{t}}\\
&\le \abs{\int_{\hbox{\boldmath$\omega$}'}^{\hbox{\boldmath$\omega$}} \abs{\jmath e^{\jmath \vect{k}^{\scriptscriptstyle\mathsf{T}}\vect{t}}} \vect{k}^{\scriptscriptstyle\mathsf{T}} d \vect{t}}\\
&\le \abs{ \vect{k}^{\scriptscriptstyle\mathsf{T}} \int_{\hbox{\boldmath$\omega$}'}^{\hbox{\boldmath$\omega$}} d \vect{t}}\\
&\le \abs{ \vect{k}^{\scriptscriptstyle\mathsf{T}} (\hbox{\boldmath$\omega$}-\hbox{\boldmath$\omega$}') }
\end{align}
and $(c)$ is due to Cauchy-Schwarz inequality.
Let $K=\sum_{k_1=-h_1}^{h_2} \sum_{k_2=-w_1}^{w_2} \norm{\vect{k}} \normf{ \mat{t}_{k_1,k_2} }$, which is a positive constant that does not depend on $\hbox{\boldmath$\omega$}$. Thus, we have
\begin{align}
\abs{\sigma_j(F(\hbox{\boldmath$\omega$}))-\sigma_j(F(\hbox{\boldmath$\omega$}'))} \le K \norm{\hbox{\boldmath$\omega$}-\hbox{\boldmath$\omega$}'}
\end{align}
for all $j$, which means that $\sigma_j(F(\hbox{\boldmath$\omega$}))$ is $K$-Lipschitz continuous, so is $\phi_j(\hbox{\boldmath$\omega$})$ by definition.
\end{proof}
Provided Lemma \ref{lemma:thm3-lip}, following the same footsteps of Proposition 2.7 in \cite{Bogoya2015}, we conclude that $Q_{\phi_j}(u)$ is also Lipschitz continuous, i.e.,
\begin{align} \label{eq:apdx-Q-phi}
\abs{Q_{\phi_j}(u_1) - Q_{\phi_j}(u_2)} \le L \abs{u_1-u_2}
\end{align}
for all $u_1,u_2\in(0,1]$.
Now, equipped with the Lipschitz continuity, by \eqref{eq:thm3-proof-Q-sigma} and \eqref{eq:thm3-proof-Q-bounds}, we have
\begin{align} \label{eq:thm3-proof-left}
\sigma_k^{(j)}(\mat{t}) &= Q_{X_n^{(j)}}(u) \le Q_{\phi_j}(u+\epsilon) \le Q_{\phi_j}(u)+L\epsilon \\
\sigma_k^{(j)}(\mat{t}) &= Q_{X_n^{(j)}}(u) \ge Q_{\phi_j}(u-\epsilon) - \delta \ge Q_{\phi_j}(u)-L\epsilon - \delta
\end{align}
for $u \in (\frac{k-1}{n^2},\frac{k}{n^2}]$.
This implies that
\begin{align}
\abs{\sigma_k^{(j)}(\mat{t})-Q_{\phi_j}(u)} \le L\epsilon + \delta \triangleq \frac{c_2}{n}
\end{align}
for all $k \in [n^2]$ and $j \in [r]$.
This completes the proof.
\end{proof}
\begin{remark}
Theorem 3 offers a better approximation method for the individual singular values of the linear transformation matrix $\mat{t}$. Although the quantile approximation approach has the same scaling law of accumulated approximation error (i.e., $O(n)$) as the circular approximation, the individual singular value approximation accuracy is somewhat guaranteed with vanishing error as $n$ tends to infinity. In contrast, this may not be guaranteed by the circular approximation. From Theorem~2, it is possible that the largest singular value by circular approximation can scale as $n$. Albeit promising from a theoretical point of view, it is challenging to characterize the exact quantile function. A compromised way is to estimate such a quantile function through the circular approximation, from which the singular value distribution can be adjusted so as to reach a relatively better approximation. The experimental results show that a naive subtle adjustment of singular values obtained by the circular approximation (i.e., uniform sampling of $F$) yields notable improvement on approximation accuracy, especially for the largest singular value.
\end{remark}
\subsection{Proof of Theorem 4}
To upper bound the spectral norm of the linear transformation matrix $\mat{t}$, we bound it by the spectral norm of its spectral representation - the spectral density matrix $F(\omega_1,\omega_2)$.
\begin{lemma}
$
\norm{\mat{t}}_2 \le \norm{F}_2.
$
\end{lemma}
\begin{proof}
Inspired by Theorem 4.1 of \cite{Tilli1998}, we extend the proof from block Toeplitz to doubly block Toeplitz matrices.
Given a singular value of $\mat{t} \in \mbox{\bb R}^{rn^2 \times sn^2}$, say $\sigma(\mat{t})$, there exist $\vect{u} \in \mbox{\bb R}^{rn^2}$ and $\vect{v} \in \mbox{\bb R}^{sn^2}$ subject to $\norm{\vect{u}}_2=\norm{\vect{v}}_2=1$ such that $\sigma(\mat{t})=\vect{u}^{{\scriptscriptstyle\mathsf{T}}} \mat{t} \vect{v}$, where $\vect{u}=[\vect{u}_{k,l}]_{k,l}$ and $\vect{v}=[\vect{v}_{k,l}]_{k,l}$, with the $(k,l)$-th block vector $\vect{u}_{k,l} \in \mbox{\bb R}^{r \times s}$ and $\vect{v}_{k,l} \in \mbox{\bb R}^{r \times s}$ corresponding to $\mat{t}_{k,l}$. According to the definition of $\mat{t}_{k,l}$ in \eqref{eq:App-T-F}, we have
\begin{align}
\sigma(\mat{t}) = \frac{1}{(2 \pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi} u(\omega_1,\omega_2)^{{\scriptscriptstyle\mathsf{T}}} F(\omega_1,\omega_2) v(\omega_1,\omega_2) d \omega_1 d \omega_2
\end{align}
where $u(\omega_1,\omega_2)$ and $v(\omega_1,\omega_2)$ are Fourier transforms of $\vect{u}_{k,l}$ and $\vect{v}_{k,l}$, respectively, i.e.,
\begin{align}
u(\omega_1,\omega_2) = \sum_{k=1}^n \sum_{l=1}^n \vect{u}_{k,l} e^{\jmath (k \omega_1 + l \omega_2)}, \\
v(\omega_1,\omega_2) = \sum_{k=1}^n \sum_{l=1}^n \vect{v}_{k,l} e^{\jmath (k \omega_1 + l \omega_2)}.
\end{align}
Thus, we have
\begin{align}
\sigma(\mat{t}) &\stackrel{(a)}{\le} \frac{1}{(2 \pi)^2} \int_{-\pi}^{\pi} \int_{-\pi}^{\pi}
\sigma_{\max}(F)
\norm{u(\omega_1,\omega_2)}_2 \norm{v(\omega_1,\omega_2)}_2 d \omega_1 d \omega_2\\
&\stackrel{(b)}{\le} \sigma_{\max}(F)
\frac{1}{(2 \pi)^2} \sqrt{\int_{-\pi}^{\pi} \int_{-\pi}^{\pi}
\Norm{u(\omega_1,\omega_2)}_2 d \omega_1 d \omega_2}
\sqrt{\int_{-\pi}^{\pi} \int_{-\pi}^{\pi}\Norm{v(\omega_1,\omega_2)}_2 d \omega_1 d \omega_2}\\
&\stackrel{(c)}{=} \sigma_{\max}(F)
\norm{\vect{u}}_2 \norm{\vect{v}}_2\\
&= \sigma_{\max}(F)
\end{align}
where $(a)$ is from the definition of the largest singular value, i.e., $\sigma_{\max}(F) = \sup \frac{u^{{\scriptscriptstyle\mathsf{T}}} F v}{\norm{u}_2\norm{v}_2}$, $(b)$ is due to Cauchy inequality, and $(c)$ is resulted directly from the computation of integrals.
Thus, it follows immediately that $\norm{\mat{t}}_2 \le \norm{F}_2$.
\end{proof}
\begin{theorem}
The spectral norm $\norm{F}_2$ can be bounded by
\begin{align}
\norm{F}_2 &\le \min \Big\{\sqrt{hw}\norm{\mat{r}}_2, \sqrt{hw}\norm{\mat{l}}_2 \Big\}\\
\norm{F}_2 &\le \max_{\hbox{\boldmath$\omega$}} \sqrt{\norm{F(\hbox{\boldmath$\omega$})}_1 \norm{F(\hbox{\boldmath$\omega$})}_{\infty}}\\
\norm{F}_2 &\le \sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} \norm{\mat{t}_{k,l}}_2
\end{align}
where $\mat{r} \in \mbox{\bb R}^{hc_{out} \times wc_{in}}$ is a $c_{out} \times c_{in}$ block matrix with $(c,d)$-th block being $\mat{k}_{c,d,:,:} \in \mbox{\bb R}^{h \times w}$ and $\mat{l} \in \mbox{\bb R}^{wc_{out} \times hc_{in}}$ is a $c_{out} \times c_{in}$ block matrix with $(c,d)$-th block being $\mat{k}_{c,d,:,:}^{\scriptscriptstyle\mathsf{T}} \in \mbox{\bb R}^{w \times h}$.
\end{theorem}
\begin{proof}
Let $z_1=e^{\jmath \omega_1}$ and $z_2 = e^{\jmath \omega_2}$. The $(c,d)$-th element of the spectral density matrix $F(\omega_1,\omega_2)$ can be rewritten as
\begin{align}
F_{c,d}(z_1,z_2)=\sum_{k=-h_1}^{h_2} \sum_{l=-w_1}^{w_2} t_{c,d}^{k,l} z_1^k z_2^l.
\end{align}
which is a polynomial with respect to $z_1$ and $z_2$.
Let $\mat{r}_{c,d}=[t_{c,d}^{k,l}]_{k,l} \in \mbox{\bb R}^{h \times w}$, $\vect{z}_1=[z_1^{-h_2},\dots,z_1^{h_1}]$ and $\vect{z}_2=[z_2^{-w_2},\dots,z_2^{w_1}]$. Thus, we can represent $F_{c,d}$ in the following two ways.
\begin{align}
F_{c,d}=\vect{z}_1 \mat{r}_{c,d} \vect{z}_2^{\scriptscriptstyle\mathsf{T}} =\vect{z}_2 \mat{r}^{{\scriptscriptstyle\mathsf{T}}}_{c,d} \vect{z}_1^{\scriptscriptstyle\mathsf{T}}.
\end{align}
Hence, the spectral density matrix $F$ can be represented as
\begin{align}
F &= (\mat{{I}}_{r} \otimes \vect{z}_1) \mat{r}
(\mat{{I}}_{s} \otimes \vect{z}_2^{{\scriptscriptstyle\mathsf{T}}})\\
&= (\mat{{I}}_{r} \otimes \vect{z}_2) \mat{l}
(\mat{{I}}_{s} \otimes \vect{z}_1^{{\scriptscriptstyle\mathsf{T}}})
\end{align}
where
\begin{align}
\mat{r} = \begin{bmatrix}
\mat{r}_{1,1} & \mat{r}_{1,2} & \cdots & \mat{r}_{1,s}\\
\mat{r}_{2,1} & \cdots & \cdots & \mat{r}_{2,s}\\
\vdots & \vdots & \vdots &\vdots \\
\mat{r}_{r,1} & \cdots & \cdots & \mat{r}_{r,s}
\end{bmatrix}, \hspace{20pt}
\mat{l} = \begin{bmatrix}
\mat{r}_{1,1}^{\scriptscriptstyle\mathsf{T}} & \mat{r}_{1,2}^{\scriptscriptstyle\mathsf{T}} & \cdots & \mat{r}_{1,s}^{\scriptscriptstyle\mathsf{T}}\\
\mat{r}_{2,1}^{\scriptscriptstyle\mathsf{T}} & \cdots & \cdots & \mat{r}_{2,s}^{\scriptscriptstyle\mathsf{T}}\\
\vdots & \vdots & \vdots &\vdots \\
\mat{r}_{r,1}^{\scriptscriptstyle\mathsf{T}} & \cdots & \cdots & \mat{r}_{r,s}^{\scriptscriptstyle\mathsf{T}}
\end{bmatrix},
\end{align}
with $\mat{r} \in \mbox{\bb R}^{rh \times sw}$ and $\mat{l} \in \mbox{\bb R}^{rw \times sh}$.
Note that
\begin{align}
{(\mat{{I}}_{r} \otimes \vect{z}_1)} (\mat{{I}}_{r} \otimes \vect{z}_1)^{\H} = h \mat{{I}}_r\\
(\mat{{I}}_{s} \otimes \vect{z}_2) (\mat{{I}}_{s} \otimes \vect{z}_2)^{\H} = w \mat{{I}}_s
\end{align}
where the columns are orthogonal. So, we have
\begin{align}
\norm{F}_2 \le \sqrt{hw}\norm{\mat{r}}_2, \quad \norm{F}_2 \le \sqrt{hw}\norm{\mat{l}}_2.
\end{align}
This gives us the first bound.
For the second bound, given any $\hbox{\boldmath$\omega$} \in [-\pi,\pi]^2$, we have
\begin{align}
\Norm{F(\hbox{\boldmath$\omega$})}_2 \le \norm{F(\hbox{\boldmath$\omega$})}_1 \norm{F(\hbox{\boldmath$\omega$})}_{\infty}.
\end{align}
As $\norm{F}_2 \le \max_{\hbox{\boldmath$\omega$}} \norm{F(\hbox{\boldmath$\omega$})}_2$, we have the second spectral norm bound.
For the third bound, we have
\begin{align}
\norm{F(\omega_1,\omega_2)}_2 &= \norm{\sum_{k} \sum_{l} \mat{t}_{k,l} e^{\jmath(k\omega_1+l\omega_2)}}_2\\
&\le \sum_{k} \sum_{l} \norm{\mat{t}_{k,l}}_2 \abs{e^{\jmath(k\omega_1+l\omega_2)}}\\
&= \sum_{k} \sum_{l} \norm{\mat{t}_{k,l}}_2,
\end{align}
where the inequality is due to Cauchy--Schwarz inequality.
\end{proof}
\section{Extensions and Discussions}
\label{sec:discussion}
Some more general cases are discussed with respect to larger stride size, higher dimensional linear convolution, and multiple convolutional layers in linear networks without activation functions and pooling layers.
\subsection{Stride Larger Than 1}
In previous sections, we were dedicated to linear convolution with stride size 1. When the stride size $g$ is larger than 1, i.e., $g > 1$, the linear transformation matrix $\mat{t}$ becomes a block $g$-Toeplitz matrix, denoted by $\mat{t}^g$. For simplicity, we consider the same stride side on both horizontal and vertical directions. Thus, we have $\mat{t}^g=[\mat{t}_{gk}]_{k=0}^{n-1}$ where $\mat{t}_{gk}=[\mat{t}_{gk,gl}]_{l=0}^{n-1}$ with $\mat{t}_{k,l}$ defined in \eqref{eq:toe-blocks}.
According to \cite{ngondiep2010spectral}, we have an analogous result to Theorem \ref{thm:szg-limit}.
Let $F:[-\pi,\pi]^2 \mapsto \mathbb{C}^{r \times s}$ be a matrix-valued function, subject to $F \in \mathcal{L}^2([-\pi,\pi]^2)$. The linear transformation matrix $\mat{t}^g$ with stride $g$ converges to the generating function $F$, i.e., $\mat{t}^g \sim_{\sigma} F(\hbox{\boldmath$\omega$},\vect{m})$, where
\begin{align}
F(\hbox{\boldmath$\omega$},\vect{m})= \sqrt{\frac{1}{g^2} \sum_{m_1=0}^{g-1} \sum_{m_2=0}^{g-1} f^2(\hbox{\boldmath$\omega$},\vect{m})}
\end{align}
if $\vect{m}=(m_1,m_2) \in [0,\frac{1}{g}]^2$ and 0 otherwise, with
\begin{align*}
f(\hbox{\boldmath$\omega$},\vect{m}) = \sum_{k}^{} \sum_{l}^{} {\mat{t}}_{gk,gl} e^{\jmath \frac{1}{g}(k (\omega_1+2 \pi m_1) + l (\omega_2 + 2 \pi m_2))}.
\end{align*}
By this, the singular value distribution of $\mat{t}^g$ can be alternatively studied on the generating function $F(\hbox{\boldmath$\omega$},\vect{m})$.
\subsection{Higher Dimensional Convolution}
According to \cite{Oudin2009}, a block multi-level Toeplitz matrix $\mat{t}=\{\mat{t}_{\vect{i}-\vect{j}}\}_{\vect{i},\vect{j}=\mat{\mathrm{1}}}^\vect{n}$ with $\vect{i}=(i_1,\dots,i_d)$, $\vect{j}=(j_1,\dots,j_d)$, and $\vect{n}=(n_1,\dots,n_d)$, it can be alternatively represented as
\begin{align}
\mat{t} = \sum_{\abs{k_1}<n_1} \dots \sum_{\abs{k_d}<n_d} [\mat{j}_{n_1}^{(k_1)} \otimes \dots \otimes \mat{j}_{n_d}^{(k_d)}] \otimes \mat{t}_{\vect{k}}
\end{align}
where $\mat{j}_{n_j}^{(k_j)}$ is a $n_j \times n_j$ binary matrix with $(p,q)$-th entry being 1 of $p-q=k_j$ and 0 elsewhere, and
\begin{align}
\mat{t}_{\vect{k}}=\frac{1}{(2\pi)^d} \int_{\Omega} F(\hbox{\boldmath$\omega$}) e^{-\jmath <\vect{k},\hbox{\boldmath$\omega$}>} d \hbox{\boldmath$\omega$}
\end{align}
with $\Omega=[-\pi,\pi]^d$, $\vect{k}=(k_1,\dots,k_d)$, $\hbox{\boldmath$\omega$}=(\omega_1,\dots,\omega_d)$ and $<\vect{k},\hbox{\boldmath$\omega$}>=\sum_{j=1}^d k_j \omega_j$. Then it follows that Theorem 1 can be generalized to $d$-dim linear convolutional layers
\begin{align} \label{eq:multi-level-theorem-1}
\MoveEqLeft \lim_{\vect{n} \to \infty} \frac{1}{N} \sum_{j=1}^{\min\{r,s\} N} \Phi(\sigma_j(\mat{t}))
= \frac{1}{(2\pi)^d} \int_{\Omega} \sum_{j=1}^{\min\{r,s\}} \Phi(\sigma_j(F(\hbox{\boldmath$\omega$}))) d\hbox{\boldmath$\omega$}
\end{align}
with $N=\prod_{i=1}^d n_i$,
for which the asymptotic singular value distribution of higher dimensional linear convolutional layers can be studied through $F: [-\pi,\pi]^d \mapsto \mbox{\bb C}^{r \times s}$.
\subsection{Multiple Linear Convolutional Layers}
The collective effect of multiple linear convolutional layers without activation function or pooling layers in CNNs can be seen as the product of the linear transformation matrices of multiple convolutional layers.
For convolutional layers, denote by $\mat{t}(F_i)$ the linear transformation matrix generated from the spectral density matrix $F_i: [-\pi,\pi]^2 \mapsto \mbox{\bb C}^{r \times s}$, for $i=1,\dots,M$. It follows from \cite{barbarino2020block}[Theorem 2.46] that
\begin{align}
\lim_{n \to \infty} \frac{1}{n^2} \norm{\prod_{i=1}^M \mat{t}(F_i) - \mat{t}(\prod_{i=1}^M F_i)}_1 = 0
\end{align}
which means that the product of Toeplitz matrices is asymptotically equal to the Toeplitz matrix generated by the product of all generating functions associated to each linear convolutional layer.
By this, the spectral analysis of $M$ linear convolutional layers can be alternatively studied on the product of generating functions $\prod_{i=1}^M F_i$. %
\section{Additional Experimental Results}
\label{sec:experiments}
\subsection{Singular Value Approximation}
\label{sec:experiments-sva}
To evaluate the accuracy of our singular value approximation method, we consider three different types of weights that are: (1) randomly generated according to uniform and Gaussian distributions (as CNNs are usually initialized), (2) extracted from pre-trained networks on ImageNet dataset (as CNNs finally converge), and (3) extracted from the training process of ResNets on CIFAR-10 dataset (as CNNs are updated with training epochs).
For simplicity, we set $h_1=h_2$ and $w_1=w_2$, and the input size $n \times n$ per channel is set to $10 \times 10$.
In what follows, the plots present the $(i-1)n+1$-th largest singular values ($i \in [n]$) of four methods with different filter sizes.
It is worth noting that each singular value shown in the figures represents a cluster of singular values with similar behavior. For instance, the first spike shows the largest singular value, and the following $n-1$ large singular values between the first and the second spikes, which have not been shown in the figure, have similar approximation behavior.
\subsubsection{Randomly Generated Weights}
\begin{figure}[t]
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{uniform_rnd_8833-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{uniform_rnd_8855-eps-converted-to}}
\vspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{normal_rnd_8833-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{normal_rnd_8855-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\vspace{0.1in}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. With input size per channel $10 \times 10$, only $10$ singular values are shown, each of which represents the behavior of a cluster of singular values. Four types of convolutional filters are considered from left to right with sizes $8 \times 8 \times 3 \times 3$ (uniform distribution), $8 \times 8 \times 5 \times 5$ (uniform distribution), $8 \times 8 \times 3 \times 3$ (Gaussian distribution), and $8 \times 8 \times 5 \times 5$ (Gaussian distribution), respectively. }
\label{fig:APP-Fig-1}
\end{center}
\vskip -0.35in
\end{figure}
We consider two distributions used for weights initialization.
It has been observed in \cite{Thoma:2017} that weight of CNN layers are located within [-0.5, 0.5]. As such, we randomly generate the weights of convolutional filters following uniform distribution in [-0.5, 0.5]. In addition, the Gaussian distribution initialized weights with zero mean and unit variance are also considered.
First, as in the main text, we illustrate the singular value approximation accuracy among three methods - circular approximation, uniform sampling, and quantile interpolation - against the exact method.
Fig. \ref{fig:APP-Fig-1} presents the $(i-1)n+1$-th largest singular values ($i \in [n]$) of four methods with four different filter sizes.
Differently from the observations in the main text, we observe that, (1) for uniformly distributed weights, quantile interpolation substantially improve over the circular approximation on the larger singular values (where each spike in the figure represents a number of them with similar behavior), while the smaller singular values obtained from both the circular approximation and quantile interpolation approach the exact values; (2) for Gaussian distributed weights, while quantile interpolation has significant improvement over the circular approximation for the larger singular values (including the largest one and those that are not shown in the figure), the improvement for small singular values is little, because the circular approximation is very inaccurate and the simple adjustment of singular value distribution using linear interpolation does not improve much the accuracy. It calls for more sophisticated nonlinear interpolation methods.
Besides the singular value illustration as above, we also compute the average accuracy of different approximation methods by Monte-Carlo simulation. We randomly generate 100 different realizations, and calculate the average accuracy over these 100 realizations.
We mainly consider three input sizes $10 \times 10$, $20 \times 20$, and $10 \times 30$ with stride 1 due to limited computing resources (i.e., HP EliteBook with Intel i5 CPU and 8GB RAM). To reduce computational complexity, the inputs with larger size usually have larger stride, which can be roughly seen as a smaller input size with stride 1. Table \ref{tab:accuracy} collects the accuracy performance of different approximation methods (CA=circular approximation, QI=quantile interpolation) compared with the exact solution. As the circular approximation is identical to the uniform sampling method, we only collect the performance of circular approximation for brevity. We mainly consider the approximation error of overall singular values and the largest one, for which the overall error is defined as $\frac{\sum_j\abs{\sigma_j(\mat{t})-\hat{\sigma}_j}}{\sum_j\abs{\sigma_j(\mat{t})}}$
and the error for the first singular value is $\frac{\abs{\sigma_1(\mat{t})-\hat{\sigma}_1}}{\abs{\sigma_1(\mat{t})}}$
with $\hat{\sigma}_j$ being the approximated value by different methods.
\begin{table}[]
\caption{Accuracy of approximation methods.}
\vskip 0.1in
\centering
\begin{tabular}{c|c|c|c}
\hline
\centering
Input size & Convolutional filter size & Overall Error & Error for 1st Singular Value\\ \hline
$10 \times 10$ & $8 \times 8 \times 3 \times 3$ & CA=10.4\%, QI=8.3\% & CA=5.6\%, QI=0.9\% \\ \hline
& $8 \times 8 \times 3 \times 5$ & CA=15.6\%, QI=12.7\% & CA=10.7\%, QI=3.9\% \\ \hline
& $8 \times 8 \times 5 \times 3$ & CA=14.5\%, QI=11.3\% & CA=10.7\%, QI=3.7\% \\ \hline
& $8 \times 8 \times 5 \times 5$ & CA=20.4\%, QI=14.8\% & CA=16.1\%, QI=3.9\% \\ \hline
& $8 \times 8 \times 7 \times 7$ & CA=30.9\%, QI=23.2\% & CA=31.4\%, QI=8.7\% \\ \hline
& $8 \times 8 \times 9 \times 9$ & CA=46.4\%, QI=31.8\% & CA=51.9\%, QI=11.3\% \\ \hline
& $8 \times 8 \times 5 \times 9$ & CA=29.9\%, QI=16.4\% & CA=32.8\%, QI=9.9\% \\\hline
& $8 \times 8 \times 9 \times 5$ & CA=30.1\%, QI=17.0\% & CA=32.7\%, QI=9.3\% \\\hline
& $16 \times 3 \times 5 \times 9$ & CA=29.4\%, QI=15.3\% & CA=32.8\%, QI=10.1\% \\\hline
& $16 \times 8 \times 5 \times 9$ & CA=31.1\%, QI=18.5\% & CA=32.8\%, QI=9.8\% \\\hline
& $16 \times 16 \times 5 \times 9$ & CA=31.9\%, QI=19.9\% & CA=32.9\%, QI=9.7\% \\\hline
$20 \times 20$ & $8 \times 8 \times 5 \times 5$ & CA=9.1\%, QI=7.7\% & CA=4.3\%, QI=0.6\% \\ \hline
& $8 \times 8 \times 7 \times 7$ & CA=14.0\%, QI=11.1\% & CA=8.4\%, QI=1.5\% \\ \hline
& $8 \times 8 \times 7 \times 9$ & CA=12.5\%, QI=9.9\% & CA=11.1\%, QI=0.8\% \\ \hline
& $8 \times 8 \times 9 \times 9$ & CA=18.4\%, QI=13.2\% & CA=13.7\%, QI=3.0\% \\ \hline
$10 \times 30$ & $8 \times 8 \times 7 \times 7$ & CA=19.9\%, QI=16.9\% & CA=16.8\%, QI=11.8\% \\ \hline
& $8 \times 8 \times 5 \times 11$ & CA=13.5\%, QI=10.3\% & CA=12.6\%, QI=1.1\% \\ \hline
& $8 \times 8 \times 7 \times 11$ & CA=24.0\%, QI=19.1\% & CA=19.9\%, QI=7.6\% \\ \hline
\end{tabular}
\label{tab:accuracy}
\end{table}
It is observed from Table \ref{tab:accuracy} that for smaller filter size, e.g., $3 \times 3$, the circular approximation looks good enough, and the improvement by quantile interpolation is not much, e.g., by 2.1\% for overall performance, and by 4.7\% for the large singular value. However, as the filter size increases, e.g., $7 \times 7$, the circular approximation can be as large as 30\% away from the exact singular values, and the improvement by quantile interpolation is significant, e.g., by 7.7\% for overall performance and by 22.7\% for the largest singular value. %
As the input size increases, the approximation accuracy for both circular approximation and quantile interpolation is improved, and therefore the improvement of the latter over the former is not substantial compared with those of the smaller input size. It appears in Table \ref{tab:accuracy} that the numbers of channels of the input and the output do not have much influence on the accuracy performance.
It appears that (1) the circular approximation is sufficiently good when the input size is large and the convolutional filter size is small, and it leaves no much room to improve by quantile interpolation; (2) the quantile interpolation approach makes a difference when convolutional filter size is large, yet there is still certain gap to the exact values. This is mainly due to the simple quantile estimation method. It is expected to have larger improvement with more accurate quantile interpolation methods. We leave it to our future work.
\subsubsection{Weights from Pre-trained Networks}
In what follows, we present more experimental results on the accuracy of singular value approximation on pre-trained network models, such as VGG16, VGG19 \cite{vgg16-paper}, AlexNet, DenseNet \cite{huang2017densely}, GoogLeNet \cite{szegedy2015going}, InceptionResNetv2, Inceptionv3, and ResNets \cite{ResNet}, which are trained on ImageNet dataset.
\begin{figure}[t]
{\includegraphics[width=0.26\columnwidth]{vgg16-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg16-conv1_2-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg16-conv2_1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg16-conv2_2-eps-converted-to}}
\\
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg19-conv1_1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg19-conv1_2-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg19-conv2_1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{vgg19-conv2_2-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. For illustration, only 10 singular values are plotted, each of which represents the behavior of a cluster of singular values. Four types of convolutional filters of pre-trained VGG16 (first row) and VGG19 (second row) networks on ImageNet dataset are considered from first to last column with sizes $64 \times 3 \times 3 \times 3$ (conv1\_1), $64 \times 64 \times 3 \times 3$ (conv1\_2), $128 \times 64 \times 3 \times 3$ (conv2\_1), and $128 \times 128 \times 3 \times 3$ (conv2\_2), respectively. }
\label{fig:Fig-2}
\end{center}
\vskip -0.35in
\end{figure}
Figure \ref{fig:Fig-2} presents the singular values of pretrained VGG models, where most convolutional layers have size $3 \time 3$ filters. With respect to singular values, VGG16 and VGG19 have similar spectral behavior. The improvement of the quantile interpolation over the circular approximation lies in small singular values, while for the largest singular value the improvement is subtle. %
It is also observed that, as the number of input/output channels increases, the singular values are decreasing.
\begin{figure}[t]
\vskip 0.1in
{\includegraphics[width=0.26\columnwidth]{alexnet-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{alexnet-conv2_2-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{densenet201-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{googlenet-inception3b-eps-converted-to}}
\\
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{InceptionResNetv2-conv2d_8-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{Inceptionv3-conv2d_8-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{Inceptionv3-conv2d_15-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{Inceptionv3-conv2d_22-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\vspace{-0.1in}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. For illustration, only 10 singular values are plotted, each of which represents the behavior of a cluster of singular values. Four types of convolutional filters of pre-trained networks on ImageNet dataset are considered from top left to bottom right with sizes $96 \times 3 \times 11 \times 11$ (AlexNet conv1), $128 \times 48 \times 5 \times 5$ (AlexNet conv2), $64 \times 3 \times 7 \times 7$ (DenseNet201 conv1), $96 \times 32 \times 5 \times 5$ (GoogLeNet Inception\_3b), $64 \times 48 \times 5 \times 5$ (InceptionResNetv2 conv2d\_8), $64 \times 48 \times 5 \times 5$ (Inceptionv3 conv2d\_8), $64 \times 48 \times 5 \times 5$ (Inceptionv3 conv2d\_15), $64 \times 48 \times 5 \times 5$ (Inceptionv3 conv2d\_22), respectively. }
\label{fig:Fig-3}
\end{center}
\vskip -0.35in
\end{figure}
\begin{figure}[t]
\vskip 0.1in
{\includegraphics[width=0.26\columnwidth]{resnet-18-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-18-res2a-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-50-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-50-res2a-eps-converted-to}}
\\
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-101-conv1-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-101-res2a-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-101-res2b-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-101-res2c-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\vspace{-0.1in}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. For illustration, only 10 singular values are plotted, each of which represents the behavior of a cluster of singular values. Four types of convolutional filters of pre-trained ResNets are considered from top left to bottom right with sizes $64 \times 3 \times 7 \times 7$ (ResNet-18 conv1), $64 \times 64 \times 3 \times 3$ (ResNet-18 res2a), $64 \times 3 \times 7 \times 7$ (ResNet-50 conv1), and $64 \times 64 \times 3 \times 3$ (ResNet-50 res2a), $64 \times 3 \times 7 \times 7$ (ResNet-101 conv1), $64 \times 64 \times 3 \times 3$ (ResNet-101 res2a), $64 \times 64 \times 3 \times 3$ (ResNet-101 res2b), $64 \times 64 \times 3 \times 3$ (ResNet-101 res2c), respectively. }
\label{fig:Fig-4}
\end{center}
\vskip -0.35in
\end{figure}
Figure \ref{fig:Fig-3} presents the singular values from another set of pre-trained networks, for which we select the filters with larger size, i.e., $h=w=5,7,11$. It is observed that the improvement of the quantile interpolation over the circular approximation is enhanced for the filters with larger size. The improvement of the largest singular values is more significant than VGG networks.
Figure \ref{fig:Fig-4} is dedicated to ResNets in which we present singular values for the convolutional layers in ResNet-18, ResNet-50, and ResNet-101. For the filters with size $3 \times 3$, the improvement of the largest singular value is subtle, while the smaller singular values contribute much on the improvement, as observed in VGG networks. For the filter with size $7 \times 7$, the major improvement of singular values occurs in the intermediate ones. The negative result is that it seems both circular approximation and quantile interpolation do not work well for ResNet-101 the convolutional layer res2c. The quantile approach with linear interpolation relies much on the circular approximation - if the latter does not work well, so does the former very likely. It may require the nonlinear interpolation.
Table \ref{tab:running} summarizes the accuracy and running time of different singular value computation methods on various convolutional layers of pre-trained networks on ImageNet dataset. The experiments have been conducted in MATLAB on an HP EliteBook (Intel i5 CPU with 8G RAM). The sizes of different filters can be referred as above. The numbers ``$a/b$'' should read as $a\%$ difference from the exact method and the running time is $b$ seconds. Note that for the accuracy we consider the sum of all singular values and use the exact method as the reference. For instance, for the convolutinal layer named ``conv1\_1'' in VGG16 model, the running time of the exact method is 0.1223 seconds, compared with 0.0106 seconds and 0.0245 seconds for uniform sampling and quantile interpolation methods, respectively. For both the circular approximation and uniform sampling, the accuracy of all singular values is 7.51\% larger than the exact value computed by the exact method, while the quantile interpolation reduces such a difference to 1.47\%.
It can be observed from experimental results that quantile approximation always outperforms the circular approximation by more than 5\% in overall accuracy for most cases at the expense of extra running time. The running time is negligible compared with that using SVD in the exact method.
The most significant improvement in approximation accuracy is for AlexNet conv1 with filter size $96\times3\times11\times11$. This confirms our observation earlier that the quantile interpolation has more substantial improvement over the circular approximation for the larger filter size. The least improvement happens for ResNet-101 Res2b/c with filter size $3 \times 3$.
\begin{table}[t]
\caption{Comparison of approximation accuracy and running time.}
\label{tab:running}
\vskip -0.1in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccc}
\toprule
Filter & Exact & Sampling & Quantile\\
\midrule
VGG16 conv1\_1 ($64\times3\times3\times3$) & -/0.1233 & 7.51\%/0.0106 & 1.47\%/0.0245 \\
VGG16 conv1\_2 ($64\times64\times3\times3$) & -/120.53 & 7.79\%/0.1043 & 1.96\%/0.2269 \\
VGG16 conv2\_1 ($128\times64\times3\times3$)& -/145.10 & 7.31\%/0.3483 & 3.32\%/0.5894 \\
VGG16 conv2\_2 ($128\times128\times3\times3$) & -/958.84 & 7.86\%/0.5402 & 3.60\%/0.9414 \\
VGG19 conv1\_1 ($64\times3\times3\times3$) &-/0.1591 & 7.37\%/0.0079 & 1.36\%/0.0173 \\
VGG19 conv1\_2 ($64\times64\times3\times3$) &-/120.87 & 7.81\%/0.0956 & 2.08\%/0.2482 \\
VGG19 conv2\_1 ($128\times64\times3\times3$) & -/141.39 & 7.32\%/0.2218 & 3.47\%/0.4453 \\
VGG19 conv2\_2 ($128\times128\times3\times3$) & -/964.5 & 7.88\%/0.4840 & 3.90\%/0.7565 \\
AlexNet conv1 ($96\times3\times11\times11$) & -/0.3429 & 22.47\%/0.1218 & 10.56\%/0.1252 \\
AlexNet conv2 ($128\times48\times5\times5$) & -/44.03& 9.65\%/0.2835 & 3.90\%/0.4551\\
DenseNet201 conv1 ($64\times3\times7\times7$) & -/0.1481 & 11.79\%/0.0347 & 5.11\%/0.0429 \\
GoogLeNet conv1 ($64\times3\times7\times7$) & -/0.1352 & 14.13\%/0.0248 & 6.71\%/0.0340 \\
GoogLeNet Inception\_3a ($32\times16\times5\times5$) & -/2.6050 & 10.56\%/0.0341 & 5.30\%/0.0686 \\
GoogLeNet Inception\_3b ($96\times32\times5\times5$) & -/15.869 & 13.53\%/0.1138 & 8.68\%/0.1764 \\
InceptionResNetv2 conv2d\_8 ($64\times48\times5\times5$) & -/22.997 & 10.15\%/0.1218 & 4.81\%/0.2149 \\
Inceptionv3 conv2d\_8 ($64\times48\times5\times5$) & -/18.055 & 9.36\%/0.1105 & 4.99\%/0.2049 \\
Inceptionv3 conv2d\_15 ($64\times48\times5\times5$) & -/18.281 & 10.32\%/0.1143 & 6.40\%/0.2054 \\
Inceptionv3 conv2d\_22 ($64\times48\times5\times5$) & -/18.656 & 10.17\%/0.1808 & 6.50\%/0.2931 \\
ResNet-18 conv1 ($64\times3\times7\times7$) & -/0.1416 & 12.23\%/0.0240 & 5.74\%/0.0358 \\
ResNet-18 res2a ($64\times64\times3\times3$) & -/122.55 & 5.59\%/0.0886 & 2.46\%/0.2086 \\
ResNet-50 conv1 ($64\times3\times7\times7$) & -/0.1463 & 13.39\%/0.0291 & 6.78\%/0.0383 \\
ResNet-50 res2a ($64\times64\times3\times3$) & -/133.45 & 6.26\%/0.0918 & 3.08\%/0.2106 \\
ResNet-101 conv1 ($64\times3\times7\times7$) & -/0.1412 & 13.63\%/0.0274 & 7.25\%/0.0395 \\
ResNet-101 res2a ($64\times64\times3\times3$) & -/131.003 & 6.04\%/0.0984 & 2.94\%/0.2163 \\
ResNet-101 res2b ($64\times64\times3\times3$) & -/126.58 & 6.66\%/0.0929 & 4.47\%/0.2147 \\
ResNet-101 res2c ($64\times64\times3\times3$) & -/120.56 & 7.18\%/0.0946 & 5.49\%/0.2144 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.25in
\end{table}
\subsubsection{Weights from Training Process}
Figure \ref{fig:Fig-5} presents the singular value approximation for the weights extracted from the training process of ResNet-20 on CIFAR-10 dataset. We consider the filters of four convolutional layers after 10 and 100 training epochs. It is observed that, as the larger singular values increase with training epochs, the improvement of quantile approach over circular approximation is enlarged, while the improvement of small singular values is moderate during the training.
It suggests that for ResNet models, while the accuracy of circular approximation is relatively reasonable for smaller singular values, it calls for more accurate approximation methods for larger singular values.
\begin{figure}[t]
\vskip 0.1in
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer1-conv1-10epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer1-conv1-100epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer1-conv2-10epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer1-conv2-100epoch-eps-converted-to}}
\hspace{-12pt}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer2-conv1-10epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer2-conv1-100epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer3-conv2-10epoch-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.26\columnwidth]{resnet-20-layer3-conv2-100epoch-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\caption{Exact and approximated singular values of linear convolutional layers arranged in descending order. For illustration, only 10 singular values are plotted, each of which represents the behavior of a cluster of singular values. Four types of convolutional filters of the ResNet-20 network trained on CIFAR-10 dataset are considered with sizes $16 \times 16 \times 3 \times 3$ (layer1-conv1), $16 \times 16 \times 3 \times 3$ (layer1-conv2), $32 \times 16 \times 3 \times 3$ (layer2-conv1), and $64 \times 64 \times 3 \times 3$ (layer3-conv2), respectively. For each filter, the plot after 10 training epochs comes first, followed by the one after 100 training epochs.}
\label{fig:Fig-5}
\end{center}
\vskip -0.35in
\end{figure}
\subsection{Spectral Norm Bounding}
\label{sec:experiments-snr}
\subsubsection{Accuracy vs. Running Time}
As did in the main text, we evaluate the accuracy of spectral norm bound \eqref{eq:upper-bound-F-2norm}, \eqref{eq:upperibound-oneinfnorm}, and \eqref{eq:upperibound-2norm} against the running time for different pre-trained convolutional layers with input size $10 \times 10$. The experiments are conducted on an HP EliteBook with Intel i5 CPU.
Table \ref{tab:running-spectral} presents the accuracy and the running time for different convolutional layers, where $a/b$ reads as the spectral norm bound is $a$ times of the circular approximation and the computation takes $b$ seconds.
We have the similar observations as those in the main text. In particular, the spectral norm bound \eqref{eq:upperibound-2norm} has comparable accuracy as \eqref{eq:upper-bound-F-2norm} but the computation of the former takes much less time than that of the latter. The computation of matrix norms uses the function ``norm'' in MATLAB.
\begin{table}[t]
\caption{Comparison of spectral norm bounding accuracy and running time.}
\label{tab:running-spectral}
\vskip -0.1in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{lcccc}
\toprule
Filter & \eqref{eq:upper-bound-F-2norm} & \eqref{eq:upperibound-oneinfnorm} & \eqref{eq:upperibound-2norm}\\
\midrule
VGG16 conv1\_1 ($64\times3\times3\times3$) & 1.3974/0.0407 & 1.6218/0.0221 & 1.9398/0.0011 \\
VGG16 conv1\_2 ($64\times64\times3\times3$) & 1.3726/0.0954 & 2.5652/0.0600 & 1.6452/0.0037 \\
VGG16 conv2\_1 ($128\times64\times3\times3$) & 1.3146/0.1336 & 3.1937/0.0771 & 2.0145/0.0064 \\
VGG16 conv2\_2 ($128\times128\times3\times3$) & 1.3680/0.2517 & 4.4693/0.1715 & 1.9320/0.0136 \\
VGG19 conv1\_1 ($64\times3\times3\times3$) & 1.4092/0.0200 & 1.6601/0.0194 & 1.9479/0.0012 \\
VGG19 conv1\_2 ($64\times64\times3\times3$) & 1.3791/0.0601 & 2.6451/0.0520 & 1.6618/0.0026 \\
VGG19 conv2\_1 ($128\times64\times3\times3$) & 1.3995/0.1239 & 3.1582/0.0908 & 2.0804/0.0057 \\
VGG19 conv2\_2 ($128\times128\times3\times3$) & 1.3692/0.2485 & 4.2432/0.1524 & 1.9724/0.0122 \\
AlexNet conv1 ($96\times3\times11\times11$) & 2.9577/0.0230 & 2.3555/0.1005 & 4.9880/0.0032\\
AlexNet conv2 ($128\times48\times5\times5$) & 2.0185/0.1143 & 3.7714/0.1284 & 2.6927/0.0105\\
DenseNet201 conv1 ($64\times3\times7\times7$) & 2.3323/0.0114 & 2.0559/0.0421 & 3.4654/0.0009 \\
GoogLeNet conv1 ($64\times3\times7\times7$) & 2.4652/0.0136 & 2.3505/0.0328 & 3.8639/0.0007 \\
GoogLeNet Inception\_3a ($32\times16\times5\times5$) & 1.6482/0.0123 & 2.0477/0.0383 & 2.6716/0.0012 \\
GoogLeNet Inception\_3b ($96\times32\times5\times5$) & 1.3553/0.0546 & 2.5794/0.0806 & 2.1185/0.0035 \\
InceptionResNetv2 conv2d\_8 ($64\times48\times5\times5$) & 1.3767/0.0643 & 2.6342/0.0958 & 1.7666/0.0046 \\
Inceptionv3 conv2d\_8 ($64\times48\times5\times5$) & 1.6534/0.0665 & 2.8253/0.0728 & 2.3093/0.0051 \\
Inceptionv3 conv2d\_15 ($64\times48\times5\times5$) & 2.0562/0.0568 & 3.2000/0.0891 & 2.9470/0.0052 \\
Inceptionv3 conv2d\_22 ($64\times48\times5\times5$) & 2.2887/0.0552 & 4.1094/0.0840 & 3.8964/0.0071 \\
ResNet-18 conv1 ($64\times3\times7\times7$) & 3.0015/0.0130 & 2.1353/0.0251 & 4.3294/0.0009 \\
ResNet-18 res2a ($64\times64\times3\times3$) & 1.6284/0.0905 & 3.2053/0.0543 & 2.1965/0.0035 \\
ResNet-50 conv1 ($64\times3\times7\times7$) & 2.9348/0.0162 & 2.1886/0.0336 & 3.9626/0.0004 \\
ResNet-50 res2a ($64\times64\times3\times3$) & 1.3949/0.0609 & 2.9974/0.0451 & 2.0898/0.0044 \\
ResNet-101 conv1 ($64\times3\times7\times7$) & 2.9633/0.0095 & 1.9349/0.0302 & 3.8502/0.0004 \\
ResNet-101 res2a ($64\times64\times3\times3$) & 1.4508/0.0615 & 2.9684/0.0500 & 2.1092/0.0032 \\
ResNet-101 res2b ($64\times64\times3\times3$) & 1.7636/0.0631 & 3.3880/0.0481 & 2.5061/0.0026 \\
ResNet-101 res2c ($64\times64\times3\times3$) & 1.6225/0.0643 & 3.5631/0.0449 & 2.3516/0.0027 \\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.15in
\end{table}
\subsubsection{Regularization}
We use spectral norm bounds as regularizers during the training of ResNet-20 model on CIFAR-10 dateset.
According to the accuracy and running time of different spectral norm bounds in Table \ref{tab:running-spectral}, we place our focus on the first \eqref{eq:upper-bound-F-2norm} and the third bounds \eqref{eq:upperibound-2norm} for spectral regularization.
Given the training data samples $\{(\vect{x}_i,y_i)\}_{i=1}^N$ drown from an unknown distribution of $(\vect{x},y)$ for training an $L$-layer deep neural network model $y=f_{\Theta}(\vect{x})$ with parameters $\Theta$, the spectral regularization is to minimize the following objective function
\begin{align}
\min_{\Theta} \ \E_{(\vect{x},y)} \ell (f_{\Theta}(\vect{x}),y) + \beta \sum_{j=1}^{L} R^u_j
\end{align}
where $\ell(f)$ is the loss function of the model for training, $R^u_j$ is a regularization term using the spectral norm upper bounds of the $j$-th layer, e.g., \eqref{eq:upper-bound-F-2norm}-\eqref{eq:upperibound-2norm}, and $\beta>0$ is a constant to balance between the loss function and the spectral norm regularizer.
In the experiments, the cross entropy function is chosen as the loss function. For the $j$-th convolutional layer, the regularization term $R^u_j$ is the spectral norm upper bounds chosen from \eqref{eq:upper-bound-F-2norm} with $R^u_j=\sqrt{hw}\min\{\norm{\mat{r}}_2, \norm{\mat{l}}_2\}$ and \eqref{eq:upperibound-2norm} with $R^u_j=\sum_k \sum_l \norm{\mat{t}_{k,l}}_2$, respectively. For the fully-connected layers, $R^u_j$ is directly chosen as the exact spectral norm of the weight matrices. As both the upper bounds in \eqref{eq:upper-bound-F-2norm} and \eqref{eq:upperibound-2norm} are in the form of spectral norm, we adopt power method to compute it in the forward propagation.
As shown in the proof of Theorem \ref{theorem:bounding-norm}, $\mat{r}$ and $\mat{l}$ are reshaped matrices of the convolutional filter $\mat{k}$ with sizes $hc_{out} \times wc_{in}$ and $wc_{out} \times hc_{in}$, respectively, in contrast to the set of $hw$ matrices $\{\mat{t}_{k,l}\}$ with size $c_{out} \times c_{in}$ each rearranged from $\mat{k}$. For a matrix $\mat{a} \in \mbox{\bb R}^{m \times n}$, the computational complexity of power method is $O(mn)$. While both bounds \eqref{eq:upper-bound-F-2norm} and \eqref{eq:upperibound-2norm} have the same level of computational complexity $O(hwc_{out}c_{in})$, it turns out computing \eqref{eq:upperibound-2norm} with power method is much faster as the matrices has smaller size.
In the backward propagation, the derivative of spectral norms of a matrix $\mat{a}$ can be computed as
$
\nabla_{\mat{a}} \norm{\mat{a}}_2 = \vect{v}_1 \vect{u}_1^{\scriptscriptstyle\mathsf{T}}
$
where $\vect{u}_1$ and $\vect{v}_1$ are the left and right singular vectors corresponding to the largest singular value, respectively.
Such a derivative is used to update weights for SGD in the backward propagation.
Due to limited computing resource, we focus on the training and testing of the ResNet-20 model on CIFAR-10 dataset. The ResNet-20 model has 20 convolutional layers, most of which have a $3 \times 3$ filter. The CIFAR-10 dataset consists of 50,000 training and 10,000 testing images with size $32 \times 32$ in 10 classes.
The batch size is 128, and the learning rate is initialized as 0.1 and changed to 0.01 after 100 training epochs. The weight decay is set to 0, and the momentum is 0.9.
The final prediction accuracy is collected after in total 150 training epochs.
\begin{figure}[t]
\vskip 0.1in
\hspace{-12pt}
{\includegraphics[width=0.55\columnwidth]{trainingloss-eps-converted-to}}
\hspace{-12pt}
{\includegraphics[width=0.55\columnwidth]{prec-eps-converted-to}}
\hspace{-12pt}
\begin{center}
\vspace{-0.1in}
\caption{The training loss (left) and test accuracy (right) versus the number of training epochs for ResNet-20 on CIFAR-10 dataset with or without regularization using spectral norm bounds \eqref{eq:upper-bound-F-2norm} and \eqref{eq:upperibound-2norm} with $\beta=0.0014$.}
\label{fig:Fig-6}
\end{center}
\vskip -0.35in
\end{figure}
For comparison, we use the case with no regularization ($\beta=0$), which has a test accuracy 89.67\%, and the case with regularization ($\beta=0.0014$) using spectral norm bound \eqref{eq:upper-bound-F-2norm}, which has a test accuracy 90.77\%, as references.
Figure \ref{fig:Fig-6} presents the training loss and test accuracy versus the number of training epochs for ResNet-20 on CIFAR-10 dataset. The training loss keeps decreasing and becomes stable after 120 epochs with a smaller learning rate. Note that the original case with no regularization term has the smallest training loss, and the upper bound \eqref{eq:upperibound-2norm} has a larger training loss because it is less tighter than \eqref{eq:upper-bound-F-2norm}. The test accuracy has a similar behavior, and the regularizer using \eqref{eq:upper-bound-F-2norm} has a higher accuracy (0.3\%) than \eqref{eq:upperibound-2norm}, due to the more tighter upper bound. Both spectral norm regularizers have improvement, 1.1\% with \eqref{eq:upper-bound-F-2norm} as the regularizer and 0.8\% with \eqref{eq:upperibound-2norm} as the regularizer, over the the one with no regularizer, which demonstrates the effective of spectral regularization in enhancing generalization performance.
\begin{table}[t]
\caption{Comparison of test accuracy with spectral norm regularization.}
\label{tab:prediction}
\vskip -0.1in
\begin{center}
\begin{small}
\begin{sc}
\begin{tabular}{c|c|c|c|c}
\toprule
$\beta$ & 0.0008 & 0.001 & 0.0014 & 0.0018\\
\midrule
Accuracy & 90.40\% & 90.35\% & {\bf 90.48\%} & 90.24\%\\
\bottomrule
\end{tabular}
\end{sc}
\end{small}
\end{center}
\vskip -0.25in
\end{table}
Table \ref{tab:prediction} collects the test accuracy with regularization using the spectral norm bound \eqref{eq:upperibound-2norm} with different values of $\beta$.
In addition to the observations in the main text, we observe that different values of $\beta$ make different trade-off between loss and spectral regularization, and the choice of $\beta=0.0014$ as that in \cite{Singla2019} for ResNet-34 yields the best generalization performance.
| 2024-02-18T23:39:56.775Z | 2020-06-15T02:15:13.000Z | algebraic_stack_train_0000 | 905 | 21,828 |
|
proofpile-arXiv_065-4451 | \section{Introduction}\label{s:intro}
The identification of characteristic parameters for bifurcations is one of the key goals of bifurcation theory. For the Andronov-Hopf bifurcation from equilibria to periodic orbits, the most relevant characteristic parameter is the first Lyapunov coefficient, $\sigma_s\in\mathbb{R}$. If it is non-zero, it determines the scaling and the direction of bifurcation relative to real part of the critical eigenvalues of the linearization at the equilibrium. It is well known that the truncated normal form of the radial component on the center manifold reads
\begin{equation}
\dot{u} = \mu u +\sigma_s u^3,
\label{Npitchfork}
\end{equation}
with parameter $\mu\in\mathbb{R}$. {An excellent exposition of Andronov-Hopf bifurcation theory and applications can be found in \cite{Marsden76}, see also \cite{Guckenheimer, Kuznetsov}.}
In Figure~\ref{f:Hopf}(a,b) we plot the associated pitchfork bifurcation for different signs of $\sigma_s$.
\begin{figure}
[bt]
\begin{tabular}{cccc}
\includegraphics[width= 0.15\textwidth]{supercritical_smooth3.pdf}\hspace*{1cm}
&\includegraphics[width= 0.15\textwidth]{subcritical_smooth3n.pdf}\hspace*{1.5cm}
&\includegraphics[width= 0.15\textwidth]{supercritical_nonsmooth3.pdf}\hspace*{1cm}
& \includegraphics[width= 0.15\textwidth]{subcritical_nonsmooth3n.pdf}\\
(a) & (b) &(c) &(d)
\end{tabular}
\caption{(a) Supercritical, $\sigma_s=-1$, and (b) subcritical, $\sigma_s=1$, pitchfork bifurcation of \eqref{Npitchfork} with stable (green) and unstable (red dashed) equilibria. In (c,d) we plot the analogous `degenerate' pitchforks for the non-smooth case \eqref{Dpitchfork}, with $\sigma_{_\#}=-1 ,1$, respectively. }
\label{f:Hopf}
\end{figure}
Generically, $\sigma_s\neq0$ and the bifurcating periodic orbits either coexist with the more unstable equilibrium -- the supercritical case, $\sigma_s<0$ -- or with the more stable equilibrium -- the subcritical case, $\sigma_s>0$. This distinction is relevant for applications since the transition induced by the bifurcation is a `soft' first order phase transition in the supercritical case, while it is `hard' in the subcritical one. Indeed, the transition is `safe' in a control sense in the supercritical case and `unsafe' in the subcritical one, where the local information near the equilibrium is insufficient to determine the dynamics near the unstable equilibrium.
Therefore, a formula for the first Lyapunov coefficient is important from a theoretical as well as applied viewpoint. In generic smooth bifurcation theory, such a formula is well known in terms of quantities derived from the Taylor expansion to order three in the equilibrium at bifurcation, {e.g.,} \cite{Kuznetsov}. However, this cannot be applied in non-smooth systems. Non-smooth terms appear in models for numerous phenomena and their study has gained momentum in the past decades,
as illustrated by the enormous amount of literature, see \cite{BookAlan,KuepperHoshamWeiss2013,NonsmoothSurvey2012,TianThesis} and the references therein to hint at some; below we discuss literature that relates to our situation.
In this paper, we provide explicit formulas for the analogues of the first Lyapunov coefficient in systems with regular linear
term and Lipschitz continuous, but only piecewise smooth nonlinear terms, with jumps in derivatives across switching surfaces. We also discuss codimension-one degeneracies and the second Lyapunov coefficient. To the best of our knowledge, this analysis is new.
Such systems can be viewed as mildly non-smooth, but occur in various models, e.g., for ship maneuvering
\cite{InitialPaper,FossenHandbook,ToxopeusPaper}, which motivated the present study. Here the hydrodynamic drag force
at high-enough Reynolds number is a non-smooth function of the velocity $u$. More specifically, a dimensional and
symmetry analysis with $\rho$ being the density of water, $C_D$ the drag coefficient and $A$ the effective drag area, yields
$$F_D = -\frac{1}{2}\rho C_D A u\abs{u}.$$
Effective hydrodynamic forces among velocity components $u_i, u_j$, $1\leq i,j\leq n$, with $n$ depending on the model type, are often likewise modeled by second order modulus terms: $u_i\abs{u_j}$, cf.\ \cite{FossenHandbook}.
For illustration, let us consider the corresponding non-smooth version of \eqref{Npitchfork},
\begin{equation}
\dot{u} = \mu u +\sigma_{_\#} u\abs{u},
\label{Dpitchfork}
\end{equation}
where the nonlinear term has the odd symmetry of the cubic term in \eqref{Npitchfork} and is once continuously differentiable, but not twice. We note that in higher dimensions, the mixed nonlinear terms $u_i\abs{u_j}$ for $i\neq j$, are differentiable at the origin only.
In Figure \ref{f:Hopf}(c,d) we plot the resulting bifurcation diagrams. Compared with \eqref{Npitchfork},
the amplitude scaling changes from $\sqrt{\mu}$ to $\mu$ and the rate of convergence to the bifurcating state changes from $-2\mu$ to $-\mu$.
Indeed, in this scalar equation case, the singular coordinate change $u=\tilde u^2$ transforms \eqref{Dpitchfork} into \eqref{Npitchfork} up to time rescaling by $2$. However, there is no such coordinate change for general systems of equations with non-smooth terms of this kind.
More generally, we consider $n$-dimensional systems of ordinary differential equations (ODE) of the form
\begin{equation}\label{e:abstract0}
\dot \textbf{u}= A(\mu)\textbf{u}+G(\textbf{u}),
\end{equation}
with matrix $A(\mu)$ depending on a parameter $\mu\in\mathbb{R}$, and Lipschitz continuous nonlinear $G(\textbf{u})=\mathcal{O}(|\textbf{u}|^2)$. We shall assume the nonlinearity is smooth away from the smooth hypersurfaces $H_j$, $j=1,\ldots,n_H$, the \emph{switching surfaces}, which intersect pairwise transversally at the equilibrium point $\textbf{u}_*=0$. We assume {further that} the smoothness of $G$ extends to the boundary within each component of the complement of $\cup_{j=1}^{n_H} H_j\subset \mathbb{R}^n$.
The bifurcation of periodic orbits is -- as in the smooth case -- induced by the spectral structure of $A(\mu)$, which is (unless stated otherwise) hyperbolic except for a simple complex conjugate pair that crosses the imaginary axis away from the origin as $\mu$ crosses zero.
\medskip
Our main results may be summarized informally as follows.
We infer from the result in \cite{IntegralManifold} that the center manifold of the smooth case is replaced by a Lipschitz invariant manifold (Proposition \ref{prop:inv_man}), and directly prove that a unique branch of periodic orbits emerges at the bifurcation (Theorem \ref{t_per_orb}). Moreover, we prove that the quadratic terms of $G$ are of generalized second order modulus type if $G$ is piecewise $C^2$ smooth (Theorem \ref{t:abstractnormal}). Here the absolute value in the above terms is replaced by
\begin{align}
[u]_{\pn}^{\pp} =
\begin{cases}
p_{_+} u, & u\geq 0, \\
p_{_-} u, & u<0,
\end{cases}
\label{gen_abs_val}
\end{align}
where $p_{_-},p_{_+}\in\mathbb{R}$ are general different slopes left and right of the origin, respectively.
This already allows to express the first Lyapunov coefficient in an integral form, but its explicit evaluation is somewhat involved, so that we defer it to \S\ref{Gen_linear_part}. Instead, we start with the simpler case when $A$ is in block-diagonal form, in normal form on the center eigenspace, and of pure second order modulus form ($p_{_+}=-p_{_-}=1$). For the planar situation, we derive a normal form of the bifurcation equation with rather compact explicit coefficients using averaging theory (Theorem \ref{t_averaging}). Beyond the first Lyapunov coefficient $\sigma_{_\#}$, this includes the second Lyapunov coefficient $\sigma_2$, which becomes relevant when $\sigma_{_\#}=0$, and which explains how the smooth quadratic and cubic terms interact with the non-smooth ones in determining the bifurcation's criticality.
For refinement and generalization, and to provide direct self-contained proofs, we proceed using the Lyapunov-Schmidt reduction for the boundary value problem of periodic solutions, and refer to this as the `direct method' (\S\ref{s:direct}). We also include a discussion of the Bautin-type bifurcation in this setting, when $\sigma_{_\#}=0$. Concluding the planar case, we {generalize} the results to arbitrary $p_{_+}, p_{_-}$ (\S\ref{s:appplanar}).
These results of the planar case readily generalize to higher dimensions, $n>2$, with additional hyperbolic directions (\S\ref{s:3D}, \ref{s:nD}). In addition, we apply the direct method to the situation with an additional non-hyperbolic direction {in the sense that the linearization at bifurcation has three eigenvalues on the imaginary axis: one zero eigenvalue and a complex conjugate pair.}
{In this case we show that either no periodic solutions bifurcate or two curves bifurcate (Corollaries \ref{c:3D}, \ref{c:nD}), depending on the sign of a particular combination of coefficients of the system that is assumed to be non-zero.}
{Concluding the summary of main results,} in \S\ref{Gen_linear_part}, we discuss the modifications in case the linear part is not in normal form.
\medskip
For illustration, we consider the planar case with linear part in normal form, so \eqref{e:abstract0} with $\textbf{u}=(v,w)$ reads
\begin{equation}\label{e:planar0}
\begin{aligned}
\dot{v} &= \mu v - \omega w + f\left( v, w \right), \\
\dot{w} &= \omega v + \mu w + g\left( v, w \right),
\end{aligned}
\end{equation}
and the case of purely quadratic nonlinearity with second order modulus terms gives
\begin{equation}\label{e:quadnonlin}
\begin{aligned}
f\left( v, w \right) &= a_{11}v\abs{v} + a_{12}v\abs{w} + a_{21}w\abs{v} + a_{22}w\abs{w},\\
g\left( v, w \right) &= b_{11}v\abs{v} + b_{12}v\abs{w} + b_{21}w\abs{v} + b_{22}w\abs{w},
\end{aligned}
\end{equation}
where $a_{ij}$, $b_{ij}$, $1\leq i,j\leq 2$, are real parameters. In this simplest situation, our new first Lyapunov coefficient reads
\begin{equation}\label{sigma1}
\sigma_{_\#} = 2a_{11}+a_{12}+b_{21}+2b_{22}
\end{equation}
and we plot samples of bifurcation diagrams in Figure~\ref{f:auto} computed by numerical continuation with the software \texttt{Auto} \cite{auto}. {For these} we used numerically computed Jacobians and avoided evaluation too close to the non-smooth point by choosing suitable step-sizes and accuracy.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width= 0.4\linewidth]{subcrit3D.pdf} &
\includegraphics[width= 0.4\linewidth]{supcrit3D.pdf}\\
(a) & (b)
\end{tabular}
\caption{Plotted are bifurcation diagrams of \eqref{e:planar0} with \eqref{e:quadnonlin} computed by numerical continuation. Blue curves are periodic orbits, black lines the equilibrium at the origin, and orange the extrema in $w$, showing the non-smooth bifurcation. In (a) we use $a_{ij}=b_{ij}=1$ except
$b_{21}=-1$, so that $\sigma_{_\#}=4>0$ (subcritical). In (b) $b_{22}=-3$, so that $\sigma_{_\#}=-4$ (supercritical).}
\label{f:auto}
\end{figure}
In comparison, the classical first Lyapunov coefficient for purely cubic nonlinearity, i.e., $|\cdot |$ replaced by $(\cdot)^2$, reads
\[
\sigma_s = 3a_{11}+a_{12}+b_{21}+3b_{22}.
\]
The leading order expansion of the radius $r_{_\#}$ and $r_s$ of bifurcating periodic solutions in these cases read, respectively,
\[
r_{_\#}(\mu) = -\frac{3\pi}{2\sigma_{_\#}}\mu + \mathcal{O}\left(\mu^2\right), \qquad
r_s(\mu) = 2\sqrt{-\frac{2}{\sigma_s}\mu} + \mathcal{O}\left(\mu\right).
\]
We show that for $\sigma_{_\#}=0$ the bifurcation takes the form
\[
r_0=\sqrt{\frac{2\pi\omega}{\sigma_2}\mu}+\mathcal{O}\left(\mu\right),
\]
analogous to the smooth case, but with second Lyapunov coefficient in this setting given by
\begin{equation}\label{sigma2}
\begin{aligned}
\sigma_2 =& \frac{1}{3}( b_{12}a_{11}-b_{21}a_{22}-a_{21}b_{22}+a_{12}b_{11}-2a_{11}a_{22}-2a_{12}a_{21}+2b_{12}b_{21}+2b_{11}b_{22} ) \\
+\, & \frac{\pi}{4}( b_{12}b_{22}-a_{12}a_{22}-a_{11}a_{21}+b_{21}b_{11}+2a_{11}b_{11}-2b_{22}a_{22} ).
\end{aligned}
\end{equation}
In presence of smooth quadratic and cubic terms, the latter is modified with the classical terms, as we present in \S\ref{AV_S}.
Despite the similarity of $\sigma_{_\#}$ and $\sigma_s$, it turns out that there is no fixed smoothening of the absolute value function that universally predicts the correct criticality of Hopf bifurcations in these systems (\S\ref{s:smooth}).
For exposition of this issue, consider the $L^\infty$-approximations, with regularization parameter $\varepsilon> 0$, of the absolute value function $f_1(x) = \abs{x}$, given by $f_2(x)=\frac{2}{\pi}\arctan\left(\frac{x}{\varepsilon}\right)x$ (cf.\ \cite{Leine}), and $f_3(x)=\frac{2}{\pi}\arctan\left(\frac{x}{\varepsilon}(x-1)(x+1)\right)x$, a convex and a non-convex approximation, respectively.
This last function approximates the absolute value for large (absolute) values of $x$. We plot the graphs in Figure \ref{3_Cases}(a) and the bifurcation diagrams for $\dot{x}=\mu x-f_i(x)x$, $i\in\{1,2,3\}$, in Figure \ref{3_Cases}(b). In particular, $f_3$ gives a `microscopically' wrong result, which is nevertheless correct `macroscopically'.
\begin{figure}
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width= 0.6\linewidth]{3_Cases_abs_fun.pdf}
\subcaption{In blue $f_1(x)$, in green $f_2(x)$ and in red $f_3(x)$.}
\end{subfigure}
\begin{subfigure}{.47\textwidth}
\centering
\includegraphics[width= 0.6\linewidth]{multiplots.pdf}
\subcaption{Bifurcation diagrams respect to $\mu$.}
\end{subfigure}
\caption{Comparison of bifurcation diagrams for $f_1, f_2$ and $f_3$ as in the text.}
\label{3_Cases}
\end{figure}
Indeed, non-smooth terms in models typically stem from passing to a macro- or mesoscopic scale such that microscopic and smooth information is lost. Hence, the bifurcations in such models carry a macroscopic character and it is not surprising that an arbitrary smoothening changes this nature microscopically: a macroscopically supercritical bifurcation might show a subcritical behaviour on the microscopic level. However, the relevant information for the model is the macroscopic character, and -- for the class of models considered -- this is given by our newly derived Lyapunov coefficients.
The basic idea of proof is to change coordinates to a non-autonomous system for which the lack of smoothness is in the time variable only, so that averaging and the `direct method' can be applied.
We remark that in standard ODE literature on existence and bifurcations, smoothness of the time variable is often assumed, for example \cite{ChowHale}, but it is not needed in parts relevant for us. Indeed, merely continuity in time is for instance considered in \cite{CoddLev,Hartman,BookRasmussen}.
\medskip
{In order to demonstrate how to apply our method in a concrete case, we discuss in \S\ref{s:shim} the 3D model of a shimmying wheel from \cite{SBeregi}. This systems is of the form \eqref{e:abstract0} with pure second order modulus nonlinearity,
but linear part not in normal form, though it has a non-zero real eigenvalue as well as a pair of complex conjugate eigenvalues that crosses the imaginary axis upon parameter change.
We fully characterize the resulting bifurcations in Theorem \ref{t:shym}.}
\medskip
We briefly discuss related literature. As mentioned, piecewise smooth vector fields have been widely investigated in many different applications as well as from a theoretical point of view, leading to a broad analysis in terms of bifurcation theory, cf.\ \cite{ReviewAlan,KuepperHoshamWeiss2013,NonsmoothSurvey2012}.
Herein theory of continuous as well as discontinuous vector fields, e.g., \cite{Filippov1988, Kunze2000}, is used and further developed. A major distinction between our case and the systems studied in the literature is that we assume a separation a priori of a linear part and a non-smooth nonlinear part. Broadly studied are the more general switching differential systems that are discontinuous across a switching surface or piecewise linear. These have been analyzed in various different forms, and we refer to \cite{TianThesis} for an exhaustive list of references; a typical case of discontinuity across the switching manifolds arises from the Heaviside step functions in biology neural models, e.g., \cite{Amari1977,Coombes2005,Harris2015}.
In analogy to center manifolds, the existence of invariant manifolds and sets
has been investigated in \cite{IntegralManifold} for Carath\'eodory vector fields, and in \cite{KuepperHosham2010,KuepperHoshamWeiss2012} for vector fields with one switching surface.
The bifurcation of periodic orbits in planar vector fields with one axis as the switching line has been studied in \cite{CollGasullProhens,GasTorr2003} via one-forms, and characteristic quantities have been determined, though the aforementioned Lyapunov coefficients are not included. Planar Hopf bifurcations for piecewise linear systems have been studied via return maps for one switching line in \cite{KuepperMoritz}, for several switching lines meeting at a point in \cite{BuzMedTor2018,Simpson2019,ZouKuepper2005},
and for non-intersecting switching manifold using Li\'enard forms in \cite{LlibrePonce}. Higher dimensional Filippov-type systems with a single switching manifold are considered in \cite{ZouKuepperBeyn}, which allows to abstractly study the occurrence of a Hopf bifurcation also for our setting; see also \cite{KuepperHoshamWeiss2013}.
An approach via averaging with focus on the number of bifurcating periodic orbits for discontinuous systems is discussed in \cite{LlibreEtAl2017}.
Nevertheless, we are not aware of results in the literature that cover our setting and our results on the explicit derivation of Lyapunov coefficients and the leading order analysis of bifurcating periodic solutions.
\medskip
This paper is organized as follows. In \S\ref{s:abstract} we discuss the abstract setting and provide basic results for the subsequent more explicit analysis. This is conducted in \S\ref{Planar_Section} for the planar case with linear part in normal form and nonlinear part with pure second order modulus terms for the non-smooth functions, together with quadratic and cubic smooth functions. In \S\ref{s:general} we generalize the absolute value to arbitrary slopes, the system to higher space dimensions, and consider the linear part not being in normal form. Finally, in \S\ref{s:shim} we illustrate the application of our method and results to a concrete model.
\section{Abstract viewpoint}\label{s:abstract}
In this section we discuss the abstract starting point for our setting and motivate the specific assumptions used in the following sections. We consider an $n$-dimensional system of autonomous ODEs in an open set $U\subset\mathbb{R}^n$, with $0\in U$, of the form
\begin{equation}\label{e:abstract}
\dot \textbf{u}= A(\mu)\textbf{u}+G(\textbf{u}),
\end{equation}
with matrix $A(\mu)$ depending on a parameter $\mu\in\mathbb{R}$, and Lipschitz continuous nonlinear $G(\textbf{u})$.
We are interested in a detailed analysis of Hopf-type bifurcations at the equilibrium point $\textbf{u}_*=0$. This requires control over the linear part, which is separated a priori in \eqref{e:abstract} from the potentially non-differentiable nonlinear part -- note that $G$ is differentiable at $\textbf{u}_*$ but not necessarily elsewhere. As usual for Hopf bifurcations, we assume that a pair of simple complex conjugate eigenvalues of $A(\mu)$ crosses the imaginary axis upon moving $\mu\in\mathbb{R}$ through zero. We collect the structural hypotheses on $A$ and $G$ without further loss of generality to our leading order analysis.
\begin{hypothesis}\label{h:AG}
The eigenvalues of $A(\mu)$ are given by $\mu\pm \mathrm{i}\omega(\mu)$ with smooth non-zero $\omega(\mu)\in\mathbb{R}$ and all other eigenvalues have non-zero real part at $\mu=0$. The nonlinearity $G$ is Lipschitz continuous and satisfies $G(\textbf{u})=\mathcal{O}(|\textbf{u}|^2)$.
\end{hypothesis}
We denote by $E^\mathrm{c}$ the center eigenspace of $A(0)$ of the eigenvalues $\pm\mathrm{i}\omega(0)$, and first note the following result on invariant manifolds due to \cite{IntegralManifold}, which corresponds to center manifolds in the smooth case.
\begin{proposition}\label{prop:inv_man}
Under Hypothesis~\ref{h:AG}, for $0\leq |\mu|\ll1$ there exist $2$-dimensional Lipschitz continuous invariant manifolds $\mathcal{M}_\mu$ in an open neighborhood $U_*\subset U$ of $\textbf{u}_*$, which contain $\textbf{u}_*$ and all solutions that stay in $U_*$ for all time. Furthermore, if at $\mu=0$ all eigenvalues other than $\pm i\omega(0)$ have strictly negative real part, then each $\mathcal{M}_\mu$ is (transversally) exponentially attractive. In addition, each $\mathcal{M}_\mu$ is a Lipschitz continuous graph over $E^\mathrm{c}$ that depends Lipschitz continuously on $\mu$.
\end{proposition}
\begin{proof}
The statements follow directly from \cite{IntegralManifold} upon adding a trivial equation for the parameter, as usual in center manifolds. As for center manifolds, the proof relies on cutting off the vector field near $\textbf{u}_*$, cf.\ \cite[Remark 6.2]{IntegralManifold}, and we infer the existence of $\mathcal{M}_\mu$ from \cite[Corollary 6.4]{IntegralManifold}. The assumptions are satisfied since $G$ is of quadratic order, which means the Lipschitz constant of $G$ becomes arbitrarily small on small balls centered at $\textbf{u}_*$. The stability statement follows from \cite[Corollary 6.5]{IntegralManifold}.
\end{proof}
More refined stability information and estimates can be found in \cite{IntegralManifold}.
Next, we present a variant of the standard Andronov-Hopf bifurcation theorem, cf.\ \cite{ChowHale}, which does not use any additional smoothness assumption. Here the uniqueness part relies on Proposition~\ref{prop:inv_man}, but the existence is independent of it. As mentioned, in case of a single switching surface, the abstract bifurcation of periodic solutions without smoothness statement concerning the branch follows from the results in \cite{ZouKuepperBeyn}, see also \cite{KuepperHoshamWeiss2013}.
\begin{theorem}
\label{t_per_orb}
Assume Hypothesis~\ref{h:AG}.
A locally unique branch of periodic solutions to \eqref{e:abstract} bifurcates from $\textbf{u}_*=0$ at $\mu=0$. Specifically, there is a neighborhood $V\subset U$ of $\textbf{u}_*$, such that for $0<|\mu|\ll1$ periodic solutions to \eqref{e:abstract} in $V$ are given (up to phase shift) by a Lipschitz continuous one-parameter family of $\tilde\omega(a)$-periodic solutions $\textbf{u}_{\rm per}(t;a)$, $\mu=\mu(a)$ for $0\leq a\ll1$,
$\tilde\omega(0)=\omega(0)$, $\mu(0)=0$, whose projections into $E^\mathrm{c}$ have the complexified form
$a \mathrm{e}^{\mathrm{i} \tilde\omega(a) t} + o(|a|)$. Moreover, we have the estimate $\mathrm{dist}(\textbf{u}_{\rm per}(\cdot;a),E^{\mathrm{c}}) =\mathcal{O}(a^{2})$.
\end{theorem}
This bifurcation is typically `degenerate' compared to the generic smooth Hopf bifurcation as in the example \eqref{Dpitchfork}, where {the bifurcating branch} is not $C^1$ through $u=0$.
\begin{proof}
We change coordinates such that $A(\mu)$ is in block-diagonal form with upper left 2-by-2 block for the eigenspace $E^\mathrm{c}$ having diagonal entries $\mu$ and anti-diagonal $\pm\omega(\mu)$, and remaining lower right $(n-2)$-dimensional block invertible at $\mu=0$; the modified $G$ remains of quadratic order and is Lipschitz continuous. Upon changing to cylindrical coordinates with vertical component $u=(u_3,\ldots,u_n)$, where $u_j$ are the scalar components of $\textbf{u}$, we obtain
\begin{equation}\label{e:cylindrical0}
\begin{aligned}
\dot{r} &= \mu r + \mathcal{R}_1(r,u;\mu),\\
r\dot{\varphi} &= \omega(0)r + \mathcal{R}_2(r,u;\mu),\\
\dot u &= \tilde A u + \mathcal{R}_3(r,u;\mu).
\end{aligned}
\end{equation}
Here $\tilde A$ is the invertible right lower block at $\mu=0$ and we suppress the dependence on $\varphi$ of $\mathcal{R}_j$, $j=1,2,3$. Due to the Hypothesis~\ref{h:AG} in these coordinates we have the estimates $\mathcal{R}_1(r,u;\mu) = \mathcal{O}(r^2 +|\mu|(r^2 + |u|) + |u|^2)$, $\mathcal{R}_{j}(r,u;\mu) = \mathcal{O}(r^2 +|\mu|(r + |u|) + |u|^2)$, $j=2,3$. We seek initial conditions $r_0,u_0,\varphi_0$ and a parameter $\mu$ that permit a periodic solution near the trivial solution $r=u=0$.
By Proposition~\ref{prop:inv_man} any such periodic orbit is a Lipschitz graph over $E^\mathrm{c}$ so that there is a periodic function ${\tilde u}$ with $u=r{\tilde u}$.
Let $T>0$ denote the period and suppose $r(t)=0$ for some $t\in[0,T]$. Then $u(t)=0$ and therefore $\textbf{u}(t)=\textbf{u}_*$, so that $\textbf{u}=\textbf{u}_*$ is the trivial solution. Hence, we may assume that $r$ is nowhere zero and thus ${\tilde u}$ solves
\[
\dot {\tilde u} = \tilde A{\tilde u} + \widetilde{\mathcal{R}}_3(r,{\tilde u};\mu),
\]
where $\widetilde{\mathcal{R}}_3(r,{\tilde u};\mu) = \mathcal{O}\big(r+|\mu|+|{\tilde u}|(|\mu|+r|{\tilde u}|)\big)$. By variation of constants we solve this for given $r, \varphi$ as
\begin{equation}\label{e:tv}
{\tilde u}(t) = e^{\tilde A t}{\tilde u}_0 + \int_0^t e^{\tilde A(t-s)}\widetilde{\mathcal{R}}_3(r(s),{\tilde u}(s);\mu) ds,
\end{equation}
with initial condition ${\tilde u}(0)={\tilde u}_0$. $T$-periodic solutions solve in particular the boundary value problem
\begin{align}
0&= {\tilde u}(T)-{\tilde u}(0) = \int_0^{T} \dot {\tilde u}(s)ds
\nonumber\\
&= \int_0^{T}\tilde A e^{\tilde A s}{\tilde u}_0 ds + \int_0^{T}\left(\tilde A \int_0^s e^{\tilde A(s-\tau)}\widetilde{\mathcal{R}}_3(r(\tau),{\tilde u}(\tau);\mu) d\tau + \widetilde{\mathcal{R}}_3(r(s),{\tilde u}(s);\mu) \right)ds \nonumber\\
&= \left(e^{\tilde A T} - \mathrm{Id}\right){\tilde u}_0 + \widetilde{\mathcal{R}}_4(r,{\tilde u};\mu), \label{e:bvp0}
\end{align}
where $e^{\tilde A T} - \mathrm{Id}$ is invertible since $\tilde A$ is invertible.
We have $\widetilde{\mathcal{R}}_4(r,{\tilde u};\mu)= \mathcal{O}\big(r_\infty +|\mu| + {\tilde u}_\infty(|\mu| + r_\infty {\tilde u}_\infty)\big)$ with $r_\infty = \sup\{r(t)\;|\; t\in[0,T]\}$, ${\tilde u}_\infty = \sup\{|{\tilde u}(t)|\;|\; t\in[0,T]\}$ and by \eqref{e:tv} there is a $C>0$ depending on $T$ with
\[
{\tilde u}_\infty \leq C \big(|{\tilde u}_0| + r_\infty + |\mu| + {\tilde u}_\infty(|\mu|+r_\infty {\tilde u}_\infty)\big)
\;\Leftrightarrow\;\big(1-C(|\mu|+r_\infty {\tilde u}_\infty)\big){\tilde u}_\infty \leq C(|{\tilde u}_0| + r_\infty + |\mu|),
\]
so that for $0\leq |{\tilde u}_0|, r_\infty, |\mu|\ll1$ it follows $\frac{1}{2}\leq \big(1-C(|\mu|+r_\infty {\tilde u}_\infty)\big)$ and therefore ${\tilde u}_\infty \leq 2C(|{\tilde u}_0| + r_\infty + |\mu|)$. Thus,
\[
\widetilde{\mathcal{R}}_4(r,{\tilde u};\mu)= \mathcal{O}(r_\infty +|\mu| + |{\tilde u}_0|(|\mu| + r_\infty|{\tilde u}_0|)).
\]
Based on this, the uniform Banach contraction principle applies upon rewriting \eqref{e:bvp0} as
\[
{\tilde u}_0 = \left(e^{\tilde A T} - \mathrm{Id}\right)^{-1}\widetilde{\mathcal{R}}_4(r,{\tilde u};\mu),
\]
which yields a locally unique Lipschitz continuous solution ${\tilde u}_0(r,\varphi;\mu) = \mathcal{O}(r_\infty+|\mu|)$.
Note that together with the aforementioned, this implies the estimate ${\tilde u}_{\infty}=\mathcal{O}(r_{\infty}+|\mu|)$.
Substituting ${u}(t) = r(t){\tilde u}(t)$ with initial condition ${\tilde u}_0(r,\varphi;\mu)$ for ${\tilde u}$
into the first two equations of \eqref{e:cylindrical0} gives
\begin{equation}\label{e:cylindrical01}
\begin{aligned}
\dot{r} &= \mu r + \mathcal{R}_5(r;\mu),\\
\dot{\varphi} &= \omega(0) + \mathcal{R}_6(r;\mu),
\end{aligned}
\end{equation}
where we have divided the equation for $\varphi$ by $r$, since we look for non-zero solutions, and
\[
\mathcal{R}_5(r;\mu)=r\mathcal{O}\big(r+|\mu|r+|{\tilde u}|(|\mu|+r|{\tilde u}|)\big)=r\mathcal{O}(r_\infty + |\mu|r_\infty)=r\mathcal{O}(r_\infty),
\quad \mathcal{R}_6(r;\mu) = \mathcal{O}(r_\infty+|\mu|).
\]
Since $\omega(0)\neq 0$, for $0\leq r,|\mu|\ll 1$ we may normalize the period to $T=2\pi$ and obtain
\begin{equation}\label{e:absper}
\frac{d r}{d\varphi} = \frac{\mu r + \mathcal{R}_5(r;\mu)}{\omega(0) + \mathcal{R}_6(r;\mu)}
= r\left(\frac{\mu}{\omega(0)} + \mathcal{R}_7(r;\mu)\right),
\end{equation}
where $\mathcal{R}_7(r;\mu) = \mathcal{O}(r_\infty + |\mu|r_\infty) = \mathcal{O}(r_\infty)$ follows from direct computation.
Analogous to ${\tilde u}$ above, the boundary value problem $r(2\pi)=r(0)$ can be solved by the uniform contraction principle, which yields a locally unique and Lipschitz continuous solution $\mu(r_0)= \mathcal{O}(r_0)$. Since $\varphi$ is $2\pi$-periodic, any periodic solution has a period $2\pi m$ for some $m\in \mathbb{N}$, and the previous computation gives a unique solution for any $m$, from which we took the one with minimal period, i.e., $m=1$.
Finally, the statement of the form of periodic solutions directly proceeds with $a=r_0$ from changing back to the original time scale and coordinates. Notice that $r_{\infty}=\mathcal{O}(r_{0})$ holds true since we are integrating an ODE over a bounded interval, such that the ratio between $r_\infty$ and $r_0$ is a bounded quantity, which is uniform because the vector field goes to zero when $r$ goes to zero. Therefore, and together with $\mu(r_0)= \mathcal{O}(r_0)$, the previous estimate ${\tilde u}_{\infty}=\mathcal{O}(r_{\infty}+|\mu|)$ becomes ${\tilde u}_{\infty}=\mathcal{O}(r_0)$. Moreover, applying the supremum norm on both sides of $u=r{\tilde u}$ one gets $u_\infty = r_\infty{\tilde u}_\infty$, which is precisely of order $\mathcal{O}(r_0^2)$, as we wanted to prove.
\end{proof}
While this theorem proves the existence of periodic orbits, it does not give information about their location in parameter space, scaling properties and stability; the problem is to control the leading order part of $\mathcal{R}_7$ in \eqref{e:absper}, which -- in contrast to the smooth case -- turns out to be tedious. Consequently, we next aim to identify a suitable setting analogous to the center manifold reduction, and normal form transformations for a smooth vector field. In particular, we seek formulas for the analogue of the first Lyapunov coefficient from the smooth framework, whose sign determines whether the bifurcation is sub- or supercritical.
\medskip
In order to specify a setting that allows for such an analysis, and is also relevant in applications, we will assume additional regularity away from sufficiently regular hypersurfaces $H_j$, $j=1,\ldots,n_H$, and denote $H:=\cup_{j=1}^{n_H} H_j$. We refer to these hypersurfaces as \emph{switching surfaces} and assume these intersect pairwise transversally at the equilibrium point $\textbf{u}_*=0$.
\begin{hypothesis}\label{h:Ck}
The switching surfaces $H_j$, $j=1,\ldots,n_H$, are $C^k$ smooth, $k\geq1$ and intersect transversally at $\textbf{u}_*=0$. In each connected component of $U\setminus H$ the function $G$ is $C^k$ smooth and has a $C^k$ extension to the component's boundary.
\end{hypothesis}
For simplicity, and with applications in mind, we consider only two switching surfaces, $n_H=2$. In order to facilitate the analysis, we first map $H_1, H_2$ locally onto the axes by changing coordinates.
\begin{lemma}\label{l:cyl}
Assume Hypotheses~\ref{h:AG} and \ref{h:Ck} and let $n_H=2$. There is a neighborhood $V\subset U$ of $\textbf{u}_*$ and a diffeomorphism $\Psi$ on $V$ such that $\Psi(H_j\cap V) = \{ u_j=0\}\cap \Psi(V)$, $j=1, 2$; in particular $\Psi(\textbf{u}_*)=0$.
In subsequent cylindrical coordinates $(r,\varphi,u)\in \mathbb{R}_+ \times[0,2\pi)\times\mathbb{R}^{n-2}$ with respect to
the $(u_1,u_2)$-coordinate plane,
the vector field is $C^k$ with respect to $(r,u)$.
\end{lemma}
\begin{proof}
The smoothness of $H_j$, $j=1, 2$, and their transverse intersection allow for a smooth change of coordinates that straighten $H_1$, $H_2$ locally near $\textbf{u}_*$ and maps these onto the coordinate hypersurfaces $\{u_1=0\}$, $\{u_2=0\}$, respectively. The assumed smoothness away from the switching surfaces implies the smoothness in the radial direction.
\end{proof}
A concrete analysis of the nature of a Hopf bifurcation requires additional information on the leading order terms in $G$. As shown subsequently, a sufficient condition to identify the structure of the quadratic terms is Hypothesis~\ref{h:Ck} with $k=2$.
\begin{theorem}
\label{t:abstractnormal}
Assume Hypotheses~\ref{h:AG} and \ref{h:Ck}
for $k\geq 2$ and let $n_H=2$. In the coordinates of Lemma~\ref{l:cyl}, the non-smooth quadratic order terms in a component $G_j$, $j=1,\ldots, n$, of $G$ are of the form $u_\ell[u_i]_{\pn}^{\pp}$, $1\leq \ell\leq n$, $i=1,2$, where $p_{_+},p_{_-}\in \mathbb{R}$ depend on $i,\ell,j$ and are the limits of second derivatives of $G$ on the different connected components of $\mathbb{R}^n\setminus H$.
\end{theorem}
\begin{proof}
Consider a coordinate quadrant and let $\widetilde{G}$ be the extension of $G$ to its closure. By assumption, we can Taylor expand $\widetilde{G}(\textbf{u})= \frac 1 2 D^2\widetilde{G}(0)[\textbf{u},\textbf{u}] + o(|\textbf{u}|^2)$ since $G(0)=0$ as well as $D\widetilde{G}(0)=0$. However, for different coordinate quadrants the second order partial derivates may differ. By the form of $H$ in Lemma~\ref{l:cyl}, one-sided derivatives transverse to the coordinate axes might be distinct only for the $u_1, u_2$ axes. Hence, at $\textbf{u}=0$ second order derivatives involving $u_1, u_2$ may differ, and we denote by $p_{j\ell i_\pm}$ the partial derivatives $\frac{\partial^2}{\partial u_i \partial u_\ell}G_j(0)$, $1\leq \ell\leq n$, that are one-sided with respect to $i=1,2$ as indicated by the sign. The functions $\ABS{u_i}{p_{j\ell i}}$ thus provide a closed formula for the quadratic terms of $G_j$ as claimed.
\end{proof}
Even with explicit quadratic terms in these coordinates, an analysis based on the coordinates of Lemma~\ref{l:cyl} remains a challenge.
\begin{remark}\label{e:arrangement}
In cylindrical coordinates relative to $E^\mathrm{c}$, cf.\ \eqref{e:cylindrical0}, the vector field is generally not smooth in the radial direction. In general, smoothness cannot be achieved by changing coordinates as this typically modifies $H$ to be non-radial. In particular, we cannot assume, without loss of generality, that the linear part in the coordinates of Lemma~\ref{l:cyl} is in block-diagonal form or in Jordan normal form as in \eqref{e:cylindrical0}.
\end{remark}
For exposition, we consider the planar situation $n=2$,
where $H_1, H_2$ are the $u_1$- and $u_2$-axes, respectively. In contrast to \eqref{e:planar0} (and \eqref{e:cylindrical0}), the linear part is generally not in normal form, i.e., we have
\begin{equation}\label{e:abstractplanar}
\begin{pmatrix}
\dot u_1\\
\dot u_2
\end{pmatrix} =
\begin{pmatrix}
m_1 & m_2\\
m_3 & m_4
\end{pmatrix}\begin{pmatrix}
u_1\\
u_2
\end{pmatrix}+\begin{pmatrix}
f_1\left( u_1, u_2 \right)\\
f_2\left( u_1, u_2 \right)
\end{pmatrix},
\end{equation}
where $G=(f_1,f_2)$ is nonlinear.
Based on Hypothesis~\ref{h:AG} the linear part satisfies $\mu=\frac 1 2 (m_1+m_4)$, with $\mu=0$ at the bifurcation point, and the determinant at $\mu=0$ is positive so that we get together $m_1^2+m_2m_3<0$ and $m_2 m_3<0$.
Upon changing to polar coordinates we obtain, generally different from \eqref{e:cylindrical01},
\begin{equation}
\begin{cases}
\dot{r} = M(\varphi)r+\chi_2(\varphi)r^2 + \mathcal{O}(r^3),\\
\dot{\varphi} = W(\varphi) + \Omega_1(\varphi)r + \mathcal{O}(r^2),
\end{cases}
\label{Sys_Polar_NoNF}
\end{equation}
where $M, \chi_2, W, \Omega_1$ are $2\pi$-periodic in $\varphi$. Abbreviating $c:=\cos{\varphi}$ and $s:=\sin{\varphi}$, we have explicitly
\begin{align*}
M(\varphi) &= m_1c^2 + (m_2+m_3)sc + m_4s^2, \\
W(\varphi)&= m_3c^2 + (m_4-m_1)sc - m_2s^2,
\end{align*}
where $\chi_2, \Omega_1$ are continuous but in general non-smooth in $\varphi$ as a combination of generalized absolute value terms \eqref{gen_abs_val}.
Due to the conditions at $\mu=0$ we have $W(\varphi)\neq 0$ for any $\varphi$ so that $\dot{\varphi}\neq 0$ for $0\leq r,|\mu|\ll1$.
This allows to rescale time in \eqref{Sys_Polar_NoNF} analogous to \eqref{e:absper} and gives
\begin{equation}\label{new_time}
{r}' := \frac{dr}{d\varphi} = \frac{M(\varphi)r+\chi_2(\varphi)r^2}{W(\varphi) + \Omega_1(\varphi)r} + \mathcal{O}(r^3)
= \frac{M(\varphi)}{W(\varphi)}r+\left(\frac{\chi_2(\varphi)}{W(\varphi)} - \frac{M(\varphi)\Omega_1(\varphi)}{W(\varphi)^2}\right) r^2 +\mathcal{O}(r^3).
\end{equation}
Using averaging theory, as it will be discussed in detail in \S\ref{AV_S}, periodic orbits of \eqref{new_time} are generically in 1-to-1 correspondence with equilibria of the averaged form of \eqref{new_time} given by
\begin{align}\label{r_bar}
\bar{r}' &= {\Lambda}\bar{r}+{\Sigma}\bar{r}^2+\mathcal{O}(\bar{r}^3),
\end{align}
where $\Lambda, \Sigma\in\mathbb{R}$ are the averages of the linear and quadratic coefficients, respectively:
\begin{align}
\Lambda &= \frac{1}{2\pi} \int_0^{2\pi}\frac{M(\varphi)}{W(\varphi)}\mathrm{d} \varphi=
\frac{m_1+m_4}{\sqrt{-4m_2m_3-(m_1-m_4)^2}}, \label{check_mu} \\
\Sigma &= \frac{1}{2\pi} \int_0^{2\pi}\frac{\chi_2(\varphi)}{W(\varphi)}
- \frac{M(\varphi)\Omega_1(\varphi)}{W(\varphi)^2}\mathrm{d} \varphi.\label{check_sigma}
\end{align}
The explicit expression in \eqref{check_mu} follows from a straightforward but tedious calculation;
note that $\Lambda\in \mathbb{R}$ for $0\leq |\mu|\ll1$ due to the above conditions at bifurcation.
For $\Sigma\neq 0$, equilibria of \eqref{r_bar} are $\bar{r}=0$ and $\bar{r}=-\Lambda/{\Sigma}$, which gives a branch of non-trivial periodic orbits parameterized by $\Lambda$. The direction of branching, and thus the super- and subcriticality, is determined by the sign of $\Sigma$, which therefore is a generalized first Lyapunov coefficient.
However, this is still unsatisfying as it does not readily provide an explicit algebraic formula for $\Sigma$ in terms of the coefficients of $A(\mu)$ and $G$.
In order to further illustrate this issue, let $f_1,f_2$ be purely quadratic and built from second order modulus terms as in \eqref{e:quadnonlin}. In this case we explicitly have
\begin{align}
\chi_2(\varphi) &= c\abs{c}(a_{11}c+b_{11}s) + c\abs{s}(a_{12}c+b_{12}s) + s\abs{c}(a_{21}c+b_{21}s) + s\abs{s}(a_{22}c+b_{22}s), \label{chi}\\
\Omega_1(\varphi) &= -\Big[c\abs{c}(a_{11}s-b_{11}c) + c\abs{s}(a_{12}s-b_{12}c) + s\abs{c}(a_{21}s-b_{21}c) + s\abs{s}(a_{22}s-b_{22}c) \Big],\label{Omega}
\end{align}
which are continuous but not differentiable due to the terms involving $|c|,|s|$.
Clearly, the building blocks of the integrals in \eqref{check_sigma} are rational trigonometric functions with denominator $W$ of degree $2$ and numerators of degree $3$ and $5$. However, explicit formulas based on this appear difficult to obtain, so that we instead change to linear normal form as discussed in \S\ref{Gen_linear_part}, with the caveat that the nonlinear terms are in general not smooth in the radius.
Indeed, in the normal form case $m_1=\mu,\, m_2=-\omega,\, m_3=\omega,\, m_4=\mu$ the situation becomes manageable: in \eqref{Sys_Polar_NoNF} we have constant $M(\varphi)= \mu$ and $W(\varphi)=\omega(\mu)$, and we will show below that then $\Sigma=\frac{2}{3\pi\omega}\sigma_{_\#}$, with $\sigma_{_\#}$ as defined in \S\ref{s:intro}.
Therefore, until \S\ref{Gen_linear_part} we will assume that the linear part is in normal form in the coordinates of Lemma~\ref{l:cyl}, which also occurs in applications as mentioned in \S\ref{s:intro}.
\section{Planar normal form case with absolute values}
\label{Planar_Section}
In this section we discuss two approaches to {prove}
existence and bifurcation of periodic orbits in our mildly non-smooth setting. First, we provide details for the aforementioned approach by averaging, and second discuss a direct approach that provides a detailed unfolding by Lyapunov-Schmidt reduction, and that can also be used in some non-generic cases.
While we focus here on the planar case, both methods readily generalize to higher dimensional settings. For averaging one needs normal hyperbolicity in general, and for the direct approach we present higher dimensional cases in upcoming sections.
Without change in the leading order result, for simplicity we fix the imaginary part $\omega\neq 0$ independent of $\mu$.
To simplify the exposition in this section, we assume the linear part is in normal form and the non-smooth terms are of second order modulus type, i.e., with absolute value $|\cdot| = [\cdot]_{-1}^1$. The general case will be discussed in \S\ref{s:general}.
With the linear part in normal form and including smooth quadratic and cubic terms we thus consider the form of \eqref{e:abstract} given by, cf.\ {\eqref{e:planar0}},
\begin{align}
\begin{cases}
\dot{v} &= \mu v - \omega w + f\left( v, w \right) + f_q\left( v, w\right) + f_c\left( v, w \right), \\
\dot{w} &= \omega v + \mu w + g\left( v, w \right) + g_q\left( v, w \right) + g_c\left( v, w \right),
\end{cases}
\label{General2D_AV}
\end{align}
where $f,g$ are as in \eqref{e:quadnonlin}, and
\begin{align*}
f_q\left( v, w\right) &= a_1 v^2 + a_2 vw + a_3 w^2,
&f_c\left( v, w \right) &= c_{a1} v^3 + c_{a2} vw^2 + c_{a3} v^2w + c_{a4} w^3,\\
g_q\left( v, w\right) &= b_1 v^2 + b_2 vw + b_3 w^2,
&g_c\left( v, w\right) &= c_{b1} v^3 + c_{b2} vw^2 + c_{b3} v^2w + c_{b4} w^3,
\end{align*}
and $\mu, \omega \in \mathbb{R}$ with $\omega\neq 0$, and $a_{ij}, b_{ij}, a_k, b_k, c_{ah}, c_{bh},$ $\forall i,j\in \{1,2\}, \forall k\in\{1,2,3\}, \forall h\in\{1,2,3,4\}$ are real constants, all viewed as parameters.
\subsection{Averaging}\label{AV_S}
We next show how to apply averaging theory to \eqref{General2D_AV} in polar coordinates. In addition to $\sigma_{_\#}, \sigma_2$ from \eqref{sigma1}, \eqref{sigma2}, the following appear as normal form coefficients:
\begin{align*}
S_q :=\, & a_1a_2 +a_2a_3 - b_1b_2 - b_2b_3 -2a_1b_1 + 2a_3b_3,\nonumber\\
S_c :=\, & 3c_{a1} + c_{a2} + c_{b3} + 3c_{b4}.\nonumber
\end{align*}
Notice that $\sigma_{_\#}, \sigma_2$ depend only on $f, g$, i.e., the non-smooth terms, while $S_q$ depends on the smooth quadratic terms $f_q, g_q$; and $S_c$ on the cubic ones $f_c,g_c$.
\begin{theorem}
\label{t_averaging}
For $0<|\mu| \ll1$ periodic solutions to \eqref{General2D_AV} are locally in 1-to-1 correspondence with equilibria of the averaged normal form in polar coordinates $v=r\cos{\varphi},\, w=r\sin{\varphi}$ of \eqref{General2D_AV} given by
\begin{equation}
{\bar{r}}' = \frac{\mu}{\omega} \bar{r} + \frac{2}{3\pi\omega}\sigma_{_\#} \bar{r}^2 + \left( \frac{1}{8\omega^2}S_q + \frac{1}{8\omega}S_c - \frac{1}{2\pi\omega^2}\sigma_2 \right) \bar{r}^3 + \mathcal{O}\left(\bar{r}^4+\abs{\mu}\bar{r}^2\right).
\label{NormalFormAV}
\end{equation}
\end{theorem}
Remark that in accordance with the smooth Hopf bifurcation, the quadratic term in $\bar r'$ vanishes for vanishing non-smooth terms $f=g=0$, so that the leading order nonlinear term in the normal form is cubic. Before giving the proof we note and discuss an important corollary. For this recall the pitchfork bifurcation of \eqref{Dpitchfork} which is degenerate in that the bifurcating branch is non-smooth.
\begin{corollary}
\label{c_averaging}
If $\sigma_{_\#}\neq 0$, then at $\mu=0$ \eqref{NormalFormAV} undergoes a degenerate pitchfork bifurcation in $\mu$, where non-trivial equilibria are of the form
\begin{equation}\label{periodic_orbit}
r_0(\mu) = -\frac{3\pi}{2\sigma_{_\#}}\mu + \mathcal{O}\left(\mu^2\right).
\end{equation}
In this case, \eqref{General2D_AV} undergoes a degenerate Hopf bifurcation in the sense that for $0<|\mu| \ll1$ periodic solutions to \eqref{General2D_AV} are locally in 1-to-1 correspondence with $r_0(\mu)$, which is also the expansion of the radial component of the periodic solutions.
In particular, this Hopf bifurcation is subcritical if {$\sgn(\sigma_{_\#})>0$} and supercritical if {$\sgn(\sigma_{_\#})<0$}.
%
Moreover, the bifurcating periodic orbits of \eqref{General2D_AV} are of the same stability as the {corresponding} equilibria in \eqref{NormalFormAV}.
\end{corollary}
\begin{proof} (Corollary \ref{c_averaging})
The bifurcation statement follows directly from Theorem \ref{t_averaging} and the statement about stability follows from \cite[Thm.\ 6.3.3]{Averaging}. Since $r_0\geq 0$ we must have $\frac{\mu}{\sigma_{_\#}}\geq 0$. Hence, the sign of $\sigma_{_\#}$ determines the criticality of the bifurcation.
\end{proof}
The radial components $r(\varphi;\mu)$ of the periodic orbits are in general not constant in $\varphi$, but this dependence is of order $\mu^2$. We thus consider \eqref{periodic_orbit} as the leading order amplitude of the periodic solutions.
\begin{remark}\label{1st_Lyap}
Since the criticality of the Hopf bifurcation is given by the sign of $\sigma_{_\#}$, it is an analogue of the first Lyapunov coefficient in this non-smooth case. For the smooth case $f=g=0$, where $\sigma_{_\#}=\sigma_2=0$, the classical first Lyapunov coefficient is $\sigma_s:=\frac{1}{8\omega}S_q + \frac{1}{8}S_c$. In \S\ref{s:smooth} we show that there is no canonical way to infer the sign of $\sigma_{_\#}$ from smoothening a priori.
\end{remark}
\begin{remark}\label{2nd_Lyap}
In case $\sigma_{_\#}=0$ but non-zero cubic coefficient in \eqref{NormalFormAV}, the bifurcating branch is a quadratic function of $\mu$ to leading order. This readily gives an analogue of the second Lyapunov coefficient in this non-smooth case. An explicit statement in absence of smooth terms is given in Theorem~\ref{2ndPart} below. Notably, in the smooth case, vanishing first Lyapunov coefficient, but non-zero second Lyapunov coefficient yields a quartic bifurcation equation. Hence, the scaling laws {for the radius} are $\mu^{1/j}$ with $j=1,2$ in the non-smooth {case} and $j=2,4$ in the smooth case, respectively.
\end{remark}
Next we give the proof of Theorem \ref{t_averaging}.
\begin{proof} (Theorem \ref{t_averaging})
Taking polar coordinates $(v,w)=(r\cos{\varphi},r\sin{\varphi})$ system \eqref{General2D_AV}, cf.\ \eqref{Sys_Polar_NoNF}, becomes
\begin{align}
\begin{cases}
\dot{r} &= \mu r + r^2\chi_2(\varphi) + r^3\chi_3(\varphi), \\
\dot{\varphi} &= \omega + r\Omega_1(\varphi) + r^2\Omega_2(\varphi),
\end{cases}
\label{System2DpolarNF}
\end{align}
where $\chi_2(\varphi)$ and $\Omega_1(\varphi)$ are as in \eqref{chi} and \eqref{Omega}, respectively, but adding now the contributions of the smooth quadratic terms of $f_q, g_q$:
\begin{align*}
\chi_2(\varphi) &= c\abs{c}(a_{11}c+b_{11}s) + c\abs{s}(a_{12}c+b_{12}s) + s\abs{c}(a_{21}c+b_{21}s) + s\abs{s}(a_{22}c+b_{22}s) \\
&+ (a_1-b_2-a_3)c^3 + (b_1+a_2-b_3)sc^2 + (b_2+a_3)c + b_3s, \\
\Omega_1(\varphi) &= -\Big[c\abs{c}(a_{11}s-b_{11}c) + c\abs{s}(a_{12}s-b_{12}c) + s\abs{c}(a_{21}s-b_{21}c) + s\abs{s}(a_{22}s-b_{22}c) \Big] \\
&+ (b_1+a_2-b_3)c^3 + (-a_1+b_2+a_3)sc^2 + (-a_2+b_3)c - a_3s,
\end{align*}
and $\chi_3(\varphi)$ and $\Omega_2(\varphi)$ are smooth functions of $\varphi$ and the coefficients of $f_c, g_c$:
\begin{align*}
\chi_3(\varphi) &= (c_{a1}-c_{a2}-c_{b3}+c_{b4})c^4 + (c_{a3}-c_{a4}+c_{b1}-c_{b2})sc^3 \\
&+ (c_{a2}+c_{b3}-c_{b4})c^2 + (c_{a4}+c_{b2})sc + c_{b4}s^2,\\
\Omega_2(\varphi) &= (c_{a3}-c_{a4}+c_{b1}-c_{b2})c^4 + (c_{a2}-c_{a1}+c_{b3}-c_{b4})sc^3 - c_{a3}c^2 \\
&+ (c_{b4}-c_{a2})sc - c_{a4}s^2 + (c_{b2}+c_{a4})c^2.
\end{align*}
To simplify the notation we write, as before, $c:=\cos{\varphi}$, $s:=\sin{\varphi}$.
Analogous to \eqref{new_time}, we change parametrization such that the return time to $\varphi=0$ is equal for all orbits starting on this half-axis with initial radius $r_0>0$ to get
\begin{align*}
{r}':= \dv{r}{\varphi} &= \frac{\mu r + r^2\chi_2(\varphi) + r^3\chi_3(\varphi)}{\omega + r\Omega_1(\varphi) + r^2\Omega_2(\varphi)}.
\end{align*}
Expanding the right-hand side of $r'$ in small $r$ and $\mu$ gives
\begin{equation}
{r}' =
\frac{\mu}{\omega} r + \frac{\chi_2}{\omega} r^2 + \left( \frac{\chi_3}{\omega} - \frac{\chi_2\Omega_1}{\omega^2} \right) r^3 + \mathcal{O}\left( r^4+\abs{\mu}r^2 \right).
\label{DoNormalForm_r}
\end{equation}
In order to follow the method of averaging (e.g., \cite{Guckenheimer,Averaging}), we write $r=\epsilon x$ and $\mu=\epsilon m$ for $0<\epsilon\ll 1$, such that \eqref{DoNormalForm_r} in terms of $x$ and $m$ becomes
\begin{align}
x' &= \epsilon \left( \frac{m}{\omega} x + \frac{\chi_2}{\omega} x^2 \right) + \epsilon^2\left( \frac{\chi_3}{\omega} - \frac{\chi_2\Omega_1}{\omega^2} \right) x^3 + \epsilon^2\mathcal{O}\left( \epsilon x^4+\abs{m}x^2 \right).
\label{x_form_for_av}
\end{align}
Following \cite{Averaging}, there is a near-identity transformation which
maps solutions of the truncated averaged equation
\begin{equation}
y' = \epsilon\bar{f}_1(y) + \epsilon^2\bar{f}_{2}(y)+ \epsilon^3\bar{f}_{[3]}(y,\varphi,\epsilon)
\label{truncated_av}
\end{equation}
to solutions of \eqref{x_form_for_av}, where its detailed derivation is given in Appendix~\ref{NIT}, as well as an explanation of the computation of the following integrals:
\begin{equation}
\label{averaging_integrals}
\begin{aligned}
\bar{f}_1(y) &=
\frac{1}{2\pi} \int_0^{2\pi} \left( \frac{m}{\omega}z + \frac{\chi_2(\varphi)}{\omega}z^2 \right)\mathrm{d}\varphi = \frac{m}{\omega}y +\frac{2}{3\pi\omega}\sigma_{_\#}y^2, \\
\bar{f}_{2}(y) &=
\frac{1}{2\pi} \int_0^{2\pi} \left( \frac{\chi_3(\varphi)}{\omega} - \frac{\chi_2(\varphi)\Omega_1(\varphi)}{\omega^2} \right)y^3 \mathrm{d}\varphi = \left( \frac{1}{8\omega^2}S_q + \frac{1}{8\omega}S_c - \frac{1}{2\pi\omega^2}\sigma_2 \right)y^3.
\end{aligned}
\end{equation}
We obtain the averaged equation \eqref{NormalFormAV} from \eqref{DoNormalForm_r} by the change of coordinates $y=\frac{\bar{r}}{\epsilon}$ and $m=\frac{\mu}{\epsilon}$ applied to \eqref{truncated_av} {with \eqref{averaging_integrals}}; this becomes \eqref{NormalFormAV} since all terms involving $\epsilon$ cancel out.
Finally, from \cite[Thm.\ 6.3.2]{Averaging} the existence of a periodic orbit in the averaged system implies the existence of a periodic orbit in the original system.
\end{proof}
\subsection{Smoothening and the first Lyapunov coefficient}\label{s:smooth}
From Remarks \ref{1st_Lyap} and \ref{2nd_Lyap} on the first and second Lyapunov coefficients, it is natural to ask in what way the non-smooth first Lyapunov coefficient
\[
\sigma_{_\#} = 2a_{11}+a_{12}+b_{21}+2b_{22}
\]
from \eqref{sigma1} differs from the first Lyapunov coefficient of a smoothened version of \eqref{e:planar0}.
More specifically, the question is whether one can smoothen the vector field in such a way that the sign of the {resulting} first Lyapunov coefficient is the same as that of the non-smooth one, $\sigma_{_\#}$, in all cases. We shall prove that this is not possible \emph{without using the formula for $\sigma_{_\#}$} -- {with the help of} this formula we can find suitable smoothening.
\medskip
Clearly, non-convex approximations of the absolute value $|\cdot|$ can change criticality compared to the non-smooth case (see Figure \ref{3_Cases}). More generally, we have the following.
\begin{lemma}
\label{Convex_Approx}
For any $f,g$ with a sign change in the coefficients $a_{11}, a_{12}, b_{21}, b_{22}$, there are smooth approximations $f_\varepsilon, g_\varepsilon$ with $(f_\varepsilon,g_\varepsilon)\to (f,g)$ in $L^\infty$ such that the criticality of the smoothened Hopf bifurcation is opposite that of the non-smooth case. Moreover, $f_\varepsilon, g_\varepsilon$ can be chosen as symmetric smooth convex approximations of the absolute values in $f,g$.
\end{lemma}
\begin{proof}
Without loss of generality, we consider system \eqref{e:planar0}. For given $f,g$ we can choose a smooth approximation of $|\cdot|$ in the terms with coefficients $a_{11}, a_{12}, b_{21}, b_{22}$ that have quadratic terms with positive coefficients of the form $\varepsilon^{-1} \tilde a_{11}, \varepsilon^{-1} \tilde a_{12}, \varepsilon^{-1} \tilde b_{21}, \varepsilon^{-1} \tilde b_{22}$, respectively. Then the (smooth) first Lyapunov coefficient reads
\[
\sigma_{s,\varepsilon}:= \varepsilon^{-1}\left(3\tilde a_{11} a_{11}+\tilde a_{12} a_{12}+\tilde b_{21} b_{21}+3\tilde b_{22} b_{22} \right),
\]
which is the same as $S_c$ in \S\ref{AV_S} when replacing accordingly coefficients of $f,g$ and $f_c,g_c$, respectively.
Suppose now $\sigma_{_\#}<0$. In this case, the sign change within $(a_{11}, a_{12}, b_{21}, b_{22})$ allows to choose $(\tilde a_{11}, \tilde a_{12}, \tilde b_{21}, \tilde b_{22})>0$ such that $\sigma_{s,\varepsilon}>0$. Likewise for $\sigma_{_\#}>0$ we can arrange $\sigma_{s,\varepsilon}<0$.
\end{proof}
\begin{remark}
If all of $a_{11}, a_{12}, b_{21}, b_{22}$ have the same sign, then any convex smoothening of the absolute value with non-zero quadratic terms will yield a first Lyapunov coefficient of the same sign as $\sigma_{_\#}\neq 0$.
Moreover, having derived the formula for $\sigma_{_\#}$, we can -- a posteriori -- identify a smoothening that preserves {the} criticality for all $f,g$. With the notation of Lemma \ref{Convex_Approx} this is
$\tilde a_{11} = \tilde b_{22} = 2/3, \tilde a_{12} = \tilde b_{21} = 1$.
\end{remark}
\begin{lemma}
There is no smooth approximation of the absolute value function with non-zero quadratic term that preserves the criticality of the non-smooth case for all $f,g$.
\end{lemma}
\begin{proof}
In contrast to Lemma \ref{Convex_Approx}, here all absolute value terms in $f,g$ are approximated in the same way so that in the notation of the proof of Lemma \ref{Convex_Approx} we have $\tilde a_{11}=\tilde a_{12}= \tilde b_{21}= \tilde b_{22} > 0$. Without loss of generality we can assume {these coefficients are all equal $1$} due to the prefactor $\varepsilon^{-1}$, so that the first Lyapunov coefficient is
\[
\sigma_{s,\varepsilon}= \varepsilon^{-1}\left(3 a_{11}+ a_{12}+ b_{21}+3 b_{22} \right),
\]
and we readily find examples of $(a_{11}, a_{12}, b_{21}, b_{22})$ such that the signs of $\sigma_{_\#}$ and $\sigma_{s,\varepsilon}$ differ.
\end{proof}
The discrepancies shown here for the absolute value function readily carry over to the generalized absolute value function \eqref{gen_abs_val}.
\subsection{Direct method}\label{s:direct}
In Theorem \ref{t_averaging}, the conclusion for \eqref{General2D_AV} does not cover the bifurcation point $\mu=0$ so that we cannot infer uniqueness of the branch of bifurcating periodic orbits directly.
In order to directly include $\mu=0$ in the bifurcation analysis and to facilitate the upcoming generalizations, we present a `direct' method for a general (possibly) non-smooth planar system. This does not rely on the existence of an invariant manifold as in Proposition~\ref{prop:inv_man} or results from averaging theory.
The basic result is the following bifurcation of periodic solutions for a radial equation with quadratic nonlinear terms, which cannot stem from a smooth planar vector field, but occurs in our setting as in \eqref{Sys_Polar_NoNF}.
\begin{proposition}
\label{Thm_Gen}
Consider a planar system in polar coordinates $(r,\varphi)\in \mathbb{R}_+ \times [0,2\pi)$ periodic in $\varphi$ {of the form}
\begin{align}
\begin{cases}
\dot{r} &= r\mu + r^2\chi_2(\varphi), \\
\dot{\varphi} &= \omega + r\Omega_1(\varphi),
\end{cases}
\label{System2Dpolar}
\end{align}
where $\mu\in\mathbb{R}$, $\omega\neq 0$ and continuous $\chi_2(\varphi),\Omega_1(\varphi)$ with minimal period $2\pi$.
If $\int_0^{2\pi}\chi_2(\varphi)\mathrm{d} \varphi\neq 0$, then a locally unique branch of periodic orbits bifurcates at $\mu=0$. These orbits have period $2\pi + \mathcal{O}(\mu)$ and constant radius satisfying
\begin{equation} \label{General_Result}
r_0=\frac{-2\pi}{\int_0^{2\pi}\chi_2(\varphi)\mathrm{d}\varphi}\mu+\mathcal{O}\left(\mu^2\right).
\end{equation}
In particular, since $r_0\geq 0$, the criticality of the bifurcation is determined by the sign of $\int_0^{2\pi}\chi_2(\varphi)\mathrm{d} \varphi$.
\end{proposition}
For later reference we present a rather detailed proof.
\begin{proof}
As in the proof of Theorem \ref{t_averaging}, for small $r$ the radius satisfies
\begin{align}
{r}':= & \frac{r\mu + r^2\chi_2(\varphi)}{\omega + r\Omega_1(\varphi)} =: \Psi(r,\varphi).
\label{System2Dparam}\end{align}
We fix the initial time at $\varphi_0=0$ and for any initial $r(0)=r_0$, a unique local solution is guaranteed from the Picard-Lindel\"of theorem with continuous time dependence, e.g., \cite{Hartman}.
This also guarantees existence on any given time interval for sufficiently small $|\mu|, r_0$. Moreover, the solution $r(\varphi;r_0)$ can be Taylor expanded with respect to $r_0$ due to the smoothness of $\Psi(r,\varphi)$ in $r$ and continuity in the time component using the uniform contraction principle for the derivatives, cf.\ \cite{Hartman}.
On the one hand, we may thus expand $r(\varphi)= r(\varphi; r_0)$ as
\begin{equation*}
r(\varphi)=\alpha_1(\varphi)r_0 + \alpha_2(\varphi)r_0^2 + \mathcal{O}\left(r_0^3\right),
\end{equation*}
and differentiate with respect to $\varphi$,
\begin{equation}
r'(\varphi)=\alpha_1'(\varphi)r_0 + \alpha_2'(\varphi)r_0^2 + \mathcal{O}\left(r_0^3\right),
\label{Expansion_r'}
\end{equation}
where $\alpha_1(0)=1$ and $\alpha_2(0)=0$ since $r(0)=r_0$.
On the other hand, we Taylor expand $\Psi(r,\varphi)$ in $r=0$ from \eqref{System2Dparam}, using $\Psi(0,\varphi)=0$, as
\begin{align}
r' &= \Psi(r,\varphi) = \Psi(0,\varphi) + \partial_r\Psi(0,\varphi)r + \frac{1}{2}\partial^2_r\Psi(0,\varphi) r^2 + \mathcal{O}\left(r^3\right)
= k_1 r + k_2 r^2 + \mathcal{O}\left(r^3\right),
\label{Expansion2_r'}
\end{align}
where we denote $\partial^i_r\Psi(0,\varphi)=\frac{\partial^i\Psi(r,\varphi)}{\partial r^i}\big\rvert_{r=0}$, $i\in\mathbb{N}$, and set
\begin{equation}\label{ks}
k_1 := \partial_r\Psi(0,\varphi) = \frac{\mu}{\omega}, \hspace*{4mm} k_2(\varphi) := \frac{1}{2}\partial^2_r\Psi(0,\varphi) = \frac{\omega\chi_2(\varphi) - \mu\Omega_1(\varphi)}{\omega^2}.
\end{equation}
Matching the coefficients of $r_0$ and $r_0^2$ in \eqref{Expansion_r'} and \eqref{Expansion2_r'} gives the ODEs $\alpha_1' = k_1 \alpha_1$ and $\alpha_2' = k_1 \alpha_2 +k_2\alpha_1^2$.
The solutions with $\alpha_1(0)=1$ and $\alpha_2(0)=0$ read
\begin{align*}
\alpha_1(\varphi) &= e^{k_1 \varphi},
&\alpha_2(\varphi) &=
\int_0^\varphi e^{k_1(\varphi+s)}k_2(s)\mathrm{d} s.
\end{align*}
Periodic orbits necessarily have period $2\pi m$ for some $m\in\mathbb{N}$, which yields the condition
\begin{equation}\label{per_orb_r}
0 = r(2\pi m)-r(0) = \int_0^{2\pi m} r' \mathrm{d}\varphi = r_0\int_0^{2\pi m} \alpha_1'(\varphi)\mathrm{d} \varphi + r_0^2\int_0^{2\pi m} \alpha_2'(\varphi)\mathrm{d} \varphi + \mathcal{O}\left(r_0^3\right).
\end{equation}
Using the series expansion of $e^{2\pi m k_1}$ in $\mu=0$ we have
$$ \int_0^{2\pi m} \alpha_1'(\varphi)\mathrm{d} \varphi = \alpha_1(2\pi m)-\alpha_1(0) = e^{2\pi m k_1}-1 = 2\pi m k_1 + \mathcal{O}\left(\mu^2\right), $$
and similarly,
\begin{align*}
\int_0^{2\pi m} \alpha_2'(\varphi)\mathrm{d} \varphi &= \alpha_2(2\pi m)-\alpha_2(0) = e^{2\pi m k_1}\int_0^{2\pi m} e^{k_1\varphi}k_2(\varphi)\mathrm{d} \varphi - 0 \\
&= (1+2\pi m k_1)\int_0^{2\pi m}k_2(\varphi)\mathrm{d}\varphi + k_1\int_0^{2\pi m}\varphi k_2(\varphi)\mathrm{d}\varphi + \mathcal{O}\left(\mu^2\right).
\end{align*}
For non-trivial periodic orbits, $r_0\neq 0$, we divide \eqref{per_orb_r} by $r_0$, which provides the bifurcation equation
\begin{equation*}
0 = 2\pi m k_1 + r_0\left( (1+2\pi m k_1)\int_0^{2\pi m}k_2(\varphi)\mathrm{d}\varphi + k_1\int_0^{2\pi m}\varphi k_2(\varphi)\mathrm{d}\varphi \right) + \mathcal{O}\left(\mu^2\right),
\end{equation*}
where the factor of $r_0$ is non-zero at $\mu=0$ by assumption. Hence, the implicit function theorem applies and gives a unique solution. Since the solution for $m=1$ is a solution for any $m$, this is the unique periodic solution. Solving the bifurcation equation for $m=1$ yields
\begin{equation*}
r_0 = \frac{- 2\pi \mu}{(\omega+2\pi \mu)\int_0^{2\pi }k_2(\varphi)\mathrm{d}\varphi + \mu\int_0^{2\pi m}\varphi k_2(\varphi)\mathrm{d}\varphi} + \mathcal{O}\left(\mu^2\right),
\end{equation*}
whose expansion in $\mu=0$ gives the claimed \eqref{General_Result} and in particular the direction of branching.
Finally, the exchange of stability between the trivial equilibrium and the periodic orbit follows from the monotonicity of the $1$-dimensional Poincar\'e Map on an interval that contains $r=0$ and $r=r_0(\mu)$ by uniqueness of the periodic orbit.
\end{proof}
We next note that higher order perturbations do not change the result to leading order.
\begin{corollary}\label{hot2D}
The statement of Proposition \ref{Thm_Gen} holds for a planar system in polar coordinates $(r,\varphi)\in \mathbb{R}_+ \times [0,2\pi)$ periodic in $\varphi$, of the form
\begin{align}
\begin{cases}
\dot{r} &= r\mu + r^2\chi_2(\varphi) + r^3\chi_3(r,\varphi), \\
\dot{\varphi} &= \omega + r\Omega_1(\varphi) + r^2\Omega_2(r,\varphi),
\end{cases}
\label{System_Polar_General}
\end{align}
where $\mu\in\mathbb{R}$, $\omega\neq 0$ and $\chi_{j+1}$, $\Omega_j$, $j=1,2$, are continuous in their variables.
\end{corollary}
Note that system \eqref{System_Polar_General} is a generalization of \eqref{System2DpolarNF} in which $\chi_3$ and $\Omega_2$ depend now on $r$.
\begin{proof}
Following the proof of Proposition \ref{Thm_Gen} we write system \eqref{System_Polar_General} analogous to \eqref{System2Dparam} with $$\Psi(r,\varphi) = \frac{r\mu + r^2\chi_2(\varphi) + r^3\chi_3(r,\varphi)}{\omega + r\Omega_1(\varphi) + r^2\Omega_2(r,\varphi)}.$$
Upon subtracting the leading order part of \eqref{Expansion2_r'}, a direct computation produces a remainder term of order $\mathcal{O}(r^3)$, which leads to the claimed result.
\end{proof}
Next we show how these results can be directly used to determine the Hopf bifurcation and its super- or subcriticality.
Starting with the simplest model, we return to system \eqref{General2D_AV} with $f_q, g_q, f_c, g_c \equiv 0$, i.e., \eqref{e:planar0}.
Recall $\sigma_{_\#}=2a_{11}+a_{12}+b_{21}+2b_{22}$ from \eqref{sigma1} was identified as determining the criticality in Corollary \ref{c_averaging}. With the direct method we obtain the following.
\begin{theorem}
\label{1stPart}
If $\sigma_{_\#}\neq 0$, then there exists an interval $I$ around $\mu=0$ such that at $\mu=0$ system \eqref{e:planar0} with $f,g$ from \eqref{e:quadnonlin} undergoes a degenerate Hopf bifurcation in $\mu$ where the leading order amplitudes of the locally unique periodic orbits is given by \eqref{periodic_orbit}.
In particular, the unique bifurcating branch of periodic solutions emerges subcritically if {$\sgn(\sigma_{_\#})>0$} and supercritically if {$\sgn(\sigma_{_\#})<0$}. Moreover, the bifurcating periodic orbits have exchanged stability with the equilibrium at $r=0$, i.e., are stable if they exist for $\mu>0$ and unstable if this is for $\mu<0$.
\end{theorem}
\begin{proof}
Taking polar coordinates $(v,w)=(r\cos{\varphi},r\sin{\varphi})$ for system \eqref{e:planar0} gives \eqref{System2Dpolar}, where $\chi_2(\varphi)$ and $\Omega_1(\varphi)$ are as in \eqref{chi} and \eqref{Omega}, respectively.
Applying Proposition \ref{Thm_Gen}
and computing the integral of $\chi_2$
in each quadrant as in the proof of Theorem~\ref{t_averaging}, we obtain
\eqref{periodic_orbit}, and the criticality follows as in Corollary~\ref{c_averaging}.
Finally, the exchange of stability is due to the monotonicity of the $1$-dimensional Poincar\'e Map.
\end{proof}
We next note that, proceeding as for Corollary \ref{hot2D}, the coefficients from the quadratic and cubic terms do not affect the bifurcation to leading order.
\begin{corollary}
\label{thm_cubic}
If $\sigma_{_\#}\neq 0$, then the statement of Theorem \ref{1stPart} holds for the more general system \eqref{General2D_AV}. In particular, $f_q$, $g_q$, $f_c$, $g_c$ do not affect $\sigma_{_\#}$ and the leading order bifurcation.
\end{corollary}
Having investigated $\sigma_{_\#}\neq 0$, we next consider the degenerate case $\sigma_{_\#}= 0$. For that, recall Remark~\ref{2nd_Lyap} and {$\sigma_2$ from} \eqref{sigma2}.
\begin{theorem}
\label{2ndPart}
If $\sigma_{_\#}= 0$ and $\sigma_2\neq 0$, then there exists an interval $I$ around $\mu=0$ such that at $\mu=0$ system \eqref{General2D_AV} undergoes a degenerate Hopf bifurcation in $\mu$ where the leading order amplitude of the locally unique periodic orbit is given by
\begin{equation}\label{r_2nd_HB}
r_0=\sqrt{\frac{2\pi\omega}{\sigma_2}\mu}+\mathcal{O}\left(\mu\right).
\end{equation}
In particular, the unique bifurcating branch of periodic solutions emerges subcritically if\\
$\sgn(\omega\sigma_2)<0$ and supercritically if $\sgn(\omega\sigma_2)>0$. Moreover, the bifurcating periodic orbits have exchanged stability with the equilibrium at $r=0$, i.e., are stable if they exist for $\mu>0$ and unstable if this is for $\mu<0$.
\end{theorem}
\begin{proof}
Proceeding as before, we write \eqref{General2D_AV} in polar coordinates $(v,w)=(r\cos{\varphi},r\sin{\varphi})$ and change the time parametrization to obtain the form \eqref{System2Dparam} for the radial equation.
On the one hand, we expand the solution $r(\varphi)=r(\varphi;r_0)$ with $r(0)=r_0$ as
\begin{equation}
r'(\varphi)=\alpha_1'(\varphi)r_0 + \alpha_2'(\varphi)r_0^2 + \alpha_3'(\varphi)r_0^3 + \mathcal{O}\left(r_0^4\right),
\label{2Expansion_r}
\end{equation}
where $\alpha_1(0)=1$ and $\alpha_2(0)=\alpha_3(0)=0$.
On the other hand, we compute the Taylor expansion of $r'$, from \eqref{System2Dparam}, up to third order in $r=0$ as
\begin{equation*}
r' = \Psi(r,\varphi) = k_1 r + k_2 r^2 + k_3 r^3 + \mathcal{O}\left(r^4\right),
\end{equation*}
where we use $\Psi(0,\varphi)=0$ and the notation \eqref{ks} as well as
\begin{equation*}
k_3(\varphi) := \frac{1}{3!}\partial^3_r\Psi(0,\varphi)= \frac{-\omega\chi_2(\varphi)\Omega_1(\varphi) + \mu\Omega_1(\varphi)^2}{\omega^3}.
\end{equation*}
Analogous to the proof of Proposition \ref{Thm_Gen}, using \eqref{2Expansion_r} and its derivate, and comparing coefficients, we obtain the ODEs
\begin{align*}
\alpha_1' &= k_1\alpha_1,
& \alpha_2' &= k_1\alpha_2 + k_2\alpha_1^2,
& \alpha_3' &= k_1\alpha_3 + 2k_2\alpha_1\alpha_2 + k_3\alpha_1^3.
\end{align*}
We solve these by variation of constants, using $\alpha_1(0)=1$ and $\alpha_2(0)=\alpha_3(0)=0$ as
\begin{align*}
\alpha_1(\varphi) &= e^{k_1 \varphi}, \\
\alpha_2(\varphi) &= \int_0^\varphi e^{k_1(\varphi+s)}k_2(s)\mathrm{d} s, \\
\alpha_3(\varphi) &= e^{k_1\varphi}\left[2\int_0^\varphi k_2(s)\alpha_2(s)\mathrm{d} s + \int_0^\varphi e^{2k_1s}k_3(s)\mathrm{d} s \right].
\end{align*}
Periodic orbits are the solutions with $r_0\neq 0$ of
\begin{equation}\label{e:2ndper}
0 = r(2\pi)-r(0) = r_0\int_0^{2\pi}\alpha_1'(\varphi)\mathrm{d}\varphi + r_0^2\int_0^{2\pi}\alpha_2'(\varphi)\mathrm{d}\varphi + r_0^3\int_0^{2\pi}\alpha_3'(\varphi)\mathrm{d}\varphi + \mathcal{O}\left(r_0^4\right).
\end{equation}
Straightforward computations give $\alpha_j(2\pi)-\alpha_j(0) = \Gamma_j + \mathcal{O}\left(\mu\right)$, $j=2,3$,
where $\Gamma_2 = \frac{4}{3\omega}\sigma_{_\#}=0$ and $\Gamma_3 = \frac{1}{\omega^2}\left( \frac{32}{9}\sigma_{_\#}^2-\sigma_2 \right)=-\frac{\sigma_2}{\omega^2}$ since $\sigma_{_\#}=0$.
Substitution into the equation \eqref{e:2ndper} for periodic orbits and dividing out $r_0\neq 0$ yields the bifurcation equation
\begin{equation}\label{Thm2_eq}
0 = \frac{2\pi}{\omega}\mu +
\Gamma_3 r_0^2 + \mathcal{O}\left(\mu^2 + \mu r_0 + r_0^3\right).
\end{equation}
Here the implicit function theorem applies a priori to provide a unique branch $\mu(r_0)$ with
\begin{equation}\label{e:bifeq2}
\mu =
\frac{\sigma_2}{2\pi\omega}r_0^2 + \mathcal{O}\left(r_0^3\right).
\end{equation}
Solving this for $r_0$ provides \eqref{r_2nd_HB}, where the square root to be real requires $\mu\omega\sigma_2 > 0$, which gives the claimed sub/supercriticality.
\end{proof}
This last theorem readily extends to the analogue of the so-called Bautin bifurcation for smooth vector fields, also called generalized Hopf bifurcation, which unfolds from zero first Lyapunov coefficient and identifies a curve of fold points. From \eqref{Thm2_eq} we directly derive the loci fold points in the $(\mu, \sigma_{_\#})$-parameter plane as
\[
\mu = -\frac{2\omega^2}{9\sigma_2} \sigma_{_\#}^2
\]
to leading order with respect to $\sigma_{_\#}$.
Notably, the loci of fold points for the smooth Bautin bifurcation also lie on a quadratic curve in terms of the first Lyapunov coefficient. This last similarity is due to the fact that the ODE of the smooth case has no even terms in the radial component, $\dot{r} = \mu r + \sigma_s r^3 + \sigma_l r^5$, leading to $\mu = -\frac{\sigma_s^2}{4\sigma_{l}}$. In the $(\mu,\sigma_{_\#})$-parameter plane, the origin corresponds to the Bautin point and the vertical axis, $\mu=0$, to the sub- and supercritical Hopf bifurcations for positive- and negative values of $\sigma_{_\#}$, respectively.
\section{Generalizations}\label{s:general}
In this section we discuss analogous bifurcation results for the generalization from the absolute value, \eqref{gen_abs_val}, and then turn to higher dimensional systems as well as general linear form of the linear part.
\subsection{Generalization from the absolute value}\label{s:appplanar}
Recall our notation for different left and right slopes \eqref{gen_abs_val}, and consider the generalized canonical equation
\begin{equation}\label{e:genscalar}
\dot{u} = \mu u+\sigma_{_\#} u^j [u]_{\pn}^{\pp},
\end{equation}
with left slope $p_-$, right slope $p_+$ and $j\in\mathbb{N}$ measuring the degree of smoothness such that the right-hand side is $C^j$ but not $C^{j+1}$ smooth. Sample bifurcation diagrams for $j=1$ and $j=2$ are plotted in Figure \ref{f:genslopes} for $\sigma_{_\#}=-1$.
\begin{figure}
\centering
\begin{tabular}{cc}
\includegraphics[width= 0.25\linewidth]{1D_generalization_1.pdf}\hspace*{1cm}
&\includegraphics[width= 0.25\linewidth]{1D_generalization_2.pdf}\\
(a) & (b)
\end{tabular}
\caption{Degenerated supercritical pitchfork bifurcation of \eqref{e:genscalar} for {$p_-=-1$, $p_+=5$} of degree $j=1$ (a) and $j=2$ (b).}\label{f:genslopes}
\end{figure}
The case $j=2$ highlights that also lack of smoothness in the cubic terms impacts the bifurcation in general. We do not pursue this further here, but analogous to the following discussion, it is possible to derive a modified normal form coefficient $S_c$.
For the Hopf bifurcation analysis, we analogously replace the absolute value in \eqref{e:quadnonlin} by \eqref{gen_abs_val}, and thus replace $f,g$ in \eqref{e:planar0} by
\begin{align}
f\left( v, w; \alpha \right) &= a_{11}v\ABS{v}{\alpha_1} + a_{12}v\ABS{w}{\alpha_2} + a_{21}w\ABS{v}{\alpha_3} + a_{22}w\ABS{w}{\alpha_4},\label{f_general}\\
g\left( v, w; \beta \right) &= b_{11}v\ABS{v}{\beta_1} + b_{12}v\ABS{w}{\beta_2} + b_{21}w\ABS{v}{\beta_3} + b_{22}w\ABS{w}{\beta_4},\label{g_general}
\end{align}
where $\alpha = (\alpha_{1_\pm},\alpha_{2_\pm},\alpha_{3_\pm},\alpha_{4_\pm})$, $\beta = (\beta_{1_\pm},\beta_{2_\pm},\beta_{3_\pm},\beta_{4_\pm}) \in\mathbb{R}^8$.
This generalization leads to the generalized non-smooth first Lyapunov coefficient given by
\begin{equation}\label{e:tildesig}
\widetilde\sigma_{_\#}:=a_{11}(\alpha_{1_+}-\alpha_{1_-})+\frac 1 2 a_{12}(\alpha_{2_+}-\alpha_{2_-})+
\frac 1 2 b_{21}(\beta_{3_+}-\beta_{3_-})+ b_{22}(\beta_{4_+}-\beta_{4_-}).
\end{equation}
Notably, in the smooth case, where the left- and right slopes coincide, we have $\widetilde\sigma_{_\#}=0$, and if left- and right slopes are $-1$ and $1$, respectively, we recover $\sigma_{_\#}$.
\begin{theorem}\label{Thm0_General}
If $\widetilde\sigma_{_\#}\neq 0$, then the statement of Theorem~\ref{1stPart} holds true for \eqref{e:planar0} with $f,g$ from \eqref{f_general}, \eqref{g_general}, respectively, with $\sigma_{_\#}$ replaced by $\widetilde\sigma_{_\#}$.
\end{theorem}
\begin{proof}
Taking polar coordinates
we obtain \eqref{System2Dpolar},
where
\begin{align*}
\chi_2(\varphi) =& \; c^2\left( a_{11}\ABS{c}{\alpha_1}+a_{12}\ABS{s}{\alpha_2} \right) + s^2\left( b_{21}\ABS{c}{\beta_3}+b_{22}\ABS{s}{\beta_4} \right)\\ &+ sc\left( a_{21}\ABS{c}{\alpha_3} + a_{22}\ABS{s}{\alpha_4} + b_{11}\ABS{c}{\beta_1} + b_{12}\ABS{s}{\beta_2} \right),
\end{align*}
{again with $s:=\sin(\varphi)$, $c:=\cos(\varphi)$.}
Applying Proposition \ref{Thm_Gen}
we compute $\int_0^{2\pi}\chi_2(\varphi)\mathrm{d}\varphi$, which gives $\frac{4}{3} \widetilde\sigma_{_\#} $.
Indeed, from \eqref{gen_abs_val} we obtain
\begin{align*}
\int_0^{2\pi} c^2 a_{11}\ABS{c}{\alpha_1} \mathrm{d}\varphi &= \int_0^{\frac{\pi}{2}} c^3 a_{11}\alpha_{1_+} \mathrm{d}\varphi + \int_{\frac{\pi}{2}}^{\frac{3\pi}{2}} c^3 a_{11}\alpha_{1_-} \mathrm{d}\varphi + \int_{\frac{3\pi}{2}}^{2\pi} c^3 a_{11}\alpha_{1_+} \mathrm{d}\varphi \\
&= \frac{4}{3}a_{11}\left( \alpha_{1_+}-\alpha_{1_-} \right), \\
\int_0^{2\pi} c^2 a_{12}\ABS{s}{\alpha_2} \mathrm{d}\varphi &= \int_0^\pi c^2s\ a_{12}\alpha_{2_+} \mathrm{d}\varphi + \int_\pi^{2\pi} c^2s\ a_{12}\alpha_{2_-} \mathrm{d}\varphi \\
&= \frac{2}{3}a_{12}\left( \alpha_{2_+}-\alpha_{2_-} \right),
\end{align*}
and similarly for the other terms. Note that the integral of the {third term on the} right-hand side of $\chi_2$ vanishes due to the symmetry of $sc$.
Thus, we get \eqref{periodic_orbit} with $\sigma_{_\#}$ replaced by $\widetilde\sigma_{_\#}=0$.
\end{proof}
\subsection{$3$D system}\label{s:3D}
In this section we extend the previous results to higher dimensional systems. Recall that Proposition~\ref{prop:inv_man} and Theorem~\ref{t_per_orb} rely on hyperbolicity of the spectrum of $A(0)$ from \eqref{e:abstract} except for a simple pair of complex conjugate eigenvalues. Analogously, averaging theory can be used in this setting to obtain a normal form as in Theorem~\ref{t_averaging}. Here we follow the `direct method' and obtain bifurcation results also without normal hyperbolicity.
To simplify the exposition, we start with the absolute value $\abs{\cdot}$ and consider an extension of the planar quadratic case \eqref{e:planar0}, \eqref{e:quadnonlin}, motivated by the example in \cite{InitialPaper}, which is a simplification of a model used for ship maneuvering. As discussed in \S\ref{s:abstract}, we first assume the linear part is in normal form -- a general linear part will be considered in \S\ref{Gen_linear_part} -- which gives
\begin{equation}
\begin{pmatrix}
\dot{u}\\
\dot{{v}}\\
\dot{{w}}\\
\end{pmatrix} =
\begin{pmatrix}
c_1 u + c_2 u^2 + { c_{3} uv + c_{4} uw } + c_5vw + h\left( v, w \right)\\
\mu v - \omega w + c_6 uv + c_7 uw + f\left( v, w \right)\\
\omega v + \mu w + c_{8} uv + c_{9} uw + g\left( v, w \right)
\end{pmatrix},
\label{3DAbstractSystem}
\end{equation}
where $f,g$ are as in \eqref{e:quadnonlin}, $h\left( v, w \right) = h_{11}v\abs{v} + h_{12}v\abs{w} + h_{21}w\abs{v} + h_{22}w\abs{w}$
and $h_{ij}, c_k$, $\forall i,j\in\{1,2\}$, $\forall k\in\{1,\ldots 9\}$, are real constants, all viewed as parameters. Again we assume $\omega\neq 0$ and take $\mu$ as the bifurcation parameter.
With linear part in normal form in the coordinates of Lemma \ref{l:cyl}, the vector field is actually smooth in the additional variable $u$. It turns out that in the generic case $c_1\neq 0$, this additional smoothness will not be relevant for the leading order analysis, while we make use of it in the degenerate case $c_1=0$.
We define the following quantities that appear in the upcoming results:
\begin{equation}\begin{aligned}
{\overline{\gamma}}_{10} &= e^{\frac{2\pi c_1}{\omega}}-1,
&{\overline{\gamma}}_{20} &= \frac{e^{\frac{2\pi c_1}{\omega}}\left(e^{\frac{2\pi c_1}{\omega}}-1\right)}{c_1} \left( c_2-\frac{c_1\rho_2}{\omega(c_1^2+4\omega^2)} \right),\\[1em]
{\overline{\gamma}}_{02} &= \frac{1}{\omega}e^{\frac{2\pi c_1}{\omega}} \int_0^{2\pi} e^{s\frac{2\mu-c_1}{\omega}}\Upsilon(s) \mathrm{d} s,
&{\overline{\gamma}}_{11} &= \frac{2}{3\omega^2}e^{\frac{2\pi c_1}{\omega}}
\left[ c_1\left(2\tau_2 + \frac{P}
{3\omega}\mu\right) {-3\pi c_4\mu} \right]
+ \mathcal{O}(\mu^2),\\[1em]
{\overline{\delta}}_{01} &= \frac{2\pi\mu}{\omega} + \mathcal{O}(\mu^2),
&{\overline{\delta}}_{02} &= \frac{2}{3\omega}\left[ \sigma_{_\#}\left(2 + \frac{6\pi}{\omega}\mu\right) + \frac{Q
}{3\omega}\mu \right] + \mathcal{O}(\mu^2),\\[1em]
&&{\overline{\delta}}_{11} &= \frac{e^{\frac{2\pi c_1}{\omega}}-1}{c_1}\frac{1}{\omega(c_1^2+4\omega^2)} \big[ \omega\rho_1+R\mu \big] + \mathcal{O}(\mu^2),
\end{aligned}\label{ogammas_odeltas}\end{equation}
where we shortened the notation by lumping the weighted sums of coefficients from $f,g$, and from the smooth quadratic terms, respectively, given by
\begin{align*}
\tau_1&=4a_{22}+5a_{21}-5b_{12}-4b_{11}, \quad
\tau_2=2a_{22}+a_{21}-b_{12}-2b_{11}, \quad \tau_3=a_{11}-a_{12}-b_{21}+b_{22},\\
P& =3\pi(2\tau_2-a_{11}+b_{21})+4\tau_3, \quad Q =-3\pi(b_{11}+a_{21})+2\tau_1, \quad\quad R=2\pi\rho_1-\rho_2,\\
\rho_1&=c_6c_1^2-c_7c_1\omega +2c_6\omega^2-c_8c_1\omega+2c_9\omega^2,
\quad\rho_2=c_8c_1^2-c_9c_1\omega+2c_8\omega^2+c_6c_1\omega-2c_7\omega^2,
\end{align*}
as well as the $h$-dependent
\[
\Upsilon(\varphi) = c_5cs+h_{11}c\abs{c}+h_{12}c\abs{s}+h_{21}s\abs{c}+h_{22}s\abs{s}.
\]
The explicit form of ${\overline{\gamma}}_{02}$ can be found in Appendix \ref{3D_gammas_deltas}.
\begin{theorem}\label{thm3D}
In cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$, up to time shifts, periodic solutions to \eqref{3DAbstractSystem} with $r(0)=r_0, u(0)=u_0$ for $0\leq |\mu| \ll 1$ near $r=u=0$ are in 1-to-1 correspondence with solutions to the algebraic equation system
\begin{align}
0 &= {\overline{\gamma}}_{10} u_0 + {\overline{\gamma}}_{20} u_0^2 + {\overline{\gamma}}_{02} r_0^2 + {\overline{\gamma}}_{11} u_0r_0 + \mathcal{O}\left(3\right),\label{Periodic_R-a}\\
0 &= {\overline{\delta}}_{01} r_0 + {\overline{\delta}}_{02} r_0^2+ {\overline{\delta}}_{11} u_0r_0 + \mathcal{O}\left(3\right),\label{Periodic_R-b}
\end{align}
where $\mathcal{O}(3)$ are terms of at least cubic order in $u_0,r_0$.
\end{theorem}
\begin{proof}
In cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$ system \eqref{3DAbstractSystem} becomes
\begin{align}
\label{polar_system_3D}
\begin{cases}
\dot{u} &= c_1 u + c_2 u^2 + { (c_3 c u + c_4 s u)r } + \Upsilon(\varphi)r^2,\\
\dot{r} &= \left(\mu+\chi_1(\varphi)u\right)r + \chi_2(\varphi) r^2,\\
\dot{\varphi} &= \omega + \Omega_0(\varphi)u + \Omega_1(\varphi) r,
\end{cases}
\end{align}
where
$\chi_1(\varphi) = c_6c^2+(c_7+c_8)cs+c_9s^2$, $\Omega_0(\varphi) = c_8c^2+(c_9-c_6)cs-c_7s^2$,
and the non-smooth functions $\chi_2(\varphi)$ and $\Omega_1(\varphi)$ are as in \eqref{chi} and \eqref{Omega}, respectively.
Upon rescaling time the equations for $u$ and $r$ of the previous system become
\begin{align}\label{eq_u_r}
\begin{cases}
{u}' &= \frac{du/dt}{d\varphi /dt} = \frac{c_1 u + c_2 u^2 + { (c_3 c u + c_4 s u)r } + \Upsilon(\varphi)r^2}{\omega + \Omega_0(\varphi)u + \Omega_1(\varphi) r} =:\Psi_u(u,r,\varphi),\\[10pt]
{r}' &= \frac{dr/dt}{d\varphi /dt} = \frac{\left(\mu+\chi_1(\varphi)u\right)r + \chi_2(\varphi) r^2}{\omega + \Omega_0(\varphi)u + \Omega_1(\varphi) r} =:\Psi_r(u,r,\varphi).
\end{cases}
\end{align}
Taylor expansion of $u'$ and $r'$ in $(u,r)=(0,0)$ up to third order gives:
\begin{subequations}
\begin{align}
\begin{split} \label{UP-a}
u' &= \Psi_u(0,0,\varphi) + \partial_u\Psi_u(0,0,\varphi)u + \partial_r\Psi_u(0,0,\varphi)r \\
&+ \frac{1}{2}\partial_u^2\Psi_u(0,0,\varphi)u^2 + \frac{1}{2}\partial_r^2\Psi_u(0,0,\varphi)r^2 + \partial_{ur}^2\Psi_u(0,0,\varphi)ur + \mathcal{O}\left(3\right),
\end{split}\\[5pt]
\begin{split} \label{UP-b}
r' &= \Psi_r(0,0,\varphi) + \partial_u\Psi_r(0,0,\varphi)u + \partial_r\Psi_r(0,0,\varphi)r \\
&+ \frac{1}{2}\partial_u^2\Psi_r(0,0,\varphi)u^2 + \frac{1}{2}\partial_r^2\Psi_r(0,0,\varphi)r^2 + \partial_{ur}^2\Psi_r(0,0,\varphi)ur + \mathcal{O}\left(3\right).
\end{split}
\end{align}
\end{subequations}
On the other hand, and similarly to the procedure of the $2$-dimensional case, we write $u(\varphi)$ and $r(\varphi)$ as the following expansions with coefficients $\gamma_{ij}, \delta_{ij}$:
\begin{equation}\begin{aligned}
u(\varphi) &= \gamma_{10}(\varphi)u_0 + \gamma_{20}(\varphi)u_0^2 + \gamma_{01}(\varphi)r_0 + \gamma_{02}(\varphi)r_0^2 + \gamma_{11}(\varphi)u_0r_0 + \mathcal{O}\left(3\right), \\
r(\varphi) &= \delta_{10}(\varphi)u_0 + \delta_{20}(\varphi)u_0^2 + \delta_{01}(\varphi)r_0 + \delta_{02}(\varphi)r_0^2 + \delta_{11}(\varphi)u_0r_0 + \mathcal{O}\left(3\right),
\label{UE_RE}
\end{aligned}\end{equation}
with the initial conditions $u(0)=u_0$ and $r(0)=r_0$, which imply $\gamma_{10}(0)=\delta_{01}(0)=1$ and the rest zero.
Substituting (\ref{UE_RE}) into (\ref{UP-a}) and (\ref{UP-b}) and matching the coefficients of the powers of $u_0$ and $r_0$ we get to solve a set of ODEs in order to obtain the expressions for $\gamma_{ij}$ and $\delta_{ij}$ (see Appendix \ref{3D_gammas_deltas} for the details). Using these, the system of boundary value problems $0 = u(2\pi) - u(0)$, $0 = r(2\pi) - r(0)$ for periodic solutions precisely yields
\eqref{Periodic_R-a}, \eqref{Periodic_R-b}, where ${\overline{\gamma}}_{ij}=\gamma_{ij}(2\pi)-\gamma_{ij}(0)$ and ${\overline{\delta}}_{ij}=\delta_{ij}(2\pi)-\delta_{ij}(0)$.
\end{proof}
The solution structure of \eqref{Periodic_R-a}, \eqref{Periodic_R-b} strongly depends on whether $c_1=0$ or not. If not, then the transverse direction is hyperbolic and Theorem~\ref{1stPart} implies a locally unique branch of periodic solutions. In the non-hyperbolic case the situation is different and we note that if $c_1=0$, then
with
$\gamma_{_\#} := 2h_{21} + c_5 + \pi h_{22}$, we have
\begin{equation}\label{e:c20}
{\overline{\gamma}}_{10}=0, \hspace*{0.5cm} {\overline{\gamma}}_{11}={-\frac{2\pi c_4}{\omega^2}\mu+}\mathcal{O}(\mu^2), \hspace*{0.5cm} {\overline{\gamma}}_{20}=\frac{2\pi c_2}{\omega}, \hspace*{0.5cm}
{\overline{\gamma}}_{02} = -\frac{\pi\gamma_{_\#}}{\omega^2}\mu + \mathcal{O}(\mu^2).
\end{equation}
\begin{corollary}\label{c:3D}
Consider system \eqref{3DAbstractSystem} in cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$. If $c_1\neq0$, then $u=u(\varphi;\mu) = \mathcal{O}\left(\mu^2\right)$ and the statement of Theorem~\ref{1stPart} holds true.
If $c_1=0$ and $\omega c_2\gamma_{_\#} \mu>0$, then precisely two curves of periodic solutions bifurcate at $\mu=0$ for $\mu\sigma_{_\#}\leq 0$, each in the sense of Theorem \ref{t_per_orb}, and their initial conditions $r(0)=r_0$, $u(0)=u_0^\pm$ satisfy
\begin{align}
u_0^\pm &= u_0^\pm(\mu) =
\mp \frac{3\pi}{2\sigma_{_\#}}\sqrt{\frac{\gamma_{_\#}}{2\omega c_2} \mu^3} + \mathcal{O}(\mu^{2}) =\mathcal{O}(|\mu|^{3/2}),\label{e:u3D} \\
r_0 &= r_0(\mu)= -\frac{3\pi}{2\sigma_{_\#}}\mu + \mathcal{O}(|\mu|^{3/2}). \label{e:3Dr2}
\end{align}
In case $c_1=0$ and $\omega c_2\gamma_{_\#} \mu<0$, there is no bifurcation through $\mu$.
\end{corollary}
\begin{proof}
In the (transversely) hyperbolic case $c_1\neq 0$ we have ${\overline{\gamma}}_{10}\neq 0$, and thus one may solve \eqref{Periodic_R-a} for $u_0$ by the implicit function theorem as $u_0=u_0(r_0) = \mathcal{O}(r_0^2)$.
Substitution into \eqref{Periodic_R-b} changes the higher order term only, so that to leading order we obtain the same problem as in Theorem~\ref{1stPart} with solution given by \eqref{periodic_orbit}. The stability statement of Theorem~\ref{1stPart} holds true from the existence of a $2$-dimensional Lipschitz continuous invariant manifold given by Proposition \ref{prop:inv_man}.
We now consider $c_1=0$. Using \eqref{e:c20} we can cast \eqref{Periodic_R-a}, \eqref{Periodic_R-b} as
\begin{align}
0 &= \frac{2\pi c_2}{\omega} u_0^2 - \frac{\pi\gamma_{_\#}}{\omega^2} \mu r_0^2 {-\frac{2\pi c_4}{\omega^2}\mu u_0r_0} + \mathcal{O}\left(\mu^2 r_0^2\right) + \mathcal{O}\left(3\right),\label{Periodic_R-aa}\\
0 &= {\overline{\delta}}_{01} r_0 + \frac{4\sigma_{_\#}}{3\omega} r_0^2 + \mathcal{O}\big(|u_0r_0| + |\mu r_0| (|u_0|+ |r_0|)\big) + \mathcal{O}\left(3\right),\label{Periodic_R-bb}
\end{align}
so that we may solve \eqref{Periodic_R-aa} to leading order as
\begin{equation}
\label{exp_u0_of_r0}
u_0 = u_0^\pm(r_0;\mu) = \frac{c_4}{\omega c_2}\mu r_0 \pm r_0\sqrt{\frac{c_4^2}{4c_2^2}\mu^2+\frac{\gamma_{_\#}}{2\omega c_2} \mu} + \mathcal{O}(|\mu|) = \pm r_0\sqrt{\frac{\gamma_{_\#}}{2\omega c_2} \mu} + \mathcal{O}(|\mu|).
\end{equation}
Substitution into \eqref{Periodic_R-b} gives a factor $r_0$ corresponding to the trivial solution $u_0=r_0=0$. For non-trivial solutions we divide by $r_0\neq 0$ and solve the leading order part as
$$ r_0 = - \frac{{\overline{\delta}}_{01}}{\frac{4\sigma_{_\#}}{3\omega} + \mathcal{O}(\sqrt{\mu})} = - \frac{3\pi}{2\sigma_{_\#}}\mu + \mathcal{O}(|\mu|^{3/2}).$$
Next, we substitute this into \eqref{exp_u0_of_r0} and note that perturbation by the higher order terms yields \eqref{e:u3D}, \eqref{e:3Dr2}. These give positive $r_0$ in case $\mu\sigma_{_\#}<0$ and therefore real valued $u_0$ in case $\omega c_2\gamma_{_\#} \mu>0$. However, if $\omega c_2\gamma_{_\#} \mu<0$ then for any $0<|\mu|\ll 1$ either $r_0<0$ or $u_0$ is imaginary.
\end{proof}
We subsequently consider the degenerate case $\sigma_{_\#}=0$, but assume $c_1\neq 0$,
which generalizes Theorem~\ref{2ndPart} to the present $3$-dimensional setting.
We will show that the generalization of $\sigma_2$ is given by $-\omega^2\widetilde{\Gamma}_3$, where
\begin{equation}\label{tilde_o}
\widetilde{\Gamma}_3 := \tilde{\delta}_{03} - \tilde{\delta}_{11}\frac{\tilde{\gamma}_{02}}{{\overline{\gamma}}_{10}},
\end{equation}
with ${\overline{\gamma}}_{10}$ from \eqref{ogammas_odeltas}, and
\begin{equation*}
\begin{aligned}
\tilde{\delta}_{11} &:= {\overline{\delta}}_{11}|_{\mu=0} =\frac{e^{\frac{2\pi c_1}{\omega}}-1}{c_1}\frac{\rho_1}{c_1^2+4\omega^2}, \hspace*{1cm} \tilde{\gamma}_{02} := {\overline{\gamma}}_{02}|_{\mu=0} =\frac{1}{\omega}e^{\frac{2\pi c_1}{\omega}} \int_0^{2\pi}e^{-s\frac{c_1}{\omega}}\Upsilon(s) \mathrm{d} s,\\
\tilde{\delta}_{03} &:= \Gamma_3 + \frac{1}{\omega^2}\int_0^{2\pi} \chi_1(s)\int_0^s e^{c_1\frac{s-\tau}{\omega}}\Upsilon(\tau) \mathrm{d}\tau\mathrm{d} s.
\end{aligned}
\end{equation*}
Here we use the same notation for $\Gamma_3$ as in \eqref{Thm2_eq}, i.e., $\Gamma_3=\frac{2}{\omega^2}\int_0^{2\pi}\chi_2(s) \int_0^{s}\chi_2(\tau)\mathrm{d}\tau \mathrm{d} s-\frac{1}{\omega^2}\int_0^{2\pi}\chi_2(s)\Omega_1\mathrm{d} s$.
Comparing $\widetilde{\Gamma}_3$ with $\Gamma_3$, we thus expect $\widetilde{\Gamma}_3 \neq \Gamma_3$, as a results of the coupling with the additional variable $u$. We omit here the fully explicit approach for $\widetilde{\Gamma}_3$, since the expressions become too lengthy for practical uses. However, for illustration, we consider the simpler case $h=0$ in \eqref{3DAbstractSystem}, which yields
\begin{equation*}
\tilde{\delta}_{03} = \Gamma_3 + \frac{ c_5\pi(c_6+c_9)}{c_1^2+4\omega^2}\left(e^{\frac{2\pi}{\omega}c_1}-1\right), \hspace*{1cm}
\tilde{\delta}_{11}\frac{\tilde{\gamma}_{02}}{{\overline{\gamma}}_{10}} = \frac{c_5\rho_1\omega}{c_1(c_1^2+4\omega^2)^2} \left(e^{\frac{2\pi}{\omega}c_1}-1\right),
\end{equation*}
and thus,
\begin{equation*}
\widetilde{\Gamma}_3 = \Gamma_3 + \frac{c_5\left(e^{\frac{2\pi}{\omega}c_1}-1\right)}{c_1^2+4\omega^2}\left[ \pi (c_6+c_9) - \frac{\rho_1\omega}{c_1(c_1^2+4\omega^2)} \right].
\end{equation*}
\begin{corollary}\label{Second:c:3D}
Consider \eqref{3DAbstractSystem} in cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$ and $\sigma_{_\#}=0$. If $c_1\neq0$, then $u=u(\varphi;\mu) = \mathcal{O}\left(\mu^2\right)$ and the statement of Theorem~\ref{2ndPart} holds true {with} $\sigma_2$ {replaced} by $-\omega^2\widetilde{\Gamma}_3$.
\end{corollary}
\begin{proof}
Upon rescaling time the equations for $u,r$ in cylindrical coordinates of \eqref{3DAbstractSystem} become \eqref{eq_u_r}. Similarly to the proof of Theorem \ref{thm3D}, we compute the Taylor expansion of $u'$ and $r'$ in $(u,r)=(0,0)$ up to forth order (see Appendix \ref{3D_gammas_deltas_second} for the details) and we write $u(\varphi)$ and $r(\varphi)$ as the following expansions:
\begin{equation}\begin{aligned} u(\varphi) &= \gamma_{10}(\varphi)u_0 + \gamma_{20}(\varphi)u_0^2 + \gamma_{30}(\varphi)u_0^3 + \gamma_{01}(\varphi)r_0 + \gamma_{02}(\varphi)r_0^2 + \gamma_{03}(\varphi)r_0^3 \\
&+ \gamma_{11}(\varphi)u_0r_0 + \gamma_{21}(\varphi)u_0^2r_0 + \gamma_{12}(\varphi)u_0r_0^2 + \mathcal{O}\left(4\right), \\[0.5em]
r(\varphi) &= \delta_{10}(\varphi)u_0 + \delta_{20}(\varphi)u_0^2 + \delta_{30}(\varphi)u_0^3 + \delta_{01}(\varphi)r_0 + \delta_{02}(\varphi)r_0^2 + \delta_{03}(\varphi)r_0^3 \\
&+ \delta_{11}(\varphi)u_0r_0 + \delta_{21}(\varphi)u_0^2r_0 + \delta_{12}(\varphi)u_0r_0^2 + \mathcal{O}\left(4\right),
\label{UE_RE_Second}
\end{aligned}\end{equation}
with the initial conditions $u(0)=u_0$ and $r(0)=r_0$, which imply $\gamma_{10}(0)=\delta_{01}(0)=1$ and the rest zero. With these expressions we compute, as before, the functions $\gamma_{ij}$ and $\delta_{ij}$ $\forall i,j\in\mathbb{N}_0$ such that $i+j=3$. Note that the others are the same as for Theorem \ref{thm3D}. The periodic solutions with $r(0)=r_0$, $u(0)=u_0$ for $0\leq |\mu| \ll 1$ near $r=u=0$ are in 1-to-1 correspondence with solutions to the algebraic equation system
\begin{align}
0 &= {\overline{\gamma}}_{10} u_0 + {\overline{\gamma}}_{20} u_0^2 + {\overline{\gamma}}_{30} u_0^3 + {\overline{\gamma}}_{02} r_0^2 + {\overline{\gamma}}_{03} r_0^3 + {\overline{\gamma}}_{11} u_0r_0 + {\overline{\gamma}}_{21} u_0^2r_0 + {\overline{\gamma}}_{12} u_0r_0^2 + \mathcal{O}\left(4\right),\label{Periodic_R-a3}\\
0 &= {\overline{\delta}}_{01} r_0 + {\overline{\delta}}_{02} r_0^2 + {\overline{\delta}}_{03} r_0^3 + {\overline{\delta}}_{11} u_0r_0 + {\overline{\delta}}_{21} u_0^2r_0 + {\overline{\delta}}_{12} u_0r_0^2 + \mathcal{O}\left(4\right),\label{Periodic_R-b3}
\end{align}
where $\mathcal{O}(4)$ are terms of at least fourth order in $u_0$, $r_0$, and ${\overline{\gamma}}_{ij}=\gamma_{ij}(2\pi)-\gamma_{ij}(0)$, ${\overline{\delta}}_{ij}=\delta_{ij}(2\pi)-\delta_{ij}(0)$.
Moreover, since $c_1\neq 0$ we have ${\overline{\gamma}}_{10}\neq 0$. Therefore, we may solve \eqref{Periodic_R-a3} for $u_0$ by the implicit function theorem as $u_0=-\frac{{\overline{\gamma}}_{02}}{{\overline{\gamma}}_{10}}r_0^2+\mathcal{O}\left(r_0^3\right)=\mathcal{O}\left(r_0^2\right)$.
Substitution into \eqref{Periodic_R-b3} and dividing out $r_0\neq 0$ yield
\begin{equation*}
0 = {\overline{\delta}}_{01} + {\overline{\delta}}_{02} r_0 + \left({\overline{\delta}}_{03} - {\overline{\delta}}_{11}\frac{{\overline{\gamma}}_{02}}{{\overline{\gamma}}_{10}}\right)r_0^2 + \mathcal{O}\left(3\right),
\end{equation*}
which we rewrite, to leading order and similarly to \eqref{Thm2_eq} in Theorem~\ref{2ndPart}, as
\begin{equation}\label{Cor_deg}
0 = \frac{2\pi}{\omega}\mu + \widetilde{\Gamma}_2 r_0 + \widetilde{\Gamma}_3 r_0^2 + \mathcal{O}\left(\mu^2 + \mu r_0 + r_0^3\right),
\end{equation}
where $\widetilde{\Gamma}_2={\overline{\delta}}_{02}|_{\mu=0}$, which vanishes for $\sigma_{_\#}=0$ analogous to $\Gamma_2$ in \eqref{Thm2_eq}, and $\widetilde{\Gamma}_3$ is as defined in \eqref{tilde_o}; the expression for ${\overline{\delta}}_{03}$ stems from \eqref{delta_03}.
Hence, the solution for \eqref{Cor_deg} is given by \eqref{r_2nd_HB} replacing $\sigma_2$ by $-\omega^2\widetilde{\Gamma}_3$, which is assumed to be non-zero.
The stability statement of Theorem~\ref{1stPart} holds true from the existence of a $2$-dimensional Lipschitz continuous invariant manifold given by Proposition \ref{prop:inv_man}.
\end{proof}
Lastly, we use these results to extend system \eqref{3DAbstractSystem} to a higher order model with the generalized absolute value \eqref{gen_abs_val} as follows
\begin{equation}
\begin{pmatrix}
\dot{u}\\
\dot{{v}}\\
\dot{{w}}\\
\end{pmatrix} =
\begin{pmatrix}
c_1 u + c_2 u^2 + { c_3 uv + c_4 uw } + c_5 vw + h(v, w; \gamma)\\
\mu v - \omega w + c_6 uv + c_7 uw + f\left( v, w; \alpha \right) + f_q\left( v, w\right) + f_c\left( v, w\right) \\
\omega v + \mu w + c_8 uv + c_9 uw + g\left( v, w; \beta \right) + g_q\left( v, w\right) + g_c\left( v, w \right) \\
\end{pmatrix},
\label{3DAbstractSystem_General}
\end{equation}
where $f\left( v, w; \alpha \right)$ and $g\left( v, w; \beta \right)$ are \eqref{f_general} and \eqref{g_general}, respectively, and the functions $f_q,g_q,f_c,g_c$ are as in system \eqref{General2D_AV}. The expression of $h$ is analogous to $f,g$. {We recall also $\widetilde\sigma_{_\#}$ from \eqref{e:tildesig}.}
\begin{corollary}
\label{Generalization_3D}
If $\widetilde\sigma_{_\#}\neq 0$, the statement of Corollary \ref{c:3D} for system \eqref{3DAbstractSystem_General} holds true with $\sigma_{_\#}$ replaced by $\widetilde\sigma_{_\#}$.
\end{corollary}
\begin{proof}
The proof follows from Theorems \ref{Thm0_General} and \ref{thm3D} and Corollary \ref{thm_cubic}.
\end{proof}
This concludes our analysis for the $3$-dimensional case, which paves the way for the $n$-dimensional case discussed thereafter.
\subsection{$n$D system}\label{s:nD}
We consider the $n$-dimensional generalization of system \eqref{3DAbstractSystem} with additional component $u=(u_1,\cdots,u_{n-2})\in\mathbb{R}^{ n-2}$ given by
\begin{equation}
\begin{pmatrix}
\dot{u}\\
\dot{{v}}\\
\dot{{w}}\\
\end{pmatrix} =
\begin{pmatrix}
\tilde A u + U(u,v,w)\\
\mu v - \omega w + \sum_{i=1}^{n-2}({c_6}_i u_iv + {c_7}_i u_iw) + \tilde f\left( v, w \right)\\
\omega v + \mu w + \sum_{i=1}^{n-2}({c_8}_i u_iv + {c_9}_i u_iw) + \tilde g\left( v, w \right)
\end{pmatrix},
\label{nDAbstractSystem}
\end{equation}
where $\tilde A=({c_1}_{ij})_{1\leq i,j\leq n-2}$ is a $(n-2)\times(n-2)$ matrix
and $U: \mathbb{R}^{n-2}\times\mathbb{R}\times\mathbb{R} \longrightarrow \mathbb{R}^{n-2}$ is a nonlinear function, smooth in $u$ and possibly non-smooth in $v,w$ with absolute values as in \eqref{3DAbstractSystem}. Hence, $U(u,v,w) = \mathcal{O}(2)$,
where $\mathcal{O}(2)$ are terms of at least second order in $u_i,v,w$. The constants ${c_1}_{ij}, {c_6}_i, {c_7}_i, {c_8}_i, {c_9}_i$ are all real $\forall i,j\in\{1,\cdots, n-2\}$, and the functions $\tilde f, \tilde g$ are of the same form as the nonlinear part of system \eqref{General2D_AV}.
We present now analogous results as before for this $n$-dimensional case.
However, we refrain from explicitly determining the coefficients involved.
\begin{theorem}
\label{Thm_nD}
Consider \eqref{nDAbstractSystem} in cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$
analogous to Theorem \ref{thm3D} with $u\in\mathbb{R}^{n-2}$. Up to time shifts, periodic solutions to \eqref{nDAbstractSystem} with $r(0)=r_0$, $u(0)=u_0\in\mathbb{R}^{n-2}$,
for $0\leq |\mu|, r_0, |u_0| \ll 1$ are in 1-to-1 correspondence with solutions to the algebraic $(n-1)$-dimensional system given by equations analogous to \eqref{Periodic_R-a} and \eqref{Periodic_R-b}, where ${\overline{\delta}}_{01},{\overline{\delta}}_{02}$ are scalars and ${\overline{\gamma}}_{10}, {\overline{\gamma}}_{11}, {\overline{\gamma}}_{02}, {\overline{\gamma}}_{20}, {\overline{\delta}}_{11}$ are linear maps and quadratic forms in $n-2$ dimensions.
\end{theorem}
\begin{proof}
The proof is analogous to that of Theorem \ref{thm3D}, now by setting up a boundary value problem with $n-2$ equations for $0=u(2\pi)-u(0)$ and one for $0=r(2\pi)-(0)$. This results in a system of $n-1$ equations formed by direct analogues to \eqref{Periodic_R-a} and \eqref{Periodic_R-b}, where $\mathcal{O}(3)$ contains all terms of at least cubic order in ${u_0}_i, r_0$,
and ${\overline{\gamma}}_{20}u_0^2$ is a quadratic form in $n-2$ dimensions.
\end{proof}
Similar to the $3$-dimensional case, the solution structure of the $(n-1)$-dimensional system \eqref{Periodic_R-a}, \eqref{Periodic_R-b} depends on whether the matrix $\tilde A$ is invertible (i.e., {the full linear part} $A$ satisfies Hypothesis~\ref{h:AG}) or not, as shown in the next result.
\begin{corollary}
\label{c:nD}
Consider \eqref{nDAbstractSystem} in cylindrical coordinates $(u,v,w)=(u,r\cos{\varphi},r\sin{\varphi})$. If $\tilde A$ is invertible,
then the solution vector $u=u(\varphi;\mu)$ is of order $\mathcal{O}\left(\mu^2\right)$ and the statement of Theorem \ref{1stPart} holds true. If $\tilde A$ is not invertible with $1$-dimensional generalized kernel, then there are constants $c_2$, $\gamma_{\#}$ such that the statements of Corollary \ref{c:3D} for $c_1=0$ hold true.
\end{corollary}
\begin{proof}
From Theorem \ref{Thm_nD} we have the corresponding equations \eqref{Periodic_R-a}, \eqref{Periodic_R-b} for the $n$-dimensional system \eqref{nDAbstractSystem}, where ${\overline{\gamma}}_{20}u_0^2$ is a quadratic form in $n-2$ dimensions. If $\tilde A$ is invertible, so is the $(n-2)\times(n-2)$ matrix ${\overline{\gamma}}_{10}=e^{2\pi \tilde A/\omega}-\mathrm{Id}$. Solving the $(n-1)$-dimensional system gives the same as in the proof of Corollary \ref{c:3D} to leading order.
If $\tilde A$ is not invertible, then by assumption it has a $1$-dimensional generalized kernel. In this case, we {change coordinates in the analogue of} \eqref{Periodic_R-a} such that the matrix ${\overline{\gamma}}_{10}$ is block-diagonal with the kernel in the top left, and an invertible $(n-3)\times(n-3)$ block ${\overline{\gamma}}'_{10}$ on the lower right of the matrix. Thus, we split \eqref{Periodic_R-a} into a scalar equation and a $(n-3)$-dimensional system. By the implicit function theorem we solve the equations corresponding to ${\overline{\gamma}}'_{10}$ and substitute the result into the other two equations: the one with the $1$-dimensional kernel and the corresponding \eqref{Periodic_R-b} with ${\overline{\delta}}_{01} = \frac{2\pi\mu}{\omega} + \mathcal{O}(\mu^2)$, ${\overline{\delta}}_{02} = \frac{4\sigma_{_\#}}{3\omega} + \mathcal{O}(\mu)$. We obtain then two scalar equations of the same type as in Corollary \ref{c:3D} for the case $c_1=0$.
\end{proof}
We omit explicit formulas for $c_2, \gamma_{\#}$, but note that these can be provided in terms of data from $\tilde A$.
Before concluding this section, we note that these results directly extend to the more general non-smooth terms \eqref{gen_abs_val} and to additional higher order functions as in \eqref{General2D_AV}.
\begin{corollary}\label{c:nD:gen_abs}
Consider system \eqref{nDAbstractSystem} with $\tilde f, \tilde g$ as the nonlinear part of \eqref{General2D_AV}, but with $f,g$ as in \eqref{f_general}, \eqref{g_general}, respectively. If $\tilde A$ is invertible and $\widetilde\sigma_{_\#}\neq 0$, cf.\ \eqref{e:tildesig}, then the statement of Corollary \ref{c:nD} holds true with $\sigma_{_\#}$ replaced by $\widetilde\sigma_{_\#}$.
\end{corollary}
Recall from \S\ref{s:abstract} that we have presented results for systems where the linear part is in block-diagonal form and normal form for the oscillatory part, while the nonlinear part is smooth in the radial direction. For completeness, we next discuss the case of general linear part, i.e., not necessarily in normal form.
\subsection{General linear part}\label{Gen_linear_part}
Here we show that our analysis also applies to systems with general linear part. First, we consider the planar case \eqref{e:abstractplanar} with
\begin{align*}
f_1(u_1,u_2)&=a_{11}u_1|u_1|+a_{12}u_1|u_2|+a_{21}u_2|u_1|+a_{22}u_2|u_2| +\mathcal{O}(3),\\
f_2(u_1,u_2)&=b_{11}u_1|u_1|+b_{12}u_1|u_2|+b_{21}u_2|u_1|+b_{22}u_2|u_2| +\mathcal{O}(3).
\end{align*}
Under Hypothesis~\ref{h:AG}, changing the linear part of \eqref{e:abstractplanar} to normal form by the associated matrix $\textbf{T}$,
i.e., $\textbf{T}\cdot(v_1,v_2)^T=(u_1,u_2)^T$, the system becomes
\begin{equation}\label{e:abstractplanar:2}
\begin{pmatrix}
\dot v_1\\
\dot v_2
\end{pmatrix} =
\begin{pmatrix}
\mu & -\omega\\
\omega & \mu
\end{pmatrix}\begin{pmatrix}
v_1\\
v_2
\end{pmatrix}+\textbf{T}^{-1}\begin{pmatrix}
g_1\left( v_1, v_2 \right)\\
g_2\left( v_1, v_2 \right)
\end{pmatrix},
\end{equation} where $g_i(v_1,v_2)=f_i\left(\textbf{T}\cdot (v_1,v_1)^T\right)$ for $i\in\{1,2\}$ and with $\textbf{T}=(z_{ij})_{1\leq i,j\leq 2}$, {as well as the shorthand $[[\cdot]]:=\cdot|\cdot|$},
we have
\begin{align*}
g_1(v_1,v_2)&=a_{11} [[z_{11}v_1+z_{12}v_2]] +a_{12}(z_{11}v_1+z_{12}v_2)|z_{21}v_1+z_{22}v_2|\\
&+a_{21}(z_{21}v_1+z_{22}v_2)|z_{11}v_1+z_{12}v_2|+a_{22} [[z_{21}v_1+z_{22}v_2]] +\mathcal{O}(3),\\[0.5em]
g_2(v_1,v_2)&=b_{11} [[z_{11}v_1+z_{12}v_2]] +b_{12}(z_{11}v_1+z_{12}v_2)|z_{21}v_1+z_{22}v_2|\\
&+b_{21}(z_{21}v_1+z_{22}v_2)|z_{11}v_1+z_{12}v_2|+b_{22} [[z_{21}v_1+z_{22}v_2]] +\mathcal{O}(3).
\end{align*}
{We use} polar coordinates for $(v_1,v_2)=(r\cos(\varphi),r\sin(\varphi))$ {as before,} and
\[
(z_{11},z_{12})=(C\cos(\phi),C\sin(\phi)), \hspace*{1cm} (z_{21},z_{22})=(D\cos(\vartheta),D\sin(\vartheta)),
\] where $C,D\in\mathbb{R}$, $\phi,\vartheta\in[0,2\pi)$ are fixed constants.
System \eqref{e:abstractplanar:2} can be written as
\begin{equation}
\begin{cases}
\dot{r} = \mu r+\chi_2(\varphi)r^2 + \mathcal{O}(r^3),\\
\dot{\varphi} = \omega + \Omega_1(\varphi)r + \mathcal{O}(r^2),
\end{cases}
\label{e:abstract:polar}
\end{equation}
where, using trigonometric identities, we have
{\begin{align*}
\chi_2(\varphi) =& \frac{1}{\det(\textbf{T})}\Big( [[\cos(\varphi-\phi)]]C\abs{C}(a_{11}R+b_{11}S) + \cos(\varphi-\phi)|\cos(\varphi-\vartheta)|C\abs{D}(a_{12}R+b_{12}S) \\
&+ \cos(\varphi-\vartheta)|\cos(\varphi-\phi)|\abs{C}D(a_{21}R+b_{21}S) + [[\cos(\varphi-\vartheta)]]D\abs{D}(a_{22}R+b_{22}S) \Big).
\end{align*}}
{with $R:=D\sin(\vartheta-\varphi)$, $S:=C\sin(\varphi-\phi)$.}
By assumption, $\omega\neq 0$ so that
rescaling time in \eqref{e:abstract:polar} analogous to \eqref{e:absper} gives \eqref{new_time} with $M(\varphi)=\mu$ and $W(\varphi)=\omega$.
Following the approach described in \S\ref{s:abstract}, for the analogue of \eqref{r_bar} we obtain
\begin{align}
\Lambda &= \frac{1}{2\pi} \int_0^{2\pi}\frac{\mu}{\omega}\mathrm{d} \varphi= \frac{\mu}{\omega}, \\
\Sigma &= \frac{1}{2\pi} \int_0^{2\pi}\frac{\chi_2(\varphi)}{\omega} \mathrm{d}\varphi,
\label{check_sigma_nnf}
\end{align}
where we set $\mu=0$ in \eqref{check_sigma_nnf} (unlike in \eqref{check_sigma})
and the expression for $\Sigma$ can be determined explicitly. For instance, the first term of $\chi_2(\varphi)$ can be integrated as
\begin{equation*}
\frac{C{\abs{C}D}}{{\det(\textbf{T})}}a_{11}\int_0^{2\pi} [[\cos(\varphi-\phi)]] {\sin(\vartheta-\varphi)} \mathrm{d}\varphi = \frac{8C{\abs{C}D}}{3{\det(\textbf{T})}}a_{11}{\sin(\vartheta-\phi)=\frac{8}{3}\abs{C}a_{11}},
\end{equation*}
{with last equality due to $\det(\textbf{T})=CD\sin(\vartheta-\phi)$.} Computing the integral of $\chi_2(\varphi)$, {equation \eqref{check_sigma_nnf} turns into}
{\begin{equation}\begin{aligned}\label{generalized_sigma}
\Sigma
=\frac{2}{3\pi\omega}&\Big[ 2\abs{C}a_{11} + \abs{D}a_{12} +\abs{C}b_{21} + 2\abs{D}b_{22} \\
&+\cos(\vartheta-\phi)\big(\sgn(C)Da_{21}+\sgn(D)Cb_{12}\big)\Big].
\end{aligned}\end{equation}}
In case $\phi=0$ and $\vartheta=\frac{\pi}{2}$,
{we have $\cos(\vartheta-\phi)=0$ so that the last few terms in \eqref{generalized_sigma} vanish}
and for $C=D=1$ the same expression as in \eqref{averaging_integrals} is obtained, i.e., $\Sigma=\frac{2}{3\pi\omega}\sigma_{_\#}$. Notice that this set of parameters gives $z_{11}=z_{22}=1$, $z_{12}=z_{21}=0$, i.e., $\textbf{T}$ is the identity.
Moreover, we can
{derive} the analogue of \eqref{generalized_sigma} for the generalized non-smooth function \eqref{gen_abs_val} and compute the integrals involved in the generalized $\chi_2(\varphi)$ as in the proof of Theorem \ref{Thm0_General}. For instance, some of them read, omitting the factor $\det(T)^{-1}$,
\begin{align*}
C{\abs{C}}\int_0^{2\pi} \cos(\varphi-\phi)&\left( \ABS{\cos(\varphi-\phi)}{\alpha_1} {D\sin(\vartheta-\varphi)}a_{11}+\ABS{\cos(\varphi-\phi)}{\beta_1}{C\sin(\varphi-\phi)}b_{11}\right) \mathrm{d}\varphi \\
=\frac{4}{3}C{\abs{C}} &{D\sin(\vartheta-\phi)}a_{11}\left(\alpha_{1_+}-\alpha_{1_-}\right),
\end{align*}
\begin{align*}
C{\abs{D}}\int_0^{2\pi} \cos(\varphi-\phi)&\left(\ABS{\cos(\varphi-\vartheta)}{\alpha_2}{D\sin(\vartheta-\varphi)}a_{12}+\ABS{\cos(\varphi-\vartheta)}{\beta_2}{C\sin(\varphi-\phi)}b_{12}\right) \mathrm{d}\varphi \\
= \frac{1}{3}C{\abs{D}}\bigg[ &2{D\sin(\vartheta-\phi)}a_{12}\left(\alpha_{2_+}-\alpha_{2_-}\right)
+{C\sin\big(2(\vartheta-\phi)\big)}b_{12}\left(\beta_{2_+}-\beta_{2_-}\right) \bigg].
\end{align*}
{The full expression can be simplified to }
{\begin{equation*}
\begin{aligned}
\widetilde\Sigma :=
\frac{2}{3\pi\omega} \bigg[&\abs{C}a_{11}\left(\alpha_{1_+}-\alpha_{1_-}\right)+\frac{\abs{D}}{2}a_{12}\left(\alpha_{2_+}-\alpha_{2_-}\right)\\
&+\frac{\abs{C}}{2}b_{21}\left(\beta_{3_+}-\beta_{3_-}\right)
+\abs{D}b_{22}\left(\beta_{4_+}-\beta_{4_-}\right) \\
&+ \frac{\cos(\vartheta-\phi)}{2}\Big(\sgn(C)Da_{21}\left(\alpha_{3_+}-\alpha_{3_-}\right)+\sgn(D)Cb_{12}\left(\beta_{2_+}-\beta_{2_-}\right)\Big) \bigg].
\end{aligned}
\end{equation*}
As above, for $\phi=0$ and $\vartheta=\frac{\pi}{2}$ the last few terms vanish, and for $C=D=1$ we have $\widetilde\Sigma = \frac{2}{3\pi\omega}\widetilde\sigma_{_\#}$ with $\widetilde\sigma_{_\#}$ from \eqref{e:tildesig}.}
\medskip
Furthermore, we can extend these results for the case $n>2$ in the form of a coupled system similar to \eqref{e:cylindrical0} using the approach presented in the proof of Theorem~\ref{t_per_orb}. This gives an integral expression for the generalized first Lyapunov coefficient which provides an explicit algebraic formula for an adjusted $\widetilde\sigma_{_\#}$. We comprise this in the following result.
\begin{theorem}\label{Thm_Gen_Lin_Part}
Consider system \eqref{e:abstract} with general linear part $A(\mu)$, and
satisfying the hypotheses of Theorem \ref{t:abstractnormal}.
The statement of Corollary \ref{c:nD:gen_abs}
holds true with $\widetilde\sigma_{_\#}$ replaced by $\frac{3\pi\omega}{2}\widetilde{\Sigma}$.
\end{theorem}
In particular, this theorem covers system \eqref{e:abstractplanar} with general matrix $A=(m_{ij})_{1\leq i,j\leq 2}$. We also remark that the system considered here is neither of the form of \eqref{3DAbstractSystem} nor \eqref{nDAbstractSystem} in terms of smoothness of the $u$ variable.
\begin{proof}
We proceed as before to get the analogue of system \eqref{e:abstractplanar:2}, i.e., transforming the linear part into a block-diagonal matrix and normal form in the center eigenspace $E^\mathrm{c}$. From Theorem \ref{t:abstractnormal} the nonlinear terms are second order modulus terms, which in this case are of the form $(L_i(v,w)+K_i(u))\ABS{L_j(v,w)+K_j(u)}{p}$, $i,j\in\{1,2\}$, where the functions $L_i(v,w)$ are linear combinations of $v,w$; $K_i(u)$ are linear combinations of the components of the vector $u$, i.e., $u_l$, $\forall l\in\{1,\cdots,n-2\}$; and $p_+,p_-\in\mathbb{R}$ are as in \eqref{gen_abs_val}. Note that $L_1, K_1$ are not necessarily equal to $L_2, K_2$, respectively.
The previous product can be expanded as
\begin{equation}\label{gen_expansion}
(L_i(v,w)+K_i(u))\ABS{L_j(v,w)+K_j(u)}{p} = L_i\ABS{L_j}{p} + \mathcal{O}(L_iK_j+L_jK_i+K_iK_j),
\end{equation}
since the error term $p_+ L_i L_j - L_i\ABS{L_j}{p}$ (resp. $p_- L_i L_j - L_i\ABS{L_j}{p}$ ) is of order $|u|^2$, i.e., contained in the higher order terms of \eqref{gen_expansion}. More precisely, consider the case $L_j+K_j \geq 0$. Then, the error term is $p_+L_i L_j - L_i\ABS{L_j}{p}$, which is zero for $L_j \geq 0$, and otherwise $(p_+ - p_-)L_iL_j$. However, in order to have both $L_j+K_j \geq 0$ and $L_j<0$, the signs of $L_j$ and $K_j$ have to differ, which happens only if these magnitudes are comparable. Hence, $\mathcal{O}(L_j)=\mathcal{O}(K_j)$. For the case $L_j+K_j < 0$ we proceed analogously.
In particular, $\mathcal{O}(L_iK_j+L_jK_i+K_iK_j) = \mathcal{O}(K(\check{K}+L))$, where $K, \check{K}$ are linear combinations of the components of $u$, and $L$ of $v, w$.
Following the proof of Theorem \ref{t_per_orb}, we write $u=r{\tilde u}$ and, together with the change of polar coordinates from above, $L_i=r\cos(\varphi-\zeta_i)$ (where $\zeta_i$ is either $\phi$ or $\vartheta$), so that
$$(L_i(v,w)+rK_i({\tilde u}))\ABS{L_j(v,w)+rK_j({\tilde u})}{p} = r^2\cos(\varphi-\zeta_i)\ABS{\cos(\varphi-\zeta_j)}{p} + r^2\mathcal{O}({\tilde u}).$$
From Theorem \ref{t_per_orb} we have ${\tilde u}=\mathcal{O}(r_0)$ and thus $r^2\mathcal{O}({\tilde u})$ is of higher order. We can then integrate explicitly the leading order as done for \eqref{check_sigma_nnf}.
\end{proof}
We implement now these results to an applied $3$-dimensional model in the field of land vehicles.
\section{A 3D example: shimmying wheel}\label{s:shim}
For illustration of the theory and its practice, we consider as an example the model of a shimmying wheel with contact force analyzed in \cite{SBeregi}, where a towed caster with an elastic tyre is studied.
The equations of motion of the towed wheel can be written as follows:
\begin{equation}
\begin{pmatrix}
\dot{\Omega}\\
\dot{\psi}\\
\dot{q}
\end{pmatrix}=
\mathbf{J}\begin{pmatrix}
{\Omega}\\
{\psi}\\
{q}
\end{pmatrix} + {\tilde{c}_4}
\begin{pmatrix}
q\abs{q}\\
0\\
0
\end{pmatrix},\quad
\textbf{J}:=
\begin{pmatrix}
\tilde{c}_1 & \tilde{c}_2 & \tilde{c}_3\\
1 & 0 & 0\\
\tilde{c}_5 & \tilde{c}_6 & \tilde{c}_7
\end{pmatrix},
\label{System1Beregi}
\end{equation}
where $\psi$ is the yaw angle, $q$ is the deformation angle of the tyre due to the contact with the ground and $\Omega=\dot{\psi}$, and the parameters $\tilde{c}_i\in\mathbb{R}$ are constants determined by the system. We can readily see that there is only one switching surface in this case, namely $\{q=0\}$. Here $\mathbf{J}$ is the Jacobian matrix at the equilibrium point $(\Omega,\psi,q)=(0,0,0)$.
The system is of the form \eqref{e:abstract} and suitable parameter choices yield a pair of complex conjugate eigenvalues crossing the imaginary axis, as well as one non-zero real eigenvalue. The resulting bifurcations were studied in \cite{SBeregi} and termed `dynamic loss of stability'. Here we expound how our approach applies to this system.
Clearly, Theorem~\ref{t_per_orb} applies for any Hopf bifurcation eigenvalue configuration, which proves that a unique branch of periodic solutions bifurcates. In order to identify the direction of bifurcation, we first aim to apply the results of \S\ref{s:3D} and therefore attempt to bring the nonlinear part into a second order modulus form, while also bringing the linear part into Jordan normal form.
We thus suppose the parameters are such that the Jacobian matrix has a pair of complex conjugate eigenvalues $\lambda_{\pm}=\mu\pm i\omega$, where $\mu,\omega,\lambda_3\in\mathbb{R}$, $\omega,\lambda_3\neq 0$, with the corresponding eigenvectors $\textbf{s}_1=\textbf{u}+i\textbf{v}$, $\textbf{s}_2=\textbf{u}-i\textbf{v}$ and $\textbf{s}_3$, where $\textbf{u},\textbf{v},\textbf{s}_3\in\mathbb{R}^3$. Such parameter choices are possible as it can be seen from inspecting the characteristic equation with the Routh-Hurwitz criterion; we omit details and refer to \cite{SBeregi}.
The transformation $\textbf{T}=(\textbf{u} | \textbf{v} |\textbf{s}_3)$
with the new state variables
$(\xi_1,\xi_2,\xi_3)^T=\textbf{T}^{-1}(\Omega,\psi,q)^T$ turns \eqref{System1Beregi} into
\begin{equation}
\begin{pmatrix}
\dot{\xi_1}\\
\dot{\xi_2}\\
\dot{\xi_3}
\end{pmatrix}=
\mathbf{A}
\begin{pmatrix}
\xi_1\\
\xi_2\\
\xi_3
\end{pmatrix}+\textbf{h}_2(\xi_1,\xi_2,\xi_3), \quad
\mathbf{A} = \begin{pmatrix}
\mu & \omega & 0\\
-\omega & \mu & 0\\
0 & 0 & \lambda_3
\end{pmatrix},
\label{system_xi}
\end{equation}
where $\textbf{h}_2$ contains the quadratic terms and reads, using the shorthand $[[\cdot]]:=\cdot|\cdot|$,
\begin{equation}\textbf{h}_2(\xi_1,\xi_2,\xi_3)=
\left(
\tilde{T_{1}},
\tilde{T_{2}},
\tilde{T_{3}}
\right)^T
[[u_3\xi_1+v_3\xi_2+s_3\xi_3]],
\label{h_2_xi}
\end{equation}
{where $u_j, v_j, s_j$, $j\in\{1,2,3\}$ are the} components of the vectors $\textbf{u}, \textbf{v}, \textbf{s}_3$, respectively, and $$\tilde{T}_1:=\tilde{c}_4\frac{v_2s_3-v_3s_2}{\det(\textbf{T})}, \hspace*{0.5cm} \tilde{T}_2:=\tilde{c}_4\frac{s_2u_3-s_3u_2}{\det(\textbf{T})}, \hspace*{0.5cm} \tilde{T}_3:=\tilde{c}_4\frac{u_2v_3-u_3v_2}{\det(\textbf{T})}.$$
If $u_3=v_3=0$, then the nonlinear term $\mathbf{h}_{2}$ in \eqref{h_2_xi} is of second order modulus form:
\begin{equation}\textbf{h}_2(\xi_{1},\xi_{2},\xi_{3})=
s_3\abs{s_3}\left( \tilde{T}_1, \tilde{T}_2, 0\right)^T
\xi_{3}\abs{\xi_{3}},
\label{h_3_d0}
\end{equation}
where $\det(\textbf{T})\neq 0$ implies $s_3\neq 0$.
Here we need no further theory as we can directly solve {\eqref{system_xi}}: the equation for $\xi_{3}$ reads $\dot \xi_{3} = \lambda_3 \xi_{3}$ so that periodic solutions require $\xi_{3}(t)\equiv0$, i.e., $\xi_{3}(0)=0$.
The remaining system for $\xi_{1},\xi_{2}$ is then the purely linear part
\begin{equation*}
\begin{pmatrix}
\dot{\xi_{1}}\\
\dot{\xi_{2}}
\end{pmatrix}=\begin{pmatrix}
\mu & \omega\\
-\omega & \mu
\end{pmatrix}\begin{pmatrix}
\xi_{1}\\
\xi_{2}
\end{pmatrix},
\end{equation*}
and consists of periodic solutions (except the origin) for $\mu=0$. The unique branch of bifurcating periodic solutions is thus vertical, i.e., has $\mu=0$ constant.
Next, we consider the case when one of $u_{3}, v_{3}$ is non-zero. In order to simplify the nonlinear term, we apply a rotation $\mathbf{R}_{\theta}$ about the $\xi_{3}$-axis with angle $\theta$,
which keeps the Jordan normal form matrix invariant, and in the new variables {$(v,w,u)^T=\mathbf{R}_{\theta}^{-1}(\xi_1,\xi_2,\xi_3)^T$, in particular $\xi_{3}=u$,} the nonlinear term from
\eqref{h_2_xi} reads
\begin{align}
\abs{u_3(v\cos{\theta}-w\sin{\theta})+v_3(v\sin{\theta}+w\cos{\theta})+s_3u}
= \abs{\tilde{d} v +w(v_3\cos{\theta}-u_3\sin{\theta})+s_3u},
\label{abs_eqs}
\end{align}
where $\tilde{d}=u_3\cos{{\theta}}+v_3\sin{{\theta}}$.
We select $\theta$ to simplify \eqref{abs_eqs}: if $u_3\neq 0$ we choose $\theta=\tilde{\theta}=\arctan\left(\frac{v_3}{u_3}\right)$ such that the coefficient of $w$ in \eqref{abs_eqs} vanishes, i.e., $v_3\cos{\tilde{\theta}}-u_3\sin{\tilde{\theta}} = 0$. Note that $\tilde{d}\neq 0$ since otherwise $v_{3}\tan\tilde\theta=-u_{3}$, but $\tan \tilde\theta=v_{3}/u_{3}$ so together $v_{3}^2=-u_{3}^{2}$ and thus $u_{3}=v_{3}=0$ (which has been discussed above).
If $u_3=0$ and $v_3\neq 0$ we choose $\theta=\tilde{\theta}=\arctan\left(-\frac{u_3}{v_3}\right)$ such that the coefficient of $v$ vanishes, i.e., $u_3\cos{\tilde{\theta}}+v_3\sin{\tilde{\theta}} = 0$, and the following computation is analogous.
Hence, in case $u_3\neq 0$, system \eqref{system_xi} becomes
\begin{equation}
\begin{pmatrix}
\dot{v}\\
\dot{w}\\
\dot{u}
\end{pmatrix}=
\mathbf{A}
\begin{pmatrix}
v\\
w\\
u
\end{pmatrix}+\textbf{h}_3(v,w,u), \quad
\label{after_rotation}
\textbf{h}_3(v,w,u)=
\begin{pmatrix}
\tilde{T}_1\cos{\tilde{\theta}}+\tilde{T}_2\sin{\tilde{\theta}}\\
-\tilde{T}_1\sin{\tilde{\theta}}+\tilde{T}_2\cos{\tilde{\theta}}\\
\tilde{T}_3
\end{pmatrix}
[[\tilde{d}v+s_3u]].
\end{equation}
Notably, since $\tilde{d}\neq 0$, the nonlinear term is of second order modulus form for $s_3=0$, and we consider this degenerate situation first; as mentioned, the case $u_{3}=0, v_{3}\neq 0$ is analogous.
If $s_3=0$ (which means that the third component of the third eigenvector of the matrix
$\mathbf{T}$ is zero) the nonlinear term in \eqref{after_rotation} is of second order modulus form. We can write system (\ref{after_rotation}) in the notation of system (\ref{3DAbstractSystem}):
\begin{equation}
\begin{pmatrix}
\dot{u}\\
\dot{v}\\
\dot{w}
\end{pmatrix} = \begin{pmatrix}
c_1u + h_{11}v\abs{v}\\
\mu v - \omega w + a_{11}v\abs{v}\\
\omega v + \mu w + b_{11}v\abs{v}
\end{pmatrix},
\end{equation}
where we changed $\omega$ to $-\omega$ and set $c_1:=\lambda_3$, $h_{11}:=\tilde{T}_3\tilde{d}|\tilde{d}|$, $a_{11}:=\left( \tilde{T}_1\cos{\tilde{\theta}}+\tilde{T}_2\sin{\tilde{\theta}} \right)\tilde{d}|\tilde{d}|$ and $b_{11}:=\left( -\tilde{T}_1\sin{\tilde{\theta}}+\tilde{T}_2\cos{\tilde{\theta}} \right)\tilde{d}|\tilde{d}|$. Since $s_{3}=0$ we have $a_{11}=0$ by choice of $\tilde{\theta}$, which implies $\sigma_{_\#}=0$. Furthermore, $\sigma_2=0$ holds so that Theorem \ref{2ndPart} does not apply.
However, at $\mu=0$ we have
$ \ddot{v} = -\omega^2v-\omega b_{11}v\abs{v} = -\frac{\mathrm{d}}{\mathrm{d} v}P$ with potential energy
$$ P(v) = \frac{\omega^2}{2}v^2+\frac{\omega b_{11}}{3}v^2\abs{v}, $$
which is globally convex if $\omega b_{11}\geq 0$ and otherwise convex in an interval around zero and concave outside of it. In both cases there is a vertical branch of periodic solutions, which is either unbounded or bounded by heteroclinic orbits.
\medskip
Let us now come back to \eqref{after_rotation} for $s_{3}\neq 0$, where the nonlinearity is of the form $\mathbf{h}_3=(h_{31}, h_{32}, h_{33})^{T}[[\tilde{d}v+s_3u]]$. We first note that in the cylindrical coordinates from \eqref{e:cylindrical0} with the rescaled $u=r{\tilde u}$ {for $r\neq 0$} we have
\begin{align*}
\dot r &= \mu r + r^{2}[[\tilde d \cos(\varphi) + s_{3}{\tilde u}]](h_{31}\cos(\varphi) + h_{32}\sin(\varphi)),\\
\dot \varphi &= \omega + r[[\tilde d \cos(\varphi) + s_{3}{\tilde u}]](h_{32}\cos(\varphi) - h_{31}\sin(\varphi)),\\
\dot{\tilde u} &= \lambda_{3}{\tilde u} + \tilde{T}_{3} r[[\tilde d \cos(\varphi) + s_3{\tilde u}]].
\end{align*}
Following the notation of the proof of Theorem~\ref{t_per_orb} we have the estimate $|{\tilde u}_{\infty}| = \mathcal{O}(r_{\infty})$ and together with the expansion of the $[[\cdot]]$ terms from proof of Theorem \ref{Thm_Gen_Lin_Part}, we can write
\[
\dot r = \mu r + r^{2}[[{\tilde{d}}\cos(\varphi)]](h_{31}\cos(\varphi) + h_{32}\sin(\varphi)) + \mathcal{O}(r^{2} r_{\infty}).
\]
In the notation of Proposition~\ref{Thm_Gen}, in this case $\chi_{2}(\varphi)= [[{\tilde{d}}\cos(\varphi)]](h_{31}\cos(\varphi) + h_{32}\sin(\varphi))$, and according to Corollary \ref{hot2D} the bifurcating branch ist given by \eqref{General_Result} with
\[
\int_{0}^{2\pi} \chi_{2}(\varphi) \mathrm{d} \varphi = {\tilde{d}|\tilde{d}|} h_{31} \int_{0}^{2\pi} \cos^{2}(\varphi)|\cos(\varphi)| \mathrm{d} \varphi
= \frac 8 3 {\tilde{d}|\tilde{d}|} h_{31} = \frac 8 3{|\tilde{d}|} \frac{{\tilde{d}}s_3\tilde{c}_{4}}{\det(\mathbf{T})}.
\]
Since $\tilde{d}\neq 0$ the direction of bifurcation is determined by the sign of ${\tilde{d}}s_3\tilde{c}_{4}\det(\mathbf{T})$.
Note that {$\tilde{d}$, $s_3$, $\det(\mathbf{T})$ are independent of $\tilde{c}_4$, and} ${\tilde{d}}s_3\tilde{c}_{4}\det(\mathbf{T})=0$ requires $s_{3}=0$ as discussed above, or $\tilde{c}_{4}=0$, which implies vanishing nonlinearity. Thus, in all degenerate cases the branch is vertical and we have proven the following.
\begin{theorem}\label{t:shym}
Any Hopf bifurcation in \eqref{System1Beregi} yields either a vertical branch of periodic solutions, or is super- or subcritical as in {Proposition \ref{Thm_Gen}}.
Using the above notation, the branch is vertical if and only if $\tilde{d}s_{3}\tilde{c}_{4}=0$, where $\tilde{d}=0$ means $u_{3}=v_{3}=0$. The bifurcation is supercritical if ${\tilde{d}}s_3\tilde{c}_{4}\det(\mathbf{T})<0$ and subcritical for positive sign. {In particular, reversing the sign of $\tilde{c}_4$ switches the criticality of the bifurcation.}
\end{theorem}
This conclusion is consistent with the results in \cite{SBeregi}.
\section{Discussion}
In this {paper} we have analyzed Hopf bifurcations in mildly non-smooth systems with piecewise smooth nonlinearity for which standard center manifold reduction and normal form computations cannot be used. By averaging and a direct approach we have derived explicit analogues of Lyapunov coefficients and have discussed some codimension-one degeneracies as well as the modified scaling laws.
In an upcoming paper we will apply these results to models for controlled ship maneuvering, where stabilization by p-control induces a Hopf bifurcation.
We believe this is an interesting class of equations from a theoretical as well as applied viewpoint, arising in a variety of models for macroscopic laws that lack smoothness in the nonlinear part. Among the perspectives, there is an analysis of normal forms for coefficients for other bifurcations in these models, such as
Bogdanov-Takens points. Particularly interesting is the impact on scaling laws, including exponentially small effects
for smooth vector fields.
| 2024-02-18T23:39:56.833Z | 2020-10-13T02:01:58.000Z | algebraic_stack_train_0000 | 908 | 20,520 |
|
proofpile-arXiv_065-4641 | \section{Introduction}
Passage retrieval is an important component in applications
like ad-hoc information retrieval,
open-domain question answering ~\citep{Karpukhin2020DensePR},
retrieval-augmented generation ~\citep{Lewis2020RetrievalAugmentedGF}
and fact verification ~\citep{Thorne2018TheFE}.
Sparse retrieval methods such as BM25 were the dominant approach for several decades,
and still play a vital role nowadays.
With the emergence of large-scale pre-trained language models (PLM) ~\citep{Devlin2019BERTPO},
increasing attention is being paid to neural dense retrieval methods ~\citep{Lin2021PretrainedTF}.
Dense retrieval methods map both queries and passages into a low-dimensional vector space,
where the relevance between the queries and passages are measured by the dot product or cosine similarity between their respective vectors.
\begin{table}[ht]
\caption{Inconsistent performance trends between different models on retrieval task and NLU tasks.
We report MRR@10 on the dev set of MS-MARCO passage ranking dataset
and test set results on GLUE benchmark.
Details are available in the Appendix ~\ref{sec:inconsistent}.}
\centering
\scalebox{0.9}{\begin{tabular}{@{}ccc@{}}
\hline
PLM & MS-MARCO & GLUE \\ \hline
BERT & \textbf{33.7} & 80.5 \\
RoBERTa & 33.1 & 88.1 \\
ELECTRA & 31.9 & \textbf{89.4} \\ \hline
\end{tabular}}
\label{tab:inconsistent_perf}
\end{table}
Like other NLP tasks, dense retrieval benefits greatly from a strong general-purpose pre-trained language model.
However,
general-purpose pre-training does not solve all the problems.
As shown in Table ~\ref{tab:inconsistent_perf},
improved pre-training techniques that are verified by benchmarks like GLUE ~\citep{Wang2018GLUEAM}
do not result in consistent performance gain for retrieval tasks.
Similar observations are also made by ~\citet{Lu2021LessIM}.
We hypothesize that
to perform robust retrieval,
the [CLS] vector used for computing matching scores
should encode all the essential information in the passage.
The next-sentence prediction (NSP) task in BERT
introduces some supervision signals for the [CLS] token,
while RoBERTa ~\citep{Liu2019RoBERTaAR} and ELECTRA do not have such sequence-level tasks.
In this paper,
we propose SimLM to pre-train a representation bottleneck with replaced language modeling objective.
SimLM consists of a deep encoder and a shallow decoder
connected with a representation bottleneck,
which is the [CLS] vector in our implementation.
Given a randomly masked text segment,
we first employ an ELECTRA-style generator to sample replaced tokens for masked positions,
then use both the deep encoder and shallow decoder to predict the original tokens at \emph{all} positions.
Since the decoder only has limited modeling capacity,
it must rely on the representation bottleneck to perform well on this pre-training task.
As a result,
the encoder will learn to compress important semantic information into the bottleneck,
which would help training biencoder-based ~\footnote{Also called dual-encoder / two-tower encoder / Siamese networks in different contexts.} dense retrievers.
Compared to existing pre-training approaches such as Condenser ~\citep{Gao2021CondenserAP} or coCondenser ~\citep{Gao2022UnsupervisedCA},
our method has several advantages.
First,
it does not have any extra skip connection between the encoder and decoder,
thus reducing the bypassing effects and simplifying the architecture design.
Second,
similar to ELECTRA pre-training,
our replaced language modeling objective can back-propagate gradients at all positions
and does not have [MASK] tokens in the inputs during pre-training.
Such a design increases sample efficiency and
decreases the input distribution mismatch between pre-training and fine-tuning.
To verify the effectiveness of our method,
we conduct experiments on several large-scale web search and open-domain QA datasets:
MS-MARCO passage ranking ~\citep{Campos2016MSMA},
TREC Deep Learning Track datasets,
and the Natural Questions (NQ) dataset ~\citep{Kwiatkowski2019NaturalQA}.
Results show substantial gains over other competitive methods using BM25 hard negatives only.
When combined with mined hard negatives and cross-encoder based re-ranker distillation,
we can achieve new state-of-the-art performance.
\section{Related Work}
\noindent
\textbf{Dense Retrieval }
The field of information retrieval (IR) ~\citep{Manning2005IntroductionTI}
aims to find the relevant information given an ad-hoc query
and has played a key role in the success of modern search engines.
In recent years,
IR has witnessed a paradigm shift from traditional BM25-based inverted index retrieval
to neural dense retrieval ~\citep{Lin2021PretrainedTF,Karpukhin2020DensePR}.
BM25-based retrieval, though efficient and interpretable,
suffers from the issue of lexical mismatch between the query and passages.
Methods like document expansion ~\citep{Nogueira2019DocumentEB} or query expansion ~\citep{Azad2019QueryET}
are proposed to help mitigate this issue.
In contrast,
neural dense retrievers first map the query and passages to a low-dimensional vector space,
and then perform semantic matching.
Popular methods include DSSM ~\citep{Huang2013LearningDS}, C-DSSM ~\citep{Shen2014LearningSR},
and DPR ~\citep{Karpukhin2020DensePR} etc.
Inference can be done efficiently with approximate nearest neighbor (ANN) search algorithms
such as HNSW ~\citep{Malkov2020EfficientAR}.
Some recent works ~\citep{Chen2021SalientPA, Reimers2021TheCO, Sciavolino2021SimpleEQ} show that
neural dense retrievers may fail to capture some exact lexical match information.
To mitigate this issue,
~\citet{Chen2021SalientPA} proposes to use BM25 as a complementary teacher model,
ColBERT ~\citep{Khattab2020ColBERTEA} instead replaces simple dot-product matching
with a more complex token-level MaxSim interaction,
while COIL ~\citep{Gao2021COILRE} incorporates lexical match information
into the scoring component of neural retrievers.
Our proposed pre-training method aims to adapt the underlying text encoders for retrieval tasks,
and can be easily integrated with existing approaches.
\newline
\noindent
\textbf{Pre-training for Dense Retrieval }
With the development of large-scale language model pre-training ~\citep{Dong2019UnifiedLM, Clark2020ELECTRAPT},
Transformer-based models such as BERT ~\citep{Devlin2019BERTPO} have become
the de facto backbone architecture for learning text representations.
However,
most pre-training tasks are designed without any prior knowledge of downstream applications.
~\citet{Chang2020PretrainingTF} presents three heuristically constructed pre-training tasks tailored for text retrieval:
inverse cloze task (ICT), body first selection (BFS), and wiki link prediction (WLP).
These tasks exploit the document structure of Wikipedia pages
to automatically generate contrastive pairs.
Other related pre-training tasks include representative words prediction ~\citep{Ma2021PROPPW}
and contrastive span prediction ~\citep{Ma2022PretrainAD} etc.
Another line of research builds upon the intuition
that the [CLS] vector should encode all the important information in the given text for robust matching,
which is also one major motivation for this paper.
Such methods include Condenser ~\citep{Gao2021CondenserAP}, coCondenser ~\citep{Gao2022UnsupervisedCA},
SEED ~\citep{Lu2021LessIM}, DiffCSE ~\citep{Chuang2022DiffCSEDC}, and RetroMAE ~\citep{Liu2022RetroMAEPR} etc.
Compared with Condenser and coCondenser,
our pre-training architecture does not have skip connections between the encoder and decoder,
and therefore forces the [CLS] vector to encode as much information as possible.
RetroMAE is a concurrent work by ~\citet{Liu2022RetroMAEPR} that
combines a bottleneck architecture and the masked auto-encoding objective.
\section{SimLM}
\subsection{Pre-training}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.0\linewidth]{./pt_procedure.pdf}
\caption{Pre-training architecture of SimLM.
Replaced tokens are underlined.}
\label{fig:pretrain}
\end{center}
\end{figure}
For pre-training,
there is a collection of passages $\mathbb{C} = \{\mathbf{x}_i\}_{i=1}^{|\mathbb{C}|}$,
where $\mathbf{x}$ denotes a single passage.
Since our motivation is to have a general pre-training method,
we do not assume access to any query or human-labeled data.
The overall pre-training architecture is shown in Figure ~\ref{fig:pretrain}.
Given a text sequence $\mathbf{x}$,
its tokens are randomly replaced with probability $p$ by two sequential operations:
random masking with probability $p$ denoted as
$\mathbf{x}' = \text{Mask(}\mathbf{x}, p\text{)}$,
and then sampling with an ELECTRA-style generator $g$
denoted as $\text{Sample(}g, \mathbf{x}'\text{)}$.
Due to the randomness of sampling,
a replaced token can be the same as the original one.
The above operations are performed twice with potentially
different replace probabilities $p_\text{enc}$ and $p_\text{dec}$
to get the encoder input $\mathbf{x}_\text{enc}$ and decoder input $\mathbf{x}_\text{dec}$.
\begin{equation}
\begin{aligned}
\mathbf{x}_{\text{enc}} & = \text{Sample(}g,\ \text{Mask(}\mathbf{x},\ p_{\text{enc}}\text{))} \\
\mathbf{x}_{\text{dec}} & = \text{Sample(}g,\ \text{Mask(}\mathbf{x},\ p_{\text{dec}}\text{))}
\end{aligned}
\end{equation}
We also make sure that any replaced token
in $\mathbf{x}_\text{enc}$ is also replaced in $\mathbf{x}_\text{dec}$
to increase the difficulty of the pre-training task.
The encoder is a deep multi-layer Transformer that can be initialized with pre-trained models like BERT ~\citep{Devlin2019BERTPO}.
It takes $\mathbf{x}_\text{enc}$ as input
and outputs the last layer [CLS] vector $\mathbf{h}_\text{cls}$ as a representation bottleneck.
The decoder is a 2-layer shallow Transformer with a language modeling head
and takes $\mathbf{x}_\text{dec}$ and $\mathbf{h}_\text{cls}$ as inputs.
Unlike the decoder component in autoregressive sequence-to-sequence models,
the self-attention in our decoder is bi-directional.
The pre-training task is replaced language modeling for both the encoder and decoder,
which predicts the tokens before replacement at \emph{all} positions.
The loss function is the token-level cross-entropy.
The encoder loss $L_\text{enc}$ is shown as follows:
\begin{equation}
\min\ \ L_{\text{enc}} = -\frac{1}{|\mathbf{x}|}\sum_{i=1}^{|\mathbf{x}|} \log p(\mathbf{x}[i]\ |\ \mathbf{x}_\text{enc})
\end{equation}
Similarly for the decoder loss $L_\text{dec}$.
The final pre-training loss is their simple sum: $L_\text{pt} = L_\text{enc} + L_\text{dec}$.
We do not fine-tune the parameters of the generator
as our preliminary experiments do not show any performance gain.
It is often reasonable to assume access to the target retrieval corpus
before seeing any query.
Therefore,
we directly pre-train on the target corpus similar to coCondenser ~\citep{Gao2022UnsupervisedCA}.
After the pre-training finishes,
we throw away the decoder and
only keep the encoder for supervised fine-tuning.
Since the decoder has very limited modeling capacity,
it needs to rely on the representation bottleneck to perform well on the pre-training task.
For the encoder,
it should learn to compress all the semantic information and pass it to the decoder through the bottleneck.
\subsection{Fine-tuning}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=1.0\linewidth]{./ft_procedure.pdf}
\caption{Illustration of our supervised fine-tuning pipeline.
Note that we only use SimLM to initialize the biencoder-based retrievers.
For cross-encoder based re-ranker,
we use off-the-shelf pre-trained models such as ELECTRA$_\text{base}$.}
\label{fig:finetune}
\end{center}
\end{figure}
Compared to training text classification or generation models,
training state-of-the-art dense retrieval models requires a relatively complicated procedure.
In Figure ~\ref{fig:finetune},
we show our supervised fine-tuning pipeline.
In contrast to previous approaches,
our proposed pipeline is relatively straightforward
and does not require joint training ~\citep{Ren2021RocketQAv2AJ}
or re-building index periodically ~\citep{Xiong2021ApproximateNN}.
Each stage takes the outputs from the previous stage as inputs
and can be trained in a standalone fashion.
\newline
\noindent
\textbf{Retriever$_\text{1}$ }
Given a labeled query-passage pair ($q^+, d^+$),
we take the last-layer [CLS] vector of the pre-trained encoder
as their representations ($\mathbf{h}_{q^+}, \mathbf{h}_{d^+}$).
Both the in-batch negatives and BM25 hard negatives
are used to compute the contrastive loss $L_\text{cont}$:
\begin{equation} \label{equ:infonce}
\min\ \ L_{\text{cont}} = -\log \frac{\phi(q^+, d^+)}{\phi(q^+, d^+) + \displaystyle\sum_{n_i \in \mathbb{N}}(\phi(q^+, n_i) + \phi(d^+, n_i))}
\end{equation}
Where $\mathbb{N}$ denotes all the negatives,
and $\phi(q, d)$ is a function to compute the matching score between query $q$ and passage $d$.
In this paper,
we use temperature-scaled cosine similarity function:
$\phi(q, d) = \text{exp}(\frac{1}{\tau}\cos(\mathbf{h}_q, \mathbf{h}_d))$.
$\tau$ is a temperature hyper-parameter and
set to a constant $0.02$ in our experiments.
\newline
\noindent
\textbf{Retriever$_\text{2}$ }
It is trained in the same way as Retriever$_\text{1}$ except that
the hard negatives are mined based on a well-trained Retriever$_\text{1}$ checkpoint.
\newline
\noindent
\textbf{Re-ranker }
is a cross-encoder that re-ranks the top-$k$ results of Retriever$_\text{2}$.
It takes the concatenation of query $q$ and passage $d$ as input
and outputs a real-valued score $\theta(q, d)$.
Given a labeled positive pair ($q^+, d^+$)
and $n - 1$ hard negative passages randomly sampled
from top-$k$ predictions of Retriever$_\text{2}$,
we adopt a listwise loss to train the re-ranker:
\begin{equation}
-\log \frac{\text{exp}(\theta(q^+, d^+))}{\text{exp}(\theta(q^+, d^+)) + \sum_{i=1}^{n-1}\text{exp}(\theta(q^+, d_{i}^{-}))}
\end{equation}
The cross-encoder architecture can model the full interaction between the query and the passage,
making it suitable to be a teacher model for knowledge distillation.
\newline
\noindent
\textbf{Retriever$_\text{distill}$ }
Although cross-encoder based re-ranker is powerful,
it is not scalable enough for first-stage retrieval.
To combine the scalability of biencoder and the effectiveness of cross-encoder,
we can train a biencoder-based retriever by distilling the knowledge from the re-ranker.
The re-ranker from the previous stage is employed to
compute scores for both positive pairs and mined negatives from Retriever$_\text{2}$.
These scores are then used as training data for knowledge distillation.
With $n - 1$ mined hard negatives,
we use KL (Kullback-Leibler) divergence $L_\text{kl}$
as the loss function for distilling the soft labels:
\begin{equation}
L_\text{kl} = \sum_{i=1}^{n} p_\text{ranker}^i \log \frac{p_\text{ranker}^i}{p_\text{ret}^i}
\end{equation}
where $p_\text{ranker}$ and $p_\text{ret}$ are normalized probabilities from
the re-ranker teacher and Retriever$_\text{distill}$ student.
For training with the hard labels,
we use the contrastive loss $L_\text{cont}$ as defined in Equation ~\ref{equ:infonce}.
The final loss is their linear interpolation: $L = L_\text{kl} + \alpha L_\text{cont}$.
Our pre-trained SimLM model is used to initialize all three biencoder-based retrievers
but not the cross-encoder re-ranker.
Since our pre-training method only affects model initialization,
it can be easily integrated into other more effective training pipelines.
\section{Experiments}
\begin{table}[ht]
\centering
\caption{Main results on MS-MARCO passage ranking and TREC datasets.
Results with * are from our reproduction with public checkpoints.
$\dagger$: from Pyserini ~\citep{Lin2021PyseriniAP}.}
\scalebox{0.95}{\begin{tabular}{@{}lccccccc@{}}
\hline
\multirow{2}{*}{Model} & \multirow{2}{*}{+distill} & \begin{tabular}[c]{@{}c@{}}\\ single\\ vector?\end{tabular} & \multicolumn{3}{c}{MS MARCO dev} & TREC DL 19 & TREC DL 20 \\
& & & MRR@10 & R@50 & R@1k & nDCG@10 & nDCG@10 \\ \hline
\textbf{Sparse retrieval} & & & & & & & \\
BM25 & & \cmark & 18.5 & 58.5 & 85.7 & 51.2$^*$ & 47.7$^*$ \\
DeepCT ~\citep{Dai2019ContextAwareST} & & \cmark & 24.3 & 69.0 & 91.0 & 57.2 & - \\
docT5query ~\citep{nogueira2019doc2query} & & \cmark & 27.7 & 75.6 & 94.7 & 64.2 & - \\ \hline
\textbf{Dense retrieval} & & & & & & & \\
ANCE ~\citep{Xiong2021ApproximateNN} & & \cmark & 33.0 & - & 95.9 & 64.5$^\dagger$ & 64.6$^\dagger$ \\
SEED ~\citep{Lu2021LessIM} & & \cmark & 33.9 & - & 96.1 & - & - \\
TAS-B ~\citep{Hofsttter2021EfficientlyTA} & \cmark & \cmark & 34.0 & - & 97.5 & 71.2 & 69.3 \\
RetroMAE ~\citep{Liu2022RetroMAEPR} & & \cmark & 35.0 & - & 97.6 & - & - \\
COIL ~\citep{Gao2021COILRE} & & & 35.5 & - & 96.3 & 70.4 & - \\
ColBERT ~\citep{Khattab2020ColBERTEA} & & & 36.0 & 82.9 & 96.8 & - & - \\
Condenser ~\citep{Gao2021CondenserAP} & & \cmark & 36.6 & - & 97.4 & 69.8 & - \\
RocketQA ~\citep{Qu2021RocketQAAO} & \cmark & \cmark & 37.0 & 85.5 & 97.9 & - & - \\
PAIR ~\citep{Ren2021PAIRLP} & \cmark & \cmark & 37.9 & 86.4 & 98.2 & - & - \\
coCondenser ~\citep{Gao2022UnsupervisedCA} & & \cmark & 38.2 & 86.5$^*$ & 98.4 & \textbf{71.7}$^*$ & 68.4$^*$ \\
RocketQAv2 ~\citep{Ren2021RocketQAv2AJ} & \cmark & \cmark & 38.8 & 86.2 & 98.1 & - & - \\
AR2 ~\citep{zhang2021adversarial} & \cmark & \cmark & 39.5 & \textbf{87.8} & 98.6 & - & - \\
ColBERTv2 ~\citep{Santhanam2021ColBERTv2EA} & \cmark & & 39.7 & 86.8 & 98.4 & - & - \\ \hline
\our{} & \cmark & \cmark & \textbf{41.1} & \textbf{87.8} & \textbf{98.7} & 71.2 & \textbf{69.7} \\ \hline
\end{tabular}}
\label{tab:ir_results}
\end{table}
\subsection{Setup}
\noindent
\textbf{Datasets and Evaluation }
We use MS-MARCO passage ranking ~\citep{Campos2016MSMA},
TREC Deep Learning (DL) Track 2019 ~\citep{craswell2020overview}
and 2020 ~\citep{Craswell2020OverviewOT},
Natural Questions (NQ) ~\citep{Kwiatkowski2019NaturalQA, Karpukhin2020DensePR} datasets for training and evaluation.
The MS-MARCO dataset is based on Bing search results
and consists of about $500k$ labeled queries and $8.8M$ passages.
Since the test set labels are not publicly available,
we report results on the development set with $6980$ queries.
The NQ dataset is targeted for open-domain QA
with about $80k$ question-answer pairs in the training set
and $21M$ passages based on Wikipedia.
For evaluation metrics,
we use MRR@10, Recall@50, and Recall@1000 for MS-MARCO,
nDCG@10 for TREC DL,
and Recall@20, Recall@100 for the NQ dataset.
\noindent
\textbf{Implementation Details } ~\label{sec:impl_details}
For pre-training,
we initialize the encoder with BERT$_\text{base}$ (uncased version).
The decoder is a two-layer Transformer
whose parameters are initialized with the last two layers of BERT$_\text{base}$.
The generator is borrowed from the ELECTRA$_\text{base}$ generator,
and its parameters are frozen during pre-training.
We pre-train for $80k$ steps for MS-MARCO corpus
and $200k$ steps for NQ corpus,
which roughly correspond to $20$ epochs.
Pre-training is based on $8$ V100 GPUs.
With automatic mixed-precision training,
it takes about $1.5$ days and $3$ days for the MS-MARCO and NQ corpus respectively.
For fine-tuning on the MS-MARCO dataset,
we train for $3$ epochs with a peak learning rate $2 \times 10^{-5}$.
Each batch consists of $16$ queries,
each query has $1$ positive passage and $15$ randomly sampled hard negatives.
One shared encoder is used to encode both the query and passages.
We start with the official BM25 hard negatives in the first training round
and then change to mined hard negatives.
During inference,
given a query,
we use brute force search to rank all the passages for a fair comparison with previous works.
For more implementation details,
please check out the Appendix section ~\ref{sec:more_impl_details}.
\subsection{Main Results}
\begin{table}[ht]
\caption{Results on the test set of Natural Questions (NQ) dataset.
Listed results of SimLM are based on Retriever$_\text{2}$
and do not use knowledge distillation.}
\centering
\scalebox{0.95}{\begin{tabular}{@{}lcc@{}}
\toprule
\multirow{2}{*}{Model} & \multicolumn{2}{c}{NQ} \\
& R@20 & R@100 \\ \hline
BM25 & 59.1 & 73.7 \\
DPR$_\text{single}$ ~\citep{Karpukhin2020DensePR} & 78.4 & 85.4 \\
ANCE ~\citep{Xiong2021ApproximateNN} & 81.9 & 87.5 \\
RocketQA ~\citep{Qu2021RocketQAAO} & 82.7 & 88.5 \\
Condenser ~\citep{Gao2021CondenserAP} & 83.2 & 88.4 \\
PAIR ~\citep{Ren2021PAIRLP} & 83.5 & 89.1 \\
RocketQAv2 ~\citep{Ren2021RocketQAv2AJ} & 83.7 & 89.0 \\
coCondenser~\citep{Gao2022UnsupervisedCA} & \textbf{84.3} & 89.0 \\ \hline
\our{} & \textbf{84.3} & \textbf{89.3} \\ \hline
\end{tabular}}
\label{tab:dpr_result}
\end{table}
\begin{table}[ht]
\caption{Comparison with ColBERTv2 ~\citep{Santhanam2021ColBERTv2EA}
in terms of index storage cost (w/o any compression)
and complexity of index search algorithms.}
\centering
\scalebox{0.95}{\begin{tabular}{lcc}
\hline
& Index size & Index search \\ \hline
ColBERTv2 & 150GB & Two-stage \\
\our{} & 27GB & One-stage \\ \hline
\end{tabular}}
\label{tab:compare_colbert}
\end{table}
We list the main results in Table ~\ref{tab:ir_results} and ~\ref{tab:dpr_result}.
For the MS-MARCO passage ranking dataset,
the numbers are based on the Retriever$_\text{distill}$ in Figure ~\ref{fig:finetune}.
Our method establishes new state-of-the-art with MRR@10 41.1,
even outperforming multi-vector methods like ColBERTv2.
As shown in Table ~\ref{tab:compare_colbert},
ColBERTv2 has a 6x storage cost as
it stores one vector per token instead of one vector per passage.
It also requires a customized two-stage index search algorithm during inference,
while our method can utilize readily available vector search libraries.
The TREC DL datasets have more fine-grained human annotations,
but also much fewer queries (less than 100 labeled queries).
We find that using different random seeds could have a 1\%-2\% difference in terms of nDCG@10.
Though our model performs slightly worse on the 2019 split compared to coCondenser,
we do not consider such difference as significant.
For passage retrieval in the open-domain QA setting,
a passage is considered relevant if it contains the correct answer for a given question.
In Table ~\ref{tab:dpr_result},
our model achieves R@20 84.3 and R@100 89.3 on the NQ dataset,
which are comparable to or better than other methods.
For end-to-end evaluation of question answering accuracy,
we will leave it as future work.
\begin{table}[ht]
\caption{Re-ranker performance w/ different pre-trained models on the dev set of MS-MARCO passage ranking dataset.}
\centering
\scalebox{0.9}{\begin{tabular}{lc}
\hline
Model & MRR@10 \\ \hline
BERT$_\text{base}$ & 42.3 \\
ELECTRA$_\text{base}$ & \textbf{43.7} \\ \hline
\our{} & 42.9 \\ \hline
\end{tabular}}
\label{tab:compare_rerank}
\end{table}
Though SimLM achieves substantial gain for biencoder-based retrieval,
its success for re-ranking is not as remarkable.
In Table ~\ref{tab:compare_rerank},
when used as initialization for re-ranker training,
SimLM outperforms BERT$_\text{base}$ by 0.6\%
but still lags behind ELECTRA$_\text{base}$.
\begin{table}[ht]
\caption{Comparison with state-of-the-art dense retriever coCondenser
under various settings on the dev set of MS-MARCO passage ranking dataset.
Results with * are from our reproduction.}
\centering
\begin{tabular}{@{}lcc@{}}
\hline
& MRR@10 & R@1k \\ \hline
\textbf{coCondenser} & & \\
BM25 negatives & 35.7 & 97.8 \\
+ mined negatives & 38.2 & 98.4 \\
+ distillation & 40.2$^*$ & 98.3$^*$ \\ \hline
\textbf{\our{}} & & \\
BM25 negatives (Retriever$_\text{1}$) & 38.0 & 98.3 \\
+ mined negatives (Retriever$_\text{2}$) & 39.1 & 98.6 \\
+ distillation (Retriever$_\text{distill}$) & \textbf{41.1} & \textbf{98.7} \\ \hline
Cross-encoder re-ranker & 43.7 & 98.6 \\ \hline
\end{tabular}
\label{tab:cocon_comparison}
\end{table}
Next,
we zoom in on the impact of each stage in our training pipeline.
In Table ~\ref{tab:cocon_comparison},
we mainly compare with coCondenser ~\citep{Gao2022UnsupervisedCA}.
With BM25 hard negatives only,
we can achieve MRR@10 38.0,
which already matches the performance of many strong models like RocketQA ~\citep{Qu2021RocketQAAO}.
Model-based hard negative mining and re-ranker distillation can bring further gains.
This is consistent with many previous works ~\citep{Xiong2021ApproximateNN, Ren2021RocketQAv2AJ}.
We also tried an additional round of mining hard negatives
but did not observe any meaningful improvement.
Based on the results of Table ~\ref{tab:cocon_comparison},
there are many interesting research directions to pursue.
For example,
how to simplify the training pipeline of dense retrieval systems
while still maintaining competitive performance?
And how to further close the gap between biencoder-based retriever
and cross-encoder based re-ranker?
\section{Analysis}
\subsection{Variants of Pre-training Objectives}
\begin{table}[ht]
\caption{Different pre-training objectives.
Reported numbers are MRR@10 on the dev set of MS-MARCO passage ranking.
We finetune the pre-trained models with official BM25 hard negatives.}
\centering
\scalebox{0.9}{\begin{tabular}{cccccccc}
\hline
& \our{} & Enc-Dec MLM & Condenser & MLM & Enc-Dec RTD & AutoEncoder & BERT$_\text{base}$ \\ \hline
MRR@10 & \textbf{38.0} & 37.7 & 36.9 & 36.7 & 36.2 & 32.8 & 33.7 \\ \hline
\end{tabular}}
\label{tab:objectives}
\end{table}
Besides our proposed replaced language modeling objective,
we also tried several other pre-training objectives as listed below.
\newline
\noindent
\textbf{Enc-Dec MLM }
uses the same encoder-decoder architecture as in Figure ~\ref{fig:pretrain} but without the generator.
The inputs are randomly masked texts and
the pre-training objective is masked language modeling (MLM) over the masked tokens only.
The mask rate is the same as our method for a fair comparison,
which is 30\% for the encoder and 50\% for the decoder.
In contrast,
RetroMAE ~\citep{Liu2022RetroMAEPR} uses a specialized decoding mechanism
to derive supervision signal from all tokens in the decoder side.
\noindent
\textbf{Condenser }
is a pre-training architecture proposed by ~\citet{Gao2021CondenserAP}.
Here we pre-train Condenser with a 30\% mask rate on the target corpus.
\noindent
\textbf{MLM }
is the same as the original BERT pre-training objective with a 30\% mask rate.
\noindent
\textbf{Enc-Dec RTD }
is the same as our method in Figure ~\ref{fig:pretrain} except that
we use replaced token detection (RTD) ~\citep{Clark2020ELECTRAPT}
as pre-training task for both the encoder and decoder.
This variant shares some similarity with DiffCSE ~\citep{Chuang2022DiffCSEDC}.
The main difference is that the input for DiffCSE encoder is the original text,
making it a much easier task.
Our preliminary experiments with DiffCSE pre-training
do not result in any improvement.
\noindent
\textbf{AutoEncoder }
attempts to reconstruct the inputs based on the bottleneck representation.
The encoder input is the original text without any mask,
and the decoder input only consists of [MASK] tokens
and [CLS] vector from the encoder.
\noindent
\textbf{BERT$_\text{base}$} just uses off-the-shelf checkpoint published by ~\citet{Devlin2019BERTPO}.
It serves as a baseline to compare against various pre-training objectives.
\newline
The results are summarized in Table ~\ref{tab:objectives}.
Naive auto-encoding only requires memorizing the inputs
and does not need to learn any contextualized features.
As a result,
it becomes the only pre-training objective that underperforms BERT$_\text{base}$.
Condenser is only slightly better than simple MLM pre-training,
which is possibly due to the bypassing effects of the skip connections in Condenser.
Enc-Dec MLM substantially outperforms Enc-Dec RTD,
showing that MLM is a better pre-training task than RTD for retrieval tasks.
This is consistent with the results in Table ~\ref{tab:inconsistent_perf}.
Considering the superior performance of RTD pre-trained models on benchmarks like GLUE,
we believe further research efforts are needed to investigate the reason behind this phenomenon.
\subsection{Effects of Replace Rate}
\begin{table}[ht]
\caption{MS-MARCO passage ranking performance w.r.t different token replace rates.
Here the replace rate is the percentage of masked tokens fed to the generator.}
\centering
\begin{tabular}{ccc}
\hline
encoder & decoder & MRR@10 \\ \hline
15\% & 15\% & 37.6 \\
15\% & 30\% & 37.5 \\
30\% & 30\% & 37.9 \\
30\% & 50\% & \textbf{38.0} \\
40\% & 60\% & \textbf{38.0} \\
30\% & 100\% & 36.6 \\ \hline
\end{tabular}
\label{tab:replace_ratio}
\end{table}
In the experiments,
we use fairly large replace rates (30\% for the encoder and 50\% for the decoder).
This is in stark contrast to the mainstream choice of 15\%.
In Table ~\ref{tab:replace_ratio},
we show the results of pre-training with different replace rates.
Our model is quite robust to a wide range of values
with 30\%-40\% encoder replace rate performing slightly better.
Similar findings are also made by ~\citet{Wettig2022ShouldYM}.
One interesting extreme scenario is a 100\% replace rate on the decoder side.
In such a case,
the decoder has no access to any meaningful context.
It needs to predict the original texts solely based on the representation bottleneck.
This task may be too difficult and has negative impacts on the encoder.
\subsection{Effects of Pre-training Steps}
\begin{figure}[ht]
\begin{center}
\includegraphics[width=0.5\linewidth]{./convergence.pdf}
\caption{Our pre-training objective converges faster and consistently
outperforms vanilla masked language model pre-training.
The y-axis shows the MRR@10 on the dev set of MS-MARCO dataset.}
\label{fig:converge}
\end{center}
\end{figure}
Since pre-training can be costly in terms of both time and carbon emission,
it is preferred to have an objective that converges fast.
Our proposed method shares two advantages of ELECTRA ~\citep{Clark2020ELECTRAPT}.
First,
the loss is computed over \emph{all} input tokens instead of a small percentage of masked ones.
Second,
the issue of input distribution mismatch is less severe than MLM,
where the [MASK] token is seen during pre-training but not for supervised fine-tuning.
In Figure ~\ref{fig:converge},
our method achieves competitive results with only $10k$ training steps
and converges at $60k$,
while MLM still slowly improves with more steps.
\subsection{On the Choice of Pre-training Corpus}
\begin{table}[ht]
\caption{Fine-tuning performance w.r.t different pre-training corpora.
We use BM25 negatives for MS-MARCO and mined negatives for NQ.
``Wikipedia'' refers to the corpus with $21M$ 100-word Wikipedia passages released by ~\citet{Karpukhin2020DensePR}.
``none'' use BERT$_\text{base}$ as the foundation model.}
\centering
\scalebox{0.85}{\begin{tabular}{lcccc}
\hline
\multirow{2}{*}{Corpus} & \multicolumn{2}{c}{MS-MARCO} & \multicolumn{2}{c}{NQ} \\
& MRR@10 & R@1k & R@20 & R@100 \\ \hline
none & 33.7 & 95.9 & 82.9 & 88.0 \\
MS-MARCO & \textbf{38.0} & \textbf{98.3} & 83.3 & 88.6 \\
Wikipedia & 36.3 & 97.4 & \textbf{84.3} & \textbf{89.3} \\ \hline
\end{tabular}}
\label{tab:choice_pretrain_corpus}
\end{table}
For a typical retrieval task,
the number of candidate passages is much larger than the number of labeled queries,
and many passages are never seen during training.
Take the NQ dataset as an example,
it has $21M$ candidate passages but only less than $80k$ question-answer pairs for training.
In the experiments,
we directly pre-train on the target corpus.
Such pre-training can be regarded as implicit memorization of the target corpus
in a query-agnostic way.
One evidence to support this argument is that,
as shown in Table ~\ref{tab:objectives},
simple MLM pre-training on target corpus can have large performance gains.
An important research question to ask is:
will there be any benefits of our method when pre-training on non-target corpus?
In Table ~\ref{tab:choice_pretrain_corpus},
the largest performance gains are obtained
when the corpus matches between pre-training and fine-tuning.
If we pre-train on the MS-MARCO corpus and fine-tune on the labeled NQ dataset
or the other way around,
there are still considerable improvements over the baseline.
We hypothesize that
this is due to the model's ability to compress information into a representation bottleneck.
Such ability is beneficial for training robust biencoder-based retrievers.
\subsection{Case Analysis}
\begin{table}[ht]
\caption{Some (cherry-picked) examples from the dev set of MS-MARCO passage ranking dataset.
We show the query, top retrieved passages, and their binary relevance labels.
Relevant text snippets are shown in italic.}
\centering
\scalebox{0.88}{\begin{tabular}{ll}
\hline
\small query & \small was winnie the pooh a boy \\ \hline
\small BERT$_\text{base}$ & \small\begin{tabular}[c]{@{}l@{}} \textbf{Rank}: 1, \textbf{Relevant}: \xmark \\ \textbf{Passage}: The little boy who talks to the animals in the Winnie-the-Pooh stories is called Christopher Robin, \\ which is the name of A. A. Milne's real-life son, who was born in 1920. On August 21, 1921, \\ the real-life Christopher Robin Milne received a stuffed bear from Harrods for his first birthday \ldots \end{tabular} \\ \hline
\small \our{} & \small\begin{tabular}[c]{@{}l@{}} \textbf{Rank}: 1, \textbf{Relevant}: \cmark \\ \textbf{Passage}: So, it looks like we were lied to our entire childhood! \emph{Winnie the Pooh is not a boy.} SHE is a girl \\ and she's from Canada, not England. Really! In a new picture book called Finding Winnie: \\ The True Story of the World's Most Famous Bear, we learn that Winnie is actually named after \ldots \end{tabular} \\ \hline \hline
\small query & \small colorado routing number loveland colorado \\ \hline
\small BERT$_\text{base}$ & \small\begin{tabular}[c]{@{}l@{}} \textbf{Rank}: 1, \textbf{Relevant}: \xmark \\ \textbf{Passage}: Loveland, CO is currently served by one area code which is area code 970. In addition to Loveland, \\ CO area code information read more about area code 970 details and Colorado area codes. \ldots \end{tabular} \\ \hline
\small \our{} & \small\begin{tabular}[c]{@{}l@{}} \textbf{Rank}: 2, \textbf{Relevant}: \cmark \\ \textbf{Passage}: \emph{107006787 Routing Transit Number (RTN) for Advantage Bank Main Office located at} \\ \emph{Loveland, Colorado, CO}, 80538, United States, Street Address 1475 NORTH DENVER AVENUE, \\ Telephone Number 970-613-1982 \ldots \end{tabular} \\ \hline
\end{tabular}}
\label{tab:case_analysis}
\end{table}
To qualitatively understand the gains brought by pre-training,
we show several examples in Table ~\ref{tab:case_analysis}.
The BERT$_\text{base}$ retriever can return passages with high lexical overlap
while missing some subtle but key semantic information.
In the first example,
the retrieved passage by BERT$_\text{base}$ contains keywords like ``boy'', ``Winnie the Pooh'',
but does not answer the question.
In the second example,
there is no routing number in the BERT$_\text{base}$ retrieved passage,
which is the key intent of the query.
Our proposed pre-training can help to learn better semantics to answer such queries.
For more examples,
please check out Table ~\ref{tab:more_cases} in the Appendix.
\section{Conclusion}
This paper proposes \our{},
a novel pre-training method for dense passage retrieval.
It follows an encoder-decoder architecture with a representation bottleneck in between.
The encoder learns to compress all the semantic information into a dense vector
and passes it to the decoder to perform well on the replaced language modeling task.
When used as initialization in a dense retriever training pipeline,
our model achieves competitive results on
several large-scale passage retrieval datasets.
We also provide detailed ablation analyses to show the key ingredients behind its success.
For future work,
we would like to increase the model size and the corpus size
to examine the scaling effects.
It is also interesting to explore other pre-training mechanisms
to support unsupervised dense retrieval and multilingual retrieval.
| 2024-02-18T23:39:57.608Z | 2022-07-07T02:13:48.000Z | algebraic_stack_train_0000 | 948 | 5,548 |
|
proofpile-arXiv_065-4714 | \section{Introduction}
Recently there has been observed a big resurgence of interests in
a special class of Einstein field equation solutions representing tunnel-like structures connecting spatially separated regions or even more different Universes,
nowadays called wormholes.
\red{These fascinating objects are not only important for popular culture, but also gain a lot of scientific attention as their properties allow them to be black hole mimickers.}
From historical point of view,
the first description of such kind of objects begins with the issue of \cite{fla16}, devoted to spatial part of Schwarzschild solution studies.
The prototype of wormhole emerged from the studies devoted to particle model, where the mathematical construction which tried to eliminate
coordinate or curvature singularities, dubbed as Einstein-Rosen bridge, was proposed in \cite{ein35}.
Later on, the Kruskal-Szekeres coordinates were implemented for the description of Schwarzschild wormhole \cite{whe55},
while
the Euclidean form of wormhole solution was obtained in
\cite{haw88}. One should remark that all these concepts were postulated at quantum scale.
The current understanding of wormholes was revealed in \cite{mor88}, where the conditions for traversability for Lorentzian wormholes were defined by the survivability of human travellers.
\red{This redefinition was not only of great importance to physics, but also to futurology and is still seen as a main way to travel at large distances in space by humans.}
On the other hand, models of a wormhole, possessing no event horizon and physical singularities, were elaborated in
\cite{ell73}-\cite{ell79}.
In order to obtain such kind of wormhole solutions one should invoke phantom field (exotic matter), whose energy momentum tensor violates the null, weak and strong energy conditions, as well as, its kinetic energy term is of the reversed sign.
However, traversability requires also stability of the wormhole solution, except small acceleration and tidal forces. To achieve this goal we may consider
a generalized
Einstein gravity theories,
like Gauss-Bonnet-dilaton theory. Moreover in this theory
wormholes can be built with no use of such exotic kind of matter \cite{kan11}-\cite{har13}.
On the other hand, the method of constructing traversable wormholes by applying duality rotation and complex transformations was proposed \cite{gib16,gib17}.
By assuming that the dilaton field constitutes a phantom one,
an electrically charged traversable wormhole solution in Einstein-Maxwell-phantom dilaton gravity, has been revealed \cite{gou18}.
Soon after the rotating wormhole solutions were paid attention to \cite{teo98}-\cite{bro13}.
There were also conceived perturbative and numerical attempts to construct spinning generalization of static wormhole solutions \cite{kas08}-\cite{che16}.
It was claimed that the rotating wormholes
would be with a higher possibility stable \cite{mat06} and therefore traversable.
The other interesting problem in wormhole physics is their classification.
Having in mind classification delivered by the black hole uniqueness theorem,
the first work in this direction was provided in \cite{rub89}, delivering the uniqueness theorem for wormhole spaces with vanishing Ricci scalar.
Further, the uniqueness of Ellis-Bronikov wormhole with phantom field was found in \cite{yaz17}, while the uniqueness
for four-dimensional case of the Einstein-Maxwell-dilaton wormholes with the dilaton coupling constant equal to one, was presented in \cite{laz17}.
The case of higher-dimensional generalization of wormhole solution, valid from the point of view of the unification theories like string/M-theory
attracts also attention.
The uniqueness theorem for higher-dimensional case of the static spherically symmetric phantom wormholes was treated in \cite{rog18}, while
the case of of static spherically symmetric traversable wormholes with two asymptotically flat ends, subject to the higher-dimensional solutions
of Einstein-Maxwell-phantom dilaton field equations with an arbitrary dilaton coupling constant, was elaborated in \cite{rog18a}.
Various other aspects of physics of these objects were under intensive studies (for a detailed review of the blossoming subject the reader may consult \cite{worm}).
Wormholes being a fascinating subject of their possible impact on space and time travels, may also be regarded as potential
astrophysical objects, that can be observationally search for.
From the astrophysical point of view, it is persuasive to consider rotating wormholes. The problem that arises is how to observationally distinguish
rotating wormholes from stationary axisymmetric black holes of Kerr-type. Remarkable attention to the aforementioned problem was paid to after
the Even Horizon Telescope observed the black hole shadow in the center of the galaxy M87.\\
The first studies to what extent wormholes can imitate the observational characteristics of black holes were conducted in \cite{dam07},
where the simple generalization of Schwarzschild-like line element was revealed.
The considered metric differs from the static general relativity one by introducing the dimensionless parameter ${\lambda}$. The value of the parameter equal to zero is
responsible for the ordinary Schwarzschild black hole solution.
Of course one should be aware that for non-zero values of the parameter the presented line element is no longer the static solution of Einstein equations
and changes the structure of the manifold. Therefore the matter with almost vanishing energy density ought to be required to maintain the aforementioned
gravitational configuration (for the discussion of the influence of the parameter ${\lambda}$ on the static manifold structure see, e.g., \cite{bue18}).
Further generalization of the idea given in \cite{dam07} to describe Kerr-like wormhole spacetime as a toy model, was achieved by applying
a modification on the Kerr metric similar to the procedure performed in \cite{dam07}.
The embedding diagrams, geodesic structure, as well as, shadow characteristics of the obtained Kerr-like wormhole were given in \cite{ami19}.
On the other hand, the throat-like effects on the shadow of Kerr-like wormholes were elaborated in \cite{kas21}.
However,
the problem of the structure at the horizon scale of black hole which gives rise to echoes of the gravitational wave signal bounded with the postmeger
ring-down phase in binary coalescences, in the case of static and rotating toy models of traversable wormholes, has been
elucidated in \cite{bue18}.
The other subject acquiring much attention in contemporary astrophysics and physics is the unrelenting search for finding {\it dark matter}
sector particles. The nature of this elusive ingredient of our Universe is a mystery and several models try to explain it and constitute the possible guidance for the future experiments.
The main aim of our work will be to investigate the behavior of axion-like particle {\it dark matter} model clouds, around the mimickers of rotating
black holes, stationary axially symmetric wormholes. The work will provide some continuity with our previous studies \cite{kic21}, where we have paid
attention to the main features of axionic clouds {\it dark matter} in the vicinity of magnetized rotating black holes.
The principal goal of the investigations will be to find the possible differences in characteristic features of the axion-like condensate, between those two classes of compact objects,
i.e., rotating black holes and black hole mimickers. Our studies will constitute the first glimpse at the problem in question. Namely, we restrict our consideration to the probe limit case,
when one has the complete separation of the degrees of freedom, i.e., matter fields do not backreact on wormhole spacetime.
The organization of the paper is as follows. In Sec. II we deliver the basic facts about the axion-like {\it dark matter} model. Sec. III
will be devoted to the description of the rotating wormholes models surrounded by {\it dark matter} clouds, in the considered model
of axion-like {\it dark matter}. In Sec. IV we describe the numerical results of the studies, while in Sec. V we conclude our investigations and aim the possible
problems for the future investigations.
\section{Model of axion-like {\it dark matter} sector}
The explanation of astronomical and cosmological observations require {\it dark matter} existence, whose nature is one of the most tantalizing questions confronting
contemporary physics and cosmology. A large number of ongoing or planned experimental searches for its detection and understanding of the {\it dark sector} role
in a fundamental description of the Universe.
Axions are among the strongest candidates for the possible explanation of the existence of {\it hidden sector} \cite{pre83}-\cite{din83}.
Their existence has been postulated to explain the apparent lack of violation of charge conjugate parity \cite{pec77}-\cite{wil78} and in the strong interaction
motivated the absence of observable electric dipole moment of the neutron \cite{pen15}. Axionlike particles are also widely spotted in the realm of string theories \cite{svr06}.
In what follows, we shall study axionlike scalar particles coupled to the
Maxwell $U(1)$-gauge field. The non-trivial coupling of axion field to the Maxwell field strength invariant plays the crucial role in the model in question.
The field equations of motion are provided by the variation procedure with respect to the action given by
\begin{equation}
\mathcal{S} = \int d^4 x \sqrt{-g} \left[R - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
- \frac{1}{2} \nabla_\mu \Psi \nabla^\mu \Psi - \frac{\mu^2}{2} \Psi^2
- \frac{k}{2} \Psi \ast F^{\mu \nu} F_{\mu \nu} \right],
\end{equation}
where we set $R$ for the Ricci scalar, $F_{\mu \nu} = 2 \nabla_{[\mu} A_{\nu]}$, while $\Psi$ stands for the scalar field (axion) with mass $\mu$.
$\ast F^{\mu \nu} = 1/2 \epsilon_{\mu \nu \alpha \beta} F^{\alpha \beta}$ is the dual to Maxwell field strength.
The equation of motion for the scalar field $\Psi$, which constitutes a covariant Klein-Gordon equation with a source term of the dual Maxwell field invariant, implies
\begin{equation}
\nabla_\mu \nabla^\mu \Psi - \mu^2 \Psi
- \frac{k}{2} ~\ast F^{\mu \nu} F_{\mu \nu} = 0,
\label{eq:field_eqn}
\end{equation}
while the $U(1)$-gauge field is subject to the relation as follows:
\begin{equation}
\nabla_\mu F^{\nu \mu} + 2 k~\ast F^{\nu \mu} \nabla_{\mu }\Psi = 0.
\end{equation}
We refer to the $\Psi$ field as axionlike, because the axions (originating from QCD) have adequate constrains on both mass and coupling parameter. Here however we consider particles with physics given by an analogical Lagrangian yet with arbitrary values of physical parameters. However for simplicity we might refer to the studied axionlike particles as simply axions.
The {\it dark matter} model in question was widely elaborated in studies of black hole superradiance
and light polarization effects, possible experimental signals of {\it dark sector} around these objects \cite{pla18}-\cite{car18}, \cite{kic21},
and neutron stars \cite{gar18}-\cite{gra15}, as well as,
the influence of axionic {\it dark matter} on the physics on early Universe and primordial black holes \cite{fed19}-\cite{ros18}.
The form of the relation (\ref{eq:field_eqn}) envisages the fact that the presence of the non-zero source term, containing the dual invariant, given by
\begin{equation}
\mathcal{I} = ~\ast F^{\mu \nu} F_{\mu \nu} \neq 0,
\end{equation}
is crucial.
In the opposite case, when the invariant is equal to zero, the axion-like scalar field equation of motion reduces to the simple massive Klein-Gordon case,
without any self-interaction potential. It means that no scalar hair configuration on the studied line element can emerge.
Although it has been shown that in Kerr spacetime scalar hair may emerge in certain situations \cite{herd14}, here we pick a different ansatz (see below) as we focus on stationary configurations, which appear to be magnetically induced in this approach.
On the other hand, it can be noticed that the discussed invariant, $\ast F_{\mu \nu} F^{\mu \nu}$,
is equal to zero in the case when $F_{\mu \nu} =0$, or for spherically symmetric spacetime. However, it has a non-trivial form, $\ast F_{\mu \nu} F^{\mu \nu} \neq 0$,
when both rotation and magnetic $U(1)$-gauge field components are present in the spacetime under consideration.
To introduce the magnetic field we use the method proposed by Wald \cite{wal74}, where the vector potential is sourced by Killing vectors of the rotating spacetime.
In general it has a form
\begin{equation}
A_\mu = \frac{1}{2}B (m_\mu + 2 a k_\mu),
\end{equation}
where $k_\mu$ and $m_\mu$ are the Killing vectors connected with temporal invariance and $\phi$ rotation respectively.
As in \cite{kic21}, where we have studied rotating magnetized black holes submerged into axionic {\it dark matter} cloud, one can introduce a static magnetic
field to the system, which will be oriented along the rotation axis. It seems to be plausible from the point of astrophysical perspective
and can be regarded as a starting point for studies of the magnetic field influence of the system in question. Because of the fact that our investigations
focus on static magnetic field, parallel to the wormhole rotation axis, the gauge potential may be rewritten in the form as $
A_\mu dx^\mu = B/2~ g_{\mu \nu} m^\nu dx^\mu.$
For our considerations we choose a static, time independent ansatz. The symmetry of the problem enables us to elaborate
the axion field in the form provided by
\begin{equation}
\Psi = \psi(r, \theta),
\label{eq:ansatz}
\end{equation}
which will be plugged into the equation \eqref{eq:field_eqn}, for the considered line element.
\section{Rotating wormhole metrics}
The simplicity of the static line element describing a wormhole may suggest that the spinning generalization can be achieved analytically
and ought to be globally regular. But in vain, it happens that finding the stationary solution with an extended source is far more complicated (see
for the recent aspects of this problem \cite{vol21}). However, the rotating wormhole solutions are widely discussed in literature \cite{teo98}-\cite{che16},
but one should be aware that they do not constitute the exact solutions of the equations of motion but rather comprise some model of geometries.
In this section, we shall study two kinds of rotating wormhole model metrics. First one accounts for the extension of the regular black hole Kerr metric \cite{bue18,ami19}. The other is the Teo class wormhole \cite{teo98}, a rotating generalization of Morris-Thorne wormhole, which serves us as comparison to a bit more realistic Kerr-like wormhole.
\subsection{Kerr-like wormhole}
To begin with, we consider the metric of Kerr-like rotating wormhole.
It is constructed by a slight modification of stationary axisymmetric line element with a parameter ${\lambda}$.
For the first time, such construction was proposed in \cite{dam07}, where the static Schwarzschild black hole was considered.
Then, it was generalized to the case of stationary axisymmetric line element \cite{bue18,ami19}.
The Kerr-like wormhole line element yields
\begin{eqnarray}
ds^2 &=& - \left( 1 - \frac{2 M r}{\Sigma} \right)dt^2 - \frac{4 M ar \sin^2 \theta}{\Sigma} dt d\phi + \frac{\Sigma}{\tilde{\Delta}} dr^2 + \Sigma d\theta^2\\ \nonumber
&+& \Big(r^2 + a^2 + \frac{2 M a^2 r \sin^2 \theta}{\Sigma} \Big) \sin^2 \theta d\phi^2,
\end{eqnarray}
where we set
\begin{align}
\Sigma(r, \theta) = r^2 + a^2 cos^2 \theta, \\
\tilde{\Delta}(r) = r^2 + a^2 - 2M(1 + \lambda^2)r.
\end{align}
The parameters $M$ and $a M$ correspond to mass and angular momentum of a wormhole.
For a small deviation parameter ${\lambda}$, one achieves
almost indistinguishable from of Kerr black hole line element. These three parameters describe the system as seen from the outside.
Moreover its Arnowitt-Deser-Misner (ADM) mass, as seen by the observer at asymptotic spatial infinity, is given by $M_{ADM} = M (1 + {\lambda}^2)$.
The largest root of $\tilde{\Delta}(r) = 0$, establishes the surface provided by
\begin{equation}
r_+ = M ( 1 + \lambda^2 ) + \sqrt{M^2 ( 1 + \lambda^2)^2 - a^2}.
\end{equation}
For the model in question it does not constitute a radius of the event horizon, but describes the radius of the throat of the rotating wormhole, which
connects two asymptotically flat regions of the spacetime. It can be explicitly seen by the adequate changes of variables \cite{bue18,ami19}. The points
with the condition $r<r_+$ do not exist.
Consequently the axion field equation written in the Kerr-like wormhole spacetime implies the following:
\begin{align}
\tilde{\Delta} \partial_r^2 \psi +
\frac{2(r - M)\tilde{\Delta} - M \lambda^2 (r^2 + a^2)}{\Delta} \partial_r \psi
+ \partial_{\theta}^2 \psi
+ \cot \theta \partial_{\theta} \psi
- \mu^2 \Sigma \psi = \frac{k \Sigma}{2} \mathcal{I}_{KWH},
\label{eqn:kwh_axion}
\end{align}
where
the electromagnetic field invariant is provided by
\begin{align}
\mathcal{I}_{KWH} = - \frac{a B^2 M \tilde{\Delta} \sin^2 \theta \cos \theta}{2 \Delta \Sigma^4} \big[ 3 a^6 + 2 a^4 M r - 5 a^4 r^2 - 8 a^2 M r^3 - 32 a^2 r^4 - 24 r^6 \nonumber \\
+ 4 a^2 (a^4 - a^2 r^2 + 2(M - r)r^3 ) \cos 2\theta + a^4 (a^2 - 2 M r + r^2) \cos 4\theta \big].
\end{align}
The equation \eqref{eqn:kwh_axion} undergoes a following scaling transformation
\begin{equation}
r \rightarrow \eta r, \quad a \rightarrow \eta a, \quad M \rightarrow \eta M, \quad
B \rightarrow B/\eta, \quad \mu^2 \rightarrow \mu^2 / \eta^2, \quad r_+ \rightarrow \eta r_+,
\end{equation}
\red{which allows us to fix one of model parameters to unity. For this we pick $M = 1$.}
\subsection{Teo rotating wormhole}
The well-known Morris-Thorne metric, introduced in Ref. \cite{mor88},
describes a traversable wormhole spacetime, which is stabilised by exotic matter in the area of its throat.
That solution was achieved by using reverse engineering of general relativity, namely the metric was postulated first and with a help of Einstein equations
the suitable matter components were found.
Generalization of the aforementioned solution, by including the rotation into the consideration, was performed in \cite{teo98}.
The resulting metric of the rotating wormhole has a following form:
\begin{equation}
ds^2 = -N^2 dt^2 + \frac{dr^2}{1 - \frac{b}{r}} + K^2 r^2 \left[ d \theta^2 + \sin^2 \theta (d \phi - \omega dt)^2 \right],
\end{equation}
where, as in the Morris-Thorne case, one has a lot of freedom in choosing the shape of $N$, $b$, $K$ and $\omega$ functions, as long as they meet specific requirements.
Firstly, all the functions can be functions of $r$ and $\theta$ and should be regular on the symmetry axis $\theta =0, \pi$.
Secondly, $N$, the gravitational redshift function, ought to be finite and nonzero, $b$ as the shape function determining the shape of the wormhole throat,
should satisfy $b \leqslant r$.
$K$ accounts for the radial distance with respect to the coordinate origin and $\omega$ stands for the angular velocity of the wormhole.
The embedding of constant $t$ and $\theta$-cross sections in the three-dimensional Euclidean space reveals the well-recognizable form of the wormhole
spacetime. The constructed geometry describes two regions, where the radial coordinates are given by $r \in [r_+,~\infty)$, which are joined together
at the wormhole throat $r=r_+$. At spatial infinity, the requirement of asymptotic flatness regions provides that the metric coefficients ought to satisfy
the following expansions:
\begin{equation}
N = 1 - \frac{M}{r} + {\cal O} \Big(\frac{1}{r^2}\Big), \qquad K = 1 + {\cal O}\Big(\frac{1}{r}\Big),
\qquad \frac{b}{r} = {\cal O}\Big(\frac{1}{r}\Big), \qquad \omega = \frac{2 J}{r^3} + {\cal O}\Big(\frac{1}{r^4}\Big),
\label{eq:twh_asympt}
\end{equation}
where we have denoted by $M$ the mass of the wormhole and by $J$ its angular momentum.
In general, one encounters the whole range of functions, which fulfil the aforementioned conditions and constitute a regular rotating wormhole solution.
For the numerical calculations, we pick a set of functions which appear to be quite popular in the literature of the subject,
and were previously used by different authors \cite{shaikh18, nedkova13, abdujabbarov16, harko09, bambi13}
\begin{equation}
N = \exp\left[- \frac{r_+}{r} \right], \qquad b(r) = r_+ \left( \frac{r_+}{r} \right)^\gamma, \qquad
\omega = \frac{2 a r_+}{r^3}, \qquad K=1,
\label{eq:twh_metric_fun}
\end{equation}
where we use the $r_+$ symbol, for denoting the wormhole throat radius.
\red{The angular momentum parameter is defined in the standard way $a = J/M$.
Using the asymptotic relations \eqref{eq:twh_asympt} we find that for the picked set of functions \eqref{eq:twh_metric_fun} $M = r_+$.}
Thus, the family of the above
solutions is described by three parameters, i.e., the throat radius $r_+$, angular momentum parameter $a$ and the shape parameter $\gamma$.
After putting the ansatz \eqref{eq:ansatz} and the metric into the field equation \eqref{eq:axion_only_action} we arrive at the equation of motion
\begin{align}
\left[ r^2 - r_+ r \left( \frac{r_+}{r} \right)^\gamma \right] \partial_r^2 \psi + \left[ 2r + r_+ + \left(\frac{r_+}{r} \right)^\gamma \left(\frac{1}{2}r_+ \gamma - \frac{r_+^2}{r} -\frac{3}{2} r_+ \right) \right] \partial_r \psi \nonumber \\
+ \partial_{\theta}^2 \psi + \cot \theta \partial_{\theta} \psi - \mu^2 r^2 \psi = \frac{1}{2} k r^2 \mathcal{I}_{TWH},
\label{eqn:twh_axion}
\end{align}
which radial part is strongly dependent on $\gamma$.
The Maxwell field invariant related to uniform magnetic field in this spacetime implies
\begin{equation}
\mathcal{I}_{TWH} = \frac{12 a B^2 r_+ \cos \theta \sin^2 \theta}{r^{5/2}} \sqrt{\frac{r - r_+ \left(\frac{r_+}{r} \right)^\gamma}{\exp \left[ -\frac{2 r_+}{r} \right]}}.
\end{equation}
The equation \eqref{eqn:twh_axion} follows a scaling transformation
\begin{equation}
r \rightarrow \eta r, \quad r_+ \rightarrow \eta r_+, \quad a \rightarrow \eta a, \quad B \rightarrow B/ \eta, \quad \mu^2 \rightarrow \mu^2 / \eta^2.
\end{equation}
\red{Using this transformation we fix $r_+ = 1$.}
\subsection{Free energy}
As a benchmark for the thermodynamical preference of the obtained states we use free energy by evaluating the on-shell action of the axion dependent part of the theory
\begin{equation}
\mathcal{S}_{axion} = \int d^4 x \sqrt{-g} \left[- \frac{1}{2} \nabla_\mu \Psi \nabla^\mu \Psi - \frac{\mu^2}{2} \Psi^2
- \frac{k}{2} \Psi \ast F^{\mu \nu} F_{\mu \nu} \right].
\label{eq:axion_only_action}
\end{equation}
By substituting the equations of motion into the action and imposing the ansatz of the field we arrive to the formula for the free energy
\begin{equation}
F = - 2 \pi \int_\mathcal{M} dr d\theta ~\sqrt{-g} \bigg[ (\partial_r \psi)^2 g^{rr} + (\partial_\theta \psi)^2 g^{\theta \theta} + \mu^2 \psi^2 \bigg].
\label{eq_freeenergy}
\end{equation}
The straightforward integration of the equation \eqref{eq_freeenergy} appears to be problematic. It is because both considered backgrounds have singular metric determinant at the throat, which makes simple integration from throat to infinity impossible in these coordinates.
It should be noted that this singularity is merely a coordinate singularity, as the curvature of both wormholes is regular and finite at the throat.
In
the case of Kerr-like wormhole metric,
we have
\begin{equation}
\sqrt{-g} = \sqrt{\frac{\Delta}{\tilde{\Delta}}} \Sigma \sin^2 \theta,
\end{equation}
where for the case of ${\lambda}$ equal to zero we obtain that $\Delta = {\tilde{\Delta}}$.
This fact
naturally eradicates the singularity problem in the black hole scenario. Here, however, as we radially fall toward the wormhole, the root of ${\tilde{\Delta}}$ comes first and creates the singularity.
On the other hand, for the Teo rotating line element we get
\begin{equation}
\sqrt{-g} = \frac{\exp \left( - \frac{r_+}{r} \right) r^2 \sin \theta}{\sqrt{1 - \left( \frac{r_+}{r} \right)^{\gamma + 1}}},
\end{equation}
with the denominator naturally generating the infinity.
To deal with the integration in such spacetimes we use energy differences instead.
Also we introduce a cutoff to the lower integration bound, so we start from $r_+ + \epsilon$ rather than simply $r_+$.
In this way we ensure the finiteness of energy differences and give them straightforward physical interpretation.
With the change of the background parameter the solution becomes more or less thermodynamically stable with respect to some \textit{ground} solution.
\section{Results}
In this section we pay attention to the solutions of the equations of motion for the previously described two toy models of rotating wormholes.
Due to the complications of the relations \eqref{eqn:kwh_axion} and \eqref{eqn:twh_axion},
we solve them numerically by virtue of spectral methods. Firstly the adequate equation is discretized on Gauss-Lobato grid \cite{matlabnum} and
next translated into a system of algebraic equations with spectral differentiation matrices.
The method in question
has already been implemented in Python and tested for the numerical stability.
The technical details, especially convergence tests of the numerical method are described in the Appendix of \cite{kic21}, where we studied
the problem of axionlike particle clouds in the spacetimes of rotating magnetized black holes.
The spectral nature of the numerical scheme requires remapping the coordinates onto the $[-1, 1]$ intervals.
It can be achieved by the coordinate transformation provided by
\begin{align}
z = 1 - \frac{2 r_+}{r}, \\
u = \frac{4 \theta}{\pi} - 1,
\end{align}
where $r_+$ is the wormhole throat radius.
After such operation, our numerical domain may be written in the form $[-1, 1]\times[-1, 1]$.
For $z$-coordinate, the boundaries are the wormhole throat ($z = -1$) and spatial infinity ($z=1$), while for $u = -1$, one talks about \textit{north pole} of a wormhole and the \textit{equator} with $u = 1$.
Consequently after the coordinate transformation in the underlying equations, one shall impose the adequate boundary conditions.
Namely, on the throat surface we demand that the axion field should be regular, therefore $\partial_r \psi = 0$ provides a desirable conduct of the field.
\red{Alternatively, setting the field to a constant value, such as zero in a wormhole scenario, is also a possible choice. However we wish to explore the Kerr-like solution for different values of $\lambda$ parameter, including its zeroing when it simplifies to the Kerr black hole.
Given that for the consistency between these two kinds of solutions we use the Neumann boundary condition.}
At the spatial infinity, we take a look on the asymptotic behaviour of the equation itself and the source term $\mathcal{I}$.
It appears that the Maxwell field invariants in both backgrounds are vanishing functions. As $r \rightarrow \infty$, we have
\begin{equation}
I_{KWH} = \mathcal{O}\left(\frac{1}{r^4}\right),
\end{equation}
\begin{equation}
I_{TWH} = \mathcal{O}\left( \frac{1}{r^2} \right).
\end{equation}
Which means that both equations \eqref{eqn:kwh_axion} and \eqref{eqn:twh_axion} reach a simple, asymptotic form, to the leading order
\begin{equation}
\partial^2_r \psi + \frac{2}{r} \partial_r \psi - \mu^2 \psi = 0.
\end{equation}
This simple equation has a solution
\begin{equation}
\psi = A \frac{\exp(\mu r)}{r} + B \frac{\exp(-\mu r)}{r},
\label{eq:psi_asympt}
\end{equation}
where $A$ and $B$ are constants.
Naturally the field ought to decay for the sake of asymptotic flatness of the spacetime.
Given that we are allowed to choose $A = 0$, with arbitrary $B$.
This means that a boundary condition $\psi(r \rightarrow \infty) = 0$ is an adequate and mathematically motivated choice.
On the other hand, the boundary conditions for the angular dependency are built on the basis of the spacetime symmetry. Both considered spacetimes are rotating, therefore we demand $\partial_{\theta} \psi = 0$ on the \textit{north pole}.
On the \textit{equator}, the presence of magnetic field combined with the spacetime symmetry implies that $\psi = 0$.
\subsection{Kerr-like wormhole}
To commence with, we solve the equation \eqref{eqn:kwh_axion} for the Kerr-like background metric.
A portion of obtained distributions is depicted in Fig. \ref{fig_kw_maps}.
In the following panels we see the increasing mass of the axionic field.
In the panel (a) the field is ultralight, subsequently in (b) $\mu^2 = 0.01$, (c) $\mu^2 = 0.1$ and finally in (d) $\mu^2 = 1$.
In every panel we have $a = 0.99$ and $\lambda = 0.5$. We can clearly see how the mass of the axionlike field changes the angular distribution of it around the wormhole.
For little masses the clouds are concentrated around the poles of the wormhole and spread in the space for several throat radii.
As we increase the axion mass we see that the polar regions of the wormhole become depleted and the field drifts towards the equator. The largest concentration is visible on the latitude $\theta \simeq \pi/4$.
Second important effect is the influence of the field mass on the magnitude of the field.
Inspection of
the colorbars reveals that the larger the mass the smaller the field.
The spatial tail of the field is also much shorter, when the mass of the field is larger.
Intuitively, in the asymptotic solution \eqref{eq:psi_asympt} $\mu$ enters the suppressing exponential term. The field decays faster for larger masses, which means the massive fields are localized in the vicinity of the throat surface.
Another important thing that stands out in relation to the black hole solutions is the repulsion of the axion cloud from the wormhole throat surface.
While in the case of the black hole, the field had non-zero values on the surface of the event horizon, and its radial character was monotonically decreasing, here we have a completely different situation.
For the wormhole, the field vanishes or at least has a significantly smaller value on the throat.
Then it grows with the radius as it reaches the maximum and finally decreases.
This effect is particularly visible for the high values of the angular momentum.
\begin{figure}[h]
\centering
\subfloat[$\mu^2 \rightarrow 0^+, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m0_HD.pdf}
}
\qquad
\subfloat[$\mu^2 = 0.01, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m001_HD.pdf}
}
\vspace{0.5cm}
\subfloat[$\mu^2 = 0.1, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m01_HD.pdf}
}
\qquad
\subfloat[$\mu^2 = 1, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m1_HD.pdf}
}
\caption{Axion field distribution around Kerr-like wormholes for given sets of parameters.
The blank space in the vicinity of wormhole throat distinguishes the solution from the black hole counterpart, where the field appears to be non-zero on the event horizon.
Subsequent panels for each mass parameter show how the angular distribution of the field is affected.}
\label{fig_kw_maps}
\end{figure}
The radial behaviour of the axionic field can be seen more precisely in Fig. \ref{fig:kw_slices}.
We present there a slice of $\psi$ as a function of $r$ in throat radius units, for constant $\theta=\pi/4$ and few different values of the $\lambda$ parameter.
In contrast, we also plot the behaviour of axions in Kerr black hole metric (that is $\lambda = 0$).
What we can see is the increasing $\lambda$ consequently extinguishes the axionic hair.
In the foreground a structural change in the field profile is visible as we compare it to the black hole scenario.
An axionic field over a black hole has a maximum value on the event horizon.
The opposite is true for a wormhole, on the throat the field vanishes, then grows to its maximum and fades away with the radius.
Then, the bigger is $\lambda$ the smaller are the maxima and overall magnitude of the axionic hair.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{psi_pi4_a99_m0.pdf}
\caption{A closer look on the axion cloud gap near the wormhole throat. Here we show slices of $\psi$ for constant $\theta = \pi/4$, with parameters $a = 0.99, \mu^2 = 0^+$.
Increasing of $\lambda$ decreases the magnitude of the axions and cuts off its tail.}
\label{fig:kw_slices}
\end{figure}
In the next step we investigate the free energy of the obtained axion cloud configurations.
It is interesting to see how the parameters describing the spacetime geometry around the wormhole influence the thermodynamics of the axion clouds.
Due to the previously mentioned difficulties in computing the free energy in these metrics, we rather talk about energy differences, than the exact values.
In Fig. \ref{fig:kw_fe} we present the differences of the free energy versus angular momentum $a$, with respect to the $\lambda=0$ level, which constitutes a plain Kerr black hole.
It is clearly visible that the larger value of the distortion parameter ${\lambda}$ one takes into account, the higher value of the free energy of the cloud we achieve.
It turns out, that the more the gravitational background deviates from from the black hole metric, the less thermodynamically desirable axion clouds are.
This effect works together with the diminishing magnitude of the field on the previously discussed Fig. \ref{fig:kw_slices}.
Additionally the increasing angular momentum of the wormhole also increases the free energy difference.
This means that for \textit{extreme} Kerr-like wormhole axion hair is the least favourable.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fediff_a_lambs_m0.pdf}
\caption{Free energy differences as a function of angular momentum (with $\lambda = 0$ as the ground curve) for different values of $\lambda$. With axion mass $\mu^2 = 0^+$ we see that the cloud
thermodynamical favourability decreases with growth of both angular momentum and $\lambda$ parameter.}
\label{fig:kw_fe}
\end{figure}
\subsection{Teo rotating wormhole}
Teo class wormhole has a different set of parameters and it does not simply transform into a black hole solution, just like a Kerr-like wormhole does.
Here the throat radius is independent of the other parameters and is imposed manually in $g_{tt}$ and $g_{rr}$.
With the particular choice of functions \eqref{eq:twh_metric_fun}, we can only steer with the shape of metric components via $\gamma$ parameter.
Therefore, let us consider values of $\gamma$ in the interval $(-1, 1]$,
where for $\gamma = -1$ the function $g_{rr}$ is singular, so we can only approach this value.
\red{As it was mentioned, the Kerr-like wormhole can be reduced to a black hole solution by setting $\lambda = 0$.
Teo solution does not share this feature, but is a well-known wormhole metric, just like its non-rotating counterpart the archetypical Morris-Thorne wormhole.
While it can not serve as a testing field for differences between axion clouds around wormholes and black holes, one can treat it as a benchmark for behaviours of the axion hair in another wormhole environment.
Using this background might help us to see if the obtained axion solutions share similar features.
}
In Fig. \ref{fig_teo_maps} one can see the distribution of the axionic cloud around a Teo wormhole for different axionlike field masses.
In the panel (a) we have an ultralight field, then it takes values $0.1$, $0.5$ and $1.0$ for (b), (c) and (d) respectively.
In all panels we use $\gamma = -0.99$, which gives us tiny value of the axionic field (see the colorbars).
The angular distribution has similar features to the Kerr-like metric.
For ultralight axions the field is localized in the majority of the wormhole surroundings.
As the mass increases the hair tightens spatially and disappears from the polar regions.
For the large mass case the axionic clouds are drifting toward the equator with the polar caps left almost empty. Moreover the radial reach is very short - around one throat radius.
One can clearly notice that for the negative gamma value the analogous effect to the Kerr-like scenario is observed.
The axionic cloud is also pushed away from the throat surface - in its vicinity the field acquires small values, reaches the maximum and the descends monotonically to zero.
However in this gravitational background this effect is not as dramatic as in case of Kerr-like wormhole.
The weakening of the axionic field in the vicinity of the throat is easily visible in the distributions, although it is not that large.
Increasing $\gamma$ up to zero and beyond causes the rise of $\psi$ field. It grants bigger values, but the spatial qualitative characteristic remains intact.
\begin{figure}[h]
\centering
\subfloat[$\mu^2 \rightarrow 0^+, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m0.pdf}
}
\qquad
\subfloat[$\mu^2 = 0.1, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m01.pdf}
}
\vspace{0.5cm}
\subfloat[$\mu^2 = 0.5, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m05.pdf}
}
\qquad
\subfloat[$\mu^2 = 1, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m1.pdf}
}
\caption{Axion cloud distribution around the magnetized Teo type wormhole. The negative value of $\gamma$ resembles the results for Kerr-like wormhole, with
large $\lambda$ value, i.e., the decreasing of the field magnitude.
Each panel depicts the field distribution for different axionlike masses.
Once again a big mass results with a localized field, concentrated closely to the throat with depleted polar region. For bigger values of $\gamma$ we observe similar influence of $\mu$ on the spatial distribution.}
\label{fig_teo_maps}
\end{figure}
The next figure brings a closer look on the field drop near the throat.
In Fig. \ref{fig_teo_slice} we present radial slices of the field distribution with $\theta = \pi/4$. In this particular figure we depict the behaviour for ultralight field, however a similar tendencies are shown by more massive fields.
First of all, we observe a significant amplification of the field with the growth of $\gamma$.
The field does not acquire new features however, but it seems that the curves follow some kind of scaling related to gamma.
Additionally the profiles resemble the results obtained for Kerr-like wormholes. As we have previously mentioned, Teo class wormhole cannot be simply transformed into a black hole by a simple choice of parameter value.
However the features like a drop near the throat surface, then maximum and monotonic fall show that these might be more general wormhole related behaviours of axionic hair.
\begin{figure}[h]
\centering
\includegraphics[width=0.7 \textwidth]{teowh_psi_pi4_a99_m0.pdf}
\caption{Radial slices of the axion field for constant angle $\theta = \pi/4$.
The growth of $\gamma$ increases the maximum value of the field.
However in this metric the significant growth of the blank space near the throat is not present.
Also the growth of $\gamma$ does not seem to greatly affect the tail of the field, away from the throat, which is different from the Kerr-like wormhole results.}
\label{fig_teo_slice}
\end{figure}
If we consider the free energy, it appears that the axion clouds for the background with negative $\gamma$, are definitely less thermodynamically favourable.
In Fig. \ref{fig_teo_fe} we plot the free energy difference as a function of angular momentum for several values of the gamma parameter.
We use the curve with $\gamma = 0$ as a baseline for calculating energy differences.
Free energy difference curves for $\gamma < 0$ are positive, especially the $\gamma = -0.99$ curve reaches relatively big values.
Therefore thermodynamically speaking wormholes with $\gamma$ close to $-1$ have least chances to hold axionic hair.
With the growth of the parameter, the free energy of the cloud decreases, which makes the axions more thermodynamically favourable.
However this fall is rather moderate comparing to the rise of the top curve.
In both cases the increase of angular momentum amplifies the tendencies of the curves.
Curves for negative gamma grow, while the positive fall.
A consequent growth of $\gamma$ parameter leads the hair to some limit characteristics which can be seen in Fig. \ref{fig_teo_slice} and Fig. \ref{fig_teo_fe}.
\begin{figure}[h]
\centering
\includegraphics[width=0.7 \textwidth]{fediff_a_gam_m0.pdf}
\caption{Free energy differences vs. angular momentum for different values of $\gamma$. The curve of $\gamma = 0$ is the reference level. We see that free energy increases for negative values of gamma and slightly drops for positive ones, as angular momentum grows.}
\label{fig_teo_fe}
\end{figure}
Finally let us conduct qualitative comparison of the axionic clouds in considered metrics.
The solutions have undoubtedly similar features, especially when one takes a look on the $\psi$ slices.
We also observe the separation of the cloud from the surface of the throat in both cases.
This allows us to notice some general wormhole related phenomena, which are not present around the black holes \cite{kic21}.
Naturally we cannot speak in a fully general manner, as we only considered here merely two distinct gravitational backgrounds, which on the other hand should be treated as toy models.
\section{Conclusion}
In our paper, we have considered the problem of the distribution of axionlike particles, being regarded as {\it dark matter} sector model, around
the toy models of rotating wormholes. We have investigated the Kerr-like wormhole line element with the distortion parameter ${\lambda}$ and
Teo model of a rotating axisymmetric wormhole. The models under inspection were characterized by the mass, angular momentum, distortion parameter
(for Kerr-like wormhole) and the shape parameter $\gamma$ (for the Teo model). We numerically solve the equations of motion for the underlying cases, using spectral methods.
Among all we have found that the axion clouds are pushed forward from the wormhole throat, especially for the case of large value of the rotation
parameter $a$. The voids in the vicinity of the wormhole throat appear for the larger value of the distortion parameter ${\lambda}$. This phenomenon distinguishes the studied
system from the previously elaborated black holes \cite{kic21}. On the other hand, for the larger ${\lambda}$, one achieves higher value of the free energy, and therefore
this solution is less thermodynamically favoured.
As far as the Teo class of rotating wormholes is concerned, we have for the negative value of $\gamma$ the analogous effect as in the latter case is obtained. However
for the positive value, the behavior of axionic clouds resembles features of the {\it dark matter} clouds around Kerr black hole in a uniform magnetic field.
The solution with negative $\gamma$ is not thermodynamically favourable, as it has been revealed in free energy studies. However when $\gamma$ increases, the free energy
of the axionic cloud decreases.
We have found that the behavior of the axionic clouds significantly differs from the black hole scenario, which we discussed in our previous work \cite{kic21}.
This fact will account for the possible guidance, enabling one to distinguish between these two classes of compact objects.
Nevertheless, the search for astronomically observable criteria require far more complex approach. A more realistic dynamical gravitational model is needed,
when the time dependence of the studied fields is taken into account, as well as, the direct mathematical proofs of the stabilities of rotating wormhole spacetimes ought to be found.
These subjects impose a real mathematical challenge and require also solid numerical relativity machinery.
These problems shall be investigated elsewhere.
\section{Introduction}
Recently there has been observed a big resurgence of interests in
a special class of Einstein field equation solutions representing tunnel-like structures connecting spatially separated regions or even more different Universes,
nowadays called wormholes.
\red{These fascinating objects are not only important for popular culture, but also gain a lot of scientific attention as their properties allow them to be black hole mimickers.}
From historical point of view,
the first description of such kind of objects begins with the issue of \cite{fla16}, devoted to spatial part of Schwarzschild solution studies.
The prototype of wormhole emerged from the studies devoted to particle model, where the mathematical construction which tried to eliminate
coordinate or curvature singularities, dubbed as Einstein-Rosen bridge, was proposed in \cite{ein35}.
Later on, the Kruskal-Szekeres coordinates were implemented for the description of Schwarzschild wormhole \cite{whe55},
while
the Euclidean form of wormhole solution was obtained in
\cite{haw88}. One should remark that all these concepts were postulated at quantum scale.
The current understanding of wormholes was revealed in \cite{mor88}, where the conditions for traversability for Lorentzian wormholes were defined by the survivability of human travellers.
\red{This redefinition was not only of great importance to physics, but also to futurology and is still seen as a main way to travel at large distances in space by humans.}
On the other hand, models of a wormhole, possessing no event horizon and physical singularities, were elaborated in
\cite{ell73}-\cite{ell79}.
In order to obtain such kind of wormhole solutions one should invoke phantom field (exotic matter), whose energy momentum tensor violates the null, weak and strong energy conditions, as well as, its kinetic energy term is of the reversed sign.
However, traversability requires also stability of the wormhole solution, except small acceleration and tidal forces. To achieve this goal we may consider
a generalized
Einstein gravity theories,
like Gauss-Bonnet-dilaton theory. Moreover in this theory
wormholes can be built with no use of such exotic kind of matter \cite{kan11}-\cite{har13}.
On the other hand, the method of constructing traversable wormholes by applying duality rotation and complex transformations was proposed \cite{gib16,gib17}.
By assuming that the dilaton field constitutes a phantom one,
an electrically charged traversable wormhole solution in Einstein-Maxwell-phantom dilaton gravity, has been revealed \cite{gou18}.
Soon after the rotating wormhole solutions were paid attention to \cite{teo98}-\cite{bro13}.
There were also conceived perturbative and numerical attempts to construct spinning generalization of static wormhole solutions \cite{kas08}-\cite{che16}.
It was claimed that the rotating wormholes
would be with a higher possibility stable \cite{mat06} and therefore traversable.
The other interesting problem in wormhole physics is their classification.
Having in mind classification delivered by the black hole uniqueness theorem,
the first work in this direction was provided in \cite{rub89}, delivering the uniqueness theorem for wormhole spaces with vanishing Ricci scalar.
Further, the uniqueness of Ellis-Bronikov wormhole with phantom field was found in \cite{yaz17}, while the uniqueness
for four-dimensional case of the Einstein-Maxwell-dilaton wormholes with the dilaton coupling constant equal to one, was presented in \cite{laz17}.
The case of higher-dimensional generalization of wormhole solution, valid from the point of view of the unification theories like string/M-theory
attracts also attention.
The uniqueness theorem for higher-dimensional case of the static spherically symmetric phantom wormholes was treated in \cite{rog18}, while
the case of of static spherically symmetric traversable wormholes with two asymptotically flat ends, subject to the higher-dimensional solutions
of Einstein-Maxwell-phantom dilaton field equations with an arbitrary dilaton coupling constant, was elaborated in \cite{rog18a}.
Various other aspects of physics of these objects were under intensive studies (for a detailed review of the blossoming subject the reader may consult \cite{worm}).
Wormholes being a fascinating subject of their possible impact on space and time travels, may also be regarded as potential
astrophysical objects, that can be observationally search for.
From the astrophysical point of view, it is persuasive to consider rotating wormholes. The problem that arises is how to observationally distinguish
rotating wormholes from stationary axisymmetric black holes of Kerr-type. Remarkable attention to the aforementioned problem was paid to after
the Even Horizon Telescope observed the black hole shadow in the center of the galaxy M87.\\
The first studies to what extent wormholes can imitate the observational characteristics of black holes were conducted in \cite{dam07},
where the simple generalization of Schwarzschild-like line element was revealed.
The considered metric differs from the static general relativity one by introducing the dimensionless parameter ${\lambda}$. The value of the parameter equal to zero is
responsible for the ordinary Schwarzschild black hole solution.
Of course one should be aware that for non-zero values of the parameter the presented line element is no longer the static solution of Einstein equations
and changes the structure of the manifold. Therefore the matter with almost vanishing energy density ought to be required to maintain the aforementioned
gravitational configuration (for the discussion of the influence of the parameter ${\lambda}$ on the static manifold structure see, e.g., \cite{bue18}).
Further generalization of the idea given in \cite{dam07} to describe Kerr-like wormhole spacetime as a toy model, was achieved by applying
a modification on the Kerr metric similar to the procedure performed in \cite{dam07}.
The embedding diagrams, geodesic structure, as well as, shadow characteristics of the obtained Kerr-like wormhole were given in \cite{ami19}.
On the other hand, the throat-like effects on the shadow of Kerr-like wormholes were elaborated in \cite{kas21}.
However,
the problem of the structure at the horizon scale of black hole which gives rise to echoes of the gravitational wave signal bounded with the postmeger
ring-down phase in binary coalescences, in the case of static and rotating toy models of traversable wormholes, has been
elucidated in \cite{bue18}.
The other subject acquiring much attention in contemporary astrophysics and physics is the unrelenting search for finding {\it dark matter}
sector particles. The nature of this elusive ingredient of our Universe is a mystery and several models try to explain it and constitute the possible guidance for the future experiments.
The main aim of our work will be to investigate the behavior of axion-like particle {\it dark matter} model clouds, around the mimickers of rotating
black holes, stationary axially symmetric wormholes. The work will provide some continuity with our previous studies \cite{kic21}, where we have paid
attention to the main features of axionic clouds {\it dark matter} in the vicinity of magnetized rotating black holes.
The principal goal of the investigations will be to find the possible differences in characteristic features of the axion-like condensate, between those two classes of compact objects,
i.e., rotating black holes and black hole mimickers. Our studies will constitute the first glimpse at the problem in question. Namely, we restrict our consideration to the probe limit case,
when one has the complete separation of the degrees of freedom, i.e., matter fields do not backreact on wormhole spacetime.
The organization of the paper is as follows. In Sec. II we deliver the basic facts about the axion-like {\it dark matter} model. Sec. III
will be devoted to the description of the rotating wormholes models surrounded by {\it dark matter} clouds, in the considered model
of axion-like {\it dark matter}. In Sec. IV we describe the numerical results of the studies, while in Sec. V we conclude our investigations and aim the possible
problems for the future investigations.
\section{Model of axion-like {\it dark matter} sector}
The explanation of astronomical and cosmological observations require {\it dark matter} existence, whose nature is one of the most tantalizing questions confronting
contemporary physics and cosmology. A large number of ongoing or planned experimental searches for its detection and understanding of the {\it dark sector} role
in a fundamental description of the Universe.
Axions are among the strongest candidates for the possible explanation of the existence of {\it hidden sector} \cite{pre83}-\cite{din83}.
Their existence has been postulated to explain the apparent lack of violation of charge conjugate parity \cite{pec77}-\cite{wil78} and in the strong interaction
motivated the absence of observable electric dipole moment of the neutron \cite{pen15}. Axionlike particles are also widely spotted in the realm of string theories \cite{svr06}.
In what follows, we shall study axionlike scalar particles coupled to the
Maxwell $U(1)$-gauge field. The non-trivial coupling of axion field to the Maxwell field strength invariant plays the crucial role in the model in question.
The field equations of motion are provided by the variation procedure with respect to the action given by
\begin{equation}
\mathcal{S} = \int d^4 x \sqrt{-g} \left[R - \frac{1}{4} F_{\mu \nu} F^{\mu \nu}
- \frac{1}{2} \nabla_\mu \Psi \nabla^\mu \Psi - \frac{\mu^2}{2} \Psi^2
- \frac{k}{2} \Psi \ast F^{\mu \nu} F_{\mu \nu} \right],
\end{equation}
where we set $R$ for the Ricci scalar, $F_{\mu \nu} = 2 \nabla_{[\mu} A_{\nu]}$, while $\Psi$ stands for the scalar field (axion) with mass $\mu$.
$\ast F^{\mu \nu} = 1/2 \epsilon_{\mu \nu \alpha \beta} F^{\alpha \beta}$ is the dual to Maxwell field strength.
The equation of motion for the scalar field $\Psi$, which constitutes a covariant Klein-Gordon equation with a source term of the dual Maxwell field invariant, implies
\begin{equation}
\nabla_\mu \nabla^\mu \Psi - \mu^2 \Psi
- \frac{k}{2} ~\ast F^{\mu \nu} F_{\mu \nu} = 0,
\label{eq:field_eqn}
\end{equation}
while the $U(1)$-gauge field is subject to the relation as follows:
\begin{equation}
\nabla_\mu F^{\nu \mu} + 2 k~\ast F^{\nu \mu} \nabla_{\mu }\Psi = 0.
\end{equation}
We refer to the $\Psi$ field as axionlike, because the axions (originating from QCD) have adequate constrains on both mass and coupling parameter. Here however we consider particles with physics given by an analogical Lagrangian yet with arbitrary values of physical parameters. However for simplicity we might refer to the studied axionlike particles as simply axions.
The {\it dark matter} model in question was widely elaborated in studies of black hole superradiance
and light polarization effects, possible experimental signals of {\it dark sector} around these objects \cite{pla18}-\cite{car18}, \cite{kic21},
and neutron stars \cite{gar18}-\cite{gra15}, as well as,
the influence of axionic {\it dark matter} on the physics on early Universe and primordial black holes \cite{fed19}-\cite{ros18}.
The form of the relation (\ref{eq:field_eqn}) envisages the fact that the presence of the non-zero source term, containing the dual invariant, given by
\begin{equation}
\mathcal{I} = ~\ast F^{\mu \nu} F_{\mu \nu} \neq 0,
\end{equation}
is crucial.
In the opposite case, when the invariant is equal to zero, the axion-like scalar field equation of motion reduces to the simple massive Klein-Gordon case,
without any self-interaction potential. It means that no scalar hair configuration on the studied line element can emerge.
Although it has been shown that in Kerr spacetime scalar hair may emerge in certain situations \cite{herd14}, here we pick a different ansatz (see below) as we focus on stationary configurations, which appear to be magnetically induced in this approach.
On the other hand, it can be noticed that the discussed invariant, $\ast F_{\mu \nu} F^{\mu \nu}$,
is equal to zero in the case when $F_{\mu \nu} =0$, or for spherically symmetric spacetime. However, it has a non-trivial form, $\ast F_{\mu \nu} F^{\mu \nu} \neq 0$,
when both rotation and magnetic $U(1)$-gauge field components are present in the spacetime under consideration.
To introduce the magnetic field we use the method proposed by Wald \cite{wal74}, where the vector potential is sourced by Killing vectors of the rotating spacetime.
In general it has a form
\begin{equation}
A_\mu = \frac{1}{2}B (m_\mu + 2 a k_\mu),
\end{equation}
where $k_\mu$ and $m_\mu$ are the Killing vectors connected with temporal invariance and $\phi$ rotation respectively.
As in \cite{kic21}, where we have studied rotating magnetized black holes submerged into axionic {\it dark matter} cloud, one can introduce a static magnetic
field to the system, which will be oriented along the rotation axis. It seems to be plausible from the point of astrophysical perspective
and can be regarded as a starting point for studies of the magnetic field influence of the system in question. Because of the fact that our investigations
focus on static magnetic field, parallel to the wormhole rotation axis, the gauge potential may be rewritten in the form as $
A_\mu dx^\mu = B/2~ g_{\mu \nu} m^\nu dx^\mu.$
For our considerations we choose a static, time independent ansatz. The symmetry of the problem enables us to elaborate
the axion field in the form provided by
\begin{equation}
\Psi = \psi(r, \theta),
\label{eq:ansatz}
\end{equation}
which will be plugged into the equation \eqref{eq:field_eqn}, for the considered line element.
\section{Rotating wormhole metrics}
The simplicity of the static line element describing a wormhole may suggest that the spinning generalization can be achieved analytically
and ought to be globally regular. But in vain, it happens that finding the stationary solution with an extended source is far more complicated (see
for the recent aspects of this problem \cite{vol21}). However, the rotating wormhole solutions are widely discussed in literature \cite{teo98}-\cite{che16},
but one should be aware that they do not constitute the exact solutions of the equations of motion but rather comprise some model of geometries.
In this section, we shall study two kinds of rotating wormhole model metrics. First one accounts for the extension of the regular black hole Kerr metric \cite{bue18,ami19}. The other is the Teo class wormhole \cite{teo98}, a rotating generalization of Morris-Thorne wormhole, which serves us as comparison to a bit more realistic Kerr-like wormhole.
\subsection{Kerr-like wormhole}
To begin with, we consider the metric of Kerr-like rotating wormhole.
It is constructed by a slight modification of stationary axisymmetric line element with a parameter ${\lambda}$.
For the first time, such construction was proposed in \cite{dam07}, where the static Schwarzschild black hole was considered.
Then, it was generalized to the case of stationary axisymmetric line element \cite{bue18,ami19}.
The Kerr-like wormhole line element yields
\begin{eqnarray}
ds^2 &=& - \left( 1 - \frac{2 M r}{\Sigma} \right)dt^2 - \frac{4 M ar \sin^2 \theta}{\Sigma} dt d\phi + \frac{\Sigma}{\tilde{\Delta}} dr^2 + \Sigma d\theta^2\\ \nonumber
&+& \Big(r^2 + a^2 + \frac{2 M a^2 r \sin^2 \theta}{\Sigma} \Big) \sin^2 \theta d\phi^2,
\end{eqnarray}
where we set
\begin{align}
\Sigma(r, \theta) = r^2 + a^2 cos^2 \theta, \\
\tilde{\Delta}(r) = r^2 + a^2 - 2M(1 + \lambda^2)r.
\end{align}
The parameters $M$ and $a M$ correspond to mass and angular momentum of a wormhole.
For a small deviation parameter ${\lambda}$, one achieves
almost indistinguishable from of Kerr black hole line element. These three parameters describe the system as seen from the outside.
Moreover its Arnowitt-Deser-Misner (ADM) mass, as seen by the observer at asymptotic spatial infinity, is given by $M_{ADM} = M (1 + {\lambda}^2)$.
The largest root of $\tilde{\Delta}(r) = 0$, establishes the surface provided by
\begin{equation}
r_+ = M ( 1 + \lambda^2 ) + \sqrt{M^2 ( 1 + \lambda^2)^2 - a^2}.
\end{equation}
For the model in question it does not constitute a radius of the event horizon, but describes the radius of the throat of the rotating wormhole, which
connects two asymptotically flat regions of the spacetime. It can be explicitly seen by the adequate changes of variables \cite{bue18,ami19}. The points
with the condition $r<r_+$ do not exist.
Consequently the axion field equation written in the Kerr-like wormhole spacetime implies the following:
\begin{align}
\tilde{\Delta} \partial_r^2 \psi +
\frac{2(r - M)\tilde{\Delta} - M \lambda^2 (r^2 + a^2)}{\Delta} \partial_r \psi
+ \partial_{\theta}^2 \psi
+ \cot \theta \partial_{\theta} \psi
- \mu^2 \Sigma \psi = \frac{k \Sigma}{2} \mathcal{I}_{KWH},
\label{eqn:kwh_axion}
\end{align}
where
the electromagnetic field invariant is provided by
\begin{align}
\mathcal{I}_{KWH} = - \frac{a B^2 M \tilde{\Delta} \sin^2 \theta \cos \theta}{2 \Delta \Sigma^4} \big[ 3 a^6 + 2 a^4 M r - 5 a^4 r^2 - 8 a^2 M r^3 - 32 a^2 r^4 - 24 r^6 \nonumber \\
+ 4 a^2 (a^4 - a^2 r^2 + 2(M - r)r^3 ) \cos 2\theta + a^4 (a^2 - 2 M r + r^2) \cos 4\theta \big].
\end{align}
The equation \eqref{eqn:kwh_axion} undergoes a following scaling transformation
\begin{equation}
r \rightarrow \eta r, \quad a \rightarrow \eta a, \quad M \rightarrow \eta M, \quad
B \rightarrow B/\eta, \quad \mu^2 \rightarrow \mu^2 / \eta^2, \quad r_+ \rightarrow \eta r_+,
\end{equation}
\red{which allows us to fix one of model parameters to unity. For this we pick $M = 1$.}
\subsection{Teo rotating wormhole}
The well-known Morris-Thorne metric, introduced in Ref. \cite{mor88},
describes a traversable wormhole spacetime, which is stabilised by exotic matter in the area of its throat.
That solution was achieved by using reverse engineering of general relativity, namely the metric was postulated first and with a help of Einstein equations
the suitable matter components were found.
Generalization of the aforementioned solution, by including the rotation into the consideration, was performed in \cite{teo98}.
The resulting metric of the rotating wormhole has a following form:
\begin{equation}
ds^2 = -N^2 dt^2 + \frac{dr^2}{1 - \frac{b}{r}} + K^2 r^2 \left[ d \theta^2 + \sin^2 \theta (d \phi - \omega dt)^2 \right],
\end{equation}
where, as in the Morris-Thorne case, one has a lot of freedom in choosing the shape of $N$, $b$, $K$ and $\omega$ functions, as long as they meet specific requirements.
Firstly, all the functions can be functions of $r$ and $\theta$ and should be regular on the symmetry axis $\theta =0, \pi$.
Secondly, $N$, the gravitational redshift function, ought to be finite and nonzero, $b$ as the shape function determining the shape of the wormhole throat,
should satisfy $b \leqslant r$.
$K$ accounts for the radial distance with respect to the coordinate origin and $\omega$ stands for the angular velocity of the wormhole.
The embedding of constant $t$ and $\theta$-cross sections in the three-dimensional Euclidean space reveals the well-recognizable form of the wormhole
spacetime. The constructed geometry describes two regions, where the radial coordinates are given by $r \in [r_+,~\infty)$, which are joined together
at the wormhole throat $r=r_+$. At spatial infinity, the requirement of asymptotic flatness regions provides that the metric coefficients ought to satisfy
the following expansions:
\begin{equation}
N = 1 - \frac{M}{r} + {\cal O} \Big(\frac{1}{r^2}\Big), \qquad K = 1 + {\cal O}\Big(\frac{1}{r}\Big),
\qquad \frac{b}{r} = {\cal O}\Big(\frac{1}{r}\Big), \qquad \omega = \frac{2 J}{r^3} + {\cal O}\Big(\frac{1}{r^4}\Big),
\label{eq:twh_asympt}
\end{equation}
where we have denoted by $M$ the mass of the wormhole and by $J$ its angular momentum.
In general, one encounters the whole range of functions, which fulfil the aforementioned conditions and constitute a regular rotating wormhole solution.
For the numerical calculations, we pick a set of functions which appear to be quite popular in the literature of the subject,
and were previously used by different authors \cite{shaikh18, nedkova13, abdujabbarov16, harko09, bambi13}
\begin{equation}
N = \exp\left[- \frac{r_+}{r} \right], \qquad b(r) = r_+ \left( \frac{r_+}{r} \right)^\gamma, \qquad
\omega = \frac{2 a r_+}{r^3}, \qquad K=1,
\label{eq:twh_metric_fun}
\end{equation}
where we use the $r_+$ symbol, for denoting the wormhole throat radius.
\red{The angular momentum parameter is defined in the standard way $a = J/M$.
Using the asymptotic relations \eqref{eq:twh_asympt} we find that for the picked set of functions \eqref{eq:twh_metric_fun} $M = r_+$.}
Thus, the family of the above
solutions is described by three parameters, i.e., the throat radius $r_+$, angular momentum parameter $a$ and the shape parameter $\gamma$.
After putting the ansatz \eqref{eq:ansatz} and the metric into the field equation \eqref{eq:axion_only_action} we arrive at the equation of motion
\begin{align}
\left[ r^2 - r_+ r \left( \frac{r_+}{r} \right)^\gamma \right] \partial_r^2 \psi + \left[ 2r + r_+ + \left(\frac{r_+}{r} \right)^\gamma \left(\frac{1}{2}r_+ \gamma - \frac{r_+^2}{r} -\frac{3}{2} r_+ \right) \right] \partial_r \psi \nonumber \\
+ \partial_{\theta}^2 \psi + \cot \theta \partial_{\theta} \psi - \mu^2 r^2 \psi = \frac{1}{2} k r^2 \mathcal{I}_{TWH},
\label{eqn:twh_axion}
\end{align}
which radial part is strongly dependent on $\gamma$.
The Maxwell field invariant related to uniform magnetic field in this spacetime implies
\begin{equation}
\mathcal{I}_{TWH} = \frac{12 a B^2 r_+ \cos \theta \sin^2 \theta}{r^{5/2}} \sqrt{\frac{r - r_+ \left(\frac{r_+}{r} \right)^\gamma}{\exp \left[ -\frac{2 r_+}{r} \right]}}.
\end{equation}
The equation \eqref{eqn:twh_axion} follows a scaling transformation
\begin{equation}
r \rightarrow \eta r, \quad r_+ \rightarrow \eta r_+, \quad a \rightarrow \eta a, \quad B \rightarrow B/ \eta, \quad \mu^2 \rightarrow \mu^2 / \eta^2.
\end{equation}
\red{Using this transformation we fix $r_+ = 1$.}
\subsection{Free energy}
As a benchmark for the thermodynamical preference of the obtained states we use free energy by evaluating the on-shell action of the axion dependent part of the theory
\begin{equation}
\mathcal{S}_{axion} = \int d^4 x \sqrt{-g} \left[- \frac{1}{2} \nabla_\mu \Psi \nabla^\mu \Psi - \frac{\mu^2}{2} \Psi^2
- \frac{k}{2} \Psi \ast F^{\mu \nu} F_{\mu \nu} \right].
\label{eq:axion_only_action}
\end{equation}
By substituting the equations of motion into the action and imposing the ansatz of the field we arrive to the formula for the free energy
\begin{equation}
F = - 2 \pi \int_\mathcal{M} dr d\theta ~\sqrt{-g} \bigg[ (\partial_r \psi)^2 g^{rr} + (\partial_\theta \psi)^2 g^{\theta \theta} + \mu^2 \psi^2 \bigg].
\label{eq_freeenergy}
\end{equation}
The straightforward integration of the equation \eqref{eq_freeenergy} appears to be problematic. It is because both considered backgrounds have singular metric determinant at the throat, which makes simple integration from throat to infinity impossible in these coordinates.
It should be noted that this singularity is merely a coordinate singularity, as the curvature of both wormholes is regular and finite at the throat.
In
the case of Kerr-like wormhole metric,
we have
\begin{equation}
\sqrt{-g} = \sqrt{\frac{\Delta}{\tilde{\Delta}}} \Sigma \sin^2 \theta,
\end{equation}
where for the case of ${\lambda}$ equal to zero we obtain that $\Delta = {\tilde{\Delta}}$.
This fact
naturally eradicates the singularity problem in the black hole scenario. Here, however, as we radially fall toward the wormhole, the root of ${\tilde{\Delta}}$ comes first and creates the singularity.
On the other hand, for the Teo rotating line element we get
\begin{equation}
\sqrt{-g} = \frac{\exp \left( - \frac{r_+}{r} \right) r^2 \sin \theta}{\sqrt{1 - \left( \frac{r_+}{r} \right)^{\gamma + 1}}},
\end{equation}
with the denominator naturally generating the infinity.
To deal with the integration in such spacetimes we use energy differences instead.
Also we introduce a cutoff to the lower integration bound, so we start from $r_+ + \epsilon$ rather than simply $r_+$.
In this way we ensure the finiteness of energy differences and give them straightforward physical interpretation.
With the change of the background parameter the solution becomes more or less thermodynamically stable with respect to some \textit{ground} solution.
\section{Results}
In this section we pay attention to the solutions of the equations of motion for the previously described two toy models of rotating wormholes.
Due to the complications of the relations \eqref{eqn:kwh_axion} and \eqref{eqn:twh_axion},
we solve them numerically by virtue of spectral methods. Firstly the adequate equation is discretized on Gauss-Lobato grid \cite{matlabnum} and
next translated into a system of algebraic equations with spectral differentiation matrices.
The method in question
has already been implemented in Python and tested for the numerical stability.
The technical details, especially convergence tests of the numerical method are described in the Appendix of \cite{kic21}, where we studied
the problem of axionlike particle clouds in the spacetimes of rotating magnetized black holes.
The spectral nature of the numerical scheme requires remapping the coordinates onto the $[-1, 1]$ intervals.
It can be achieved by the coordinate transformation provided by
\begin{align}
z = 1 - \frac{2 r_+}{r}, \\
u = \frac{4 \theta}{\pi} - 1,
\end{align}
where $r_+$ is the wormhole throat radius.
After such operation, our numerical domain may be written in the form $[-1, 1]\times[-1, 1]$.
For $z$-coordinate, the boundaries are the wormhole throat ($z = -1$) and spatial infinity ($z=1$), while for $u = -1$, one talks about \textit{north pole} of a wormhole and the \textit{equator} with $u = 1$.
Consequently after the coordinate transformation in the underlying equations, one shall impose the adequate boundary conditions.
Namely, on the throat surface we demand that the axion field should be regular, therefore $\partial_r \psi = 0$ provides a desirable conduct of the field.
\red{Alternatively, setting the field to a constant value, such as zero in a wormhole scenario, is also a possible choice. However we wish to explore the Kerr-like solution for different values of $\lambda$ parameter, including its zeroing when it simplifies to the Kerr black hole.
Given that for the consistency between these two kinds of solutions we use the Neumann boundary condition.}
At the spatial infinity, we take a look on the asymptotic behaviour of the equation itself and the source term $\mathcal{I}$.
It appears that the Maxwell field invariants in both backgrounds are vanishing functions. As $r \rightarrow \infty$, we have
\begin{equation}
I_{KWH} = \mathcal{O}\left(\frac{1}{r^4}\right),
\end{equation}
\begin{equation}
I_{TWH} = \mathcal{O}\left( \frac{1}{r^2} \right).
\end{equation}
Which means that both equations \eqref{eqn:kwh_axion} and \eqref{eqn:twh_axion} reach a simple, asymptotic form, to the leading order
\begin{equation}
\partial^2_r \psi + \frac{2}{r} \partial_r \psi - \mu^2 \psi = 0.
\end{equation}
This simple equation has a solution
\begin{equation}
\psi = A \frac{\exp(\mu r)}{r} + B \frac{\exp(-\mu r)}{r},
\label{eq:psi_asympt}
\end{equation}
where $A$ and $B$ are constants.
Naturally the field ought to decay for the sake of asymptotic flatness of the spacetime.
Given that we are allowed to choose $A = 0$, with arbitrary $B$.
This means that a boundary condition $\psi(r \rightarrow \infty) = 0$ is an adequate and mathematically motivated choice.
On the other hand, the boundary conditions for the angular dependency are built on the basis of the spacetime symmetry. Both considered spacetimes are rotating, therefore we demand $\partial_{\theta} \psi = 0$ on the \textit{north pole}.
On the \textit{equator}, the presence of magnetic field combined with the spacetime symmetry implies that $\psi = 0$.
\subsection{Kerr-like wormhole}
To commence with, we solve the equation \eqref{eqn:kwh_axion} for the Kerr-like background metric.
A portion of obtained distributions is depicted in Fig. \ref{fig_kw_maps}.
In the following panels we see the increasing mass of the axionic field.
In the panel (a) the field is ultralight, subsequently in (b) $\mu^2 = 0.01$, (c) $\mu^2 = 0.1$ and finally in (d) $\mu^2 = 1$.
In every panel we have $a = 0.99$ and $\lambda = 0.5$. We can clearly see how the mass of the axionlike field changes the angular distribution of it around the wormhole.
For little masses the clouds are concentrated around the poles of the wormhole and spread in the space for several throat radii.
As we increase the axion mass we see that the polar regions of the wormhole become depleted and the field drifts towards the equator. The largest concentration is visible on the latitude $\theta \simeq \pi/4$.
Second important effect is the influence of the field mass on the magnitude of the field.
Inspection of
the colorbars reveals that the larger the mass the smaller the field.
The spatial tail of the field is also much shorter, when the mass of the field is larger.
Intuitively, in the asymptotic solution \eqref{eq:psi_asympt} $\mu$ enters the suppressing exponential term. The field decays faster for larger masses, which means the massive fields are localized in the vicinity of the throat surface.
Another important thing that stands out in relation to the black hole solutions is the repulsion of the axion cloud from the wormhole throat surface.
While in the case of the black hole, the field had non-zero values on the surface of the event horizon, and its radial character was monotonically decreasing, here we have a completely different situation.
For the wormhole, the field vanishes or at least has a significantly smaller value on the throat.
Then it grows with the radius as it reaches the maximum and finally decreases.
This effect is particularly visible for the high values of the angular momentum.
\begin{figure}[h]
\centering
\subfloat[$\mu^2 \rightarrow 0^+, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m0_HD.pdf}
}
\qquad
\subfloat[$\mu^2 = 0.01, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m001_HD.pdf}
}
\vspace{0.5cm}
\subfloat[$\mu^2 = 0.1, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m01_HD.pdf}
}
\qquad
\subfloat[$\mu^2 = 1, \quad \lambda = 0.5$]{
\includegraphics[width=0.45 \textwidth]{wh_a09_la05_m1_HD.pdf}
}
\caption{Axion field distribution around Kerr-like wormholes for given sets of parameters.
The blank space in the vicinity of wormhole throat distinguishes the solution from the black hole counterpart, where the field appears to be non-zero on the event horizon.
Subsequent panels for each mass parameter show how the angular distribution of the field is affected.}
\label{fig_kw_maps}
\end{figure}
The radial behaviour of the axionic field can be seen more precisely in Fig. \ref{fig:kw_slices}.
We present there a slice of $\psi$ as a function of $r$ in throat radius units, for constant $\theta=\pi/4$ and few different values of the $\lambda$ parameter.
In contrast, we also plot the behaviour of axions in Kerr black hole metric (that is $\lambda = 0$).
What we can see is the increasing $\lambda$ consequently extinguishes the axionic hair.
In the foreground a structural change in the field profile is visible as we compare it to the black hole scenario.
An axionic field over a black hole has a maximum value on the event horizon.
The opposite is true for a wormhole, on the throat the field vanishes, then grows to its maximum and fades away with the radius.
Then, the bigger is $\lambda$ the smaller are the maxima and overall magnitude of the axionic hair.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{psi_pi4_a99_m0.pdf}
\caption{A closer look on the axion cloud gap near the wormhole throat. Here we show slices of $\psi$ for constant $\theta = \pi/4$, with parameters $a = 0.99, \mu^2 = 0^+$.
Increasing of $\lambda$ decreases the magnitude of the axions and cuts off its tail.}
\label{fig:kw_slices}
\end{figure}
In the next step we investigate the free energy of the obtained axion cloud configurations.
It is interesting to see how the parameters describing the spacetime geometry around the wormhole influence the thermodynamics of the axion clouds.
Due to the previously mentioned difficulties in computing the free energy in these metrics, we rather talk about energy differences, than the exact values.
In Fig. \ref{fig:kw_fe} we present the differences of the free energy versus angular momentum $a$, with respect to the $\lambda=0$ level, which constitutes a plain Kerr black hole.
It is clearly visible that the larger value of the distortion parameter ${\lambda}$ one takes into account, the higher value of the free energy of the cloud we achieve.
It turns out, that the more the gravitational background deviates from from the black hole metric, the less thermodynamically desirable axion clouds are.
This effect works together with the diminishing magnitude of the field on the previously discussed Fig. \ref{fig:kw_slices}.
Additionally the increasing angular momentum of the wormhole also increases the free energy difference.
This means that for \textit{extreme} Kerr-like wormhole axion hair is the least favourable.
\begin{figure}
\centering
\includegraphics[width=0.8\textwidth]{fediff_a_lambs_m0.pdf}
\caption{Free energy differences as a function of angular momentum (with $\lambda = 0$ as the ground curve) for different values of $\lambda$. With axion mass $\mu^2 = 0^+$ we see that the cloud
thermodynamical favourability decreases with growth of both angular momentum and $\lambda$ parameter.}
\label{fig:kw_fe}
\end{figure}
\subsection{Teo rotating wormhole}
Teo class wormhole has a different set of parameters and it does not simply transform into a black hole solution, just like a Kerr-like wormhole does.
Here the throat radius is independent of the other parameters and is imposed manually in $g_{tt}$ and $g_{rr}$.
With the particular choice of functions \eqref{eq:twh_metric_fun}, we can only steer with the shape of metric components via $\gamma$ parameter.
Therefore, let us consider values of $\gamma$ in the interval $(-1, 1]$,
where for $\gamma = -1$ the function $g_{rr}$ is singular, so we can only approach this value.
\red{As it was mentioned, the Kerr-like wormhole can be reduced to a black hole solution by setting $\lambda = 0$.
Teo solution does not share this feature, but is a well-known wormhole metric, just like its non-rotating counterpart the archetypical Morris-Thorne wormhole.
While it can not serve as a testing field for differences between axion clouds around wormholes and black holes, one can treat it as a benchmark for behaviours of the axion hair in another wormhole environment.
Using this background might help us to see if the obtained axion solutions share similar features.
}
In Fig. \ref{fig_teo_maps} one can see the distribution of the axionic cloud around a Teo wormhole for different axionlike field masses.
In the panel (a) we have an ultralight field, then it takes values $0.1$, $0.5$ and $1.0$ for (b), (c) and (d) respectively.
In all panels we use $\gamma = -0.99$, which gives us tiny value of the axionic field (see the colorbars).
The angular distribution has similar features to the Kerr-like metric.
For ultralight axions the field is localized in the majority of the wormhole surroundings.
As the mass increases the hair tightens spatially and disappears from the polar regions.
For the large mass case the axionic clouds are drifting toward the equator with the polar caps left almost empty. Moreover the radial reach is very short - around one throat radius.
One can clearly notice that for the negative gamma value the analogous effect to the Kerr-like scenario is observed.
The axionic cloud is also pushed away from the throat surface - in its vicinity the field acquires small values, reaches the maximum and the descends monotonically to zero.
However in this gravitational background this effect is not as dramatic as in case of Kerr-like wormhole.
The weakening of the axionic field in the vicinity of the throat is easily visible in the distributions, although it is not that large.
Increasing $\gamma$ up to zero and beyond causes the rise of $\psi$ field. It grants bigger values, but the spatial qualitative characteristic remains intact.
\begin{figure}[h]
\centering
\subfloat[$\mu^2 \rightarrow 0^+, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m0.pdf}
}
\qquad
\subfloat[$\mu^2 = 0.1, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m01.pdf}
}
\vspace{0.5cm}
\subfloat[$\mu^2 = 0.5, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m05.pdf}
}
\qquad
\subfloat[$\mu^2 = 1, \quad \gamma = -0.99$]{
\includegraphics[width=0.45 \textwidth]{psidist_g-099_a99_m1.pdf}
}
\caption{Axion cloud distribution around the magnetized Teo type wormhole. The negative value of $\gamma$ resembles the results for Kerr-like wormhole, with
large $\lambda$ value, i.e., the decreasing of the field magnitude.
Each panel depicts the field distribution for different axionlike masses.
Once again a big mass results with a localized field, concentrated closely to the throat with depleted polar region. For bigger values of $\gamma$ we observe similar influence of $\mu$ on the spatial distribution.}
\label{fig_teo_maps}
\end{figure}
The next figure brings a closer look on the field drop near the throat.
In Fig. \ref{fig_teo_slice} we present radial slices of the field distribution with $\theta = \pi/4$. In this particular figure we depict the behaviour for ultralight field, however a similar tendencies are shown by more massive fields.
First of all, we observe a significant amplification of the field with the growth of $\gamma$.
The field does not acquire new features however, but it seems that the curves follow some kind of scaling related to gamma.
Additionally the profiles resemble the results obtained for Kerr-like wormholes. As we have previously mentioned, Teo class wormhole cannot be simply transformed into a black hole by a simple choice of parameter value.
However the features like a drop near the throat surface, then maximum and monotonic fall show that these might be more general wormhole related behaviours of axionic hair.
\begin{figure}[h]
\centering
\includegraphics[width=0.7 \textwidth]{teowh_psi_pi4_a99_m0.pdf}
\caption{Radial slices of the axion field for constant angle $\theta = \pi/4$.
The growth of $\gamma$ increases the maximum value of the field.
However in this metric the significant growth of the blank space near the throat is not present.
Also the growth of $\gamma$ does not seem to greatly affect the tail of the field, away from the throat, which is different from the Kerr-like wormhole results.}
\label{fig_teo_slice}
\end{figure}
If we consider the free energy, it appears that the axion clouds for the background with negative $\gamma$, are definitely less thermodynamically favourable.
In Fig. \ref{fig_teo_fe} we plot the free energy difference as a function of angular momentum for several values of the gamma parameter.
We use the curve with $\gamma = 0$ as a baseline for calculating energy differences.
Free energy difference curves for $\gamma < 0$ are positive, especially the $\gamma = -0.99$ curve reaches relatively big values.
Therefore thermodynamically speaking wormholes with $\gamma$ close to $-1$ have least chances to hold axionic hair.
With the growth of the parameter, the free energy of the cloud decreases, which makes the axions more thermodynamically favourable.
However this fall is rather moderate comparing to the rise of the top curve.
In both cases the increase of angular momentum amplifies the tendencies of the curves.
Curves for negative gamma grow, while the positive fall.
A consequent growth of $\gamma$ parameter leads the hair to some limit characteristics which can be seen in Fig. \ref{fig_teo_slice} and Fig. \ref{fig_teo_fe}.
\begin{figure}[h]
\centering
\includegraphics[width=0.7 \textwidth]{fediff_a_gam_m0.pdf}
\caption{Free energy differences vs. angular momentum for different values of $\gamma$. The curve of $\gamma = 0$ is the reference level. We see that free energy increases for negative values of gamma and slightly drops for positive ones, as angular momentum grows.}
\label{fig_teo_fe}
\end{figure}
Finally let us conduct qualitative comparison of the axionic clouds in considered metrics.
The solutions have undoubtedly similar features, especially when one takes a look on the $\psi$ slices.
We also observe the separation of the cloud from the surface of the throat in both cases.
This allows us to notice some general wormhole related phenomena, which are not present around the black holes \cite{kic21}.
Naturally we cannot speak in a fully general manner, as we only considered here merely two distinct gravitational backgrounds, which on the other hand should be treated as toy models.
\section{Conclusion}
In our paper, we have considered the problem of the distribution of axionlike particles, being regarded as {\it dark matter} sector model, around
the toy models of rotating wormholes. We have investigated the Kerr-like wormhole line element with the distortion parameter ${\lambda}$ and
Teo model of a rotating axisymmetric wormhole. The models under inspection were characterized by the mass, angular momentum, distortion parameter
(for Kerr-like wormhole) and the shape parameter $\gamma$ (for the Teo model). We numerically solve the equations of motion for the underlying cases, using spectral methods.
Among all we have found that the axion clouds are pushed forward from the wormhole throat, especially for the case of large value of the rotation
parameter $a$. The voids in the vicinity of the wormhole throat appear for the larger value of the distortion parameter ${\lambda}$. This phenomenon distinguishes the studied
system from the previously elaborated black holes \cite{kic21}. On the other hand, for the larger ${\lambda}$, one achieves higher value of the free energy, and therefore
this solution is less thermodynamically favoured.
As far as the Teo class of rotating wormholes is concerned, we have for the negative value of $\gamma$ the analogous effect as in the latter case is obtained. However
for the positive value, the behavior of axionic clouds resembles features of the {\it dark matter} clouds around Kerr black hole in a uniform magnetic field.
The solution with negative $\gamma$ is not thermodynamically favourable, as it has been revealed in free energy studies. However when $\gamma$ increases, the free energy
of the axionic cloud decreases.
We have found that the behavior of the axionic clouds significantly differs from the black hole scenario, which we discussed in our previous work \cite{kic21}.
This fact will account for the possible guidance, enabling one to distinguish between these two classes of compact objects.
Nevertheless, the search for astronomically observable criteria require far more complex approach. A more realistic dynamical gravitational model is needed,
when the time dependence of the studied fields is taken into account, as well as, the direct mathematical proofs of the stabilities of rotating wormhole spacetimes ought to be found.
These subjects impose a real mathematical challenge and require also solid numerical relativity machinery.
These problems shall be investigated elsewhere.
| 2024-02-18T23:39:57.950Z | 2022-07-07T02:13:30.000Z | algebraic_stack_train_0000 | 959 | 14,182 |
|
proofpile-arXiv_065-4745 | \section{Introduction}
\label{sect:introduction}
The origin of planetary systems remains a major challenge to
astrophysical theory. Aside from the quandary of planet formation,
planet survival is also a problem. Planet formation occurs when the
gas disk is still present, and by exchanging angular momentum with the
gas, planets start to migrate. This occurs
as the planet excites perturbations in the disk that, in turn, exert
torques on the planet. The asymmetry of these perturbations on either
side of the planet determines the strength of the torques, and thus the
direction of migration.
The main components of the excited perturbations are the one-armed spirals
launched at the Lindblad resonances and the librating material on horseshoe orbits in the planet's corotation region.
For low-mass planets, with weak wakes, both the shape of the spiral wake and the
resulting Lindblad torque can be treated by linear analysis under some
assumptions, particularly of local isothermality. The
analytical prediction (\citealp{Goldreich,Ward,Tanaka}), confirmed by numerical simulations
(\citealp{Nelson,D'Angelo,Bate}), is that the inner Lindblad resonances lead
to positive torques, whereas the outer ones lead to negative
torques. The outer Lindblad resonances lie closer to the planet, and
thus produce stronger torques, so the planet
migrates inwards. This migration mode, referred to as Type I migration,
occurs on timescales between $\ttimes{4}$ and $\ttimes{5}$ yr. This is a serious problem
for planet formation since these timescales are much shorter than the lifetimes of
disks ($\ttimes{6}$-$\ttimes{7}$ yr). Halting or slowing Type I migration is imperative if planets
are to survive at all. Indeed, planet population synthesis models (\citealp[e.g.,][]{Alibert,Mordasini})
have to assume a reduction factor for Type I migration of
30--1000 in order to match the observed distribution of planetary semimajor axes.
\citet{Paardekooper06} made a major step toward the solution
of this problem. They found that when the locally isothermal
approximation usually assumed in the literature was relaxed, the
planets migrated outwards. This behavior was explained by \citet{Baruteau}
and \citet{Paardekooper08} as
resulting from an entropy-related torque exerted by material on
horseshoe orbits in the corotation region.
This mechanism operates in regions of the disk
that have a negative entropy gradient and
inefficient radiative cooling, where sustenance of the torque
requires some
viscous and thermal diffusion (\citealp{Paardekooper08}).
\citet{Kley08} and \citet{Kley09} showed that outward migration indeed occurs in
disks with realistic heating and cooling.
This outward migration remains rapid. Planets migrate toward the
outer disk, reach an equilibrium radius of zero torque, and stay put
thereafter. As emphasized by \citet{Paardekooper10}, slow inward
migration then occurs as disk evolution shifts the equilibrium radius
inwards. The situation becomes similar to that of gap-opening planets
(Type II migration), with the planet migrating in lockstep with the
gas as the gas accretes.
This scenario poses a subtle problem that we examine in this
Letter. If the planet is moving together with the gas as the disk
depletes, at some point in the evolution the disk may reach a
thermodynamic state such that inward migration resumes. This brings
the problem back to square one, because, if the planet is to survive,
the remaining disk lifetime must be shorter than the Type I migration
timescale in that evolutionary state. We examine this possibility
using one-dimensional evolutionary models of protoplanetary disks
including heating and cooling. We describe the model in the following
section, present our results in \S~3, and give concluding remarks in
\S~4.
\section{The model}
\label{sect:model}
\subsection{Gas evolution}
We consider non-irradiated disks evolving by viscous
diffusion and photoevaporation (\citealp{Lynden-Bell,Lin}),
\begin{equation}
\pderiv{\varSigma}{t}=\frac{3}{r}\pderiv{}{r}\left[r^{1/2}\pderiv{\varSigma\nu{r^{1/2}}}{r}\right]-\dot{\varSigma}_{\rm w}(r,t),\label{eq:density}
\end{equation}
where $\varSigma$ is the surface density, $r$
is the radius, and $\nu$ is the effective viscosity. We take the
photoevaporation rate to be (\citealp{Veras,Mordasini})
\begin{equation}
\dot\varSigma_{\rm w}=\left\{\begin{array}{ll}
0,&\mbox{~for~$r<R_g$},\\
\dot{M}_{\rm{w}}/[2\pi(r_{\rm ext}-R_g)r],&\mbox{~for~}r\geq{R_g},\end{array}\right.
\end{equation}which is valid for external irradiation.
For temperature evolution, we use a model without shock heating (\citealp{Nakamoto})
\begin{equation}
2\sigma{T^4}=\tau_{\rm eff}\left(\frac{9}{4}\varSigma\nu\varOmega^2\right)+2\sigma{T_b^4},\label{eq:temperature}
\end{equation}
where $T$ and $T_b$ are the midplane and background temperatures, respectively,
$\varOmega$ is the Keplerian frequency, and $\sigma$ is the Stefan--Boltzmann constant. We take
the effective optical depth at
the midplane (\citealp{Hubeny,Kley08}):
\begin{equation}
\tau_{\rm eff}=\frac{3\tau}{8}+\frac{\sqrt{3}}{4}+\frac{1}{4\tau}.\label{eq:taueff}
\end{equation}
Equation~(\ref{eq:temperature}) states that the emerging flux is the result of an equilibrium
between viscous heating, background radiation, and radiative cooling. The optical depth
is $\tau=\kappa\varSigma/2$, and
the opacities $\kappa$ are taken from \citet{Bell}. We assume that although
dust growth and planet formation lock away refractory material,
fragmentation efficiently replenishes small grains, keeping
the disks opaque during their evolution (\citealp{Birnstiel}).
\subsection{Planet evolution}
The planet's orbital radius evolves as
\begin{equation}
\frac{d{r_p}}{dt}=\frac{2\Gamma}{m_pr_p\varOmega_p},\label{eq:planet}
\end{equation}where $m_p$ is the planet's mass and $\Gamma$ is the torque
from the gas. We assume a circular orbit and constant planet mass.
We only consider low-mass planets, and so can ignore the back reaction of the planet onto
the gas (\citealp{Alexander}). The torques
are modeled with analytical fits
valid for the fully unsaturated case (\citealp{Paardekooper10}). Using $b/h$=0.4, where $h$
is the disk's aspect ratio and $b$ is the gravitational softening of the
planet's potential (necessary in models with less than three dimensions), the torques are
\begin{eqnarray}
\Gamma_{\rm{iso}}/\Gamma_0&=&-0.85-\alpha-0.9\beta,\\
\gamma\Gamma_{\rm{ad}}/\Gamma_0&=&-0.85-\alpha-1.7\beta+7.9\ksi/\gamma
\end{eqnarray}for the locally isothermal and adiabatic equations of state,
respectively. The adiabatic index $\gamma=1.4$, and $\alpha$,
$\beta$, and $\ksi$ are the negative of the local density, temperature, and
entropy gradients:
\begin{equation}
\alpha=-\pderiv{\ln\varSigma}{\ln{r}};\qquad\beta=-\pderiv{\ln{T}}{\ln{r}};\qquad\ksi=\beta-(\gamma-1)\alpha.
\end{equation}
The torques are normalized by
\begin{equation}
\Gamma_0=(q/h)^2\varSigma_pr_p^4\varOmega_p^2,\label{eq:torque-norm}
\end{equation}
where $q$ is the ratio of planetary to stellar mass and $\varSigma_p$
is the surface density at the position of the planet. We interpolate between the
two torque regimes to get
\begin{equation}
\Gamma=\frac{\Gamma_{\rm ad}\varTheta^2+\Gamma_{\rm{iso}}}{(\varTheta+1)^2}.\label{eq:torque}
\end{equation}
Here, $\varTheta=t_{\rm{rad}}/t_{\rm{dyn}}$, where $t_{\rm{rad}}$ and $t_{\rm{dyn}}$ are
the radiative and dynamical timescales. If $\varTheta\ll{1}$, radiative processes
can restore the temperature
quickly compared to the horseshoe turnover time,
leading to isothermal horseshoe turns. It was shown in \citet{Paardekooper09} that the
timescale to establish the full horseshoe drag ($t_{\rm{drag}}$) is a fraction of the
libration timescale (typically 10 $t_{\rm{dyn}}$ for a 5 \mearthp planet). However,
as long as $t_{\rm{rad}} > t_{\rm{dyn}}$, the torque is affected by nonlinear effects,
and since the nonlinear torque is so much stronger than its linear counterpart
(\citealp{Paardekooper10}), the parameter governing linearity (and therefore
isothermality) is $\varTheta$ rather than
$t_{\rm{rad}}/t_{\rm{drag}}$.
Since $t_{\rm{drag}}$ depends on $q$, using $t_{\rm{drag}}$ would lead to different
migration behavior for different planet masses. In practice, the transition between
isothermal and adiabatic regions occurs on such a small length scale that this would
not change the qualitative outcome of the models. Mass segregation is more likely to
come from effects of saturation, which we do not consider in this Letter, since it is
as yet poorly understood for non-barotropic disks. We note, however, that for viscously
heated, optically thick disks close to thermal equilibrium,
radiative thermal diffusion will approach the viscous heating rate. Therefore, even in the
very optically thick inner regions of the disk, a sizeable fraction of the unsaturated
torque can be sustained as long as the disk remains viscous.
To calculate $\varTheta$, we consider $t_{\rm rad}=E/\dot{E}$, where $E$ is
the internal energy. The cooling $\dot{E} =\Div{\v{F}}$, where $\v{F}$ is
the flux. Using $E=\cv\rho{T}$, $\rho=\varSigma/2H$, and
$|\v{F}|=\sigma{T}_{\rm eff}^4$, we have
\begin{equation}
t_{\rm rad}=\frac{\cv\varSigma\tau_{\rm eff}}{6\sigma{T^3}}.
\end{equation}
The dynamical time $t_{\rm dyn}=2\pi/\varOmega$, so
\begin{equation}
\varTheta=\frac{\cv\varSigma\varOmega\tau_{\rm eff}}{12\pi\sigma{T}^3}.
\end{equation}
\subsection{Simulation parameters}
We use a one-dimensional linear grid covering 0.1--30~AU with 200
points. The surface density is specified by the initial mass
accretion rate $\dot{M}_0$, and the viscosity parameter
$\alpha_{\rm SS}$ (\citealp{Shakura}), following the analytical fits of \citet{Papaloizou}
for $\varSigma$--$\nu$ relations. We use
$\dot{M}_0=\ttimes{-7}$\,M$_{\odot}$~yr$^{-1}$ and $\alpha_{\rm
SS}=\ttimes{-2}$. The wind is modeled with $\dot{M}_{\rm
w}=\ttimes{-8}$\,M$_{\odot}$~yr$^{-1}$ and $R_g=5$\,AU. For a
given surface density,
Equation~(\ref{eq:temperature}) specifies the temperature, with $T_b=10$\,K. Because the optical depth
depends on temperature, we solve Equation~(\ref{eq:temperature}) with a
Newton--Raphson root-finding algorithm (using 0.01\,K precision). We
examine planets of mass 0.1, 1, and 10\mearth. Planet--planet
interactions are ignored. Boundary conditions are taken as outflow. We
compute the derivatives as in the {\sc Pencil Code}{\footnote{See
http://www.nordita.org/software/pencil-code}}, with sixth-order
spatial derivatives and a third-order Runge--Kutta time integrator.
\section{Results}
\Figure{fig:evolution} shows the evolution of our fiducial disk model.
The changes in density slope initially located at $\sim$0.4 and 21\,AU
correspond to opacity transitions (\citealp{Papaloizou}). The
constant temperature plateau initially at 4--5\,AU corresponds to the
opacity transition at $\sim$130\,K where ice grains sublimate (see
\Figure{fig:evolution}c). Initially, the ratio of radiative to dynamical timescales
$\varTheta>1$ through most of the disk ($\varTheta<1$ only at
$r>70$\,AU, beyond the model grid). Thus, the torque in Equation~(\ref{eq:torque}) is mostly
adiabatic, and largely independent of the
interpolation procedure.
\begin{figure*}
\begin{center}
\resizebox{.9\textwidth}{!}{\includegraphics{fig1.ps}}
\end{center}
\caption[]{Evolution of (a) surface density, (b) midplane temperature,
(c) optical depth, and (d) $\varTheta$, the ratio of radiative to dynamical
timescales. The disk is drained after 4.8\,Myr, with mass dropping
below \mearth.
The temperature shows a plateau at $\approx$130\,K, produced by the opacity
transition when ice grains sublimate that moves inward as the disk cools.
Through most of the evolution, $\varTheta>1$, so torques are
adiabatic.}
\label{fig:evolution}
\end{figure*}
Total disk mass and accretion rate decay nearly linearly with time. A
disk of 0.08 M$_{\odot}$ initially accreting at
$\ttimes{-7}\,M_{\odot}$~yr$^{-1}$ gets depleted in 4.8 Myr. At
that time, the total disk mass is $<1$\mearth, and we terminate the
simulation. The model gives a power-law decline of the density.
An inner hole (\citealp{Clarke}) never forms, because of
the shallow radial slope of unity for the wind driven by external photoevaporation.
As the surface density drops, so does the viscous heating
rate (Equation~\ref{eq:temperature}), and the temperature falls accordingly,
down to $T=T_b$. During the evolution, the
isothermal plateau
shifts inwards from 4\,AU to 1\,AU by 4.5\,Myr.
The optical depth only drops to unity at 10\,AU after 4\,Myr,
and at 5\,AU after 4.4\, Myr, so the planet formation region is optically
thick through most of the disk evolution. To check our method, we also ran models with
Equation~(\ref{eq:temperature}),
and models where the temperature was derived using (1$+$1)-dimensional models, following
\citet{Papaloizou}. The
differences were minor, consisting of a slightly larger isothermal plateau (extending to 6\,AU instead
of 5\,AU), and an outer disk $\sim$10\,K hotter.
To understand our results for planet migration in evolving disks, we
first examine the behavior of planets in stationary models. In
\Figure{fig:torques}, we show the torque as a function of radius.
Migration halts at stable equilibrium radii where
$\Gamma=0$ and $d\Gamma/dr<0$, that is, where the torque is
positive within (corresponding to outward migration) and
negative outside (inward migration). Two such equilibrium
radii occur,
at 4\,AU and 21\,AU, corresponding to the inner boundaries of the isothermal plateau and the outer
isothermal region, where $T=T_b$. (There is also a thin region
around an equilibrium radius at 0.6~AU that we do not consider here.)
A negative torque acts on planets migrating from
$r_p>21$\,AU, bringing them to that radius, while a positive torque acts on
planets with $4<r_p/{\rm AU}< 21$, also bringing them
to that radius. Planets with $r_p<4$\,AU migrate to the inner
equilibrium radius.
We
released planets of varying mass at several radii in the disk, and see precisely this behavior
(\Figure{fig:planet_gasevolution_linear}a).
The migration time to
the equilibrium radius is inversely proportional to the
planet mass, because the torque depends quadratically on mass
(Equation~\ref{eq:torque-norm}), so
$\dot{r}_p\propto{1/m_p}$ (Equation~\ref{eq:planet}). Planets of 10\,\mearthp
reach the
equilibrium radii in $\leq0.1$\,Myr. In the isothermal
case, planets migrate inwards at comparable speeds.
\begin{figure}
\begin{center}
\resizebox{.9\hfwidthsingle}{!}{\includegraphics{fig2.ps}}
\end{center}
\caption[]{Radial distribution of torque over time (Equation~\ref{eq:torque}).
Positive torques drive outward migration. The
equilibrium radii of zero torque shift inwards as the disk
evolves. The isothermal plateau corresponds to the well of
negative torques initially at 4--5\,AU. Discontinuities
correspond to opacity jumps. Torques are scaled by
$\varSigma/T$ to aid visualization.}
\label{fig:torques}
\end{figure}
\begin{figure*}
\begin{center}
\resizebox{.9\hfwidth}{!}{\includegraphics{fig3a.ps}}
\resizebox{.9\hfwidth}{!}{\includegraphics{fig3b.ps}}
\resizebox{.9\textwidth}{!}{\includegraphics{fig3c.ps}}
\end{center}
\caption[]{(a) Orbital migration in stationary disks. Planets
migrate at mass-dependent rates toward equilibrium radii of zero torque (see
\Figure{fig:torques}).
(b) Orbital migration in
evolving disks. Equilibrium radii migrate inwards on the
accretion timescale, but the planets eventually migrate too slowly
to remain in equilibrium. In the lower panels (c)--(e)
we show the same as in panel (b) but separated by planet mass.
The tracks of the equilibrium
radii (dotted lines) are shown for comparison.}
\label{fig:planet_gasevolution_linear}
\end{figure*}
In \Figure{fig:planet_gasevolution_linear}b, we show trajectories of
planet migration in an evolving disk. Now
the radii of zero torque shift inwards as the
disk evolves. The planets migrate to these radii on the Type~I
timescale of $\ttimes{5}$\,yr, and then
couple to disk evolution. Subsequently, they migrate inwards on the slow accretion
timescale of $\ttimes{6}$\,yr, comparable to Type~II migration, as
predicted by \citet{Paardekooper10}.
\Figure{fig:planet_gasevolution_linear}b has some intriguing features that
give pause. There is a clear effect of mass on the later evolution
of the planets. In the stationary case, the only effect of mass is to
determine the speed of migration to the equilibrium radii. In the
evolving case, the mass also
helps determine the final location of the planet. To show this, we
separately plot the tracks of planets of different masses in
Figures~\ref{fig:planet_gasevolution_linear}c--e, as well as
the trajectories of the equilibrium radii. The planets
follow these equilibrium radii until late times, when they decouple.
We can understand the process of decoupling by comparing the viscous
accretion timescale to the migration timescale. As the disk thins, the surface density
reaches a value so low that the gas cannot transfer
sufficient angular momentum to the planet for its
orbital radius to evolve as fast the equilibrium radius.
Another way of understanding this is that a perturbation of the
planet away from the equilibrium radius will only be corrected
if the torque is sufficiently strong to return the planet
before the radius moves a substantial distance. As the torque is a
function of planet mass, the time and radius when decoupling occurs
are also functions of the planet mass.
We can estimate this time and radius by comparing the migration timescale
($t_{\rm mig}$=$r_p/|\dot{r}_p|$) to the disk accretion timescale
($t_{\nu}$=$\varSigma/|\dot{\varSigma}|$). As long as $t_{\rm mig}<
t_{\nu}$, the planet can keep up with the evolution of the
disk. When this is no longer true,
decoupling occurs, releasing the planet.
We plot the quantity $t_{\rm mig}/t_{\nu}$ in \Figure{fig:timescales},
for the planet of 1\,\mearthp at different times. The structure in the plot
comes from the radial derivatives of $\varSigma\nu$ that define the mass
accretion flow, and from the torques $\Gamma$ that define the migration rate.
The spikes in the figure correspond to the equilibrium locations,
where $\Gamma=0$ (and therefore formally $t_{\rm mig}/t_{\nu}=\infty$).
The decoupling in \Figure{fig:planet_gasevolution_linear}d starts to occur at
$\approx4.0$\,Myr. In \Figure{fig:timescales}, we see that it roughly corresponds
to the time when the vicinity of the equilibrium radius crosses the line of
$t_{\rm mig}/t_{\nu}=1$.
\begin{figure}
\begin{center}
\resizebox{.9\hfwidthsingle}{!}{\includegraphics{fig4.ps}}
\end{center}
\caption[]{Comparison of the timescales of planet migration ($t_{\rm mig}$) and
disk accretion ($t_\nu$) for a planet of 1\mearth.}
\label{fig:timescales}
\end{figure}
Evolution after decoupling proceeds as follows. The inner equilibrium
radius corresponds to the inner edge of the isothermal plateau. Once
the planet decouples, it is released inside the plateau, itself a
region of inward migration since the temperature gradient vanishes. As
disk evolution continues, the decoupled planet soon finds itself at
the outer edge of the isothermal plateau, and starts migrating
outwards. For planets of 1\mearth, decoupling occurs at $r_p$=1\,AU at
4.5\,Myr (\Figure{fig:planet_gasevolution_linear}b). The planet then
rapidly descends the temperature gradient until it reaches the outer
equilibrium radius. However, this radius too moves inward faster than
the planet can migrate, so the planet enters the isothermal outer disk
(where $T\simeq{T_b}$), another region of slow inward migration. At
this stage, a planet of 1\,\mearthp still has time to migrate from
$r_p=2.0$~AU to $r_p=1.5$~AU before the density drops too low to drive
further migration (\Figure{fig:planet_gasevolution_linear}d).
The planet of 10\,\mearthp is strongly coupled and would follow the outer equilibrium
radius until it hit the inner boundary of our model at 0.1\,AU. The halt at 2\,AU seen in
\Figure{fig:planet_gasevolution_linear}e represents an artificial termination
of the simulation. At that point, two criteria had been fulfilled. First, the scale
height had become smaller than the Hill radius of the planet, so gap formation
should have occurred. This is not sufficient to terminate the simulation, because
the inward motion of the equilibrium radius itself occurs at the timescale of
Type II migration. However, the second criterion was that the mass parameter determining Type
II migration, $\varSigma{r^2}$ (\citealp{Mordasini}), had become smaller than the
planet's mass. At this stage, Type II migration becomes planet dominated and we consider
that it comes to a halt. The smaller planets never carve gaps.
We investigate migration in disks with different values of $\dot{M}_{\rm w}$ and $\dot{M_0}$,
yet constrained by a lifetime of 1-10\,Myr, without finding qualitative differences.
A change in migration behavior is only seen for hotter disks. For
$\alpha_{\rm SS}$=0.1, the disks show $\varTheta \approx 1$
throughout. Migration then shows a mix of isothermal and adiabatic behavior,
being mostly inwards yet with equilibrium points present. The torque only becomes
isothermally dominated for $\alpha_{\rm SS}\simeq$1, which yields an unrealistically
high accretion rate.
\section{Conclusions}
\label{sect:conclusions}
In this Letter, we examine the trajectories of planets undergoing
Type I migration in evolving, radiative disks with initially nearly
adiabatic midplanes, where outward migration can occur.
Planets migrate toward equilibrium radii, where the torque
acting on them vanishes. These radii correspond to opacity
jumps and to the transition, in the outer disk, to an isothermal state.
Because of viscous accretion and photoevaporation, these equilibrium radii
themselves move inwards on disk accretion timescales.
As long as the torques are strong enough to keep the planets coupled
to disk evolution, the planets migrate in lockstep with the gas at the
accretion timescale. However, as the disk surface density drops, the
timescales of orbital migration and disk accretion eventually become
comparable. At this stage, if the planet is perturbed from an inner
equilibrium radius, the equilibrium radius moves inward faster than
the torques can return the planet, so it decouples.
If the continuing migration is outwards, the planet encounters another
equilibrium radius, and the same process of locking and decoupling
occurs. The outermost equilibrium radius lies where the disk reaches
the background temperature, and from there the planet can only migrate
inwards. However, by the time that the planet decouples even from that
equilibrium radius, the disk is already so severely depleted that the
ensuing inward migration is feeble, and soon comes to a halt, as the
remaining disk mass cannot exert a substantial torque. In no
case did a planet released beyond 1\,AU migrate all the way to the
star.
We stress that we only use a single model for the opacities,
which may change as planet formation progresses. The dust size
distribution must depend on the balance between coagulation and
fragmentation, which remains poorly understood. We also neglect
stellar irradiation, which will become important in the late stages of
the disk evolution. Irradiation will maintain high temperatures in the
inner disk ($\approx$100\,K; \citealp{Chambers}), preventing gap
formation for 10\mearthp planets. On the other hand, irradiation should
lead to a hole-forming photoevaporating wind (\citealp{Clarke,Alexander}),
which quickly depletes the disk, thus possibly
bringing migration to an even earlier halt. Future work should
self-consistently address these issues.
As a consequence of the independence of equilibrium radii on
planet mass, all planets migrate to these equilibrium
locations. Ensembles of planets reaching them may become
violently unstable due to $N$-body interactions. Nevertheless, even if
scattered away, migration will invariably drive the planets back toward
these radii. The final outcome may well be collisions driving
further planet growth, aiding rapid giant planet formation
or forming planets in 1:1 resonance.
If this is the case, however, it raises the question of why the solar
system has a set of neatly spaced planets as opposed to only two, as
the two equilibrium radii of the model might naively suggest. One
possible solution is that $\Gamma$ (and thus any equilibrium radius)
shows a dependence on the planet-to-star mass ratio $q$ at the verge
of gap opening when $q\approx{h^3}$ (\citealp{Masset}). Another is
that we only consider the fully unsaturated torque, whereas saturation
depends on the width of the horseshoe region and therefore on the
planet's mass. However, the level of saturation in radiative disks is
not fully understood at present, and we cannot easily add it to our
study. Future models should include effects of saturation to study possible mass segregation.
Finally, in view of the long migration timescales for
$M\lesssim{0.1}$\mearth, such planets may just not have the time to
migrate back to the equilibrium location before the disk vanishes if
scattered far enough. This scattered population of small planets
could provide the initial conditions for the terrestrial planets
of our own solar system.
Our results provide qualitative and quantitative justification for
the reduction of Type I migration rates assumed in planetary population
synthesis models (\citealp[e.g.,][]{Alibert,Ida,Mordasini}).
Instead of migrating on the fast, mass-dependent,
timescale $t_{\rm mig}$, we find that planets spend their first Myr near
equilibrium radii that change only on the slow accretion timescale
$t_{\nu}$. We show in \Figure{fig:timescales} that $t_{\rm mig}/t_{\nu}\sim0.1$
for a 1\mearthp planet during most of the evolution of the disk. Examining
the same figure for different masses shows a linear dependence on mass, $t_{\rm mig} /
t_{\nu} \sim 0.1 (M / \mbox{\mearth})$, consistent with the population
synthesis assumptions.
\acknowledgments W.L. is partly supported by a Kalbfleisch Research
Fellowship from the AMNH. M.-M.M.L. is partly supported by NSF CDI grant AST-0835734,
and NASA OSS grant NNX07AI74G. S.-J.P. is an STFC postdoctoral fellow. We acknowledge useful
discussions with C. Mordasini and T. Birnstiel.
| 2024-02-18T23:39:58.062Z | 2010-04-14T02:00:50.000Z | algebraic_stack_train_0000 | 964 | 4,215 |
|
proofpile-arXiv_065-5173 | \section{Introduction}
An invariant theory of surfaces in the four-dimensional Euclidean
space $\mathbb R^4$ was developed in \cite{GM1} and \cite{GM3} on the
base of the Weingarten map similarly to the classical theory of
surfaces in $\mathbb R^3$.
Let $M^2$ be a surface in $\mathbb R^4$ with tangent space $T_pM^2$ at
any point $p \in M^2$. In \cite{GM1} we introduced an invariant
linear map $\gamma$ of Weingarten-type at any $T_pM^2$, which
plays a similar role in the theory of surfaces in $\mathbb R^4$ as the
Weingarten map in the theory of surfaces in $\mathbb R^3$. This map
$\gamma$ generates two invariant functions $k$ and $\varkappa$,
analogous to the Gauss curvature and the mean curvature in $\mathbb R^3$.
The sign of $k$ is a geometric invariant and the sign of
$\varkappa$ is invariant under the motions in $\mathbb R^4$. However, the
sign of $\varkappa$ changes under symmetries with respect to a
hyperplane in $\mathbb R^4$. The invariant $\varkappa$ is the curvature
of the normal connection of the surface $M^2$ in $\mathbb R^4$.
As in the classical case, the invariants $k$ and $\varkappa$
divide the points of $M^2$ into four types: flat, elliptic,
parabolic and hyperbolic. The surfaces consisting of flat points
satisfy the conditions $k=0, \,\, \varkappa=0$. These surfaces are
either planar surfaces ($M^2$ lies in a hyperplane $\mathbb R^3$ in
$\mathbb R^4$) or developable ruled surfaces in $\mathbb R^4$ \cite{GM1}.
The map $\gamma$ generates the second fundamental form at any
point $p \in M^2$. The first fundamental form and the second
fundamental form generate principal tangents and principal lines,
as in $\mathbb R^3$. Using the principal tangents, we obtained a
geometrically determined moving frame field at each point $p \in
M^2$. Writing derivative formulas of Frenet-type for this frame
field, we found eight invariant functions. In \cite{GM3} we proved
a fundamental theorem of Bonnet-type for surfaces without minimal
points, stating that these eight invariants determine the surface
up to a motion.
The minimal surfaces in $\mathbb R^4$ are characterized by the equality
$\varkappa^2 - k = 0.$ We proved in \cite{GM2} that on any minimal
surface $M^2$ the Gauss curvature $K$ and the normal curvature
$\varkappa$ satisfy the following inequality $K^2-\varkappa^2\geq
0.$ This inequality generates two geometric classes of minimal
surfaces:
\begin{itemize}
\item the class of minimal super-conformal surfaces characterized by
$K^2 - \varkappa^2 =0$; \item the class of minimal surfaces of
general type characterized by $K^2-\varkappa^2>0$.
\end{itemize}
The class of minimal super-conformal surfaces in $\mathbb R^4$ is locally
equivalent to the class of holomorphic curves in $\mathbb C^2 \equiv
\mathbb R^4$ \cite{Ein}.
A fundamental theorem of Bonnet-type for strongly regular minimal
surfaces of general type was proved in \cite{GM2} in terms of four
invariant functions.
In the present paper we apply the invariant theory of surfaces on
a special class of surfaces in $\mathbb R^4$, which are general
rotational surfaces in the sense of C. Moore.
\section{Preliminaries}
Let $M^2: z = z(u,v), \, \, (u,v) \in {\mathcal D}$ (${\mathcal D}
\subset \mathbb R^2$) be a surface in $\mathbb R^4$ with tangent space $T_pM^2 =
{\rm span} \{z_u, z_v\}$ at an arbitrary point $p \in M^2$. We
choose an orthonormal normal frame field $\{e_1, e_2\}$ of $M^2$
so that the quadruple $\{z_u, z_v, e_1, e_2\}$ is positive
oriented in $\mathbb R^4$. Then the following derivative formulas hold:
$$\begin{array}{l}
\vspace{2mm} \nabla'_{z_u}z_u=z_{uu} = \Gamma_{11}^1 \, z_u +
\Gamma_{11}^2 \, z_v
+ c_{11}^1\, e_1 + c_{11}^2\, e_2,\\
\vspace{2mm} \nabla'_{z_u}z_v=z_{uv} = \Gamma_{12}^1 \, z_u +
\Gamma_{12}^2 \, z_v
+ c_{12}^1\, e_1 + c_{12}^2\, e_2,\\
\vspace{2mm} \nabla'_{z_v}z_v=z_{vv} = \Gamma_{22}^1 \, z_u +
\Gamma_{22}^2 \, z_v
+ c_{22}^1\, e_1 + c_{22}^2\, e_2,\\
\end{array}$$
where $\Gamma_{ij}^k$ are the Christoffel's symbols and $c_{ij}^k,
\,\, i,j,k = 1,2$ are functions on $M^2$.
Let $g$ be the standard metric in $\mathbb R^4$ and $\nabla'$ its flat
Levi-Civita connection. We use the standard denotations
\;$E=g(z_u,z_u), \; F=g(z_u,z_v), \; G=g(z_v,z_v)$ for the
coefficients of the first fundamental form and set
$W=\sqrt{EG-F^2}$. Denoting by $\sigma$ the second fundamental
tensor of $M^2$, we have
$$\begin{array}{l}
\sigma(z_u,z_u)=c_{11}^1\, e_1 + c_{11}^2\, e_2,\\
[2mm]
\sigma(z_u,z_v)=c_{12}^1\, e_1 + c_{12}^2\, e_2,\\
[2mm] \sigma(z_v,z_v)=c_{22}^1\, e_1 + c_{22}^2\,
e_2.\end{array}$$
The three pairs of normal vectors $\{\sigma(z_u,z_u),
\sigma(z_u,z_v)\}$, $\{\sigma(z_u,z_u), \sigma(z_v,z_v)\}$,
$\{\sigma(z_u,z_v), \sigma(z_v,z_v)\}$ form three parallelograms
with oriented areas
$$\Delta_1 = \left|%
\begin{array}{cc}
\vspace{2mm}
c_{11}^1 & c_{12}^1 \\
c_{11}^2 & c_{12}^2 \\
\end{array}%
\right|, \quad
\Delta_2 = \left|%
\begin{array}{cc}
\vspace{2mm}
c_{11}^1 & c_{22}^1 \\
c_{11}^2 & c_{22}^2 \\
\end{array}%
\right|, \quad
\Delta_3 = \left|%
\begin{array}{cc}
\vspace{2mm}
c_{12}^1 & c_{22}^1 \\
c_{12}^2 & c_{22}^2 \\
\end{array}%
\right|,$$ respectively. These oriented areas determine three
functions
$$\displaystyle{L = \frac{2 \Delta_1}{W}, \quad M = \frac{ \Delta_2}{W}, \quad N = \frac{2 \Delta_3}{W}},$$
which change in the same way as the coefficients $E, F, G$ under
any change of the parameters $(u,v)$.
Using the functions $L$, $M$, $N$ and $E$, $F$, $G$ we introduce
the linear map $\gamma$ in the tangent space at any point of $M^2$
$$\gamma: T_pM^2 \rightarrow T_pM^2$$
similarly to the theory of surfaces in $\mathbb R^3$.
The linear map $\gamma$ of Weingarten-type is invariant with
respect to changes of parameters on $M^2$ as well as to motions in
$\mathbb R^4$. Thus the functions
$$k = \frac{LN - M^2}{EG - F^2}, \qquad
\varkappa =\frac{EN+GL-2FM}{2(EG-F^2)}$$ are invariants of the
surface $M^2$.
The map $\gamma$ determines a second fundamental form of the
surface as follows. Let $X = \alpha z_u + \beta z_v, \,\,
(\alpha,\beta) \neq (0,0)$ be a tangent vector at a point $p \in
M^2$. The second fundamental form of $M^2$ at $p$ is defined by
$$II(\alpha,\beta) = - g(\gamma(X),X) = L\alpha^2 + 2M\alpha\beta + N\beta^2, \quad \alpha,\beta \in \mathbb R.$$
The notions of a normal curvature of a tangent, conjugate and
asymptotic tangents are introduced in the standard way by means of
$II$. The asymptotic tangents are characterized by zero normal
curvature.
A tangent $g: X = \alpha z_u + \beta z_v$ is said to be
\textit{principal} if it is perpendicular to its conjugate. A line
$c: u=u(q), \; v=v(q); \; q\in J \subset \mathbb R$ on $M^2$ is said to
be a \textit{principal line} if its tangent at any point is
principal. The surface $M^2$ is parameterized with respect to the
principal lines if and only if $F=0, \,\, M=0.$
Let $M^2$ be parameterized with respect to the principal lines and
denote the unit vector fields $\displaystyle{x=\frac{z_u}{\sqrt
E}, \; y=\frac{z_v}{\sqrt G}}$. The equality $M = 0$ implies that
the normal vector fields $\sigma(x,x)$ and $\sigma(y,y)$ are
collinear. We denote by $b$ a unit normal vector field collinear
with $\sigma(x,x)$ and $\sigma(y,y)$, and by $l$ the unit normal
vector field such that $\{x,y,b,l\}$ is a positive oriented
orthonormal frame field of $M^2$. Thus we obtain a geometrically
determined orthonormal frame field $\{x,y,b,l\}$ at each point $p
\in M^2$. With respect to the frame field $\{x,y,b,l\}$ we have
the following Frenet-type formulas:
$$\begin{array}{ll}
\vspace{2mm} \nabla'_xx=\quad \quad \quad \gamma_1\,y+\,\nu_1\,b;
& \qquad
\nabla'_xb=-\nu_1\,x-\lambda\,y\quad\quad \quad +\beta_1\,l;\\
\vspace{2mm} \nabla'_xy=-\gamma_1\,x\quad \quad \; + \; \lambda\,b
\; + \mu\,l; & \qquad
\nabla'_yb=-\lambda\,x - \; \nu_2\,y\quad\quad \quad +\beta_2\,l;\\
\vspace{2mm} \nabla'_yx=\quad\quad \;-\gamma_2\,y \; + \lambda\,b
\; +\mu\,l; & \qquad
\nabla'_xl= \quad \quad \quad \;-\mu\,y-\beta_1\,b;\\
\vspace{2mm} \nabla'_yy=\;\;\gamma_2\,x \quad\quad\quad+\nu_2\,b;
& \qquad \nabla'_yl=-\mu\,x \quad \quad \quad \;-\beta_2\,b,
\end{array}\leqno{(2.1)}$$
where $\gamma_1, \gamma_2, \nu_1, \nu_2, \lambda, \mu, \beta_1,
\beta_2$ are geometric invariant functions.
The invariants $k$, $\varkappa$, and the Gauss curvature $K$ are
expressed by the functions $\nu_1$, $\nu_2$, $\lambda$, $\mu$ as
follows:
$$k = - 4 \nu_1 \nu_2 \mu^2, \qquad \varkappa = (\nu_1 - \nu_2) \mu, \qquad K = \nu_1 \nu_2 - (\lambda^2 + \mu^2). \leqno{(2.2)}$$
\vskip 5mm
\section{General rotational surfaces}
Considering general rotations in $\mathbb R^4$, C. Moore introduced
general rotational surfaces \cite{M} \, (see also \cite{MW1, MW2})
as follows. Let $c: x(u) = \left( x^1(u), x^2(u), x^3(u),
x^4(u)\right); \,\, u \in J \subset \mathbb R$ be a smooth curve in
$\mathbb R^4$, and $\alpha$, $\beta$ are constants. A general rotation of
the meridian curve $c$ in $\mathbb R^4$ is given by
$$X(u,v)= \left( X^1(u,v), X^2(u,v), X^3(u,v), X^4(u,v)\right),$$
where
$$\begin{array}{ll}
\vspace{2mm} X^1(u,v) = x^1(u)\cos\alpha v - x^2(u)\sin\alpha v; &
\qquad
X^3(u,v) = x^3(u)\cos\beta v - x^4(u)\sin\beta v; \\
\vspace{2mm} X^2(u,v) = x^1(u)\sin\alpha v + x^2(u)\cos\alpha v;&
\qquad X^4(u,v) = x^3(u)\sin\beta v + x^4(u)\cos\beta v.
\end{array}$$
In the case $\beta = 0$ the $X^3X^4$-plane is fixed and one gets
the classical rotation about a fixed two-dimensional axis.
\vskip 2mm We consider a surface $\mathcal{M}^2$ in $\mathbb R^4$,
defined by the vector-valued function
$$z(u,v) = \left( f(u) \cos\alpha v, f(u) \sin \alpha v, g(u) \cos \beta v, g(u) \sin \beta v \right);
\quad u \in J \subset \mathbb R, \,\, v \in [0; 2\pi), \leqno{(3.1)}$$
where $f(u)$ and $g(u)$ are smooth functions, satisfying $\alpha^2
f^2(u)+ \beta^2 g^2(u) > 0$, $f'\,^2(u)+ g'\,^2(u) > 0$, $u \in
J,$ and $\alpha, \beta$ are positive constants. The surface
$\mathcal{M}^2$, given by (3.1) is a general rotational surface
whose meridians lie in two-dimensional planes. In our case the
meridian is given by $m: x(u) = \left( f(u), 0, g(u), 0\right);
\,\, u \in J \subset \mathbb R$.
Each parametric curve $u = u_0 = const$ of $\mathcal{M}^2$ is
given by
$$c_v: z(v) = \left( a \cos \alpha v, a \sin \alpha v, b \cos \beta v, b \sin \beta v \right);
\quad a = f(u_0), \,\, b = g(u_0)$$ and its Frenet curvatures are
$$\varkappa_{c_v} = \displaystyle{\sqrt{\frac{a^2 \alpha^4 + b^2 \beta^4}{a^2 \alpha^2 + b^2 \beta^2}}}; \quad
\tau_{c_v} = \displaystyle{\frac{ab \alpha \beta (\alpha^2 - \beta^2)}
{\sqrt{a^2 \alpha^4 + b^2 \beta^4}\sqrt{a^2 \alpha^2 + b^2
\beta^2}}}; \quad \sigma_{c_v} = \displaystyle{\frac{\alpha \beta \sqrt{a^2
\alpha^2 + b^2 \beta^2}} {\sqrt{a^2 \alpha^4 + b^2 \beta^4}}}.$$
Hence, in case of $\alpha \neq \beta$ each parametric $v$-line is
a curve in $\mathbb R^4$ with constant curvatures (helix in $\mathbb R^4$
\cite{CDV}), and in case of $\alpha = \beta$ each parametric
$v$-line is a circle. We shall consider the case $\alpha \neq
\beta$.
Each parametric curve $v = v_0 = const$ of $\mathcal{M}^2$ is
given by
$$c_u: z(u) = \left(\, A_1 f(u), A_2 f(u), B_1 g(u), B_2 g(u) \, \right),$$
where $A_1 = \cos \alpha v_0, \, A_2 = \sin \alpha v_0, \,B_1 =
\cos \beta v_0, \,B_2 = \sin \beta v_0$. The Frenet curvatures of
$c_u$ are expressed as follows:
$$\varkappa_{c_u} = \displaystyle{\frac{|g' f'' - f' g''|}{(\sqrt{f'\,^2 + g'\,^2})^3}}; \quad \tau_{c_u} = 0.$$
Obviously, $c_u$ is a plane curve with curvature $\varkappa_{c_u}
= \displaystyle{\frac{|g' f'' - f' g''|} {(\sqrt{f'\,^2 + g'\,^2})^3}}$. So,
for each $v = const$ the parametric curves $c_u$ are congruent in
$\mathbb R^4$. These curves are the \textit{meridians} of
$\mathcal{M}^2$.
We shall call the surface, defined by (3.1) in the case $\alpha \neq \beta$, a \emph{general rotational surface}.
\vskip 2mm Calculating the tangent vector fields $z_u$ and $z_v$
we find the coefficients of the first fundamental form: $E =
f'\,^2(u)+ g'\,^2(u)$; $F = 0$; $G = \alpha^2 f^2(u)+ \beta^2
g^2(u).$ We consider the following orthonormal tangent frame field
$$\begin{array}{l}
\vspace{2mm}
x = \displaystyle{\frac{1}{\sqrt{f'\,^2 + g'\,^2}}\left(f' \cos \alpha v, f' \sin \alpha v, g' \cos \beta v, g' \sin \beta v \right)};\\
\vspace{2mm} y = \displaystyle{\frac{1}{\sqrt{\alpha^2 f^2 + \beta^2
g^2}}\left( - \alpha f \sin \alpha v, \alpha f \cos \alpha v, -
\beta g \sin \beta v, \beta g \cos \beta v \right)};
\end{array}$$
and the following orthonormal normal frame field
$$\begin{array}{l}
\vspace{2mm}
n_1 = \displaystyle{\frac{1}{\sqrt{f'\,^2 + g'\,^2}}\left(g' \cos \alpha v, g' \sin \alpha v, - f' \cos \beta v, - f' \sin \beta v \right)};\\
\vspace{2mm} n_2 = \displaystyle{\frac{1}{\sqrt{\alpha^2 f^2 + \beta^2
g^2}}\left( - \beta g \sin \alpha v, \beta g \cos \alpha v, \alpha
f \sin \beta v, - \alpha f \cos \beta v \right)}.
\end{array}$$
It is easy to verify that $\{x, y, n_1, n_2\}$ is a positive
oriented orthonormal frame field in $\mathbb R^4$.
Finding the second partial derivatives $z_{uu}$, $z_{uv}$,
$z_{vv}$, we calculate the functions $c_{ij}^k, \,\, i,j,k = 1,2$:
$$\begin{array}{ll}
\vspace{2mm} c_{11}^1 = g(z_{uu}, n_1) = \displaystyle{\frac{g' f'' - f'
g''}{\sqrt{f'\,^2 + g'\,^2}}};
\quad \quad & c_{11}^2 =g(z_{uu}, n_2) = 0;\\
\vspace{2mm} c_{12}^1 = g(z_{uv}, n_1) = 0;
\quad \quad & c_{12}^2 = g(z_{uv}, n_2) = \displaystyle{\frac{\alpha \beta (g f' - f g')}{\sqrt{\alpha^2 f^2 + \beta^2 g^2}}};\\
\vspace{2mm} c_{22}^1 = g(z_{vv}, n_1) = \displaystyle{\frac{\beta^2 g f' -
\alpha^2 f g'}{\sqrt{f'\,^2 + g'\,^2}}}; \quad \quad & c_{22}^2 =
g(z_{vv}, n_2) = 0.
\end{array} \leqno{(3.2)}$$
Therefore the coefficients $L$, $M$ and $N$ of the second
fundamental form of $\mathcal{M}^2$ are expressed as follows:
$$L = \displaystyle{\frac{2 \alpha \beta (g f' - f g') (g' f'' - f' g'')}{(\alpha^2 f^2 + \beta^2 g^2) (f'\,^2 + g'\,^2)}}; \qquad M = 0; \qquad
N = \displaystyle{\frac{- 2\alpha \beta (g f' - f g') (\beta^2 g f' -
\alpha^2 f g')}{(\alpha^2 f^2 + \beta^2 g^2) (f'\,^2 +
g'\,^2)}}.$$
Consequently, the invariants $k$, $\varkappa$ and $K$ of
$\mathcal{M}^2$ are:
$$k = \displaystyle{\frac{- 4 \alpha^2 \beta^2 (g f' - f g')^2 (g' f'' - f' g'') (\beta^2 g f' - \alpha^2 f g')}{(\alpha^2 f^2 + \beta^2 g^2)^3 (f'\,^2 + g'\,^2)^3}};$$
$$\varkappa = \displaystyle{\frac{\alpha \beta (g f' - f g')}{(\alpha^2 f^2 + \beta^2 g^2)^2 (f'\,^2 + g'\,^2)^2} \,
[(\alpha^2 f^2 + \beta^2 g^2)(g' f'' - f' g'') - (f'\,^2 +
g'\,^2) (\beta^2 g f' - \alpha^2 f g') ]};$$
$$K = \displaystyle{\frac{(\alpha^2 f^2 + \beta^2 g^2)(\beta^2 g f' - \alpha^2 f g')(g' f'' - f' g'') - \alpha^2 \beta^2 (f'\,^2 + g'\,^2) (g f' - f g')^2}{(\alpha^2 f^2 + \beta^2 g^2)^2 (f'\,^2 + g'\,^2)^2} \,}.$$
In \cite{GM3} we found all general rotational surfaces consisting
of parabolic points, i.e. $k = 0$.
\vskip 2mm Now we shall apply the invariant theory of surfaces in
$\mathbb R^4$ finding the geometric invariant functions $\gamma_1,
\gamma_2, \nu_1, \nu_2, \lambda, \mu, \beta_1, \beta_2$ in the
Frenet-type formulas of $\mathcal{M}^2$.
The positive oriented orthonormal frame field $\{x, y, n_1, n_2\}$
defined above is the geometric frame field of $\mathcal{M}^2$, the
$u$-lines and $v$-lines of $\mathcal{M}^2$ are principal lines.
Let $\sigma$ be the second fundamental tensor of $\mathcal{M}^2$.
Using (3.2) we obtain
$$\begin{array}{l}
\vspace{2mm}
\sigma(x, x) = \displaystyle{\frac{g' f'' - f' g''}{\left(\sqrt{f'\,^2 + g'\,^2}\right)^3}\,\, n_1};\\
\vspace{2mm}
\sigma(x, y) = \displaystyle{\frac{\alpha \beta (g f' - f g')}{\sqrt{f'\,^2 + g'\,^2}(\alpha^2 f^2 + \beta^2 g^2)}\,\, n_2};\\
\vspace{2mm} \sigma(y, y) = \displaystyle{\frac{\beta^2 g f' - \alpha^2 f
g'}{\sqrt{f'\,^2 + g'\,^2}(\alpha^2 f^2 + \beta^2 g^2)}\,\, n_1}.
\end{array}$$
The partial derivatives of the normal vector field $n_1$ are:
$$\begin{array}{l}
\vspace{2mm}
(n_1)_u = \displaystyle{\frac{f' g'' - g' f''}{\left(\sqrt{f'\,^2 + g'\,^2}\right)^3}\left(f' \cos \alpha v, f' \sin \alpha v, g' \cos \beta v, g' \sin \beta v \right)};\\
\vspace{2mm} (n_1)v = \displaystyle{\frac{1}{\sqrt{f'\,^2 + g'\,^2}}\left( -
\alpha g' \sin \alpha v, \alpha g' \cos \alpha v, \beta f' \sin
\beta v, - \beta f' \cos \beta v \right)}.
\end{array}$$
The last equalities imply
$$\langle (n_1)_u, n_2 \rangle = 0; \qquad \langle (n_1)_v, n_2 \rangle = \displaystyle{\frac{\alpha \beta
(f f' + g g')}{\sqrt{f'\,^2 + g'\,^2}\sqrt{\alpha^2 f^2 + \beta^2
g^2}}}.$$
So, the invariants in the Frenet-type derivative formulas (2.1)
for the surface $\mathcal{M}^2$ are given by
$$\begin{array}{ll}
\vspace{2mm}
\gamma_1 = 0; & \qquad \nu_1 = \displaystyle{\frac{g' f'' - f' g''}{(f'\,^2 + g'\,^2)^{\frac{3}{2}}}};\\
\vspace{2mm} \gamma_2 = \displaystyle{- \frac{\alpha^2 f f' + \beta^2 g
g'}{\sqrt{f'\,^2 + g'\,^2}(\alpha^2 f^2 + \beta^2 g^2)}}; & \qquad
\nu_2 = \displaystyle{\frac{\beta^2 g f' - \alpha^2 f g'}{\sqrt{f'\,^2 + g'\,^2}(\alpha^2 f^2 + \beta^2 g^2)}};\\
\vspace{2mm}
\lambda = 0; & \qquad \beta_1 = 0;\\
\vspace{2mm} \mu = \displaystyle{\frac{\alpha \beta (g f' - f
g')}{\sqrt{f'\,^2 + g'\,^2}(\alpha^2 f^2 + \beta^2 g^2)}}; &
\qquad \beta_2 = \displaystyle{\frac{\alpha \beta (f f' + g
g')}{\sqrt{f'\,^2 + g'\,^2}\sqrt{\alpha^2 f^2 + \beta^2 g^2} }}.
\end{array} \leqno{(3.3)}$$
Thus we obtain that for the class of general rotational surfaces
only five invariants from the general theory are essential (they
may not be zero). These five invariant functions determine the
general rotational surfaces up to a motion in $\mathbb R^4$.
\vskip 5mm
\section{Minimal super-conformal general rotational surfaces}
In this section we shall find all minimal super-conformal general
rotational surfaces.
\vskip 2mm
Let us recall that the \textit{ellipse of normal curvature} at a
point $p$ of a surface $M^2$ in $\mathbb R^4$ is the ellipse in the
normal space at the point $p$ given by $\{\sigma(x,x): \, x \in
T_pM^2, \, g(x,x) = 1\}$ \cite{MW1, MW2}. Let $\{x,y\}$ be an
orthonormal base of the tangent space $T_pM^2$ at $p$. Then, for
any $v = \cos \psi \, x + \sin \psi \, y$, we have
$$\sigma(v, v) = H + \displaystyle{\cos 2\psi \, \frac{\sigma(x,x) - \sigma(y,y)}{2}
+ \sin 2 \psi \, \sigma(x,y)},$$ where $H = \displaystyle{\frac{\sigma(x,x)
+ \sigma(y,y)}{2}}$ \, is the mean curvature vector of $M^2$ at
$p$. So, when $v$ goes once around the unit tangent circle, the
vector $\sigma(v,v)$ goes twice around the ellipse centered at
$H$. Obviously, $M^2$ is minimal if and only if for each point $p
\in M^2$ the ellipse of curvature is centered at $p$.
A surface $M^2$ in $\mathbb R^4$ is called \textit{super-conformal}
\cite{BFLPP} if at any point of $M^2$ the ellipse of normal
curvature is a circle. An explicit construction of any simply
connected super-conformal surface in $\mathbb R^4$ that is free of
minimal and flat points was given in \cite{DT}.
\vskip 2mm
In \cite{GM2} it was proved that the class of minimal
super-conformal surfaces is characterized by the equalities
$$ \varkappa^2 - k = 0; \qquad K^2 - \varkappa^2 = 0. \leqno{(4.1)}$$
Without loss of generality we assume that the meridian $m$ is
given by
$$\begin{array}{ll}
\vspace{1mm}
f = u;\\
\vspace{1mm} g = g(u).
\end{array}$$
Then, from (3.3) we obtain that the functions $\nu_1$, $\nu_2$,
$\mu$ are expressed by the function $g(u)$ and its derivatives as
follows:
$$\nu_1 = \displaystyle{\frac{- g''}{(1 + g'\,^2)^{\frac{3}{2}}}}; \quad
\nu_2 = \displaystyle{\frac{\beta^2 g - \alpha^2 u g'}{\sqrt{1 +
g'\,^2}(\alpha^2 u^2 + \beta^2 g^2)}}; \quad
\mu = \displaystyle{\frac{\alpha \beta (g - ug')}{\sqrt{1 + g'\,^2}(\alpha^2 u^2 + \beta^2 g^2)}}. \leqno{(4.2)}$$
Now, using (2.2), (4.1) and (4.2) we get that the general
rotational surface $\mathcal{M}^2$, given by (3.1), is minimal
super-conformal if and only if $g(u)$ satisfies the equation:
$$\alpha \beta (g - u g') = \varepsilon (\alpha^2 u g' - \beta^2 g), \qquad \varepsilon = \pm 1. \leqno{(4.3)}$$
All solutions of equation (4.3) are given by
$$g(u) = \displaystyle{c\,u^{\varepsilon \frac{\beta}{\alpha}}},\qquad c = const.$$
Thus we obtained that the class of all minimal super-conformal
general rotational surfaces is described as follows:
\begin{prop}
Let $\mathcal{M}^2$ be a general rotational surface in $\mathbb R^4$.
Then $\mathcal{M}^2$ is minimal super-conformal if and only if
the meridian is determined by
$$g(u) = c\,u^{\varepsilon \frac{\beta}{\alpha}},\qquad c = const.$$
\end{prop}
\vskip 3mm Denoting by $k = \varepsilon \displaystyle{\frac{\beta}{\alpha}}$
\, ($k \neq \pm 1$), we obtain that all minimal super-conformal
general rotational surfaces are given by
$$\mathcal{M}^2: z(u,v) = \left( u \cos\alpha v,\, u \sin \alpha v, \,c\, u^k \cos \beta v,\, c\, u^k \sin \beta v \right).$$
The invariants $k$, $\varkappa$ and $K$ of $\mathcal{M}^2$ are
expressed as follows:
$$k = 4 \displaystyle{\frac{c^4 k^4(1-k)^4 u^{4(k-2)}}{\left(1 + c^2 k^2\,u^{2(k-1)}\right)^6}}; \quad
\varkappa = 2\, \varepsilon \displaystyle{\frac{c^2 k^2(1-k)^2
u^{2(k-2)}}{\left(1 + c^2 k^2\,u^{2(k-1)}\right)^3}}; \quad K = -
2 \displaystyle{\frac{c^2 k^2(1-k)^2 u^{2(k-2)}}{\left(1 + c^2
k^2\,u^{2(k-1)}\right)^3}}.$$
\vskip 5mm
| 2024-02-18T23:40:00.035Z | 2010-03-02T11:38:54.000Z | algebraic_stack_train_0000 | 1,051 | 3,922 |
|
proofpile-arXiv_065-5213 | \section{Introduction}
It is not feasible anymore to expect performance gains for sequential codes by means of continuously
increasing processor clock speeds. Nowadays, processor vendors have been
concentrated on developing systems that group two or more processors onto a
single socket, sharing or not the same memory resources. This technology, called
\textit{multi-core}, has been successfully employed to different application
domains ranging from computer graphics to scientific computing, and in these
times it is commonly seen on high performance clusters, desktop
computers, notebooks, and even mobile devices. The spread of such architecture
has consequently stimulated an increasing number of researches on parallel
algorithms.
To obtain efficient implementations of parallel algorithms, one must consider
the underlying architecture on which the program is supposed to be run. In
fact, even processors belonging to the multi-core family may present different
hardware layouts, which can make an implementation to perform poorly on one
platform, while running fast on another. As an example of such issue,
multi-core processors may have different memory subsystems for each core, therefore
forcing programmers to take care of thread and memory affinity.
The finite element method is usually the first choice for numerically solving
integral and partial differential equations. Matrices arising from finite
element discretizations are usually sparse, i.e., most of its entries are
zeros. Effectively storing sparse matrices requires the use of compressed data
structures. Commonly employed approaches are the \textit{element-based},
\textit{edge-based} and \textit{compressed} data
structures. Since the later provides the best compromise between space
complexity and performance \cite{RC05a}, it was chosen as the primary data
structure of our implementation.
The \textit{compressed sparse row} (CSR) data structure stores contiguously in memory non-zero entries belonging
to the same row of a matrix. While in a dense representation any
element can be randomly accessed through the use of its row and
column indices, the CSR explicitly stores in memory the combinatorial
information for every non-zero entry. Given an $n \times n$ matrix $A$ with $nnz$
non-zero coefficients, the standard version of the CSR \cite{Saa95a} consists
of three arrays: two integer arrays $ia(n+1)$ and $ja(nnz)$ for storing
combinatorial data, and one floating-point array $a(nnz)$ containing the
non-zero coefficients. The value $ia(i)$ points to the first element of row $i$
in the $a$ array, i.e., row $i$ is defined as the subset of $a$ starting and
ending at $ia(i)$ and $ia(i+1)-1$, respectively. The column index of each
non-zero entry is stored in $ja$. There is also a transpose version,
called \textit{compressed sparse column} (CSC) format.
This representation supports matrices of arbitrary shapes and symmetry
properties. In the context of the finite element method, however, the
generality provided by the CSR is underused as most matrices are structurally symmetric.
In this case, it would be sufficient to store, roughly, half
of the matrix connectivity. The \textit{compressed sparse row-column} (CSRC)
format was designed to take benefit from this fact \cite{RF07a}. Basically, it
stores the column indices for only half of the off-diagonal entries. As the
working set size has a great impact on the performance of CSR-like data
structures, the running time of algorithms such as the matrix-vector product is expected
to be improved when using the CSRC. Also, solvers based on oblique projection methods can efficiently
access the transpose matrix, since it is implicitly defined.
The performance of finite element codes using iterative
solvers is dominated by the computations associated with the matrix-vector
multiplication algorithm. In this algorithm, we are given an $n \times n$
sparse matrix $A$ containing $nnz$ non-zeros, and a dense $n$-vector $x$,
called the \textit{source} vector. The output is an $n$-vector $y$, termed the
\textit{destination} vector, which stores the result of the $Ax$ operation.
Performing this operation using the CSR format is trivial, but it was observed
that the maximum performance in Mflop/s sustained by a na\"ive implementation
can reach only a small part of the machine peak performance \cite{GKKS99a}. As
a means of transcending this limit, several optimization techniques have been
proposed, such as reordering \cite{Tol97a,PH99a,WS97a,TJ92a}, data compression \cite{MGMM05a,WL06a},
blocking \cite{TJ92a,IYV04a,Tol97a,VM05a,PH99a,AGZ92a,NVDY07a}, vectorization \cite{AFM05a,BHZ93a}, loop unrolling
\cite{WS97a} and jamming \cite{MG04a}, and software prefetching \cite{Tol97a}.
Lately, the dissemination of multi-core computers have promoted
multi-threading as an important tuning
technique, which can be further combined with purely sequential methods.
\subsection{Related work}
Parallel sparse matrix-vector multiplication using CSR-like data structures on
multi-processed machines has been the focus of a number of researchers since
the 1990s. Early attempts to date include the paper by {\c C}ataly\"urek and Aykanat \cite{CA96a}, on
hypergraph models applied to the matrix partitioning problem, Im and Yelick \cite{IY99a},
who analysed the effect of register/cache blocking and reordering, and
Geus and R{\"{o}}llin \cite{GR01a}, considering prefetching, register blocking and reordering for
symmetric matrices. Kotakemori et al.~\cite{KHKNSN05a} also examined several storage formats on
a ccNUMA machine, which required the ability of dealing with page allocation
mechanisms.
Regarding modern multi-core platforms, the work of Goumas et al.~\cite{GKAKK08a} contains a
thorough analysis of a number of factors that may degrade the performance of
both sequential and multi-thread implementations. Performance tests were
carried out on three different platforms, including SMP, SMT and ccNUMA
systems.
Two partitioning schemes were implemented, one guided by the number of rows
and the other by the number of non-zeros per thread. It was
observed that the later approach contributes to a better load balancing,
thus improving significantly the running time.
For large matrices, they obtained average speedups of 1.96 and 2.13 using 2 and
4 threads, respectively, on an Intel Core 2 Xeon.
In this platform, their code reached about 1612 Mflop/s for 2 threads,
and 2967 Mflop/s when spawning 4 threads.
This performance changes
considerably when considering matrices whose working set sizes are far from fitting in cache.
In particular, it drops to around 815 Mflop/s and 849 Mflop/s, corresponding to the 2-
and 4-threaded cases.
Memory contention is viewed as the major bottleneck of implementations of the
sparse matrix-vector product. This problem was tackled by Kourtis et al.~\cite{KGK08a} via
compression techniques, reducing both the matrix connectivity and
floating-point numbers to be stored. Although leading to good scalability,
they obtained at most a 2-fold speedup on 8 threads, for matrices out of cache.
The experiments were conducted on two Intel Clovertown with 4MB of L2
cache each. In the same direction, Belgin et al.~\cite{BBR09a} proposed a
pattern-based blocking scheme for reducing the index overhead.
Accompanied by software prefetching and vectorization techniques, they attained
an average sequential speedup of 1.4. Their multi-thread implementation required the
synchronization of the accesses to the $y$ vector. In brief, each thread
maintains a private vector for storing partial values, which are summed up in a
reduction step into the global destination vector. They observed average
speedups around 1.04, 1.11 and 2.3 when spawning 2, 4, and 8 threads,
respectively. These results were obtained on a 2-socket Intel Harpertown 5400
with 8GB of RAM and 12MB L2 cache per socket.
Different row-wise partitioning methods were considered by Liu et al.~\cite{LZSQ09a}.
Besides evenly splitting non-zeros among threads, they evaluated
the effect of the automatic scheduling mechanisms provided by OpenMP, namely,
the \textit{static}, \textit{dynamic} and \textit{guided} schedules. Once
more, the non-zero strategy was the best choice. They also
parallelized the block CSR format. Experiments were run on four AMD Opteron 870
dual-core processors, with 16GB of RAM and $2 \times 1$MB L2 caches. Both
CSR and block CSR schemes resulted in poor scalability for large matrices,
for which the maximum speedup was approximately 2, using 8 threads.
Williams et al.~\cite{WOVSYD09a} evaluated the sparse matrix-vector kernel using the CSR
format on several up-to-date chip multiprocessor systems, such as the
heterogeneous STI Cell. They examined the effect of various optimization techniques
on the performance of a multi-thread CSR, including software
pipelining, branch elimination, SIMDization, explicit prefetching, 16-bit
indices, and register, cache and translation lookaside buffer (TLB) blocking.
A row-wise approach was employed for
thread scheduling. As regarding finite element matrices and in comparison to OSKI~\cite{VDY05a},
speedups for the fully tuned parallel code ranged
from 1.8 to 5.5 using 8 threads on an Intel Xeon E5345.
\begin{figure}[!t]
\centering
\includegraphics{images/csrc_scheme}
\caption{The layout of CSRC for an arbitrary 9$\times$9 non-symmetric matrix.}
\label{fig:csrc_scheme}
\end{figure}
More recently, Bulu{\c{c}} et al.~\cite{BFFGL09a} have presented a block structure that allows
efficient computation of both $Ax$ and $A^{\mathsf{T}}x$ in parallel. It can
be roughly seen as a dense collection of sparse blocks, rather than a sparse
collection of dense blocks, as in the standard block CSR format. In
sequential experiments carried out on an ccNUMA machine featuring AMD Opteron 8214
processors, there were no improvements over the standard CSR.
In fact, their data structure was always slower for band matrices. Concerning
its parallelization, however, it was proved that it yields a
parallelism of $\Theta(nnz/\sqrt{n}\log n)$.
In practice, it scaled up to 4 threads on an Intel Xeon X5460, and
presented linear speedups on an AMD Opteron 8214 and an Intel Core i7 920. On
the later, where the best results were attained, it reached speedups of 1.86,
2.97 and 3.71 using 2, 4 and 8 threads, respectively. However,
it does not seem to straightly allow the simultaneous
computation of $y_i \leftarrow y_i + a_{ij} x_j$ and $y_j \leftarrow y_j +
a_{ij} x_i$ in a single loop, as CSRC does.
\subsection{Overview}
The remainder of this paper is organized as follows. Section \ref{sec:csrc}
contains a precise definition of the CSRC format accompanied with a description
of the matrix-vector multiplication algorithm using such structure. Its
parallelization is described in Section \ref{sec:parallel-csrc}, where we
present two strategies for avoiding conflicts during write accesses to
the destination vector. Our results are shown in Section \ref{sec:results},
supplemented with some worthy remarks. We finally draw some conclusions in
Section \ref{sec:conclusion}.
\section{The CSRC storage format}
\label{sec:csrc}
The \textit{compressed sparse row-column} (CSRC) format is a specialization of
CSR for structurally symmetric matrices arising in finite element modelling
\cite{RF07a}, which is the target domain application of this work. Given an
arbitrary $n \times n$ global matrix $A = (a_{ij})$, with $nnz$
non-zeros, the CSRC decomposes $A$ into the sum $A_D + A_L + A_U$, where $A_D$,
$A_L$, and $A_U$ correspond to the diagonal, lower and upper parts of $A$,
respectively. The sub-matrix $A_L$ (resp.~$A_U$) is stored in a row-wise
(resp.~column-wise) manner.
In practice, the CSRC splits the off-diagonal coefficients into two
floating-point arrays, namely, $al(k)$ and $au(k)$, $k = \frac{1}{2}(nnz - n)$,
where the lower and upper entries of $A$ are stored. In other words, if $j <
i$, then $a_{ij}$ is stored in $al$, and $au$ contains its transpose $a_{ji}$.
The diagonal elements are stored in an array $ad(n)$. Other two integer
arrays, $ia(n+1)$ and $ja(k)$, are also maintained. These arrays can be defined
in terms of either the upper or lower coefficients. The $ia$ array
contains pointers to the beginning of each row (resp.~column) in $al$
(resp.~$au$), and $ja$ contains column (resp.~row) indices for those non-zero
coefficients belonging to $A_L$ (resp.~$A_U$). Another
interpretation is that $A_L$ is represented using CSR, while $A_U$ is stored using
CSC. We illustrate the CSRC data structure for an arbitrary 9$\times$9 non-symmetric matrix
consisting of 33 non-zeros in Figure \ref{fig:csrc_scheme}.
Notice that the CSRC could be viewed as the sparse skyline (SSK) format
restricted to structurally symmetric matrices \cite{Saa95a,GR01a}.
However, as shown in Section \ref{sec:rectextension}, we made it capable of
representing rectangular matrices after minor modifications. Furthermore, to
our knowledge, this is the first evaluation of such structure on modern
multi-processed machines.
\subsection{Extension to rectangular matrices}
\label{sec:rectextension}
The way the CSRC is defined would disallow us handling matrices with different
aspect ratios other than square. In the overlapping strategy
implemented in any distributed-memory finite element code using a
subdomain-by-subdomain approach
\cite{RF07a,ARM09a}, it is normal the
occurrence of rectangular matrices with a remarkable property.
An $n \times m$ matrix $A$,
with $m > n$, can always be written as the sum $A_S + A_R$, where
$A_S$ and $A_R$ are of order $n \times n$ and $n \times k$, respectively, with
$k = m - n$. In addition, the $A_S$ matrix has symmetric non-zero pattern, and
it is occasionally numerically symmetric. Therefore, it can be represented by the
CSRC definition given before, while $A_R$ can be stored using an auxiliary CSR
data structure.
\begin{figure}[!t]
\centering
\subfloat[]{\label{fig:matvec_csrc}%
\lstinputlisting[boxpos=b]{matvec_csrc.f}}
\hfil
\subfloat[]{\label{fig:matvec_csrcr}%
\lstinputlisting[boxpos=b]{matvec_csrcr.f}}
\caption{Code snippets for the non-symmetric matrix-vector multiplication
algorithm using CSRC for (a) square and
(b) rectangular matrices.}
\label{fig:SpMV}
\end{figure}
\subsection{Sequential matrix-vector product}
The sequential version of the CSRC matrix-vector multiplication algorithm has
the same loop structure as for CSR. The input matrix $A$ is
traversed by rows, and row $i$ is processed from the left to the right up to
its diagonal element $a_{ii}$. Because we assume $A$ is structurally
symmetric, its upper part can be simultaneously traversed. That is, we are
allowed to compute both $y_i \leftarrow y_i + a_{ij} x_j$ and $y_j \leftarrow
y_j + a_{ji} x_i$, in the $i$-th loop. If $A$ is also numerically symmetric, we can further
eliminate one load instruction when retrieving its upper entries. For
rectangular matrices, there is another inner loop to process the coefficients
stored in the auxiliary CSR. Figure \ref{fig:SpMV} contains Fortran implementations of
the sparse matrix-vector product using CSRC for square and rectangular matrices.
\section{Parallel implementation}
\label{sec:parallel-csrc}
To parallelize the sparse matrix-vector product using the CSRC, one can basically spawn threads at either the inner or the outer loop.
This means adding a \texttt{parallel do} directive just above line 1 or 4 of Figure \subref{fig:matvec_csrc} (and 9, for Figure \subref{fig:matvec_csrcr}).
As the amount of computations per row is usually low, the overhead due to the inner parallelization would counteract any parallelism.
On the other hand, recall that the CSRC matrix-vector product has the property that
the lower and upper parts of the input matrix are simultaneously traversed.
Thus spawning threads at line 1 requires the synchronization of writings into the destination vector.
That is, there exists a race condition on the access of the vector $y$.
If two threads work on different rows,
for example, rows $i$ and $j$, $j > i$, it is not unlikely that both threads
require writing permission to modify $y(k)$, $k \leq i$.
In short, our data structure is required to support concurrent reading and
writing on the vector $y$. These operations need to be thread-safe, but at
the same time very efficient, given the fine granularity of the operations.
Common strategies to circumvent this problem would employ atomic primitives,
locks, or the emerging transactional memory model. However, the overheads incurred by
these approaches are rather costly, compared to the total cost of accessing
$y$. A more promising solution would be to determine subsets of rows
that can be handled by distinct threads in parallel. In this paper, we have
considered two of such solutions, here termed \textit{local buffers} and
\textit{colorful} methods.
Our algorithms were analyzed using the concepts of \textit{work} and
\textit{span} \cite[Ch.~27]{CLRS09a}. The \textit{work} $T_{1}$ of an
algorithm is the total cost of running it on exactly one processor, and the
\textit{span} $T_{\infty}$ is equal to its cost when running on an infinite
number of processors. The \textit{parallelism} of a given algorithm is then
defined as the ratio $T_{1}/T_{\infty}$. So, the greater the parallelism of
an algorithm, the better the theoretical guarantees on its performance. The
work of the matrix-vector multiply using the CSRC is clearly $\Theta(nnz)$. To
calculate its span, we need to consider our partitioning strategies
separately.
\subsection{Local buffers method}
One way to avoid conflicts at the $y$ vector is to assign different
destination vectors to each thread. That is, thread $t_i$ would compute its
part of the solution, store it in a local buffer $y_i$, and then accumulate
this partial solution into the $y$ vector. This method, here called
\textit{local buffers method}, is illustrated in
Figure \subref{fig:partitioning-simple}, which shows the distribution of rows for an
arbitrary non-symmetric $9 \times 9$ matrix. In the example, the matrix is
split into three regions to be assigned to three different threads. The number of non-zeros per thread is 7, 5 and
21.
The main drawback of this method is the introduction of two additional steps: initialization and accumulation.
The accumulation step is performed to compute the final destination vector resultant from merging partial values stored in local buffers.
Threads must initialize their own buffers, because of this accumulation, otherwise they would store wrong data.
For convenience, we define the \textit{effective range} of a thread as the set of rows in $y$ that it indeed needs to modify.
We consider four ways of implementing both steps:
\begin{enumerate}
\item \textit{All-in-one}: threads initialize and accumulate in parallel the buffers of the whole team.
\item \textit{Per buffer}: for each buffer, threads initialize and accumulate in parallel.
\item \textit{Effective}: threads initialize and accumulate in parallel over the corresponding effective ranges.
\item \textit{Interval}: threads initialize and accumulate in parallel over intervals of $y$ defined by the intersection of their effective ranges.
\end{enumerate}
The spans of the \textit{all-in-one} and \textit{per buffer} methods are $\Theta(p + \log n)$ and $\Theta(p\log n)$, respectively.
If the number of threads is $\Theta(n)$, then their respective parallelism are $O(nnz / n)$ and $O(nnz/n\log n)$.
The platforms considered herein,
however, feature at most four processors. Our experiments will show that
these methods can still provide reasonable scalability for such systems.
In this case, their parallelism would be better approximated by $O(nnz / \log n)$, as for CSR.
In fact, the problem with the first two methods is that they treat all buffers as dense vectors, which is rarely true in practice as we are dealing with sparse matrices.
The \textit{effective} and \textit{interval} methods try to mitigate this issue by performing computations only on effective ranges.
For narrow band matrices, which is usually the case of finite element matrices, we can assume the effective range is $\Theta(n/p)$.
Hence the span of both methods is $\Theta(p \log (n/p))$.
Since the work per thread strongly depends on the number of non-zeros per row,
a partitioning technique based just on the number of rows may result in load
imbalance. A more efficient way is to consider the number of non-zeros per
thread, because the amount of floating point operations becomes balanced. The
results presented herein were obtained using such a non-zero guided implementation,
in which the deviation from the average number of non-zeros per row
is minimized.
\begin{figure}[t]
\centering
\subfloat[]{\includegraphics{images/partitioning-simple}
\label{fig:partitioning-simple}}
\hfil
\subfloat[]{\includegraphics{images/partitioning-colorful}
\label{fig:partitioning-colorful}}
\hfil
\subfloat[]{\includegraphics{images/conflicts}
\label{fig:conflicts}}
\caption{Illustration of the (a) local buffers and the (b) colorful
partitioning methods for 3 threads on a $9 \times 9$ matrix along with its (c) conflict graph.}
\label{fig:partitioning}
\end{figure}
\subsection{Colorful method}
The \textit{colorful method} partitions a matrix into sets of pairwise
conflict-free rows.
Here we distinguish between two kinds of conflicts.
If a thread owns row $i$ and a second thread owning row $j$, $j > i$, requires
to modify $y(k)$, $k < i$, it is called an \textit{indirect} conflict.
If $k = i$, we call such conflict \textit{direct}.
The \textit{conflict graph} of a matrix $A$ is the graph $G[A] = (V,E)$,
where each vertex $v \in V$ corresponds to a row in $A$, and the edges in $E$
represent conflicts between vertices.
Figure \subref{fig:conflicts} shows the conflict graph for the matrix in
Figure \ref{fig:csrc_scheme}.
Direct and indirect conflicts are indicated by solid and dashed lines,
respectively. In the graph, there are 12 direct and 7 indirect conflicts.
The direct
conflicts of row $i$ are exactly the rows corresponding to the column indices of
the non-zero entries at that row, i.e., the indices $ja(k)$, $k \in [ia(i), ia(i+1))$. They can be computed in a
single loop through the CSRC structure. The computation of indirect conflicts
is more demanding. In our implementation, these are determined with the aid of
the induced subgraph $G'[A]$ spanned by the edges in $G[A]$ associated with
direct conflicts. Given two vertices $u, v \in V$, if the intersection of
their neighborhood in $G'[A]$ is non-empty, then they are indirectly in
conflict.
We color the graph $G[A]$ by applying a standard sequential
coloring algorithm \cite{CM83a}.
The color classes correspond to
conflict-free blocks where the matrix-vector product can be safely carried
out in parallel. Observe that coloring rectangular matrices
is the same as coloring only its square part, since
the rectangular part is accessed by rows. The layout of a 5-colored
matrix is depicted in Figure \subref{fig:partitioning-colorful}.
Let $k$
denote the number of colors used by the coloring algorithm.
Suppose that the color classes are evenly sized, and
that the loop over the rows is implemented as a divide-and-conquer recursion.
Under such hypothesis, the span of the colorful method can be approximated by $\Theta(k\log(n/k))$.
Thus, the colorful matrix-vector product has a parallelism of $O(nnz / k\log(n/k) )$.
Although $k < p$ would lead to a better scalability when compared to the local buffers strategy,
the possibility of exploiting systems based on cache hierarchies decreases,
which affects considerably the code performance.
Furthermore, the number of processors used in our experiments was always smaller than the number of colors.
\section{Experimental results}
\label{sec:results}
Our implementation was evaluated on two Intel processors, including an Intel
Core~2 Duo E8200 (codenamed \textit{Wolfdale}) and an Intel i7 940 (codenamed
\textit{Bloomfield}). The Wolfdale processor runs at 2.66GHz, with L2 cache of
6MB and 8GB of RAM, and the Bloomfield one runs at 2.93GHz with $4\times$256KB
L2 caches, 8MB of L3 cache and 8GB of RAM. Our interest on Intel Core~2 Duo
machines lies on the fact that our finite element simulations are carried out
on a dedicated 32-nodes cluster of such processors.
The code was parallelized using OpenMP directives, and compiled
with Intel Fortran compiler (\texttt{ifort}) version 11.1 with level 3
optimizations (\texttt{-O3} flag) enabled. Machine counters were accessed
through the PAPI 3.7.1 library API \cite{BDGHM00a}. The measurements of
speedups and Mflop/s were carried out with PAPI instrumentation disabled.
The tests were performed on a data set comprised of 60 matrices, from which 32 are
numerically symmetric. There is one non-symmetric dense matrix of order 1K, 50
matrices selected from the University of Florida sparse matrix collection
\cite{Dav97a}, and 3 groups of 3 matrices each, called angical, tracer, and
cube2m, of our own devise. Inside these groups, matrices correspond to
one global finite element matrix output by our sequential finite element code,
and two global matrices for both of the adopted domain partitioning schemes,
overlapping (suffix ``\_o32'') and non-overlapping (suffix ``\_n32''),
where 32 stands for the number of sub-domains. Our benchmark computes the sparse
matrix-vector product a thousand times for each matrix in Table
\ref{tab:matrices-details}, which is a reasonable value for iterative solvers
like the preconditioned conjugate gradient method and the generalized
minimum residual method. All results correspond to
median values over three of such runs.
\begin{table}[!t]
\caption{Details of the matrices used in our experiments.}
\label{tab:matrices-details}
\centering
{\scriptsize
\begin{tabularx}{0.485\textwidth}{@{}ll@{\ \ }rrrr@{}}
\toprule
Matrix & Sym. & $n$ & $nnz$ & $nnz/n$ & $ws$ (KB)\\
\midrule
thermal & no & 3456 & 66528 & 19 & 710 \\
ex37 & no & 3565 & 67591 & 18 & 722 \\
flowmeter5 & no & 9669 & 67391 & 6 & 828 \\
piston & no & 2025 & 100015 & 49 & 1012 \\
SiNa & yes & 5743 & 102265 & 17 & 1288 \\
benzene & yes & 8219 & 125444 & 15 & 1598 \\
cage10 & no & 11397 & 150645 & 13 & 1671 \\
spmsrtls & yes & 29995 & 129971 & 4 & 1991 \\
torsion1 & yes & 40000 & 118804 & 2 & 2017 \\
minsurfo & yes & 40806 & 122214 & 2 & 2069 \\
wang4 & no & 26068 & 177196 & 6 & 2188 \\
chem\_master1 & no & 40401 & 201201 & 4 & 2675 \\
dixmaanl & yes & 60000 & 179999 & 2 & 3046 \\
chipcool1 & no & 20082 & 281150 & 14 & 3098 \\
t3dl & yes & 20360 & 265113 & 13 & 3424 \\
poisson3Da & no & 13514 & 352762 & 26 & 3682 \\
k3plates & no & 11107 & 378927 & 34 & 3895 \\
gridgena & yes & 48962 & 280523 & 5 & 4052 \\
cbuckle & yes & 13681 & 345098 & 25 & 4257 \\
bcircuit & no & 68902 & 375558 & 5 & 4878 \\
angical\_n32 & yes & 20115 & 391473 & 19 & 4901 \\
angical\_o32 & no & 18696 & 732186 & 39 & 4957 \\
tracer\_n32 & yes & 33993 & 443612 & 13 & 5729 \\
tracer\_o32 & no & 31484 & 828360 & 26 & 5889 \\
crystk02 & yes & 13965 & 491274 & 35 & 5975 \\
olafu & yes & 16146 & 515651 & 31 & 6295 \\
gyro & yes & 17361 & 519260 & 29 & 6356 \\
dawson5 & yes & 51537 & 531157 & 10 & 7029 \\
ASIC\_100ks & no & 99190 & 578890 & 5 & 7396 \\
bcsstk35 & yes & 30237 & 740200 & 24 & 9146 \\
\bottomrule
\end{tabularx}
\hfill
\begin{tabularx}{0.485\textwidth}{@{}l@{\ \ }l@{\ \ }r@{\ \ }r@{\ \ }r@{\ \ }r@{}}
\toprule
Matrix & Sym. & $n$ & $nnz$ & $nnz/n$ & $ws$ (KB)\\
\midrule
dense\_1000 & no & 1000 & 1000000 & 1000 & 9783 \\
sparsine & yes & 50000 & 799494 & 15 & 10150 \\
crystk03 & yes & 24696 & 887937 & 35 & 10791 \\
ex11 & no & 16614 & 1096948 & 66 & 11004 \\
2cubes\_sphere & yes & 101492 & 874378 & 8 & 11832 \\
xenon1 & no & 48600 & 1181120 & 24 & 12388 \\
raefsky3 & no & 21200 & 1488768 & 70 & 14911 \\
cube2m\_o32 & no & 60044 & 1567463 & 26 & 16774 \\
nasasrb & yes & 54870 & 1366097 & 24 & 16866 \\
cube2m\_n32 & no & 65350 & 1636210 & 25 & 17127 \\
venkat01 & no & 62424 & 1717792 & 27 & 17872 \\
filter3D & yes & 106437 & 1406808 & 13 & 18149 \\
appu & no & 14000 & 1853104 & 132 & 18342 \\
poisson3Db & no & 85623 & 2374949 & 27 & 24697 \\
thermomech\_dK & no & 204316 & 2846228 & 13 & 31386 \\
Ga3As3H12 & yes & 61349 & 3016148 & 49 & 36304 \\
xenon2 & no & 157464 & 3866688 & 24 & 40528 \\
tmt\_sym & yes & 726713 & 2903837 & 3 & 45384 \\
CO & yes & 221119 & 3943588 & 17 & 49668 \\
tmt\_unsym & no & 917825 & 4584801 & 4 & 60907 \\
crankseg\_1 & yes & 52804 & 5333507 & 101 & 63327 \\
SiO2 & yes & 155331 & 5719417 & 36 & 69451 \\
bmw3\_2 & yes & 227362 & 5757996 & 25 & 71029 \\
af\_0\_k101 & yes & 503625 & 9027150 & 17 & 113656 \\
angical & yes & 546587 & 11218066 & 20 & 140002 \\
F1 & yes & 343791 & 13590452 & 39 & 164634 \\
tracer & yes & 1050374 & 14250293 & 13 & 183407 \\
audikw\_1 & yes & 943695 & 39297771 & 41 & 475265 \\
cube2m & no & 2000000 & 52219136 & 26 & 545108 \\
cage15 & no & 5154859 & 99199551 & 19 & 1059358 \\
\bottomrule
\end{tabularx}}
\end{table}
\subsection{Sequential performance}
We have compared the sequential performance of CSRC to the standard CSR.
For symmetric matrices, we have chosen the OSKI implementation \cite{LVDY04a}
as the representative of the symmetric CSR algorithm, assuming that only the
lower part of $A$ is stored.
In the sparse matrix-vector product, each element of the matrix is accessed
exactly once. Thus, accessing these entries incurs only on compulsory misses.
On the other hand, the elements of $x$ and $y$ are accessed multiple times.
This would enable us to take advantage of cache hierachies by
reusing recently accessed values. In the CSR, the access pattern of the $x$
vector is known to be the major hindrance to the exploitation of data reuse,
because arrays $y$, $ia$, $ja$ and $a$ all have stride-1 accesses. Since the
$y$ vector is not traversed using unit stride anymore in the CSRC, one could argue
that there would be an increase in the number of cache misses. As
presented in Figure \ref{fig:missratio}, experiments on L2 data cache misses
suggest just the converse, while the ratio of TLB misses is roughly constant.
\begin{figure}[t]
\centering
\includegraphics[height=0.22\textheight]{images/wolfdale-matvec_bench-sequential-missratio}
\caption{Percentages of L2 and TLB cache misses
using CSRC and CSR on the
Wolfdale processor.}
\label{fig:missratio}
\end{figure}
\begin{figure}[t]
\centering
\includegraphics[height=0.22\textheight]{images/matvec_bench-sequential-mflops}
\caption{Sequential performance in Mflop/s of the matrix-vector
product using CSR and CSRC on both Wolfdale and Bloomfield processors.}
\label{fig:matvec_bench-sequential-mflops}
\end{figure}
The performance of the algorithm considered herein is memory bounded, because the
number of load/store operations is at least as greater as the number of
floating-point multiply-add instructions. In a dense matrix-vector product, we
need to carry out $O(n^2)$ operations on $O(n^2)$ amount of data, while for sparse
matrices, these quantities are both $O(n)$.
In particular, observe that the computation
of the square $Ax$ product using the CSRC requires the execution of $n$ multiply and $nnz - n$
multiply-add operations, whereas the CSR algorithm requires $nnz$ multiply-add
operations. On systems without fused multiply-add operations, the CSR and CSRC
algorithms would perform $2nnz$ and $2nnz - n$ floating-point instructions,
respectively. On the other hand, the number of load instructions for CSR is
$3nnz$, and $\frac{5}{2}nnz - \frac{1}{2}n$ for the CSRC format. Hence the
ratio between loadings and flops is approximately 1.26 for CSRC and exactly 1.5
for CSR. This bandwidth mitigation may be the most relevant reason for the
efficiency of CSRC shown in Figure \ref{fig:matvec_bench-sequential-mflops}.
It is also worth noting the advantage of the augmented CSRC on
matrices whose square part is numerically symmetric,
i.e., the matrices angical\_o32 and tracer\_o32.
\subsection{Multi-thread version}
Our parallel implementation was evaluated with up to 4 threads on Bloomfield
with Hyper-Threading technology disabled.
The values of speedup
are relative to the pure sequential CSRC algorithm, and not to the one thread case.
One would expect that the colorful method is best suited to matrices with
few conflicts, e.g., narrow band matrices,
because the lower is the maximum degree in the conflict graph, the larger is its parallelism.
As shown in Figure
\ref{fig:matvec_bench-local_buffers_effective_nonzeros_vs_colorful},
it was more efficient only on the matrices torsion1, minsurfo and dixmaanl, which have the smallest bandwidth among all matrices.
Nonetheless, according to Figures \subref{fig:wolfdale-matvec_bench-colorful-speedup} and \subref{fig:bloomfield-matvec_bench-colorful-speedup}, small matrices can still benefit from some parallelism.
An important deficiency of the colorful strategy, which contributes to its lack of
locality, is the variable-size stride access to the source and destination
vectors.
Inside each color, there not exist rows
sharing neither $y$ nor $x$ positions, because if they do share there will be a
conflict, therefore they must have different colors.
We claim that there must be an optimal color size to compensate such irregular accesses.
\begin{figure}[!t]
\centering
\includegraphics[height=0.22\textheight]{images/matvec_bench-local_buffers_effective_nonzeros_vs_colorful}
\caption{Performance comparison between the colorful method and the fastest local buffers implementation
on the Wolfdale and Bloomfield systems.}
\label{fig:matvec_bench-local_buffers_effective_nonzeros_vs_colorful}
\end{figure}
\begin{figure}[!t]
\centering
\begin{tabular}{@{}>{\footnotesize}lm{0.9\textwidth}@{}}
(a) & \subfloat{\includegraphics[height=0.21\textheight]{images/wolfdale-matvec_bench-colorful-speedup}\label{fig:wolfdale-matvec_bench-colorful-speedup}}\\
(b) & \subfloat{\includegraphics[height=0.21\textheight]{images/bloomfield-matvec_bench-colorful-speedup}\label{fig:bloomfield-matvec_bench-colorful-speedup}}
\end{tabular}
\caption{Speedups for the colorful method on the (a) Wolfdale and (b) Bloomfield processors.}
\label{fig:matvec_bench-colorful-speedups}
\end{figure}
Figures \ref{fig:wolfdale-matvec_bench-local_buffers-speedups}
and \ref{fig:bloomfield-matvec_bench-local_buffers-speedups} show the outcomes of speedups attained
by all four implementations of the local buffers strategy.
The
overheads due to the initialization and accumulation steps become
notorious when using just one thread. This can be easily overcome by
checking the number of threads at runtime. If there exists only one thread
in the working team, the global destination vector is used instead.
Although all four implementations reached reasonable speedup peaks, the effective method has been more stable over the whole data set.
On the average, it is the best choice for 93\% of the cases on the Wolfdale, and for 80\% and 78\% on Bloomfield with 2 and 4 threads, respectively.
\begin{figure}[p]
\centering
\begin{tabular}{@{}>{\footnotesize}lm{0.9\textwidth}@{}}
(a) & \subfloat{\includegraphics[height=0.21\textheight]{images/wolfdale-matvec_bench-local_buffers_static_nonzeros-speedup}\label{fig:wolfdale-matvec_bench-local_buffers_static_nonzeros-speedup}}\\
(b) & \subfloat{\includegraphics[height=0.21\textheight]{images/wolfdale-matvec_bench-local_buffers_full_nonzeros-speedup}\label{fig:wolfdale-matvec_bench-local_buffers_full_nonzeros-speedup}}\\
(c) & \subfloat{\includegraphics[height=0.21\textheight]{images/wolfdale-matvec_bench-local_buffers_effective_nonzeros-speedup}\label{fig:wolfdale-matvec_bench-local_buffers_effective_nonzeros-speedup}}\\
(d) & \subfloat{\includegraphics[height=0.21\textheight]{images/wolfdale-matvec_bench-local_buffers_strict_nonzeros-speedup}\label{fig:wolfdale-matvec_bench-local_buffers_strict_nonzeros-speedup}}\\
\end{tabular}
\caption{Speedups achieved by the local buffers strategy using the (a) all-in-one, (b) per buffer, (c) effective and (d) interval methods of initialization/accumulation on the Wolfdale processor.}
\label{fig:wolfdale-matvec_bench-local_buffers-speedups}
\end{figure}
\begin{figure}[p]
\centering
\begin{tabular}{@{}>{\footnotesize}lm{0.9\textwidth}@{}}
(a) & \subfloat{\includegraphics[height=0.21\textheight]{images/bloomfield-matvec_bench-local_buffers_static_nonzeros-speedup}\label{fig:bloomfield-matvec_bench-local_buffers_static_nonzeros-speedup}}\\
(b) & \subfloat{\includegraphics[height=0.21\textheight]{images/bloomfield-matvec_bench-local_buffers_full_nonzeros-speedup}\label{fig:bloomfield-matvec_bench-local_buffers_full_nonzeros-speedup}}\\
(c) & \subfloat{\includegraphics[height=0.21\textheight]{images/bloomfield-matvec_bench-local_buffers_effective_nonzeros-speedup}\label{fig:bloomfield-matvec_bench-local_buffers_effective_nonzeros-speedup}}\\
(d) & \subfloat{\includegraphics[height=0.21\textheight]{images/bloomfield-matvec_bench-local_buffers_strict_nonzeros-speedup}\label{fig:bloomfield-matvec_bench-local_buffers_strict_nonzeros-speedup}}\\
\end{tabular}
\caption{Speedups achieved by the local buffers strategy using the (a) all-in-one, (b) per buffer, (c) effective and (d) interval methods of initialization/accumulation on the Bloomfield processor.}
\label{fig:bloomfield-matvec_bench-local_buffers-speedups}
\end{figure}
To better illustrate the performance of different initialization/accumulation algorithms, Table \ref{tab:matvec_bench-acctime} presents average values of the running time consumed by these algorithms considering two classes of matrices, the ones that fit in cache and the others that do not.
As expected, both all-in-one and per buffer strategies have similar performance.
The effective and interval methods have demonstrated to be very feasible for practical use, although the later may incur a higher overheard because the number of intervals is at least as great as the number of threads.
In general, the running time is influenced by the working set size
and the band structure of the matrix. When the arrays used by the CSRC
fit or nearly fit into cache memory, better speedups were
obtained with almost linear scalability, reaching up to 1.87 on Wolfdale. For some matrices
from the University of Florida collection it was observed a poor
performance, e.g., tmt\_sym, tmt\_unsym, cage15 and F1. In the case of cage15 and F1, this may be attributed to the
absence of a band structure. On
the other hand, there seems to be a bandwidth lower bound for preserving performance.
In particular, the quasi-diagonal profile of the matrices tmt\_sym and tmt\_unsym
have contributed to amplify indirection overheads.
Our code has been 63\% more efficient on Bloomfield using 2 threads than on Wolfdale.
Taking a closer view, however, we see that Wolfdale is faster on 80\% of matrices with working set sizes up to 8MB, while Bloomfield beats the former on 94\% of the remaining matrices.
Notice that Wolfdale requires less cycles than Bloomfield to access its outer most cache, which would explain its superiority on small matrices.
Analysing the performance with 4 threads on the Bloomfield processor shown in Figure \subref{fig:bloomfield-matvec_bench-local_buffers_effective_nonzeros-speedup}, speedups indicate that large working sets drastically degrades the efficiency of the implementation, compared to the 2-threaded case.
On smaller matrices, speedups seem to grow linearly, with peaks of 1.83 and 3.40 using 2 and 4 threads, respectively.
\begin{table}[t]
\caption{Average values of the maximum running time among all threads spent during the initialization and accumulation steps using four different approaches.}
\label{tab:matvec_bench-acctime}
\centering
\renewcommand{\arraystretch}{1.1}
{\footnotesize
\begin{tabular}{@{}l@{\ \ }c@{\ \ }cc@{\ \ }cc@{\ \ }c@{}}
\toprule
\multirow{3}{*}{Method} & \multicolumn{2}{c}{Wolfdale} & \multicolumn{4}{c}{Bloomfield}\\\cmidrule(r){2-3}\cmidrule(l){4-7}
& $ws < 6$MB & $ws > 6$MB & \multicolumn{2}{c}{$ws < 8$MB} & \multicolumn{2}{c}{$ws > 8$MB}\\\cmidrule(r){2-2}\cmidrule(r){3-3}\cmidrule(lr){4-5}\cmidrule(l){6-7}
& 2 & 2 & 2 & 4 & 2 & 4\\
\midrule
all-in-one & 0.0455 & 4.3831 & 0.0370 & 0.0475 & 1.3127 & 2.5068\\
per buffer & 0.0455 & 4.3876 & 0.0320 & 0.0393 & 1.8522 & 3.8299\\
effective & \textbf{0.0215} & \textbf{1.8785} & \textbf{0.0176} & \textbf{0.0234} & \textbf{0.8094} & \textbf{1.2575}\\
interval & 0.0858 & 2.9122 & 0.0748 & 0.0456 & 1.3920 & 1.4939\\
\bottomrule
\end{tabular}}
\end{table}
\section{Conclusion}
\label{sec:conclusion}
We have been concerned with the parallelization of the matrix-vector multiplication
algorithm using the CSRC data structure, focusing on multi-core
architectures. It has been advocated that multi-core parallelization alone can
compete with purely sequential optimization techniques. We could observe that,
provided sufficient memory bandwidth, our implementation has demonstrated to be fairly scalable.
The main deficiency of the colorful method is due to variable size stride
accesses, which can destroy any locality provided by matrix reordering
techniques. We claim that it could be improved by fixing the maximum allowed
stride size inside each color class. This will be the objective of our future
investigations.
Computing the transpose matrix-vector multiplication is
considered costly when using the standard CSR. An easy but still expensive solution
would be to convert it into the CSC format before spawning threads.
This operation is very straightforward using the CSRC,
as we just need to swap the addresses of $al$ and $au$, and we are done.
Clearly, the computational costs remain the same.
Our results extend previous work
on the computation of the sparse matrix-vector product for
structurally symmetric matrices to multi-core
architectures.
The algorithms hereby presented are now part of a distributed-memory implementation
of the finite element method \cite{RF07a}.
Currently,
we conduct experiments on the effect of coupling both coarse- and fine-grained parallelisms.
\section*{Acknowledgment}
We would like to thank Prof.~Jos{\'e} A.~F.~Santiago and Cid S.~G.~Monteiro for
granting access to the Intel i7 machines used in our experiments. We are also
grateful to Ayd{\i}n Bulu\c{c} and the anonymous reviewers
for the helpful comments.
\bibliographystyle{abbrv}
| 2024-02-18T23:40:00.217Z | 2010-06-01T02:00:23.000Z | algebraic_stack_train_0000 | 1,060 | 6,516 |
|
proofpile-arXiv_065-5250 | \section{Introduction}\label{aba:sec1}
Electromagnetic properties of neutrino are among the key items of
modern particle physics (see \cite{GiuStu09} for a recent review).
It seems quite natural that a massive neutrino would have nonzero
diagonal or transition magnetic moment. If a neutrino has
non-trivial electromagnetic properties, then neutrino coupling to
photons is possible and several important, for instance for
astrophysics, processes can exist ~\cite{Raf_book96_RafPR99} .
Recently we have proposed a new mechanism of neutrino radiation of
photons that is realized when a relativistic neutrino with nonzero
magnetic moment moves in dense matter. This mechanism was termed
the ``spin light of neutrino" ($SL\nu$)~\cite{LobStuPLB03} . The
quantum theory of this phenomenon was developed
in~\cite{StuTerPLB05} (see also ~\cite{Lob05} and
~\cite{StuJPA_06_08} ).
In this paper we extend our studies of $SL\nu$ ~\cite{StuTerPLB05}
and consider the $SL\nu$ in a more general case when the photon is
emitted in the neutrino radiative decay. The $SL\nu$ considered in
~\cite{StuTerPLB05} was investigated under condition of equal
masses for the initial and final neutrino states. Here below we
examine the case when the neutrino transition between two
different neutrino mass states is realized. Thus, we consider the
$SL\nu$ mode in the neutrino radiative decay in matter originated
due to the neutrino transition magnetic moment.
It should be noted that the neutrino radiative decay was
considered before by several authors ~\cite{Smi78} . It was shown
that the process characteristics are substantially changed if the
presence of a medium is taken into account. In these calculations,
the influence of the background matter was considered only in the
electromagnetic vertex. Here we are going to discuss the impact of
the medium also onto the state of neutrino itself. At the same
time we will be interested in the another aspect of the problem
and consider it from the point of view of light emission. Under
the condition of equal initial and final particle masses the
process becomes equivalent to the $SL\nu$ in matter . With
different masses for the initial and final neutrino states, the
spin light becomes only the constituting channel for the overall
process corresponding to the change of the neutrino helicity. The
mechanism of $SL\nu$ is based on helicity states energy difference
of the particle arising due to the weak interaction with the
background matter. Hence, our study makes sense, obviously, if the
scale of neutrino mass difference is of the order of spin energy
splitting owing to the interaction with matter.
Let us specify now the process under consideration. We are
considering the decay of one neutrino mass state $\nu_1$ into
another mass state $\nu_2$ assuming that $m_1>m_2$, and restrict
ourselves with only these two neutrino species and accordingly
with two flavour neutrinos. Having in mind that conditions for the
most appropriate application of the process under study can be
found in the vicinity of neutron stars we will take for the
background a neutron-rich matter. In this case a process with
participation of antineutrinos is more appropriate and thus will
study here. However for the convenience in what follows we will
still refer to the particles as to neutrinos. Since the
interactions of flavour neutrinos with neutron star matter are the
same and governed by the neutron density we will take equal
interactions for the initial and final massive neutrinos with the
matter.
\vspace{-0.4cm}
\section{Modified Dirac equation}
The system ``neutrino $\Leftrightarrow$ dense matter" depicted
above can be circumscribed mathematically in different ways. Here
we use the powerful method of exact solutions, discussed in a
series of our papers ~\cite{StuJPA_06_08} . This method is based
on solutions ~ \cite{StuTerPLB05} of the modified Dirac equation
for neutrino in the background medium
\begin{equation}
\{i\gamma_{\mu}\partial^{\mu}-\frac{1}{2}\gamma_{\mu}(1+\gamma^{5})f^{\mu}-m\}\Psi(x)=0,
\label{eq:dirac}
\end{equation}
where in the case of unpolarized and nonmoving matter
$f^{\mu}=G_{f}/\sqrt{2} \ (n,\textbf{0})$ with $n$ being matter
number density. At this, the energy spectrum of neutrino is given
by
\begin{equation}
E_\varepsilon=\varepsilon\sqrt{(p-s\alpha
m_{\nu})^{2}+m_{\nu}^{2}}+\alpha m_{\nu}, \ \ \alpha =
\frac{1}{2\sqrt{2}}G_F\frac{n}{m_{\nu}} \label{eq:spektr}
\end{equation}
where $\varepsilon=\pm1 $ defines the positive and negative-energy
branches of the solutions, $s$ is the helicity of neutrino, $p$
is the neutrino momentum. The exact form of the solutions
$\Psi_{\varepsilon,p,s}(\textbf{r},t)$ can be found in
~\cite{StuTerPLB05} and ~\cite{StuJPA_06_08} .
\vspace{-0.4cm}
\section{Spin light mode of massive neutrino decay}
The S-matrix element for the decay has the standard form that of
the magnetic moment radiation process:
\begin{equation}
S_{fi}=-(2\pi)^4\mu\sqrt{\frac{\pi}{2wL^3}}\delta(E_2-E_1+w)
\delta^{3}({\bf p}_2-{\bf p}_1+{\bf k})
\overline{u}_{f}({{\bf e}},{\bf \Gamma}_{fi})u_i.
\label{eq:amplitude}
\end{equation}
Here ${\bf \Gamma}=i\omega\big\{\big[{\bf \Sigma} \times {\bm
\varkappa}\big]+i\gamma^{5}{\bf \Sigma}\big\}$, $u_{i,f}$ are the
spinors for the initial and final neutrino states, ${\bf e}$ is
the photon polarization vector, $\mu$ is the transitional magnetic
moment~\cite{GiuStu09} and $L$ is the normalization length.
In the process, we have the following conservation laws:
\begin{equation}
E_1=E_2+\omega; \ \ \bf{p_1}=\bf{p_2}+\bf{k}.
\label{eq:conservation}
\end{equation}
It is useful to carry out our computations through non-dimensional
terms. For that purpose we introduce the following notations:
$\gamma=\frac{m_1}{p_1};\kappa=\frac{\alpha
m_1}{p_1}=\frac{\tilde{n}}{p_1};\delta=\frac{\triangle
m^2}{p_{1}^{2}}=\frac{m_{1}^{2}-m_{2}^{2}}{p_{1}^{2}}$. To single
out the the spin light part of the radiative decay process we
should choose different helicities for the initial and final
neutrinos. Keeping the analogy with the usual process of $SL\nu$
we take the helicity quantum numbers as $s_1=-1$, $s_2=1$. Then
the solution of the kinematic system (\ref{eq:conservation}) can
be written in the form
\begin{equation}
w=\frac{-(KD+x\kappa^2)+\sqrt{(KD+x\kappa^2)^2-(K^2-\kappa^2)(D^2-\kappa^2)}}{(K^2-\kappa^2)}
\label{eq:freq}
\end{equation}
with the notations $D=s_{1}\kappa-\delta$;
$\tilde{n}=\frac{1}{2\sqrt{2}}G_F n$,
$K=\sqrt{(1-s_{1}\cdot\kappa)^2+\gamma^2}-x$, here $x$ stands for
$\cos\theta$, $\theta$ is the angle between ${\bf p}_1$ and $\bf
k$.
Performing all the calculations we obtain angle distribution of
the probability for the investigated process:
\begin{equation}
\frac{d\Gamma}{dx}=\mu^2 p_1^3\frac{(K-w+x)(wK-\kappa-\delta)w^3S'}
{\sqrt{(KD-w+x)^2-(K^2-\kappa^2)(D^2-\kappa^2)}},
\label{eq:density}
\end{equation}
where $S'= (1+\beta_1 \beta_2)(1-\frac{w-x-w \cdot x + w \cdot
x^2}{\sqrt{1+w^2-2w \cdot x}}x)-(\beta_1 +
\beta_2)(x-\frac{w-x}{\sqrt{1+w^2-2w \cdot x}})$ and
$\beta_1=\frac{1+\kappa}{\sqrt{(1+\kappa)^2+\gamma^2}}$,
$\beta_2=\frac{\sqrt{1+w^2-2w \cdot x}-\kappa}{K-w+x}$.
The total probability can be computed from the equation
(\ref{eq:density}) by taking the integration over the angle
$\theta$ range. However, manual calculations are not quite simple
to carry through them. Even though the integral can be calculated
exactly, the final expression is enormously complex and its
explicit form is optional to be given here.
\vspace{-0.4cm}
\section{Results and discussion}
It is worth to investigate the asymptotical behavior of the
probability $\Gamma$ in three most significant relativistic
limiting cases keeping only the first infinitesimal order of small
parameters. On this way we have,
\begin{equation}
\Gamma=4\mu^2\tilde{n}^3(1+\frac{3}{2}\frac{m_1^2-m_2^2}{\tilde{n}p_1}+\frac{p_1}{\tilde{n}}), \
({\text {ultrahigh density:}}\ 1 \ll \frac{p_1}{m_1} \ll \frac{\tilde{n}}{p_1});
\label{ultrahighdensity}
\end{equation}
\begin{equation}
\Gamma=4\mu^2\tilde{n}^2p_1(1+\frac{\tilde{n}}{p_1}+\frac{m_1^2-m_2^2}
{\tilde{n}p_1}+\frac{3}{2}\frac{m_1^2-m_2^2}{p_1^2}), ({\text {high density:}}\
\frac{m_1^2}{p_1^2}\ll \frac{\tilde{n}}{p_1} \ll 1);
\label{highdensity}
\end{equation}
\begin{equation}
\Gamma\approx\mu^2 \frac{m_1^6}{p_1^3}, ({\text {quasi-vacuum case:}}
\frac{\tilde{n}}{p_1} \ll \frac{m_1}{p_1}\ll 1, m_1 \gg m_2).
\label{lowdensity}
\end{equation}
The obtained results (\ref{ultrahighdensity}) and
(\ref{highdensity}) exhibit the power of the method of exact
solutions since they establish clear connection between the case
of massive weak-interacting particles when the masses of the
initial and final particles differ with the previously
investigated equal mass case. Indeed, it is easy to verify that
these results transforms exactly into the results of $SL\nu$
calculation~\cite{LobStuPLB03}. The asymptotic estimation
(\ref{lowdensity}) can not be reduced to the $SL\nu$ case and thus
it is a new result, which is characteristic feature for the decay
process under study. The above-mentioned asymptotical cases
(\ref{ultrahighdensity}), (\ref{highdensity}) and
(\ref{lowdensity}) where calculated with the assumption that the
initial neutrino is relativistic $(\gamma=\frac{m_1}{p_1}\ll 1)$.
In particular the relativistic character of the initial neutrino
propagation influences strongly on the emitted $SL\nu$ photon
energy because of increase of the part of neutrino energy in it.
It is also interesting to investigate the spin light mode in the
radiative decay of slowly moving massive neutrino (or even
stationary initial neutrino). This process has been calculated
several times \cite{Smi78} . We consider the vacuum case to find
the interrelation of our results obtained using the method of
exact solutions with the results of previous works. So, taking
into account $\gamma=\frac{m_1}{p}\ll 1,
\kappa=\frac{\tilde{n}}{p_1}, \delta\equiv\frac{\gamma^2}{2}$ for
the probability of the process we finally get:
\begin{equation}
\Gamma\approx \frac{7}{24}\mu^2 {m_1^3} \sim m_1^5.
\label{vacuum}
\end{equation}
We obtain here the same dependency of the probability from the
mass of the decaying neutrino as in the classical papers on the
radiative neutrino decay. By this means we justify usage of the
modified Dirac equation exact solutions method.
{\it Acknowledgements.} One of the authors (A.S.) is thankful to
Kim Milton for the invitation to attend the 9th Conference on
Quantum Field Theory Under the Influence of External Conditions
(Oklahoma, USA) and for the kind hospitality provided in Norman.
\vspace{-0.4cm}
| 2024-02-18T23:40:00.447Z | 2010-03-02T17:58:12.000Z | algebraic_stack_train_0000 | 1,071 | 1,783 |
|
proofpile-arXiv_065-5273 | \section{Introduction}
Conformal mapping is often presented as a method for simplifying mathematical problems in two dimensions. Liouville's theorem \cite{DEB} in fact shows that higher dimensional conformal maps are possible but must be composed of translations, similarities, orthogonal transformations and inversions. The purpose of this paper is to present an application of a higher-dimensional conformal transformation in the simplification of the Schr\"{o}dinger equation for the three-dimensional harmonic oscillator.
Suppose an oscillator of energy E consists of a single particle with a spatial displacement $x_i$ $(i=1,2,3)$ from the source of the potential confining it at time t. The goal of section 2 of this paper is to introduce an isometric conformal mapping of the form $z_i= x_i,s=t+\imath f(|x_i|,E)$ where $f$ is a real function and $\imath=\sqrt{-1}$. It is clear from inspection that this passive transformation does no more than introduce an imaginary shift in the time of the two related coordinate systems.
It is convenient to write the complex conjugate form of the $(z_i,s)$-coordinates as $(z_i^*,s^*)$ even though $z_i^*=z_i$ since $z_i^*$ and $z_i$ still belong to different coordinate systems. This distinction is shown to be most evident in the computation of the partial derivatives $\partial / \partial z_i$ and $\partial / \partial z_i^*$ from the chain rule of partial differentiation as these evaluate differently in their conjugate coordinate systems. One further topic to be introduced in section 2 is the Cauchy-Riemann equations that are necessary to determine if a function transformed into the $(z_i,s)$-coordinate system has well defined partial derivatives.
The Schr\"{o}dinger equation for the three-dimensional harmonic oscillator is presented in section 3 alongside a complete set of eigensolutions. It is shown that both these results simplify in terms $(z_i,s)$-coordinates and that the eigensolutions are holomorphic. This argument is used to demonstrate that harmonic interactions may be introduced into the free-field Schr\"{o}dinger equation by adding an imaginary component to the world time of the particle. The imaginary part of time has a confining effect on the particle equivalent to including an oscillator potential in the hamiltonian.
In section 4, it is demonstrated that $\kappa \partial / \partial z_i$ and $-\kappa \partial / \partial z_i^*$ are respectively the operators for lowering and raising the eigenstates of the harmonic oscillator where $\kappa$ is a scaling constant. These results are compared to both the Bargmann \cite{VB} and conjugate form of the Bargmann representation \cite{RNA} .
\section{Conformal Mapping}
The task ahead is to present a isometric conformal transformation relating a real $(x_i,t)$-coordinate system and a complex $(z_i,s)$ coordinate system. This mapping is to be applied in section 3 to a harmonic oscillator consisting of a single particle of mass m and total energy E. It is convenient to express it in the form
\begin{equation} \label{eq: conftrans1}
z_{i} = x_{i}, \quad s = t - \imath \frac{ m\omega}{2 E}x^2
\end{equation}
where $\omega$ is the spring constant of the oscillator . In the $(x_i,t)$-coordinate system, the particle has a spatial displacement $x_i$ from the source but shares the same world time t. Similarly, in the $(z_i,s)$-coordinate system, the particle has a dispacement $z_i = x_i$ from the source but shares the same complex time s. The isometric nature of the transformation therefore follows from the result $|z_i|=|x_i|$. It is also clear that the complex time $s$ is translated through an imaginary displacement $\imath \frac{ m\omega}{2 E}x^2$ from the real time $t$.
In the application of complex coordinates to express physical problems, there is generally going to be both a complex and a complex conjugate coordinate representation for each individual problem. In the present case, the complex conjugate of eq. (\ref{eq: conftrans1}) is
\begin{equation} \label{eq: conftrans2}
z_{i}^* = x_{i}, \quad s^*= t + \imath \frac{m\omega}{2E}x^2
\end{equation}
Naturally, there must also be inverse transformations mapping the complex and complex conjugate representations of the problem back into a single physical coordinate system. The inverses of the transformations (\ref{eq: conftrans1}) and (\ref{eq: conftrans2}) are readily shown to be
\begin{equation} \label{eq: inv_ict1}
x_{i} = z_{i}, \quad t = s + \imath \frac{m\omega}{2E}z^2
\end{equation}
\begin{equation} \label{eq: inv_ict2}
x_{i} = z_{i}^*, \quad t = s^* - \imath \frac{m\omega}{2E}z^2
\end{equation}
respectively.
It is now interesting to investigate properties of derivatives with respect to complex 4-position coordinates. In particular, the chain rule of partial differentiation gives
\begin{equation} \label{eq: complexDiff1}
\frac{\partial}{\partial s}
= \frac{\partial t}{\partial s} \frac{\partial}{\partial t}
+ \frac{\partial x_{i}}{\partial s} \frac{\partial}{\partial x_{i}}
= \frac{\partial}{\partial t}
\end{equation}
\begin{equation} \label{eq: complexDiff2}
\frac{\partial}{\partial s^*}
= \frac{\partial t}{\partial s^*} \frac{\partial}{\partial t}
+ \frac{\partial x_{i}}{\partial s^*} \frac{\partial}{\partial x_{i}}
= \frac{\partial}{\partial t}
\end{equation}
\begin{equation} \label{eq: complexDiff3}
\frac{\partial}{\partial z_i}
= \frac{\partial x_i}{\partial z_i} \frac{\partial}{\partial x_i}
+ \frac{\partial t}{\partial z_i} \frac{\partial}{\partial t}
= \frac{\partial}{\partial x_i} + \imath \frac{m \omega x_i}{E} \frac{\partial}{\partial t}
\end{equation}
\begin{equation} \label{eq: complexDiff4}
\frac{\partial}{\partial z_i^*}
= \frac{\partial x_i}{\partial z_i^*} \frac{\partial}{\partial x_i}
+ \frac{\partial t}{\partial z_i^*} \frac{\partial}{\partial t}
= \frac{\partial}{\partial x_i} - \imath \frac{m \omega x_i}{E} \frac{\partial}{\partial t}
\end{equation}
Note, eqs. (\ref{eq: complexDiff1}) and (\ref{eq: complexDiff3}) have been obtained using eq. (\ref{eq: inv_ict1}); eqs. (\ref{eq: complexDiff2}) and (\ref{eq: complexDiff4}) are based on eq. (\ref{eq: inv_ict2}). It has also been assumed in deriving eqs. (\ref{eq: complexDiff1}) through (\ref{eq: complexDiff4}) that
\begin{equation} \label{eq: complexDiff5}
\frac{\partial z_\nu}{\partial s}=\frac{\partial s}{\partial z_\mu} = \frac{\partial z_\nu^*}{\partial s^*}=\frac{\partial s^*}{\partial z_\mu^*}=0
\end{equation}
indicating that the coordinates $z_i$ and $s$ are independent of each other as are the complex conjugate coordinates $z_i^*$ and $s^*$. This assumption is readily validated using eqs. (\ref{eq: complexDiff1}) through (\ref{eq: complexDiff4}) to directly evaluate each of the derivatives in eq. (\ref{eq: complexDiff5}) in $(x_i,t)$-coordinates.
In further consideration of eqs. (\ref{eq: conftrans1}), it is convenient to write $s=t+iy$ where $y = -(m \omega / 2E) x^2$. The requirement for a continuously differentiable function $\tau(s)=g(t,y)+ih(t,y)$ to be holomorphic is then for the real functions g and h to satisfy the set of Cauchy-Riemann equations
\begin{equation}
\frac{\partial g}{\partial y}=\frac{\partial h}{\partial t} \quad
\frac{\partial h}{\partial y}=-\frac{\partial g}{\partial t}
\end{equation}
or equivalently
\begin{equation} \label{eq: creq}
\frac{\partial^2 \tau}{\partial t^2} + \frac{4E^2}{m^2 \omega^2} \frac{\partial^2 \tau}{\partial x^4} = 0
\end{equation}
It is thus concluded that a function $\psi(x_i,t)$ will also have an equivalent holomorphic form $\theta(z_i)\tau(s)$ in the complex $(z_\mu, s)$-coordinate system providing it is separable and $\tau$ satisfies eq. (\ref{eq: creq}). Here, it is understood that the domain of the Cauchy-Riemann equations in this problem is the complex plane containing s. The Cauchy-Riemann equations put no restriction at all on the form of the function $\theta(z_i)$ since $z_i$ and $s$ are independent coordinates and $z_i$ belongs to a real three-dimensional space.
\section{The Harmonic Oscillator}
The Schr\"{o}dinger equation determining the wavefunction $\psi(x_i, t)$ for a single particle confined in a 3-dimensional harmonic oscillator potential can be expressed in the form
\begin{equation} \label{eq: schrod1}
-\frac{\hbar^2}{2 m} \frac{\partial^2 \psi}{\partial x_i^2} + \frac{1}{2} m \omega^2 x^2\psi = E\psi
\end{equation}
where $\hbar$ is Planck's constant divided by 2$\pi$ and
\begin{equation} \label{eq: schrod2}
E\psi = \imath \hbar \frac{\partial \psi}{\partial t}
\end{equation}
gives the total energy of the particle.
The solution \cite{DFL} to eqs. (\ref{eq: schrod1}) and (\ref{eq: schrod2}) takes the separable form
\begin{eqnarray} \label{eq: psi1}
\psi(x_i,t) = \phi_{l_1}(x_1)\phi_{l_2}(x_2)\phi_{l_3}(x_3)\exp(-\imath Et / \hbar)
\end{eqnarray}
where
\begin{eqnarray} \label{eq: phi1}
\phi_l(x_i) = k_l H_{l}(\xi_i) \exp \left(-\frac{\xi^2}{2} \right)
\end{eqnarray}
$\xi_i=\sqrt{\frac{m \omega}{\hbar}}x_i$, $H_{l_j}$ are Hermite polynomials and $l_1,l_2,l_3$ are positive integers. The normalization constant is
\begin{equation}
k_l = \left( \frac{2 m \omega}{\hbar} \right)^{1/4} \left( \frac{1}{2^l l!} \right)^{1/2}
\end{equation}
In developing the connection between complex $(z_i,s)$-coordinates and the quantum harmonic oscillator, eqs. (\ref{eq: complexDiff3}), (\ref{eq: complexDiff4}) and (\ref{eq: schrod2}) can be combined to give
\begin{equation} \label{eq: complexDiff6}
\frac{\partial}{\partial z_i}
= \frac{\partial}{\partial x_i} + \frac{m \omega}{\hbar} x_i
\end{equation}
\begin{equation} \label{eq: complexDiff7}
\frac{\partial}{\partial z_i^*}
= \frac{\partial}{\partial x_i} - \frac{m \omega}{\hbar} x_i
\end{equation}
These results lead to the operator relationship
\begin{equation}\label{eq: qprop1}
-\frac{\hbar^2}{2m}\frac{\partial}{\partial z_i^* \partial z_i} + \frac{3}{2}\hbar \omega = -\frac{\hbar^2}{2m}\frac{\partial}{\partial x_i^2}+ \frac{1}{2} m \omega^2 x^2
\end{equation}
enabling the Schr\"{o}dinger equation (\ref{eq: schrod1}) for the harmonic oscillator to be expressed in the concise form
\begin{equation}\label{eq: complexSchrod1}
-\frac{\hbar^2}{2m}\frac{\partial \psi}{\partial z_i^* \partial z_i} + \frac{3}{2}\hbar \omega \psi= E \psi
\end{equation}
It is also readily shown using eqs. (\ref{eq: complexDiff1}) and (\ref{eq: schrod2}) that
\begin{equation} \label{eq: complexSchrod2}
E\psi = \imath \frac{\partial \psi}{\partial s}
\end{equation}
Eqs. (\ref{eq: complexSchrod1}) and (\ref{eq: complexSchrod2}) together, therefore, constitute a complete description of the quantum harmonic oscillator in terms of $(z_i,s)$-coordinates. On comparing eq. (\ref{eq: schrod1}) and (\ref{eq: complexSchrod1}), it is clear that the harmonic oscillator potential term in the original Schr\"{o}dinger equation is replaced by a constant term in complex coordinates.
The oscillator function (\ref{eq: psi1}) is readily transformed into $(z_i,s)$-coordinates using eqs. (\ref{eq: conftrans1}) to give
\begin{eqnarray} \label{eq: psi3}
\psi(z_i,s) = \theta_{l_1}(z_1)\theta_{l_2}(z_2)\theta_{l_3}(z_3)\tau(s)
\end{eqnarray}
where
\begin{eqnarray} \label{eq: psi4}
\theta_l(z_i) = k_l H_{l}(\zeta_i), \quad \tau(s) = \exp(-\imath Es / \hbar)
\end{eqnarray}
and $\zeta_i = \sqrt{\frac{m \omega}{\hbar}}z_i$. It is notable that eq. (\ref{eq: psi1}) and (\ref{eq: psi3}) are similar except that eq. (\ref{eq: psi3}) does not contain a gaussian term. It is also notable that $\tau(s)$ is a continuously differentiable solution of eq.(\ref{eq: creq}) thus demonstrating that the oscillator function $\psi(z_i, s)$ is holomorphic.
In consideration of the foregoing arguments, it is of interest that eqs. (\ref{eq: conftrans1}) reduces to the form $z_i=x_i, s=t$ on setting $\omega=0$. It also apparant that eq. (\ref{eq: complexSchrod1}) reduces to the free field form of the Schr\"{o}dinger equation under these same conditions. The converse of this argument is that harmonic interactions may be introduced into the free-field Schr\"{o}dinger equation through the replacement $t \rightarrow t - \imath \frac{ m\omega}{2 E}x^2$ exactly equivalent to the more usual approach of adding the oscillator potential into the hamiltonian for the oscillator.
\section{Ladder Operators}
As is well known, the Schr\"{o}dinger equation (\ref{eq: schrod1}) can be simplified in terms of the non-relativistic raising and lowering operators:
\begin{equation} \label{lower_nr}
\hat{a}_i = \sqrt{\frac{\hbar}{2 m\omega}} \frac{\partial}{\partial x_i} + \sqrt{\frac{m\omega}{2 \hbar}} x_i
\end{equation}
\begin{equation} \label{raise_nr}
\hat{a}_i^\dag = -\sqrt{\frac{\hbar}{2 m\omega}} \frac{\partial}{\partial x_i} + \sqrt{\frac{m\omega}{2 \hbar}} x_i
\end{equation}
to give
\begin{equation} \label{schrod3}
\hat{a}_i^\dag \hat{a}_i \psi + \frac{3}{2}\psi = \frac{E}{\omega \hbar} \psi
\end{equation}
On comparing eq. (\ref{eq: complexDiff6}) and (\ref{eq: complexDiff7}) with eq. (\ref{lower_nr}) and (\ref{raise_nr}), it can be seen that
\begin{equation} \label{ladder_nr}
\hat{a}_i = \frac{1}{\sqrt{2}}\frac{\partial}{\partial \zeta_i}, \quad \hat{a}_i^\dag = -\frac{1}{\sqrt{2}}\frac{\partial}{\partial \zeta_i^*}
\end{equation}
It is clear therefore that raising and lowering operators have a more concise representation in complex $(z_i,s)$-coordinates than in real $(x_i,t)$-coordinates.
Applying ladder operators to the eigenfunctions for the harmonic oscillator gives
\begin{equation} \label{eq: lower1}
\hat{a}_1\psi(l_1, l_2, l_3) = \sqrt{l_1} \psi(l_1-1, l_2, l_3)
\end{equation}
\begin{equation} \label{eq: lower2}
\hat{a}_2\psi(l_1, l_2, l_3) = \sqrt{l_2} \psi(l_1, l_2-1, l_3)
\end{equation}
\begin{equation} \label{eq: lower3}
\hat{a}_3\psi(l_1, l_2, l_3) = \sqrt{l_3} \psi(l_1, l_2, l_3-1)
\end{equation}
for lowering the state of the oscillator, alongside the conditions
\begin{equation} \label{eq: raise1}
\hat{a}_1^\dag \psi(l_1, l_2, l_3) = \sqrt{l_1+1} \psi(l_1+1, l_2, l_3)
\end{equation}
\begin{equation} \label{eq: raise2}
\hat{a}_2^\dag \psi(l_1, l_2, l_3) = \sqrt{l_2+1} \psi(l_1, l_2+1, l_3)
\end{equation}
\begin{equation} \label{eq: raise3}
\hat{a}_3^\dag \psi(l_1, l_2, l_3) = \sqrt{l_3+1} \psi(l_1, l_2, l_3+1)
\end{equation}
for raising it.
It is interesting now to compare the results of conformal transformation in this paper with two other complex formulations of the quantum harmonic oscillator. These other formulations are based on integral transformations. One of these is the Bargmann representation obtained using the Segal-Bargmann transformation:
\begin{equation} \label{sb_trans}
\theta_{l}(a_i) = \pi^{-1/4}\int \phi_{l}(\xi_i) \exp \left(\frac{-\xi_i^2-a_i^2}{2} \right) \exp \left( -\sqrt{2} a_i \xi_i \right) d\xi_i
\end{equation}
The other is the conjugate of the Bargmann representation given by
\begin{equation} \label{conjugate_trans}
\theta_{l}(b_i) = \pi^{-1/4}\int \theta_{l}(a_i)\exp \left( - a_i b_i \right) da_i
\end{equation}
The results of the transformations (\ref{sb_trans}) and (\ref{conjugate_trans}) are presented in table 1 including both ladder operators and eigenfunctions. The ladder operators can be inserted into eq. (\ref{schrod3}) and the eigenfunctions validated as the solution.
\begin{table}[h]
\centering
\begin{tabular}{|c|c|c|c|}
\hline
Space & $\hat{a}_i$ & $\hat{a}_i^\dag $ & $\theta_{l}$ \\ \hline
Bargmann & $\partial / \partial a_i$ &$a_i$ & $a_i^l / \sqrt{l!}$ \\ \hline
Conjugate & $b_i$ & -$\partial / \partial b_i$ & $\sqrt{l!} / b_i^{l+1}$ \\ \hline
Conformal & $2^{-1 / 2} \partial / \partial \zeta_i$ & -$2^{-1 / 2} \partial / \partial \zeta_i^*$ & $k_l H_l(\zeta_i)$\\ \hline
\end{tabular}
\caption{Ladder operators and eigenfunctions for the harmonic oscillator in different complex spaces.}
\label{tab:}
\end{table}
\section{Concluding Remarks}
It has been shown the mathematical description of the non-relativistic quantum harmonic oscillator can be simplified through the use of a conformal transformation. In the transformed coordinate system, time is a complex quantity. The real part of this complex time is the world time; the imaginary part represents the harmonic interaction confining the particle.
\newpage
| 2024-02-18T23:40:00.560Z | 2010-03-03T02:18:14.000Z | algebraic_stack_train_0000 | 1,076 | 2,578 |
|
proofpile-arXiv_065-5534 | \section{Introduction}
\label{sec1}
\setcounter{equation}{0}
\setcounter{theorem}{0}
Let $n>m$ and let $A$ be a real $(m\times n)$-matrix
of rank $m$. The kernel of such a matrix has the
dimension $n-m$ and hence can, in dependence of the
dimensions, be a large subspace of the $\mathbb{R}^n$.
Nevertheless, the set of all $x$ for which
\begin{equation} \label{eq1.1}
\|Ax\|\geq\delta\,\|A\|\|x\|
\end{equation}
holds fills, in the high-dimensional case, often
almost the complete $\mathbb{R}^n$ once $\delta$
falls below a certain bound; the involved norms
are here and throughout the paper the euclidean
norm on the $\mathbb{R}^m$ and the $\mathbb{R}^n$
and the assigned spectral norm of matrices. Let
$\chi$ be the characteristic function of the
set of all $x$ for which
\mbox{$\|Ax\|<\delta\,\|A\|\|x\|$} holds, and
let $\nu_n$ be the volume of the unit ball in
$\mathbb{R}^n$. The normed area measure
\begin{equation} \label{eq1.2}
\frac{1}{n\nu_n}\int_{S^{n-1}}\!\chi(\eta)\,\diff{\eta}
\end{equation}
of the subset of the unit sphere on which the
condition (\ref{eq1.1}) is violated takes in
such cases an extremely small value, which
conversely again means that (\ref{eq1.1}) holds
on an overwhelmingly large part of the unit
sphere and with that of the full space. The
aim of the present paper is to study this
phenomenon in dependence of characteristic
quantities like the ratio of the dimensions
$m$ and $n$ and the extremal singular values
of the matrices qualitatively and, as far as
possible, also quantitatively.
The described effect is best understood for orthogonal
projections. For matrices of this kind, this observation
is a direct consequence of the random projection theorem
(see Lemma~5.3.2 in \cite{Vershynin}, for example), which
is in close connection with the Johnson--Lindenstrauss
theorem \cite{Johnson-Lindenstrauss}. The random projection
theorem deals with orthogonal projections from the
$\mathbb{R}^n$ onto random subspaces of lower dimension
$m$, but equally one can consider orthogonal projections
$Px$ of random vectors $x$ to the~$\mathbb{R}^m$. The
theorem states that with probability greater than
$1-2\exp(-c\,\varepsilon^2m)$
\begin{equation} \label{eq1.3}
(1-\varepsilon)\sqrt{\frac{m}{n}}\,\|x\|
\leq \|Px\| \leq
(1+\varepsilon)\sqrt{\frac{m}{n}}\,\|x\|
\end{equation}
holds for all $x$ on the unit sphere and thereby
also in the full space. The random projection
theorem is a manifestation of the concentration
of measure phenomenon, which plays a fundamental
role in the analysis of very many problems in
high space dimensions and became a backbone of
high-dimensional probability theory and modern
data science. The interest in the concentration
of measure phenomenon arose in the early 1970s
in the study of the asymptotic theory of Banach
spaces. Classical texts are \cite{Ledoux} and
\cite{Ledoux-Talagrand}. An up-to-date exposition
containing a lot of information on random vectors
and matrices is \cite{Vershynin}. In the present
article, we carefully reconsider the random
projection theorem. Among other things, we
calculate the normed area measure (\ref{eq1.2})
for projection matrices of the described kind
for even-numbered differences of the two
dimensions exactly and derive a very sharp
inclusion for the odd-numbered case. The
results for these projection matrices serve
as a basis for the examination of general
matrices in dependence of their singular values.
One might ask what is known about these. The
simple answer is that this depends on the
class of matrices one considers. Graph theory
\cite{Brouwer-Haemers},
\cite{Cvetkovic-Rowlinson-Simic} is an
important source of information. There is
an extensive literature about the extremal
singular values of random matrices with
independent, identically distributed
entries. An early breakthrough was Edelman's
thesis \cite{Edelman}. Other significant
contributions are
\cite{Rudelson-Vershynin}, \cite{Tao-Vu},
and, very recently,
\cite{Livshyts-Tikhomirov-Vershynin}.
Random matrices play an important role
in fields like compressed sensing
\cite{Foucart-Rauhut} and all sorts
of data acquisition and compression
techniques. Our interest in the problem
originates from the attempt \cite{Yserentant}
to extend the applicability of modern tensor
product methods \cite{Hackbusch} to more
general problem classes. Assume that we
are looking for the solution
$u:\mathbb{R}^m\to\mathbb{R}$ of the
Laplace-like equation
\begin{equation} \label{eq1.4}
-\Delta u+\mu u=f
\end{equation}
that vanishes at infinity, where $\mu>0$ is
constant and the right-hand side $f$ is,
for instance, a product of functions depending
only on a single component of $x$ or on the
difference of two such components. The
question is how well such structures are
reflected in the solution of the equation.
Let us assume that the right-hand side is
of the form $f(x)=F(Tx)$, with a function
$F:\mathbb{R}^n\to\mathbb{R}$, $n>m$, with
an integrable Fourier transform and with $T$
a matrix of full rank that is determined by
the structure of the underlying problem. As
shown in \cite{Yserentant}, the solution
is then the trace $u(x)=U(Tx)$ of the
function
\begin{equation} \label{eq1.5}
U(y)=\Big(\frac{1}{\sqrt{2\pi}}\Big)^n\!\int
\frac{1}{\mu+\|T^t\omega\|^2}\,\fourier{F}(\omega)\,
\mathrm{e}^{\,\mathrm{i}\,\omega\cdot y}\,\diff{\omega}.
\end{equation}
The function (\ref{eq1.5}) is in the domain
of the operator $\mathcal{L}$ given by
\begin{equation} \label{eq1.6}
(\mathcal{L} U)(y)=\Big(\frac{1}{\sqrt{2\pi}}\Big)^n\!\int
\big(\mu+\|T^t\omega\|^2\big)\fourier{U}(\omega)\,
\mathrm{e}^{\,\mathrm{i}\,\omega\cdot y}\,\diff{\omega}
\end{equation}
and solves by definition the degenerate
second-order elliptic equation $\mathcal{L} U=F$.
It can be calculated approximately by
means of the iteration
\begin{equation} \label{eq1.7}
U_{k+1}=(I-\alpha\mathcal{L})U_k+\alpha F,
\end{equation}
where the operator $\alpha$ is given by
\begin{equation} \label{eq1.8}
(\alpha F)(y)=\Big(\frac{1}{\sqrt{2\pi}}\Big)^n\!\int
\frac{1}{\mu+\|T^t\|^2\|\omega\|^2}\,
\fourier{F}(\omega)\,
\mathrm{e}^{\,\mathrm{i}\,\omega\cdot y}\,\diff{\omega}.
\end{equation}
Provided that $\|T^t\omega\|\geq\delta\,\|T^t\|\|\omega\|$
holds on the support of $\fourier{F}$, the $L_1$-norm
of the Fourier transform of the error is in every
iteration step reduced by the factor \mbox{$1-\delta^2$},
or by more than the factor $1-\delta$ with polynomial
acceleration. If this condition is only violated on a
very small set, the additional error can in general be
neglected without hard conditions to $\fourier{F}$ or
$F$ itself. The idea is to approximate the kernel in
(\ref{eq1.8}) by a linear combination of Gauss functions.
If $F$ is as in the example above the product of
lower-dimensional functions depending only on small
groups of components of $Tx$, the iterates are then
composed of functions of the same type.
\section{Reformulations as volume integrals and first estimates}
\label{sec2}
\setcounter{equation}{0}
\setcounter{theorem}{0}
The surface integrals (\ref{eq1.2}) are not easily
accessible and are difficult to calculate and estimate.
We reformulate them therefore at first as volume
integrals and draw some first conclusions from
these representations. The starting point is the
decomposition
\begin{equation} \label{eq2.1}
\int_{\mathbb{R}^n}f(x)\,\diff{x}\,=
\int_{S^{n-1}}\left(\int_0^\infty\!f(r\eta)r^{n-1}\,\diff{r}\right)\!\,\diff{\eta}
\end{equation}
of the integrals of functions in $L_1$ into
an inner radial and an outer angular part.
Inserting the characteristic function of the
unit ball, one recognizes that the area of
the $n$-dimensional unit sphere is $n\nu_n$,
with $\nu_n$ the volume of the unit ball. If
$f$ is rotationally symmetric, $f(r\eta)=f(re)$
holds for every $\eta\in S^{n-1}$ and every
fixed, arbitrarily given unit vector $e$.
In this case, (\ref{eq2.1}) reduces
therefore to
\begin{equation} \label{eq2.2}
\int f(x)\,\diff{x}\,=n\nu_n\int_0^\infty\!f(re)r^{n-1}\,\diff{r}.
\end{equation}
The volume measure on the $\mathbb{R}^n$ will
in the following be denoted by $\lambda$.
\begin{lemma} \label{thm2.1}
Let $A$ be an arbitrary matrix of dimension $m\times n$,
$m<n$, let $\chi$ be the characteristic function of
the set of all $x\in\mathbb{R}^n$ for which
$\|Ax\|<\delta\,\|A\|\|x\|$ holds, and let
$W:\mathbb{R}^n\to\mathbb{R}$ be a rotationally
symmetric function with integral
\begin{equation} \label{eq2.3}
\int W(x)\,\diff{x}\,=\,1.
\end{equation}
The weighted surface integral \rmref{eq1.2}
then takes the value
\begin{equation} \label{eq2.4}
\int \chi(x)W(x)\,\diff{x}.
\end{equation}
\end{lemma}
\begin{proof}
Let $e$ be a given unit vector. For $\eta\in S^{n-1}$ and
$r>0$ then $\chi(r\eta)=\chi(\eta)$ and $W(r\eta)=W(re)$
holds and the integral (\ref{eq2.4}) can by (\ref{eq2.1})
be written as
\begin{displaymath}
\int\chi(x)W(x)\,\diff{x}\,=
\int_{S^{n-1}}\chi(\eta)\left(\int_0^\infty\!W(re)r^{n-1}\,\diff{r}\right)\!\,\diff{\eta}.
\end{displaymath}
Because the inner integral takes by (\ref{eq2.2})
and (\ref{eq2.3}) the value
\begin{displaymath}
\int_0^\infty\!W(re)r^{n-1}\,\diff{r}=\frac{1}{n\nu_n},
\end{displaymath}
this proves the proposition.
\end{proof}
An obvious choice for the weight function $W$,
which will later still play an important role
and will be used at several places, is the
normed Gauss function
\begin{equation} \label{eq2.5}
W(x)=\Big(\frac{1}{\sqrt{\pi}}\Big)^n\exp\big(-\|x\|^2\big).
\end{equation}
Another possible choice is the characteristic
function of the ball of radius $R$ around the
origin divided by the volume of this ball.
It leads to the following lemma.
\begin{lemma} \label{thm2.2}
Let $A$ be a matrix of dimension $m\times n$, $m<n$,
and let $\lambda$ be the volume measure on the
$\mathbb{R}^n$. The weighted integral \rmref{eq1.2}
over the surface of the unit ball is then
independent of the radius $R$ equal to the volume
ratio
\begin{equation} \label{eq2.6}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}.
\end{equation}
\end{lemma}
Because the euclidean length of a vector and
the volume of a set are invariant to orthogonal
transformations, the surface ratio (\ref{eq1.2})
and the volume ratio (\ref{eq2.6}) as well
depend only on the singular values of the
matrix under consideration.
\begin{lemma} \label{thm2.3}
Let $A$ be a matrix of dimension $m\times n$, $m<n$,
with singular value decomposition $A=U\Sigma V^t$.
The volume ratios \rmref{eq2.6} are then equal to
the volume ratios
\begin{equation} \label{eq2.7}
\frac
{\lambda\big(\big\{x\,\big|\,\|\Sigma x\|<\delta\,\|\Sigma\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)};
\end{equation}
that is, they depend exclusively on the singular
values of the matrix $A$.
\end{lemma}
\begin{proof}
As the multiplication with the orthogonal matrices
$U$ and $V^t$, respectively, does not change the
euclidean norm of a vector, the set of all
$x\in\mathbb{R}^n$ for which
\begin{displaymath}
\|Ax\|<\delta\,\|A\|\|x\|,\quad \|x\|\leq R,
\end{displaymath}
holds coincides with the set of all $x$ for which
we have
\begin{displaymath}
\|\Sigma V^tx\|<\delta\,\|\Sigma\|\|V^tx\|,
\quad \|V^tx\|\leq R.
\end{displaymath}
As the volume is invariant to orthogonal
transformations, the proposition follows.
\end{proof}
Orthogonal projections, or in other words matrices
with one as the only singular value, represent one of
the few cases for which the volume ratios (\ref{eq2.6})
can be more or less explicitly calculated. Orthogonal
projections are of particular importance and will,
as said, serve as the anchor for many of our estimates.
Again, it suffices to consider the corresponding
diagonal matrices $\Sigma$, denoted in the
following by $P$.
\begin{theorem} \label{thm2.4}
Let $P$ be the $(m\times n)$-matrix that extracts
from a vector in $\mathbb{R}^n$ its first $m$
components. For $0\leq\delta<1$ and all radii
$R>0$, then
\begin{equation} \label{eq2.8}
\frac{\lambda\big(
\big\{x\,\big|\,\|Px\|<\delta\,\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
=\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
\end{equation}
holds, where the function $\psi$ is defined
by the integral expression
\begin{equation} \label{eq2.9}
\psi(\varepsilon)=
\frac{2\,\Gamma(n/2)}{\Gamma(m/2)\Gamma((n-m)/2)}
\int_0^{\varepsilon}\frac{t^{m-1}}{(1+t^2)^{n/2}}\,\diff{t}.
\end{equation}
\end{theorem}
\begin{proof}
Differing from the notation in the theorem
but consistent within the proof, we split
the vectors in $\mathbb{R}^n$ into parts
$x\in\mathbb{R}^m$ and $y\in\mathbb{R}^{n-m}$.
The set whose volume has to be calculated
consists then of the points in the given
ball for which
\begin{displaymath}
\|x\|<\delta\,\Big\|\Big(\,\begin{matrix}x\\y\end{matrix}\,\Big)\Big\|
\end{displaymath}
or, resolved for the norm of the component
$x\in\mathbb{R}^m$,
\begin{displaymath}
\|x\|<\varepsilon\,\|y\|, \quad
\varepsilon=\frac{\delta}{\sqrt{1-\delta^2}},
\end{displaymath}
holds. For homogeneity reasons, that is, by
Lemma~\ref{thm2.2}, we can restrict ourselves
to the ball of radius $R=1$. The volume can
then be expressed as double integral
\begin{displaymath}
\int\bigg(\int H\big(\varepsilon\|y\|-\|x\|\big)
\chi\big(\|x\|^2+\|y\|^2\big)\,\diff{x}\bigg)\,\diff{y},
\end{displaymath}
where $H(t)=0$ for $t\leq0$, $H(t)=1$ for $t>0$,
$\chi(t)=1$ for $t\leq 1$, and $\chi(t)=0$ for
arguments $t>1$. In terms of polar coordinates,
that is, by (\ref{eq2.2}), it reads as
\begin{displaymath}
(n-m)\nu_{n-m}\int_0^\infty\bigg(
m\nu_m\int_0^{\varepsilon s}\chi\big(r^2+s^2\big)r^{m-1}\,\diff{r}
\bigg)s^{n-m-1}\,\diff{s},
\end{displaymath}
with $\nu_d$ the volume of the $d$-dimensional unit
ball. Substituting $t=r/s$ in the inner integral,
the upper bound becomes independent of $s$ and
the integral can be written~as
\begin{displaymath}
(n-m)\nu_{n-m}\int_0^\infty\bigg(m\nu_m\,s^m\!
\int_0^{\varepsilon}\chi\big(s^2(1+t^2)\big)t^{m-1}\,\diff{t}
\bigg)s^{n-m-1}\,\diff{s},
\end{displaymath}
and interchanging the order of integration,
it attains finally the value
\begin{displaymath}
\frac{(n-m)\nu_{n-m}\,m\nu_m}{n}
\int_0^{\varepsilon}\frac{t^{m-1}}{(1+t^2)^{n/2}}\,\diff{t}.
\end{displaymath}
Dividing this by the volume $\nu_n$ of the
unit ball itself and remembering that
\begin{displaymath}
\nu_d=\frac{2}{d}\,\frac{\pi^{d/2}}{\Gamma(d/2)},
\end{displaymath}
this completes the proof of the theorem.
\end{proof}
The following lemma describes the dependence of the
volume ratio (\ref{eq2.8}) on the dimensions $m$ and
$n$. In conjunction with Theorem~\ref{thm3.2} below
it can be used to enclose the volume ratio from
both sides for uneven differences of the dimensions.
\begin{lemma} \label{thm2.5}
The volume ratio \rmref{eq2.8} decreases, for $n$ kept
fixed, when $m$ increases, and it increases, for $m$
kept fixed, when $n$ increases.
\end{lemma}
\begin{proof}
The set in the numerator on the left-hand side of (\ref{eq2.8})
gets smaller when $m$ gets larger. This proves the first
proposition. The argumentation for increasing dimension $n$
is more involved. It is based on the representation from
Lemma~\ref{thm2.1} with the weight function (\ref{eq2.5}).
For $x\in\mathbb{R}^n$ and $y\in\mathbb{R}^p$, let
\begin{displaymath}
\chi(x,y)=
\begin{cases}
\,1,& \text{if $\|Px\|^2<\delta^2\big(\|x\|^2+\|y\|^2\big)$}\\
\,0,& \text{otherwise}.
\end{cases}
\end{displaymath}
By Lemma~\ref{thm2.1}, the volume ratio (\ref{eq2.8})
can then be written as the integral
\begin{displaymath}
\Big(\frac{1}{\sqrt{\pi}}\Big)^n\!
\int\chi(x,0)\exp\big(-\|x\|^2\big)\,\diff{x}
\end{displaymath}
over the $\mathbb{R}^n$. This integral takes
by Fubini's theorem the same value as the
integral
\begin{displaymath}
\Big(\frac{1}{\sqrt{\pi}}\Big)^{n+p}\!
\int\chi(x,0)\exp\big(-\big(\|x\|^2+\|y\|^2\big)\big)\,\diff{(x,y)}
\end{displaymath}
over the $\mathbb{R}^n\times\mathbb{R}^p$ and
can be estimated from above by the integral
\begin{displaymath}
\Big(\frac{1}{\sqrt{\pi}}\Big)^{n+p}\!
\int\chi(x,y)\exp\big(-\big(\|x\|^2+\|y\|^2\big)\big)\,\diff{(x,y)}.
\end{displaymath}
This integral takes by Lemma~\ref{thm2.1} the same value
as the volume ratio (\ref{eq2.8}) , with the original
dimension $m$, but with the dimension $n$ now
replaced by $n'=n+p$.
\end{proof}
\pagebreak
The volume ratios (\ref{eq2.6}) can be bounded
from both sides in terms of the, as we will see,
more or less explicitly known volumes ratios
(\ref{eq2.8}), i.e., of the function (\ref{eq2.9}).
\begin{theorem} \label{thm2.6}
Let $n>m$, let $A$ be an $(m\times n)$-matrix of full
rank, and let $\kappa$ be the condition number of
$A$, the ratio of its maximum to its minimum singular
value. If $\kappa\delta$ is less than one, then for
all radii $R>0$ the upper estimate
\begin{equation} \label{eq2.10}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\leq
\psi\bigg(\frac{\kappa\delta}{\sqrt{1-\kappa^2\delta^2}}\bigg)
\end{equation}
holds, where $\psi$ is the function \rmref{eq2.9}.
Without further conditions to $\delta<1$, conversely
\begin{equation} \label{eq2.11}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\geq
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg).
\end{equation}
holds. For orthogonal projections, that is,
if $\kappa=1$, in both cases equality holds.
\end{theorem}
\begin{proof}
We can restrict ourselves in the proof to the diagonal
matrices $\Sigma$ from Lemma~\ref{thm2.3}. The proposition
follows then rather immediately from the inequalities
\begin{displaymath}
\sigma_1\|Px\|\leq\|\Sigma x\|\leq\sigma_m\|Px\|
\end{displaymath}
and the fact that $\|\Sigma\|=\sigma_m$ comparing
the corresponding volumes.
\end{proof}
Undoubtedly, (\ref{eq2.10}) and (\ref{eq2.11})
are despite their generality in many cases
rather poor estimates because they largely ignore
the underlying geometry. If the singular values
$\sigma_1\leq\sigma_2\leq\cdots\leq\sigma_m$
of the matrix $A$ cluster around $\sigma_m>0$
or are, up to very few, even equal to $\sigma_m$,
the following lemma opens a way out and
provides a remedy.
\begin{lemma} \label{thm2.7}
Let $\sigma_k=\sigma_m$ for all $k>m_0$ and let
$P'$ be the matrix that extracts from a vector
$x\in\mathbb{R}^n$ its components $x_k$,
$m_0<k\leq m$. The volume ratio \rmref{eq2.6}
is then less than or at most equal to the
volume ratio
\begin{equation} \label{eq2.12}
\frac
{\lambda\big(\big\{x\,\big|\,\|P'x\|<\delta\,\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}.
\end{equation}
\end{lemma}
\begin{proof}
This follows by Lemma~\ref{thm2.3} from
$\sigma_m\|P'x\|\leq\|\Sigma x\|$
and $\|\Sigma\|=\sigma_m$.
\end{proof}
The volume ratio (\ref{eq2.12}) possesses then
a representation like that in Theorem~\ref{thm2.4},
where $m'=m-m_0$ replaces the dimension $m$. But
above all the potentially disastrous influence of
the condition number vanishes. As indicated, the
argumentation can be generalized to the case that
the ratio $\kappa'=\sigma_m/\sigma_k$ is small in
comparison to $\kappa$ for an index $k=m_0+1$
that is small in comparison to $m$.
The example that we have here in mind arises
in connection with the iterative solution of
high-dimensional elliptic partial differential
equations as sketched in the introduction.
The dimensions of the matrices $A=T^t$ under
consideration are
\begin{equation} \label{eq2.13}
m=3\times N, \quad n=3\times\frac{N(N+1)}{2}.
\end{equation}
The vectors $x$ in $\mathbb{R}^{m}$ and
$\mathbb{R}^{n}$, respectively, are partitioned
into subvectors $x_i\in\mathbb{R}^{3}$. The
matrices $T$ map the parts $x_i$ of
$x\in\mathbb{R}^m$ first to themselves and
then to the differences $x_i-x_j$, $i<j$.
If one thinks of the Schr\"odinger equation,
the $x_i$ are associated with the positions
of $N$ electrons or other particles. The
structure of $T$ reflects then that approximate
solutions are sought that are composed of
products of orbitals, depending only on a
single component $x_i$, and of geminals,
functions of the differences $x_i-x_j$.
The euclidean norm of the vector
$Tx\in\mathbb{R}^n$ is given by
\begin{equation} \label{eq2.14}
\|Tx\|^2=\sum_{i=1}^N\|x_i\|^2+
\frac12\sum_{i=1}^N\sum_{j=1}^N\|x_i-x_j\|^2
\end{equation}
or, after rearrangement, with the rank
three map $T_0x=x_1+x_2+\cdots+x_m$ by
\begin{equation} \label{eq2.15}
\|Tx\|^2=(N+1)\|x\|^2-\|T_0x\|^2.
\end{equation}
The square matrix $T^tT$ therefore has the
eigenvalues $1$ and $N+1$ and only the
first three singular values of the matrix
$T^t$ differ from the last one. If only some
of the differences $x_i-x_j$ are taken into
account, the minimum singular value of the
resulting matrix remains $\sigma_1=1$ and
the maximum singular value and the ratio
of the dimensions as well can be bounded
in terms of the degrees of the vertices
of the underlying graph \cite{Yserentant}.
The spectral theory of graphs is itself
a large field \cite{Brouwer-Haemers},
\cite{Cvetkovic-Rowlinson-Simic} of great
importance and has numerous applications.
\section{Exact representations for orthogonal projections}
\label{sec3}
\setcounter{equation}{0}
\setcounter{theorem}{0}
One of the primary aims of this paper is a detailed
study of the volume ratio (\ref{eq2.8}),
\begin{equation} \label{eq3.1}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg),
\quad 0\leq\delta<1,
\end{equation}
and of its limit behavior when the dimensions tend
to infinity. The starting point is at first an
integral representation of this expression, which
can also serve as a basis for its approximate
calculation via a quadrature formula.
\begin{theorem} \label{thm3.1}
The expression \rmref{eq3.1} possesses for
$0\leq\delta<1$ the representation
\begin{equation} \label{eq3.2}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
=\frac{2\,\Gamma(n/2)}{\Gamma(m/2)\Gamma((n-m)/2)}\,
\int_0^\delta(1-t^2)^\alpha t^{m-1}\,\diff{t},
\end{equation}
where the exponent $\alpha\geq -1/2$ is given by
\begin{equation} \label{eq3.3}
\alpha=\frac{n-m-2}{2}
\end{equation}
and takes nonnegative values for dimensions
$n\geq m+2$.
\end{theorem}
\begin{proof}
For abbreviation, we introduce the function
\begin{displaymath}
f(\delta)=g\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg),
\quad
g(\varepsilon)=
\int_0^{\varepsilon}\frac{t^{m-1}}{(1+t^2)^{n/2}}\,\diff{t},
\end{displaymath}
on the interval $0\leq\delta<1$. Its derivative
is the continuous function
\begin{displaymath}
f'(\delta)=(1-\delta^2)^\alpha\delta^{m-1}.
\end{displaymath}
Because $f(0)=0$, it possesses therefore the
representation
\begin{displaymath}
f(\delta)=\int_0^\delta(1-t^2)^\alpha t^{m-1}\,\diff{t}.
\end{displaymath}
This already proves the proposition.
\end{proof}
The integral (\ref{eq3.2}) can by means of the
substitution $t=\sin\varphi$ be transformed
into an integral over a trigonometric polynomial
and can thus be calculated in terms of elementary
functions. This does, however, not help too much
because of the inevitably arising cancellation
effects as soon as one tries to evaluate the
result numerically. Such problems can be avoided
if $\alpha$ is an integer, that is, if $m$ and
$n$ are both even or both odd. The volume ratio
(\ref{eq3.1}) is then a polynomial in $\delta$
that is composed of positive terms, which can as
such be summed up in a numerically stable way.
\begin{theorem} \label{thm3.2}
If the difference $n-m$ of the dimensions
is even, the function
\begin{equation} \label{eq3.4}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
=\;\sum_{j=0}^{k}\,
\frac{\Gamma(k+l+1)}{\Gamma(k-j+1)\Gamma(l+j+1)}\,
(1-\delta^2)^{k-j}\,\delta^{2(l+j)}
\end{equation}
is a polynomial of degree $n-2$ in $\delta$,
where $k$ and $l$ are given by
\begin{equation} \label{eq3.5}
k=\frac{n-m-2}{2},\quad l=\frac{m}{2}.
\end{equation}
\end{theorem}
\begin{proof}
Let $\nu<k+1$ be a nonnegative integer and
set for $j=0,\ldots,\nu$
\begin{displaymath}
a_j=\frac{\Gamma(k+1)\Gamma(l)}{2\,\Gamma(k-j+1)\Gamma(l+j+1)}.
\end{displaymath}
Because of $2la_0=1$ and $(l+j)a_j-(k-j+1)a_{j-1}=\,0$
for $j=1,\ldots,\nu$, then
\begin{displaymath}
\frac{\diff{}}{\diff t}\,\bigg\{
\sum_{j=0}^\nu a_j\,(1-t^2)^{k-j}\,t^{2(l+j)}\bigg\} =\,
(1-t^2)^k\,t^{2l-1}+R_\nu(t)
\end{displaymath}
holds on the set of all $t$ between $-1$ and $1$,
where the remainder is given by
\begin{displaymath}
R_\nu(t)=2\,(\nu-k)\,a_\nu\,(1-t^2)^{k-\nu-1}\,t^{2(l+\nu)+1}.
\end{displaymath}
If $k$ is, as in the present case, itself an
integer and $\nu=k$ is chosen, this remainder
vanishes. As the function possesses, in terms
of the given $k$ and $l$, the representation
\begin{displaymath}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg) =\;
\frac{2\,\Gamma(k+l+1)}{\Gamma(k+1)\Gamma(l)}\,
\int_0^{\delta}(1-t^2)^k\,t^{2l-1}\,\diff{t},
\end{displaymath}
its derivative and that of the right-hand side
of (\ref{eq3.4}) thus coincide. As both sides
of this equation take at $\delta=0$ the value
zero, this proves the proposition.
\end{proof}
For \mbox{$m=128$} and \mbox{$n=256$}, for example, the
function (\ref{eq3.4}) takes for \mbox{$\delta\leq 1/4$}
values less than \mbox{$1.90\cdot 10^{-42}$}, and even
for \mbox{$\delta\leq 1/2$} still values less than
\mbox{$6.95\cdot 10^{-10}$}. For \mbox{$m=1024$} and
\mbox{$n=2048$}, these values fall to
\mbox{$2.68\cdot 10^{-66}$} and
\mbox{$3.54\cdot 10^{-325}$}, that is, de facto to
zero. This clearly demonstrates the announced effect.
The coefficients in (\ref{eq3.4}) are rational
numbers. They can be calculated recursively
starting from the last one, which takes
independent of $m$ and $n$ or $k$ and $l$
the value one.
Things become particularly simple when $m$
and $n$ are both even and $k$ and $l$ are
then both integers. The representation
(\ref{eq3.4}) then turns into the sum
\begin{equation} \label{eq3.6}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
=\;\sum_{j=l}^{k+l}\,\binom{k+l\,}{j}
(1-\delta^2)^{k+l-j}\,\delta^{2j}
\end{equation}
of Bernstein polynomials of order $k+l=n/2-1$
in the variable $\delta^2$. One can even go
a step further. Let $\chi$ be a step function
with values $\chi(t)=0$ for $t<m/n$ and
$\chi(t)=1$ for $t>m/n$. The representation
can then be considered as the approximation
\begin{equation} \label{eq3.7}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
=\;\sum_{j=0}^{k+l}\,
\chi\bigg(\frac{j}{k+l}\bigg)\binom{k+l\,}{j}
(1-\delta^2)^{k+l-j}\,\delta^{2j}
\end{equation}
of $\chi$ by the Bernstein polynomial of order
$n/2-1$ in the variable $\delta^2$. If the
ratio of $m$ and $n$ is kept fixed, these
polynomials tend at all points $\delta$
less than
\begin{equation} \label{eq3.8}
\delta_0=\sqrt{\frac{m}{n}}
\end{equation}
to zero and at all points $\delta>\delta_0$ to one.
The convergence is even uniform outside every open
neighborhood of jump position $\delta_0$. This
follows from the theory of Bernstein polynomials
\cite{Lorentz}, but also from the considerations in
the next section and is from the random projection
theorem a known fact. Figure~\ref{fig1} reflects
this behavior.
\begin{figure}[t] \label{fig1}
\includegraphics[width=0.93\textwidth]{fig_1.pdf}
\caption{The volume ratio \rmref{eq2.8} as function of
$0\leq\delta<1$ for $m=2^k$, $k=1,\ldots,16$, and $n=2m$}
\end{figure}
If the difference $n-m$ of the dimensions is odd,
the arguments from the proof of Theorem~\ref{thm3.2}
lead to a representation with an integral remainder
that can, however, in the given context almost
always be neglected.
\begin{theorem} \label{thm3.3}
If the difference $n-m\geq 3$ of the two dimensions
is odd, if the quantities $k$ and~$l$ are
defined as in the previous theorem, and
if $\nu=k-1/2$ is set,
\begin{equation} \label{eq3.9}
\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg)
=\;\sum_{j=0}^{\nu}\,
\frac{\Gamma(k+l+1)}{\Gamma(k-j+1)\Gamma(l+j+1)}\,
(1-\delta^2)^{k-j}\,\delta^{2(l+j)}
+\Delta(\delta)
\end{equation}
holds, where the remainder possesses the
integral representation
\begin{equation} \label{eq3.10}
\Delta(\delta)=
\frac{2}{\sqrt{\pi}}\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}
\int_0^\delta\frac{t^{n-2}}{\sqrt{1-t^2}}\,\diff{t}
\end{equation}
and satisfies the estimate
$0\leq\Delta(\delta)\leq\delta^{n-1}$ on its
interval of definition.
\end{theorem}
\begin{proof}
By a simple substitution one obtains
\begin{displaymath}
\frac{\Delta(\delta)}{\delta^{n-1}}=
\frac{2}{\sqrt{\pi}}\frac{\Gamma(n/2)}{\Gamma((n-1)/2)}
\int_0^1\frac{t^{n-2}}{\sqrt{1-\delta^2t^2}}\,\diff{t}
\leq \Delta(1).
\end{displaymath}
Because of $\Delta(1)=1$, the estimate
$\Delta(\delta)\leq\delta^{n-1}$ follows.
\end{proof}
\section{On the limit behavior for high space dimensions}
\label{sec4}
\setcounter{equation}{0}
\setcounter{theorem}{0}
\setcounter{figure}{1}
In this section, we derive bounds for the
area ratios (\ref{eq1.2}) and the volume ratios
(\ref{eq2.6}), respectively, with the aim to
understand their behavior when the dimensions
tend to infinity. The starting point is a result
on general matrices. Its proof is based on the
Markov inequality and once again on the
separability of Gauss functions.
\begin{theorem} \label{thm4.1}
Let $0<\sigma_1\leq\sigma_2\leq\cdots\leq\sigma_m$
be the singular values of the matrix~$A$ under
consideration. The volume ratio \rmref{eq2.6} can
then be estimated as
\begin{equation} \label{eq4.1}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\leq \min_t X(t)
\end{equation}
by the minimum of the strictly convex function
\begin{equation} \label{eq4.2}
X(t)=\bigg(\,\prod_{k=1}^m
\frac{1}{1-\delta^2\sigma_m^2\,t+\sigma_k^2\,t}\bigg)^{1/2}
\bigg(\frac{1}{1-\delta^2\sigma_m^2\,t}\bigg)^{(n-m)/2}
\end{equation}
over its interval $0\leq t<1/(\delta^2\sigma_m^2)$
of definition.
\end{theorem}
\begin{proof}
We can restrict ourselves by Lemma~\ref{thm2.3} as
before again to the diagonal matrix $\Sigma$ with the
entries $\sigma_1,\ldots,\sigma_m$. The characteristic
function $\chi$ of the set of all $x$ for which
\mbox{$\|\Sigma x\|<\delta\,\|\Sigma\|\|x\|$} holds
satisfies, for any $t>0$, the crucial estimate
\begin{displaymath}
\chi(x)<
\exp\big(\,t\,\big(\delta^2\|\Sigma\|^2\|x\|^2-\|\Sigma x\|^2\big)\big)
\end{displaymath}
by a product of univariate functions.
By Lemma~\ref{thm2.1}, the subsequent remark, and
Lemma~\ref{thm2.2}, the volume ratio (\ref{eq2.7})
can therefore be estimated by the integral
\begin{displaymath}
\Big(\frac{1}{\sqrt{\pi}}\Big)^n\!\int
\exp\big(\,t\,\big(\delta^2\|\Sigma\|^2\|x\|^2-\|\Sigma x\|^2\big)\big)
\exp\big(-\|x\|^2\big)\,\diff{x}
\end{displaymath}
that remains finite for all $t$ in the given interval.
It splits into a product of one-dimensional integrals
and takes, for given $t$, the value $X(t)$. All
even-order order derivatives of the function $X(t)$
are greater than zero, as follows by differentiation
under the integral sign. The function is therefore,
in particular, strictly convex.
\end{proof}
Because the second-order derivative of $X(t)$ is
greater than zero and $X'(t)$ tends to infinity as
$t$ approaches the right endpoint of the interval,
the first-order derivative of the function $X(t)$
possesses a then also unique zero $t^*>0$ if and
only if
\begin{equation} \label{eq4.3}
X'(0)=\frac{n}{2}\,\delta^2\sigma_m^2-\frac{1}{2}\sum_{k=1}^m\sigma_k^2
\end{equation}
is less than zero, or equivalently if $\delta$
satisfies the condition
\begin{equation} \label{eq4.4}
\bar{\kappa}\delta<\sqrt{\frac{m}{n}}, \quad
\frac{1}{\bar{\kappa}^2}=
\frac{1}{m}\sum_{k=1}^m\Big(\frac{\sigma_k}{\sigma_m}\Big)^2.
\end{equation}
The strictly convex function $X(t)$ attains then and
only then its minimum at a point in the interior of
the interval and there then takes a value less than
$X(0)=1$. Otherwise, the estimate (\ref{eq4.1}) is
worthless and $X(t)\geq 1$ for all $t$ in the interval.
\pagebreak
The minimum of the function (\ref{eq4.2}) can in
general only be calculated numerically, say by
some variant of Newton's method, and cannot be
given in closed form. It is, however,
comparatively simple to estimate this minimum
from above and below.
\begin{lemma} \label{thm4.2}
Let $\kappa=\sigma_m/\sigma_1$ again
be the condition number of the matrix and let~$\xi$
be the square root of the dimension ratio $m/n$.
If $\kappa\delta<\xi$, then the estimate
\begin{equation} \label{eq4.5}
\min_t X(t)\leq\bigg(\,\frac{\kappa\delta}{\xi}\,
\bigg(\frac{1-\kappa^2\delta^2}{1-\xi^2}\bigg)^\gamma\;
\bigg)^m,
\quad \gamma=\frac{1-\xi^2}{2\xi^2},
\end{equation}
holds for the minimum of the function \rmref{eq4.2}.
Under the for condition numbers $\kappa>1$ weaker
condition \rmref{eq4.4}, conversely the lower estimate
\begin{equation} \label{eq4.6}
\min_t X(t)\geq\bigg(\,\frac{\bar{\kappa}\delta}{\xi}\,
\bigg(\frac{1-\bar{\kappa}^2\delta^2}{1-\xi^2}\bigg)^\gamma\;
\bigg)^m
\end{equation}
holds. For orthogonal projections, in both cases
equality holds and it is
\begin{equation} \label{eq4.7}
\min_t X(t)=\bigg(\,\frac{\delta}{\xi}\,
\bigg(\frac{1-\delta^2}{1-\xi^2}\bigg)^\gamma\;
\bigg)^m.
\end{equation}
\end{lemma}
\begin{proof}
The function (\ref{eq4.2}) reads in the case
of orthogonal projections, that is, if all
singular values of the matrix take the value
$\sigma_k=1$, as
\begin{displaymath}
X(t)=
\bigg(\frac{1}{1-\delta^2 t+t}\bigg)^{m/2}
\bigg(\frac{1}{1-\delta^2 t}\bigg)^{(n-m)/2}.
\end{displaymath}
It attains its minimum at the point
\begin{displaymath}
t=\frac{\xi^2-\delta^2}{(1-\delta^2)\delta^2}
\end{displaymath}
in its interval $0<t<1/\delta^2$ of definition
and takes there the value (\ref{eq4.7}). In
the general case, the function (\ref{eq4.2})
satisfies, because of $\sigma_1\leq\sigma_k$,
the estimate
\begin{displaymath}
X(t)\,\leq\,\bigg(
\frac{1}{1-\delta^2\sigma_m^2\,t+\sigma_1^2\,t}\bigg)^{m/2}
\bigg(\frac{1}{1-\delta^2\sigma_m^2\,t}\bigg)^{(n-m)/2}.
\end{displaymath}
The upper estimate (\ref{eq4.5}) thus follows
minimizing the right-hand side as above as a
function of $t'=\sigma_1^2\,t$. The proof of
the lower estimate is a little bit more involved.
Since the geometric mean can be estimated by
the arithmetic mean,
\begin{displaymath}
\bigg(\,\prod_{k=1}^m
\big(1-\delta^2\sigma_m^2\,t+\sigma_k^2\,t\big)\bigg)^{1/m}
\!\leq\;
\frac{1}{m}\sum_{k=1}^m
\big(1-\delta^2\sigma_m^2\,t+\sigma_k^2\,t\big)
\end{displaymath}
holds.
By the definition (\ref{eq4.4}) of $\bar{\kappa}$,
this leads to the estimate
\begin{displaymath}
X(t)\,\geq\,
\bigg(
\frac{1}{1-\delta^2\sigma_m^2\,t+\bar{\kappa}^{-2}\sigma_m^2\,t}\bigg)^{m/2}
\bigg(\frac{1}{1-\delta^2\sigma_m^2\,t}\bigg)^{(n-m)/2}.
\end{displaymath}
Minimizing the right-hand side as a function
of $t'=\bar{\kappa}^{-2}\sigma_m^2\,t$,
one gets (\ref{eq4.6}).
\end{proof}
The question is how tight the derived inclusion
for the minimum of the function (\ref{eq4.2})
is in nontrivial cases, for condition numbers
$\kappa>1$. The answer is that there is practically
no room for improvement without additional
conditions to the singular values. Consider a
sequence of matrices with fixed dimension ratios
$\xi^2=m/n$ and fixed condition number $\kappa$
and let $\kappa\delta<\xi$. Assume that
$\bar{\kappa}$ tends to $\kappa$ as $m$ goes
to infinity. This is, for example, the case
if $\sigma_k=\sigma_1$ for $k=1,\ldots,m-1$.
The rates
\begin{equation} \label{eq4.8}
\frac{\bar{\kappa}\delta}{\xi}\,
\bigg(\frac{1-\bar{\kappa}^2\delta^2}{1-\xi^2}\bigg)^\gamma,
\quad \frac{\kappa\delta}{\xi}\,
\bigg(\frac{1-\kappa^2\delta^2}{1-\xi^2}\bigg)^\gamma
\end{equation}
then approach each other arbitrarily as
$m$ goes to infinity. If additionally
\begin{equation} \label{eq4.9}
\bar{\kappa}=\kappa-\frac{\kappa_1}{m}+o\Big(\frac{1}{m}\Big)
\end{equation}
holds with some positive constant $\kappa_1$,
the ratio of the two bounds enclosing the
minimum of the function (\ref{eq4.2}) tends
to a limit value greater than zero.
The bound (\ref{eq4.5}) can be simplified, and
the minimum of the function (\ref{eq4.2}) be
further estimated in terms of the function
\begin{equation} \label{eq4.10}
\phi(\vartheta) =
\vartheta\,\exp\bigg(\frac{1-\vartheta^2}{2}\bigg),
\end{equation}
which increases on the interval $0\leq\vartheta\leq 1$
strictly, attains at the point $\vartheta=1$ its
maximum value one, and decreases from there again
strictly.
\begin{lemma} \label{thm4.3}
As long as $\kappa\delta$ is less than
the square root $\xi$ of $m/n$, one has
\begin{equation} \label{eq4.11}
\min_t X(t)\leq\phi\bigg(\frac{\kappa\delta}{\xi}\bigg)^m.
\end{equation}
\end{lemma}
\begin{proof}
Set $\kappa\delta/\xi=\vartheta$ for abbreviation.
The logarithm
\begin{displaymath}
\ln\bigg(\bigg(\frac{1-\kappa^2\delta^2}{1-\xi^2}\bigg)^\gamma\;\bigg)
=\,\frac{1-\xi^2}{2\xi^2}\,\ln\bigg(\frac{1-\vartheta^2\xi^2}{1-\xi^2}\bigg)
\end{displaymath}
then possesses, because of $\vartheta^2\xi^2<1$ and
$\xi^2<1$, the power series expansion
\begin{displaymath}
\frac {1-\vartheta^2}{2}-\frac12\,\sum_{k=1}^\infty
\bigg(\frac{1-\vartheta^{2k}}{k}-\,\frac{1-\vartheta^{2k+2}}{k+1}\bigg)\xi^{2k}.
\end{displaymath}
Because the series coefficients are for all
$\vartheta\geq 0$ greater than or equal to
zero and, by the way, polynomial multiples of
$(1-\vartheta^2)^2$, the proposition follows
from (\ref{eq4.5}).
\end{proof}
In the following, we will use the estimate
(\ref{eq4.11}) for the minimum of the
function given by (\ref{eq4.2}). The next
theorem is then a trivial conclusion from
Theorem~\ref{thm4.1}.
\begin{theorem} \label{thm4.4}
Let $n>m$, let $A$ be an $(m\times n)$-matrix of full
rank $m$, let $\kappa$ be the condition number of
$A$, and let $\xi$ be the square root of the
dimension ratio $m/n$. If $\kappa\delta$ is less
than $\xi$, then for all radii $R>0$ one has
\begin{equation} \label{eq4.12}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\leq \phi\bigg(\frac{\kappa\delta}{\xi}\bigg)^m.
\end{equation}
\end{theorem}
Consider a sequence of matrices with dimension
ratios $m/n\geq\delta_0^2$ and condition
numbers $\kappa\leq\kappa_0$. The volume
ratios (\ref{eq2.6}) tend then for
$\kappa_0\delta<\delta_0$ not slower than
\begin{equation} \label{eq4.13}
\sim\,\phi\bigg(\frac{\kappa_0\delta}{\delta_0}\bigg)^m
\end{equation}
to zero as the dimensions go to infinity.
Under suitable conditions to the singular
values, considerable improvements are possible.
In extreme cases, such as in Lemma~\ref{thm2.7},
the volume ratios (\ref{eq2.6}) can essentially
be estimated as those for orthogonal projections
and the potentially devastating influence of
the condition number vanishes.
\begin{theorem} \label{thm4.5}
Let $n>m$ and let $A$ be a nonvanishing $(m\times n)$-matrix
with singular values $\sigma_k=\sigma_m$ for $k>m_0$. If
one sets $m'=m-m_0$ and $\xi'$ is the square root of $m'/n$,
the volume ratio \rmref{eq2.6} satisfies then for
$0\leq\delta<\xi'$ the estimate
\begin{equation} \label{eq4.14}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|<\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\leq \phi\bigg(\frac{\delta}{\xi'}\bigg)^{m'}.
\end{equation}
\end{theorem}
The proof results from Lemma~\ref{thm2.7},
Theorem~\ref{thm4.1}, and Lemma~\ref{thm4.3}.
Theorem~\ref{thm4.4} possesses a counterpart that deals
with values $\delta$ greater than the square root of
the ratio of the dimensions $m$ and $n$.
\begin{theorem} \label{thm4.6}
Let $A$ be a nonvanishing $(m\times n)$-matrix and
let $\xi$ be the square root of the dimension
ratio $m/n$. For $\xi<\delta\leq 1$ then one has
\begin{equation} \label{eq4.15}
\frac
{\lambda\big(\big\{x\,\big|\,\|Ax\|\geq\delta\,\|A\|\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}
\leq \phi\bigg(\frac{\delta}{\xi}\bigg)^m.
\end{equation}
\end{theorem}
\begin{proof}
We can restrict ourselves again to diagonal matrices
$A=\Sigma$. Let $P$ be the matrix that extracts from
a vector in $\mathbb{R}^n$ its first $m$ components.
As $\|\Sigma x\|\leq\|\Sigma\|\|Px\|$ and
$\|\Sigma\|>0$, the given volume ratio can
then be estimated by the volume ratio
\begin{displaymath}
\frac
{\lambda\big(\big\{x\,\big|\,\|Px\|\geq\delta\,\|x\|,\,\|x\|\leq R\big\}\big)}
{\lambda\big(\big\{x\,\big|\,\|x\|\leq R\big\}\big)}.
\end{displaymath}
As in the proof of Theorem~\ref{thm4.1}, we can
estimate this volume ratio for sufficiently small
positive values $t$ by the integral
\begin{displaymath}
\Big(\frac{1}{\sqrt{\pi}}\Big)^n\!\int
\exp\big(\,t\,\big(\|Px\|^2-\delta^2\|x\|^2\big)\big)
\exp\big(-\|x\|^2\big)\,\diff{x}.
\end{displaymath}
This integral splits into a product of
one-dimensional integrals and takes
the value
\begin{displaymath}
\bigg(\frac{1}{1+\delta^2 t-t}\bigg)^{m/2}
\bigg(\frac{1}{1+\delta^2 t}\bigg)^{(n-m)/2},
\end{displaymath}
which attains, for $\delta<1$, on the interval
$0<t<1/(1-\delta^2)$ its minimum at
\begin{displaymath}
t=\frac{\delta^2-\xi^2}{(1-\delta^2)\delta^2}.
\end{displaymath}
It takes at this point $t$ again the value
\begin{displaymath}
\bigg(\,\frac{\delta}{\xi}\,
\bigg(\frac{1-\delta^2}{1-\xi^2}\bigg)^\gamma\;
\bigg)^m, \quad \gamma=\frac{1-\xi^2}{2\xi^2}.
\end{displaymath}
This leads as in the proof of Lemma~\ref{thm4.3}
to the estimate (\ref{eq4.15}). As the set of
all $x$ for which $\|Px\|=\|x\|$ holds has
measure zero, (\ref{eq4.15}) remains true for
$\delta=1$.
\end{proof}
For $(m\times n)$-matrices $A$ with dimension
ratio $m/n\leq\delta_0^2$, the volume ratios
(\ref{eq2.6}) tend therefore on the interval
$\delta_0<\delta\leq 1$ pointwise and on its
closed subintervals uniformly and exponentially
to one as $m$ goes to infinity. For sequences
of matrices for which the ratio $m/n$ of their
dimensions tends to zero, the volume ratios
(\ref{eq2.6}) hence tend, for all $\delta>0$,
pointwise to one. This has, however, often
less severe implications than it might first
appear. This is demonstrated by the example
of the matrices $A=T^t$ from section~\ref{sec2},
whose dimensions (\ref{eq2.13}) were
\begin{equation} \label{eq4.16}
m=3\times N, \quad n=3\times\frac{N(N+1)}{2}.
\end{equation}
Figure~\ref{fig2} shows the bounds for the
volume ratios (\ref{eq2.6}) resulting from
the application of Lemma~\ref{thm2.7} to
these matrices as functions of $\delta<1$
for $N$ ranging from $4$ to $32$, or, in
the framework of quantum mechanics, for
systems with up to $32$ electrons.
\begin{figure}[t] \label{fig2}
\includegraphics[width=0.93\textwidth]{fig_2.pdf}
\caption{The bounds for the volume ratios \rmref{eq2.6} for the
example from section~{\rm \ref{sec2}} for $N=4,\ldots,32$.}
\end{figure}
We finally consider the case of orthogonal projections
from the $\mathbb{R}^n$ to the $\mathbb{R}^m$, that
is, of $(m\times n)$-matrices $P$ with one as the only
singular value. The condition number of such matrices
is $\kappa=1$, and their norm is $\|P\|=1$. If the
dimension ratios $m/n$ tend to $\delta_0^2$, or even
remain as in Figure~\ref{fig1} fixed, the corresponding
volume ratios (\ref{eq2.8}) tend therefore to a step
function with jump discontinuity at~$\delta_0$. This
observation is widely equivalent to the random projection
theorem. Let $\xi$ be again the square root of $m/n$.
For a randomly chosen vector $x$, the probability that
\begin{equation} \label{eq4.17}
(1-\varepsilon)\xi\,\|x\|\leq\|Px\|<(1+\varepsilon)\xi\,\|x\|
\end{equation}
holds is then $F((1+\varepsilon)\xi)-F((1-\varepsilon)\xi)$,
with the at least for even-numbered differences of the
dimensions explicitly known distribution function
\begin{equation} \label{eq4.18}
F(\delta)=\psi\bigg(\frac{\delta}{\sqrt{1-\delta^2}}\bigg),
\end{equation}
and tends exponentially to one as $m$ goes to
infinity. This means that the orthogonal projection
of a randomly chosen unit vector $x\in\mathbb{R}^n$
onto a given subspace of high dimension $m$
possesses with high probability a norm
\begin{equation} \label{eq4.19}
\approx\sqrt{\frac{m}{n}}.
\end{equation}
A lower bound for this probability depending only
on the dimension $m$ but not on the dimension $n$
can be derived from the estimates (\ref{eq4.12})
and (\ref{eq4.15}). Because of
\begin{equation} \label{eq4.20}
\phi(1\pm\varepsilon)<\exp(-c\,\varepsilon^2),
\quad c=-\ln(\phi(2)),
\end{equation}
for values $0<\varepsilon<1$, the probability that
(\ref{eq4.17}) holds is in any case greater than
\begin{equation} \label{eq4.21}
1-2\exp(-c\,\varepsilon^2m)
\end{equation}
and the random projection theorem recovered.
\bibliographystyle{siamplain}
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proofpile-arXiv_065-5642 | \section{Introduction}
Quantum technologies are a fast developing scientific and industrial field \cite{DowlingPTRSA2003}. They have been already implemented in several different platforms, as for instance photonic circuits \cite{OBrienNatPhot2009,WangNatPhot2020}, but also Rydberg atoms \cite{AdamsReview2020}, superconducting devices \cite{devoret2004superconducting} and others. Likely, the most promising quantum technology is represented by quantum computers, i.e., quantum devices for quantum computing, among which it is worth mentioning superconducting circuits \cite{clarke2008superconducting,WuPRL2021}, trapped-ions quantum computers \cite{wineland2003quantum,PogorelovPRXQuantum2021}, photonic chips \cite{spring2013boson,metcalf2014quantum} and topological qubits \cite{freedman2003topological}. Both academic laboratories and industrial companies are devoting lots of effort and funding to boost research and technological improvements, towards the so-called \textit{quantum supremacy} \cite{AruteNature2019}, i.e., a quantum advantage to solve (numerical) problems that no classical computer will never solve. The actual drawback of these devices is the absence of a standard hardware (and thus even software) architecture, on which research activities may be jointly coordinated. For each of these platforms, indeed, ad hoc solutions are proposed and then realized, and this makes such a technologies still very expensive and incompatible from a device to another.
However, in quantum computing, the main issue to be still solved is the unavoidable presence of external noise sources that dramatically limit the accuracy of quantum computations, as well as the large-scale realization of quantum circuits and algorithms. The negative impact of noise on quantum computing is so noticeable that the acronym \emph{Noisy Intermediate-Scale Quantum} (NISQ) technology has been recently introduced \cite{preskill2018quantum}. Furthermore, commercial quantum devices as for example the quantum computers by the companies Q-IBM\textsuperscript{\textregistered} \cite{ibmq} and Rigetti\textsuperscript{\textregistered} \cite{rigetti}, albeit they have been made available by anyone who creates a free account on their database, are not physically accessible and several specifications on the chip's parameters are not made public.
In the paper \cite{martinaArXiv2021Learning}, we have recently observed on some IBM quantum computers that main features of the noise sources affecting the devices are specific of each single computer and have a clear time-dependence. For such a purpose, a \emph{testbed quantum circuit} -- composed by a fixed number of qubits -- is designed, then made run for a sufficient number of times and finally locally measured in correspondence of each qubit. From the measurements of the qubits (the measurement observable was the Pauli matrix $\sigma_z$), a set of measurement outcomes is recorded, collected, and then used to train a \emph{machine learning} (ML) algorithm \cite{BishopPRML2006,HastieESL2009}. However, it is worth noting that in \cite{martinaArXiv2021Learning} the features of the quantum noise are not reconstructed but just classified from a quantum device to another. Specifically, the classification task was successfully carried out by means of a \emph{support vector machine} (SVM) \cite{HastieESL2009,BishopPRML2006}, with a classification accuracy equal or greater than $99\%$. Hence, thanks to our procedure, one just needs to collect an informative statistics of quantum measurement outcomes (that are \emph{quantum data}) from the testbed quantum circuit, and subsequently train ML (classical) algorithms. In fact, no quantum noise modelling is required nor, in principle, the testbed circuit has to be controlled by time-dependent pulses \cite{MuellerArXiv2020}. Also for these reasons, the use of a ML technique is the most natural choice to perform classification, since it naturally provides a black-box model with predictive outcomes. In this regard, we recall that in the current literature ML has been already adopted to distinguish open quantum dynamics \cite{YoussryArXiv,luchnikov2020machine,fanchini2020estimating} and to perform quantum sensing tasks \cite{niu2019learning,HarperNatPhys2020,MartinaArXiv2021,wise2021using}, as for example the learning and classification of non-Markovian noise \cite{niu2019learning,MartinaArXiv2021} or the detection of qubits correlations \cite{HarperNatPhys2020}.
\begin{figure}[t!]
\centering
\includegraphics[width=0.9\textwidth]{functionalGraph.pdf}
\caption{Pictorial figure showing the structure of the software architecture and how its different parts depend each other. On the left, one can observe the part of the software that is designed for the generation of the testbed quantum circuit. Specifically, \code{createCircuit.py} is used to launch the quantum circuit on the IBM cloud services with the aim to get the raw data from the measurement procedure in each execution of the circuit. The file \code{extractExecutions.py} computes the probabilities to get the measurement outcomes from the execution of the testbed quantum circuit. Either \code{createDataset.py} and \code{createDatasetTimeSeries.py} is used to pack in datasets the measurement outcomes probabilities. The former creates datasets with data collected on two or more machines, while the latter collects data taken on the same machine but at different times. On the right side of the figure, we represent the workflow for the training of ML methods. Specifically, the files \code{runSvmDatFile.py}, \code{runSvmTable.py}, \code{runSvmTableHoriz.py} and \code{runSvmTableTriang.py} are employed to generate the output data that we have shown in the tables and plots in \cite{martinaArXiv2021Learning}. All these scripts call the function \code{runSVM} in the file \code{runSvm.py} that contains the main code for the definition, training and evaluation of the SVM models. Note that the function is called with one of the configuration names (\code{config} in the picture) that is listed in the config file \code{configurations.py}. The configuration name denotes what is the generated dataset that is used for the training of the ML models.}
\label{fig:functionalGraph}
\end{figure}
As depicted in Fig.\,\ref{fig:functionalGraph}, our software architecture adopted in \cite{martinaArXiv2021Learning} has two distinct parts: The one on the left of the figure generates the testbed quantum circuit (see Sec.\,\ref{sec:circuit_generation}), while the other is designed for the implementation of the ML models that classify quantum noise fingerprints (refer to Sec.\,\ref{sec:ML_models}).
\section{Testbed quantum circuit}\label{sec:circuit_generation}
For our experiments of quantum noise classification in \cite{martinaArXiv2021Learning}, we made use of the IBM Quantum cloud services to remotely run quantum circuits on different machines. In particular, to interact with the cloud services, one can use the Qiskit sdk \cite{qiskit} that is an open-source python package, useful both to \textit{simulate} quantum dynamics and to \textit{program} a given set of operations on a real quantum computer. Currently, one has at disposal up to $11$ superconducting quantum computers, ranging from a single qubit up to $15$ qubits, with different topology and calibration routines. For all the available devices and their specifications, we direct the reader to the IBM documentation \cite{ibmq}.
\begin{algorithm}[t!]
\caption{Generation of the testbed quantum circuit (baseline version)}\label{alg:cap}
\begin{algorithmic}
\Require IBM-Q backend (specific device to fingerprint)
\Ensure $|0\rangle_i$ $\forall i \in 0,...,3$
\For{number of repetitions}
\State $0 \gets H$
\Comment Hadamard gate on the $0^{th}$ qubit
\State $1 \gets H$
\State ${\rm CNOT}(0 \to 2)$ \Comment Controlled NOT gate on the $2^{nd}$ qubit conditioned on the qubit $0$
\State ${\rm CNOT}(1 \to 3)$
\State $0 \gets X$
\Comment X gate on the $0^{th}$ qubit
\State $1 \gets X$
\State ${\rm Toffoli}(0,1 \to 2)$
\Comment Toffoli gate on the $2^{nd}$ qubit conditioned on the qubits $0,1$
\EndFor
\State Measure(2)
\Comment Projective measurements of the $i^{th}$ qubit
\State Measure(3)
\State\Return 1000 shots from the measurements
\end{algorithmic}
\end{algorithm}
\begin{figure}[h!]
\begin{subfigure}[b]{0.5\textwidth}
\centering
\includegraphics[height=4.25cm]{walker1.pdf}
\caption{}
\label{fig:circuit1}
\end{subfigure}
\begin{subfigure}[b]{0.5\textwidth}
\includegraphics[height=4.25cm]{walker2.pdf}
\caption{}
\label{fig:circuit2}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[height=4.25cm]{walker3.pdf}
\caption{}
\label{fig:circuit3}
\end{subfigure}
\begin{subfigure}[b]{\textwidth}
\centering
\includegraphics[height=4.25cm]{walker4.pdf}
\caption{}
\label{fig:circuit4}
\end{subfigure}
\caption{Pictorial representation of the first 4 measurement steps (the panels of the figure) applied to the testbed quantum circuit designed in Ref.\,\cite{martinaArXiv2021Learning}. The 4 panels have to be read from left to right, and from top to bottom. The baseline version of the testbed quantum circuit (provided by Algorithm 1) is the one depicted in panel (c), in correspondence of the third measurement step. The full set of measurement outcomes is obtained by repeating 3 times the baseline circuit and then performing a total of 9 measurement steps, each of them acting on the qubits 2 and 3. The outcomes collected in the 9 measurement steps come from executing incrementally the testbed quantum circuit in different runs, where the qubits 2 and 3 are measured only at the end of the implemented circuits.}
\label{fig:circuit}
\end{figure}
In Algorithm 1 we provide the pseudo-code for the generation of the testbed quantum circuit in its baseline version (see also panel (c) in Fig.\,\ref{fig:circuit} for a pictorial representation of the circuit) realized in \cite{martinaArXiv2021Learning} to carry out the classification of noise fingerprints.
For the quantum computation in the aforementioned circuit, we made use of standard gates whose mathematical definitions
is given in terms of matrices that one can easily find in quantum computing textbooks \cite{nielsen2002quantum}. The baseline version of the testbed quantum circuit is repeated 3 times overall for a total of 9 measurements (also denoted as measurement steps) of both qubits 2 and 3. Indeed, operationally, the 9 measurement steps are not performed all in the same run (i.e., sequentially), but on consecutive runs by implementing incremental parts of the quantum circuit. To clarify this aspect as much as possible, in Fig.\,\ref{fig:circuit} we have represented pictorially the first 4 measurement steps of the testbed quantum circuit, whose baseline version (returned by Algorithm 1) is depicted in the panel (c) of the figure. Specifically, first we execute the part of the circuit that is obtained by cutting the testbed quantum circuit after the first measurement step, i.e., after the measurement of qubit 2 and 3 following the Hadamard gates on the $0^{th}$ and $1^{th}$ qubits and the CNOT gate from qubit $0$ to $2$). Then, the measurement outcomes are recorded. Subsequently, we execute part of the testbed quantum circuit until the second measurement step (measurements of qubits 2 and 3 included), thus by also taking into account the $X$-gate on the $0^{th}$ qubit and the CNOT gate from qubit $1$ to $3$, and again we record the measurement outcomes. The procedure is then repeated for all the 9 measurement steps. Before proceeding, it is worth stressing that, for each implemented quantum circuit, the measurements of the qubits 2 and 3 are performed only at the end of the circuits.
Overall, in \cite{martinaArXiv2021Learning}, several experiments have been conducted on different IBM chips that have different physical specifications, as the architecture of the qubits or the quantum volume \cite{cross2019validating}.
Some quantum machines, indeed, are inherently noisier than other, and even single qubits inside a machine can have a distinctive noise profile. All these peculiar differences in noise and topology contribute to the fingerprint that we aim to classify using our ML method.
\subsection{Data acquisition}
The pipeline designed for the creation of the dataset, set as input of the ML models, is constituted of several scripts that can be customized according to the needs of the user. First, for each implemented quantum circuit, the script \code{createCircuit.py} adopts Qiskit to interact with the IBM quantum services for the measurement of a predefined number of circuit executions. Specifically, such a script is parameterized to launch the runs of the circuits on several quantum machines with a specific amount of parallel tasks. The runs are executed in two different modalities that generates the datasets that we called FAST and SLOW in \cite{martinaArXiv2021Learning}. In the former dataset, the aim was to collect as many runs as possible in the shortest time interval. For this purpose, the script launches 20 parallel processes, each of them adds to the IBM queue a predefined number of runs with 8000 execution-shots. After that, each batch of 8000 execution-shots is split into 8 batches of 1000 shots that are then employed to compute the outcomes' probabilities.
Instead, for the dataset named SLOW, we collect a sequence of measurement outcomes that are uniformly distributed over time. To obtain such dataset, the script launches only one run at a time with 1000 execution-shots and waits two minutes from a run to another.
The second script in the pipeline is \code{extractExecutions.py}, whose objective is to compute the probabilities of the measurement outcomes from the raw data returned by the calls to the IBM quantum services (this is ensured by the previous script). After that, either \code{createDataset.py} and \code{createDatasetTimeSeries.py} pack the probability distributions in datasets. The difference between such scripts is the following. The former builds binary or multiclass classification dataset using data from at least two quantum machines, while the latter builds classification datasets with data collected in a single machine and labelled by the time interval in which the testbed quantum circuit is executed.
In the github repository at \url{https://github.com/trianam/learningQuantumNoiseFingerprint} and on CodeOcean at \url{https://codeocean.com/capsule/fa6e1d85-c99f-4a38-9c16-ac204da85040/}, we release the source code of all the scripts and all the data obtained from the execution of \code{createCircuit.py} on each quantum machine we employed.
\section{Machine Learning models}\label{sec:ML_models}
\subsection{Support Vector Machine}
In \cite{martinaArXiv2021Learning} we have successfully classified the noise fingerprints on several IBM quantum computers by training Support Vector Machine models \cite{HastieESL2009,BishopPRML2006}. SVM is a machine learning technique that is usually used to solve binary classification tasks. Generally speaking, a SVM model is trained on a dataset composed of pairs $(\ve{x}_i,y_i)$, where $\ve{x}_i$ are points in a certain space $\RN^n$ of dimension $n$, and $y_i$ is equal to $1$ or $-1$ depending the corresponding point belongs to one or the other class. The Support Vector Machine is trained to find the hyperplane that divides the space representation of the two classes, by ensuring the maximum distance from the points. When the points of the two classes are not linearly separable, a common solution is to resort to the so-called ``kernel trick'', i.e., the points $\ve{x}_i$ are mapped to a larger dimension space until they become linearly separable. The most common kernels are polynomial functions with varying degree number and the so-called Radial Basis Functions (RBF) \cite{HastieESL2009, BishopPRML2006}. Finally, SVM models can also be extended to multiclass classification tasks using the strategies One-Versus-All (OVA), or One-Versus-One (OVO) \cite{HastieESL2009, BishopPRML2006}.
In our work, to implement and train the SVM models, we leveraged the \emph{scikit-learn} python library \cite{scikit-learn}.
\subsection{Data interpretation}
The code that implements and train the SVM is defined by the functions in the file \code{runSVM.py}. Specifically, the main function is called \code{runSVM}: it requires a configuration object that \emph{(i)} identifies what model has to be used, and \emph{(ii)} set optional arguments to tune the number of hyperparameters (\code{mask}) and to control if the method is verbose (\code{verbose}) and if the results have to be written in an output file (\code{writeToFile}). Practically, the function \code{runSVM} first calls \code{extractData} whose purpose is to load the dataset file, extract the data in the desired time steps and split them in \emph{training}, \emph{validation} and \emph{test} sets. After that, \code{runSVM} proceeds to train a set of possible SVM models on the training set, by then evaluating them on the validation set and computing on the test set the resulting accuracy of the model that performed better on the validation set. The possible models that can be employed are: \emph{(i)} Standard linear SVM (using two different libraries), \emph{(ii)} SVM with \emph{polynomial} kernel with degree from 2 to 4, and \emph{(iii)} SVM with RBF kernel.
The file \code{runSVM.py} is provided with a \code{main} method. Thus, it can be directly called as a script using the configuration name as argument. We have also designed some useful methods that call \code{runSVM}, build directly the latex table with the results and calculate the points for the figures shown in \cite{martinaArXiv2021Learning}.
\section{Impacts}
In this paper, we have explained in great detail the software architecture of the ML method, introduced in \cite{martinaArXiv2021Learning}, to carry out quantum noise classification. Such a tools are intended to be applied to quantum technologies, as e.g., quantum computers.
The main impact of our software lies in its ability in classifying the fingerprint left by quantum noise sources on devices that have identical technical specifications and are thus expected to provide the same outcomes. Unfortunately, in quantum machines, the influence of the environment is so relevant that different noise fingerprints can be identified depending on the type of quantum computer (as previously explained, quantum computers can differ, e.g., on the number of qubits and/or the quantum volume), on the time period in which the single machine has worked, and on environmental changes mainly due to temperature fluctuations. However, thanks to our quantum-classical machine learning method, one can \emph{(i)} distinguish the noise fingerprints in different quantum devices; \emph{(ii)} classify the noise fingerprint on the same quantum devices but in different times; \emph{(iii)} learn if and how a given noise fingerprint changes over time.
In \cite{martinaArXiv2021Learning} our method is proved to be very \emph{accurate} (more than $99\%$ of effectiveness) in classifying a clear machine-related noise fingerprint in each of the analysed IBM quantum computers, and even \emph{robust} since any noise fingerprint is highly predictable over time in windows of consecutive runs. Also an evident time-dependence of the noise fingerprints has been classified, by observing changes over time after few hours from the first execution of the testbed quantum circuit.
Another important feature of our software architecture is that the ML models do not require a complete set of measurement outcomes as input data, but conversely the outcomes from a sequence of repeated measurements of a single observable. For an example, for the experiments in \cite{martinaArXiv2021Learning}, the chosen observable was the tensor product of $\sigma_z$ Pauli matrices locally applied on each qubit of the testbed circuit. Furthermore, the proposed method is able to distinguish and classify noise fingerprints, even without knowing the microscopic model that describes the (real or effective) interaction between the device and the external noise fields. This important aspect allows the user to employ our quantum-classical machine learning algorithm to classify the noise fingerprints of even \emph{inaccessible} quantum machines.
\subsection{Applications}
The experimental evidences in \cite{martinaArXiv2021Learning} lead us to conclude that different quantum devices exhibit distinctive, and thus distinguishable, noise fingerprints that one can classify and predict. Therefore, in principle, our method could be adopted \emph{to identify} from which specific quantum device certain data (a collection of measurement outcomes) are generated, just looking at the noise fingerprint of the device. Moreover, the proposed solution might be employed \emph{to certify} the time-scheduling in which a given quantum computation is executed. Both these applications are expected to play a key role for diagnostics purposes -- especially in all those contexts where quantum computations cannot be error-corrected \cite{deutsch2020harnessing} -- and to accomplish benchmarking and certification \cite{WrightNatureComm2019,eisert2020quantum} of quantum noise sources within a default error threshold.
\subsection{Outlook}
The proposed methodology, aimed to learn the noise fingerprint of quantum devices from time-ordered measurements of a testbed quantum circuit, may be in principle applied to any quantum devices, and thus not only to the IBM quantum computers as done in \cite{martinaArXiv2021Learning}. The possibility to predict on which device, and at which time, a given quantum operation (even time-varying) has been executed is expected to help the mitigation of quantum computational errors (e.g., by means of calibration routines), and to assist the application of ad-hoc error corrections.
Furthermore, instead of SVMs, one could employ deep learning techniques, as for example Recurrent Neural Networks (RNN) \cite{BishopPRML2006,GoodfellowDL2016,schmidhuber2015deep}, to make more efficient the classification of quantum noise fingerprints. In such a case, the software architecture should be modified a bit, but not in the part that concerns the generation of the quantum data placed in input to the ML model. What should be different, indeed, is the way the input data would be processed.
Finally, we are also confident that, thanks to specific modifications, it is possible to carry out even the reconstruction of some quantum noise features. However, for such a purpose, a minimal knowledge of the way noise sources affect the quantum device under investigation will be required.
\section*{Acknowledgements}
We acknowledge the access to advanced services provided by the IBM Quantum Researchers Program.\\
This work was financially supported from Fondazione CR Firenze through the project QUANTUM-AI, from University of Florence through the project Q-CODYCES, and from the European Union’s Horizon 2020 research and innovation programme under FET-OPEN Grant Agreement No.\,828946 (PATHOS).
\bibliographystyle{unsrt}
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proofpile-arXiv_065-5700 | \section{Introduction}
Swampland conjectures provide predictions for theories that are consistently coupled to gravity \cite{Vafa:2005ui,Ooguri:2006in,Palti:2019pca}. They can be seen as the imprint of the underlying quantum theory of gravity on the effective gravitational description at low energies. These predictions become trivial when gravity decouples, i.e., in the limit of vanishing Planck length $L_p =M_p^{-1}=\sqrt{8\pi \hbar G_N}\rightarrow 0$, that is either for small Newton constant $G_N$ or vanishing Planck constant $\hbar$. Instead they tend to become more relevant in extremal regimes of the parameter space of the theory, for example when gauge couplings vanish, or scalar fields approach an infinite distance point in moduli space.
In this paper, we explore limiting regions of thermodynamic quantities of gravitational backgrounds and connect them to more familiar Swampland constraints. Most prominently, our investigation will focus on regions of large as well as small entropy $\mathcal{S}$ and temperature $\mathcal{T}$ of gravitational solutions within effective theories. Since we are interested in non-compact spacetimes, we restrict to situations where the temperature is associated to a horizon. We show that the Unruh temperature of Rindler spacetime does not lead to Swampland constraints. Black hole solutions, however, allow for a connection to Swampland constraints in certain limits of the thermodynamic properties captured by the Hawking temperature and entropy.
Our strategy is as follows. We describe black hole solutions in effective theories such as Einstein-Maxwell-dilaton theory and $\mathcal{N}=2$ supergravity that allow for limiting cases of vanishing and diverging $\mathcal{S}$ and $\mathcal{T}$. In situations where the effective theory allows for a string theory embedding we can also associate the limiting values of the thermodynamic order parameters with properties of the internal space, such as its volume. It turns out that in the string theory sub-class of these limiting solutions the moduli fields of the theory will traverse an infinite field distance from spatial infinity to the horizon.
In this way we are able to identify the tower of light string states in the vicinity of the black hole horizon, which is also suggested by the Swampland Distance Conjecture in the cases of infinite field distances.
Hence these limiting solutions need to be taken with care in view of the Swampland Distance Conjecture and are problematic within the effective description. This infinite field distance precisely happens in the limit of vanishing and diverging entropy of the black hole for arbitrary temperatures and thus offers a generalization of the investigations in \cite{Bonnefoy:2019nzv} for large entropies, see also \cite{DeBiasio:2020xkv,Luben:2020wix,Hamada:2021yxy}. We identify four generic limits in the thermodynamic quantities:
\begin{itemize}
\item{${\cal S}\rightarrow0$ and ${\cal T}\rightarrow \infty$: \, The moduli field distance diverges in this limit. The small horizon size and large temperature also suggests that quantum gravity effects become important. It was argued that this limit can be associated to elementary particles \cite{Holzhey:1991bx}.}
\item{${\cal S}\rightarrow\infty$ and ${\cal T}\rightarrow 0$: \, The moduli field distance diverges in this limit. This has been discussed in \cite{Bonnefoy:2019nzv} where the large number of black hole microstates was associated to a light tower of states.}
\item{${\cal S}\rightarrow0$ and ${\cal T}\rightarrow 0$: \, The moduli field distances diverges in this limit. Once more the vanishing horizon area suggests the importance of quantum gravity corrections to avoid the violation of entropy bounds, see \cite{Hamada:2021yxy}. As we will show this behavior is general for arbitrary finite temperatures $\mathcal{T}$ and the associated non-extremal black holes.}
\item{${\cal T}\rightarrow 0$ and $\mathcal{S}$ finite: \, The moduli field distance is finite in this limit. Particularly this regime includes the charged extremal black holes, which can be described within the effective theory. Note, however, that in the non-supersymmetric case it was argued that there still might be large quantum corrections for the thermodynamic quantities \cite{Preskill:1991tb, Maldacena:1998uz, Page:2000dk, Heydeman:2020hhw}.}
\end{itemize}
These results confirm the importance of quantum gravity corrections in these situations and thus provides an interesting correlation between black hole thermodynamics and Swampland Constraints, suggesting a notion of {\it entropy} and {\it temperature-distance}\footnote{See also \cite{Agrawal:2020xek}, which suggests that the limit of high temperature should be at infinite distance.}. Note that for this observation it is crucial that the string theory black hole geometries are described by warped products, as opposed to fibrations, and therefore we do expect the modes arising from compactification to be stable\footnote{For a non-trivial fibration it is often the case that winding modes can unwrap and the Kaluza-Klein momentum is not preserved. In such situations we do not expect a stable tower of states, see e.g. \cite{Draper:2019utz, Lanza:2021udy}}.
We also investigate the black hole solutions above in terms of a distance in the space of metrics (see for example \cite{PhysRev.160.1113}), which diverges in all the limiting cases. An explanation for this can be the formation of an infinite throat in the black hole geometry (see also \cite{Li:2021gbg,Li:2021utg} for a related discussion and \cite{Elander:2020rgv} for an holographic application). We therefore conclude that the metric distance as a Swampland parameter in these non-compact backgrounds needs to be taken with care.
Moreover, the black hole formulae suggest that some of the limits might be connected by a form of {\it thermodynamic dualities} which for example act as $\mathcal{T} \rightarrow 1/\mathcal{T}$ or $\mathcal{S} \rightarrow 1/\mathcal{S}$. We regard this as a formal result, which is expected not to hold in its original form once corrections are taken into account (e.g. in the limit of low entropy, one expects the contribution of higher derivative terms). Nevertheless, in the string embedding, we will be able to relate these to a combination of T- and S-dualities affecting the internal volume as well as the coupling constants. From the point of view of the Swampland program, the emergence of these dualities provides additional testament that duality itself should be a general principle of quantum gravity. While this is one of main lessons of string theory, our investigation suggests a novel manifestation of this in terms of the thermodynamic properties within a bottom-up description.
The paper is organized as follows. In Section \ref{sec:BHgen} we start by discussing temperature limits in gravity, for Rindler space and also for the Schwarzschild black hole. Then we review relevant aspects of the Reissner-Nordstrom black hole as well as the notion of distance in the space of black hole geometries. In Section \ref{sec:EMDblackholes} we investigate in detail non-extremal dyonic black holes, coupled to a dilaton field. In this context we study the $\mathcal{T}-\mathcal{S}$ phase diagram of the charged black holes and the various extremal limits. In Section \ref{SG-blackholes} we turn to the ${\cal N}=2$ supergravity realization of general four-dimensional, non-extremal black holes together with their behavior under a variation of thermodynamic quantities. Focussing on the $STU$ model we can explicitly identify the tower of states. We further show the supergravity embedding of the Einstein-Maxwell-dilation black hole into ${\cal N}=2$ supergravity. In Section \ref{sec:concl} we present our conclusions. Some generalities about $\mathcal{N} = 2$ supergravity are collected in the Appendix.
\section{Distance, temperature and entropy in gravity}
\label{sec:BHgen}
In the context of the Swampland Program \cite{Vafa:2005ui,Ooguri:2006in,Palti:2019pca}, large variations of parameters in an effective theory become problematic when consistently coupling the theory to gravity. In particular, the Swampland Distance Conjecture states that a large variation of field values for a scalar field\footnote{The correct quantity is the geodesic distance in the scalar moduli space.} $\phi$ is accompanied by a light tower of an infinite number of states. These light states in turn invalidate the effective description by lowering the quantum gravity cut-off. Specifically, the masses of the states in the tower decrease exponentially in the field distance $\Delta (\phi)$
\begin{align}
{m L_p} \sim e^{-\lambda \Delta(\phi)} \,,
\label{gendisca}
\end{align}
where $\lambda \sim \mathcal{O}(1)$ is a positive parameter. For this to be a meaningful Swampland constraint the tower needs to disappear when decoupling gravity, i.e., for $L_p \rightarrow 0$. This is realized if the masses of the states in the tower remain positive when measured in Planck units, i.e.,
\begin{align}
m L_p = \frac{m}{M_p} > 0 \quad \text{for} \quad L_p \rightarrow 0 \,.
\label{eq:qgconst}
\end{align}
If this is not the case, the related tower cannot be identified with the states postulated by the SDC.
In this section, we explore whether also the variation of thermodynamic quantities such as temperature $\mathcal{T}$ and entropy $\mathcal{S}$ can lead to Swampland constraints, similar to the SDC. For that we first investigate flat space at finite temperature and then turn to the analysis of black hole geometries.
\subsection{Temperature in flat space}
One natural situation where the SDC becomes relevant are field theories derived by compactification. When the internal space becomes large the corresponding momentum modes, the Kaluza-Klein (KK) states, become light constituting the predicted tower. In the opposite limit, when the internal space becomes small, one expects dual winding modes to appear. This seems to have direct implications for field theories at finite temperature. However, in the following we will see that the temporal circle in flat space is not associated to quantum gravity constraints imposed by Swampland conjectures.
To describe the finite temperature regime one performs a Wick rotation of the time direction, $t \rightarrow i \tau$, and compactifies the Euclidean time on a circle. The radius of this circle $R_{\tau}$ is related to the temperature $\mathcal{T}$ of the system\footnote{We set the Boltzmann constant to one.}
\begin{align}
\mathcal{T} = \frac{\hbar}{\beta} = \frac{\hbar}{2 \pi R_{\tau}} \,.
\end{align}
In analogy to the KK states in a circle compactification, we can associate a tower of states with frequencies
\begin{align}
\omega_n = \frac{2 \pi n}{\beta} = \frac{2 \pi n}{\hbar} \mathcal{T} \,,
\end{align}
to the time-like circle. These are the so-called {\it Matsubara modes}. One then might be tempted to predict a tower of states with mass scale determined by
\begin{align}
m = \omega_1 = \frac{2 \pi}{\hbar} \mathcal{T} \,.
\label{eq:Matfreq}
\end{align}
This would follow from a distance associated to a variation of the temperature given by
\begin{align}
\Delta_{\mathcal{T}} \sim \bigg| \frac{1}{\lambda} \text{log} \Big( \frac{2 \pi L_p}{\hbar} \, \mathcal{T} \Big) \bigg| \,.
\end{align}
The SDC then demands the appearance of a light tower for $\Delta_{\mathcal{T}} \rightarrow \infty$. One realization of this limit is given by $\mathcal{T} \rightarrow 0$, with the Matsubara modes above becoming light. Another realization can be obtained by $\mathcal{T} \rightarrow \infty$, which would demand a dual tower of thermal winding modes, see \cite{Atick:1988si,Kounnas:1989dk,Angelantonj:2008fz}. Therefore, these considerations show that the temperature might indeed be an interesting parameter to explore from a quantum gravity perspective.
However, a gravitational system at finite temperature is highly problematic \cite{Atick:1988si}. The finite temperature induces a finite energy density which, in a large enough region, leads to gravitational (Jeans) instabilities. Since we are still interested in non-compact backgrounds, we explore other gravitational systems that allow for a well-defined notion of temperature. In these setups the temperature appears at a horizon. For these backgrounds, we investigate the appearance of light states, similar to the Matsubara modes above, in the extremal temperature regime $\mathcal{T} \rightarrow 0$ and $\mathcal{T} \rightarrow \infty$. Importantly, all meaningful towers need to satisfy \eqref{eq:qgconst}.
As a warm-up example we can study the temperature observed by an accelerated observer, i.e., the Unruh effect in Rindler space. Rindler space can be obtained from Minkowski spacetime with coordinates $(x_1, x_2, x_3, x_4)$ by the following identification
\begin{align}
x_0=\rho\sinh (a\eta)\, ,\qquad x_1=\rho\cosh(a \eta) \,.
\end{align}
In these hyperbolic coordinates the four-dimensional metric is given by
\begin{align}
ds^2=(-a^2\rho^2d\eta^2+d\rho^2)+(dx_2)^2+(dx_3)^2 \,,
\label{Rindler}
\end{align}
where $a$ is the acceleration parameter of the observer. Due to the Unruh effect, the accelerated observer experiences a thermal radiation, and the corresponding Unruh temperature can be read off from the Rindler metric by switching to Euclidean time, i.e., by replacing $\eta= i\tau$. To avoid a conical singularity at the origin, the Euclidean time $\tau$ must have periodicity $\tau\sim\tau+\beta_U$ and the Unruh temperature is given by
\begin{align}
{\cal T}_U = {\hbar \over\beta_U}={\hbar a\over 2\pi} \,.
\end{align}
In analogy to the Matsubara frequency \eqref{eq:Matfreq}, we define the Unruh frequency
\begin{align}
\omega_U = \frac{2 \pi}{\hbar} \, \mathcal{T}_U = a \,.
\end{align}
We see that $\omega_U$ and therefore the associated mass scale does not depend on $L_p$. Consequently, the decoupling condition \eqref{eq:qgconst} is not satisfied for finite $a$ and one does not obtain an SDC tower by varying $\mathcal{T}_U$. This is indeed expected, since Rindler space describes part of flat space. However, it shows that the condition \eqref{eq:qgconst} is crucial in order to draw conclusions concerning Swampland constraints associated to the temperature of a system.
Therefore, we turn to different gravitational backgrounds that exhibit a temperature and have a more direct relation to quantum gravity, namely black hole metrics. We further include a second thermodynamic quantity, the entropy, which allows the study of a two-parameter family of gravitational solutions.
\subsection{Schwarzschild black holes and Hawking temperature}
We start by considering the simplest black hole solution, namely the Schwarzschild black hole, whose metric is given by
\begin{equation}
\text ds^2 = -f(r)\text dt^2+ f(r)^{-1} \text dr^2+r^2\text d\Omega_{2}^2\;,\qquad f(r) = 1-\frac{2M G_N}{r} \,,
\label{staticbh}
\end{equation}
with mass $M$ and horizon size $r_S=2MG_N$. The associated Bekenstein-Hawking entropy reads
\begin{equation}
\label{entropySchw}\boxed{
{\cal S}= {8 \pi^2 r_S^2 \over L_p^2} = {32 \pi^2 G_N^2 M^2 \over L_p^2} = \frac{L_p^2 M^2}{2 \hbar^2} \,,}
\end{equation}
and the Hawking temperature is
\begin{equation}\boxed{
{\cal T} = { \hbar\kappa\over 2\pi}={\hbar \over 8\pi G_N M} = {\hbar^2\over L_p^2 M} \,,}
\label{HawkingT}
\end{equation}
where $\kappa = (2 r_S)^{-1}$ is the surface gravity.
For the Schwarzschild black hole, the entropy and the Hawking temperature are not independent quantities but are related as (setting $\hbar=L_p=1$)
\begin{equation}
{\cal S}{\cal T}^2 = \frac{1}{2} \,.
\label{STrel}
\end{equation}
Anticipating some later results we further note that the transformation
\begin{equation}
\sqrt{\cal S}\,\longleftrightarrow\,{\cal T}
\label{STex}
\end{equation}
exchanges small with large black holes, namely acts on the mass as $M\longleftrightarrow{1\over M}$, while leaving eq.~\eqref{STrel} invariant. Using the relation (\ref{STrel}) we can express the transformation eq.~\eqref{STex} (up to a constant factor) equivalently as
\begin{equation}
{\cal T}\,\longleftrightarrow\,{1\over {\cal T}} \quad{\rm and}\quad {\cal S}\,\longleftrightarrow\,{1\over {\cal S}}\, .
\label{STdual}
\end{equation}
In section (\ref{thlimits}) we will discuss how these transformations act on the internal KK and winding spectrum in supergravity and in string compactifications.
It is instructive to recall how the Hawking temperature can be derived from the near horizon geometry. For this purpose, we introduce the coordinate ${x^2\over 8G_NM}=r-r_S$ and for small $x^2$ we obtain the metric in the near horizon limit and for Euclidean time $\tau=it$,
\begin{equation}
\text ds^2=(\kappa x)^2\text d\tau^2+ dx^2+(2\kappa)^{-2}\text d\Omega_2^2\,.
\label{nearHorizonMetricSchwarzschild}
\end{equation}
The two-dimensional part of this metric for $x$ and $\tau$ is the Rindler space metric \eqref{Rindler} and we can immediately read off the Hawking temperature ${\cal T}$ in agreement with \eqref{HawkingT}. Notice that the derivation is completely analogous to that of the Unruh temperature, i.e., we require the absence of a conical singularity at the origin. However, while in the case of the Unruh temperature the acceleration $a$ is a free parameter, in the case of the Hawking temperature the surface gravity $\kappa$ fixes also the size of the $S^2$ part of the space. As such, $\kappa$ is not a free parameter, but it is determined by the geometry of the full four-dimensional space. This has important consequences and in fact amounts to saying that, for the case of Schwarzschild black hole, entropy and temperature are interdependent quantities.
The two potentially interesting temperature regimes $\mathcal{T} \rightarrow \{ 0, \infty\}$ can be rephrased in terms of the mass $M$ of the black hole:\footnote{In addition there are also two kinds of classical limits:
\vskip0.2cm\noindent
Semiclassical limit: $M\rightarrow\infty$ and $G_N\rightarrow 0$ while keeping $\hbar$ and $r_S$ finite. One has that $L_p \rightarrow 0$. In this limit, ${\cal T}$ stays finite and the Hawking radiation is exactly thermal.
\vskip0.2cm\noindent
Classical limit: $\hbar \rightarrow 0$ while keeping $M$ and $G_N$ finite, and hence also $r_S$ finite. Again, one has that $L_p \rightarrow 0$. In this limit, the Hawking temperature goes to zero, ${\cal T}\rightarrow 0$, and therefore there is no Hawking radiation.
\vskip0.2cm\noindent
For these two cases the Planck length goes to zero, since either gravity decouples or because quantum effects vanish. Therefore, for these two cases Swampland arguments should not be applicable and hence we do not generically expect a massless tower of states.}
\vskip0.2cm
(I) Small black hole limit, $\hbar$ finite, $G_N$ finite, $M\rightarrow 0$: \, One has $r_S\rightarrow 0$ ($\kappa\to\infty$). In this limit, the Hawking temperature goes to infinity, ${\cal T}\rightarrow \infty$, and at the same time the entropy becomes very small, i.e. ${\cal S}\rightarrow 0$. For $M=0$ one is dealing with flat Minkowski spacetime.
\vskip0.2cm
(II) Large black hole limit, $\hbar$ finite, $G_N$ finite, $M\rightarrow \infty$: \, One has $r_S\rightarrow \infty$ ($\kappa\to0$). In this limit, the Hawking temperature goes to zero, ${\cal T}\rightarrow 0$, and therefore there is no Hawking radiation. The entropy becomes very large, ${\cal S}\rightarrow\infty$, and the near-horizon geometry approaches Rindler space with zero temperature, which is again flat Minkowski spacetime.
\vskip0.2cm
\noindent
Since in both of these limits $L_p$ remains finite, we expect quantum gravity effects to be important and Swampland constraints to be applicable.
Let us once more use the analogy to finite temperature systems giving rise to Matsubara modes, by defining the Hawking frequency via the Hawking temperature
\begin{align}
\omega_H={2\pi\over \hbar }{\cal T}=\kappa={2 \pi \hbar\over L_p^2M} \,.
\label{eq:Hfreq}
\end{align}
In the limit $\mathcal{T} \rightarrow 0$, which is limit (II) above, also the Hawking frequency vanishes. However, as opposed to the Unruh temperature the consistency condition \eqref{eq:qgconst} is satisfied and the mass scale decouples for $L_p \rightarrow 0$. This indicates that the temperature of a black hole solution might be a viable Swampland parameter. Utilizing \eqref{STrel} we can further determine the relevant quantities in terms of the entropy $\mathcal{S} = \frac{L_p^2 M^2}{2}$ of the Schwarzschild black hole
\begin{align}
\omega_H=\frac{\sqrt{2} \pi}{L_p \sqrt{\cal S}} \,.
\end{align}
This demonstrates that the vanishing of the Hawking frequency corresponds to the limit of infinite entropy $\mathcal{S} \rightarrow \infty$ which was also discussed in \cite{Bonnefoy:2019nzv}. Assuming that a tower of the mass scale \eqref{eq:Hfreq} indeed appears, this can be used to define a distance for Schwarzschild black hole geometries given by
\begin{align}
\Delta_{BH}=\left|{1\over \lambda}\log\biggl({2\pi L_p\over \hbar}{\cal T}\biggr)\right|=\left|{1\over \lambda}\log \biggl(\frac{\sqrt{2} \pi}{ \sqrt{\cal S}}\biggr)\right| \,.
\label{BHDIST}
\end{align}
In \cite{Bonnefoy:2019nzv} it was argued that at infinite entropy, i.e., for $\Delta_{BH} \rightarrow \infty$ an infinite number of black hole microstates becomes degenerate. Interpreting $\omega_H$ as a natural mass gap between those, therefore also leads to an interpretation in terms of a light tower of states.
However a priori there is nothing quantum-mechanical or gravitational about this: the black hole becomes large, so its characteristic frequency goes to zero.
Therefore, as already mentioned, we like to consider black hole solutions, for which ${\cal S}$ and ${\cal T}$ are related to moduli parameters and to the physical towers of scalar fields within an underlying string theory
realization.
This infinite entropy limit can also be discussed from the point of view of the near horizon geometry, which is ${\cal R}^{1,1}\times S^2$, where ${\cal R}^{1,1}$ is the two-dimensional Rindler space and $S^2$ is the 2-sphere. In the limit ${\cal S}\rightarrow \infty$ (${\cal T}\rightarrow 0$), the volume of the $S^2$ becomes infinite and the corresponding Kaluza-Klein modes becomes massless. This could be a signal of a breakdown of the two-dimensional effective field theory on ${\cal R}^{1,1}$. However, it is unclear if one should apply Swampland considerations in two (and three) dimensions, and it would be safer to derive Swampland arguments signaling a breakdown of the entire four-dimensional effective field theory.
The black hole distance $\Delta_{BH}$ in \eqref{BHDIST} also diverges in the \emph{dual} limit of small entropy and large temperature ${\cal S}\rightarrow 0$, ${\cal T}\rightarrow \infty$, corresponding to case (I). We could thus wonder if there are certain modes which become massless in this limit. We will come back to this question in the context of charged dilatonic and supergravity black holes.
\subsection{Reissner--Nordstrom black hole, temperature, and entropy}
Motivated by the previous discussion, we extend our consideration in this section to black holes with two parameters, mass and charge, i.e., Reissner-Nordstrom black holes. For a black hole with non-vanishing charge, we can introduce the concept of extremality. A black hole is extremal if it has the lowest possible mass for a fixed charge, while avoiding naked singularities. The Reissner--Nordstrom (RN) solution is a charged black hole with metric
\begin{equation}
\label{RNbh}
\text ds^2 = -f(r) \text dt^2 + f(r)^{-1} \text dr^2 + r^2 \text d \Omega_2^2, \qquad f(r) = 1 - \frac{2M G_N}{r} + \frac{Q^2 G_N}{r^2} \,,
\end{equation}
where $M$ is the mass and $Q$ the (electric) charge. This black hole has two horizons located at the zeroes of the function $f(r)$,
\begin{equation}
f(r_\pm) = 0, \qquad r_\pm = G_N( M \pm c) \,,
\end{equation}
where we introduced the extremality parameter $c$ such that
\begin{equation}
G_N\,c = \sqrt{G_N^2 M^2- G_NQ^2} \geq0 \,, \qquad M\geq c \geq 0 \,.
\end{equation}
In order to avoid naked singularities, we have to require $M^2\geq Q^2$. The entropy of the black hole is given by
\begin{equation}
\label{SRN}\boxed{
\mathcal{S} = \frac{8 \pi^2 r_+^2}{L_p^2} \,.}
\end{equation}
It is important to notice that the regimes $c>0$ and $c=0$ are qualitatively different, so we discuss them separately below.
For $c >0$, we can introduce the coordinate $\frac{G_N\, c \, x^2}{2 r_+^2} = r-r_+$. For small $x^2$ we obtain the metric in the near horizon limit and with Euclidean time $\tau = i t$,
\begin{equation}
\label{RNnh}
\text d s^2 = (\kappa x)^2 \text d \tau^2 + \text d x^2 + (r_+)^2 \text d \Omega_2^2 \,,
\end{equation}
where the surface gravity is $\kappa = \frac{G_N\, c}{r_+^2}$.
By imposing the absence of conical singularities, we find the temperature
\begin{equation}
\label{TRN}\boxed{
{\cal T}= \frac{\hbar\kappa}{2 \pi} = \frac{\hbar c}{2 \pi G_N (M+c)^2} \,.}
\end{equation}
In terms of the Hawking temperature ${\cal T}$ and the entropy ${\cal S}$, the extremality parameter can be expressed as
\begin{equation}
\label{c=2ST}\boxed{
c = 2 \mathcal{S} {\cal T} \,,}
\end{equation}
and the metric \eqref{RNnh} can be rewritten as
\begin{equation}
\label{RNnh1}
\text d s^2 = \Big({4\pi^2\over \hbar^2}\Big)({\cal T} x)^2 \text d \tau^2 + \text d x^2 + \Big({L_p^2\over 8 \pi^2}\Big){\cal S} \,\text d \Omega_2^2 \,.
\end{equation}
Therefore, the radius of the Euclidean time-circle is given in terms of ${\cal T}$, whereas the area of the $S^2$ is determined by the entropy ${\cal S}$. This has to be contrasted with the Schwarzschild black hole, where both these quantities were governed by the same (unique) parameter $M$ of the solution. The Hawking frequency of the RN black hole is given by
\begin{equation}
\label{matsRN}
\omega_H = \frac{2\pi}{\hbar} {\cal T} = \kappa = \frac{G_N\, c}{r_+^2} \, ,
\end{equation}
and again the condition \eqref{eq:qgconst} is satisfied. Therefore, the variation of the temperature of a RN black hole might lead to interesting constraints.
For the extremal case, $c=0$, the coordinate $x^2$ introduced above becomes ill-defined. Thus, we cannot take the limit $c \rightarrow 0$ in the near horizon metric \eqref{RNnh}. The correct strategy is to first take the extremal limit and then the near horizon limit. The RN metric \eqref{RNbh} then reduces to
\begin{equation}
\text d s^2 = - \Big(\frac{r-r_h}{r}\Big)^2 \text d t^2 + \Big(\frac{r-r_h}{r}\Big)^{-2} \text dr^2 + r^2 \text d \Omega_2^2,
\end{equation}
where $r_h = G_N M = G_N^{1/2} Q$ is the horizon radius (we will set $M_p=1$ from now on). The near horizon coordinate is now $x = r-r_h$ and for small $x$ we get the metric
\begin{equation}
\label{RNextnh}
\text ds^2 = - \frac{x^2}{r_h^2} \text d t^2 + \frac{r_h^2}{x^2} \text d x^2 + r_h^2 \text d \Omega_2^2.
\end{equation}
We recognize the $AdS_2$ metric along the $(t,x)$ directions with radius $r_h$. This metric is clearly regular everywhere, so there is no temperature. Equivalently, taking ${\cal T} = \frac{\hbar \kappa}{2 \pi}$ as a definition of Hawking temperature, we can calculate the surface gravity for the extremal RN black hole and find that $\kappa=0$, implying ${\cal T}=0$.\footnote{We recall that for Schwarzschild and RN black holes the surface gravity is given by $\kappa = \frac12 f^\prime(r_+)$}
For the purposes of our analysis, it is convenient to choose a basis for the parameter space of RN solutions in which the independent coordinates are ${\cal T}$ and $\mathcal{S}$. We can then study the limit of small or large temperature or entropy. Using, \eqref{SRN}, \eqref{TRN} and \eqref{c=2ST}, we can rewrite the full RN metric in terms of ${\cal T}$ and $\mathcal{S}$ as
\begin{equation}
\text ds^2 = - \frac{(r-r_+)(r-r_-)}{r^2} dt^2+\left(\frac{(r-r_+)(r-r_-)}{r^2}\right)^{-1} dr^2 + r^2 d\Omega_2^2,
\end{equation}
where
\begin{equation}
r_+ = \sqrt{\frac{\mathcal{S}}{8 \pi^2}},\qquad r_- = \sqrt{\frac{\mathcal{S}}{8 \pi^2}} - \frac{1}{2 \pi} \mathcal{S}{\cal T}.
\end{equation}
In the small temperature limit, ${\cal T} = 0$, the expressions simplify. Clearly, this basis illustrates that temperature and entropy of a RN black hole are different order parameters. In a ${\cal T}-\mathcal{S}$ diagram, charged RN black holes populate the region bounded by uncharged Schwarzschild black holes, as shown in Figure \ref{fig:BHpopulation}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=1.25]
\draw[->, line width=.5mm] (0, 0) -- (4.2, 0) node[below] {${\cal T}$};
\draw[->,line width=.5mm] (0, 0) -- (0,4.33) node[left] {${\cal S}$};
\draw[scale=1.5,domain=0.6:2.7,smooth, variable=\x,blue, thick,line width=.6mm] plot ({\x},{1/((\x)*(\x))});
\fill [blue!25, domain=0.1:4.2, variable=\x]
(0.02, 4.17)
-- plot[scale=1.5,domain=0.6:2.7] ({\x},{1/((\x)*(\x))})
-- (0.02, 0.02);
\fill [blue!25] (0.02,0.02) rectangle (4.05,0.2);
\node () at (3,3) {excluded region};
\draw[scale=1.5,domain=1.04:2.7,smooth, variable=\x,red,line width=.6mm] plot ({\x},{1/(\x)});
\draw[scale=1.5,domain=0.36:1.1,smooth, variable=\x,red, dashed,line width=.6mm] plot ({\x},{1/(\x)});
\draw[dashed] (1.55,1.4)--(1.55,0);
\node () at (1.55,-.3) {${\cal T}_0$};
\end{tikzpicture}
\end{center}
\caption{Population of the ${\cal T}-\mathcal{S}$ plane by RN black holes. The boundary ${\cal S}=(2 {\cal T}^2)^{-1}$ between the allowed region in blue (with $Q^2>0$)
and the excluded region (where
$Q^2<0$) is given by uncharged ($Q=0$) Schwarzschild black holes (blue line). The red line is the line of constant extremality parameter $c=2 \mathcal{S} {\cal T}$. For ${\cal T}>{\cal T}_0={1\over c}$, one cannot keep $c$ constant while staying in the allowed region.}
\label{fig:BHpopulation}
\end{figure}
While the condition \eqref{eq:qgconst} is satisfied for the Hawking temperature, it is not clear whether a distance defined by \eqref{BHDIST} and its related version for the RN black hole is appropriate for an interpretation in terms of the SDC. The reason for this is that we employed an analogy to the Matsubara and therefore KK momentum states that appear if the underlying geometry contains a circle. However, in the black hole geometries there is in general no such a circle and one has to be more careful. The distance \eqref{BHDIST} would for example demand that whenever a black hole becomes extremal there should be an associated light tower of states. This seems to be unlikely, since for supersymmetric black holes we even control the microscopic details and small deformations away form extremality seem to be well-behaved, see e.g. \cite{Saraikin:2007jc}. Moreover, it was argued that the semi-classical thermodynamic quantities in non-supersymmetric setups are potentially subject to large quantum corrections in the extremal limit \cite{Preskill:1991tb, Maldacena:1998uz, Page:2000dk, Heydeman:2020hhw}.
Our strategy is therefore to relate the black hole limits of large/small temperature and entropy to field excursions of scalar fields, which are under better control and have a direct relation to the SDC. In order to do so, we need to extend our theories by coupling to scalar fields. This is why we turn to Einstein-Maxwell-dilaton theory in Section \ref{sec:EMDblackholes} and to the $STU$ model in $\mathcal{N} = 2$ supergravity in Section \ref{SG-blackholes}.
Before we start our discussion with scalar fields, we want to analyze the two-parameter black hole solutions above from the perspective of a distance in the space of metrics.
\subsection{Distance in the space of metrics}
One possible definition of a distance between distinct solutions to the gravitational equations of motion is given by the metric distance as discussed in \cite{Bonnefoy:2019nzv}. In practice, we can consider these distances directly in the near horizon geometries. We have seen that starting from the RN black hole \eqref{RNbh} and performing first the near horizon limit, we get the geometry \eqref{RNnh}, which is locally ${\cal R}^{1,1} \times S^2$, with non-vanishing Hawking temperature. The extremal limit is then ill-defined, since the near horizon coordinate becomes singular for $c=0$. On the other hand, performing first the extremal limit and then the near horizon limit, we arrive at the metric \eqref{RNextnh}, which is locally $AdS_2 \times S^2$ and which has vanishing temperature.
The question is then if the spaces ${\cal R}^{1,1} \times S^2$ and $AdS_2 \times S^2$ could be at infinite distance to one another, with the Hawking modes \eqref{matsRN} being the tower predicted by the SDC.
Given a family of metrics with a set of parameters which we collectively denote by $\alpha$, the distance in the parameter space is defined via the formula
\begin{equation}
\label{distmetric}
\Delta_g(\alpha) \sim \left|\int_{\alpha_i}^{\alpha_f} \text d \alpha \left(\frac{1}{\text{Vol}} \int d^4 x \sqrt{g} \, g^{\mu\nu}g^{\rho\sigma}\frac{\partial g_{\mu\rho}}{\partial \alpha} \frac{\partial g_{\nu\sigma}}{\partial \alpha}\right)^{\frac 12}\right|,
\end{equation}
where $\text{Vol} = \int \text d^4x \sqrt{g}$ is the volume of spacetime. We can now use this formula for different choices of the parameters characterizing the solution.
The simplest choice is to consider the metric \eqref{RNnh} as a function of the extremality parameter $c$. We can compute the integral above and find
\begin{equation}
\Delta_{RN,nh} (c) \sim |\log c|\, \qquad \text{for} \quad c \to 0 \,.
\end{equation}
Therefore, taking the extremal limit in the near horizon RN metric \eqref{RNnh} corresponds to going to infinite distance in the space of metrics. One possible explanation for this divergence in the space of metrics is the formation of an infinitely long throat. Indeed, if one calculates the distance to the outer horizon from a reference radius $r_*$ outside the black hole one finds (along a slice of constant time)
\begin{align}
\ell = \int_{r_+}^{r_*} \sqrt{g_{rr}} \,dr =\int_{r_+}^{r_*} dr \sqrt{\frac{r^2}{(r - r_+) (r - r_-)}} \, ,
\end{align}
which is finite for non-extremal black holes, but it diverges logarithmically in the extremal case. This can be seen explicitly in the shifted radial coordinate $y = r - r_+$, giving
\begin{align}
\ell = \int_{0}^{r_* - r_+} dy \sqrt{ \frac{(y + r_+)^2}{y \big(y + \tfrac{c}{8 \pi} \big)}} \,,
\end{align}
leading to a logarithmic contribution $\sim|\log c \, |$ close to the outer horizon, just as for the metric distance. Therefore, a conceivable physical interpretation is that the infinite metric distance is induced by the stretching of the throat region when sending $c \rightarrow 0$.
We can also consider ${\cal S}$ and ${\cal T}$ as independent parameters and repeat a similar analysis. The metric distance with two parameters $\alpha_i=(\alpha_1,\alpha_2)$ along a path $\alpha(\alpha_i)$ is
\begin{align}
\label{distance}
\Delta(g) \sim \left|\int_{\alpha_i}^{\alpha_f}d\alpha \left(
\frac{1}{\text{Vol}}\int d^4 x \sqrt{g} \, g^{\mu\nu}g^{\rho\sigma}\partial_i g_{\mu\rho}\partial_j g_{\nu\sigma}\right)^{1/2}\right|
\end{align}
It is now convenient to employ the Kruskal extension of the black hole metric at the outer horizon
\begin{align}
\label{Kruskal-RN-outer-hor}
ds^2_{Kruskal}&=- \frac{1}{2\pi^2 {\cal T}^2} dU dV + \frac{\cal S}{8 \pi^2} d\Omega_{2}^2 \ .
\end{align}
Using these coordinates and for $\alpha_1 = {\cal T}$, $\alpha_2=\mathcal{S}$, we find
\begin{equation}
\Phi_{ij} [g_{\mu\nu}, \alpha^k]\equiv \frac{1}{\rm Vol}\int d^4x \sqrt{g} \, g^{\mu\nu} \partial_ig^{\rho\sigma}\partial_{j}g_{\mu\rho} =
\left(\begin{array}{cc}
\frac{8}{{\cal T}^2}&0 \\
0 & \frac{2}{{\cal S}^2}
\end{array}
\right)
\end{equation}
and thus for a path parametrized by $\tau \in [0,1]$
\begin{align}\label{metricdistanceRN}
\Delta(g)&\sim \int_0^1 d\tau \sqrt{\Phi_{ij} \frac{d \alpha^i}{d\tau} \frac{d \alpha^j}{d \tau}} = \int_0^1 d \tau \sqrt{8 \Big(\frac{d \, \text{log} \mathcal{T}}{d\tau}\Big)^2 + 2 \Big( \frac{d \, \text{log} \mathcal{S}}{d \tau} \Big)^2 } \,,
\end{align}
for the metric in the two parameter space of the RN black hole. This distance\footnote{The specific distance depends on the path chosen.} diverges in the limits $\mathcal{S}, \mathcal{T} \rightarrow \{ 0, \infty\}$.
The metric distance \eqref{metricdistanceRN} in fact diverges in any of the five limits discussed in the introduction of this paper. Nevertheless the identification of the corresponding tower of states represents in general a challenge. While the stretching of a throat at first might have some similarity to a decompactification process, which leads to a tower of light KK modes, the direct correlation to the SDC is more subtle, see also \cite{Li:2021utg}. For the special limit $\mathcal{S} \rightarrow \infty$ there are other checks that suggest the appearance of light modes, i.e. the high ground state degeneracy. The same logic, however, cannot be applied for finite $\mathcal{S}$. Therefore, we do not conclude that extremal $\mathcal{T} = 0$ black holes are at infinite distance for finite entropy, but rather try to explore these limits in systems with scalar fields. This points towards limitations of the metric distance as a Swampland parameter, resonating with the additional caveat that \eqref{distmetric} for the metric distance is not diffeomorphism invariant\footnote{A procedure to deal with this technical obstruction has been proposed in \cite{Bonnefoy:2019nzv}, to which we refer the reader.}.
\section{Dyonic black holes in Einstein-Maxwell-dilaton theory}
\label{sec:EMDblackholes}
In this section, we investigate black holes with electric and/or magnetic charges in Einstein-Maxwell-dilaton (EMd) theory. We start with a review of these solutions, following \cite{Ivashchuk:1999jd, Abishev:2015pqa, Loges:2019jzs} (see also \cite{Gibbons:1987ps, Garfinkle:1990qj, Cheng:1993wp, Heidenreich:2015nta}). The action is given by
\begin{align}
S_{\text{EMd}} = \int d^4 x \sqrt{- g} \Big( \tfrac{1}{2} M^2_p \big( R - \partial_{\mu} \phi \partial^{\mu} \phi \big) - \tfrac{1}{4} e^{-2 \lambda \phi} F_{\mu \nu} F^{\mu \nu} \Big) \,,
\end{align}
with dilaton field $\phi$ and its coupling to the U$(1)$ gauge field determined by $\lambda$. In particular, the previously considered RN black hole corresponds to $\lambda=0$.
It is convenient to introduce a rescaled dilaton coupling,
\begin{align}
h = \frac{2}{1+ 2 \lambda^2} \leq 2 \,,
\end{align}
with respect to which the static, spherically symmetric black holes solutions can be given as
\begin{equation}
\begin{split}
ds^2 &= - f(r) dt^2 + \frac{1}{f(r)} dr^2 + r^2 \big( H_e H_m \big)^h d S_2^2 \,, \\
f(r) &= \big( H_e H_m \big)^{-h} \Big( 1 - \frac{c}{4 \pi r} \Big) \,, \\
e^{-2 \lambda \phi} &= \Big( \frac{H_e}{H_m} \Big)^{2 - h} \,, \\
F &= \frac{q}{r^2} H_e^{-2} H_m^{2 - 2h} dt \wedge dr + p \, \text{sin} (\theta) d\theta \wedge d \varphi \,,
\end{split}
\label{eq:EMdBH}
\end{equation}
with $q,p$ electric and magnetic charges, respectively.
These solutions are well-defined for the coordinate values
\begin{align}
r \in \big( \tfrac{c}{4 \pi} \,, \infty\big) \,,
\label{eq:rrange}
\end{align}
where the extremality parameter ${c}$ defines the location of the outer horizon
\begin{align}
r_+ = \tfrac{c}{4 \pi} \,.
\end{align}
(The inner horizon is at $r_-=0$.)
The two undetermined functions $H_i (r)$, $i=e,m$, are solutions to the differential equation
\begin{align}
\label{diffH}
r^2 \frac{d}{d r} \bigg( r^2 \Big( 1 - \frac{c}{4 \pi r} \Big) \frac{H'_e (r)}{H_e (r)} \bigg) = - \frac{q^2}{h} H_e^{-2} H_m^{2 - 2h} \,,
\end{align}
and similarly for $e \leftrightarrow m$, $q \leftrightarrow p$. They are subject to the boundary conditions
\begin{align}
H_i (r) \underset{r \rightarrow \infty}{\longrightarrow} 1 \,, \quad \text{and} \quad H_i (r) > 0 \,, \quad \text{for} \enspace r \rightarrow r_+ \,,
\end{align}
guaranteeing that the solutions are well-behaved in the region \eqref{eq:rrange}. These differential equations do not have a closed form for solutions at arbitrary value of $\lambda$. Nevertheless, solutions can still be parametrized as a power-series
\begin{align}
H_i (r) = 1 + \frac{\xi^{(1)}_i}{r} + \frac{\xi^{(2)}_i}{r^2} + \dots \,, \qquad i=e,m,
\label{eq:Hexpa}
\end{align}
in terms of constants $\xi^{(j)}_i$, and they converge within \eqref{eq:rrange}. Moreover, the system admits an integral of motion, which in the $r\to \infty$ limit reads
\begin{equation}
\label{1stint}
(\xi_e^{(1)})^2 + (\xi_m^{(1)})^2 + 2(h-1) \xi_e^{(1)} \xi_m^{(1)} + \tfrac{c}{4\pi} (\xi_e^{(1)}+\xi_m^{(1)}) - h^{-1}(q^2 + p^2) = 0,
\end{equation}
and it can be solved explicitly for some interesting values of the dilaton coupling $h$.
The above solutions approach extremality in the limit $c \rightarrow 0$. The entropy and temperature of the corresponding dilatonic black holes are given by
\begin{equation}\boxed{
\begin{split}
{\cal T} &= \tfrac{1}{{c}} \big( H_e \big( \tfrac{c}{4 \pi}\big) H_m \big( \tfrac{c}{4 \pi}\big) \big)^{-h} \,, \\
{\cal S} &= \tfrac{1}{2} {c}^2 \big( H_e \big( \tfrac{c}{4 \pi}\big) H_m \big( \tfrac{c}{4 \pi}\big) \big)^h \,.
\end{split}}
\label{eq:TSEMd}
\end{equation}
From which one again finds the extremality parameter
\begin{align}
2 {\cal S} {\cal T} = c \,.
\end{align}
The mass of the black hole is given by
\begin{align}
M = c + 4 \pi h \big(\xi^{(1)}_e + \xi^{(1)}_m \big) \,,
\end{align}
which can be obtained from the metric in the limit of large $r$.
\subsection{Universal behavior in the extremal limit}
\label{subsec:univ}
Before analyzing some specific models, we want to explore the universal behavior of the above solutions in the extremal limit, $c \rightarrow 0$.
First, we notice that in this limit the product $\mathcal{S} \mathcal{T}$ needs to vanish. The asymptotic behavior for ${\cal T}$ and $\mathcal{S}$ individually depends on the functions $H_i (r)$ in the vicinity of the outer horizon as well as the dilaton coupling $h$. With \eqref{eq:TSEMd}, one finds for $c \rightarrow 0$ the following possibilities.
\begin{equation}
\begin{split}
{\cal T} \rightarrow 0 \, : \, \text{if}& \quad H_e \big( \tfrac{c}{4 \pi} \big) H_m \big( \tfrac{c}{4 \pi} \big) \rightarrow \infty \quad \text{faster than} \quad {c}^{- \frac{1}{h}} \,, \\
{\cal T} \rightarrow \infty \, : \, \text{if}& \quad H_e \big( \tfrac{c}{4 \pi} \big) H_m \big( \tfrac{c}{4 \pi} \big) \rightarrow \infty \quad \text{slower than} \quad {c}^{- \frac{1}{h}} \,, \\
\mathcal{S} \rightarrow \infty \, :\, \text{if}& \quad H_e \big( \tfrac{c}{4 \pi} \big) H_m \big( \tfrac{c}{4 \pi} \big) \rightarrow \infty \quad \text{faster than} \quad {c}^{- \frac{2}{h}} \,, \\
\mathcal{S} \rightarrow 0 \, :\, \text{if}& \quad H_e \big( \tfrac{c}{4 \pi} \big) H_m \big( \tfrac{c}{4 \pi} \big) \rightarrow \infty \quad \text{slower than} \quad {c}^{- \frac{2}{h}} \,.
\end{split}
\end{equation}
This identifies three generic regions, I, II, and III, separated by threshold values $t_1$ and $t_2$ which allow for finite temperature and entropy, respectively. They are depicted in Figure \ref{fig:para_behav}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=1.]
\draw[->, line width=.5mm] (0, 0) node[left] {$\left(H_e \left(\frac{c}{4\pi}\right) H_m\left(\frac{c}{4\pi}\right)\right)^h\bigg|_{c\to 0}$} -- (9, 0) node[below] {};
\draw[line width = .4mm, color=red] (3,-.4)--(3,3);
\draw[line width = .4mm, color=red] (6,-.4)--(6,3);
\node () at (3,-.8) {$\sim 1/c$};
\node () at (3,3.3) {$t_1$};
\node () at (6,3.3) {$t_2$};
\node () at (6,-.8) {$\sim 1/c^2$};
\node () at (1.5,1.5) {$\begin{array}{c}\, \, \,\ \,{\cal T}\to \infty \\ \ \ \mathcal{S} \to 0\end{array}$};
\node () at (4.5,1.5) {$\begin{array}{c} \ \, {\cal T} \to 0 \\ \ \ \mathcal{S} \to 0\end{array}$};
\node () at (7.5,1.5) {$\begin{array}{c} \ \, {\cal T}\to 0 \\ \ \ \,\, \, \mathcal{S} \to \infty\end{array}$};
\node () at (1.5,2.5) {I};
\node () at (4.5,2.5) {III};
\node () at (7.5,2.5) {II};
\end{tikzpicture}
\end{center}
\caption{Depending on the behavior of $H_e(r) H_m (r)$ for $r \rightarrow 0$, there are three generic regions of the black hole thermodynamics, which are separated by threshold values.}
\label{fig:para_behav}
\end{figure}
In the parameter region I, the extremal limit leads to a state at infinite temperature with vanishing entropy. This is a small black hole limit. The specific realization of these states likely needs a UV complete description of the model and might be related to particle states as discussed in \cite{Holzhey:1991bx}. At the threshold $t_1$, the entropy still vanishes in the extremal limit, but one can achieve finite temperatures by balancing the parameters in the setup. The region II leads to infinitely large extremal black holes with vanishing temperature. These solutions are a priori not problematic from the viewpoint of the low energy effective description. In the parameter region III, both the temperature and the entropy of the extremal state vanishes. This is another small black hole limit. These charged extremal black holes of vanishing size are problematic in view of entropy arguments as those discussed in \cite{Hamada:2021yxy}. Once more, a UV description is necessary, whose higher curvature corrections induce a small but finite size of the extremal black holes. The appearance of small black holes in the effective theory can therefore be used to indicate that higher order effects become important. At the threshold $t_2$, the extremal black holes have vanishing temperature but a non-vanishing horizon area and entropy and thus they differ from the problematic small black holes of region III. A priori these black holes seem to be well-behaved in the effective field theory description. However, it has been argued that the semiclassical description of their thermodynamical properties can receive large corrections in the non-supersymmetric setup, see e.g.~\cite{Preskill:1991tb, Maldacena:1998uz, Page:2000dk, Heydeman:2020hhw} and references therein.
To illustrate the different universal regions, we refer to the diagram of possible charged black holes in Figure \ref{fig:BHpopulationandlimits}.
\begin{figure}
\begin{center}
\begin{tikzpicture}[scale=1.25]
\draw[->, line width=.5mm] (0, 0) -- (4.2, 0) node[below] {${\cal T}$};
\draw[->,line width=.5mm] (0, 0) -- (0,4.33) node[left] {${\cal S}$};
\draw[scale=1.5,domain=0.6:2.7,smooth, variable=\x,blue, thick,line width=.6mm] plot ({\x},{1/((\x)*(\x))});
\fill [blue!25, domain=0.1:4.2, variable=\x]
(0.02, 4.17)
-- plot[scale=1.5,domain=0.6:2.7] ({\x},{1/((\x)*(\x))})
-- (0.02, 0.02);
\fill [blue!25] (0.02,0.02) rectangle (4.05,0.2);
\node () at (3,3) {excluded region};
\draw[->,line width=.4mm,rounded corners=10pt, color=red] (2.6,.3)--(3,.2)--(3.4,.15);
\draw[->,line width=.4mm,rounded corners=10pt,color=red] (.6,.6)--(0.2,0.2);
\draw[->,line width=.4mm,rounded corners=10pt,color=red] (.5,3.2)--(.45,3.5)--(0.42,3.9);
\node () at (2.3,.4) {I};
\node () at (.8,.8) {III};
\node () at (.6,2.9) {II};
\draw[->,line width=.4mm,rounded corners=7pt,color=black] (.8,1.8)--(.4,1.9)--(0.2,2);
\draw[->,line width=.4mm,rounded corners=7pt,color=black] (1.5,.7)--(1.6,.4)--(1.7,.2);
\node () at (1.5,.9) {$t_1$};
\node () at (1.05,1.8) {$t_2$};
\end{tikzpicture}
\end{center}
\caption{Representation of the limits of Figure \ref{fig:para_behav} in the ${\cal T}- \mathcal{S}$ plane. The red lines correspond to the three different generic regions in Figure \ref{fig:para_behav}, while the black lines correspond to the threshold values for which the function $(H_e H_m)^h$ vanishes as $\sim 1/c$ and $\sim 1/c^2$ respectively. In particular, the limit $t_1$ leads to ${\cal T} \text{ finite}$, $\mathcal{S} \to 0$, while $t_2$ leads to ${\cal T} \to 0$, $\mathcal{S} \text{ finite}$.}
\label{fig:BHpopulationandlimits}
\end{figure}
The three generic limits I, II, and III are described by asymptotic points on the axes, whereas the threshold values potentially populate the axes as well. Next, we turn to the analysis of some special cases where the functions $H_i$ can be determined explicitly.
\subsection{Single-charged solutions}
EMd black holes with either electric charge $q$ or magnetic charge $p$ are arguably the simplest example and thus we discuss them first, before turning to the analysis of dyonic ones. Indeed, in this case one can find explicit solutions for $H_i$ for arbitrary dilaton coupling.
We start from the case with magnetic charges only, for which one finds the solution
\begin{align}
H_e (r) = 1 \,, \quad H_m (r) = 1 + \frac{\xi_m}{r} \,,
\end{align}
with \eqref{1stint} reducing to
\begin{align}
h \xi_m \big(\xi_m + \tfrac{c}{4 \pi} \big) = p^2 \,.
\end{align}
With this, we can derive the asymptotic behaviour for the extremal limit $c \rightarrow 0$:
\begin{align}
H_e \big(\tfrac{c}{4 \pi} \big) H_m \big( \tfrac{c}{4 \pi} \big) \sim \tfrac{1}{{c}} \,.
\end{align}
We see that the temperature goes to zero for $h > 1$ and the entropy vanishes in the extremal limit for $h < 2$. This means that for a non-trivial dilaton coupling, $\lambda > 0$, the extremal limit for a single-charged black hole always leads to a small black hole. The case at the $t_2$ threshold, $h = 2$ ($\lambda=0$), corresponds to a decoupling of the dilaton and reproduces the results for magnetically charged RN solutions, for which the entropy remains finite for $c \rightarrow 0$ and is given by
\begin{align}
\mathcal{S} = 4 \pi^2 p^2 \,, \quad \text{for} \enspace h = 2 \quad \text{($\lambda=0$)}\,.
\end{align}
This is the first example we analyse of a black hole solution coupled to a scalar field. Thus, we are interested in studying the profile of the dilaton. For $\lambda > 0$, one has (see \eqref{eq:EMdBH})
\begin{align}
\left(g(\phi)\right)^{-2} \equiv e^{- 2 \lambda \phi} = \left( 1 + \frac{\xi_m}{r} \right)^{h - 2} \,,
\end{align}
where $g$ denotes the electric gauge coupling. The above quantity, which is in fact the (squared) magnetic gauge coupling, vanishes at the outer horizon $r_+ = \tfrac{c}{4\pi}$ in the extremal limit (for $h < 2$). This indicates that the dilaton diverges
\begin{align}\label{eq:dilat-sing-charge}
\phi \rightarrow \infty \,.
\end{align}
Since the asymptotic value of the dilaton in this solution is fixed to $\phi |_{r \rightarrow \infty} = 0$ , we see that for purely magnetically charged extremal black holes the dilaton traverses an infinite field distance and, at the same time, a global symmetry is restored. This infinite field distance (or global symmetry restoration) gives an alternative perspective in terms of the SDC on our intuition that the small black hole limit is outside the validity of the effective description, which should be related to a light tower of states close to the black hole horizon.
For the coupling $h=1$, the single charge black hole realizes the threshold case $t_1$ of Figure \ref{fig:para_behav}, see also \cite{Holzhey:1991bx}. This corresponds to the charged dilatonic black hole in string theory, with only one charge. In the extremal case the solution has zero horizon but finite temperature. In addition to the singular behavior of the dilaton \eqref{eq:dilat-sing-charge}, in the $t_1$ limit we also have a singular geometry due to the vanishing size of the horizon. Again we expect quantum gravity corrections to take over and smooth out the classical singularity.
The case of purely electrically charged black holes can be deduced from the electric-magnetic duality of the action, under which
\begin{align}
\left( \begin{array}{c} F_{\mu\nu} \\ G_{\mu\nu} \end{array} \right) \rightarrow \left( \begin{array}{c} G_{\mu\nu} \\ -F_{\mu\nu} \end{array} \right) \,, \quad \phi \rightarrow - \phi,
\end{align}
where $G_{\mu \nu} = \frac{\delta\mathcal{L}}{\delta F_{\mu\nu}}=\tfrac{1}{2} e^{-2 \lambda \phi} \epsilon_{\mu \nu \rho \sigma} F^{\rho \sigma}$. Once more, the entropy vanishes in the extremal case unless $\lambda = 0$. Since the dilaton changes sign one now has
\begin{align}
\phi \rightarrow - \infty \,,
\end{align}
at the outer horizon and again the traversed field distance in the black hole background is infinite. A recent discussion on this class of black holes in the Swampland context can be found in \cite{Hamada:2021yxy}. In particular, it is argued that the vanishing size of extremal EMd black holes with charge leads to a violation of entropy bounds. Additionally, one finds that in these situations the dilaton traverses infinite distance. The SDC is thus assumed to avoid such a problematic situation by invalidating the effective theory with the appearance of a tower of light states. RN black holes evade this conclusion since they still have a finite size in the extremal limit. Thus, the role of the scalars (dilaton) in this class of model is crucial for understanding properties of quantum gravity.
To summarize, we see that for non-trivial dilaton coupling the limit of extremal single-charge solutions is problematic since it is described by the parameter region II and is beyond the regime of validity for the Einstein-Maxwell-dilaton effective theory. The usual expectation is that the full UV-complete theory contains higher derivative corrections that lead to a finite horizon size. Note that in all of the above cases the problematic regime ${\cal S} \rightarrow 0$ is accompanied by an infinite field distance traversed by the dilaton field from spatial infinity to the outer horizon relating it to the predictions of the SDC.
\subsection{Dyonic solutions}
\label{subsec:dyon}
We now turn to solutions with both electric and magnetic charges, dyonic black holes. Unfortunately, these cannot be discussed in full generality for arbitrary $h$ with one exception, for which the electric and magnetic charges are equal
\begin{align}
\label{p=q}
p = q \,.
\end{align}
The corresponding solution is given by
\begin{align}
H_e (r) = H_m (r) = \Big( 1 + \frac{\xi}{r} \Big)^{1/h} \,, \quad \xi \big(\xi + \tfrac{c}{4 \pi} \big) = q^2 \,,
\end{align}
leading to the asymptotic behavior in the extremal limit
\begin{align}
H_e \big( \tfrac{c}{4\pi} \big) H_m \big( \tfrac{c}{4\pi} \big) \sim {c}^{-2/h} \,,
\end{align}
located precisely at the threshold $t_2$.
In fact, in the extremal limit one finds that the entropy
\begin{align}
\mathcal{S} = 8 \pi^2 q^2 \,,
\end{align}
is finite and independent of $h$. At the same time the dilaton profile is constant. It therefore seems that for these black holes one can reach the extremal limit within the effective description and correspondingly one does not traverse an infinite distance in the moduli space of the dilaton field. Moreover, also in the infinite entropy limit, $q^2\rightarrow \infty$, $e^{-2\lambda\phi}$ stays constant and one again does not traverse an infinite distance in the moduli space of $\phi$. Note, however, that especially in non-supersymmetric cases, these black holes might receive large quantum corrections affecting their thermodynamical properties \cite{Preskill:1991tb, Maldacena:1998uz, Page:2000dk, Heydeman:2020hhw}.
For the analysis at arbitrary charges, we focus on two different interesting cases for which explicit solutions of $H_i (r)$ are known (the case $h = 2$ is fully covered by the considerations above):
\begin{itemize}
\item{$h = 1$: Low-energy effective action of heterotic string theory}
\item{$h = \tfrac{1}{2}$: Kaluza-Klein theory, with the dilaton as radion field}
\end{itemize}
We will see that in these examples the extremal limit for dyonic black holes corresponds to the threshold $t_2$ and thus the entropy remains finite and the dilaton does not traverse to infinite distance.
\subsubsection*{String theory coupling ($h = 1$, $\lambda=1/\sqrt2$)}
For the typical string theory coupling of the dilaton, i.e., $h = 1$, one finds the following solution
\begin{align}
H_i = 1 + \frac{\xi_i}{r} \,, \quad \xi_i \big(\xi_i + \tfrac{c}{4\pi} \big) = q_i^2 \,,
\end{align}
where $q_e = q$ and $q_m = p$.
When both charges are nonzero, in the extremal limit one recovers
\begin{align}
H_e \big( \tfrac{c}{4\pi} \big) H_m\big( \tfrac{c}{4\pi} \big) \sim {c}^{-2} \,.
\end{align}
Again, this is at the threshold $t_2$, where
\begin{align}
{\cal S} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace 8 \pi^2 p q \,.
\end{align}
The dilaton field further has the asymptotic value
\begin{align}
e^{- 2 \lambda \phi} \Big|_{r_+} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace \frac{q}{p} \,,
\end{align}
at the outer horizon. The field distance in moduli space is therefore finite, for finite $p$ and $q$ and the extremal limit seems to be within the validity of the effective description.
When only one of the charges is nonzero, we recover the $t_1$ threshold states discussed above.
\subsubsection*{Kaluza-Klein coupling ($h = \tfrac{1}{2}$, $\lambda=\sqrt{3\over2}$)}
For the Kaluza-Klein value of the dilaton coupling one has the solutions
\begin{align}
H_i (r) = 1 + \frac{\xi_i^{(1)}}{r} + \frac{\xi_i^{(2)}}{r^2} \,,
\end{align}
with
\begin{align}
2 q_i^2 = \frac{\xi_i^{(1)} \big( \xi_i^{(1)} + \tfrac{c}{4 \pi} \big) \big( \xi_i^{(1)} + \tfrac{c}{2 \pi} \big)}{\xi_e^{(1)} + \xi_m^{(1)} + \tfrac{c}{2\pi}} \,, \quad \xi_i^{(2)} = \frac{\xi_e^{(1)} \xi_m^{(1)} \big( \xi_i^{(1)} + \tfrac{c}{4 \pi} \big)}{2 \big( \xi_e^{(1)} + \xi_m^{(1)} + \tfrac{c}{2 \pi} \big)} \,.
\end{align}
and the extremal limit is characterized by
\begin{align}
H_e \big( \tfrac{c}{4\pi} \big) H_m \big( \tfrac{c}{4\pi} \big)\sim {c}^{-4} \,,
\end{align}
which is again at the threshold $t_2$.
Once more the entropy in the extremal limit is determined by the electric and magnetic charges
\begin{align}
{\cal S} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace 8 \pi^2 p q \,,
\end{align}
as is the asymptotic value of the dilaton field
\begin{align}
e^{- 2 \lambda \phi} \Big|_{r_+} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace \frac{q}{p} \,.
\end{align}
Therefore, for finite values of the charges the extremal black holes have a finite entropy and do not lead to an infinite field distance in the moduli space of the dilaton.
\subsection{Infinite entropy at threshold}
We have seen that the dyonic black holes are realized at the threshold value $t_2$ of Figure \ref{fig:BHpopulationandlimits}. This implies that for finite charges their entropy remains non-vanishing and finite in the extremal limit. There are, however, limits in which one sends the charges to infinity that also lead to infinite entropy. These limits further affect the dilaton field profile, which we want to analyze next.
\subsubsection*{Infinite entropy and infinite scalar field limit}
For the first limit we consider dyonic black holes with electric and magnetic charges given by $q$ and $p$, respectively. The entropy in the cases analyzed above is proportional to their product. Therefore, one can generate infinite entropy by keeping one of the charges fixed and sending the other one to infinity. This generates a hierarchy in the two charges and induces a large field limit for the dilaton, since its value diverges at the outer horizon. Concretely, for these two cases the entropy can be expressed in terms of the scalar field as
\begin{eqnarray}
p \text{ fixed}:\quad&{~}&{\cal S} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace 8 \pi^2 p^2 e^{-2 \lambda \phi} \Big|_{r_+}
\quad {\rm or}\nonumber\\
q \text{ fixed}:\quad&{~}&{\cal S} \enspace \underset{{c} \rightarrow 0}{\longrightarrow} \enspace 8 \pi^2 q^2 e^{ 2 \lambda \phi} \Big|_{r_+}
\end{eqnarray}
So for large entropy and fixed magnetic or fixed electric charge we get
\begin{equation}
\begin{split}
p \text{ fixed}:\quad{\cal S} \rightarrow \infty:& \quad \phi \rightarrow - \infty \,, \\
q \text{ fixed}:\quad{\cal S} \rightarrow \infty:& \quad \phi \rightarrow \infty \,.
\end{split}
\end{equation}
We see that the infinite entropy limit corresponds to an infinite distance limit in the scalar field moduli space, again, demonstrating the parallels with the SDC and its predictions\footnote{Note that if the spacetime region, at which the infinite distance limit is achieved, has vanishing volume, one in general does not expect an infinite tower of states. An example for this situation is the Taub-NUT geometry. This often seems to be associated with the fact that these backgrounds are non-trivial fibrations, in which the charge labelling the infinite tower, such as the KK charge, is not conserved. In our case, we consider geometries which are warped products and, in particular, are trivial fibrations. Moreover, once corrections are taken into account, a finite horizon is generated. We therefore expect the SDC to be applicable in our set-up. We thank the referee for pointing out this issue.}. Moreover, these two limits are dual to each other, in the sense that the infinite entropy limit either corresponds to very weak or very strong $U(1)$ gauge coupling.
In a string theory embedding, small dilatonic black holes should correspond to regions of strong/weak string coupling limits ($g_s\rightarrow \infty/0$) and then to an infinite tower of light strings \cite{Lee:2019wij}.
\subsubsection*{Infinite entropy and finite scalar field limit}
Next, we consider the case, where the electric and magnetic charges scale in the same way, i.e.,
\begin{align}
p \sim {q} \,.
\end{align}
This case was already discussed in Section (\ref{subsec:dyon}). The value of the dilaton field stays finite in the infinite entropy limit:
\begin{equation}
\begin{split}
{\cal S} \rightarrow \infty: \quad e^{ 2 \lambda \phi} \rightarrow {\rm const}\,.
\end{split}
\end{equation}
So the infinite entropy limit corresponds to a finite field distance in the moduli space of the scalar field. We will come back to these ``scaling'' solutions in the context of $\mathcal{N} = 2$ supergravity.
\subsubsection*{Finite entropy and infinite scalar field limit}
The dyonic black hole solutions further allow for a parametric solution with
\begin{align}
p \sim \frac{1}{q} \,,
\end{align}
which for $q \rightarrow \infty$, seems to lead to a finite entropy black hole in the extremal limit. At the same time the value of the dilaton at the outer horizon takes one of its singular values
\begin{equation}
\begin{split}
q \rightarrow \infty:& \quad \phi \rightarrow - \infty \,, \\
p \rightarrow \infty:& \quad \phi \rightarrow \infty \,.
\end{split}
\end{equation}
Therefore they seem to correspond to finite entropy black holes, which nevertheless induce an infinite field distance in the moduli space of the dilaton and are problematic from an effective field theory point of view.
\subsection{Infinite temperature limit}
The infinite temperature limit in the extremal case is captured already by the discussion in Section \ref{subsec:univ} and corresponds to the asymptotic behavior in the parameter region I, i.e.,
\begin{align}
H_e \big( \tfrac{c}{4\pi} \big) H_m \big( \tfrac{c}{4\pi} \big) \rightarrow \infty \quad \text{slower than} \quad {c}^{- \frac{1}{h}} \,.
\end{align}
For non-vanishing ${c}$ we find that ${\cal T} \rightarrow \infty$ demands
\begin{align}
H_e \big( \tfrac{c}{4\pi} \big) H_m \big( \tfrac{c}{4\pi} \big) \rightarrow 0 \,.
\end{align}
Since the boundary conditions are such that $H_i (r) > 0$ for $r \rightarrow \tfrac{c}{4\pi}$, we see that this is not possible. To be more precise for finite ${c}$ this demands that some of the terms in expansion \eqref{eq:Hexpa} are negative. However, these will dominate in the extremal limit and violate the boundary condition. Therefore, all infinite temperature limits of the black holes above necessarily are extremal and therefore captured by the asymptotic behavior above. This can also be seen in the Figure \ref{fig:BHpopulation} and \ref{fig:BHpopulationandlimits}, where in the limit ${\cal T} \rightarrow \infty$ one approaches the axis, i.e., the extremal black holes, and all finite values of $c$ exit the allowed region at a certain point.
\vskip0.4cm \noindent
We see that already in the simplest model of gravity coupled to a scalar field, Einstein-Maxwell-dilaton theory, we can relate many of the thermodynamic limits to infinite field ranges of the dilaton field in the black hole background. The SDC then predicts a tower of light states to appear in the vicinity of the black holes leading to large corrections and necessitate an understanding within a more fundamental description. In order to explicitly study these light states we therefore turn to an effective theory closely related to string theory, namely $\mathcal{N} = 2$ supergravity.
\section{Multi-charge black holes in ${\cal N}=2$ supergravity}
\label{SG-blackholes}
The behavior of black holes solutions across the parameter space explored in the previous section can also be found for more general, multi-charge black holes in four-dimensional $\mathcal{N}=2$ supergravity coupled to vector multiplets.
The main example we will concentrate on is the ${\it STU}$ model arising from type IIA compactifications, but our analysis can be easily adapted to different Calabi-Yau spaces or dualized to type IIB.\footnote{An exhaustive review on ${\cal N}=2$ supergravity in the context of black holes is \cite{Andrianopoli:2006ub}. For BPS black holes and their relation to string theory compactified on Calabi-Yau we refer to \cite{Behrndt:1996jn,Greene:1996cy,Bohm:1999uk,Denef:1999idt,Shmakova:1996nz}. See also \cite{Cvetic:1995uj,Cvetic:1995bj} for $\mathcal{N}=4$ black holes.}
The four-dimensional effective $\mathcal{N}=2$ supergravity theory arising from type IIA string theory compactified on a Calabi-Yau threefold $Y$ is specified by a prepotential
\begin{align}
F(X^\Lambda)=\frac16 d_{ijk}\frac{X^{i}X^{j}X^{k}}{X^0} \ ,
\end{align}
where $X^\Lambda=(X^0,X^i)$, and the indices $i,j,k$ run over $1,..,h^{1,1} \equiv n_V$.
The bosonic sector contains the graviton, $n_V+1$ vectors and $n_V \equiv h^{1,1}$ complex scalars $z^i$ spanning a special K\"ahler manifold. A convenient formulation of special geometry is in terms of projective coordinates $X^\Lambda(z^i)$ which, together with $F_\Lambda = \frac{\partial F}{ \partial X^\Lambda} $ (valid only if a prepotential exists), form holomorphic symplectic sections $(X^\Lambda,F_\Lambda)$. They arise from a choice of symplectic basis of 2- and 4-cycles $\{A^\Lambda\,, B_\Lambda\}$ on $Y$.
Finally, $d_{ijk}$ gives the triple intersection numbers of the Calabi-Yau 4-cycles.
Four-dimensional black holes are solutions of the supergravity action and, upon a constraint on the parameters, at zero temperature they can preserve supersymmetry. We choose a microscopic configuration obtained from 10d as a superposition of $h^{1,1}$ D4-branes, each of them wrapped $p_i$-times around the corresponding 4-cycles in $Y$, times a stack of $q_0$ D0-branes.
In four dimensions, they yield a black hole with electric charge $q_0$ and magnetic charges $p_i$ with respect to the abelian ${h^{1,1}+1}$ gauge fields. Analogous results to those we present can be dualized to the D2-D6 setup, where the black hole has instead one magnetic $p^0$ and electric $q_i$ charges. We refer the reader to Appendix \ref{appendix-supergravity} for more details on four-dimensional $\mathcal{N}=2$ supergravity and for definitions of the quantities further introduced in this section.
The explicit setup we will focus on is the so-called ${\it STU}$ model, corresponding to the case where $d_{123}=1$ and zero otherwise. We have three scalars $z^1=S\,,z^2=T\,,z^3=U$, defined as $z^i=\frac{X^i}{X^0}$, which enter the action as a nonlinear sigma model with target space $\left[SU(1,1)/U(1)\right]^3$. The ${\it STU}$ model can be interpreted as a truncation of $\mathcal{N}=8$ supergravity arising from type IIA compactified on $T^6$, in which case the imaginary parts of the three scalar fields parametrize the volume moduli of three two-tori. It can also arise as a compactification of heterotic string theory on $T^2\times K_3$, which is dual to type IIA compactified on a CY with $h^{1,1}=3$ \cite{Behrndt:1996jn}.
In the limits $S,T,U\rightarrow 0,\infty$ (neglecting the axions), the volumes of these two-tori will either shrink to zero or grow to infinite sizes.
It is known that these wo limits are at infinite distance in the internal moduli space. In fact the overall volume of the Calabi-Yau is given by
\begin{equation} \label{volumeCY3}
\mathcal{V} = -i z^1 z^2 z^3 =\,-iSTU\,,
\end{equation}
where we implicitly set $\mathop{\rm Re}\nolimits z^i=0$.
The overall KK modes have masses (in string units)
\begin{equation}
m_{KK}= \left(\frac{1}{\mathcal{V}}\right)^{1/6}\, .
\end{equation}
In case also string winding modes are present, as for example on the torus $T^6$, their masses are given by
\begin{equation}
m_{wind}={\mathcal{V}}^{1/6}\, .
\end{equation}
Therefore, in the limits ${\mathcal{V}}\rightarrow \infty$ or ${\mathcal{V}}\rightarrow 0$, either the tower of KK or winding modes (if present) become massless, in agreement with the infinite distance in the internal moduli spaces.
Specifically, the moduli space distance in the limit of large or small volume behaves as
\begin{equation}
\Delta_{\cal V}\simeq |\log({\cal V})|\, .
\end{equation}
In the next sections, we will see how the internal volume is mapped to the entropy and the temperature of the non-extremal black hole solutions.
\subsection{Non-extremal black holes \label{sec:non-extremal}}
Black hole solutions with electric and magnetic charges at finite temperature can be described in supergravity by the metric\footnote{In the present Section \ref{SG-blackholes} we set $G_N=1$, as we are referring to the supergravity black hole literature which uses these conventions. This will affect the Schwarzschild curve in the ${\cal T}-{\cal S}$ plane defined as ${\cal S}{\cal T}^2=\frac{1}{16\pi G_N}$, which here will appear as ${\cal S}{\cal T}^2=\frac{1}{16\pi}$. One can restore the explicit $G_N$ dependence by setting $c \to G_N c$ everywhere in the solution presented in this Section; thermodynamic quantities can be compared with those in the previous sections by transforming \eqref{S-T-BH} as
\begin{align}
{\cal S}(c)&\to \frac1{G_N}{\cal S}(G_N c) \ ,
\qquad \qquad
{\cal T}(c) \to {\cal T}(G_N c) \ ,
\end{align}
and then chosing $G_N=\frac{1}{8\pi}$.
}
\begin{align}
\label{metric}
ds^2&=-e^{2U}dt^2+e^{-2U}\left[ \frac{c^4 d\rho^2}{\sinh^4(c\rho)} + \frac{c^2}{\sinh^2(c\rho)} \left(d\theta^2+\sin^2\theta d\phi^2\right) \right] \ ,
\end{align}
where the warp factor $e^{2U}$ represents the coupling to scalar fields \cite{Breitenlohner:1987dg}. This metric interpolates between the extremal ($c=0$) and non-extremal $(c> 0)$ geometries.
The black hole is characterized by a mass $M$ and by electric and magnetic charges, $q_\Lambda$ and $p^\Lambda$, with $\Lambda \in \{0,1,..,n_V\}$. Due to the coupling to the scalars, $M$ is fixed by $c$, the black hole charges and also by the values of the scalar fields at asymptotic infinity. The latter determine the nature of the specific compactification and we decide to work with a fixed choice of moduli at infinity, i.e., we fix the low energy theory. Our conclusions will not depend on this specific choice, as long as we remain inside the moduli space. Besides the metric, the black hole solution is specified also by the field strengths
\begin{align}
\label{Fmunu-spher-symm-4d}
F^\Lambda&=p^\Lambda \sin\theta \, d\theta \wedge d\phi - e^{2U} (\mathcal{I}^{-1})^{\Lambda\Sigma}\left(q_\Sigma-\mathcal{R}_{\Sigma\Delta}p^\Delta\right) d\rho \wedge dt \ ,
\end{align}
where $\mathcal{I}_{\Lambda\Sigma}(z^i,\bar{z}^{\imath})\,,\mathcal{R}_{\Lambda\Sigma}(z^i,\bar{z}^{\imath})$ are scalar couplings to the gauge fields in the theory entering the action in front of the gauge kinetic and theta-term, respectively (see Appendix \ref{appendix-supergravity}).
The \emph{STU} model admits black holes with up to four electric and four magnetic charges \cite{Galli:2011fq,Bellucci:2008sv}. For simplicity, we work with a configuration with only nonvanishing $q_0,p^1,p^2,p^3$ charges, all assumed to be positive. The solution can be written in terms of four functions $I_0(\rho)\,, I^1(\rho)\,, I^2(\rho)\,, I^3(\rho)$, in such a way that the scalar fields are
\begin{equation}
\label{scalars-stu}
z^i=-i\frac{I_0I^i}{\sqrt{\mathcal{I}_4}} \ , \qquad \mathcal{I}_4(I)=I_0 I^1I^2I^3 \ ,
\end{equation}
and the warp factor is
\begin{equation}
\label{warp-fact-nonextr}
e^{-2U}= e^{-2U_e+2c\rho}\ , \qquad \text{with} \qquad e^{-2U_e}=4\sqrt{\mathcal{I}_4}\ .
\end{equation}
The $I$-functions take the form
\begin{equation}\label{I-func}
I_0=a_0+b_0 e^{2c\rho} \ , \qquad \qquad I^i=a^i+b^i e^{2c\rho} \ ,
\end{equation}
with
\begin{align}
\label{a0b0}
\begin{pmatrix} a_0 \\ b_0 \end{pmatrix} &= \, \frac{1 }{8\sqrt2 L^0_{\infty}}
\left[1\pm \frac{1}{c}\sqrt{c^2+16q_0^2(L_{\infty}^0)^2}\right] \ , \\ \label{a1b1}
\begin{pmatrix} a^i \\ b^i \end{pmatrix} &= - \, \frac{1}{8\sqrt2 M_{i\, \infty}}\left[1\pm\frac{1}{c}\sqrt{c^2+ 16 (p^i)^2 M^2_{i\,\infty} } \right]\,,
\end{align}
and
\begin{align}
\label{simpl-secs-infty}
L^0_\infty&= \frac{1}{2\sqrt2}\frac{1}{\sqrt{\lambda^1_\infty\lambda^2_\infty\lambda^3_\infty}}\ , \qquad \qquad M_{i\,\infty }= -\frac{\sqrt{\lambda^1_\infty\lambda^2_\infty\lambda^3_\infty}}{2\sqrt2 \lambda^i_\infty}\ ,
\end{align}
where we denoted $z^i_\infty=-i\lambda^i_\infty$ the values of the scalar fields at infinity, that will be kept fixed in all our analysis. Notice that the horizon is located at $\rho\to-\infty$, so the value of the scalars at the horizon are in principle dependent on the charges, the moduli at infinity, and on $c$.
From the expression of the metric in \eqref{metric}, we derive that the black hole entropy and temperature are
\begin{equation}
\label{S-T-BH}
\boxed{
\begin{split}
{\cal S}=& 16 \pi \sqrt{a_0a^1a^2a^3} \, c^2 \ ,
\\
{\cal T}=& \frac{\kappa}{2\pi} = \frac{1}{2\pi} \left[ \frac12 \left(\frac{\sinh(c\rho)^2}{c^2}\frac{d e^{2U(\rho)}}{d\rho}\right)
\right]_{\rho\to-\infty} = \frac{1}{32\pi}\frac{1}{c\sqrt{a_0a^1a^2a^3}}\, ,
\end{split} }
\end{equation}
and the extremality parameter is
\begin{equation}
\label{nonextremalityrelation}
c=2{\cal S}{\cal T} \ .
\end{equation}
This solution generalizes non-extremal black holes in the presence of scalar fields running from asymptotic infinity to their values at the horizon. When the scalars are constant throughout the whole four-dimensional spacetime, the solution effectively reduces to a non-extremal Reissner-Nordstrom black hole.
The volume of the Calabi-Yau for this solution is given by
\begin{equation}
\label{volumeCY3-stu}
\mathcal{V} =- i S T U = \frac{I_0^2}{\sqrt{\mathcal{I}_4}}\,,
\end{equation}
with the $I$-functions defined in \eqref{I-func}.
\subsection{Extremal BPS black holes}
Supersymmetric black holes are an important subset of the solutions of the previous section. Supersymmetry requires the solution to be extremal, meaning $c=0$, and thus \eqref{nonextremalityrelation} implies $\mathcal{T}=0$ for a regular horizon. Therefore, according to our classification of limits, we are dealing here with the threshold case $t_2$.
This class of solutions is described by a metric of the form\footnote{Notice that the coordinates $r$ and $\rho$ are related by $dr=-\frac{c^2}{\sinh(c\rho)^2}d\rho $, which in the extremal case simply reduces to $dr=-\frac{d\rho}{\rho^2}$.}
\begin{equation}
\label{dsextr1}
ds^2= -e^{2U} dt^2 + e^{-2U} ( dr^2 + r^2 d\Omega_{2}^2) \ .
\end{equation}
Not all possible choices of charges in the theory allow for a BPS solution, i.e., the extremal limit of a multi-charge black hole in supergravity may lead to non-BPS solutions with finite entropy. However, the non-extremal configuration of the \emph{STU} model with charges $q_0\,, p^1\,,p^2\,,p^3$ considered so far does reduce to a supersymmetric black hole for $c\to 0$.\footnote{We refer to \cite{Bellucci:2008sv} for a review of BPS and non-BPS black holes in the \emph{STU} model.} This BPS black hole is given by the scalar fields $z^i$ of \eqref{scalars-stu} and the warp factor $e^{-2U}\to e^{-2U_e}$ of \eqref{warp-fact-nonextr}, with
\begin{equation}
I_0= \frac{1}{4\sqrt2 L^0_\infty}-\frac{1}{\sqrt2} \frac{q_0}{r} \ ,\qquad I^i= -\frac{1}{4\sqrt2 M_{i\,\infty}}-\frac{1}{\sqrt2}\frac{p^i}{r} \ .
\end{equation}
Being extremal, BPS black holes in $\mathcal{N}=2$ supergravity have a near horizon geometry of the form $AdS_2\times S^2$, with the anti-de Sitter radius equal to the sphere radius and proportional to the black hole entropy. These black holes are governed by the central charge function $Z(z^i,\bar z^{\bar \imath},q_0,p^i)$ defined in Appendix \ref{appendix-supergravity}, and the horizon corresponds to an attractor point for the flow of the scalar fields determined by $Z$ via the \emph{attractor equations}
\begin{align}
&\partial_i Z(z^i_{h},\bar z^{\bar \imath}_{h},q_0,p^i)=0\ , \\
&{\cal S}_{BPS}=\pi |Z(z^i_{h},\bar z^{\bar \imath}_{h},q_0,p^i)|=2\pi \sqrt{q_0p^1p^2p^3}\ .
\end{align}
For the BPS black hole of the $STU$ model with charges $q_0\,, p^1\,,p^2\,,p^3$, this yields
\begin{align}\label{BPS-sol}
z^i_{h}&=-i\frac{q_0 p^i}{\sqrt{q_0p^1p^2p^3}} \ , \\
{\cal S}_{BPS}&=2\pi \sqrt{q_0p^1p^2p^3}\ ,
\end{align}
corresponding to the $c\to0$ limit of \eqref{S-T-BH}. We see that in the BPS case the scalars at the horizon are completely fixed by the charges, differently from the non-extremal solution where the horizon values still depend on the moduli at infinity, in addition to the extremality parameter $c$. The mass of the BPS black hole is also determined in terms of the central charge as $M~=~|Z(z^i_{h},\bar z^{\bar \imath}_{h},q_0,p^i)|$.
We will now explore the parameter space of this general class of $STU$ black holes in the ${\cal T}-{\cal S}$ space.
\subsection{Thermodynamic limits and tower of states}\label{thlimits}
In this section, we investigate the limits of large and small temperature or entropy for extremal and non-extremal black holes in supergravity,
by studying the behaviour of the CY volume \eqref{volumeCY3} at the horizon, whose divergence or singular behaviour may signal the appearance of infinite towers of states. We will specialize our discussion to the $STU$ black holes presented above, for concreteness, but our results can be easily generalized to other configurations.
\subsubsection{Extremal black holes}\label{Sec.Extremal}
Let us first consider the BPS black hole case. The thermodynamic quantities in the extremal limit are, from \eqref{BPS-sol}
\begin{equation}
\boxed{
\begin{split}
{\cal S}&=2\pi \sqrt{q_0 p^1 p^2 p^3} \ , \\
{\cal T} &= 0\,.
\end{split}}
\end{equation}
The volume (\ref{volumeCY3-stu}) at the horizon is thus given in terms of the charges as
\begin{equation}\boxed{
\mathcal{V}_h = \frac{(q_0)^{3/2}}{\sqrt{p^1 p^2 p^3}}\,.}
\end{equation}
The KK modes have masses
\begin{equation}
m_{KK}= \left(\frac{1}{\mathcal{V}_h}\right)^{1/6}=\frac{(p^1 p^2 p^3)^{1/12}}{(q_0)^{1/4}}\,.
\end{equation}
Let us now consider the exchange of a small extremal black hole with a large extremal one, which can be realized by the inversion of the charges (and of the mass):
\begin{equation}
q_0\,\longleftrightarrow\,{1\over q_0}\,, \quad
p^i\,\longleftrightarrow\,{1\over p^i} \quad
\Longrightarrow
\quad
{\cal S}\,\longleftrightarrow\,{1\over{\cal S}}\, .
\end{equation}
This transformation also inverts the volume of the internal space, i.e.,
\begin{equation}
{\cal V}_h\,\longleftrightarrow\,{1\over{\cal V}_h}\, ,
\end{equation}
and exchanges the KK modes with the winding modes.
So this exchange of large and small extremal black holes is induced by a T-duality transformation on the internal moduli fields.
Now, let us see if the large or small entropy limits also induce a tower of light states and hence leads to an infinite distance limit in the moduli space. For this purpose, we will focus on two types of black holes, which will exhibit a kind of T-dual behavior with respect to each other:
\begin{itemize}
\item {\bf Type A}: Black holes with \emph{fixed electric charge $q_0$}, varying magnetic charges $p^i$ and varying mass $M$. The entropy will therefore be a function of the $p^i$ and the mass $M$.
\item {\bf Type B}: Black holes with \emph{fixed magnetic charges $p^i$}, varying electric charge $q_0$ and varying mass $M$. Then the entropy will be a function of $q_0$ and the mass $M$.
\end{itemize}
\noindent
For A-type black holes, the volume scales as a {\it negative} power of the entropy
\begin{equation}\label{Vextremalq}
\boxed{
\mathcal{V}_h = 2\pi\frac{q_0^2}{\cal S}\,.}
\end{equation}
This implies that for fixed electric charge $q_0$ the KK masses grow together with the entropy as
\begin{equation}
m_{KK}=\frac{1}{(2\pi)^{1/6}} \frac{{\cal S}^{1/6}}{q_0^{1/3}}\,
\end{equation}
and thus become light in the limit of small entropy. For toroidal compactification on $T^6$, there are also T-dual winding modes with masses of the order
\begin{equation}
m_{wind}=\mathcal{V}_h^{1/6}\simeq \frac{{q_0}^{1/3}}{{\cal S}^{1/6}}\,,
\end{equation}
which become light in the limit of large entropy.
\vskip0.2cm
On the other hand for the B-type black holes the volume scales as a {\it positive} power of the entropy
\begin{equation}\label{Vextremalp}\boxed{
\mathcal{V}_h = \frac{1}{(2\pi)^3}\frac{{\cal S}^3} {(p^1 p^2 p^3)^2}\,,}
\end{equation}
and the KK masses decrease in the large entropy limit as
\begin{equation}
m_{KK}=(2\pi)^{1/2} \frac{(p^1 p^2 p^3)^{1/3}}{{\cal S}^{1/2}}\,.
\end{equation}
In case there are T-dual winding modes they have masses
\begin{equation}
m_{wind}\simeq\frac{{\cal S}^{1/2}}{(p^1 p^2 p^3)^{1/3}}\,
\end{equation}
and become light in the limit of small entropy.
\vskip0.2cm
The A- and B-type black holes, i.e.~fixed $q_0$ or $p^i$, behave T-dual to each other, since sending ${\cal S}\rightarrow0$ or $\mathcal{S}\to\infty$ implies opposite behavior for the volume, i.e.~$\mathcal{V}_h\rightarrow 0$ or $\mathcal{V}_h\rightarrow\infty$, and in turn for the KK masses, i.e., $m_{KK}\rightarrow\infty$ or $m_{KK}\rightarrow 0$. In the large entropy limit, for the A-type the relevant light tower is that of the winding modes, while for the B-type the relevant one is that of KK modes. On the other hand, in the small entropy limit the situation is reserved. All this finds nice agreement with the results of \cite{Bonnefoy:2019nzv}, which investigated the limit of large entropy of $\mathcal{N}=2$ extremal black holes, for fixed magnetic charges.
It is however important to emphasise that there also exists other types of black holes, with different combinations of electric and magnetic charges being kept fixed. They behave in a different way and do not necessarily lead to massless towers of KK or winding modes for large or small entropy. These particular examples are black holes with charges such that the volume in \eqref{Vextremalp} is kept fixed (see Section (\ref{rescalingcharges}) for more details). Then, the limits ${\cal S}\rightarrow0$ or $\mathcal{S} \to\infty$ do not lead to a light tower of states.
\subsubsection{Non-extremal black holes}
We consider now thermodynamic limits of non-extremal black holes. Before discussing the A- and B-type, a somehow special case of non-extremal black hole is given by the Schwarzschild solution, where all electric and magnetic charges are turned off. Then, one is precisely moving along the Schwarzschild hyperbola ${\cal S}=1/(16\pi{\cal T}^2)$, which is the boundary of the colored region in Figure \ref{fig:BHpopulation}. This is expected, since for a neutral black hole in ${\cal N}=2$ supergravity the scalar fields are constant and independent of ${\cal S}$ and ${\cal T}$. Hence, the volume of the internal space is also constant and equal to its value at infinity, $ \mathcal{V}= \mathcal{V}_\infty = ( 2\sqrt{2} L_\infty^0)^{-2}$. In the limits of large or small asymptotic values, $L_\infty^0\to0$ or $L_\infty^0\to\infty$, one can get light KK or winding modes and the distance conjecture with respect to the internal moduli fields applies.
We now move to black holes with non-vanishing electric and magnetic charges and focus in particular on the type A and B introduced in the previous section. As before, other cases of combinations of charges to be hold fixed are possible.
\subsubsection*{A-type black holes}
At the horizon, $\rho\to-\infty$, the formula \eqref{volumeCY3} of the internal volume reduces to
\begin{equation}\boxed{
\mathcal{V}_h= \frac{a_0^2}{\sqrt{a_0 a^1 a^2 a^3}} = 64 \pi a_0^2 {\cal S} {\cal T}^2.}
\end{equation}
The dependence of the volume in terms of ${\cal S}$ and $\cal{T}$ is given once we substitute the expression for $a_0$ in eq.~\eqref{a0b0} and use eq.~\eqref{nonextremalityrelation}. We obtain
\begin{equation}\label{Vnonextremal}\boxed{
\mathcal{V}_h= \frac{4\pi \mathcal{V}_\infty}{\cal S} \bigg({\cal S}{\cal T}+\sqrt{{\cal S}^2 {\cal T}^2 + \frac{2q_0^2}{{\cal V}_\infty}}\bigg)^2\,,}
\end{equation}
which is the expression of the internal volume, when we keep the electric charge $q_0$ constant, while entropy and temperature vary together with the black hole mass and the magnetic charges.
First, we consider two different limits for the temperature, for which the internal volume either becomes small or large:
\begin{itemize}
\item In the {\it small temperature} limit (case (II) in Figure \ref{fig:BHpopulationandlimits}), ${\cal T}\rightarrow 0$, we recover eq.~\eqref{Vextremalq} of the extremal case. If we follow a trajectory such as ${\cal S}({\cal T})=1/(8\pi n {\cal T}^2)$, with $n>2$, then we are sure to be within the validity regime of charged black holes, that is below the Schwarzschild hyperbola ${\cal S}=1/(16\pi {\cal T}^2)$. In the small temperature limit, along this trajectory we have
\begin{equation}
{\cal T}\rightarrow0\, :\qquad \mathcal{V}_h\simeq{q_0^2{\cal T}^2}\rightarrow0\, ,\quad m_{KK}\simeq (q_0{\cal T})^{-1/3}\rightarrow\infty\, .\label{smallT}
\end{equation}
The T-dual winding modes behave in an opposite way:
\begin{equation}
{\cal T}\rightarrow0\, :\qquad m_{wind}\simeq (q_0{\cal T})^{1/3}\rightarrow0\, .\label{smallTwind}
\end{equation}
\item The {\it large temperature} limit (case (I) in Figure \ref{fig:BHpopulationandlimits}), ${\cal T}\rightarrow \infty$, is somewhat more delicate to treat. In this limit, the relevant contributions to the volume are
\begin{equation} \label{VTinf}
\mathcal{V}_h = 16 \pi \mathcal{V}_\infty {\cal S} {\cal T}^2+\frac{4 \pi q_0^2}{\cal S}\, .
\end{equation}
It is important to note that this limit necessarily corresponds to small entropy, ${\cal S}\rightarrow 0$, for a charged black hole, since the maximum entropy for a given temperature is bounded from above by the hyperbola ${\cal S}=1/(16\pi{\cal T} ^{2})$, corresponding to a Schwarzschild black hole (see Figure \ref{fig:BHpopulation}). This implies that the first term of eq.~\eqref{VTinf} approaches a constant, namely the asymptotic value of the volume at infinity, while the second term diverges. We can follow again a trajectory below the limiting hyperbola, such as ${\cal S}({\cal T})=1/(8\pi n {\cal T}^2)$.
Then, the large temperature limit provides
\begin{equation}
{\cal T}\rightarrow\infty\, :\quad \mathcal{V}_h\simeq{q_0^2{\cal T}^2}\rightarrow\infty\, ,\quad m_{KK}\simeq (q_0{\cal T})^{-1/3}\rightarrow0\, ,
\quad m_{wind}\simeq (q_0{\cal T})^{1/3}\rightarrow\infty\, .\label{largeT}
\end{equation}
We see that the limit of large temperature always corresponds to decompactification with $\mathcal{V}_h \rightarrow \infty$. The KK masses will then be vanishing, thus leading to the breakdown of the effective field theory.
\end{itemize}
\vskip0.3cm
Then, we consider similar limits for the entropy:
\begin{itemize}
\item For {\it small entropy} (case (I) in Figure \ref{fig:BHpopulationandlimits}), we recover the extremal limit with eq.~\eqref{Vextremalq} for the volume $\mathcal{V}_h$, similarly to the small temperature limit. However, in this case the volume will become large scaling as
\begin{equation}
\mathcal{V}_h\simeq 2\pi\frac{q_0^2}{\cal S} \rightarrow \infty\,.
\end{equation}
Then, for constant electric charge, the limit of small entropy necessarily corresponds to a light tower of KK modes, with masses $m_{KK}\rightarrow 0$.
\item In limit of {\it large entropy} (case (II) in Figure \ref{fig:BHpopulationandlimits}), the volume at leading order is
\begin{equation}
\mathcal{V}_h = 16 \pi \mathcal{V}_\infty {\cal S} {\cal T}^2\,,
\end{equation}
which matches the first term of eq.~\eqref{VTinf}. We note that the limit ${\cal S}\rightarrow \infty$ necessarily corresponds to the limit of small temperature, ${\cal T}\rightarrow 0$, since both values are upper bounded by the Schwarzschild hyperbola ${\cal T}\sim {\cal S}^{-1/2}$ . This is analogous and parallel to the situation we had for the limit of large temperature. Fixing again a trajectory ${\cal S}={\cal S}({\cal T})$ below the limiting Schwarzschild hyperbola, the volume will become small in the limit of large entropy. The relevant light tower invalidating the effective description is represented thus by winding modes.
\end{itemize}
\subsubsection*{B-type black holes}
\noindent
At the horizon, $\rho\to-\infty$, we can write the internal volume as
\begin{equation}\boxed{
\mathcal{V}_h= \frac{1}{(64 \pi)^3 (a^1 a^2 a^3)^2 {\cal S}^3 {\cal T}^6}\,.}
\end{equation}
We can obtain the dependence of the volume entirely in terms of the entropy ${\cal S}$ and the temperature ${\cal T}$ by using eq.~\eqref{a1b1} and eq.~\eqref{nonextremalityrelation}. We have
\begin{equation}\label{Vnonextremalp}\boxed{
\mathcal{V}_h= \frac{ 8(M^1_\infty M^2_\infty M^3_\infty)^2 {\cal S}^3}{\pi ^3 \prod_{i=1}^3\left(\sqrt{4(M^i_\infty)^2 (p^i)^2+ {\cal S}^2 {\cal T}^2}+ {\cal S} {\cal T}\right)^2}\,,}
\end{equation}
which is the expression of the internal volume at \emph{fixed magnetic charges}.
\vskip0.2cm
\noindent
Now, we can again consider the following limits for the temperature:
\begin{itemize}
\item In the limit ${\cal T} \rightarrow 0$ (case (II) in Figure \ref{fig:BHpopulationandlimits}) the volume at leading order precisely agrees with \eqref{Vextremalp}. We can follow a trajectory ${\cal S}\approx 1/{\cal T}^2$, which stays below the Schwarzschild hyperbola, and obtain
\begin{equation}
{\cal T}\rightarrow0\, :\qquad \mathcal{V}_h \simeq \frac{1}{(2\pi)^3}\frac{1} {{\cal T}^6(p^1 p^2 p^3)^2} \rightarrow\infty\, ,\quad m_{KK}\rightarrow0\, ,\quad m_{wind}\rightarrow\infty\, .\label{smallTm}
\end{equation}
\item The limit of large temperature ${\cal T} \rightarrow \infty$ (case (I) in Figure \ref{fig:BHpopulationandlimits})
necessarily implies ${\cal S} \rightarrow 0$. Being close to the hyperbola ${\cal T}=(16\pi{\cal S})^{-1/2}$, we get that the internal volume becomes infinitesimally small, $\mathcal{V}_h\rightarrow 0$, and thus an infinite tower of light winding modes:
\begin{equation}
{\cal T}\rightarrow\infty\, :\qquad \mathcal{V}_h \simeq \frac{1}{(2\pi)^3}\frac{1} {{\cal T}^6(p^1 p^2 p^3)^2} \rightarrow0\, ,\quad m_{KK}\rightarrow\infty\, ,\quad m_{wind}\rightarrow0\, .\label{largeTm}
\end{equation}
\end{itemize}
The analogous limits for the entropy are:
\begin{itemize}
\item For {\it small entropy} (case (I) in Figure \ref{fig:BHpopulationandlimits}), the volume goes to zero:
\begin{equation}
\mathcal{S}\to 0:, \qquad \mathcal{V}_h\propto \frac{1}{(2\pi)^3}\frac{{\cal S}^3} {(p^1 p^2 p^3)^2}
\rightarrow 0\, ,
\end{equation}
thus leading to a light tower of winding modes with masses approaching zero.
\item For {\it large entropy} (case (II) in Figure \ref{fig:BHpopulationandlimits}), ${\cal S} \rightarrow \infty$, the volume has an identical expansion as in the limit of large temperature considered above. After fixing a proper trajectory in the ${\cal T}-{\cal S}$ plane, we obtain that the limit of large entropy corresponds to large volume $\mathcal{V}_h\rightarrow \infty$ and a light tower of KK modes. This result is somehow analogous to the case of extremal black-holes, with fixed magnetic charges, seen in Sec.~\ref{Sec.Extremal} and investigated in \cite{Bonnefoy:2019nzv}.
\end{itemize}
In summary, we see that the limit (I) in Figure \ref{fig:BHpopulationandlimits} (for the chosen combinations of charges) leads either to a tower of light KK particles or a tower of light winding modes, measured at the horizon of the black hole. This is the limit of small black holes with large temperatures, which behave like particles states. Due to the presence of infinite light modes, in accordance with the SDC we conclude that the effective field theory on the horizon breaks down. A similar conclusion can be drawn for the limit (II) of large black holes with large entropy and small temperature.
It is interesting to notice that the volume of the internal manifold ${\cal V}$ also appears in the supergravity Lagrangian \eqref{S4dSG} as the gauge coupling of the U(1) gauge fields in the matrix $\mathcal{I}_{\Lambda\Sigma}$, as one can check from \eqref{symm-matr-N}. One finds in fact that the $F^0_{\mu\nu}$ gauge field has coupling equal the internal volume ${\cal I}_{00}=-{\cal V}$. From the analysis in this section we see that near the horizon, in addition to the KK masses and winding modes, the limits discussed above also affect the U(1) couplings. This observation connects our results to yet another swampland conjecture, namely the absence of global symmetries in quantum gravity.
\subsubsection{Rescaling of all charges}\label{rescalingcharges}
As we already mentioned before, the light towers of states appear in the limit of large/small temperature and entropy in case certain combinations of electric or magnetic charges are kept fixed. However, if all charges are increased or decreased in the same way, the internal volume stays constant. Below, we investigate this particular scaling limit in some detail.
Let us consider a black hole of the ${\it STU}$ model with charges $q_0,p^1,p^2,p^3$, and let us study the transformation that rescales all charges in the same way. We start from the extremal case, where the metric is
\begin{align}\label{metric-extremal-tau}
ds^2=- e^{2U} dt^2 + e^{-2U}\left(\frac{d\rho^2}{\rho^4} + \frac{1}{\rho^2}d\Omega_{2}^2\right)\ .
\end{align}
A rescaling of all charges $q_0\to \lambda q_0$ and $p^i\to \lambda p^i$, together with the coordinates rescaling $\rho\to \lambda^{-1}\rho$, leaves the functions $I_0,I^i$ invariant, thus also $e^{2U}\to e^{2U}$, and $z^i(r)\to z^i(r)$. If we also redefine the time coordinate by $t \to \lambda t$, the metric becomes
\begin{align}
ds^2 \to \lambda^2 \left( - e^{2U} dt^2 + \left(e^{-2U}\frac{d\rho^2}{\rho^4} + \frac{1}{\rho^2}d\Omega_{2}^2\right)\right),
\end{align}
namely the charge rescaling and coordinate redefinition result in a Weyl rescaling of the metric.
The same also happens for the non-extremal solution. Considering the family of rescalings
\begin{align}
\label{eq:rescaling-nonex}
q_0 \to \lambda q_0 \ , \qquad p^i \to \lambda p^i \ , \qquad \rho \to \lambda^{-1} \rho \ , \qquad c \to \lambda c \ , \qquad t\to \lambda t \ ,
\end{align}
the quantities $a_0$, $a^i$ and $b_0$, $b^i$ are left invariant, so again $e^{2U}\to e^{2U}$, and $z^i(\rho)\to z^i(\rho)$, while
\begin{align}
\mathcal{S} \to \lambda^2 \mathcal{S} \ , \qquad\mathcal{T} \to \lambda^{-1}\mathcal{T} \ .
\end{align}
On the metric \eqref{metric}, this corresponds to
\begin{align}
ds^2 \to \lambda^2 \left( - e^{2U}dt^2+e^{-2U} \left[ \frac{c^4 d\rho^2}{\sinh^4(c\rho)} + \frac{c^2}{\sinh^2(c\rho)} \left(d\theta^2+\sin^2\theta d\phi^2\right) \right] \right),
\end{align}
which is again a Weyl rescaling. Now notice that, since the scalars are left invariant, the volume is also invariant \cite{Bonnefoy:2019nzv}. If one considers this Weyl rescaling within a ten-dimensional embedding of the four-dimensional black hole geometry, then the transformation \eqref{eq:rescaling-nonex} can be interpreted as a rescaling of the Planck Mass.
Notice that the black hole ADM mass, by simple background subtraction at infinity, is
\begin{align}
M= \frac14 \left(\sqrt{c^2+(4q_0L^0_\infty)^2}+\sum_{i=1}^3 \sqrt{c^2+(4p^iM_{i\infty})^2} \right)\ ,
\end{align}
which also gets rescaled as $M\to \lambda M$, to preserve the BPS bound of the theory $M\geq |Z|$ (the central charge scales together with the charges, $Z \to \lambda Z$).
\subsection{Thermodynamic dualities}
In this section, we would like to investigate further the existence of new temperature and entropy dualities, ${\cal T}\leftrightarrow 1/{\cal T}$ or ${\cal S}\leftrightarrow 1/{\cal S}$, acting as geometric duality on the compact internal space, namely exchanging KK modes with winding modes. We will point out situations in which large and small temperature limits are precisely dual to each other.
Let us pick a fixed symplectic frame for the charges, and let us investigate possible dualities between the large and the small temperature regime of supergravity black holes. In particular, for the {\it STU} model, we consider black holes with four non-vanishing charges $q_0,p^1,p^2,p^3$.
We start from the temperature. Consider A-type black holes and compare the high temperature limit \eqref{largeT} with the small temperature limit \eqref{smallT}. It is easy to see that the transformation
\begin{equation}
{\cal T}\,\longleftrightarrow\, {1\over q_0^2{\cal T}}\, ,\qquad \qquad \quad\qquad q_0\longrightarrow q_0\ ,
\end{equation}
leaving the electric charge $q_0$ fixed, inverts the internal volume, ${\cal V}_h\longleftrightarrow{1\over{\cal V}_h}$ and hence exchanges KK with winding modes. This is a realization of a ${\cal T}\leftrightarrow 1/{\cal T}$ \textit{temperature duality}.\\
\noindent Then, consider B-type black holes. We see immediately that there is again a {\it high-low temperature duality}, inducing ${\cal V}_h\longleftrightarrow{1\over{\cal V}_h}$ and exchanging the internal KK and winding modes among each other. This has the following form:
\begin{equation}
{\cal T}\,\longleftrightarrow\, {1\over (p^1 p^2 p^3)^{2/3}{\cal T}}\, ,\qquad\qquad p^i\longrightarrow p^i\, ,
\end{equation}
where the magnetic charges $p^i$ are kept fixed.
Similarly to what done for the temperature, we can also point out an entropy duality. For the A-type black holes this is of the form
\begin{equation}
{\cal S}\,\longleftrightarrow\,{q_0^{4}\over{\cal S}}\, ,\qquad\qquad\qquad \qquad q_0\longrightarrow q_0\, ,
\end{equation}
while for B-type black holes we have
\begin{equation}
{\cal S}\,\longleftrightarrow\,{(p^1 p^2 p^3)^{4/3}\over{\cal S}}\, ,\qquad \qquad p^i\longrightarrow p^i\, .
\end{equation}
Note that there is yet another class of entropy/temperature transformations, which however keep the volume ${\cal V}_h$ invariant, but rescale the electric charge $q_0$ (case A) or the magnetic charges $p^i$ (case B) in a non-trivial way:
\begin{equation}
\sqrt{\cal S}\,\longleftrightarrow\,{\cal T} \quad{\rm with}\quad q_0\,\longrightarrow\,{{\cal T}\over\sqrt {\cal S}}~q_0\,\, \,\,(A)
\quad {\rm or} \quad p^i\,\longrightarrow\,{{\cal T}\over\sqrt {\cal S}}~p^i \,\,\,\,(B)\, .
\label{STex1}
\end{equation}
These transformations act like in the Schwarzschild case (see eq.(\ref{STex})) and exchange small with large black holes. Moving on a Schwarzschild like hyperbola, these transformations look like
\begin{equation}
{\cal S}\,\longleftrightarrow\,{1\over {\cal S}} \quad{\rm and}\quad{\cal T}\,\longleftrightarrow\,{1\over {\cal T}} \quad{\rm with}\quad q_0\,\longrightarrow\,{ {\cal T}^2}~q_0 \,\,\,\, (A)
\quad {\rm or} \quad p^i\,\longrightarrow\,{ {\cal T}^2}~p^i \,\,\,\,(B) \, .
\label{STdual2}
\end{equation}
\subsection{Embedding of Einstein-Maxwell-dilaton in ${\cal N}$=2 Supergravity}
\label{EMdSUGRA}
The Einstein-Maxwell-dilaton (EMd) black holes of Section \ref{sec:EMDblackholes}, for specific values of parameter $\lambda$, are captured by the $p^0,q_0$ configuration of ${\cal N}=2$ matter coupled supergravity \cite{Ferrara:1996dd, Ceresole:2007rq}, and correspond to the KK black hole in the reduction from five to four dimensions\cite{Gibbons:1985ac,Horowitz:2011cq}. To discuss the supergravity embedding of the KK black hole we consider the ${\it STU}$ model introduced in Section \ref{SG-blackholes}; we set $z^1=z^2=z^3=- i e^{- \sqrt{2/3}\phi}$ and only take the vector field strength $F^0_{\mu\nu}$ to be nonzero \cite{Duff:1999gh}. One can see then that the Lagrangian of this model is of the EMd form
\begin{align}
\mathcal{I}_{\Lambda\Sigma} F^{\Lambda}_{\mu\nu} F^{\Sigma\, \mu\nu} \to \mathcal{I}_{00} (F^{0}_{\mu\nu})^2 &= e^{-\sqrt6 \phi} (F^0_{\mu\nu})^2 \ ,\\
g_{i\bar\jmath}\partial_\mu z^i\partial^\mu \bar z^{\bar \jmath} &\to \frac12 \partial_\mu \phi \partial^\mu \phi\ ,
\end{align}
for a value $\lambda = \sqrt{3/2}$.
In the ${\it STU}$ model, the properties of extremal black holes are captured by the U-duality group quartic invariant \cite{Bellucci:2008sv}
\begin{align}
\label{quartic-inv}
\mathcal{J}_4 = - (p^0q_0+p^i q_i)^2 +4 \sum_{i<j} p^iq_i p^j q_j-4 p^0 q_1 q_2 q_3 + 4 q_0 p^ 1 p^ 2 p^ 3\ .
\end{align}
One sees that for a choice of charges $p^0,q_0$, this invariant is negative definite, thus it corresponds to a configuration which does not preserve supersymmetry in the extremal limit (non-BPS) (see \cite{Bellucci:2008sv} and references therein). For such choice of charges, the extremal black hole we obtain is given by (see \cite{Ceresole:2009jc,Ferrara:2006em} and references therein)
\begin{align}
z^1 z^2 z^3|_{h} &= i \frac{q_0}{p^0} \ , \\
e^{-\sqrt{2/3}\phi}|_{h} &= \left( \frac{q_0}{p^0} \right)^{1/3} \ , \\
\frac{{\cal S}}{4\pi}&= \sqrt{\mathcal{J}_4(p^0,q^0)} =|p^0 q_0| \ ,
\end{align}
which has a residual $\left(SO(1,1)\right)^2$ moduli space \cite{Bellucci:2008sv}. This reproduces the extremal black hole behavior of the EMd models.
The complete flow from asymptotic infinity to the horizon is given by the extremal metric \eqref{metric-extremal-tau} with
\begin{align}
e^{-2U(\rho)}&=\sqrt{I_0(\rho) \tilde I_0(\rho)}. \\
z^i &= - i \left(\frac{q_0}{p^0} \right) ^{1/3} \sqrt{\frac{\tilde I_0(\rho)}{I_0(\rho)}}, \\
I_0 &= \left(a-\sqrt{q_0 p^0}\rho\right)^2-b \ , \\
\tilde I_0 &= \left(a-\sqrt{p^0 q_0}\rho\right)^2+b \ ,
\end{align}
where $a\neq0$ and $b$ are real constants.
Notice that this is a case where the extremal solution has running scalars, since the values of the scalars vary throughout the radial direction. This black hole is the static extremal limit of the rotating black hole constructed in \cite{Bena:2009ev}.
\subsection{Small black holes}
The configuration with only one of $p^0$ or $q_0$ turned on corresponds to \emph{small black hole} solutions of ${\cal N}=2$ matter coupled supergravity. However, there are also more general small black hole configurations in ${\cal N}=2$ supergravity.
Small black holes are a class of extremal black hole solutions of ${\cal N}=2$ supergravity arising for a choice of charges such that the quartic U-duality invariant of the theory (given in \eqref{quartic-inv}) is ${\cal J}_4=0$. They are relevant for our discussion since first the scalar fields for this solutions never reach a fixed point at finite distance in moduli space\footnote{Contrary to the attractor mechanism for finite area extremal black holes in ${\cal N}=2$ supergravity.} \cite{Ceresole:2010nm}, and second they arise as a $I_4\to 0$ limit of finite-area black holes, for an appropriate tuning of the charges. Thus, they appear as solutions naturally associated to a notion of infinite distance in moduli space.
In the case of the $STU$ model, also the configurations $p^0, q_1,q_2,q_3$ and $q_0,p^1,p^2,p^3$ become small black holes when $p^0 \to 0$ and $q_0 \to 0$ respectively, as can be seen from \eqref{quartic-inv}. To illustrate what happens let us consider the limit $q_0\to 0$ in the non-extremal solution considered for the $STU$ model in Section \ref{sec:non-extremal} (which corresponds to setting $b_0=0$ in the simpler solution \eqref{a0b0}). Entropy and temperature are
\begin{align}
\mathcal{S}|_{q_0=0} \ & \sim \sqrt{ c} \ \prod_{i=1}^3\sqrt{c+\sqrt{c^2+\left(4 p^iM_{i\infty}\right)^2}} \ , \\
\mathcal{T}|_{q_0=0} \ & \sim \frac{\sqrt c}{ \prod_{i=1}^3\sqrt{c+\sqrt{c^2+\left(4p^iM_{i\infty}\right)^2}} \ ,}
\end{align}
while scalars run from their value at infinity $z^i=-i\lambda_\infty^i$ to the horizon value:
\begin{align}
z^i_{h}=-i \, \sqrt{2c}\, \lambda_\infty^1 \left( \frac {c+\sqrt{c^2+(4 p^1 M_{1\infty})^2}} { \left( c+\sqrt{c^2+ (4 p^2 M_{2\infty})^2} \right) \left( c+\sqrt{c^2+ (4 p^3 M_{3\infty})^2} \right) } \right)^\frac12 \ .
\end{align}
Thus, we see that in the extremal limit $c\to 0$ (corresponding again to ${\cal T}\to0$)
\begin{align}
\mathcal{S}|_{q_0=0} &\to 0 \ ,
\end{align}
and the scalars run from a constant value at infinity to $z_h^i = 0$ at the horizon. In the simpler solution of the previous section, this limit corresponds to
\begin{align}
e^{-\sqrt{2/3}\phi}\to 0 \,, \qquad \phi \to \infty \
\end{align}
and it is again a limit at infinite distance in moduli space. Thus, we can extend to this setup the conclusion that small black holes are outside of the regime of validity of the supergravity effective theory, as we discussed for the EMd case in the previous section.
\subsubsection{Doubly small black holes}
There is another interesting configuration that one can analyse, starting from the $STU$ black hole with charges $q_0,p^1,p^2,p^3$ (or $p^0,q_1,q_2,q_3$, analogously). Let us consider again the non-extremal solution of Section \ref{sec:non-extremal} and now turn off both $q_0$ and one of the magnetic charges, say $p^1$. In this case, entropy and temperature read
\begin{align}
\mathcal{S}|_{q_0=0,p^1=0}\ \ & \sim \ \ c \left(\left(c+\sqrt{c^2+(4{p}^2 M_{2\infty})^2}\right)\left(c+\sqrt{c^2+(4{p}^3M_{3\infty})^2}\right)\right)^\frac12 \ , \\
\mathcal{T}|_{q_0=0,p^1=0} \ \ & \sim \ \ \frac{1}{\left(\left(c+\sqrt{c^2+(4{p}^2 M_{2\infty})^2}\right)\left(c+\sqrt{c^2+(4{p}^3M_{3\infty})^2}\right)\right)^\frac12}\ ,
\end{align}
while the horizon value of the scalars is
\begin{align}
z^i_{h}&=-i \, \lambda_\infty^i \frac{2c}{\left(\left(c+\sqrt{c^2+(4{p}^2 M_{2\infty})^2}\right)\left(c+\sqrt{c^2+(4{p}^3M_{3\infty})^2}\right)\right)^\frac12} \ .
\end{align}
Again, these are regular solutions at finite temperature but, as $c\to0$, they have vanishing entropy and finite temperature, and the scalars at the horizon $z_{h}^i\to 0$. This limit is once more at infinite distance in moduli space. These solutions are thus supergravity examples of the extremal limit $t_1$ illustrated in Figure \ref{fig:BHpopulationandlimits}. Let us stress once more that this limit corresponds to a breakdown of the effective description associated with a light tower of states predicted by the SDC.
\section{Conclusion}
\label{sec:concl}
In this work, we have studied thermodynamic properties of non-extremal charged black holes in relation to the Swampland program. The ${\cal T}-{\cal S}$ phase diagram, in terms of the temperature and entropy, has been a main focus of our investigation. Specifically, the analysis of asymptotic regions, as shown in Figure \ref{fig:BHpopulationandlimits}, has turned out to provide novel insight into the relation between the Swampland Distance Conjecture and the physics of black holes.
While an application of the metric distance in these non-compact backgrounds turns out to be problematic in the case of finite entropy extremal black holes, the field distance traversed by the moduli fields from spatial infinity to the black hole horizon successfully identifies the problematic regions. This was also confirmed by embedding the effective descriptions, Einstein-Maxwell-dilaton theory as well as $\mathcal{N} = 2$ supergravity, in string theory and identifying the light towers suggested by the Swampland Distance Conjecture in the vicinity of the black hole.
Our findings are visually summarized in Figure \ref{fig:BHpopulationcompact}, which proposes again the temperature-entropy phase diagram of non-extremal black holes, but now in a ``compactified" form. The points at infinity are shrunk to $\frac{\pi}{2}$ from the origin and the validity region for RN black holes becomes a finite triangle. The loci where massless towers of states are expected to emerge are highlighted in red.
\begin{figure}[h]
\begin{center}
\begin{tikzpicture}[scale=1.25]
\draw[->, line width=.5mm] (0, 0) -- (4.2, 0) node[below] {${\cal X}$};
\draw[->,line width=.5mm] (0, 0) -- (0,4.33) node[left] {${\cal Y}$};
\draw[scale=1.5,domain=0.:2.,smooth, variable=\x,blue, thick,line width=.6mm] plot ({\x},{-(\x)+2)});
\fill [blue!25, domain=0.:2., variable=\x] (0.0, 4.15) -- plot[scale=1.5,domain=0.:2.] ({\x},{-(\x)+2)})
-- (0.02, 0.02);
\draw[red, line width=.7mm] (0, 0) -- (3, 0);
\node () at (3,-.3) {$\frac \pi2$};
\node () at (-.3,3) {$\frac \pi2$};
\node () at (3,3) {excluded region};
\draw[red,fill=red] (0,3) circle[radius=2pt];
\draw[red,fill=red] (3,0) circle[radius=2pt];
\end{tikzpicture}
\end{center}
\caption{Version of the diagram in Figure \ref{fig:BHpopulation} with the point at infinity included (red dots). The coordinates are related as $\mathcal{Y} =\arctan \mathcal{S}$ and $\mathcal{X} = \arctan (2 \mathcal{T}^2)$. The red points and line are regions where towers of infinitely-many massless states are expected to emerge. The blue line $\mathcal{Y} = - \mathcal{X} + \frac{\pi}{2}$ is the Schwarzschild hyperbola $\mathcal{S} = (2\mathcal{T}^2)^{-1}$ in the new coordinates.}
\label{fig:BHpopulationcompact}
\end{figure}
The large/small entropy limit, which induces the infinite field distance, is also closely related to the AdS distance conjecture \cite{Lust:2019zwm} and the scale separation conjecture \cite{Tsimpis:2012tu,Gautason:2015tig}, which state that in the limit of small AdS curvature a massless tower of states should emerge. For four-dimensional extremal black holes, the near horizon geometry has the form $AdS_2\times S^2\times M_6$, where $M_6$ is an internal space of volume ${\cal V}_h$. The problem of scale separation for these kind of $AdS_2$ geometries was recently investigated in \cite{Lust:2020npd}. In the large entropy limit, ${\cal S}\rightarrow\infty$, the radii of $AdS_2$ and $S^2$ become large and according to the AdS distance conjecture a massless tower of states would be expected.\footnote{Again, we want to advise caution in the direct application of Swampland constraints in two and three dimensions.} This tower can be identified with the KK modes of the $S^2$. In this paper, we are rather investigating the question of scale separation between the four-dimensional black hole and the internal space $M_6$.
As we have seen, this scale separation is possible if all electric and magnetic charges scale in the same way (since the volume remains constant while the entropy can vary - see Sec.~\ref{rescalingcharges}), but it is not possible if either the electric or the magnetic charges are kept fixed. For non-extremal black holes, the near horizon geometry is instead ${\cal R}^{1,1}\times S^2\times M_6$. The scale of the 2-dimensional Rindler space ${\cal R}^{1,1}$ is determined by the surface gravity $\kappa=2\pi{\cal T}$. Then, the question of scale separation and the emergence of massless tower of states can be rephrased in terms of a {\it temperature distance} and one can study the limits ${\cal T} \rightarrow \{0,\infty\}$.
Another intriguing question addressed in our paper regards the existence of thermodynamic dualities between small and large temperature as well as entropy, i.e. between small and large black holes. Our motivation comes from the fact that temperature dualities are present in several statistical models and that black holes allow in fact for a statistical interpretation. Already at the level of a Schwarzschild black holes, we have observed the existence of such dualities (expressed in terms of $\cal T$ and $\cal S$), which exchange large and small black holes, thus acting on the mass as $M \leftrightarrow 1/M$.
More interestingly, in the context of non-extremal black holes in ${\cal N}=2$ supergravity, we have shown that temperature- and entropy-dualities act on the internal volume $\cal V$ by exchanging light KK and winding states.
With this observation at hand, we do not claim that large and small black holes are physically equivalent to each other but, when coupled to scalar field of the effective field theory, they can correspond to an equivalent internal (dual) spectrum. We interpret such thermodynamic dualities as the effective remnants of the string dualities (such as T- or S-dualities).
Clearly, also the existence of such dualities, like the exchange of small and large entropy (or small and large mass), implies having control over higher derivative corrections, which mainly come into play for small ${\cal S}$, that is when the supergravity regime breaks down. We leave the question whether our findings are stable against such corrections for future work.
Finally, let us mention a few more questions and possible generalizations of the present work.
First, as already mentioned, the limit of small entropy ${\cal S}\rightarrow 0$ is very subtle, requires very small electric or magnetic charges, and most likely needs some refinement of the effective two-derivative supergravity action.
Here we expect that higher derivative corrections to the effective action
could change the relation between the thermodynamic black hole properties and the internal moduli fields.
Second, the discussion could be extended to rotating non-extremal solutions, using the stationary solutions in ${\cal N}=2$ of \cite{Behrndt:1997ny} or the non-BPS black rings, which are considered in \cite{Bena:2007kg,Bena:2009ev}. Another interesting problem is to investigate the similarity of black hole potentials and the scalar field flux potentials in extended supergravity. In this context, there might arise some interesting analogies between the formalism of the black hole limits considered in this paper and of the limits for supersymmetry breaking in the context of the gravitino conjecture \cite{Cribiori:2021gbf,Castellano:2021yye}.
\vskip0.5cm
\vspace{10px}
{\bf Acknowledgements}
\vskip0.1cm
\noindent
We thank Mirjam Cveti\v{c} and Gary Shiu
for useful discussions.
The work of N.C.~is supported by the Alexander von Humboldt foundation.
The work of D.L.~is supported by the "Origins" Excellence Cluster and also by the German-Israelian-Project (DIP) "Holography and the Swampland".
| 2024-02-18T23:40:02.394Z | 2022-10-24T02:02:23.000Z | algebraic_stack_train_0000 | 1,154 | 19,493 |
|
proofpile-arXiv_065-5718 | \section{Introduction}
\label{sec:introduction}
Recent technological advances have empowered computers to reason and perform activities once attributed to human beings only, such as writing, speaking, and even helping with decisions like the best route on a trip or the most appropriate drugs for treating a specific disease. Nevertheless, such reasoning is limited to the domain upon which the algorithm is trained, i.e., the actions and decisions adopted by the algorithm are based on patterns somehow encoded in the dataset. This approach seems unnatural if considering the learning processes performed by the biological brain, in which stimuli are provided by a set of different sensors, e.g., vision and hearing, and this multimodal information is combined in such a way that redundant information is essential to reinforcing and improving noisy, ambiguous, and imperfect signals from distinct sources.
Multimodal learning approaches are be beneficial to improve one modality feature representation~\cite{singh2022watmif,iqbal2022ff}. To illustrate the idea, consider, for instance, a boisterous speech. If the speaker only resorted his voice to convey its message, the comprehension of the subject may be considerably impaired by the noise. On the other hand, if the images are also available, it is possible to try to complement the corrupted information with insights provided by this secondary source, such as following the speaker's hands and body movements or trying to read his lips.
Despite the availability of contextual information provided by different input signals, usually such tasks also rely on temporal information for reasoning. Revisiting the speech example, it is easy to infer that words being said at the present moment are probably strongly correlated with the last few pronounced words. Even though some works addressed the problem using recurrent networks~\cite{mai2021analyzing,li2020multimodal}, most of them perform target-driven supervised learning, which usually requires a considerable number of labeled samples for training.
In this context, unsupervised or self-supervised algorithms show themselves capable of extracting strongly correlated features, which are highly desired for two main reasons: (i) their representations are usually more general-purpose than target-driven features extracted with supervised algorithms, and (ii) they do not require labeled instances for training, which are usually limited and costly. Regarding the proposed approaches for correlated feature extraction, one can refer to energy- and mutual information-based methods, such as the Deep Graph Infomax~\cite{velivckovic2018deep}, which relies on mutual information maximization and graph neural networks (GNN) for leveraging information propagation in a graph.
With the advent of deep learning, GNNs emerged as an elegant solution to extract in-depth dependencies from such intricate relationships. Moreover, it also presents itself as a powerful alternative to convolutional neural networks suitable for datasets composed of non-imagery data. Despite the success obtained by~\cite{velivckovic2018deep}, Zhang et al.~\cite{zhang2021canonical} pointed towards a set of drawbacks in the model: (i) reliance on negative samples - corrupting the graph structure by selecting arbitrary negative examples may lead to large variance for stochastic gradients and slow training convergence; (ii) require a parameterized estimator to approximate mutual information between two views; and (iii) it contrasts node embeddings with graph embedding leading to a higher complexity. To tackle such problems, they propose the GNNs with Canonical Correlation Analysis (CCA-GNN), which aims at maximizing the correlation between two augmented views of the same input while decorrelating different dimensions of such views. Besides, Dwivedi et al.~\cite{dwivedi2021graph} exposed another shortcoming regarding GNNs message-passing mechanism, which builds node representation by aggregating feature space-based local neighborhood information and leads representations dependants on the local structure of the graph and proposed using positional encoding to solve the problem.
Therefore, this paper aims to redesign the GNNs with Canonical Correlation Analysis (CCA-GNN) to deal with the challenging context of multimodal representation learning. Specifically, it formulates a parallel CCA-GNN architecture for each input channel, i.e., audio and visual. The new audio-visual (AV) CCA-GNN model minimizes both the canonical correlation between the augmented samples of the same channel as well as between the augmented samples of the other mode. Additionally, it introduces graph modeling that considers a time-frame sequence distance positional encoding to compute the node's neighborhood. The idea is to introduce temporal information through the samples' connectivity in the embeddings. Figure~\ref{f.intro} depicts the whole pipeline.
\begin{figure*}[!ht]
\centerline{
\begin{tabular}{c}
\includegraphics[width=\textwidth]{intro-eps-converted-to.pdf}
\end{tabular}}
\caption{Proposed pipeline. Features are extracted from the noisy audio input using Fourier transformation followed by logarithmic compression, while the visual inputs employ an encoder-decoder approach followed by Viola-Jones~\cite{viola2001rapid} for lip-regions identification. Both sets are converted into graphs using the proposed prior-frame-based positional encoding to maintain the temporal information. Such graphs are used to generate two augmented versions per epoch, which feed a set of graph convolutions. The model minimizes canonical correlation analysis of convolutions' output, aiming to generate a set of correlated features in a self-supervised fashion. Such features are finally used to feed a supervised dense layer responsible for reconstructing the clean audio signal.}
\label{f.intro}
\end{figure*}
Experiments conducted over the AV ChiME3 dataset compare the proposed approach against a CCA-based multilayer perceptron (MLP). Results show that (i) the multimodal CCA-GNNs produce more representative features than the standard unimodal version, leading to lower errors over clean audio data reconstruction; (ii) the proposed prior-frame approach for sequential-time modeling in graphs outperform the standard feature-space distance-based neighborhood connections; and (iii) CCA-GNNs deliver better results than the CCA-MLP model in the context of feature extraction for data reconstruction, requiring a considerably reduced rate of firing neurons, indicating the new model is more suitable for energy-constrained environments, such as AV hearing aid devices. To the best of our knowledge, this study is the first to demonstrate the application of GNN-CCA for energy-efficient rich real-world multi-modal data for a benchmark AV speech enhancement problem, where multiple real-world noises corrupt speech in real-world conditions. Therefore, the main contributions of this paper are threefold:
\begin{enumerate}
\item Demonstrate the application of a novel self-supervised energy-efficient CCA-GNN-based model that considers the fusion between different sources of information from the environment to improve sound quality.
\item A novel GNN-CCA model with integrated positional node encoding considering a prior-frame sequence distance instead of a feature-space representation when computing the node’s nearest neighbors, introducing temporal information in the embeddings through the neighborhood’s connectivity.
\item The proposed method is evaluated with the benchmark AV Grid and ChiME3 corpora, with $4$ different real-world noise types (cafe, street junction, public transport (BUS), pedestrian area) and compared with standard GNNs and MLP models for unsupervised AV speech processing tasks. Comparative results show that our new method demonstrates superior energy consumption and generalization performance in all experimental conditions. Other comparative models fail to reconstruct speech-in-noise with a similar number of neurons.
\end{enumerate}
The remainder of this paper is described as follows. Section~\ref{s.proposed} provides a brief background regarding GNN-CCA and introduces the proposed approaches. Further, Section~\ref{s.methodology} provides the necessary information regarding the dataset and the evaluation metrics, while Section~\ref{s.setup} describes the experimental setup. Finally, Sections~\ref{s.experiments} and~\ref{s.conclusions} state the results and conclusions, respectively.
\section{Related Work}
\label{sec:related}
Ngiam et al.~\cite{ngiam2011multimodal} showed that multimodal learning approaches could be beneficial to improve one modality feature representation. The approach was recently applied to a wide variety of applications, such as emotion analysis~\cite{jia2021multimodal}, scene change detection~\cite{santana2019novel} and medicine~\cite{venugopalan2021multimodal}, to cite a few. Regarding audio-visual (AV) data processing, Adeel et al.~\cite{adeel2018real} suggested an integration of Internet of Things (IoT) and 5G Cloud-Radio Access Network to create a chaotic encryption-based lightweight model for lip-reading driven hearing aids. In further work~\cite{adeel2020novel} the model was improved to transmit encrypted compressed audio-visual (AV) information and receive encrypted enhanced reconstructed speech in real-time. Recent works comprise a deep learning-based framework for speech enhancement that exploits AV cues concerning different operating conditions to estimate clean audio~\cite{adeel2020contextual}, as well as the CochleaNet~\cite{gogate2020cochleanet}, which integrates noisy audio and visuals from distinct language speakers.
Regarding the proposed approaches for correlated feature extraction using energy-based approaches, one can refer to~\cite{hinton2002training,passos2017fine,passos2019kappa,passos2018temperature}. In the context of mutual information, Belghazi et al.~\cite{belghazi2018mine} proposed the so-called MINE, a mutual information neural estimation, while Hjelm et al.~\cite{hjelm2018learning} proposed a model for Learning deep representations by mutual information estimation and maximization. Further, Veli{\v{c}}kovi{\'c} et al.~\cite{velivckovic2018deep} proposed a graph-based solution called Deep Graph Infomax, which combines on mutual information maximization with graph neural networks.
Graph theory describes strong architectures capable of modelling complex relationships, with applications ranging from small world design~\cite{newman2000models} to oversampling~\cite{passos20202pf,passos2022handling}. Considering GNNs applications for correlated feature extraction, aside from~\cite{velivckovic2018deep}, Zhang et al.~\cite{zhang2021canonical} introduced canonical correlation analysis for self-supervised feature extraction using GNNs, providing a more efficient alternative for the task since it does not rely on negative pairs, does not require learning parameters of additional components such as an estimator. Further, the complexity of the model is considerably smaller since it requires O($F^2$) space cost against O($N$) from the Deep Graph Infomax, where $F$ denotes the feature space size and $N$ denotes the number of nodes.
\section{Graph Neural Network with Canonical Correlation Analysis}
\label{s.CCAGNN}
Consider a single graph $G = (X, A)$ where $X\in \mathbb{R}^{N\times F}$ denotes the node's feature vectors and $A \in \mathbb{R}^{N \times N}$ stands for the adjacency matrix. The CCA-GNN~\cite{zhang2021canonical} is composed of three main parts: (i) a random graph generator $T$, (ii) a graph neural network encoder $f_\theta$, where $\theta$ stands for the learnable parameters, and (iii) a Canonical Correlation Analysis-based objective function. The idea is to present two augmented versions of the same graph to the network and maximize the canonical correlation between their outputs. Such an approach aims at preserving correlated components while discarding decorrelated ones, i.e., maintaining the relevant information present in both augmented versions and avoiding particular behaviors, such as anomalies and noise. Figure~\ref{f.CCAGNN} depicts the Graph Neural Network with Canonical Correlation Analysis.
\begin{figure*}[!ht]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.9\textwidth]{graph_canonical_correlation-eps-converted-to.pdf}
\end{tabular}}
\caption{Graph Neural Network with Canonical Correlation Analysis. The dataset is converted into a graph where each sample represents a node, and the edges denote the nodes' connection. Two augmented versions of this graph are generated and used to feed a GNN. Finally, the canonical correlation between the output of both versions is computed and used as the cost function to optimize the GNN parameters.}
\label{f.CCAGNN}
\end{figure*}
Regarding the graph augmentation, CCA-GNN employs the same approach used in~\cite{zhu2020deep,thakoor2021bootstrapped}, which basically performs a random edge dropping and feature masking. Thus, each $t\sim T$ stands for a transformed version of $G$. Notice that those augmented versions are sampled at each iteration.
Concerning the encoder, the model employs a simple two-layered graph neural network, which can be easily replaced by more complex or sophisticated architectures.
Finally, the objective function aims at modeling the learning problem as a canonical correlation-based~\cite{chang2018scalable} self-supervised approach in which the two randomly augmented versions of the graph yields two normalized views of the input data, $\bm{Z}_A$ and $\bm{Z}_B$, and their correlation is maximized. The objective function is described as follows:
\begin{equation}
{\cal L}(\bm{Z}_A, \bm{Z}_B) = ||\bm{Z}_A - \bm{Z}_B||_{F}^2+\lambda\left(||\bm{Z}_A^T\bm{Z}_A-\bm{I}||_F^2+||\bm{Z}_B^T\bm{Z}_B-\bm{I}||_F^2\right),
\label{e.canonicalCorrelation}
\end{equation}
where $I$ is the identity matrix and $\lambda$ is a non-negative trading-off hyperparameter. The first term, namely the invariance term, is responsible for the minimization of the invariance between the two views, which is essentially the same as maximizing the correlation between them. The second is the decorrelation term, which seeks a regularization that encourages distinct features to capture different semantics.
Further, the authors provided a variance-covariance perspective~\cite{tian2021understanding} of the objective function. Suppose $\bm{s}$ as an augmented version sampled from an input $\bm{x}$, and $\bm{z_s}$ is the representation of $\bm{s}$ obtained through a decoder. The invariance term can be minimized using expectation as follows:
\begin{eqnarray}
{\cal L}_{inv} &=& ||\bm{Z}_A - \bm{Z}_B||_{F}^2= \sum_{i=1}^N\sum_{k=1}^D(z_{i,j}^A-z_{i,j}^B)^2\nonumber\\
&\cong& \mathbb{E}_{\bm{x}}\left[\sum_{k=1}^D\mathbb{V}_{\bm{s}|\bm{x}}[\bm{z_s},k]\right]*2N,
\label{e.loss_inv}
\end{eqnarray}
where $\mathbb{V}$ is the variance. In a similar fashion, the decorrelation term is written as follows:
\begin{eqnarray}
{\cal L}_{dec} &=& ||\bm{Z}_S^T\bm{Z}_S-\bm{I}||_F^2= ||\bm{Cov}_{\bm{s}}[\bm{z}]-I||_F^2\nonumber\\
&\cong& \sum_{i\neq j}(\rho_{i,j}^{\bm{z_s}})^2,\forall \bm{Z}_S\in\{\bm{Z}_A,\bm{Z}_B\},
\label{e.loss_dec}
\end{eqnarray}
where $\bm{Cov}$ is the covariance matrix and $\rho$ is the Pearson correlation coefficient.
\section{Proposed Approach}
\label{s.proposed}
This section presents a multimodal extension for the Canonical Correlation Analysis Graph Neural Network. Further, it also introduces the idea of modeling the temporal information of sequence data as node relationships in a graph.
\subsection{Multimodal Canonical Correlation Analysis Graph Neural Network for Audio-Visual Embedding Learning}
\label{ss.CCAGNN_av}
The proposed extension of the Canonical Correlation Analysis Graph Neural Network for multimodal data comprises a pair of networks, each of them fed with a modality of data, e.g., audio and visual, running in parallel. At the output layer, the canonical correlation analysis is performed considering both the intra-channel correlation, i.e., the two augmented versions of the same data modality, as well as inter-channels correlation. Figure~\ref{f.CCAGNN_av} depicts the model.
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=\textwidth]{graph_nets_multimodal-eps-converted-to.pdf}
\end{tabular}}
\caption{Multimodal Canonical Correlation Analysis Graph Neural Network for Audio-Visual Embedding Learning.}
\label{f.CCAGNN_av}
\end{figure*}
To accommodate the intra- and inter-channel computations of the canonical correlation analysis in the objective function, one can firstly consider two randomly augmented versions of normalized views for each channel, namely $\bm{Z}_1$ and $\bm{Z}_2$ for audio data, and $\bm{Z}_3$ and $\bm{Z}_4$ for visual data. The individual losses for both channels are computed using Equation~\eqref{e.canonicalCorrelation}:
\begin{equation}
\begin{array}{c}
{\cal L}_{\text{Audio}} = {\cal L}(\bm{Z}_1, \bm{Z}_2),\\
{\cal L}_{\text{Visual}} = {\cal L}(\bm{Z}_3, \bm{Z}_4).
\end{array}
\label{e.AudioVisual_alternative}
\end{equation}
Further, all the possible combinations of audio and visual data are computed, namely \textit{Audio1Visual1} ({$Z_1,Z_3$}), \textit{Audio1Visual2} ({$Z_1,Z_4$}), \textit{Audio2Visual1} ({$Z_2,Z_3$}), and \textit{Audio2Visual2} ({$Z_2,Z_4$}), as follows:
\begin{equation}
\begin{array}{c}
{\cal L}_{\text{Audio1Visual1}} = {\cal L}(\bm{Z}_1, \bm{Z}_3),\\
{\cal L}_{\text{Audio1Visual2}} = {\cal L}(\bm{Z}_1, \bm{Z}_4),\\
{\cal L}_{\text{Audio2Visual1}} = {\cal L}(\bm{Z}_2, \bm{Z}_3),\\
{\cal L}_{\text{Audio2Visual2}} = {\cal L}(\bm{Z}_2, \bm{Z}_4).
\end{array}
\label{e.combinedLosses}
\end{equation}
Finally, the objective function of the multimodal Canonical Correlation Analysis Graph Neural Network is given by:
\begin{eqnarray}
{\cal L} &=& \alpha{\cal L}_{\text{Audio}} + \beta{\cal L}_{\text{Visual}}\nonumber\\
&& + \gamma({\cal L}_{\text{Audio1Visual1}}+{\cal L}_{\text{Audio1Visual2}}\nonumber\\
&& +{\cal L}_{\text{Audio2Visual1}}+{\cal L}_{\text{Audio2Visual2}}),
\label{e.finalLoss}
\end{eqnarray}
where $\alpha$, $\beta$, and $\gamma$ are constants that control the influence of audio, video, and the combined canonical correlation, respectively.
\subsection{Modelling Temporal Information as Graph Nodes Relationships}
\label{ss.temporalRelationship}
The usual approach for modeling a dataset into a graph structure consists of representing its samples as the graph's nodes, whose edges connect the adjacent instances inserted into a $D$-dimensional feature space. A common approach is to connect each node to its $k$ nearest neighbors only, where $k$ is a hyperparameter, presenting two main advantages: (i) it reduces the computational burden since it considers only $k$ operations per node instead of $N$, and (ii) it enhances the influence of the neighborhood of the node, avoiding the effect of uncorrelated samples to the local process. Figure~\ref{f.sequentialGraph}(a) depicts the idea.
This paper proposes a novel approach for modeling the nodes' connectivity considering temporal information propagation instead of the distance in the feature space. The strategy conducts the positional encoding of the instances by connecting each vertex to its $k$ previous nodes, e.g., frames in a video sequence. Figure~\ref{f.sequentialGraph}(b) illustrates the process.
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{standardGraph-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{priorGraph-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Node neighborhood modeling considering (a) the standard feature-space-based $k$-nearest neighbors approach and (b) the proposed $k$ prior frames with $k=2$.}
\label{f.sequentialGraph}
\end{figure*}
Notice that the edges between each pair of connected nodes are weighted accordingly to their distances in this temporal representation, i.e., the first prior frame of a node is more strongly connected to it than the second prior, and so on consecutively. The edge weight $w_{ij}$ connecting a node $i$ to a previous $j$ is computed as follows:
\begin{equation}
w_{ij}=k+1 - d_{ij},
\label{e.weight_distribution}
\end{equation}
where $d_{ij}$ is the distance from $i$ to $j$ in prior frames steps, i.e., $d_{ij}=1$ means $j$ is the first prior frame of $i$, while $d_{ij}=2$ means $j$ is two prior frames away from $i$, and so on. Notice each node is also connected to itself through a self-reference edge $w_{ii} = k+1$ since $d_{ii}=0$. Moreover, the experiments also consider weighting self-reference connections $w_{ii} = 1$, increasing the influence of neighborhood in the GNN decision process. After defining the distance between each pair of connected samples, such values are stored in a distance matrix employed to compute nodes' normalized positional encoding, replacing the adjacency matrix in the factorization of the graph Laplacian.
In a nutshell, the multimodal approach and the positional encoding complement each other with specific helpful information. The multimodal architecture aims to maximize the canonical correlation between audio and visual channels, whose objective is amplifying the flow of correlated information associated with the speech and suppressing uncorrelated information related to noise. In other words, it focuses on the synchronicity of the sounds and the mouth movements. On the other hand, positional encoding aims to introduce a temporal dependence in this information since meaning in speech is associated with prior sounds, words, and sentences.
\section{Methodology}
\label{s.methodology}
This section describes the dataset considered for the task of audio/visual correlated embedding learning for the task of clean sound reconstruction and the process employed for feature extraction. Further, it also exposes the evaluation metrics considered in the experiments.
\subsection{AV ChiME3 Dataset}
\label{ss.dataset}
This paper employs a dataset composed of pairs of image and noisy audio signals for input and clean audio signals for output, aiming to provide an efficient tool capable of enhancing and cleaning the relevant audio signal considering environmental information fusion. The dataset comprises a combination of clean videos from the Grid~\cite{cooke2006audio} dataset with noises (pedestrian area, public transport, street junction, cafe) with signal to noise ratios (SNR) ranging from -12 to 12dB extracted from ChiME3~\cite{barker2015third}, composing the AV ChiME3~\cite{adeel2019lip} dataset. The preprocessing comprises sentence alignment, which is conducted to prevent the model from learning redundant or insignificant information and removing silent takes from data, as well as incorporating prior multiple visual frames used to include temporal data, thus improving the mapping between audio and visual characteristics. The dataset comprises $5$ speakers (one black male, two white males, and two white females) selected from Grid corpus reciting $989$ sentences each.
\subsubsection{Audio feature extraction}
\label{sss.audioDataset}
The audio features are extracted using log-FB vectors, which are computed by sampling the input audio signal at $22,050$kHz and segmented into $M$ $16$ms frames with $800$ samples per frame. Notice that each instance is sampled using a hamming window function~\cite{bojkovic2017hamming} with $62.5\%$ of the frame size ($500$ samples), therefore performing some frame overlapping during the procedure steps. Further, the Fourier transform is computed to produce a 2048-bin power spectrum. Finally, a logarithmic compression is applied to obtain a $22$-dimensional log-FB signal.
\subsubsection{Visual feature extraction}
\label{sss.visualDataset}
The visual features were extracted from the Grid Corpus dataset through a simple encoder-decoder setup approach. After extracting a sequence of individual frames, the lip-regions are identified using Viola-Jones~\cite{viola2001rapid} and tracked across a sequence of frames using a method proposed in~\cite{ross2008incremental}. The sentences are manually inspected using a random approach to ensure good lip tracking and delete sentences with misclassified lip regions~\cite{abel2016data}. Finally, the encoder-decoder approach is employed to produce vectors of pixel intensities, whose first $50$ components are vectorized in a zigzag order and then interpolated to match the equivalent audio sequence.
Finally, the dataset employed in this paper is composed of three subsets, i.e., clean audio, noisy audio, and visual features. Both clean and noisy audio subsets comprise $22$ features each, while the visual features are represented by a $50$-dimensional vector. The subsets contain $750$ out of $989$ sequences with $48$ frames each, summing up to a total of $36,000$ synchronized samples per subset. Figure~\ref{f.dataset} illustrates a simplified schema of the feature extraction process.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.6\textwidth]{datasets-eps-converted-to.pdf}
\end{tabular}}
\caption{Simplified feature extraction process schema.}
\label{f.dataset}
\end{figure}
\subsection{Evaluation Metrics}
\label{ss.metrics}
This section provides a brief description of the metrics employed to evaluate the experiments, i.e., Perceptual Evaluation of Speech Quality (PESQ), Mean Square Error (MSE), Area under the Curve, and energy efficiency.
\subsubsection{PESQ}
\label{sss.PESQ}
Perceptual Evaluation of Speech Quality (PESQ) comprises a set of mechanisms for automated speech quality evaluation~\cite{recommendation2001perceptual}. It was developed for objective voice quality testing by telecom operators, equipment vendors, and phone manufacturers.
Its testing topology depends on the available information, and the method is divided into two classes:
\begin{itemize}
\item Full Reference (FR): uses the original signal as a reference for comparison. This approach compares each reference sample to the corresponding noisy signal, thus delivering more accurate results.
\item No Reference (NR): uses only the noisy signal for quality estimation and has no information regarding the reference signal.
\end{itemize}
In a nutshell, PESQ is a full-reference algorithm that analyzes the speech signal sample-by-sample after a temporal alignment of corresponding excerpts of reference and test signal. This paper employs the FR version.
\subsubsection{MSE}
\label{sss.MSE}
The Mean Squared Error measures the average of the squares of the errors, i.e., the average squared difference between the estimated values and the actual value. The MSE formulation is described as follows:
\begin{equation}
MSE = \frac{1}{n}\sum_{i=1}^n(y_i-\hat{y}_i)^2,
\label{e.mse}
\end{equation}
where $n$ stands for the number of data points, $y$ is the expected value, and $\hat{y}$ is the predicted value.
\subsubsection{Area under the Curve}
\label{sss.AuC}
This work employs a discrete version of the area under the curve, which denotes the sum of the outputs of a given function $f(x)$ such that $\{x\in\mathbb{R}|a\leq x \leq b\}$, where $a$ and $b$ denote the lower and upper bounds, respectively. One can define the area under the curve as follows:
\begin{equation}
Area = \sum_{x=a}^bf(x).
\label{e.auc}
\end{equation}
\subsubsection{Energy Efficiency}
\label{sss.energyEfficiency}
This paper employs the term ``energy efficiency'' to describe the rate of active neurons during execution. This approach considers the idea that neurons that do not ``fire'' during the execution of the model are not performing any costly operation, thus saving energy. In a nutshell, the algorithm computes the sum of the activations whose output is greater than zero and divides it by the total number of activations.
\section{Experimental Setup}
\label{s.setup}
The experiments provided in the next section were conducted considering a graph neural network and a multilayer perceptron network as the backbone. Both networks share a similar architecture for comparison purposes, i.e., two hidden layers with $512$ neurons in each layer, using the Adam optimizer with a learning rate of $0.001$. The models are trained with the objective of maximizing the canonical correlation for coherent features extraction during $5,000$ epochs considering a trading-off parameter $\lambda=0.0001$ in Equation~\eqref{e.canonicalCorrelation}, while Equation~\eqref{e.finalLoss} is set with the values $\alpha=0.5$, $\beta=0.25$, and $\gamma=0.0625$. Notice that these values were empirically chosen using a grid search in the range $[0, 1]$ with step $0.0625 (1/16)$. These values make sense since they give more importance to the receptive input than to the context provided by the visual information and a combination of both, and the most relevant information to reconstruct the clean audio in this context is the noisy audio. The data augmentation step considers both an edge dropping rate and a feature masking rate of $0.5$. Moreover, the graphs are generated considering three distinct neighborhood scenarios, i.e., with $3$, $10$, and $30$ neighbors. Notice this neighborhood is defined in two manners, i.e., the standard feature-space-based approach and the proposed temporal-information-based relationship, as described in Section~\ref{ss.temporalRelationship}.
After training, the networks' outputs are used to feed a dense layer, which is responsible for reconstructing the clean signal given the features extracted from noisy audio for the single modality and noisy audio and clean video for the proposed multimodal extension. The dense layer is optimized during $600$ epochs using the Adam optimizer with a learning rate of $0.005$ and a weight decay of $0.0004$ using the minimization of the mean squared error as the objective function.
The dataset was divided into $15$ folds to provide an in-depth statistical analysis. Each fold comprises $50$ sequences of $48$ frames each, summing up to a total of 2,400 samples per fold. As stated in Section~\ref{ss.dataset}, the dataset is formed by three subsets: (i) clean audio, (ii) noisy audio, and (iii) clean visual. The noisy audio is used to train the standard unimodal approach, while the proposed multimodal approach employs both the noisy audio and clean visual in the training process. The clean audio is used as a reconstruction target for both cases. Finally, each fold is split into train, validation, and test sets, following the proportions of $60\%$, $20\%$, and $20\%$, respectively. For statistical evaluation, the Wilcoxon signed-rank test~\cite{Wilcoxon:45} with $5\%$ of significance was considered.
\section{Experiments}
\label{s.experiments}
This section presents the experimental results considering three tasks: (i) feature extraction driven by canonical correlation analysis maximization, (ii) clean audio data reconstruction based only on noisy audio or noisy audio and clean visual data fusion, and (iii) energy efficiency analysis in terms of neuron activation rate.
\subsection{Feature Extraction Analysis}
\label{ss.featureExtraction}
The experiments presented in this section compare the performance of graph neural networks with canonical correlation analysis for the task of self-supervised relevant feature extraction in three distinct neighborhood scenarios, i.e., $3$, $10$, and $30$ neighbors, namely GNN 3, GNN 10, and GNN 30, respectively. Moreover, two distinct neighborhood strategies are compared: (i) the standard (Std.) feature-space distance and (ii) the proposed prior frames-based connections for time-sequence (Seq.). Finally, a multilayer perceptron network with similar architecture (same number of layers, neurons per layer, optimizer, and learning rate) was trained using an identical self-supervised approach with the canonical correlation analysis maximization as the target function, denoted the baseline.
Figure~\ref{f.feature_extraction_audioOnly}(a) presents the convergence of the unimodal models considering the task of feature extraction through CCA maximization over noisy audio data. In this context, one can observe CCA-MLP obtained the highest results considering CCA maximization itself, even though such results do not necessarily imply better data representation for the specific task of clean audio reconstruction since the prior frames-based CCA-GNN (Seq. and Seq.* for $w_{ii} = k+1$ and $w_{ii} = 1$, respectively) obtained better reconstruction rates, as presented in Table~\ref{t.reconstruction_audioOnly}. Notice that CCA-GNN Seq.* obtained similar results to CCA-GNN Seq. and thus is overlapped in the image.
Regarding the multimodal extension conducted over noisy audio and visual features, one can observe in Figure~\ref{f.feature_extraction_audioOnly}(b) that the proposed CCA-GNN Seq. obtained the best results overall considering a neighborhood (number of prior frames) of $30$. Such a behavior is expected since a more significant number of prior frames reinforces the coherent information shared between the two channels during a longer period of time, providing more robust and connected features. Such robustness is in fact propagated to the reconstruction task since CCA-GNN Seq. with $k=30$ also provided the best results available in Table~\ref{t.reconstruction_audioOnly}.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{featureExtraction_noisyOnly-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{featureExtraction_AudioVisual-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Feature extraction for (a) noisy audio only and (b) noisy audio and clean visual.}
\label{f.feature_extraction_audioOnly}
\end{figure}
\subsection{Clean Signal Reconstruction}
\label{ss.reconstruction}
Table~\ref{t.reconstruction_audioOnly} presents the clean audio data reconstruction error considering the unimodal approach given the noisy data signal as input. Notice Standard denotes the common approach using the feature space to represent the nodes adjacency, while Sequential and Sequential* stand for the proposed prior frame-based approach for the positional encoding using $w_{ii} = k+1$ and $w_{ii} = 1$, respectively. From these results, one can observe that (i) a more significant number of nodes' neighbors lead to better reconstruction errors; (ii) the proposed sequential approach with prior frame connections outperformed the standard feature space distance-based neighborhood modeling and the MLP, and (iii) Sequential and Sequential* did not present significative differences, showing that the influence of the node self-connection is not relevant for the task. Notice that the better results according to the Wilcoxon signed-rank test are presented in bold. In this context, none of the other approaches obtained statistically similar results to the proposed sequential CCA-GNN with $k=30$. Further, Figure~\ref{f.reconstruction_audioOnly} shows that all techniques present similar convergence considering the reconstruction task, even though the MLP presents a slightly slower convergence, as depicted in the zoomed frame.
\begin{table}[!htb]
\caption{Average Mean Squared Error and standard deviation over unimodal CCA-MLP and CCA-GNN considering clean audio reconstruction given noisy audio input.}
\begin{center}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $0.0206\pm0.0012$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $0.0238\pm0.0009$ & $0.0220\pm0.0025$& $0.0220\pm0.0025$ \\
& 10 & $0.0235\pm0.0008$ & $0.0218\pm0.0025$& $0.0218\pm0.0025$ \\
& 30 & $0.0234\pm0.0008$ & \bm{$0.0187\pm0.0026$}& \bm{$0.0186\pm0.0025$}\\
\bottomrule
\end{tabular}}
\label{t.reconstruction_audioOnly}
\end{center}
\end{table}
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.60\textwidth]{testReconstruction_noisyOnly-eps-converted-to.pdf}
\end{tabular}}
\caption{Clean audio reconstruction error convergence considering the unimodal architecture based on the noisy audio.}
\label{f.reconstruction_audioOnly}
\end{figure}
Figure~\ref{f.signal_reconstruction_audio_only} depicts a randomly selected clean audio sample reconstruction regarding the unimodal architecture trained over noisy audio data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. Regarding the standard approach, one can notice in Figure~\ref{f.signal_reconstruction_audio_only}(a) that none of the models performed significantly well, especially for the first $8$ features. On the other hand, the proposed approach for frame-based positional encoding obtained much better results in this context considering the sequential CCA-GNN with $k=30$, as illustrated in Figure~\ref{f.signal_reconstruction_audio_only}(b).
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{reconstruction_audio_shuffled_17-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{reconstruction_audio_sequential_17-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Clean audio signal reconstruction regarding the unimodal architecture trained over noisy audio data considering the (a) standard approach and (b) the proposed prior frame-based positional encoding for time sequence modeling.}
\label{f.signal_reconstruction_audio_only}
\end{figure}
Table~\ref{t.reconstruction_audioVisual} present very similar results in the context of the multimodal architecture considering the noisy data and clean video signal as input, in which the proposed prior frame-based neighborhood outperformed the standard feature distance node's connection and the CCA-GNN with $k=30$ obtaining the better results overall considering the Wilcoxon signed-rank test. This table also shows that the multimodal approach is, in fact, capable of providing more representative features for clean audio reconstruction since all methods outperformed the respective unimodal versions. Figure~\ref{f.reconstruction_audioVisual}, which depicts the reconstruction error convergence considering the multimodal architectures, also presents very similar results to Figure~\ref{f.reconstruction_audioOnly}, in which all techniques perform similarly, with CCA-MLP showing a slower convergence in the first $50$ iterations.
\begin{table}[!htb]
\caption{Average Mean Squared Error and standard deviation over unimodal CCA-MLP and CCA-GNN considering clean audio reconstruction given noisy audio and visual inputs.}
\begin{center}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $0.0189\pm0.0009$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $0.0238\pm0.0009$ & $0.0220\pm0.0025$& $0.0220\pm0.0024$ \\
& 10 & $0.0233\pm0.0009$ & $0.0216\pm0.0025$& $0.0216\pm0.0026$ \\
& 30 & $0.0225\pm0.0009$ & \bm{$0.0179\pm0.0025$} & \bm{$0.0180\pm0.0026$}\\
\bottomrule
\end{tabular}}
\label{t.reconstruction_audioVisual}
\end{center}
\end{table}
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.55\textwidth]{testReconstruction_AudioVisual-eps-converted-to.pdf}
\end{tabular}}
\caption{Clean audio reconstruction error convergence considering the proposed multimodal architecture based on the noisy audio and the clean visual data.}
\label{f.reconstruction_audioVisual}
\end{figure}
Similar to Figure~\ref{f.signal_reconstruction_audio_only}, Figure~\ref{f.signal_reconstruction_audio_visual} presents a randomly selected clean audio sample reconstruction regarding the multimodal architecture trained over noisy audio and clean visual data. Again, Figure~\ref{f.signal_reconstruction_audio_visual}(a) shows the standard approach did not performed significantly well for the first five or eight features, while the proposed approach with $k=30$ provided much more accurate results, as depicted in Figure~\ref{f.signal_reconstruction_audio_visual}(b).
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{reconstruction_audio_visual_shuffled_17-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{reconstruction_audio_visual_sequential_17-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Clean audio signal reconstruction regarding the multimodal architecture trained over noisy audio and clean visual data considering the (a) standard approach and (b) the proposed prior frame-based positional encoding for time sequence modeling.}
\label{f.signal_reconstruction_audio_visual}
\end{figure}
\textcolor{white}{}
\subsection{Neuronal Activation Analysis}
\label{ss.neuronActivation}
This section provides an energy-efficiency analysis based on neuronal activation behavior. Such an analysis is fundamental since using such models in real-world applications, such as hearing devices embedded systems, for instance, is constrained to energy issues. It is to be noted that here the energy-efficiency is referred to the number of neurons with zero activity. In hardware (e.g., FPGA) zero signal does not propagate in the network and contributes nothing to the dynamic power consumption because of no switching activity. Figure~\ref{f.activation_audioOnly} describes the average rate of the first hidden layer's neurons activation during the training of the unimodal approach, while Table~\ref{t.activation_audioOnly} provides the area under the curve. Notice that the better values are underlined and some less relevant results were omitted from the plot for better visualization. In this context, one can observe that, even though GNNs and the MLP comprise the same architecture, i.e., two hidden layers with $512$ neurons each, around $100$ neurons in the MLP architecture fired $100\%$ of the time, while the same pattern is observed over a range between $10$ and $20$ neurons concerning the GNNs. Moreover, although the best performing model for the clean audio reconstruction task, i.e., the sequential CCA-GNN with $k=30$, is the most energy-consuming approach between all the CCA-GNN-based methods, it still presents a nominal neuron activation rate if compared to the MLP.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.55\textwidth]{activation_noisyOnly-eps-converted-to.pdf}
\end{tabular}}
\caption{Neuron activation rate considering the unimodal architecture.}
\label{f.activation_audioOnly}
\end{figure}
\begin{table}[!htb]
\caption{Area under the curve considering the unimodal architecture neuron activation rate.}
\begin{center}
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $147.27$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $\underline{10.17}$ & $11.73$& $11.73$ \\
& 10 & $21.64$ & $26.56$& $26.56$ \\
& 30 & $29.99$ & $47.71$& $47.71$\\
\bottomrule
\end{tabular}}
\label{t.activation_audioOnly}
\end{center}
\end{table}
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{activation_AudioVisual_audio-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{activation_AudioVisual_visual-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Neuron activation rate considering the multimodal architecture considering (a) audio and (b) visual channels.}
\label{f.activation_audioVisual}
\end{figure*}
Figures~\ref{f.activation_audioVisual}(a) and~\ref{f.activation_audioVisual}(b) present the neural activation analysis considering the multimodal approach for the audio and the visual channels, respectively, whose values are reflected in Table~\ref{t.activation_audioVisual}. The figures present a similar behavior to the one observed in Figure~\ref{f.activation_audioOnly}, in which the MLP presents a neuron activation rate substantially larger than the MLP-based approaches. Moreover, one can also observe that the prior frame-based approach has different behaviors for each channel since their neuron activation conduct is more intense in the audio channel but produces a reduced fire rate considering the visual channel. Such behavior may suggest that using the prior frame-based approach implies a network inclined to extract more relevant features considering the temporal flow as a context, thus becoming less dependant on the visual context itself.
\begin{table}[!htb]
\caption{Area under the curve considering the multimodal architecture neuron activation rate.}
\begin{center}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{ccccccccc}
\toprule
&&\multicolumn{3}{c}{\textbf{Audio}}& & \multicolumn{3}{c}{\textbf{Visual}}\\
\cmidrule{3-5} \cmidrule{7-9}
Model & Neighbors & Standard & Sequential & Sequential* & & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $142.07$ & - & - & & $203.01$ & -& -\\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $\underline{11.77}$ & $14.88$& $14.88$ & & $19.05$ & $\underline{17.36}$& $\underline{17.36}$\\
& 10 & $19.29$ & $24.01$& $24.01$ & & $38.52$ & $32.28$& $32.28$ \\
& 30 & $27.09$ & $45.80$& $45.80$ & & $55.85$ & $36.66$& $36.66$\\
\bottomrule
\end{tabular}}
\label{t.activation_audioVisual}
\end{center}
\end{table}
\subsection{Speech Enhancement Framework and Results}
For speech enhancement, state-of-the-art enhanced visually derived Wiener filter (EVWF) is used as shown in Figure~\ref{f.VWF}~\cite{adeel2019lip}. The EVWF uses an audio or AV driven regression model to estimate clean audio features using noisy audio or AV temporal features. The low dimensional estimated clean audio features are transformed back into high dimensional clean audio power spectrum using inverse filter bank transformation to calculate Wiener filter. The Wiener filter is applied to the magnitude spectrum of the noisy input audio signal, followed by the inverse fast Fourier transform, overlap, and combining processes to produce enhanced magnitude spectrum. More details are comprehensively presented in our previous work \cite{adeel2019lip}\cite{adeel2020contextual}\cite{adeel2020novel}. For objective testing and comparison, perceptual evaluation of speech quality (PESQ) is used as shown in Tables~\ref{t.PESQ_audio} and~\ref{t.PESQ_audioVisual} for audio only and audio-visual data, respectively. PESQ is one of the most reliable methods to evaluate the quality of restored speech. The PESQ score is computed as a linear combination of the average disturbance value and the average asymmetrical disturbance values. Scores range from $-0.5$ to $4.5$, corresponding with low to high speech quality. It can be seen that at low SNR levels (-6dB), EVWF with AV-GNN30* performs comparably to the state-of-the-art LSTM model and significantly outperforms spectral subtraction (SS) and Log-Minimum Mean Square Error (LMMSE), and MLP based speech enhancement methods using far few neurons.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.7\textwidth]{VWFnew.png}
\end{tabular}}
\caption{Enhanced visually-derived Wiener filtering. Lip reading refers to audio-visual clean log-FB features estimation.}
\label{f.VWF}
\end{figure}
\begin{table}[!htb]
\caption{Speech Enhancement Results: Perceptual evaluation of speech quality (PESQ) for the unimodal architecture trained over
noisy audio data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. It is to be noted that only GNN30 and GNN30* with prior frame-based positional encoding perform reasonably well compared to all other models.}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{6pt}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\cellcolor[HTML]{EFEFEF}&\multicolumn{4}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Standard}}&\multicolumn{5}{c}{\cellcolor[HTML]{EFEFEF}\textbf{Prior frame-based}}\\\hhline{~---------}
\multirow{-2}{*}{\cellcolor[HTML]{EFEFEF}{\textbf{SNR}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30*}}} \\ \hline
-12dB & 0.83 & 0.86& 0.80 & 0.88 & 0.80 & 0.91& 0.90 & 1.24 & 1.27\\ \hline
-6dB & 0.81 & 0.84& 0.85 & 0.87 & 0.82 & 0.83& 0.99 & 1.26 & 1.28\\ \hline
-3dB & 0.82 & 0.88& 0.89 & 0.94 & 0.86 & 0.87& 0.85 & 1.26 & 1.30\\ \hline
0dB & 0.85 & 0.85& 0.87& 0.88 & 0.90 & 0.96& 0.88 & 1.33 & 1.34\\ \hline
3dB & 0.88 & 0.82& 0.91 & 0.86 & 0.86 & 0.92&0.92 & 1.36 & 1.38\\ \hline
6dB & 0.84 & 0.90& 0.88 & 0.82 & 0.87 & 0.84& 0.89 & 1.40 & 1.42\\ \hline
12dB & 0.91 & 0.89 & 0.92 & 0.95 & 0.82 & 0.87& 0.81 & 1.48 & 1.55\\ \hline\hline
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\end{tabular}}
\label{t.PESQ_audio}
\end{center}
\end{table}
\begin{table}[!htb]
\caption{Speech Enhancement Results: Perceptual evaluation of speech quality (PESQ) for the multimodal architecture trained over
noisy audio and clean visual data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. Here only GNN30 without prior frame-based positional encoding and GNN30 and GNN30* with prior frame-based positional encoding perform reasonably well compared to all other models. These models also perform slightly better than audio-only models.}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{6pt}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\cellcolor[HTML]{EFEFEF}&\multicolumn{4}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Standard}}&\multicolumn{5}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Prior frame-based}}&\multicolumn{2}{c}{\cellcolor[HTML]{EFEFEF}\textbf{Obtained from~\cite{adeel2018real}}}\\\hhline{~-----------}
\multirow{-2}{*}{\cellcolor[HTML]{EFEFEF}{\textbf{SNR}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30*}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{SS}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{LMMSE}}} \\ \hline
-12dB & 0.84 & 0.88& 0.87 & 1.10 & 0.86 & 0.94& 0.80 & 1.25 & 1.29 & 0.9& 0.95\\ \hline
-6dB & 0.88 & 0.80& 0.86 & 1.15 & 0.91 & 0.80& 0.89 & 1.27 & 1.29& 1.01 &1.03\\ \hline
-3dB & 0.90 & 0.87& 0.89 & 1.15 & 0.83 & 0.86& 0.88 & 1.28 & 1.31 & 1.17 &1.18\\ \hline
0dB & 0.82 & 0.86& 0.81& 1.18 & 0.80 & 0.84& 0.84 & 1.34 & 1.36 & 1.21& 1.20\\ \hline
3dB & 0.86 & 0.89& 0.93 & 1.20 & 0.92& 0.88&0.88 & 1.40 & 1.42 & 1.25 &1.34\\ \hline
6dB & 0.89 & 0.87& 0.88 & 1.22 & 0.87 & 0.83& 0.81 & 1.44 & 1.48 & 1.26 & 1.39\\ \hline
12dB & 0.84 & 0.91 & 0.84 & 1.25 & 0.88 & 0.81& 0.89 & 1.51 & 1.60 &1.54 &1.60\\
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\end{tabular}}
\label{t.PESQ_audioVisual}
\end{center}
\end{table}
\section{Conclusions}
\label{s.conclusions}
This paper proposes an energy-efficient approach for improving and boosting sound signals through environmental information fusion. The model extends Graph Neural Networks with Canonical Correlation Analysis for multimodal data integration and further incorporates a prior frame-based node's positional encoding that considers the temporal sequence in data to establish the information similarity instead of the usual feature space distance. Experiments conducted over the AV Grid and ChiME3 corpora considering the task of clean audio reconstruction based on the fusion of noisy audio and clean video data show that the proposed approaches are capable of outperforming a baseline composed of an MLP/ LSTM with similar architecture trained under the same conditions, i.e., in a self-supervised fashion using canonical correlation analysis maximization as the target function. Finally, the proposed approach provides a considerable gain regarding energy efficiency, given that the CCA-GNN neuron firing rates are dramatically lower than MLP and LSTM. It is worth mentioning that it is impossible to quantify the energy saving at this stage, but the ongoing work includes implementing these models on FPGA in which neurons with zero activity will not propagate in the network and will therefore contribute nothing to the dynamic power consumption.
The experiments also indicate that the multimodal approach can produce better results than the unimodal architecture, leading to minor reconstruction errors. Such behavior is expected since the visual data acts as a context to introduce some additional meaning and improve the noisy audio signal. Additionally, the prior frame-based approach provided better results than the standard model, showing the importance of the temporal information as an additional context to the noisy signal. Finally, one could notice by the neuron's firing behavior that using prior frame node connection reinforces the information present in the noisy audio channel, making it less dependant on the visual data context for clean audio reconstruction. Future work involves developing a more biologically realistic neuronal model, introducing a concept of memory to improve communication mechanisms between the channels. Further, Graph Neural Networks with Canonical Correlation Analysis will be used to improve cross channels communication blocks within convolutional neural networks.
\section*{Acknowledgments}
\begin{sloppypar}
This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant Ref. EP/T021063/1. J. Del Ser would like to thank the Spanish Centro para el Desarrollo Tecnologico Industrial (CDTI, Ministry of Science and Innovation) through the ``Red Cervera'' Programme (AI4ES project), as well as by the Basque Government through EMAITEK and ELKARTEK (ref. 3KIA) funding grants and the Consolidated Research Group MATHMODE (ref. IT1456-22).
\end{sloppypar}
\section*{References}
\section{Introduction}
\label{sec:introduction}
Recent technological advances have empowered computers to reason and perform activities once attributed to human beings only, such as writing, speaking, and even helping with decisions like the best route on a trip or the most appropriate drugs for treating a specific disease. Nevertheless, such reasoning is limited to the domain upon which the algorithm is trained, i.e., the actions and decisions adopted by the algorithm are based on patterns somehow encoded in the dataset. This approach seems unnatural if considering the learning processes performed by the biological brain, in which stimuli are provided by a set of different sensors, e.g., vision and hearing, and this multimodal information is combined in such a way that redundant information is essential to reinforcing and improving noisy, ambiguous, and imperfect signals from distinct sources.
Multimodal learning approaches are be beneficial to improve one modality feature representation~\cite{singh2022watmif,iqbal2022ff}. To illustrate the idea, consider, for instance, a boisterous speech. If the speaker only resorted his voice to convey its message, the comprehension of the subject may be considerably impaired by the noise. On the other hand, if the images are also available, it is possible to try to complement the corrupted information with insights provided by this secondary source, such as following the speaker's hands and body movements or trying to read his lips.
Despite the availability of contextual information provided by different input signals, usually such tasks also rely on temporal information for reasoning. Revisiting the speech example, it is easy to infer that words being said at the present moment are probably strongly correlated with the last few pronounced words. Even though some works addressed the problem using recurrent networks~\cite{mai2021analyzing,li2020multimodal}, most of them perform target-driven supervised learning, which usually requires a considerable number of labeled samples for training.
In this context, unsupervised or self-supervised algorithms show themselves capable of extracting strongly correlated features, which are highly desired for two main reasons: (i) their representations are usually more general-purpose than target-driven features extracted with supervised algorithms, and (ii) they do not require labeled instances for training, which are usually limited and costly. Regarding the proposed approaches for correlated feature extraction, one can refer to energy- and mutual information-based methods, such as the Deep Graph Infomax~\cite{velivckovic2018deep}, which relies on mutual information maximization and graph neural networks (GNN) for leveraging information propagation in a graph.
With the advent of deep learning, GNNs emerged as an elegant solution to extract in-depth dependencies from such intricate relationships. Moreover, it also presents itself as a powerful alternative to convolutional neural networks suitable for datasets composed of non-imagery data. Despite the success obtained by~\cite{velivckovic2018deep}, Zhang et al.~\cite{zhang2021canonical} pointed towards a set of drawbacks in the model: (i) reliance on negative samples - corrupting the graph structure by selecting arbitrary negative examples may lead to large variance for stochastic gradients and slow training convergence; (ii) require a parameterized estimator to approximate mutual information between two views; and (iii) it contrasts node embeddings with graph embedding leading to a higher complexity. To tackle such problems, they propose the GNNs with Canonical Correlation Analysis (CCA-GNN), which aims at maximizing the correlation between two augmented views of the same input while decorrelating different dimensions of such views. Besides, Dwivedi et al.~\cite{dwivedi2021graph} exposed another shortcoming regarding GNNs message-passing mechanism, which builds node representation by aggregating feature space-based local neighborhood information and leads representations dependants on the local structure of the graph and proposed using positional encoding to solve the problem.
Therefore, this paper aims to redesign the GNNs with Canonical Correlation Analysis (CCA-GNN) to deal with the challenging context of multimodal representation learning. Specifically, it formulates a parallel CCA-GNN architecture for each input channel, i.e., audio and visual. The new audio-visual (AV) CCA-GNN model minimizes both the canonical correlation between the augmented samples of the same channel as well as between the augmented samples of the other mode. Additionally, it introduces graph modeling that considers a time-frame sequence distance positional encoding to compute the node's neighborhood. The idea is to introduce temporal information through the samples' connectivity in the embeddings. Figure~\ref{f.intro} depicts the whole pipeline.
\begin{figure*}[!ht]
\centerline{
\begin{tabular}{c}
\includegraphics[width=\textwidth]{intro-eps-converted-to.pdf}
\end{tabular}}
\caption{Proposed pipeline. Features are extracted from the noisy audio input using Fourier transformation followed by logarithmic compression, while the visual inputs employ an encoder-decoder approach followed by Viola-Jones~\cite{viola2001rapid} for lip-regions identification. Both sets are converted into graphs using the proposed prior-frame-based positional encoding to maintain the temporal information. Such graphs are used to generate two augmented versions per epoch, which feed a set of graph convolutions. The model minimizes canonical correlation analysis of convolutions' output, aiming to generate a set of correlated features in a self-supervised fashion. Such features are finally used to feed a supervised dense layer responsible for reconstructing the clean audio signal.}
\label{f.intro}
\end{figure*}
Experiments conducted over the AV ChiME3 dataset compare the proposed approach against a CCA-based multilayer perceptron (MLP). Results show that (i) the multimodal CCA-GNNs produce more representative features than the standard unimodal version, leading to lower errors over clean audio data reconstruction; (ii) the proposed prior-frame approach for sequential-time modeling in graphs outperform the standard feature-space distance-based neighborhood connections; and (iii) CCA-GNNs deliver better results than the CCA-MLP model in the context of feature extraction for data reconstruction, requiring a considerably reduced rate of firing neurons, indicating the new model is more suitable for energy-constrained environments, such as AV hearing aid devices. To the best of our knowledge, this study is the first to demonstrate the application of GNN-CCA for energy-efficient rich real-world multi-modal data for a benchmark AV speech enhancement problem, where multiple real-world noises corrupt speech in real-world conditions. Therefore, the main contributions of this paper are threefold:
\begin{enumerate}
\item Demonstrate the application of a novel self-supervised energy-efficient CCA-GNN-based model that considers the fusion between different sources of information from the environment to improve sound quality.
\item A novel GNN-CCA model with integrated positional node encoding considering a prior-frame sequence distance instead of a feature-space representation when computing the node’s nearest neighbors, introducing temporal information in the embeddings through the neighborhood’s connectivity.
\item The proposed method is evaluated with the benchmark AV Grid and ChiME3 corpora, with $4$ different real-world noise types (cafe, street junction, public transport (BUS), pedestrian area) and compared with standard GNNs and MLP models for unsupervised AV speech processing tasks. Comparative results show that our new method demonstrates superior energy consumption and generalization performance in all experimental conditions. Other comparative models fail to reconstruct speech-in-noise with a similar number of neurons.
\end{enumerate}
The remainder of this paper is described as follows. Section~\ref{s.proposed} provides a brief background regarding GNN-CCA and introduces the proposed approaches. Further, Section~\ref{s.methodology} provides the necessary information regarding the dataset and the evaluation metrics, while Section~\ref{s.setup} describes the experimental setup. Finally, Sections~\ref{s.experiments} and~\ref{s.conclusions} state the results and conclusions, respectively.
\section{Related Work}
\label{sec:related}
Ngiam et al.~\cite{ngiam2011multimodal} showed that multimodal learning approaches could be beneficial to improve one modality feature representation. The approach was recently applied to a wide variety of applications, such as emotion analysis~\cite{jia2021multimodal}, scene change detection~\cite{santana2019novel} and medicine~\cite{venugopalan2021multimodal}, to cite a few. Regarding audio-visual (AV) data processing, Adeel et al.~\cite{adeel2018real} suggested an integration of Internet of Things (IoT) and 5G Cloud-Radio Access Network to create a chaotic encryption-based lightweight model for lip-reading driven hearing aids. In further work~\cite{adeel2020novel} the model was improved to transmit encrypted compressed audio-visual (AV) information and receive encrypted enhanced reconstructed speech in real-time. Recent works comprise a deep learning-based framework for speech enhancement that exploits AV cues concerning different operating conditions to estimate clean audio~\cite{adeel2020contextual}, as well as the CochleaNet~\cite{gogate2020cochleanet}, which integrates noisy audio and visuals from distinct language speakers.
Regarding the proposed approaches for correlated feature extraction using energy-based approaches, one can refer to~\cite{hinton2002training,passos2017fine,passos2019kappa,passos2018temperature}. In the context of mutual information, Belghazi et al.~\cite{belghazi2018mine} proposed the so-called MINE, a mutual information neural estimation, while Hjelm et al.~\cite{hjelm2018learning} proposed a model for Learning deep representations by mutual information estimation and maximization. Further, Veli{\v{c}}kovi{\'c} et al.~\cite{velivckovic2018deep} proposed a graph-based solution called Deep Graph Infomax, which combines on mutual information maximization with graph neural networks.
Graph theory describes strong architectures capable of modelling complex relationships, with applications ranging from small world design~\cite{newman2000models} to oversampling~\cite{passos20202pf,passos2022handling}. Considering GNNs applications for correlated feature extraction, aside from~\cite{velivckovic2018deep}, Zhang et al.~\cite{zhang2021canonical} introduced canonical correlation analysis for self-supervised feature extraction using GNNs, providing a more efficient alternative for the task since it does not rely on negative pairs, does not require learning parameters of additional components such as an estimator. Further, the complexity of the model is considerably smaller since it requires O($F^2$) space cost against O($N$) from the Deep Graph Infomax, where $F$ denotes the feature space size and $N$ denotes the number of nodes.
\section{Graph Neural Network with Canonical Correlation Analysis}
\label{s.CCAGNN}
Consider a single graph $G = (X, A)$ where $X\in \mathbb{R}^{N\times F}$ denotes the node's feature vectors and $A \in \mathbb{R}^{N \times N}$ stands for the adjacency matrix. The CCA-GNN~\cite{zhang2021canonical} is composed of three main parts: (i) a random graph generator $T$, (ii) a graph neural network encoder $f_\theta$, where $\theta$ stands for the learnable parameters, and (iii) a Canonical Correlation Analysis-based objective function. The idea is to present two augmented versions of the same graph to the network and maximize the canonical correlation between their outputs. Such an approach aims at preserving correlated components while discarding decorrelated ones, i.e., maintaining the relevant information present in both augmented versions and avoiding particular behaviors, such as anomalies and noise. Figure~\ref{f.CCAGNN} depicts the Graph Neural Network with Canonical Correlation Analysis.
\begin{figure*}[!ht]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.9\textwidth]{graph_canonical_correlation-eps-converted-to.pdf}
\end{tabular}}
\caption{Graph Neural Network with Canonical Correlation Analysis. The dataset is converted into a graph where each sample represents a node, and the edges denote the nodes' connection. Two augmented versions of this graph are generated and used to feed a GNN. Finally, the canonical correlation between the output of both versions is computed and used as the cost function to optimize the GNN parameters.}
\label{f.CCAGNN}
\end{figure*}
Regarding the graph augmentation, CCA-GNN employs the same approach used in~\cite{zhu2020deep,thakoor2021bootstrapped}, which basically performs a random edge dropping and feature masking. Thus, each $t\sim T$ stands for a transformed version of $G$. Notice that those augmented versions are sampled at each iteration.
Concerning the encoder, the model employs a simple two-layered graph neural network, which can be easily replaced by more complex or sophisticated architectures.
Finally, the objective function aims at modeling the learning problem as a canonical correlation-based~\cite{chang2018scalable} self-supervised approach in which the two randomly augmented versions of the graph yields two normalized views of the input data, $\bm{Z}_A$ and $\bm{Z}_B$, and their correlation is maximized. The objective function is described as follows:
\begin{equation}
{\cal L}(\bm{Z}_A, \bm{Z}_B) = ||\bm{Z}_A - \bm{Z}_B||_{F}^2+\lambda\left(||\bm{Z}_A^T\bm{Z}_A-\bm{I}||_F^2+||\bm{Z}_B^T\bm{Z}_B-\bm{I}||_F^2\right),
\label{e.canonicalCorrelation}
\end{equation}
where $I$ is the identity matrix and $\lambda$ is a non-negative trading-off hyperparameter. The first term, namely the invariance term, is responsible for the minimization of the invariance between the two views, which is essentially the same as maximizing the correlation between them. The second is the decorrelation term, which seeks a regularization that encourages distinct features to capture different semantics.
Further, the authors provided a variance-covariance perspective~\cite{tian2021understanding} of the objective function. Suppose $\bm{s}$ as an augmented version sampled from an input $\bm{x}$, and $\bm{z_s}$ is the representation of $\bm{s}$ obtained through a decoder. The invariance term can be minimized using expectation as follows:
\begin{eqnarray}
{\cal L}_{inv} &=& ||\bm{Z}_A - \bm{Z}_B||_{F}^2= \sum_{i=1}^N\sum_{k=1}^D(z_{i,j}^A-z_{i,j}^B)^2\nonumber\\
&\cong& \mathbb{E}_{\bm{x}}\left[\sum_{k=1}^D\mathbb{V}_{\bm{s}|\bm{x}}[\bm{z_s},k]\right]*2N,
\label{e.loss_inv}
\end{eqnarray}
where $\mathbb{V}$ is the variance. In a similar fashion, the decorrelation term is written as follows:
\begin{eqnarray}
{\cal L}_{dec} &=& ||\bm{Z}_S^T\bm{Z}_S-\bm{I}||_F^2= ||\bm{Cov}_{\bm{s}}[\bm{z}]-I||_F^2\nonumber\\
&\cong& \sum_{i\neq j}(\rho_{i,j}^{\bm{z_s}})^2,\forall \bm{Z}_S\in\{\bm{Z}_A,\bm{Z}_B\},
\label{e.loss_dec}
\end{eqnarray}
where $\bm{Cov}$ is the covariance matrix and $\rho$ is the Pearson correlation coefficient.
\section{Proposed Approach}
\label{s.proposed}
This section presents a multimodal extension for the Canonical Correlation Analysis Graph Neural Network. Further, it also introduces the idea of modeling the temporal information of sequence data as node relationships in a graph.
\subsection{Multimodal Canonical Correlation Analysis Graph Neural Network for Audio-Visual Embedding Learning}
\label{ss.CCAGNN_av}
The proposed extension of the Canonical Correlation Analysis Graph Neural Network for multimodal data comprises a pair of networks, each of them fed with a modality of data, e.g., audio and visual, running in parallel. At the output layer, the canonical correlation analysis is performed considering both the intra-channel correlation, i.e., the two augmented versions of the same data modality, as well as inter-channels correlation. Figure~\ref{f.CCAGNN_av} depicts the model.
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=\textwidth]{graph_nets_multimodal-eps-converted-to.pdf}
\end{tabular}}
\caption{Multimodal Canonical Correlation Analysis Graph Neural Network for Audio-Visual Embedding Learning.}
\label{f.CCAGNN_av}
\end{figure*}
To accommodate the intra- and inter-channel computations of the canonical correlation analysis in the objective function, one can firstly consider two randomly augmented versions of normalized views for each channel, namely $\bm{Z}_1$ and $\bm{Z}_2$ for audio data, and $\bm{Z}_3$ and $\bm{Z}_4$ for visual data. The individual losses for both channels are computed using Equation~\eqref{e.canonicalCorrelation}:
\begin{equation}
\begin{array}{c}
{\cal L}_{\text{Audio}} = {\cal L}(\bm{Z}_1, \bm{Z}_2),\\
{\cal L}_{\text{Visual}} = {\cal L}(\bm{Z}_3, \bm{Z}_4).
\end{array}
\label{e.AudioVisual_alternative}
\end{equation}
Further, all the possible combinations of audio and visual data are computed, namely \textit{Audio1Visual1} ({$Z_1,Z_3$}), \textit{Audio1Visual2} ({$Z_1,Z_4$}), \textit{Audio2Visual1} ({$Z_2,Z_3$}), and \textit{Audio2Visual2} ({$Z_2,Z_4$}), as follows:
\begin{equation}
\begin{array}{c}
{\cal L}_{\text{Audio1Visual1}} = {\cal L}(\bm{Z}_1, \bm{Z}_3),\\
{\cal L}_{\text{Audio1Visual2}} = {\cal L}(\bm{Z}_1, \bm{Z}_4),\\
{\cal L}_{\text{Audio2Visual1}} = {\cal L}(\bm{Z}_2, \bm{Z}_3),\\
{\cal L}_{\text{Audio2Visual2}} = {\cal L}(\bm{Z}_2, \bm{Z}_4).
\end{array}
\label{e.combinedLosses}
\end{equation}
Finally, the objective function of the multimodal Canonical Correlation Analysis Graph Neural Network is given by:
\begin{eqnarray}
{\cal L} &=& \alpha{\cal L}_{\text{Audio}} + \beta{\cal L}_{\text{Visual}}\nonumber\\
&& + \gamma({\cal L}_{\text{Audio1Visual1}}+{\cal L}_{\text{Audio1Visual2}}\nonumber\\
&& +{\cal L}_{\text{Audio2Visual1}}+{\cal L}_{\text{Audio2Visual2}}),
\label{e.finalLoss}
\end{eqnarray}
where $\alpha$, $\beta$, and $\gamma$ are constants that control the influence of audio, video, and the combined canonical correlation, respectively.
\subsection{Modelling Temporal Information as Graph Nodes Relationships}
\label{ss.temporalRelationship}
The usual approach for modeling a dataset into a graph structure consists of representing its samples as the graph's nodes, whose edges connect the adjacent instances inserted into a $D$-dimensional feature space. A common approach is to connect each node to its $k$ nearest neighbors only, where $k$ is a hyperparameter, presenting two main advantages: (i) it reduces the computational burden since it considers only $k$ operations per node instead of $N$, and (ii) it enhances the influence of the neighborhood of the node, avoiding the effect of uncorrelated samples to the local process. Figure~\ref{f.sequentialGraph}(a) depicts the idea.
This paper proposes a novel approach for modeling the nodes' connectivity considering temporal information propagation instead of the distance in the feature space. The strategy conducts the positional encoding of the instances by connecting each vertex to its $k$ previous nodes, e.g., frames in a video sequence. Figure~\ref{f.sequentialGraph}(b) illustrates the process.
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{standardGraph-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{priorGraph-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Node neighborhood modeling considering (a) the standard feature-space-based $k$-nearest neighbors approach and (b) the proposed $k$ prior frames with $k=2$.}
\label{f.sequentialGraph}
\end{figure*}
Notice that the edges between each pair of connected nodes are weighted accordingly to their distances in this temporal representation, i.e., the first prior frame of a node is more strongly connected to it than the second prior, and so on consecutively. The edge weight $w_{ij}$ connecting a node $i$ to a previous $j$ is computed as follows:
\begin{equation}
w_{ij}=k+1 - d_{ij},
\label{e.weight_distribution}
\end{equation}
where $d_{ij}$ is the distance from $i$ to $j$ in prior frames steps, i.e., $d_{ij}=1$ means $j$ is the first prior frame of $i$, while $d_{ij}=2$ means $j$ is two prior frames away from $i$, and so on. Notice each node is also connected to itself through a self-reference edge $w_{ii} = k+1$ since $d_{ii}=0$. Moreover, the experiments also consider weighting self-reference connections $w_{ii} = 1$, increasing the influence of neighborhood in the GNN decision process. After defining the distance between each pair of connected samples, such values are stored in a distance matrix employed to compute nodes' normalized positional encoding, replacing the adjacency matrix in the factorization of the graph Laplacian.
In a nutshell, the multimodal approach and the positional encoding complement each other with specific helpful information. The multimodal architecture aims to maximize the canonical correlation between audio and visual channels, whose objective is amplifying the flow of correlated information associated with the speech and suppressing uncorrelated information related to noise. In other words, it focuses on the synchronicity of the sounds and the mouth movements. On the other hand, positional encoding aims to introduce a temporal dependence in this information since meaning in speech is associated with prior sounds, words, and sentences.
\section{Methodology}
\label{s.methodology}
This section describes the dataset considered for the task of audio/visual correlated embedding learning for the task of clean sound reconstruction and the process employed for feature extraction. Further, it also exposes the evaluation metrics considered in the experiments.
\subsection{AV ChiME3 Dataset}
\label{ss.dataset}
This paper employs a dataset composed of pairs of image and noisy audio signals for input and clean audio signals for output, aiming to provide an efficient tool capable of enhancing and cleaning the relevant audio signal considering environmental information fusion. The dataset comprises a combination of clean videos from the Grid~\cite{cooke2006audio} dataset with noises (pedestrian area, public transport, street junction, cafe) with signal to noise ratios (SNR) ranging from -12 to 12dB extracted from ChiME3~\cite{barker2015third}, composing the AV ChiME3~\cite{adeel2019lip} dataset. The preprocessing comprises sentence alignment, which is conducted to prevent the model from learning redundant or insignificant information and removing silent takes from data, as well as incorporating prior multiple visual frames used to include temporal data, thus improving the mapping between audio and visual characteristics. The dataset comprises $5$ speakers (one black male, two white males, and two white females) selected from Grid corpus reciting $989$ sentences each.
\subsubsection{Audio feature extraction}
\label{sss.audioDataset}
The audio features are extracted using log-FB vectors, which are computed by sampling the input audio signal at $22,050$kHz and segmented into $M$ $16$ms frames with $800$ samples per frame. Notice that each instance is sampled using a hamming window function~\cite{bojkovic2017hamming} with $62.5\%$ of the frame size ($500$ samples), therefore performing some frame overlapping during the procedure steps. Further, the Fourier transform is computed to produce a 2048-bin power spectrum. Finally, a logarithmic compression is applied to obtain a $22$-dimensional log-FB signal.
\subsubsection{Visual feature extraction}
\label{sss.visualDataset}
The visual features were extracted from the Grid Corpus dataset through a simple encoder-decoder setup approach. After extracting a sequence of individual frames, the lip-regions are identified using Viola-Jones~\cite{viola2001rapid} and tracked across a sequence of frames using a method proposed in~\cite{ross2008incremental}. The sentences are manually inspected using a random approach to ensure good lip tracking and delete sentences with misclassified lip regions~\cite{abel2016data}. Finally, the encoder-decoder approach is employed to produce vectors of pixel intensities, whose first $50$ components are vectorized in a zigzag order and then interpolated to match the equivalent audio sequence.
Finally, the dataset employed in this paper is composed of three subsets, i.e., clean audio, noisy audio, and visual features. Both clean and noisy audio subsets comprise $22$ features each, while the visual features are represented by a $50$-dimensional vector. The subsets contain $750$ out of $989$ sequences with $48$ frames each, summing up to a total of $36,000$ synchronized samples per subset. Figure~\ref{f.dataset} illustrates a simplified schema of the feature extraction process.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.6\textwidth]{datasets-eps-converted-to.pdf}
\end{tabular}}
\caption{Simplified feature extraction process schema.}
\label{f.dataset}
\end{figure}
\subsection{Evaluation Metrics}
\label{ss.metrics}
This section provides a brief description of the metrics employed to evaluate the experiments, i.e., Perceptual Evaluation of Speech Quality (PESQ), Mean Square Error (MSE), Area under the Curve, and energy efficiency.
\subsubsection{PESQ}
\label{sss.PESQ}
Perceptual Evaluation of Speech Quality (PESQ) comprises a set of mechanisms for automated speech quality evaluation~\cite{recommendation2001perceptual}. It was developed for objective voice quality testing by telecom operators, equipment vendors, and phone manufacturers.
Its testing topology depends on the available information, and the method is divided into two classes:
\begin{itemize}
\item Full Reference (FR): uses the original signal as a reference for comparison. This approach compares each reference sample to the corresponding noisy signal, thus delivering more accurate results.
\item No Reference (NR): uses only the noisy signal for quality estimation and has no information regarding the reference signal.
\end{itemize}
In a nutshell, PESQ is a full-reference algorithm that analyzes the speech signal sample-by-sample after a temporal alignment of corresponding excerpts of reference and test signal. This paper employs the FR version.
\subsubsection{MSE}
\label{sss.MSE}
The Mean Squared Error measures the average of the squares of the errors, i.e., the average squared difference between the estimated values and the actual value. The MSE formulation is described as follows:
\begin{equation}
MSE = \frac{1}{n}\sum_{i=1}^n(y_i-\hat{y}_i)^2,
\label{e.mse}
\end{equation}
where $n$ stands for the number of data points, $y$ is the expected value, and $\hat{y}$ is the predicted value.
\subsubsection{Area under the Curve}
\label{sss.AuC}
This work employs a discrete version of the area under the curve, which denotes the sum of the outputs of a given function $f(x)$ such that $\{x\in\mathbb{R}|a\leq x \leq b\}$, where $a$ and $b$ denote the lower and upper bounds, respectively. One can define the area under the curve as follows:
\begin{equation}
Area = \sum_{x=a}^bf(x).
\label{e.auc}
\end{equation}
\subsubsection{Energy Efficiency}
\label{sss.energyEfficiency}
This paper employs the term ``energy efficiency'' to describe the rate of active neurons during execution. This approach considers the idea that neurons that do not ``fire'' during the execution of the model are not performing any costly operation, thus saving energy. In a nutshell, the algorithm computes the sum of the activations whose output is greater than zero and divides it by the total number of activations.
\section{Experimental Setup}
\label{s.setup}
The experiments provided in the next section were conducted considering a graph neural network and a multilayer perceptron network as the backbone. Both networks share a similar architecture for comparison purposes, i.e., two hidden layers with $512$ neurons in each layer, using the Adam optimizer with a learning rate of $0.001$. The models are trained with the objective of maximizing the canonical correlation for coherent features extraction during $5,000$ epochs considering a trading-off parameter $\lambda=0.0001$ in Equation~\eqref{e.canonicalCorrelation}, while Equation~\eqref{e.finalLoss} is set with the values $\alpha=0.5$, $\beta=0.25$, and $\gamma=0.0625$. Notice that these values were empirically chosen using a grid search in the range $[0, 1]$ with step $0.0625 (1/16)$. These values make sense since they give more importance to the receptive input than to the context provided by the visual information and a combination of both, and the most relevant information to reconstruct the clean audio in this context is the noisy audio. The data augmentation step considers both an edge dropping rate and a feature masking rate of $0.5$. Moreover, the graphs are generated considering three distinct neighborhood scenarios, i.e., with $3$, $10$, and $30$ neighbors. Notice this neighborhood is defined in two manners, i.e., the standard feature-space-based approach and the proposed temporal-information-based relationship, as described in Section~\ref{ss.temporalRelationship}.
After training, the networks' outputs are used to feed a dense layer, which is responsible for reconstructing the clean signal given the features extracted from noisy audio for the single modality and noisy audio and clean video for the proposed multimodal extension. The dense layer is optimized during $600$ epochs using the Adam optimizer with a learning rate of $0.005$ and a weight decay of $0.0004$ using the minimization of the mean squared error as the objective function.
The dataset was divided into $15$ folds to provide an in-depth statistical analysis. Each fold comprises $50$ sequences of $48$ frames each, summing up to a total of 2,400 samples per fold. As stated in Section~\ref{ss.dataset}, the dataset is formed by three subsets: (i) clean audio, (ii) noisy audio, and (iii) clean visual. The noisy audio is used to train the standard unimodal approach, while the proposed multimodal approach employs both the noisy audio and clean visual in the training process. The clean audio is used as a reconstruction target for both cases. Finally, each fold is split into train, validation, and test sets, following the proportions of $60\%$, $20\%$, and $20\%$, respectively. For statistical evaluation, the Wilcoxon signed-rank test~\cite{Wilcoxon:45} with $5\%$ of significance was considered.
\section{Experiments}
\label{s.experiments}
This section presents the experimental results considering three tasks: (i) feature extraction driven by canonical correlation analysis maximization, (ii) clean audio data reconstruction based only on noisy audio or noisy audio and clean visual data fusion, and (iii) energy efficiency analysis in terms of neuron activation rate.
\subsection{Feature Extraction Analysis}
\label{ss.featureExtraction}
The experiments presented in this section compare the performance of graph neural networks with canonical correlation analysis for the task of self-supervised relevant feature extraction in three distinct neighborhood scenarios, i.e., $3$, $10$, and $30$ neighbors, namely GNN 3, GNN 10, and GNN 30, respectively. Moreover, two distinct neighborhood strategies are compared: (i) the standard (Std.) feature-space distance and (ii) the proposed prior frames-based connections for time-sequence (Seq.). Finally, a multilayer perceptron network with similar architecture (same number of layers, neurons per layer, optimizer, and learning rate) was trained using an identical self-supervised approach with the canonical correlation analysis maximization as the target function, denoted the baseline.
Figure~\ref{f.feature_extraction_audioOnly}(a) presents the convergence of the unimodal models considering the task of feature extraction through CCA maximization over noisy audio data. In this context, one can observe CCA-MLP obtained the highest results considering CCA maximization itself, even though such results do not necessarily imply better data representation for the specific task of clean audio reconstruction since the prior frames-based CCA-GNN (Seq. and Seq.* for $w_{ii} = k+1$ and $w_{ii} = 1$, respectively) obtained better reconstruction rates, as presented in Table~\ref{t.reconstruction_audioOnly}. Notice that CCA-GNN Seq.* obtained similar results to CCA-GNN Seq. and thus is overlapped in the image.
Regarding the multimodal extension conducted over noisy audio and visual features, one can observe in Figure~\ref{f.feature_extraction_audioOnly}(b) that the proposed CCA-GNN Seq. obtained the best results overall considering a neighborhood (number of prior frames) of $30$. Such a behavior is expected since a more significant number of prior frames reinforces the coherent information shared between the two channels during a longer period of time, providing more robust and connected features. Such robustness is in fact propagated to the reconstruction task since CCA-GNN Seq. with $k=30$ also provided the best results available in Table~\ref{t.reconstruction_audioOnly}.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{featureExtraction_noisyOnly-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{featureExtraction_AudioVisual-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Feature extraction for (a) noisy audio only and (b) noisy audio and clean visual.}
\label{f.feature_extraction_audioOnly}
\end{figure}
\subsection{Clean Signal Reconstruction}
\label{ss.reconstruction}
Table~\ref{t.reconstruction_audioOnly} presents the clean audio data reconstruction error considering the unimodal approach given the noisy data signal as input. Notice Standard denotes the common approach using the feature space to represent the nodes adjacency, while Sequential and Sequential* stand for the proposed prior frame-based approach for the positional encoding using $w_{ii} = k+1$ and $w_{ii} = 1$, respectively. From these results, one can observe that (i) a more significant number of nodes' neighbors lead to better reconstruction errors; (ii) the proposed sequential approach with prior frame connections outperformed the standard feature space distance-based neighborhood modeling and the MLP, and (iii) Sequential and Sequential* did not present significative differences, showing that the influence of the node self-connection is not relevant for the task. Notice that the better results according to the Wilcoxon signed-rank test are presented in bold. In this context, none of the other approaches obtained statistically similar results to the proposed sequential CCA-GNN with $k=30$. Further, Figure~\ref{f.reconstruction_audioOnly} shows that all techniques present similar convergence considering the reconstruction task, even though the MLP presents a slightly slower convergence, as depicted in the zoomed frame.
\begin{table}[!htb]
\caption{Average Mean Squared Error and standard deviation over unimodal CCA-MLP and CCA-GNN considering clean audio reconstruction given noisy audio input.}
\begin{center}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $0.0206\pm0.0012$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $0.0238\pm0.0009$ & $0.0220\pm0.0025$& $0.0220\pm0.0025$ \\
& 10 & $0.0235\pm0.0008$ & $0.0218\pm0.0025$& $0.0218\pm0.0025$ \\
& 30 & $0.0234\pm0.0008$ & \bm{$0.0187\pm0.0026$}& \bm{$0.0186\pm0.0025$}\\
\bottomrule
\end{tabular}}
\label{t.reconstruction_audioOnly}
\end{center}
\end{table}
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.60\textwidth]{testReconstruction_noisyOnly-eps-converted-to.pdf}
\end{tabular}}
\caption{Clean audio reconstruction error convergence considering the unimodal architecture based on the noisy audio.}
\label{f.reconstruction_audioOnly}
\end{figure}
Figure~\ref{f.signal_reconstruction_audio_only} depicts a randomly selected clean audio sample reconstruction regarding the unimodal architecture trained over noisy audio data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. Regarding the standard approach, one can notice in Figure~\ref{f.signal_reconstruction_audio_only}(a) that none of the models performed significantly well, especially for the first $8$ features. On the other hand, the proposed approach for frame-based positional encoding obtained much better results in this context considering the sequential CCA-GNN with $k=30$, as illustrated in Figure~\ref{f.signal_reconstruction_audio_only}(b).
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{reconstruction_audio_shuffled_17-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{reconstruction_audio_sequential_17-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Clean audio signal reconstruction regarding the unimodal architecture trained over noisy audio data considering the (a) standard approach and (b) the proposed prior frame-based positional encoding for time sequence modeling.}
\label{f.signal_reconstruction_audio_only}
\end{figure}
Table~\ref{t.reconstruction_audioVisual} present very similar results in the context of the multimodal architecture considering the noisy data and clean video signal as input, in which the proposed prior frame-based neighborhood outperformed the standard feature distance node's connection and the CCA-GNN with $k=30$ obtaining the better results overall considering the Wilcoxon signed-rank test. This table also shows that the multimodal approach is, in fact, capable of providing more representative features for clean audio reconstruction since all methods outperformed the respective unimodal versions. Figure~\ref{f.reconstruction_audioVisual}, which depicts the reconstruction error convergence considering the multimodal architectures, also presents very similar results to Figure~\ref{f.reconstruction_audioOnly}, in which all techniques perform similarly, with CCA-MLP showing a slower convergence in the first $50$ iterations.
\begin{table}[!htb]
\caption{Average Mean Squared Error and standard deviation over unimodal CCA-MLP and CCA-GNN considering clean audio reconstruction given noisy audio and visual inputs.}
\begin{center}
\resizebox{0.7\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $0.0189\pm0.0009$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $0.0238\pm0.0009$ & $0.0220\pm0.0025$& $0.0220\pm0.0024$ \\
& 10 & $0.0233\pm0.0009$ & $0.0216\pm0.0025$& $0.0216\pm0.0026$ \\
& 30 & $0.0225\pm0.0009$ & \bm{$0.0179\pm0.0025$} & \bm{$0.0180\pm0.0026$}\\
\bottomrule
\end{tabular}}
\label{t.reconstruction_audioVisual}
\end{center}
\end{table}
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.55\textwidth]{testReconstruction_AudioVisual-eps-converted-to.pdf}
\end{tabular}}
\caption{Clean audio reconstruction error convergence considering the proposed multimodal architecture based on the noisy audio and the clean visual data.}
\label{f.reconstruction_audioVisual}
\end{figure}
Similar to Figure~\ref{f.signal_reconstruction_audio_only}, Figure~\ref{f.signal_reconstruction_audio_visual} presents a randomly selected clean audio sample reconstruction regarding the multimodal architecture trained over noisy audio and clean visual data. Again, Figure~\ref{f.signal_reconstruction_audio_visual}(a) shows the standard approach did not performed significantly well for the first five or eight features, while the proposed approach with $k=30$ provided much more accurate results, as depicted in Figure~\ref{f.signal_reconstruction_audio_visual}(b).
\begin{figure}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{reconstruction_audio_visual_shuffled_17-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{reconstruction_audio_visual_sequential_17-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Clean audio signal reconstruction regarding the multimodal architecture trained over noisy audio and clean visual data considering the (a) standard approach and (b) the proposed prior frame-based positional encoding for time sequence modeling.}
\label{f.signal_reconstruction_audio_visual}
\end{figure}
\textcolor{white}{}
\subsection{Neuronal Activation Analysis}
\label{ss.neuronActivation}
This section provides an energy-efficiency analysis based on neuronal activation behavior. Such an analysis is fundamental since using such models in real-world applications, such as hearing devices embedded systems, for instance, is constrained to energy issues. It is to be noted that here the energy-efficiency is referred to the number of neurons with zero activity. In hardware (e.g., FPGA) zero signal does not propagate in the network and contributes nothing to the dynamic power consumption because of no switching activity. Figure~\ref{f.activation_audioOnly} describes the average rate of the first hidden layer's neurons activation during the training of the unimodal approach, while Table~\ref{t.activation_audioOnly} provides the area under the curve. Notice that the better values are underlined and some less relevant results were omitted from the plot for better visualization. In this context, one can observe that, even though GNNs and the MLP comprise the same architecture, i.e., two hidden layers with $512$ neurons each, around $100$ neurons in the MLP architecture fired $100\%$ of the time, while the same pattern is observed over a range between $10$ and $20$ neurons concerning the GNNs. Moreover, although the best performing model for the clean audio reconstruction task, i.e., the sequential CCA-GNN with $k=30$, is the most energy-consuming approach between all the CCA-GNN-based methods, it still presents a nominal neuron activation rate if compared to the MLP.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.55\textwidth]{activation_noisyOnly-eps-converted-to.pdf}
\end{tabular}}
\caption{Neuron activation rate considering the unimodal architecture.}
\label{f.activation_audioOnly}
\end{figure}
\begin{table}[!htb]
\caption{Area under the curve considering the unimodal architecture neuron activation rate.}
\begin{center}
\resizebox{0.6\textwidth}{!}{
\begin{tabular}{cccccc}
\toprule
Model & Neighbors & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $147.27$ & -& - \\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $\underline{10.17}$ & $11.73$& $11.73$ \\
& 10 & $21.64$ & $26.56$& $26.56$ \\
& 30 & $29.99$ & $47.71$& $47.71$\\
\bottomrule
\end{tabular}}
\label{t.activation_audioOnly}
\end{center}
\end{table}
\begin{figure*}[!htb]
\centerline{
\begin{tabular}{cc}
\includegraphics[width=.44\textwidth]{activation_AudioVisual_audio-eps-converted-to.pdf} &
\includegraphics[width=.44\textwidth]{activation_AudioVisual_visual-eps-converted-to.pdf} \\
(a) & (b)
\end{tabular}}
\caption{Neuron activation rate considering the multimodal architecture considering (a) audio and (b) visual channels.}
\label{f.activation_audioVisual}
\end{figure*}
Figures~\ref{f.activation_audioVisual}(a) and~\ref{f.activation_audioVisual}(b) present the neural activation analysis considering the multimodal approach for the audio and the visual channels, respectively, whose values are reflected in Table~\ref{t.activation_audioVisual}. The figures present a similar behavior to the one observed in Figure~\ref{f.activation_audioOnly}, in which the MLP presents a neuron activation rate substantially larger than the MLP-based approaches. Moreover, one can also observe that the prior frame-based approach has different behaviors for each channel since their neuron activation conduct is more intense in the audio channel but produces a reduced fire rate considering the visual channel. Such behavior may suggest that using the prior frame-based approach implies a network inclined to extract more relevant features considering the temporal flow as a context, thus becoming less dependant on the visual context itself.
\begin{table}[!htb]
\caption{Area under the curve considering the multimodal architecture neuron activation rate.}
\begin{center}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{ccccccccc}
\toprule
&&\multicolumn{3}{c}{\textbf{Audio}}& & \multicolumn{3}{c}{\textbf{Visual}}\\
\cmidrule{3-5} \cmidrule{7-9}
Model & Neighbors & Standard & Sequential & Sequential* & & Standard & Sequential & Sequential* \\
\midrule
\textbf{MLP} & - & $142.07$ & - & - & & $203.01$ & -& -\\
\midrule
\multirow{3}{*}{\textbf{GNN}} & 3 & $\underline{11.77}$ & $14.88$& $14.88$ & & $19.05$ & $\underline{17.36}$& $\underline{17.36}$\\
& 10 & $19.29$ & $24.01$& $24.01$ & & $38.52$ & $32.28$& $32.28$ \\
& 30 & $27.09$ & $45.80$& $45.80$ & & $55.85$ & $36.66$& $36.66$\\
\bottomrule
\end{tabular}}
\label{t.activation_audioVisual}
\end{center}
\end{table}
\subsection{Speech Enhancement Framework and Results}
For speech enhancement, state-of-the-art enhanced visually derived Wiener filter (EVWF) is used as shown in Figure~\ref{f.VWF}~\cite{adeel2019lip}. The EVWF uses an audio or AV driven regression model to estimate clean audio features using noisy audio or AV temporal features. The low dimensional estimated clean audio features are transformed back into high dimensional clean audio power spectrum using inverse filter bank transformation to calculate Wiener filter. The Wiener filter is applied to the magnitude spectrum of the noisy input audio signal, followed by the inverse fast Fourier transform, overlap, and combining processes to produce enhanced magnitude spectrum. More details are comprehensively presented in our previous work \cite{adeel2019lip}\cite{adeel2020contextual}\cite{adeel2020novel}. For objective testing and comparison, perceptual evaluation of speech quality (PESQ) is used as shown in Tables~\ref{t.PESQ_audio} and~\ref{t.PESQ_audioVisual} for audio only and audio-visual data, respectively. PESQ is one of the most reliable methods to evaluate the quality of restored speech. The PESQ score is computed as a linear combination of the average disturbance value and the average asymmetrical disturbance values. Scores range from $-0.5$ to $4.5$, corresponding with low to high speech quality. It can be seen that at low SNR levels (-6dB), EVWF with AV-GNN30* performs comparably to the state-of-the-art LSTM model and significantly outperforms spectral subtraction (SS) and Log-Minimum Mean Square Error (LMMSE), and MLP based speech enhancement methods using far few neurons.
\begin{figure}[!htb]
\centerline{
\begin{tabular}{c}
\includegraphics[width=.7\textwidth]{VWFnew.png}
\end{tabular}}
\caption{Enhanced visually-derived Wiener filtering. Lip reading refers to audio-visual clean log-FB features estimation.}
\label{f.VWF}
\end{figure}
\begin{table}[!htb]
\caption{Speech Enhancement Results: Perceptual evaluation of speech quality (PESQ) for the unimodal architecture trained over
noisy audio data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. It is to be noted that only GNN30 and GNN30* with prior frame-based positional encoding perform reasonably well compared to all other models.}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{6pt}
\resizebox{0.9\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\cellcolor[HTML]{EFEFEF}&\multicolumn{4}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Standard}}&\multicolumn{5}{c}{\cellcolor[HTML]{EFEFEF}\textbf{Prior frame-based}}\\\hhline{~---------}
\multirow{-2}{*}{\cellcolor[HTML]{EFEFEF}{\textbf{SNR}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30*}}} \\ \hline
-12dB & 0.83 & 0.86& 0.80 & 0.88 & 0.80 & 0.91& 0.90 & 1.24 & 1.27\\ \hline
-6dB & 0.81 & 0.84& 0.85 & 0.87 & 0.82 & 0.83& 0.99 & 1.26 & 1.28\\ \hline
-3dB & 0.82 & 0.88& 0.89 & 0.94 & 0.86 & 0.87& 0.85 & 1.26 & 1.30\\ \hline
0dB & 0.85 & 0.85& 0.87& 0.88 & 0.90 & 0.96& 0.88 & 1.33 & 1.34\\ \hline
3dB & 0.88 & 0.82& 0.91 & 0.86 & 0.86 & 0.92&0.92 & 1.36 & 1.38\\ \hline
6dB & 0.84 & 0.90& 0.88 & 0.82 & 0.87 & 0.84& 0.89 & 1.40 & 1.42\\ \hline
12dB & 0.91 & 0.89 & 0.92 & 0.95 & 0.82 & 0.87& 0.81 & 1.48 & 1.55\\ \hline\hline
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|}
\end{tabular}}
\label{t.PESQ_audio}
\end{center}
\end{table}
\begin{table}[!htb]
\caption{Speech Enhancement Results: Perceptual evaluation of speech quality (PESQ) for the multimodal architecture trained over
noisy audio and clean visual data considering the standard approach and the proposed prior frame-based positional encoding for time sequence modeling. Here only GNN30 without prior frame-based positional encoding and GNN30 and GNN30* with prior frame-based positional encoding perform reasonably well compared to all other models. These models also perform slightly better than audio-only models.}
\begin{center}
\renewcommand{\arraystretch}{1.5}
\setlength{\tabcolsep}{6pt}
\resizebox{\textwidth}{!}{
\begin{tabular}{c|c|c|c|c|c|c|c|c|c|c|c}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\cellcolor[HTML]{EFEFEF}&\multicolumn{4}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Standard}}&\multicolumn{5}{c|}{\cellcolor[HTML]{EFEFEF}\textbf{Prior frame-based}}&\multicolumn{2}{c}{\cellcolor[HTML]{EFEFEF}\textbf{Obtained from~\cite{adeel2018real}}}\\\hhline{~-----------}
\multirow{-2}{*}{\cellcolor[HTML]{EFEFEF}{\textbf{SNR}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{MLP}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 3}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 10}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{GNN 30*}}}& {\cellcolor[HTML]{EFEFEF}{\textbf{SS}}}&
{\cellcolor[HTML]{EFEFEF}{\textbf{LMMSE}}} \\ \hline
-12dB & 0.84 & 0.88& 0.87 & 1.10 & 0.86 & 0.94& 0.80 & 1.25 & 1.29 & 0.9& 0.95\\ \hline
-6dB & 0.88 & 0.80& 0.86 & 1.15 & 0.91 & 0.80& 0.89 & 1.27 & 1.29& 1.01 &1.03\\ \hline
-3dB & 0.90 & 0.87& 0.89 & 1.15 & 0.83 & 0.86& 0.88 & 1.28 & 1.31 & 1.17 &1.18\\ \hline
0dB & 0.82 & 0.86& 0.81& 1.18 & 0.80 & 0.84& 0.84 & 1.34 & 1.36 & 1.21& 1.20\\ \hline
3dB & 0.86 & 0.89& 0.93 & 1.20 & 0.92& 0.88&0.88 & 1.40 & 1.42 & 1.25 &1.34\\ \hline
6dB & 0.89 & 0.87& 0.88 & 1.22 & 0.87 & 0.83& 0.81 & 1.44 & 1.48 & 1.26 & 1.39\\ \hline
12dB & 0.84 & 0.91 & 0.84 & 1.25 & 0.88 & 0.81& 0.89 & 1.51 & 1.60 &1.54 &1.60\\
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\hhline{-|-|-|-|-|-|-|-|-|-|-|-|}
\end{tabular}}
\label{t.PESQ_audioVisual}
\end{center}
\end{table}
\section{Conclusions}
\label{s.conclusions}
This paper proposes an energy-efficient approach for improving and boosting sound signals through environmental information fusion. The model extends Graph Neural Networks with Canonical Correlation Analysis for multimodal data integration and further incorporates a prior frame-based node's positional encoding that considers the temporal sequence in data to establish the information similarity instead of the usual feature space distance. Experiments conducted over the AV Grid and ChiME3 corpora considering the task of clean audio reconstruction based on the fusion of noisy audio and clean video data show that the proposed approaches are capable of outperforming a baseline composed of an MLP/ LSTM with similar architecture trained under the same conditions, i.e., in a self-supervised fashion using canonical correlation analysis maximization as the target function. Finally, the proposed approach provides a considerable gain regarding energy efficiency, given that the CCA-GNN neuron firing rates are dramatically lower than MLP and LSTM. It is worth mentioning that it is impossible to quantify the energy saving at this stage, but the ongoing work includes implementing these models on FPGA in which neurons with zero activity will not propagate in the network and will therefore contribute nothing to the dynamic power consumption.
The experiments also indicate that the multimodal approach can produce better results than the unimodal architecture, leading to minor reconstruction errors. Such behavior is expected since the visual data acts as a context to introduce some additional meaning and improve the noisy audio signal. Additionally, the prior frame-based approach provided better results than the standard model, showing the importance of the temporal information as an additional context to the noisy signal. Finally, one could notice by the neuron's firing behavior that using prior frame node connection reinforces the information present in the noisy audio channel, making it less dependant on the visual data context for clean audio reconstruction. Future work involves developing a more biologically realistic neuronal model, introducing a concept of memory to improve communication mechanisms between the channels. Further, Graph Neural Networks with Canonical Correlation Analysis will be used to improve cross channels communication blocks within convolutional neural networks.
\section*{Acknowledgments}
\begin{sloppypar}
This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC) Grant Ref. EP/T021063/1. J. Del Ser would like to thank the Spanish Centro para el Desarrollo Tecnologico Industrial (CDTI, Ministry of Science and Innovation) through the ``Red Cervera'' Programme (AI4ES project), as well as by the Basque Government through EMAITEK and ELKARTEK (ref. 3KIA) funding grants and the Consolidated Research Group MATHMODE (ref. IT1456-22).
\end{sloppypar}
\section*{References}
| 2024-02-18T23:40:02.478Z | 2022-09-19T02:17:11.000Z | algebraic_stack_train_0000 | 1,157 | 16,230 |
|
proofpile-arXiv_065-5781 | \section{Introduction}
Shape changes (morphodynamics) are one of the principal mechanisms through which individual cells interact with their environment \cite{yin2014cells, bodor2020cell}. These dynamics arise from the interplay between a multitude of molecules and complex signalling pathways that often organise with emergent simplicity to carry out critical cellular functions, including division and migration. T cells, specialised cells of the adaptive immune system, are highly dependent on global morphodynamics to squeeze through gaps in the extracellular matrix (ECM), in contrast to the ECM-degrading strategies other cells use (e.g. tumour cells). Despite plasticity for adjusting the mode of migration to environmental conditions, the migration of T cells is often characterised as amoeboid: fast (up to 25 \textmu m min\textsuperscript{-1} \cite{friedl1994locomotor}), with low adhesion and polarised morphologies arising due to the segregation of different cytoskeletal networks to specific subcellular compartments \cite{weninger2014leukocyte}. In this mode of locomotion, dynamic F-actin forms pseudopods at the leading edge and an actomyosin-rich uropod at the rear generates contractile forces \cite{dupre2015t} (see Fig. \ref{fig:figure1}a for a schematic). However, this canonical migration mechanism is not fixed and T cells adapt their motility to their immediate environment.
T cells are thought to toggle between exploration and exploitation states, balancing surface receptor cues for interacting with antigen-presenting or target cells (`stop') with chemokine-driven or purely exploratory searches (`run') \cite{krummel2016t}. The specific morphodynamics and force-generating mechanisms behind these states are not well understood, in part due to their large variety and adaptability in different environments \cite{fowell2021spatio}. Proposed methods for propulsion include leading edge extension and intercalation with the ECM (using either low-adhesion integrin connections or surface texture for friction), followed by contraction of the uropod for small pore sizes \cite{soriano2011vivo, fowell2021spatio}. In addition to creating friction for moving forward, the rearward flow of actin waves from the leading edge may connect with the ECM like a paddle \cite{reversat2020cellular, abercrombie1970locomotion}. However, the extent to which these methods are used in complex 3D ECM environments, and their precise organisation, are far from well-characterised.
Accurate characterisation is important as dysregulation of T cell migration processes can be highly deleterious. T cells differentiate into different effector states. For instance, antigen-specific CD4\textsuperscript{+} `helper' T cells amplify the immune response, while CD8\textsuperscript{+} `cytotoxic' T cells seek out and neutralise infected or cancerous cells \cite{nino2020cytotoxic}. Inadequate migration leaves infected and cancerous cells free to proliferate, while over-stimulation can cause inflammation-based diseases like asthma and arthritis \cite{xia2009recent}. While there are exciting immunotherapeutic avenues manipulating the migration process, these have so far disappointed \cite{rafiq2020engineering}. With quantitative representations of T cell morphodynamics, their statistics can be interpreted with high-precision and compared across conditions for potentially improved immunotherapeutic development, and mechanistic models can be developed \cite{keren2008mechanism,tweedy2013distinct,tweedy2019screening}.
One of the main challenges for analysing morphodynamics is that cells do not have obvious landmarks (e.g. legs, eyes, wings of animals), and so the important degrees of freedom must be inferred from the data itself. Where there is important landmark-like information (e.g. polarisation that can manifest as subtle morphological features), this is typically diffuse rather than precisely-locatable, which further complicates quantification. Current methods therefore do not explicitly include this information. 2D cell morphologies are often quantified using Fourier descriptors. This method decomposes the cell outline coordinates as functions of rotation around the centroid in terms of Fourier coefficients, which then represent the morphology. This approach has revealed that amoeboid migrating cells in 2D, including epithelial keratocytes and \textit{Dictyostelium} amoebae \cite{keren2008mechanism, tweedy2013distinct}, explore only a small subspace of the shapes that might be thought possible from qualitative inspection (i.e. low-dimensionality of morphology). Furthermore, the morphodynamics within this space are composed primarily of frequently-used, or `stereotyped', motifs (i.e. low-dimensionality of morphodynamics).
Imaging of 3D cell dynamics at sufficiently high spatio-temporal resolution has only recently become available through lattice-light sheet microscopy \cite{chen2014lattice}. Whether T cells navigating complex 3D ECM environments similarly have low-dimensional morphology and morphodynamics remains to be understood. Such questions in 3D necessitate automated analysis even more than in 2D, both because such datasets are inherently harder to visualise and interpret, and because 3D environments typically induce a richer variety of morphodynamics \cite{driscoll2015quantifying}. Spherical harmonic descriptors (SPHARM), a 3D analogue of Fourier descriptors, are a promising method for quantifying 3D cell shape and connecting with motion \cite{heryanto2021integrated}. However, the representations are typically too uninterpretable for exploring morphodynamics with high precision, and use so far is primarily limited to classification or the detection of established shape changes \cite{ducroz2012characterization, medyukhina2020dynamic}. It therefore remains an open question how best to quantify 3D cell shapes without clear landmarks and interpret high spatiotemporal dynamics.
Here, we sought to combine lattice light-sheet microscopy \cite{chen2014lattice} with quantitative image analysis to explore the 3D morphodynamics of cytotoxic T cells migrating in the absence of chemoattractant cues through 3D collagen matrices \cite{galeano2016antigen}. We first created a new compact shape descriptor, based on SPHARM, but better connected to key polarisation information than current approaches. We found that T cells explore a low-dimensional morphological space, and that run-and-stop migration emerges at long timescales. We explored the morphodynamic compositions of these two modes using multiscale wavelet analysis, previously used to explore the structure of fruit fly behaviour \cite{berman2014mapping, berman2016predictability}, uncovering a global set of largely discrete stereotyped motifs. Focusing ultimately on the run mode, due to its key role in active translocation and polarised morphologies that are well-suited for analysis with our descriptor, we found that periodically oscillating morphodynamics (every $\sim$100 s) sustain forward motion. These can be understood as a biphasic process integrating previously hypothesised propulsion mechanisms \cite{reversat2020cellular, abercrombie1970locomotion}, namely: front-widening and retraction of the uropod (rear moves forward), and rearward surface motion with forward extension (front moves forward).
\section{Results}
\subsection{T Cell Shape is Low-Dimensional}
\begin{figure}[!htb]
\center{\includegraphics[]
{figures/figure1.pdf}}
\caption{\label{fig:figure1} \textbf{T cell shape can be quantified by spherical harmonic descriptors in 3D.} \textbf{(a)} Schematic of a T cell employing an amoeboid migration strategy to navigate through the extracellular matrix (ECM) in 3D. Actin polymerisation at the front results in the formation of pseudopods, and a complex of actomyosin at the rear forms the uropod, important for stability and generating contractile forces. \textbf{(b)} Complex spherical harmonic functions, $Y_{l}^{m}(\theta, \phi)$, (real parts shown for $m\geq0$) form a basis on the surface of a sphere. \textbf{(c)} Cartesian coordinates of the cell surface, $\{x, y, z\}$, are mapped to the surface of a sphere, as parameterised by polar coordinates $\{\theta, \phi\}$. The three resulting functions $\{x(\theta, \phi), y(\theta, \phi), z(\theta, \phi)\}$ are decomposed in terms of the spherical harmonic functions and transformed to be translation, scale and rotation invariant. This yields the final shape representation, $D_{l}$, based on the harmonics at each energy level, $l$, with the exclusion of $l=0$ giving the translation invariance. \textbf{(d)} Truncation of the representation at different degrees of $l$ leads to different levels of smoothing, with $l=1$ describing the ellipsoid part of the shape. \textbf{(e)} An additional descriptor, $D_{0}$, for accounting for cell orientation, with the landmark-like smooth uropod at the rear and dynamic protrusions at the leading edge. Without this additional variable, the two cells shown have very similar descriptors. The standard deviation of $D_{0}$ across all datasets is 0.31, and the standard deviations of the remaining $D_{l}$ are all lower. \textbf{(f)} For cases where the uropod vanishes, the landmark-like rear can still be identified by its smoothness and stationarity, compared with the dynamic leading edge, as shown in the example. For simplicity, we refer to this region at the rear as the uropod for all frames.}
\end{figure}
We imaged primary mouse effector CD8\textsuperscript{+} cytotoxic T cells in 3D collagen matrices without chemical cues, with a lattice light-sheet microscope (LLSM) \cite{chen2014lattice} at spatial resolution of 0.145, 0.145, 0.4 \textmu m and temporal resolution of $\sim$2-5 s (see Methods for details on the imaging and pre-processing and Supplementary Fig. 1 for a representative snapshot and 3D trajectories). Spherical harmonics (Fig. \ref{fig:figure1}b) can be used to quantify 3D cell shapes, as shown in Fig. \ref{fig:figure1}c \cite{ducroz2012characterization, medyukhina2020dynamic, brechbuhler1995parametrization, kazhdan2003rotation}. The spherical harmonic functions, $Y_{l}^{m}(\theta, \phi)$, form a basis over the sphere, where $l$ is the function degree (related to frequency) and $m$ is the order (rotations at each degree). The full approach is detailed in Methods and summarised here. The Cartesian coordinates describing the cell surface are each mapped to a sphere, so as polar coordinates $\{\theta, \phi\}$ move over the sphere surfaces, the cell surface is traced out in object space. Analogous to a Fourier decomposition, the functions describing the cell surface can be decomposed into a set of spherical harmonic coefficients, $c_{l, i}^{m}$ with $i\in \{x, y, z\}$. The $l=0$ coefficients describe the centroid location, the $l=1$ coefficients describe the ellipsoid part of the shapes, and so on, with increasing levels of detail. Truncation of the representation at a certain $l_{max}$ therefore leads to a representation of a smoothed version of the original morphology, where higher-frequency features are filtered out (Fig. \ref{fig:figure1}d). Translation invariance is achieved by omitting the $l=0$ coefficient, scale invariance is achieved by dividing all coefficients by $V^{-\frac{1}{3}}$ where $V$ is the volume \cite{zhao2017application}, and rotational invariance is achieved by transforming to a new representation, $\{D_{l}\}_{l>0}$, with
\begin{equation}
D_{l} = \sum_{i\in (x,y,z)}\sum^{l}_{m=0}c_{l,i}^{m}c_{l,i}^{m*},
\label{eq:rotinv}
\end{equation}
analogous to how rotational invariance can be achieved by extracting the power spectrum from Fourier descriptors of 2D cell shapes \cite{tweedy2013distinct}. There are two key problems with the descriptor in its current form, and we made two modifications to remedy these.
\begin{figure}
\center{\includegraphics[]
{figures/figure2.pdf}}
\caption{\label{fig:figure2} \textbf{T cell shape is low-dimensional as quantified with 3 principal components.} \textbf{(a)} Principal components (PCs) 1, 2 and 3 capture 74\%, 12\% and 9.8\% (total of 96\%) of the variance in $D_{l}$, respectively. \textbf{(b)} Shape changes associated with each PC ($l_{max}=3$ reconstructions), found by splitting the PCA space into 7 equal-length bins along each axis and plotting the T cell within each bin with the lowest value for the other PCs. An increasing PC 1 represents elongation and front-widening, a decreasing PC 2 represents contraction with front-widening, and an increasing PC3 represents elongation (forward or sideways) with the centroid moving towards the uropod. \textbf{(c)} Correspondence between the principal components (PCs) and $D_{l}$ is found by inverting the minimum, mean and maximum of each PC, with the other two PCs set to zero. Red and blue indicate decreasing and increasing descriptors, respectively, as the PCs are increased. $D_{0}$ represents the closeness between the uropod and centroid, $D_{1}$ the ellipsoidal aspects, and higher descriptors represent higher-frequency shape features. \textbf{(d)} Cell reconstructions with $l_{max}=3$ at their positions in PCA space. Darkness of colour indicates increasing PC 2.}
\end{figure}
First, the coefficients are not linearly related to the spatial extent of different features. We therefore took the square root of each element, i.e. $\{D_{l}\} \to \{D_{l}^\frac{1}{2}\}$, which yields a descriptor more representative than the power spectrum \cite{shen2009modeling}. Without this operation, almost all variance is contained in the first (ellipsoid) coefficient. Second, we added an element to the shape representation to capture key polarisation information lost in a purely global shape representation. At the cell rear is the uropod, a smooth round appendage that stabilises the cell and generates contractile forces, and at the leading edge emerge dynamic, higher-frequency protrusions. The cells in frames A and B in Fig. \ref{fig:figure1}e have very similar descriptors under a regular spherical harmonic representation, reflecting the similarity of their ellipsoid components, but this misses the polarisation conveyed in subtler features. We therefore added an extra descriptor, linearly related to the distance between the uropod and centroid, $D_{0}$ (see Methods for the full expression). The standard deviation of $D_{0}$ across all datasets is 0.31, and the standard deviations of the remaining $D_{l}$ are all lower, showing that frames A and B in Fig. \ref{fig:figure1}e are approximately two standard deviations apart along the $D_{0}$ dimension. While most cells have a well-defined uropod that can be readily identified (e.g. frames A and B in Fig. \ref{fig:figure1}e), some can exhibit more spherical shapes, as shown in Fig. \ref{fig:figure1}f. However, even for these cells there is still an identifiable smooth rear opposite a dynamic leading edge, and temporal information can reveal where the uropod transiently forms. For simplicity, we refer to this region at the rear as the uropod for all frames. The ultimate representation of T cell shape is therefore $\{D_{l}\}_{l=0}^{l_{max}=15}$ with $D_{0}$ as described above and $D_{l}$ for $l>0$ the square root of the expression in Eq. \ref{eq:rotinv}.
We used principal component analysis (PCA) to identify a set of uncorrelated linear features, or principal components (PCs), from the initial high-dimensional shape representation, $\{D_{l}\}$. Despite the lack of obvious constraints from manual inspection, Fig. \ref{fig:figure2}a shows only three PCs are required to capture $\sim$96\% of the variance in the data (74\%, 12\% and 9.8\% for PCs 1, 2 and 3, respectively). The rotational invariance means that the PCA coordinates are not invertible to unique shapes. To better isolate what features each PC describes, we therefore split the PCA space into 7 equal-length bins along each axis and plotted the T cell within each bin with the lowest value for the other PCs, shown in Fig. \ref{fig:figure2}b for $l_{max}=3$ reconstructions and Supplementary Fig. 2a for full cells (and Supplementary Fig. 2b shows the PC values of these plotted cells). Fig. \ref{fig:figure2}c shows what $D_{l}$ transitions these PCs correspond to, with the minimum, mean and maximum inverted for each PC (with the other PCs set to zero), and Supplementary Fig. 2c shows the vector composition of each PC. An increasing PC 1 represents elongation and front-widening, a decreasing PC 2 represents contraction with front-widening, and an increasing PC3 represents elongation (forward or sideways) with the centroid moving towards the uropod. Fig. \ref{fig:figure2}d shows a sample of cells (with $l_{max}=3$) at their locations in the PC space. Supplementary Fig. 2d shows that along the main axis of variation (PC 1), dimensionality is relatively constant, and Supplementary Fig. 2e shows only modest differences in the spherical harmonic spectra of the low and high PC 1 populations.
Uncertainty in the uropod label, a diffuse region rather than a precisely-locatable point, can be quantified and propagated to downstream variables of interest (Supplementary Fig. 3 and Methods). Uropod uncertainty was found using the curvature around the labelled point (Supplementary Fig. 3a-b), and then PC uncertainties were calculated by re-computing $D_{0}$ using each point on the cell rear within this uncertainty (Supplementary Fig. 3c). The mean percent uncertainty in $D_{0}$ is 1.5\%, which is lower than the uropod uncertainty since cell rears are typically perpendicular to the axis defined by the centroid and uropod. The percentage uncertainties of the PCs (relative to their standard deviations) are 4.4\%, 0.30\% and 9.2\% for PCs 1-3, respectively.
\subsection{Run-and-stop migration emerges over long timescale}
To connect morphodynamics with migration strategies, variables describing cell motion are required. There are two landmark-like features of the cell that move through the ECM, the uropod and the centroid, and we calculated velocity vectors for both, invariant to cell scale (i.e. units of s\textsuperscript{-1}; see Methods). To ensure uropod velocities have adequate signal-to-noise ratio (SNR), where noise arises from uropod labelling uncertainty, we found for each dataset the mean time taken for the uropod to move a significant distance, $\tau_{sig}$, and then computed velocities using running means over position with a time window of $\tau_{sig}$ (see Methods and Supplementary Fig. 3d for details). We then calculated speeds, which are 1D and rotationally-invariant, unlike velocities. The uropod and centroid speeds alone cannot separate distinct behaviours at small timescales, like translation and rotation (Supplementary Fig. 4a), and so we searched for a biologically-meaningful reference frame. We found that long-timescale migration is typically along the axis defined by the uropod and centroid (the UC axis), rather than the ellipsoid major axis (Supplementary Fig. 4b). The speeds of the uropod and centroid along this axis then better differentiate distinct motifs (Supplementary Fig. 4c) and Supplementary Fig. 4d shows these describe largely irreversible motion. The former has lower variance and fewer reversals (Supplementary Fig. 4d), and Fig. \ref{fig:figure1}f and Supplementary video 9 show dynamics that we observed in some datasets, where the cell appears to test routes with multiple extensions and retractions but a relatively static uropod, before committing with the uropod. We therefore selected uropod speed along the UC axis as the variable for cell motion (Fig. \ref{fig:figure3}a), and henceforth refer to it simply as speed.
\begin{figure}
\center{\includegraphics[width = 0.7\linewidth]
{figures/figure3.pdf}}
\caption{\label{fig:figure3} \textbf{Run-and-stop migration emerges over long timescales.} \textbf{(a)} Speed is defined as the uropod speed along the uropod-centroid (UC) axis, $\| (\Delta\textnormal{uropod}/\Delta t) \|\cos{\varphi}$, with smoothed uropod and centroid locations, and a further operation for invariance to cell scale (see Methods). \textbf{(b)} Cumulative speed plots show some cells have repeated phases of high speed (e.g. Cell A) while others have much lower speeds (e.g. Cell B). Lines are coloured by the maximum distance traveled divided by total dataset duration. \textbf{(c)} This is despite significant uropod motion in some cases. Meshes are shown every $\sim$104 s and 103 s for Cells A and B, respectively. \textbf{(d)} Histograms of speed with different running mean windows ($t_{win}$). At small timescales, differences in speed between cells can be indistinguishable because most exhibit phases of low speed, highlighted for Cells A and B. However, bimodality into two modes (run-and-stop) emerges at around 150 s.}
\end{figure}
Two migration modes separate out at long timescales, as shown in plots of cumulative speed (Fig. \ref{fig:figure3}b): repeated phases of high speed, making significant progress forward, e.g. Cell A; and lower speeds, yielding little progress, despite significant uropod motion in some cases, e.g. Cell B (Fig. \ref{fig:figure3}c). Fig. \ref{fig:figure3}d shows that, while at small times the dynamics can be indistinguishable (both modes have phases of near-zero speed), run-and-stop bimodality emerges at approximately 150 s. This bimodality is consistent with conclusions from lower-resolution experiments, where long-timescale trajectories of single cells have been modelled with Lévy-type random walks (characteristic of switching between stop and run modes) \cite{harris2012generalized}. Interestingly, another study suggested more complex statistics, with cells divided into sub-populations described by distinct random walk models \cite{banigan2015heterogeneous}. PCs 1 and 2 have a stronger correlation with run-and-stop mode than speed, indicating that shape is specialised more for migration mode than instantaneous speed, with cells in the run mode longer and thinner than those in the stop mode (Supplementary Fig. 5a). We next explored the morphodynamics behind these migration modes.
\subsection{Stereotyped morphodynamics underlie migration modes}
We analysed longer duration datasets for each of the run and stop modes to investigate how they differ (Supplementary videos 1-4 and 5-8 for the run and stop modes, respectively). We first computed the autocorrelation functions of the shape (PCs 1-3) and speed dynamics (using high SNR timeseries; see Methods for details). The autocorrelation function (ACF) is the correlation of a timeseries with a lagged version of itself, as a function of the lag, which can reveal the presence of latent variables preserving information across time. We found an autocorrelation decay time, $\tau\textsubscript{ACF}$, by fitting an exponential decay model to the peaks of the oscillating ACFs (Supplementary Fig. 5b), and these decay times are indicative of the timescales over which processes are likely guided more by internal cytoskeletal machinery than stochastic external cues. For the stop mode, PC 3 is more autocorrelated than the other variables (mean $\tau\textsubscript{ACF} \sim$250 s compared with $\sim$150 s of the other variables; Supplementary Fig. 5b). PC 3 dynamics are suggestive of sensing: forward extension with a tentative rearward centroid, and reaching sideways. See Supplementary videos 5-9, with the three included in the PC 3 ACF analysis coloured by PC 3. For the run mode, the main differences are a decrease in the PC 3 autocorrelation (to $\sim$150 s) and an increase in the speed and PC 2 (contraction with front-widening) autocorrelations (to $\sim$225 s). The power spectra in Supplementary Fig. 5c show the run mode has larger oscillations in speed and PC 2, particularly for 0.005-0.01 Hz. The run mode is therefore associated with faster oscillations in speed and PC 2 that typically remain autocorrelated for longer than those of the stop mode. These ACFs give a global perspective on morphodynamics, and the presence of long timescales suggest that the morphodynamics are, as with the morphologies, low-dimensional. We therefore next zoomed in on the PC timeseries to interpret the organisation of local morphodynamics, or `behaviours', that underlies this low-dimensionality.
\begin{figure}
\center{\includegraphics[]
{figures/figure4.pdf}} \caption{\label{fig:figure4}\textbf{Morphodynamics are organised in stereotyped motifs.} \textbf{(a)} 11 stereotyped motifs are local peaks in a probability density function (PDF) over the spectrogram embeddings in morphodynamic space, where different locations represent different local morphodynamics. These form more of a discrete set than continuum, and some examples of the local PC series are shown (red, blue and green lines for PCs 1-3, respectively). \textbf{(b)} Example stereotyped motifs, with frames evenly-spaced across 150 s. \textbf{(c)} Utilisation of motifs in the stop and run modes. Red and blue indicate the speed running mean is above 0.005 s$^{-1}$ (run mode) and below 0.0025 s$^{-1}$ (stop mode), respectively (selected from the bimodal distribution in Fig. \ref{fig:figure3}d), and grey indicates it is in between these values (e.g. transitions). \textbf{(d)} Transition probability matrices reveal how cells move around the morphodynamic space, counting transitions only once the cell moves to a different motif.}
\end{figure}
The continuous wavelet transform is a method for finding local morphodynamics (behaviours) from a timeseries of morphologies, and has been used to map stereotyped behaviour in fruit flies \cite{berman2014mapping, berman2016predictability}. See Methods for the full pipeline and Supplementary Fig. 6a for a schematic. Wavelets are used to transform the timeseries into a spectrogram with multiscale dynamic information. Dimensionality reduction with t-SNE \cite{van2008visualizing} can then be performed to map the spectrogram to an interpretable 2D morphodynamic space, where different locations represent different local morphodynamics (Fig. \ref{fig:figure4}a), and Supplementary Fig. 6b shows the dimensionality reduction is robust across different hyperparameters. Stereotyped motifs are those that are frequently performed, and so correspond to peaks in the probability density function (PDF) of spectrogram embeddings in this space. We used wavelets with a maximum width of influence of 150 s, the approximate timescale of organisation found from the autocorrelation analysis.
We found that behaviours are organised into more of a discrete set rather than a continuum (Fig. \ref{fig:figure4}a): `islands' between which cells jump, and we could therefore categorise and interpret these individually. Fig. \ref{fig:figure4}b shows key examples, with frames evenly-spaced over a 150 s interval (with the remainder and further examples in Supplementary Fig. 7), and Supplementary Fig. 8 shows the PC dynamics of three examples from each motif. Fig. \ref{fig:figure4}c shows how these are utilised differently in the run and stop modes. Red and blue indicate the speed running mean is above 0.005 s$^{-1}$ (run mode) and below 0.0025 s$^{-1}$ (stop mode), respectively (selected from the bimodal distribution in Fig. \ref{fig:figure3}d), and grey indicates it is in between these values (e.g. transitions). In the stop mode, stereotyped motifs include static shape or minor reaching with centroid towards rear (4); forward lengthen with centroid towards front (5); and edge centroid forward (9) (Fig. \ref{fig:figure4}b). In the run mode, stereotyped motifs include front-widen then streamline and extend (8); and retract and front-widen then extend (2). A probability matrix for transitions between the stereotyped motifs is shown in Fig. \ref{fig:figure4}d, with rows and columns corresponding to the start and end motifs, respectively. We assigned points to the closest stereotyped motif and counted transitions only once the cell moves to a different motif (i.e. diagonal entries are zero). Frequent transitions include from 3 to 7 (retract to reach to one side) and from 8 to 1 (front-widen then streamline and extend to front-widening).
\begin{figure}
\center{\includegraphics[]
{figures/figure5.pdf}} \caption{\label{fig:figure5}\textbf{Periodic oscillations in PC 2 underlie the run mode.} \textbf{(a)} Entropy of the run mode marginal dynamics for each PC (5 repeats for each) shows a minimum for PC 2, and therefore that these dynamics are the most stereotyped, consistent with the autocorrelation results of Supplementary Fig. 5b. Markov chain entropies were calculated for transitions on grids over morphodynamic spaces for each PC, found by repeating the wavelet analysis with each PC on its own. \textbf{(b)} Dynamics in the PC 2 morphodynamic space for the run mode, where different locations represent different local PC 2 morphodynamics (and decreasing PC 2 represents contraction and front-widening). Tracking the trajectories of the longer duration datasets reveals periodic oscillations of varying amplitude. Local PC2 dynamics in 150 s windows are shown inset at key points in the morphodynamic space, showing outer rings represent higher-amplitude oscillations, with a region for particularly large PC 2 decreases, bottom right. The top left corner represents rearward surface motion and the bottom right corner represents contraction and front-widening. Regions in between represent transitions between these motifs. Maximum uropod speeds correspond to contraction and front-widening. \textbf{(c)} These results are suggestive of the following cyclic morphodynamic propulsion mechanism: the leading edge widens, likely intercalating with the ECM and contracting the uropod; the leading edge then extends forward, as the previously-widened leading edge regions undergo a rearward motion that may connect with the ECM like a paddle. This cycle repeats every $\sim$100 s. \textbf{(d)} An example showing these oscillations in Cell 1, coloured by PC 2.}
\end{figure}
We next looked in detail at the run mode, of particular interest as this is when cells use global morphodynamics for active translocation through the ECM, and because it is in all cases defined by polarised morphologies, for which our descriptor was designed. First, we repeated the wavelet analysis with the longer duration datasets of the run mode, finding more of a continuum than the global morphodynamic space, but for which stereotyped motifs can still be categorised (Supplementary Fig. 9a-b, with PC timeseries of three examples from each motif in Supplementary Fig. 10). Supplementary Fig. 9c-d shows the speeds and transition probability matrix. Aside for a turning motif, all fall into two categories: compression, and a rearward surface motion with extension forward (rearward with respect to the cell frame of reference, and relatively static in the lab frame). The precise motifs are then variants on these base behaviours, e.g. whether there is also widening. These underlying morphodynamics, omitting the distracting variations, are most characteristic of PC 2 dynamics, and this connection between PC 2 dynamics and migration is certainly consistent with the increased autocorrelation timescales and power of PC 2 relative to the stop mode (Supplementary Fig. 5b-c).
To test this theory, we calculated the entropy of each PC's morphodynamics. We did this by repeating the wavelet analysis for each PC on its own and calculating the Markov chain entropy for transitions on a grid over the resulting morphodynamic space (see Methods and Supplementary Fig. 9e). We used grids since these dynamics formed continuums rather than discrete, categorisable morphodynamic spaces. We found an entropy minimum for PC 2 (Fig. \ref{fig:figure5}a), confirming that PC 2 dynamics are the most stereotyped. Fig. \ref{fig:figure5}b shows how all four cells follow the same circular oscillations of varying radius in the space of PC 2 morphodynamics. Supplementary videos 1-4 are labelled as Cells 1-4. Outer and inner rings represent high and low amplitude oscillations, respectively, and there is a region for particularly large decreases in PC 2 (but not for large increases). These results, in conjunction with Supplementary videos 1-4, coloured by PC 2, suggest the following morphodynamic propulsion mechanism (sketched in Fig. \ref{fig:figure5}b-c): the leading edge widens, likely intercalating with the ECM to contract the uropod (PC 2 decreases); the leading edge then extends forward, as the previously-widened leading edge regions undergo a rearward flow that may connect with the ECM like a paddle, ultimately streamlining (PC 2 increases). This cycle is repeated every $\sim$100 s, and explains the oscillations in (uropod) speed observed in Fig. \ref{fig:figure3}d. Fig. \ref{fig:figure5}d shows an example section of Supplementary video 1, coloured by PC 2. These results suggest T cells utilise a highly periodic internal machinery to generate a sustained migration effort, alternating between two previously proposed propulsion mechanisms to move the uropod then leading edge forward \cite{reversat2020cellular, fowell2021spatio, abercrombie1970locomotion}. A plausible mechanistic basis for the rearward morphodynamic flow is retrograde cortical actin flow, a process that has been implicated in amoeboid migration in a number of cells, including T cells \cite{abercrombie1970locomotion, reversat2020cellular}. However, further investigations of internal actin dynamics are needed to explore this connection.
\section{Discussion}
T cells are a key part of the adaptive immune system, migrating through the extracellular matrix (ECM) to neutralise infected and cancerous cells. However, their morphodynamics have not yet been completely quantitatively mapped in 3D. Here, we used lattice light-sheet microscopy (LLSM) to acquire datasets of primary mouse cytotoxic T cells migrating through a collagen matrix with high spatiotemporal resolution. Using a novel shape descriptor that incorporates key polarisation information with a uropod label, we found that shape was low-dimensional. Run-and-stop migration emerges at long timescales ($\sim$150 s), and global morphodynamics are stereotyped, forming a discrete set rather than continuum. Stop mode morphodynamics primarily involve oscillations in centroid movement towards the uropod, with extension forwards or sideways (PC 3 dynamics), and these remain autocorrelated for long timescales (decay time, $\tau\textsubscript{ACF} \sim$250 s). The run mode (i.e. active translocation) arises from periodic oscillations in PC 2, with a period of $\sim$100 s and $\tau\textsubscript{ACF} \sim$225 s: the leading edge widens, likely using intercalation with the ECM to contract the uropod (PC 2 decreases); the leading edge then extends forward, as the previously-widened leading edge regions undergo a rearward motion that may connect with the ECM like a paddle, ultimately streamlining (PC 2 increases). These results indicate periodicity in the cellular machinery help sustain forward motion during active translocation.
Uropod tracking proved vital for differentiating key morphological and morphodynamic states. Uropod uncertainties were then required to ensure analysis was at sufficient signal-to-noise ratio (SNR), because the uropod is a diffuse region rather than a precisely-locatable point. In analogy to the role of the Hessian matrix in parameter fitting, we found this could be achieved relatively simply by quantifying uropod uncertainty through the curvature of the cell rear, then propagating this to downstream variables of interest. The inclusion of landmark-like but diffuse features will likely become more important as methods for tracking intracellular structures at high spatiotemporal resolution continue to improve, meaning spatial regions can be associated with specific internal organisation and activity \cite{mckayed2013actin}. In a small number of cases (e.g. Supplementary video 7), thin fluid-like protrusions extend out of the uropod, which cause dynamics in $D_{0}$ that are unlikely to be important for migration. To reduce these effects, in future work we will explore labelling uropods based on smoothed reconstructions (with e.g. $l_{max}=15$). We found uropod definition reduced for some cells in a long-lived stop mode (and therefore had high uncertainties for some PCs, meaning they were omitted from analyses). This may be indicative of loss of polarisation, so for these modes alternative shape descriptors may be more appropriate.
Internal retrograde actin flow has been a hallmark of cell migration models for decades, since Abercrombie first observed centripetal flow of particles on fibroblast surfaces \cite{dupre2015t, abercrombie1970locomotion}. However, Abercrombie also proposed a second propulsion mechanism, where rearward flows of surface deformation might push the cell forward like a paddle. Such morphodynamic flows (or `waves') have recently been observed in 2D migrating \textit{Dictyostelium} cells \cite{driscoll2012cell}, and in T cells embedded in microfluidic channels where they can enable migration without any adhesion \cite{reversat2020cellular}. To our knowledge, however, they have not been characterised in 3D ECM environments. Through inhibition at obstacles and activation on the opposite side, flows may also aid turning as has been described in neutrophils \cite{weiner2007actin}, and the lateral protrusions likely serve as an anchor in confined geometries \cite{tozluouglu2013matrix, mandeville1997dynamic}. Analysis of actomyosin dynamics, as well as tracking of the ECM fibres (perhaps with a contact map over the cell surfaces), would help test the connection between the rearward surface motion and internal actin dynamics, and the specific nature of how these interact with the ECM for anchoring and propulsion. The analysis would also reveal the extent to which decreasing PC2 (contraction with front-widening) is driven by contact with fibres, although the periodic PC 2 dynamics across all run mode cells suggests this may predominantly be internally regulated.
Exciting areas for future work include the extension of the analysis to the timescale of hours, where the statistics and morphodynamics of switching between run and stop modes could be interpreted at the single-cell level, and the hierarchical organisation of the stereotyped motifs could be mapped. There are technical challenges, however: individual cells would have to be followed and migration distances would exceed the scales of current LLSM fields of view. Dataset sizes might also become problematic, given that a 20 min video corresponds to 1 TB of data (with one colour). Furthermore, non-stationary issues such as aging, differentiation and activation may come into effect \cite{metzner2015superstat}. It would also be interesting to build statistical models of T cell morphodynamics \cite{tweedy2019screening}, which may then enable the development of mechanistic models \cite{zhu2016comp}, connecting morphodynamics to both extra and intra-cellular processes.
Ultimately, we hope quantitative morphodynamic analyses of T cells navigating the complex ECM environment will aid comparison of migration across different conditions (e.g. tissues, drugs and cell mutants). In particular, the prevalence and switching between human-labelled modes of migration such as chimneying, mesenchymal, amoeboid (blebbing), finger-like, and rear-squeezing could be put on firm objective grounds \cite{zhu2016comp, yamada2019mechanisms}. These advanced morphodynamic analyses will in turn help the development of mechanistic models, with a view to enhanced understanding of, and more effective, immunotherapeutics.
\section{Methods}
\begin{small}
\subsection{LLSM imaging and pre-processing}
\subsubsection{Lattice light-sheet microscopy (LLSM)}
LLSM experiments were either performed on a custom-built system described in \cite{geoghegan20214d}, or on a Zeiss Lattice Light Sheet 7 microscope (Zeiss, Oberkochen, Germany). OT1-Lifeact-GFP T cells \cite{galeano2020lifeact} labelled with CellTracker Deep Red dye were excited at 642 nm, OT1-mT/mG at 561 nm and plain OT1-Lifeact-GFP T cells at 488 nm. For all experiments performed on the home-built system, Point Spread Functions were measured using 200 nm Tetraspeck beads. The acquired datasets were deskewed and deconvolved using LLSpy, a Python interface for processing of LLSM data. Deconvolution was performed using a Richardson-Lucy algorithm using the PSFs generated for each excitation wavelength. Datasets acquired on the Zeiss system were deskewed using the Zeiss Zen (blue edition) software. All data were acquired at 37$^{\circ}$C and 5\% humidified CO2. The voxel size was 0.1x01x0.2 \textmu m\textsuperscript{3} for the home-built system and 0.145x0.145x0.4 \textmu m\textsuperscript{3} for the Zeiss system. The temporal resolution was 2.5 s per frame for the OT1-mT/mG datasets, 5.6 s per frame for the OT1-Lifeact-GFP-CellTracker Deep Red datasets (both imaged on the home-built system) and 4.17 s per frame for the plain OT1-Lifeact-GFP datasets imaged on the Zeiss system. We collected 29 datasets with 2,850 frames altogether and a mean and standard deviation across datasets of 98 and 78, respectively.
\subsubsection{Image segmentation}
Before further processing, membrane Tomato signal was denoised using a deep-learning approach based on Content-Aware Image Reconstruction (CARE) \cite{weigert2018content}. CellTracker Deep Red and membrane Tomato signal were bleach-corrected using FIJI \cite{schindelin9714}. Cell surfaces were segmented using Imaris 8.4.1 (Bitplane, Zurich, Switzerland). To minimise the occurrence of holes in the surfaces, depending on the signal to noise ratio smoothing factors between 0.35 \textmu m and 0.8 \textmu m were applied. Cell surface triangulations were exported using custom Matlab code and again analysed for surface holes. If required, surface holes were eliminated by custom Matlab code based on closing operations.
\subsubsection{Sample preparation}
Primary murine OT1-Lifeact-GFP and OT1-mT/mG cytotoxic T cells were isolated and cultured as previously described \cite{galeano2020lifeact}. All imaging was done with T cells cultured over 6 or 7 days. For imaging on the home-built system, OT1-Lifeact-GFP T cells were labelled with 100 nM CellTracker Deep Red dye (ThermoFisher Scientific, Waltham, USA).
Keeping all components on ice, collagen matrix solution was prepared by adding 10 \textmu l of 10x PBS, 1.15 \textmu l 1N NaOH and 39 \textmu l T cell medium (TCM), consisting of phenol-free RPMI 1640, 10\% foetal calf serum, 1 mM sodium pyruvate, 10 mM HEPES, 100 U/ml penicillin, 100 \textmu g/ml streptomycin, 2 mM L-glutamine and 50 \textmu M $\beta$2-mercaptoethanol (all from Gibco, ThermoFisher Scientific, Waltham, USA), to 50 \textmu l liquid-phase rat-tail collagen I ($\sim$3 mg/ml; Corning, New York, USA). Coverslip and imaging dish glass surfaces were treated with 2\% (3-aminopropyl) triethoxysilane in ethanol and 6\% glutaraldehyde to facilitate firm attachment of collagen gels. For imaging on the home-built LLSM, 6 \textmu l of collagen mix were placed onto surface-treated round 5 mm coverslips (Warner Instruments, Hamden, USA) and polymerised at 37$^{\circ}$C for 15 min. After polymerisation, 10\textsuperscript{5} T cells in phenol-free TCM were seeded on top of the gel and allowed to infiltrate over 3 h before imaging. For imaging on the Zeiss LLS system, 10\textsuperscript{5} T cells were added to TCM during collagen matrix mix preparation. 70 \textmu l of collagen mix were added to well of 35 mm imaging dishes (Mattek, Ashland, USA) and polymerised at 37$^{\circ}$C for 30 min. After polymerisation, 1 ml of pre-warmed phenol-free TCM was added to the dish and cells were allowed to recover for 1 h before imaging.
\subsection{Quantifying 3D cell morphology}
Cell morphologies were quantified using SPHARM. First, the cell surface, described with 3 Cartesian coordinates, $\{x, y, z\}$, is mapped to the unit sphere, described with polar coordinates $\{\theta, \phi\}$, such that the three Cartesian coordinates are functions of the polar coordinates: $\{x(\theta, \phi), y(\theta, \phi), z(\theta, \phi)\}$. $\{x(\theta, \phi), y(\theta, \phi), z(\theta, \phi)\}$ are then be decomposed in terms of the spherical harmonics, $Y_{l}^{m}(\theta, \phi)$, and only $m\geq0$ functions are required \cite{styner2006framework}. For $x$ for example,
\begin{equation}
x(\theta, \phi) = \sum_{l=0}^{\infty}\sum_{m=0}^{l}c_{l, x}^{m}Y_{l}^{m}(\theta, \phi),
\label{eq:decomposition}
\end{equation}
and the (in general complex) coefficients, $c_{l, i}^{m}$ with $i\in \{x, y, z\}$, represent the morphology. We used the SPHARM-PDM software package \cite{styner2006framework} to find the coefficients for the T cells with $l_{max}=15$ and cell meshes converted to voxel grids with a spatial resolution of 0.5 $\mu m$ for computational speed. The additional variable for capturing polarisation information was $D_{0}$. This was the distance between the uropod and centroid multiplied by $\frac{3}{2}$, with the numerator reflecting the fact that the harmonics are summed over 3 spatial coordinates and the denominator accounting for the fact that the coefficients have a spatial extent double their magnitude. The uropod was manually selected (aiming for its center) in alternating frames and linearly interpolated. PCA is a dimensionality reduction method that finds a set of uncorrelated linear features (the principal components, PCs), which are the eigenvectors of the data covariance matrix (which for $D_{l}$ has dimensions 16 $\times$ 16) \cite{wold1987principal}. Supplementary Fig. 2c shows the vector composition of each PC. As explored through the main text, PC 1 is largely associated with transitions between run and stop mode morphologies, PC 2 is largely associated with morphological transitions in the run mode, and PC 3 is largely associated with morphological transitions in the stop mode. For implementing PCA, we used the Scikit-learn Python package.
\subsection{Uncertainty quantification}
The uncertainty in the uropod label depends on the curvature of the cell rear, which we quantified using the mean curvature averaged across the 15 closest mesh vertices to the labelled point (with a sub-sampled mesh for computational speed). We then defined the positional uncertainty as the cord length associated with a 20$^{\circ}$ rotational perturbation. To convert this to PC uncertainties, we found the set of possible $D_{0}$ values using mesh vertices within this uncertainty (i.e. within one cord length of the uropod label), calculated the standard deviation, and converted to PC uncertainties by multiplying by the cosine of the angle between the $D_{0}$ and PC vectors in $\{D_{l}\}$ space. This process was repeated for every 10 frames in each dataset to get a single characteristic uncertainty for each PC (the mean) for each dataset. $\tau_{sig}$ was calculated as the mean time take for the uropod to move twice the cord length. Some cells have uropods that are near-stationary, and therefore have a $\tau_{sig}$ comparable with the full dataset duration. To account for such cases, we used a maximum $\tau_{sig}$ of 100 s, in order that these could be plotted for comparison with dynamic cells, but we excluded them from quantitative analysis.
\subsection{Finding a motion variable for small timescales}
We calculated uropod and centroid velocities by finding the displacements between consecutive positions (smoothed with running means over $\tau_{sig}$ for both, for consistency) and dividing by the time step and cube root of cell volume (for invariance to cell scale). The ellipsoid major axis was calculated as the eigenvector with the largest eigenvalue of $A^{T}A$ where A is a matrix of the $l=1$ spherical harmonic coefficients \cite{brechbuhler1995parametrization}: $\frac{\sqrt{3}}{3\sqrt{2\pi}}(\mathbf{c}_{1}^{-1}-\mathbf{c}_{1}^{1}, i(\mathbf{c}_{1}^{-1}-\mathbf{c}_{1}^{1}), \sqrt{2}\mathbf{c}_{1}^{0})$. For comparing the uropod-centroid (UC) and ellipsoid axes, we used running means for uropod and centroid with a time window of 100 s for long-timescale behaviour. We compared cells where the uropod speed was above 0.0025 s$^{-1}$, i.e. moving more than a quarter cell length in 100 s, and the distance between the uropod and centroid velocity vectors was within half the uropod speed, i.e. they were aligned.
\subsection{Timeseries autocorrelation functions and power spectra}
Autocorrelation functions and power spectra were computed for longer duration datasets for each of the run and stop mode. We removed timeseries with low SNR: PC timeseries where the ratio of the signal standard deviation to the PC uncertainty was below 2.5 and speed timeseries where $t_{sig}$ was of a similar scale to the full dataset duration. There was one removal for each of PC 1, 3 and speed, across different cells. We calculated the autocorrelation on de-trended timeseries, in order to only capture statistically significant correlations, removing trends with frequencies lower than 0.0025 Hz (corresponding to a period of approximately half the total dataset duration) with a Butterworth highpass filter \cite{butterworth1930theory}. We then found a decay time, $\tau\textsubscript{ACF}$, by fitting an exponential decay model, $y=e^{-\frac{x}{\tau\textsubscript{ACF}}}$, to the peaks of the ACF (rather than the full ACF, which is more appropriate for non-oscillatory patterns).
\subsection{Continuous wavelet transform}
The continuous wavelet transform was used to find local morphodynamics (or `behaviours') from the PC timeseries. A wavelet that decays to zero either side of a central peak is convolved with the timeseries, which produces a new timeseries where each element now represents local morphodynamics. Repeating this process with dilated versions of the wavelet and stacking the resulting set of timeseries yields a spectrogram with multiscale dynamic information, where high-frequency components are analysed close in time, but lower frequency information bleeds in from afar. This spectrogram is then mapped to an interpretable 2D space using t-SNE \cite{van2008visualizing}, and a PDF can be computed with kernel density estimation \cite{davis2011remarks}. t-SNE (t-distributed stochastic neighbour embedding) is a non-linear dimensionality reduction method that uses machine learning. Two similarity metrics between datapoints are defined for each of the two representations, the initial (high-dimensional) representation and the target (lower-dimensional) representation. The difference between the distributions of these similarities across all data pairs is minimised. For implementing t-SNE, we used the Scikit-learn Python package with default parameters: perplexity (analogous to the number of neighbours in other algorithms) of 30, learning rate of 200, and 1000 iterations.
We identified stereotyped motifs (PDF peaks) using adaptive binarisation, a method that thresholds pixels in an image with a threshold value that depends on the local statistics: the mean over a surrounding square of pixels with an added bias (we used square dimensions of 7 and a bias of 20, found with a grid search). We used adaptive rather than pure binarisation so that regions with high-density peaks and high PDF between then (`superhighways' representing common transitions) could be separated, while lower peaks in absolute terms could also be captured. We used two simple wavelets, the `mexican hat' wavelet and Gaussian 1\textsuperscript{st} derivative wavelet, with the combination of the two required to capture symmetric and antisymmetric features. For organisms where the morphodynamics of interest are organised in repeating bouts, e.g. high-frequency wing-beating of fruit flies, complex wavelets that enable the removal of phase information can be useful. However, over the timescales analysed here, T cell morphodynamics are slower-changing, and phase information is important. We used six equally-spaced frequencies for each wavelet from double the Nyquist limit up to the (wavelet-specific) frequency with width of influence corresponding to 150 s, the approximate timescale of organisation found from the autocorrelation analysis. The width of influence was found by convolving each wavelet with a square pulse to find where edge effects begin. When repeating this method for only the four run mode datasets, we used for the adaptive binarisation parameters square dimensions of 15 and a bias of 50, again found with a grid search.
\subsection{Comparing marginal morphodynamics of the run mode}
The marginal morphodynamics form continuums, and so transition matrices over stereotyped PDF peaks cannot be defined. Instead, we defined transition matrices over points on a grid. We then quantified the entropy for the transition dynamics of each PC (and compared with that of their combined dynamics). The entropy is $-\sum{\pi_{i}p_{ij}log_{2}p_{ij}}$, where $\pi_{i}$ is the equilibrium distribution and $p_{ij}$ is the probability that the next motif to be visited after $i$ will be $j$. For plotting the PC 2 dynamics of the four cells, we perturbed the wavelets slightly to further improve the interpretability. This was done by searching locally across options for the maximum wavelet width (keeping 150 s as an upper bound) and finding combinations with reduced entropy. Reduced entropy was associated with reducing the Gaussian wavelet maximum width to 100 s, but with the same Mexican hat wavelets as before.
\end{small}
\vspace{5mm}
\noindent \textbf{Author Contributions} \\
DK, JM and MB acquired and segmented the cell surface segmentation data; HC and RE performed the downstream morphological and morphodynamic analysis. \\
\noindent \textbf{Data accessibility} \\
The cell surface segmentation data that support the findings of this study have been deposited on Dryad\\
(DOI: \url{https://doi.org/10.5061/dryad.tdz08kq1r}). The code used is available at \url{https://github.com/hcbiophys/tcells_paper_code}. \\
\noindent \textbf{Funding Statement} \\
This work was funded by the Biotechnology and Biological Sciences Research Council (grant number BB/M011178/1) to RE and the Australian Research Council (Discovery Project grant DP180102458) to MB.\\
\noindent \textbf{Acknowledgements} \\
The authors thank ND Geoghegan, KL Rogers, N Tubau and LW Whitehead (WEHI, Melbourne, Australia) for technical assistance with LLS microscopy and image denoising. MB acknowledges Bitplane AG for an Imaris Developer licence, and HC and RE thank Suhail Islam for invaluable computational suppport. \\
\noindent \textbf{Competing Interests} \\
The authors declare no competing interests. \\
\printbibliography
\end{document}
\section{Supplementary Figures}
\pagenumbering{arabic}
\subsection{Supplementary Figure 1: 3D T Cell Migration}
\begin{figure}[H]
\center{\includegraphics[]
{figures/3d.pdf}}
\caption{\label{fig:3d} \textbf{3D T Cell Migration.} \textbf{(a)} Representative snapshot of a T cell migrating in a 3D collagen gel. Scale bar: 5 \textmu m. \textbf{(b)} Migration tracks of T cells embedded in a 3D collagen gel. Left: xy view. Right: xz view. Starting point of tracks was translated to origin of coordinate system for visualisation purposes.}
\end{figure}
\subsection{Supplementary Figure 2: Full meshes and principal components of the sampled frames}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_pca.pdf}}
\caption{\label{fig:SI_pca} \textbf{Full meshes and principal components of the sampled frames.} \textbf{(a)} Shape changes associated with each PC are shown, found by splitting the PCA space into 7 equal-length bins along each axis and plotting the T cell within each bin with the lowest value for the other PCs ($l_{max}=3$ reconstructions in the main text, full cells here). An increasing PC 1 represents elongation and front-widening, a decreasing PC 2 represents contraction with front-widening, and an increasing PC3 represents elongation (forward or sideways), with the centroid moving towards the uropod. \textbf{(b)} Normalised PC values of the displayed cells, where black colouring indicates which PC is being sampled. \textbf{(c)} Vector composition of each PC, in terms of the spherical harmonic descriptors, $D_{l}$. \textbf{(d)} Dimensionality across the main mode of variation (PC 1) is relatively constant. For data both below the mean along PC 1 (red) and above it (green), the explained variance ratios by a new set of PCs, PC$'$, are plotted, and these decay at similar rates. \textbf{(e)} Difference in the spherical harmonic spectra (expressed through the descriptors, $D_{l}$) between the low and high PC 1 (below and above the PC 1 mean, respectively) populations.}
\end{figure}
\subsection{Supplementary Figure 3: Uropod uncertainty quantification and propagation to downstream variables of interest}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_uncertainty.pdf}}
\caption{\label{fig:SI_uncertainty} \textbf{Uropod uncertainty quantification and propagation to downstream variables of interest.} \textbf{(a)} Mean curvature across the closest 15 mesh vertices to the uropod label was calculated (with a sub-sampled mesh for computational speed). \textbf{(b)} Uropod uncertainty is defined as the cord length associated with a 20$^{\circ}$ rotational perturbation. \textbf{(c)} PC uncertainties for each video were found by recalculating $D_{0}$ with all mesh vertices within this uropod uncertainty, calculating the standard deviation, and multiplying this by the cosine of the angle between the $D_{0}$ and PC vectors in $\{D_{l}\}$ space for every 10 frames in each video, then calculating the mean. \textbf{(d)} Uropods were also tracked for connecting morphodynamics to motion. To ensure these are at sufficient signal-to-noise ratio, we found for each video the mean time taken for the uropod to move a significant distance (defined as twice the cord length), $\tau_{sig}$, and then computed velocities using uropod running means with a time window of $\tau_{sig}$. We used a maximum $\tau_{sig}$ of 100s to ensure near-stationary cells were included in visualisations, but these insignificant velocities were omitted from quantitative analysis.}
\end{figure}
\subsection{Supplementary Figure 4: Finding a motion variable for small timescales}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_speed.pdf}}
\caption{\label{fig:SI_speed} \textbf{Finding a motion variable for small timescales.} \textbf{(a)} We tracked two variables that can be used to link morphodynamics with motion: the uropod and the centroid. Their velocities (dividing by the cube root of volume for scale invariance) are 3D and not rotationally invariant, and a simple description in terms of the speed of either does not adequately separate distinct motifs, like turning (motif 1) and moving forward (motif 2). An internal reference frame is needed. \textbf{(b)} There are two options: the centroid-uropod (UC) axis and ellipsoid axis. A histogram shows that at times when the whole cell moves in unison, this happens more along the UC axis than the ellipsoid axis, with an example cell where the two differ shown. Running means for uropods and centroids with a time window of 100s were used for long-timescale behavior. Motion in unison was taken to be when uropod speed was above 0.0025 s$^{-1}$, i.e. moving more than a quarter cell length in 100s, and the distance between the uropod and centroid velocity vectors was within half the uropod speed, i.e. they were aligned. \textbf{(c)} Descriptors in terms of the speed of the uropod and centroid along the UC axis (speed\textsubscript{uropod,UC} and speed\textsubscript{centroid,UC}) can then differentiate motifs 1 and 2. \textbf{(d)} These describe largely irreversible motion (now with running means using the time windows from the uncertainty analysis, to include shorter-timescale behavior). speed\textsubscript{uropod, UC} has lower variance and fewer reversals, and Supplementary video 9 and Fig. 1f show examples where speed\textsubscript{centroid, UC} is much more oscillatory. We therefore selected speed\textsubscript{uropod, UC} as the cell motion variable, and referred to it simply as speed.}
\end{figure}
\subsection{Supplementary Figure 5: Link between run-and-stop modes and shape}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_correlations.pdf}}
\caption{\label{fig:SI_correlations} \textbf{Link between run-and-stop modes and shape.} \textbf{(a)} The link between cell shape and speed is shown through correlations between the shape PCs, raw speed, and the running mean of speed with a window size of 150s. PCs 1 and 2 have a stronger correlation with long rather than short-timescale speed, indicating that shape is specialised more for migration mode than instantaneous speed, with cells in the run mode longer and thinner than those in the stop mode. $p$-values and $r$-values (Pearson correlation coefficients) are shown. \textbf{(b)} The autocorrelations (ACFs) were calculated for four long videos from each of the stop and run modes. Decay timescales, $\tau\_{ACF}$, were found using exponential decay models fitted to the peaks of the oscillating ACFs. For cells in the stop mode, PC 3 is the most strongly autocorrelated, followed by PC 2. For cells in the run mode, the main differences are a large drop in the autocorrelation of PC 3, meaning PC 2 becomes the most autocorrelated shape variable, and an increase in the autocorrelation of speed. \textbf{(c)} The power spectra of the PC and speed time series. The run mode is also associated with larger oscillations in PC 2 and speed. Only powers above the mean variance of associated with PC uncertainty from the uropod labelling are shown.}
\end{figure}
\subsection{Supplementary Figure 6: Continuous wavelet transform used to map stereotyped motifs}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_wavelets_method.pdf}}
\caption{\label{fig:SI_wavelets_method} \textbf{Continuous wavelet transform used to map stereotyped motifs.} \textbf{(a)} The continuous wavelet transform was used to acquire a spectrogram capturing local multi-scale dynamic information from the PC time series, with two wavelet types to ensure both symmetric and antisymmetric features are captured (`mexican hat' and Gaussian 1\textsuperscript{st} derivative). We used 6 frequencies per wavelet, from double the Nyquist limit up to the frequencies associated with widths of influence of approximately 150s, as found from the autocorrelation analysis to be the timescale of morphodynamic organisation. The spectrogram, which represents morphodynamics at each time point, was embedded in an interpretable 2D morphodynamic space using t-SNE. \textbf{(b)} Robustness of the morphodynamic space found with t-SNE. The perplexity parameter is analogous to the number of neighbours in alternative dimensionality reduction methods, and the value used was 30 (the default for the Scikit-learn Python package). Colouring four of the motifs (top) shows that the embeddings are similar across the perplexity range suggested in the original t-SNE paper \cite{van2008visualizing}. Re-running the algorithm with a perplexity of 30 coloured by cell (bottom) also shows the embeddings are robust across random initialisations. }
\end{figure}
\subsection{Supplementary Figure 7: Global morphodynamic space with further examples.}
\begin{figure}[H]
\center{\includegraphics[width = 0.9\linewidth]
{figures/SI_2_motifs.pdf}}
\caption{\label{fig:SI_2_motifs} \textbf{Two examples for each stereotyped motif from Fig. 4a.} }
\end{figure}
\subsection{Supplementary Figure 8: PC dynamics of the stereotyped motifs}
\begin{figure}[H]
\center{\includegraphics[width = \linewidth]
{figures/SI_local_PCs_combined.pdf}}
\caption{\label{fig:SI_local_PCs_combined} \textbf{PC dynamics of the stereotyped motifs.} Three principal component (PC) time series for each of the stereotyped motifs are shown, with a 150s time window, and aligned in the $y$ direction so the middle times coincide. Colours indicate the PC and different PC series have different line styles.}
\end{figure}
\subsection{Supplementary Figure 9: Run mode morphodynamics}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_wavelets_run.pdf}}
\caption{\label{fig:SI_wavelets_run} \textbf{Run mode morphodynamics.} \textbf{(a)} A morphodynamic space now formed exclusively from the four long videos of cells in the run mode shows the morphodynamic composition with higher resolution. \textbf{(b)} Examples of the stereotyped motifs, each over a 150s period. \textbf{(c)} Colouring by speed shows that motif 3 (compress and front-widen) is associated with the highest speeds. \textbf{(d)} The transition probability matrix (sequential stereotyped peaks). \textbf{(e)} Marginal dynamics of each PC. These form continuums, and so transition matrices were defined over points on a grid. We then quantified the entropy for the dynamics of each PC (and compared with that of their combined dynamics) and found an entropy minimum in PC 2 (consistent with the autocorrelation results). The entropy is $-\sum{\pi_{i}p_{ij}log_{2}p_{ij}}$, where $\pi_{i}$ is the equilibrium distribution and $p_{ij}$ is the probability that the next motif to be visited after $i$ will be $j$.}
\end{figure}
\subsection{Supplementary Figure 10: PC dynamics of the stereotyped motifs in the run mode}
\begin{figure}[H]
\center{\includegraphics[]
{figures/SI_local_PCs_split.pdf}}
\caption{\label{fig:SI_local_PCs_split} \textbf{PC dynamics of the stereotyped motifs in the run mode.} Three principal component (PC) time series for each of the stereotyped motifs are shown, with a 150s time window, and aligned in the $y$ direction so the middle times coincide. Colours indicate the PC and different PC series have different line styles.}
\end{figure}
\printbibliography
\end{document}
| 2024-02-18T23:40:02.761Z | 2022-04-11T02:01:53.000Z | algebraic_stack_train_0000 | 1,170 | 10,152 |
|
proofpile-arXiv_065-5790 | \section{Introduction}\label{sec:Intro}
Most galaxies, including our own, host a MBH at its center \citep[e.g.][]{Kormendy+95,Ghez+00,Ghez+08,Kormendy+Ho}. Around these MBHs are the incredibly dense structures composed of stars and stellar remnants, called nuclear star clusters (NSCs).
Both theoretical and observational evidence suggest that binaries are common in NSCs. On the theoretical side, \citet{Stephan+16} showed that as many as 70\% of the binaries formed in an initial star formation episode survive after a few Myrs. Furthermore, \citet{Rose+20} showed that these surviving young (few Myrs) binaries can have a wide range of orbital configurations and separations. \citet{Naoz+18} showed that a high binary fraction can explain many puzzling aspects of the young stellar disk in our own Galactic Center \citep[e.g.,][]{Yelda+14}.
On the observational side, recent works have suggested that binaries are prevalent in our Galaxy for massive stars (about $70\%$ for ABO spectral type stars) \citep[e.g.][]{Raghavan+10,Moe+17}. Thus, we can reasonably infer a comparably large binary fraction for massive stars in NSCs. Furthermore, the large number of observed X-ray sources in the central 1 pc suggest relatively large numbers of binaries containing BHs and NSs in the Galactic Center \citep{Muno+05,Hailey+18}. Moreover, there are three confirmed binaries in the inner $\sim 0.2$~pc \citep[e.g.,][]{Ott+99,Martins+06,Rafelski+07,Pfuhl+14}, with possibly more candidates \citep[e.g.,][]{Jia+19,Gautam+19}.
For massive binaries, a potentially important effect is the ``natal" kick that neutron star progenitors (and debatedly, black hole progenitors), receive from supernova explosions. Observations of pulsar proper motions have shown that neutron stars (NSs) receive large kicks of hundreds of km/s \citep[e.g.][]{Hansen+Phinney97,Lorimer+Bailes97,Cordes+Chernoff98,Fryer+99,Hobbs+04}. Natal kicks have been shown to explain the spin-orbit misalignment of pulsar binaries \citep{Lai+95,Kalogera96,Kaspi+96,Kalogera+98,Kalogera00}. Furthermore, it has been suggested that hypervelocity stars \citep{Zubovas+13,Bortolas+17,Fragione+17,Lu+19}, and EMRIs \citep[e.g.,][]{Lu+19,Bortolas+19}, can result from natal kicks.
Recently, \citet{Lu+19} explored natal kicks in hierachical triple systems, including stellar-mass binaries near an MBH \citep[see also][]{Hamers+18}. They found that the natal kicks in these systems can lead to the inner binary shrinking as well as expanding, leading to many possible observable phenomena. In this work, we explore the dynamical and observable consequences of two successive natal kicks on a population of stellar binaries in the Galactic Center. Below we give a brief summary of the steps we take in our study and the structure of this paper:
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{Schematic.jpeg}
\caption{A simplified schematic of our analysis process. Note that this schematic does not illustrate every single possible outcome of natal kicks on a binary.}
\label{fig:schematic}
\end{figure*}
\begin{enumerate}
\item We generate a population of massive stellar binaries. This is described in Section \ref{sec:MC}.
\item We evolve the stellar binaries up to the $m_1$'s natal kick and eliminate any binaries that undergo Roche Lobe overflow before $m_1$'s natal kick. This process is described in Section \ref{sec:preSN}.
\item We apply the natal kick to $m_1$ and recalculate the orbits. We check the new orbits for any natal kick-induced observable phenomena (e.g. X-ray Binary, hypervelocity star). We then apply the natal kick to $m_2$ and repeat the process of checking for outcomes. We describe how we apply the natal kicks in more detail in Section \ref{sec:SNkicks}. We describe the criteria we used to classify the outcomes and describe the results of our Monte Carlo simulations in Sections \ref{sec:single_outcomes} and \ref{sec:binary_outcomes}.
\end{enumerate}
A simplified schematic of our steps is given in Figure \ref{fig:schematic}, and a summary of the outcomes of our simulations is given in bar chart form in Figure \ref{fig:AggregateStats}.
\section{Methodology}
We employ simple Monte Carlo simulations to explore the outcomes of two natal kicks on massive stellar binaries orbiting a MBH within $0.1$~pc. Inwards to this limit $\sim 1.5\times 10^5$, stars are expected to reside, estimated from the M-$\sigma$ relation \citep[e.g.,][]{Tremaine+02}. Thus, this limit is used to estimate the effects of natal kicks on the system close to the MBH.
\subsection{Monte Carlo Birth Distributions}\label{sec:MC}
We run a total of 300,000 Monte Carlo simulations of a binary star (the ``inner binary") orbiting a MBH (the ``outer binary") in a hierarchical triple configuration system. The inner binary comprises of main sequence stars with birth masses $m_1$ and $m_2$, and the MBH has a mass of $m_{\rm MBH} = 4 \times 10^6~\ensuremath {\rm M}_{\odot}$. The inner (outer) orbit is characterized by the following orbital parameters: the semi-major axis $a_1$($a_2$), the eccentricity $e_1$($e_2$), the argument of periapsis $\omega_1$ ($\omega_2$), the longitude of the ascending node $\Omega_1$ ($\Omega_2$), and the true anomaly $f_1$($f_2$). The inner and outer orbits have a mutual inclination of $i_{\rm tot}$.
For our Monte Carlo simulations we set $m_1$ as the more massive star and therefore is always the one that undergoes the first supernova in the system. The birth value of $m_1$ is chosen from a Kroupa mass distribution ranging from $8-100~\ensuremath {\rm M}_{\odot}$ \citep{Kroupa01}, and $m_2/m_1= q$, where the mass ratio $q$ is chosen from a uniform distribution ranging from $0.1 -1$. Note that because of this mass distribution, some $m_2$'s will be white dwarf (WD) progenitors, and a small percentage will not undergo SN at all. Among the systems that are not eliminated in the pre-SN evolution (see Section \ref{sec:preSN}), \NScount \% of stars become NSs, \BHcount \% become BHs, WD \WDcount \% become WDs, and \STcount \% never become a CO.
The inner (outer) orbital eccentricities are chosen from a uniform (thermal) distribution ranging from $0-1$. We distribute the mutual inclination $i_{\rm tot}$ isotropically. Further the arguments of periapsis, true anomalies, and the inner binary longitude of the ascending node $\Omega_1$ are chosen from a uniform distribution between $0$ and $2 \pi$. Since we are working in the invariable plane reference plane, $\Omega_2 = \Omega_1 - \pi$.
The outer semi-major axis $a_2$ is chosen from a number density distribution that goes as $n \propto r^{-2}$ with a minimum value of $1000$~au, which is roughly the semi-major axis of the well-studied star S0-2 in the S-cluster in the Galactic Center, and a maximum value of $0.1$~pc (see above). We choose the inner binary semi-major axis $a_1$ from the period distribution $dn/dP \propto {\rm log}(P)^{-0.45}$ \citep{Sana+13}, with the minimum and maximum values of $a_1$ chosen for each system according to the following criteria. First we require the inner orbit pericenter to be greater than 2 times the Roche limit of the system:
\begin{equation}\label{eqn:pericenter_Roche}
a_1 (1 - e_1) > 2~a_{\rm Roche}.
\end{equation}
The Roche limit $a_{{\rm Roche},ij}$ for either star in the inner binary\footnote{For Equation \ref{eqn:pericenter_Roche}, we take $a_{\rm Roche} = {\rm max}[a_{{\rm Roche},12},a_{{\rm Roche},21}]$} is defined as:
\begin{equation}\label{eq:aRoche}
a_{{\rm Roche},ij} = \frac{R_j}{\mu_{{\rm Roche},ji}},
\end{equation}
where $R_j$ is the radius of the star of mass $m_j$, found by the ZAMS mass radius relation:
\begin{equation}\label{eq:radius}
R(m) = 1.01~m^{0.57}~\ensuremath {\rm R}_{\odot}
\end{equation}
\citep{Demircan+Kahraman}, and $\mu_{{\rm Roche},ji}$ is defined by:
\begin{equation}
\mu_{{\rm Roche},ji} = \frac{0.49(m_j/m_i)^{2/3}}{0.6 (m_j/m_i)^{2/3} + {\rm ln}(1 + (m_j/m_i)^{1/3})}
\end{equation}
\citep{Eggleton83}. To find the upper limit of $a_1$, we require that each triple system is dynamically stable \citep[][]{NaozReview}:
\begin{equation}
\frac{a_1}{a_2}\frac{e_2}{1 - e^2_2} < 0.1
\end{equation}
Finally, we require that the inner binary does not cross the Roche limit of the central MBH, and thus eliminate systems that do not fulfill the criterion below:
\begin{equation}\label{eq:RocheCrossing}
a_2 (1 - e_2) > a_1 (1 + e_1)\Big(\frac{3~m_{\rm MBH}}{m_1 + m_2}\Big)^{1/3}.
\end{equation}
\citep[e.g.][]{Naoz+Silk}. We show the initial distributions of $m_1$, $m_2$, $a_1$, $a_2$, $e_1$, and $e_2$ in Figure \ref{fig:ICs}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{InitialConditions.png}
\caption{Distributions of the initial triple parameters.}
\label{fig:ICs}
\end{figure*}
\subsection{Pre-SN Evolution}\label{sec:preSN}
For each star in the inner binary, we find the time at which it becomes a compact object and its mass before and after this transition with {\tt SSE} \citep{Hurley+00}. The stars experience mass loss due to main sequence evolution between birth and the first supernova, and between the first supernova and the second supernova. Between birth and the first supernova the inner binary will expand due to mass loss (the outer binary will also expand, but by a negligible amount due to the large mass of the MBH). We find the inner binary semi-major axis immediately before the first supernova, $a_{1,{\rm pre-SN}}$ by adopting adiabatic expansion, which conserved specific angular momentum.:
\begin{equation}\label{eq:massloss}
a_{1,{\rm pre-SN}} = \frac{m_1 + m_2}{m_{1,{\rm pre-SN}} + m_{2,{\rm pre-SN}}} a_1 \ .
\end{equation}
We note that we do not perform a numerical integration of the triple system either before the first kick or between the first and second kick. Eccentricity oscillations in the inner binary driven by the Eccentric Kozai-Lidov effect \citep[EKL,][]{Naoz16}, coupled with stellar evolution, can cause the inner stellar binary to become tidally locked, collide, or overflow its Roche Lobe, long before either of the stars undergo a supernova \citep[e.g.,][]{Stephan+16,Stephan+19}. We cannot account for all of these systems without a numerical simulation. However, we use an analytical criteria to identify systems where EKL causes the stellar binary to overflow its Roche Lobe. The EKL timescale on which the inner binary experiences eccentricity oscillations is approximately \citep[e.g.,][]{Antognini15}:
\begin{equation}\label{eq:tEKL}
t_{\rm EKL} \sim \frac{16}{15}\frac{a^3_2 (1 - e^2_2)^{3/2} \sqrt{m_1 + m_2}}{a^{3/2}_1 m_{\rm MBH} k} \ .
\end{equation}
This timescale needs to be shorter than the inner binary General Relativity (GR) precession timescale $t_{\rm GR}$ in order for EKL to excite the inner binary eccentricity\footnote{However, in some cases, when the GR timescale is comparable or longer than the EKL timescale, the GR precession can trigger eccentricity excitation even in regimes that supress the large eccentricity excitations \citep[e.g.,][]{Naoz+2013,Hansen+20}.}. The GR precession timescale of the inner binary is defined as:
\begin{equation}\label{eq:tGR}
t_{\rm GR, inner} \sim 2 \pi \frac{a^{5/2}_1 c^2 (1 - e^2_1)}{3 G^{3/2} (m_1 + m_2)^{3/2}} \ ,
\end{equation}
where $c$ is the speed of light.
For systems with $t_{\rm ELK} < t_{\rm GR, inner}$ and mutual inclination between $40^{\circ}$ and $140^{\circ}$ the maximal eccentricity $e_{1, {\rm max}}$ achieved by the inner binary during one Kozai-Lidov cycle can be approximated analytically \citep[e.g.][]{Miller+Hamilton,Wen,Fabrycky+Tremaine,Liu+15}. We approximate $e_{1, {\rm max}}$ using the method outlined in \citet{Wen}.
If the minimum pericenter distance achieved during $t_{\rm EKL}$ (we also require that $t_{\rm EKL}$ is less than the timescale to the first SN) is less than the binary Roche Limit $a_1 (1 - e_{1,{\rm max}}) < a_{\rm Roche}$, the stellar binary will undergo Roche Lobe overflow and mass transfer. We found that roughly 10\% of initial stellar binaries fit these criteria. The subsequent binary evolution is out of the scope of our methodology. Thus, we eliminate these systems from our simulations and do not evolve them to the supernova stage. We note that the approach here uses the three-body Hamiltonian up to the quadrupole order, and cannot account for extreme eccentricities excited by the octupole term \citep[e.g.][]{NaozReview}. Moreover, for inner and outer eccentric orbits, the EKL mechanism can excite high eccentricity for wide range of inclinations \citep[e.g.,][]{LN,Naoz+11sec,Li+13,Li+14Chaos}. Therefore, our approach here eliminates conservative fraction of systems. However, we do expect that the binary orbital distribution from this approach will capture the distribution of the binaries post EKL, just before the first SN \citep[as was highlighted in][]{Rose+19}.
We note that even though we eliminate these systems in the pre-SN stage, the statistics that we give later in the paper are with respect to the initial triple population we generated in Section \ref{sec:MC}. This is so that the event formation fractions we give later on in the paper can be simply multiplied by a star formation rate.
\subsection{Supernova Kicks}\label{sec:SNkicks}
We then apply the supernova kick to $m_1$.
We assume instantaneous supernova kicks that are isotropically distributed in the inner binary orbital frame. For NS progenitors, we assume kicks drawn from a normal distribution with an average of 400 km/s and standard deviation of 265 km/s \citep[e.g.,][]{Hansen+Phinney97,Arzoumanian+02,Hobbs+04}. For WDs, which are present in the second set of kicks, we assume a normal distribution with an average of 0.8 km/s and standard deviation of 0.5 km/s \citep[e.g.,][]{ElBadry+Rix,Hamers+Thompson}. There is much uncertainty about whether BHs receive natal kicks, and if they do, what the distribution of kick velocities would be \citep[e.g.,][]{Gualandris+05BHkick,Willems+05,Fragos+09,Repetto+12}. Thus, we adopt three different kick distributions for BH progenitors:
\begin{enumerate}
\item ``Fast" BH kicks that are drawn from the same distribution as NS kicks.
\item Following \citet{Bortolas+Mapelli}, ``slow" BH kicks that assume BHs receive the same linear momentum as NSs (i.e. we draw a kick velocity from the NS distribution, then multiply this by ($1.4~ \ensuremath {\rm M}_{\odot}/m_{\rm BH}$), where $m_{\rm BH}$ is the mass of the BH progenitor, and 1.4 $\ensuremath {\rm M}_{\odot}$ is the typical NS mass).
\item No BH kicks.
\end{enumerate}
We run 100,000 simulations for each scenario, for a total of 300,000 simulations. We present statistics from all three scenarios in Sections \ref{sec:single_outcomes} and \ref{sec:binary_outcomes}, and in Figure \ref{fig:AggregateStats}.
To apply the the natal kick, we simply add the Cartesian velocity kick vector to the velocity vector of $m_1$, and change $m_1$ to the post-SN mass found with SSE. We then use new velocity vector and mass to recalculate new orbital parameters.
\begin{table}
\makegapedcells
\begin{tabular}{|P{2cm}||P{1cm}|P{1cm}|P{1cm}|P{1cm}|}
\hline
\multirow{2}{2cm}{\textbf{BH kick distribution}}
& \multicolumn{2}{P{2.365cm}|}{\textbf{Inner binary disrupted (\%)}}
& \multicolumn{2}{P{2.365cm}|}{\textbf{Inner binary survived (\%)}} \\
\cline{2-5}
& post-$m_1$ kick
& post-$m_2$ kick
& post-$m_1$ kick
& post-$m_2$ kick
\\
\hline \hline
fast
& 88.2 & 94.5 & 11.8 & 5.5 \\
\hline
slow
& 73.0 & 88.1 & 27.0 & 11.9 \\
\hline
none
& 66.7 & 80.1 & 33.3 & 19.9 \\
\cline{1-5}
\end{tabular}
\caption{\label{tab:innerbinary} \textbf{Percentage of inner binaries disrupted/ survived} after $m_1$'s and $m_2$'s kick. As expected, the percentage of inner binaries that survive both kicks is inversely correlated to the magnitude of BH natal kicks.}
\end{table}
The two main scenarios resulting from this first kick are the \textbf{``inner binary survived"} and the \textbf{``inner binary disrupted"} cases (see Figure \ref{fig:schematic}). In the case that the inner binary survives the first kick, we have two further branching scenarios. The inner binary can either stay bound to the MBH (ellitical orbit), or become unbound from the MBH (hyperbolic orbit). In the case that the inner binary is disrupted by the first kick, the component masses form separate binaries with the MBH (i.e. $m_1$-MBH and $m_2$-MBH). These binaries can either be bound or unbound. We then evolve $m_2$ up to its own natal kick, and recalculate the orbits after $m_2$'s kick. Between $m_1$'s and $m_2$'s kicks we evolve the orbits as follows:
\begin{enumerate}
\item We adiabatically expand the orbits, due to $m_2$'s mass loss, using Equation \ref{eq:massloss}
\item For hyperbolic orbits, we solve the hyperbolic Kepler Equation using the HKE-SDG package \citep{HKE-SDG} to find the hyperbolic anomaly at the time of $m_2$'s kick, from which we calculate the true anomaly. Using the newly calculated true anomaly with the Keplerian elements of the orbit, we calculate the Cartesian coordinates of the objects in the orbit.
\item For elliptical orbits, how we calculate the true anomaly at the time of $m_2$'s kick depends on the timescales of the system. If the timescale between the first and second natal kick is longer than 10 times the orbital period, we simply choose the eccentric anomaly at the time of $m_2$'s kick randomly from a uniform distribution between 0 and $2\pi$. Otherwise, we solve the elliptical Kepler's Equation iteratively using Newton's Method to find the eccentric anomaly. From this we calculate the true anomaly, and then the Cartesian coordinates of the orbit.
\end{enumerate}
After each kick we check whether the natal kick had induced any observable phenomena (e.g., X-ray binary, GW merger, etc.). We describe our criteria in Section \ref{sec:single_outcomes} for the cases where the inner binary is disrupted, and in Section \ref{sec:binary_outcomes} for the cases where the inner binary survived. We also present results and statistics from our simulations for each outcome in the the following two sections. We include a simplified schematic of our process in Figure \ref{fig:schematic}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{orb_config.png}
\caption{\textbf{Summary of orbital configuration resulting from each natal kick}. In the top (bottom) panels we show the results from after the first (second) natal kick. In the left (right) panels we show the orbital configurations where the inner binary survived (is disrupted by) the natal kicks. In the left panels, we use the crossed pattern to represent the binaries that survived the natal kick, but will be disrupted by crossing the MBH Roche limit upon pericenter passage (Section \ref{sec:bin_disruption}). We note that for the ``Single Bound to MBH" cases, we have numbers greater than 1 because this number is taken relative to number of initial binaries, and each binary contains two singles.}
\label{fig:orb_config}
\end{figure*}
\begin{figure*}[!htb]
\centering
\includegraphics[width = \linewidth]{observables.png}
\caption{\textbf{Number of observables per initial stellar binary}\textsuperscript{\ref{ratecalc}}, triggered by the first natal kick (top panel), and by the second natal kick (bottom panel). For details about how we classify the different observables, see the following sections: X-ray binaries (\ref{sec:XrayBin}); HVB candidates (hypervelocity binaries, \ref{sec:HVB}); GW candidates (\ref{sec:GWmergers}); HVS candidates (hypervelocity stars, \ref{sec:HVS}); EMRIs (extreme mass ratio inspirals, \ref{sec:EMRI}); direct plunges (\ref{sec:Plunge}); and TDEs (tidal disruption events, \ref{sec:TDE}).}
\label{fig:AggregateStats}
\end{figure*}
\section{Inner Binary Disrupted by Natal Kicks}\label{sec:single_outcomes}
In the case that the inner binary is disrupted by either $m_1$'s or $m_2$'s natal kick, the result is two separate binaries with each composed of a stellar mass object orbiting the MBH ($m_1-{\rm MBH}$ and $m_2-{\rm MBH}$). These orbits can either be bound (elliptical orbit), or unbound (hyperbolic orbit). However, natal kicks are less likely to result in unbound objects, as we find that the majority of objects remain bound to the MBH. See Table \ref{tab:singles}, for percentages of bound and unbound orbits as a function of stellar type.
The orbital configurations and observables resulting from these systems is described below. We also give rates of each orbital configuration and observables from our Monte Carlo simulations. These rates are summarized in bar chart form in Figures \ref{fig:orb_config} and \ref{fig:AggregateStats}, which also break down the rates by BH kick distribution.
\begin{table*}
\centering
\makegapedcells
\begin{tabular}{|P{2cm}||P{1.5cm}|P{1.5cm}|P{1.5cm}|P{1.5cm}|P{1.5cm}|P{1.5cm}|}
\hline
\multirow{2}{2cm}{\textbf{BH kick distribution}}
& \multicolumn{2}{P{3.365cm}|}{\textbf{NS (\%)}}
& \multicolumn{2}{P{3.365cm}|}{\textbf{BH (\%)}}
& \multicolumn{2}{P{3.365cm}|}{\textbf{WD (\%)}} \\
\cline{2-7}
& bound
& unbound
& bound
& unbound
& bound
& unbound
\\
\hline \hline
fast
& 67.1 & 32.9 & 67.7 & 32.3 & 91.7 & 8.3\\
\hline
slow
& 65.9 & 34.1 & 96.3 & 3.7 & 91.5 & 8.5 \\
\hline
none
& 65.7 & 34.3 & 98.3 & 1.7 & 91.1 & 8.9\\
\cline{1-7}
\end{tabular}
\caption{\label{tab:singles} \textbf{Percentage of single compact object orbits that are bound to/unbound from MBH} after both kicks, for different types of compact objects. Whereas majority of single BHs and WDs remain bound to the MBH after the natal kicks (except for the case where BHs receive ``fast" kicks), about a third of single NSs are unbound from the MBH after natal kicks.}
\end{table*}
\subsection{Single objects bound to MBH}\label{sec:single_bound}
\begin{figure*}[!htb]
\centering
\includegraphics[width = \linewidth]{BoundSingles.png}
\caption{\textbf{Distribution of bound single compact object orbits after two successive natal kicks} in the $(a, 1 -e)$ plane ($a$ and $e$ are the semi-major axis and eccentricity with respect to the MBH). These objects were components of the stellar binaries that were disrupted by the natal kicks. For the NSs and WDs, we show data only from the ``fast" BH kicks simulation set as the NS and WD kick distributions do not change between simulation sets and the results from any one set is fairly representative of the others. For the BHs, we show data from both the ``fast" (in blue) and no BH kicks (in red) simulation sets to demonstrate two extremes. We plot the distribution (with respect to the MBH) of the initial stellar binaries that contained the compact object progenitors in yellow for comparison. In addition, we plot the distribution of EMRI progenitors (empty star symbol) and direct plunge progenitors (empty circles).}
\label{fig:BoundSingles}
\end{figure*}
Single objects bound to the MBH is the most common resulting orbital configuration in our simulations. We find 0.9-1.2 single objects per initial binary bound to the MBH after the first natal kick, and 1.1-1.3 after both natal kicks (see Figure \ref{fig:orb_config}). The vast majority of these orbit are long term stable orbits, with a very small number becoming EMRIs (Section \ref{sec:EMRI}), direct plunges (Section \ref{sec:Plunge}), and TDEs (Section \ref{sec:TDE}).
In Figure \ref{fig:BoundSingles} we show the distribution of NSs, BHs, and WDs with respect to the MBH after two successive natal kicks. Note that each triple is evolved in its own simulation, and each system undergoes natal kicks at different times, thus Figure \ref{fig:BoundSingles} does not accurately show the distribution of a contemporaneous population of COs. However, Figure \ref{fig:BoundSingles} is a good approximation of NS and BH distributions between a few 10 Myrs and $\sim 1 {\rm Gyr}$ after the star formation episode. This is because NS natal kicks occur between $\sim 10-45$ Myr and BH natal kicks occur between $\sim 4-10$ Myr, and the two-body relaxation timescale in the Galactic Center is $\sim 0.1 -1~{\rm Gyr}$ \citep[][]{Rose+20}. WD natal kicks occur between 45 and $1.4 \times 10^4$ Myr, so Figure \ref{fig:BoundSingles} cannot be taken as representation of a comtemporaneous population of WDs at any given time.
Figure \ref{fig:BoundSingles} shows that after two successive natal kicks, the cusp of single NSs is steeper than the initial stellar binary cusp. This is also true for the BHs, if we assume fast BH kicks. WDs and BHs with no kicks more or less follow the initial stellar binary distribution.
\begin{figure*}
\centering
\includegraphics[width=0.9\linewidth]{pk1_bound_slowBHkicks.png}
\caption{\textbf{Distribution of single objects and binaries bound to the MBH after the first natal kick} in the $(a, 1 -e)$ plane (bottom panels), and the cumulative distribution of $a$ (top panels), for the slow BH kicks scenario. The variables $a$ and $e$ are the semi-major axis and eccentricity with respect to the MBH.}
\label{fig:pk1_bound}
\end{figure*}
In fact, a steeper NS cusp is already present after just one natal kick. In Figure \ref{fig:pk1_bound} we show the distribution of single NSs, BHs, stars, and binaries bound to the MBH after the first natal kick, for the slow BH kicks scenario. We see a steeper disribution of NSs than for BHs and stars, and a steeper distribution of binaries with a NS primary than for binaries with a BH primary. Thus, for a relatively young stellar population, before mass segregation has taken place, we expect a steeper cusp of NSs relative to the stellar and BH cusp. This is of course excepting the case in which BHs receive fast kicks similar to NSs, in which case the BH cusp will be as steep as the NS cusp after the first natal kick (not shown in figure).
\subsubsection{Extreme Mass Ratio Inspirals (EMRIs)}\label{sec:EMRI}
Extreme Mass Ratio Inspirals (EMRIs) are gradual GW inspirals of stellar-mass COs onto MBHs, and are one of the main targets for the future Laser Interferometer Space Antenna (LISA) \citep{LISAL3}. The timescale on which a CO inspirals onto an MBH is \citep{Peters64}:
\begin{equation}
t_{\rm GW, EMRI} \sim \frac{5}{64}\frac{c^5 a^4}{G^3 m^2_{\rm MBH} m}(1 - e^2)^{7/2} \ .
\end{equation}
In a dense environment such as the Galactic Center, we also have to take into account the effects of two-body relaxation on the potential EMRI. While two-body relaxation from other stars in the cluster can create EMRIs by pushing COs onto high eccentricity orbits, it can also push COs onto more circular orbits, such that $t_{\rm GW, EMRI}$ increases to longer than a Hubble time. Two-body relaxation changes an orbit's angular momentum by an order of itself on a timescale of $\sim (1 - e)t_{\rm relax}$, where $t_{\rm relax}$ is:
\begin{equation}
t_{\rm relax} = \frac{0.34}{{\rm ln} \Lambda}\frac{\sigma^3(r)}{G^2 \rho(r) m},
\end{equation}
where $\rm ln~\Lambda = 15$ is the Coulomb algorithm, $\sigma(r)$ is the velocity dispersion, given by \citep{Kocsis+11}:
\begin{equation}\label{eq:sigma}
\sigma(r) = 280~{\rm km~s^{-1}}~\sqrt{\frac{0.1~{\rm pc}}{r}},
\end{equation}
and $\rho(r)$ is the Galactic Center stellar density \citep{Genzel+10}:
\begin{equation}\label{eq:rho}
\rho(r) = 1.35 \times 10^6~\ensuremath {\rm M}_{\odot}~{\rm pc}^{-3} (\frac{r}{0.25~{\rm pc}})^{-1/3} \ .
\end{equation}
Thus, we require that $t_{\rm GW,EMRI} < (1 - e) t_{\rm relax}$ to be classified as an EMRI \citep[e.g.,][]{EMRIReview}.
EMRI formation is a rare event in our simulations. In particular, we find \EMRI\ EMRIs per initial stellar binary\footnote{\label{ratecalc}Here we remind the reader that we generated 100,000 initial conditions per set of Monte Carlo simulations. To arrive at the ``EMRIs (or any other outcome) per initial stellar binary" figure, we simply count the number of EMRIs (or any other outcome) in each set of simulations and divide this by 100,000.}. This is roughly consistent with results from \citet{Bortolas+Mapelli}, which studied EMRIs triggered by natal kicks on single stars in the Galactic Center and and found a rate of $10^{-7} - 10^{-4}$ EMRIs per SN. Assuming a star formation rate of $10^3~\ensuremath {\rm M}_{\odot} {\rm Myr}^{-1}$ in the Galactic Center \citep[e.g.][]{Lu+09}, and an average binary mass of $30 \ensuremath {\rm M}_{\odot}$ (from our Monte Carlo simulations), we obtain an EMRI rate of $(1 - 2.7) \times 10^{-9}~{\rm yr}^{-1}$. Assuming a galaxy density of $0.02~{\rm Mpc}^{-3}$, and that half of galaxies contain a MBH at the center, we calculate an EMRI volume rate of $0.01-0.027~{\rm Gpc}^{-3}~{\rm yr}^{-1}$ from our simulations\footnote{Note that these rates are about one to two orders of magnitude lower than the EMRI rate estimated via two-body relaxation processes \citep[e.g.,][]{Hopman+Alexander06}, and three to four orders or magnitude lower than the estimated rate of MBH binaries \citep{Naoz+22}. }. As shown by \citet{Bortolas+19}, these rates are sensitive to the inner edge of the initial binary distribution around the MBH.
Interestingly, the three different BH kick cases give very similar EMRI formation rates. This is because in our simulations most EMRIs involve a NS inspiraling into the MBH (see Figure \ref{fig:BoundSingles}). Out of EMRIs from all three BH kick cases, \NSemri\ \% are NS-EMRIs, \BHemri\ \% are BH-EMRIs, and \WDemri\ \% are WD-EMRIs.
Note that the WDs will not behave like a NS or BH EMRI because while we have a GW signal detectable by LISA while the WD is inspiralling, this signal will stop when the WD is tidally disrupted by the MBH and and replaced with an EM counterpart \citep[e.g.][]{Sesana+08}.
\subsubsection{Direct Plunges}\label{sec:Plunge}
Natal kicks can results in orbits with such a high eccentricity that they cause the orbiting object to plunge directly into the MBH instead of gradually inspiralling like EMRIs. Plunging orbits satisfy:
\begin{equation}
e > 1 - 4\frac{R_s}{a},
\end{equation}
where $a$ and $e$ are respectively the semi-major axis and eccentricity of the post-kick orbit, and $R_s = 2 G m_{\rm MBH}/c^2$ is the Schwarzchild radius of the MBH \citep[e.g.][]{Amaro+07}. Unlike EMRIs, which emit detectable GWs for thousands of orbits, creating a coherent signal, plunges emit a brief
burst of GW radiation \citep[e.g.,][]{Yunes+08,Berry+13}. Therefore direct plunges are not expected to be detected by LISA unless originating from the Milky Way Galactic Center \citep{Hopman+Alexander05}. We find, in our simulations, \Plunge\ direct plunges per initial stellar binary. The distribution of direct plunge-progenitors is shown in Figure \ref{fig:BoundSingles}.
\subsubsection{Tidal Disruption Events (TDEs)}\label{sec:TDE}
Tidal disruption events (TDEs) will happen when the pericenter of the $m_2-{\rm MBH}$ orbit drops below the MBH tidal radius as a result from natal kick of $m_1$. We also require that $m_2$ pass within the tidal radius before it undergoes its own natal kick to qualify as a TDE.
The tidal radius is defined as:
\begin{equation}
r_t \sim R_* \Big(\frac{m_{\rm MBH}}{m_*}\Big)^{1/3},
\end{equation}
where $R_*$ is the radius of the star, given by Equation (\ref{eq:radius}), and $m_*$ is its mass \citep[e.g.][]{Rees88}. We also require that $r_t$ lies outside of the MBH Schwarzchild radius so that the star is not directly swallowed instead of being tidally disrupted. TDEs are a a rare outcome in our simulations. We find \TDE\ TDEs per initial stellar binary for the ``fast" and ``slow" kicks scenarios.
\subsection{Single objects unbound from MBH}\label{sec:unbound_single}
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{post_kick1_slowBHkicks.png}
\caption{\textbf{Objects on unbound orbits after $m_1$'s natal kick} for the ``slow" BH kicks case. Here we show the \textit{immediately post kick} position and velocity of unbound objects (middle row), and the cumulative distribution of the velocities (top row). For the unbound stars and binaries, we also show the time at which $m_2$ will become a CO (bottom row). Note that this time is measured with respect to the initial stellar formation and not with respect to $m_1$'s SN. We also show the population of unbound NS X-ray Binaries in the third column.}
\label{fig:HVS}
\end{figure*}
Natal kicks can result in objects that are unbound from the MBH (i.e.) on hyperbolic orbits. In our simulations we find 0.27-0.36 single objects per initial stellar binary are unbound from the MBH after the first kick, and 0.35-0.45 after both kicks, making this the second most common orbital configuration resulting from our simulations (see Figure \ref{fig:orb_config}). Most objects are ejected from the inner 0.1 pc with velocities between 1000-10,000 km/s in our simulations, as shown by Figure \ref{fig:HVS}. Despite deceleration by the combined gravitational potential of the MBH, nuclear cluster, and galaxy, objects ejected from the Galactic Center with velocities in excess of 1000 km/s should reach $r > 8$ kpc with ease, and retain their high velocities \citep[e.g.][]{Kenyon+08}.
Unfortunately, hypervelocity NSs and BHs are not observable. However, a large number of $m_2$'s are ejected by $m_1$'s natal kick (see Figure \ref{fig:HVS}). Since $m_2 < m_1$ and thus all the $m_2$'s are still stars at the time of $m_1$'s kick, some $m_2$'s may be observed as hypervelocity stars. We discuss this possibility further in Section \ref{sec:HVS}.
\subsubsection{Hypervelocity stars}\label{sec:HVS}
If either the two members of the binary or one member of the binary becomes unbound to the MBH, it may be observed as a hyper-velocity star.
A large number of stars have been detected with velocities larger than the the escape velocity from the Galaxy and with trajectories that are consistent with a Galactic Center origin \citep[e.g.,][]{Brown+05,Brown+07,Brown+09,Brown+09MMT,Brown+10,Brown+12,Brown+18,Hirsch+05,Kollmeier+10,Boubert+18}.
The leading channel to explain these systems has involved a binary star crossing the tidal limit and breaks up, ejecting in the process one of the members \citep[known as the Hills mechanism,][]{Hills88}. The consequences of this mechanism have been explored in the literature in great details \citep[e.g.,][]{Gualandris+05,Bromley+06,Perets+07,Kenyon+08,Antonini+10,Sari+10,Kobayashi+12,Zhang+13,Rossi+14}. Additionally, it was suggested that supernova kick could also contribute to hyper-velocity population \citep[e.g.,][]{Lu+19,Bortolas+19}. As highlighted in Figure \ref{fig:HVS}, we expect to have a population of hypervelocity stars as well as hypervelocity COs after the first natal kick. The first natal kick takes place between 4 and 45 Myr after the star formation episode, and as shown in the bottom row of Figure \ref{fig:HVS}, over half of the ejected stars will not become COs until 100 Myr to 10 Gyr after star formation. Combined with the fact that the majority of these stars are ejected from the inner 0.1 pc with velocities between 1000 and 10,000 km/s, a large fraction of them can be observed as hypervelocity stars outside of the Galactic Center.
\begin{figure*}[t]
\centering
\includegraphics[width=\linewidth]{bound_binary.png}
\caption{\textbf{Distribution of triples that remain after natal kicks} in the ($a_1$,$1-e_1$) plane (top plots, $a_1$ and $e_1$ are the inner binary semi-major axis and eccentricity), and the ($a_2$,$1-e_2$) plane (bottom plots, $a_2$ and $e_2$ are the outer binary semi-major axis and eccentricity). We show X-ray binaries created by $m_1$'s natal kick in red, stable triples in blue, and binary disruptions (these are binaries that survive the natal kicks but will be disrupted by the MBH at the next pericenter passage) in purple. A fraction of the binaries in the stable triples will become GW sources.}
\label{fig:bound_binary}
\end{figure*}
\section{Inner Binary Survived Natal Kicks}\label{sec:binary_outcomes}
In the case that the inner binary survives $m_1$'s or $m_2$'s natal kick (or both), we have a stellar-mass binary orbiting the MBH. The orbital configurations and observables resulting from these systems is described below. We also give rates of each orbital configuration and observables from our Monte Carlo simulations. These rates are summarized in bar chart form in Figures \ref{fig:orb_config} and \ref{fig:AggregateStats}, which also break down the rates by BH kick distribution.
\subsection{Binaries bound to MBH}\label{sec:bin_bound}
We find 0.1-0.29 binaries per initial binary remain bound to the MBH after one natal kick, and 0.04-0.17 after both natal kicks. As highlighted in Figure \ref{fig:orb_config}, assuming no kicks for the BHs increases the survivable rate of binaries around the MBH. Furthermore, the number of BH-BH binaries after two kicks is inversely correlated with the strength of BH kicks. For the ``fast" BH kicks case, BH-BH comprise only 4.2\% of the survived binaries. This percentage increases to 39.5\% and 58.1\% for the ``slow" and no BH kicks case, respectively. As expected, due to high natal kick magnitudes, NS-NS binaries are relatively rare in our simulations. They comprise $\sim 1$\% of all surviving binaries for the fast BH kicks case, and $\sim 0.2$\% for the two other BH kick cases.
In Figure \ref{fig:CO_binaries} we show the distribution of orbital parameters for the surviving CO binaries for the no BH kicks case, compared to the initial binaries. We choose to highlight the no BH kicks because this is the case with the most surviving binaries. In general we find that the surviving binaries have larger semi-major axes ($a_1$) than the initial binaries. Surprisingly, despite the large NS kick magnitudes, the $a_1$ distribution for binaries containing at least one NS (i.e NS-WD, NS-NS, and BH-NS), peaks at a smaller value than for binaries that don't (i.e. BH-WD, BH-BH). Furthermore, the binaries containing at least one NS generally have much large eccentricities ($e_1$) than binaries that don't. We find similar trends with the semi-major axis and eccentricity of the binaries' orbits around the MBH ($a_2$ and $e_2$). In general, NS containing binaries have slightly larger $a_2$ and $e_2$ than the other types of CO binaries.
These surviving binaries can undergo EKL eccentricity excitations from the MBH which may result in GW mergers \citep[e.g.,][]{Hoang+18,Stephan+19}. We explore this effect in Section \ref{sec:GWmergers}. We also find X-ray binaries among the surviving binaries, which we discuss in Section \ref{sec:XrayBin}.
\begin{figure*}
\centering
\includegraphics[width=\linewidth]{CO_binaries.png}
\caption{\textbf{Distributions of CO binaries bound to MBH after both natal kicks} for the no BH kicks case. The initial triple probability densities are shaded in gray for comparison. Notably, CO binaries containing at least one NS tend to be more eccentric than binaries with a BH primary due to the large NS kicks.}
\label{fig:CO_binaries}
\end{figure*}
\subsubsection{X-ray Binary}\label{sec:XrayBin}
Natal kicks can actually tighten the inner binary, causing its pericenter to drop below the binary Roche Limit, creating a X-ray binary. Note that we will only get X-ray binaries after $m_1$'s kick and not after $m_2$'s kick, since one member of the inner binary needs to be still a star. We classify a system as a potential X-ray binary if the inner binary post-kick pericenter is below $a_{\rm Roche}$, given by Equation (\ref{eq:aRoche}).
In our simulations we find \XrayBin\ X-ray Binaries per initial stellar binary, for the no BH kicks and ``fast'' BH kicks, respectively.
In Figure \ref{fig:XrayBinary} we show the distribution of X-ray binaries for the different kick models. Unlike Figure \ref{fig:bound_binary} here we divide the notation to BH X-ray binaries and NS X-ray binaries. As highlighted in Figure \ref{fig:XrayBinary}, the no BH kicks case has zero BH X-ray binaries. Thus, at face value, this implies that detecting BH X-ray binaries at the center of the galaxy may imply that BH may have some kicks\footnote{Note that other formation channels for X-ray binaries, such as EKL, \citep[e.g.,][]{Naoz+16,Stephan+19}, may form BH X-ray binaries even in the absence of BH kicks. }.
Interestingly, our simulations predict some X-ray binaries binaries ejected from the GC (see Figure \ref{fig:HVS}). These comprise about 20\% of all X-ray binaries, for all BH kick scenarios. See Section \ref{sec:HVB} for further discussion.
\subsubsection{GW Candidates}\label{sec:GWmergers}
Natal kicks can shrink the inner binary orbit or increase the eccentricity, or nudge the binary into a part of the parameter space where EKL can excite the binary eccentricity, both scenarios leading to a shorter GW merger time. However, before a binary can merge, it can become unbound due to cumulative encounters with other stars in the cluster \citep{BT}. This takes place on an ``evaporation" timescale of:
\begin{equation}\label{eq:evap}
t_{ev} = \frac{\sqrt{3}\sigma(r)}{32\sqrt{\pi} G\rho(r) a_1\ln \Lambda}\frac{m_{1}+m_{2}}{m_{\rm m_b}} \ ,
\end{equation}
where $\rm ln~\Lambda = 15$ is the Coulomb algorithm, $m_b = 1~\ensuremath {\rm M}_{\odot}$ is the average mass of background stars, and $\sigma(r)$ and $\rho(r)$ are given by Equations \ref{eq:sigma} and \ref{eq:rho} respectively. Note that our orbits are very eccentric, however, as shown by \citet{Rose+20}, the eccentricity may only change the evaporation timescale by a factor of a few.
There are two pathways to merger, depending on whether EKL plays a significant part. In the case where $t_{\rm EKL} > t_{\rm GR,inner}$ (given by Equations \ref{eq:tEKL} and \ref{eq:tGR}), i.e. systems where GR effects dominate over EKL effects and the inner binary does not experience EKL-induced eccentricity excitations, we calculate the inner binary gravitation wave merger timescale following \citet{Peters64}:
\begin{equation}
t_{\rm GW} \sim \frac{5}{265} \frac{c^5 a_1^4}{G^3 (m_1 + m_2) m_1 m_2}(1 - e^2)^{7/2}.
\end{equation}
If $t_{\rm GW} < t_{\rm Evap}$, we denote the system as a GW merger candidate.
In the case where $t_{\rm EKL} < t_{\rm GR,inner}$, we analytically estimate the maximal EKL-induced eccentricity, $e_{\rm 1,max}$ using the method detailed in \citet{Wen}, and estimate the EKL-induced GW merger time as:
\begin{equation}
t_{\rm GW,EKL} \sim \frac{5}{265} \frac{c^5 a_1^4}{G^3 (m_1 + m_2) m_1 m_2}(1 - e_{\rm 1, max}^2)^3,
\end{equation}
\citep[e.g.][]{Liu+Lai18,Randall+Xianyu}. Using the above method, we find \GWmerger\ GW merger candidates per initial stellar binary (see Figure \ref{fig:AggregateStats} for break down by BH kick scenario). Note that this is likely an underestimate because the calculation of $e_{\rm 1,max}$ in \citet{Wen} uses the three-body Hamilton up to only the quadrupole order, and may miss octupole effects, which increases the merger rate \citep{Hoang+18}. Using the same method we used to calculate the volume rate of EMRIs as in Section \ref{sec:EMRI}, we find a volume rate for GW mergers facilitated by natal kicks and EKL of 0.06-0.4 Gpc$^{-3}$ yr$^{-1}$.
\subsection{Binaries unbound from MBH}\label{sec:HVB}
Natal kicks can leave the inner binary bound while unbinding it from the MBH. In our simulations we find 0.007-0.01 binaries ejected from the MBH per initial stellar binary after one natal kick, and 0.005-0.006 after both natal kicks, making this the least common orbital configuration resulting from our simulations (see Figure \ref{fig:orb_config}). After both natal kicks, these unbound CO binaries are not observable. However, similarly to the ejected single objects from Section \ref{sec:unbound_single}, binaries ejected from the Galactic Center after $m_1$'s natal kick (these binaries contain one CO and one star), are observable. In the third column of Figure \ref{fig:HVS} we show the post-kick positions and velocities of these binaries after the first natal kick for the ``slow" BH kick case. Most of these binaries are ejected from the GC with velocities between 1000 - 10,000 km/s. These binaries are ejected from the Galactic Center between 4 and 45 Myr after the star formation episode, and $\sim 38$\% of these binaries contain a star that will not evolve to the CO phase until 100 Myr - 10 Gyr after the star formation episode, giving up to a few Gyr's window to observe them.
Interestingly, $\sim 15$\% (17 \%) (22 \%) of the ejected binaries are X-ray binaries, for the no BH kicks (``slow" BH kicks) (``fast" BH kicks) scenario. For the ``fast" BH kicks scenario, 86\% (14 \%) of the ejected X-ray binaries have a NS (BH) primary. For both of the other BH kicks scenarios all the ejected X-ray binaries have a NS primary.
Of the ejected binaries that are not X-ray binaries, $\sim 76\%$ (89\%)(96\%) have an NS primary, for the no BH kicks (``slow" BH kicks) (``fast" BH kicks) scenario. The rest have a BH primary.
We note that some previous theoretical works have predicted the existence of hypervelocity binaries \citep[e.g.][]{Fragione+17,Fragione+18,Wang+19}. Furthermore, a hypervelocity binary candidate has recently been observed \citep{Nemeth+16}.
\begin{figure*}[!htb]
\centering
\includegraphics[width=\linewidth]{XrayBinary.png}
\caption{\textbf{Distribution of X-ray binaries} in the ($a_1$, $1 - e_1$) plane (bottom three panels), and the kernel density estimation (KDE) of $a_1$ (top three panels). X-ray binaries with a NS (BH) primary is shown in blue (black). We show X-ray binaries from all three BH kick distribution cases. We find that the set receiving ``fast" BH kicks resulted in the most BH-star X-ray binaries, with the set receiving no BH kicks resulting in no BH-star X-ray binaries at all. Furthermore, ``Fast" and ``slow" BH kicks result in different semi-major axis distributions for BH-star X-ray binaries, as shown by the KDEs in the top panels. Thus, the type and semi-major axis distribution of a statistically significant population of X-ray binaries in the GC can be a diagnostic for the hitherto uncertain distribution of natal kicks received by BH progenitors.}
\label{fig:XrayBinary}
\end{figure*}
\subsection{Binary Disruption by MBH}\label{sec:bin_disruption}
\begin{figure}
\centering
\includegraphics[width=\linewidth]{binary_disruption.png}
\caption{\textbf{The semi-major axis distribution of binaries disrupted by the MBH post-natal kick} (top panel), and the semi-major axis distribution of the binary component captured by the MBH (bottom panel).}
\label{fig:bin_disr}
\end{figure}
In our Monte Carlo runs, we make sure that the inner binary birth distribution does not cross the Roche Limit of the MBH (Equation \ref{eq:RocheCrossing}). However, sometimes the natal kick will push the inner binary onto an orbit around the MBH that does violate the condition given in Equation (\ref{eq:RocheCrossing}). In this case the inner binary will be disrupted at the next pericenter passage with the MBH. In our simulations we find \BinDisr\ binaries disrupted by the MBH, per initial stellar binary.
Typically, the heavier member of the inner binary is captured by (ejected from) the MBH for a bound (unbound) outer orbit, and vice versa for the lighter member \citep{Hills88,Yu+03,Sari+10,Kobayashi+12}. Previous works have shown that captured COs and stars from these binary disruption events can results in EMRIs, plunges, or TDEs \citep[e.g.][]{Sari+Fragione19,Miller+05}. We check whether this is the case for any of our binary disruptions. The inner binary is tidally disrupted by the MBH when it passes inside the tidal radius:
\begin{equation}
r_t = a_{1} \Big(\frac{m_{\rm MBH}}{m_1 + m_2}\Big)^{1/3}.
\end{equation}
The newly captured object's orbit has a pericenter equal to this tidal radius. Its new semi-major axis is given by:
\begin{equation}
a_{\rm capt} \sim a_1 \Big(\frac{m_{\rm MBH}}{m_1 + m_2}\Big)^{2/3}
\end{equation}
We show the probability density of the post-kick inner binary semi-major axis, and the the probability density of the captured objects in Figure \ref{fig:bin_disr}. Because the binaries disrupted by the MBH are wide by definition (their distribution peaks at $\sim 100 {\rm AU}$), the captured objects have relatively large semi-major axes with respect to the MBH (their distribution peaks at $\sim 1$ pc). As a result, we find that none of the captured objects in our simulations result in EMRIs, plunges, and TDEs. However, they might have an effect on the stellar cusp in the inner parts of the Galactic Center \citep[e.g.][]{Fragione+Sari18}.
The other binary member is typically ejected from the MBH at high velocities \citep[e.g.][]{Hills88}, and may add to the number of hypervelocity stars in our simulations. However, this out of the scope of this paper.
\section{Discussion}
We have conducted an analysis of the consequences of natal kicks in massive binaries at the Galactic Center. When a star undergoes supernova to become BH or NS it is expected to receive a large natal kick \citep[e.g.][]{Hansen+Phinney97,Lorimer+Bailes97,Cordes+Chernoff98,Fryer+99,Hobbs+04,Hobbs+05,Beniamini+Piran16}.
We run three large sets of Monte-Carlo simulations ($100,000$ for each set), calculating the dynamical outcome after supernova kick. The three sets correspond to three different kick distributions for BH progenitors: fast kicks (for which the kicks are similar to NS kicks), slow kicks (for which the kicks are normalized by the BH mass), and no kicks. The application of the kicks are done by simple vector analysis. The consist form of the equations are given in \citet{Lu+19}.
Generally, after each kick there are two main outcomes, where the inner binary becomes unbound or stays bound. The second kick may also unbind a binary in the latter case. After two kicks took place we find that most of the binaries were disrupted, e.g., between $94.5\% - 80.1\%$, for the fast to no kicks models respectively. See Table \ref{tab:innerbinary} for the full details. However, even in the cases of inner-binary disruption, the majority of the systems remain bound to the MBH (see Table \ref{tab:singles}). These consequences yield the following predictions and observational signatures:
\begin{itemize}
\item \textbf{ Natal kicks create X-ray binaries}, at a rate of $(6.3-9.9)\times 10^{-3}$ per initial stellar binary. In general we find that faster kicks create more X-ray binaries. In particular, we find that no BH X-ray binaries are created if we assume no BH kicks. This suggests that detecting BH X-ray binaries in the Galactic Center implies some degree of BH kicks. However, other formation channels can create BH X-ray binaries in the absence of kicks \citep[e.g.,][]{Naoz+16,Stephan+19}.
\item \textbf{ Hypervelocity stars and COs are a common outcome.} We find that after the two kicks have taken place, a fairly significant fraction of the systems became unbound to the MBH (slightly more than $30\%$ of NSs, and between $32.3\%-1.7\%$ of BHs for fast to no kicks, respectively). See Table \ref{tab:singles}, for details. After the first kick, a number of stars are ejected from the inner 0.1 pc with velocities between 1000 - 10,000 km/s. Some of these stars may be observed as hypervelocity stars outside of the Galactic Center.
\item \textbf{Hypervelocity binaries} are created in addition to the ejected COs and stars, about 20\% of which are X-ray binaries. These binaries are ejected from the inner 0.1 pc after the first kick with velocities between 1000 - 10,000 km/s. In particular, after the first natal kick, we find $\sim (7.4 - 9.5)\times 10^{-3}$ binaries (CO + star) ejected from the inner 0.1 pc. These may be observed as hypervelocity binaries and hypervelocity x-ray binaries outside of the Galactic Center.
\item \textbf{Kicks result in a steeper distribution of the bound single NSs and BHs about $\sim 0.04$~pc from the MBH} compared to the initial distribution. The progenitors began on \citet{Bahcall+76} density profile, however, after the two kicks both the NS and the BHs (with fast kicks) result in a steeper distribution inwards to $0.04$~pc. At this distance from the MBH, orbital velocity around the MBH is comparable to the average kick velocity of $\sim 200$~km~s$^{-1}$. Beyond this radius, we find that the distribution of the NS and BHs (with fast kicks) have a {\it shallower} density distribution compared to their birth distribution. BHs with no kicks and WD (without kicks) follow the initial density distribution. This behavior is depicted in Figure \ref{fig:BoundSingles}.
\item {\bf Kicks result in a slightly shallower binary density distribution about MBH.} Unlike the steeper segregation of singles toward the MBH, the bound inner binary post kicks result in a somewhat shallower density distribution, with a long tail of systems reaching large densities. Additionally, the inner binary semi-major axis distribution after two natal kicks has a greater spread and peaks at a larger value, relative to the initial distribution.
\item {\bf Exotic events} from our simulation include binary GW mergers, EMRIs, direct plunges, and TDEs. Of these, the most common are binary mergers, which occur in our simulations at a rate of $\sim$ \GWmerger\ per initial binary. EMRIs, direct plunges, and TDEs are much rarer events in our simulations. For each of these, we find a few times $10^{-5}$ events per initial binary.
\end{itemize}
In summary, we showed that natal kick have a significant effect on the density distribution of COs, and of binaries in the galactic center. For example, it produces a steeper cusp of NSs about the MBH. Additionally, natal kicks naturally give rise to hypervelocity stars, hypervelocity binaries and X-ray binaries. Interestingly, we found that X-ray binaries can serve as a discriminator between BH kicks distribution models. Lastly, natal kicks can also result in a non-negligible rate of exotic events such as TDEs, LIGO, and LISA sources.
\acknowledgments
BMH and SN acknowledge the partial support from NASA No. 80NSSC19K0321, ATP-80NSSC20K0505 and partial support from the NSF through grant No. AST-1739160. BMH thanks the University of California Office of the President Dissertation Year Fellowship. SN thanks Howard and Astrid Preston for their generous support.
| 2024-02-18T23:40:02.818Z | 2022-04-11T02:00:23.000Z | algebraic_stack_train_0000 | 1,173 | 9,218 |
|
proofpile-arXiv_065-6040 | \section{Introduction}
\label{intro}
\subsection{Measurements of Rapidity Gaps at the Double Diffraction Dissociation}
The recent measurement of diffraction gaps in ATLAS \cite{atlas} has shown that the behavior of their distribution has different character in the different gap ranges. The histogram at the large values of gap indicates some discrete states of gaps. In this area only the process of double diffraction dissociation (DD) gives the contribution. Of course, the discrete pattern can be initiated by poor statistics. Nevertheless, we have to learn here the diagrams that lead to discrete levels of DD gap.
\subsection{Topological Expansion and Pomeron Exchange}
Pomeron exchange is presented with the topological QCD diagram that is responsible for multi particle production in p-p collisions at LHC energies. The quark-gluon content was drawn in the topological expansion \cite{topologyexp} as the the cylindrical net of gluon exchanges with the random amount of quark-antiquark loops inserted. The topological expansion gives the chance to classify the contributions from general diagrams of multi-particle production in the hadron interactions. This expansion has practically allowed us to develope the Quark-Gluon String Model (QGSM) \cite{kaidalov,qgsm,baryon}. Few orders in topological expansion are graphically presented in the figure from my PhD thesis figure~\ref{myphd}, where the third order is named pomeron with handle. Double diffraction dissociation in this presentation looks like the cylinder of one pomeron exchange with the toroidal handle that left uncut, so no particles have been produced in the central rapidity region.
\begin{figure}[htpb]
\centering
\includegraphics[width=7.0cm, angle=0]{myPhD.eps}
\caption{The fragment of graphical presentation of pomeron exchange in the topological expansion, where b is the number of boundaries and h is the number of handles.}
\label{myphd}
\end{figure}
\section{ Double Diffractive Dissociation as an Exchange with Pomeron Torus}
Double diffraction dissociation (DD) is a next order in the topological expansion after the pomeron exchange and should be presented as one pomeron diagram with two-pomeron loop in the center, see figure~\ref{DD}. Actually, the DD configuration is similar to cylinder with a handle that takes $(1/9)^2$ from the single pomeron exchange cross section (1.2 percents of $\sigma_{prod}$) at high energies.
\begin{figure}[htpb]
\centering
\includegraphics[width=2.0cm, angle=0]{pomeronloop.eps}
\caption{Pomeron loop in the center of one-pomeron exchange.}
\label{DD}
\end{figure}
If the central pomeron loop was not cut, we are having the DD spectra of produced hadrons: two intervals at the ends of rapidity range, which are populated with multi particle production, and the valuable gap in the center of rapidity.
Looking at two-pomeron loop in this diagram, we are realizing that it is torus in 3D topology. This interesting object should be considered separately in order to reveal some remarkable features for the experimental detection.
\section{Baryon Junction-Antijunction Hexagon Net and Discrete Dynamics of DD Gaps}
As we remember the pomeron cylinder is built by gluon exchange net, let us consider only three-gluon-connections on the surface of torus (or pomeron loop). This String-Junction type of gluon vertices has been studied in our early researches \cite{baryon,baryonasymmetry} and plays the important role in multiple production of baryons. Since this object brings the baryon charge, the anti-SJ also exists and brings the charge of antibaryon. The only charge-neutral way to construct the net from string-junctions and anti-string-junctions is hexagon where antibaryon charge is following the baryon one as it is shown in the figure~\ref{onecell}.
\begin{figure}[htpb]
\centering
\includegraphics[width=2.0cm, angle=0]{onecell.eps}
\caption{One cell of hexagon net with the SJ and antiSJ.}
\label{onecell}
\end{figure}
The closed net of six hexagons on the torus is shown in the figure~\ref{torus}.
\begin{figure}[htpb]
\centering
\includegraphics[width=5.0cm, angle=0]{SixbeeTorus.eps}
\caption{Closed net on the surface of torus: a) six hexagon construction and b) torus covered with six hexagon net.}
\label{torus}
\end{figure}
If people are trying to match the eligible number of hexagons, it becomes clear that there is a discrete row: Hexnumb=4, 6, 8, 12, 16, 24, 32, 48, 64 etc, see figure~\ref{torus}. It means that the pomeron torus has certain levels of energy. It leads to discrete gap states at DD \cite{myatlastalk} and to some other signatures in multi-particle production spectra \cite{torusasDD}. The similar construction was presended in \cite{kharzeev} as a complicate fullerene sphere that is build with SJ's.
\section{More Suggestions on the Pomeron Torus}
It is time to imagine, where to the pomeron torus could contribute. What we have, if our gluon-junction construction that looks like a "compactificated" pomeron string would be released as metastable particle? It is charge neutral QCD cluster with the certain potential energy, which is determined by number of hexagons, Hexnumb. If such cluster would be stable, this is appropriate candidate for the dark matter (DM)\cite{ICPPA}. The masses of these states are suspected similar to the very mass sequence of heavy neutral hadron states invented in \cite{ICHEP}. The reason, why this object is hardly dissipated in the collisions with matter, is following: the atomic numbers of elements in the space are too small in comparison with the number of SJ-antiSJ vertices in our toroidal constructions (let us name them "baryoniums"), therefore compact torus leaves intact after the collision with the less dense light atoms. Since each high energy proton collision in the space, wherever it takes place, contributes 1.2 percents of energy into DM, the valuable DM mass has been accumulated in the Universe, even though some amount of low mass baryoniums dissipates back into baryons and mesons at the collisions with the interstellar light atoms. It seems \cite{ICPPA}, nevertheless, that stable baryonium DM should be mostly concentrated near Supermassive Black Holes due to the huge gravitation pressure. Such a way, DM particles mostly appear in space as the result of the giant jets radiation and the partial distruction of BHs. This idea has to be verified with more observation of SMBH.
\section{Conclusion}
The topological presentation of pomeron exchange at the proton-proton collision of high energy is cylinder that is covered with quark-gluon net \cite{topologyexp}. I suggest that the process of double diffraction (DD) can be presented as one pomeron exchange with the central loop of two uncut pomeron cylinder loop or torus \cite{qgsm,myPhD}. Taking into account that the junction of three gluons (SJ) has the positive baryon number, as well as the antijunction is of negative baryon charge, our neutral pomeron construction can be covered by only a certain number of hexagons with 3 string junction and 3 antijunction vertices each \cite{torusasDD}. It is reasonable to expect that the dynamics of rapidity gaps in DD should be determined by the number of hexagons on the surface of pomeron torus. Therefore, the gap distribution in DD events has the discrete structure in the region of large gaps \cite{atlas}. The positive baryon production asymmetries that have been measured at LHC are the demonstration of string junction participation in proton-proton interactions of high energy \cite{baryonasymmetry}. Moreover, the string-junction torus can be released in the course of pp interaction as metastable particle (baryonium) and is getting suspected as "baryonium" Dark Matter candidate \cite{ICPPA}. The possibility of production of the states with many string junctions has been discussed recently by G.C. Rossi and G. Veneziano \cite{newveneziano}.
\section{Acknowledgments}
Author would like to express her gratitude to Oleg Kancheli for numerous discussions and to Vladimir Tskhay for designing the figure with torus.
| 2024-02-18T23:40:03.996Z | 2019-09-19T02:17:33.000Z | algebraic_stack_train_0000 | 1,228 | 1,286 |
|
proofpile-arXiv_065-6089 | \section{Introduction}
Planning under uncertainty is a central problem in robotics. The space of
current methods includes several contenders, each with different simplifying
assumptions, approximations, and domains of applicability. This is a natural
consequence of the fact that the challenge of dealing with the continuous
state, control and observation space problems, for non-linear systems and
across long-time horizons with significant noise, and potentially multiple
agents, is fundamentally intractable.
Model Predictive Control is one popular means for tackling optimal control
problems~\cite{Mayne_1,Mayne_2}. The MPC approach solves a finite horizon
``deterministic'' optimal control problem at every time step given the current
state of the process, performs only the first control action and then repeats
the planning process at the next time step. In terms of computation, this is a
costly endeavor. When a stochastic control problem is well approximated by the
deterministic problem, namely when the noise is meager, much of this
computation is simply superfluous. In this paper we consider a recently
proposed method~\cite{D2C1.0}, grounded on a decoupling result, that uses a local
feedback to control noise induced deviations from the deterministic (that we term
the ``nominal'') trajectory. When the deviation is too large for the feedback
to manage, replanning is triggered and it computes a fresh nominal. Otherwise,
the feedback tames the perturbations during execution and no computation is
expended in replanning. Figure~\ref{fig:timeplot} illustrates this: the
areas under the respective curves give the total computational resources
consumed---the savings are seen to be considerable.
This paper presents an empirical investigation of this decoupling approach,
exploring dimensions that are important in characterizing its performance.
The primary focus is on understanding the performance across a wide range of
noise conditions.
\begin{figure}
\centering
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/time_comp1.pdf}
\caption{A single agent.}
\label{1_agent_replan_time}
\end{subfigure}
%
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/time_comp3.pdf}
\caption{Three agents.}
\label{3_agent_replan_time}
\end{subfigure}
\caption{Computation time expended by MPC (in blue) and the algorithms we describe (in green), at each time step for a sample experiment involving navigation. Both cases result in nearly identical motions by the robot.
The peaks in T-LQR2 and MT-LQR2 happen only when replanning takes place. Computational effort decreases for both methods because the horizon diminishes as the agent(s) reach their goals.
(To relate to subsequent figures: noise parameter $\epsilon = 0.4$ and the replan threshold = 2\% of cost deviation.)}
\label{fig:timeplot}
\end{figure}%
\subsection{Related Work}
Robotic planning problems under uncertainty can be posed as a stochastic optimal control problem that requires the solution of an associated Dynamic Programming (DP) problem, however, as the state dimension $d$ increases, the computational complexity
goes up exponentially \cite{bertsekas1}, Bellman's infamous ``curse of dimensionality". There has been recent success using sophisticated (Deep) Reinforcement Learning (RL) paradigm to solve DP problems, where deep neural networks are used as the function approximators \cite{RLHD1, RLHD2,RLHD3, RLHD4, RLHD5}, however, the training time required for these approaches is still prohibitive to permit real-time robotic planning that is considered here.
In the case of continuous state, control and observation space problems, the
Model Predictive Control \cite{Mayne_1, Mayne_2} approach has been used with a
lot of success in the control system and robotics community. For deterministic
systems, the process results in solving the original DP problem in a recursive
online fashion. However, stochastic control problems, and the control of
uncertain systems in general, is still an unresolved problem in MPC. As
succinctly noted in \cite{Mayne_1}, the problem arises due to
the fact that in stochastic control problems, the MPC optimization at every
time step cannot be over deterministic control sequences, but rather has to be
over feedback policies, which is, in general, difficult to
accomplish because a compact, tractable parametrization of such policies to perform the optimization is, in general, unavailable. Thus, the tube-based MPC approach, and its stochastic counterparts,
typically consider linear systems \cite{T-MPC1, T-MPC2,T-MPC3} for which a
linear parametrization of the feedback policy suffices but the methods require
expensive offline computation when dealing with nonlinear systems. In recent work, we have introduced a ``decoupling principle" that allows us to tractably solve such stochastic optimal control problems in a near optimal fashion, with applications to highly efficient RL and MPC implementations \cite{D2C1.0,T-PFC}. However, this prior work required a small noise assumption. In this work, we relax this small noise assumption to show, via extensive empirical evaluation, that even when the noise is not small, a ``replanning" modification of the decoupled planning algorithms suffice to keep the planning computationally efficient while retaining performance comparable to MPC.
The problem of multiple agents further and severely compounds the planning
problem since now we are also faced with the issue of a control space that
grows exponentially with the number of agents in the system. Moreover, since
the individual agents never have full information regarding the system state,
the observations are partial. Furthermore, the decision making has to be done
in a distributed fashion which places additional constraints on the networking
and communication resources. In a multi-agent setting, the stochastic optimal
problem can be formulated in the space of joint policies. Some variations of
this problem have been successfully characterized and tackled based on the
level of observability, in/dependence of the dynamics, cost functions and
communications
\cite{seuken2008formal,oliehoek2016concise,pynadath2002communicative}. This has
resulted in a variety of solutions from fully-centralized
\cite{boutilier1996planning} to fully-decentralized approaches with many
different subclasses \cite{amato2013decentralizedB,oliehoek2012decentralized}.
The major concerns of the
multi-agent problem are tractability of the solution and the level of
communication required during the execution of the policies. In this paper, we shall consider a generalization of the decoupling principle to a multi-agent, fully observed setting. We show that this leads to a spatial decoupling between agents in that they do not need to communicate for long periods of time during execution. Albeit, we do not consider the problem of when and how to replan in this paper, assuming that there exists a (yet to be determined) distributed mechanism that can achieve this, we nonetheless show that there is a highly significant increase in planning efficiency over a wide range of noise levels.
\subsection{Outline of Paper}
The rest of the document is organised as follows: Section~\ref{section:prob} states the problem, \ref{section:decoupling} gives background on the decoupling principle, \RomanNumeralCaps{4} explains the planning algorithms used, \RomanNumeralCaps{5} discusses the results and observations and \ref{section:conclusion} concludes.
\section{Problem Formulation}
\label{section:prob}
The problem of robot planning and control under noise can be formulated as a stochastic optimal control problem in the space of feedback policies. We assume here that the map of the environment is known and state of the robot is fully observed. Uncertainty in the problem lies in the system's actions. %
\subsection{System Model:}
For a dynamic system, we denote the state and control vectors by $\V{x}_t \in \ \mathbb{X} \subset \ \mathbb{R}^{n_x}$ and $\V{u}_t \in \ \mathbb{U} \subset \ \mathbb{R}^{n_u}$ respectively at time $t$. The motion model $f : \mathbb{X} \times \mathbb{U} \times \mathbb{R}^{n_u} \rightarrow \mathbb{X} $ is given by the equation
\begin{equation}
\V{x}_{t+1}= f(\V{x}_t, \V{u}_t, \epsilon\V{w}_t); \ \V{w}_t \sim \mathcal{N}(\V{0}, {\mathbf \Sigma}_{\V{w}_t})
\label{eq:model},
\end{equation}
where \{$\V{w}_t$\} are zero mean independent, identically distributed (i.i.d) random sequences with variance ${\mathbf\Sigma}_{\V{w}_t}$, and $\epsilon$ is a small parameter modulating the noise input to the system.
\subsection{Stochastic optimal control problem:} %
The stochastic optimal control problem for a dynamic system with initial state $\V{x}_0$ is defined as:
\begin{equation}
J_{\pi^{*}}(\V{x}_0) = \min_{\pi} \ \Exp{}{\sum^{T-1}_{t=0} c(\V{x}_t, \pi_t (\V{x}_t)) + c_T(\V{x}_T)},
\end{equation}
\begin{equation}
s.t.\ \V{x}_{t+1} = f(\V{x}_t, \pi_t (\V{x}_t), \epsilon\V{w}_t),
\end{equation}
where:
\begin{itemize}
\item the optimization is over feedback policies $\pi := \{ \pi_0, \pi_1, \ldots, \pi_{T-1} \} $ and $\pi_t(\cdot)$: $\mathbb{X} \rightarrow \mathbb{U}$ specifies an action given the state, $\V{u}_t = \pi_t(\V{x}_t)$;
\item $J_{\pi^{*}}(\cdot): \mathbb{X} \rightarrow \mathbb{R}$ is the cost function when the optimal policy $\pi^{*}$ is executed;
\item $c_t(\cdot,\cdot): \mathbb{X} \times \mathbb{U} \rightarrow \mathbb{R} $ is the one-step cost function;
\item $c_T(\cdot): \mathbb{X} \rightarrow \mathbb{R}$ is the terminal cost function;
\item $T$ is the horizon of the problem;
\item the expectation is taken over the random variable $\V{w}_t$.
\end{itemize}
\section{A Decoupling Principle}
\label{section:decoupling}
Now, we give a brief overview of a ``decoupling principle'' that allows us to substantially reduce the complexity of the stochastic planning problem given that the parameter $\epsilon$ is small enough. We only provide an outline here and the relevant details can be found in our recent work \cite{D2C1.0}. We shall also present a generalization to a class of multi-robot problems. Finally, we preview the results in the rest of the paper.
\subsection{Near-Optimal Decoupling in Stochastic Optimal Control}
Let $\pi_t(\V{x}_t)$ denote a control policy for the stochastic planning problem above, not necessarily the optimal policy. Consider now the control actions of the policy when the noise to the system is uniformly zero, and let us denote the resulting ``nominal'' trajectory and controls as $\overline{\V{x}}_t$ and $\overline{\V{u}}_t$ respectively, i.e., $\overline{\V{x}}_{t+1} = f(\overline{\V{x}}_t, \overline{\V{u}}_t, 0)$, where $\overline{\V{u}}_t = \pi_t(\overline{\V{x}}_t)$. Note that this nominal system is well defined. \\
Further, let us assume that the closed-loop (i.e., with $\V{u}_t = \pi_t(\V{x}_t)$), system equations, and the feedback law are smooth enough that we can expand the feedback law about the nominal as $\pi_t(\V{x}_t) = \overline{\V{u}}_t + \M{K}_t\delta \V{x}_t + \M{R}_t^{\pi}(\delta \V{x}_t)$, where $\delta \V{x}_t = \V{x}_t - \overline{\V{x}}_t$, i.e., the perturbation from the nominal, $\M{K}_t$ is the linear gain obtained by the Taylor expansion about the nominal in terms of the perturbation $\delta \V{x}_t$, and $\M{R}_t^{\pi}(\cdot)$ represents the second and higher order terms in the expansion of the feedback law about the nominal trajectory. Further we assume that the closed-loop perturbation state can be expanded about the nominal as: $\delta \V{x}_t = \M{A}_t \delta \V{x}_t + \M{B}_t \M{K}_t \delta \V{x}_t + \M{R}_t^f (\delta \V{x}_t) + \epsilon \M{B}_t \V{w}_t$, where the $\M{A}_t$, $\M{B}_t$ are the system matrices obtained by linearizing the system state equations about the nominal state and control, while $\M{R}_t^f(\cdot)$ represents the second and higher order terms in the closed-loop dynamics in terms of the state perturbation $\delta \V{x}_t$. Moreover, let the nominal cost be given by $\overline{J}^{\pi} = \sum_{t=0}^T \overline{c}_t$, where $\overline{c}_t = c(\overline{\V{x}}_t,\overline{\V{u}}_t)$, for $t\leq T-1$, and $\overline{c}_T = c_T(\overline{\V{x}}_T,\overline{\V{u}}_T)$. Further, assume that the cost function is smooth enough that it permits the expansion $J^{\pi} = \overline{J} + \sum_t \M{C}_t \delta \V{x}_t + \sum_t \M{R}_t^c(\delta \V{x}_t)$ about the nominal trajectory, where $\M{C}_t$ denotes the linear term in the perturbation expansion and $\M{R}_t^c(\cdot)$ denote the second and higher order terms in the same. Finally, define the exactly linear perturbation system $\delta \V{x}_{t+1}^\ell = \M{A}_t \delta \V{x}_t^\ell + \M{B}_t\M{K}_t \delta \V{x}_t^\ell + \epsilon \M{B}_t \V{w}_t$. Further, let $\delta J_1^{\pi,\ell}$ denote the cost perturbation due to solely the linear system, i.e., $\delta J_1^{\pi,\ell} = \sum_t \M{C}_t \delta \V{x}_t^\ell$. Then, the Decoupling result states the following \cite{D2C1.0}:
\begin{theorem}
The closed-loop cost function $J^{\pi}$can be expanded as $J^{\pi} = \overline{J}^{\pi} + \delta J_1^{\pi,\ell} + \delta J_2^{\pi}$. Furthermore, $\Exp{}{J^{\pi}} = \overline{J}^{\pi} + O(\epsilon^2)$, and $\Var[J^{\pi}] = \Var[\delta J_1^{\pi,\ell}] + O(\epsilon^4)$, where $\Var[\delta J_1^{\pi,\ell}]$ is $O(\epsilon^2)$.
\end{theorem}
Thus, the above result says the mean value of the cost is determined almost solely by the nominal control actions while the variance of the cost is almost solely determined by the linear closed-loop system. Thus, decoupling result says that the feedback law design can be decoupled into an open-loop and a closed-loop problem.\\
\textit{Open-Loop Problem:} This problem solves the deterministic/ nominal optimal control problem:
\begin{equation}
J= \min_{\overline{\V{u}}_t} \sum_{t=0} ^{T-1} c(\overline{\V{x}}_t,\overline{\V{u}}_t) + c_T(\overline{\V{x}}_T),
\end{equation}
subject to the nominal dynamics: $\overline{\V{x}}_{t+1} = f(\overline{\V{x}}_t, \overline{\V{u}}_t)$. \\
\textit{Closed-Loop Problem:} One may try to optimize the variance of the linear closed-loop system
\begin{equation}
\min_{\M{K}_t} \Var[\delta J_1^{\pi,\ell}]
\end{equation}
subject to the linear dynamics $\delta \V{x}_{t+1}^\ell = \M{A}_t \delta \V{x}_t^\ell + \M{B}_t \M{K}_t \delta \V{x}_t^\ell + \epsilon \M{B}_t \V{w}_t$.
However, the above problem does not have a standard solution but note that we are only interested in a good variance for the cost function and not the optimal one. Thus, this may be accomplished by a surrogate LQR problem that provides a good linear variance as follows.\\
\textit{Surrogate LQR Problem:} Here, we optimize the standard LQR cost:
\begin{equation}
\delta J_{\textsc{lqr}} =\min_{\V{u}_t} \Exp{\V{w}_t}{\sum_{t=0}^{T-1} \delta {\tr{\V{x}}_t} \M{Q} \delta \V{x}_t + \delta \tr{\V{u}}_t\M{R}\delta \V{u}_t + \delta \tr{\V{x}}_T \M{Q}_f \delta \V{x}_T},
\label{LQRcost}
\end{equation}
subject to the linear dynamics $\delta \V{x}_{t+1}^\ell = \M{A}_t \delta \V{x}_t^\ell + \M{B}_t \delta \V{u}_t + \epsilon \M{B}_t \V{w}_t$. In this paper, this decoupled design shall henceforth be called the trajectory-optimized LQR (T-LQR) design.
\subsection{Multi-agent setting}
Now, we generalize the above result to a class of multi-agent problems.
We consider a set of agents that are transition independent, i.e, their dynamics are independent of each other. For simplicity, we also assume that the agents have perfect state measurements. Let the system equations for the agents be given by:
$
\V{x}_{t+1}^j = f(\V{x}_t^j) + \M{B}^j_t(\V{u}_t^j + \epsilon \V{w}_t^j),
$
where $j = 1,2,\dots,M$ denotes the $j\ensuremath{{}^{\textrm{th}}}$ agent. (We have assumed the control affine dynamics for simplicity). Further, let us assume that we are interested in the minimization of the joint cost of the agents given by $\mathcal{J} = \sum_{t=0}^{T-1} c(\M{X}_t,\M{U}_t) + \Phi(\M{X}_T)$, where $\M{X}_t = [\V{x}_t^1,\dots,\V{x}_t^M]$, and $\M{U}_t = [\V{u}_t^1,\dots, \V{u}_t^M]$ are the joint state and control action of the system. The objective of the multi-agent problem is minimize the expected value of the cost $\Exp{}{\mathcal{J}}$ over the joint feedback policy $\M{U}_t(\cdot)$. The decoupling result holds here too and thus the multi-agent planning problem can be separated into an open and closed-loop problem. The open-loop problem consists of optimizing the joint nominal cost of the agents subject to the individual dynamics.\\
\textit{Multi-Agent Open-Loop Problem:}\\
\begin{align}\label{OL-MA}
\overline{\mathcal{J}} = \min_{\overline{\M{U}}_t} \sum_{t=0}^{T-1} c(\overline{\M{X}}_t,\overline{\M{U}}_t) + \Phi(\overline{\M{X}}_T),
\end{align}
subject to the nominal agent dynamics
$
\overline{\V{x}}_{t+1}^j = f(\overline{\V{x}}_t^j) + \M{B}^j_t\overline{\V{u}}_t^j.
$
The closed-loop, in general, consists of optimizing the variance of the cost $\mathcal{J}$, given by $\Var[\delta \mathcal{J}^\ell_1]$, where $\delta \mathcal{J}_1^\ell = \sum_t \M{C}_t \delta \M{X}_t^l$ for suitably defined $\M{C}_t$, and $\delta \V{X}_t^\ell = [\delta \V{x}_t^1,\dots, \delta \V{x}_t^M]$, where the perturbations $\delta \V{x}_t^j$ of the $j\ensuremath{{}^{\textrm{th}}}$ agent's state is governed by the decoupled linear multi-agent system $\delta \V{x}_t^j = \M{A}_t\delta \V{x}_t^j + \M{B}_t^j \delta \V{u}^j_t + \epsilon \M{B}_t^j \V{w}_t^j.$ This design problem does not have a standard solution but recall that we are not really interested in obtaining the optimal closed-loop variance, but rather a good variance. Thus, we can instead solve a surrogate LQR problem given the cost function $\delta \mathcal{J}_{\textsc{mtlqr}} = \sum_{t=0}^{T-1} \sum_j \delta {\tr{\V{x}_t^j}} \M{Q}^j \delta \V{x}_t^j + \delta \tr{\V{u}_t^j}\M{R}\delta \V{u}_t^j + \sum_j\delta \tr{\V{x}_T^j} \M{Q}^j_f \delta \V{x}_T^j$. Since the cost function itself is decoupled, the surrogate LQR design degenerates into a decoupled LQR design for each agent.\\
\textit{Surrogate Decoupled LQR Problem:}
\begin{equation}
\delta \mathcal{J}^j =\min_{\V{u}_t^j} \Exp{\V{w}_t^j}{\sum_{t=0}^{T-1} \delta {\tr{\V{x}_t^j}} \M{Q}^j \delta \V{x}_t^j + \delta \tr{\V{u}_t^j}\M{R}\delta \V{u}_t^j + \delta \tr{\V{x}_T^j} \M{Q}^j_f \delta \V{x}_T^j},
\end{equation}
subject to the linear decoupled agent dynamics $\delta \V{x}_t^j = \M{A}_t\delta \V{x}_t^j + \M{B}_t^j \delta \V{u}^j_t + \epsilon \M{B}_t^j \V{w}_t^j.$\\
\begin{remark}
Note that the above decoupled feedback design results in a spatial decoupling between the agents in the sense that, at least in the small noise regime, after their initial joint plan is made, the agents never need to communicate with each other in order to complete their missions.
\end{remark}
\subsection{Planning Complexity versus Uncertainty}
The decoupling principle outlined above shows that the complexity of planning can be drastically reduced while still retaining near optimal performance for sufficiently small noise (i.e., parameter $\epsilon \ll 1$). Nonetheless, the skeptical reader might argue that this result holds only for low values of $\epsilon$ and thus, its applicability for higher noise levels is suspect. Still, because the result is second order, it hints that near optimality might be over a reasonably large $\epsilon$. Naturally, the question is \textsl{`will it hold for medium to higher levels of noise?'}\\
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/cost_comp_all5_1_agent_c7_p7_02.pdf}
\caption{Full noise spectrum.}
\label{1 agent cost full}
\end{subfigure}%
%
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/cost_comp_all5_1_agent_c7_p7_02_low.pdf}
\caption{Enhanced detail: $0\leq\epsilon\leq0.4$.}
\label{1 agent cost low}
\end{subfigure}
\caption{Cost evolution of the different algorithms for varying noise for a single agent. Control Horizon ($H_c$) used for MPC-SH and T-LQR2-SH was 7. $J_{\textrm{thresh}}$ = 2\% was the replanning threshold used. $J/\overline{J}$ is the ratio of the cost incurred during execution to the nominal cost and is used as the performance measure throughout the paper. The nominal cost $\overline{J}$ which is calculated by solving the deterministic OCP for the total time horizon, just acts as a normalizing factor here. (T-LQR2-SH is not shown in (b) since it skews the graph and is not important in low noise cases.)}
\label{1_agent_cost}
\end{figure}%
\textit{Preview of our Results.} In this paper, we illustrate the
degree to which the above result still holds when we allow periodic replanning
of the nominal trajectory in T-LQR in an event triggered fashion, dubbed
T-LQR2. Here, we shall use MPC as a ``gold standard'' for comparison since the
true stochastic control problem is intractable, and it was shown by Fleming in
a seminal paper~\cite{fleming1971stochastic} that, effectively speaking, the MPC policy is
$O(\epsilon^4)$ near-optimal compared to the true stochastic policy. We show
that though the number of replanning operations in T-LQR2 increases the
planning burden over T-LQR, it is still much reduced when compared to MPC,
which replans continually. The ability to trigger replanning means that T-LQR2
can always produce solutions with the same quality as MPC, albeit by demanding
the same computational cost as MPC in some instances. But for moderate levels
of noise, T-LQR2 can produce comparable quality output to MPC with substantial
computational savings.
In the high noise regime, replanning is more frequent but we
shall see that there is another consideration at play. Namely, that the
effective planning horizon decreases and there is no benefit in planning all
the way to the end rather than considering only a few steps ahead, and in fact, in some cases, it can harmful to consider the distant future. Noting that
as the planning horizon decreases, planning complexity decreases, this
helps recover tractability even in this regime.\\
Thus, while lower levels of noise render the planning problem tractable due to
the decoupling result, planning under even medium and higher levels of noise
can be practical because the planning horizon should shrink as uncertainty
increases. When noise inundates the system, long-term predictions become so
uncertain that the best-laid plans will very likely run awry, then it would be wasteful to
invest significant time thinking very far ahead.
To examine this widely-recognized truth more quantitatively,
the parameter $\epsilon$ will be a knob we adjust, exploring these
aspects in the subsequent analysis.
\section{The Planning Algorithms}
The preliminaries and the algorithms are explained below
\subsection{Deterministic Optimal Control Problem:}
Given the initial state $\V{x}_0$ of the system, the solution to the deterministic OCP is given as:%
\begin{equation}
J^{*}(\V{x}_0) = \min_{\V{u}_{0:T-1}} \left[\sum^{T-1}_{t=0} c_t(\V{x}_t, \V{u}_t) + c_T(\V{x}_T)\right],
\label{DOCP}
\end{equation}%
\begin{align*}
s.t. \ \V{x}_{t+1} = f(\V{x}_t) + \M{B}_t \V{u}_t,\\
\V{u}_{\text{min}} \leq \V{u}_t \leq \V{u}_{\text{max}},\\
| \V{u}_{t} - \V{u}_{t-1}| \leq \Delta \V{u}_{\text{max}}.
\end{align*}
The last two constraint model physical limits that impose upper bounds and lower bounds on control inputs and rate of change of control inputs.
The solution to the above problem gives the open-loop control inputs $\overline{\V{u}}_{0:T-1}$ for the system.
For our problem, we take a quadratic cost function for state and control as
$
c_t(\V{x}_t,\V{u}_t) = \tr{\V{x}}_t \M{W}^x\V{x}_t + \tr{\V{u}}_t\M{W}^u \V{u}_t,
$
$
c_T(\V{x}_T) = \tr{\V{x}}_T\M{W}^x_f\V{x}_T,
$
where $\M{W}^x,\ \M{W}^x_f \succeq \M{0}$ and $\M{W}^u \ \succ \ \M{0}$.\\
\subsection{Model Predictive Control (MPC):}
We employ the non-linear MPC algorithm due to the non-linearities associated
with the motion model. The MPC algorithm implemented here solves the
deterministic OCP~\eqref{DOCP} at every time step, applies the control inputs
computed for the first instant and uses the rest of the solution as an initial
guess for the subsequent computation. In the next step, the current state of
the system is measured and used as the initial state and the process is
repeated.
\subsection{Short Horizon MPC (MPC-SH):}
We also implement a variant of MPC which is typically used in practical
applications where it solves the OCP only for a short horizon rather than the
entire horizon at every step. At the next step, a new optimization is solved
over the shifted horizon. This implementation gives a greedy solution but is
computationally easier to solve. It also has certain advantageous properties in
high noise cases which will be discussed in the results section. We denote the short planning horizon as $H_c$ also called as the control horizon, upto which the controls are computed. A generic algorithm for MPC is shown in
Algorithm~\ref{MPC_algo}.
\begin{algorithm}[h]
\SetAlgoLined
\KwIn{$\V{x}_0$ -- initial state, $\V{x}_g$ -- final state, $T$ -- time horizon, $H_c$ -- control horizon, $\Delta t$ -- time step, $\mathcal{P}$ -- system and environment parameters.}
\For{$t \leftarrow 0$ \KwTo $T-1$}{
$\V{u}_{t:t+H_c-1} \leftarrow$ \phantom{xxxxxxx}OCP($\V{x}_{t},\V{x}_g, \min(H_c, T\!-\!t),\V{u}_{t-1}, \V{u}_{\textrm{guess}},\mathcal{P}$)
$\V{x}_{t+1} \leftarrow$ $f(\V{x}_{t}) + \M{B}_t(\V{u}_t + \epsilon\V{w}_t)$
}
\caption{MPC algorithm\label{MPC_algo}.}
\end{algorithm}
\subsection{Trajectory Optimised Linear Quadratic Regulator~\mbox{(T-LQR)}:}
\label{sec:lqr_gains}
As discussed in Section~\ref{section:decoupling}, stochastic optimal control problem can be decoupled and solved by designing an optimal open-loop (nominal) trajectory and a decentralized LQR policy to track the nominal.
\\
\textit{Design of nominal trajectory}: The nominal trajectory is generated by first finding the optimal open-loop control sequence by solving the deterministic OCP~\eqref{DOCP} for the system. Then, using the computed control inputs and the noise-free dynamics, the sequence of states traversed $\overline{\V{x}}_{0:T}$ can be calculated.\\
\textit{Design of feedback policy:} In order to design the LQR controller, the system is first linearised about the nominal trajectory ($\overline{\V{x}}_{0:T}$, $\overline{\V{u}}_{0:T-1}$). Using the linear time-varying system, the feedback policy is determined by minimizing a quadratic cost as shown in~\eqref{LQRcost}.
The linear quadratic stochastic control problem~\eqref{LQRcost} can be easily solved using the algebraic Riccati equation and the resulting policy is $\delta{\V{u}}_{t} = -\M{L}_t\delta{\V{x}}^\ell_t$. The feedback gain and the Riccati equations are given by
\begin{equation}
\M{L}_t = \inv{(\M{R} + \M{B}^T_t\M{P}_{t+1} \M{B}_t)} \tr{\M{B}}_t \M{P}_{t+1} \M{A}_t,
\label{LQR_gain}
\end{equation}
\begin{equation}
\M{P}_{t} = \tr{\M{A}}_t \M{P}_{t+1} \M{A}_t - \tr{\M{A}}_t \M{P}_{t+1} \M{B}_t \M{L}_t + \M{Q},
\label{Riccati}
\end{equation}
respectively where $\M{Q}_f, \M{Q} \succeq \V{0}, \M{R} \ \succ \V{0}$ are the weight matrices for states and control. Here~\eqref{Riccati} is the discrete-time dynamic Riccati
equation which can be solved by backward iteration using the terminal condition
$\M{P}_{T} = \M{Q}_f $.
\subsection{T-LQR with Replanning (\mbox{T-LQR2}):}
T-LQR performs well at low noise levels, but at medium and high noise levels the system tends to deviate from the nominal. So, at any point during the execution if the deviation is beyond a threshold $J_{\textrm{thresh}}= \frac{J_t - \overline{J}_t}{\overline{J}_t}$, where $J_t$ denotes the actual cost during execution till time $t$ while $\overline{J}_t$ denotes the nominal cost. The factor $J_{\textrm{thresh}}$ measures the percentage deviation of the online trajectory from the nominal, and replanning is triggered for the system from the current state for the remainder of the horizon. Note that if we set $J_{\textrm{thresh}}=0$, T-LQR2 reduces to MPC. The calculation of the new nominal trajectory and LQR gains are carried out similarly to the explaination in Section~\ref{sec:lqr_gains}. A generic algorithm for T-LQR and T-LQR2 is shown in Algorithm~\ref{TLQR_algo}.
\subsection{Short Horizon T-LQR with Replanning (T-LQR2-SH):}
A T-LQR equivalent of MPC-SH is also implemented where the nominal is planned only for a short horizon and it is tracked with a feedback policy as described in T-LQR. It also inherits the replanning property of T-LQR2.\\
The implementations of all the algorithms are available at \url{https://github.com/MohamedNaveed/Stochastic_Optimal_Control_algos/}.
\begin{algorithm}[h]
\SetAlgoLined
\SetKwProg{Fn}{Function}{ is}{end}
\SetKwComment{Comment}{/*}{*/}
\KwIn{$\V{x}_0$ -- initial state, $\V{x}_g$ -- final state, $T$ -- time horizon, $J_{\textrm{thresh}}$ -- replan threshold, $\Delta t$ -- time step, $\mathcal{P}$ -- system and environment parameters.}
\Fn{Plan($\V{x}_{0},\V{x}_g, T, \V{{u}}_{\textrm{init}}$, $\V{u}_{\textrm{guess}}$,$\mathcal{P}$)}{
$\overline{\V{{u}}}_{0:T-1}$ $\leftarrow$ OCP($\V{x}_{0},\V{x}_g, T,\V{{u}}_{\textrm{init}}$, $\V{u}_{\textrm{guess}}$,$\mathcal{P}$)
\For{$t \leftarrow 0$ \KwTo $T-1$}{
$\overline{\V{x}}_{t+1} \leftarrow f(\overline{\V{x}}_{t}) + \M{B}_t\overline{\V{u}}_t$
}
$\M{L}_{0:T-1} \leftarrow $ $Compute\_LQR\_Gain(\overline{\V{x}}_{0:T-1},\overline{\V{u}}_{0:T-1}$)
return $\overline{\V{x}}_{0:T},\overline{\V{u}}_{0:T-1},\M{L}_{0:T-1}$
}
\Fn{Main()}{
$\overline{\V{x}}_{0:T}$,$\overline{\V{u}}_{0:T-1}$,$\M{L}_{0:T-1}$ $\leftarrow$ $\textrm{Plan}(\V{x}_{0},\V{x}_g, T,\mathbf{0}, \V{u}_{\textrm{guess}},\mathcal{P})$
\For{$t \leftarrow 0$ \KwTo $T-1$}{
$\V{u}_{t} \leftarrow \overline{\V{u}}_{t} - \M{L}_{t}(\V{x}_{t} - \overline{\V{x}}_{t})$
$\V{u}_{t} \leftarrow \textrm{Constrain}(\V{u}_{t})$ \tcp*[f]{ Enforce limits}
$\V{x}_{t+1} \leftarrow f(\V{x}_{t}) + \M{B}_t(\V{u}_t + \epsilon\V{w}_t)$
\If(\tcp*[f]{Replan?}){$ (J_t - \overline{J}_t)/\overline{J}_t > J_{\textrm{thresh}}$}{
$\overline{\V{x}}_{t+1:T}, \overline{\V{u}}_{t+1:T-1}, \M{L}_{t+1:T-1} \leftarrow \textrm{\phantom{xxxxxx}Plan}(\V{x}_{t+1},\V{x}_g, T\!-\!t\!-\!1,\V{u}_t, \V{u}_{\textrm{guess}},\mathcal{P})$
}
}
}
\caption{T-LQR algorithm with replanning}\label{TLQR_algo}
\end{algorithm}
\subsection{Multi-Agent versions}
\begin{figure}[t]
\centering
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/cost_comp_all5_3_agent_c7_p7_02.pdf}
\caption{Full noise spectrum.}
\label{3 agent cost full}
\end{subfigure}
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/cost_comp_all5_3_agent_c7_p7_02_low.pdf}
\caption{Enhanced detail: $0\leq\epsilon\leq0.4$.}
\label{3 agent cost low}
\end{subfigure}
\caption{Cost evolution of the different algorithms for varying noise for 3 agents. Control Horizon ($H_c$) used for MPC-SH and MT-LQR2-SH was 7. $J_{\textrm{thresh}}$ = 2\% was the replanning threshold used.}
\label{3_agent_cost}
\end{figure}%
The MPC version of the multi-agent planning problem is reasonably straightforward except that the complexity of the planning increased (exponentially) in the number of agents. Also, we note that the agents have to always communicate with each other in order to do the planning.\\
The Multi-agent Trajectory-optimised LQR (MT-LQR) version is also relatively straightforward in that the agents plan the nominal path jointly once, and then the agents each track their individual paths using their decoupled feedback controllers. There is no communication whatsoever between the agents during this operation.\\
The MT-LQR2 version is a little more subtle. The agents have to periodically
replan when the total cost deviates more than $J_{\textrm{thresh}}$ away from the
nominal, i.e., the agents do not communicate until the need to replan arises.
In general, the system would need to detect this in a distributed fashion, and
trigger replanning. We postpone consideration of this aspect of the problem to a
subsequent paper more directly focused on networking considerations. We will
assume that there exists a (yet to be determined) distributed strategy that
would perform the detection and replanning.
\subsection{Analysis of the High Noise Regime}
In this section, we perform a rudimentary analysis of the high noise regime. The medium noise case is more difficult to analyze and is left for future work, along with a more sophisticated treatment of the high noise regime.\\
First, recall the Dynamic Programming (DP) equation for the backward pass to determine the optimal time varying feedback policy:
J_t(\V{x}_t) = \min_{\V{u}_t}\left\{c(\V{x}_t,\V{u}_t) + \Exp{}{J_{t+1}(\V{x}_{t+1})}\right\},
where $J_t(\V{x}_t)$ denotes the cost-to-go at time $t$ given the state is $\V{x}_t$, with the terminal condition $J_T(\cdot) = c_T(\cdot)$ where $c_T$ is the terminal cost function, and the next state $\V{x}_{t+1} = f(\V{x}_t) + \M{B}_t(\V{u}_t + \epsilon \V{w}_t)$. Suppose now that the noise is so high that $\V{x}_{t+1} \approx \M{B}_t \epsilon \V{w}_t$, i.e., the dynamics are completely swamped by the noise.\\
Consider now the expectation $\Exp{}{c_T(\V{x}_{t+1})}$ given some control $\V{u}_t$ was taken at state $\V{x}_t$. Since $\V{x}_{t+1}$ is determined entirely by the noise, $\Exp{}{c_T(\V{x}_{t+1})} = \int c_T(\M{B}_t\epsilon \V{w}_t)\mathbf{p}(\V{w}_t) d\V{w}_t = \overline{c_T}$, where $\overline{c_T}$ is a constant regardless of the previous state and control pair $\V{x}_t, \V{u}_t$. This observation holds regardless of the function $c_T(\cdot)$ and the time $t$.\\
Next, consider the DP iteration at time $T-1$. Via the argument above, it follows that $\Exp{}{J_T(\V{x}_T)}=\Exp{}{c_T(\V{x}_T)} = \overline{c_T}$, regardless of the state control pair $\V{x}_{T-1},\V{u}_{T-1}$ at the $(T-1)^{th}$ step,
and thus, the minimization reduces to
$J_{T-1}(\V{x}_{T-1}) = \min_{\V{u}} \left\{c(\V{x}_{T-1},\V{u}) + \overline{c_T}\right\}$,
and thus, the minimizer is just the greedy action $\V{u}^*_{T-1} = \argmin_{\V{u}} c(\V{x}_{T-1},\V{u})$ due to the constant bias $\overline{c_T}$.
The same argument holds for any $t$ since, although there might be a different $J_{t}(\cdot)$ at every time $t$, the minimizer is still the greedy action that minimizes $c(\V{x}_t,\V{u})$ as the cost-to-go from the next state is averaged out to simply some $\bar{J}_{t+1}$.\\
\begin{figure}
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/1_agent_mpc_test_cases.pdf}
\label{test_cases_high_mpc}
\caption{MPC}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/1_agent_replan_test_cases1.pdf}
\label{test_cases_high_tlqr_replan}
\caption{T-LQR2}
\end{subfigure}
\newline
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/1_agent_MPCfast_test_cases.pdf}
\label{test_cases_high_shmpc}
\caption{MPC-SH}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/1_agent_tlqr_test_cases.pdf}
\label{test_cases_high_tlqr}
\caption{T-LQR}
\end{subfigure}
\caption{Performance of the algorithms for varying levels of noise.}
\label{fig:test_cases_high}
\end{figure}
\section{Simulation Results:}
\begin{figure}[h]
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/costvsHcvsdelta_1agents_1surf1.pdf}
\caption{}
\label{costvsHc_1_1}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/timevsHcvsdelta_1agents_1surf1.pdf}
\caption{}
\label{timevsHc_1_1}
\end{subfigure}%
\newline
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/costvsHcvsdelta_1agents_7surf1.pdf}
\caption{}
\label{costvsHc_1_7}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/timevsHcvsdelta_1agents_7surf1.pdf}
\caption{}
\label{timevsHc_1_7}
\end{subfigure}%
\caption{Variation seen in cost incurred and computation time by changing the $J_{\textrm{thresh}}$ and control horizon ($H_c$) in T-LQR2 and MPC for a single agent case. (a) and (b) show the performance in terms of cost and computation time respectively for the same experiment at $\epsilon = 0.1$. Similarly, (c) and (d) show for $\epsilon=0.7$. Though MPC doesn't have a threshold for replanning, it is plotted at $J_{\textrm{thresh}} = 0\%$ since it replans at every time step.}
\label{fig:1_agent_3d}
\end{figure}
\begin{figure}[h]
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/costvsHcvsdelta_3agents_1surf1.pdf}
\caption{}
\label{fig:costvsHc_3_1}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/timevsHcvsdelta_3agents_1surf1.pdf}
\caption{}
\label{fig:timevsHc_3_1}
\end{subfigure}%
\newline
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/costvsHcvsdelta_3agents_7surf1.pdf}
\caption{}
\label{fig:costvsHc_3_7}
\end{subfigure}%
%
\begin{subfigure}[b]{0.25\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/timevsHcvsdelta_3agents_7surf1.pdf}
\caption{}
\label{fig:timevsHc_3_7}
\end{subfigure}%
\caption{Variation seen in cost incurred and computational time by changing the $J_{\textrm{thresh}}$ and control horizon ($H_c$) in MT-LQR2 and MPC for 3 agents. (a) and (b) show for $\epsilon = 0.1$, (c) and (d) show for $\epsilon=0.7$.}
\label{fig:3_agent_3d}
\end{figure}
We test the performance of the algorithms in a car-like robot model. Numerical optimization is carried out using \texttt{CasADi} framework \cite{Andersson2018} with \texttt{Ipopt} \cite{Ipopt} NLP solver in \texttt{Python}. To provide a good estimate of the performance the results presented were averaged from 100 simulations for every value of noise considered. Simulations were carried out in parallel across 100 cores in a cluster equipped with Intel Xeon 2.5GHz E5-2670 v2 10-core processors. The experiments chosen were done with a time horizon T = 35.
\subsection*{Car-like robot model:}
The car-like robot considered in our work has the following motion model:
\begin{align*}
x_{t+1} &= x_t + v_{t}\cos(\theta_t)\Delta t, & \theta_{t+1} &= \theta_{t} + \frac{v_t}{L}\tan(\phi_t)\Delta t, \\
y_{t+1} &= y_t + v_{t}\sin(\theta_t)\Delta t, & \phi_{t+1} &= \phi_{t} + \omega_t \Delta t,
\end{align*}
where $\tr{(x_t, y_t, \theta_t, \phi_t)}$ denote the robot's state vector
namely, robot's $x$ and $y$ position, orientation and steering angle at time $t$.
Also, $\tr{(v_t, \omega_t)}$ is the control vector and denotes the robot's linear
velocity and angular velocity (i.e., steering). Here $\Delta t$ is the
discretization of the time step. The values of the parameters used in the
simulation were $L = \SI{0.5}{\meter}$ and $\Delta t = \SI{0.1}{\second}$.
\subsection*{Noise characterization:}
We add zero mean independent identically distributed (i.i.d), random sequences ($\V{w}_t$) as actuator noise to test the performance of the control scheme. The standard deviation of the noise is $\epsilon$ times the maximum value of the corresponding control input, where $\epsilon$ is a scaling factor which is varied during testing, that is:
$
\V{w}_t = \V{u}_{\textrm{max}} \bm{\nu}; \quad \bm{\nu} \sim \mathcal{N}(\V{0}, \M{I})
$
and the noise is added as $\epsilon \V{w}_t$. Note that, we enforce the constraints in the control inputs before the addition of noise, so the controls can even take a value higher after noise is added.
\subsection{Single agent setting:}
A car-like robot is considered and is tasked to move from a given initial pose
to a goal pose. The environment of the robot is shown in
Figure~\ref{fig:test_cases_high}. The experiment is done for all the control
schemes discussed and their performance for different levels of noise are shown
in Figure~\ref{1_agent_cost}.
\subsection{Multi-agent setting:}
A labelled point-to-point transition problem with 3 car-like robots is considered where each agent is assigned a fixed destination which cannot be exchanged with another agent. The performance of the algorithms is shown in Figure~\ref{3_agent_cost}. The cost function involves the state and control costs for the entire system similar to the single agent case. One major addition to the cost function is the penalty function to avoid inter-agent collisions which is given by
$
\Psi^{(i,j)} = \textrm{M}\exp\left(-(\Vert \V{p}_t^i - \V{p}_t^j\Vert_2^2 - r_{\textrm{thresh}}^2)\right)
$
where $\textrm{M} > 0$ is a scaling factor, $\V{p}^i_t = (x^i_t, y^j_t)$ and $r_{\textrm{thresh}}$ is the desired minimum distance the agents should keep between themselves.
\begin{figure}[h]
\centering
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/1_agent_replan_wo_obs_02.pdf}
\caption{A single agent.}
\label{1_agent_replan}
\end{subfigure}
%
\begin{subfigure}[b]{0.225\textwidth}
\centering
\includegraphics[width=\textwidth]{images_pdf/3_agent_replan_wo_obs_02.pdf}
\caption{Three agents.}
\label{3_agent_replan}
\end{subfigure}
\caption{Replanning operations vs. $\epsilon$ for $J_{\textrm{thresh}} = 2\%$}
\label{fig:replan plot}
\vspace*{-12pt}
\end{figure}%
\subsection{Interpretation of the results:}
From Figures~\ref{1 agent cost low} and~\ref{3 agent cost low} it can be clearly seen that the decoupled feedback law (T-LQR and MT-LQR) shows near-optimal performance compared to MPC at low noise levels~($\epsilon \ll 1$). At medium noise levels, replanning (T-LQR2 and MT-LQR2) helps to constrain the cost from deviating away from the optimal. Figure~\ref{fig:replan plot} shows the significant difference in the number of replans, which determines the computational effort, taken by the decoupled approach compared to MPC. Note that the performance of the decoupled feedback law approaches MPC as we decrease the value of $J_{\textrm{thresh}}$. The significant difference in computational time between MPC and T-LQR2 can be seen from Figure~\ref{timevsHc_1_1} which shows results for $\epsilon=0.1$. For $H_c = 35$ (i.e. we plan for the entire time horizon), and $J_{\textrm{thresh}}$= .2\% there is not much difference in the cost between them in~\ref{costvsHc_1_1} (both are in the dark green region), while there is a significant change in computation time as seen in~\ref{timevsHc_1_1}. The trend is similar in the multi-agent case as seen in Figures~\ref{fig:costvsHc_3_1} and ~\ref{fig:timevsHc_3_1} which again shows that the decoupling feedback policy is able to give computationally efficient solutions which are near-optimal in low noise cases by avoiding frequent replanning.
At high noise levels, Figures~\ref{1 agent cost full} and~\ref{3 agent cost full} show that T-LQR2 and MT-LQR2 are on a par with MPC. Additionally, we also claimed that planning too far ahead is not beneficial at high noise levels. It can be seen in Figure~\ref{fig:costvsHc_3_7} that the performance for MPC as well as MT-LQR2 is best at $H_c = 20$. Planning for a shorter horizon also eases the computation burden as seen in Figure~\ref{fig:timevsHc_3_7}. Though not very significant in the single agent case, we can still see that there is no difference in the performance as the horizon is decreased in Figure~\ref{costvsHc_1_7}. It can also be seen in Figure~\ref{3 agent cost full} where MPC-SH and MT-LQR2-SH both with $H_c=7$ outperform MPC with $H_c=35$ at high noise levels which again show that the effective planning horizon decreases at high noise levels.
\section{CONCLUSIONS}
\label{section:conclusion}
In this paper, we have considered a class of stochastic motion planning problems for robotic systems over a wide range of uncertainty conditions parameterized in terms of a noise parameter $\epsilon$. We have shown extensive empirical evidence that a simple generalization of a recently developed ``decoupling principle" can lead to tractable planning without sacrificing performance for a wide range of noise levels. Future work will seek to treat the medium and high noise systems, considered here, analytically and look to establish the near-optimality of the scheme. Further, we shall consider the question of ``when and how to replan'' in a distributed fashion in the multi-agent setting, as well as relax the requirement of perfect state observation.
\printbibliography
\end{document}
| 2024-02-18T23:40:04.214Z | 2019-09-19T02:18:46.000Z | algebraic_stack_train_0000 | 1,242 | 7,513 |
|
proofpile-arXiv_065-6470 | \section{Introduction}
The complete automation of the 1-loop calculations for LHC and ILC physics
is nowadays a feasible task~\cite{nlopapers}.
The advent of the OPP reduction method~\cite{opp},
together with the concept of multiple cuts~\cite{britto},
allowed to revitalize the
Unitarity Techniques~\cite{unitarity}, by reducing the computation of
1-loop amplitudes to a problem with the same conceptual
complexity of a tree level calculation, resulting in achievements that
were inconceivable only a few years ago~\cite{proof}.
The main idea of the OPP based techniques is directly extracting,
from the 1-loop amplitude,
the coefficients of the (known) scalar loop functions. This task can
be reached in a completely numerical way
by {\em opening the loop} and transforming the 1-loop amplitude
in a tree level object with 2 more legs, that can be calculated,
at the {\em integrand level}, by using the same
recursion relations~\cite{recursions} that allow a very efficient
computation of complicated multi-leg tree
level processes~\cite{alpgen,helac}.
A second possible
option is that one of the so-called Generalized Unitarity
methods~\cite{genunit},
where tree-level amplitudes are {\em glued together}.
Both procedures only allow the extraction of
the Cut Constructible (CC) part of the amplitude in 4 dimensions, while a
left over piece, the rational part ${\rm R}$, needs to be derived separately.
In the Generalized Unitarity approaches, that is achieved by computing
the amplitude in different numbers of space-time dimensions, or
via bootstrapping techniques~\cite{boot}, while,
in the OPP approach, ${\rm R}$ is split in 2 pieces ${\rm R= R_1+R_2}$. The first
piece, ${\rm R_1}$, is derivable in the same framework used to reconstruct the
CC part of the amplitude, while ${\rm R_2}$ is computable
through a special set Feynman rules for the theory at hand~\cite{rational},
to be used in a tree level-like computation.
Such a set of ${\rm R_2}$ Feynman rules has been already derived for QED
in~\cite{rational} and for QCD in~\cite{qcdrational}, and it is the main aim of the present paper
to present the complete set of the ${\rm R_2}$ Feynman rules for the Standard Model (SM)
of the Electroweak (EW) interactions.
In addition, as a by-product, we use the derived formulae to explicitly check
the gauge invariance of the Four Dimensional Helicity regularization
scheme in the EW sector at 1-loop, the motivation being that this is a very
well studied subject in QCD~\cite{fdhqcd}, but, in our knowledge, very little can be found in the literature for the full EW Standard Model.
The outline of the paper is as follows. In section~\ref{sec:2} we remind some
facts on the origin of ${\rm R}$ and on the splitting ${\rm R}= {\rm R_1}+{\rm R_2}$. Section~\ref{sec:3} contains the complete
list of all possible special ${\rm R_2}$ EW SM vertices
in the 't Hooft-Feynman gauge and, in section~\ref{sec:4},
we describe the tests we performed on our formulae and our findings.
Finally, our conclusions are drawn in section~\ref{sec:5} and, in the appendix,
we collect a list of Ward identities.
\section{Theory, facts and conjectures on ${\rm R}$, ${\rm R_1}$ and ${\rm R_2}$ \label{sec:2}}
Before carrying out our program, we spend a few words on
the origin of ${\rm R}$. Our starting point is the general expression for the
{\it integrand} of a generic $m$-point
one-loop (sub-) amplitude
\begin{eqnarray}
\label{eq:1}
\bar A(\bar q)= \frac{\bar N(\bar q)}{\db{0}\db{1}\cdots \db{m-1}}\,,~~~
\db{i} = ({\bar q} + p_i)^2-m_i^2\,,
\end{eqnarray}
where ${\bar q}$ is the integration momentum and where dimensional
regularization is assumed, so that
a bar denotes objects living
in $n=~4+\epsilon$ dimensions and a tilde represents $\epsilon$-dimensional
quantities.
When a $n$-dimensional index is contracted with a 4-dimensional vector
$v_\mu$, the $4$-dimensional part is automatically selected.
For example
\begin{eqnarray}
\label{noeps}
\bar q \cdot v &\equiv& (q+ {\tld q}) \cdot v\,= q \cdot v\,,~~~
\rlap/ {\bar v} \equiv \bar \gamma_{\bar \mu}\, v^\mu = \rlap /v\,
~~~{\rm and}~~~ {\bar q}^2 = q^2 + {\tld q}^2\,.
\end{eqnarray}
The numerator function $\bar{N}(\bar q)$ can be
split into a $4$-dimensional plus an $\epsilon$-dimensional part
\begin{eqnarray}
\label{eq:split}
\bar{N}(\bar q) = N(q) + \tld{N}(\tld{q}^2,q,\epsilon)\,.
\end{eqnarray}
$N(q)$ lives in $4$ dimensions, and can be therefore expanded in terms of
$4$-dimensional denominators
\begin{eqnarray}
\d{i} = ({q} + p_i)^2-m_i^2 = \db{i} - {\tld q}^2\,.
\end{eqnarray}
Some among the coefficients in this expansion are interpreted, in the OPP
method, as the desired coefficients of the 1-loop scalar integrals and
can be determined numerically, while
the mismatch between this expansion in terms of $4$-dimensional denominators,
and the $n$-dimensional denominators appearing in eq.~\ref{eq:1},
is the origin of the rational terms ${\rm R_1}$. There exist at least
two ways~\cite{sixphoton,cuttools} to compute ${\rm R_1}$,
which allow to determine it by means of a purely numerical knowledge
of the 4-dimensional CC part of the amplitude, while this does not seem to
be possible
for ${\rm R_2}$, whose origin is the term
$\tld{N}(\tld{q}^2,q,\epsilon)$ in eq.~\ref{eq:split}, after
integration over the loop momentum:
\begin{eqnarray}
\label{eqr2}
{\rm R_2} \equiv \frac{1}{(2 \pi)^4}\int d^n\,\bar q
\frac{\tld{N}(\tld{q}^2,q,\epsilon)}{\db{0}\db{1}\cdots \db{m-1}} \,.
\end{eqnarray}
However, ${\rm R_2}$ can be computed by extracting
$\tld{N}(\tld{q}^2,q,\epsilon)$ from any given {{\em integrand}
$\bar A (\bar q)$, which can be achieved
by splitting, in the analytic expression of the numerator function,
the $n$-dimensional
integration momentum
${\bar q}$, the $n$-dimensional gamma matrices
$\bar \gamma_{\bar \mu}$ and the $n$-dimensional metric tensor
$\bar g^{\bar \mu \bar \nu}$ into a $4$-dimensional
component plus remaining pieces:
\begin{eqnarray}
\label{qandg}
\bar q &=& q + \tld{q}\,, \nonumber \\
\bar \gamma_{\bar \mu} &=& \gamma_{\mu}+ \tld{\gamma}_{\tld{\mu}}\,,\nonumber \\
\bar g^{\bar \mu \bar \nu} &=& g^{\mu \nu}+ \tld{g}^{\tld{\mu} \tld{\nu}}\,.
\end{eqnarray}
Therefore, a practical way to determine ${\rm R_2}$ is computing
analytically, by means of Feynman diagrams, once for all
and with the help of eq.~\ref{qandg}, tree-level like Feynman rules,
namely effective vertices, by calculating the ${\rm R_2}$ part coming from
all possible one-particle irreducible Green functions of the theory at hand,
up to four external legs.
The fact that four external legs are enough to account for ${\rm R_2}$
is guaranteed by the ultraviolet nature of the rational terms,
proved in~\cite{directcomp1}.
This property does not hold, instead, for ${\rm R_1}$, that, diagram
by diagram, can give non vanishing contributions to any
one-particle irreducible $m$-point function, because, even when finite, the
tensor integrals generating ${\rm R_1}$ are eventually expressed, via tensor
reduction, in terms of linear combinations of 1-loop scalar functions that can
be ultraviolet divergent. This fact prevents the possibility
of calculating a finite set of effective vertices reproducing ${\rm R_1}$.
Eq.~\ref{eqr2} generates a set of simple basic integrals with up to 4 denominators, containing powers of $\tld{q}$ and $\epsilon$ in the numerator.
A list that exhausts all possibilities in the $\xi = 1$ 't Hooft-Feynman gauge
can be found in~\cite{qcdrational}.
Notice, however, that, according to the chosen regularization scheme,
results may differ. In eq.~\ref{eqr2} we assume the
't Hooft-Veltman (HV) scheme, while in the Four Dimensional Helicity scheme
(FDH), any explicit $\epsilon$ dependence in the numerator function
is discarded before integration, such that
\begin{eqnarray}
\label{eqr2fdh}
{\rm R_2} \Bigl |_{FDH} = \frac{1}{(2 \pi)^4}\int d^n\,\bar q
\frac{\tld{N}(\tld{q}^2,q,\epsilon= 0)}{\db{0}\db{1}\cdots \db{m-1}} \,.
\end{eqnarray}
The asymmetric role played by ${\rm R_1}$ and ${\rm R_2}$
is somewhat annoying. As we have seen, ${\rm R_1}$ is directly
connected with the
($4$-dimensional) CC part of the amplitude, and can be computed, even numerically, without any reference to Feynman diagrams, while
${\rm R_2}$ requires an analytic determination in terms of Feynman diagrams,
so that one would like
to be able to put both pieces on the same footing.
Unfortunately, no easy direct
connection between ${\rm R_2}$ and the CC part of the amplitude has been found so far (at least within our treatment at the {\em integrand level}) and,
in the rest of this paragraph, we speculate a bit on this subject.
Reconstructing ${\rm R_2}$ numerically would require to detect ``signs'' of it in the CC part.
For example, one could naively think that, by
looking at any $q^2$ in the CC part, the $\tld{q}^2$ dependence
could be inferred via the replacement
\begin{eqnarray}
\label{eq:repl}
q^2 \to q^2 + \tld{q}^2\,.
\end{eqnarray}
However, such a dependence is impossible to reconstruct numerically, when remaining in 4 dimensions, as it can be illustrated by considering the
following simple 3-point sub-amplitude:
\begin{eqnarray}
\label{eqex}
{\rm A } \equiv \frac{1}{(2 \pi)^4}\int d^n\,\bar q
\frac{(q \cdot \ell_3)(q \cdot \ell_4) }{\db{0}\db{1}\db{2}} \,,
\end{eqnarray}
where
\begin{eqnarray}
\ell_3^\mu = <\ell_1| \gamma^\mu | \ell_2]\,,~~
\ell_4^\mu = <\ell_2| \gamma^\mu | \ell_1]\,
~~{\rm with}~~
\ell_{1,2}^2= 0\,.
\end{eqnarray}
From the one hand, the $4$-dimensional numerator $(q \cdot \ell_3)(q \cdot \ell_4)$ in eq.~\ref{eqex} does not
contain any $q^2$ to be continued through the replacement of
eq.~\ref{eq:repl}. On the other hand, it can be
manipulated as follows
\begin{eqnarray}
(q \cdot \ell_3)(q \cdot \ell_4)= 4\,(q \cdot \ell_1)(q \cdot \ell_2)-
2\,q^2\,(\ell_1 \cdot \ell_2)\,,
\end{eqnarray}
and now the shift of eq.~\ref{eq:repl} would produce a $\tld{q}^2$ contribution,
in disagreement with our previous finding. We therefore conclude that
not enough information is present in the 4-dimensional part to reconstruct
${\rm R_2}$. This is the reason why one is forced to work analytically
in $n$ dimensions to reconstruct the ${\rm R_2}$ contribution
\footnote{
In other approaches~\cite{genunit}, a numerical determination of the
whole ${\rm R}$ contribution can be achieved, but at the price of
explicitly computing numerically the amplitude in 4, 6 and 8 dimensions.}.
Nevertheless, based on a simple reasoning, one argues
that some gauge invariance properties of the $4$-dimensional
part of the amplitude should be transferred to ${\rm R_2}$.
In fact, for physical processes, the sum of ${\rm R_1} + {\rm R_2}$ is
gauge invariant. On the other hand, ${\rm R_1}$
can be fully reconstructed from the $4$-dimensional, gauge invariant,
CC part of the amplitude, meaning that, by changing gauge,
the same expressions for ${\rm R_1}$ should be found, and, as a consequence,
also {\em the same result for ${\rm R_2}$}.
This should be off course only true for amplitudes with
physical external particles, because different gauges may have, in general,
a different content in terms unphysical external fields. Therefore
one can conjecture that
\begin{center}
{\em The ${\rm R_2}$ part of a physical amplitude gives the same result when computed in any gauge \footnote{This does not mean that the ${\rm R_2}$ part
of the Green functions satisfy the Ward identities
separately from ${\rm R_1}$, as we have checked explicitly.}
.}
\end{center}
This conjecture, being rather strong, should be proved with an actual
calculation. Unfortunately, such a calculation would require to extend the
set of basic integrals in~\cite{qcdrational} to be able to deal with
non-renormalizable gauges. That is beyond the scope of this work, and
we leave it for a future publication.
In the present paper, we fix the gauge to be the
the 't Hooft-Feynman one and we derive all of the effective Electroweak
${\rm R_2}$ Feynman rules by applying the splittings of eq.~\ref{qandg}
Feynman diagram by Feynman diagram. For the interested reader,
explicit examples of this technique can be found in ~\cite{qcdrational}.
\section{Results \label{sec:3}}
In this section, we give the complete list of the
effective Electroweak vertices contributing to ${\rm R_2}$ in
the 't Hooft-Feynman gauge
\footnote{They can be also found as a {\tt FORM}~\cite{form} output in
http://www.ugr.es/local/pittau/CutTools.}.
A parameter $\lambda_{HV}$ is introduced in our formulae such that
$\lambda_{HV}= 1$ corresponds to the 't Hooft-Veltman scheme
and $\lambda_{HV}= 0$ to the FDH scheme of eq.~\ref{eqr2fdh}.
We used the Feynman rules given in~\cite{denner} and
our notations are
as follows: $l_1= e$, $l_2= \mu$, $l_3= \tau$, $l_4= \nu_e$, $l_5= \nu_\mu$,
$l_6= \nu_\tau$ and $q_1= d$, $q_2= u$, $q_3= s$, $q_4= c$, $q_5= b$, $q_6= t$.
In addition,
$e_1= e$,
$e_2= \mu$,
$e_3= \tau$,
$\nu_1= \nu_e$,
$\nu_2= \nu_\mu$,
$\nu_3= \nu_\tau$ and
$u_1= u$, $u_2= c$, $u_3= t$, $d_1= d$, $d_2= s$, $d_3= b$.
When appearing as external particles, $l$, $\nu_l$, $u$ and $d$
stand for the three charged leptons, the three (massless) neutrinos,
the three up-type quarks and
the three down-type quarks, respectively.
Effective vertices with external quarks are always
understood to be diagonal in the color space.
Finally, $N_{col}$ is the number
of colors and $V_{u_i d_j}$ are CKM matrix elements.
Occasionally, combinations such as
\begin{eqnarray}
\sum_{i,j=1}^{3}
\left(
V_{u_i d_j} V_{d_j u_i}^\dagger
\right) = 3\,~~~{\rm and}~~~ \sum_{i=1}^{3} 1 = 3
\nonumber
\end{eqnarray}
appear in our formulae. In such cases, we do not explicitly work out
the sum in order to make our results also readable family by family.
A last comment is in order with respect to our
treatment of $\gamma_5$ in vertices
containing fermionic lines. When computing all contributing Feynman diagrams,
we pick up a ``special'' vertex in the loop and anticommute all $\gamma_5$'s
to reach it before performing the $n$-dimensional algebra, and, when a trace
is present, we start reading it from this vertex. This treatment
produces, in general, a term proportional to the totally antisymmetric $\epsilon$
tensor, whose coefficient may be different
depending on the choice of the ``special''
vertex. However, when summing over all quantum numbers of each fermionic
family, we checked that all contributions proportional to $\epsilon$ cancel.
In addition, we explicitly verified that our results satisfy
the large set of Ward identities given in appendix~\ref{appa}.
\subsection{\underline{Electroweak effective vertices with 2 external legs}}
In this section, we give the complete list of the non vanishing
2-point ${\rm R_2}$ effective vertices.
\subsubsection{{Scalar-Scalar effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(200,50)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\SetOffset(0,55)
\Vertex(142,-62){6.0}
\DashLine(56,-62)(142,-62){8}
\DashLine(142,-62)(228,-62){8}
\Text(124,-31)[]{$S_2$}
\Text(18,-31)[]{$S_1$}
\Text(138,-31)[l]{$\displaystyle = \frac{ie^2}{16 \pi^2 s_w^2} C$}
\end{picture}
}
\end{center}
with the actual values of $S_1$, $S_2$ and $C$
\begin{eqnarray}
H\chi~~:~~C & = & 0
\nonumber \\
\nonumber \\
HH~~:~~C & = &
\frac{m_\phi^2}{4} + \frac{m_\chi^2}{8 c_w^2}
+\frac{1-12\lambda_{HV}}{4}\left(1+\frac{1}{2c_w^4}\right) m_W^2
- \left(1+\frac{1}{2 c_w^2} \right) \frac{p^2}{12} + K
\nonumber \\
\nonumber \\
\chi\chi~~:~~C & = &
\frac{m_\phi^2}{4} + \frac{m_H^2}{8 c_w^2}
+\frac{1-4\lambda_{HV}}{4} \left(1+\frac{1}{2c_w^4}\right) m_W^2 -
\left(1+\frac{1}{2 c_w^2} \right) \frac{p^2}{12} + K
\nonumber \\
\nonumber \\
\phi^-\phi^+ ~~:~~C & = &
\frac{m_H^2+m_\chi^2}{8}
+ \frac{\left(3-4\lambda_{HV}\right) c_w^4 -2 c_w^2 + \left(\frac{1}{2}-2\lambda_{HV}\right)}{c_w^4}\frac{m_W^2}{4}
+ \frac{m_\phi^2}{8 c_w^2}
\nonumber \\
&-& \left(1 + \frac{1}{2 c_w^2} \right)
\frac{p^2}{12}
+
\frac{1}{2 m_W^2 } \left[
\sum_{i= 1}^{3}
\left(m_{e_i}^2
\left(m_{e_i}^2-\frac{p^2}{3} \right)\right)
\right. \nonumber \\
& + & \left. N_{\mathrm{col}} \sum_{i,j=1}^{3}
\left(
V_{u_i d_j} V_{d_j u_i}^\dagger
\left(m_{u_i}^2 + m_{d_j}^2\right)
\left(m_{u_i}^2 + m_{d_j}^2-\frac{p^2}{3}\right)
\right)
\right]
\end{eqnarray}
where
\begin{eqnarray}
K &=&
\frac{1}{m_W^2}\left[
\sum_{i=1}^{6} \left( m_{l_i}^2\left(m_{l_i}^2 - \frac{p^2}{6}
\right)\right)
+ N_{\mathrm{col}}
\sum_{i=1}^{6} \left( m_{q_i}^2\left(m_{q_i}^2 - \frac{p^2}{6}
\right)
\right)
\right]
\end{eqnarray}
\subsubsection{{Vector-Vector effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(250,50)
\SetOffset(0,42)
\SetScale{0.5}
\SetWidth{0.5}
\SetColor{Black}
\Photon(65,-34)(133,-34){5.5}{4}
\Vertex(134,-34){6.0}
\Photon(135,-34)(199,-34){5.5}{4}
\Text(50,-3)[]{{\Black{$p$}}}
\LongArrow(75,-20)(110,-20)
\Text(21,-17)[]{$V_{1\alpha}$}
\Text(113,-17)[]{$V_{2\beta}$}
\Text(130,-14)[l]{$\displaystyle
= \frac{ie^2}{\pi^2}
\left(C_1 p_\alpha p_\beta + C_2 g_{\alpha \beta}\right)
$}
\end{picture}
}
\end{center}
with the actual values of $V_1$, $V_2$, $C_1$ and $C_2$
\begin{eqnarray}
AA~~:~~C_1 & = & - \frac{1}{24} \lambda_{HV} \nonumber \\
C_2 & = & \frac{1}{8}\left[p^2\left(
\frac{1}{6}+\frac{\lambda_{HV}}{3}\right) - m_W^2 \right]
- \frac{1}{4} \left[
\sum_{i=1}^{6}\left(Q_{l_i}^2
\left(m_{l_i}^2 - \frac{1}{6} p^2\right)\right)
\right. \nonumber \\
& + & \left.
N_{\mathrm{col}}\sum_{i=1}^{6}\left(Q_{q_i}^2
\left(m_{q_i}^2 - \frac{1}{6} p^2\right)\right)
\right]
\nonumber \\
\nonumber \\
AZ~~:~~C_1 & = & \frac{1}{24} \frac{c_w}{s_w} \lambda_{HV}\, \nonumber \\
C_2 & = & - \frac{1}{8}\frac{c_w}{s_w}\left[p^2\left(
\frac{1}{6}+\frac{\lambda_{HV}}{3}\right) - m_W^2 \right]
+ \frac{1}{4 c_w} \left[ \sum_{i=1}^{6}\left(
\left(\frac{Q_{l_i} I_{3l_i}}{2 s_w} - Q_{l_i}^2 s_w\right)
\right. \right.
\nonumber \\
&\times& \left. \left.
\left(m_{l_i}^2 - \frac{1}{6} p^2\right)\right)
+ N_{\mathrm{col}} \sum_{i=1}^{6}\left(
\left(\frac{Q_{q_i} I_{3q_i}}{2 s_w} - Q_{q_i}^2 s_w\right)
\left(m_{q_i}^2 - \frac{1}{6} p^2\right)\right)
\right]
\nonumber \\
\nonumber \\
ZZ~~:~~C_1 & = & - \frac{1}{24} \frac{c_w^2}{s_w^2} \lambda_{HV}\nonumber \\
C_2 & = & \frac{1}{8}\frac{c_w^2}{s_w^2}\left[p^2\left(\frac{1}{6}+\frac{\lambda_{HV}}{3}\right)- m_W^2 \right] +
\frac{1}{4 c_w^2}
\left[
\sum_{i=1}^{6}\left(
\left(Q_{l_i} I_{3l_i} - \frac{I_{3l_i}^2}{2 s_w^2} - Q_{l_i}^2 s_w^2\right)
\right. \right.
\nonumber \\
&\times& \left. \left.
\left(m_{l_i}^2 - \frac{1}{6} p^2\right)\right)
+N_{\mathrm{col}}\sum_{i=1}^{6}\left(
\left(Q_{q_i} I_{3q_i} - \frac{I_{3q_i}^2}{2 s_w^2} - Q_{q_i}^2 s_w^2\right)
\left(m_{q_i}^2 - \frac{1}{6} p^2\right)\right)
\right]
\nonumber \\
\nonumber \\
W^-W^+~~:~~C_1 & = & - \frac{1}{24 s_w^2} \lambda_{HV}\nonumber \\
C_2 & = & \frac{1}{8 s_w^2}\left[p^2\left(\frac{1}{6}+\frac{\lambda_{HV}}{3}\right) - m_W^2 \right]
- \frac{1}{32 s_w^2}
\left[ \sum_{i=1}^3 \left(m_{e_i}^2 - \frac{p^2}{3}\right)
\right.
\nonumber \\
& + & \left. N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left(
V_{u_i d_j} V_{d_j u_i}^\dagger
\left(m_{u_i}^2 + m_{d_j}^2 - \frac{p^2}{3}\right)
\right)
\right]
\end{eqnarray}
\subsubsection{{Fermion-Fermion effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(320,50)
\SetOffset(0,55)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\ArrowLine(62,-62)(138,-62)
\ArrowLine(141,-62)(221,-62)
\Vertex(142,-63){6.0}
\Text(50,-18)[]{{\Black{$p$}}}
\LongArrow(75,-50)(110,-50)
\Text(22,-31)[]{$F_1$}
\Text(123,-31)[]{$\bar F_2$}
\Text(135,-31)[l]{$\displaystyle = \frac{ie^2}{\pi^2}
\left[\left(C_- \Omega^- + C_+\Omega^+\right)\rlap/p + C_0
\right]\lambda_{HV}
$}
\end{picture}
}
\end{center}
with the actual values of $F_1$, $\bar F_2$, $C_-$, $C_+$ and $C_0$
\begin{eqnarray}
u \bar u ~~:~~C_- & = & \frac{1}{16} \frac{Q_u^2}{c_w^2} \nonumber\\
C_+ & = & \frac{1}{16} \left(\frac{I_{3u}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{u} I_{3u}}{c_w^2} + \frac{Q_u^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger\right)
\right) \nonumber \\
C_0 & = & \frac{m_{u} Q_{u} }{8 c_w^2} \left( Q_{u} - I_{3u} \right)
\nonumber \\
\nonumber \\
d \bar d ~~:~~C_- & = & \frac{1}{16} \frac{Q_{d}^2}{c_w^2} \nonumber \\
C_+ & = & \frac{1}{16} \left(
\frac{I_{3{d}}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{d} I_{3d}}{c_w^2} + \frac{Q_{d}^2}{c_w^2}
+ \frac{1}{2 s_w^2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger\right)
\right) \nonumber \\
C_0 & = & \frac{m_{d} Q_{d}}{8 c_w^2} \left( Q_{d} - I_{3d} \right)
\nonumber \\
\nonumber \\
l \bar l ~~:~~C_- & = & \frac{1}{16} \frac{Q_{l}^2}{c_w^2} \nonumber \\
C_+ & = & \frac{1}{16} \left(\frac{I_{3l}^2}{s_w^2 c_w^2}
- \frac{ 2 Q_{l} I_{3l}}{c_w^2} + \frac{Q_{l}^2}{c_w^2}
+ \frac{1}{2 s_w^2} \right) \nonumber \\
C_0 & = & \frac{m_{l}Q_{l}}{8 c_w^2} \left( Q_{l} - I_{3l} \right)
\nonumber \\
\nonumber \\
\nu_l \bar \nu_l ~~:~~C_- & = & 0 \nonumber \\
C_+ & = & \frac{1}{32 s_w^2} \left(\frac{1}{2 c_w^2}
+ 1 \right) \nonumber \\
C_0 & = & 0
\end{eqnarray}
\subsection{\underline{Electroweak effective vertices with 3 external legs}}
We list here the 3-point ${\rm R_2}$ effective vertices.
\subsubsection{{Scalar-Fermion-Fermion effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(210,65)
\SetOffset(-20,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.0}
\DashLine(72,15)(144,15){8}
\ArrowLine(200,57)(146,15)
\ArrowLine(146,15)(200,-27)
\Text(33, 7.5)[r]{$S$}
\Text(102, 30)[l]{$F_1$}
\Text(102,-15)[l]{$\bar F_2$}
\Text(115,7.5)[l]{$\displaystyle = \frac{e^3}{\pi^2}
(C_{-}\Omega^{-} + C_{+}\Omega^{+})$}
\end{picture}
}
\end{center}
with the actual values of $S$, $F_1$, $\bar F_2$, $C_-$ and $C_+$
\begin{eqnarray}
H u \bar u ~~:~~C_- &=& \frac{i m_u}{8 m_W s_w} \left[
\frac{\left(1+\lambda_{HV} \right)Q_u^2}{2c_w^2}
+\frac{1}{16 s_w^2}\sum_{j=1}^3
\left(
V_{u d_j}V_{d_ju}^\dagger
\right)
+\frac{I_{3u}}{c_w^2}
\left(
\frac{I_{3u}}{8 s_w^2}
\right. \right. \nonumber \\
&-& \left. \left. \frac{\left(1+\lambda_{HV} \right)Q_u}{2}
\right)
+\frac{1}{16 m_W^2 s_w^2}
\sum_{j=1}^3
\left(m_{d_j}^2 V_{ud_j}V_{d_ju}^\dagger\right)
\right]
\nonumber \\
C_+ &=& C_-
\nonumber \\
\nonumber \\
H d \bar d ~~:~~C_- &=& \frac{i m_d}{8 m_W s_w} \left[
\frac{ \left(1+\lambda_{HV} \right)Q_d^2}{2c_w^2}
+\frac{1}{16 s_w^2}\sum_{i=1}^3
\left(
V_{u_i d}V_{d u_i}^\dagger
\right)
+\frac{I_{3d}}{c_w^2}
\left(
\frac{I_{3d}}{8 s_w^2}
\right. \right. \nonumber \\
&-& \left. \left. \frac{\left(1+\lambda_{HV} \right)Q_d}{2}
\right)
+\frac{1}{16 m_W^2 s_w^2}
\sum_{i=1}^3
\left(m_{u_i}^2 V_{u_id}V_{du_i}^\dagger\right)
\right]
\nonumber \\
C_+ &=& C_-
\nonumber \\
\nonumber \\
H l \bar l ~~:~~C_- &=& \frac{i m_l}{8 m_W s_w} \left[
\frac{ \left(1+\lambda_{HV} \right)Q_l^2}{2c_w^2}
+\frac{1}{16 s_w^2}
+\frac{I_{3l}}{c_w^2}
\left( \frac{I_{3l}}{8 s_w^2} - \frac{\left(1+\lambda_{HV}
\right)Q_l}{2} \right)
\right]
\nonumber \\
C_+ &=& C_-
\nonumber \\
\nonumber \\
H \nu_l \bar \nu_l ~~:~~C_- &=& 0
\nonumber \\
C_+ &=& 0
\nonumber \\
\nonumber \\
\chi u \bar u ~~:~~C_- & = & -\frac{m_u}{4 m_W s_w} \left[
\frac{ \left(1+\lambda_{HV} \right)Q_u^2 I_{3u}}{2c_w^2}
+\frac{1}{32 s_w^2}\sum_{j=1}^3
\left(
V_{u d_j}V_{d_ju}^\dagger
\right)
+\frac{I_{3u}}{c_w^2}
\left(
\frac{1}{32 s_w^2}
\right. \right. \nonumber \\
&-& \left. \left. \frac{\left(1+\lambda_{HV} \right)Q_u I_{3u}}{2}
\right)
- \frac{1}{16 m_W^2 s_w^2}
\sum_{j=1}^3
\left(m_{d_j}^2 I_{3d_j} V_{ud_j}V_{d_ju}^\dagger\right)
\right]
\nonumber \\
C_+ &=& -C_-
\nonumber \\
\nonumber \\
\chi d \bar d ~~:~~C_- & = & -\frac{m_d}{4 m_W s_w} \left[
\frac{\left(1+\lambda_{HV} \right)Q_d^2 I_{3d}}{2c_w^2}
-\frac{1}{32 s_w^2}\sum_{i=1}^3
\left(
V_{u_i d}V_{d u_i}^\dagger
\right)
+\frac{I_{3d}}{c_w^2}
\left(
\frac{1}{32 s_w^2}
\right. \right. \nonumber \\
&-& \left. \left. \frac{\left(1+\lambda_{HV} \right)Q_d I_{3d}}{2}
\right)
-\frac{1}{16 m_W^2 s_w^2}
\sum_{i=1}^3
\left(m_{u_i}^2 I_{3u_i} V_{u_id}V_{d u_i}^\dagger\right)
\right]
\nonumber \\
C_+ &=& -C_-
\nonumber \\
\nonumber \\
\chi l \bar l ~~:~~C_- & = & -\frac{m_l}{4 m_W s_w} \left[
\frac{\left(1+\lambda_{HV} \right)Q_l^2 I_{3l}}{2c_w^2}
-\frac{1}{32 s_w^2}
+\frac{I_{3l}}{c_w^2}
\left(
\frac{1}{32 s_w^2} - \frac{\left(1+\lambda_{HV} \right)Q_l I_{3l}}{2}
\right) \right.\nonumber\\
&-& \left. \frac{m_l^2 I_{3l}}{8 m_W^2 s_w^2}
\left(
-\frac{1}{4} + I_{3l}^2
\right)
\right]
\nonumber\\
C_+ &=& -C_-
\nonumber \\
\nonumber \\
\chi \nu_l \bar \nu_l ~~:~~C_- & = & 0
\nonumber\\
C_+ &=& 0
\nonumber \\
\nonumber\\
\phi^- u \bar{d} ~~:~~
C_- & = & -\frac{im_d V_{du}^\dagger}{4 \sqrt{2} m_W s_w} \left[
\frac{1}{c_w^2}
\left(
\frac{-1}{16} - \frac{\left(1+\lambda_{HV} \right)Q_u Q_d}{2}
\right)
- \frac{3}{32 s_w^2}
\right. \nonumber\\
&-&\left.
\frac{m_u^2}{16 s_w^2 m_W^2}
+\frac{I_{3u}}{c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_d}{2} + \frac{1}{16} \right)
\right]
\nonumber \\
C_+ & = & \frac{im_u V_{du}^\dagger}{4 \sqrt{2} m_W s_w} \left[
\frac{1}{c_w^2}
\left(
\frac{-1}{16} - \frac{\left(1+\lambda_{HV} \right)Q_u Q_d}{2}
\right)
- \frac{3}{32s_w^2}
\right. \nonumber\\
&-&\left.
\frac{m_d^2}{16 s_w^2 m_W^2}
+\frac{I_{3d}}{c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_u}{2} - \frac{1}{16} \right)
\right]
\nonumber \\
\nonumber \\
\phi^+ d \bar u ~~:~~
C_- & = & - \frac{ im_u V_{ud}}{4 \sqrt{2} s_w m_W} \left[
\frac{1}{c_w^2}
\left(
\frac{1}{16} + \frac{\left(1+\lambda_{HV} \right)Q_u Q_d}{2}
\right)
+ \frac{3}{32 s_w^2}
\right. \nonumber\\
&+&\left.
\frac{m_d^2}{16 s_w^2 m_W^2}
-\frac{I_{3d}}{c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_u}{2} - \frac{1}{16} \right)
\right] \nonumber \\
C_+ & = & \frac{im_d V_{ud}}{4 \sqrt{2} m_W s_w} \left[
\frac{1}{c_w^2}
\left(
\frac{1}{16} + \frac{\left(1+\lambda_{HV} \right)Q_u Q_d}{2}
\right)
+ \frac{3}{32 s_w^2}
\right. \nonumber\\
&+&\left.
\frac{m_u^2}{16 s_w^2 m_W^2}
-\frac{I_{3u}}{c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_d}{2} + \frac{1}{16} \right)
\right]
\nonumber \\
\nonumber \\
\phi^- \nu_{l} \bar{l} ~~:~~
C_- & = & -\frac{im_l}{4 \sqrt{2} m_W s_w} \left[
\frac{Q_l}{16 c_w^2}
- \frac{3}{32 s_w^2}
+\frac{I_{3\nu_l}}{c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_l}{2} + \frac{1}{16} \right)
\right] \nonumber \\
C_+ & = & 0
\nonumber \\
\nonumber \\
\phi^+ l \bar{\nu}_{l} ~~:~~
C_- & = & 0 \nonumber \\
C_+ & = & \frac{im_l}{4 \sqrt{2} m_W s_w} \left[
-\frac{Q_l}{16 c_w^2}
+ \frac{3}{32 s_w^2}
-\frac{I_{3\nu_l}}{ c_w^2}
\left(\frac{\left(1+\lambda_{HV} \right)Q_l}{2} + \frac{1}{16} \right)
\right]
\end{eqnarray}
\subsubsection{{Vector-Fermion-Fermion effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(230,65)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.}
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\ArrowLine(146,15)(200,-27)
\Photon(72,15)(146,15){5.5}{4}
\Text(30,7.5)[r]{\Black{$V_\mu$}}
\Text(102, 30)[l]{$F_1$}
\Text(102,-15)[l]{$\bar F_2$}
\Text(115,7.5)[l]{$\displaystyle =\frac{i e^3}{\pi^2} (C_{-}\Omega^{-}+C_{+}\Omega^{+}) \gamma_\mu$}
\end{picture}
}
\end{center}
with the actual values of $V$, $F_1$, $\bar F_2$, $C_-$ and $C_+$
\begin{eqnarray}
A u \bar u ~~:~~
C_- &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_u^3}{4 c_w^2}
+\frac{m_u^2}{8 s_w^2 m_W^2}
\left(\frac{1}{2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger Q_{d_j}\right)
\right. \right. \nonumber \\
&+& \left. \left. \frac{Q_u}{4} + Q_u I_{3u}^2
\right)
\right]
\nonumber \\
C_+ &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_u^3}{4 c_w^2} - \frac{\left(1+\lambda_{HV}\right)Q_u^2 I_{3u}}{2c_w^2} +
\frac{\left(1+\lambda_{HV}\right)Q_u I_{3u}^2}{4 s_w^2 c_w^2} \right.\nonumber \\
& &+ \left.\frac{1}{4 s_w^2}
\left(\frac{1}{4 m_W^2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger m_{d_j}^2 Q_{d_j}\right)
\right.\right.\nonumber \\
& & \left.\left. + \frac{m_u^2 Q_u \left(1+4 I_{3u}^2\right)}{8 m_W^2}
+ \sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger \left(
1 + Q_{d_j}
\right)\frac{\left(1+\lambda_{HV}\right)}{2}\right)
\right)
\right]
\nonumber \\
\nonumber \\
A d \bar d ~~:~~
C_- &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_d^3}{4 c_w^2}
+\frac{m_d^2}{8 s_w^2 m_W^2}
\left(\frac{1}{2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger Q_{u_i}\right)
\right. \right. \nonumber \\
&+& \left. \left. \frac{Q_d}{4} + Q_d I_{3d}^2
\right)
\right]
\nonumber \\
C_+ &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_d^3}{4 c_w^2} - \frac{\left(1+\lambda_{HV}\right)Q_d^2 I_{3d}}{2c_w^2} +
\frac{\left(1+\lambda_{HV}\right)Q_d I_{3d}^2}{4 s_w^2 c_w^2} \right.\nonumber \\
& &+\left.\frac{1}{4 s_w^2}
\left(\frac{1}{4 m_W^2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger m_{u_i}^2 Q_{u_i}\right)
\right.\right.\nonumber \\
& & \left.\left.
+ \frac{m_d^2 Q_d \left(1+4 I_{3d}^2\right)}{8 m_W^2}
+ \sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger \left(
Q_{u_i} -1
\right)\frac{\left(1+\lambda_{HV}\right)}{2}\right)
\right)
\right]
\nonumber \\
\nonumber \\
A l \bar l ~~:~~
C_- &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_l^3}{4 c_w^2}
+\frac{m_l^2}{8 s_w^2 m_W^2}
\left(\frac{Q_l}{4} + Q_l I_{3l}^2
\right)
\right]
\nonumber \\
C_+ &=& \frac{1}{4} \left[
\frac{\left(1+\lambda_{HV}\right)Q_l^3}{4 c_w^2}
- \frac{\left(1+\lambda_{HV}\right)Q_l^2 I_{3l}}{2c_w^2}
+\frac{\left(1+\lambda_{HV}\right)Q_l I_{3l}^2}{4 s_w^2 c_w^2} \right.\nonumber \\
&& +\left.\frac{1}{4 s_w^2}
\left(
\frac{m_l^2 Q_l \left(1+4 I_{3l}^2\right)}{8 m_W^2}
-\frac{\left(1+\lambda_{HV}\right)}{2}\right)\right]
\nonumber \\
\nonumber \\
A \nu_l \bar \nu_l ~~:~~
C_- &=& 0
\nonumber \\
C_+ &=&
\frac{1}{32 s_w^2}
\left[
\frac{m_{l}^2Q_{l}}{2 m_W^2}
+ \left(Q_{l}+1\right)\left(1+\lambda_{HV}\right)
\right]
\nonumber \\
\nonumber \\
Z u \bar u ~~:~~
C_- &=& \frac{1}{8 c_w} \left\{
\frac{\left(1+\lambda_{HV}\right)Q_u^3 s_w}{2c_w^2}
+\frac{m_u^2}{8 s_w m_W^2}
\left[
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger
\left(Q_{d_j}-\frac{I_{3{d_j}}}{s_w^2}\right)
\right)
\right.\right.\nonumber \\
& & \left.\left.
+ \left(Q_u-\frac{I_{3u}}{s_w^2}\right)
\right]
\right\}
\nonumber \\
C_+ &=& \frac{1}{8 c_w} \left\{
\frac{\left(1+\lambda_{HV}\right)Q_u^3 s_w}{2c_w^2} - \frac{\left(1+\lambda_{HV}\right)Q_u^2 I_{3u} (1 + 2 s_w^2)}{ 2s_w c_w^2} \right.\nonumber \\
&&+3 \left.\frac{\left(1+\lambda_{HV}\right)Q_u I_{3u}^2}{2s_w c_w^2} -
\frac{\left(1+\lambda_{HV}\right)I_{3u}^3}{2s_w^3 c_w^2} \right. \nonumber \\
&+&
\frac{1}{2s_w }
\left[\frac{1}{4 m_W^2}
\left(
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger m_{d_j}^2 Q_{d_j}\right)
+ \frac{m_u^2 Q_u \left(1+4 I_{3u}^2\right)}{2}
\right)
\right.\nonumber \\
&+& \left.\left.
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger \frac{\left(1+\lambda_{HV}\right)}{2}
\left(Q_{d_j}- \frac{c_w^2 + I_{3d_j}}{s_w^2}
\right)
\right)
\right]
\right\}
\nonumber \\
\nonumber \\
Z d \bar d ~~:~~
C_- &=& \frac{1}{8 c_w} \left\{
\frac{\left(1+\lambda_{HV}\right)Q_d^3 s_w}{2c_w^2}
+\frac{m_d^2}{8 s_w m_W^2}
\left[
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger
\left(Q_{u_i}-\frac{I_{3{u_i}}}{s_w^2}\right)
\right)
\right.\right.\nonumber \\
& & \left.\left.
+ \left(Q_d-\frac{I_{3d}}{s_w^2}\right)
\right]
\right\}
\nonumber \\
C_+ &=& \frac{1}{16 c_w} \left\{\left(1+\lambda_{HV}\right)\left(
\frac{Q_d^3 s_w}{c_w^2} - \frac{Q_d^2 I_{3d} (1 + 2 s_w^2)}{ s_w c_w^2}
+3\frac{Q_d I_{3d}^2}{s_w c_w^2} -
\frac{I_{3d}^3}{s_w^3 c_w^2} \right)\right. \nonumber \\
&+& \left. \frac{1}{s_w}
\left[\frac{1}{4 m_W^2}
\left(\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger m_{u_i}^2 Q_{u_i}\right)
\right.\right.\right.
\left. + \frac{m_d^2 Q_d \left(1+4 I_{3d}^2\right)}{2}
\right)
\nonumber \\
&+& \left.\left.
\sum_{i=1}^3 \left(\frac{1+\lambda_{HV}}{2}\right)\left(V_{u_i d} V_{d u_i}^\dagger
\left(Q_{u_i}+\frac{c_w^2 - I_{3u_i}}{s_w^2}\right)
\right)
\right]
\right\}
\nonumber \\
\nonumber \\
Z l \bar l ~~:~~
C_- &=& \frac{1}{8 c_w} \left\{
\frac{\left(1+\lambda_{HV}\right)Q_l^3 s_w}{2c_w^2}
+\frac{m_l^2}{4 s_w m_W^2}
\left[
-\frac{1}{4s_w^2}
+ \frac{1}{2} \left(Q_l-\frac{I_{3l}}{s_w^2}\right)
\right]
\right\} \nonumber \\
C_+ &=& \frac{1}{16 c_w} \left\{\left(
\frac{Q_l^3 s_w}{c_w^2} - \frac{Q_l^2 I_{3l} (1 + 2 s_w^2)}{ s_w c_w^2} \right.\right.\nonumber \\
&&+3 \left.\left.\frac{Q_l I_{3l}^2}{s_w c_w^2} -
\frac{I_{3l}^3}{s_w^3 c_w^2}\right)\left(1+\lambda_{HV}\right)
+\frac{1}{2 s_w }
\left[
\frac{m_l^2 Q_l}{2 m_W^2}
\right.\right. \nonumber \\
&&+ \left.\left.\frac{1}{s_w^2} \left(1+\lambda_{HV}\right)
\left(c_w^2 - I_{3\nu_{l}}\right)
\right]
\right\}
\nonumber\\
\nonumber\\
Z \nu_l \bar \nu_l ~~:~~
C_- &=& 0
\nonumber \\
C_+ &=& \frac{1}{16 c_w} \left\{
-\frac{\left(1+\lambda_{HV}\right)I_{3\nu_l}^3}{s_w^3 c_w^2}
+\frac{1}{2 s_w}
\left[\frac{m_{l}^2 Q_{l}}{2 m_W^2}
\right.\right.\nonumber \\
&& +\left. \left.\left(1+\lambda_{HV}\right)\left(Q_{l} - \frac{c_w^2 + I_{3l}}{s_w^2}\right)
\right]
\right\}
\nonumber\\
\nonumber\\
W^- u \bar{d} ~~:~~
C_- & = & 0
\nonumber \\
C_+ &=& \frac{V_{du}^\dagger}{16\sqrt{2}s_w}
\left[
\frac{Q_d I_{3u}+ Q_u I_{3d} - Q_u Q_d}{ c_w^2}
- \frac{1}{s_w^2}
+ \frac{1}{4 s_w^2 c_w^2}
\right] \left(1+\lambda_{HV}\right)
\nonumber\\
\nonumber\\
W^+ d \bar{u} ~~:~~
C_- & = & 0
\nonumber \\
C_+ &=& \frac{V_{ud}}{16\sqrt{2}s_w}
\left[
\frac{Q_d I_{3u}+ Q_u I_{3d} - Q_u Q_d}{ c_w^2}
- \frac{1}{s_w^2}
+ \frac{1}{4 s_w^2 c_w^2}
\right] \left(1+\lambda_{HV}\right)
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$W^- \nu_l \bar{l}$ \\
$W^+ l \bar \nu_l$
\end{tabular}\right\}~~:~~
C_- & = & 0
\nonumber \\
C_+ &=& \frac{1}{16\sqrt{2} s_w}
\left[
\frac{Q_l I_{3\nu_l}}{c_w^2}
- \frac{1}{s_w^2}
+ \frac{1}{4 s_w^2 c_w^2}
\right] \left(1+\lambda_{HV}\right)
\nonumber \\
\end{eqnarray}
\subsubsection{{Scalar-Scalar-Scalar effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
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\begin{picture}(161,65)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
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\Text(33,7.5)[r]{\Black{$S_1$}}
\Text(107, 30)[l]{$S_2$}
\Text(107,-15)[l]{$S_3$}
\Text(120,7.5)[l]{$\displaystyle =\frac{ie^3}{\pi^2} C $}
\end{picture}
}
\end{center}
with the actual values of $S_1$, $S_2$, $S_3$, and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$HH\chi$ \\
$\chi\chi\chi$ \\
$\chi\phi^+\phi^-$
\end{tabular}\right\}~~:~~
C & = & 0
\nonumber\\
\nonumber\\
HHH~~:~~C & = &
\frac{3}{32 s_w^3}\left[
\frac{1-4\lambda_{HV}}{2} m_W + \frac{1}{m_W^3} \left(
\sum_{i=1}^6 m_{l_i}^4 + N_{\mathrm{col}} \sum_{i=1}^6 m_{q_i}^4
\right) \right. \nonumber \\
&+& \left.
\frac{1}{4}\left(1+\frac{1}{2 c_w^2}\right)\frac{m_H^2}{m_W}
+\frac{\left(1-4\lambda_{HV}\right) m_W}{4 c_w^4}
\right]
\nonumber\\
\nonumber\\
H\chi\chi~~:~~C & = &
\frac{1}{8 s_w^3}\left[
\frac{1-4\lambda_{HV}}{8} m_W + \frac{1}{4 m_W^3} \left(
\sum_{i=1}^6 m_{l_i}^4
+ N_{\mathrm{col}} \sum_{i=1}^6 m_{q_i}^4
\right) \right. \nonumber \\
&+& \left.
\frac{1}{16}\left(1+\frac{1}{2 c_w^2}\right)\frac{m_H^2}{m_W}
+\frac{\left(1-4\lambda_{HV}\right) m_W}{16 c_w^4}
\right]
\nonumber\\
\nonumber\\
H\phi^+\phi^-~~:~~C & = &
\frac{1}{32 s_w^3}\left[
\frac{1}{m_W^3}\left(
\sum_{i=1}^3 m_{e_i}^4
+ N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger
(m_{u_i}^4+m_{d_j}^4)
\right) \right)
\right.\nonumber\\
&+&\left.
\frac{(1+2c_w^2)}{8 c_w^2}\frac{m_H^2}{m_W}
+\frac{3\left(1-4\lambda_{HV}\right)}{4}m_W
+\frac{1-4\lambda_{HV}}{4} \frac{s_w^2 \left(1+c_w^2\right)}{c_w^4} m_W
\right]
\nonumber \\
\end{eqnarray}
\subsubsection{{Vector-Scalar-Scalar effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(210,70)
\SetOffset(-10,28)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
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\Text(75,29)[lb]{{\Black{$p_1$}}}
\Text(75,-13)[lt]{{\Black{$p_2$}}}
\Text(33,7.5)[r]{\Black{$V_\mu$}}
\Text(107, 30)[l]{$S_1$}
\Text(107,-15)[l]{$S_2$}
\Text(125,7.5)[l]{$\displaystyle = \frac{e^3}{\pi^2} C (p_1-p_2)_\mu$}
\end{picture}
}
\end{center}
with the actual values of $V$, $S_1$, $S_2$, and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$AHH$ \\
$ZHH$ \\
$A\chi\chi$ \\
$Z\chi\chi$
\end{tabular}\right\}:~~C & = & 0
\nonumber\\
\nonumber\\
A\chi H~~:~~C & = &
\frac{5}{192 s_w^2}
\nonumber\\
\nonumber\\
Z\chi H~~:~~C & = &
-\frac{1}{96 s_w c_w}\left[
\frac{1+ 2 c_w^2 + 20 c_w^4}{8 s_w^2 c_w^2}
+ \frac{1}{s_w^2 m_W^2}\left(
\sum_{i=1}^6 \left(m_{l_i}^2 +N_{\mathrm{col}} m_{q_i}^2 \right)
\right)
\right]
\nonumber\\
\nonumber\\
A\phi^+ \phi^-:~~C & = &
\frac{i}{48 s_w^2}\left[
\frac{1+12c_w^2}{8c_w^2}
+ \frac{1}{m_W^2} \left(-\sum_{i=1}^3
\left(m_{e_i}^2 Q_{e_i}\right) \right. \right. \nonumber \\
&+& \left. \left. N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger
(m_{u_i}^2 + m_{d_j}^2)
\right)
\right)
\right]
\nonumber\\
\nonumber\\
Z\phi^+ \phi^-:~~C & = &
\frac{i}{48 s_w c_w}\left\{
\frac{1-24c_w^4}{16 c_w^2 s_w^2}
+ \frac{1}{m_W^2} \left(-\sum_{i=1}^3\left(
m_{e_i}^2 \left(Q_{e_i} + \frac{I_{3\nu_{i}}}{s_w^2}\right)
\right) \right.\right.\nonumber\\
&+& \left.\left.
N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger \left[
(m_{u_i}^2 + m_{d_j}^2)
+ \frac{m_{u_i}^2 I_{3d_j} - m_{d_i}^2 I_{3u_i}}{s_w^2}
\right]
\right)
\right)
\right\}
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$W^+\phi^- H$ \\
$W^- H \phi^+$
\end{tabular}\right\}:~~C & = &
\frac{i}{96 s_w^3}\left[
\frac{1+22c_w^2}{8 c_w^2}
+ \frac{1}{m_W^2} \left(\sum_{i=1}^3
m_{e_i}^2
\right. \right. \nonumber \\
&+& \left. \left. N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger
(m_{u_i}^2 + m_{d_j}^2)
\right)
\right)
\right]
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$W^+\phi^- \chi$ \\
$W^-\phi^+ \chi$
\end{tabular}\right\}:~~C & = &
\frac{1}{48 s_w^3}\left[
-\frac{1+22c_w^2}{16 c_w^2}
+ \frac{1}{m_W^2} \left(\sum_{i=1}^3
\left(m_{e_i}^2 I_{3e_i}\right)
\right. \right. \nonumber \\
&-& \left. \left. N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger
(m_{u_i}^2 I_{3u_i}- m_{d_j}^2 I_{3d_j})
\right)
\right)
\right]
\nonumber \\
\end{eqnarray}
\subsubsection{{Scalar-Vector-Vector effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
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\begin{picture}(190,70)
\SetOffset(-10,28)
\SetWidth{0.5}
\SetScale{0.5}
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\Photon(146,15)(200,57){5.5}{4}
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\Text(33,7.5)[r]{\Black{$S$}}
\Text(107, 30)[l]{$V_{1\mu}$}
\Text(107,-15)[l]{$V_{2\nu}$}
\Text(125,7.5)[l]{$\displaystyle = \frac{ie^3}{\pi^2} C g_{\mu\nu}$}
\end{picture}
}
\end{center}
with the actual values of $S$, $V_1$, $V_2$ and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$\chi AA$ \\
$\chi AZ$ \\
$\chi ZZ$ \\
$\chi W^- W^+$
\end{tabular}\right\}:~~C & = & 0
\nonumber\\
\nonumber\\
H AA ~~:~~ C &=&
- \frac{1}{8 s_w}
\left[\frac{1}{m_W}
\left(
\sum_{i=1}^6 \left(m_{l_i}^2 Q_{l_i}^2\right)
+ N_{\mathrm{col}} \sum_{i=1}^6 \left(m_{q_i}^2 Q_{q_i}^2\right)
\right)
+ \frac{m_W}{2}
\right]
\nonumber\\
\nonumber\\
H AZ ~~:~~ C &=&
\frac{1}{8 c_w}
\left\{\frac{1}{m_W}
\left[
\sum_{i=1}^6 \left(m_{l_i}^2 Q_{l_i} \left(\frac{I_{3l_i}}{2 s_w^2}-Q_{l_i} \right)
\right) \right. \right. \nonumber \\
&+& \left. \left. N_{\mathrm{col}} \sum_{i=1}^6 \left(m_{q_i}^2 Q_{q_i}
\left(\frac{I_{3q_i}}{2 s_w^2} - Q_{q_i}
\right)
\right)
\right]
+ \frac{m_W \left(1+2 c_w^2\right)}{4 s_w^2}\right\}
\nonumber \\
\nonumber \\
H ZZ ~~:~~ C &=&
\frac{1}{8}
\left\{\frac{1}{m_W c_w^2}
\left[
\sum_{i=1}^6 \left(m_{l_i}^2 \left(
\frac{Q_{l_i} I_{3l_i}}{s_w}-Q_{l_i}^2 s_w
-\frac{I_{3l_i}^2}{s_w^3}
\right)
\right) \right. \right. \nonumber \\
&+& \left. \left. N_{\mathrm{col}} \sum_{i=1}^6 \left(m_{q_i}^2 \left(
\frac{Q_{q_i} I_{3q_i}}{s_w} - Q_{q_i}^2 s_w
-\frac{I_{3q_i}^2}{s_w^3}
\right)
\right)
\right]
+ \frac{m_W \left(s_w^2 - 2\right)}{2 s_w^3}
\right\}
\nonumber \\
\nonumber \\
H W^-W^+ ~~:~~ C &=&
-\frac{1}{8 s_w^3}
\left[\frac{1}{4 m_W}
\left(
\sum_{i=1}^3 m_{e_i}^2
\right. \right. \nonumber\\
&+& \left. \left. N_{\mathrm{col}} \sum_{i,j=1}^3
\left(V_{u_id_j}V_{d_ju_i}^\dagger
\left(m_{u_i}^2 + m_{d_j}^2 \right) \right)
\right)
+ m_W
\right]
\nonumber \\
\nonumber \\
\left.
\begin{tabular}{l}
$\phi^- A W^+$ \\
$\phi^+ W^- A$
\end{tabular}\right\}:~~C & = &
\frac{1}{32 s_w^2} K
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$\phi^- Z W^+$ \\
$\phi^+ W^- Z$
\end{tabular}\right\}:~~C & = &
\frac{1}{32 s_w c_w} K
\end{eqnarray}
where
\begin{eqnarray}
K &=& m_W +
\frac{N_{\mathrm{col}}}{m_W}
\sum_{i,j=1}^3 \left(V_{u_id_j}V_{d_ju_i}^\dagger
\left(Q_{u_i} m_{d_j}^2 - Q_{d_j} m_{u_i}^2 \right)\right)
\end{eqnarray}
\subsubsection{{Vector-Vector-Vector effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(390,80)
\SetOffset(-6,34)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.0}
\Photon(72,15)(146,15){5.5}{4}
\Photon(146,15)(200,57){5.5}{4}
\Photon(200,-27)(146,15){5.5}{4}
\LongArrow(85,28)(120,28)
\Text(52,18)[b]{{\Black{$p_1$}}}
\LongArrow(180,62.3)(156,43)
\LongArrow(180,-31)(156,-12)
\Text(75,29)[lb]{{\Black{$p_2$}}}
\Text(75,-13)[lt]{{\Black{$p_3$}}}
\Text(33,7.5)[r]{\Black{$V_{1\alpha}$}}
\Text(107, 30)[l]{$V_{2\mu}$}
\Text(107,-15)[l]{$V_{3\nu}$}
\Text(125,7.5)[l]{$\displaystyle = \frac{ie^3}{\pi^2}\,C\,
\left[g_{\alpha\mu} (p_2-p_1)_\nu
+ g_{\mu\nu} (p_3-p_2)_\alpha
+ g_{\nu\alpha}(p_1-p_3)_\mu
\right]$}
\end{picture}
}
\end{center}
with the actual values of $V_1$, $V_2$, $V_3$ and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$AAA$ \\
$AAZ$ \\
$AZZ$ \\
$ZZZ$ \\
\end{tabular}\right\}:~~C & = & 0
\nonumber\\
\nonumber\\
AW^+W^- ~~:~~ C &=& K
\nonumber \\
\nonumber \\
ZW^+W^- ~~:~~ C &=& -\frac{c_w}{s_w} K
\end{eqnarray}
where
\begin{eqnarray}
K &=& \frac{7+4\lambda_{HV}}{96 s_w^2}
+\frac{1}{48 s_w^2}
\left[ \sum_{i=1}^3 1
+ N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left(V_{u_id_j}V_{d_ju_i}^\dagger \right)
\right]
\end{eqnarray}
\subsection{\underline{Electroweak effective vertices with 4 external legs}}
In this section, we give all possible contributing
4-point ${\rm R_2}$ effective vertices.
\subsubsection{{Scalar-Scalar-Scalar-Scalar effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(130,73)
\SetOffset(-8,31.5)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\DashLine(57,-36)(161,54){8}
\DashLine(57,54)(161,-36){8}
\Vertex(110,9){6.0}
\Text(26,-18)[tr]{$S_1$}
\Text(26, 27)[br]{$S_2$}
\Text(84, 27)[bl]{$S_3$}
\Text(84,-18)[tl]{$S_4$}
\Text(95,5)[l]{$\displaystyle = \frac{ie^4}{\pi^2} \,C $}
\end{picture}
}
\end{center}
with the actual values of $S_1$, $S_2$, $S_3$, $S_4$ and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$HHH\chi $ \\
$H\chi\chi\chi$ \\
$H\chi\phi^-\phi^+$ \\
\end{tabular}\right\}:~~C & = & 0
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$HHHH$ \\
$\chi\chi\chi\chi$ \\
\end{tabular}\right\}:~~C & = & \frac{1}{64 s_w^4} K_1
\nonumber\\
\nonumber\\
HH\chi\chi ~~:~~ C &=&
\frac{1}{192 s_w^4} K_1
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$HH\phi^-\phi^+$ \\
$\chi\chi\phi^-\phi^+$ \\
\end{tabular}\right\}:~~C & = & \frac{1}{64 s_w^4} K_2
\nonumber\\
\nonumber\\
\phi^-\phi^+\phi^-\phi^+ ~~:~~ C &=&
\frac{1}{32 s_w^4} K_3
\end{eqnarray}
where
\begin{eqnarray}
K_1 &=&
\frac{1}{m_W^2}\left[
\frac{5}{m_W^2}
\sum_{i=1}^{6} \left (m_{l_i}^4
+N_{\mathrm{col}} m_{q_i}^4 \right )
+\frac{3}{2}m_H^2\left(1+\frac{1}{2 c_w^2}\right)
\right]
+\frac{1-12\lambda_{HV}}{2}
\left(1+\frac{1}{2 c_w^4}\right)
\nonumber \\
\nonumber \\
K_2 &=&
\frac{1}{m_W^2}\left[
\frac{5}{3 m_W^2}\left(
\sum_{i=1}^{3} m_{e_i}^4
+N_{\mathrm{col}}\sum_{i,j=1}^{3} V_{u_id_j}V_{d_ju_i}^\dagger
\left(m_{u_i}^4 + m_{d_j}^4\right)\right)
+\frac{1}{2}m_H^2\left(1+\frac{1}{2 c_w^2}\right)
\right]
\nonumber\\
&+&\frac{1-12\lambda_{HV}}{4}\left(1+\frac{s_w^2}{3c_w^2}\left(1+\frac{1}{c_w^2}\right)\right)
\nonumber \\
\nonumber \\
K_3 &=&
\frac{1}{m_W^2}\left[\frac{5}{3 m_W^2}\left(
\sum_{i=1}^{3} m_{e_i}^4
+N_{\mathrm{col}}\sum_{i,j,k,l=1}^{3}
\left(V_{u_id_j}V_{d_ju_k}^\dagger V_{u_kd_l}V_{d_lu_i}^\dagger
\left(m_{u_i}^2m_{u_k}^2 + m_{d_j}^2m_{d_l}^2\right)\right)\right)
\right.\nonumber\\
&+& \left.
\frac{1}{2}m_h^2\left(1+\frac{1}{2 c_w^2}\right)
\right]
+\left( \left(\frac{1}{4}-3\lambda_{HV}\right)
\left(1+s_w^4\right)
\right. \nonumber \\
&+& \left.\left(\frac{1}{6}-2\lambda_{HV}\right) \left(s_w^2+\frac{2s_w^6}{c_w^2}\right)+\left(\frac{1}{12}-\lambda_{HV}\right)\frac{s_w^8}{c_w^4}\right)
\nonumber \\
\end{eqnarray}
\subsubsection{{Vector-Vector-Vector-Vector effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(300,73)
\SetOffset(0,31.5)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Photon(57,-36)(161,54){5.5}{8}
\Photon(57,54)(161,-36){5.5}{8}
\Vertex(110,9){6.0}
\Text(26,-18)[tr]{$V_{1\alpha}$}
\Text(26, 27)[br]{$V_{2\beta}$}
\Text(84, 27)[bl]{$V_{3\mu}$}
\Text(84,-18)[tl]{$V_{4\nu}$}
\Text(95,5)[l]{$\displaystyle =
\frac{ie^4}{\pi^2} [C_1 g_{\alpha\beta}g_{\mu\nu}+
C_2 g_{\alpha\mu}g_{\beta\nu}+
C_3 g_{\alpha\nu}g_{\beta\mu}]
$}
\end{picture}
}
\end{center}
with the actual values of $V_1$, $V_2$, $V_3$, $V_4$
$C_1$, $C_2$ and $C_3$
\begin{eqnarray}
AAAA ~~:~~ C_1 &=&
\frac{1}{12}\left(-1+\sum_{i=1}^{6} Q_{l_i}^4
+N_{\mathrm{col}}\sum_{i=1}^{6} Q_{q_i}^4\right)
\nonumber \\
C_2 &=& C_1
\nonumber \\
C_3 &=& C_1
\nonumber\\
\nonumber\\
AAAZ ~~:~~ C_1 &=&
\frac{1}{12}
\left[
\frac{c_w}{s_w}+\sum_{i=1}^{6}\left(\frac{s_w}{c_w} Q_{l_i}^4 -
\frac{1}{2 s_w c_w} Q_{l_i}^3 I_{3l_i}\right) \right. \nonumber \\
&+& \left. N_{\mathrm{col}}\sum_{i=1}^{6}\left(\frac{s_w}{c_w} Q_{q_i}^4 -
\frac{1}{2 s_w c_w} Q_{q_i}^3 I_{3q_i}
\right)
\right]
\nonumber \\
C_2 &=& C_1
\nonumber \\
C_3 &=& C_1
\nonumber\\
\nonumber\\
AAZZ ~~:~~ C_1 &=&
\frac{1}{12}
\left[- \frac{c_w^2}{s_w^2}
+ \frac{1}{2}\sum_{i=1}^{6}\left(
\frac{s_w^2}{c_w^2} Q_{l_i}^4 + \left(
\frac{s_w}{c_w}Q_{l_i}^2 - \frac{1}{s_w c_w}
Q_{l_i} I_{3l_i}\right)^2
\right) \right.\nonumber\\
&+ &\left.
\frac{N_{\mathrm{col}}}{2}\sum_{i=1}^{6}\left(
\frac{s_w^2}{c_w^2} Q_{q_i}^4 + \left(
\frac{s_w}{c_w}Q_{q_i}^2 - \frac{1}{s_w c_w}
Q_{q_i} I_{3q_i}\right)^2
\right)
\right]
\nonumber \\
C_2 &=& C_1
\nonumber \\
C_3 &=& C_1
\nonumber\\
\nonumber\\
AZZZ ~~:~~ C_1 &=&
\frac{1}{12}
\left[
\frac{c_w^3}{s_w^3}+
\sum_{i=1}^{6}\left(
\frac{s_w^3}{c_w^3} Q_{l_i}^4
-\frac{3}{2}\frac{s_w}{c_w^3}Q_{l_i}^3 I_{3l_i}
\right. \right. \nonumber \\
&+& \left. \left. \frac{3}{2}\frac{1}{s_w c_w^3}Q_{l_i}^2 I_{3l_i}^2
- \frac{1}{2 s_w^3 c_w^3} Q_{l_i} I_{3l_i}^3
\right)
\right. \nonumber \\
&+& N_{\mathrm{col}}\sum_{i=1}^{6}\left(
\frac{s_w^3}{c_w^3} Q_{q_i}^4
-\frac{3}{2}\frac{s_w}{c_w^3}Q_{q_i}^3 I_{3q_i} \right. \nonumber \\
&+& \left. \left. \frac{3}{2}\frac{1}{s_w c_w^3}Q_{q_i}^2 I_{3q_i}^2
- \frac{1}{2 s_w^3 c_w^3} Q_{q_i} I_{3q_i}^3
\right)
\right]
\nonumber \\
C_2 &=& C_1
\nonumber \\
C_3 &=& C_1
\nonumber\\
\nonumber\\
ZZZZ ~~:~~ C_1 &=&
\frac{1}{12}
\left[ -\frac{c_w^4}{s_w^4}
+ \sum_{i=1}^{6} \left(
\frac{s_w^4}{c_w^4} Q_{l_i}^4
- 2 \frac{s_w^2}{c_w^4} Q_{l_i}^3 I_{3l_i}
\right. \right. \nonumber \\
&+& \left. \left.
\frac{3}{c_w^4} Q_{l_i}^2 I_{3l_i}^2
- \frac{2}{s_w^2 c_w^4} Q_{l_i} I_{3l_i}^3
+ \frac{1}{2 s_w^4 c_w^4} I_{3l_i}^4
\right)
\right.\nonumber\\
&+&\left.
N_{\mathrm{col}}\sum_{i=1}^{6} \left(
\frac{s_w^4}{c_w^4} Q_{q_i}^4
- 2 \frac{s_w^2}{c_w^4} Q_{q_i}^3 I_{3q_i}
\right. \right. \nonumber \\
&+& \left. \left.
\frac{3}{c_w^4} Q_{q_i}^2 I_{3q_i}^2
- \frac{2}{s_w^2 c_w^4} Q_{q_i} I_{3q_i}^3
+ \frac{1}{2 s_w^4 c_w^4} I_{3q_i}^4
\right)
\right]
\nonumber \\
C_2 &=& C_1
\nonumber \\
C_3 &=& C_1
\nonumber\\
\nonumber\\
AAW^-W^+ ~~:~~ C_1 &=&
\frac{1}{16 s_w^2} \left[ \frac{10+4\lambda_{HV}}{3} + \sum_{i=1}^3
1
+ \frac{25}{27} N_{\mathrm{col}}\sum_{i,j=1}^3\left(V_{u_id_j}V_{d_ju_i}^\dagger
\right)
\right]
\nonumber \\
C_2 &=&
- \frac{1}{16 s_w^2} \left[ \frac{7+2\lambda_{HV}}{3} + \frac{1}{3}\sum_{i=1}^3
1
+\frac{11}{27} N_{\mathrm{col}}\sum_{i,j=1}^3 \left(V_{u_id_j}V_{d_ju_i}^\dagger
\right)
\right]
\nonumber \\
C_3 &=& C_2
\nonumber\\
\nonumber\\
AZW^-W^+ ~~:~~ C_1 &=&
\frac{1}{16 s_w c_w} \left[
- \frac{\left(10+4\lambda_{HV}\right) c_w^2}{3 s_w^2} +
\left( 1 - \frac{11}{12 s_w^2} \right) \sum_{i=1}^3 1
\right. \nonumber \\
&+&\left.
N_{\mathrm{col}}\sum_{i,j=1}^3 \left(V_{u_id_j}V_{d_ju_i}^\dagger
\left(
\frac{25}{27}-\frac{11}{12s_w^2}
\right)
\right)
\right]
\nonumber \\
C_2 &=&
\frac{1}{16 s_w c_w} \left[
\frac{7+2\lambda_{HV}}{3} \frac{c_w^2}{s_w^2}
+\left(
\frac{5}{12 s_w^2}- \frac{1}{3}
\right) \sum_{i=1}^3 1
\right. \nonumber \\
&+&\left.
N_{\mathrm{col}}\sum_{i,j=1}^3 \left(V_{u_id_j}V_{d_j u_i}^\dagger
\left(
\frac{5}{12 s_w^2}-\frac{11}{27}
\right)
\right)
\right]
\nonumber \\
C_3 &=& C_2
\nonumber\\
\nonumber\\
ZZW^-W^+ ~~:~~ C_1 &=&
\frac{\left(5+2\lambda_{HV}\right) c_w^2}{24 s_w^4}+ \frac{1}{16 c_w^2}
\left[
\left(
1-\frac{11}{6s_w^2}+\frac{11}{12s_w^4}
\right) \sum_{i=1}^3 1
\right.\nonumber\\
&+& \left. N_{\mathrm{col}}
\left(
\frac{25}{27}-\frac{11}{6s_w^2}+\frac{11}{12s_w^4}
\right)
\sum_{i,j=1}^3 \left(
V_{u_i d_j}V_{d_j u_i}^\dagger
\right) \right]
\nonumber \\
C_2 &=&
-\frac{\left(7+2\lambda_{HV}\right) c_w^2}{48 s_w^4}
+ \frac{1}{16 c_w^2}
\left[
\left(
-\frac{1}{3}+\frac{5}{6s_w^2}-\frac{5}{12s_w^4}
\right) \sum_{i=1}^3 1
\right.\nonumber\\
&+& \left. N_{\mathrm{col}}
\left(
-\frac{11}{27}+\frac{5}{6s_w^2}-\frac{5}{12s_w^4}
\right)
\sum_{i,j=1}^3 \left(
V_{u_i d_j}V_{d_j u_i}^\dagger
\right) \right]
\nonumber \\
C_3 &=& C_2
\nonumber\\
\nonumber\\
W^-W^+W^-W^+ ~~:~~ C_1 &=&
\frac{1}{16 s_w^4} \left[
\frac{3+2\lambda_{HV}}{3}+\frac{1}{2}\sum_{i=1}^3 1
\right. \nonumber \\
&+& \left .
\frac{N_{\mathrm{col}}}{2}\sum_{i,j,k,m=1}^3
\left(
V_{u_id_j}V_{d_ju_k}^\dagger V_{u_kd_m}V_{d_mu_i}^\dagger
\right)
\right]
\nonumber \\
C_2 &=&
-\frac{1}{8 s_w^4} \left[
\frac{7+2\lambda_{HV}}{3}+\frac{5}{12}\sum_{i=1}^3 1
\right. \nonumber \\
&+& \left. \frac{5}{12} N_{\mathrm{col}}
\sum_{i,j,k,m=1}^3
\left(
V_{u_id_j}V_{d_ju_k}^\dagger V_{u_kd_m}V_{d_mu_i}^\dagger
\right)
\right]
\nonumber \\
C_3 &=& C_1
\end{eqnarray}
\subsubsection{{Scalar-Scalar-Vector-Vector effective vertices}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(149,70)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\DashLine(135,20)(81,-24){8}
\DashLine(135,20)(81,64){8}
\Photon(135,20)(193,64){5.5}{4}
\Photon(135,20)(193,-24){5.5}{4}
\Text(38,-13)[tr]{$S_{1}$}
\Text(38, 32)[br]{$S_{2}$}
\Text(99, 32)[bl]{$V_{1\mu}$}
\Text(99,-13)[tl]{$V_{2\nu}$}
\Vertex(135,20){6}
\Text(105,10)[l]{$\displaystyle = \frac{ie^4}{\pi^2} C g_{\mu\nu}$}
\end{picture}
}
\end{center}
with the actual values of $S_1$, $S_2$, $V_1$, $V_2$ and $C$
\begin{eqnarray}
\left.
\begin{tabular}{l}
$H\chi AA $ \\
$H\chi AZ $ \\
$H\chi ZZ $ \\
$H\chi W^+W^- $
\end{tabular}\right\}:~~C & = & 0
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$H H AA$ \\
$\chi \chi AA$
\end{tabular}\right\}:~~C & = &
\frac{1}{16 s_w^2}\left\{\frac{1}{12}
-\frac{1}{m_W^2}\left[
\sum_{i=1}^{6} \left(Q_{l_i}^2 m_{l_i}^2\right)
+N_{\mathrm{col}}\sum_{i=1}^{6} \left(Q_{q_i}^2 m_{q_i}^2\right)
\right]
\right\}
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$H H AZ$ \\
$\chi \chi AZ$
\end{tabular}\right\}:~~C & = &
\frac{1}{16 s_w}\left\{\frac{4 + s_w^2}{12 s_w^2 c_w}
+\frac{1}{m_W^2 c_w}\left[
\sum_{i=1}^{6} \left(Q_{l_i} m_{l_i}^2
\left(\frac{I_{3l_i}}{2 s_w^2}-Q_{l_i}\right)
\right) \right.\right. \nonumber\\
&+& \left.\left.
N_{\mathrm{col}}\sum_{i=1}^{6} \left(Q_{q_i} m_{q_i}^2
\left(\frac{I_{3q_i}}{2 s_w^2}-Q_{q_i}\right)
\right)
\right]
\right\}
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$H H ZZ$ \\
$\chi \chi ZZ$
\end{tabular}\right\}:~~C & = &
-\frac{1}{16 c_w^2}\left\{\frac{1 + 2 c_w^2 + 40 c_w^4 - 4 c_w^6}
{48 s_w^4 c_w^2} \right. \nonumber \\
&+& \left. \frac{1}{m_W^2}\left[
\sum_{i=1}^{6} \left(m_{l_i}^2
\left(Q_{l_i}^2 +\frac{4 I_{3l_i}^2}{3 s_w^4}-
\frac{Q_{l_i} I_{3l_i}}{s_w^2}\right)
\right) \right.\right.\nonumber\\
&+& \left.\left.
N_{\mathrm{col}}\sum_{i=1}^{6} \left(m_{q_i}^2
\left(Q_{q_i}^2 + \frac{ 4 I_{3q_i}^2}{3 s_w^4}
-\frac{Q_{q_i} I_{3q_i}}{s_w^2}\right)
\right)
\right]
\right\}
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$H H W^-W^+$ \\
$\chi \chi W^-W^+$
\end{tabular}\right\}:~~C & = &
-\frac{1}{48 s_w^4}\left\{\frac{1 + 38 c_w^2}{16 c_w^2}
\right. \nonumber \\
&+& \left. \frac{1}{m_W^2}\left[
\sum_{i=1}^{3} m_{e_i}^2
+ N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left(V_{u_id_j}V_{d_ju_i}^\dagger
\left(m_{u_i}^2 + m_{d_j}^2\right)
\right)
\right]
\right\}
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$ H \phi^+ W^- A $ \\
$\phi^- H A W^+ $
\end{tabular}\right\}:~~C & = & K_1
\nonumber\\
\nonumber\\
\chi\phi^+ W^- A ~~:~~ C &=&
-i K_1
\nonumber\\
\nonumber\\
\phi^- \chi A W^+ ~~:~~ C &=&
i K_1
\nonumber\\
\nonumber\\
\left.
\begin{tabular}{l}
$ H \phi^+ W^- Z $ \\
$\phi^- H Z W^+ $
\end{tabular}\right\}:~~C & = & K_2
\nonumber\\
\nonumber\\
\chi\phi^+ W^- Z ~~:~~ C &=&
- i K_2
\nonumber\\
\nonumber\\
\phi^- \chi Z W^+ ~~:~~ C &=&
i K_2
\nonumber\\
\nonumber\\
\phi^- \phi^+ A A ~~:~~ C &=&
-\frac{1}{12 s_w^2}\left\{\frac{1 + 21 c_w^2}{16 c_w^2}
+\frac{1}{m_W^2}
\left[
\sum_{i=1}^{3}
m_{e_i}^2
\right.\right. \nonumber\\
&+&\left.\left.
\frac{5}{6} N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left(V_{u_id_j}V_{d_ju_i}^\dagger
(m_{u_i}^2 + m_{d_j}^2)
\right)
\right]
\right\}
\nonumber\\
\nonumber\\
\phi^- \phi^+ A Z ~~:~~ C &=&
\frac{1}{12 s_w c_w}\left\{\frac{42 c_w^4-10 c_w^2-1}{32 s_w^2 c_w^2}
\right. \nonumber \\
&-& \left. \frac{1}{m_W^2}
\left[
\sum_{i=1}^{3}
\left(m_{e_i}^2 Q_{e_i}
\left(Q_{e_i}
+\frac{5}{8}\frac{I_{3\nu_i}}{s_w^2}
\right)
\right)
\right.\right. \nonumber\\
&+&\left.\left.
N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left[V_{u_id_j}V_{d_ju_i}^\dagger
\left(
m_{u_i}^2
\left(\frac{5}{6} -
\frac{I_{3d_i}}{s_w^2}
\left(Q_{d_j} -\frac{5}{8}Q_{u_i}\right)
\right)
\right.\right.\right.\right.\nonumber\\
&+& \left.\left.\left.\left.
m_{d_j}^2
\left(\frac{5}{6} -
\frac{I_{3u_i}}{s_w^2}
\left(Q_{u_i} -\frac{5}{8}Q_{d_j}\right)\right)
\right)
\right]
\right]
\right\}
\nonumber\\
\nonumber\\
\phi^- \phi^+ Z Z ~~:~~ C &=&
\frac{1}{12 c_w^2}\left\{\frac{
- 1 + 2 c_w^2 + 44 c_w^4 - 84 c_w^6}{64 s_w^4 c_w^2}
\right. \nonumber \\
&-& \left. \frac{1}{m_W^2}
\left[
\sum_{i=1}^{3}
\left(m_{e_i}^2
\left(Q_{e_i}^2
+\frac{5}{4}\frac{Q_{e_i}I_{3\nu_i}}{s_w^2}
+\frac{I_{3\nu_i}^2}{s_w^4}
\right)
\right)
\right.\right. \nonumber\\
&+&\left.\left.
N_{\mathrm{col}}\sum_{i,j=1}^{3}
\left[V_{u_id_j}V_{d_ju_i}^\dagger
\left(
m_{u_i}^2
\left(\frac{5}{6} -
\frac{I_{3d_i}}{s_w^2}
\left(2Q_{d_j} -\frac{5}{4}Q_{u_i}\right)
+\frac{I_{3d_i}^2}{s_w^4}
\right)
\right.\right.\right.\right.\nonumber\\
&+& \left.\left.\left.\left.
m_{d_j}^2
\left(\frac{5}{6}-
\frac{I_{3u_i}}{s_w^2}
\left(2 Q_{u_i} -\frac{5}{4}Q_{d_j}\right)\right)
+\frac{I_{3u_i}^2}{s_w^4}
\right)
\right]
\right]
\right\}
\nonumber\\
\nonumber\\
\phi^-\phi^+ W^- W^+ ~~:~~ C &=&
- \frac{1}{48 s_w^4}\left\{
\frac{1}{m_W^2}\left[\left(
\sum_{i=1}^{3} m_{e_i}^2 \right. \right. \right. \nonumber \\
&+& \left. \left. \left. N_{\mathrm{col}}\sum_{i,j,k,l=1}^{3}
\left(V_{u_id_j}V_{d_ju_k}^\dagger V_{u_kd_l}V_{d_lu_i}^\dagger
\left(m_{u_i} m_{u_k} + m_{d_j} m_{d_l}\right)\right)\right)\right]
\right.\nonumber\\
&+& \left.
\frac{38c_w^2+1}
{16c_w^2}
\right\}
\nonumber \\
\end{eqnarray}
with
\begin{eqnarray}
K_1 &=& \frac{1}{24 s_w^3}\left\{\frac{1 + 22 c_w^2}{32 c_w^2}
+K \right\}\nonumber \\
K_2 &=& \frac{1}{24 s_w^2 c_w}\left\{\frac{1 + 21 c_w^2 -22 c_w^4}{32 c_w^2s_w^2}
+K \right\} \nonumber \\
K &=& \frac{1}{8 m_W^2} \left[
\sum_{i=1}^{3} m_{e_i}^2
+N_{\mathrm{col}} \sum_{i,j=1}^{3}
\left(V_{u_id_j}V_{d_ju_i}^\dagger
\left( 3 m_{d_j}^2 + 2 m_{u_i}^2
\right)
\right)
\right]
\end{eqnarray}
\subsection{\underline{Mixed Electroweak/QCD corrections}}
In~\cite{qcdrational}, all mixed ${\rm R_2}$ QCD/Electroweak vertices
with internal QCD particle and external weak fields are presented.
For completeness, we give here the only contributing Mixed Electroweak/QCD ${\rm R_2}$
effective vertex, with internal EW particles and external colored states.
\subsubsection{{Gluon-Quark-Quark effective vertex}}
The generic effective vertex is
\vspace{0.3cm}
\begin{center}
\fbox{
\begin{picture}(250,65)
\SetOffset(-10,25)
\SetWidth{0.5}
\SetScale{0.5}
\SetColor{Black}
\Vertex(146,15){6.}
\ArrowLine(200,57)(146,15)
\ArrowLine(146,15)(200,-27)
\Gluon(72,15)(146,15){5.5}{5}
\Text(30,7.5)[r]{\Black{$G^a_\mu$}}
\Text(102, 30)[l]{$Q_l$}
\Text(102,-15)[l]{$\bar Q_k$}
\Text(115,7.5)[l]{$\displaystyle=
\frac{i g_s e^2}{\pi^2} t_{k l}^a
(C_{-}\Omega^{-}+C_{+}\Omega^{+}) \gamma_\mu $}
\end{picture}
}
\end{center}
with the actual values of $Q$, $\bar Q$, $C_-$ and $C_+$
\begin{eqnarray}
u \bar u ~~:~~ C_- &=&
\frac{1}{16} \left[
(1+\lambda_{HV}) \frac{Q_u^2}{c_w^2}
+\frac{m_u^2}{2 s_w^2 m_W^2}
\left(\frac{1}{2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger \right)
+ \frac{1}{4} + I_{3u}^2
\right)
\right]
\nonumber\\
C_+ &=&
\frac{1}{16} \left[
\left(1+\lambda_{HV}\right)
\left(
\frac{1}{c_w^2}\left(Q_u^2+\frac{I_{3u}^2}{s_w^2} -2 Q_u I_{3u}
\right)
+\frac{1}{2 s_w^2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger\right)
\right)
\right.\nonumber \\
& & \left.
+ \frac{1}{2 m_W^2 s_w^2}
\left(
\frac{1}{2}
\sum_{j=1}^3 \left(V_{u d_j} V_{d_j u}^\dagger m_{d_j}^2 \right)
+ m_u^2 \left(\frac{1}{4} + I_{3u}^2\right)
\right)
\right]
\nonumber\\
\nonumber\\
d \bar d ~~:~~ C_- &=&
\frac{1}{16} \left[
(1+\lambda_{HV}) \frac{Q_d^2}{c_w^2}
+\frac{m_d^2}{2 s_w^2 m_W^2}
\left(\frac{1}{2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger \right)
+ \frac{1}{4} + I_{3d}^2
\right)
\right]
\nonumber\\
C_+ &=&
\frac{1}{16} \left[
\left(1+\lambda_{HV}\right)
\left(
\frac{1}{c_w^2}\left(Q_d^2+\frac{I_{3d}^2}{s_w^2} -2 Q_d I_{3d}
\right)
+\frac{1}{2 s_w^2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger\right)
\right)
\right.\nonumber \\
& & \left.
+ \frac{1}{2 m_W^2 s_w^2}
\left(
\frac{1}{2}
\sum_{i=1}^3 \left(V_{u_i d} V_{d u_i}^\dagger m_{u_i}^2 \right)
+ m_d^2 \left(\frac{1}{4} + I_{3d}^2\right)
\right)
\right]
\end{eqnarray}
\section{Tests and findings \label{sec:4}}
We performed several checks on our formulae. First of all, we
derived them by means of two independent calculations, secondly,
we explicitly checked the gauge invariance of our results with the help of
the Ward Identities listed in app.~\ref{appa}, that we derived,
by using the Background Field Method described in~\cite{bfm1}, in the way
we detail in the appendix. Given the fact
that only ${\rm R= R_1+R_2}$ is gauge invariant,
we adopted the following strategy. The terms proportional
to $\lambda_{HV}$ in our effective vertices are expected to be gauge invariant
by themselves. Such terms can only be generated
by ${\rm R_2}$, so that we could
explicitly check, by using {\tt FORM}, that this part of our results
fulfills all of the Ward identities of app.~\ref{appa}, both in
the 't Hooft-Feynman gauge and in the Background Field Method approach.
This provides an explicit test of the gauge invariance of the
Four Dimensional Helicity regularization scheme in the complete Standard Model
at 1-loop, and we consider this result as a by-product of our calculation.
To also test the parts not proportional to $\lambda_{HV}$, we
computed analytically ${\rm R_1}$
\footnote{We extracted the ${\rm R_1}$ part
of the contributing tensor integrals by using the
Passarino-Veltman~\cite{pasve} reduction technique and by further checking numerically the expressions with the
help of CutTools~\cite{cuttools}.}, we added it to ${\rm R_2}$ and checked that
the quantity ${\rm R_1} + {\rm R_2}$ fulfills all of the 2-point and 3-point
Ward identities listed in the appendix. In the 4-point case, many new vertices
are present in ${\rm R_1}$ that do not contribute to ${\rm R_2}$, such as VVVS, and, given
the fact that, after all, we just need to check ${\rm R_2}$, we limited
ourselves to verify the first six 4-point Ward identities given
in app.~\ref{appa4}, which are the only ones including both the
VVVV and VVV vertices, but not VVVS.
The described gauge invariance test on ${\rm R_1+R_2}$ is a very powerful
and non trivial one. In fact,
the analytic expressions for ${\rm R_1}$ are, in general, much
more complicated than the ones for ${\rm R_2}$, involving a huge amount of
terms with different combinations/powers of Gram determinants.
\section{Conclusions \label{sec:5}}
In the last few years, new techniques have been developed to
efficiently deal with the problem of computing the
radiative corrections needed to cope with the complicated phenomenology
expected at LHC and ILC. Nowadays, thanks to the OPP technique, the
so called Cut Constructible part of the virtual 1-loop amplitudes can be obtained, in a purely numerical way, by means of a calculation of
the same conceptual complexity of a tree level one.
However, the determination of the remaining rational part ${\rm R}$
of the amplitude requires a different strategy.
In the treatment at the {\em integrand level}, that we follow in this paper, a
piece of ${\rm R}$, called ${\rm R_1}$, can be directly
linked to the Cut Constructible part of the amplitude, and it is therefore
numerically and automatically produced, in the OPP framework, by codes like CutTools. The remaining part of ${\rm R}$, called
${\rm R_2}$, cannot be determined numerically in 4 dimensions, and requires an
explicit computation in terms of the vertices of the theory at hand,
up to four external legs. From the knowledge of these vertices,
a finite set of effective tree level Feynman rules can be extracted to
be used to compute ${\rm R_2}$ for processes with an arbitrary
number of external legs. Such effective ${\rm R_2}$ Feynman rules
have been already given, in the literature, for QED and QCD and,
in this paper, we completed the list by computing and presenting the set of ${\rm R_2}$ Feynman rules
for the Electroweak sector, which was the last missing piece for
completely automatizing, in the framework of the OPP method, the 1-loop
calculations in the $SU(3) \times SU(2) \times U(1)$ Standard Model.
In addition, since ${\rm R_2}$ is the only part of the amplitude sensitive to the choice of the regularization scheme, we explicitly proved, by checking a large set of Ward identities, the gauge invariance of the
Four Dimensional Helicity regularization scheme in the full Electroweak
sector at 1-loop.
\section*{Acknowledgments}
Many thanks to Huasheng Shao for having recomputed, independently of us, all
of the ${\rm R_2}$ effective vertices. Thanks to his help and by comparing
with an independent computation we performed in a general $R_\xi$ gauge,
we could fix, in the present version, a few problems in our formulae.
We would like to thank Costas Papadopoulos, Ronald Kleiss and Stefan
Dittmaier for useful discussions. R.P. is also indebted with Mauro Moretti
and Fulvio Piccinini for discussing the gauge invariance arguments presented
in section~\ref{sec:2}.
R.P.'s and I.M.'s research was partially supported by the RTN
European Programme MRTN-CT-2006-035505 (HEPTOOLS, Tools and Precision
Calculations for Physics Discoveries at Colliders).
M.V.G.'s research was supported by INFN.
The research of R.P. and M.V.G. was also supported
by the MEC project FPA2008-02984.
\section*{Appendices}
| 2024-02-18T23:40:05.664Z | 2010-10-04T02:02:18.000Z | algebraic_stack_train_0000 | 1,311 | 11,630 |
|
proofpile-arXiv_065-6472 | \section{Introduction}
Nematic elastomers are formed by crosslinking nematic polymers. Such molecules order orientationally
because of rod-like elements incorporated in their main chains (MC polymers) or pendant as side
chains (SC). In both cases orientational order induces the backbones to change their mean shapes
from spherical to elongated (prolate) or flattened (oblate) forms. Since chain shape and the
macroscopic shape of a network are intimately related, there is a strong coupling between the nematic
order and the shape of such solids. This obtains for both prolate and oblate chain shapes -- our
conclusions will hold for both types of polymer. Several elastic phenomena, unique to nematic
elastomers, arise from this coupling. The most relevant in a discussion of changing the magnitude
of the order $Q$, is that such changes induce huge elongations and contractions (of many 100s\%). The
order can be changed by changing temperature, or by illumination if at least some of the rods contain
chromophores. The effect is reversible and is described in a monograph \cite{warnerbook:03} that
summarises the work of many research groups. The magnitude of the spontaneous elongation,
$\lambda_m$, is proposed (as explained below) as an indicator of the shape anisotropy of the
polymers, $\lambda_m = (\ell_{\parallel}/\ell_{\perp})^{1/3}$, where $\ell_{\parallel}$ and
$\ell_{\perp}$ are the Flory effective step lengths parallel and perpendicular to the nematic
ordering direction, $\vec{n}$.
Our purpose here is to clarify the way in which, at constant temperature, the nematic can perhaps
change the magnitude as well as the direction of its order as distortions are imposed. For instance,
close to the nematic-isotropic transition, or in regions of extreme distortion close to the core of a
disclination, $Q$ may reduce. Some investigations, particularly from the mathematical community,
have had understandable concerns that the standard nematic rubber elastic free energy apparently has
terms unbounded from below in the limit $Q \rightarrow 1$. To retain a physical description, authors
have instead rigidly constrained such terms to be constant. We point
out that (i) realistic chain models do have the critical terms varying, and (ii) that there are more
practical and physically-founded ways to deal with this apparent difficulty.
A simple extension\cite{warnerbook:03} to nematic elastomers of the classical Gaussian theory of
rubber elasticity yields a model, ideal free energy density:
\be f_{\rm el} =\half \mu \left[ {\rm Tr} \left[
\ten{\ell}_0\cdot \ten{\lambda}\sp{T}\cdot \ten{\ell}_\vec{n}^{-1}\cdot \ten{\lambda}\right]
+ \ln\left(\frac{\det{\ten{\ell}_0}}{\det{\ten{\ell}}}\right) \right]
\label{eqn:single} \end{equation}
where $\mu$ is the shear modulus of the rubber in the isotropic state. $\ten{\ell}_0$ and
$\ten{\ell}_\vec{n}$ are the Flory effective step length tensors that give the mean square
dimensions of a nematic Gaussian polymer, and there
characterise the distribution of chain shapes. The former is at formation and the latter is that currently pertaining,
that is after any director rotation or changes in $Q$ induced by strain, or due to temperature or
illumination change. Thus:
\be \langle R_{\parallel}R_{\parallel}\rangle = \frac{1}{3} \ell_{\parallel} L, \;\;\;\;
\langle R_{\perp}R_{\perp}\rangle = \frac{1}{3} \ell_{\perp} L, \;\;\;\; \label{eq:step_def} \end{equation}
with $\ell_{\parallel}$ and $\ell_{\perp}$ the effective step lengths along and and perpendicular to
the director, and where $L$ is the chemical arc length of the polymer.
$\ten{\ell}_0$ and
$\ten{\ell}_\vec{n}$ depend on the order at formation and that currently pertaining ($Q$). The
deformation gradient tensor is $\ten{\lambda}$ and takes the body from its formation state to that
current. Since rubber is a soft solid, deformations are strictly at constant volume and thus $\det{\ten{\lambda}\sp{r}} =
1$.
Such a description of the network chains presumes they are long enough to be Gaussian, albeit
anisotropic. It is reasonable to make this restriction for otherwise chains would not be dominated
by their random configurations and nor would they be so highly extensible as their experimental
response clearly demands. With this generic assumption, it then does not matter greatly what kind of
model one adopts for chains -- the Flory philosophy is that local molecular structure is washed out
by randomness and can be encoded by $\ell$ (here two numbers $\ell_{\parallel}$ and $\ell_{\perp}$).
The simplest model is that of the freely jointed chain with links of length $a$ whereupon
orientational order induces:
\be
\frac{\ell_{\parallel}}{a} = (1+2Q), \;
\frac{\ell_{\perp}}{a} = (1-Q), \; \frac{\det{\ten{\ell}}}{a^3} = (1+2Q)(1-Q)^2 \label{eq:free_joints} \end{equation}
For elastomers over a large range of order parameters and hence also extensions the freely jointed
chain model has proven highly accurate. It is found \cite{GreveFW} that spontaneous deformations
arising from
eqn~(\ref{eqn:single}), $\lambda_m = \left( (1+2Q)/(1-Q)\right)^{1/3}$ correlates perfectly with
independent measurements of $Q$ from optical anisotropy which, along with $\lambda_m$ varies as
temperature is changed. Here, distortions $\lambda_m$ are with respect to a high temperature
reference state where $Q \rightarrow 0$ and $\lambda_m \rightarrow 1$. Chains can be highly extended by
high nematic order, attaining for instance a hairpin state which has a much more rapid increase of
$\ell_{\parallel}$ with $Q$ and hence with $T$ \cite{Sixou}. In any event the picture of the effective
step lengths and hence $\det{\ten{\ell}}$ varying with $Q$ persists for all known chain models. It is unphysical to constrain
the determinant to a fixed value. A supplementary argument that has been advanced for the fixing of
$\det{\ten{\ell}}$ is that this is effectively proportional to the volume in spaced covered by the
chain and that chains are incompressible, i.e. the $\det{\ten{\ell}}$ cannot change. The expression
refers to the extent of the chain, a volume proportional to $N^{3/2}$ since it is a random walk of
lineal dimension proportional to $n^{1/2}$. The actual volume occupied by the monomers of the chain
scales like $N$ (times $a^3$) and thus the density of segments of a particular chain is $\sim
N^{-1/2}a^{-3}$ -- chains are very dilute in a melt and most of the space they span is occupied by
other chains. There is accordingly no volume constraint presented to a given chain. In fact this
argument is at the heart of why chains adopt ideal statistics in a melt \cite{deGennes79}.
What then stops the free energy density (\ref{eqn:single}) minimising at $f = -\infty$ by tending to
$Q = 1$ where $\det{\ten{\ell}} = 0$ and hence $\ln\left(\det{\ten{\ell}_0}/\det{\ten{\ell}}\right) \rightarrow -\infty$?
One must consider the dominant free energy in the problem, namely that of the underlying nematic
phase rather than the weaker rubber part. The nematic chains have, in the absence of linking, a
nematic free energy density $f_N(Q)$ that one can show to be additive to the rubber part arising on
linkage. Model forms arise from simple Landau-de Gennes phenomenology, or from particular molecular
models, for instance the extension of Maier-Saupe nematic theory to worm-like chains. In fact a
freely jointed rod model of nematic chains \cite{Abramchuk:87} would have exactly the Maier-Saupe
nematic free energy since the rods are independent, except through their nematic interactions. As
with the chains, it is not vital for our argument what model one adopts. They all have in common a
free energy of order $k_{\rm B}T_{\rm ni}$ \textit{per monomer} where $T_{\rm ni}$ is a
characteristic temperature where nematic order is lost. The rubber free energy by contrast is of the
order of $k_{\rm B}T$ per network strand since the prefactor in the free energy density
eqn~(\ref{eqn:single}) is the modulus $\mu = n_{\rm s}k_{\rm B}T$ where $n_{\rm s}$ is the number
density of strands. Thus the rubber component of the free energy density is of order $1/N$ smaller
than the nematic part. $N$ is the number of effective step lengths separating crosslink points along
a chain. $N$ has to be large, $\sim 10 -- 100$ in order that chains are Gaussian and the network is
rubbery, that is highly extensible and dominated by entropy. The overall dependence of the free
energy on $Q$ is dominated by the minimum determining the stable state of the liquid under the same
conditions. It is little perturbed by the $Q$-dependent terms arising from the rubber, including the
$\ln\det{\dots}$ term. The rubber adds terms like $\frac{1}{N}(-Q_0^2Q^2 +Q^4)$
\cite{Abramchuk:87,WGV,warnerbook:03} for the ideal case and where spontaneous extension/contraction
has been allowed to occur. ($Q_0$ is the order at crosslinking.) These terms cause minor shifts in
the transition temperature and latent heat. In the non-ideal case, involving memory of orientation
pertaining at formation, a term like $-\frac{1}{N}Q$ arises, which resembles the form of an external
field and removes the isotropic high temperature phase $Q$ identically equal to zero altogether.
Strictly speaking, even though the liquid nematic free energy dominates the nematic contributions
from the rubber, the $\ln\det{\dots}$ term would still yield a separate minimum at $Q=1$. However
the use of the result $\ell_{\perp} = a (1-Q)$ near the point $Q=1$ is also unphysical. This limit
of perfect orientational order would suggest that $\langle R_{\perp}^2 \rangle \rightarrow 0$. Only
for chains mathematically narrow and without any directional fluctuations (at $T=0$) could attain
this limit. In any case the Gaussian approximation of this result will also have long since failed.
How should one proceed? Physically, one is constrained to being near in $Q$ to a modified nematic
minimum in the free energy. One can thus reasonably ignore any minimum near $Q=1$ and rely on the
nematic free energy to naturally constrain $Q$-dependence arising in $f_{\rm el}$. Practically,
unless one is close to the N-I transition where the minimum in $f$ near $Q=0$ is not stiff, one can
take $Q$ to be fixed in magnitude. If strains are extreme, then one can let $Q$ have the freedom to
respond to perturbations arising from $f_{\rm el}(\ten{\lambda},Q)$ by taking the nematic free energy
to be $\half f''_Q\cdot (Q-Q_{\rm min})^2$ (with $f''_Q$ the curvature of the potential at $Q_{\rm
min}$), which was an approached followed in \cite{GreveFW}. These authors were concerned changes in
the linear modulus from applied strains inducing changes in $Q$. This work also considered the role
of $\ten{\lambda}$ in inducing changes in the biaxial order if the principal stretch direction was
not aligned with $\vec{n}_0$. Such changes too must be accounted for in the changing
$\ln\det{\dots}$ term that is the subject of this note.
In summary we have argued physically as to why another source of $Q$-dependent free energy terms from
rubber elasticity must not be constrained to take a constant value. In fact their form relates
closely to well tested models of chain statistics and for the Gaussian theory of nematic rubber
elasticity to hold, these terms must be retained.
| 2024-02-18T23:40:05.670Z | 2009-10-20T18:36:03.000Z | algebraic_stack_train_0000 | 1,312 | 1,883 |
|
proofpile-arXiv_065-6510 | \subsection{Gersten-Bloch-Ogus-Kato complexes}\label{sect0-1}
For a scheme $X$ and a positive integer $n$ invertible on $X$, denote by $\bZ/n(1) = \mu_n$ the \'etale sheaf on $X$ of $n$-th roots of unity, and let $\bZ/n(r) = \mu_n^{\otimes r}$ be the $r$-fold Tate twist, defined for $r \in \mathbb Z$. As usual, we let
\[ \bQ_p/\bZ_p(r) = \varinjlim{}_{n \ge 1} \ \bZ/p^n(r) \qquad \hbox{(for $p$ invertible on $X$).} \]
For a smooth variety $X$ over a perfect field of positive characteristic $p>0$ and integers $n>0$ and $r \ge 0$, $\logwitt X n r$ denotes the \'etale subsheaf of the logarithmic part of the $r$-th Hodge-Witt sheaf $\witt X n r$ (\cite{B}, \cite{il}), which are $\bZ/p^n$-sheaves. It is also noted $\nu^r_{n,X}$ in the literature. We denote
\[ \logwitt X \infty r = \varinjlim{}_{n \ge 1} \ \logwitt X n r \,. \]
Let $X$ be a noetherian excellent scheme, and let $y$ and $x$ be points on $X$ such that $x$ has codimension $1$ in the closure $\ol {\{ y \}} \subset X$. Then for a prime number $p$, Kato (\cite{kk:hasse} \S1) defined `residue maps'
\begin{align}
& H^{i+1}(y,\mu_{p^n}^{\otimes r+1}) \longrightarrow H^i(x,\mu_{p^n}^{\otimes r}) & \hbox{(if $\text{\rm ch}(x)\not = p$)} \notag\\
& H^i(y,\logwitt y n {r+1}) \longrightarrow H^i(x,\logwitt x n r) & \hbox{(if $\text{\rm ch}(y)=\text{\rm ch}(x)=p$)} \label{eq-0.1.1} \\
& H^{i+m+1}(y,\mu_{p^n}^{\otimes r+1}) \longrightarrow H^i(x,\logwitt x n r) \quad & \hbox{(if $\text{\rm ch}(y)=0$ and $\text{\rm ch}(x)=p$)}, \notag
\end{align}
where the maps of second and third type have non-zero target only for $i = 0,1$, and in case $i=1$ they are only defined if $[\kappa(x):\kappa(x)^p] \le p^r$. For a point $x \in X$, we wrote $H^*(x,-)$ for \'etale cohomology of $x = \text{\rm Spec}(\kappa(x))$, so this is just the Galois cohomology of $\kappa(x)$, the residue field at $x$. These maps are defined via the Galois cohomology of discrete valuation fields, symbol maps on Milnor $K$-theory, and the valuation (see \S\ref{sect0-6} below). Therefore we will write $\dval_{y,x}$ for these maps, and denote sheafified variants in the same way. In particular, for $i=r=0$ the first and the last maps via Kummer theory correspond to the map
\[ \kappa(y)^\times/(\kappa(y)^\times)^{p^n} \longrightarrow \bZ/p^n \]
induced by the discrete valuations on the normalization of $\mathscr O_{\overline{\{y\}},x}$.
It has become customary to denote
\[ \bZ/p^n(r) := \logwitt X n r [-r]\]
for an (essentially) smooth scheme over a perfect field of characteristic $p$. With this notation, all maps above have the form
\[ \dval_{y,x}: H^{i+1}(y,\bZ/p^n(r+1)) \longrightarrow H^i(x,\bZ/p^n(r))\,. \]
Denote by $X_q$ the set of points $x \in X$ of dimension $q$ (i.e., whose Zariski closure $\ol{\{x\}}$ has dimension $q$). In \cite{kk:hasse}, Kato showed that, for each triple of integers $i, j$ and $n>0$, the sequence
\begin{align*}
\dotsb \longrightarrow \bigoplus_{x\in X_r} H^{r+i}(x,\mathbb Z/n(r+j)) & \longrightarrow \bigoplus_{x\in X_{r-1}}H^{r+i-1}(x,\mathbb Z/n(r+j-1)) \longrightarrow \\ \dotsb & \longrightarrow \bigoplus_{x\in X_0}H^i(x,\mathbb Z/n(j))\,,
\end{align*}
whose maps have the components $\dval_{y,x}$\,, forms a complex $C_n^{i,j}(X)$. It was a major motivation for this paper
to understand the maps $\dval_{y,x}$ and these complexes in
terms of \'etale duality.
\smallskip
\subsection{\'Etale duality}\label{sect0-2}
A very general duality for constructible \'etale torsion sheaves has been established in \cite{sga4}. This is a {\it relative} duality, encoded in an adjunction
\begin{equation}\label{eq-0.2.1}
\text{\rm Hom}_X(\mathscr F,Rf^!\mathscr G) \cong \text{\rm Hom}_S(Rf_!\mathscr F,\mathscr G)
\end{equation}
for a separated morphism of finite type $f: X \longrightarrow S$ and bounded complexes of \'etale torsion sheaves $\mathscr F$ (on $X$) and $\mathscr G$ (on $S$) with constructible cohomology sheaves. There is also a derived version, replacing $\text{\rm Hom}$ by $R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om$. To obtain an {\it absolute} duality for the cohomology groups of sheaves on $X$, in the spirit of Poincar\'e duality, one
needs an additional duality on the base scheme $S$. For arithmetic applications one is interested in schemes $X$ of finite type over $\mathbb Z$. Therefore we may assume that $S = \text{\rm Spec}(\mathfrak o_k)$, where $\mathfrak o_k$ is the ring of integers in a number field $k$. Here one has the Artin-Verdier duality
\begin{equation}\label{eq-0.2.2}
H^m_c(S,\mathscr F) \times \text{\rm Ext}^{3-m}_S(\mathscr F,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \longrightarrow H^3_c(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) = \bQ/\bZ \,,
\end{equation}
where $H^m_c$ denotes the `cohomology with compact support' \cite{kk:hasse} which takes care of the archimedean places of $k$. But the figuring `dualizing sheaf' ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}$ is not torsion, so the relative duality above, for a scheme $X/S$, does not apply. Nevertheless, for such a higher-dimensional arithmetic scheme $X$, various absolute duality theorems have been obtained (cf.\ \cite{de2}, \cite{sp}, \cite{moser}, \cite{milne:adual}, \cite{Ge}), although always under some restrictions. For example $n$-torsion sheaves for $n$ invertible on $X$ have been considered, or $X$ was assumed to be smooth over $S$, or $X$ was assumed to be a scheme over a finite field.
Our approach is to introduce a complex of torsion sheaves $\bQ/\bZ(1)'_S$ on $S$ (see Definition \ref{def:DC-S}, \eqref{eq4-1-3}) so that one has a perfect duality as in \eqref{eq-0.2.2} when replacing ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,S}$ by $\bQ/\bZ(1)'_S$. Next we define the dualizing `sheaf' (it is really a complex of sheaves) on $S$ as
\[ \mathscr D_S = \bQ/\bZ(1)'_S[2]\,, \]
and on every separated $S$-scheme $X$ of finite type as
\[ \mathscr D_X = Rf^!\mathscr D_S \,, \]
where $f: X \to S$ is the structural morphism. Then, by using \eqref{eq-0.2.1}, \eqref{eq-0.2.2} and additional arguments, one gets a duality (cf.\ \S\ref{sect4-3})
\[ H^m_c(X,\mathscr F) \times \text{\rm Ext}^{1-m}_X(\mathscr F, \mathscr D_X) \longrightarrow H^1_c(X,\mathscr D_X) \longrightarrow \bQ/\bZ \,. \]
This is more or less formal, but we make the following three points. First, the duality is completely general: $X$ and the constructible complex $\mathscr F$ can be arbitrary. Hence $X$ may be highly singular, and we may consider $p$-torsion sheaves even if $p$ is not invertible on $X$ (so in particular, if $X$ is an algebraic scheme over $\mathbb F_p$), and the approach connects this `$p$-case' and the case `away from $p$' in a nice way. Secondly, we have a lot of information on the complex $\mathscr D_X$. Thirdly, it is this information that we need for the applications we have in mind, cf.\ \cite{js}.
We describe the information on $\mathscr D_X$ separately for each $p$-primary part $\mathscr D_{X,p^\infty}$, where $p$ is a prime. Put \[ \mathbb Z/p^\infty := \bQ_p/\bZ_p \quad\hbox{ and }\quad \mu_{p^\infty} := \bigcup_{n \ge 1} \ \mu_{p^n} \,. \]
In the rest of this \S\ref{sect0-2}, suppose $n \in \mathbb N \cup \{ \infty \}$. First we describe $\bZ/p^n(1)'_S$.
\par\medskip
(i)\; Let $S=\text{\rm Spec}(\mathfrak o_k)$ be as before. The complex $\bZ/p^n(1)'_S$ is, by definition, the mapping fiber of a morphism
\[ \delta_S^\text{\rm val} = \delta^\text{\rm val}_{S,p^n} : Rj_*\mu_{p^n} \longrightarrow i_* \bZ/p^n[-1] \,. \]
Here $j: U= \text{\rm Spec}(\mathfrak o_k[p^{-1}]) \hookrightarrow S$ is the open immersion, $i: Z = S \smallsetminus U \hookrightarrow S$ is the closed immersion of the complement, $\bZ/p^n$ is the constant sheaf on $Z$, and $\mu_{p^n}$ is the sheaf of $p^n$-th roots of unity on $U$ (note that $p$ is invertible on $U$). One has $R^q j_*\mu_{p^n} = 0$ for $q \ge 2$, and hence $\delta_S^\text{\rm val}$ is determined by the morphism $R^1j_*\mu_{p^n} \to i_*\bZ/p^n$ it induces, and by adjunction and localization, this is in turn completely described by the induced morphisms
\[ \partial_x: k^\times/p^n = H^1(k,\bZ/p^n(1)) \longrightarrow H^0(x,\bZ/p^n) = \bZ/p^n \]
for each closed point $x\in Z=S \smallsetminus U$. Then $\delta_S^\text{\rm val}$ is completely determined by defining $\partial_x$ to be the residue map \eqref{eq-0.1.1}, i.e., as $\text{\rm ord}_x\otimes \bZ/p^n$, where $\text{\rm ord}_x: k^\times \to \mathbb Z$ is the normalized discrete valuation corresponding to $x$.
Moreover, we will show that the mapping fiber of $\delta_S^\text{\rm val}$ is unique up to unique isomorphism in the derived category of sheaves on $S_\text{\rm \'{e}t}$. In other words, $\bZ/p^n(1)'_S$ is the unique complex $\mathscr F$ with $\mathscr F|_U = \bZ/p^n(1)$, $Ri^!\mathscr F = \bZ/p^n[-2]$, and for which the canonical morphism $Rj_* \mathscr F|_U \to i_* Ri^!\mathscr F [1]$ is the morphism $\delta_S^\text{\rm val}$ described above. See the remarks after Definition \ref{def:DC-S} for details.
\par\medskip
Now we list the properties of $\mathscr D_{X,p^n} = Rf^!\mathscr D_{S,p^n} = Rf^! \bZ/p^n(1)'_S[2]$ for $f: X \to S$ separated and of finite type.
\par\medskip
(ii)\; For $p$ invertible on $X$, $\mathscr D_{X,p^\infty}$ is the usual dualizing sheaf for the `prime-to-$p$ theory' over $\mathfrak o_K[p^{-1}]$. In particular, $\mathscr D_{X,p^\infty} = \bQ_p/\bZ_p(d)[2d]$ if $X$ is regular of pure dimension $d$.
\par\medskip
(iii)\; For $X$ of characteristic $p$, i.e., of finite type over the prime field $\mathbb F_p$, and of dimension $d$, $\mathscr D_{X,p^\infty}$ is represented by the explicit complex
\[ \mathscr M_X \; : \; \bigoplus_{x\in X_d} \logwitt x \infty d \longrightarrow \bigoplus_{x\in X_{d-1}} \ \logwitt x \infty {d-1} \longrightarrow \dotsb \longrightarrow \bigoplus_{x\in X_0} \ \bQ_p/\bZ_p \]
introduced by Moser \cite{moser} (except that we put the rightmost term in degree zero, while Moser rather considers the complex $\wt{\nu}{}^d_{\infty,X} := \mathscr M_X[-d]$). In fact, we generalize Moser's duality over finite fields \[ H^m_c(X,\mathscr F) \times \text{\rm Ext}^{d+1-m}_X(\mathscr F,\mathscr M_X[-d]) \to H^{d+1}_c(X,\mathscr M_X[-d]) = \bQ_p/\bZ_p \] in the following way: We extend the duality to arbitrary perfect ground fields $k$ of characteristic $p$, and show that $\mathscr M_X$ is in fact $Rg^!\bQ_p/\bZ_p$, where $g: X \to \text{\rm Spec}(\mathbb F_p)$ is the structural morphism. (Together with the well-known duality of finite fields, this immediately gives back Moser's theorem). By Gros and Suwa \cite{gs}, one has $\mathscr M_X = \logwitt X \infty d [d]$, if $X$ is regular.
\par\medskip
(iv)\; Finally, for $X$ flat over $S = \text{\rm Spec}(\mathfrak o_K)$, consider the closed immersion
\[\xymatrix{ i : Y := X\otimes_{\mathbb Z} \mathbb F_p \, \ar@{^{(}->}[r] & X }\]
and the open immersion
\[\xymatrix{ j: U := X[p^{-1}] \, \ar@{^{(}->}[r] & X }\]
of the complement. There is a morphism
\[ \delta_X^{S\text{-}\val} = \delta_{X,p^\infty}^{S\text{-}\val} : Rj_*\mathscr D_{U,p^\infty} \longrightarrow i_* \mathscr D_{Y,p^\infty}[1] \]
obtained from $\delta_S^\text{\rm val}$ (cf.\ (i)) via $Rf^!$, where $f : X \to S$ denotes the structural map. The source and target are studied in (ii) and (iii) above, respectively, and it is clear from the definitions that $\mathscr D_{X,p^\infty}$ is a mapping fiber of $\delta_X^{S\text{-}\val}$. In general, such a mapping fiber is not unique (for the lack of the unicity of isomorphisms), but one of our main results is the following:
$\mathscr D_{X,p^\infty}$ is a unique mapping fiber of $\delta_X^{S\text{-}\val}$ up to unique isomorphism (cf.\ Theorem \ref{thm.cone})
and moreover when $U$ is smooth, $\delta_X^{S\text{-}\val}$ is uniquely characterized by the property that, for every generic point $y \in Y$ and every generic point $\xi \in U$ which specializes to $y$, the induced map
\[ H^d(\xi,\bQ_p/\bZ_p(d)) \longrightarrow H^{d-1}(y,\bQ_p/\bZ_p(d-1)) = H^0(y,\logwitt y {\infty} {d-1}) \]
coincides with the residue map in \eqref{eq-0.1.1}, cf.\ Theorem \ref{Th.1-1}\,(3). When $X$ is proper (but $U$ arbitrary), we have a similar uniqueness property.
\par
There is another morphism
\[ \delta_{U,Y}^\text{\rm loc}(\mathscr D_{X,p^\infty}) : Rj_*\mathscr D_{U,p^\infty} \longrightarrow i_* \mathscr D_{Y,p^\infty}[1]\,, \]
the connecting morphism of localization theory for $\mathscr D_{X,p^\infty}$. We will also prove that this morphism agrees with $\delta_X^{S\text{-}\val}$ up to a sign, cf.\ \eqref{eq3-9-5}.
\smallskip
\subsection{\'Etale homology}\label{sect0-3}
Let $k$ be a perfect field, and let $X$ be a separated scheme of finite type over $k$. For integers $n>0$, $a$ and $b$, we define the \'etale homology of $X$ by
\[ H_a(X,\bZ/n(b)) = H^{-a}(X,Rf^!\bZ/n(-b))\,, \]
where $f: X \to \text{\rm Spec}(k)$ is the structural morphism. Note that for $\text{\rm ch}(k) = p > 0$, we have $\bZ/p^r(-b) = \logwitt k r {-b}[b]$, which is the constant sheaf $\bZ/p^r$ for $b=0$, zero for $b<0$ (because $k$ is perfect), and zero by definition for $b>0$. Therefore we will either assume that $n$ is invertible in $k$, or that $b=0$. These groups satisfy all properties of a (Borel-Moore type) homology theory, cf.\ \cite{Bloch-Ogus}, \cite{js}. Thus the method of Bloch and Ogus provides a converging niveau spectral sequence (loc. cit.)
\begin{equation}\label{eq-0.3.1}
E^1_{s,t}(X,\bZ/n(b)) = \bigoplus_{x \in X_s} \ H_{s+t}(x,\bZ/n(b)) \Longrightarrow H_{s+t}(X,\bZ/n(b)).
\end{equation}
Here we put
\[ H_a(x,\bZ/n(b)) = \varinjlim_{V \subset \ol{\{x\}}} \ H_a(V,\bZ/n(b)) \]
and the limit is taken over all non-empty open subvarieties $V\subset \overline{\{x\}}$. If $V$ is smooth of pure dimension $d$ over $k$, then one has a canonical purity isomorphism
\[ H_a(V,\bZ/n(b)) \cong H^{2d-a}(V,\bZ/n(d-b)) \]
between homology and cohomology. This is one of the main results of the Artin-Verdier duality \cite{sga4} in the case $n$ is invertible in $k$, and follows from our results in \S\ref{sect2} for the other case. As a consequence, one has canonical isomorphisms
\[ \bigoplus_{x\in X_s} \ H_{s+t}(x,\bZ/n(b)) \cong \bigoplus_{x\in X_s} \ H^{s-t}(x,\bZ/n(s-b))\,, \]
and the complex $E^1_{*,t}$ of $E^1$-terms of the spectral sequence can be identified with a complex
\begin{align}
\notag
\dotsb \longrightarrow \bigoplus_{x\in X_s} \ H^{s-t}(x,\mathbb Z/n(s-b)) & \longrightarrow \bigoplus_{x\in X_{s-1}} \ H^{s-t-1}(x,\mathbb Z/n(s-b-1)) \longrightarrow \\
\label{eq-0.3.2}
\dotsb & \longrightarrow \bigoplus_{x\in X_0} \ H^{-t}(x,\mathbb Z/n(-b))\,,
\end{align}
where we place the last term in degree zero. Another main result of this paper is that this complex coincides with the Kato complex $C_n^{-t,-b}(X)$ mentioned in \S\ref{sect0-1}, up to well-defined signs. In \S\ref{sect1} we also give an absolute variant of this result, for the case that $X$ is a regular excellent noetherian scheme and $n$ is invertible on $X$.
Finally let $X$ be a separated scheme of finite type over $S = \text{\rm Spec}(\mathfrak o_K)$, where $K$ is a number field, and let $n$, $a$ and $b$ be integers. If $n$ is invertible on $X$, we define the \'etale homology as
\[ H_a(X,\bZ/n(b)) = H^{-a}(X,Rf^!\bZ/n(-b)), \]
where $f: X \to S[n^{-1}]$ is the structural morphism. If $n$ is not invertible on $X$, we just consider the case $b=-1$ and define
\[ H_a(X,\bZ/n(-1)) = H^{-a}(X,Rf^!\mathbb Z/n(1)'_S)\,, \]
where $f: X \to S$ is the structural morphism, and $\mathbb Z/n(1)'_S$ has the $p$-primary components $\mathbb Z/p^{r_p}(1)'_S$ from (i) for $n = \prod \, p^{r_p}$. Again, in both cases this defines a homology theory in the sense of \cite{Bloch-Ogus} and \cite{js}, and one gets a niveau spectral sequence with exactly the same numbering as in \eqref{eq-0.3.1}. By the purity isomorphisms explained above, the complex of $E^1$-terms is identified with a complex
\begin{align}
\notag
\dotsb \longrightarrow \bigoplus_{x\in X_s} \ H^{s-t-2}(x,\mathbb Z/n(s-b-1)) & \longrightarrow \bigoplus_{x\in X_{s-1}} \ H^{s-t-3}(x,\mathbb Z/n(s-b-2)) \longrightarrow \\
\label{eq-0.3.3}
\dotsb & \longrightarrow \bigoplus_{x\in X_0} \ H^{-t-2}(x,\mathbb Z/n(-b-1)),
\end{align}
cf.\ \cite{js}. The difference in numbering between \eqref{eq-0.3.2} and \eqref{eq-0.3.3} is explained by the purity results for the inclusion of the fibers $X_P \hookrightarrow X$ over closed points $P\in S$. A third main result of this paper is that, also in this mixed characteristic case, this complex coincides with a Kato complex, viz., $C_n^{-t-2,-b-1}(X)$. In fact, this gives an alternative definition of the Kato complexes under consideration, which is very useful for working with them.
\smallskip
\subsection{Notations and conventions}\label{sect0-4}
For an abelian group $M$ and a positive integer $n$, $M/n$ (resp.\ ${}_n M$) denotes the cokernel (resp.\ the kernel) of the map $M \os{\times n}{\to} M$. \par
In this paper, unless indicated otherwise, all cohomology groups of schemes are taken for the \'etale topology. \par
For a scheme $X$, we will use the following notation. For a non-negative integer $q$, $X_q$ denotes the set of points on $X$ whose closure in $X$ has dimension $q$. If $X$ is pure-dimensional, then we will often write $X^q$ for the set of points on $X$ of codimension $q$. For a point $x \in X$, $\kappa(x)$ denotes its residue field, and $\ol x$ denotes $\text{\rm Spec}(\ol {\kappa(x)})$, the spectrum of a separable closure of $\kappa(x)$. For a point $x \in X$ and an \'etale sheaf $\mathscr F$ on $X$, we will often write $H^*_x(X,\mathscr F)$ for $H^*_x(\text{\rm Spec}(\mathscr O_{X,x}),\mathscr F)$. \par
Let $X$ be a scheme and let $n$ be a non-negative integer. Let $i:Z \hookrightarrow X$ be a closed immersion, and let $j:U \hookrightarrow X$ be the open complement $X \smallsetminus Z$. For an object $\mathscr K \in D^+(X_\text{\rm \'{e}t},\mathbb Z/n)$, we define the morphism
\[ \delta_{U,Z}^{\text{\rm loc}}(\mathscr K) : Rj_*j^*\mathscr K \longrightarrow Ri_*Ri^!\mathscr K[1] \quad \hbox{ in } \; D^+(X_\text{\rm \'{e}t},\mathbb Z/n) \]
as the connecting morphism associated with the semi-splitting short exact sequence of complexes
\[ 0 \longrightarrow i_*i^!I^\bullet \longrightarrow I^\bullet \longrightarrow j_*j^*I^\bullet \longrightarrow 0 \]
(\cite{sga4.5} Cat\'egories D\'eriv\'ees I.1.2.4), where $I^\bullet$ is a resolution of $\mathscr K$ by injective $\mathbb Z/n$-sheaves on $X_\text{\rm \'{e}t}$. It induces the usual connecting morphisms
\[ \delta_{U,Z}^\text{\rm loc}(\mathscr K) : R^qj_* j^*\mathscr K \longrightarrow i_* R^{q+1}i^!\mathscr K\,,\]
or the connecting morphisms in the localization sequence for $(X,Z,U)$:
\[ \delta_{U,Z}^\text{\rm loc}(\mathscr K) : H^q(U,j^* \mathscr K) \longrightarrow H^{q+1}_Z(X,\mathscr K)\,. \]
The morphism $\delta_{X,U}^\text{\rm loc}(\mathscr K)$ is functorial in $\mathscr K$, but does not commute with shift functors in general. In fact, we have
\begin{equation}\label{eq-0.4.1}
\delta_{U,Z}^{\text{\rm loc}}(\mathscr K)[q] = (-1)^q \cdot \delta_{U,Z}^{\text{\rm loc}}(\mathscr K[q]) \quad \hbox{for } \; q \in \mathbb Z.
\end{equation}
By the convention in \cite{sga4} XVII.1.1.1 (which we follow and is usually taken, but which is opposite to the one in \cite{sga4.5} Cat\'egories D\'eriv\'ees I.1.2.1), the following triangle is distinguished in $D^+(X_\text{\rm \'{e}t},\mathbb Z/n)$:
\begin{equation}\label{eq-0.4.2}
\xymatrix{ Ri_*Ri^!\mathscr K \ar[r]^-{i_*} & \mathscr K \ar[r]^-{j^*} & Rj_*j^*\mathscr K \ar[rr]^-{-\delta_{U,Z}^{\text{\rm loc}}(\mathscr K)} && Ri_*Ri^!\mathscr K[1]\,, }
\end{equation}
where the arrow $i_*$ (resp.\ $j^*$) denotes the adjunction morphism $Ri_*Ri^! \to \text{\rm id}$ (resp.\ $\text{\rm id} \to Rj_*j^*$). We generalize the above definition of connecting morphisms to the following situation. Let $y$ and $x$ be points on $X$ such that $x$ has codimension $1$ in the closure $\ol {\{ y \}} \subset X$. Put $Y:=\text{\rm Spec}(\mathscr O_{\ol {\{ y \}},x})$, and let $i_x$ (resp.\ $i_y$, $i_Y$) be the natural map $x \to X$ (resp.\ $y \to X$, $Y \to X$). Then we define a morphism
\[ \delta_{y,x}^{\text{\rm loc}}(\mathscr K) : Ri_{y*}Ri_y^!\mathscr K \longrightarrow Ri_{x*}Ri_x^!\mathscr K[1] \quad \hbox{ in } \; D^+(X_\text{\rm \'{e}t},\mathbb Z/n) \]
as $Ri_{Y*}(\delta_{y,x}^{\text{\rm loc}}(Ri_Y^!\mathscr K))$.
\smallskip
\subsection{Derived categories}\label{sect0-5}
We shall often use the following facts. Let $\mathscr A$ be an abelian category, and let $D^*(\mathscr A)$ be its derived category with boundary condition $*\in \{\emptyset, +, -, b\}$.
\subsubsection{}\label{sect0-5-1}
A sequence $A \os{\alpha}\to B \os{\beta}\to C \os{\gamma}\to A[1]$ in $D^*(\mathscr A)$ is a distinguished triangle if and only if $B \os{\beta}\to C \os{\gamma}\to A[1] \os{-\alpha[1]}\to B[1]$ is a distinguished triangle. (This is the axiom (TR2) for triangulated categories, \cite{sga4.5} Cat\'egories Deriv\'ees I.1.1.)
\subsubsection{}\label{sect0-5-2}
Given a diagram
\[\xymatrix{
A \ar[r]^a \ar[d]_f & B \ar[r]^{b} & C \ar[r] \ar[d]^h & A[1] \ar[d]^{f[1]} \\ A' \ar[r]^{a'} & B' \ar[r]^{b'} & C' \ar[r] & A'[1]
}\]
in which the rows are distinguished triangles and the last square commutes, there is a morphism $g: B \to B'$ making the remaining squares commutative, i.e., giving a morphism of distinguished triangles. Moreover one has
\addtocounter{thm}{2}
\begin{lem}\label{lem.derived}
The morphism $g$ is unique in the following three cases{\rm:}
\begin{enumerate}
\item[{\rm (1)}] $\text{\rm Hom}_{D(\mathscr A)}(B,A') = 0$.
\item[{\rm (2)}] $\text{\rm Hom}_{D(\mathscr A)}(C,B') = 0$.
\item[{\rm (3)}] $\text{\rm Hom}_{D(\mathscr A)}(C,A') = 0$ \, and \, $\text{\rm Hom}^{-1}_{D(\mathscr A)}(A,C') = 0$.
\end{enumerate} \end{lem}
\begin{pf}
There is an induced commutative diagram with exact rows and columns
\[\xymatrix{
& \text{\rm Hom}(C,A') \ar[r] \ar[d] & \text{\rm Hom}(B,A') \ar[d] \\ & \text{\rm Hom}(C,B') \ar[r]^{b^*} \ar[d]_{b'_*} & \text{\rm Hom}(B,B') \ar[r]^{a^*} \ar[d]_{b'_*} & \text{\rm Hom}(A,B') \\ \text{\rm Hom}^{-1}(A,C') \ar[r] & \text{\rm Hom}(C,C') \ar[r]^{b^*} & \text{\rm Hom}(B,C')\,.\hspace{-3pt}
}\]
Suppose $g_1$ and $g_2$ both make the previous diagram commutative. Then the element $g_1 - g_2 \in \text{\rm Hom}(B,B')$ is mapped
to zero in $\text{\rm Hom}(A,B')$ and $\text{\rm Hom}(B,C')$. Under conditions (1) and (2), either the right hand $b'_*$ or $a^*$ is injective, so the claim follows. Under condition (3), the left hand $b'_*$ and the lower $b^*$ are both injective, and again we get $g_1 - g_2 = 0$.
\end{pf}
\par
\stepcounter{subsubsection}
\subsubsection{}\label{sect0-5-4}
Let $q,r$ be integers, and let $M$ be an object in $D(\mathscr A)$ which is concentrated in degrees $\leq r$. Let $N$ be an object in $D(\mathscr A)$ which is concentrated in degrees $\ge 0$. Then we have
\[ \text{\rm Hom}_{D(\mathscr A)}(M,N[-q]) =
\begin{cases}
\text{\rm Hom}_{\mathscr A}(\mathscr H^q(M),\mathscr H^0(N)) & \quad \hbox{(if $q=r$)} \ \ \dotsb\dotsb (1)\\
0 & \quad \hbox{(if $q>r$)} \ \ \dotsb\dotsb (2)
\end{cases}\]
Here for $s \in \mathbb Z$, $\mathscr H^s(M)$ denotes the $s$-th cohomology object of $M$. These facts are well-known and easily proved, using \cite{pervers} 1.3.2 and \cite{sga4.5} Cat\'egories D\'eriv\'ees I.1.2.
\smallskip
\subsection{Kato's residue maps}\label{sect0-6}
We recall Kato's definition of the residue maps in \eqref{eq-0.1.1}. Consider a noetherian excellent scheme $X$ and points $x,y\in X$ such that $x$ lies in $Z=\overline{\{y\}}$ and has codimension 1 in $Z$. The construction only depends on $Z$ (with the reduced subscheme structure). Put $A:=\mathscr O_{Z,x}$, a local domain of dimension $1$. We may further replace $Z$ with $\text{\rm Spec}(A)$.
\par
\medskip
(I)\; {\it Regular case}.\;
First consider the case that $A$ is regular, i.e., a discrete valuation ring. Then $K:= \kappa(y) = \text{\rm Frac}(A)$ is a discrete valuation field and $k:=\kappa(x)$ is the residue field of $A$, i.e., of the valuation. The residue map
\[ \partial^\val = \dval_{y,x} : H^{i+1}(K,\bZ/p^n(r+1)) \longrightarrow H^i(k,\bZ/p^n(r)) \]
is obtained by restricting to the henselization $K^\text{\rm h}$ (which corresponds to restricting to the henselization $A^h = \mathscr O_{Z,x}^\text{\rm h}$) and defining a map for the discrete valuation field $K^\text{\rm h}$ which has the same residue field $k$. Hence we may and will assume that $K$ is henselian (i.e., $A=A^\text{\rm h}$ and $K=K^\text{\rm h}$). Let $K^\text{\rm sh}$ be the maximal unramified extension of $K$ (corresponding to the strict henselization $A^\text{\rm sh} = \mathscr O_{Z,\overline x}$).
\par
\medskip
(I.1)\;
If $p\neq \text{\rm ch}(k)$, we first have a map
\[ H^1(K,\bZ/p^n(1)) \hspace{10pt}{}^\sim\hspace{-17.5pt}\longleftarrow K^\times/(K^\times)^{p^n} \longrightarrow \bZ/p^n = H^0(k,\bZ/p^n)\,, \]
where the first arrow is the Kummer isomorphism, and the second is induced by the valuation. This is $\partial^\val$ for $(i,r) =(0,0)$. In general $\partial^\val$ is the composition
\[ H^{i+1}(K,\bZ/p^n(r+1)) \longrightarrow H^i(k,H^1(K^\text{\rm sh},\bZ/p^n(r+1))) \longrightarrow H^i(k,\bZ/p^n(r))\,. \]
Here the first map is an edge morphism from the Hochschild-Serre sequence for $K^\text{\rm sh}/K$ (note that $\text{\rm cd}(K^\text{\rm sh})=1$), and the second map is induced by (the Tate twist of) the previously defined map.
\par
\medskip
(I.2)\;
Now let $p=\text{\rm ch}(k)$ (and recall that $K$ is henselian). In this case $H^i(k,\bZ/p^n(r))$ $= 0$ for $i \ne r, r+1$. Assume that $i = r$. Then $\partial^\val$ is defined by the commutativity of the diagram
\begin{equation}\label{eq-0.6.2}
\xymatrix{
H^{r+1}(K,\bZ/p^n(r+1)) \ar[r]^-{\partial^\val} & H^r(k,\bZ/p^n(r)) \\
K^M_{r+1}(K)/p^n \ar[u]^{h^{r+1}}_{\hspace{-1.5pt}\wr} \ar[r]^-{\partial} & K^M_r(k)/p^n \,. \ar[u]^{h^r}_{\hspace{-1.5pt}\wr}
}\end{equation}
Here $K^M_r(F)$ is the $r$-th Milnor $K$-group of a field $F$, $h^r$ is the symbol map into Galois cohomology, and $\partial$ is the suitably normalized residue map for Milnor $K$-theory. By definition,
\[ h^r(\{a_1,\dotsc,a_r\}) = h^1(a_1) \cup \dotsb \cup h^1(a_r) \in H^r(F,\bZ/p^n(r))\,, \]
where $h^1: F^\times/p^n \to H^1(F,\bZ/p^n(1))$ is defined as follows: it is the Kummer isomorphism if $p$ is invertible in $F$, and it is the isomorphism $d{\text{\rm log}}: F^\times/p^n \to H^0(F,\logwitt F n 1)$ if $\text{\rm ch}(F)=p$. It is known that, under our assumptions, the symbol maps $h^i$ in \eqref{eq-0.6.2} are isomorphisms (\cite{Bloch-Kato} \S2, \S5). Finally, if $\pi$ is a prime element for $K$, then $\partial$ is determined by the property that
\[ \partial(\{\pi,a_1,\dotsc,a_r\}) = \{\ol{a_1},\dotsc,\ol{a_r}\}\,, \]
for units $a_1,\ldots,a_r \in A^\times$, where $\overline{a_i}$ denotes the residue class of $a_i$ in the residue field $k$.
\par
\medskip
(I.3)\; Now let $i=r+1$. In this case we assume $[k:k^p] \le p^r$.
Then the residue map $\partial^\val$ is defined as the composition ($\ol k$ denotes the separable closure of $k$)
\[ \xymatrix{
H^1(k,H^{r+1}(K^{\text{\rm sh}},\bZ/p^n(r+1)) \ar[r]^-{(**)} & H^1(k,H^r(\ol k,\bZ/p^n(r))) \\
H^{r+2}(K,\bZ/p^n(r+1)) \ar[u]^{(*)}_{\hspace{-1.5pt}\wr} \ar@{.>}[r]^-{\partial^\val} & H^{r+1}(k,\bZ/p^n(r)) \,. \ar[u]^{(*)}_{\hspace{-1.5pt}\wr}
}\]
Here the isomorphisms $(*)$ come from the Hochschild-Serre spectral sequences and the fact that $\text{\rm cd}_p(k)\le 1$ and
\[ H^{j+1}(K^{\text{\rm sh}}, \bZ/p^n(r+1)) = 0 = H^j(\ol k,\bZ/p^n(r)) \quad \hbox{ for \; $j>r$.} \]
The map $(**)$ is induced by the map
\[ H^{r+1}(K^{\text{\rm sh}},\bZ/p^n(r+1)) \longrightarrow H^r(\ol k,\bZ/p^n(r)) \]
defined in (I.2). In \cite{kk:hasse} the completion $\wh K$ is used instead of the henselization $K^\text{\rm h}$, but this gives the same, because the map
\[ H^j(K^{\text{\rm sh}}, \bZ/p^n(r+1)) \longrightarrow H^j(\wh K^{\text{\rm sh}}, \bZ/p^n(r+1))\]
is an isomorphism (\cite{katokuzumaki} proof of Theorem 1). Indeed $\wh K^{\text{\rm sh}}/K^{\text{\rm sh}}$ is separable by excellency of $X$.
\par \medskip
(II)\; {\it General case}.\;
Now consider the case that $A$ is not necessarily regular. In this case let $Z' \to Z = \text{\rm Spec}(A)$ be the normalization. Note that $Z'$ is finite over $Z$ because the latter is excellent. Then we define
\[\dval_{y,x}(a) \, = \, \sum_{x'|x} \ \text{\rm Cor}_{\kappa(x')/\kappa(x)}(\partial^\val_{y,x'}(a)) \qquad (a \in H^{i+1}(y,\bZ/p^n(r+1)))\]
where the sum is taken over all points $x' \in Z'$ lying over $x$,
\[ \partial^\val_{y,x'}: H^{i+1}(y,\bZ/p^n(r+1)) \longrightarrow H^i(x',\bZ/p^n(r)) \]
is the residue map defined for the discrete valuation ring $\mathscr O_{Z',x'}$, and
\begin{equation}\label{eq-0.6.5}
\text{\rm Cor}_{\kappa(x')/\kappa(x)}: H^i(x',\bZ/p^n(r)) \longrightarrow H^i(x,\bZ/p^n(r))
\end{equation}
is the corestriction map in Galois cohomology. For $p$ invertible in $\kappa(x)$ this last map is well-known. For $\kappa(x)$ of characteristic $p$ and $i=r$, this corestriction map is defined as the composition
\begin{align}
\label{eq-0.6.6}
H^0(x',\logwitt {x'} n r) & \os{(h^r)^{-1}}{\hspace{9pt}{}^\sim\hspace{-16.5pt}\lra} K^M_r(\kappa(x'))/p^n \os{N_{x'/x}}\longrightarrow K^M_r(\kappa(x))/p^n \\
\notag
& \;\, \os{h^r}\longrightarrow \; H^0(x,\logwitt x n r) \,.
\end{align}
This implies that the diagram \eqref{eq-0.6.2} is also commutative in this case. For the remaining case $i=r+1$ we may proceed as follows. It is easy to see that the map \eqref{eq-0.6.6} is compatible with \'etale base-change in $\kappa(x)$. Therefore we get an induced corestriction or trace map
\begin{equation}\label{eq-0.6.7}
\text{\rm tr}_{x'/x}: \pi_* \logwitt {x'} n r \longrightarrow \logwitt x n r \,.
\end{equation}
Then we define the corestriction \eqref{eq-0.6.5} for $\text{\rm ch}(\kappa(x))=p$ and $i=r+1$ as
\begin{equation}\label{eq-0.6.8}
\text{\rm tr}_{x'/x} : H^1(x',\logwitt {x'} n r) \longrightarrow H^1(x,\logwitt x n r) \,,
\end{equation}
the map induced by \eqref{eq-0.6.7}. If $\kappa(x)$ is finitely generated over a perfect field $k$, the morphisms \eqref{eq-0.6.6}, \eqref{eq-0.6.7} and \eqref{eq-0.6.8} agree with the trace map in logarithmic Hodge-Witt cohomology defined by Gros \cite{gros:purity}. See the appendix for this and further compatibilities.
\newpage
\section{The case where $p$ is invertible on the scheme $X$}\label{sect1}
\medskip
In this section, we work in the following setting. Let $X$ be a noetherian excellent regular scheme, let $n$ be a positive integer invertible on $X$ and put $\varLambda:=\mathbb Z/n$. For integers $q \in \mathbb Z$, put $\varLambda(q):=\mu_n^{\otimes q}$, the $q$-fold Tate twist of the \'etale sheaf $\varLambda$ on $X$ or $X$-schemes. Let $Z \subset X$ be a regular closed subscheme of pure codimension $c$. By Gabber's construction \cite{fujiwara}, there is a cycle class $\text{\rm cl}_X(Z)$ in the \'etale cohomology group $H^{2c}_Z(X,\varLambda(c))$ (without using the absolute purity), which satisfies the following three properties:
\begin{enumerate}
\item[(G1)] For an \'etale morphism $X' \to X$ and $Z':=Z \times_X X'$, the pull-back of $\text{\rm cl}_X(Z)$ to $H^{2c}_{Z'}(X',\varLambda(c))$ agrees with $\text{\rm cl}_{X'}(Z')$.
\item[(G2)] For regular closed subschemes $Z \subset Y \subset X$, we have $\text{\rm cl}_X(Y) \cap \text{\rm cl}_{Y}(Z) = \text{\rm cl}_X(Z)$ in $H^{2c}_{Z}(X,\varLambda(c))$.
\item[(G3)] The image of $\text{\rm cl}_X(Z)$ in $H^0(Z,R^{2c}i^!\varLambda(c))$ coincides with Deligne's cycle class \cite{sga4.5} Cycle \S2.2. Here $i=i_Z$ denotes the closed immersion $Z \hookrightarrow X$.
\end{enumerate}
\smallskip
\subsection{Gysin maps and compatibility}\label{sect1-1}
For $q,r \in \mathbb Z$, one defines the Gysin map $\text{\rm Gys}_i$ as
\[ \text{\rm Gys}_i: H^q(Z,\varLambda(r)) \longrightarrow H^{q+2c}_Z(X,\varLambda(r+c)), \quad \alpha \mapsto \text{\rm cl}_X(Z) \cup \alpha\,. \]
The main aim of this section is the following compatibility result. It will turn out that we do not need the absolute purity in its proof.
\begin{thm}\label{prop:appA}
Let $c$ be a positive integer, and let $i_x:x \hookrightarrow X$ and $i_y: y \hookrightarrow X$ be points on $X$ of codimension $c$ and $c-1$, respectively, with $x \in \ol {\{y\}}$. Then the following square commutes for integers $q, r \ge 0${\rm:}
\stepcounter{equation}
\begin{equation}\label{CD:app1}
\xymatrix{
H^{q+1}(y,\varLambda(r+1)) \ar[rr]^-{-\dval_{y,x}} \ar[d]_{\text{\rm Gys}_{i_y}} && H^q(x,\varLambda(r)) \ar[d]^{\text{\rm Gys}_{i_x}} \\
H^{q+2c-1}_y(X,\varLambda(r+c)) \ar[rr]^-{\delta^\text{\rm loc}_{y,x}(\varLambda(r+c)_X)} && H^{q+2c}_x(X,\varLambda(r+c))\,.\hspace{-5pt}
}\end{equation}
\end{thm}
To prove the theorem, we may assume that $X$ is local with closed point $x$. Put $Z:=\ol {\{ y \}} \subset X$, which has dimension $1$ and consists of two points $\{ y, x \}$. Let $i_Z$ (resp.\ $\iota_x$) be the closed immersion $Z \hookrightarrow X$ (resp.\ $x \hookrightarrow Z$). The proof will be finished in \S\ref{sect1-4} below.
\smallskip
\subsection{Regular case}\label{sect1-2}
We first prove Theorem \ref{prop:appA}, assuming that $Z$ is regular. In this case $Z$ is the spectrum of a discrete valuation ring $A$, and we have the cycle classes
\[ \text{\rm cl}_X(Z) \in H^{2(c-1)}_Z(X,\varLambda(c-1)) \qquad \hbox { and } \qquad \text{\rm cl}_Z(x) \in H^{2}_x(Z,\varLambda(1)) \]
by Gabber's construction, where $\text{\rm cl}_Z(x)$ agrees with Deligne's construction in \cite{sga4.5} Cycle \S2.2 by (G3). There is a diagram of boundary maps
\[\xymatrix{
& H^{q+1}(y,\varLambda(r+1)) \ar[r]^-{\text{\rm Gys}_{i_y}} \ar[ld]_{-\dval_{y,x}} \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+1)_Z)} & H^{q+2c-1}_y(X,\varLambda(r+c)) \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+c)_X)} \\
H^q(x,\varLambda(r)) \ar[r]_-{\text{\rm Gys}_{\iota_x}} & H^{q+2}_x(Z,\varLambda(r+1)) \ar[r]_-{\text{\rm Gys}_{i_Z}} & H^{q+2c}_x(X,\varLambda(r+c))\,.
}\]
Here $\text{\rm Gys}_{i_y}$ is the map taking the cup-product with $\text{\rm cl}_X(Z) \vert_{\text{\rm Spec}(\mathscr O_{X,y})}$, by the property (G1), and hence the right square commutes by the naturality of cup products and \eqref{eq-0.4.1}. The composite of the bottom row agrees with $\text{\rm Gys}_{i_x}$ by (G2). Thus we obtain the commutativity of the diagram \eqref{CD:app1}, once we show the left triangle commutes. But this commuting follows from (G3) and \cite{sga4.5} Cycle 2.1.3. Indeed, by noting that
\[ R^q j_*\varLambda(r+1)=0 \quad \hbox{ for $q \ge 2$, where $j : y \hookrightarrow Z$,} \]
the left triangle is induced by the following square in $D^b(x_\text{\rm \'{e}t},\varLambda)$:
\[\xymatrix{
\iota_x^*R^1 j_*\varLambda(r+1)[-1] \ar[d]_{-\partial^\val[-1]} && \tau_{\leq 1}\,\iota_x^*R j_*\varLambda(r+1) \ar[ll]_-{\text{canonical}} \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+1)_Z)} \\
\varLambda(r) [-1] \ar[rr]^-{\text{\rm Gys}_{\iota_x}} && R \iota_x^!\varLambda(r+1)[1]\,,
}\]
where $\partial^\val: \iota_x^*R^1 j_*\varLambda(r+1) \to \varLambda(r)$ denotes a map of sheaves on $x_\text{\rm \'{e}t}$ induced by the valuation of $A$. We note $R i^!\varLambda(r+1)[1]$ is concentrated in degree $1$. Therefore it suffices to show its commutativity after taking the cohomology sheaves $\mathscr H^1(-)$ in degree 1, so that we are reduced to showing the commutativity of the diagram
\[ \xymatrix{
\iota_x^*R^1 j_*\varLambda(r+1) \ar[d]_{-\partial^\val} & \iota_x^*R^1j_*\varLambda(r+1) \ar@{=}[l] \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+1)_Z)} \\
\varLambda(r) \ar[r]^-{\text{\rm Gys}_{\iota_x}} & R^2 \iota_x^!\varLambda(r+1)\,,
}\]
where $\text{\rm Gys}_{\iota_x} : \varLambda(r) \to R^2 i^!\varLambda(r+1)$ is given by $a \mapsto \text{\rm cl}_Z(x)\cup a$. By looking at the stalks, we are now reduced to the case that $A$ is strictly henselian and to showing the anti-commutativity of
\[\xymatrix{
\kappa(y)^\times/n \ar[d]_{\text{\rm ord}_A} \ar[rr]^-{h^1}_-{\sim} && H^1(y,\varLambda(1)) \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+1)_Z)} \\
\varLambda \ar[rr]^-{\text{\rm Gys}_{\iota_x}}_-{1\,\longmapsto \,\text{\rm cl}_Z(x)} && H^2_x(Z,\varLambda(1))\,,
}\]
which is a consequence of \cite{sga4.5} Cycle 2.1.3 ($h^1$ is the Kummer isomorphism). Note that we have $\partial^\val = \text{\rm ord}_A \circ (h^1)^{-1}$ by \S\ref{sect0-6}\,(I.1), and that $\text{\rm Gys}_{\iota_x} \circ \text{\rm ord}_A$ sends a prime element $\pi$ of $A$ to $\text{\rm cl}_Z(x)$ and hence agrees with the map induced the composition
\[ \kappa(y)^\times \os{\delta}\longrightarrow H^1_x(Z,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \os{\delta}\longrightarrow H^2_x(Z,\varLambda(1)) \] by loc.\ cit.\ 2.1.2.
\smallskip
\subsection{General case}\label{sect1-3}
We prove Theorem \ref{prop:appA} in the case that $Z$ is not regular. Take the normalization $f:T \to Z$. Note that $T$ is regular and semi-local and that $\nu$ is finite by the excellence of $Z$. The composite morphism $T \to Z \hookrightarrow X$ is finite (hence projective by \cite{ega2} 6.1.11), and factors as $T \hookrightarrow \mathbb P^d_X \to X$ for some integer $d \ge 1$. Let $\iota_y$ be the composite map $y \hookrightarrow T \hookrightarrow \mathbb P^d_X$. Put $\varSigma:=f^{-1}(x) \subset T$ with reduced subscheme structure, which consists of closed points on $T$.
There is a commutative diagram of schemes
\begin{equation}\label{CD:schemes}
\xymatrix{
\varSigma \; \ar@{^{(}->}[r] \ar[d]_h
& T \; \ar@{^{(}->}[r]^-{i_{T}} \ar[d]_f
& \mathbb P:=\mathbb P^d_X \ar[d]_g \\
x \; \ar@{^{(}->}[r]^-{\iota_x}
& Z \; \ar@{^{(}->}[r]^-{i_Z}
& X \,. \hspace{-5pt}
}
\end{equation}
Let $i_x$ (resp.\ $i_{\varSigma}$) be the closed immersion $x \hookrightarrow X$ (resp.\ $\varSigma \hookrightarrow \mathbb P$). Now put $c':=c+d=\text{\rm codim}_{\mathbb P}(\varSigma)$, and consider the diagram in Figure 1.
\par
\begin{figure}[htp]
\setlength{\unitlength}{.6mm} {\scriptsize
\begin{picture}
(220,90)(-131,-45) \thinlines
\put(-99,-35){\vector(2,1){24}}
\put(67,-35){\vector(-2,1){24}}
\put(67,36){\vector(-2,-1){24}}
\put(-99,36){\vector(2,-1){24}}
\put(-119,34){\line(0,-1){68}}
\put(-118,34){\line(0,-1){68}}
\put(84,34){\vector(0,-1){68}}
\put(-97,40){\vector(1,0){162}}
\put(-97,-39){\vector(1,0){162}}
\put(-65,13){\vector(0,-1){26}}
\put(32,13){\vector(0,-1){26}}
\put(-35,18){\vector(1,0){36}}
\put(-35,-18){\vector(1,0){36}}
\put(-84,-32){{\tiny $\text{\rm Gys}_{i_y}$}}
\put(40,-32){{\tiny $\text{\rm Gys}_{i_x}$}}
\put(36,31){{\tiny $\text{\rm Gys}_{i_{\varSigma}}$}}
\put(-84,31){{\tiny $\text{\rm Gys}_{\iota_y}$}}
\put(86,0){{\tiny $\text{\rm tr}_{h}$}}
\put(-22,44){{\tiny $-\partial^\val_{y,\varSigma}$}}
\put(-22,-45){{\tiny $-\dval_{y,x}$}}
\put(-63,0){{\tiny $\alpha$}}
\put(27,0){{\tiny $\beta$}}
\put(-32,12){{\tiny $\delta^{\text{\rm loc}}_{y,\varSigma}(\varLambda(r+c'))$}}
\put(-32,-14){{\tiny $\delta^{\text{\rm loc}}_{y,x}(\varLambda(r+c))$}}
\put(-18,28){\tiny {\text(1)}}
\put(-95,0){\tiny {\text(2)}}
\put(-18,0){\tiny {\text(3)}}
\put(58,0){\tiny {\text(4)}}
\put(-18,-30){\tiny {\text(5)}}
\put(70,39){$H^q(\varSigma,\varLambda(r))$}
\put(-142,39){$H^{q+1}(y,\varLambda(r+1))$}
\put(-142,-41){$H^{q+1}(y,\varLambda(r+1))$}
\put(70,-41){$H^q(x,\varLambda(r))$}
\put(8,17){$H^{q+2c'}_{\varSigma}(\mathbb P,\varLambda(r+c'))$}
\put(-92,17){$H^{q+2c'-1}_y(\mathbb P,\varLambda(r+c'))$}
\put(-92,-20){$H^{q+2c-1}_y(X,\varLambda(r+c))$}
\put(8,-20){$H^{q+2c}_x(X,\varLambda(r+c))$}
\end{picture}
\caption{A diagram for the proof in the general case} }
\end{figure}
In the diagram, the arrows $\alpha$ and $\beta$ are induced by the composite morphism
\[ \xymatrix{
\gamma : Rf_*R i_{T}^!\varLambda(r+c')_{\mathbb P}[2d] \ar[r]^-{(*)} & Ri_{Z}^!R g_* \varLambda(r+c')_{\mathbb P}[2d] \ar[rr]^-{Ri_Z^!(\text{\rm tr}_{g})} && R i_Z^!\varLambda(r+c)_X
}\]
in $D^+(Z_\text{\rm \'{e}t},\varLambda)$, where $(*)$ is the cobase-change morphism (cf.\ \cite{sga4} XVIII.3.1.13.2) for the right square of \eqref{CD:schemes}. More precisely, $\alpha$ is obtained by restricting $\gamma$ to $y$, and $\beta$ is defined as the composite
\[ \xymatrix{
\beta : Rh_*Ri_{\varSigma}^!\varLambda(r+c')_{\mathbb P}[2d] \ar[r]^-{(**)} & R\iota_x^!Rf_*R i_{T}^!\varLambda(r+c')_{\mathbb P}[2d] \ar[r]^-{R\iota_x^!(\gamma)} & R i_x^!\varLambda(r+c)_X\,,
}\]
where $(**)$ is the cobase-change morphism for the left square of \eqref{CD:schemes}. Therefore the square (3) is commutative by \eqref{eq-0.4.1}.
On the other hand, the diagram (1) is commutative by the regular case \S\ref{sect1-2}, and moreover, the outer large square of Figure 1 commutes by the definition of $\dval_{y,x}$ (cf.\ \S\ref{sect0-6} (II)) and the fact that the trace map $\text{\rm tr}_{h}$ coincides with the corestriction map of Galois cohomology groups (cf.\ \cite{sga4} XVIII.2.9 (Var 4)).
Therefore, once we show that the diagrams (2) and (4) commute, we will have obtained the commutativity of (5), i.e., Theorem \ref{prop:appA}.
In what follows, we prove only the commutativity of (4); the diagram (2) is simpler and left to the reader. Put \[ Q:=\mathbb P^d_x, \] and take a section $s : X \hookrightarrow \mathbb P$. Let $t:x \hookrightarrow Q$ be the restriction of $s$ to $x$, and let $i_Q$ (resp.\ $\iota_{\varSigma}$) be the closed immersion $Q \hookrightarrow \mathbb P$ (resp.\ $\varSigma \hookrightarrow Q$). Consider the following diagram:
\begin{equation}\label{CD:appA2}
\xymatrix{
& H^q(x,\varLambda(r)) \ar[r]^-{\text{\rm Gys}_{i_x}} \ar[d]_{\text{\rm Gys}_{t}} \ar@{}[rd]|{\text{(6)}} & H^{q+2c}_x(X,\varLambda(r+c)) \ar[d]_{\text{\rm Gys}_{s}} \\
H^q(\varSigma,\varLambda(r)) \ar[r]^-{\text{\rm Gys}_{\iota_\varSigma}} \ar[rd]_{\text{\rm tr}_h}^{\quad\;\;\text{(7)}} \ar@{.>}@/^/[ru]^{\text{\rm tr}_{h}} & H^{q+2d}(Q,\varLambda(r+d)) \ar[r]^-{\text{\rm Gys}_{i_Q}} \ar[d]_{\text{\rm tr}_{Q/x}} \ar@{}[rd]|{\text{(8)}} & H^{q+2c'}_{Q}(\mathbb P,\varLambda(r+c')) \ar[d]_{\beta'} \\
& H^q(x,\varLambda(r)) \ar[r]^-{\text{\rm Gys}_{i_x}} & H^{q+2c}_x(X,\varLambda(r+c))\,.
}\end{equation}
Here the square (6) commutes by (G2). The arrow $\beta'$ is a trace map defined in a similar way as for $\beta$. The diagram (4) in question is related to the large tetragon (7)$+$(8) in \eqref{CD:appA2} by a diagram
\begin{equation}\label{CD:appA2.2}
\xymatrix{
H^q(\varSigma,\varLambda(r)) \ar[rr]^-{\text{\rm Gys}_{i_\varSigma}} \ar[rrd]^{\hspace{-2pt}\varphi}_{\text{(7)}+\text{(8)}\qquad\qquad} \ar[d]_{\text{\rm Gys}_{i_x} \circ \text{\rm tr}_h} && H^{q+2c'}_{\varSigma}(\mathbb P,\varLambda(r+c')) \ar[d]^{\iota_{\Sigma*}} \\ H^{q+2c}_x(X,\varLambda(r+c)) && \ar[ll]_{\beta'} H^{q+2c'}_{Q}(\mathbb P,\varLambda(r+c'))\,. \hspace{-5pt}
}\end{equation}
Here the arrow $\varphi$ denotes the composite of the middle row of \eqref{CD:appA2}, and the upper right triangle of \eqref{CD:appA2.2} is obviously commutative. The composition $\beta'\circ \iota_{\Sigma*}$ is $\beta$, and the square agrees with the diagram (4) in Figure 1. To prove the commutativity of (4), it thus suffices to check that of the lower left triangle of \eqref{CD:appA2.2}, i.e., the tetragon (7)$+$(8) in \eqref{CD:appA2}. To prove this, it suffices to check the following claims concerning the diagram \eqref{CD:appA2}:
\addtocounter{thm}{3}
\begin{lem}\label{lem1-1}
\begin{enumerate}
\item[{\rm (a)}] The triangle {\rm (7)} is commutative.
\item[{\rm (b)}] The composite of the right vertical column is the identity map.
\item[{\rm (c)}] The composite of the middle vertical column is the identity map.
\item[{\rm (d)}] We have $\text{\rm Gys}_{\iota_\varSigma}= \text{\rm Gys}_t \circ \text{\rm tr}_h$, i.e., the upper triangle of \eqref{CD:appA2} commutes.
\end{enumerate}
\end{lem}
\noindent The claims (a)--(c) follow from standard arguments using \cite{sga4.5} Cycle 2.3.8\,(i),\,(ii).
The details are straight-forward and left to the reader. We prove (d) in \S\ref{sect1-4} below.
\begin{rem}
By the absolute purity and the claims {\rm(b)} and {\rm(c)}, one can easily show the commutativity of the square {\rm(8)}. However the claim {\rm(d)} enables us to show the commutativity of the large tetragon {\rm(7)}$+${\rm(8)} without the absolute purity.
\end{rem}
\smallskip
\subsection{Proof of Lemma \ref{lem1-1}\,(d)}\label{sect1-4}
To prove Lemma \ref{lem1-1}\,(d), it is enough to show the following lemma:
\begin{lem}\label{lem:proj}
Let $F$ be a field, let $d$ be a positive integer and put $Q:=\mathbb P^d_F$. Let $n$ be a positive integer prime to $\text{\rm ch}(F)$ and put $\varLambda:= \mathbb Z/n$. Let $i_x:x \hookrightarrow Q$ and $i_y:y \hookrightarrow Q$ be closed points on $Q$ with $\kappa(x)=F$, and let $f:y \to \text{\rm Spec}(F)$ be the natural map. Then for integers $q,r \ge 0$, the following diagram commutes{\rm:}
\[ \xymatrix{
H^q(y,\varLambda(r)) \ar[r]^-{\text{\rm tr}_f} \ar[rd]_{\text{\rm Gys}_{i_y}} & H^q(x,\varLambda(r)) \ar[d]^{\text{\rm Gys}_{i_x}} \\
& H^{q+2d}(Q,\varLambda(r+d))\,. \hspace{-5pt}
}\]
\end{lem}
\begin{pf}
Because the case $x=y$ is obvious, we suppose that $x$ and $y$ are distinct points. Put $L:=\kappa(y)$.
\par\medskip
(i) First assume that $L/F$ is separable. In this case, $y$ is a closed point on some linear subspace of $Q$ of dimension $1$. Because any two $F$-rational points on $Q$ are linearly equivalent, we may assume that $Q=\mathbb P^1_F$. Then there is a direct decomposition
\[ H^{q+2}(Q,\varLambda(r+1)) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^q(F,\varLambda(r)) \oplus H^{q+2}(x,\varLambda(r+1)) \]
sending $a \in H^{q+2}(Q,\varLambda(r+1))$ to $(\text{\rm tr}_{Q/F}(a),i_x^*(a))$ (cf.\ Lemma \ref{lem1-1}\,(c)). In view of Lemma \ref{lem1-1}\,(a), our task is to show that the following composite map is zero:
\stepcounter{equation}
\begin{equation}\label{comp:appA}
\xymatrix{
H^q(y,\varLambda(r)) \ar[r]^-{\text{\rm Gys}_{i_y}} & H^{q+2}(Q,\varLambda(r+1)) \ar[r]^-{i_x^*} & H^{q+2}(x,\varLambda(r+1))\,.
}
\end{equation}
We prove this claim. Indeed, $\text{\rm Gys}_{i_y}$ factors through $H^{q+2}_y(Q,\varLambda(r+1))$ and $i_x^*$ factors through $H^{q+2}(Q \smallsetminus y,\varLambda(r+1))$. Hence \eqref{comp:appA} is the zero map and we obtain the assertion.
\par\medskip
(ii) In the case that $L/F$ is inseparable, let $F'$ be the perfection of $F$, let $K$ be the separable closure of $F$ in $L$, and put $K':=K \otimes_F F' $, $Q':=Q \otimes _F F'$; $K'$ is a perfect field and $L':=L \otimes_F F'$ is a finite local $K'$-algebra with residue field $K'$. Let $u$ (resp.\ $v$) be the projection $Q' \to Q$ (resp.\ $\text{\rm Spec}(F') \to \text{\rm Spec}(F)$).
By a standard norm and limit argument, it is enough to show the commutativity of a diagram
\[\xymatrix{
H^q(z,\varLambda(r)) \ar[r]^-{v^*(\text{\rm tr}_f)} \ar[rd]_{u^*(\text{\rm cl}_Q(y)) \cup ?\quad} & H^q(x',\varLambda(r)) \ar[d]^{u^*(\text{\rm cl}_Q(x)) \cup ?} \\ & H^{q+2d}(Q',\varLambda(r+d)) \,. \hspace{-5pt}
}\]
Here $z$ is the closed point of $y':= y \otimes _F F'=\text{\rm Spec}(L')$, we put $x':=x \otimes _F F'$, and we identified $H^q(z,\varLambda(r))$ with $H^q(y',\varLambda(r))$ by the restriction for $z \hookrightarrow y'$. We compute these maps. The map $v^*(\text{\rm tr}_f)$ agrees with $[L:K]\cdot \text{\rm tr}_{z/x'}$ (cf.\ \cite{sga4} XVIII.2.9). On the other hand, we have
\[ u^*(\text{\rm cl}_Q(y))=\text{\rm cl}_{Q'}(z;L')=[L:K] \cdot \text{\rm cl}_{Q'}(z) \quad \hbox{ and } \quad u^*(\text{\rm cl}_Q(x))=\text{\rm cl}_{Q'}(x') \]
by \cite{sga4.5} Cycle 2.3.8\,(i),\,(ii), where $\text{\rm cl}_{Q'}(z;L')$ denotes the cycle class with coefficients (see loc.\ cit.). Hence the above diagram commutes by the separable case.
\end{pf}
\noindent
This completes the proof of Theorem \ref{prop:appA}.
\smallskip
\subsection{Bloch-Ogus complexes and Kato complexes}\label{sect1-5}
We use Theorem \ref{prop:appA} to identify the Kato complexes with those defined via the method of Bloch and Ogus. Keep the assumptions as in the beginning: $X$ is a noetherian excellent {\it regular} scheme, $n$ is a positive integer invertible on $X$, and $\mathbb Z/n(q)$\,($q \in \mathbb Z$) is the $q$-fold Tate twist of the \'etale sheaf $\mathbb Z/n$ on $X$ or $X$-schemes. Assume that $X$ is of pure dimension $d$, and that $Z \subset X$ is a closed subscheme. By the method of Bloch and Ogus \cite{Bloch-Ogus}, there is a niveau spectral sequence
\begin{equation}\label{eq1-5-1}
E^1_{s,t}(Z/X,\mathbb Z/n(b)) = \bigoplus_{x\in Z_s} \ H^{-s-t}_x(X,\mathbb Z/n(-b)) \Longrightarrow H^{-s-t}_Z(X,\mathbb Z/n(-b))
\end{equation}
obtained by filtering with respect to the dimension of support. More precisely, it is the niveau spectral sequence on $Z$ (cf.\ \cite{Bloch-Ogus} (3.7) and \cite{js} 2.7), for the homology theory which is defined on all subschemes $V$ on $X$ by
\[ H_a(V/X,\mathbb Z/n(b)) := H^{-a}_V(U,\mathbb Z/n(-b))\,, \]
if $V$ is a closed subscheme of an open subscheme $U\subset X$.
By definition,
\[ H^a_x(X,\mathbb Z/n(-b)) = \varinjlim_{x \in U \subset X} \ H^a_{\ol{\{x\}}\cap U}(U,\mathbb Z/n(-b))\,, \]
where the limit is taken over all open subsets $U \subset X$ containing $x$. Since $X$ is excellent, $\ol{\{x\}}$ is excellent as well, and hence there is an open $U_0$ such that $\overline{\{x\}}\cap U$ is regular for $U\subset U_0$. For such $U$, by absolute purity \cite{fujiwara}, one has a Gysin isomorphism for the closed immersion $i_U: \overline{\{x\}}\cap U \hookrightarrow U$
\[ \text{\rm Gys}_{i_U}: H^{s-t-2d}(\overline{\{x\}}\cap U,\mathbb Z/n(s-d-b)) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^{-s-t}_{\overline{\{x\}}\cap U}(U,\mathbb Z/n(-b)) \]
for $x\in X_s$, i.e., of codimension $d-s$ in $X$. Hence we get an isomorphism
\begin{equation}\label{eq1-5-3}
E^1_{s,t}(Z/X,\mathbb Z/n(b)) \cong \bigoplus_{x\in X_s} \ H^{s-t-2d}(k(x),\mathbb Z/n(s-d-b))\,.
\end{equation}
For a complex of abelian groups $C^*$ denote by $(C^*)^{(-)}$ the complex with the same components, but with the differentials multiplied by $-1$.
\addtocounter{thm}{2}
\begin{thm}\label{thm1-5-1}
The Bloch-Ogus complex $E^1_{*,t}(Z/X,\mathbb Z/n(b))$ coincides with the sign-modified Kato complex $C_n^{-t-2d,-d-b}(Z)^{(-)}$ via the Gysin isomorphisms \eqref{eq1-5-3}.
\end{thm}
\begin{pf}
By definition of the spectral sequence \eqref{eq1-5-1}, its $d^1$-differentials have the components
\[ \delta_{y,x}^{\text{\rm loc}}: H^{-s-t}_y(X,\mathbb Z/n(-b)) \longrightarrow H^{-s+1-t}_x(X,\mathbb Z/n(-b)) \]
for $y\in X_s$ and $x\in X_{s-1}$ with $x\in \overline{\{y\}}$ (cf.\ \cite{js} Remarks 2.8). Therefore the claim directly follows from Theorem \ref{prop:appA}.
\end{pf}
\begin{rem}
For $Z=X$ it is often customary in literature to renumber the spectral sequence \eqref{eq1-5-1} into a cohomological coniveau spectral sequence {\rm(}with $c=-b${\rm)}
\[ E_1^{p,q}(X,\mathbb Z/n(c)) = \bigoplus_{x\in X^p} \ H^{q-p}(k(x),\mathbb Z/n(c-p)) \Longrightarrow H^{p+q}(X,\mathbb Z/n(c))\,. \]
This does not change the differentials, and so the $E_1$-terms compare in a similar way to the Kato complexes. More precisely one obtains that $E_1^{-*,q}(X,\mathbb Z/n(c))$ coincides with $C_n^{q-d,c-d}(X)^{(-)}$.
\end{rem}
By this method, we only get Bloch-Ogus complexes for schemes $Z$ which can be globally embedded in a regular scheme $X$. But the following slight variant covers all the cases considered in \cite{js} (of course, still under the assumption that $n$ is invertible on these schemes).
Let $S$ be a noetherian excellent regular base scheme of pure dimension $d$, let $n$ be invertible on $S$, and let $b$ be an integer. Similarly as in \S\ref{sect0-3} we define a homology theory (in the sense of \cite{js} 2.1) for all separated $S$-schemes of finite type $f: X \to S$ by defining
\[ H_a(X/S,\mathbb Z/n(b)) := H^{-a}(X,Rf^!\mathbb Z/n(-b))\,. \]
Then one gets a niveau spectral sequence
\addtocounter{equation}{2}
\begin{equation}\label{eq1-5-4}
E^1_{s,t}(X/S,\mathbb Z/n(b)) = \bigoplus_{x\in X_s} H_{s+t}(x/S,\mathbb Z/n(b)) \Longrightarrow H_{s+t}(X/S,\mathbb Z/n(b))\,,
\end{equation}
where $H_a(x/S,\mathbb Z/n(b))$ is defined as the inductive limit of $H_a(V/S,\mathbb Z/n(b))$ over all non-empty open subschemes $V\subset \overline{\{x\}}$. Since $\overline{\{x\}}$, being of finite type over $S$, is again excellent, there is a non-empty open subset $V_0$ which is regular. Then, for all non-empty open $V\subset V_0$ one has a canonical purity isomorphism
\begin{equation}\label{eq1-5-5}
H_a(V/S,\mathbb Z/n(b)) \cong H^{2(s-d)-a}(V,\mathbb Z/n(s-d-b))\,,
\end{equation}
where $s=\dim(V)$ is the dimension of $x$, by the construction in \cite{fujiwara} p.\ 157. This induces an isomorphism
\begin{equation}\label{eq1-5-6}
E^1_{s,t}(X/S,\mathbb Z/n(b)) \cong \bigoplus_{x\in X_s} \ H^{s-t-2d}(x,\mathbb Z/n(s-b-d))
\end{equation}
The following theorem generalizes Theorem \ref{thm1-5-1} (which is the case $X=S$).
\addtocounter{thm}{3}
\begin{thm}\label{thm:Katocomplexprimetop}
The Bloch-Ogus complex $E^1_{*,t}(X/S,\mathbb Z/n(b))$ coincides with the sign-modified Kato complex $C_n^{-t-2d,-b-d}(X)^{(-)}$ via the isomorphisms \eqref{eq1-5-6}.
\end{thm}
\begin{pf}
The question is local on $X$ and $S$. Thus we can assume that there is a factorization $f=p\circ i$, where $p: P \to S$ is a smooth morphism of relative dimension $N$ (e.g., $P = \mathbb A^N_S$) and $i: X \hookrightarrow P$ is a closed immersion. Then there is a canonical isomorphism from Poincar\'e duality (\cite{sga4} XVIII 3.2.5) \[ Rp^!\mathbb Z/n(-b) \cong \mathbb Z/n(N-b)[2N], \] which induces an isomorphism
\begin{align*}
H_a(X/S,\mathbb Z/n(b)) & = H^{-a}(X,Rf^!\mathbb Z/n(-b)) \\
& \cong H^{-a}(X,Ri^!\mathbb Z/n(N-b)[2N]) = H^{2N-a}_X(P,\mathbb Z/n(N-b))\,.
\end{align*}
Similarly, for a locally closed subset $V\subset P$, say a closed immersion $i_V: V \hookrightarrow U$ with $j_U: U \hookrightarrow P$ open, there is an isomorphism
\stepcounter{equation}
\begin{align}
H_a(V/S,\mathbb Z/n(b)) & = H^{-a}(V,R(f\circ j_U\circ i_V)^!\mathbb Z/n(-b)) \notag \\
& \cong H^{-a}(V,R(i_V)^!\mathbb Z/n(N-b)[2N]) \label{eq1-5-8} \\
& = H^{2N-a}_V(U,\mathbb Z/n(N-b))\,. \notag
\end{align}
Moreover, this is compatible with localization sequences. If $V$ is regular and of dimension $s$ (hence of codimension $d+N-s$ in $P$), then by definition, the isomorphism \eqref{eq1-5-3} is the composition of this map with the inverse of the Gysin isomorphism
\[ \text{\rm Gys}_{i_V}: H^{2s-2d-a}(V,\mathbb Z/n(s-d-b)) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^{2N-a}_V(U,\mathbb Z/n(N-b))\,. \]
This shows the following: Via the maps \eqref{eq1-5-8}, we get an isomorphism between the homology theory $H_*(-/S,\mathbb Z/n(*'))$, restricted to subschemes of $X$, and the homology theory $H_{*-2N}(-/P,\mathbb Z/n(*'+N))$ from \eqref{eq1-5-1}, restricted to subschemes of $X$, and therefore an isomorphism of the corresponding spectral sequence. Moreover, via this isomorphisms, the isomorphisms \eqref{eq1-5-3} and \eqref{eq1-5-6} correspond. Therefore the claim follows from Theorem \ref{thm1-5-1}.
\end{pf}
\newpage
\section{The case of $p$-torsion over a perfect field of characteristic $p$}\label{sect2}
\medskip
Throughout this section, $k$ always denotes a perfect field of positive characteristic $p$ and $n$ denotes a positive integer. We will often write $s$ for $\text{\rm Spec}(k)$.
\smallskip
\subsection{Gros' Gysin map}\label{sect2-1}
Let us recall that Gros has defined Gysin morphisms
\[ \text{\rm Gys}_f: Rf_*\logwitt Y n r \longrightarrow \logwitt X n {r+c}[c] \]
for any proper morphism $f: Y \to X$ of smooth equidimensional varieties over $k$, where $c = \dim(X) - \dim(Y)$ (\cite{gros:purity} II.1). These induce maps
\[ \text{\rm Gys}_f: H^q(Y,\logwitt Y n r) \longrightarrow H^{q+c}(X,\logwitt X n {r+c})\,. \]
If $i: Y \hookrightarrow X$ is a closed immersion of smooth $k$-schemes, it also induces Gysin maps
\[ \text{\rm Gys}_i: H^q(Y,\logwitt Y n r) \longrightarrow H^{q+c}_Y(X,\logwitt X n {r+c})\,, \]
where $c$ is now the codimension of $Y$ in $X$. The following result is a $p$-analogue of Theorem \ref{prop:appA}, cf. Remark \ref{rem:smooth}.
\begin{thm}\label{thm:shiho}
Let $X$ be a smooth variety over $k$. Let $n$ and $c$ be positive integers. Let $i_x:x \hookrightarrow X$ and $i_y:y \hookrightarrow X$ be points on $X$ of codimension $c$ and $c-1$, respectively, with $x \in \ol {\{ y \}}$. Then the following diagram commutes{\rm:}
\[\xymatrix{
H^0(y,\logwitt y n {r-c+1}) \ar[rr]^-{(-1)^r \, \dval_{y,x}} \ar[d]_{\text{\rm Gys}_{i_y}}
&& H^0(x,\logwitt x n {r-c}) {\ar[d]^{\text{\rm Gys}_{i_x}}}\\
H^{c-1}_y(X,\logwitt X n r) \ar[rr]^-{\delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)}
&& H^{c}_x(X,\logwitt X n r)\,.\hspace{-5pt} }\]
\end{thm}
In \cite{shm} 5.4, Shiho proved this compatibility property assuming $r=1$, but in a more general situation. The proof of Theorem \ref{thm:shiho} given below relies on the following properties of the Gysin maps:
\begin{enumerate}
\item[(P1)] Local description of Gysin maps. See \cite{gros:purity} II.3.3.9, but we will only need the case of a regular prime divisors, where one can give a simpler proof.
\item[(P2)] Transitivity of Gysin maps \cite{gros:purity} II.2.1.1.
\item[(P3)] For a finite map $h : z \to x$ of spectra of fields which are finitely generated over $k$, the Gysin map $\text{\rm Gys}_h: H^0(z,\logwitt z n r) \to H^0(x,\logwitt x n r)$ agrees with the corestriction map \eqref{eq-0.6.5}, cf.\ Lemma \ref{lem:norm0} in the appendix.
\end{enumerate}
To prove the theorem, replacing $X$ with $\text{\rm Spec}(\mathscr O_{X,x})$, we suppose that $X$ is local with closed point $x$. The proof proceeds in three steps, which will be finished in \S\ref{sect2-4} below.
\smallskip
\subsection{Divisor case}\label{sect2-2}
We first prove Theorem \ref{thm:shiho} assuming $c=1$. In this case, $A:=\mathscr O_{X,x}$ is a discrete valuation ring. Let $\pi$ be a prime element of $A$, and put $K:=\text{\rm Frac}(A)=\kappa(y)$ and $F:=\kappa(x)$. By the Bloch-Gabber-Kato theorem \cite{Bloch-Kato} 2.1, the group $H^0(y,\logwitt y n r)$ is generated by elements of the forms
\[
\hbox{(i)} \ \ d{\text{\rm log}}(\ul{f_1}) \cdot \, \dotsb \, \cdot d{\text{\rm log}}(\ul{f_r})
\quad \hbox{ and } \quad
\hbox{(ii)} \ \ d{\text{\rm log}}(\ul{\pi}) \cdot d{\text{\rm log}}(\ul{f_1}) \cdot \, \dotsb \, \cdot d{\text{\rm log}}(\ul{f_{r-1}})\,,
\]
where each $f_j$ belongs to $A^\times$, and for $a \in A$, $\ul a \in W_n(A)$ denotes its Teichm\"uller representative. The diagram in question commutes for elements of the form (i) obviously. We consider the element
\[ \alpha := d{\text{\rm log}}(\ul {\pi}) \cdot d{\text{\rm log}}(\ul {f_1}) \cdot \ \dotsb \ \cdot d{\text{\rm log}}(\ul {f_{r-1}}) \]
with each $f_j \in A^\times$, in what follows. By \cite{CTSS} p.\ 779 Lemma 2, we have
\[ \witt X n r/\logwitt X n r \cong \witt X n r/dV^{n-1}\Omega^{r-1}_X \,, \]
which is a finitely successive extension of (locally) free $\mathscr O_X$-modules by \cite{il} I.3.9. Hence the natural map
\[ \varrho : H^1_x(X,\logwitt X n r) \longrightarrow H^1_x(X,\witt X n r) \]
is injective, and our task is to show the equality
\begin{equation}\label{xy}
(-1)^r \big(\varrho \circ \text{\rm Gys}_{i_x} \circ \dval_{y,x}\big) (\alpha) = \big(\varrho \circ \delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)\big) (\alpha)
\end{equation}
in $H^1_x(X,\witt X n r)$. We regard the complex
\[ \witt A n r \os{i_y^*}{\longrightarrow} \witt K n r \]
as a representative of $R\Gamma_x(X, \witt X n r)$, where $\witt A r n$ is placed in degree $0$, cf.\ \cite{gros:purity} II.3.3.3. This identification induces an isomorphism
\[ \varphi : \witt K n r / \witt A n r \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^1_x(X,\witt X n r). \]
Now consider a commutative diagram
\[ \xymatrix{ \witt F n {r-1} \ar[rr]^-{\omega \, \mapsto \, \wt{\omega} \cdot d{\text{\rm log}}(\ul{\pi})} \ar@{}[rrd]|{\text{(1)}} \ar@{=}[d]
&& \witt K n r / \witt A n r \ar@{}[rrd]|{\text{(2)}} \ar[d]_{\varphi}^{\hspace{-1.5pt}\wr}
&& \ar@{->>}[ll]_-{(-1) \times \, \text{natural projection}} \witt K n r \ar@{=}[d] \\
H^0(x,\witt x n {r-1}) \ar[rr]^-{\text{\rm Gys}'_{i_x}} \ar@{}[rrd]|{\text{(3)}}
&& H^1_x(X,\witt X n r) \ar@{}[rrd]|{\text{(4)}}
&& \ar[ll]_-{\delta^{\text{\rm loc}}_{y,x}(\witt X n r)} H^0(y,\witt y n r) \\
H^0(x,\logwitt x n {r-1}) \ar[rr]^-{\text{\rm Gys}_{i_x}} \ar@{^{(}->}[u]
&& H^1_x(X,\logwitt X n r) \ar@{^{(}->}[u]^{\varrho}
&& \ar[ll]_-{\delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)} \ar@{^{(}->}[u] H^0(y,\logwitt y n r)\,,\hspace{-5pt}
}\]
where for $\omega \in \witt F n {r-1}$, $\wt{\omega} \in \witt A n {r-1}$ denotes a lift of $\omega$. The square (1) commutes by the property (P1) mentioned before. The square (2) commutes by a simple (but careful) computation of boundary maps, cf.\ \cite{shm} p.\ 612. By these commutative squares we have
\begin{align*}
\hbox{RHS of \eqref{xy}} & = (-1)^{r-1} \big(\varrho \circ \delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)\big) \big(d{\text{\rm log}}(\ul{f_1}) \cdot \, \dotsb \, \cdot d{\text{\rm log}}(\ul{f_{r-1}}) \cdot d{\text{\rm log}}(\ul{\pi}) \big) \\
& \os{\text{(4)}}{=} (-1)^{r-1} \big(\delta^{\text{\rm loc}}_{y,x}(\witt X n r)\big) \big(d{\text{\rm log}}(\ul{f_1}) \cdot \, \dotsb \, \cdot d{\text{\rm log}}(\ul{f_{r-1}}) \cdot d{\text{\rm log}}(\ul{\pi}) \big) \\
& \hspace{-8.0pt}\os{\text{(2)}+\text{(1)}}{=} (-1)^r \, \text{\rm Gys}'_{i_x} \big(d{\text{\rm log}}(g_1) \cdot \ \dotsb \ \cdot d{\text{\rm log}}(g_{r-1}) \big) \os{\text{(3)}}{=} \hbox{LHS of \eqref{xy}},
\end{align*}
where $g_j \in W_n(F)^{\times}$ denotes the residue class of $\ul{f_j}$ for each $j$. We thus obtain Theorem \ref{thm:shiho} in the case $c=1$.
\smallskip
\subsection{Regular case}\label{sect2-3}
We next treat the case that $c$ is arbitrary but the closure $Z:=\ol {\{ y \}} \subset X$ is regular. Let $\iota_x: x \hookrightarrow Z$ and $i : Z \hookrightarrow X$ be the natural closed immersions. Let us consider the following diagram:
\[\xymatrix{
& H^0(y,\logwitt y n {r-c+1}) \ar[r]^-{\text{\rm Gys}_{i_y}} \ar[ld]_{(-1)^r \, \dval_{y,x} \;} \ar[d]^{(-1)^{c-1}\delta}
& H^{c-1}_y(X,\logwitt X n r) \ar[d]^{\delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)} \\
H^{0}(x,\logwitt x n {r-c}) \ar[r]_-{\text{\rm Gys}_{\iota_x}}
& H^1_x(Z,\logwitt Z n {r-c+1}) \ar[r]_-{\text{\rm Gys}_i}
& H^c_x(X,\logwitt X n r)\,, \hspace{-5pt}
}\]
where we put $\delta:=\delta^{\text{\rm loc}}_{y,x}(\logwitt Z n {r-c+1})$ for simplicity. The right upper arrow and the right lower arrow are induced by the Gysin morphism for $i$,
so the right square commutes by \eqref{eq-0.4.1}. The left triangle commutes by the previous case. The composite of the bottom row coincides with $\text{\rm Gys}_{i_x}$ by (P2). Hence the assertion follows in this case.
\smallskip
\subsection{General case}\label{sect2-4}
We finally consider the general case. The arguments here proceed similarly as for \S\ref{sect1-3}. Let $Z=\ol {\{ y \}} \subset X$ be as in the previous step. We assume that $Z$ is not regular. Take the normalization $f:T \to Z$. Note that $T$ is regular and semi-local and that $\nu$ is finite by the excellence of $Z$. Since a finite morphism is projective (\cite{ega2} 6.1.11), the map $T \to X$ factors as $T \hookrightarrow \mathbb P^e_X \to X$ for some integer $e \geq 1$. Let $\iota_y$ be the composite map $y \to T \to \mathbb P^e_X$. Put $\varSigma:=f^{-1}(x) \subset T$ with the reduced subscheme structure, i.e., the scheme consisting of the closed points on $T$. There is a commutative diagram of schemes
\begin{equation}\label{CD:schemes2}
\xymatrix{ \varSigma \; \ar@{^{(}->}[r] \ar[d]_h & T \; \ar@{^{(}->}[r]^-{i_{T}} \ar[d]_f & \mathbb P:=\mathbb P^e_X \ar[d]_g \\
x \; \ar@{^{(}->}[r]^-{\iota_x} & Z \; \ar@{^{(}->}[r]^-{i_Z} & X\,.\hspace{-5pt} }
\end{equation}
Let $i_x$ (resp.\ $i_{\varSigma}$) be the closed immersion $x \hookrightarrow X$ (resp.\ $\varSigma \hookrightarrow \mathbb P$). Now put $c':=c+e=\text{\rm codim}_{\mathbb P}(\varSigma)$, and consider the diagram in Figure 2.
\begin{figure}[htp]
\setlength{\unitlength}{.6mm} {\scriptsize
\begin{picture}
(220,90)(-127,-45) \thinlines
\put(-99,-35){\vector(2,1){24}}
\put(67,-35){\vector(-2,1){24}}
\put(67,36){\vector(-2,-1){24}}
\put(-99,36){\vector(2,-1){24}}
\put(-119,34){\line(0,-1){68}}
\put(-118,34){\line(0,-1){68}}
\put(84,34){\vector(0,-1){68}}
\put(-97,40){\vector(1,0){161}}
\put(-97,-39){\vector(1,0){161}}
\put(-65,13){\vector(0,-1){26}}
\put(32,13){\vector(0,-1){26}}
\put(-42,18){\vector(1,0){51}}
\put(-42,-18){\vector(1,0){51}}
\put(-84,-32){{\tiny $\text{\rm Gys}_{i_y}$}}
\put(40,-32){{\tiny $\text{\rm Gys}_{i_x}$}}
\put(36,31){{\tiny $\text{\rm Gys}_{i_{\varSigma}}$}}
\put(-84,31){{\tiny $\text{\rm Gys}_{\iota_y}$}}
\put(86,0){{\tiny $\text{\rm Gys}_{h}$}}
\put(-22,44){{\tiny $(-1)^r\,\partial^\val_{y,\varSigma}$}}
\put(-22,-45){{\tiny $(-1)^r\,\dval_{y,x}$}}
\put(-63,0){{\tiny $\text{\rm Gys}_g$}}
\put(20,0){{\tiny $\text{\rm Gys}_g$}}
\put(-40,12){{\tiny $(-1)^e\,\delta^{\text{\rm loc}}_{y,\varSigma}(\logwitt {\mathbb P} n {r+e})$}}
\put(-34,-14){{\tiny $\delta^{\text{\rm loc}}_{y,x}(\logwitt X n r)$}}
\put(-18,28){\tiny {\text(1)}}
\put(-95,0){\tiny {\text(2)}}
\put(-18,0){\tiny {\text(3)}}
\put(58,0){\tiny {\text(4)}}
\put(-18,-30){\tiny {\text(5)}}
\put(66,39){$H^0(\varSigma,\logwitt {\varSigma} n {r-c})$}
\put(-140,39){$H^0(y,\logwitt y n {r-c+1})$}
\put(-140,-41){$H^0(y,\logwitt y n {r-c+1})$}
\put(66,-41){$H^0(x,\logwitt x n {r-c})$}
\put(12,17){$H^{c'}_{\varSigma}(\mathbb P,\logwitt {\mathbb P} n {r+e})$}
\put(-89,17){$H^{c'-1}_y(\mathbb P,\logwitt {\mathbb P} n {r+e})$}
\put(-89,-20){$H^{c-1}_y(X,\logwitt X n r)$}
\put(12,-20){$H^{c}_x(X,\logwitt X n r)$}
\end{picture}
\caption{A diagram for the proof of the general case}}
\end{figure}
The square (3) commutes by \eqref{eq-0.4.1}. Moreover, the diagrams (2) and (4) commute by the transitivity property (P2). On the other hand, the diagram (1) is commutative by the result in the previous case. Finally, the outer large square of Figure 2 commutes by Lemma \ref{lem:norm0} in the appendix and the definition of $\dval_{y,x}$, cf.\ \S\ref{sect0-6}\,(II). Thus the diagram (5) commutes, i.e., Theorem \ref{thm:shiho}. \hfill \qed
\begin{cor}\label{cor:shiho}
Let $X$ be a smooth variety of pure dimension $d$ over $k$, and let $c$ be a positive integer. Let $i_x:x \to X$ and $i_y:y \to X$ be points on $X$ of codimension $c$ and $c-1$, respectively, with $x \in \ol {\{ y \}}$. Then the following diagram commutes{\rm:}
\[ \xymatrix{
H^1(y,\logwitt y n {d-c+1}) \ar[rr]^-{(-1)^d\,\dval_{y,x}} \ar[d]_{\text{\rm Gys}_{i_y}}
&& H^1(x,\logwitt x n {d-c}) \ar[d]^{\text{\rm Gys}_{i_x}} \\
H^c_y(X,\logwitt X n d) \ar[rr]^-{\delta^{\text{\rm loc}}_{y,x}(\logwitt X n d)}
&& H^{c+1}_x(X,\logwitt X n d)\,.\hspace{-5pt}
}\]
\end{cor}
\begin{pf}
First of all we note that $[\kappa(x):\kappa(x)^p] = p^{d-c}$, because $\kappa(x)$ has transcendence degree $d-c$ over the perfect field $k$. Therefore the upper map is well-defined. For the prove of the corollary we just have to consider the case $c=1$. In fact, the reduction to this case works as in \S\ref{sect2-3} and \S\ref{sect2-4} for Theorem \ref{thm:shiho}; we only have to consider the case $n=c$, and to raise the degrees of all cohomology groups by $1$. Furthermore we have to replace Lemma \ref{lem:norm0} by Lemma \ref{cor:norm1}. \par
In the case $c=1$ we again may replace $X$ by the spectrum of the discrete valuation ring $A=\mathscr O_{X,x}$. By the definition of Kato's residue maps (cf.\ \S\ref{sect0-6}\,(I)), and since
\[ H^2_x(X,\logwitt X n d) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^2_x(X^\text{\rm h},\logwitt {X^\text{\rm h}} n d) \]
for the henselization $X^\text{\rm h}$ of $X$ at $x$, we may furthermore replace $X$ by $X^\text{\rm h}$. Then $y=\text{\rm Spec}(K)$ for a henselian discrete valuation field $K$ with residue field $\kappa(x)$. Let $y'=\text{\rm Spec}(K^{\text{\rm sh}})$, where $K^{\text{\rm sh}}$ is the strict henselization of $K$. Put $\ol x := \text{\rm Spec}(\ol{\kappa(x)})$ for the separable closure $\ol{\kappa(x)}$ of $\kappa(x)$. Then we get a diagram in Figure 3.\par
\begin{figure}[htp]
\setlength{\unitlength}{.6mm} {\scriptsize
\begin{picture}
(220,90)(-126,-45) \thinlines
\put(-99,-35){\vector(2,1){24}}
\put(67,-35){\vector(-2,1){24}}
\put(67,36){\vector(-2,-1){24}}
\put(-99,36){\vector(2,-1){24}}
\put(-117,34){\line(0,-1){68}}
\put(-116,34){\line(0,-1){68}}
\put(84,34){\vector(0,-1){68}}
\put(-97,40){\vector(1,0){161}}
\put(-97,-39){\vector(1,0){161}}
\put(-67,13){\line(0,-1){26}}
\put(-68,13){\line(0,-1){26}}
\put(38,13){\vector(0,-1){26}}
\put(-39,18){\vector(1,0){46}}
\put(-39,-18){\vector(1,0){46}}
\put(-84,-31){\tiny $a$}
\put(-94,-28){\tiny $\simeq$}
\put(49,-31){\tiny $c$}
\put(59,-28){\tiny $\simeq$}
\put(49,30){\tiny $b$}
\put(59,28){\tiny $\simeq$}
\put(-84,30){\tiny $a$}
\put(-94,28){\tiny $\simeq$}
\put(86,0){\tiny $\text{\rm Gys}$}
\put(-26,44){\tiny $(-1)^d\,\partial^\val$}
\put(-31,-45){\tiny $\delta^\text{\rm loc}(\logwitt X n d)$}
\put(13,0){\tiny $H^1(x,\text{\rm Gys})$}
\put(-26,12){\tiny $(-1)^d\,\partial^\val$}
\put(-40,-14){\tiny $H^1(x,\delta^\text{\rm loc}(\logwitt X n d))$}
\put(-18,28){\tiny {\text(1)}}
\put(-93,0){\tiny {\text(2)}}
\put(-18,0){\tiny {\text(3)}}
\put(58,0){\tiny {\text(4)}}
\put(-18,-30){\tiny {\text(5)}}
\put(66,39){$H^1(x,\logwitt x n {d-1})$}
\put(-137,39){$H^1(y,\logwitt y n d)$}
\put(-137,-41){$H^1(y,\logwitt y n d)$}
\put(66,-41){$H^2_x(X,\logwitt X n d)$}
\put(9,17){$H^1(x,H^0(\ol x,\logwitt x n {d-1}))$}
\put(-97,17){$H^1(x,H^0(y',\logwitt y n d))$}
\put(-97,-20){$H^1(x,H^0(y',\logwitt y n d))$}
\put(9,-20){$H^1(x,H^1_{\ol x}(X^\text{\rm sh},\logwitt X n d))$}
\end{picture}
\caption{A diagram for the proof of Corollary \ref{cor:shiho}}
}
\end{figure}
\noindent
Here the isomorphisms $a, b$ and $c$ come from Hochschild-Serre spectral sequences for the pro-\'etale covering $X^\text{\rm sh} \to X\,(=X^\text{\rm h})$ given by the strict henselization of $X$. See \S\ref{sect0-6}\,(I.3) for $a$ and $b$, and note the isomorphism
\[ H^{i+1}_{\ol x}(X^\text{\rm sh},\logwitt X n d) \cong H^i(\ol x , \logwitt x n {d-1}) \; \hbox{($=0$ for $i \ne 0$)} \]
for $c$, cf.\ \cite{moser} Corollary to 2.4. Then the diagram (1) commutes by the definition of Kato's residue maps. The diagram (2) commutes trivially, and the diagrams (4) and (5) commute, because the vertical maps and the two lower horizontal maps are induced by morphisms of sheaves ($\text{\rm Gys}$ and $\delta^\text{\rm loc}$), and hence are compatible with the Hochschild-Serre spectral sequence. Finally, it follows from Theorem \ref{thm:shiho} that the square (3) commutes. This implies the commutativity of the outer square and hence the corollary.
\end{pf}
\smallskip
\subsection{The complex $\bs{\mathscr M^\bullet_{n,X}}$}\label{sect2-5}
Building on work of Moser \cite{moser}, and motivated by Theorem \ref{thm:shiho}, we introduce a complex of \'etale
sheaves and prove a duality result for it (cf.\ \S\ref{sect2-6} below).
\begin{defn}
Let $X$ be a scheme of finite type over $s$. For a point $x$ on $X$, let $i_x$ be the canonical map $x \hookrightarrow X$.
We define the complex $\mathscr M^\bullet_{n,X}$ of \'etale sheaves on $X$ as
\[ \mathscr M^\bullet_{n,X} :=
\left( \left\{ \bigoplus{}_{x \in X_{-q}} \ i_{x*} \logwitt x n {-q} \right\} {}_q, \left\{ -\partial^q \right\}_q \right), \]
where $\partial^q$ has the components $\dval_{y,x}$ with $y \in X_{-q}$ and $x \in X_{-q-1}$ {\rm(}cf.\ {\rm\S\ref{sect0-6}\rm)}. We often write $\mathscr M_{n,X}$ for the image of $\mathscr M^\bullet_{n,X}$ in $D^b(X_\text{\rm \'{e}t},\bZ/p^n)$. See Remark \ref{rem:smooth} below for the reason of the sign of the differentials.
\end{defn}
The complex $\mathscr M^\bullet_{n,X}$ coincides with the complex $\wt{\nu}{}_{r,X}$ defined in \cite{moser} up to signs of boundary operators and a shift. If $X$ is smooth over $s$ of pure dimension $d$, then, by a theorem of Gros and Suwa \cite{gs}, the embedding $\logwitt X n d \hookrightarrow \bigoplus_{x \in X_d} \, i_{x*} \logwitt x n d$ induces a canonical quasi-isomorphism
\stepcounter{equation}
\begin{equation}\label{same}
\logwitt X n d [d] \os{\text{\rm qis.}}\longrightarrow \mathscr M^\bullet_{n,X}\,.
\end{equation}
Note also the following simple facts: For a closed immersion $i:Z \hookrightarrow X$ of schemes of finite type over $s$, there is a natural map of complexes
\begin{equation}\label{gysin:DC}
i_*\mathscr M^\bullet_{n,Z} \longrightarrow \mathscr M^\bullet_{n,X}\,.
\end{equation}
If $X$ and $Z$ are smooth of pure dimension $d$ and $d'$, respectively, then this map induces a morphism
\begin{equation}\label{eq2-5-5}
\gys^\circ_i : i_*\logwitt Z n {d'}[d'] \longrightarrow \logwitt X n d [d] \quad \hbox{ in }\; D^b(X_\text{\rm \'{e}t},\bZ/p^n)
\end{equation}
via \eqref{same} for $X$ and $Z$, which we call the {\it modified Gysin morphism} for $i$.
\addtocounter{thm}{3}
\begin{rem}\label{rem:smooth}
The reason we put the sign $-1$ on the differentials of $\mathscr M^\bullet_{n,X}$ is as follows. Because of these signs, the modified Gysin map \eqref{eq2-5-5} agrees with Gros' Gysin map $\text{\rm Gys}_i$ only up to the sign $(-1)^{d-d'}$, cf.\ \cite{sato1} {\rm 2.3.1}. However by this fact, if we define $\bZ/p^n(r) := \logwitt X n r [-r]$ for {\rm(}essentially{\rm)} smooth schemes $X$ over $k$ and note property \eqref{eq-0.4.1},
Theorem {\rm\ref{thm:shiho}} for $r=d$ and Corollary {\rm\ref{cor:shiho}} become a commutative diagram
\[\xymatrix{
H^{m-c+1}(y,\bZ/p^n(d-c+1)) \ar[rr]^-{-\dval_{y,x}} \ar[d]_{\gys^\circ_{i_y}} && H^{m-c}(x,\bZ/p^n(d-c)) \ar[d]^{\gys^\circ_{i_x}} \\
H^{m+c-1}_y(X,\bZ/p^n(d)) \ar[rr]^-{\delta^{\text{\rm loc}}_{y,x}(\bZ/p^n(d))} && H^{m+c}_x(X,\bZ/p^n(d))\,.\hspace{-5pt}
}\]
Note that the groups in the top row are only non-zero for $m=d, d+1$. This shows the perfect analogy with Theorem {\rm\ref{prop:appA}}.
\end{rem}
\noindent
The following lemma shows that the complex $\mathscr M^\bullet_{n,X}$ is suitable for cohomological operations:
\begin{lem}\label{lem:moser}
Let $x$ be a point on $X$ of dimension $q \geq 0$. Then{\rm:}
\begin{enumerate}
\item[(1)]
The sheaf $\logwitt x n q$ on $x_\text{\rm \'{e}t}$ is $i_{x*}$-acyclic.
\item[(2)]
For a closed immersion $i:Z \hookrightarrow X$, the sheaf $i_{x*} \logwitt x n q$ on $X_\text{\rm \'{e}t}$ is $i^!$-acyclic.
\item[(3)]
For an $s$-morphism $f:X \to Y$, the sheaf $i_{x*} \logwitt x n q$ on $X_\text{\rm \'{e}t}$ is $f_*$-acyclic.
\end{enumerate}
\end{lem}
\begin{pf}
For (1) and (2), see \cite{moser} 2.3 and 2.4. We prove (3). For a point $y \in Y$, we have
\begin{align*}
\big(R^mf_*(i_{x*} \logwitt x n q)\big)_{\ol y} & \cong H^m(X \times_Y \text{\rm Spec}(\mathscr O_{Y,\ol y}^{\text{\rm sh}}),i_{x*} \logwitt x n q) \\
& \os{\text{(1)}}{\cong} H^m(x \times_Y \text{\rm Spec}(\mathscr O_{Y,\ol y}^\text{\rm sh}), \logwitt x n q)
\end{align*}
and the last group is zero for $m>0$ by the same argument as in loc.\ cit.\ 2.5.
\end{pf}
\begin{cor}[cf.\ \cite{moser} Corollary to Theorem 2.4]
\label{cor:moser} For a closed immersion $i : Z \hookrightarrow X$, the map \eqref{gysin:DC} induces an isomorphism
\[ \gys^\circ_i : \mathscr M_{n,Z} \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Ri^!\mathscr M_{n,X} \quad \hbox{ in } \; D^+(Z_\text{\rm \'{e}t},\bZ/p^n). \]
\end{cor}
\smallskip
\subsection{Relative duality theory}\label{sect2-6}
Let $\mathscr V_s$ be the category of schemes separated of finite type over $s$ and separated $s$-morphisms of finite type.
\begin{thm}\label{thm:moser2}
Suppose that there exists an assignment of morphisms
\[ \text{\rm Tr}: ( f : Y \to X \hbox{ in } \mathscr V_s ) \longmapsto (\text{\rm Tr}_f : Rf_!\mathscr M_{n,Y} \to \mathscr M_{n,X} \hbox{ in } D^+(X_\text{\rm \'{e}t},\bZ/p^n)) \]
which satisfy the following three conditions {\rm(i)\,--\,(iii):}
\begin{enumerate}
\item[(i)]
If $f$ is \'etale, then $\text{\rm Tr}_f$ agrees with the composite morphism
\[ Rf_!\mathscr M_{n,Y} = Rf_!f^*\mathscr M_{n,X} \os{f_!}\longrightarrow \mathscr M_{n,X}\,, \]
where the arrow $f_!$ denotes the adjunction morphism $Rf_!f^*=Rf_!Rf^! \to \text{\rm id}$ {\rm(}cf.\ \cite{sga4} {\rm XVIII.3)}.
\item[(ii)]
If $f$ is a closed immersion, then $\text{\rm Tr}_f$ agrees with the composite morphism
\[\xymatrix{
Rf_*\mathscr M_{n,Y} \ar[rr]^-{Rf_*(\gys^\circ_f)} && Rf_*Rf^!\mathscr M_{n,X} \ar[r]^-{f_*} & \mathscr M_{n,X}\,,
}\]
where the arrow $f_*$ denotes the adjunction morphism $Rf_*Rf^!=Rf_!Rf^! \to \text{\rm id}$.
\item[(iii)]
For morphisms $g:Z \to Y$ and $f:Y \to X$ with $h:=f \circ g$, $\text{\rm Tr}_h$ agrees with the composition
\[\xymatrix{
Rh_!\mathscr M_{n,Z} = Rf_!Rg_!\mathscr M_{n,Z} \ar[rr]^-{Rf_!(\text{\rm Tr}_g)} && Rf_!\mathscr M_{n,Y} \ar[r]^-{\text{\rm Tr}_f} & \mathscr M_{n,X}\,.
}\]
\end{enumerate}
Then for a map $f:Y \to X$ in $\mathscr V_s$, the adjoint morphism $\text{\rm Tr}^f$ of $\text{\rm Tr}_f$ is an isomorphism{\rm:}
\[ \text{\rm Tr}^f: \mathscr M_{n,Y} \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra R f^!\mathscr M_{n,X} \quad \hbox{ in } \; D^+(Y_\text{\rm \'{e}t},\bZ/p^n). \]
\end{thm}
\noindent
This theorem is a variant of Moser's duality \cite{moser} 5.6 (which itself generalizes Milne's duality for smooth projective varieties \cite{milne:mot}). However, because Theorem \ref{thm:moser2} looks quite different from Moser's formulation, we outline a proof of our statement below in \S\ref{sect2-7}. The main result of this section is the following theorem:
\begin{thm}\label{thm:jss}
There exists a unique assignment of morphisms
\[ \text{\rm tr}: ( f:Y \to X \hbox{ in } \mathscr V_s ) \longmapsto ( \text{\rm tr}_f : Rf_!\mathscr M_{n,Y} \to \mathscr M_{n,X} \hbox{ in } D^+(X_\text{\rm \'{e}t},\bZ/p^n) ) \]
that satisfies the conditions {\rm(i)\,--\,(iii)} in Theorem {\rm\ref{thm:moser2}} with $\text{\rm Tr}:=\text{\rm tr}$. Consequently, for a map $f:Y \to X$ in $\mathscr V_s$, the morphism $\text{\rm tr}^f:\mathscr M_{n,Y} \to Rf^!\mathscr M_{n,X}$ adjoint to $\text{\rm tr}_f$ is an isomorphism.
\end{thm}
\noindent
We will prove Theorem \ref{thm:jss} in \S\S\ref{sect2-8}--\ref{sect2-9} below.
\smallskip
\subsection{Proof of Theorem \ref{thm:moser2}}\label{sect2-7}
By the transitivity property (iii) of $\text{\rm Tr}$, the assertion is reduced to the case of a structure morphism $f:X \to s$, and moreover, by the property (i) of $\text{\rm Tr}$, we may suppose that $s= \ol s$ (i.e., $k$ is algebraically closed) and that $f$ is proper. In this situation, we claim the following:
\begin{thm}\label{thm:moser}
Let $X$ be a proper scheme of finite type over the algebraically closed field $k$ of characteristic $p>0$, with structural morphism $f: X \to \text{\rm Spec}(k)$. Then, for any constructible $\bZ/p^n$-sheaf $\mathscr F$ on $X_\text{\rm \'{e}t}$ and any integer $m$, the pairing
\[ \alpha_X(m,\mathscr F):H^m(X,\mathscr F) \times \text{\rm Ext}^{-m}_{X,\bZ/p^n}(\mathscr F,\mathscr M_{n,X}) \longrightarrow H^0(X,\mathscr M_{n,X}) \os {\text{\rm Tr}_f}\longrightarrow \bZ/p^n \]
induced by Yoneda pairing is a non-degenerate pairing of finite groups.
\end{thm}
\defj_!\hspace{1pt}\zpn{j_!\hspace{1pt}\bZ/p^n}
We first prove Theorem \ref{thm:moser2}, admitting Theorem \ref{thm:moser}: Applying \ref{thm:moser} to $\mathscr F=j_!\hspace{1pt}\zpn$ with $j:U \to X$ \'etale, and noting the isomorphisms
\[ \text{\rm Ext}^{-m}_{X,\bZ/p^n}(j_!\hspace{1pt}\zpn,\mathscr M_{n,X}) \cong \text{\rm Ext}^{-m}_{U,\bZ/p^n}(\bZ/p^n,\mathscr M_{n,U})\cong H^{-m}(U,\mathscr M_{n,U})\,, \]
we obtain isomorphisms
\[ H^{-m}(U,\mathscr M_{n,U}) \os{a}\cong \text{\rm Hom}_{\,\bZ/p^n\text{-mod}}(H^{m}(X,j_!\hspace{1pt}\zpn),\bZ/p^n) \os{b}\cong H^{-m}(U,Rg^!\bZ/p^n) \]
for any $m \in \mathbb Z$, where $g= f \circ j$, the first isomorphism comes from the pairing, and the second isomorphism comes from the adjunction between $Rg^!$ and $Rg_!$ and the fact that $\bZ/p^n$ is an injective object in the category of $\bZ/p^n$-modules. We verify that this composite map agrees with that coming from $\text{\rm Tr}^f$ --- then the morphism $\text{\rm Tr}^f$ is bijective on cohomology sheaves, and we obtain Theorem \ref{thm:moser2}. Indeed, by the definition of the pairing, the map $a$ sends $x \in H^{-m}(U,\mathscr M_{n,U})=\text{\rm Hom}_{D(X,\bZ/p^n)}(j_!\hspace{1pt}\zpn,\mathscr M_{n,X}[-m])$ to the composition \[ H^m(X,j_!\hspace{1pt}\zpn) \os{x}\longrightarrow H^0(X,\mathscr M_{n,X})=H^0(s,Rf_*\mathscr M_{n,X}) \os{\text{\rm Tr}_f}\longrightarrow \bZ/p^n, \] which, by \ref{thm:moser2}\,(i) and (iii), coincides with the map induced by
\[ Rg_!\hspace{1pt}\bZ/p^n[m] \os{x}\longrightarrow Rg_!\mathscr M_{n,U} \os{\text{\rm Tr}_g}\longrightarrow \bZ/p^n\,. \]
By definition (and functoriality of adjunction), the map $b$ sends this to the composition
\[ \bZ/p^n[m] \os{x}\longrightarrow \mathscr M_{n,U} \os{\text{\rm Tr}^{\hspace{1pt}g}}\longrightarrow Rg^!\bZ/p^n\,, \]
which shows the claim, again by \ref{thm:moser2}\,(i) and (iii). \qed
\par\medskip\smallskip
As for Theorem \ref{thm:moser}, it follows from the arguments in \cite{moser} 5.6. More precisely, it follows from the properties (i)\,--\,(iii) of $\text{\rm Tr}$, the steps (a)\,--\,(c), (f)\,--\,(k) of loc.\ cit.\ 5.6, and the following lemma:
\begin{lem}\label{lem:moser2}
Let $f : X \to s (=\ol s)$ be a proper smooth morphism with $X$ connected. Then for an integer $m$ and a positive integer $t \leq r$, the pairing $\alpha_X(m,\mathbb Z/p^t)$ {\rm(}cf.\ Theorem {\rm\ref{thm:moser})} is a non-degenerate pairing of finite groups.
\end{lem}
\begin{pf*}{\it Proof of Lemma \ref{lem:moser2}}
The problem is reduced to the case $t=r$ by \eqref{same} and \cite{moser} 5.4. Now we note that Milne duality \cite{milne:mot} 1.11 gives an isomorphism of finite groups in our case, because, with the notation of \cite{milne:mot} p.\ 305, there are no unipotent parts in the cohomology of the complex of perfect group schemes $\ul{H}^*(X,\bZ/p^n)$. Therefore it is enough to show that the composite map
\[ \text{\rm Tr}'_f : H^d(X,\logwitt X n d) \os{\eqref{same}}\cong H^0(X,\mathscr M_{n,X}) \os{\text{\rm Tr}_f}\longrightarrow \bZ/p^n \]
with $d:=\dim(X)$ coincides with the trace map $\eta_n$ in \cite{milne:mot} p.\ 308, up to a sign. But, for a closed point $i_x:x \hookrightarrow X$, $\text{\rm Tr}'_f$ sends the cycle class $\gys^\circ_{i_x}(1) \in H^d(X,\logwitt X n d)$ (= the image of $1$ under $\gys^\circ_{i_x}$) to $1$ by the properties (ii), (iii) of $\text{\rm Tr}$, and hence $\text{\rm Tr}'_f=(-1)^d \cdot \eta_n$ by Remark \ref{rem:smooth}\,(1). This completes the proof of Lemma \ref{lem:moser2}, Theorem \ref{thm:moser} and Theorem \ref{thm:moser2}.
\end{pf*}
\begin{rem}\label{rem:moser}
Note that step {\rm (j)} of \cite{moser} {\rm5.6} uses de Jong's alteration theorem \cite{dj} {\rm4.1}.
\end{rem}
\begin{cor}\label{cor:jss}
Suppose that there exist two assignments $\sigma:f \mapsto \sigma_f$ and $\tau : f \mapsto \tau_f$ satisfying {\rm(i)\,--\,(iii)} in Theorem {\rm\ref{thm:moser2}} with $\text{\rm Tr}:=\sigma$ and $\tau$, respectively. Then we have $\sigma=\tau$.
\end{cor}
\vspace{-5pt}
\begin{pf}
Let $f:Y \to X$ be a morphism in $\mathscr V_s$. We show that $\sigma_f=\tau_f$ as morphisms $Rf_!\mathscr M_{n,Y} \to \mathscr M_{n,X}$ in $D^+(X_\text{\rm \'{e}t},\bZ/p^n)$, in two steps. We first prove the case $X=s$ (hence $\mathscr M_{n,X}=\bZ/p^n$). By the properties (i) and (iii), we may suppose that $f$ is proper. Then $Rf_!\mathscr M_{n,Y}=Rf_*\mathscr M_{n,Y}$ is computed by the complex $f_*\mathscr M_{n,Y}^\bullet$ by Lemma \ref{lem:moser}\,(3), and the morphisms $\sigma_f$ and $\tau_f$ are determined by maps $f_*\mathscr M_{n,Y}^0 \to \bZ/p^n$ of sheaves on $s_\text{\rm \'{e}t}$ by \S\ref{sect0-5-4}\,(1). Hence in view of the properties (ii) and (iii) and the assumption that $s$ is perfect, the problem is reduced to the case where $f$ is \'etale, and we obtain $\sigma_f=\tau_f$ by the property (i). This completes the first step. Next we prove the general case. Let $\sigma^f$ and $\tau^f$ be the adjoint morphisms of $\sigma_f$ and $\tau_f$, respectively. By adjunction, it is enough to show $\sigma^f=\tau^f$ as morphisms $\mathscr M_{n,Y} \to Rf^!\mathscr M_{n,X}$ in $D^+(Y_\text{\rm \'{e}t},\bZ/p^n)$. Let $g:X \to s$ be the structure map and put $h:=g \circ f$. By the first step and the property (iii), we have
\[ Rf^!(\sigma^g) \circ \sigma^f = \sigma^h = \tau^h = Rf^!(\tau^g) \circ \tau^f \]
as morphisms $\mathscr M_{n,Y} \to Rh^!\bZ/p^n$. On the other hand, we have $Rf^!(\sigma^g)=Rf^!(\tau^g)$ by the first step, and these are isomorphisms in $D^+(Y_\text{\rm \'{e}t},\bZ/p^n)$ by Theorem \ref{thm:moser2}. Hence we have $\sigma^f = \tau^f$. This completes the proof of Corollary \ref{cor:jss}.
\end{pf}
\smallskip
\subsection{Covariant functoriality}\label{sect2-8}
In this subsection, we prove Lemma \ref{lem:jss2} stated below (cf.\ \cite{moser} 4.1), which is a key ingredient of Theorem \ref{thm:jss}. Let $f:Y \to X$ be a morphism in $\mathscr V_s$. Let $q$ be a non-negative integer and let $x$ (resp.\ $y$) be a point on $X$ (resp.\ on $Y$) of dimension $q$. Let $f_y$ (resp.\ $i_x$) be the composite map $y \to Y \to X$ (resp.\ $x \to X$). We define a map of sheaves on $X_\text{\rm \'{e}t}$
\[ \text{\rm tr}_{f,(y,x)}:f_{y*} \logwitt y n q \longrightarrow i_{x*}\logwitt x n q \]
as Gros' Gysin map for $y \to x$ (\cite{gros:purity} II.1.2.7), if $y$ is finite over $x$ via $f$. We define $\text{\rm tr}_{f,(y,x)}$ as zero otherwise. Collecting this map for points on $Y$ and $X$, we obtain a map of {\it graded abelian sheaves} on $X_\text{\rm \'{e}t}$
\[ \text{\rm tr}_f^\bullet : f_*\mathscr M^\bullet_{n,Y} \longrightarrow \mathscr M^\bullet_{n,X}\,. \]
By definition and \cite{gros:purity} II.2.1.1, this map of graded sheaves satisfies transitivity, that is, for morphisms $g:Z \to Y$ and $f:Y \to X$ in $\mathscr V_s$, we have the equality
\begin{equation}\label{lem:jss2:transitive}
\text{\rm tr}_f^\bullet \circ f_*(\text{\rm tr}_g^\bullet) = \text{\rm tr}_{f \circ g}^\bullet
\end{equation}
of maps of graded sheaves on $Y_\text{\rm \'{e}t}$. We prove the following lemma:
\stepcounter{thm}
\begin{lem}\label{lem:jss2}
Suppose that $f$ is proper. Then $\text{\rm tr}_f^\bullet$ is a map of complexes. Consequently, $\text{\rm tr}_f^\bullet$ induces a morphism
\[ \text{\rm tr}_f: Rf_*\mathscr M_{n,Y} \longrightarrow \mathscr M_{n,X} \quad \hbox{ in } \; D^b(X_\text{\rm \'{e}t},\bZ/p^n) \]
by Lemma {\rm\ref{lem:moser}\,(3)}.
\end{lem}
\begin{pf}
Let $q$ be a negative integer, and let $z$ and $x$ be points on $X$ of dimension $-q$ and $-q-1$, respectively. Our task is to show the commutativity of a diagram
\stepcounter{equation}
\begin{equation}\label{lem:jss2:commute2}
\xymatrix{
\displaystyle \bigoplus_{w \in Y_{-q} \cap f^{-1}(\ol {\{ z \}})}\ f_{w*}\logwitt w n {-q}
\ar@<5pt>[rr]^-{\bigoplus_{y}(\sum_w \, \dval_{w,y})} \ar[d]_-{\sum_w \, \text{\rm tr}_{f,(w,z)}}
&& \displaystyle \bigoplus_{y \in Y_{-q-1} \cap f^{-1}(\ol {\{ x \} })} \ f_{y*}\logwitt y n {-q-1} \ar[d]^-{\sum_y \, \text{\rm tr}_{f,(y,x)}} \\ i_{z*}\logwitt z n {-q} \ar[rr]^-{\dval_{z,x}}
&& i_{x*}\logwitt x n {-q-1}\,,\hspace{-5pt}
}
\end{equation}
where $\ol {\{z\} }$ and $\ol {\{ x \} }$ denote the closure of $z$ and $w$ in $X$, respectively. If $z$ does not specialize to $x$, then both horizontal maps are zero and the diagram commutes. In what follows, we assume that $z$ specializes to $x$. Let $T$ be the localization of $\ol{ \{ z \} }$ at $x$, and let $w$ be a point on $Y$ with $w \in Y_{-q} \cap f^{-1}( \ol {\{z\} })$. Put
\[ Z_w := \ol { \{ w \} } \times_X T (=\ol { \{ w \} } \cap f^{-1}(T))\,, \]
where $\ol { \{ w \} }$ denotes the closure of $w$ in $Y$, regarded as an $X$-scheme via $f$. If $y \in Y_{-q-1} \cap f^{-1}(\ol {\{z\} })$ is away from $Z_w$, then $y$ is outside of $\ol {\{ w \} } \cap f^{-1}(x)$ and hence at least one of $\dval_{w,y}$ and $\text{\rm tr}_{f,(y,x)}$ is zero. Thus the commutativity of \eqref{lem:jss2:commute2} is reduced to that of the following diagram:
\vspace{-8pt}
\begin{equation}\label{lem:jss2:commute3}
\xymatrix{
f_{w*}\logwitt w n {-q} \ar[rr]^-{\bigoplus_y\,\dval_{w,y}} \ar[d]_-{\text{\rm tr}_{f,(w,z)}}
&& \displaystyle \bigoplus_{y \in Y_{-q-1} \cap Z_w}^{\phantom{|}}\ f_{y*} \logwitt y n {-q-1} \ar[d]^-{\sum_y \, \text{\rm tr}_{f,(y,x)}} \\
i_{z*}\logwitt z n {-q} \ar[rr]^-{\dval_{z,x}} && i_{x*}\logwitt x n {-q-1}\,.\hspace{-5pt}
}
\end{equation}
Note that there happen only the following three cases:
\begin{itemize}
\item[(1)] $Z_w$ is empty (i.e., $w$ maps to a point on $\ol {\{ z \} }$ outside of $T$).
\item[(2)] $w$ maps to $x$.
\item[(3)] $w$ maps to $z$.
\end{itemize}
In the case (1), the problem is obvious, because the upper right sheaf and the left vertical map are trivial by definition. Therefore, we restrict our attention to the cases (2) and (3), in what follows. Note that $Z_w$ is integral with generic point $w$ in these cases. We claim here the following:
\par
\vspace{8pt}
\noindent
{\it Claim.}
(a) {\it In the case {\rm(2)}, $Z_w$ is a proper curve over $x$.} \par
(b) {\it In the case {\rm(3)}, the canonical morphism $f_T:Z_w \to T$ is finite.} \par
(c) {\it In both cases, the set $Y_{-q-1} \cap Z_w$ agrees with the set of all closed points on $Z_w$.}
\par
\vspace{8pt}
\noindent
{\it Proof of Claim.}
(a) Since $Z_w$ is proper over $x$ by the properness of $f$, it suffices to show that $Z_w$ is one-dimensional. This follows from \cite{hartshorne} III.9.6. \par
(b) By Zariski's Main Theorem (see e.g., \cite{milne} I.1.10) and the properness of $f_T$, it suffices to show that $f_T$ is quasi-finite. Note that $\kappa(w)$ is a finite field extension of $\kappa(z)$. Let $\nu:U \to T$ be the normalization of $T$ in $\kappa(w)$. Then $\nu$ is finite (cf.\ \cite{hartshorne} I.3.9A) and $U$ is the spectrum of a Dedekind ring, which imply that $\nu$ factors as $U \to Z_w \to T$ by the valuative criterion for proper morphisms (cf.\ loc.\ cit.\ II.4.7). Here the map $U \to Z_w$ is surjective, because it is proper and dominant. Hence $f_T$ is quasi-finite by the finiteness of $\nu$ and we obtain the assertion. \par
(c) immediately follows from the description of $Z_w$ in (a) and (b). \qed
\par
\vspace{8pt}
We turn to the proof of Lemma \ref{lem:jss2} and prove the commutativity of \eqref{lem:jss2:commute3}. We first consider the case (3). Recall that $Z_w$ is then finite over $T$ by {\it Claim} (b). Since the problem is \'etale local at $x \in T$, we assume that $T$ and $Z_w$ are strictly henselian by replacing them with $\text{\rm Spec}(\mathscr O_{T,\ol x}^\text{\rm sh})$ and a connected component of $Z_w \times_T \text{\rm Spec}(\mathscr O_{T,\ol x}^\text{\rm sh})$, respectively. Then by the Bloch-Gabber-Kato theorem (\cite{Bloch-Kato} 2.1) and Lemma \ref{lem:norm0} in the appendix, we are reduced to the commutativity of residue maps of Milnor $K$-groups (\S\ref{sect0-6}\,(I.2)) via norm maps due to Kato \cite{kk:res}, Lemma 3 (which assumes the domains concerned are normal, but is easily generalized to our situation by a standard argument using normalization):
\vspace{-6pt}
\[\xymatrix{
K^M_{-q}(\kappa(w))/p^n \ar[rr]^-{\bigoplus_y\,\partial_{w,y}} \ar[d]_-{N_{\kappa(w)/\kappa(z)}}
&&\displaystyle\bigoplus_{y \in (Z_w)_0}^{\phantom{|}}\ K^M_{-q-1}(\kappa(y))/p^n \ar[d]^-{\sum_y \, N_{\kappa(y)/\kappa(x)}}\\
K^M_{-q}(\kappa(z))/p^n \ar[rr]^-{\partial_{z,x}} && K^M_{-q-1}(\kappa(x))/p^n\,.\hspace{-5pt}
}\]
Hence \eqref{lem:jss2:commute3} commutes in this case.
\par
Finally we prove the commutativity of \eqref{lem:jss2:commute3} in the case (2). In this case, the map $\text{\rm tr}_{f,(w,z)}$ is zero by definition, and our task is to show that the composite of the top horizontal arrow and the right vertical arrow is zero. Moreover, by {\it Claim} (a) and Lemma \ref{lem:norm0}, we are reduced to the case that $Z_w$ is a projective line over $x$. The assertion then follows from {\it Claim} in the proof of Lemma \ref{lem:norm0}.
This completes the proof of Lemma \ref{lem:jss2}.
\end{pf}
\addtocounter{thm}{2}
\begin{rem}\label{rem:jss}
If $f$ is finite and \'etale, then $\text{\rm tr}_f^\bullet$ coincides with the adjunction map
\[ f_* : f_*\mathscr M^\bullet_{n,Y} = f_*f^*\mathscr M^\bullet_{n,X} \longrightarrow \mathscr M^\bullet_{n,X}\,. \]
Indeed, the claim is reduced to the case of a finite separable extension of a point, which follows from a standard base-change argument and \cite{milne} {\rm V.1.12}.
\end{rem}
\smallskip
\subsection{Proof of Theorem \ref{thm:jss}}\label{sect2-9}
By Corollary \ref{cor:jss}, it remains to show the existence of a desired assignment. For a map $f:Y \to X$ in $\mathscr V_s$, we define the morphism $\text{\rm tr}_f$ as follows. If $f$ is proper, then we define $\text{\rm tr}_f$ as that constructed in Lemma \ref{lem:jss2}. Next suppose that $f$ is not proper. Take a compactification of $f$, i.e., an open immersion $ j : Y \hookrightarrow Z$ and a proper map $ g : Z \to X$ with $f=g \circ j$, and define $\text{\rm tr}_{f,(Z,j,g)}$ as the composite morphism
\begin{equation}\label{comp2}
\text{\rm tr}_{f,(Z,j,g)}: Rf_! \mathscr M_{n,Y} = Rg_*Rj_!j^*\mathscr M_{n,Z} \os{j_!}\longrightarrow Rg_*\mathscr M_{n,Z} \os{\text{\rm tr}_g}\longrightarrow \mathscr M_{n,X}\,,
\end{equation}
where the arrow $j_!$ is defined by the adjunction morphism $Rj_!j^*\mathscr M_{n,Z} \to \mathscr M_{n,Z}$. We are going to define
\[ \text{\rm tr}_f:=\text{\rm tr}_{f,(Z,j,g)}\,. \]
To verify the well-definedness, it suffices to show the following:
\stepcounter{thm}
\begin{lem}\label{lem:jss}
Let $Y \os{\varrho}{\hookrightarrow} V \os{h}{\to} X$ be another compactification of $f$. Then we have
\[ \text{\rm tr}_{f,(Z,j,g)}=\text{\rm tr}_{f,(V,\varrho,h)}\,. \]
\end{lem}
\vspace{-8pt}
\begin{pf}
Replacing $Z$ by the closure of the image of $Y \os{(j,\varrho)}{\longrightarrow} Z \times_X V$, we may suppose that there exists a proper morphism $\pi:Z \to V$ fitting into a commutative diagram with cartesian square
\[\xymatrix{ Y \, \ar@{^{(}->}[r]^j \ar@<-1.5pt>@{=}[d] \ar@{}[rd]|{\square} & Z \ar[rd]^g \ar[d]_\pi \\
Y \, \ar@{^{(}->}[r]^\varrho & V \ar[r]^h & X\,.\hspace{-5pt} }\]
Since $\text{\rm tr}_g= \text{\rm tr}_h \circ Rh_*(\text{\rm tr}_\pi)$ by \eqref{lem:jss2:transitive} and Lemma \ref{lem:moser}\,(3), it remains to show that $\text{\rm tr}_{\varrho,(Z,j,\pi)}$ agrees with the adjunction morphism
\[ \varrho_! \; : \; \varrho_!\mathscr M_{n,Y} = \varrho_!\varrho^*\mathscr M_{n,V} \longrightarrow \mathscr M_{n,V}\,. \]
Indeed, since $\pi^{-1}(Y)=Y$, $\varrho^*(\text{\rm tr}_{\varrho,(Z,j,h)})$ is the identity morphism of $\mathscr M_{n,Y}$ (cf.\ \eqref{comp2}), which implies that $\text{\rm tr}_{\varrho,(Z,j,h)} = \varrho_!$. This completes the proof of the lemma.
\end{pf}
\medskip
Thus we obtained a well-defined assignment $f \mapsto \text{\rm tr}_f$. We show that this satisfies the conditions (i)--(iii) in Theorem \ref{thm:moser2}. The condition (ii) holds obviously by definition (cf.\ \eqref{gysin:DC}, \S\ref{sect2-8}). We next show the condition (i). Suppose that $f:Y \to X$ is \'etale. Take an open immersion $j:Y \hookrightarrow Z$ and a finite map $g:Z \to X$ with $f=g \circ j$ (cf.\ \cite{milne} 1.8). We claim that the morphism $\text{\rm tr}_{f,(Z,j,g)}$ coincides with the adjunction $f_!$, which implies (i).
Indeed, since $f$ is \'etale and $g$ is finite, $\text{\rm tr}_{f,(Z,j,g)}$ is represented by the composite map of complexes
\begin{equation}\notag
\xymatrix{
f_{!}\mathscr M^\bullet_{n,Y} = g_* j_!j^*\mathscr M^\bullet_{n,Z} \ar[r]^-{g_*(j_!)} & g_*\mathscr M^\bullet_{n,Z} \ar[r]^-{\text{\rm tr}_g^\bullet} & \mathscr M^\bullet_{n,X} \,,
}
\end{equation}
which agrees with $f_!$ by a similar argument as for Remark \ref{rem:jss} (see also \cite{milne} II.3.18).
We finally show the condition (iii), that is, for two maps $g:Z \to Y$ and $f:Y \to X$ in $\mathscr V_s$, we prove
\stepcounter{equation}
\begin{equation}\label{eq2-9-3} \text{\rm tr}_h=\text{\rm tr}_f \circ Rf_!(\text{\rm tr}_g) \quad \hbox{ with \, $h:=f \circ g$.} \end{equation}
If $f$ and $g$ are open immersions, \eqref{eq2-9-3} follows from the property (i) and the transitivity of adjunction maps for open immersions. If $f$ and $g$ are proper, \eqref{eq2-9-3} follows from \eqref{lem:jss2:transitive} and Lemma \ref{lem:moser}\,(3). Hence, if $g$ is an open immersion {\it or} $f$ is proper, then we obtain \eqref{eq2-9-3} by the previous two cases. We show the general case. Take compactifications of $f$ and $g$ as follows:
\[ \xymatrix{ & T \ar[rd]^q & & V \ar[rd]^{\pi} & \\
Z \ar[rr]_g \ar@{^{(}->}[ru]^{j} && Y \ar[rr]_f \ar@{^{(}->}[ru]^{\varrho} && X\,,\hspace{-5pt}} \]
where $j$ and $\varrho$ are open immersions and $q$ and $\pi$ are proper maps which make the triangles commutative.
Because we already know, by the previous cases, that
\[ \text{\rm tr}_h= \text{\rm tr}_{\pi} \circ R\pi_*(\text{\rm tr}_{\alpha}) \circ R(f \circ q)_!(\text{\rm tr}_j) \quad \hbox{ with \, $\alpha:=\varrho \circ q$,} \]
it remains to show the following composite morphism agrees with $\text{\rm tr}_{\alpha}$:
\[\xymatrix{
R\alpha_!\mathscr M_{n,T} \ar[rr]^-{R \varrho_!(\text{\rm tr}_q)} && R\varrho_! \mathscr M_{n,Y} \ar[r]^-{\varrho_!} & \mathscr M_{n,V}\,.
}\]
We prove this equality. Take an open immersion $\beta:T \hookrightarrow W$ with {\it dense image} and a proper map $\gamma:W \to V$ satisfying $\alpha= \gamma \circ \beta$. Then one can easily check that the square
\[\xymatrix{
T \, \ar@{^{(}->}[r]^\beta \ar@<-1.5pt>[d]_q & W \ar[d]^\gamma \\ Y \, \ar@{^{(}->}[r]^\varrho & V
}\]
is cartesian. Hence we have
\begin{align*}
\varrho^*(R\gamma_*(\beta_!)) &=\text{\rm id}_{Rq_*} \quad & \hbox{($\beta_!$ denotes $R\beta_!\beta^* \to \text{\rm id}$)} \\
\varrho^*(\text{\rm tr}_{\gamma})& =\text{\rm tr}_q \qquad & \hbox{(cf.\ \S\ref{sect2-8})}
\end{align*}
and thus $\varrho^*(\text{\rm tr}_{\alpha,(W,\beta,\gamma)})=\text{\rm tr}_q$, which implies $\varrho_! \circ R \varrho_!(\text{\rm tr}_q)=\text{\rm tr}_{\alpha,(W,\beta,\gamma)} = \text{\rm tr}_{\alpha}$. This completes the proof of Theorem \ref{thm:jss}. \qed
\smallskip
\subsection{Purity for logarithmic Hodge-Witt sheaves}\label{sect2-10}
Theorem \ref{thm:jss} implies the following purity result, whose special case was needed in \cite{js}.
\begin{cor}\label{cor:p-purity}
Let $f : X \to Y$ be a morphism of smooth varieties of pure dimension $d$ and $e$, respectively over $s=\text{\rm Spec}(k)$. Then there is a canonical Gysin isomorphism
\[ \text{\rm tr}^f : \logwitt X n d [d] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Rf^! \logwitt Y n e [e] \,. \]
Especially, for $g : X \to s$ smooth of dimension $d$ we get a canonical isomorphism
\[ \text{\rm tr}^g : \logwitt X n d [d] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Rg^!\bZ/p^n \,. \]
\end{cor}
\begin{pf}
The first claim follows from the isomorphisms
\[ \logwitt X n d [d] \; \us{\;\eqref{same}}\hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \; \mathscr M_{n,X} \; \us{\;\text{\ref{thm:jss}}}\hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \;
Rf^!\mathscr M_{n,Y} \; \us{\;\eqref{same}}\hspace{10pt}{}^\sim\hspace{-17.5pt}\longleftarrow \; Rf^!\logwitt Y n e [e] \,. \]
For the special case note that $\logwitt s n 0 = \bZ/p^n$.
\end{pf}
\begin{rem}\label{rem:p-purity}
With the notation $\bZ/p^n(r)_X := \logwitt X n r [-r]$ the purity isomorphism in Corollary {\rm\ref{cor:p-purity}} becomes
\addtocounter{equation}{2}
\begin{equation}\label{eq2-10-3}
\text{\rm tr}^f : \bZ/p^n(d)_X[2d] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Rf^!\bZ/p^n(e)_Y[2e] \,.
\end{equation}
When $f$ is a closed immersion, $\text{\rm tr}^f$ is adjoint to the modified Gysin morphism \eqref{eq2-5-5}. When $f$ is proper, $\text{\rm tr}^f$ is adjoint to Gros' Gysin morphism $\text{\rm Gys}_f$ only up to the sign $(-1)^{d-e}$, cf.\ Remark {\rm\ref{rem:smooth}}.
\end{rem}
\smallskip
\subsection{Bloch-Ogus complexes and Kato complexes}\label{sect2-11}
Finally we have the following application to Kato complexes, which is analogous to Theorem \ref{thm:Katocomplexprimetop}. Let $S$ be a smooth scheme of pure dimension $d$ over $k$. (Most interesting is the case $S=\text{\rm Spec}(k)$, $d=0$, which was needed in \cite{js}.) For a separated scheme of finite type over $S$, $f: X \to S$, define its homology with coefficients in $\bZ/p^n(-d)$ by
\begin{equation}\label{eq2-11-1}
H_a(X/S,\bZ/p^n(-d)) := H^{-a}(X, Rf^!\bZ/p^n(d)_S)\,.
\end{equation}
These groups define a homology theory on the category of separated $S$-schemes of finite type, in the sense of \cite{js} 2.1 (cf.\ loc.\ cit.\ 2.2), and in a standard way one obtains a niveau spectral sequence
\begin{equation}\label{eq2-11-2}
E^1_{q,t}(X/S,\bZ/p^n(-d)) = \bigoplus_{x\in X_q} H_{q+t}(x/S,\bZ/p^n(-d)) \Rightarrow H_{q+t}(X/S,\bZ/p^n(-d)) \hspace{-10pt} \end{equation}
for $X$ as above (cf.\ \S\ref{sect1-5} and \cite{js} 2.7).
\addtocounter{thm}{2}
\begin{thm}\label{thm:Katocomplexp-case}
Let $X$ be a separated $S$-scheme of finite type.
\begin{enumerate}
\item[(1)] There is a canonical isomorphism
\[ E^1_{q,t}(X/S,\bZ/p^n(-d)) \cong \bigoplus_{x\in X_q} \ H^{q-t-2d}(x,\bZ/p^n(q)) =
\bigoplus_{x\in X_q} \ H^{-t-2d}(x, \logwitt x n q)\,. \]
\item[(2)] Via these isomorphisms, the Bloch-Ogus complex $E^1_{*,t}(X/S,\bZ/p^n)$ coincides with the sign-modified modified Kato complex $C_{p^n}^{-t-2d,0}(X)^{(-)}$.
\item[(3)] Especially, for a separated $k$-scheme $X$ of finite type, purity induces an isomorphism $E^1_{*,t}(X/k,\bZ/p^n)\cong C_{p^n}^{-t,0}(X)^{(-)}$.
\end{enumerate}
\end{thm}
\begin{pf}
(1) follows from the purity isomorphism
\stepcounter{equation}
\begin{align}
\notag
H_a(V/S,\bZ/p^n(-d)) & = H^{-a}(V,Rf^!\bZ/p^n(d)_S) \\
\label{eq2-11-3}
& \hspace{-8.2pt}\us{\text{\eqref{eq2-10-3}}}\cong H^{-a+2q-2d}(V,\bZ/p^n(q)_V)
\end{align}
for $f: V \to S$ with $V$ smooth of pure dimension $q$.
\par
Since (3) is a special case of (2), we prove (2) in what follows, by similar arguments as in the proof of Theorem \ref{thm:Katocomplexprimetop}. The question is local in $S$ and $X$. Therefore we may assume that $f : X \to S$ factors as follows:
\[\xymatrix{ X \; \ar@{^{(}->}[r]^i & P \ar[r]^\pi & S\,, }\]
where $\pi$ is a smooth morphism of pure relative dimension $N$ and $i$ is a closed immersion. The Gysin isomorphism $\bZ/p^n(d+N)[2N] \simeq R\pi^!\bZ/p^n(d)$ from \eqref{eq2-10-3} induces an isomorphism of homology theories
\[ \gamma: H_{*-2N}(-/P,\bZ/p^n(-d-N)) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H_*(-/S,\bZ/p^n(-d)) \]
on all subschemes of $P$, and therefore an isomorphism between the corresponding spectral sequences. Moreover, for an open subscheme $j_U: U \hookrightarrow P$ and a closed subscheme $i_V: V \hookrightarrow U$ of dimension $q$, the purity isomorphism \eqref{eq2-10-3} for the composition
\[\xymatrix{ g = \pi \circ j_U \circ i_V : V \; \ar@{^{(}->}[r]^-{i_V} & U \; \ar@{^{(}->}[r]^-{j_U} & P \ar[r]^-{\pi} & S }\]
factors as
\[ \begin{CD}
\bZ/p^n(q)_V[2q] @>{\quad\text{\rm tr}^{i_V}\quad}>> Ri_V^!\bZ/p^n(d+N)_U[2(d+N)] \\
@>{Ri_V^!j_U^*(\text{\rm tr}^\pi)}>> Ri_V^!j_U^* R\pi^!\,\bZ/p^n(d)_S[2d] \\
@= Rg^!\bZ/p^n(d)_S[2d] \,.
\end{CD}\]
The first morphism here induces the modified Gysin map
\[ \gys^\circ_{i_V} : H^{m+2q}(V,\bZ/p^n(q)_V) \longrightarrow H^{m+2(d+N)}_V(U,\bZ/p^n(d+N)_U) \]
in \eqref{eq2-5-5}. Thus the compatibility facts in Remark \ref{rem:smooth} implies the claim.
\end{pf}
\newpage
\section{The case of $p$-torsion in mixed characteristic $(0,p)$}\label{sect3}
\setcounter{thm}{0}
\setcounter{equation}{0}
\medskip
Let $S$ be the spectrum of a henselian discrete valuation ring $A$ with fraction field $K$ of characteristic zero and {\it perfect} residue field $k$ of characteristic $p>0$. Consider a diagram with cartesian squares
\[\xymatrix{
X_\eta \, \ar@{^{(}->}[r]^-{j_X} \ar[d]_{f_\eta} \ar@{}[rd]|{\square} & X \ar[d]_{f\hspace{-1pt}} \ar@{}[rd]|{\square} & \ar@{_{(}->}[l]_-{i_X} \ar[d]_{f_s \hspace{-2pt}} \,X_s \\ \eta \, \ar@{^{(}->}[r]^j & S & \ar@{_{(}->}[l]_i \,s\,,\hspace{-5pt}
} \]
where $\eta$ (resp.\ $s$) is the generic point (resp.\ closed point) of $S$, and $f$ is separated of finite type. Let $n$ be a positive integer, and let $\mathcal E_\eta$ (resp.\ $\mathcal E_s$) be the \'etale sheaf $\mu_{p^n}$ on $\eta$ (resp.\ the constant \'etale sheaf $\bZ/p^n$ on $s$). We define
\begin{align*}
\mathcal E_{X_\eta} & := Rf_\eta^! \mathcal E_\eta \in D^+(X_{\eta,\text{\rm \'{e}t}},\bZ/p^n),\\
\mathcal E_{X_s} &:= Rf_s^! \mathcal E_s \in D^+(X_{s,\text{\rm \'{e}t}},\bZ/p^n).
\end{align*}
We recall some standard facts on $\mathcal E_{X_\eta}$ (compare Theorems \ref{thm:jss} and \ref{thm:shiho} for $\mathcal E_{X_s}$)
\subsubsection{}\label{rel:dual}
If $X_\eta$ is smooth over $\eta$ of pure dimension $d$, then there is a canonical isomorphism
\[ \text{\rm tr}^{f_\eta}: \mu_{p^n}^{\otimes d+1}[2d] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \mathcal E_{X_\eta} \]
in $D^+(X_{\eta,\text{\rm \'{e}t}},\bZ/p^n)$ by the relative Poincar\'e duality \cite{sga4} XVIII.3.25.
\subsubsection{}\label{rem:lam}
For points $y \in (X_\eta)_q$ and $x \in (X_\eta)_{q-1}$ with $x \in \ol { \{ y \} }\subset X_\eta$, there is a commutative diagram
\[\xymatrix{
\iota_x^*R\iota_{y*}\mu_{p^n}^{\otimes q+1}[2q] \ar[rr]^-{-\dval_{y,x}} \ar[d]_{\iota_x^*R\iota_{y*}(\tau_y)}^{\hspace{-1.5pt}\wr}
&& \mu_{p^n}^{\otimes q}[2q-1] \ar[d]_{\wr\hspace{-1.5pt}}^{\tau_x[1]} \\
\iota_x^*R\iota_{y*}R\iota_y^!\mathcal E_{X_\eta} \ar[rr]^-{\iota_x^*\{\delta^\text{\rm loc}_{y,x}(\mathcal E_{X_\eta})\}}
&& R\iota_x^!\mathcal E_{X_\eta}[1]
}\]
in $D^+(x_\text{\rm \'{e}t},\bZ/p^n)$. Here for a point $v \in (X_\eta)_m$, $\iota_v$ denotes the canonical map $v \hookrightarrow X_\eta$ and $\tau_v$ denotes the canonical isomorphism $\mu_{p^n}^{\otimes m+1}[2m] \cong R\iota_v^!\mathcal E_{X_\eta}$ obtained from \S\ref{rel:dual} for a smooth dense open subset of $\ol {\{ v \}}$. The top arrow is given by the boundary map
\[ \dval_{y,x}: R^1\iota_{y*}\mu_{p^n}^{\otimes q+1} \longrightarrow \iota_{x*}\mu_{p^n}^{\otimes q}
\quad \hbox{(cf.\ \S\ref{sect0-6})} \]
and the fact that $\iota_x^*R^u\iota_{y*}\mu_{p^n}^{\otimes q+1}=0$ for $u \geq 2$ (cf.\ \cite{sga5} I.5). One can check this commutativity in the following way. Localizing and embedding $X_\eta$ into an affine space, we may suppose that $X_\eta$ is smooth. Because $R\iota_x^!\mathcal E_{X_\eta}[1]$ (resp.\ $\iota_x^*R\iota_{y*}\mu_{p^n}^{\otimes q+1}[2q]$) is concentrated in degree $-2q+1$ (resp.\ $\leq -2q+1$), the problem is reduced, by \S\ref{sect0-5-4}\,(1), to the commutativity at the $(-2q+1)$-st cohomology sheaves, which follows from Theorem {\rm\ref{prop:appA}} and \cite{sga4.5} Cycle, 2.3.8\,(i).
\smallskip
\subsection{Condition $\bs{\mathbb K_q}$}\label{sect3-1}
The complexes $\mathcal E_{X_\eta}$ and $\mathcal E_{X_s}$ are important for the theory of duality and homology over $\eta$ and $s$, as we have seen in \S\ref{sect1} and \S\ref{sect2}. For working over $S$, we study morphisms
\[ Rj_{X*}\mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_{s}}[-1]\,, \] see \S\ref{sect3-2} and \S\ref{sect3-6} below.
In particular, we want to investigate local conditions. For a point $v \in X$, let $i_v$ be the canonical map $v \hookrightarrow X$. Let $q$ be a non-negative integer, and take points $y\in (X_\eta)_q$ and $x\in (X_s)_q$ with $x\in \overline{\{y\}} \subset X$. Put $Y:=\text{\rm Spec}(\mathscr O_{\ol { \{ y \} },x})$ and $x':=Y \times_X X_s$ and let $\pi : Y \hookrightarrow X$ be the natural map. Then we have cartesian squares
\[\xymatrix{
y \, \ar@{^{(}->}[r]^-{j_Y} \ar[d]_{\iota_y} \ar@{}[rd]|{\square}
& Y \ar[d]_{\pi\hspace{-1pt}} \ar@{}[rd]|{\square}
& \ar@{_{(}->}[l]_-{i_Y} \ar[d]_{\epsilon_{x'} \hspace{-2pt}} \,x' \\
X_\eta \, \ar@{^{(}->}[r]^{j_X}
& X
& \ar@{_{(}->}[l]_{i_X} \, X_s\,,\hspace{-5pt}
}\]
and a canonical nilpotent closed immersion $x \hookrightarrow x'$.
Now let
\[ \delta_X:Rj_{X*}\mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_s}[-1] \]
be a morphism in $D^+(X_\text{\rm \'{e}t},\bZ/p^n)$. Applying $R\pi_* R\pi^!$ to $\delta_X$, we obtain a morphism
\begin{equation}\label{mor1}
R\pi_{*}R\pi^!(\delta_X):Ri_{y*} R\iota_y^!\mathcal E_{X_\eta} \longrightarrow Ri_{x*} R\epsilon_x^! \mathcal E_{X_s}[-1]\,,
\end{equation}
where $\epsilon_x$ denotes the canonical map $x \hookrightarrow X_s$, and we have used base-change isomorphisms
\[ R\pi^!Rj_{X*}=Rj_{Y*}R\iota_y^!\quad \hbox{ and } \quad R\pi^!Ri_{X*}=Ri_{Y*}R\epsilon_{x'}^!\,, \]
and the isomorphism
\[ R\epsilon_{x'*}R\epsilon_{x'}^!=R\epsilon_{x*}R\epsilon_{x}^! \]
by the invariance of \'etale topology. Furthermore, we have $R\iota_y^!\mathcal E_{X_\eta} \cong \mu_{p^n}^{\otimes q+1}[2q]$ by \S\ref{rel:dual}, and we have $R\iota_x^! \mathcal E_{X_s} \cong \logwitt x n q [q]$ by Theorem \ref{thm:jss}. Therefore the morphism \eqref{mor1} is identified with a morphism $Ri_{y*} \mu_{p^n}^{\otimes q+1}[2q] \to Ri_{x*} \logwitt x n q [q-1]$, which induces a map of cohomology sheaves in degree $-q+1$:
\[ \delta_X(y,x): R^{q+1}i_{y*} \mu_{p^n}^{\otimes q+1} \longrightarrow i_{x*} \logwitt x n q \,. \]
We are going to compare this map of sheaves on $X_\text{\rm \'{e}t}$ with Kato's residue map (cf.\ \eqref{eq-0.1.1}):
\[ \dval_{y,x} : R^{q+1}i_{y*} \mu_{p^n}^{\otimes q+1} \longrightarrow i_{x*} \logwitt x n q \,. \]
\stepcounter{thm}
\begin{defn}\label{def:K_q}
We say that $\delta_X$ satisfies $\mathbb K_q$ if the induced map $\delta_X(y,x)$ agrees with $\dval_{y,x}$ for all points $y\in (X_\eta)_q$ and $x\in (X_s)_q$ with $x\in \overline{\{y\}}$.
\end{defn}
\begin{rem}
In view of \S{\rm\ref{sect0-5-4}\,(1)}, the morphism \eqref{mor1} is determined by $\delta_X(y,x)$. In fact, we have $R^mi_{y*} \mu_{p^n}^{\otimes q+1}=0$ for any $m>q+1$ by a similar argument as for Lemma {\rm\ref{lem1}} below.
\end{rem}
\smallskip
\subsection{Functoriality of Kato's residue maps}\label{sect3-2}
Let
\[ \delta_S^\text{\rm val}:Rj_*\mathcal E_\eta \longrightarrow Ri_*\mathcal E_s[-1]\]
be the composite morphism
\[ Rj_*\mathcal E_\eta \cong \tau_{\leq 1} Rj_*\mathcal E_\eta \longrightarrow R^1j_* \mathcal E_\eta [-1] \longrightarrow i_*\mathcal E_s[-1] \]
in $D^b(S_\text{\rm \'{e}t},\bZ/p^n)$, where the first isomorphism follows from a theorem of Lang: $\text{\rm cd}({\eta})=1$ (cf.\ Lemma \ref{lem1} below) and the last morphism is induced by Kummer theory and the normalized valuation $v_A$ on $K^{\times}$, i.e., Kato's residue map (cf.\ \S\ref{sect0-1}). By the base-change isomorphisms
\[ Rf^! Rj_* \mathcal E_\eta = Rj_{X*}\mathcal E_{X_\eta} \quad \hbox{ and } \quad Rf^! Ri_* \mathcal E_s = Ri_{X*}\mathcal E_{X_s}\,, \]
we obtain a morphism
\[ \delta_X^{S\text{-}\val}:=Rf^!(\delta_S^\text{\rm val}):Rj_{X*} \mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_s}[-1] \quad\hbox{ in }\; D^b(X_\text{\rm \'{e}t},\bZ/p^n).\]
The first main result of this section is the following theorem:
\begin{thm}\label{Th.1-1}
\begin{enumerate}
\item[(1)]
The morphism $\delta_X^{S\text{-}\val}$ satisfies $\mathbb K_q$ for all $q\ge 0$.
\item[(2)]
$\delta_X^{S\text{-}\val}$ is the only morphism that satisfies $\mathbb K_q$ for all $q\ge 0$.
\item[(3)]
If $X_\eta$ is smooth of pure dimension $d$, $\delta_X^{S\text{-}\val}$ is the only morphism satisfying $\mathbb K_d$.
\end{enumerate}
\end{thm}
\noindent
The proof of this result will be finished in \S\ref{sect3-8} below.
\smallskip
\subsection{First reductions}\label{sect3-3}
We first note that, to prove Theorem \ref{Th.1-1}, we may assume that $X$ is reduced and the closure of $X_\eta$. In fact, let $X' \subset X$ be the closure of $X_\eta$ with the reduced subscheme structure. Then we get cartesian squares
\[\xymatrix{
(X_\eta)_{\text{red}} \, \ar@{^{(}->}[r]^-{j_{X'}} \ar[d]_{\kappa_\eta\hspace{-1pt}} \ar@{}[rd]|{\square}
& X' \ar[d]_{\kappa\hspace{-1pt}} \ar@{}[rd]|{\square}
& \ar@{_{(}->}[l]_-{i_{X'}} \ar[d]_{\kappa_s\hspace{-1pt}} \, X'_s \\
X_\eta \, \ar@{^{(}->}[r]^-{j_X}
& X
& \ar@{_{(}->}[l]_-{i_X} \, X_s\,,\hspace{-5pt}
}\]
where $\kappa$ is the closed immersion. They induce a commutative diagram
\[\xymatrix{
\kappa_* Rj_{X'*}\mathcal E_{X'_\eta} \ar[rr]^-{\kappa_*R\kappa^!(\delta_X)} \ar[d]_{\kappa_{\eta*}}^{\hspace{-1.5pt}\wr}
&& \kappa_* i_{X'*}\mathcal E_{X_s'} \ar[d]^{\kappa_{s*}} \\
Rj_{X*}\mathcal E_{X_\eta} \ar[rr]^-{\delta_X}
&& i_{X*}\mathcal E_{X_s}\,,\hspace{-5pt}
}\]
for any given morphism $\delta_X$ at the bottom. The left adjunction map is an isomorphism by topological invariance of \'etale cohomology. Moreover, $R\kappa^!(\delta_X^{S\text{-}\val}) = \delta_{X'}^{S\text{-}\val}$, and evidently $\delta_X$ satisfies $\mathbb K_q$ if and only if $R\kappa^!(\delta_X)$ does. This shows that the claims of Theorem \ref{Th.1-1} hold for $X$ if and only if they hold for $X'$. We also note the following reduction:
\begin{lem}\label{Cl.1-3}
A morphism $\delta_X$ satisfies $\mathbb K_q$ if and only if for all integral closed subschemes $\iota_Z : Z \hookrightarrow X$ of dimension $q$ the morphism $R\iota_Z^!(\delta_X)$ satisfies $\mathbb K_q$. In particular, Theorem {\rm\ref{Th.1-1}\,(1)} holds for $X$ if and only if $\delta_Z^{S\text{-}\val}$ satisfies $\mathbb K_q$ for all integral subschemes $Z\subset X$ of dimension $d$.
\end{lem}
\begin{pf}
Let $X$ be arbitrary. Take a point $y \in (X_\eta)_q$ ($0 \le q \le d:=\dim(X_\eta)$), let $Z$ be its closure in $X$, and take an $x \in (X_s)_q$ with $x \in Z$. Let $\iota_Z: Z\hookrightarrow X$ be the natural inclusion. We have base-change isomorphisms
\[ R\iota_Z^!Rj_{X*}\mathcal E_{X_\eta}=Rj_{Z*}\mathcal E_{Z_\eta}\quad \hbox{ and } \quad
R\iota_Z^!Ri_{X*}\mathcal E_{X_s}=Ri_{Z*}\mathcal E_{Z_s}, \]
and it follows from the definitions in \S\ref{sect3-1} that $\delta_X(y,x) = (R\iota_Z^!(\delta_X))(y,x)$, if we regard these as maps $\iota_x^* R^{q+1}\iota_{y*} \mu^{\otimes{q+1}}_{p^n} \to \logwitt x n q$. This shows the first claim. The second claim follows, because $R\iota_Z^!(\delta_X^{S\text{-}\val}) = \delta_Z^{S\text{-}\val}$.
\end{pf}
Finally we note:
\begin{rem}\label{rem1}
To prove that $\delta_X^{S\text{-}\val}$ satisfies $\mathbb K_q$ it suffices to assume that $f:X \to S$ is proper by taking a compactification of $f$.
\end{rem}
\smallskip
\subsection{Criterion in the proper case}\label{sect3-4}
Suppose that we are given two morphisms
\begin{align*}
\delta_S & : Rj_*\mathcal E_\eta \longrightarrow Ri_*\mathcal E_s[-1] \quad \hbox{ in } \; D^+(S_\text{\rm \'{e}t},\bZ/p^n), \\
\delta_X & : Rj_{X*}\mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_s}[-1] \quad \hbox{ in } \; D^+(X_\text{\rm \'{e}t},\bZ/p^n).
\end{align*}
Assuming that $f$ is proper, we give a simple criterion as to when $\delta_X$ agrees with $Rf^!(\delta_S)$, that is, as to when the following diagram commutes in $D^+(X_\text{\rm \'{e}t},\bZ/p^n)$:
\begin{equation}\label{dg:proper}
\xymatrix{
Rj_{X*}\mathcal E_{X_\eta} \ar[rr]^-{\delta_X} \ar@{=}[d] && Ri_{X*}\mathcal E_{X_s}[-1] \ar@{=}[d] \\
Rf^!Rj_*\mathcal E_\eta \ar[rr]^-{Rf^!(\delta_S)} && Rf^!Ri_*\mathcal E_s[-1]\,,\hspace{-5pt}
}
\end{equation}
where the equalities mean the identifications by base-change isomorphisms.
\stepcounter{thm}
\begin{prop}\label{Lem.1-1}
Suppose that $f$ is proper. Then the diagram \eqref{dg:proper} commutes if and only if the following diagram is commutative$:$
\stepcounter{equation}
\begin{equation}\label{dg:proper2}
\xymatrix{
H^1(X_{\wt{\eta}},\mathcal E_{X_\eta}) \ar[rr]^-{\delta_X} \ar[d]_{f_*} && H^0(X_{\ol s},\mathcal E_{X_s}) \ar[d]^{f_*} \\
H^1(\wt{\eta},\mathcal E_\eta) \ar[rr]^-{\delta_S} && H^0(\ol s,\mathcal E_s)\,,\hspace{-5pt}
}
\end{equation}
where $\wt{\eta}$ denotes the generic point of the maximal unramified extension $\wt{S}$ of $S$ {\rm(}$\ol s$ is the closed point of $\wt{S}${\rm);} the vertical maps are defined by the adjunction map $Rf_!Rf^! \to \text{\rm id}$ and the properness of $f$, that is, $Rf_!=Rf_*$.
\end{prop}
\begin{pf}
By the adjointness between $Rf^!$ and $Rf_!$, we have the adjunction maps $f^! : \text{\rm id} \to Rf^!Rf_!$ and $f_! : Rf_!Rf^! \to \text{\rm id}$,
which satisfy the relation that the composite
\[ Rf^! \os{f^!}\longrightarrow Rf^!Rf_!Rf^! \os{f_!}\longrightarrow Rf^! \]
is the identity map. By these facts, it is easy to see that the commutativity of \eqref{dg:proper} is equivalent to that of the following diagram in $D^+(S_\text{\rm \'{e}t},\bZ/p^n)$:
\begin{equation}\label{dg:proper3}
\xymatrix{
Rf_!Rj_{X*}\mathcal E_{X_\eta} \ar[rr]^-{Rf_!(\delta_X)} \ar[d]_\alpha && Rf_!Ri_{X*}\mathcal E_{X_s}[-1] \ar[d]^\beta \\
Rj_*\mathcal E_\eta \ar[rr]^-{\delta_S} && Ri_*\mathcal E_s[-1]\,,\hspace{-5pt}
}
\end{equation}
where $\alpha$ is defined as the composite
\[\xymatrix{
\alpha : Rf_!Rj_{X*}\mathcal E_{X_\eta} \ar@{=}[rr]^-{\text{base-change}} && Rf_!Rf^!Rj_*\mathcal E_\eta \ar[r]^-{f_!} & Rj_*\mathcal E_\eta
}\]
and $\beta$ is defined in a similar way (note that we do not need the properness of $f$ for this equivalence). We prove that the commutativity of \eqref{dg:proper3} is equivalent to that of \eqref{dg:proper2}.
For this, we first show the following:
\par\medskip\noindent
{\it Claim. $i^*Rf_!Rj_{X*}\mathcal E_{X_\eta}=i^*Rf_*Rj_{X*}\mathcal E_{X_\eta}$ is concentrated in degrees $\le 1$.}
\par\medskip\noindent
{\it Proof of Claim}.
Because the stalk at $\ol s$ of the $m$-th cohomology sheaf is
\[ \mathscr H^m(i^*Rf_*Rj_{X*} \mathcal E_{X_\eta})_{\ol s} \cong H^m(X_{\wt{\eta}},\mathcal E_{X_\eta}) \]
by the properness of $f$, it suffices to show that the group on the right hand side is zero for $m>1$. Take an open subset $U_\eta\subset X_\eta$ which is smooth over $\eta$ of pure dimension $d:=\dim(X_\eta)$ and such that $\dim(Z_\eta)<d$, where $Z_\eta$ denotes the closed complement $X_\eta \smallsetminus U_\eta$. By \eqref{eq-0.4.2}, there is a localization exact sequence
\begin{equation}\label{exact:local}
\dotsb \longrightarrow H^m(Z_{\wt{\eta}},\mathcal E_{Z_\eta}) \longrightarrow H^m(X_{\wt{\eta}},\mathcal E_{X_\eta}) \longrightarrow H^m(U_{\wt{\eta}},\mathcal E_{U_\eta}) \longrightarrow \dotsb.
\end{equation}
Now we have $\mathcal E_{U_\eta} \cong \mu_{p^n}^{\otimes d+1} [2d]$ by \S\ref{rel:dual} for $U_\eta$, so that
\[ H^m(U_{\wt{\eta}},\mathcal E_{U_\eta}) \cong H^{m+2d}(U_{\wt{\eta}},\mu_{p^n}^{\otimes d+1})\,, \]
which vanishes for $m>1$ because $\text{\rm cd}(U_{\wt{\eta}}) \leq 2d+1$, cf.\ Lemma \ref{lem1} below. Thus the vanishing of $H^m(X_{\wt{\eta}},\mathcal E_{X_\eta})$ for $m>1$ is shown by induction on $\dim(X_\eta)$ and we obtain the claim.
\qed
\par\medskip \noindent
We turn to the proof of Proposition \ref{Lem.1-1}. By the above claim and \S\ref{sect0-5-4}\,(1), a morphism $i^*Rf_!Rj_{X*}\mathcal E_{X_\eta} \to \mathcal E_s[-1]$ is determined by the map of the $1$st cohomology sheaves, and thus determined by the associated map of their stalks at $\ol s$. Hence by the adjointness between $Ri_*$ and $i^*$, the diagram \eqref{dg:proper3} commutes if and only if the diagram \eqref{dg:proper2} does. This completes the proof of Proposition \ref{Lem.1-1}.
\end{pf}
\smallskip
\subsection{Result for smooth generic fiber}\label{sect3-5}
In Proposition \ref{Cl.1-2} below we obtain a first step towards part (3) of Theorem \ref{Th.1-1} which will also be used for the other parts. We first show:
\begin{lem}\label{lem1}
Let $\mathscr F$ be a torsion sheaf on $(X_\eta)_\text{\rm \'{e}t}$. Then $R^mj_{X*}\mathscr F=0$ for any $m > \dim(X_\eta)+1$.
\end{lem}
\begin{pf}
Clearly $R^mj_{X*}\mathscr F$ is trivial on $X_\eta$ if $m>0$. Hence the problem is \'etale local on $X_s$ and we may suppose that $s = \ol s$. Let $x$ be a point on $X_s$. The stalk of $R^mj_{X*}\mathscr F$ at $\ol x$ is isomorphic to $H^m(\text{\rm Spec}(\mathscr O_{X,\ol x}^{\text{\rm sh}}[p^{-1}]),\mathscr F)$, where $\text{\rm Spec}(\mathscr O_{X,\ol x}^{\text{\rm sh}}[p^{-1}])$ is written as a projective limit of affine varieties over $\eta$ of dimension $\leq \dim(X_\eta)$.
Hence the assertion follows from the affine Lefschetz theorem (\cite{sga4} XIV.3.2) and Lang's theorem: $\text{\rm cd}(\eta)=1$ (\cite{se} II.3.3).
\end{pf}
\begin{prop}\label{Cl.1-2}
If $X_\eta$ is smooth of pure dimension $d$, then there exists a unique morphism $\delta_X:Rj_{X*}\mathcal E_{X_\eta} \to Ri_{X*}\mathcal E_{X_s}[-1]$ satisfying $\mathbb K_d$.
\end{prop}
\begin{pf}
By \S\ref{sect3-3} we may assume that $\dim(X_s)\leq d$. We have $\mathcal E_{X_\eta} \cong \mu_{p^n}^{\otimes d+1}[2d]$ by \S\ref{rel:dual}, and $Rj_{X*} \mathcal E_{X_\eta}$ is concentrated in $[-2d,-d+1]$ by Lemma \ref{lem1}. On the other hand, $Ri_{X*}\mathcal E_{X_s}[-1]$ is concentrated in degree $[-d+1,1]$ by Theorem \ref{thm:jss} and the assumption $\dim(X_s) \leq d$. Hence a morphism $\delta_X:Rj_{X*}\mathcal E_{X_\eta} \to Ri_{X*}\mathcal E_{X_s}[-1]$ is determined by the map $\mathscr H^{-d+1}(\delta_X)$ of the $(-d+1)$-st cohomology sheaves by \S\ref{sect0-5-4}\,(1). Moreover, for a given $\delta_X$, there is a commutative diagram of sheaves on $X_\text{\rm \'{e}t}$:
\vspace{-10pt}
\[\xymatrix{
\mathscr H^{-d+1}(Rj_{X*}\mathcal E_{X_\eta}) \ar@{=}[r] \ar[d]_{\mathscr H^{-d+1}(\delta_X)}
& R^{d+1}j_{X*} \mu_{p^n}^{\otimes d+1} \ar[r]^-{\alpha}
& \displaystyle \bigoplus_{y \in (X_\eta)_d}^{\phantom{|^a}} \ R^{d+1}i_{y*} \mu_{p^n}^{\otimes d+1} \ar[d]_-{\gamma}\\
\us{\phantom{|^{|^|}}} \mathscr H^{-d+1}(Ri_{X*}\mathcal E_{X_s}[-1]) \ar@<5pt>@{=}[r]
& \us{\phantom{|^a}}{i_{X*}\mathscr H^{-d}(\mathcal E_{X_s})} \; \ar@<5pt>@{^{(}->}[r]^-{\beta}
& \displaystyle \bigoplus_{x \in (X_s)_d} \ i_{x*}\logwitt x n d \,,\hspace{-5pt}
}\]
where $\alpha$ is the adjunction map, $\beta$ is an inclusion obtained from Theorem \ref{thm:jss} and $\gamma$ is the sum of $\delta_X(y,x)$'s. These facts show the uniqueness of $\delta_X$ satisfying $\mathbb K_d$. Next we prove its existence. For this, let us consider the following diagram of sheaves:
\vspace{-10pt}
\[\xymatrix{
& R^{d+1}j_{X*} \mu_{p^n}^{\otimes d+1} \ar[r]^-{\alpha} \ar@{.>}[d]_{\varrho}
& \displaystyle
\bigoplus_{y \in (X_\eta)_d}^{\phantom{|^a}} \ R^{d+1}i_{y*} \mu_{p^n}^{\otimes d+1} \ar[d]_-{\partial_2} \ar[r]^-{\partial_1}
& \displaystyle \bigoplus_{w \in (X_\eta)_{d-1}}^{\phantom{|^a}} \ R^{d}i_{w*} \mu_{p^n}^{\otimes d} \ar[d]_-{\partial_3} \\
\us{\phantom{|^a}}0 \ar@<5pt>[r]
& \us{\phantom{|^a}}{i_{X*}\mathscr H^{-d}(\mathcal E_{X_s})} \; \ar@<5pt>[r]^-{\beta}
& \displaystyle \bigoplus_{x \in (X_s)_d} \ i_{x*}\logwitt x n d \ar@<5pt>[r]^-{\partial_4}
& \displaystyle \bigoplus_{z \in (X_s)_{d-1}} \ i_{z*}\logwitt z n {d-1} \,,\hspace{-5pt}
}\]
where $\alpha$ and $\beta$ are the same maps as above, and each $\partial_i$ ($i=1,\dotsc,4$) is the sum of Kato's residue maps. We have the following facts for this diagram: the right square is anti-commutative by \cite{kk:hasse} 1.7 for $X$; the upper row is a complex by \S\ref{rem:lam}; the lower row is exact by Theorem \ref{thm:jss}. Hence $\partial_2$ induces a map $\varrho$ as in the diagram, and we obtain a morphism $\delta_X$ satisfying $\mathbb K_d$ by extending this map (cf.\ \S\ref{sect0-5-4}\,(1)). This completes the proof.
\end{pf}
\smallskip
\subsection{Case of points}\label{sect3-6}
We will prove Theorem \ref{Th.1-1}\,(1) by induction on $\dim(X_\eta)$. We start with:
\begin{lem}\label{Cl.1-1}
Theorem {\rm\ref{Th.1-1}} is true for $X$ with $\dim(X_\eta)=0$.
\end{lem}
\begin{pf}
First we show \ref{Th.1-1}\,(1). By Lemma \ref{Cl.1-3} and Remark \ref{rem1} we may assume that $X$ is integral and proper. Then $f:X \to S$ is flat and finite by Zariski's main theorem, and moreover, $X_s$ is irreducible because $S$ is henselian and $X$ is irreducible. Let $j':\eta' \hookrightarrow X$ (resp.\ $i':s' \hookrightarrow X$) be the generic (resp.\ closed) point. Then $\eta'=X_\eta$ and $\eta' \to \eta$ is finite \'etale, because $X$ is integral and $\text{\rm ch}(K)=0$. On the other hand, $s'\to s$ is finite \'etale as well by the perfectness of $k$, and this map factors as the composite of a nilpotent closed immersion $s' \hookrightarrow X_s$ with $f_s:X_s \to s$. Therefore we have $\mathcal E_{X_\eta}=\mu_{p^n}$ and $\mathcal E_{X_s}=\bZ/p^n$. Now let
\[ \delta_X:Rj_{X*}\mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_s}[-1] \]
be the composite morphism $Rj_{X*}\mu_{p^n} \to R^1j_{X*}\mu_{p^n}[-1] \to i_{X*}\bZ/p^n[-1]$, where the last morphism is given by the map $\partial^\val_{\eta',s'}$. Because $\delta_X$ satisfies ${\mathbb K}_0$ by definition, our task is to show the equality $\delta_X=\delta_X^{S\text{-}\val} (:=Rf^!(\delta_S^\text{\rm val}))$. Moreover, by the finiteness of $f$ and Proposition \ref{Lem.1-1}, we have only to show the commutativity of the diagram
\begin{equation}\label{com1}
\xymatrix{
H^1(\eta',\mu_{p^n}) \ar[rd]^-{\partial^\val_{\eta',s}} \ar[d]_{\text{\rm tr}_f} \\
H^1(\eta,\mu_{p^n}) \ar[r]_-{\partial^\val_{\eta,s}} & H^0(s,\bZ/p^n)\,,\hspace{-5pt}
}\end{equation}
assuming that $s=s'=\ol s$ (that is, $k$ is algebraically closed).
We show this commutativity. Let $B_0$ be the affine ring of $X$, let $B$ be the normalization of $B_0$, let $L$ be the fraction field of $B$ and let $x$ be the closed point of $\text{\rm Spec}(B)$. By definition, $\partial^\val_{\eta',s}$ is the composite
\[ H^1(\eta',\mu_{p^n}) \longrightarrow H^0(x,\bZ/p^n) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra H^0(s,\bZ/p^n)\,. \]
where the first map is given by the normalized valuation $v_B$ on $L^{\times}$ and the second map is induced by the isomorphism $x \cong s$. On the other hand, there is a commutative diagram
\[ \xymatrix{
L^{\times}/p^n \ar[r]^-{\cong} \ar[d]_{N_{L/K}} &H^1(\eta',\mu_{p^n}) \ar[d]^{\text{\rm tr}_f} \\
K^{\times}/p^n \ar[r]^-{\cong} & H^1(\eta,\mu_{p^n})\,,\hspace{-5pt}
}\]
where $N_{L/K}$ denotes the norm map (cf.\ \cite{sga4} XVIII.2.9 (Var 4)), and the horizontal arrows are boundary maps coming from the Kummer theory for $\eta'$ and $\eta$, respectively. Therefore the commutativity of \eqref{com1} follows from the fact that $v_B = v_A \circ N_{L/K}$. Now we prove the other parts of Theorem \ref{Th.1-1} for $X$. By \S\ref{sect3-3} we may assume that $X$ is reduced. Then, since $\dim(X_\eta)=0$, $X_\eta$ is smooth, and Proposition \ref{Cl.1-2} implies that $\delta_X^{S\text{-}\val}$ is the only morphism satisfying $\mathbb K_0$.
\end{pf}
\smallskip
\subsection{Induction step}\label{sect3-7}
Consider the following situation.
Suppose that $X$ is reduced, separated of finite type over $S$, that $X_\eta$ has dimension $d \ge 1$, and that $X_\eta$ is dense in $X$. Choose a {\it smooth affine} dense open subset $U_\eta \subset X_\eta$. Let $Z_\eta:=X_\eta \smallsetminus U_\eta$ with the reduced structure, let $Z$ be the closure of $Z_\eta$ in $X$, and let $U = X \smallsetminus Z$. Then the composite morphism $f_Z: Z \to X \to S$ is flat, and hence we have
\begin{equation}\label{cond1}
(U_s)_d=(X_s)_d\,.
\end{equation}
We name the canonical immersions as follows:
\[ \xymatrix{
U_\eta \, \ar@{^{(}->}[r]^-{j_U} \ar@{_{(}->}[d]_{\phi_\eta} \ar@{}[rd]|{\square}
& \, U_{\phantom.} \ar@{_{(}->}[d]_\phi \ar@{}[rd]|{\square}
& \ar@{_{(}->}[l]_-{i_U} \ar@{_{(}->}[d]_{\phi_s} \, U_s \\ X_\eta \, \ar@{^{(}->}[r]^-{j_X}
& X
& \ar@{_{(}->}[l]_{i_X} \, X_s \\
Z_\eta \phantom{|}\hspace{-2pt} \ar@{^{(}->}[r]^-{j_Z} \ar@{^{(}->}[u]^{\psi_\eta} \ar@{}[ru]|{\square}
& Z \phantom{|}\hspace{-2pt} \ar@{^{(}->}[u]^\psi \ar@{}[ru]|{\square}
& \ar@{_{(}->}[l]_-{i_Z} \ar@{^{(}->}[u]^{\psi_s} \, Z_s\,. \phantom{|}\hspace{-5pt}
} \]
Consider a diagram of the following type in $D^+(X_\text{\rm \'{e}t},\bZ/p^n)$:
\medskip
\begin{equation}\label{com2}
\xymatrix{
R\psi_*Rj_{Z*} \mathcal E_{Z_\eta} \ar[r]^-{\psi_*} \ar[d]_{\delta_Z^S} \ar@{}[rd]|{(1)}
& Rj_{X*} \mathcal E_{X_\eta} \ar[r]^-{\phi^*} \ar@{.>}[d]_{\delta_1} \ar@{}[rd]|{(2)}
& R\phi_* Rj_{U*} \mathcal E_{U_\eta} \ar[r]^-{-\epsilon_1} \ar@{.>}[d]_{\delta_2} \ar@{}[rd]|{(3)}
& R\psi_* Rj_{Z*} \mathcal E_{Z_\eta}[1] \ar[d]_{\delta_Z^S[1]} \\
R\psi_* Ri_{Z*} \mathcal E_{Z_s}[-1] \ar[r]^-{\psi_*} & Ri_{X*} \mathcal E_{X_s}[-1] \ar[r]^-{\phi^*}
& R\phi_* Ri_{U*} \mathcal E_{U_s}[-1] \ar[r]^-{\epsilon_2[-1]} & R\psi_* Ri_{Z*} \mathcal E_{Z_s}\,.\hspace{-5pt}
}
\end{equation}
Here we put
\[ \delta_Z^S := R\psi_*(\delta_Z^{S\text{-}\val})=R\psi_* Rf_Z^!(\delta_S^\text{\rm val})\,, \quad
\epsilon_1 :=\delta_{U,Z}^{\text{\rm loc}}(Rj_{X*} \mathcal E_{X_\eta})\,, \quad \epsilon_2 :=\delta_{U,Z}^{\text{\rm loc}}(Ri_{X*} \mathcal E_{X_s})\,, \]
the horizontal rows are the distinguished triangles deduced from the obvious localization triangles (cf.\ \eqref{eq-0.4.2}) and the base-change isomorphisms
\begin{align*}
R\psi^! Rj_{X*} \mathcal E_{X_\eta} = Rj_{Z*} \mathcal E_{Z_\eta}\,,\qquad & \phi^* Rj_{X*} \mathcal E_{X_\eta} = Rj_{U*} \mathcal E_{U_\eta}\,,\\
R\psi^! Ri_{X*} \mathcal E_{X_s} = Ri_{Z*} \mathcal E_{Z_s}\,,\qquad & \phi^* Ri_{X*} \mathcal E_{X_s} = Ri_{U*} \mathcal E_{U_s}\,.
\end{align*}
\addtocounter{thm}{2}
\begin{lem}\label{lem:delta1}
If $\delta_2$ is given, there is at most one morphism $\delta_1$ making the squares {\rm(1)} and {\rm(2)} in \eqref{com2} commutative.
\end{lem}
\begin{pf}
We want to apply Lemma \ref{lem.derived}\,(3). Because $U_\eta$ is smooth and affine, we have $\mathcal E_{U_\eta} \cong \mu_{p^n}^{\otimes d+1}[2d]$ by \S\ref{rel:dual}, and $C = R (\phi j_U)_* \mathcal E_{U_\eta}$ is concentrated in $[-2d,-d+1]$ by a similar argument as for Lemma \ref{lem1}. On the other hand, because $\dim(X_s) \leq d$, $Ri_{X*}\mathcal E_{X_s}$ is concentrated in $[-d,0]$ by Theorem \ref{thm:jss} (note that $i_X$ is a closed immersion). Similarly, $A' = R\psi_* Ri_{Z*}\mathcal E_{Z_s}$ is concentrated in $[-d+1,0]$, because we have $\dim(Z_s) \leq d-1$ by the flatness of $f_Z:Z \to S$. Therefore we get
\[ \text{\rm Hom}_{D(X,\bZ/p^n)}(C,A') = 0. \]
On the other hand, for $A = R\psi_* Rj_{Z*} \mathcal E_{Z_\eta}$ and $C' = R\phi_* Ri_{U*} \mathcal E_{U_s}[-1]$ we have
{\allowdisplaybreaks
\begin{align*}
\text{\rm Hom}^{-1}_{D(X,\bZ/p^n)}(A,C') & = \text{\rm Hom}_{D(X,\bZ/p^n)}(R\psi_* Rj_{Z*} \mathcal E_{Z_\eta},R\phi_* Ri_{U*} \mathcal E_{U_s}[-2]) \\
& = \text{\rm Hom}_{D(X,\bZ/p^n)}(\phi^* R\psi_* Rj_{Z*} \mathcal E_{Z_\eta}, Ri_{U*} \mathcal E_{U_s}[-2]) & \hbox{(adjunction)} \\
& = 0 &\hbox{($\phi^* R\psi_* = 0$)}
\end{align*}
}So the lemma follows from Lemma \ref{lem.derived}\,(3).
\end{pf}
\begin{lem}\label{Cl.1-5}
Consider the diagram \eqref{com2} and assume that $\delta_2 = R\phi_*(\delta_U)$ where $\delta_U: Rj_{U*} \mathcal E_{U_\eta} \to Ri_{U*} \mathcal E_{U_s}$ denotes the morphism obtained by applying Proposition {\rm\ref{Cl.1-2}} to $U$. Assume that $\mathbb K_{d-1}$ holds for $\delta_Z^{S\text{-}\val}$.
\begin{enumerate}
\item[(i)] If $X$ is integral, then the square {\rm(3)} in $\eqref{com2}$ commutes. Consequently, there exists a morphism $\delta_1$ which makes the squares {\rm(1)} and {\rm(2)} in \eqref{com2} commutative at the same time.
\item[(ii)] If $f: X \to S$ is proper, then any morphism $\delta_1$ making the square {\rm(1)} in \eqref{com2} commutative necessarily coincides with $\delta_X^{S\text{-}\val}$.
\end{enumerate}
\end{lem}
\begin{pf*}{\it Proof of Lemma \ref{Cl.1-5}}
(i) As we have seen in the proof of lemma \ref{lem:delta1}, $R(\phi j_U)_* \mathcal E_{U_\eta}$ is concentrated in $[-2d,-d+1]$ and $R\psi_* Ri_{Z*}\mathcal E_{Z_s}$ is concentrated in $[-d+1,0]$.
By these facts, the square (3) commutes if and only if the square of the induced homomorphisms on the $(-d+1)$-st cohomology sheaves commutes. We prove this commutativity on cohomology sheaves. By Theorem \ref{thm:jss}, we have
\[ \mathscr H^{-d+1}(R\psi_* Ri_{Z*}\mathcal E_{Z_s})= \psi_* i_{Z*} \mathscr H^{-d+1} (\mathcal E_{Z_s}) \hookrightarrow
\bigoplus_{x\in (Z_s)_{d-1}} \ i_{x*}\logwitt x n {d-1} \,. \]
Hence we may suppose that $(Z_s)_{d-1}$ is not empty, and the problem is local at each point in $(Z_s)_{d-1}$. Now fix a point $x \in (Z_s)_{d-1}$, and define $B$ (resp.\ $C$, $D$) as $\text{\rm Spec}(\mathscr O_{X,x})$ (resp.\ $U \times _X B$, $Z \times _X B$), and let $\sigma$ be the open immersion $C_\eta \hookrightarrow B$. Note that $B$ is integral local of dimension two and that $D_\eta$ and $E:=(C_s)_{\text{\rm red}}$ are finite sets of points in $B^1 \subset X^1$. Our task is to show the commutativity of the following diagram on $B_\text{\rm \'{e}t}$:
\vspace{-10pt}
\addtocounter{equation}{2}
\begin{equation}\label{com3}
\xymatrix{
R^{d+1}\sigma_*\mu_{p^n}^{\otimes d+1} \ar[r]^-{\delta_3} \ar[d]_{\delta_4}
& \displaystyle \bigoplus_{z \in D_\eta}^{\phantom{|}} \ R^d i_{z*}\mu_{p^n}^{\otimes d} \ar[d]^-{\delta_5} \\
\displaystyle \bigoplus_{y \in E} \ i_{y*}\logwitt y n d \ar@<6pt>[r]^-{\delta_6}
& \us{\phantom{l}}{i_{x*}\logwitt x n {d-1}}\,,\hspace{-5pt}
}
\end{equation}
where for a point $v \in B$, we wrote $i_v$ for the map $v \to B$ and we have used the isomorphisms $\mathcal E_{Z_\eta}\vert_{D_\eta} \cong \mu_{p^n}^{\otimes d}[2(d-1)]$ (cf.\ \S\ref{rel:dual}) and $\mathcal E_{U_s}\vert_{C_s} \cong \logwitt E n d$ (cf.\ Theorem \ref{thm:jss}). Each $\delta_i$ $(i=3,\dotsc,6)$ denotes the map obtained by restricting the corresponding morphism in the square (3) of \eqref{com2}. Now let $w$ be the generic point of $B$ and let $\alpha$ be the adjunction map $R^{d+1}\sigma_* \mu_{p^n}^{\otimes d+1} \to R^{d+1}i_{w*} \mu_{p^n}^{\otimes d+1}$. We have the following facts for the maps in \eqref{com3}.
\begin{itemize}
\item $\delta_3$ factors, by \S\ref{rem:lam}, as
\[
\delta_3: R^{d+1}\sigma_* \mu_{p^n}^{\otimes d+1} \os{\alpha}
\longrightarrow R^{d+1}i_{w*} \mu_{p^n}^{\otimes d+1}
\os{\bigoplus_z \, \dval_{w,z}}{-\hspace{-5pt}-\hspace{-5pt}-\hspace{-6pt}\lra} \bigoplus_{z \in D_\eta} \ R^d i_{z*} \mu_{p^n}^{\otimes d} \,.
\]
\item $\delta_4$ factors as
\[
\delta_4: R^{d+1}\sigma_* \mu_{p^n}^{\otimes d+1}
\os {\alpha}\longrightarrow R^{d+1}i_{w*} \mu_{p^n}^{\otimes d+1}
\os{\bigoplus_y \, \dval_{w,y}}-\hspace{-5pt}-\hspace{-5pt}-\hspace{-6pt}\lra \bigoplus_{y \in E} \ i_{y*} \logwitt y n d
\]
by the construction of $\delta_U$ (cf.\ Proposition \ref{Cl.1-2}).
\item $\delta_5= \sum_{z \in D_\eta} \ \dval_{z,x}$ \; by the assumption of the lemma.
\item $\delta_6= - \sum_{y \in E} \ \dval_{y,x}$ \; by Theorem \ref{thm:jss} and the construction of $\mathscr M_{n,X_s}^\bullet$, cf.\ \S\ref{sect2-5}.
\end{itemize}
Therefore we obtain the commutativity of \eqref{com3} from a result of Kato \cite{kk:hasse} 1.7 for $B$, by noting that $B^1=(C_\eta)^1 \coprod D_\eta \coprod E$ and that $\text{\rm Im}(\alpha)$ is contained in the kernel of the map
\[
\bigoplus_v \ \partial^\val_{w,v} :
R^{d+1}i_{w*} \mu_{p^n}^{\otimes d+1} \longrightarrow \bigoplus_{v \in (C_\eta)^1} \ R^d i_{v*} \mu_{p^n}^{\otimes d}
\]
(cf.\ proof of Proposition \ref{Cl.1-2}). This completes the proof of Lemma \ref{Cl.1-5}\,(i), because its second claim follows with \S\ref{sect0-5-2}.
\par
(ii) By the properness of $f$ and Proposition \ref{Lem.1-1}, we have only to show the commutativity of the right square (1)$''$ of the following diagram, assuming that $s=\ol s$:
\begin{equation}\label{com4}
\xymatrix{
H^1(Z_\eta,\mathcal E_{Z_\eta}) \ar[r]^-{\psi_*} \ar[d]_{Rf_*(\delta_Z^S)\hspace{-1.5pt}} \ar@{}[rd]|{\text{(1)$'$}}
& H^1(X_\eta,\mathcal E_{X_\eta}) \ar[r]^-{f_*} \ar[d]_{Rf_*(\delta_1)\hspace{-1.5pt}} \ar@{}[rd]|{\text{(1)$''$}}
& H^1(\eta ,\mathcal E_\eta) \ar[d]_{\delta_S^\text{\rm val} \hspace{-1.5pt}} \\
H^0(Z_{s},\mathcal E_{Z_s}) \ar[r]^-{\psi_*}
& H^0(X_{s},\mathcal E_{X_s}) \ar[r]^-{f_*}
& H^0(s,\mathcal E_{s})\,,\hspace{-5pt}
} \end{equation}
where for a proper morphism $g$ of schemes, we wrote $g_*$ for the adjunction map $Rg_*Rg^! \to \text{\rm id}$. The outer square of this diagram commutes, because $\delta_Z^S=R\psi_* Rf_Z^!(\delta_S^\text{\rm val})$ and the composite
\[ Rf_{Z*}Rf_Z^!=Rf_*R\psi_*R\psi^!Rf^! \os{\psi_*}\longrightarrow Rf_*Rf^! \os{f_*}\longrightarrow \text{\rm id} \]
is functorial (in fact, this coincides with $f_{Z*}$). On the other hand, the square (1)$'$ commutes, because $\delta_1$ makes the square (1) in \eqref{com2} commutative. Moreover, in view of the exact sequence \eqref{exact:local}, the upper horizontal arrow $\psi_*$ in (1)$'$ is surjective, because we have
\[ H^1(U_\eta,\mathcal E_{U_\eta}) \cong H^{2d+1}(U_\eta,\mu_{p^n}^{\otimes d+1})=0 \]
by the assumptions that $s=\ol s$ and that $U_\eta$ is smooth affine of dimension $d \geq 1$ (cf.\ Lemma \ref{lem1}).
Therefore (1)$''$ is commutative, and we obtain Lemma \ref{Cl.1-5}\,(ii).
\end{pf*}
\smallskip
\subsection{Proof of Theorem \ref{Th.1-1}}\label{sect3-8}
First consider Theorem \ref{Th.1-1}\,(1). By Lemma \ref{Cl.1-3} and Remark \ref{rem1} it suffices to show:
\par\medskip\noindent
\quad\quad $(\sharp)$ \;
For integral $X$, $\delta_X^{S\text{-}\val} :=Rf^!(\delta_S^\text{\rm val})$ satisfies $\mathbb K_d$ with $d:=\dim(X_\eta)$.
\par\medskip\noindent
We show this property by induction on $d = \dim(X_\eta)$. The case $d=0$ is settled by Lemma \ref{Cl.1-1}. Now let $\dim(X_\eta)\ge 1$ and choose $U$ and $Z=X \smallsetminus U$ as in \S\ref{sect3-7}. Assume that $\delta_2 = R\phi_*(\delta_U)$ with $\delta_U$ as in Lemma \ref{Cl.1-5}. The assumption of this lemma holds because $(\sharp)$ holds for $Z$ by induction assumption. Therefore there is a morphism $\delta_1$ making \eqref{com2} commutative, and this morphism is $\delta_1 = \delta_X^{S\text{-}\val}$. We conclude that $\delta_U = \phi^*(\delta_X^{S\text{-}\val}) = \delta_U^{S\text{-}\val}$. Hence $\delta_U^{S\text{-}\val}$ satisfies $\mathbb K_q$ (by choice of $\delta_U$), and $\delta_X^{S\text{-}\val}$ satisfies $\mathbb K_q$ as well, because $(X_\eta)_d = (U_\eta)_d$ by density of $U$ in $X$, and $(X_s)_d = (U_s)_d$ as noted in \eqref{cond1}.
Theorem \ref{Th.1-1}\,(3) now follows from Proposition \ref{Cl.1-2}, because $\delta_X^{S\text{-}\val}$ satisfies $\mathbb K_d$.
Theorem \ref{Th.1-1}\,(2) follows once more by induction on $d=\dim(X_\eta)$, the case $d=0$ being given by Lemma \ref{Cl.1-1}. If $d\ge 1$ we may assume that $X$ is reduced and then again choose $U$ and $Z=X \smallsetminus U$ as in \S\ref{sect3-7}. Assume that a morphism
\[ \delta_1: Rj_{X*}\mathcal E_{X_\eta} \longrightarrow Ri_{X*}\mathcal E_{X_s}[-1] \]
satisfies $\mathbb K_q$ for all $q\geq 0$. Then $R\psi^!(\delta_1)$ satisfies $\mathbb K_q$ for all $q\geq 0$ and agrees with $\delta_Z^{S\text{-}\val}$ by induction assumption. On the other hand, $\phi^*(\delta_1)$ satisfies $\mathbb K_d$ and thus coincides with $\delta_U^{S\text{-}\val}$ by Theorem \ref{Th.1-1}\,(3) just proved. The conclusion is that $\delta_1$ makes the square (1) of \eqref{com2} commutative with $\delta_Z^S = R\psi_*(\delta_Z^{S\text{-}\val})$, and the square (2) of \eqref{com2} commutative with $\delta_2 = R\phi_*(\delta_U^{S\text{-}\val})$. Since obviously $\delta_X^{S\text{-}\val}$ makes these diagrams commutative as well, Lemma \ref{lem:delta1} implies $\delta_1 =\delta_X^{S\text{-}\val}$ as wanted. This concludes the proof of Theorem \ref{Th.1-1}. \qed
\smallskip
\subsection{Dualizing complexes}\label{sect3-9}
We apply our results to the study of dualizing complexes as indicated in part \S\ref{sect0-2} of the introduction.
Recall the following diagram:
\[\xymatrix{ \eta \; \ar@{^{(}->}[r]^j & S & \ar@{_{(}->}[l]_i \; s\,. }\]
\begin{defn}\label{def:DC-S}
For each integer $r \ge 1$ define
\[ \bZ/p^n(1)'_S := \text{\rm Cone}(\delta_S^\text{\rm val}: Rj_*\mu_{p^n} \to i_* \bZ/p^n [-1])[-1] \in D^b(S_\text{\rm \'{e}t},\bZ/p^n), \]
the mapping fiber of the morphism $\delta_S^\text{\rm val}$ defined in {\rm\S\ref{sect3-2}}.
\end{defn}
\noindent
In general, mapping cone or fiber of a morphism in a derived category is only well-defined up non-canonical isomorphism. However in our case it is well-defined up to a unique isomorphism, because we can apply the criterion of Lemma \ref{lem.derived}\,(1). Indeed the complex $Rj_*\mu_{p^n}$ is concentrated in $[0,1]$, $A[1]=i_* \bZ/p^n [-1]$ is concentrated in degree 1, and $\delta_S$ induces a surjection $R^1j_*\mu_{p^n} \twoheadrightarrow i_* \bZ/p^n$ so that the mapping fiber $B$ is concentrated in $[0,1]$ as well. Therefore $\text{\rm Hom}_{D(S,\bZ/p^n)}(B,A)=0$. (This argument should replace the reasoning in \cite{js} p.\ 497, where the criterion is misstated.)
By the above, there is a canonical exact triangle
\stepcounter{equation}
\begin{equation}\label{eq3-9-2}
i_*\bZ/p^n[-2] \os{g}\longrightarrow \bZ/p^n(1)'_S \os{t}\longrightarrow Rj_*\mu_{p^n} \os{\delta_S^\text{\rm val}}\longrightarrow i_*\bZ/p^n[-1]\,,
\end{equation}
which induces canonical isomorphisms
\begin{equation}\label{isos.DC-S}
t: j^*(\bZ/p^n(1)'_S) \cong \mu_{p^n} \quad \hbox{ and } \quad g: \bZ/p^n[-2] \cong Ri^!(\bZ/p^n(1)'_S)\,.
\end{equation}
Now let $f: X \to S$ be separated of finite type, and define
\[ \mathcal E_X := Rf^!\bZ/p^n(1)'_S \in D^+(X_\text{\rm \'{e}t},\bZ/p^n). \]
Also, let $\mathcal E_{X_\eta} = Rf^!_\eta \mu_{p^n}$ and $\mathcal E_{X_s} = Rf^!_s\bZ/p^n$, as we defined at the beginning of this secton. Then, by applying $Rf^!$ to the exact triangle \eqref{eq3-9-2} and using the base-change isomorphisms as in \eqref{dg:proper} we get a canonical isomorphism of exact triangles
\begin{equation}\label{triangles.DC-X}
\xymatrix{
i_{X *}\mathcal E_{X_s}[-2] \ar[r]^-{g_X} \ar@{=}[d]
& \mathcal E_X \ar[r]^-{t_X} \ar@{=}[d]_{\text{(def)}}
& Rj_{X*}\mathcal E_{X_\eta} \ar[r]^-{\delta^{S\text{-}\val}_X} \ar@{=}[d]
& i_{X*}\mathcal E_{X_s}[-1] \ar@{=}[d] \\
Rf^!i_*\bZ/p^n[-2] \ar[r]^-{Rf^!(g)}
& Rf^!\bZ/p^n(1)'_S \ar[r]^-{Rf^!(t)}
& Rf^!Rj_*\mu_{p^n} \ar[r]^-{Rf^!(\delta^\text{\rm val}_S)}
& Rf^!i_*\bZ/p^n[-1]\,,\hspace{-5pt}
}
\end{equation}
where $g_X$ and $t_X$ are the adjunction maps for $i_X$ and $j_X$, respectively. By Theorem \ref{Th.1-1} the morphism $\delta_X^{S\text{-}\val}$ satisfies the localization property $\mathbb K_q$ for all $q\geq 0$ (i.e., is locally given by Kato's residue maps), and is determined by this property (and just by $\mathbb K_d$ if $X_\eta$ is smooth of dimension $d$). Moreover, by Lemma \ref{lem.deltaloc} below (see also \eqref{eq4-2-1} below), we see that
\begin{equation}\label{eq3-9-5}
\delta^{S\text{-}\val}_X = -\delta^\text{\rm loc}_{X_\eta,X_s}(\mathcal E_X)\,.
\end{equation}
Because the dualizing complex is $\mathscr D_{X,p^n} = \mathcal E_X[2]$ by definition (cf.\ \S\ref{sect0-2}), this equality implies the last claim in the part (iv) of \S\ref{sect0-2}. In fact, it is easy to see that the local version treated in this section can be extended to the more global situation described in the introduction.
\addtocounter{thm}{4}
\begin{lem}\label{lem.deltaloc} Consider cartesian squares of schemes
\[ \xymatrix{
X_Z \, \ar@{^{(}->}[r]^-{i} \ar[d] \ar@{}[rd]|{\square} & X \ar[d]_f \ar@{}[rd]|{\square} & \ar@{_{(}->}[l]_-{j} \, X_U \ar[d] \\
Z \, \ar@{^{(}->}[r]^-{i'} & Y & \ar@{_{(}->}[l]_-{j'} \, U\,,\hspace{-5pt}
}\]
where $i'$ is a closed immersion and $j'$ is the open immersion of the complement $U = Y \smallsetminus Z$. Then, for any complex of torsion sheaves $\mathscr K \in D^+(Y_\text{\rm \'{e}t})$ the base-change isomorphisms give an identification
\[ Rf^!(\delta^\text{\rm loc}_{U,Z}(\mathscr K)) = \delta^\text{\rm loc}_{X_U,X_Z}(Rf^!\mathscr K)\,. \]
\end{lem}
\begin{pf}
There is a commutative diagram with distinguished rows
\[ \xymatrix{
i_* Ri^!(Rf^!\mathscr K) \ar[r]^-{i_*} \ar[d]_\beta^{\hspace{-1.5pt}\wr} & Rf^!\mathscr K \ar[r]^-{j^*} \ar@{=}[d] & Rj_* j^*(Rf^!\mathscr K) \ar[rr]^-{-\delta^\text{\rm loc}_{X_U,X_Z}(Rf^!\mathscr K)} \ar@{.>}[d]_\alpha && Ri_* Ri^!(Rf^!\mathscr K)[1] \ar[d]_{\beta[1]}^{\hspace{-1.5pt}\wr} \\
Rf^!(i'_* Ri'^!\mathscr K) \ar[r]^-{Rf^!(i'_*)} & Rf^!\mathscr K \ar[r]^-{Rf^!(j'^*)} & Rf^!(Rj'_* j'^*\mathscr K) \ar[rr]^-{-Rf^!(\delta^\text{\rm loc}_{U,Z}(\mathscr K))} && Rf^!(i'_* Ri'^!\mathscr K)[1]\,,\hspace{-5pt}
}\]
where the top row is the localization exact triangle \eqref{eq-0.4.2} for $Rf^!\mathscr K$, the bottom row is obtained by applying $Rf^!$ to a localization exact triangle for $\mathscr K$ and the arrow $\beta$ is a base-change isomorphism. By adjunction, the left hand square commutes. Therefore there exists a morphism $\alpha$ which makes the other squares commute (see \S\ref{sect0-5-2}). By the commutativity of the middle square, $\alpha$ is mapped to the identity under the canonical isomorphisms
\begin{align*}
\text{\rm Hom}_{D(X)}(Rj_* j^*Rf^!\mathscr K, Rf^!Rj'_*j'^*\mathscr K) & \cong \text{\rm Hom}_{D(U)}(j^* Rf^!\mathscr K, j^* Rf^!Rj'_* j'^*\mathscr K) \\
& = \text{\rm Hom}_{D(U)}(j^* Rf^!\mathscr K, j^* Rf^!\mathscr K)\,.
\end{align*}
But this means that $\alpha$ is the base-change isomorphism, and the claim follows.
\end{pf}
\smallskip
\subsection{Bloch-Ogus complexes and Kato complexes}\label{sect3-10}
As an application, used in \cite{js}, we deduce the following result on Kato complexes, analogous to \S\ref{sect1-5} and \S\ref{sect2-11}. As in \cite{js} 2.C we define a homology theory on all separated $S$-schemes $f : X \to S$ of finite type by letting
\[ H_a(X/S,\bZ/p^n(-1)) := H^{-a}(X,Rf^!(\bZ/p^n(1)'_S))\,, \]
and, following the method of Bloch and Ogus, a niveau spectral sequence
\begin{equation}\label{arspectralsequence}
E^1_{q,t}(X/S,\bZ/p^n(-1)) = \bigoplus_{x\in X_q} H_{q+t}(x/S,\bZ/p^n(-1)) \Rightarrow H_{q+t}(X/S,\bZ/p^n(-1)),\hspace{-10pt}
\end{equation}
where $H_a(x/S;\bZ/p^n(-1))$ is defined as the inductive limit over all $H_a(U/S,\bZ/p^n(-1))$, for all non-empty open subschemes $V\subset \overline{\{x\}}$. Then we have
\stepcounter{thm}
\begin{thm} \label{thm:katoc3}
\begin{enumerate}
\item[(1)] For $X=X_\eta$ the spectral sequence \eqref{arspectralsequence} is canonically isomorphic to the spectral sequence from \eqref{eq1-5-4}
\[ E^1_{q,t}(X_\eta/\eta,\bZ/p^n(-1)) = \bigoplus_{x\in (X_\eta)_q} \ H_{q+t}(x/\eta,\bZ/p^n(-1)) \Longrightarrow H_{q+t}(X_\eta/\eta;\bZ/p^n(-1))\,. \]
\item[(2)] For $X=X_s$ the spectral sequence \eqref{arspectralsequence} is canonically isomorphic to the
spectral sequence
\[ E^1_{q,t+2}(X_s/s,\bZ/p^n) = \bigoplus_{x\in (X_s)_q} \ H_{q+t+2}(x/s,\bZ/p^n) \Longrightarrow H_{q+t+2}(X_s/s;\bZ/p^n) \]
obtained from \eqref{eq2-11-2} after a shift in the second degree.
\item[(3)] Let $x \in X_q\cap X_s=(X_s)_q$ and $y\in X_{q+1}\cap X_\eta=(X_\eta)_q$ with $x\in\overline{\{y\}}$. Then there are canonical purity isomorphisms
\begin{align*}
H_{q+1+t}(y/S,\bZ/p^n(-1)) & \cong H^{q-t-1}(y,\bZ/p^n(q+1))\,, \\
H_{q+t}(x/S,\bZ/p^n(-1)) & \cong H^{q-t-2}(x,\bZ/p^n(q))\,.
\end{align*}
Via these isomorphisms, the $(y,x)$-component
\[ d^1_{q+1,t}(y,x): H^{q-t-1}(y,\bZ/p^n(q+1)) \to H^{q-t-2}(x,\bZ/p^n(q)) \]
of the differential $d^1_{q+1,t}$ in \eqref{arspectralsequence} coincides with $-\dval_{y,x}$.
\item[(4)]
The isomorphisms in {\rm(1)}, {\rm(2)} and {\rm(3)} induce isomorphisms
\[ E^1_{*,t}(X/S,\bZ/p^n(-1)) \cong C^{-t-2,0}_{p^n}(X)^{(-)} \]
between Bloch-Ogus complexes and sign-modified Kato complexes.
\end{enumerate}
\end{thm}
\begin{pf}
(1) and (2) are obvious from the isomorphisms \eqref{isos.DC-S}. The first claim in (3) is clear from the fact that $\overline{\{y\}}$ meets $X_s$, and the isomorphisms then follow from (1) and (2) and the purity isomorphisms \eqref{eq1-5-5} and \eqref{eq2-11-3}, respectively. For the third statement of (3) we recall that the upper exact triangle in \eqref{triangles.DC-X} induces isomorphisms \[ t_X: j_X^*\mathcal E_X \cong \mathcal E_{X_\eta} \quad \hbox{ and } \quad g_X : \mathcal E_{X_s} \cong Ri_X^!\mathcal E_X \] identifying $\delta_X^{S\text{-}\val}$ with the connecting morphism $-\delta^\text{\rm loc}_{X_\eta,X_s}(\mathcal E_X)$, cf.\ \eqref{eq3-9-5}. Since $\delta_X^{S\text{-}\val}$ induces Kato's residue maps, we get the claim. As for (4), the compatibility $d^1(y,x) = -\dval_{y,x}$ between the differentials and Kato's residue maps follow from (1) and Theorem \ref{thm:Katocomplexprimetop} for $y,x \in X_\eta$, and from (2) and Theorem \ref{thm:Katocomplexp-case} for $y,x \in X_s$. The remaining case is covered by (3).
\end{pf}
\begin{rem}
It is easy to see that this theorem proves the claims in \cite{js} {\rm 2.20} and {\rm 2.21}, except that the signs needed to be corrected. The reason for this lies in the interpretation of the connecting morphism and the resulting minus sign in \eqref{eq-0.4.2}.
\end{rem}
\smallskip
\subsection{Unicity of the cone}\label{sect3-11}
As a complement we show the following unicity result for $\mathcal E_X=Rf^!(\bZ/p^n(1)_S')$. Recall the situation at the beginning of this section
\[ \xymatrix{
X_\eta \, \ar@{^{(}->}[r]^-{j_X}\ar[d]_{f_\eta} \ar@{}[rd]|{\square}
& X \ar[d]_f \ar@{}[rd]|{\square}
& \,X_s \ar@{_{(}->}[l]_-{i_X}\ar[d]_{f_s} \\
\eta \, \ar@{^{(}->}[r]^-j
& S
& \ar@{_{(}->}[l]_-i \, s \,,\hspace{-5pt} }\]
and the associated exact triangle, cf.\ \eqref{triangles.DC-X}
\[\xymatrix{
i_{X_*}\mathcal E_{X_s}[-2] \ar[r] & \mathcal E_X \ar[r] & Rj_{X*}\mathcal E_{X_\eta} \ar[r]^-{\delta_X^{S\text{-}\val}} & i_{X*}\mathcal E_X[-1]\,.
}\]
\begin{thm}\label{thm.cone}
The object $\mathcal E_X$ is uniquely determined, up to unique isomorphism, as the mapping fiber of $\delta_X^{S\text{-}\val}$.
\end{thm}
\noindent
By Lemma \ref{lem.derived}\,(3) it suffices to show
\smallskip
\begin{enumerate}
\item[(i)] $\text{\rm Hom}^{-1}_{D(X,\bZ/p^n)}(i_{X*}\mathcal E_{X_s}[-2],Rj_{X*}\mathcal E_{X_\eta})=0$. \smallskip
\item[(ii)] $\text{\rm Hom}_{D(X,\bZ/p^n)}(Rj_{X*}\mathcal E_{X_\eta},i_{X_*}\mathcal E_{X_s}[-2])=0$.
\end{enumerate}
\smallskip \noindent
(i) follows by adjunction for $j_X$, because $j^*_Xi_{X*}=0$. As for (ii), since
\[ \text{\rm Hom}_{D(X,\bZ/p^n)}(Rj_{X*}\mathcal E_{X_\eta},\,i_{X*}\mathcal E_{X_s}[-2])= \text{\rm Hom}_{D(s,\bZ/p^n)}(Rf_{s!}i_X^{*}Rj_{X*}\mathcal E_{X_\eta},\, \bZ/p^n[-2]) \]
by adjunction for $i_X$ and $f_s$, it suffices to show
\begin{lem}\label{lem.boundmixed}
$Rf_{s!}i^*_XRj_{X*}\mathcal E_{X_\eta}$ is concentrated in $[-2d,1]$, where $d = \dim(X_\eta)$.
\end{lem}
\noindent We first show the following result, which may be of own interest.
\begin{lem}\label{lem.boundfield} Let $k$ be a field, and let $f : X \to \text{\rm Spec}(k)$ be separated of finite type, and let $n$ be a positive integer which is invertible in $k$. Then $Rf_!Rf^!\mathbb Z/n(i)$ is concentrated in $[-2d,0]$, where $d:=\dim(X)$.
\end{lem}
\begin{pf} We proceed by induction on $d=\dim(X)$. We may assume that $k$ is separably closed, that $i=0$ and that $X$ is reduced, and then the case $d=0$ is clear. Choose an affine open subset $U \subset X$ which is smooth of pure $\dim d$ and whose complement $Z := X \smallsetminus U$ has dimension $e \le d-1$. We get a commutative diagram
\[ \xymatrix{
U \,\ar@{^{(}->}[r]^\phi \ar[dr]_{f_U}
& X \ar[d]^f
& \, Z\ar@{_{(}->}[l]_\psi \ar[dl]^{f_Z} \\
& \text{\rm Spec}(k) \,, \hspace{-5pt}} \]
where $\phi$ (resp.\ $\psi$) denotes the natural open (resp.\ closed) immersion, and we defined $f_U:=f \circ \phi$ and $f_Z:=f \circ \psi$. We note that $\phi$ is affine, because $X$ is separated over $k$ (if $V\subset X$ is affine, then $\phi^{-1}(V)=U\cap V$ is affine). There is an exact triangle
\[ Rf_!\psi_* R\psi^!Rf^!\mathbb Z/n \longrightarrow Rf_!Rf^!\mathbb Z/n \longrightarrow Rf_!R\phi_*\phi^* Rf^!\mathbb Z/n \os{+1}\longrightarrow . \]
Since $U$ is smooth of pure dimension $d$, we have
\[ \phi^* Rf^!\mathbb Z/n = Rf^!_U \mathbb Z/n \cong \mathbb Z/n(d)[2d] \,. \]
Moreover we have $R\psi^!Rf^!=Rf_Z^!$. Therefore we can identify the above triangle with
\addtocounter{equation}{3}
\begin{equation}\label{triangle.localization1}
Rf_{Z!} Rf^!_Z \mathbb Z/n \longrightarrow Rf_!Rf^!\mathbb Z/n \longrightarrow Rf_!R\phi_* \mathbb Z/n(d)[2d] \os{+1}\longrightarrow .
\end{equation}
Since $Rf_{Z!}Rf^!_Z\mathbb Z/n$ is concentrated in $[-2d+2,0]$ by induction, it is enough to show that $A:=Rf_! R\phi_* \mathbb Z/n(d)[2d]$ is concentrated in $[-2d,0]$. Obviously $A$ is concentrated in degrees $\ge -2d$, because this holds for $\mathbb Z/n(d)[2d]$. On the other hand, we note that $\mathbb Z/n(d)[d]$ is a perverse sheaf on $U$ (\cite{pervers} p.\ 102), so that $R\phi_*\mathbb Z/n(d)[d]$ is perverse, because $\phi$ is an affine open immersion (and hence $t$-exact for the perverse $t$-structure loc.\ cit.\ 4.1.10\,(i)), and that $A=Rf_!R\phi_*\,\mathbb Z/n(d)[d]$ is of perversity $\le d$ (loc.\ cit.\ 4.2.4), i.e., lies in $D^{p \le d}_c(k,\mathbb Z/n)=D_c^{p\le 0}(k,\mathbb Z/n)[-d]$. This means that
\[ A \in D^{p\leq 0}_c(k,\mathbb Z/n). \]
Since the perverse $t$-structure is the classical $t$-structure on $\text{\rm Spec}(k)$, we get that $A$ is concentrated in degrees $\le 0$. Thus we obtain Lemma \ref{lem.boundfield}.
\end{pf}
\noindent{\bf Proof of Lemma \ref{lem.boundmixed}.}
We may assume that $X$ is reduced and the closure of $X_\eta$. Then we prove the lemma by induction on $d=\dim X_\eta$. The case $d=0$ is easy and left to the reader. Suppose $d>1$. Then there is a commutative diagram
\[ \xymatrix{U \, \ar@{^{(}->}[r]^-\phi \ar[dr] & X \ar[d]^f & \, Z\ar@{_{(}->}[l]_\psi \ar[dl]^g \\ & S\,,\hspace{-5pt}} \]
where $\phi$ is an open immersion, $U_\eta$ is affine, smooth over $\eta$ and has pure dimension $d$, $\psi$ is the closed immersion of the complement $Z = X \smallsetminus U$ (with reduced subscheme structure), and $\dim Z_\eta\leq d-1$. We get an exact triangle
\begin{equation}\label{triangle.localization2}
i^*_X Rj_{X*}\psi_{\eta*}R\psi_\eta^!\mathcal E_{X_\eta} \longrightarrow i^*_X Rj_{X*}\mathcal E_{X_\eta}
\longrightarrow i^*_X Rj_{X*}R\phi_{\eta*}\phi_\eta^* \mathcal E_{X_\eta} \os{+1}\longrightarrow\,,
\end{equation}
where we used morphisms in the following diagram:
\[ \xymatrix{
U_\eta \, \ar@{^{(}->}[r]^-{j_U} \ar@{_{(}->}[d]_{\phi_\eta} \ar@{}[rd]|{\square}
& \, U_{\phantom.} \ar@{_{(}->}[d]_\phi \ar@{}[rd]|{\square}
&\ar@{_{(}->}[l]_-{i_U} \ar@{_{(}->}[d]_{\phi_s} \, U_s \ar[rrd] \\
X_\eta \, \ar@{^{(}->}[r]^-{j_X}
& X
& \ar@{_{(}->}[l]_{i_X} \, X_s \ar[rr]^{f_s}
&& s \, . \\
Z_\eta \phantom{|}\hspace{-2pt} \ar@{^{(}->}[r]^-{j_Z} \ar@{^{(}->}[u]^{\psi_\eta} \ar@{}[ru]|{\square}
& Z \phantom{|}\hspace{-2pt} \ar@{^{(}->}[u]^\psi \ar@{}[ru]|{\square}
&\ar@{_{(}->}[l]_-{i_Z} \ar@{^{(}->}[u]^{\psi_s} \ar[rru]_{g_s} \, Z_s \phantom{|}\hspace{-2pt}
}\]
By the proper base-change theorem for $\psi$ we identify
\[ i^*_X Rj_{X*}\psi_{\eta*}R\psi_\eta^!\mathcal E_{X_\eta} = i^*_X \psi_{*}Rj_{Z*}\mathcal E_{Z_\eta} = \psi_{s*}i^*_ZRj_{Z*}\mathcal E_{Z_\eta} \,. \]
Because $\phi$ is \'etale and $U_\eta$ is smooth of pure dimension $d$, we have
\[ R\phi_{\eta*}\phi^*_\eta \mathcal E_{X_\eta}=R\phi_{\eta*}\mathcal E_{U_\eta}=R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[2d]\,.\]
Therefore triangle \eqref{triangle.localization2}, after application of $Rf_{s!}$, leads to an exact triangle
\[ Rg_{s!}i^*_ZRj_{Z*}\mathcal E_{Z_\eta} \longrightarrow Rf_{s!}i^*_X Rj_{X*}\mathcal E_{X_\eta}
\longrightarrow Rf_{s!}i^*_XRj_{X*} R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[2d] \os{+1}\longrightarrow. \]
Since the first term is concentrated in $[-2d+2,1]$ by induction, it suffices to show that
\begin{equation}\label{eq3-11-6}
A:=Rf_{s!}i^*_XRj_{X*} R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[2d]
\end{equation}
is concentrated in $[-2d,1]$. It is clearly concentrated in degrees $\ge -2d$, because this holds for $\mu_{p^n}^{\otimes d+1}[2d]$. We prove that $A$ is concentrated in degrees $\le 1$ in what follows. By the proof of Lemma \ref{lem.boundfield}, $R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[d]$ is a perverse sheaf, i.e.,
\[ \mathscr P^q :=\mathscr H^q(R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[d])=R^{q+d}\phi_{\eta*}\mu_{p^n}^{\otimes d+1} \]
has support in dimension $\le -q$. In particular, it is non-zero only for $-d \le q \le 0$. We will prove
\par\bigskip\noindent
{\it Claim. The sheaf $i^*_XR^mj_{X*}\mathscr P^q$ is zero for $m+q>1$.}
\par\bigskip\noindent
We see that $i^*_XRj_{X*}R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[d]$ is concentrated in degrees $\le 1$ by the claim and the Leray spectral sequence
\[ E_2^{a,b}=i^*_XR^aj_{X*}\mathscr P^b \Longrightarrow \mathscr H^{a+b}(i^*_X Rj_{X*}R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[d])\,. \]
Moreover since $\dim X_s \le d$ (see the beginning of proof of Lemma \ref{lem.boundmixed}) and $\text{\rm ch}(s)=p$,
we see that \[ A[d]=Rf_{s!}i^*_XRj_{X*}R\phi_{\eta*}\mu_{p^n}^{\otimes d+1}[d] \] is concentrated in degrees $\le d+1$, so that $A$ is concentrated in degrees $\le 1$. Thus it remains to show the above claim. By the remark before the claim, it suffices to prove
\addtocounter{thm}{3}
\begin{lem}\label{lem.specialization}
If $\mathscr F$ is an \'etale sheaf on $X_\eta$ with $\dim(\text{\rm Supp}\,\mathscr F)\le b$, then we have
\[ \dim(\text{\rm Supp} \; i^*_XR^mj_{X*}\mathscr F) \le b \quad \hbox{ for \; $m \ge 0$,} \]
and $i^*_XR^mj_{X*}\mathscr F=0$ for $m>b+1$.
\end{lem}
\begin{pf} By assumption, there is a closed subset $V \os{\iota}\hookrightarrow X_\eta$ of dimension $\le b$ such that $\mathscr F=\iota_* \mathscr G$ with $\mathscr G=\iota^*\mathscr F$. Let $Y=\ol V$, the closure of $V$ in $X$ endowed with the reduced subscheme structure. Then $V=Y_\eta$, and $Y_s$ has dimension $\le b$. We get cartesian squares
\[\xymatrix{
Y_\eta \, \ar@{^{(}->}[r]^{j_Y} \ar@{_{(}->}[d]_{\iota=\kappa_\eta} \ar@{}[rd]|{\square}
& Y{\phantom{,}}\hspace{-2pt} \ar@{_{(}->}[d]_{\kappa} \ar@{}[rd]|{\square}
& \,Y_s\ar@{_{(}->}[l]_{i_Y} \ar@{_{(}->}[d]_{\kappa_s} \\
X_\eta \, \ar@{^{(}->}[r]^{j_X}
& X
& \,X_s \ar@{_{(}->}[l]_{i_X}
}\]
with $\kappa,\kappa_\eta$ and $\kappa_s$ being closed immersions. Since $\mathscr F=\kappa_{\eta*}\mathscr G$, we get
\[ i^*_XR^aj_{X*}\mathscr F = i^*_XR^aj_{X*} \kappa_{\eta*}\mathscr G = i^*_X\kappa_* R^qj_{Y*}\mathscr G = \kappa_{s*}i^*_Y R^qj_{Y*}\mathscr G\,,\]
where the last equality is a base-change isomorphism. This shows that $i^*_XR^mj_{X*}\mathscr F$ has support in $Y_s$, i.e., in dimension $\le b$. Finallly, since $R^mj_{Y*}\mathscr G=0$ for $m>b+1$ by Lemma \ref{lem1}, we have $i^*_XR^mj_{X*}\mathscr F=\kappa_{s*}i_Y^*R^mj_{Y*}\mathscr G=0$ for $m>b+1$.
\end{pf}
\noindent
This completes the proof of Lemma \ref{lem.boundmixed} and Theorem \ref{thm.cone}.
\qed
\par\bigskip \noindent
By the above results, we obtain the following bounds for $\mathcal E_X = Rf^!(\bZ/p^n(1)'_S)$.
\begin{cor}\label{cor.boundglobal}
Put $d:=\max(\dim X_\eta,\dim X_s)$. Then{\rm:}
\begin{enumerate}
\item[(1)] $i^* Rf_!\mathcal E_X$ is concentrated in $[-2d,2]$.
\item[(2)] $j^* Rf_!\mathcal E_X$ is concentrated in $[-2d,0]$.
\end{enumerate}
In particular, $Rf_!\mathcal E_X$ is concentrated in $[-2d,2]$.
\end{cor}
\begin{pf}
(1) Consider the exact triangle
\[ \xymatrix{
i^{*}Rf_!i_{X*}\mathcal E_{X_s}[-2] \ar[r] \ar@{=}[d] & i^* Rf_!\mathcal E_X \ar[r] \ar@{=}[d] &
i^* Rf_!Rj_{X*}\mathcal E_{X_\eta} \ar[r]^-{+1} \ar@{=}[d] & \,. \\
A & B & C }\]
Here
\[ A=Rf_{s!}\mathcal E_{X_s}[-2] \quad \hbox{ and }\quad C=Rf_{s!}i_X^*Rj_{X*}\mathcal E_{X_\eta} \]
by the proper base-change theorem. Since $C$ is concentrated in $[-2d,1]$ by Lemma \ref{lem.boundmixed}, it is enough to show that $A$ is concentrated in $[-d+2,2]$. Since $\mathcal E_{X_s} \cong \mathscr M_{n,X_s}$ by Theorem \ref{thm:jss}, the complex $\mathcal E_{X_s}$ is concentrated in $[-d,0]$ and any non-zero section of $\mathscr H^q(\mathcal E_{X_s})$ has support of dimension $\le -q$.
This implies that
\[ R^mf_{s!}\mathscr H^q(\mathcal E_{X_s})=0 \quad\hbox{ for }\; m+q> 0. \]
Indeed, $Rf_{s!}$ commutes with inductive limits of sheaves, and for any separated of finite type morphism $g : Z \to s$ with $\dim(Z)=e$ and any $p$-primary torsion sheaf $\mathscr F$ on $Z_\text{\rm \'{e}t}$, the complex $Rg_!\mathscr F$ is concentrated in $[0,e]$. Therefore $Rf_{s!}\mathcal E_{X_s}$ is concentrated in $[-d,0]$, and $A$ is concentrated in $[-d+2,2]$.
\par
(2) Since
\[ j^* Rf_!\mathcal E_X=Rf_{\eta!}\mathcal E_{X_\eta} = Rf_{\eta!}Rf_\eta^!\mu_{p^n} \,, \]
the assertion follows from Lemma \ref{lem.boundfield}.
\end{pf}
\newpage
\section{Duality for arithmetic schemes}\label{sect4}
\medskip
The aim of this section is to prove a general duality for constructible sheaves on separated schemes of finite type over $\mathbb Z$. The main result of this section will be stated in \S\ref{sect4-3} below. We begin by reviewing the Artin-Verdier duality for number fields (cf.\ \cite{arver}, \cite{mazur}, \cite{milne:adual} II.2--3).
\par\smallskip
\subsection{Artin-Verdier duality}\label{sect4-1}
Let $k$ be a number field with ring of integers $\mathfrak o_k$, and let $S=\text{\rm Spec}(\mathfrak o_k)$. For an \'etale sheaf or a complex of \'etale sheaves $\mathscr F$ on $S$ let $H^m_c(S,\mathscr F)$ be its \'etale cohomology group with compact support (see e.g., \cite{milne:adual} II.2, \cite{kk:hasse} \S3 for generalities). Let ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}:={\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,S}$ be the sheaf on $S_\text{\rm \'{e}t}$ given by the multiplicative group. By global class field theory, we have
\begin{equation}\label{isom:cft}
H_c^m(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \cong
\begin{cases}
\bQ/\bZ \qquad & \hbox{($m=3$)}\\
0 \qquad & \hbox{($m=2$ or $m \ge 4$)}.
\end{cases}
\end{equation}
We normalize the isomorphism for $m=3$ as follows. For a closed point $y$ of $S$, let $G_y$ be the absolute Galois group of $\kappa(y)$, and let
\[ \text{\rm tr}_{y,\bQ/\bZ}:H^1(y,\bQ/\bZ) \longrightarrow \bQ/\bZ \]
be its trace map, i.e., the unique homomorphism that evaluates a continuous character $\chi \in \text{\rm Hom}_{\text{\rm cont}}(G_y,\bQ/\bZ) =H^1(y,\bQ/\bZ)$ at the arithmetic Frobenius substitution $\varphi_y \in G_y$.
Then for any closed point $i_y:y \hookrightarrow S$ of $S$ the composition
\[\xymatrix{
H^1(y,\bQ/\bZ) \ar[r]^-{\delta} & H^2(y,\mathbb Z) \ar[rr]^-{\text{\rm Gys}_{i_y,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}} && H_c^3(B,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \ar[r]^-{\eqref{isom:cft}} & \bQ/\bZ
}\]
coincides with $\text{\rm tr}_{y,\bQ/\bZ}$, where $\text{\rm Gys}_{i_y,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}$ denotes the Gysin map $\mathbb Z[-1] \to Ri_y^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}$ defined in \cite{sga4.5} Cycle 2.1.1 (see also Proposition \ref{prop:Kummer}\,(1) below), and the map $\delta$ is the connecting homomorphism associated with the short exact sequence
\[ 0 \longrightarrow \mathbb Z \longrightarrow \mathbb Q \longrightarrow \bQ/\bZ \longrightarrow 0. \]
The Artin-Verdier duality shows that for an integer $m$ and a constructible sheaf $\mathscr F$ on $S_\text{\rm \'{e}t}$, the pairing
\begin{equation}\label{dual_Gm}
\xymatrix{
H_c^m(S,\mathscr F) \times \text{\rm Ext}^{3-m}_S(\mathscr F,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \ar[r] & H_c^3(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \ar[r]^-{\eqref{isom:cft}} & \bQ/\bZ
}
\end{equation}
induced by Yoneda pairing is a non-degenerate pairing of finite groups. The first step for a higher-dimensional duality is to replace ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}$ by the object
\begin{equation}\label{eq4-1-3}
\bQ/\bZ(1)'_S:= \bigoplus_{p} \ \varinjlim_{n \ge 1} \ \bZ/p^n(1)_S' \in D^b(S_\text{\rm \'{e}t}),
\end{equation}
where $p$ runs through all rational prime numbers and $n$ runs through all natural numbers. See Definition \ref{def:DC-S} for $\bZ/p^n(1)_S'$, and note that for $r \ge n$ there is a unique transition map $\bZ/p^n(1)_S' \to \bZ/p^r(1)_S'$ induced by the natural inclusion $\mu_{p^n} \hookrightarrow \mu_{p^r}$ on $S[p^{-1}]$, cf.\ Lemma \ref{lem.derived}\,(1). We will explain a version of Artin-Verdier duality using $\bQ/\bZ(1)'_S$ in \S\ref{sect4-2'} below. Our main result on the higher-dimensional duality will be stated in \S\ref{sect4-3} below.
\smallskip
\subsection{Kummer theory}\label{sect4-2}
We discuss the Kummer theory for ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}={\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,S}$.
\begin{prop}\label{prop:Kummer}
Let $p$ be a prime number and let $n$ be positive integer. Let $i$ be the closed immersion $Y:= S \times_\mathbb Z \mathbb F_p \hookrightarrow S$, and let $j$ be the open immersion of the complement $U:=S[p^{-1}] \hookrightarrow S$. Finally let $\mathscr G_{p^n} := R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})$. Then{\rm:}
\begin{enumerate}
\item[(1)]
There is a canonical isomorphism $\beta_U: \mu_{p^n} \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra j^* \mathscr G_{p^n}$ on $U_\text{\rm \'{e}t}$.
\item[(2)]
For any closed subscheme $i_Z: Z \hookrightarrow S$ of codimension $1$ there are canonical Gysin isomorphisms on $Z_\text{\rm \'{e}t}$
\[ \text{\rm Gys}_{i_Z,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}} : \mathbb Z[-1] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \quad \text{and}\quad \text{\rm Gys}_{i_Z,p^n} : \bZ/p^n[-2] \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra Ri_Z^!\mathscr G_{p^n}\,. \]
\item[(3)] There is a unique isomorphism $\beta: \bZ/p^n(1)'_S \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \mathscr G_{p^n}$ completing the following diagram to an isomorphism of distinguished triangles
\stepcounter{equation}
\begin{equation}\label{eq4-2-1}
\xymatrix{
i_*\bZ/p^n[-2] \ar[r]^-g \ar[d]_{i_*(\text{\rm Gys}_{i,p^n})}^{\hspace{-1.5pt}\wr}
& \bZ/p^n(1)'_S \ar[r]^-t \ar[d]_\beta^{\hspace{-1.5pt}\wr}
& Rj_*\mu_{p^n} \ar[rr]^-{\delta_S^\text{\rm val}} \ar[d]_{Rj_*(\beta_U)}^{\hspace{-1.5pt}\wr}
&& i_*\bZ/p^n[-1] \ar[d]_{i_*(\text{\rm Gys}_{i,p^n})[1]}^{\hspace{-1.5pt}\wr} \\
i_* Ri^!\mathscr G_{p^n} \ar[r]^-{i_*}
& \mathscr G_{p^n} \ar[r]^-{j^*} & Rj_* j^* \mathscr G_{p^n} \ar[rr]^-{-\delta^\text{\rm loc}(\mathscr G_{p^n})}
&& i_* Ri^!\mathscr G_{p^n}[1]\,.\hspace{-5pt}}
\end{equation}
Here the top triangle comes from the definition of $\bZ/p^n(1)'_S$, and the bottom triangle from the localization sequence \eqref{eq-0.4.2}.
\item[(4)] There is a canonical distinguished triangle {\rm(}in $D^b(S_\text{\rm \'{e}t})${\rm)}
\[ \bZ/p^n(1)'_S \os{\gamma}\longrightarrow {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \os{\times p^n}\longrightarrow {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \longrightarrow \bZ/p^n(1)'_S[1] \,. \]
\end{enumerate}
\end{prop}
\begin{pf}
(1)
The exact sequence of sheaves $0 \to \mathbb Z \os{\times p^n}\longrightarrow \mathbb Z \to \bZ/p^n \to 0$ induces a canonical distinguished triangle
\begin{equation}\label{KummerZ}
\xymatrix{ \mathbb Z \ar[r]^-{\times p^n} & \mathbb Z \ar[r]^-{\text{\rm can}} & \bZ/p^n \ar[r]^-{\delta^\text{\rm tr}_\mathbb Z} & \mathbb Z[1]\,. }
\end{equation}
Applying the exact functor $R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om(-,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})$, we get a canonical distinguished triangle
\begin{equation}\label{KummerGm}
\xymatrix{ \mathscr G_{p^n} \ar[r]^-{\iota} & {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{\times p^n} & {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}} & \mathscr G_{p^n}[1]\,,}
\end{equation}
where $\iota:=R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om(\text{\rm can},{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})$ and $\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}:=R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om(\delta^\text{\rm tr}_\mathbb Z,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})[1]$.
On the other hand, since $p$ is invertible on $U$, there is an exact sequence
\begin{equation}\label{KummerGmU}
\xymatrix{ 0 \ar[r] & \mu_{p^n} \ar[r] & {\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,U} \ar[r]^-{\times p^n} & {\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,U} \ar[r] & 0 \,. }
\end{equation}
This gives canonical isomorphisms $j^*\mathscr G_{p^n}=R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_U(\bZ/p^n,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,U}) \cong \mu_{p^n}$ as claimed.
(2) First one notes that
\[ R^mi_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \cong \begin{cases}
\mathbb Z \qquad & \hbox{($m=1$)}\\
0 \qquad & \hbox{($m \ne 1$)}
\end{cases} \]
(cf.\ e.g., \cite{milne:adual} p.\ 185, bottom). Therefore
\[ \text{\rm Hom}_Z(\mathbb Z[-1],Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) = H^1_Z(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \cong \bigoplus_{z \in Z} \ \mathbb Z \,, \]
and to get a canonical isomorphism $\text{\rm Gys}_{i_Z,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}$ it suffices to replace $Z$ by a point $z\in Z$ and to find a canonical generator of $H^1_z(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})$. This is done by the localization sequence
\[ \mathscr O_{S,z}^\times \longrightarrow k^\times \os{\delta}\longrightarrow H^1_z(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \longrightarrow H^1(k,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})=0 \]
for the discrete valuation ring $\mathscr O_{S,z}$. Now we take $\delta(\pi)$ as a generator for any prime element $\pi$ of $\mathscr O_{S,z}$.
\par
As for the second Gysin isomorphism in (2), consider a diagram on $Z_\text{\rm \'{e}t}$
\begin{equation}\label{betaY}
\xymatrix{
\mathbb Z[-1] \ar[rr]^-{\times p^n} \ar[d]_{\text{\rm Gys}}^{\hspace{-1.5pt}\wr} \ar@{}[rrd]|{\text{(}*\text{)}}
&& \mathbb Z[-1] \ar[rr]^-{\text{\rm can}} \ar[d]_{\text{\rm Gys}}^{\hspace{-1.5pt}\wr}
&& \bZ/p^n[-1] \ar[rr]^-{-\delta^\text{\rm tr}_\mathbb Z\,[-1]} \ar@{.>}[d]_{\beta_Z}
&& \mathbb Z \ar[d]_{\text{\rm Gys}[1]}^{\hspace{-1.5pt}\wr} \\
Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[rr]^-{\times p^n}
&& Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[rr]^-{-Ri_Z^!(\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}})}
&& Ri_Z^!\mathscr G_{p^n} [1] \ar[rr]^-{Ri_Z^!(\iota)[1]}
&& Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \,,\hspace{-5pt}}
\end{equation}
where $\text{\rm Gys}$ denotes $\text{\rm Gys}_{i_Z,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}$. The top sequence is a distinguished triangle by \eqref{KummerZ} and the rule recalled in \S\ref{sect0-5-1}. The bottom distinguished triangle is obtained by applying $Ri^!$ to \eqref{KummerGm} and shifting suitably.
Now the commutativity of the square $(*)$ implies the existence of a morphism $\beta_Z$ making the diagram commutative (cf.\ \S\ref{sect0-5-2}), which then necessarily is an isomorphism. Moreover, since
\[ \text{\rm Hom}_{D(S_\text{\rm \'{e}t})}(\bZ/p^n[-1],Ri_Z^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \cong \text{\rm Hom}_{D(S_\text{\rm \'{e}t})}(\bZ/p^n,\mathbb Z[-1]) = 0
\quad \hbox{(cf.\ \S\ref{sect0-5-4}\,(2)),} \]
such $\beta_Z$ is unique by Lemma \ref{lem.derived}\,(1). So $\text{\rm Gys}_{i_Z,p^n} := \beta_Z$ gives the desired canonical isomorphism.
\par (The sign $-1$ on $Ri_Z^!(\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}})$ is motivated by the fact that the restriction $(-\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}})|_U$ is the connecting morphism ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,U} \to \mu_{p^n}[1]$ associated with the short exact sequence \eqref{KummerGmU}, which appears in the definition of Deligne's cycle class \cite{sga4.5} Cycle. In particular, by our choice, $\text{\rm Gys}_{i_Z,p^n}$ agrees with the Gysin morphism in \S\ref{sect1-1} when $Z$ is contained in $U$.)
\par
(3) Since $\bZ/p^n(1)'_S$ is concentrated in $[0,1]$, we have
\[ \text{\rm Hom}_{D(S_\text{\rm \'{e}t})}(\bZ/p^n(1)'_S,i_* Ri^!\mathscr G_{p^n}) \cong \text{\rm Hom}_{D(S_\text{\rm \'{e}t})}(\bZ/p^n(1)'_S,i_*\bZ/p^n[-2]) = 0\,. \]
In view of Lemma \ref{lem.derived}\,(1) and the fact that $\beta_U$ and $\text{\rm Gys}_{i,p^n}$ are isomorphisms, our task is to show that the right hand square of \eqref{eq4-2-1} is commutative, which we prove in what follows.
There is a commutative diagram of distinguished triangles
\begin{equation}\label{9diagram}
\xymatrix{
i_* Ri^!\mathscr G_{p^n} \ar[r]^-{i_*} \ar[d]_{\iota}
& \mathscr G_{p^n} \ar[r]^-{j^*} \ar[d]_{\iota}
& Rj_* j^*\mathscr G_{p^n} \ar[rr]^-{-\delta^\text{\rm loc}(\mathscr G_{p^n})} \ar[d]_{\iota}
&& \\
i_* Ri^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{i_*} \ar[d]_{\times p^n}
& {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{j^*} \ar[d]_{\times p^n}
& Rj_* j^*{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[rr]^-{-\delta^\text{\rm loc}({\mathbb G}_{\hspace{-.6pt}\text{\rm m}})} \ar[d]_{\times p^n}
&& \\
i_* Ri^!{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{i_*} \ar[d]_{\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}
& {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[r]^-{j^*} \ar[d]_{\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}
& Rj_* j^*{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[rr]^-{-\delta^\text{\rm loc}({\mathbb G}_{\hspace{-.6pt}\text{\rm m}})} \ar[d]_{\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}
&&\,, \\
& & &&
}
\end{equation}
where the columns are the distinguished triangles coming from \eqref{KummerGm}, and the rows are localization triangles. We now obtain the following diagram of sheaves on $S_\text{\rm \'{e}t}$:
\[ \xymatrix{
R^1j_*\mu_{p^n} \ar[rrrr]^{\delta^\text{\rm val}_S} \ar[dr]^{\beta_U} \ar@{}[rrrrd]|{\text{\scriptsize(a)}}
& & & & i_*\bZ/p^n \ar[dl]_{\text{\rm Gys}_{i,p^n}} \\
& R^1j^* \mathscr G_{p^n} \ar[rr]^{\delta^\text{\rm loc}(\mathscr G_{p^n})} \ar@{}[rrd]|{\text{\scriptsize(c)}}
& & i^* R^2i^! \mathscr G_{p^n} & \\
& j_*j^*{\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[u]^{-\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}} \ar[rr]_{\delta^\text{\rm loc}({\mathbb G}_{\hspace{-.6pt}\text{\rm m}})}_{\raisebox{-0,8cm}{\text{\scriptsize(e)}}}
& & i_* R^1i^! {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \ar[u]_{-\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}_{\hspace{1,2cm}\raisebox{-0,25cm}{\text{\scriptsize(d)}}} \\
j_*{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}{}_{,U} \ar@{->>}[uuu]^{-\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}_-{\hspace{.7cm}\raisebox{-0,25cm}{\text{\scriptsize(b)}}} \ar@{=}[ur] \ar[rrrr]_{\text{\rm ord}} & & & & i_*\mathbb Z\,. \hspace{-5pt} \ar[uuu]_{\text{\rm can}}\ar[ul]^{\text{\rm Gys}_{i,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}}
}\]
The middle square (c) with the four $\delta$'s comes from diagram \eqref{9diagram} and anti-commutes, because $\delta^\text{\rm loc}$ is functorial for the morphism $\delta^\text{\rm tr}_{{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}: {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \to \mathscr G_{p^n}[1]$ and $\delta^\text{\rm loc}(\mathscr G_{p^n}[1]) = - \delta^\text{\rm loc}(\mathscr G_{p^n})[1]$, cf.\ \eqref{eq-0.4.1}. The top arrow $\delta_S^\text{\rm val}$ is induced by residue maps, so the outer square of the diagram commutes by the remark after the proof of (2). The diagram (b) commutes by the definition of $\beta_U$, and the diagram (d) commutes by the definition of $\text{\rm Gys}_{i,p^n}$, i.e., by the commutativity of the diagram \eqref{betaY}. The bottom arrow is induced by the normalized discrete valuations for the points $y \in Y$, and the diagram (e) commutes by the definition of the Gysin map $\text{\rm Gys}_{i,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}}$.
Consequently the diagram (a) anti-commutes, and the right hand square of \eqref{eq4-2-1} commutes by \S\ref{sect0-5-4}\,(1).
Thus we obtain (3).
Finally (4) follows from \eqref{KummerGm} and the isomorphism $\beta$ in (3) by letting $\gamma := \iota \circ \beta$.
\end{pf}
\smallskip
\subsection{Artin-Verdier duality revisited}\label{sect4-2'}
We formulate a version of Artin-Verdier duality using $\bQ/\bZ(1)'_S$ defined in \eqref{eq4-1-3}. Let $p$ be a prime number and let $n$ be a positive integer. There is a commutative diagram of canonical morphisms in $D^b(S_\text{\rm \'{e}t})$
\[\xymatrix{
\bZ/p^n(1)'_S \ar[r]^-\gamma \ar[d]_{\alpha_0} & {\mathbb G}_{\hspace{-.6pt}\text{\rm m}} \\
\ar[ur]_{\tau_0} \bQ/\bZ(1)'_S\,, \hspace{-5pt}
}\]
where $\gamma$ is given in Proposition \ref{prop:Kummer}\,(4), and $\tau_0$ is the inductive limit of $\gamma$ on $r \ge 1$ and primes $p$. Concerning these morphisms, we prepare the following lemma:
\begin{lem}\label{lem4-2}
The map $\alpha_0$ induces an isomorphism
\[ \alpha : \bZ/p^n(1)'_S \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,\bQ/\bZ(1)'_S) \quad \hbox{ in } \; D^+(S_\text{\rm \'{e}t},\bZ/p^n), \]
which fits into a commutative diagram in $D^+(S_\text{\rm \'{e}t},\bZ/p^n)$
\[\xymatrix{
\bZ/p^n(1)'_S \ar[r]^-\beta_-\sim \ar[d]_\alpha^{\hspace{-1.5pt}\wr} & R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \\
\ar[ur]_\tau R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,\bQ/\bZ(1)'_S)\,. \hspace{-5pt}
}\]
Here $\beta$ denotes the isomorphism given in Proposition {\rm\ref{prop:Kummer}\,(3)}, and $\tau$ is induced by $\tau_0$. In particular, $\tau$ is an isomorphism as well.
\end{lem}
\begin{pf}
We define $\alpha$ as the composite of canonical morphisms
\begin{align*}
\alpha : \bZ/p^n(1)'_S & = R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_{S,\bZ/p^n}(\bZ/p^n,\bZ/p^n(1)'_S) \\
& \longrightarrow R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,\bZ/p^n(1)'_S) \os{\alpha_0}\longrightarrow R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,\bQ/\bZ(1)'_S)\,.
\end{align*}
By this definition, the diagram in the lemma commutes obviously. We show that $\alpha$ is an isomorphism. Put
\[ \mathscr H_{p^n}:=R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,\bQ/\bZ(1)'_S) \]
for simplicity. Applying $R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bullet,\bQ/\bZ(1)'_S)$ to the short exact sequence \eqref{KummerZ}, we get a canonical distinguished triangle
\[\xymatrix{ \mathscr H_{p^n} \ar[r] & \bQ/\bZ(1)'_S \ar[r]^-{\times p^n} & \bQ/\bZ(1)'_S \ar[r]^-{\delta_{\bQ/\bZ(1)}} & \mathscr H_{p^n}[1]\,.} \]
Applying $R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bullet,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}})$ to the short exact sequence $0 \to \mathbb Z/p^t \to \mathbb Z/p^{n+t} \to \bZ/p^n \to 0$, we get another canonical distinguished triangle from Proposition {\rm\ref{prop:Kummer}\,(3)}
\[\xymatrix{ \bZ/p^n(1)'_S \ar[r] & \mathbb Z/p^{n+t}(1)'_S \ar[r] & \mathbb Z/p^t(1)'_S \ar[r]^-{\delta_{t,n}} & \bZ/p^n(1)'_S[1]\,.}\]
These triangles form a commutative diagram
\[\xymatrix{
\bZ/p^n(1)'_S \ar[r] \ar[d]_{\alpha}
& \mathbb Z/p^{n+t}(1)'_S \ar[r] \ar[d]_{\alpha_0\;\text{for}\;r+t}
& \mathbb Z/p^t(1)'_S \ar[r]^-{\delta_{t,n}} \ar[d]_{\alpha_0\;\text{for}\;t}
& \bZ/p^n(1)'_S[1] \ar[d]_{\alpha[1]} \\
\mathscr H_{p^n} \ar[r] & \bQ/\bZ(1)'_S \ar[r]^-{\times p^n} & \bQ/\bZ(1)'_S \ar[r]^-{\delta_{\bQ/\bZ(1)}} & \mathscr H_{p^n}[1]\,. \hspace{-5pt}
}\]
Therefore we see that $\alpha$ is an isomorphism by taking the inductive limit on $t \ge 1$ of the upper row.
\end{pf}
\smallskip
The next result replaces ${\mathbb G}_{\hspace{-.6pt}\text{\rm m}}$ by $\bQ/\bZ(1)'_S$ in the duality for constructible torsion sheaves.
\begin{prop}\label{lem:modp}
For $\mathscr L \in D^-(S_\text{\rm \'{e}t},\bZ/p^n)$, there is a commutative diagram of functorial isomorphisms in $D^+(S_\text{\rm \'{e}t},\bZ/p^n)$
\[\xymatrix{
R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_{S,\bZ/p^n}(\mathscr L,\bZ/p^n(1)'_S) \ar@{=}[r]^-{\gamma_*} \ar@{=}[d]_{\alpha_{0*}} & R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\mathscr L,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \\
\ar@{=}[ur]_{\tau_{0*}} R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\mathscr L,\bQ/\bZ(1)'_S)\,.
}\]
Moreover, $\tau_0$ induces a functorial isomorphism
\[ R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\mathscr L,\bQ/\bZ(1)'_S) = R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\mathscr L,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \]
for $\mathscr L \in D^-(S_\text{\rm \'{e}t})$ with constructible torsion cohomology sheaves.
\end{prop}
\begin{pf}
Let $f$ be the natural embedding functor of sheaves: $\mathscr S(S_\text{\rm \'{e}t},\bZ/p^n) \to \mathscr S(S_\text{\rm \'{e}t})$. Then the functor $\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,?) : \mathscr S(S_\text{\rm \'{e}t}) \to \mathscr S(S_\text{\rm \'{e}t},\bZ/p^n)$ preserves injective objects, because its left adjoint $f$ is exact. Hence we have
\[ R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_{S,\bZ/p^n}(\bullet,R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bZ/p^n,?)) = R\hspace{0.5pt}{\mathscr H}\hspace{-1.2pt}om_S(\bullet,?) \]
as bifunctors from $D^-(S_\text{\rm \'{e}t},\bZ/p^n)^\text{\rm op} \times D^+(S_\text{\rm \'{e}t})$ to $D^+(S_\text{\rm \'{e}t},\bZ/p^n)$ by \cite{sga4.5} Ca\-t\'e\-go\-ries D\'e\-ri\-v\'ees II.1.2.3\,(3). The first assertion follows from this fact and Lemma \ref{lem4-2}.
To show the second assertion, we may assume that $\mathscr L$ is a constructible torsion sheaf by a standard argument using spectral sequences. Then by the constructibility, we may further assume that $\mathscr L$ is annihilated by some positive integer, which reduces the problem to the first assertion by considering its $p$-primary components for each prime $p$.
\end{pf}
\smallskip
This result implies the following variant of Artin-Verdier duality.
\begin{cor}\label{cor:AVvariant}
\begin{enumerate}
\item[(1)]
There is a canonical trace isomorphism
\[ \text{\rm tr}_S: H^3_c(S,\bQ/\bZ(1)'_S) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \bQ/\bZ\,. \]
\item[(2)]
For $\mathscr L \in D^b(S_\text{\rm \'{e}t})$ with constructible torsion cohomology sheaves, the pairing
\[ H_c^m(S,\mathscr L) \times \text{\rm Ext}^{3-m}_S(\mathscr L,\bQ/\bZ(1)'_S) \longrightarrow H^3_c(S,\bQ/\bZ(1)'_S) \os{\text{\rm tr}_S}\longrightarrow \bQ/\bZ \]
induced by Yoneda pairing is a non-degenerate pairing of finite groups.
\end{enumerate}
\end{cor}
\begin{pf}
By Proposition \ref{prop:Kummer}\,(4), we have a long exact sequence
\begin{align*}
\dotsb & \longrightarrow H_c^m(S,\bZ/p^n(1)'_S) \longrightarrow H_c^m(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \os{\times p^n}{\longrightarrow} H_c^m(S,{\mathbb G}_{\hspace{-.6pt}\text{\rm m}}) \\
& \longrightarrow H_c^{m+1}(S,\bZ/p^n(1)'_S) \longrightarrow \dotsb .
\end{align*}
By \eqref{isom:cft} and this exact sequence, we obtain $H_c^m(S,\bZ/p^n(1)'_S)=0$ for $m \geq 4$ and a trace isomorphism
\[ \text{\rm tr}_{S,p^n} : H_c^3(S,\bZ/p^n(1)'_S) \hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \bZ/p^n\,. \]
We get the trace isomorphism in (1) by passing to the limit on $r \ge 1$ and then taking the direct sum on $p$.
\par
The claim (2) follows from the non-degeneracy of \eqref{dual_Gm} and Proposition \ref{lem:modp}.
\end{pf}
\smallskip
\subsection{Higher-dimensional duality}\label{sect4-3}
Now let $X$ be a separated scheme of finite type over $S:=\text{\rm Spec}(\mathbb Z)$, with structural morphism $f: X \to S$. We define
\[ \mathscr D_S:= \bQ/\bZ(1)'_S[2] \quad \hbox{ and } \quad \mathscr D_X:= Rf^!\mathscr D_S \quad \hbox{(cf.\ \S\ref{sect0-2})}. \]
For $\mathscr L \in D^+(X_\text{\rm \'{e}t})$, we define the $m$-th \'etale cohomology group {\it with compact support} as
\[ H^m_c(X,\mathscr L) := H^m_c(S,Rf_!\mathscr L)\,, \]
where $H^m_c(S,\bullet)$ denotes the \'etale cohomology group with compact support of $S$ as recalled before. The main result of this section is the following duality (see also \cite{de2}, \cite{sp}):
\begin{thm}\label{thm:global-duality}
\begin{enumerate}
\renewcommand{\labelenumi}{(\arabic{enumi})}
\item There is a canonical trace map
\[ \text{\rm tr}_X:H_c^1(X,\mathscr D_X) \longrightarrow \bQ/\bZ\,. \]
\item For $\mathscr L \in D^b(X_\text{\rm \'{e}t})$ with constructible torsion cohomology sheaves, the pairing
\[ H_c^m(X,\mathscr L) \times {\text{\rm Ext}}^{1-m}_X(\mathscr L,\mathscr D_X) \longrightarrow H_c^1(X,\mathscr D_X) \os{\text{\rm tr}_X}\longrightarrow \bQ/\bZ \]
induced by Yoneda pairing is a non-degenerate pairing of finite groups.
\end{enumerate}
\end{thm}
\begin{pf}
(1) Let $\text{\rm tr}_f: Rf_!\mathscr D_X=Rf_!Rf^!\mathscr D_S \to \mathscr D_S$ be the canonical trace map, i.e., the adjunction morphism for the adjunction between $Rf^!$ and $Rf_!$ (\cite{sga4} XVIII.3.1.4). We then define the trace map $\text{\rm tr}_X$ as the composite
\[ \text{\rm tr}_X: H_c^1(X,\mathscr D_X) \os{\text{\rm tr}_f}\longrightarrow H_c^1(S,\mathscr D_S) = H^3_c(S,\bQ/\bZ(1)'_S) \os{\text{\rm tr}_S}\hspace{9pt}{}^\sim\hspace{-16.5pt}\lra \bQ/\bZ\,. \]
(2) There is a commutative diagram of Yoneda pairings
\[ \xymatrix{
H^m_c(X,\mathscr L) \ar@<19pt>@{=}[d] \hspace{-40pt}
& \times \; \text{\rm Ext}^{1-m}_X(\mathscr F,Rf^!\mathscr D_S) \ar@<7pt>@{=}[d] \ar[r]
& H^1_c(X,Rf^!\mathscr D_S) \ar[d]^{\text{\rm tr}_f} \\
H^m_c(B,Rf_!\mathscr L) \hspace{-28pt}
& \times \; \text{\rm Ext}^{1-m}_S(Rf_!\mathscr L,\mathscr D_S) \ar[r]
& H^1_c(S,\mathscr D_S)\,,
}\]
and the result follows from Corollary \ref{cor:AVvariant}.
\end{pf}
\newpage
| 2024-02-18T23:40:05.776Z | 2009-10-21T01:58:40.000Z | algebraic_stack_train_0000 | 1,322 | 35,056 |
|
proofpile-arXiv_065-6563 | \section{Introduction}
Jet correlation measurements at the Brookhaven Relativistic Heavy Ion Collider show a striking increase in the yield of associated particles in a narrow range in relative azimuthal angle near $\phi\approx 0$ \cite{Putschke:2007mi}. This region of enhanced particle production is called a ridge because it extends over a broad range in relative pseudorapidity. Interestingly, a similar ridge of enhanced production has been reported for two-particle correlations without a jet trigger \cite{Daugherity:2008su}.
We seek a common explanation for the jet-triggered hard ridge and the untriggered soft ridge based on particle production in an early Glasma stage followed by transverse flow \cite{Dumitru:2008wn},
\cite{Gavin:2008ev}.
Correlation measurements of high momentum particles provided the first evidence of jet quenching in nuclear collisions \cite{Arsene20051,Back200528,Adcox:2004mh,Adams:2005dq}. New jet-tagged correlation studies observe a high transverse momentum trigger particle and measure the yield of associated low $p_t$ particles \cite{Putschke:2007mi,Abelev:2009qa,Nattrass:2008tw,Wenger:2008ts}. The yield is reported as a function of azimuthal angle $\phi$ and pseudorapidity $\eta$ relative to the direction of the trigger particle in gold-gold, Au+Au, and deuteron-gold, d+Au, collisions. A jet peak near $\eta= \phi = 0$ likely consists of particles from the fragmenting jet trigger, assuming the high $p_t$ trigger particle is indeed from a jet. The hard ridge appears in Au+Au collisions as an enhancement of the yield at small $\phi$ under the jet peak. The ridge is much wider in $\eta$ than the jet peak, which has a gaussian width of $\sim 0.2$ units, and is absent from d+Au collisions.
The staggering success of such studies has inspired broader interest in correlations of all particle types, with and without triggers. Untriggered correlation studies measure the number of particle pairs over a more comprehensive momentum range dominated by typical low momentum particles of $p_t < 1$~GeV \cite{Daugherity:2006hz,Daugherity:2008su,Adams:2005aw,Adams:2004pa,Adams:2005ka,Adamova:2008sx}. Particles in this range are primarily thermal and well described by hydrodynamics in Au+Au collisions \cite{Arsene20051,Back200528,Adcox:2004mh,Adams:2005dq}. The soft ridge is an enhancement of correlations in central collisions that -- like the hard ridge -- is broad in $\eta= \eta_1-\eta_2$ and centered in the near side, i.e., $\eta= \phi = 0$. The soft ridge is greatly reduced in peripheral Au+Au collisions and absent in $pp$ collisions.
We have described the soft ridge as a consequence of early-stage correlations in concert with late-stage transverse flow \cite{Gavin:2008ev}. Ours is one of a family of models in which particles are initially correlated at the point of production \cite{Voloshin:2003ud,Dumitru:2008wn,Dusling:2009ar,Lindenbaum:2007ui,Sorensen:2008bf,Peitzmann:2009vj,Takahashi:2009na}.
Flow then boosts the correlated particles into a small opening angle in $\phi$. Flow models have also been used to describe qualitative features of the hard ridge \cite{Pruneau:2007ua}, \cite{Takahashi:2009na}.
The hard ridge has most commonly been described as a consequence of the jet passing through the flowing bulk matter produced by the nuclear collision \cite{Armesto:2004pt,Romatschke:2006bb,Majumder:2006wi,Chiu:2005ad,Hwa:2009bh,Dumitru:2007rp,Wong:2007pz,Mizukawa:2008tq,Wong:2009cx}. While in principle jets and minijets may contribute to the soft ridge \cite{Trainor:2007ny}, the overwhelming success of hydrodynamics at low $p_t$ suggests both effects play a role. Of particular interest in this regard is work by Shuryak in which jet quenching and transverse flow of the bulk play a combined role \cite{Shuryak:2007fu}. Jet quenching produces a near side bias by suppressing the away side jet. Near side jet particles are then correlated with other particles from the same jet or transversely flowing bulk particles.
Important motivation for our work comes from data from the PHOBOS collaboration, which suggests that the hard ridge and possibly the soft ridge may extend over the broad range $-4 < \eta < 2$ \cite{Wenger:2008ts}. Correlations over several rapidity units can only originate at the earliest stages of an ion collision when the first partons are produced \cite{Dumitru:2008wn,Gavin:2008ev}. Hydrodynamics, hadronization, resonance decay, and freeze out can modify these correlations, but causality limits such effects to a horizon of roughly from one to two rapidity units. Many models referenced above are challenged by this data \cite{Nagle:2009wr}.
In this paper we extend the model of Ref.\ \cite{Gavin:2008ev} to incorporate jet production and address the soft and hard ridges. Long range rapidity correlations and the insight they provide on early time dynamics are our driving concerns. We take particle production to occur through a Glasma state. Our emphasis is on how computed Glasma correlations can affect ridge measurements.
In Ref.\ \cite{Gavin:2008ev} we found excellent agreement with the peak amplitude and azimuthal width shown in current Au+Au data.
In the next section we extend this work here to include Cu+Cu systems. We then extend the model of Ref.\ \cite{Gavin:2008ev} to address varying $p_t$ ranges so that we may address the hard ridge.
In Sec.\ \ref{sec:jets} we add a contribution of jets following the model of Ref.\ \cite{Shuryak:2007fu}. We extend that model to compute both the strength and azimuthal dependence of the jet contribution to the hard and soft ridges. We then combine the flow and jet effects and find that correlations of thermally produced pairs constitute a significant contribution to the triggered measurement. We then discuss how experiments might distinguish the different contributions.
\section{Glasma Correlations}
The theory of Color Glass Condensate (CGC) predicts an early Glasma stage in a high energy collision in which particles are produced by strong longitudinal color fields. As nuclei collide, the transverse fields of each nucleus are instantaneously transformed into longitudinal fields that are approximately uniform in rapidity. These fields are essentially random over transverse distances $r_t$ larger than the saturation scale $Q_s^{-1}$, where $Q_s\sim 1-2$~GeV. We can think of such field configurations as consisting of a collection of longitudinal flux tubes. Flux tubes are ubiquitous in QCD-based descriptions of high energy collisions. In the Glasma they are closely packed and not strictly distinct due to saturation. They are, however, uncorrelated for $r_t > Q_s^{-1}$. This is their essential feature for this work. The Glasma changes to plasma as particles form from the fields and thermalize.
In the saturation regime, the number of gluons in a rapidity interval $\Delta y$ is
\be
N = ({{dN}/{dy}})\Delta y \sim {\alpha_s}^{-1}Q_s^2R_A^2,
\label{eq:Nscale}
\ee
where $R_A$ is the nuclear radius and $\alpha_s$ is the strong coupling constant at the saturation scale $Q_s$ \cite{Kharzeev:2000ph}. We understand (\ref{eq:Nscale}) as the number of flux tubes $ K \sim (Q_sR_A)^2$ times the density of gluons per flux tube $\propto \alpha_s^{-1}$.
The scale of correlations is set by
\be
{\cal R} = {{\langle N^2\rangle - \langle N\rangle^2 - \langle N\rangle}\over{\langle N\rangle^2}},
\label{eq:Rdef}
\ee
where the brackets denote an average over collision events.
This quantity vanishes for uncorrelated gluons, for which multiplicity fluctuations are necessarily Poissonian. In Ref.~\cite{Gavin:2008ev} we argued that the Glasma correlation strength is ${\cal R}\propto \langle K\rangle^{-1} = (Q_sR)^{-2}$, and found that the Glasma contribution to correlations is
\be
{\cal R}{{dN}/{dy}} \sim {\alpha_s}(Q_s)^{-1},
\label{eq:CGCscale}
\ee
a result consistent with calculations of Dumitru et al. in Ref.~\cite{Dumitru:2008wn}. Equations (\ref{eq:Nscale}) and (\ref{eq:CGCscale}) constitute initial conditions for the hydrodynamic evolution of the system.
These Glasma initial conditions affect the final state correlations in several ways. First, particles emitted from the same tube share a common origin that is localized to a very small transverse area, since $Q_s \ll R_A$. Second, the flux tubes correlate particles over a large pseudorapidity range. Some of the flux tubes can stretch across the full longitudinal extent of the system at times $< 1$~fm. These flux tubes rapidly fragment. At later times, the particles they produce can be separated by large longitudinal distances depending on their momenta. Consequently, subsequent scattering and hydrodynamic evolution cannot erase their correlations -- they are causally disconnected.
Third, the strength of the correlation depends on the number of flux tubes. The number of tubes depends on the centrality and energy of the collision, as well as the transverse area of the tube, all of which, in turn, depend on $Q_s$. Finally, as a result of the common origin, particles coming from the same tube must have the same initial radial position and feel the same effects from flow, independent if their rapidity.
During the Glasma phase, flux tubes thermalize into partons and pressure builds as the systems moves toward an equilibrated Quark Gluon Plasma. Partons initially localized in tubes are now localized in small fluid cells with a uniform azimuthal distribution of particles. Following a Hubble-like expansion of the system, the transverse fluid velocity takes the form $\gamma_t\mathbf{v}_t = \lambda \mathbf{r}_t$. All of partons in a fluid cell are boosted radially depending on their initial radial position. Consequentially, partons in any given fluid cell gain transverse momentum in the radial direction and the relative angle between any two momentum vectors in that cell becomes smaller. Furthermore, fluid cells at a larger radial position have a larger final transverse velocity and the relative angle between parton momentum vectors is narrower. This angular narrowing depends only on the initial radial position and the small transverse area of the flux tube source. In our simplistic view, all flux tubes are uniform in rapidity and extend to the same longitudinal length. This provides for a longitudinally uniform fireball that experiences the same radial flow at every longitudinal position. In this way, at every rapidity, angular correlations are enhanced in the same way and the initial state spatial correlations are both preserved through freeze out and represented in momentum correlations.
Following \cite{Gavin:2008ev} we define the momentum space correlation function at freeze out as
\be
r (\mathbf{p}_1, \mathbf{p}_2) = \rho_2 (\mathbf{p}_1, \mathbf{p}_2) - \rho_1(\mathbf{p}_1)\rho_1( \mathbf{p}_2)
\label{eq:MomCorr0}
\ee
where $\rho_2 (\mathbf{p}_1, \mathbf{p}_2) = dN/dy_1d^2p_{t1}dy_2d^2p_{t2}$ is the pair distribution, and $ \rho_1(\mathbf{p}) \equiv dN/dyd^2p_t$ is the single particle spectrum. We describe the effect of flow using the familiar blast-wave model \cite{Schnedermann:1993ws,Kiyomichi:2005zz, Barannikova:2004rp, Iordanova:2007vw, Retiere:2003kf}. In this model, the single particle spectrum is $ \rho_1(\mathbf{p}) = \int f(\mathbf{x},\mathbf{p}) \, d\Gamma$, where $f(\mathbf{x},\mathbf{p}) = {(2\pi)^{-3}}\exp\{-p^\mu u_\mu/T\}$ is the Boltzmann phase-space density and $d\Gamma = p^\mu d\sigma_{\mu}$ is the differential element of the Cooper-Frye freeze out surface. We assume a constant proper time freeze out, so that $d\Gamma = \tau_F m_t \cosh(y-\eta)d\eta d^2 r_t$, where $\eta = (1/2)\ln((t+z)/(t-z))$ is the spatial rapidity.
We argue in Ref.~\cite{Gavin:2008ev} that the final-state momentum space correlation function is
%
\be
r(\mathbf{p}_1, \mathbf{p}_2) =
\!\!\int c(\mathbf{x}_1, \mathbf{x}_2)
\frac{f(\mathbf{x}_1,\mathbf{p}_1)}{n_1(\mathbf{x}_1)}
\frac{f(\mathbf{x}_2,\mathbf{p}_2)}{n_1(\mathbf{x}_2)}
d\Gamma_1d\Gamma_2,
\label{eq:MomCorr}
\ee
where $n_1(\mathbf{x}) = \int f(\mathbf{x},\mathbf{p})d\Gamma$. The spatial correlation function $c(\mathbf{x}_1, \mathbf{x}_2)$ depends on the Glasma conditions as follows
\be
c(\mathbf{x}_1, \mathbf{x}_2)
= {\cal R}\,\delta(\mathbf{r}_t ) \rho_{{}_{FT}} (\mathbf{R}_t),
\label{eq:param}
\ee
where $\mathbf{r}_t = \mathbf{r}_{t1} - \mathbf{r}_{t2}$ is the relative transverse position, and $\mathbf{R}_t = (\mathbf{r}_{t1} + \mathbf{r}_{t2})/2$ is the average position. The delta function accounts for the fact that Glasma correlations are highly localized to $r_t < Q_s^{-1}$.
The factor $\rho_{_{FT}}(\mathbf{R}_t)$ describes the transverse distribution of the flux tubes in the collision volume, which we assume follows the thickness function of the colliding nuclei.
We comment that the form of (\ref{eq:param}) holds as long as $\cal R$ is unmodified from its initial Glasma value by particle production and hydrodynamic evolution. This is only strictly true as long as a) subsequent evolution doesn't change the relative number of particles in the rapidity interval of interest; and b) the number of observed hadrons is proportional to the initial number of gluons. Causality prevents these effects from altering $\cal R$ for truly long range correlations, $|\eta_1 - \eta_2| > 1-2$. This would hold for smaller rapidities in Glasma theory as long as boost invariance is a reasonable approximation, since $dN/dy$ is then a hydrodynamic constant of motion in each event. Moreover, assumption (b) is common in Glasma/CGC calculations. On the other hand, for $|\eta_1 - \eta_2| < 1-2$, the experimental $dN/dy$ is not flat and will change with time due to particle diffusion and number changing processes; see \cite{Gavin:2006xd}. For now, we will assume that $\cal R$ is constant and defer the hydrodynamic modification for later work.
The analysis in Ref.~\cite{Gavin:2008ev} focused on a measurement of the near side peak of the soft ridge in 200 GeV Au+Au using the observable $\Delta \rho/\sqrt{\rho} = (\rho_{sib} - \rho_{ref})/\sqrt{\rho_{ref}}$ \cite{Daugherity:2008su}. The quantity $\rho_{sib}(\phi , \eta)$ represents the distribution of ``sibling pairs" from the same event, as a function of relative pseudorapidity and relative azimuthal angle, and is comparable to our $\rho_2$. The quantity $\rho_{ref}(\phi , \eta)$ represents uncorrelated pairs from mixed events and is equivalent to the square of our $\rho_1$. The difference $(\rho_{sib} - \rho_{ref})$ is a measure of correlated pairs and is comparable to the integral of (\ref{eq:MomCorr}) over the transverse momenta and average azimuthal angle $\Phi=(\phi_1+\phi_2)/2$. It is convenient to compute the quantity
\be
\frac{\Delta\rho}{\rho_{ref}}=
\frac{\int r(\mathbf{p}_1, \mathbf{p}_2)p_{t1} p_{t2} dp_{t1} dp_{t2} d\Phi}
{\int \rho_1(p_{t1}) \rho_1(p_{t2}) d^2p_{t1} d^2p_{t2} },
\label{eq:dratio}
\ee
which is independent of the overall scale of the multiplicity. We emphasize that this quantity includes correlated pairs in which both particles can have any momentum, while measurements of the hard ridge correlate particles from different $p_t$ ranges. We will extend our approach to address such quantities below.
\begin{figure}
\centerline{\includegraphics[width=4in]{AuAmpVsNu.pdf}}
\caption[]{Au+Au amplitude as a function of centrality ($\nu=2N_{bin}/N_{part}$) for both 200 and 62 GeV. Solid lines represent Glasma initial collisions ${\cal{R}}dN/dy$ from (\ref{eq:CGCscale}) plus blast wave flow (\ref{eq:deltaRho}); parameters are unchanged from Ref.\ \cite{Gavin:2008ev}. Dashed lines replace Glasma with independent $NN$ initial conditions. }
\label{fig:fig0}\end{figure}
To construct the observed quantity $\Delta\rho/\sqrt{\rho_{ref}}$, we notice that $r(\mathbf{p}_1, \mathbf{p}_2)$ computed from eq. (\ref{eq:dratio}) is proportional to the correlations strength ${\cal R}$ (\ref{eq:Rdef}). We can write
\be
\frac{\Delta\rho}{\rho_{ref}}={\cal R}F(\phi),
\label{eq:dratioNorm}
\ee
where $F(\phi)$ is normalized such that $\int_{0}^{2\pi} F(\phi) d\phi =1$. The distribution $F(\phi)$ depends only on the blast-wave parameters $\gamma m/T$ and $v_s$, and represents the angular correlations of particles from flux tubes after hydrodynamic expansion. The factor ${\cal R}$ scales the strength of the correlations with both energy and centrality and determines the rapidity dependence (which is flat in this case).
We now combine (\ref{eq:dratio}) and (\ref{eq:Nscale}) to obtain the observed quantity
\be
\Delta \rho/\sqrt{\rho_{ref}} = \kappa{\cal R} dN/dy~F(\phi),
\label{eq:deltaRho}
\ee
where we equate the factor ${\cal R}dN/dy$ with (\ref{eq:CGCscale}), which accounts for all of the Glasma energy and centrality dependence. The scale constant $\kappa$ is independent of energy.
As described in \cite{Gavin:2008ev} we set $\kappa $ only for Au+Au 200GeV collisions such that $F(\phi)$ for the most central collisions is aligned with the most central data point. Although blast-wave parameters have some energy dependence, the Glasma factor (\ref{eq:CGCscale}) allows for strong agreement with the 62~GeV data without further adjustment of $\kappa$.
\begin{figure}
\centerline{\includegraphics[width=4in]{CuAmps.pdf}}
\caption[]{Cu+Cu amplitude as a function of centrality ($\nu=2N_{bin}/N_{part}$) for both 200 and 62 GeV. Solid and dashed lines compare Glasma and independent NN initial conditions as in Fig.\ \ref{fig:fig0}.}
\label{fig:fig1}\end{figure}
We now apply (\ref{eq:deltaRho}) and (\ref{eq:CGCscale}) to address new and forthcoming data for $\Delta\rho/\sqrt{\rho_{ref}}$ and the azimuthal width Cu+Cu for both 200 and 62 GeV as a function of centrality. No parameters are adjusted from \cite{Gavin:2008ev} -- in that sense, these results are predictions. The $Q_s^2$ has a dependence on the density of participants $\rho_{\rm part}$ that is determined in Ref.~ \cite{Kharzeev:2000ph}. The saturation scale in central Cu+Cu is then
\be
Q_s({\rm Cu})^2=Q_c({\rm Au})^2\frac{\rho_{\rm part}({\rm Cu,\, central})}{\rho_{\rm part}{(\rm Au,\, central})}.
\label{qscu}
\ee
After this scaling, we take the relative centrality dependence of $Q_s$ to be the same in the Cu and Au systems. This assumption can be refined by measuring the centrality dependence of $dN/dy$ in Cu+Cu. Similarly, we obtain the blast wave parameters $T$ and $v_s$ in Cu+Cu from the Au+Au values by assuming that they scale with the number of participants. These assumptions can be refined as single particle spectra measurements as a function of centrality become available.
In Fig.\ \ref{fig:fig0} we reproduce calculations of $\Delta\rho/\sqrt{\rho_{ref}}$ from \cite{Gavin:2008ev} using (\ref{eq:deltaRho}) for Au+Au at 200 GeV and 62 GeV. Preliminary STAR data are from \cite{Daugherity:2008su}. Here, we compare these calculations to dashed curves computed assuming initial conditions from the wounded nucleon model rather than Glasma. The wounded nucleon model implies ${\cal R} dN/dy$ in (\ref{eq:deltaRho}) is independent of centrality. We fix its value to the most peripheral 200 GeV Au data as a proxy for $pp$, and take it to be independent of energy. Note that we might have alternatively fixed the wounded nucleon model to peripheral collisions at all energies, but this would introduce new unconstrained parameters. The importance of the centrality dependence of the Glasma initial condition (\ref{eq:CGCscale}) is evident, particularly at the highest energy.
Figure \ref{fig:fig1} shows our prediction for the soft ridge amplitude in Cu+Cu 200 and 62 GeV systems as a function of centrality compared to preliminary STAR data \cite{Chanaka}. In both panels, the dashed line represents wounded nucleon model initial conditions as in Fig.\ \ref{fig:fig0}. The dashed lines are included to show how the energy and system-size dependence affects the calculation when the Glasma dependence (\ref{eq:CGCscale}) is omitted. The solid lines are the result of including the Glasma scaling (\ref{eq:CGCscale}) adjusted by (\ref{qscu}). The error band represents a $~10\%$ uncertainty in the blast-wave parameters plus an additional uncertainty in the parameterization of $Q_s$ that increases with decreasing centrality. We comment that the wounded nucleon model with energy independent ${\cal R}dN/dy$ adequately describes the data in peripheral collisions but fails in central collisions. This also makes the wounded nucleon model prediction larger than the Glasma curve in 62 GeV Cu+Cu.
In Fig.\ \ref{fig:fig2} we show the soft ridge azimuthal width in Cu+Cu systems compared to preliminary STAR measurements as compared to previously published Au+Au result \cite{Gavin:2008ev}. We have also included preliminary STAR measurement of the Cu+Cu width \cite{Chanaka} to compare with other preliminary STAR measurements of the soft ridge in Au+Au width \cite{Daugherity:2008su}. In \cite{Gavin:2008ev} we find that the azimuthal width of the near side peak of the soft ridge in Au+Au is due to radial flow, is constant with a change in energy, and is relatively uniform with change in centrality. The error band is representative of the uncertainty of fitting an offset gaussian to the angular calculation. Since the azimuthal width is completely determined by radial flow, which depends completely on the choice of centrality and blast-wave parameterizations, and all of those parameters have remained unchanged, we calculate the same enhancement in the width for Cu+Cu as Au+Au. The black line extending to $\nu=6$ is the Au+Au result from \cite{Gavin:2008ev}, and the overlaid blue line extending to $\nu=4$ with the hatched error band is the Cu+Cu result. Again, as with the Au+Au result, the Cu+Cu result is independent of energy since the measured transverse expansion does not depend on energy.
\begin{figure}
\centerline{\includegraphics[width=4in]{200Azimuth.pdf}}
\caption[]{Comparison of the previously published angular width calculations for Au+Au (black line) with the angular widths for Cu+Cu systems (blue line, hatched error band) compared with preliminary STAR Au+Au 200 and 62 GeV data as well as Cu+Cu 200 GeV data \cite{Chanaka}. Width calculations remain independent of energy and nearly independent of system}
\label{fig:fig2}\end{figure}
A key feature of the flow-based descriptions of the ridge is that it is the angular width, $\sigma_r$, of correlated pairs decreases as the mean $p_t$ of the pair increases. The greater the radial boost given to a fluid cell, the narrower the relative angle between the momentum vectors of particles in that cell. A very high momentum correlated pair is more likely to have come from a fluid cell that received a very large transverse boost. To study whether this effect is present, we compute $\Delta \rho/\sqrt{\rho_{ref}}$ for pairs of $p_t > p_{t,\,{\rm min}}$. As $p_{t,\,{\rm min}}$ increases, we also expect the amplitude of the ridge to decrease, since it is more difficult to find higher $p_t$ bulk particles. We therefore compute
\begin{eqnarray}
\left( \frac{\Delta\rho}{\rho_{ref}} \right)_{p_{t}} =
\frac
\int \limits_{p_{t2,min}}^{p_{t2,max}} \int \limits_{p_{t1,min}}^{p_{t1,max}}
r(\mathbf{p}_1, \mathbf{p}_2)
}
{\int \limits_{p_{t1,\rm{min}}}^{p_{t1,max}} \rho_1(p_{t1}
\int \limits_{p_{t2,min}}^{p_{t2,max}} \rho_1(p_{t2})
}\nonumber \\
\nonumber \\
={\cal{R}}F(\phi;p_{t1,min},p_{t1,max},p_{t2,min},p_{t2,max}),
\label{eq:ptmin}
\end{eqnarray}
where the integration measures are the same as in (\ref{eq:dratio}). To obtain the measured ratio we write
\be
\frac{(\Delta \rho/\sqrt{\rho_{ref}})_{p_t}}{\Delta \rho/\sqrt{\rho_{ref}}} =
\frac{F(\phi;p_{t,min},\infty,p_{t,min},\infty)}{F(\phi)}
\frac{\int_{p_{t1,min}}^\infty \rho_1}{\int_0^\infty \rho_1}.
\label{eq:CGCpt}
\ee
For increasing values of $p_{t,min}$ we calculate the correlation function as before using (\ref{eq:ptmin}) and (\ref{eq:CGCpt}) and find that the azimuthal width does indeed decrease as shown in the upper panel of Fig.\ \ref{fig:fig2a}.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{200AzimuthCuts.pdf}}
\caption[]{Top panel shows the angular width for Au+Au systems with increasing minimum $p_{t,min}$ limits compared with preliminary STAR Au+Au 200 GeV data. The dashed line represents the $p_{t,min}$=0 calculation shown in Fig.\ \ref{fig:fig2}. The lower panel shows the azimuthal width for most central collisions vs the $p_{t,min}$ limit for both Au+Au and Cu+Cu 200 GeV. Preliminary STAR data for Cu+Cu is also shown \cite{Chanaka}. }
\label{fig:fig2a}\end{figure}
The upper panel of Fig.\ \ref{fig:fig3} shows the correlation amplitude vs. centrality for different choices of $p_{t,min}$. The amplitude decreases with increasing $p_{t,min}$ because the number of particles contributing to correlations is reduced. The lower panel of Fig.\ \ref{fig:fig3} shows the amplitude of $\Delta\rho / \sqrt{\rho_{ref}}$ for the most central collision as a function of the choice of $p_{t,min}$ for Au+Au and Cu+Cu at 200 GeV. We compare these calculations to preliminary Cu+Cu data \cite{Chanaka}. Similarly, the blue curve in the lower panel of Fig.\ \ref{fig:fig2a} represents the azimuthal width of the soft ridge in most central collisions as a function of $p_{t,min}$.
We see that the azimuthal width of the hard ridge is smaller than that of of the soft ridge, but as the $p_{t,min}$ limit of the soft ridge is increased, the amplitude of the correlations drops and the azimuthal width narrows. In the $p_t$ range of the hard ridge, it appears that the azimuthal width could be narrow enough, but the amplitude is not directly comparable. To understand this difference, we must understand the differences in the two measurements. The most significant difference is the choice of the momentum range of the correlated particles. The hard ridge measurement analyzes the yield of associated particles per jet trigger where the associated particle $p_t$ range and the trigger range do not overlap. The soft ridge measurement, however, finds the number of correlated pairs per particle where both particles are in the same range with $p_t$ above minimum bias. The normalization of the soft ridge is found by taking the square root of the uncorrelated pair reference spectrum.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{Au200AmpCuts.pdf}}
\caption[]{Top panel: Au+Au 200 GeV amplitude calculations for increasing $p_{t,min}$ limits. The dashed line is calculation with $p_{t,min}=0$. Lower panel: the soft ridge amplitude for most central collisions plotted as function of the $p_{t,min}$ limit for both Au+Au and Cu+Cu 200 GeV. Preliminary STAR data is also shown \cite{Chanaka}.}
\label{fig:fig3}\end{figure}
STAR measures the hard ridge, or yield of associated particles per jet trigger, for Au+Au 200 GeV for $3<p_{t,trigg}<4$ GeV with $2<p_{t,assoc}<3$ \cite{Putschke:2007mi}. Identifying $p_{t1}$ with the trigger range and the associated range with $p_{t2}$, we calculate $\Delta\rho / \sqrt{\rho_{ref}}$ and transform to yield by
\begin{eqnarray}
{\rm Yield}=
\left( \frac{\Delta\rho}{\sqrt{\rho_{ref}}} \right)_{p_t}
\left( \frac{\int\rho_1(p_{t2})}{\sqrt{\int\rho_1(p_{t1})\int\rho_1(p_{t2})}} \right).
\label{eq:yield}
\end{eqnarray}
At higher ranges of $p_{t1,2}$ the contribution from jets should become more significant. It is important therefore to know the relative contribution of thermal particles and jet particles. As will be discussed in more detail later, we decompose the total particle spectrum into thermal bulk and jet fractions, and to obtain the contribution of bulk correlations to the hard ridge, we multiply (\ref{eq:yield}) by the bulk fraction $\int\rho_1(p_{t1})/\int\rho_{tot}(p_{t1})$ where $\int\rho_{tot}(p_{t1})$ is the total number of particles in the range of $p_{t1}$.
The blue curve in Fig.\ \ref{fig:fig4} represents the contribution to the hard ridge from only thermal bulk pairs. As can be seen on the figure, bulk-bulk correlations contribute significantly to the amplitude of the triggered measurement, but seems to have a somewhat narrow profile in azimuth. It was shown in \cite{Shuryak:2007fu} that a jet acquires angular correlations with flowing matter due to quenching, but the width of the correlation is wider than the data. The contribution of jet correlations with bulk particles could make up the difference between the blue curve in Fig.\ \ref{fig:fig4} and the data by increasing both the amplitude and the width of the calculation.
In the next section we combine a theory angular correlations from \cite{Shuryak:2007fu} with spatial correlations of jets and flux tubes to obtain a jet-bulk contribution to the hard ridge.
\section{Jets, Glasma, and Correlations}\label{sec:jets}
As the $p_t$ of correlated particles is increased, the contributions from jets should become prevalent, particularly for small $\eta$. At small rapidity differences, correlations of jet particles with fragments should be large, but restricted to the size of the jet cone, and the transfer of momentum from jet particles to bulk particles is causally limited to $\sim 1-2$ units in rapidity. The existence of correlations with jet particles at larger $\eta$ would require a correlation early in the collision that remains through the longitudinal expansion of the system and is still present at freeze out. Both the hard collisions and flux tubes are made in the initial moments of the nuclear collision. Assuming that the entire overlap region of the colliding nuclei is in the saturation regime, flux tubes would fill the collision volume and a jet formed at any transverse position would be accompanied by a flux tube at the same position.
Since the flux tube extends to large rapidities, the correlation of particles from the tube and the jet can extend to large $\eta$. Angular correlations arise since particles from the tube acquire a radial trajectory from flow as before, but the jet trajectory has a bias in the radial direction due to quenching \cite{Shuryak:2007fu}.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{JornRidge.pdf}}
\caption[]{Angular profile of the jet triggered ridge in a rapidity range away from the jet peak. The solid black line combines of long range correlations. Bulk-bulk (blue line) and jet-bulk (dashed red line) contributions are shown separately. The jet fraction is determined by $p_{s}=1.25$ GeV; bulk-bulk correlations make up $\sim75\%$ of the total amplitude.}
\label{fig:fig4}
\end{figure}
We construct a distribution of jets as follows. We assume that jets are produced with a hard scattering rate $f_0(\mathbf{p}_1)$ that is independent of position, multiplied by a spatial profile $P_{rod}\propto (1-r_1^2/R_A^2) $ that is roughly proportional to the density of binary collisions. The phase space density of jet particles is then
\begin{eqnarray}
f_{J}(\mathbf{x}_1,\mathbf{p}_1)=
f_0(\mathbf{p}_1)P_{prod}(r_1)S(r_1,\phi_1),
\label{eq:jet}
\label{eq:sruvive0}
\end{eqnarray}
where $S$ is the survival probability of a jet due to jet quenching. In practice $f_0(\mathbf{p}_1)$ cancels in $\Delta \rho/\rho_{ref}$ so we need not specify it. We follow Ref.~\cite{Shuryak:2007fu} and take
\begin{eqnarray}
S(r_1,\phi_1)=exp(-L(r_1,\phi_1)/l_{abs}),
\label{eq:sruvive}
\end{eqnarray}
where $l_{abs}=0.25 fm$ is the jet attenuation length. The survival of the jet depends on the path it takes out of the medium
\be
L(r_1,\phi_1)=\sqrt{R_A^2-r_1^2\sin^2(\phi_1)}-r_1\cos(\phi_1).
\label{eq:jpath}
\ee
The path (\ref{eq:jpath}) is the distance a jet would have to travel out of a circular transverse area at an angle $\phi_1$ with respect to the radial vector pointing to its position of production $r_1$ \cite{Shuryak:2007fu}. In view of (\ref{eq:jpath}) and (\ref{eq:sruvive}), the shortest path, which is the path a jet is most likely to survive, is one that is radially outward from its position of production. Although a jet production is at a minimum on the surface, $r\approx R_A$, with little material for the jet to pass through, the survival probability is maximum in any direction (not pointing into the volume). The resulting angular correlations are weak, since only a small fraction of phase space is restricted by quenching. The largest probability of production would occur at the center where $L\approx R_A$ in all directions, but there are no correlations in this case since quenching would be maximum. More concisely, jets are less likely to be produced at the surface and have a wide angular distribution. The most jets are produced the center and would have the narrowest correlation with radially flowing particles, but have the highest probability of being quenched. As the production point of the jet moves from the center toward the surface, the probability of production decreases, the probability of survival increases, and the angular correlation with radially flowing particles widens. Integration over all possibilities determines the width of the
correlations.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{NormalizedHardRidge.pdf}}
\caption[]{Comparison of the angular shape of the jet triggered ridge to long range bulk-bulk (blue line) and jet-bulk (dashed red line) correlations arbitrarily normalized to the measured peak height.}
\label{fig:fig4b}
\end{figure}
The calculation follows the analysis of (\ref{eq:MomCorr}), but with the first particle from a jet and the second from a flux tube, so that
%
\be
r_{_{JB}}(\mathbf{p}_1, \mathbf{p}_2) =
\!\!\int c_{_{JB}}(\mathbf{x}_1, \mathbf{x}_2)
\frac{f_{_{J}}(\mathbf{x}_1,\mathbf{p}_1)}{n_{1J}(\mathbf{x}_1)}
\frac{f_{_{B}}(\mathbf{x}_2,\mathbf{p}_2)}{n_{1B}(\mathbf{x}_2)}
d\Gamma_1d\Gamma_2.
\label{eq:JB}
\ee
The correlation function $c_{_{JB}}(\mathbf{x}_1, \mathbf{x}_2)$ requires that a jet and a bulk particle must come from the same radial position. The rest of the equation accounts for the different the spectra for jet and bulk particles. We write
\be
c_{_{JB}}(\mathbf{x}_1, \mathbf{x}_2)
=\frac{{\cal R}_{_{JB}}\la N_J\ra}{{\cal R} \la N_B\ra}
c(\mathbf{x}_1, \mathbf{x}_2),
\label{eq:jbcorr}
\ee
where $c$ is given by (\ref{eq:param}).
To relate the correlation strength ${\cal R}_{_{JB}}$ to the bulk correlation strength ${\cal R}$ discussed earlier, we assume that the hard scattering rate is independent of the flux tube dynamics. Recall that the bulk quantity $\cal R$ in (2) is related to the number of flux tubes; see (\ref{eq:CGCscale}). If we take the fraction of jet and bulk particles per flux tube to be independent of the number of flux tubes, we can write $\langle N_{_{J}}\rangle=\alpha \langle N\rangle$, where $\alpha$ and $\beta$ may depend on momentum, but do not vary event by event. We then follow \cite{Pruneau:2002yf} to find $\langle N_{_{B}}\rangle=\beta \langle N\rangle$, and $\langle N_{_{J}}N_{_{B}}\rangle=\alpha\beta\langle N(N-1)\rangle$, so that
\begin{eqnarray}
{\cal R}_{_{JB}}
&=&\frac{\langle N_{_{J}}N_{_{B}}\rangle-\langle N_{_{J}}\rangle\langle N_{_{B}}\rangle}
{\langle N_{_{J}}\rangle\langle N_{_{B}}\rangle}
\,\,=\,\,\frac{\alpha\beta\langle N(N-1)\rangle-\alpha\langle N\rangle\beta\langle N\rangle}
{\alpha\langle N\rangle\beta\langle N\rangle} \nonumber \\ \nonumber \\
&=&\frac{\langle N(N-1)\rangle-\langle N\rangle^2}
{\langle N\rangle^2}\,\,=\,\,{\cal R}.
\label{eq:rjb}
\end{eqnarray}
Therefore, the addition of jets to the total multiplicity doesn't change the correlation strength. In essence, the beam jet associated with the hard process is just another flux tube in the high density Glasma state.
We can now rewrite (\ref{eq:deltaRho}) for jet-bulk correlations as
\be
\Delta \rho_{_{JB}}/\sqrt{\rho_{ref}} = \kappa{\cal R} dN_{jet}/dy\,\, F_{_{JB}}(\phi).
\label{eq:dRhoJB}
\ee
Our calculation of yield and implementation of lower $p_t$ limits follows (\ref{eq:yield}) and (\ref{eq:ptmin}) but, this time, we scale by the jet fraction $\int\rho_{1,J}(p_{t1})/\int\rho_{1,tot}(p_{t1})$.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{Au200Total.pdf}}
\caption[]{Au+Au 200 GeV soft ridge most central amplitude (top panel) and azimuthal width (lower panel) vs, the $p_{t,min}$ predictions with $p_{s}=1.25$ GeV. Blue lines represent the most central points of the soft ridge calculation for varying $p_{t,min}$ as shown in Fig.\ \ref{fig:fig2} (width) and Fig.\
\ref{fig:fig3} (amplitude). Red lines represent similar calculations for jet-bulk calculations, and black lines represent the total as determined by the relative bulk and jet fractions.}
\label{fig:fig5}\end{figure}
In order to compute the amplitude and azimuthal width of the hard ridge in Fig.~\ref{fig:fig4}, we must determine the relative contributions from both bulk-bulk and jet-bulk correlations. The measured $p_t$ spectrum follows an exponential behavior at low $p_t$ and a power law behavior where jets play a larger role, see e.g.\ Ref.\ \cite{Adams:2003kv}. Our blast wave formulation describes the exponential behavior of the low $p_t$ spectrum well. The scale $p_s$ at which the spectrum begins to deviate from exponential behavior is proportional to $Q_s$ in Glasma theory, but the proportionality constant is not known. This introduces a free parameter -- $p_s$ at $\sqrt{s} = 200$~GeV -- that we fix below. We then find the number of jet particles by taking the difference between the total number of particles and the number of thermal particles $\rho_{1,J}=\rho_{1,tot}-\rho_{1,B}$. We take $\rho_{1,tot}$ from the measured spectrum in Ref.\ \cite{Adams:2003kv} and $\rho_{1,B}$ from the blast wave calculation with the appropriate normalization.
We now calculate the combined effect of Glasma, flow, and jets on the correlation function. Adding the bulk-bulk and jet-bulk contributions, we obtain
\bea
\frac{\Delta\rho}{\sqrt{\rho_{ref}}}=
\kappa{\cal R} \frac{dN}{dy}F_{_{BB}}(\phi)\frac{\int\rho_{1,B}(p_t)}{\int\rho_{1,tot}(p_t)}
+\kappa{\cal R} \frac{dN_{jet}}{dy}F_{_{JB}}(\phi)\frac{\int\rho_{1,J}(p_t)}{\int\rho_{1,tot}(p_t)}.
\label{eq:total}
\eea
In order to compare to the yield of associated particles in the hard ridge, we combine (\ref{eq:total}) and (\ref{eq:yield}) with the appropriate integration limits. We find that the agreement with the data in Fig.\ \ref{fig:fig4} requires $p_{s}=1.25$ GeV. The dashed red curve in Fig.\ \ref{fig:fig4} represents the contribution to the yield from correlations of jet and bulk thermal particles. This contribution is too wide. On the other hand, the bulk-bulk correlation function describing the effect of flow alone, which is given by the blue curve, is too narrow and the computed peak height is too small. To emphasize the disagreement of the angular shapes, we show the contributions normalized to the peak in Fig.\ \ref{fig:fig4b}. The combination of the two effects shown as the black curve in Fig.\ \ref{fig:fig4} gives nice agreement with both the amplitude and azimuthal width.
We now compare hard and soft ridge measurements directly by computing the momentum dependent correlation function (\ref{eq:total}). Results are shown in Figs.\ \ref{fig:fig5} and \ref{fig:fig6}.
At low $p_{t,min}$ the contribution from jets is negligible, therefore the amplitude and width of (\ref{eq:total}) is determined by the bulk-bulk term. As the $p_{t,min}$ is increased, both the amplitude and the azimuthal width of the bulk-bulk term decreases, while the amplitude of the jet-bulk term increases.
The azimuthal width of jet-bulk correlations is roughly independent of $p_{t,min}$; the growth in the figures for $p_{t,min} > 1$~GeV is due to the growth of the jet fraction. Jet-bulk correlations should become a more significant fraction of the total as $p_{t,min}$ is increased, and the azimuthal width of the ridge should increase toward the jet-bulk width. The trends shown in Figs.\ \ref{fig:fig2a} and \ref{fig:fig6} follow preliminary data from Ref.\ \cite{Chanaka}. The small difference may reflect the fact that our calculation omits the jet peak.
We emphasize that the decrease of the bulk-bulk contribution to $\sigma_r$ with increasing $p_t$ in Figs.\ \ref{fig:fig5} and \ref{fig:fig6} is a direct consequence of transverse flow and, consequently, is a firm prediction if the jet-bulk contribution is neglected. The role of jets and other phenomena like recombination are less clear. We have chosen the model of Ref.\ \cite{Shuryak:2007fu} because it relies only of the well-studied phenomena of jet quenching. Our calculations using this model predict that the width would increase for higher $p_t$ ranges.
\begin{figure}
\centerline{\includegraphics[width=3.2in]{Cu200Total.pdf}}
\caption[]{Same as Fig.\ \ref{fig:fig5} but for Cu+Cu 200 GeV, with data from \cite{Chanaka}.}
\label{fig:fig6}\end{figure}
\section{Summary}
We have studied the contribution to the jet-triggered hard ridge and the untriggered soft ridge from long range correlations due to particle production from a Glasma state followed by transverse flow \cite{Dumitru:2008wn}, \cite{Gavin:2008ev}. To explore the effect of jet production and quenching on the ridge we included a model of jet-bulk correlations following \cite{Shuryak:2007fu}. We found that transverse bulk flow and jet production affect the transverse momentum dependence of the ridge in different ways, as shown in Figs.\ \ref{fig:fig5} and \ref{fig:fig6}. If transverse flow is the only contribution, we predict that the azimuthal width of the near side peak $\sigma_r$ will decrease as $p_t$ increases. Jets may introduce new behavior depending on the way in which they influence the ridge. Many other effects such as recombination may influence the hard ridge. However, we emphasize that we cannot explain the magnitude of the hard ridge yield without a substantial soft component.
We have emphasized the quantitative comparison to data and the information that such comparisons can provide. That said, our current model is still schematic and there is a lot of data that we have omitted. We have not discussed the rapidity extent of the ridge. This is infinite in our approximate Glasma calculation, but is finite in experiments. We will address this rapidity dependence elsewhere, since that analysis requires quantum corrections to the glasma as well as viscous corrections to the hydrodynamic treatment \cite{Gavin:2006xd,Gavin:2008ta}. In addition, we do not address the away side region, $|\phi| > \pi/2$. The broad enhancement measured there is attributed to momentum conservation and can is described by hydrodynamic calculations \cite{Takahashi:2009na}. We also have not computed the jet peak, which contains information on the interactions of the jet with matter \cite{Wong:2009cx}. Further work is needed \cite{Nagle:2009wr}.
More generally, experiments have treated the soft ridge and the hard ridge as separate phenomenon. The soft ridge measurements intend to study properties of the medium, while the hard ridge measurement intends to study the effects of jets on the medium. The observables used in these studies are different, though not unrelated. We feel that the experimental situation may be clarified by using common techniques and observables in the different regimes.
The fundamental reason for studying these complex phenomena lies in the fact that long range correlations provide information on the very earliest stage of particle production. The overall amplitude of correlations is fixed by the CGC-Glasma dependence of ${\cal{R}}dN/dy$. We have seen that this behavior explains the energy dependence and system size dependence of the soft ridge.
Moreover, as the $p_t$ range of correlations is increased toward jet-dominated part of spectrum, long range correlations continue to influence the near side peak because flux tubes induce correlations of jets with low $p_t$ bulk particles.
\section*{Acknowledgments}
We thank L. McLerran for collaboration, and R. Bellwied, C. De Silva, A. Timmins, A. Dumitru, A. Majumder, J. Nagel, P. Sorenson, P. Steinberg, R. Venugopalan, C. Pruneau, and S. Voloshin for useful discussions. This work was supported in part by U.S. NSF grants PHY-0348559 and PHY-0855369.
\bibliographystyle{G-h-physrev5}
| 2024-02-18T23:40:06.004Z | 2009-10-30T21:45:07.000Z | algebraic_stack_train_0000 | 1,331 | 7,380 |
|
proofpile-arXiv_065-6565 | \section{Introduction}
Collimated, bipolar outflows accompany the birth of young stars from the
earliest stages of star formation to the end of their accretion phase
\citep[e.g.][]{reipurth2001}. While the birth of isolated low-mass stars is
becoming well understood, the formation of massive stars ($>10 \msun$) and clusters remains a
topic of intense study. Observations show that moderate to high-mass stars
tend to form in dense clusters \citep{lada2003}. In a clustered environment, the dynamics of
the gas and stars can profoundly impact both accretion and mass-loss processes.
Feedback from these massive clusters may play a significant role in momentum
injection and turbulence driving in the interstellar medium.
Outflows from massive stars are less studied than those from low mass stars
largely because massive stars accrete most of their mass while deeply embedded.
Therefore, unlike low mass young stars that are accessible in the optical,
massive stellar outflows can only be seen at infrared and longer wavelengths.
Direct evidence for jets from massive young stellar objects (YSOs) from \hh\ or
optical emission is generally lacking
\citep[e.g.][]{alvarez2005,kumar2002,wang2003}, although there is evidence that
massive stars are the sources of collimated molecular outflows from millimeter
observations \citep[e.g.][]{beuther2002b}. Outflows from massive stars may
allow accretion to continue after their radiation pressure would
otherwise halt accretion in a spherically symmetric system
\citep{krumholz2009}. They therefore represent a crucial component in
understanding how stars above $\sim$10 \msun\ can form.
IRAS 05358\ is a double cluster of embedded infrared sources located at a distance
of 1.8 kpc in the Auriga molecular cloud complex \citep{heyer1996} associated
with the HII regions Sh-2 231 through 235 at Galactic coordinates around $l,~b$
= 173.48,+2.45 in the Perseus arm. Sh~2-233IR~NE\ is the
collection of highly obscured and mm-bright sources slightly northeast of
Sh~2-233IR~SW, which is the location of the IRAS 05358+3543 point source and the
optical emission nebula (see Figure \ref{fig:overview_ha}). The IRAS source is
probably a blend of the three brightest infrared objects in the MSX A-band and
MIPS 24 \um\ images, which are located at Sh~2-233IR~NE, IR 41, and IR 6. For the
purpose of this paper,the whole complex including both sources is referred toas
IRAS 05358, and otherwise refer to individual objects specifically.
Early observations revealed the presence of OH \citep{Wouterloot1993}, \HtwoO\
\citep{Scalise1989, Henning1992}, and methanol \citep{Menten1991} masers about
an arcminute northeast of the IRAS source, indicating that massive stars are
likely present at that location. Near infrared observations revealed
the presence of two embedded clusters \citep{porras2000,jiang2001} labeled
Sh~2-233IR~SW\ for the southwestern cluster associated with the IRAS source, and Sh~2-233IR~NE\
for the northeastern cluster located near the OH, \HtwoO, and CH$_3$OH masers.
Stars identified in \citet{porras2000} are referred to by the designation
``IR (number)'' corresponding to the catalog number in that paper.
\citet{porras2000} also included scanning Fabry-Perot velocity measurements of
the inner $\sim1$\arcmin. CO observations revealed broad line wings indicative
of a molecular outflow \citep{casoli1986,shepherd1996}. \citet{kumar2002} and
\citet{khanzadyan2004} presented narrow band images of 2.12 \um\ \htwo\
emission that reveled the presence of multiple outflows. Interferometric
imaging of CO and SiO confirmed the presence of at least three flows emerging
from the northeast cluster centered on the masers \citep{beuther2002} having a
total mass of about 20 \msun . \citet{beuther2002} also presented MAMBO 1.2 mm
maps and a mass estimate of 610 \msun\ for the whole region.
\citet{williams2004} presented SCUBA maps and mass estimates of the clusters of
195/126\msun\ for Sh~2-233IR~NE\ and 24/12 \msun\ for Sh~2-233IR~SW\ (850 \um/450 \um).
\citet{Zinchenko1997} measured the dense gas properties using the NH\ensuremath{_3}\
(1,1) and (2,2) lines. They measure a mean density $n \approx 10^{3.60}$ \ensuremath{\textrm{cm}^{-3}},
temperature 26.5K, and a mass of 600 \msun . The total luminosity of the two
clusters is about 6300 \lsun , indicating that the region is giving birth to
massive stars \citep{porras2000}.
Millimeter wavelength interferometry with arcsecond angular resolution has
revealed a compact cluster of deeply embedded sources centered on the \HtwoO\
and methanol maser position \citep{beuther2002,beuther2007,leurini2007}.
\citet{beuther2002} identified 3 mm continuum cores, labeled mm1-mm3 (shown in
Figure \ref{fig:outflowsh2}). \citet{beuther2007} resolved these cores into
smaller objects. Source mm1a is associated with a cm continuum point source
and will be discussed in detail below.
IRAS 05358\ has previously been observed at low spatial resolution in the J=2-1 and J=3-2
transitions with the Kosma 3m telescope \citep{Mao2004}. While the general presence
of outflows was recognized and a total mass estimated, the specific outflows were not
resolved. \citet{beuther2002} observed the CO J=6-5, J=2-1, and J=1-0
transitions at moderate resolution in the inner few arcminutes.
\citet{Thomas2008} observed C$^{17}$O in the J=2-1 and J=3-2 transitions with a
single pointing using the JCMT.
\section{Observations}
A collection of data acquired by the authors and from publicly
available archives is presented. An overview of the data is presented in figure
\ref{fig:overview_ha}. The goal was to develop a complete picture of the outflows
in IRAS 05358\ and their probable sources. CO data were acquired to estimate the
total outflowing mass and to identify outflowing molecular material
unassociated with \hh\ shocks. Archival Spitzer IRAC and MIPS 24 \um\ data
were used to identify probable YSOs as candidate outflow sources.
Near-infrared spectra were acquired primarily to determine \hh\ kinematics and
develop a 3D picture of the region. Optical spectra were acquired to attempt
to identify stellar types in the unobscured Sh~2-233IR~SW\ region. Finally,
archival VLA data were used to acquire better constraints on the position and
physical properties of the known ultracompact HII (UCHII) region, and to detect
or set limits on other UCHIIs.
\subsection{Sub-millimeter Observations}
The 345 GHz J = 3-2 rotational transition of CO was observed with the James
Clerk Maxwell Telescope (JCMT) on 4 January, 2008 with the 16 element (14
functional) HARP-B heterodyne focal plane array. Two 12\arcmin\ $\times$
10\arcmin\ raster scans in R.A. and Dec. were taken with orthogonal
orientations to assure complete coverage in the region of interest; this
resulted in a useable field 11.7\arcmin\ $\times$ 11.3\arcmin\ with higher noise
along the edges. The beam size at 345 GHz is about 15\arcsec.
Observations were conducted during grade 3 conditions with the 225 GHz zenith
optical depth of the atmosphere $\tau\sim0.1$. A channel width of 488 kHz
corresponding to 0.423 \kms\ was used. The maps required a total of 1 hour
to acquire and resulted in an effective integration time of 4.6 seconds per
pixel (there are 12,000 $6\times6\arcsec$\ pixels in the final grid), resulting
in a noise per pixel of 0.36 K \kms.
The optical depth and telescope efficiency corrections were applied by the JCMT
pipeline to convert the recorded antenna temperatures to the corrected antenna
\footnote{See \\
\url{http://docs.jach.hawaii.edu/JCMT/OVERVIEW/tel\_overview/} for a discussion
of JCMT parameters}. An additional
main-beam correction has been applied, $$T_{mb}=\frac{T_A*}{\eta_{mb}}$$ where
$\eta_{mb} $ was measured by observing Mars to be $\approx0.60$ at 345
GHz. Emission in the sidelobes is expected to be small at the outflow velocities.
On September 25 and November 15, 2008 the CO, $^{13}$CO, and C$^{18}$O J=2-1
transitions were observed in the central 3\arcmin\ of IRAS 05358. The beamsize
at 220 GHz is about 23\arcsec. The sideband configuration used also includes
the \linebreak \nolinebreak{ SO~\ensuremath{5_6-4_5} } and $^{13}$CS 5-4 transitions. Conditions
during these observations were grade 5 ($\tau \sim 0.24-0.28$) and therefore
too poor to use the HARP instrument, but acceptable for the A3 detector.
Data reduction used the Starlink package following the standard routines
recommended by the JCMT support scientists \footnote{
\url{http://www.jach.hawaii.edu/JCMT/spectral\_line/data\_reduction/acsisdr/}}.
The CO 3-2 data cube was extracted over a velocity range from --50 to 10 \kms\ LSR and
spectral baselines were fit over the velocity range --50 to --40 and 0 to 10
\kms\ and subtracted. The data were re-gridded into 6\arcsec\ pixels and 2
pixel Gaussian smoothing was used to fill in the gaps left by the two bad
detectors in the 4 $\times$4 array. The data cube was cropped to remove
undersampled edges which have high noise and bad baselines. The beam efficiency
was 0.68 at 230 GHz.
The A3 data cubes were extracted over the velocity range --60 to 20 \kms\ and
baselines were calculated over --60 to --40 and 0 to 20 \kms. The data
was gridded into 10\arcsec\ pixels with 2 pixel gaussian smoothing to reduce
sub-resolution noise variations.
\subsection{Spitzer}
Spitzer IRAC bands 1 to 4 and MIPS band 1 data were retrieved from the Spitzer
Science Center archive. \citet{qiu2008} acquired the data as part of a study
of many high-mass star forming regions; they identified YSO candidates based on
IRAC colors. The version 18 post-BCD data products were used to produce images
and photometric catalog from \citet{qiu2008}, which was made from a more
carefully-reduced data set, was used for SED analysis.
\subsection{Near-IR images}
Near-infrared data were acquired using the Wide-field Infrared Camera (WIRCam) on
the Canada-France-Hawaii Telescope (CFHT) on Mauna Kea. The field of view is
20\arcmin$\times$20\arcmin\ ~and pixel scale 0.3\arcsec. Data were acquired on
November 18, 19 and December 20, 2005. The seeing was 0.5-0.7\arcsec\ during the
observations. A 0.032 \um\ wide filter centered at 2.122 \um\ was used to take images
of the \hh\ S(1) 1-0 rovibrational transition. Each \hh\ exposure was 58
seconds, and dithered images were taken for a total exposure time of 1755
seconds. The data were reduced with the WIRCam pipeline.
\subsection{Near-IR spectra}
Near-infrared spectra were acquired using the TripleSpec instrument at Apache
Point Observatory. TripleSpec simultaneously acquires J, H, and K band spectra
over a 42\arcsec\ long slit. A slit width of 1.1\arcsec\ with an
approximate spectral resolution $\lambda/\Delta\lambda=2700$ was used.
Observations were taken on the nights of December 2, 2008 and January 7, 2009.
Data on December 2 were taken in an ABBA nod pattern, but because of the need
to observe extended structure across the slit a stare strategy was selected on
January 7.
The data were reduced using the {\sc twodspec} package in IRAF.
HD31135, an A0 star, was used as a flux calibrator. Wavelength calibration was
performed using night sky lines. Lines filling the slit were subtracted
to remove atmospheric emission lines. Telluric absorption correction was {\emph
not} performed, but telluric absorption is considered in the analysis.
The transformations from the observed geocentric reference frame to $v_{LSR}$
were computed to be 0.78 \kms\ on Dec 2 and 19.74 \kms\ on Jan 8.
\subsection{Optical Spectra}
Optical spectra were acquired using the Double Imaging Spectrograph instrument
at APO. The high-resolution red and blue gratings
were centered at 6564 \AA\ and 5007 \AA\ with a coverage of about 1200 angstroms and
resolution $\lambda/\Delta\lambda \approx 5000$. Sets of three 900s exposures
and three 200s exposures were acquired on the targets and on the spectrophotometric
calibrator G191-b2b with a 1.5" slit. Observations were taken on the night
of January 17, 2009 under clear conditions.
Optical spectra were also reduced using the {\sc twodspec} package in IRAF.
Wavelength calibration was done with HeNeAr lamps and night sky lines
in the red band, and HeNeAr lamps in the blue band. Lines filling the slit
were subtracted to remove atmospheric lines, though some astrophysical lines
also filled the slit and these were measured before background subtraction.
The $v_{LSR}$ correction for this date was 24.4 \kms.
\subsection{Optical imaging}
CCD images images were obtained on the nights of 14 and 15 September 2009
NOAO Mosaic 1 Camera at the f/3.1 prime focus
of the 4 meter Mayal telescope atthe Kitt Peak National Observatory (KPNO).
The Mosaic 1 camera is a 8192$\times$8192 pixel array (consisting of eight
2048$\times$4096 pixel CCD chips) with a pixel scale of 0.26$''$ pixel$^{-1}$ and
a field of view 35.4$'$ on a side. Narrow-band filters centered on
6569\AA\ and 6730\AA\ both with a FWHM of 80\AA\ were use to obtain
H$\alpha$ and [SII] images. An SDSS i' filter which is centered on
7732\AA\ with a FWHM of 1548\AA was used for continuum imaging.
A set of five dithered 600 second exposures were obtained in
H$\alpha$ and [SII] using the standard MOSDITHER pattern
to eliminate cosmic rays and the gaps between the
individual chips in Mosaic. A dithered set of five 180 second
exposures were obtained in the in the broad-band SDSS i-band filter to
discriminate between H$\alpha$, [SII], and continuum emission.
Images were reduced in the standard manner by the NOAO Mosaic reduction
pipeline \citep{valdes2007}.
\Figure{f1.eps}{The CFHT \hh\ (bold), CO 3-2 HARP
(thin), and CO 2-1 A3 (dashed) fields overlaid on the KPNO \ensuremath{\textrm{H}\alpha}\ mosaic
with selected objects identified by their SIMBAD names. Sh~2-233IR~SW\ is coincident
with IRAS 05358+3543.
}{fig:overview_ha}{1.0}
\subsection{VLA data}
VLA archival data from projects AR482, AR513, AS831, and AM697 were re-reduced
to perform a deeper search for UCHII regions and aquire more data points on the
known UCHII's SED. Data from AR482 were previously published in
\citet{beuther2007}, the other data are unpublished. The data were reduced
using the VLA pipeline in AIPS ({\sc vlarun}). The observations used and
sensitivities and beam sizes achieved are listed in Tables \ref{tab:vlatimes}
and \ref{tab:vla}. There appeared to be calibration errors in the AR482
observations (the phase calibrator was 2-3 times brighter than in all other
observations) and this data were therefore not used in the final analysis, but
it produced consistent pointing results.
\Table{cccccccc}
{VLA Observation Program Names, Dates, and Times}
{\colhead{VLA } & \colhead{Observation} & \colhead{Time } & \colhead{Array} &
\colhead{Band} & \colhead{Fluxcal} & \colhead{Phase cal} & \colhead{Phase cal } \\
Observation & Date &on&&&&& Percent \\
Name &&Source&&&&& Uncertainty \\}
{tab:vlatimes}
{
AR482 & August 2 2001 & 2580s & B & X &3c286 & 0555+398 & 22 \\
AR513 & June 21 2003 & 7770s & A & X &3c286 & 0555+398 & 0.8 \\
AS831 & February 26 2005 & 2640s & B & X &3c286 & 0555+398 & 0.7 \\
AS831 & August 5 2005 & 2660s & C & X &3c286 & 0555+398 & 0.3 \\
AS831 & May 11 2006 & 2610s & A & X &3c286 & 0555+398 & 3.0 \\
AL704 & August 7 2007 & 6423s & A & Q &3c273 & 0555+398 & 18 \\
AL704 & September 1 2007 & 6423s & A & Q &3c273 & 0555+398 & 13 \\
AM697 & November 26 2001 & 2880s & D & Q &3c286 & 0555+398 & 2.2 \\
AM697 & November 28 2001 & 1530s & D & K &3c286 & 0555+398 & 2.1 \\
AM697 & November 28 2001 & 1530s & D & U &3c286 & 0555+398 & 5.8 \\
}{}
\section{Results}
\subsection{Near Infrared Imaging: Outflows and Stars}
Eleven distinct outflows have been identified in IRAS 05358\ in the images.
Outflows are identified from a combination of J=3-2 CO data, shock excited
\htwo\ emission, and published interferometric maps \citep{beuther2002}.
Suspected CO outflows were identified by the presence of wings on the CO J=3-2
emission lines that extended beyond the typical velocity range of emission
associated with the line core. The single dish data were compared to the
interferometric maps of \citet{beuther2002}. The CFHT \hh\ image was then used
to search for shock-excited emission associated with the outflow lobes.
\begin{figure*}[htpb]
\center
\epsscale{0.75}
\hspace{-1.2in}
\plotone{f2.eps}
\caption{The outflows described in section \ref{sec:outflows} overlaid on
the CFHT \hh\ image. Numbers followed by {\it r} and {\it b} (red and
blue), {\it n} and {\it s} (north and south), or {\it e} and {\it w} (east
and west) are thought to be counterflows. Red and blue vectors indicate
red and blue doppler shifts. Green vectors indicate where the doppler
shift is ambiguous or cannot be determined. Magenta circles are Spitzer
24\um\ sources. Red squares are \citet{beuther2002} mm sources (from left
to right, mm1, mm2, mm3). The blue diamond is a YSO candidate detected
only in IRAC bands. The length of the vectors corresponds to the
approximate length of the outflows. Source 1 and 6 correspond to
\citet{porras2000} IR 6 and IR 41 respectively, and they are discussed under
these names in sections \ref{sec:outflows}. The bows of Outflow 1n and 4n
are detected in \ensuremath{\textrm{H}\alpha}\ and [S II] emission and are therefore as identified
as Herbig-Haro objects HH 993 and 994 respectively.
\label{fig:outflowsh2}}
\end{figure*}
\Figure{f3.eps} {CO contours integrated from $v_{LSR}=$ -13 to -4
\kms\ (red) and -34 to -21 \kms\ (blue) at levels of 2,4,6,10,20,30,40,50 K
\kms\ overlaid on the \hh\ image. Specific outflows are labeled in Figure
\ref{fig:outflowsh2} on the same scale.} {fig:COonH2}{0.75}
\begin{figure*}[htpb]
\hspace{-0.6in}
\includegraphics[scale=0.40,clip=true]{f4.eps}
\includegraphics[scale=0.40,clip=true]{f5.eps}
\caption{(a) \hh\ image with SO~\ensuremath{5_6-4_5} \ peak flux contours at 0.5-1.4 K in intervals of 0.15 K
overlaid. With a critical density $\sim3.5\ee{6}$ \citep{leidendb}, this
transition is a dense gas tracer. (b) The [S II] image with outflow vectors overlaid.
Diffuse emission can be seen at the north ends of Outflows 1, 4, and 6 and around the
reflection nebula near source IR 41.}
\label{fig:so_on_h2}
\end{figure*}
\label{sec:outflows}
Figure \ref{fig:outflowsh2} shows the \htwo\ S(1) 1-0 2.1218 \um\ (a rovibrational
transition in the electronic ground state from the $v=1$, $J=3$ to the $v=0$,
$J=1$ state) emission in the vicinity of IRAS 05358\ with outflows and possible
outflow sources labeled. The mm cores from \citet{beuther2002} are identified
by red squares.
The flow vectors in figure \ref{fig:outflowsh2} were chosen on the basis of the
\htwo\ bow shock morphologies and orientations of chains of \htwo\ features,
association with arcsecond-scale CO features on the \citet{beuther2002} Figure
8 CO map, and/or association with lobes of Doppler-shifted CO emission in the
CO 3-2 data. The color of the vector indicates the suspected Doppler shift;
red and blue correspond to red and blueshifts and green vectors indicate that
the Doppler shift is uncertain.
{\it IRAS 05358\ outflow 1:} The most prominent flow in \htwo\ is associated with
the bright bow-shocks N1 and N6 \citep{khanzadyan2004} located towards PA $\approx$ 345\arcdeg\
and 170\arcdeg\ respectively from the sub-mm source mms1b \citep{beuther2002}.
This flow, \citet{beuther2002} outflow A, is associated with redshifted and
blueshifted CO emission. The northern shock is seen in \ensuremath{\textrm{H}\alpha}\ and [S II]
emission (figure \ref{fig:so_on_h2}b) and is given a Herbig-Haro designation
HH 993.
This flow is indicated by oppositely directed green vectors from the vicinity
of smm1, 2, and 3. It is listed as ``Jet 1'' in \citet{qiu2008}. \citet{kumar2002}
identified the knot immediately behind the bow shock as a Mach disk. In the
\citet{beuther2002} interferometric maps, the north flow contains redshifted features
and the south flow contains primarily blueshifted features. There are also blueshifted
CO features to the west of the \hh\ knots that are probably part of a different flow
that is not seen in \hh\ emission.
The velocity of the flow as measured from \hh\ emssion is blueshifted as much
as 80 \kms (LSR), but one component is blueshifted only 14 \kms\ (see table
\ref{tab:OutflowH2}), which is consistent with the cloud velocity. A
redshifted SiO lobe is present in the south counterflow. The presence of \ensuremath{\textrm{H}\alpha},
[S II], and [O III] emission in the north shock and corresponding
nondetections in the south shock suggest that there is substantially greater
extinction towards the south knot. While the velocities in three of the four
apertures picked along the TripleSpec slit are blueshifted, there are also knots
with velocities consistent with the cloud velocity. \citet{porras2000} measure
the velocity of the counterflow to be -17.3 \kms, which is consistent with
the cloud velocity. Outflow 1 is propagating very nearly in
the plane of the sky.
A line connecting the two bow shocks in Outflow 1 goes directly through
\citet{beuther2007} source mm2a despite the clear association in the
\citet{beuther2002} interferometric CO map (their Figure 8) with mm1a. The
currently available data do not clarify which is the source of the outflow:
while the bent CO outflow appears to trace Outflow 1 back to mm1a, there are
additional parallel CO outflows towards the confused central region that could
originate from either mm1a or mm2a.
A Spitzer 4.5 \um\ and 24 \um\ source is barely detected in \hh\ 2.5\arcmin\ to
the north of Outflow 1. It is only apparent when the \hh\ image is smoothed
and would have been dismissed as noise except for the association with
a probably 4.5 \um\ extended source. It is labeled 24\um\ source 7 in figure
\ref{fig:outflowsh2}. It appears to be slightly resolved at 4.5\um, and is
therefore likely shocked emission. The object may be a protostellar source
with an associated outflow, but its proximity to the projected path of Outflow
1 suggests that it may be an older outflow knot.
{\it IRAS 05358\ Outflow 2:} The second brightest \htwo\ features trace a bipolar
flow emerging from the immediate vicinity of the sub-mm cluster at PA $\approx$
135\arcdeg\ (red lobe) and 315\arcdeg\ (blue lobe). It is listed as ``Jet 2'' in
Figure 6 of \citet{qiu2008}.
The counterflow probably overlaps in the line of sight with the counterflow
from Outflow 3. It is shorter on the counterflow side either because it has
already penetrated the cloud and is no longer impacting any ambient gas or,
more likely, it has slowly drilled its way out of the molecular cloud and has
not been able to propagate as quickly as the northwest flow.
The \hh\ velocities measured for these knots are $\sim$ 30 \kms\ blueshifted, or
marginally blue of the cloud LSR velocity.
The disk identified in \citet{Minier2000} is approximately perpendicular to the
measured angle of Outflow 2 assuming that mm1a is the source of this flow. It
is therefore an excellent candidate for the outflow source. A diagram of the
mm1a region is shown in figure \ref{fig:mm1adiagram}. See Section
\ref{sec:vlaresults} for detailed discussion.
{\it IRAS 05358\ outflow 3:} The \citet{beuther2002} CO and SiO maps reveal a
third flow, their outflow B at PA $\approx$ 135\arcdeg\ (red lobe) and
315\arcdeg\ (blue lobe). A chain of \htwo\ features, \citet{khanzadyan2004}
features N3D and N3E, are probably shocks in this flow. It is listed as ``Jet
3'' in \citet{qiu2008}. The two chains of \htwo\ emission indicate that
outflows 2 and 3 are distinct. There also appears to be a counterflow at a
shorter distance from the mm cores similar to counterflow 2.
Outflows 2 and 3 may be associated with either redshifted or blueshifted
features in the \citet{beuther2002} CO and SiO maps. High velocity flows with both
parities are present near both the northwest (\citet{beuther2002} outflow C)
and southeast flow for these jets, but the resolution of the millimeter
observations is inadequate to determine which flow is in which direction.
\citet{porras2000}
measures $v_{LSR} = -7.5$ \kms\ for their knot 4A, which corresponds to the
blended southeast counterflow of outflows 2 and 3. Their Figure 7 shows a wide
line that is probably better represented by two or three blended lines, one
consistent with the cloud velocity and the other(s) redshifted. Since Outflow 2
has a measured blueshift and outflow 3 is significantly fainter, the redshifted
counterflow emission is probably associated with Outflow 2 and the blueshifted
with outflow 3.
{\it IRAS 05358\ outflow 4:} The JCMT CO data and \htwo\ images reveal a large
outflow lobe consisting of blue lobes 1 and 4 that form a tongue of blueshifted
emission propagating to the northeast at PA $\approx$ 20\arcdeg\ (Figure
\ref{fig:outflowsh2}) from the cluster of sub-mm cores. A faint chain of
\htwo\ features runs along the axis of the CO tongue and terminates in a bright
\htwo\ bow shock located at the northern edge of \ref{fig:outflowsh2}. Several
\htwo\ knots lie along the expected counterflow direction, but that portion of
the field contains multiple outflows and is highly confused. If the
counterflow is symmetric with the northeast knot, it extends 2.1 parsecs on the
sky.
The bow shock of Outflow 4 is seen in the HII and [S II] images, implying that
the extinction is much lower than in the cluster. Two apertures placed along
the bow shock reveal that it is blueshifted about 70\kms\ and may be extincted
by as little as $A_V\sim.5$. It is designated HH 994.
{\it IRAS 05358\ outflow 5:} Figure \ref{fig:outflowsh2} shows a bright chain of
\htwo\ knots and bow shocks starting about 10\arcsec\ west of mm3 and
propagating south at PA $\approx$ 190\arcdeg. The SiO maps of
\citet{beuther2002} show a tongue of blueshifted emission along this chain
(their Outflow C). The outflow projects back to H$^{13}$CO$^+$ source 3,
which is also a weak mm source. A lack of obvious counterflow and the
possibility that the knots identified with Outflow 5 could be associated with a
number of different crossing flows makes this identification very tentative.
Higher spatial resolution observations will be required to determine the
association of this outflow.
{\it IRAS 05358\ outflow 6:} The fourth brightest source in the Spitzer 24\um\
data is located at J(2000) = 05:39:08.5, +35:46:38 (source 5 in the IRAS 05358\
section of the \citet{qiu2008} catalog, referred to in table \ref{tab:OutflowH2}
as Q5) in the middle of the
molecular ridge that extends from IRAS 05358\ towards the northwest (24\um\
object 4 in figure \ref{fig:outflowsh2}). The star is
located at the northwest end of the tongue of 1.2mm emission mapped by
\citep{beuther2002} with the MAMBO instrument on the IRAM telescope. This
part of the cloud is also seen in silhouette against brighter surrounding
emission at 8\um. At wavelengths below 2\um, it is fainter than 14-th
magnitude and therefore is not listed in the 2MASS catalog, and it is not
detected in \citet{yan2009} down to 19th magnitude in K.
Spitzer data indicates very red colors between 3.6 and 70 \um, indicating that
this object is likely to be a Class I protostar. The SED is fit using the
online tool provided by \citet{robitaille2007}. Unfortunately, a wide variety
of parameters all achieved equally good fits, so no conclusions are drawn
about the stellar mass or other very uncertain parameters. However, the top
models all had $A_V > 20$ and many in the range 30-50, indicating that the
line of sight is probably through a thick envelope or disk towards this
source.
This source lies at the base of the tongue of blueshifted CO 3-2
emission that extends northwest of IRAS 05358\ at PA $\approx$ 345\arcdeg\ and
has mass $\sim .5\msun$. A pair of \htwo\ features,
\citet{khanzadyan2004} N12A and N12B are located 30 and 55\arcsec\ from the
suspected YSO, forming a chain along the axis of the blueshifted CO
tongue. \citet{khanzadyan2004} \htwo\ knot N3F lies along the flow axis in
the redshifted direction.
{\it IRAS 05358\ outflow 7:} The 20\arcsec\ long chain of \htwo\ knots labeled
\citet{khanzadyan2004} N11 appears to trace part of a jet at PA $\approx$
345\arcdeg\ that propagated parallel to outflow 6 about 20\arcsec\ to the east.
The northwest portion of Outflow C in the \citet{beuther2002} SiO map is in
approximately the same direction as Outflow 7, and it may represent a redshifted
counterflow to the northwest-pointing \hh\ knots.
The jet axis passes within a few arc-seconds of a faint and red YSO located at
J(2000) = 05 39 10.0, +35 46 27 (blue diamond in figure \ref{fig:outflowsh2}
about 35\arcsec\ south of the southern end of the \htwo\ feature). It may be a
24\um\ source but is lost in the PSF of the bright source at the center of
Sh~2-233IR~NE. This object is also undetected down to 19th magnitude in the
\citet{yan2009} K-band image.
{\it IRAS 05358\ outflow 8:} A prominent jet-like \htwo\ feature protrudes from
the vicinity of Sh~2-233IR~SW\ at PA $\approx$ 335\arcdeg\ and ends in
bright knot
N9. The feature N5B is is
located just outside the ring of \htwo\ emission that surrounds the IRAS source
at the base of the jet. Towards the southeast,
knot N6 is located opposite knot N9 with respect to the southwest cluster. IR 41,
the \ensuremath{\textrm{H}\alpha}\ emission source, labeled 24\um\ source 6 in figure \ref{fig:outflowsh2},
is probably the source of this outflow.
{\it IRAS 05358\ outflow 9:} In the Spitzer and K$_s$ images, an infrared
reflection nebula opens towards the southwest at PA $\approx$ 245\arcdeg\ and
points towards a blueshifted CO region. The reflection nebula is also seen in
\ensuremath{\textrm{H}\alpha}. It is likely that the CO emission in CO Region 1 (table \ref{tab:comeas})
traces a fossil cavity whose walls provide the scattering surface of the
reflection nebula.
{\it IRAS 05358\ outflow 10 and IR 6:} A bright \htwo\ filament protrudes at PA $\approx$
15\arcdeg\ towards the northeast of IR 6 (24\um\ source 1, \citet{qiu2008}
source 8). The star is the third brightest 24\um\ source in the IRAS 05358\
region. Since it is visible at visual wavelengths, it is not heavily
embedded. Its H$\alpha$ emission and association with an outflow lobe and
\htwo\ emission suggest that it is a moderate mass Herbig AeBe star associated
with the IRAS 05358\ complex. The optical spectrum confirms this hypothesis: the
star has \ensuremath{\textrm{H}\alpha}\ absorption wings on either side of a very bright, asymmetric \ensuremath{\textrm{H}\alpha}\
emission profile (see section \ref{sec:dis}).
IR 6 is seen to be the source of Outflow 10. Data for this source is available
from $\sim$0.45-24\um, so the \citet{robitaille2007} spectral fitter puts
strong constraints on the star's mass and luminosity. The measured mass and
luminosity are $M=4.5\pm0.5$ \msun\ and $L = 10^{2.3\pm.25} L_\odot$, parameters
consistent with a B7V ($\pm 1$ spectral class) main sequence star. The range
of ages in the models covers $10^4-10^7$ years but favors stars in the range
$10^5-10^6$ years.
While there is a small clump of redshifted CO emission to the northeast of the
object, the \htwo\ spectrum shows that the north flow is blueshifted $v_{LSR}\sim-40
$\kms, and the lack of a visible counterflow suggests that the counterflow may
be masked behind an additional extincting medium. The counterflow drawn in
figure \ref{fig:outflowsh2} is not seen in emission but is identified as a
probable location for a counterflow because of the confident association of
outflow 10n with source IR 6.
{\it IRAS 05358\ outflow 11:} A chain of \hh\ knots is seen at 2.12\um\ and in the
Spitzer 4.5\um\ image. They trace back to either IR 78 or 24\um\ source 4.
There is a tongue of redshifted CO 3-2 emission in the same direction as this
flow that suggests it may be redshifted.
{\it IR 41}:
There is an arc-like \hh\ emission feature surrounding the \ensuremath{\textrm{H}\alpha}\ emission line
star IR 41. This implies that the star is probably a late B-type star with too
little Lyman continuum emission to generate a photon-dominated region (PDR) but
enough soft UV to excite \hh. From the measured \ensuremath{\textrm{H}\alpha}\ and nondetection of \ensuremath{\textrm{H}\beta}\
at the star's location down to a 5-$\sigma$ limit of 1\ee{-17} erg s$^{-1}$
\ensuremath{\textrm{cm}^{-2}} \AA$^{-1}$, a lower limit on the extinction column $A_V=15$ is derived.
The \citet{robitaille2007} fitter yields a mass estimate of 7.4$\pm 0.6$\msun
and luminosity $L=10^{2.97\pm.16}L_\odot$ among the 222 best fits out of a grid
of 200,000 model SEDs (fits with $\chi^2<5000$). The luminosity is very well
constrained, varying only modestly to $L=10^{2.99\pm.15}L_\odot$ for the 904
best fits ($\chi^2 < 10000$), while the mass shifts down to $6.5\pm1.0\msun$.
The mass estimate may be biased by the lower number of high-mass models
computed. The star's mass is most compatible with a main sequence B4V star,
though its luminosity is closer to a B5V star. The disk mass is constrained to
be $>10^3 \msun$. The age is reasonably well constrained to be $T =
10^{5.78\pm.12}$ for the best 904 models, but is essentially unconstrained for
the best 222. Similarly, the stellar temperature is entirely unconstrained by
the fitting process.
The very high values of $\chi^2$ would normally be worrisome, but the $\chi^2$
statistic only represents statistical error, while the data is dominated by
various systematic errors including calibration offsets in the optical/NIR and
poor resolution in the far-IR. Therefore, it is not possible to find a perfect
model fit, but still possible to put constraints on the physical properties of
the source.
\Figure{f6.eps}{The \ensuremath{\textrm{H}\alpha}\ image with CO contours at
redshifted, blueshifted, and middle velocities in red, blue, and green
respectively. Contours are at 2,4,8,12,20 K \kms\ for the red and blue,
and 20,25,30,40,50,60,70 K \kms\ for the green. Red is integrated from
-12 to -4 \kms, Blue from -31 to -21 \kms, and green from -21 to -12 \kms.
}{fig:HA_with_CO}{1.0}
{\it South of IRAS 05358}:
There is a symmetric flow with one faint \hh\ knot and a bright central source
about 4\arcmin\ south of IRAS 05358. The \hh\ knot is at J(2000) = 05:39:15.63 +35:42:13.2.
The flow has a clear red and blue region as identified in figure
\ref{fig:cofig}; the red flow extends from -9 to -14 \kms\ and the blue from -19
to -23 \kms\ (the outflow is swamped by ambient emission in the intermediate
velocity range). The outflow is $\sim 2\arcmin$ long, though the probable
source identified is not directly between the two lobes. The ellipses used are
labeled in table \ref{tab:comeas} as Red S and Blue S.
\subsection{Imaging results: Optical}
Deep [S II] images show that some of the outflows have pierced through the
obscuring dust layers or excited extremely bright sulfur emission.
\citet{khanzadyan2004} knot N1 at the end of Outflow 1 is visible [S II] emission
The bow of outflow 4 and the northwest end of outflow 6 are
detected in [S II]. Only the Outflow 1 and 4 bows are detected in \ensuremath{\textrm{H}\alpha}\
emission, indicating that the emission is most likely from shock heating, not
external photoionizing radiation. If the shocks were externally irradiated, we
would expect the emission to be dominated by the recombination lines. Because
they have been detected in the optical, these two flows can be classified
as Herbig-Haro objects.
\subsection{CO results}
IRAS 05358\ is located at the center of the CO 3-2 integrated velocity maps
(Figure \ref{fig:cofig}). The parent molecular cloud, centered at $v_{LSR} =
-17.5$ \kms , extends from the southeast towards the northwest with the
brightest emission coming from the core associated with Sh~2-233IR~SW, while the
highest integrated emission is associated with Sh~2-233IR~NE. Sh~2-233IR~NE\ has a central
velocity of $\sim -16.0$ \kms\ from the optically thin \ensuremath{\textrm{C}^{18}\textrm{O}}\ 2-1 measurements.
Material that has been swept up and accelerated by jets and outflows can be
seen at velocities $v_{LSR} < -21$ \kms and $v_{LSR} > -12$ \kms\ (Figure
\ref{fig:cofig}). The integrated CO 3-2 map peaks at J(2000) = 05:39:12.8
+35:45:55, while the highest observed brightness temperature is at J(2000) =
5:39:09.4 +35:45:12. This offset is discussed in the context of CO isotopologues in
section \ref{sec:co21} and in section \ref{sec:discussion-outflows}.
Regions with line wings relative to the ambient cloud within 5\arcmin\ of the
northeast cluster were assumed to be associated with outflows from the cluster.
Further than 5\arcmin, it is likely that the high velocity wings are accelerated by
neighboring HII regions (see section \ref{sec:surroundings}). These line
wings were integrated over the velocity range -34 to -21 \kms\ (blue) and -12 to
1 \kms\ (red) to acquire estimates of the outflowing mass under the assumption
that outflowing gas is optically thin. The extracted regions are displayed in
Figure \ref{fig:cofig}b and measurements in table \ref{tab:comeas}. The line
wings in the central arcminute and central 5 arcminutes were measured for
comparison with lower resolution data and to compute a total outflow mass in
the central region.
The objects in Table \ref{tab:comeas} labeled CO Region 1, 2, and 3 have
uncertain associations with outflows. CO Region 1 is tentatively associated
with outflow 11. CO region 2 may be associated with Outflow 3 but is in a
highly confused region and may have many contributors. CO region 3 is probably
associated with outflow 10. In contrast, the associations with outflows 4 and
6/7 are more certain because they are further from the central region and less
confused. Outflow 1 is seen at high velocities in \citet{beuther2002}
interferometer maps. Outflow 9 is selected primarily based on CO emission.
\Figure{f7.eps}
{
The JCMT HARP CO J=3-2 map integrated over all velocities with significant
emission (-34 \kms\ to -4 \kms) shown in gray log scale from 0 to 150 K \kms.
The elliptical regions over which line wings were integrated are shown with
blue and red circles corresponding to blue and red line wings. The
measurements are presented in table \ref{tab:comeas}.}{fig:cofig}{1.0}
\Figure{f8.eps}{SCUBA 850\um\ image in linear grayscale from
-1 to +10 mJy/beam, with a saturated peak of 24 mJy/beam, with \ensuremath{^{12}\textrm{CO}}\ 2-1
(orange solid, contours at 45,60,85,100,115,130,145 K \kms) and \ensuremath{^{13}\textrm{CO}}\
2-1 (green dashed, contours at 20,30,40,45 K \kms) integrated contours. The
box shows the region plotted in Figure \ref{fig:co21map}.}{fig:scuba_co21}{1.0}
\Figure{f9.eps}{CO spectra of Sh~2-233IR~NE\ in \ensuremath{^{12}\textrm{CO}}\ (blue),
\ensuremath{^{13}\textrm{CO}}\ (green), and \ensuremath{\textrm{C}^{18}\textrm{O}}\ (red). The top-left plot is the pixel
centered at J(2000) = 5:39:13.67 +35:46:26.0 and each pixel is 10\arcsec\ on a side.
The region mapped here is shown with a box in Figure \ref{fig:scuba_co21}.
Redshifted self-absorption, a possible infall tracer, is evident in the \ensuremath{^{12}\textrm{CO}}\
spectra in the outer pixels. The inner pixels show self-absorption only at
central velocities: this may be an indication that emission from outflows
dominates any infall signature, or simply that there is no bulk infall towards
Sh~2-233IR~NE.}{fig:co21map}{0.75} \Figure{f10.eps}{CO spectra of inner
12\arcsec\ centered on Sh~2-233IR~NE\ for all observed CO lines. The CO 3-2 and
2-1 beams are not matched, but in both cases the area integrated over is 1-2
resolution elements across. The divisions demarcating the red and blue line
wings are shown with vertical dashed lines at $v_{LSR}=-21$ and -12 \kms.
}{fig:co21_all3}{1.0}
\begin{deluxetable}{lccccc}
\tabletypesize{\footnotesize}
\centering
\tablecaption{Measured properties of CO flows
\label{tab:comeas}
}
\tablehead{
\colhead{\tablenotemark{a}Region Name} & \colhead{$\int T_{mb}*$} & \colhead{$M (M_\odot)$} & \colhead{p ($M_\odot$ \kms)} &
\colhead{N (\ensuremath{\textrm{cm}^{-2}})} & \colhead{E ($10^{42}$ erg)}}
\startdata
\tablenotemark{b}A. Outflow 4a & 4.27 & .022 & .15 & 1.4\ee{19} & 11 \\
\tablenotemark{b}B. Outflow 4b & 4.60 & .032 & .21 & 1.5\ee{19} & 13 \\
\tablenotemark{b}C. Outflow 1n & 14.5 & .088 & .71 & 4.8\ee{19} & 66 \\
\tablenotemark{b}D. Outflow 6/7 & 4.45 & .045 & .30 & 1.5\ee{19} & 29 \\
\tablenotemark{r}E. CO Region 3 & 1.31 & .016 & .112 & 4.3\ee{18} & 8.5 \\
\tablenotemark{b}F. Sh~2-233IR~NE\ & 41.8 & .464 & 3.72 & 1.4\ee{20} & 330 \\
\tablenotemark{m}F. Sh~2-233IR~NE\ & 132.9& 1.47 & - & 4.4\ee{20} & -\\
\tablenotemark{r}F. Sh~2-233IR~NE\ & 30.0 & .333 & 2.03 & 9.9\ee{19} & 135 \\
\tablenotemark{b}G. Outflow1s & 14.6 & .064 & .48 & 4.8\ee{19} & 40 \\
\tablenotemark{r}H. CO Region 2 & 4.54 & .012 & .074 & 1.5\ee{19} & 5 \\
\tablenotemark{b}I. Outflow 9 & 6.33 & .039 & .39 & 2.1\ee{19} & 43\\
\tablenotemark{b}J. CO Region 1 & 3.61 & .015 & .12 & 1.2\ee{19} & 11\\
\tablenotemark{r}K. Red S & 5.26 & .051 & .34 & 1.7\ee{19} & 26 \\
\tablenotemark{b}L. Blue S & 3.66 & .053 & .47 & 1.2\ee{19} & 47 \\
\tablenotemark{b}1\arcmin\ aperture\tablenotemark{c}& 15.1 & .96 & 7.6 & 5.0\ee{19} & 670\\
\tablenotemark{b}3\arcmin\ aperture& 2.7 & 1.6 & 12 & 9.0\ee{18} & 1000\\
\tablenotemark{b}5\arcmin\ aperture& 1.7 & 2.7 & 20 & 5.6\ee{18} & 1600 \\
\tablenotemark{r}1\arcmin\ aperture & 11.8 & 0.75 & 4.7 & 3.9\ee{19} & 320\\
\tablenotemark{r}3\arcmin\ aperture & 1.9 & 1.1 & 6.8 & 6.2\ee{18} & 460\\
\tablenotemark{r}5\arcmin\ aperture & 0.96 & 1.5 & 10 & 3.2\ee{18} & 640\\
\tablenotemark{b}1\arcmin\ $^{12}$CO 2-1 & 10.4 & .94 & 7.1 & 4.9\ee{19} & 590 \\
\tablenotemark{m}1\arcmin\ $^{12}$CO 2-1 & 97.78 & 8.83 & - & 4.6\ee{20} & - \\
\tablenotemark{r}1\arcmin\ $^{12}$CO 2-1 & 9.17 & 0.83 & 5.52 & 4.3\ee{19} & 430 \\
\tablenotemark{m}1\arcmin\ $^{13}$CO 2-1 & 41.12 & 211 & - & 1.1\ee{22} & - \\
\tablenotemark{m}1\arcmin\ C$^{18}$O 2-1 & 5.31 & 271 & - & 1.4\ee{22} & - \\
\enddata
\tablenotetext{a}{Unless labeled otherwise, regions are extracted from CO 3-2 data as shown in figure \ref{fig:cofig}b }
\tablenotetext{b}{Blue integration over velocity range -34 to -21 \kms}
\tablenotetext{c}{Apertures are centered on J(2000) = 05:39:11.238 +35:45:41.80 in Sh~2-233IR~NE}
\tablenotetext{r}{Red integration over velocity range -13 to -4 \kms}
\tablenotetext{m}{Middle range integration over -21 \kms\ to -13 \kms. Assumed not to be outflowing, so no momentum is computed}
\end{deluxetable}
\subsection{Near-infrared spectroscopy: Velocities}
\label{sec:tspecresults}
The slit positions used and apertures extracted from those slits are displayed
in Figure \ref{fig:tspecslits}. Position-velocity diagrams of the 1-0 S(1)
line are displayed in Figure \ref{fig:outflows_h2_pv}. Velocity measurements
are presented in Table \ref{tab:OutflowH2}.
\Figure{f11.eps}{TripleSpec slits (blue) overlaid on the
\hh\ image. The red boxes indicate the apertures extracted from those slits to
fit and measure \hh\ properties. The apertures are also indicated in the
position-velocity diagrams.}{fig:tspecslits}{0.75}
The near-IR spectrum of Outflow 1 has the largest signal. All of the K-band
\hh\ lines except the 2-1 S(0) 2.3556 \um\ (too weak) and 1-0 S(4) 1.8920 \um
(poor atmospheric transmission) lines were detected (see Table
\ref{tab:nirmeas}). Velocities from gaussian fits to each line are reported.
In the central portion of Sh~2-233IR~NE, outflowing \hh\ emission at
$v_{LSR}\approx-30$ \kms\ is detected. This material may be associated with a
line-of-sight flow, or may originate from the base of the already identified
flows 1-3. In source IR 58, Br$\gamma$ and He I 2.05835\um are detected,
indicating that there is an embedded PDR in this source. There is a hint of a
second, fainter star adjacent to IR 58. IR 93 is observed to be a double
source in the TripleSpec spectrum, but the spectrum is too weak to do any
identification. Br$\gamma$ and possibly He I are detected at fainter levels.
Table \ref{tab:nirmeas} shows the measured line strengths (when detected) of
all \hh\ lines in each aperture. The errors listed are statistical errors
that do not include the systematics errors introduced by a failure to correct
for narrow atmospheric absorption lines.
\Figure{f12.eps}{Position-velocity diagrams of the \hh\ 2.1218 \um\ line in Outflows
1,2, 4, IR 6, and IR93/IR58. The velocity range is from -340 to 190 \kms.}{fig:outflows_h2_pv}{1.0}
\Table{cccccc}
{TripleSpec fitted \htwo\ outflow velocities}
{Outflow Number & Aperture Number & \tablenotemark{a}v$_{LSR}$ (\kms) &\tablenotemark{b}v$_{LSR} (\kms)$ }
{tab:OutflowH2}
{
1 & 1 & -33.54 (0.15) & -31.85 (0.32) \\
1 & 2 & -13.60 (0.57) & -13.56 (0.96) \\
1 & 3 & -40.51 (0.41) & -36.13 (0.81) \\
1 & 4 & -88.7 (2.8) & -83.7 (7.9) \\
2 & 1 & -82.6 (7.6) & -81 (21) \\
2 & 2 & -30.41 (0.57) & -28.9 (1.4) \\
2 & 3 & -33.89 (0.62) & -35.2 (3.7) \\
4 & 1 & -73.34 (0.48) & -70.2 (1.1) \\
4 & 2 & -64.08 (0.61) & -67.8 (2.2) \\
IR6 & 1 & -39.4 (1.6) & -39.4 (4.2) \\
IR93 & 2 & -26.07 (0.43) & -26.85 (0.97) \\
IR93 & 3 & -30.6 (1.5) & -32.0 (2.5) \\
IR93 & 4 & -29.14 (0.77) & -30.3 (2.1) \\
IR93 & 6 & -47.7 (7.9) & -71 (37) \\
}{
\tablenotetext{a}{Measured from \hh\ 1-0 S(1) 2.1218 \um\ line
\tablenotetext{b}{Measured from all detected \hh\ lines fit with model described in section \ref{sec:tspecresults}}
}
\clearpage
\begin{deluxetable}{ccccccccc}
\centering
\tabletypesize{\scriptsize}
\tablecaption{Measured properties of \hh\ flows}
\tablehead{
Outflow & \tablenotemark{a}Center & \tablenotemark{b}PA & \tablenotemark{c}Length &
\tablenotemark{d}Source & \tablenotemark{e}Flow & \tablenotemark{e}Counterflow & \tablenotemark{f}Age & \tablenotemark{g}LOS \\
& & & & & Length & Length & (50 \kms) & Velocity \\
}
\startdata
1 & 05:39:13.023 +35:45:38.66 & -13.3 &142.3" &mm2? &58 &84.2 &1.4e4 &- \\
2 & 05:39:13.058 +35:45:51.28 & -47.0 &44.6" &mm1a &44.6 &- &6.6e3 &Blue \\
3 & 05:39:12.48 +35:45:54.9 & -62 &44" &mm3? &44 &- &6.5e3 &Red \\
4 & ambiguous & 17.8-21.8 &141-144" & ? &141-144 &- &2.1e4 &Blue \\
5 & 05:39:12 +35:45:51 & 170 &38-48 &mm3? &38-48 &- &6.5e3 &Blue \\
6 & 05:39:09.7 +35:45:17 & 14.5 &197 &Q5 &197 &- &2.9e4 &Blue \\
8 & 05:39:10.002 +35:45:10.87 & -154.6 &105.5" &IR41? &54.7 &52.9 &7.9e3 &- \\
\enddata
\tablenotetext{a}{Midpoint of bipolar outflow if symmetric, position of jet source candidate if asymmetric}
\tablenotetext{b}{Position angle uncertainties are $\sim 5\ensuremath{^{\circ}}$ because they are not perfectly collimated,
causing an ambiguity in their true directions. The exact angles used to draw vectors in figure \ref{fig:outflowsh2}
are listed for reproducibility.}
\tablenotetext{c}{Total length of outflow on the sky, including counterflow}
\tablenotetext{d}{Candidate jet source object. Outflows 2 and 6 have clear associations,
the others are weaker candidates.}
\tablenotetext{e}{Flow length is the distance from the CENTER position to the last \hh\ knot
in the position angle direction as listed. Counterflow length is the distance from the CENTER position
to the opposite far knot.}
\tablenotetext{f}{Timescale of jet assuming it is propagating at 50 \kms, an effective lower limit to see
\hh\ emission. If two lengths are available, uses the longer of the two. These are lower limits to the
true timescale \citep{parker1991}.}
\tablenotetext{g}{The parity of the outflow along the line of sight. Outflow 1 and 8 have counterflows with
parities as indicated in figure \ref{fig:outflowsh2}}
\end{deluxetable}
\clearpage
\begin{deluxetable}{ccccccccccccc}\setlength\tabcolsep{3pt}
\scriptsize
\tabletypesize{\tiny}
\tablewidth{0pt}
\centering
\tablecaption{Measured \hh\ line strengths
\label{tab:nirmeas}}
\tablehead{
&1-0 S(0)&1-0 S(1)&1-0 S(2)&1-0 S(3)&1-0 S(6)&1-0 S(7)&1-0 S(8)&1-0 S(9)&1-0 Q(1)&1-0 Q(2)&1-0 Q(3)&1-0 Q(4) \\
aperture&2.2233&2.12183&2.03376&1.95756&1.78795&1.74803&1.71466&1.68772&2.40659&2.41344&2.42373&2.43749}
\startdata
outflow1ap1 & 3.60E-15& 9.80E-15& 5.50E-15& 1.20E-14& 4.70E-15& 3.10E-15& 8.60E-16& 1.10E-15& 9.20E-15& 6.10E-15& 1.10E-14& 6.90E-15 \\
& ( 2.4e-17)&( 3.4e-17)&( 6.8e-17)&( 2e-16)&( 2e-16)&( 2.8e-17)&( 2.8e-17)&( 2.7e-17)&( 1.4e-16)&( 7.4e-17)&( 8.8e-17)&( 7.8e-17) \\
outflow1ap2 & 7.10E-16& 1.80E-15& 9.90E-16& 1.80E-15& -& -& -& -& 3.00E-15& 1.90E-15& 3.20E-15& 2.00E-15 \\
& ( 2.1e-17)&( 2.7e-17)&( 6.8e-17)&( 1.7e-16)& -& -& -& -&( 1.3e-16)&( 4e-17)&( 7.8e-17)&( 3.8e-17) \\
outflow1ap3 & 1.60E-15& 4.10E-15& 2.20E-15& 4.70E-15& -& 8.30E-16& -& -& 5.60E-15& 3.70E-15& 6.60E-15& 4.80E-15 \\
& ( 2.4e-17)&( 3.4e-17)&( 6.3e-17)&( 1.8e-16)& -&( 2.8e-17)& -& -&( 1.4e-16)&( 5.9e-17)&( 8.2e-17)&( 5.2e-17) \\
outflow1ap4 & -& 9.00E-16& -& -& -& -& -& -& -& -& -& - \\
& -&( 3e-17)& -& -& -& -& -& -& -& -& -& - \\
outflow2ap1 & -& 3.60E-16& -& -& -& -& -& -& -& -& -& - \\
& -&( 1.5e-17)& -& -& -& -& -& -& -& -& -& - \\
outflow2ap2 & 9.40E-16& 2.40E-15& 1.50E-15& 1.80E-15& -& 4.00E-16& -& -& 3.00E-15& -& 3.70E-15& - \\
& ( 1.7e-17)&( 2.2e-17)&( 5.8e-17)&( 1.1e-16)& -&( 2.3e-17)& -& -&( 4.7e-17)& -&( 7.9e-17)& - \\
outflow2ap3 & 2.10E-15& 1.90E-15& 1.80E-15& 2.20E-15& -& 6.70E-16& -& -& 5.70E-15& -& 7.30E-15& - \\
& ( 1.7e-17)&( 2.2e-17)&( 5.1e-17)&( 1.3e-16)& -&( 2.9e-17)& -& -& -8.00E-16& -& -8.00E-16& - \\
outflow4ap1 & 5.50E-16& 2.00E-15& 8.50E-16& 2.00E-15& -& 9.40E-16& 1.90E-16& 3.40E-16& 1.40E-15& -& 1.40E-15& - \\
& ( 2e-17)&( 2e-17)&( 5e-17)&( 1.3e-16)& -&( 2.8e-17)&( 1.8e-17)&( 2.3e-17)& -4.00E-16& -&( 6.9e-17)& - \\
outflow4ap2 & 5.60E-16& 2.00E-15& 5.30E-16& 2.10E-15& -& 5.80E-16& -& 1.10E-16& -& -& 2.00E-15& - \\
& ( 2e-17)&( 2.2e-17)&( 2.4e-17)&( 1.2e-16)& -&( 2.3e-17)& -&( 1.8e-17)& -& -& -2.00E-16& - \\
IR6ap1 & -& 1.10E-15& -& 9.30E-16& -& 4.30E-16& -& -& -& -& -& - \\
& -&( 3e-17)& -&( 1.4e-16)& -&( 3.2e-17)& -& -& -& -& -& - \\
IR93ap1 & -& 6.60E-15& -& 2.70E-15& -& -& -& -& -& -& 5.80E-15& - \\
& -&( 3.5e-17)& -&( 1e-16)& -& -& -& -& -& -&( 7.4e-17)& - \\
IR93ap2 & 4.40E-15& 6.60E-15& 3.90E-15& 3.30E-15& -& 1.10E-15& -& -& 7.60E-15& 5.10E-15& 6.90E-15& 5.50E-15 \\
& ( 3.2e-17)&( 3.7e-17)&( 9.2e-17)&( 1.4e-16)& -&( 2.7e-17)& -& -&( 8e-17)&( 5.2e-17)&( 7.4e-17)&( 6.1e-17) \\
IR93ap3 & 1.00E-15& 1.70E-15& -& 9.00E-16& -& -& -& -& 2.00E-15& 1.70E-15& 1.90E-15& - \\
& ( 2.3e-17)&( 3.6e-17)& -&( 1.2e-16)& -& -& -& -&( 8e-17)&( 3.8e-17)&( 8.8e-17)& - \\
IR93ap4 & 2.60E-15& 3.70E-15& -& -& -& -& -& -& 4.30E-15& 3.50E-15& 4.70E-15& - \\
& ( 3.2e-17)&( 3.6e-17)& -& -& -& -& -& -&( 8.5e-17)&( 5.2e-17)&( 7.4e-17)& - \\
IR93ap5 & -& 1.90E-15& -& 1.00E-15& -& -& -& -& -& -& -& - \\
& -&(2.4e-17) & -&(1.00e-16)& -& -& -& -& -& -& -& - \\
IR93ap6 & -& 4.10E-16& -& -& -& -& -& -& -& -& -& - \\
& -&( 3e-17)& -& -& -& -& -& -& -& -& -& - \\
\enddata
\tablecomments{Fluxes are in units erg s$^{-1} \ensuremath{\textrm{cm}^{-2}} $\AA$^{-1}$. Errors are listed on
the second row for each aperture. Errors of (0) indicate that the line was detected, but that
the fluxes should not be trusted because the background was probably oversubtracted.}
\end{deluxetable}\addtocounter{table}{-1}
\begin{deluxetable}{ccccccccccccccc}
\scriptsize
\tabletypesize{\tiny}
\tablewidth{0pt}
\centering
\tablecaption{Measured \hh\ line strengths (cont'd)
\label{tab:nirmeas2}}
\tablehead{
&2-1 S(1)&2-1 S(3)&3-2 S(3)&3-2 S(4)&4-3 S(5) & [Fe II] & [Fe II] \\
&2.24772&2.07351&2.2014&2.12797&2.20095 & 1.6435 & 1.2567 \\ }
\startdata
outflow1ap1 & 2.00E-15& 1.20E-15& 6.20E-16& 2.60E-16& 7.10E-16 & 4.4e-15 &3.5e-15 \\
& ( 2.5e-17)&( 3.5e-17)&( 2.2e-17)&( 1.6e-17)&( 1.9e-17) & ( 7.9e-17) &( 4e-17) \\
outflow1ap2 & -& -& -& -& - & 6.7e-16 &3.1e-16 \\
& -& -& -& -& - & ( 7.8e-17) &( 3.3e-17) \\
outflow1ap3 & 9.90E-16& -& 5.70E-16& 2.50E-16& 6.40E-16 & 1.3e-15 &5.7e-16 \\
& ( 2.6e-17)& -&( 0)&( 1.2e-17)&( 0) & ( 8.9e-17) &( 4e-17) \\
outflow1ap4 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
outflow2ap1 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
outflow2ap2 & 6.40E-16& -& -& -& - & - & - \\
& ( 1.9e-17)& -& -& -& - & - & - \\
outflow2ap3 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
outflow4ap1 & 4.30E-16& 4.20E-16& -& -& 1.60E-16 & - & - \\
& ( 1.9e-17)& (0) & -& -& (0) & - & - \\
outflow4ap2 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR6ap1 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR93ap1 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR93ap2 & 3.80E-15& -& 3.10E-15& -& - & - & - \\
& ( 2.2e-17)& -& (0) & -& - & - & - \\
IR93ap3 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR93ap4 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR93ap5 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
IR93ap6 & -& -& -& -& - & - & - \\
& -& -& -& -& - & - & - \\
\enddata
\tablecomments{Fluxes are in units erg s$^{-1} \ensuremath{\textrm{cm}^{-2}} $\AA$^{-1}$. Errors are listed on
the second row for each aperture. Errors of (0) indicate that the line was detected, but that
the fluxes should not be trusted because the background was probably oversubtracted.}
\end{deluxetable}
\clearpage
\subsection{Spectroscopic Results: Optical}
\label{sec:dis}
IR 6 and IR 41 (objects 1 and 6 in Figure \ref{fig:outflowsh2}) both show
\ensuremath{\textrm{H}\alpha}\ in emission. IR 41 is close to the reflection nebula in the southeast
portion of IRAS 05358\ and is probably the reflected star. The reflection
nebula's spectrum is very similar to IR 41's spectrum at \ensuremath{\textrm{H}\alpha}\ in both width
and brightness (see Figure \ref{fig:outflow10_pv}).
\Figure{f13.eps}{A position-velocity diagram of IR 6 and 41
including the reflection nebula near IR 41 ($\approx 7.4\msun$). IR 6 shows a
two-peaked \ensuremath{\textrm{H}\alpha}\ emission profile, but is the less massive ($\approx 4.5\msun$) of
the pair. The separation between the two sources is 55\farcs3, and each pixel
is 0\farcs4.}{fig:outflow10_pv}{0.25}
There are three
components in the \ensuremath{\textrm{H}\alpha}\ profile of IR 6: a broad absorption feature seen far
($\sim400\kms$ from the line center) on the wings and two emission peaks. The
peaks are separated by 190 \kms\ and the blueshifted peak is weaker than the
redshifted (Table \ref{tab:IR6}). The H$\beta$ profile shows much deeper
absorption and weaker emission but with similar characteristics. The presence of the \ensuremath{\textrm{H}\alpha}\ emission
makes identification of the stellar type from the \ensuremath{\textrm{H}\alpha}\ line profile uncertain.
The derived extinction to IR 6 is at least $A_V=7$ from an assumed \ensuremath{\textrm{H}\alpha}/\ensuremath{\textrm{H}\beta}\
ratio of 2.87 \citep{agnsquared}. The \ensuremath{\textrm{H}\beta}\ flux was measured from zero to the
peaks of the emission profile and therefore probably overestimates the
\ensuremath{\textrm{H}\beta}\ flux and underestimates the extinction.
\Table{cccccccc}
{IR 6 Deblended Profiles}
{& \tablenotemark{a} Blue & \tablenotemark{b}Blue & \tablenotemark{a} Red & \tablenotemark{b}Red & Absorption &
Gaussian / & \tablenotemark{b}Absorption \\
&Emission & Wavelength & Emission & Wavelength & & Lorentzian FWHM & Wavelength \\
}
{tab:IR6}
{\ensuremath{\textrm{H}\alpha}\ & 4.4\ee{-14} & 6559.79 & 1.3\ee{-13} & 6564.23 & -2.6\ee{-14} & 1.5 / 0.19 & 6563.02 \\
\ensuremath{\textrm{H}\beta} & \tablenotemark{c}2.4\ee{-14} & 4857.68 & \tablenotemark{c}1.8\ee{-14} &
4864.28 & \tablenotemark{d} -4.6\ee{-14} & 0.17 / 16.5 & 4861.91 \\}
{
\linebreak
\tablecomments{Measurements are made using a Voigt profile fit in the IRAF {\sc splot} task.}
\tablenotetext{a}{Flux measurements are in units of erg s$^{-1}$ \ensuremath{\textrm{cm}^{-2}} \AA$^{-1}$}
\tablenotetext{b}{Wavelengths are in Geocentric coordinates. Subtract 0.53\AA\ from \ensuremath{\textrm{H}\alpha}\
and 0.39\AA\ from \ensuremath{\textrm{H}\beta}\ to put in LSR coordinates.}
\tablenotetext{c}{\ensuremath{\textrm{H}\beta}\ emission was measured assuming a continuum of zero and therefore represents
an upper limit in the \ensuremath{\textrm{H}\beta}\ emission}
\tablenotetext{d}{\ensuremath{\textrm{H}\beta}\ deblending may contain systematic errors
from a guessed subtraction of the \ensuremath{\textrm{H}\beta}\ emission}
}
\Table{cccccccc}
{Lines observed in the optical spectra}
{Source & \ensuremath{\textrm{H}\alpha}\ & \ensuremath{\textrm{H}\beta}\ & [S II] 6716\AA & [S II] 6731\AA & [O I] 6300\AA & [O I] 6363\AA & [O I] 5577\AA }
{tab:optical}
{
Outflow1 ap1 & 4.3\ee{-16} & - & 5.7\ee{-16} & 6.3\ee{-16} & 5.3\ee{-16} & 1.8\ee{-16} & - \\
& 6561.49 & - & 6715.3 & 6729.6 & 6299.7 & 6363.3 & - \\
Outflow1 ap2 & 4.5\ee{-16} & - & 4.5\ee{-16} & 4.6\ee{-16} & 3.1\ee{-16} & 1.2\ee{-16} & - \\
& 6561.22 & - & 6714.9 & 6729.3 & 6299.4 & 6363.2 & - \\
Ambient Medium - slit 1 & 6.7\ee{-17} & 5.3\ee{-18} & 1.0\ee{-17} & 7.9\ee{-18} & 3.5\ee{-16} & 1.2\ee{-16} & 4.8\ee{-17} \\
& 6562.87 & 4861.7 & 6716.7 & 6731.2 & 6300.3 & 6363.8 & 5578.0 \\
IR 41 nebula & 2.6\ee{-15} & - & - & - & 4.4\ee{-16} & 1.9\ee{-16} & - \\
& 6562.85 & - & - & - & 6300.3 & 6363.9 & - \\
IR 41 & 6.5\ee{-15} & - & - & - & 1.1\ee{-16} & 7\ee{-17} & - \\
& 6562.9 & - & - & - & 6300.0 & 6363.3 & - \\
IR 6 & 1.76\ee{-13} & \tablenotemark{a} 4.1\ee{-14} &-&-&-&- & \\
& - & - &-&-&-&- & \\
}{
\linebreak
\tablecomments{
Wavelengths listed are in \AA\ and are geocentric. To convert to LSR
velocities, subtract 24.35 \kms. The ambient medium fluxes represent averages
across the slit.
Fluxes are in erg s$^{-1} \ensuremath{\textrm{cm}^{-2}} $\AA$^{-1}$.
}
\tablenotetext{a}{\ensuremath{\textrm{H}\beta}\ measurement in IR 6 is an upper limit}
}
\subsection{Radio Interferometry}
\label{sec:vlaresults}
A point source was detected in the X, U, K and Q band VLA maps with high
significance at the same location as the X-band point source reported in
\citet{beuther2007}. Seven-parameter gaussians were fit to each image to measure
the beam sizes and positions and flux densities. The measurements are listed
in Table \ref{tab:vla}. The locations of the point source and the shape of the
beams from the re-reduced X and Q band images are displayed in figure
\ref{fig:mm1adiagram}.
A Class II 6.7 GHz methanol maser was detected in IRAS 05358\ by \citet{Menten1991}.
It was observed with the European VLBI Network (EVN) by \citet{Minier2000} and
seen to consist of a linear string of maser spots that trace a probable disk in
addition to maser spots scattered around a line perpendicular to the proposed
disk (see Figure \ref{fig:mm1adiagram}).
The VLA source is more than a VLA beam away from the VLBI CH$_3$OH maser disk
identified by \citet{Minier2000}. It is to the southeast in the opposite
direction of Outflow 2. Outflow 2 is at position angle -47$\ensuremath{^{\circ}}$, while the
disk is at PA 25\ensuremath{^{\circ}}. The 8$\ensuremath{^{\circ}}$ difference from being perpendicular is
well within the error associated with determining the angle of the outflow in
this confused region, so the VLBI disk is a strong candidate for the source of Outflow 2.
\begin{figure*}[htpb]
\epsscale{0.75}
\plotone{f14.eps}
\caption{A diagram of the region surrounding mm1a from \citet{beuther2007}.
The ellipses are centered at the measured source centers and their sizes
represent the beam sizes of the Plateau de Bure interferometer at 1.2mm
\citep[blue,][]{beuther2007}, Gemini MICHELLE at 7.9\um\
\citep[red,][]{Longmore2006}, the VLA at 3.6cm (green), and the VLA at 7mm
(orange). The maser disk was measured with the European VLBI Network by
\citet{Minier2000}, so the size and direction of the disk are very well
constrained. The black circle is centered on the pointing center of the
VLBI observation and represents the absolute pointing uncertainty. The
arrow pointing in the direction of Outflow 2 traces clumps along the
outflow back to the mm emission region. The vector is not to scale -
Outflow 2 is about 45\arcsec\ long.
\label{fig:mm1adiagram}}
\end{figure*}
The astrometric uncertainty in VLA measurements are typically
$\lesssim0.1$\arcsec. Different epochs of high-resolution X-band and Q-band
data confirmed that the pointing accuracy is substantially better than
0.1\arcsec\ in this case. The VLBI absolute pointing uncertainty is reported
to have an upper limit of $\sim 0.03$\arcsec\ \citep{Minier2000}. The
separation between the VLA Q-band center and the VLBI disk center is
0.22\arcsec, whereas the separation between the combined X and Q band pointing
centers is only 0.027\arcsec, which can be viewed as a characteristic
uncertainty. This is evidence for at least two distinct massive stars in a
binary separated by $\sim$ 400 AU. While the statistical significance of the
binary separation is quite high using formal errors, the systematic errors
cannot be constrained nearly as well. This object is a candidate binary system
but is not yet confirmed.
\begin{deluxetable}{lllll}
\tablecolumns{7}
\tablecaption{VLA measurements near IRAS 05358\ mm1a
\tabletypesize{\footnotesize}
\label{tab:vla}}
\footnotesize
\tablehead{
\colhead{Frequency } & \colhead{Beam major /} & \colhead{RA (error)} &
\colhead{Peak flux } & \colhead{Map RMS } \\
\colhead{Observed} & \colhead{minor / PA} & \colhead{Dec (error)} & \colhead{(error)} & \colhead{(mJy/beam)} \\ }
\startdata
43.3 GHz & 0.022\arcsec\ / 0.029\arcsec\ / -10.4 \ensuremath{^{\circ}} & 05:39:13.065425 (0.000015) & 1.319 (0.027) & 0.179 \\
&&35:45:51.14732 (0.00031) & \\
22.5 GHz & 1.52\arcsec\ / 1.28\arcsec\ / 232 \ensuremath{^{\circ}} & 05:39:13.05521 (0.0029) & 1.26 (0.04) & 0.091 \\
&&35:45:51.378 (0.046) &\\
15.0 GHz & 1.58\arcsec\ / 2.00\arcsec\ / 0 \ensuremath{^{\circ}} & 05:39:13.062 (0.005) & 1.274 (0.065) & 0.124 \\
&&35:45:51.4 (0.1) &\\
8.4 GHz & 0.107\arcsec\ / 0.122\arcsec\ / 7.9 \ensuremath{^{\circ}} & 05:39:13.064548 (0.000036) & 0.506 (0.003) & 0.015 \\
&&35:45:51.170356 (0.000613) &\\
\enddata
\tablenotetext{.}{Errors reported here are fit errors. Absolute flux calibration
errors are negligible for the X-band data but are about equivalent to measurement errors
for the K, and U bands and dominant in the Q band}
\end{deluxetable}
\section{Analysis}
\subsection{Near-Infrared Spectroscopic Extinction Measurements}
Extinction along a line of sight can be calculated using the 1-0 Q(3) / 1-0 S(1) line ratio.
\begin{equation}
A_\lambda = 1.09 \left[ -\textrm{ln}
\frac{S_\nu(S)/S_\nu(Q)}{A_{ul}(S)\lambda_Q/A_{ul}(Q) \lambda_S} \right] \left[
\left(\frac{\lambda_S}{\lambda_Q}\right)^{-1.8} -1 \right]^{-1}
\end{equation}
Because they are from the same upper state,
their intensity ratio should be set by their Einstein A values times the
relative energies of the transitions.
However, as shown by
\citet{luhman1998}, narrow atmospheric absorption lines in the long wavelength
portion of the K band, where the Q branch lines lie, can create a significant
bias. Because the lines have not been corrected for atmospheric absorption,
the Q branch fluxes should actually be lower limits. Since the 1-0 S(1)
transition at 2.1818 microns is affected very little by atmospheric absorption,
and the exinction measured is proportional to the Q/S line ratio, the measured
extinction should be a lower limit.
The [Fe II] 1.6435 and 1.2567 \um\ lines were detected in Outflow 1, allowing for
another direct measurement of the extinction. The measured ratio $FR$ =
1.26\um/1.64\um\ in Outflow 1 was .8, while the true value is at least 1.24 but
may be as high as 1.49 \citep{Smith2006,luhman1998,Giannini2008}. The
extinction measured from this ratio ranges from $A_V = 4.1$ ($FR=1.24$) to 5.8
($FR=1.49$). The S(1)/Q(3) ratio uncorrected for telluric absorption is .91,
which yields an extinction lower limit of $A_V = 18.7$, is inconsistent with
the measurement from [Fe II]. The \ensuremath{\textrm{H}\alpha}\ detection and \ensuremath{\textrm{H}\beta}\ upper limit give a
lower limit on the extinction of $A_V = 6.6$, which is consistent with both of
the other methods to within the calibration uncertainty.
It is possible that the two measurements come from unresolved regions with
different levels of extinction, though a factor of at least 3 change over an
area $\sim 100 AU$ far from the millimeter cores seems unlikely. A strong IR
radiation field could plausibly change the line ratio from the expected
Einstein A value. The question is not resolved but may be possible to address
with near-IR observations of nearby bright HH flows with more careful
atmospheric calibration.
\subsection{Optical Spectra}
\subsubsection{Stellar Type}
IR 6 is suspected to be the source of the bright \hh\ finger at PA $\approx$
15\arcdeg.
IR 6 is also
a 24\um\ source and was detected by MSX (designation G173.4956+02.4218). We
identify this star as a Herbig Ae/Be star.
\subsubsection{Density and Extinction Measurements}
The spectrum of knot N1 (the bow of Outflow 1) allowed a measurement of
electron density in the shocks from the [S II] 6716/6731 line ratio
. Densities were determined to be $n=$ 700 \ensuremath{\textrm{cm}^{-3}}\ in the forward lump
and $n=$500 \ensuremath{\textrm{cm}^{-3}}\ in the second lump. \ensuremath{\textrm{H}\alpha}, [N II] 6583, [O I] 6300, and [O
III] 6363 were also detected, but no lines were detected in the blue portion of
the spectrum presumably because of extinction. The measured velocities from [S
II] are faster than the \hh\ velocity measurements at about $v_{LSR} = -68
\pm 5$ \kms.
There is also an ambient ionized medium that uniformly fills the slit with a
[S II]-measured density $n_e=120\ \ensuremath{\textrm{cm}^{-3}}$. Evidently, nearby massive stars are
ionizing the low-density ISM located in front of IRAS 05358. This material is
moving at velocity $v_{LSR} = -7 \pm 5$ \kms\ and is extincted by $A_V=1.5$ as
determined from \ensuremath{\textrm{H}\alpha}/H$_\beta=2.87$ assuming the gas is at 10$^4$ K.
\subsection{UCHII region measurement}
A uniform density, ideal HII region will have an intensity curve $I = I_0
( 1 - e^{-\tau_\nu})$ where
\begin{equation} \tau = 8.235\times10^{-2}
\left(\frac{T_e}{K}\right)^{-1.35} \left(\frac{\nu}{\textrm{GHz}}\right)^{-2.1}
\left(\frac{\textrm{EM}}{\textrm{pc~cm}^{-6}}\right) a(\nu,T)
\end{equation}
following \citet{rohlfs2004} equation 9.35, where $a(\nu,T) \approx 1$ is a
correction factor. By assuming an excitation temperature $T_{ex} = 8500$K,
blackbody with a turnover to an optically thin thermal source was fit to the
centimeter SED. The turnover frequency from this fit is $\tau=$15.5 GHz,
corresponding to an emission measure $EM=7.4 \times 10^8$ pc cm$^{-6}$. This
turnover frequency is lower than the $\sim35$ GHz reported by
\citet{beuther2007}. The turnover is clearly visible in the U, K and Q data
points in figure \ref{fig:HIIregionfit}.
By assuming the X-band emission is optically thick, a source size can be derived.
\begin{equation}
\label{eqn:uchiirad}
2 r = \left[\frac{S_\nu}{2 k_B T_{ex}} \lambda^2 D^2\right]^{1/2}
\end{equation}
where D is the distance to the source. Assuming a spherical UCHII region and a
distance of 1.8 kpc, the source has radius $r=$30 AU (for comparison, the
Q band beam minor axis is $\sim$90 AU, so the region could in principle be
resolved by the VLA + Pie Town configuration).
The measured density is $n=(EM / r)^{1/2} = 2.2\times10^6$ \ensuremath{\textrm{cm}^{-3}}, with a
corresponding emitting mass $M = n \mu m_H 4/3 \pi r^3 = 1.0\times 10^{-6}$
\msun\ using $\mu=1.4$. Using \citet{kurtz1994} equation 1,
\begin{equation}
N_{Lyc} = (8.04\times10^{46} s^{-1}) T_e^{-0.85} \left(\frac{r}{pc} \right)^3 n_e^2
\end{equation}
the number of Lyman continuum photons per second required to maintain
ionization is estimated to be $N_{Lyc} = 5.9\times10^{44}$
, a factor of $\sim4$ lower than measured by \citet{beuther2007} and
closer to a B2 ZAMS star ($\sim11\msun$) than B1 using Table 2 of
\citet{panagia1973}. If the star has not yet reached the main sequence, it
could be significantly more massive \citep{hosokawa2009}, so our stellar mass
estimate is a lower limit.
The gravitational binding radius of a 11 \msun\ star is $r_g = 2 G M / c_s^2 \approx
190$ AU (the HII region is assumed to be supported entirely by thermal
pressure, which provides an upper limit on the binding radius since turbulent
pressure can exceed thermal pressure). The UCHII region radius of 30 AU is
much smaller, indicating that, under the assumption of spherical symmetry, the
HII region is bound.
\citet{leurini2007} noted that the CH$_3$CN line profile around this source
could be fit with a binary system with separation $<1100$AU and a total mass
of 7-22 \msun . This is entirely consistent with our picture of a massive binary
system with a 11 \msun\ star in a UCHII region and another high mass star with
a maser disk.
There are no other sources in the IRAS 05358\ region to a 5$\sigma$ limit of 0.075
mJy in the X-band, which provides the strictest upper limit. From equation
\ref{eqn:uchiirad}, this corresponds to an optically thick source size of 24
AU. The maser disk has a spatial extent of around 140 AU, so it is quite unlikely
that either an undetected UCHII region or the observed UCHII are associated with
the maser disk.
Assuming the same turnover point for undetected sources,
an upper limit is set on $N_{Lyc}$ for undetected sources:
\begin{equation}
N_{Lyc} = (8.04\times10^{46} s^{-1}) \left(\frac{S_\nu }{ 2 k_B T_{ex}^{1.85} } \lambda_{cm}^2 D_{pc}^2\right) EM
\end{equation}
Our 5$\sigma$ upper limit is $N_{Lyc} = 1.38\ee{44}$ s$^{-1}$, indicating that
any stars present must be a later class than B3, or lower than about 8 \msun .
For an emission measure as much as 3 times higher, the corresponding stellar
mass would be less than 10 \msun . It is likely that no other massive
stars have formed in Sh~2-233IR~NE.
After independently determining the best-fit UCHII model to the VLA data, we
included the PdBI data points from \citet{beuther2007} and fit a power-law to
both data sets. If the emission measure was allowed to vary, the derived
parameters were $EM=6.3\ee{8}$ and $\beta=0.7$. However, doing this visibly
worsened the UCHII region fit without significantly improving the power-law
fit, so the fit was repeated holding a fixed emission measure, yielding
$\beta=0.8$ (plotted in Figure \ref{fig:HIIregionfit}b). This power-law is
much shallower than the $\beta=1.6$ measured by \citet{beuther2007} without
access to the 44 GHz data point, and suggests that there is a significant
population of large grains in source mm1a. However, we caution that the fits
were performed only accounting for statistical errors, not the significant and
unknown systematic errors that are likely to be present in mm interferometric
data. The PdBI beams are much larger than the VLA beams, so the larger beams
could be systematically shifted up by including additional emission, which
would reduce $\beta$. Nonetheless, the new VLA data constrains the UCHII
emission to contribute no more than 10\% of the 3.1mm flux.
\begin{figure*}[htpb]
\epsscale{0.75}
\plotone{f15.eps}
\plotone{f16.eps}
\caption{(a) The HII region fit to measured X, K, U, and Q band data. Error bars
represent statistical error in the flux measurement. The Q band error is
dominated by flux calibration uncertainty (see Table \ref{tab:vlatimes}). The
measured turnover is at 9.5 GHz. (b) A fit to both the VLA data presented in this paper
and the (sub)mm points from \citet{beuther2007}. The best fit spectral index for the
dust emission is $\alpha=2.8$ ($\beta=0.8$), significantly lower than the $\alpha=3.6$
measured by \citet{beuther2007} without the 0.7 mm data point.}
\label{fig:HIIregionfit}
\end{figure*}
\subsection{Mass, Energy, and Momentum estimates from CO}
\subsubsection{Equations}
The column density for CO J=3-2 is estimated using the equation
\begin{equation}
\label{eqn:column}
N_{\hh} =
\frac{\hh}{\textrm{CO}}\frac{8\pi\nu^3k_B}{3c^3hB_eA_{ul}}(1-e^{h\nu/k_BT_{ex}})^{-1}
\frac{1}{\eta_{mb}} \int T_A^*(v) dv
\end{equation}
where $A_{ul}=A_{32}=2.5\times10^{-6}\textrm{s}^{-1}$ and
$A_{21}=6.9\ee{-7}$s$^{-1}$\citep{turner1977}, the rotational constant $B_e =
57.64$, 55.10, and 55.89 GHz for \ensuremath{^{12}\textrm{CO}}, \ensuremath{^{13}\textrm{CO}}, and \ensuremath{\textrm{C}^{18}\textrm{O}}\ respectively,
$\eta_{mb} = .68$, and $T_{ex}$ is assumed to be 20K. The partition function
is approximated as
\begin{equation}
Z=\sum_{J=1}^\infty (2J+1) exp
\left(\frac{-J(J+1)hB_e}{k_B T_{ex}}\right) \approx \int_0^\infty (2J+1)exp
\left(\frac{-J(J+1)hB}{k_bT_{ex}}\right) dJ
\end{equation}
which is valid when $T_{ex} >> hB_e/k_B \sim 2.8 $K.
Equation \ref{eqn:column} becomes
\begin{equation}
N_{\hh} = ( 3.27\times10^{18} \ensuremath{\textrm{cm}^{-2}}) \frac{1}{\eta_{mb}} \int T_A^*(v) dv
\end{equation}
where the integrand is in units K \kms. The mass is then
\begin{equation}
M = \mu\ m_{\hh}\ A\ N_{\hh} = 1.42\times10^{-5} A \frac{1}{\eta_{mb}} \int
T_A^*(v) dv
\end{equation}
where A is the area in cm$^2$, $\mu=1.4$ is a constant to account for the presence of
helium, and again velocity is in \kms.
\subsubsection{CO J = 2-1 Isotopologue Comparison}
\label{sec:co21}
\citet{Thomas2008} observed C$^{17}$O in the J=2-1 and 3-2 transitions each with a
single pointing using the JCMT centered at J(2000) = 05:39:10.8 +35:45:16 and measured a
column density $N_{\hh}=4.03\times10^{22}$ cm$^{-2}$. The peak column
density is $1.7\ee{22}$\ensuremath{\textrm{cm}^{-2}}\ in \ensuremath{^{13}\textrm{CO}}\ and $2.2\ee{22}$ in \ensuremath{\textrm{C}^{18}\textrm{O}}\ at J(2000) = 5:39:10.2
+35:45:26, which is reasonably consistent with the C$^{17}$O measurement
considering abundance uncertainties. The peaks of the integrated
spectra for \ensuremath{\textrm{C}^{18}\textrm{O}}\ and \ensuremath{^{13}\textrm{CO}}\ are coincident, but the \ensuremath{^{12}\textrm{CO}}\
integrated peak is at J(2000) = 5:39:12.6 +35:45:46 (Figure \ref{fig:scuba_co21},
discussed more in section \ref{sec:discussion-outflows}).
Measurements of the column density, mass, momentum, and energy are performed as
in Equation \ref{eqn:column}. Assuming a \ensuremath{^{12}\textrm{CO}}/\ensuremath{^{13}\textrm{CO}}\ ratio of 60
\citep{lucas1998} and optically thin \ensuremath{^{13}\textrm{CO}}, the mean column density
across the region is $N_{\hh} = 4.5\times10^{21}$ \ensuremath{\textrm{cm}^{-2}}. The resulting total
mass of the central $\sim3$\arcmin\ is about 320 \msun, which is substantially
smaller than the 600 \msun\ measured by \citet{beuther2002} and
\citet{Zinchenko1997}, but it is nearly consistent with 870\um\ and NH$_3$
estimates of 450 and 400 \msun\ from \citet{Mao2004} and is within the
systematic uncertainties of these measurements. Assuming \ensuremath{\textrm{C}^{18}\textrm{O}}\ is
optically thin and the \ensuremath{\textrm{C}^{18}\textrm{O}}/\ensuremath{^{13}\textrm{CO}}\ ratio is 10, the
column density is 5.2\ee{21} \ensuremath{\textrm{cm}^{-2}}\ and the mass is 360 \msun, which is
consistent with the \ensuremath{^{13}\textrm{CO}}\ measurements, indicating that optical depth
effects are probably not responsible for the discrepancy with the dust mass
estimate.
\subsubsection{CO Mass and Energy Measurements for Specific Outflows}
Table \ref{tab:comeas} lists measurements of mass and momentum in
apertures shown in figure \ref{fig:cofig}. Where red and blue masses are
listed, there is an outflow in the red and blue along the line of sight. Where
only one is listed, an excess to one side of the cloud rest velocity was
detected and assumed to be accelerated gas from a protostellar outflow. Blue
velocities are integrated from -33 to -21 \kms. Red velocities are integrated
from -12 \kms\ to 1 \kms. All masses are computed assuming CO is optically
thin in the outflow, which leads to a lower bound on the mass; \ensuremath{^{13}\textrm{CO}}\ 2-1 was
measured to have an optical depth of 0.1 in 7 very high velocity outflows in
\citet{choi1993}, so if a relative abundance \ensuremath{^{12}\textrm{CO}}/\ensuremath{^{13}\textrm{CO}} =
60 is assumed\citep{lucas1998}, masses increase by a factor of 6.
It is not possible to completely distinguish outflowing matter from the ambient
medium. While the outflowing matter is generally at higher velocities, the
outflow and ambient line profiles are blended. A uniform selection of high
velocities was applied across the region but this may include some matter from
the cloud, biasing the mass measurements upward. Outflows in the plane of the
sky and low-velocity components of outflows will be blended with the cloud
profile, which would lead to underestimates of the outflowing mass. The
momentum measurements, however, should be more robust because they are weighted
by velocity, and higher velocity material is more certainly outflowing. The
momentum measurements are referenced to the central velocity of Sh~2-233IR~NE,
$v_{LSR}=-16.0$ \kms.
\section{Discussion}
\subsection{Outflow Mass and Momentum}
\label{sec:discussion-outflows}
\citet{beuther2002} reported a total outflowing mass of 20 \msun\ in Sh~2-233IR~NE. We
measure a significantly lower outflow mass of 2 \msun\ under the assumption
that the gas is optically thin, but this assumption is not valid: a lower limit
can be set from the weak \ensuremath{^{13}\textrm{CO}}\ 2-1 outflow detection (lower limit
because not all of the outflowing material is detected) on the outflowing mass
of $\sim 4$ \msun . \citet{choi1993} measure an optical depth of \ensuremath{^{13}\textrm{CO}}\
2-1 $\tau \approx 0.1$ in 7 very high velocity outflows.
Our \ensuremath{^{13}\textrm{CO}}\ data suggests that the optical depth is somewhat lower,
$\tau\approx0.07$. The abundance \ensuremath{^{12}\textrm{CO}}/\ensuremath{^{13}\textrm{CO}} = 60 is used \citep{lucas1998}
to derive a total outflowing mass estimate $M\approx25$ \msun . The total outflowing mass is
therefore $\sim 4\%$ of the total cloud mass, though most
of the outflowing material is coming from Sh~2-233IR~NE, so as much as 13\% of the
material in Sh~2-233IR~NE\ may be outflowing.
The most prominent outflow in IRAS 05358, Outflow 1, is primarily along
the plane of the sky, so the high velocity CO is likely associated
with the other outflows that have significant components along the line of
sight. As pointed out in \citet{beuther2002}, the integrated and peak CO are
aligned with the main mm core. High-velocity \hh\ near the mm cores and the
blueshifted outflows 2 and 4 all suggest that there are many distinct outflows
that together are responsible for the high velocity CO gas.
The offset between the integrated \ensuremath{^{13}\textrm{CO}}\ peak and \ensuremath{^{12}\textrm{CO}}\ peak in the
J=2-1 integrated maps, which corresponds with an offset in the peak of the
integrated CO 3-2 map and the peak temperature observed in CO 3-2, suggests
that the gas mass is largely associated with Sh~2-233IR~SW, but the outflowing gas
is primarily associated with Sh~2-233IR~NE. The integrated and maximum brightness
temperatures in \ensuremath{^{13}\textrm{CO}}\ and \ensuremath{\textrm{C}^{18}\textrm{O}}\ are also centered near Sh~2-233IR~SW,
which rules out optical depth as the cause of this offset. CO may be depleted
in the dense mm cores, which would help account for the lower mass estimate
from CO isotopologues relative to dust mass and NH$_3$. Alternately, the gas
temperature in Sh~2-233IR~SW\ may be significantly higher than in Sh~2-233IR~NE\
except in the outflows, which are probably warm. In this case, the outflowing
\ensuremath{^{12}\textrm{CO}}\ enhances the integrated intensity because of its high temperature
and reduced effective optical depth, but it does not set the peak brightness
because of the low filling-factor of the high-temperature gas.
Because the outflows are seen in \hh, which requires shock velocities $\sim30$
\kms\ to be excited \citep{bally2007}, and because the association between the
high-velocity CO and the plane-of-the-sky \hh\ is unclear, a velocity of 30
\kms\ is used when estimating the dynamical age.
Assuming the outflow is about 0.5 pc long in one direction (e.g. Outflow 1),
the dynamical age is 1.6\ee{4} years. Outflow 4, which is around 1 pc
long, is also seen at a velocity of -70 \kms\ LSR, or about -50 \kms\ with
respect to the cloud, and therefore has a dynamic age 2\ee{4} years, which is
consistent.
\subsection{Energy Injection / Ejection}
Using an assumed outflow lifetime of $5\times10^3$ years for $v=100\ \kms$ as a
lower limit (because the full extent of the flows is not necessarily observed)
and $1\times10^5$ as an upper limit (for the CO velocities $\sim10\ \kms$ and
the longest $\sim1$ pc flows), mechanical luminosities of the outflows $L=E/t$
are derived. The summed mechanical luminosity of the outflows is compared to the
turbulent decay luminosity within a 12\arcsec, 1\arcmin, and 5\arcmin\ radius
centered on Sh~2-233IR~NE\ in Table \ref{tab:turb}. \setlength\tabcolsep{3pt}
\Table{ccccccc}{Comparison of turbulent decay and outflow injection}
{Radius (pc) & $t_{turb}$\tablenotemark{a} (yr) & $L_{turb} (\lsun)$ & $L_{outflows}$\tablenotemark{b} $(\lsun) $
& Binding Energy (ergs) \tablenotemark{c} & Outflow Energy (ergs) & Turbulent Energy (ergs) \tablenotemark{d}}
{tab:turb}
{
0.10 & 2\ee{4} & 20 & 0.03-0.6 & 3.4 \ee{46} & 3.5 \ee{44} & 5.0\ee{46}\\
0.52 & 1\ee{5} & 12 & 0.6 - 9.4 & 5.9 \ee{46} & 5.9 \ee{45} & 1.5\ee{47}\\
2.62 & 5\ee{5} & 2.3 & 1-22 & 1.2 \ee{46} & 1.4 \ee{46} & 1.5\ee{47}\\
}
{
\tablenotetext{a}{Masses are assumed to be 600 \msun\ for the 1\arcmin\ and 5\arcmin\
apertures, and 200\msun\ for the 12\arcsec\ aperture.}
\tablenotetext{b}{Outflow luminosities are given as a range with a lower limit
$L=E_{out} / 10^5 \textrm{yr}$ and upper limit $L=E_{out}/ 5\times10^3 \textrm{yr}$,
where $E_{out}$ is from Table \ref{tab:comeas} multiplied by 6 to correct
for outflow opacity. }
\tablenotetext{c}{Binding energy is the order-of-magnitude estimate GM$^2$/R}
\tablenotetext{d}{Turbulent energy is computed using the measured 5 \kms\ line
width as the turbulent velocity.}
}
The rate of turbulent decay can be estimated from the crossing-time of the
region, $L / v$, where $L$ is the length scale and $v$ is the the typical
turbulent velocity. On the largest ($\sim$ few pc) scales, the mechanical
luminosity from high-velocity outflowing material is approximately capable of
balancing turbulent decay and upholding the cloud against collapse. However,
at the size scales of the Sh~2-233IR~NE\ clump ($\sim 0.1$ pc), turbulent decay
occurs on more than an order of magnitude faster timescales than outflow energy
injection. On the smallest scales, outflow energy can be lost from the cluster
through collimated outflows, though wide-angle flows and wrapped up magnetic
fields will not propagate outside of the core region. Once collimated flows
impact the local interstellar medium in a bow shock, their energy and momentum
are distributed more isotropically and again contribute to turbulence. The
imbalance on a small size scale is consistent with the observed infall
signature (Figure \ref{fig:co21_all3}) in the inner 12\arcsec\ around
Sh~2-233IR~NE\ and the lack of a similar profile elsewhere.
\subsection{Comparison to other clumps}
The classification scheme laid out in \citet{klein2005} is used to identify
Sh~2-233IR~NE\ as a Protocluster and Sh~2-233IR~SW\ as a Young Cluster. \citet{maury2009}
performed a similar analysis of the Early Protocluster NGC 2264-C. They also
found that the outflow mechanical luminosity could provide the majority of the
turbulent energy $L_{turb}\sim1.2 \lsun$ within the protocluster in a radius of
0.7 pc with a mass 2300\msun . \citet{williams2003} performed an outflow
study of the OMC 2/3 region with radius 1.2 pc and mass 1100 \msun, which is also an Early
Protocluster, and concluded that $L_{turb} \sim L_{flow} \sim 1.3 \lsun$.
While all three regions have nearly the same turbulent decay luminosities and
outflow mechanical luminosities, Sh~2-233IR~NE\ in IRAS 05358\ is significantly more
compact and lower mass than the Early Clusters, and is the only one of the three
that contains signatures of massive star formation.
\subsection{Surrounding Regions}
\label{sec:surroundings}
About 8\arcmin\ to the southeast of IRAS 05358\ is another embedded star forming
region, G173.58+2.45. Interferometric and stellar population studies have been
performed by \citet{shepherd1996} and \citet{shepherd2002}. The bipolar
outflow detected in their interferometric maps is also cleanly resolved in our
figure \ref{fig:cofig}. In our wide-field \hh\ maps, there is a complex of
outflows similar to that of IRAS 05358, but fainter.
The large HII region Sharpless 231 to the northeast can be seen in the \ensuremath{\textrm{H}\alpha}\
image (figure \ref{fig:overview_ha}). The expanding HII region is pushing against
the molecular ridge that includes IRAS 05358\ and accelerating the CO gas in the
blue direction (e.g. the northern blueshifted clumps in figures \ref{fig:cofig}
and \ref{fig:HA_with_CO}). It can be seen from the IRAC 8\um\ data
that UV radiation from the HII region reaches to
the IRAS 05358\ clusters. The expanding HII region's pressure on the molecular
ridge may be responsible for triggering the collapse of IRAS 05358\ and G173. The
size gradient from S232 ($\sim 30\arcmin$\ across) to S231 ($\sim 10\arcmin$)
to S233 ($\sim 2-3\arcmin$) is suggestive of an age gradient assuming uniform
HII region expansion velocities and a common distance. Investigation of this
hypothesis will require detailed stellar population studies in the HII regions
with proper regard for eliminating foreground and background sources.
\subsection{Massive Star Binary}
Our identification of a probable massive star binary with an associated outflow
contributes to a very small sample of known maser disks with \hh\ emission
perpendicular to the disk. \citet{debuizer2003} observed 28 methanol maser
sources with linear distributions of maser spots in the \hh\ 2.12 \um\ line,
and he identified only 2 sources with \hh\ emission perpendicular to the maser
lines. None of the outflows identified in his survey were as collimated as
Outflow 2, so the methanol disk / outflow combination presented here may be the
most convincing association of a massive protostellar disk with a collimated
outflow.
The association of a massive star with an UCHII region and a methanol maser
disk and the very small size of the UCHII region both suggest that the massive
stellar system is very young. \citet{walsh1998} suggested that the development
of a UCHII region leads to the destruction of maser emission regions. Their
conclusion is consistent with our interpretation of mm1a as a binary system.
\section{Summary \& Conclusion}
We have presented a multiwavelength study of the IRAS 05358\ star forming region.
IRAS 05358\ contains an embedded cluster of massive stars and is surrounded by
outflows. The outflows were linked to probable sources and determined that at
least one outflow is probably associated with a massive ($\sim 10 \msun$) star.
Added kinematic information and a wide field view of the infrared outflows has
been used to develop a more complete picture of the region.
\begin{itemize}
\item Sh~2-233IR~NE\ is a Protocluster and Sh~2-233IR~SW\ is a Young Cluster
\item Energy injection on the scales of IRAS 05358\ can maintain turbulence, but on
the small scales of the Sh~2-233IR~NE\ protocluster, is inadequate by $\sim 2$ orders
of magnitude. Sh~2-233IR~NE\ is collapsing.
\item there are 11 candidate outflows, 7 of which have candidate counterflows, in the
IRAS 05358\ complex
\item there is a probable massive binary with one member of mass 12 \msun\ in
mm1a, and the other which is the source of Outflow 2
\item there are at least two moderate-mass ($\sim$5\msun) young stars in IRAS 05358\
\end{itemize}
We have identified additional middle- and high-mass young stars with outflows, and
presented a case for a high-mass binary system within the millimeter core mm1a.
\section{Acknowledgements}
We would like to thank Vincent Minier for providing us with the positions of
the VLBI maser spots and Steve Myers and George Moellenbrock for their assistance with
VLA data reduction.
We would also like to thank Cara Battersby, Devin Silvia, Mike Shull, and Jeremy
Darling for helpful comments on early drafts.
This work made use of SAOIMAGE DS9 (\url{http://hea-www.harvard.edu/RD/ds9/}),
IRAF (\url{http://iraf.net/}, scipy (\url{http://www.scipy.org}), and APLpy
(\url{http://aplpy.sourceforge.net/}).
| 2024-02-18T23:40:06.021Z | 2009-10-16T00:59:04.000Z | algebraic_stack_train_0000 | 1,332 | 15,256 |
|
proofpile-arXiv_065-6620 | \section{Introduction}
Let $A_{i}, i=1,\ldots,n$ be positive semidefinite matrices, $\lambda_{i}>0$
with $\sum_{i=1}^{n}\lambda_{i}=1$ and $\ensuremath{\operatorname{Id}}:\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\mathbb R}^N$ be the identity mapping. For
$$\ensuremath{\boldsymbol{A}}=(A_{1},\ldots,A_{n}),\quad
\ensuremath{\boldsymbol{\lambda}}=(\lambda_{1},\ldots, \lambda_{n}),$$ we define
\begin{equation}\label{thestart}
\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} =\big[\lambda_{1}(A_{1}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}+\cdots+\lambda_{n}(A_{n}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big]^{-1}
-\mu^{-1}\ensuremath{\operatorname{Id}},
\end{equation}
and call it the \emph{resolvent average} of $\ensuremath{\boldsymbol{A}}$. This is motivated
from the fact that when $\mu=1$
\begin{equation}\label{original}
\big(\ensuremath{\mathcal{R}_{1}(\bA,\fettla)}+\ensuremath{\operatorname{Id}}\big)^{-1}=\lambda_{1}\big(A_{1}+\ensuremath{\operatorname{Id}}\big)^{-1}+\cdots+\lambda_{n}\big(
A_{n}+\ensuremath{\operatorname{Id}}\big)^{-1},
\end{equation}
which says that the resolvent of $\ensuremath{\mathcal{R}_{1}(\bA,\fettla)}$ is the (arithmetic)
average of resolvents of the $A_{i}$, with weight
$\ensuremath{\boldsymbol{\lambda}}=(\lambda_{1},\ldots,\lambda_{n})$. The resolvent average provides
a novel averaging technique, and having the parameter $\mu$
in $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$ will allow us to take limits later on.
We denote the well known
\emph{harmonic average} and \emph{arithmetic average} by
$$\ensuremath{\mathcal{H}(\bA,\fettla)}=(\lambda_{1}A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1})^{-1},$$
$$\ensuremath{\mathcal{A}(\bA,\fettla)}=\lambda_{1}A_{1}+\cdots+\lambda_{n}A_{n},$$
respectively.
In the literature, $(A_{1}^{-1}+\cdots +A_{n}^{-1})^{-1}$ is called the
\emph{parallel sum} of the matrices $A_{1},\ldots,
A_{n}$; see, e.g.,
\cite{anderson,attouch,urruty1,urruty2,mazure,Mou1,Mou2,passty}.
\emph{The goal of this note is to study relationships among the
resolvent average, the harmonic average and the arithmetic average
of matrices. Our proofs are based on convex analytical techniques
and on the proximal average, instead of the more commonly employed
matrix diagonalizations.}
The plan of the paper is as follows. After proving some elementary
properties of $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$ in Section~\ref{basic}, we gather some basic
properties of proximal averages and general convex functions in
Section~\ref{keyfacts}. The main results, which are given
in Section~\ref{takinglimit}, state that
$$\ensuremath{\mathcal{H}(\bA,\fettla)}\preceq\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\preceq\ensuremath{\mathcal{A}(\bA,\fettla)},$$
$$\lim_{\mu\rightarrow 0^+}\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}=\ensuremath{\mathcal{A}(\bA,\fettla)}, \quad \lim_{\mu\rightarrow +\infty}\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}=\ensuremath{\mathcal{H}(\bA,\fettla)},$$
and that $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$ enjoys self-duality, namely $\big[\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\big]^{-1}=\ensuremath{\mathcal{R}_{\mu^{-1}}(\bA^{-1},\fettla)}.$
In Section~\ref{compare}, we show that the resolvent average and geometric
mean have strikingly similar properties, even though they are different.
\noindent {\bf Notation}:
Throughout, $\ensuremath{\mathbb R}^N$ is the standard $N$-dimensional Euclidean space.
For $\lambda >0$,
\begin{equation}\label{regularization}
J_{A}=(\ensuremath{\operatorname{Id}}+A)^{-1}, \quad \ensuremath{ \; {}^}{\lambda}A=\lambda^{-1}(\ensuremath{\operatorname{Id}}-J_{\lambda A}),
\end{equation}
are called the \emph{resolvent} of $A$ and \emph{Yosida $\lambda$-regularization } of $A$. A function
$f:\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]} = \ensuremath{\mathbb R}\cup\{\ensuremath{+\infty}\}$
is said to be convex if its domain is convex and
\begin{equation}\label{convex}
f(\lambda x+(1-\lambda)y)\leq \lambda f(x)+(1-\lambda)f(y)\quad \forall \ x,y \in\ensuremath{\mathbb R}^N, 0<\lambda <1,
\end{equation}
with $f$ being strictly convex if \eqref{convex} becomes a strict inequality whenever $x\neq y$.
The function $f$ is proper
if $f(x)>-\infty\ \forall x\in\ensuremath{\mathbb R}^N$ and $f(x_{0})<\ensuremath{+\infty}$
for some $x_{0}\in\ensuremath{\mathbb R}^N$.
The class of proper
lower semicontinuous convex functions from $\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ will be denoted by $\ensuremath{\Gamma}$. For $f\in\ensuremath{\Gamma}$, $\partial f$ denotes its convex subdifferential:
$\partial f(x)=\{x^*\in\ensuremath{\mathbb R}^N:\ f(y)\geq f(x)+\scal{x^*}{y-x}\ \forall y\in \ensuremath{\mathbb R}^N\}.$
If $f$ is differentiable at $x$, then $\partial f(x)=\{\nabla f(x)\}$.
$f^*$ denotes its \emph{Fenchel conjugate} given by $(\forall x^*\in\ensuremath{\mathbb R}^N)\ f^*(x^*)=\sup_{x}\{\scal{x^*}{x}-f(x)\}$.
For $\alpha>0$, $\alpha\star f=\alpha f(\cdot/\alpha)$. If $f,g\in\ensuremath{\Gamma}$, $f\Box g$ stands for the infimal convolution
of $f,g$ given by
$(f\Box g)(x)
=\inf\{f(x_{1})+g(x_{2}):\ x_{1}+x_2=x\}$ $\forall x\in \ensuremath{\mathbb R}^N$.
When $A:\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\mathbb R}^N$ is linear, the quadratic form $q_{A}:\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\mathbb R}$ is defined by
$$q_{A}(x)=\ensuremath{\tfrac{1}{2}}\scal{Ax}{x} \ \forall x\in\ensuremath{\mathbb R}^N,$$
and we also use $q_{\ensuremath{\operatorname{Id}}}=j$ interchangeably. For convex functions $f_{1},\ldots, f_{n}$, we write
$$\ensuremath{\boldsymbol{f}}=(f_{1},\ldots, f_{n}), \qquad \ensuremath{\boldsymbol{f}}^*=(f_{1}^*,\ldots, f_{n}^*).$$
In the space $\ensuremath{\mathbb S}^N$ of $N\times N$ real symmetric matrices,
$\ensuremath{\mathbb S}^N_+$ (resp. $\ensuremath{\mathbb S}^N_{++}$) denotes the set of $N\times N$ positive semidefinite matrices (resp. positive definite matrices).
For $X,Y\in \ensuremath{\mathbb S}^N$, we write $Y\preceq X$ if $X-Y\in \ensuremath{\mathbb S}^N_+$ and
$Y\prec X$ if $X-Y\in \ensuremath{\mathbb S}^N_{++}$.
\section{Basic properties}\label{basic}
In this section, we give some basic properties of $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$.
\begin{proposition}\label{differentreform}
We have
\begin{equation}\label{resolventidentity}
J_{\mu \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}=\lambda_{1}J_{\mu A_{1}}+\cdots +\lambda_{n}J_{\mu A_{n}},
\end{equation}
\begin{equation}\label{t:averageyosida}
\ensuremath{ \; {}^}{\mu}\big(\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\big)=\lambda_{1}\ensuremath{ \; {}^}{\mu}A_{1}+\cdots +\lambda_{n}\ensuremath{ \; {}^}{\mu}A_{n}.
\end{equation}
\end{proposition}
\begin{proof}
Multiplying \eqref{thestart} both sides by $\mu$ gives
$$
\mu\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} +\ensuremath{\operatorname{Id}} =\big[\lambda_{1}(\mu A_{1}+\ensuremath{\operatorname{Id}})^{-1}+\cdots+\lambda_{n}(\mu A_{n}+\ensuremath{\operatorname{Id}})^{-1}\big]^{-1}.
$$
Then \eqref{resolventidentity} follows by taking inverse both sides and using \eqref{regularization}.
By \eqref{resolventidentity}, we obtain that
$$(\ensuremath{\operatorname{Id}}-J_{\mu \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}})=\lambda_{1}(\ensuremath{\operatorname{Id}} -J_{\mu A_{1}})+\cdots +\lambda_{n}(\ensuremath{\operatorname{Id}} -J_{\mu A_{n}}).$$
Dividing both sides by $\mu$,
$$\mu^{-1}(\ensuremath{\operatorname{Id}}-J_{\mu \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}})=\lambda_{1}\mu^{-1}(\ensuremath{\operatorname{Id}} -J_{\mu A_{1}})+\cdots +\lambda_{n}\mu^{-1}(\ensuremath{\operatorname{Id}} -J_{\mu A_{n}}).$$
It remains to use \eqref{regularization}.
\end{proof}
\begin{proposition}\label{primalinverse}
Let $\ensuremath{\boldsymbol{A}}=(A_{1},A_{1}^{-1},\ldots, A_{m}, A_{m}^{-1})$,
$\lambda=(\tfrac{1}{2m},\tfrac{1}{2m},\ldots, \tfrac{1}{2m})$,
and $\mu=1$.
Then $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} =\ensuremath{\operatorname{Id}}$.
\end{proposition}
\begin{proof} This follows from \eqref{original} and the
identity
$(A+\ensuremath{\operatorname{Id}})^{-1}+(A^{-1}+\ensuremath{\operatorname{Id}})^{-1}=\ensuremath{\operatorname{Id}}.$
\end{proof}
\begin{proposition}\label{sameoperator} Let $\ensuremath{\boldsymbol{A}}=(A_{1},\ldots, A_{1})$.
Then $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} =A_{1}$.
\end{proposition}
\begin{proof}
We have
\begin{align*}
\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} & =\big((\lambda_{1}+\cdots+\lambda_{n})(A_{1}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big)^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}}\\
& =
\big((A_{1}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big)^{-1}-\mu\ensuremath{\operatorname{Id}}=
A_{1}+\mu^{-1}\ensuremath{\operatorname{Id}}-\mu^{-1}\ensuremath{\operatorname{Id}}=A_{1},
\end{align*}
which proves the result.
\end{proof}
Note that for $A,B\in \ensuremath{\mathbb S}^N_{++}$, we have
\begin{equation}\label{comparenew}
A\succeq B\quad \Leftrightarrow \quad A^{-1}\preceq B^{-1}
\end{equation}
and
\begin{equation}\label{compareneu}
A\succ B\quad \Leftrightarrow \quad A^{-1}\prec B^{-1};
\end{equation}
see, e.g., \cite[Corollary~7.7.4.(a)]{HoJo} and
\cite[Section~16.E]{MO} or \cite[page~55]{borwein}.
\begin{proposition}\label{monotonecompare}
Assume that $(\forall\ i)\ A_{i}, B_{i}\in \ensuremath{\mathbb S}^N_+$ and $A_{i}\succeq B_{i}$.
Then
\begin{equation}\label{monotoneorder}
\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\succeq \ensuremath{\mathcal{R}_{\mu}(\bB,\fettla)}.
\end{equation}
Furthermore, if additionally some $A_j\succ B_j$, then $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\succ \ensuremath{\mathcal{R}_{\mu}(\bB,\fettla)}$.
\end{proposition}
\begin{proof}
Note that $\forall \ \mu>0$,
$$A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}}\succeq B_{i}+\mu^{-1}\ensuremath{\operatorname{Id}}\succ 0,$$
so that
$$0 \prec (A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\preceq (B_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1},$$
by \eqref{comparenew}. As $\ensuremath{\mathbb S}^N_+$ and $\ensuremath{\mathbb S}^N_{++}$
are convex cones, we obtain that
\begin{equation}\label{keeppositive}
0\prec \sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\preceq \sum_{i=1}^{n}\lambda_{i}(B_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}.
\end{equation}
Using \eqref{comparenew} on \eqref{keeppositive}, followed by subtracting $\mu^{-1}\ensuremath{\operatorname{Id}}$, gives
$$\big[\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big]^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}}\succeq \big[\sum_{i=1}^{n}\lambda_{i}(B_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big]^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}},$$
which establishes \eqref{monotoneorder}.
The ``Furthermore'' part follows analogously using \eqref{compareneu}.
\end{proof}
\begin{theorem}
\label{t:blabla}
Assume that $(\forall\ i)\ A_{i}\in \ensuremath{\mathbb S}^N_+$.
Then $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\in \ensuremath{\mathbb S}^N_+$.
Furthermore, if additionally some $A_j\in\ensuremath{\mathbb S}^N_{++}$, then
$\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\in \ensuremath{\mathbb S}^N_{++}$.
\end{theorem}
\begin{proof}
This follows from Proposition~\ref{monotonecompare} (with each $B_i=0$) and
Proposition~\ref{sameoperator}.
\end{proof}
We end this section with a recursion formula that may be
verified directly using the definitions.
\begin{proposition}[recursion]
We have
$$\ensuremath{\mathcal{R}_{\mu}}(A_{1},\ldots,A_n;\lambda_{1},\ldots, \lambda_{n})=
\ensuremath{\mathcal{R}_{\mu}}\Big(\ensuremath{\mathcal{R}_{\mu}}\big(A_1,\ldots,A_{n-1};\tfrac{\lambda_1}{1-\lambda_n},
\ldots,\tfrac{\lambda_{n-1}}{1-\lambda_n}\big),
A_n;1-\lambda_n,\lambda_n\Big).$$
\end{proposition}
\section{Auxiliary results and facts}\label{keyfacts}
The key tool in this note is the \emph{proximal average of convex
functions}, which finds its roots in \cite{matou,Minty,Moreau},
and which has been further systematically studied in
\cite{bwang,bglw,BLT,byw}.
\begin{definition}[proximal average] \label{d:proxaverage} Let $(\forall i) \ f_{i}\in\ensuremath{\Gamma}$.
The $\lambda$-weighted proximal average of $\ensuremath{\boldsymbol{f}}=(f_1, \ldots, f_{n})$ with parameter $\mu$ is
defined by
\begin{equation}
\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}=
\bigg(\lambda_{1} (f_1+\tfrac{1}{\mu}j)^* +
\lambda_{2}(f_2+\tfrac{1}{\mu}j)^*+\cdots +\lambda_{n}(f_{n}+\tfrac{1}{\mu}j)^{*}\bigg)^* -\tfrac{1}{\mu}
j.
\end{equation}
\end{definition}
The function
$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}$ is a proper lower semicontinuous convex function on $\ensuremath{\mathbb R}^N$, and
it inherits many desirable properties from each underlying function
$f_{i}$; see \cite{bglw,BLT}.
A fundamental property of proximal average is:
\begin{fact}\emph{(\cite[Theorem 5.1]{bglw})}\label{t:Fenchel}
$\big(\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}\big)^* = p_{\mu^{-1}}({\ensuremath{\boldsymbol{f}}^*,\ensuremath{\boldsymbol{\lambda}}})$.
\end{fact}
To give new proofs of Fact~\ref{t:ineqs} and Fact~\ref{t:mono} below,
we shall need reformulations of $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}$.
\begin{proposition}\label{domapp1} Let $f_{1},\ldots,
f_{n}\in \ensuremath{\Gamma}$ and $\lambda_{1},\ldots, \lambda_{n}> 0$ with
$\sum_{i=1}^{n}\lambda_{i}=1$.
Then for every $x\in \ensuremath{\mathbb R}^n$, \\
$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)$
\begin{align}
&=\inf_{x_{1}+\cdots+x_{n}=x}\left\{\lambda_{1}(f_{1}+\frac{1}{\mu}j)(\frac{x_{1}}{\lambda_{1}})+\cdots
+\lambda_{n}(f_{n}+\frac{1}{\mu}j)(\frac{x_{n}}{\lambda_{n}})\right\}-\frac{1}{\mu}j(x)\label{onerep}\\
&=\inf_{x_{1}+\cdots+x_{n}=x}\left\{\lambda_{1}f_{1}(\frac{x_{1}}{\lambda_{1}})+\cdots
+\lambda_{n}f_{n}(\frac{x_{n}}{\lambda_{n}})+\frac{1}{4\mu}\sum_{i=1}^{n}
\sum_{j=1}^{n}\lambda_{i}\lambda_{j}\|\frac{x_{i}}{\lambda_{i}}-
\frac{x_{j}}{\lambda_{j}}\|^2\right\}\label{tworep}\\
&=\inf_{\lambda_{1}y_1+\cdots +\lambda_{n}y_{n}=x}
\left\{\lambda_{1} f_{1}(y_{1})+\cdots
+\lambda_{n}f_{n}(y_{n})+\frac{1}{\mu}[\lambda_{1}j(y_{1})+\cdots
+\lambda_{n}j(y_{n})-j(\lambda_{1}y_{1}+\cdots
+\lambda_{n}y_{n})]\right\}\label{fourrep}\\
&= \inf_{\lambda_{1}y_1+\cdots
+\lambda_{n}y_{n}=x}\left\{\lambda_{1}f_{1}(y_{1})+\cdots+\lambda_{n}f_{n}(y_{n})+
\frac{1}{4\mu}\sum_{i=1}^{n}\sum_{j=1}^{n}
\lambda_{i}\lambda_{j}\|y_{i}-y_{j}\|^2\right\}\label{threerep}\\
&=\inf_{x_{1}+\cdots
+x_{n}=x}\left\{\lambda_{1}f_{1}(\frac{x_{1}}{\lambda_{1}})+\cdots
+\lambda_{n}f_{n}(\frac{x_{n}}{\lambda_{n}})+
\frac{1}{\mu}[\lambda_{1}j(x-\frac{x_{1}}{\lambda_{1}}))+
\cdots +\lambda_{n}j(x-\frac{x_{n}}{\lambda_{n}})]\right\}.\label{fiverep}
\end{align}
Furthermore, the infimal convolutions in
(\ref{onerep})--(\ref{fiverep}) are exact.
\end{proposition}
\begin{proof} Indeed, as
$$\left(f_{i}+\frac{1}{\mu}j\right)^*=f_{i}^*\Box (\mu j),$$
it is finite-valued everywhere, we write
$$
f=\lambda_{1}\star(f_{1}+\frac{1}{\mu}j)\Box \cdots \Box
\lambda_{n}\star(f_{n}+\frac{1}{\mu}j)- \frac{1}{\mu}j,$$ by
\cite[Theorem~16.4]{Rocky}. That is, for every $x$,
$$f(x)=\inf\left\{\lambda_{1}(f_{1}+\frac{1}{\mu}j)(\frac{x_{1}}{\lambda_{1}})
+\cdots
+\lambda_{n}(f_{n}+\frac{1}{\mu}j)(\frac{x_{n}}{\lambda_{n}}): \;
x_{1}+\cdots +x_{n}=x\right\}-\frac{1}{\mu}j(x),$$ and the infimum
is attained. Hence (\ref{onerep}) holds.
Now rewrite (\ref{onerep}) as
\begin{align}
&\inf_{x_{1}+\cdots+x_{n}=x}\left\{\lambda_{1}f_{1}(\frac{x_{1}}{\lambda_{1}})+\cdots
+\lambda_{n}f_{n}(\frac{x_{n}}{\lambda_{n}})+\frac{1}{\mu}[\lambda_{1}j(\frac{x_{1}}{\lambda_{1}})
+\cdots +\lambda_{n}j(\frac{x_{n}}{\lambda_{n}})-j(x_{1}+\cdots+
x_{n})]\right\},\label{whom}\\
&=\inf_{\lambda_{1}y_1+\cdots +\lambda_{n}y_{n}=x}
\left\{\lambda_{1} f_{1}(y_{1})+\cdots
+\lambda_{n}f_{n}(y_{n})+\frac{1}{\mu}[\lambda_{1}j(y_{1})+\cdots
+\lambda_{n}j(y_{n})-j(\lambda_{1}y_{1}+\cdots
+\lambda_{n}y_{n})]\right\}.\nonumber
\end{align}
Thus, (\ref{tworep})--(\ref{threerep}) follow by using the
identity
$$\sum_{i=1}^{n}\lambda_{i}j(y_{i})-j(\sum_{i=1}^{n}\lambda_{i}y_{i})=
\frac{1}{4}\sum_{i=1}^{n}\sum_{j=1}^{n}\lambda_{i}\lambda_{j}\|y_{i}-y_{j}\|^2.$$
Observe that
\begin{align*}
&\lambda_{1}j(x_{1}+\cdots
+x_{n}-\frac{x_{1}}{\lambda_{1}})+\cdots
+\lambda_{n}j(x_{1}+\cdots+x_{n}-\frac{x_{n}}{\lambda_{n}})\\
&=\lambda_{1}j(\frac{x_{1}}{\lambda_{1}})+\cdots+\lambda_{n}j(\frac{x_{n}}{\lambda_{n}})-j(x_{1}+\cdots+x_{n}),
\end{align*}
we have (\ref{fiverep}) by (\ref{whom}).
\end{proof}
\begin{fact}\emph{(\cite[Theorem 5.4]{bglw})}\label{t:ineqs}
$(\lambda_1f_1^* + \cdots + \lambda_n f_n^*)^{*} \leq \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}
\leq \lambda_1 f_1+\cdots +\lambda_n f_n$.
\end{fact}
\begin{proof}
This follows from \eqref{tworep} or \eqref{threerep}.
\end{proof}
\begin{fact}\emph{(\cite[Theorem 8.5]{bglw})}\label{t:mono}
Let $x\in \ensuremath{\mathbb R}^N$. Then the function
\begin{equation} \label{e:hike:0}
\ensuremath{\,\left]0,+\infty\right[}\to\ensuremath{\,\left]-\infty,+\infty\right]}\colon\mu \mapsto \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)\quad\text{is decreasing.}
\end{equation}
Consequently, $\lim_{\mu\to 0^+}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)$ and $\lim_{\mu\to\ensuremath{+\infty}}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)$ exist. In fact,
\begin{equation} \label{e:hike:a}
\lim_{\mu\to 0^+}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x) = \sup_{\mu>0}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=
\big(\lambda_1f_1+\cdots+\lambda_nf_n\big)(x)
\end{equation}
and
\begin{equation} \label{e:hike:b}
\lim_{\mu\to \ensuremath{+\infty}}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x) =\inf_{\mu>0} \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)
= \big(\lambda_1\star f_1\Box
\cdots\Box\lambda_n\star f_n\big)(x).
\end{equation}
\end{fact}
\begin{proof}
\eqref{e:hike:0} follows from \eqref{tworep}. \eqref{e:hike:b} also follows from \eqref{tworep}.
To see \eqref{e:hike:a},
by \eqref{fiverep}, $\forall x\in \ensuremath{\mathbb R}^N$,
\begin{align*}
\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x) &\geq
\lambda_{1}\inf_{x_{1}}\bigg(f_{1}(x_{1}/\lambda_{1})+\frac{1}{\mu}j(x-x_{1}/\lambda_{1})\bigg)
+\cdots
+\lambda_{n}\inf_{x_{n}}\bigg(f_{n}(x_{n}/\lambda_{n})+\frac{1}{\mu}j(x-x_{n}/\lambda_{n})\bigg)\\
&=\lambda_{1}e_{\mu}f_{1}(x)+\cdots+\lambda_{n}e_{\mu}f_{n}(x),
\end{align*}
where $e_{\mu}f_{i}=f_{i}\Box (1/\mu j)$.
Then
$$ \lambda_{1}e_{\mu}f_{1}+\cdots+\lambda_{n}e_{\mu}f_{n}\leq \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}} \leq
\lambda_{1}f_{1}+\cdots+\lambda_{n}f_{n},
$$
so that
$$\lim_{\mu\rightarrow 0^{+}}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}=\lambda_{1}f_{1}+\cdots+\lambda_{n}f_{n},$$
since $\lim_{\mu\rightarrow 0^+} e_{\mu}f_{i}=f_{i}$
by \cite[Theorem~2.26 and Theorem~1.25]{RockWets}.
\end{proof}
\begin{fact}\emph{(\cite[Theorem 25.7]{Rocky})}\label{rockstar}
Let $C$ be a nonempty open convex subset of $\ensuremath{\mathbb R}^N$,
and let $f$ be a convex function
which is finite and differentiable on $C$. Let $f_{1},f_{2},\ldots,$ be a sequence of convex functions finite
and differentiable on $C$ such that $\lim_{i\rightarrow\infty} f_{i}(x)=f(x)$ for every
$x\in C$. Then $$\lim_{i}\nabla f_{i}(x)=\nabla f(x), \quad \forall x\in C.$$
In fact, the sequence of gradients
$\nabla f_{i}$ converges to $\nabla f$ uniformly on every compact
subset of $C$.
\end{fact}
\begin{fact}\emph{(\cite[page 108]{Rocky})}\label{qconjugate}
Let $Q\in \ensuremath{\mathbb S}^N_{++}$. Then $(q_{Q})^*=q_{Q^{-1}}.$
\end{fact}
\begin{fact}\emph{(\cite[Theorem 23.5]{Rocky})}\label{conjgrad}
Let $f:\ensuremath{\mathbb R}^N\rightarrow\ensuremath{\,\left]-\infty,+\infty\right]}$ be a proper lower semicontinuous convex function. Then $\partial f^*=(\partial f)^{-1}.$
\end{fact}
\section{Main results}\label{takinglimit}
We start by computing the proximal average of general linear-quadratic
functions thereby extending
\cite[Example~4.5]{bglw} and \cite[Example~7.4]{BLT}.
\begin{lemma}\label{quadraticprox}
Let $A_{i}\in \ensuremath{\mathbb S}^N_+$, $b_{i}\in\ensuremath{\mathbb R}^N$, $r_{i}\in \ensuremath{\mathbb R}$. If each $f_{i}=q_{A_{i}}+\scal{b_{i}}{\cdot}+r_{i}$, i.e.,
linear-quadratic,
then $\forall x^*$,
\begin{align}\label{quadratic}
\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x^*)& =q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}(x^*)+\scal{x^*}{\big(\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big)^{-1}\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}
+\nonumber\\
& q_{\big(\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\big)^{-1}}(\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i})-\sum_{i=1}^{n}\lambda_{i}\big(q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-r_{i}\big).
\end{align}
In particular,
if $(\forall i)\ f_{i}$ is quadratic, i.e., $\ b_{i}=0, r_i=0$, then $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}$ is quadratic with
$$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}} =q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}};$$
If $(\forall i)\ f_{i}$ is affine, i.e., $\ A_{i}=0$, then
$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}$ is affine.
\end{lemma}
\begin{proof}
We have
$f_{i}+\mu^{-1}j=q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})}+\scal{b_{i}}{\cdot}+r_{i}$ and by Fact~\ref{qconjugate}
\begin{align*}
(f_{i}+\mu^{-1}j)^*(x^*)& =q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(x^*-b_{i})-r_{i}\\
&=q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(x^*)-\scal{x^*}{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}+q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-
r_{i}.
\end{align*}
Then
$\big(\lambda_{1}(f_{1}+\mu^{-1}j)^{*}+\cdots+\lambda_{n}(f_{n}+\mu^{-1}j)^*\big)(x^*)=$
\begin{align*}
&\sum_{i=1}^{n}\lambda_{i}\bigg(q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(x^*)-\scal{x^*}{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}+q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-
r_{i}\bigg)
\\
&= q_{{\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}}(x^*)-
\scal{x^*}{\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}
+\sum_{i=1}^{n}\lambda_{i}\big(q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-
r_{i}\big).
\end{align*}
It follows that
$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x^*)=$
$$q_{[{\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}]^{-1}}(x^*+\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i})
-\sum_{i=1}^{n}\lambda_{i}\big(q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-
r_{i}\big)-q_{\mu^{-1}\ensuremath{\operatorname{Id}}}(x^*).$$
As
\begin{align*}
& q_{{[{\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}]^{-1}}}(x^*+
\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i})
\\
&=q_{[{\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}]^{-1}}(x^*)
+\scal{x^*}{[\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}]^{-1}\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}+\\
& q_{[\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}]^{-1}}(\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}),
\end{align*}
we obtain that $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x^*)=$
\begin{align*}
&
q_{[{\sum_{i=1}^{n}\lambda_{i}(A_{i}+
\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}]^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}}}(x^*)
+\scal{x^*}{[\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}]^{-1}\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i}}+
\\
& q_{[\sum_{i=1}^{n}\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}]^{-1}}(\sum_{i=1}^{n}
\lambda_{i}(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}b_{i})-
\sum_{i=1}^{n}\lambda_{i}\big(q_{(A_{i}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}}(b_{i})-
r_{i}\big),
\end{align*}
which is \eqref{quadratic}. The remaining claims are immediate from \eqref{quadratic} and
that $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}=0$ when $(\forall\ i)\ A_{i}=0$ by Proposition~\ref{sameoperator}.
\end{proof}
We are ready for our main result:
\begin{theorem}[harmonic-resolvent-arithmetic average inequality and
limits]\label{matrixequ} ~\\
Let $A_{1},\ldots,A_{n}\in \ensuremath{\mathbb S}^N_{++}$. We have
\begin{enumerate}
\item
\begin{equation}\label{harmarithmetic}
\ensuremath{\mathcal{H}(\bA,\fettla)}\preceq \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} \preceq \ensuremath{\mathcal{A}(\bA,\fettla)};
\end{equation}
In particular, $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} \in \ensuremath{\mathbb S}^N_{++}$.
\item $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\rightarrow \ensuremath{\mathcal{A}(\bA,\fettla)}$ when $\mu\rightarrow 0^{+}$.
\item $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\rightarrow\ensuremath{\mathcal{H}(\bA,\fettla)}$ when $\mu\rightarrow +\infty$.
\end{enumerate}
\end{theorem}
\begin{proof}
(i). According to Fact~\ref{t:ineqs},
\begin{equation}
\label{key}
(\lambda_{1}f_{1}^*+\cdots +\lambda_{n}f_{n}^*)^{*}\leq\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}\leq \lambda_{1}f_{1}+\cdots+\lambda_{n}f_{n}.
\end{equation}
Let $f_{i}=q_{A_{i}}$. Using $(q_{A_{i}})^*=q_{A_{i}^{-1}}$ (by Fact~\ref{qconjugate})
and Lemma~\ref{quadraticprox} we have
\begin{align}
(\lambda_{1}f_{1}^*+\cdots +\lambda_{n}f_{n}^*)^{*}&=(\lambda_{1}q_{A^{-1}_{1}}+\cdots+\lambda_{n}q_{A^{-1}_{n}})^*
=(q_{\lambda_{1}A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1}} )^{*}\nonumber\\
& =q_{(\lambda_{1}A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1})^{-1}}
=q_{\ensuremath{\mathcal{H}(\bA,\fettla)}}.\label{thanksgiving1}\\
\lambda_{1}f_{1}+\cdots+\lambda_{n}f_{n} & =q_{\lambda_{1}A_{1}+\cdots +\lambda_{n}A_{n}}=q_{\ensuremath{\mathcal{A}(\bA,\fettla)}},\label{thanksgiving2}\\
\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}} &=q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}. \label{thanksgiving3}
\end{align}
Then \eqref{key} becomes
$$q_{\ensuremath{\mathcal{H}(\bA,\fettla)}}\leq q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}\leq q_{\ensuremath{\mathcal{A}(\bA,\fettla)}}.$$
As $q_{X}\leq q_{Y} \Leftrightarrow X\preceq Y$, \eqref{harmarithmetic} is established.
Since $A_{i}\in \ensuremath{\mathbb S}^N_{++}, A_{i}^{-1}\in \ensuremath{\mathbb S}^N_{++}, \lambda_{1}A_{1}^{-1}+\cdots
+\lambda_{n}A_{n}^{-1}\in \ensuremath{\mathbb S}^N_{++}$,
we have $\ensuremath{\mathcal{H}(\bA,\fettla)}=(\lambda_{1}A_{1}^{-1}+\cdots
+\lambda_{n}A_{n}^{-1})^{-1}\in \ensuremath{\mathbb S}^N_{++}$, thus
$\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\in \ensuremath{\mathbb S}^N_{++}$ by \eqref{harmarithmetic}. (Alternatively,
apply Theorem~\ref{t:blabla}.)
(ii) and (iii):
Observe that ($\forall i$)
$\big(\lambda_i\star f_i)^*=\lambda_{i}f_{i}^*=\lambda_{i}q_{A_{i}^{-1}}$ has full domain, by \cite[Theorem~16.4]{Rocky},
$$(\lambda_{1}f_{1}^*+\cdots +\lambda_{n}f_{n}^*)^{*}=\big(\lambda_1\star f_1\Box
\cdots\Box\lambda_n\star f_n\big).$$
By Fact~\ref{t:mono}, $\forall x\in\ensuremath{\mathbb R}^N$ one has
$$\lim_{\mu\rightarrow 0^{+}}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=(\lambda_{1}f_{1}+\cdots +\lambda_{n}f_{n})(x),$$
$$\lim_{\mu\rightarrow +\infty}\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=(\lambda_{1}f_{1}^*+\cdots+\lambda_{n}f_{n}^{*})^*(x).$$
Since $(\forall i) \ f_{i}, f_{i}^*$
are differentiable on $\ensuremath{\mathbb R}^N$,
so is $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}$ by \cite[Corollary~7.7]{bglw}.
According to Fact~\ref{rockstar}, $\forall x$
\begin{equation}\label{martial1}
\lim_{\mu\rightarrow 0^{+}}\nabla \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=\lambda_{1}\nabla f_{1}(x)+\cdots +\lambda_{n} \nabla f_{n}(x),
\end{equation}
\begin{equation}\label{martial2}
\lim_{\mu\rightarrow +\infty}\nabla\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=\nabla (\lambda_{1}f_{1}^*+\cdots+\lambda_{n}f_{n}^{*})^*(x).
\end{equation}
Moreover, the convergences in \eqref{martial1}-\eqref{martial2} are uniform on every closed bounded subset of
$\ensuremath{\mathbb R}^N$.
Now it follows from ~\eqref{thanksgiving1}-\eqref{thanksgiving3} that
$\nabla\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}=\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$, $\nabla (\lambda_{1} f_{1}+\cdots +\lambda_{n} f_{n})=\ensuremath{\mathcal{A}(\bA,\fettla)}$,
$\nabla (\lambda_{1}f_{1}^*+\cdots+\lambda_{n}f_{n}^{*})^*
=\ensuremath{\mathcal{H}(\bA,\fettla)}.$
\eqref{martial1}-\eqref{martial2} transpire to
\begin{equation}\label{julian1}
\lim_{\mu\rightarrow 0^{+}}\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} x =\ensuremath{\mathcal{A}(\bA,\fettla)} x,
\end{equation}
\begin{equation}\label{julian2}
\lim_{\mu\rightarrow + \infty}\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)} x=\ensuremath{\mathcal{H}(\bA,\fettla)} x,
\end{equation}
where the convergences are uniform on every closed bounded subset of
$\ensuremath{\mathbb R}^N$.
Hence (ii) and (iii) follow from \eqref{julian1} and \eqref{julian2}.
\end{proof}
Note that in Theorem~\ref{matrixequ}(ii),(iii), there is no ambiguity since all norms in finite dimensional spaces
are equivalent.
\begin{definition} A function $g\colon \mathbb{D}\to\ensuremath{\mathbb S}^N$,
where $\mathbb{D}$ is a convex subset of $\ensuremath{\mathbb S}^N$, is matrix convex if
$\forall A_{1}, A_{2}\in \mathbb{D}$, $\forall \lambda\in [0,1]$,
$$g(\lambda A_1+(1-\lambda) A_{2}) \preceq\lambda g(A_{1})+(1-\lambda)g(A_{2}).$$
Matrix concave functions are defined similarly.
\end{definition}
It is easy to see that a symmetric matrix valued function $g$ is matrix concave (resp. convex) if and only if
$\forall x\in \ensuremath{\mathbb R}^N$ the function
$A\mapsto q_{g(A)}(x)$ is concave (resp. convex).
Some immediate consequences of Theorem~\ref{matrixequ} on matrix-valued functions are:
\begin{corollary}\label{known1}
Assume that $(\forall i)\ A_{i}\in \ensuremath{\mathbb S}^N_{++}$ and $\sum_{i=1}^{n}\lambda_{i}=1$ with $\lambda_{i}>0$. Then
$$(\lambda_{1}A_{1}+\cdots +\lambda_{n}A_{n})^{-1}\preceq \lambda_{1}
A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1}.$$
Consequently, the matrix function $X\mapsto X^{-1}$
is matrix convex on $\ensuremath{\mathbb S}^N_{++}$.
\end{corollary}
\begin{proof}
Apply Theorem~\ref{matrixequ} equation~\eqref{harmarithmetic} for
$\ensuremath{\boldsymbol{A}} =(A_{1}^{-1},\cdots, A_{n}^{-1}).$
\end{proof}
\begin{corollary}\label{known2}
For every $\mu>0$, the resolvent average matrix function
$\ensuremath{\boldsymbol{A}}\mapsto \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$ given by
\begin{align}\label{amanda}
&(A_{1},\cdots, A_{n})\mapsto [\lambda_{1}(A_{1}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}+\cdots+\lambda_{n}(A_{n}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}]^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}} \nonumber \\
& \mbox{ is matrix concave on $\ensuremath{\mathbb S}^N_{++}\times\cdots\times \ensuremath{\mathbb S}^N_{++}$.}
\end{align}
For each $\lambda=(\lambda_{1}, \cdots, \lambda_{n})$ with $\sum_{i=1}^{n}\lambda_{i}=1$ and
$\lambda_{i}>0\ \forall i$,
the harmonic average matrix function
\begin{equation}\label{known1.5}
(A_{1},\cdots, A_{n})\mapsto (\lambda_{1}A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1})^{-1} \mbox{ is matrix concave}
\end{equation}
on $\ensuremath{\mathbb S}^N_{++}\times\cdots\times \ensuremath{\mathbb S}^N_{++}$.
Consequently, the harmonic average function
\begin{equation}\label{known3}
(x_{1},\cdots, x_{n})\mapsto \frac{1}{x_{1}^{-1}+\cdots +x_{n}^{-1}} \mbox{ is concave}
\end{equation}
on $\ensuremath{\mathbb R}_{++}\times\cdots\times\ensuremath{\mathbb R}_{++}$.
\end{corollary}
\begin{proof} Set $f_{i}=q_{A_{i}}$.
Then $\forall x\in \ensuremath{\mathbb R}^N$, we have $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=$
$$\min_{\lambda_{1}x_{1}+\cdots +\lambda_{n}x_{n}=x}\bigg(\big(\lambda_{1}q_{A_{1}}(x_{1})+
\cdots +\lambda_{n}q_{A_{n}}(x_{n})\big)+\big(\mu^{-1}\lambda_{1}q_{\ensuremath{\operatorname{Id}}}(x_{1})+\cdots+\mu^{-1}\lambda_{n}
q_{\ensuremath{\operatorname{Id}}}(x_{n})\big)\bigg)
-\mu^{-1}q_{\ensuremath{\operatorname{Id}}}(x).$$
Since for each fixed $(x_{1},\ldots,x_{n})$,
$$(A_{1},\cdots, A_{n})\mapsto
\big(\lambda_{1}q_{A_{1}}(x_{1})+
\cdots +\lambda_{n}q_{A_{n}}(x_{n})\big)+\big(\mu^{-1}q_{\ensuremath{\operatorname{Id}}}(x_{1})+\cdots+\mu^{-1}q_{\ensuremath{\operatorname{Id}}}(x_{n})\big)
,$$
is affine, being the infimum of affine functions we have that $\forall x$ the function
$$(A_{1},\cdots, A_{n})\mapsto \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x),$$
is concave. As
$\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}(x)=q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}(x)$ by Lemma~\ref{quadraticprox},
this shows that $\forall x\in\ensuremath{\mathbb R}^N$ the function
$$\ensuremath{\boldsymbol{A}}=(A_{1},\cdots,A_{n})\mapsto q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}(x) \mbox{ is concave,}$$ so
$\ensuremath{\boldsymbol{A}} \mapsto \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}$
is matrix concave.
Now by Theorem~\ref{matrixequ}(iii), $\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\rightarrow\ensuremath{\mathcal{H}(\bA,\fettla)}$ when $\mu\rightarrow +\infty$.
This and \eqref{amanda}
implies that
$$\ensuremath{\boldsymbol{A}}\mapsto\ensuremath{\mathcal{H}(\bA,\fettla)},$$
is also matrix concave. \eqref{known3} follows from \eqref{known1.5}
by setting $N=1$ and $\lambda_{1}=\cdots=\lambda_{n}=1/n$.
\end{proof}
\begin{remark} Corollary~\ref{known1} is well-known,
cf.\ \cite[Proposition~2.56 on page~73]{RockWets}.
Corollary~\ref{known2} \eqref{known3} is also well-known,
cf.\ \cite[Exercise~3.17 on page~116]{boyd}.
\end{remark}
We proceed to show that resolvent averages of matrices enjoy self-duality.
\begin{theorem}[self-duality]\label{mainresult}
Let $(\forall \ i)\ A_{i}\in \ensuremath{\mathbb S}^N_{++}$ and $\mu>0$. Assume that
$\sum_{i=1}^{n}\lambda_{i}=1$ with $\lambda_{i}>0$. Then
\begin{equation}\label{niceselfdual}
\big[\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\big]^{-1}=\ensuremath{\mathcal{R}_{\mu^{-1}}(\bA^{-1},\fettla)}, \mbox{ i.e.,}
\end{equation}
\begin{align*}
&\bigg[\bigg(\lambda_{1}(A_1+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}+\cdots +\lambda_{n}(A_{n}+\mu^{-1}\ensuremath{\operatorname{Id}})^{-1}\bigg)^{-1}-\mu^{-1}\ensuremath{\operatorname{Id}}\bigg]^{-1}=\\
& \bigg(\lambda_{1}(A_1^{-1}+\mu\ensuremath{\operatorname{Id}})^{-1}+\cdots +\lambda_{n}(A_{n}^{-1}+\mu\ensuremath{\operatorname{Id}})^{-1}\bigg)^{-1}-\mu\ensuremath{\operatorname{Id}}.
\end{align*}
In particular, for $\mu=1$, $[\ensuremath{\mathcal{R}_{1}(\bA,\fettla)}]^{-1}=\ensuremath{\mathcal{R}_{1}(\bA^{-1},\fettla)}.$
\end{theorem}
\begin{proof} Let $f_{i}=q_{A_{i}}$.
By Fact~\ref{t:Fenchel},
$\big(\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}\big)^* = p_{\mu^{-1}}({\ensuremath{\boldsymbol{f}}^*,\ensuremath{\boldsymbol{\lambda}}}),$
taking subgradients both sides, followed by using Fact~\ref{conjgrad}, we obtain that
$$\partial \big(\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}\big)^*= \big(\partial \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}\big)^{-1}=\partial \big(p_{\mu^{-1}}({\ensuremath{\boldsymbol{f}}^*,\ensuremath{\boldsymbol{\lambda}}})\big).$$
By Lemma~\ref{quadraticprox}, $\ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}}=q_{\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}}, p_{\mu^{-1}}({\ensuremath{\boldsymbol{f}}^*,\ensuremath{\boldsymbol{\lambda}}})=q_{\ensuremath{\mathcal{R}_{\mu^{-1}}(\bA^{-1},\fettla)}},$ we have
$$\partial \ensuremath{p_\mu({\ensuremath{\boldsymbol{f}},\fettla)}} = \ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}, $$
$$\partial p_{\mu^{-1}}({\ensuremath{\boldsymbol{f}}^*,\ensuremath{\boldsymbol{\lambda}}})=\ensuremath{\mathcal{R}_{\mu^{-1}}(\bA^{-1},\fettla)}.$$
Hence
$$\big[\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}\big]^{-1}=\ensuremath{\mathcal{R}_{\mu^{-1}}(\bA^{-1},\fettla)},$$
as claimed.
\end{proof}
\begin{remark} Although the harmonic and arithmetic average
lack self-duality, they are dual to each other:
$$[\ensuremath{\mathcal{H}(\bA,\fettla)}]^{-1}=\lambda_{1}A_{1}^{-1}+\cdots+\lambda_{n}A_{n}^{-1}=\ensuremath{\mathcal{A}(\bA^{-1},\fettla)},$$
$$[\ensuremath{\mathcal{A}(\bA,\fettla)}]^{-1}=\big[\lambda_{1}(A_{1}^{-1})^{-1}+\cdots+\lambda_{n}(A_{n}^{-1})^{-1}\big]^{-1}
=\ensuremath{\mathcal{H}(\bA^{-1},\fettla)}.$$
\end{remark}
\section{A comparison to weighted geometric means}\label{compare}
If $A,B\in \ensuremath{\mathbb S}^N_{++}$, the geometric mean is defined by
$$A\sharp B=A^{1/2}\big(A^{-1/2}BA^{-1/2}\big)^{1/2}A^{1/2}.$$
In general, the geometric mean of $A_{1},\ldots, A_{n}\in \ensuremath{\mathbb S}^N_+$ for $n\geq 3$ is defined either as the limit of an inductive procedure or by the Riemannian distance without a closed form \cite{ando,petz,moakher,kubo}.
To compare the resolvent average with the well-known geometric mean, we restrict our attention to
non-negative real numbers ($1\times 1$ matrices).
When $\ensuremath{\boldsymbol{A}}=\ensuremath{\boldsymbol{x}} =(x_{1},\ldots, x_{n})$ with $x_{i}\in\ensuremath{\mathbb R}_{+}$ and $\mu=1$, we write
$$\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=\ensuremath{\mathcal{R}_{\mu}(\bA,\fettla)}=\big(\lambda_{1}(x_{1}+1)^{-1}+\cdots+\lambda_{n}(x_{n}+1)^{-1}\big)^{-1}-1,
$$
and $\ensuremath{\boldsymbol{x}}^{-1}=(1/x_{1},\ldots, 1/x_{n})$ when $(\forall i)\ x_{i}\in\ensuremath{\mathbb R}_{++}$.
\begin{proposition} Let $(\forall\ i)\ x_{i}>0, y_{i}>0$. We have
\begin{enumerate}
\item[(i)]\emph{({\bf harmonic-resolvent-arithmetic mean inequality})}:
\begin{equation}\label{hram}
\big(\lambda_{1}x_{1}^{-1}+\cdots+\lambda_{n} x_{n}^{-1}\big)^{-1}\leq
\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}\leq \lambda_{1}x_{1}+\cdots +\lambda_{n}x_{n}.
\end{equation}
Moreover, $\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=\lambda_{1}x_{1}+\cdots+\lambda_{n}x_{n}$ if and only if $x_{1}=\cdots=x_{n}$.
\item[(ii)]\emph{({\bf self-duality})}: $[\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}]^{-1}=\ensuremath{\mathcal{R}_{}(\fettx^{-1},\fettla)}$.
\item[(iii)] If $\ensuremath{\boldsymbol{x}}=(x_{1},\ldots,x_{1})$, then $\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=x_{1}$.
\item[(iv)]If $\ensuremath{\boldsymbol{x}}=(x_{1},x_{1}^{-1}, x_{2},x_{2}^{-1},\ldots,
x_{n},x_{n}^{-1})$ and $\ensuremath{\boldsymbol{\lambda}}=(\tfrac{1}{2n},\ldots,\tfrac{1}{2n})$,
then $\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=1$.
\item[(v)] The function $\ensuremath{\boldsymbol{x}}\mapsto \ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$ is concave on $\ensuremath{\mathbb R}_{++}\times\cdots\times\ensuremath{\mathbb R}_{++}$.
\item[(vi)] If $\ensuremath{\boldsymbol{x}}\succeq \ensuremath{\boldsymbol{y}}$, then $\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}\geq \ensuremath{\mathcal{R}_{}(\fetty,\fettla)}$.
\end{enumerate}
\end{proposition}
\begin{proof}
(i): For \eqref{hram}, apply Theorem~\ref{matrixequ}(i) with $\mu=1$. Now
$\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=\lambda_{1}x_{1}+\cdots+\lambda_{n}x_{n}$ is equivalent to
\begin{equation}\label{newform}
\big(\lambda_{1}(x_{1}+1)^{-1}+\cdots+\lambda_{n}(x_{n}+1)^{-1}\big)^{-1}=\lambda_{1}x_{1}+\cdots+\lambda_{n}x_{n}+1,
\end{equation}
As $\sum_{i=1}^{n}\lambda_{i}=1$, \eqref{newform} is the same as
$$\lambda_{1}\frac{1}{(x_{1}+1)}+\cdots+\lambda_{n}\frac{1}{(x_{n}+1)}=\frac{1}{\lambda_{1}(x_{1}+1)+\cdots+
\lambda_{n}(x_{n}+1)}.$$
Since the function $x\mapsto 1/x$ is strictly convex on
$\ensuremath{\mathbb R}_{++}$, we must have $x_{1}=\cdots=x_{n}$.
(ii): Theorem~\ref{mainresult}. (iii): Proposition~\ref{sameoperator}.
(iv): Proposition~\ref{primalinverse}. (v): Corollary~\ref{known2}. (vi): Proposition~\ref{monotonecompare}.
\end{proof}
Recall the \emph{weighted geometric mean}:
$$\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=x_{1}^{\lambda_{1}}x_{2}^{\lambda_{2}}\cdots x_{n}^{\lambda_{n}}.$$
$\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}$ always has the following properties:
\begin{fact} Let $(\forall\ i)\ x_{i}>0, y_{i}>0$. We have
\begin{enumerate}
\item[(i)] \emph{({\bf harmonic-geometric-arithmetic mean inequality})}:
$$\big(\lambda_{1}x_{1}^{-1}+\cdots+\lambda_{n} x_{n}^{-1}\big)^{-1}\leq
\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}\leq \lambda_{1}x_{1}+\cdots +\lambda_{n}x_{n}.$$
Moreover, $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=\lambda_{1}x_{1}+\cdots +\lambda_{n}x_{n}$ if and only $x_{1}=\cdots=x_{n}$.
\item[(ii)]\emph{({\bf self-duality})}: $[\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}]^{-1}=\ensuremath{\mathcal{G}_{}(\fettx^{-1},\fettla)}$.
\item[(iii)] If $\ensuremath{\boldsymbol{x}}=(x_{1},\ldots,x_{1})$, then $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=x_{1}$.
\item[(iv)] If $\ensuremath{\boldsymbol{x}}=(x_{1},x_{1}^{-1}, x_{2},x_{2}^{-1},\ldots,
x_{n},x_{n}^{-1})$ and $\ensuremath{\boldsymbol{\lambda}}=(\tfrac{1}{2n},\ldots, \tfrac{1}{2n})$, then $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=1$.
\item[(v)] The function $\ensuremath{\boldsymbol{x}}\mapsto \ensuremath{\mathcal{G}_{}(\fettx,\fettla)}$ is concave on $\ensuremath{\mathbb R}_{++}\times\cdots\times\ensuremath{\mathbb R}_{++}$.
\item[(vi)] If $\ensuremath{\boldsymbol{x}}\succeq \ensuremath{\boldsymbol{y}}$, then $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}\geq \ensuremath{\mathcal{G}_{}(\fetty,\fettla)}$.
\end{enumerate}
\end{fact}
\begin{proof} (i): See \cite[page 29]{Rocky}. (ii)-(iv) and (vi) are simple. (v): See \cite[Example 2.53]{RockWets}.
\end{proof}
The means $\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$ and $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}$ have strikingly
similar properties. Are they the same?
\begin{example}
(i). Let $\lambda=(\ensuremath{\tfrac{1}{2}},\ensuremath{\tfrac{1}{2}})$. When $x=(0,1)$, $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=0$ but
$\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}=\tfrac{1}{3}$, so $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}
\neq\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$.
(ii). \emph{Is it right that $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}\leq \ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$
$\forall\ x\in \ensuremath{\mathbb R}_{++}^2$?}
The answer is also no.
Assume to the contrary that
$\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}\leq \ensuremath{\mathcal{R}_{}(\fettx,\fettla)}, \ \forall \ \ensuremath{\boldsymbol{x}} \in \ensuremath{\mathbb R}_{++}\times \ensuremath{\mathbb R}_{++}$. Taking inverse both sides, followed by applying the self-duality
of $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}, \ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$, gives
$$\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}^{-1}\geq \ensuremath{\mathcal{R}_{}(\fettx,\fettla)}^{-1}=\ensuremath{\mathcal{R}_{}(\fettx^{-1},\fettla)}\geq \ensuremath{\mathcal{G}_{}(\fettx^{-1},\fettla)}=\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}^{-1},$$
and this gives that $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}^{-1}=\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}^{-1}$ so that $\ensuremath{\mathcal{G}_{}(\fettx,\fettla)}=\ensuremath{\mathcal{R}_{}(\fettx,\fettla)}$.
This is a contradiction to (i).
\end{example}
Finally, we note that the resolvent average can be defined for general monotone operators and that
Theorem~\ref{mainresult} holds even when $A_1,\ldots,A_n$
are monotone operators
(not necessarily positive semi-definite matrices), in that situation one needs to use \emph{set-valued
inverses}.
This and further details on the resolvent average for
general monotone operators will appear in the forthcoming paper
\cite{bws}.
\section*{Acknowledgment}
Heinz Bauschke was partially supported by the Canada Research
Chair program and by the Natural Sciences and Engineering Research
Council of Canada.
Sarah Moffat was partially supported by the Natural Sciences and
Engineering Research Council of Canada.
Xianfu Wang was partially supported by the Natural Sciences and
Engineering Research Council of Canada.
| 2024-02-18T23:40:06.265Z | 2009-10-19T23:24:26.000Z | algebraic_stack_train_0000 | 1,345 | 7,220 |
|
proofpile-arXiv_065-6665 | \section{Introduction}
A real number $\xi$ is called a \textit{Liouville number}, if for any positive real number $w$ there exist infinitely many rational numbers $p/q$, with $q\geq 1$, such that
\begin{center}
$0<\left|\xi - \dfrac{p}{q}\right|<\dfrac{1}{q^w}$.
\end{center}
Transcendental number theory began in 1844 when Liouville \cite{lio} showed that all Liouville numbers are transcendental establishing thus the first examples of such numbers. For instance, the number
\begin{center}
$L=\displaystyle\sum_{j=1}^{\infty}10^{-j!}=0.11000100000000000000001000...$,
\end{center}
which is known as \textit{Liouville's constant}, is a Liouville number and therefore transcendental. In 1962, Erd$\ddot{\mbox{o}}$s \cite{erdos} proved that every nonzero real number can be written as the sum and the product of two Liouville numbers.
In 1932, Mahler \cite{mah} splited the set of the transcendental numbers in three disjoint sets named \textit{$S$-,\ $T$-} and {\it $U$-numbers}. Particularly, the $U$-numbers generalizes the concept of Liouville numbers. We denote by $w^*_n(\xi)$ as the supremum of the real numbers $w^*$ for which there exist infinitely many real algebraic numbers $\alpha$ of degree $n$ satisfying
\begin{center}
$0<|\xi - \alpha| < H(\alpha)^{-w^*-1}$,
\end{center}
where $H(\alpha)$ (so-called the \emph{height} of $\alpha$) is the maximum of absolute value of coefficients of the minimal polynomial of $\alpha$ (over $\mathbb{Z}$). The number $\xi$ is said to be a \emph{$U^*_m$-number} (according to LeVeque \cite{le}) if $w^*_m(\xi)=\infty$ and $w^*_n(\xi)<\infty$ for $1\leq n< m$ ($m$ is called the \emph{type} of the $U$-number). We point out that we actually have defined a Koksma $U^*_m$-number instead of a Mahler $U_m$-number. However, it is well-known that they are the same \cite[cf. Theorem 3.6]{bugeaud} and \cite{al2}. We observe that the set of $U_1$-numbers is precisely the set of Liouville numbers.
The existence of $U_m$-numbers, for all $m\geq 1$, was proved by LeVeque \cite{le}. In 1993, Pollington \cite{pol} showed that for any positive integer $m$, every real number can be expressed as a sum of two $U_m$-numbers.
Since two algebraically dependent numbers must belong to the same Mahler's class \cite[Theorem 3.2]{bugeaud}, then $\alpha L$ and $\alpha+L$ are $U$-numbers, for any nonzero algebraic number $\alpha$. But what are their types?
In this note, we use G$\ddot{\mbox{u}}$tting's method \cite{gut} for proving that the sum and the product of every $m$-degree algebraic number $\alpha$ by $L$ is a $U_m$-number. Moreover, we obtain an upper bound for $w_n^{\ast}$.
\begin{theorem}
Let $\alpha$ be an algebraic number of degree $m$ and let $L$ be the Liouville's constant. Then $\alpha L$ and $\alpha+L$ are $U_m$-numbers, with
\begin{equation}\label{in}
\max\{w^{\ast}_n(\alpha L),w^{\ast}_n(\alpha + L)\}\leq 2m^2n+m-1,\ for\ n=1,...,m-1.
\end{equation}
\end{theorem}
\section{Auxiliary Results}
Before the proof of the main result, we need two technical results. The first one follows as an easy consequence of the triangular inequality and binomial identities.
\begin{lemma}\label{1}
Given $P(x)\in \mathbb{Z}[x]$ with degree $m$ and $a/b\in \mathbb{Q} \backslash \{0\}$. If $Q_1(x)=a^mP(\frac{b}{a}x)$ and $Q_2(x)=b^mP(x-\frac{a}{b})$, then
\begin{itemize}
\item[\textup{(i)}] $H(Q_1)\leq \max\{|a|,|b|\}^mH(P)$;
\item[\textup{(ii)}] $H(Q_2)\leq 2^{m+1}\max\{|a|,|b|\}^mH(P)$.
\end{itemize}
\end{lemma}
\begin{proof}
(i) If $P(x)=\sum_{j=0}^ma_jx^j$, then $Q_1(x)=\sum_{j=0}^ma_jb^ja^{m-j}x^j$. Supposing, without loss of generality, that $|a|\geq |b|$, we have $|a|^m|a_j|\geq |a|^{m-j}|a_j||b|^j$ for $0\leq j\leq m$. Hence, we are done. For (ii), write $Q_2(x)=\sum_{i=0}^mc_ix^i$, where
\begin{center}
$c_i=b^m\displaystyle\sum_{j=i}^ma_j{j\choose j-i}(-1)^{j-i}\left(\frac{a}{b}\right)^{j-i}$
\end{center}
Therefore
\begin{center}
$|c_i|\leq H(P)\displaystyle\sum_{k=0}^{m-i}\displaystyle{k+i\choose k}|a|^k|b|^{m-k}\leq \max\{|a|,|b|\}^mH(P)\displaystyle\sum_{k=0}^{m-i}\displaystyle{k+i\choose k}$.
\end{center}
Since $\sum_{k=0}^{m-i}{k+i\choose k}={m+1\choose m-i}\leq 2^{m+1}$, we finally have
\begin{center}
$|c_i|\leq 2^{m+1}\max\{|a|,|b|\}^mH(P)$,
\end{center}
which completes our proof.
\end{proof}
In addition to Lemma 1, we use the fact that algebraic numbers are not well aproximable by algebraic numbers.
\begin{lemma}[Cf. Corollary A.2 of \cite{bugeaud}]\label{2}
Let $\alpha$ and $\beta$ be two distinct nonzero algebraic numbers of degree $n$ and $m$, respectively. Then we have
\begin{center}
$|\alpha-\beta|\geq (n+1)^{-m/2}(m+1)^{-n/2}\max\{2^{-n}(n+1)^{-(m-1)/2},2^{-m}(m+1)^{-(n-1)/2}\}$\\
$\times H(\alpha)^{-m}H(\beta)^{-n}$.
\end{center}
\end{lemma}
\begin{proof}
A sketch of the proof can be found in the Appendix A of \cite{bugeaud}.
\end{proof}
\section{Proof of the Theorem}
For $k\geq 1$, set
\begin{center}
$p_k=10^{k!}\displaystyle\sum_{j=1}^k10^{-j!}$, $q_k=10^{k!}$ and $\alpha_k=\dfrac{p_k}{q_k}$.
\end{center}
We observe that $H(\alpha_{k-1})<H(\alpha_k)=10^{k!}=H(\alpha_{k-1})^{k}$ and
\begin{equation}\label{lp}
|L - \alpha_k| < \frac{10}{9}H(\alpha_k)^{-k-1}.
\end{equation}
Thus, setting $\gamma_k=\alpha \alpha_k$, we obtain of (\ref{lp})
\begin{equation}\label{11}
|\alpha L - \alpha \alpha_k|\leq cH(\alpha_k)^{-k-1},
\end{equation}
where $c=10|\alpha|/9$. It follows by the Lemma \ref{1} (i) that $H(\alpha_k)^m\geq H(\alpha)^{-1}H(\gamma_k)$ and thus we conclude that
\begin{equation}\label{opo}
|\alpha L - \alpha \alpha_k|\leq cH(\alpha)^{k+1}H(\gamma_k)^{-k-1}.
\end{equation}
Consequently, $\alpha \beta$ is a $U$-number with type at most $m$ (since $\gamma_k$ has degree $m$).
Again, we use Lemma \ref{1} (i) for obtaining
\begin{equation}
H(\gamma_{k+1})\leq H(\alpha)H(\alpha_{k+1})^m=H(\alpha)H(\alpha_k)^{(k+1)m}\leq H(\alpha)H(\gamma_k)^{(k+1)m}
\end{equation}
Now, let $\gamma$ be an $n$-degree real algebraic number, with $n<m$ and $H(\gamma)\geq H(\gamma_1)$. Thus, one may ensure the existence of a sufficient large $k$ such that
\begin{equation}\label{key}
H(\gamma_k)< H(\gamma)^{2m^2}< H(\gamma_{k+1})\leq H(\alpha)H(\gamma_k)^{(k+1)m}.
\end{equation}
So, by Lemma \ref{2}, it follows that
\begin{equation}\label{eq1}
|\gamma_k - \gamma|\geq f(m,n)H(\gamma)^{-m}H(\gamma_k)^{-n},
\end{equation}
where $f(m,n)$ is a positive number which does not depend on $k$ and $\gamma$. Therefore by (\ref{key})
\begin{equation}\label{22}
|\gamma_k - \gamma|\geq f(m,n)H(\alpha)^{-1/2m}H(\gamma_k)^{-(k+1)/2-n}.
\end{equation}
By taking $H(\gamma)$ large enough, the index $k$ satisfies
\begin{equation}\label{33}
H(\gamma_k)^{(k+1)/2-n}\geq 2cf(m,n)^{-1}H(\alpha)^{k+1/2m}.
\end{equation}
Thus, it follows from (\ref{opo}), (\ref{22}) and (\ref{33}) that $|\gamma_k-\gamma|\geq 2|\alpha L - \gamma_k|$. Therefore, except for finitely many algebraic numbers $\gamma$ of degree $n$ strictly less than $m$, we have
\begin{eqnarray*}
|\alpha L - \gamma| & \geq & |\gamma_k - \gamma|-|\alpha L-\gamma_k|\nonumber\\
& \geq & \frac{1}{2}|\gamma_k - \gamma| \nonumber\\
& \geq & \frac{f(m,n)}{2}H(\gamma)^{-m}H(\gamma_k)^{-n} > \dfrac{f(m,n)}{2}H(\gamma)^{-2m^2n-m},\nonumber\\
\end{eqnarray*}
where we use the left-hand side of (\ref{key}). It follows that $w_n^*(\alpha L)\leq 2m^2n+m-1$ which finishes our proof.
The case $\alpha+L$ follows the same outline, where we use Lemma \ref{1} (ii) rather than (i).
\qed
\section{The general case and further comments}
Let $\beta$ be a Liouville number. Since that a $U$-number keeps its type when multiplied by any nonzero rational number, we can consider $0<\beta < 1$. Set
\begin{center}
$S_{\beta}=\{(\frac{p_k}{q_k})_{k\geq 1}\in \mathbb{Q}^{\infty}\ :\ |\beta-\frac{p_k}{q_k}|<\frac{1}{q_k^{k+1}},\ k=1,2,...\}$.
\end{center}
By the assumption on $\beta$, we may suppose $1\leq p_k\leq q_k$ and then $H(p_k/q_k)=q_k$, for all $k$. Note that $S_{\beta}$ is an infinite set.
As is customary, the symbols $\ll$, $\gg$ mean that there is an implied constant in the inequalities $\leq$, $\geq$, respectively. In our process for proving the Theorem 1, the key step happens when holds an inequality like in (\ref{key}). Thus it follows that
\begin{theorem}
Let $\alpha$ be an $m$-degree algebraic number and let $\beta$ be a Liouville number. If there exists a sequence $(p_k/q_k)_{k\geq 1} \in S_{\beta}$ such that $q_k\ll q_{k+1}\ll q_{k}^{k+1}$ for all $k\gg 1$, then the numbers $\alpha \beta$ and $\alpha+\beta$ are $U_m$-numbers and
\begin{equation}\label{in2}
\max\{w^{\ast}_n(\alpha \beta),w^{\ast}_n(\alpha + \beta)\}\leq 2m^2n+m-1,\ for\ n=1,...,m-1.
\end{equation}
\end{theorem}
\begin{exe}
For any integer number $m\geq 2$ and any $a_j\in \{1,...,9\}$, the number $\sum_{j=1}^{\infty}a_jm^{-j!}$ is a Liouville number satisfying the hypothesis of the previous theorem.
\end{exe}
\begin{corollary}
For any $m\geq 1$, there exists an uncountable collection of Liouville numbers that are expressible as sum of two algebraically dependent $U_m$-numbers.
\end{corollary}
\begin{proof}
Set $\beta=\displaystyle\sum_{j=1}^{\infty}a_j10^{-j!}$, where $a_j\in \{1,2\}$. The result follows immediately of Theorem 2 and of writing $\beta = (\frac{\beta+\sqrt[m]{2}}{2})+(\frac{\beta-\sqrt[m]{2}}{2})$.
\end{proof}
There exist several lower estimates for the distance between two distinct algebraic numbers, e.g., Liouville's inequality and Lemma \ref{2}. A too-good-to-be-true Conjecture due to Schmidt \cite{sch} states that
\begin{conjecture}
For any number field $\mathbb{K}$ and any positive real number $\epsilon$, we have
\begin{center}
$|\alpha-\beta| > c(\mathbb{K},\epsilon)(\max\{H(\alpha),H(\beta)\})^{-2-\epsilon}$,
\end{center}
for any distinct $\alpha,\beta \in \mathbb{K}$, where $c(\mathbb{K},\epsilon)$ is some constant depending only on $\mathbb{K}$ and on $\epsilon$.
\end{conjecture}
We conclude by pointing that if the Schmidt's conjecture is true, then the sum and the product of any $m$-degree algebraic number $\alpha$ by any Liouville number $\beta$ is a $U_m$-number and the inequality (\ref{in2}) can be considerable improved for
\begin{center}
$\max\{w^{\ast}_n(\alpha \beta),w^{\ast}_n(\alpha + \beta)\}\leq 1$.
\end{center}
\section*{Acknowledgement}
The second author thanks to the Department of Mathematics of Universidade Federal do Cear\' a for its hospitality. We also thank Yann Bugeaud for nice discussions on the subject.
| 2024-02-18T23:40:06.468Z | 2009-10-20T00:28:24.000Z | algebraic_stack_train_0000 | 1,355 | 1,887 |
|
proofpile-arXiv_065-6736 | \section{Introduction}\label{intro}
All topological spaces mentioned here are Hausdorff. As is traditional
in general Banach space theory, Banach spaces we mention
are considered as real Banach spaces even though nothing essential changes in the
context of complex spaces. Throughout let $K$ stand for an infinite compact topological space.
Let $X$ be a Banach space and $C$ a closed
convex subset of $X$.
A point $x_0\in C$ is a {\em point of support for} $C$ if there
is a functional $\varphi\in X^\ast$ such that $\varphi(x_0)\le\varphi(x)$
for all $x\in C$, and $\varphi(x_0)<\varphi(x')$ for some $x'\in C$.
Rolewicz \cite{Ro} proved in 1978 that every separable closed convex
subset $Y$ of a Banach space contains a point
which is not a point of support for $Y$, and asked if every
non-separable Banach space must contain a closed convex set
containing only points of support. In fact, this topic was already considered
by Klee \cite{Klee} in 1955 and the above theorem follows from 2.6
in that paper, by the same proof and taking $x_i$s to form
a dense set in $C$ \footnote{We thank Libor Vesely for pointing
out this, not widely recognised, connection.}. However it was Rolewicz's paper
which started a whole series of articles on this topic, and his question
has not yet been settled completely. It is known that the answer to
Rolewicz's question is independent of ZFC, and it is still not known if
the negative answer follows from $CH$. In \S\ref{semi} we
construct a $CH$ example of a nonseparable Banach space of the form $C(K)$ which violates
a strenghtening of the requirements in the Rolewicz's question.
The proof in \S\ref{semi} uses certain systems of pairs of points
of $K$, whose structure seems to us to be of independent interest.
They appear implicitly in many proofs about biorthogonal systems
in spaces of the form $C(K)$, see \cite{Hajekbook}, but their
existence is in fact entirely a property of the compact space $K$.
We call such systems {\em bidiscrete systems}. They are studied in
\S \ref{bidiscr}. Specifically, we prove in Theorem \ref{lemma2}
that if $K$ is an infinite compact Hausdorff space then $K$ has a
bidiscrete system of size $d(K)$, the density of $K$. This theorem
has not been stated in this form before, but we note that an
argument by Todor\v cevi\'c in \cite{StevoMM} can be easily
extended to give this result.
We now give some historical background. Mathematical background will be presented in
Section \ref{background}.
Borwein and Vanderwerff \cite{Bv} proved in 1996 that, in a Banach
space $X$, the existence of a closed convex set all of whose
points are support points is equivalent to the existence of an
uncountable semi-biorthogonal sequence for $X$, where
semi-biorthogonal sequences are defined as follows:
\begin{Definition}\label{biorthognal} Let $X$ be a Banach space. A sequence
$\langle (f_\alpha, \varphi_\alpha):\,\alpha<\alpha^\ast\rangle$ in $X\times X^\ast$ is said
to be
a {\em semi-biorthogonal sequence} if for all $\alpha,\beta<\alpha^\ast$ we have:
\begin{itemize}
\item $\varphi_\alpha(f_\alpha)=1$,
\item $\varphi_\alpha (f_\beta)=0$ if $\beta<\alpha$,
\item $\varphi_\alpha (f_\beta)\ge 0$ if $\beta>\alpha$.
\end{itemize}
\end{Definition}
We remind the reader of the better known notion of {\em a biorthogonal system}
$\{(f_\alpha, \varphi_\alpha):\,\alpha<\alpha^\ast\}$ in $X\times X^\ast$ which is defined to satisfy the first
item of the Definition \ref{biorthognal} with the last two items are strengthened to
\begin{itemize}
\item $\varphi_\alpha (f_\beta)=0$ if $\beta\neq\alpha$.
\end{itemize}
Notice that the requirements of a semi-biorthogonal sequence make it clear that we really need a well ordering
of the sequence
in the definition, but that the definition of a biorthogonal system does not require an underlying well-ordering.
There is nothing special about the values 0 and 1 in the above definitions, of course, and we could
replace them by any pair $(a,b)$ of distinct values in $\mathbb R$ and even let $b=b_\alpha$ vary with
$\alpha$. Equally, we could require the range of all $f_\alpha$ to be in $[0,1]$ or some other fixed nonempty
closed interval.
Obviously, any well-ordering of a biorthogonal system gives a
semi-biorthogonal sequence. On the other hand, there is an example
by Kunen under $CH$ of a nonmetrizable compact scattered space $K$
for which $X=C(K)$ does not have an uncountable biorthogonal
system, as proved by Borwein and Vanderwerff in \cite{Bv}. Since
$K$ is scattered, it is known that $X$ must have an uncountable
semi-biorthogonal sequence (see \cite{Hajekbook} for a
presentation of a similar example under $\clubsuit$ and a further
discussion). Let us say that a Banach space is a {\em Rolewicz
space} if it is nonseparable but does not have an uncountable
semi-biorthogonal sequence.
In his 2006 paper \cite{StevoMM}, Todor\v cevi\'c proved that
under Martin's Maximum (MM) every non-separable Banach space has
an uncountable biorthogonal system, so certainly it has an
uncountable semi-biorthogonal sequence. Hence, under MM there are
no Rolewicz spaces. On the other hand, Todor\v cevi\'c informed us
that he realized in 2004 that a forcing construction in \cite{bgt}
does give a consistent example of a Rolewicz space. Independently,
also in 2004 (published in 2009), Koszmider gave a similar forcing construction in
\cite{Kosz}. It is still not known if there has to be a Rolewicz
space under CH.
Our motivation was to construct a Rolewicz space of the form
$X=C(K)$ under CH. Unfortunately, we are not able to do so, but we
obtain in Theorem \ref{CH} a space for which we can at least show
that it satisfies most of the known necessary conditions for a
Rolewicz space and that it has no uncountable semi-bidiscrete
sequences of the kind that are present in the known failed
candidates for such a space, for example in $C(S)$ where $S$ is
the split interval.
Specifically, it is known that if $K$ has a non-separable Radon
measure or if it is scattered then $C(K)$ cannot be Rolewicz
(\cite{GJM}, \cite{Hajekbook}) and our space does not have either
of these properties. Further, it is known that a compact space $K$
for which $C(K)$ is a Rolewicz space must be both HS and HL
(\cite{Lazar}, \cite{Bv}) while not being metrizable, and our
space has these properties, as well. It follows from the
celebrated structural results on Rosenthal compacta by Todor{\v
c}evi\'c in \cite{StevoRos} that a Rosenthal compactum cannot be a
Rolewicz space, and our space is not Rosenthal compact. Finally,
our space is not metric but it is a 2-to-1 continuous preimage of
a metric space. This is a property possessed by the forcing
example in \cite{Kosz} and it is interesting because of a theorem
from \cite{StevoRos} which states that every non-metric Rosenthal
compact space which does not contain an uncountable discrete
subspace is a 2-to-1 continuous preimage of a metric space. Hence
the example in \cite{Kosz} is a space which is not Rosenthal
compact and yet it satisfies these properties, and so is our
space.
\section{Background}\label{background}
\begin{Definition}\label{nice} Let $X=C(K)$ be the Banach space of continuous functions
on a compact space $K$. We say that a sequence $\langle
(f_\alpha,\phi_\alpha):\,\alpha<\alpha^\ast\rangle$ in $X\times
X^\ast$ is a {\em nice} semi-biorthogonal sequence if it is a
semi-biorthogonal sequence and there are points $\langle
x^l_\alpha:\,l=0,1,\alpha<\alpha^\ast\rangle$ in $K$ such that
$\phi_\alpha=\delta_{x^1_\alpha}-\delta_{x^0_\alpha}$, where
$\delta$ denotes the Dirac measure. We similarly define nice
biorthogonal systems.
\end{Definition}
As Definition \ref{nice} mentions points of $K$ and $C(K)$ does
not uniquely determine $K$~\footnote{see e.g. Miljutin's theorem
\cite{Mil1}, \cite{Mil2} which states that for $K$, $L$ both
uncountable compact metrizable, the spaces $C(K)$ and $C(L)$ are
isomorphic.}, the definition is actually topological rather than
analytic. We shall observe below that the existence of a nice
semi-biorthogonal sequence of a given length or of a nice
biorthogonal system of a given size in $C(K)$ is equivalent to the
existence of objects which can be defined in terms which do not
involve the dual $C(K)^\ast$.
\begin{Definition}\label{nicetop} (1) A system
$\{(x_\alpha^0, x_\alpha^1):\,\alpha<\kappa\}$ of pairs of points
in $K$ (i.e. a subfamily of $K^2$) is called {\em a bidiscrete
system in} $K$ if there exist functions $\{
f_\alpha:\,\alpha<\kappa\} \subseteq C(K)$ satisfying that for every
$\alpha,\beta<\kappa$:
\begin{itemize}
\item $f_\alpha(x_\alpha^l)=l$ for $l\in \{0,1\}$,
\item if $\alpha\neq\beta$ then $f_\alpha(x_\beta^0)=f_\alpha(x_\beta^1)$.
\end{itemize}
\end{Definition}
(2) We similarly define semi-bidiscrete sequences in $K$ as
sequences $\langle (x_\alpha^0,
x_\alpha^1):\,\alpha<\alpha^\ast\rangle$ of points in $K^2$ that
satisfy the first requirement of (1) but instead of the second the
following two requirements:
\begin{itemize}
\item if $\alpha>\beta$ then $f_\alpha(x_\beta^0)=f_\alpha(x_\beta^1)$,
\item if $\alpha<\beta$ then $f_\alpha(x_\beta^0)=1\implies f_\alpha(x_\beta^1)=1$.
\end{itemize}
\begin{Observation}\label{nice=bidiscrete} For a compact space $K$,
$\{(x_\alpha^0, x_\alpha^1):\,\alpha<\alpha^*\} \subseteq K^2$ is a
bidiscrete system iff there are $\{ f_\alpha:\,\alpha<\alpha^*\}
\subseteq C(K)$ such that $\{(f_\alpha,
\delta_{x^\alpha_1}-\delta_{x^\alpha_0}):\,\alpha<\alpha^*\}$ is a
nice biorthogonal system for the Banach space $X=C(K)$. The
analogous statement holds for nice semi-bidiscrete sequences.
\end{Observation}
\begin{Proof} We only prove the statement for nice biorthogonal systems, the proof for the nice
semi-biorthogonal sequences is the same. If we are given a system
exemplifying (1), then
$\delta_{x^1_\alpha}(f_\beta)-\delta_{x^0_\alpha}(f_\beta)=
f_\beta(x^1_\alpha)-f_\beta(x^0_\alpha)$ has the values as
required. On the other hand, if we are given a nice biorthogonal
system of pairs $\{(f_\alpha,
\delta_{x^1_\alpha}-\delta_{x^0_\alpha}) : \alpha < \alpha^*\}$
for $X$, define for $\alpha<\alpha^*$ the function $g_\alpha \in
C(K)$ by $g_\alpha(x)=f_\alpha(x)-f_\alpha(x^0_\alpha)$. Then $\{
(x_\alpha^0, x_\alpha^1):\,\alpha<\alpha^\ast\}$ satisfies (1), as
witnessed by $\{ g_\alpha:\,\alpha<\alpha^\ast\}$.
$\eop_{\ref{nice=bidiscrete}}$
\end{Proof}
In the case of a 0-dimensional space $K$ we are often able to make
a further simplification by requiring that the functions
$f_\alpha$ exemplifying the bidiscreteness of $(x_\alpha^0,
x_\alpha^1)$ take only the values 0 and 1. This is clearly
equivalent to asking for the existence of a family $\{
H_\alpha:\,\alpha<\alpha^\ast\}$ of clopen sets in $K$ such that
each $H_\alpha$ separates $x^0_\alpha$ and $x^1_\alpha$ but not
$x^0_\beta$ and $x^1_\beta$ for $\beta\neq\alpha$. We call such
bidiscrete systems {\em very nice}. We can analogously define a
{\em very nice} semi-bidiscrete sequence, where the requirements
on the clopen sets become $x^l_\alpha\in H_\alpha\iff l=1$,
$\beta<\alpha\implies [x^0_\beta\in H_\alpha\iff x^{1}_\beta\in
H_\alpha]$ and $[\beta>\alpha\, \wedge \, x^0_\beta\in H_\alpha]
\implies x^1_\beta\in H_\alpha$.
We shall use the expression {\em very nice (semi-)biorthogonal
system (sequence)} in $C(K)$ to refer to a nice
(semi-)biorthogonal system (sequence) obtained as in the proof of
Claim \ref{nice=bidiscrete} from a very nice (semi-)bidiscrete
system (sequence) in $K$.
\begin{Example}
\label{splitinterval} (1) Let $K$ be the split interval (or double
arrow) space, namely the ordered space $K=[0,1]\times\{0,1\}$,
ordered lexicographically. Then $$\{\big((x,0), (x,1)\big) : x
\in [0,1]\}$$ forms a very nice bidiscrete system in $K$. This is
exemplified by the two-valued continuous functions $\{f_x : x \in
[0,1]\}$ defined by $f_x(r)=0$ if $r\le (x, 0)$ and $f_x(r)=1$
otherwise.
(2) Suppose that $\kappa$ is an infinite cardinal and
$K=2^\kappa$. For $l\in \{0,1\}$ and $\alpha<\kappa$ we define
$x^l_\alpha\in K$ by letting $x^l_\alpha(\beta)=1$ if
$\beta<\alpha$, $\,x^l_\alpha(\beta)=0$ if $\beta> \alpha$, and
$x^l_\alpha(\alpha)=l$. The clopen sets $H_\alpha=\{f\in
K:\,f(\alpha)=1\}$ show that the pairs $\{(x^0_\alpha,x^1_\alpha)
: {\alpha<\kappa}\}$ form a very nice bidiscrete system in the
Cantor cube $K = 2^\kappa$.
\end{Example}
In \cite{StevoMM}, Theorem 10, it is proved under
$MA_{\omega_1}\,$ that every Banach space of the kind $X=C(K)$ for
a nonmetrizable compact $K$ admits an uncountable nice
biorthogonal system. Moreover, at the end of the proof it is
stated that for a 0-dimensional $K$ this biorthogonal system can
even be assumed to be very nice (in our terminology).
As nice semi-biorthogonal sequences may be defined using only $K$
and $X=C(K)$ and do not involve the dual $X^\ast$, in
constructions where an enumerative tool such as $CH$ is used it is
easier to control nice systems than the general ones. In our CH
construction below of a closed subspace $K$ of $2^{\omega_1}$ we
would at least like to destroy all uncountable nice
semi-biorthogonal sequences by controlling semi-bidiscrete
sequences in $K$. We are only able to do this for semi-bidiscrete
sequences which are not already determined by the first
$\omega$-coordinates, in the sense of the following Definition
\ref{supernice} : In our space $K$ any uncountable nice
semi-biorthogonal sequence must be $\omega$-determined.
\begin{Definition}\label{supernice}
A family $\{(x^0_\alpha, x^1_\alpha):\,\alpha<\alpha^*\} \subseteq
2^{\omega_1} \times 2^{\omega_1}$ is said to be {\em
$\omega$-determined} if
\[
(\forall s\in
2^\omega)\,\{\alpha:\,x^0_\alpha\rest\omega=x^1_\alpha\rest\omega=s\}\mbox{
is countable}.
\]
For $K \subseteq 2^{\omega_1}$ we define an {\em $\omega$-determined
semi-biorthogonal sequence in $C(K)$} to be any nice
semi-biorthogonal sequence $\langle
(f_\alpha,\delta_{x^1_\alpha}-\delta_{x^0_\alpha}):\,\alpha<\alpha^\ast\rangle$
for which the associated semi-bidiscrete sequence $\langle
(x^0_\alpha, x^1_\alpha):\alpha<\alpha^\ast \rangle$ forms an
$\omega$-determined family.
\end{Definition}
\section{The $CH$ construction}\label{semi}
\begin{Theorem}\label{CH} Under $CH$, there is a compact space $K \subseteq 2^{\omega_1}$ with the following properties:
\begin{itemize}
\item $K$ is not metrizable, but is a 2-to-1 continuous preimage of a metric space,
\item $K$ is HS and HL,
\item\label{treci} every Radon measure on $K$ is separable,
\item $K$ has no isolated points,
\item $K$ is not Rosenthal compact,
\item any uncountable nice semi-biorthogonal sequence in $C(K)$ is $\omega$-determined.
\end{itemize}
\end{Theorem}
\begin{proof} We divide the proof into two parts. In the first we give various requirements on the construction,
and show that if these requirements are satisfied the space meeting the claim of the theorem can be constructed. In the second part we show that
these requirements can be met.
\subsubsection{The requirements}
Our space will be a closed subspace of $2^{\omega_1}$. Every such space can be viewed as the limit of an inverse system of spaces, as we now explain.
\begin{Definition}
For $\alpha\leq\beta\le\omega_1$, define $\piba : 2^{\beta}\rightarrow
2^{\alpha}$ by $\piba (f) = f\rest\alpha $.
\end{Definition}
Suppose that $K$ is a closed subspace of $2^{\omega_1}$, then
for $\alpha \le \omega_1$ we let $K_\alpha=\pi^{\omega_1}_\alpha(K)$. So, if $\alpha \le \beta$ then
$K_\alpha$ is the $\pi^\beta_\alpha$-projection of $K_\beta$. For $\alpha<\omega_1$ let
\[
A_\alpha=\pi^{\alpha+1}_\alpha(\{x\in K_{\alpha+1}:x(\alpha)=0\}), B_\alpha=\pi^{\alpha+1}_\alpha(\{x\in K_{\alpha+1}:x(\alpha)=1\}).
\]
The following statements are then true:
\begin{description}
\item{\bf R1}.\label{1.1} $K_\alpha$ is a closed subset of $2^\alpha$, and
$\pi_\alpha^\beta(K_\beta) = K_\alpha$ whenever
$\alpha \le \beta \le \omega_1$.
\item{{\bf R2}.}\label{1.2} For $\alpha < \omega_1$,
$A_\alpha$ and $B_\alpha$ are closed in $K_\alpha$,
$A_\alpha \cup B_\alpha = K_\alpha$, and
$K_{\alpha + 1} = A_\alpha \times \{0\} \cup B_\alpha \times \{1\}$.
\end{description}
Now $K$ can be viewed as the limit of the inverse system $\mathcal
K=\{K_\alpha:\,\alpha<\omega_1, \piba\rest K_\beta:\,\alpha\le
\beta<\omega_1\}$. Therefore to construct the space $K$ it is
sufficient to specify the system ${\cal K}$, and as long as the
requirements {\bf R1} and {\bf R2} are satisfied, the resulting
space $K$ will be a compact subspace of $2^{\omega_1}$. This will
be our approach to constructing $K$, that is we define $K_\alpha$
by induction on $\alpha$ to satisfy various requirements that we
list as {\bf Rx}.
The property HS+HL will be guaranteed by a use of irreducible
maps, as in \cite{DzK}. Recall that for spaces $X,Y$, a map
$f:\,X\rightarrow Y$ is called {\em irreducible} on $A\subseteq X$ iff
for any proper closed subspace $F$ of $A$ we have that $f(F)$ is a
proper subset of $f(A)$. We shall have a special requirement to
let us deal with HS+HL, but we can already quote Lemma 4.2 from
\cite{DzK}, which will be used in the proof. It applies to any
space $K$ of the above form.
\begin{Lemma}\label{Lemma4.2} Assume that $K$ and $K_\alpha$ satisfy {\bf R1} and {\bf R2} above.
Then $K$ is HL+HS iff for all closed $H \subseteq K$,
there is an $\alpha <\omega_1$ for which
$\pi^{\omega_1}_\alpha$ is irreducible on $(\pi^{\omega_1}_\alpha )^{-1} (\pi^{\omega_1}_\alpha (H))$.
\end{Lemma}
In addition to the requirements given above
we add the following basic requirement {\bf R3} which assures that $K$ has no isolated points.
\begin{description}
\item{{\bf R3}.}\label{1.3} For $n < \omega$,
$K_n = A_n = B_n = 2^n$. For $\alpha \ge \omega$, $A_\alpha$ and
$B_\alpha$ have no isolated points.
\end{description}
Note that the requirement {\bf R3} implies that for each $\alpha
\ge \omega$, $K_\alpha$ has no isolated points; so it is easy to
see that the requirements guarantee that $K$ is a compact subspace
of $2^{\omega_1}$ and that it has no isolated points. Further,
$K_\omega = 2^\omega$ by {\bf R1} and {\bf R3}. The space $K$ is
called {\em simplistic} if for all $\alpha$ large enough
$A_\alpha\cap B_\alpha$ is a singleton. For us `large enough' will
mean `infinite', i.e. during the construction we shall obey the
following:
\begin{description}
\item{{\bf R4}.}\label{1.4} For all $\alpha\in [\omega,\omega_1)$ we
have $A_\alpha\cap B_\alpha=\{s_\alpha\}$ for some $s_\alpha\in K_\alpha$.
\end{description}
By {\bf R4} we can make the following observation which will be useful later:
\begin{Observation}\label{delta} Suppose that $x\in K_\alpha, y\in K_\beta$ for some $\omega\le\alpha\le\beta$ and $x\nsubseteq y$, $y\nsubseteq x$ with $\Delta(x,y)\ge\omega$. Then $x\rest\Delta(x,y)=y\rest\Delta(x,y)=s_{\Delta(x,y)}$.
\end{Observation}
As usual, we used here the notation $\Delta(x,y) = \min \{\alpha :
x(\alpha) \ne y(\alpha)\}$.
Requirement {\bf R4} implies that $K$ is not 2nd countable, hence not
metrizable.
The following is folklore in the subject, but one can also see \cite{Piotr} for a detailed explanation
and stronger theorems:
\begin{Fact} Every Radon measure on a simplistic space is separable.
\end{Fact}
Now we come back to the property HS+HL. To assure this we shall
construct an auxiliary Radon measure $\mu$ on $K$. This measure
will be used, similarly as in the proof from Section \S 4 in
\cite{DzK}, to assure that for every closed subset $H$ of $K$ we
have $H = (\pi^{\omega_1}_\alpha )^{-1} (\pi^{\omega_1}_\alpha (H))$ for some countable coordinate
$\alpha$. In fact, what we need for our construction is not the
measure $\mu$ itself but a sequence $\langle
\mu_\alpha:\,\alpha<\omega_1 \rangle$ where each $\mu_\alpha$ is a
Borel measure on $K_\alpha$ and these measures satisfy that for
each $\alpha\le\beta<\omega_1$ and Borel set $B\subseteq K_\beta$,
we have $\mu_\beta(B)=\mu_\alpha(\piba(B))$. As a side remark
the
sequence $\langle \mu_\alpha:\,\alpha<\omega_1 \rangle$ will
uniquely determine a Radon measure $\mu = \mu_{\omega_1}$ on $K$.
To uniquely determine each Borel (=Baire)
measure $\mu_\alpha$ it is sufficient to decide its values on the
clopen subsets of $K_\alpha$. We formulate a requirement to
encapsulate this discussion:
\begin{description}
\item{{\bf R5}.} For
$\alpha\le\omega_1$, $\mu_\alpha$ is a finitely additive probability
measure on the clopen subsets of $K_\alpha$, and
$\mu_\alpha = \mu_\beta (\piba )^{-1}$ whenever
$\omega\le\alpha \le \beta \le \omega_1$. For $\alpha \le \omega$,
$\mu_\alpha$ is the usual product measure on the clopen subsets of $K_\alpha=2^\alpha$.
\end{description}
Let $\widehat{\mu}_\alpha$ be the Borel measure on $K_\alpha$
generated by $\mu_\alpha$. It is easy to verify that {\bf R1}-{\bf
R5} imply that for $\alpha \le \omega$, $\widehat {\mu}_\alpha$ is
the usual product measure on $K_\alpha=2^\alpha$, and that for any
$\alpha$, $\widehat {\mu}_\alpha$ gives each non-empty clopen set
positive measure and measure 0 to each point in $K_\alpha$. We
shall abuse notation and use $\mu_\alpha$ for both
$\widehat{\mu}_\alpha$ and its restriction to the clopen sets.
Note that by the usual Cantor tree argument these properties
assure that in every set of positive measure there is an
uncountable set of measure 0; this observation will be useful
later on.
The following requirements will help us both to obtain HS+HL and
to assure that $K$ is not Rosenthal compact. To formulate these
requirements we use $CH$ to enumerate the set of pairs $\{(\gamma,
J):\,\gamma<\omega_1\,,\, J \subseteq 2^\gamma \mbox{ is Borel}
\}$ as $\{(\delta_\alpha, J_\alpha):\,\omega\le \alpha
<\omega_1\}$ so that $\delta_\alpha \le\alpha$ for all $\alpha$
and each pair appears unboundedly often.
Suppose that $\omega \le \alpha < \omega_1$ and $K_\alpha$ and $\mu_\alpha$ are defined.
We define the following subsets of $K_\alpha$:
{\parindent= 40pt
$C_\alpha = (\pi^\alpha_{\delta_\alpha} )^{-1} (J_\alpha )$, if $J_\alpha\subseteq
K_{\delta_\alpha}$; $C_\alpha = \emptyset$ otherwise.
$L_\alpha = C_\alpha$ if $C_\alpha$ is closed;
$L_\alpha = K_\alpha$ otherwise.
$Q_\alpha = L_\alpha \setminus\bigcup\{ O:
O \hbox{\ is open and \ } \mu_\alpha (L_\alpha \cap O) = 0\}$
$N_\alpha =(L_\alpha\setminus Q_\alpha)\cup C_\alpha$, if
$\mu_\alpha(C_\alpha ) =0$;
$N_\alpha =(L_\alpha\setminus Q_\alpha)$ otherwise.
\medskip
}
Let us note that $L_\alpha$ is a closed subset of $K_\alpha$ and
that $Q_\alpha\subseteq L_\alpha$ is also closed and satisfies
$\mu_\alpha(Q_\alpha)=\mu_\alpha(L_\alpha)$, and hence
$\mu_\alpha(N_\alpha)=0$. Also observe that $Q_\alpha$ has no
isolated points, as points have $\mu_\alpha$ measure 0.
We now recall from \cite{DzK} what is meant by $A$ and $B$ being
{\em complementary regular closed subsets} of a space $X$: this
means that $A$ and $B$ are both regular closed with $A\cup B=X$,
while $A\cap B$ is nowhere dense in $X$. Finally, we state the
following requirements:
\begin{description}
\item{\bf R6}.\label{4.1}
For any $\beta\geq\alpha\geq\omega$, $s_\beta\notin
(\piba)^{-1}( N_\alpha)$;
\item{\bf R7}.\label{2.3} For any $\beta\geq\alpha\geq\omega$, $A_\beta\cap (\piba)^{-1}(Q_\alpha)$ and
$B_\beta\cap (\piba)^{-1} (Q_\alpha)$ are complementary regular closed
subsets of $(\piba )^{-1} (Q_\alpha)$.
\end{description}
The following claim and lemma explain our use of irreducible maps,
and the use of measure as a tool to achieve the HS+HL properties
of the space. The proof is basically the same as in \cite{DzK} but
we give it here since it explains the main point and also to show
how our situation actually simplifies the proof from \cite{DzK}.
For any $\alpha$, we use the notation $[s]$ for a finite partial
function $s$ from $\alpha$ to 2 to denote the basic clopen set
$\{f\in 2^{\alpha}:\,s\subseteq f\}$, or its relativization to a
subspace of $2^\alpha$, as it is clear from the
context.~\footnote{The notation also does not specify $\alpha$ but
again following the tradition, we shall rely on $\alpha$ being
clear from the context.}
\begin{Claim}\label{induction} Assume the requirements {\bf R1}-{\bf R5} and {\bf R7}. Then
for each $\beta\in [\alpha,\omega_1]$ the projection $\piba$ is irreducible on
$(\piba )^{-1} (Q_\alpha )$.
\end{Claim}
\begin{Proof of the Claim}
We use induction on $\beta\ge\alpha$.
The step $\beta=\alpha$ is clear. Assume that we know that the projection $\piba$ is irreducible on
$(\piba )^{-1} (Q_\alpha )$ and let us prove that $\pi^{\beta+1}_\alpha$ is irreducible on
$(\pi^{\beta+1}_\alpha )^{-1} (Q_\alpha )$. Suppose that $F$ is a proper closed subset of $(\pi^{\beta+1}_\alpha )^{-1}
(Q_\alpha )$ satisfying $\pi^{\beta+1}_\alpha(F)=Q_\alpha$. Then by the inductive assumption
$\pi^{\beta+1}_\beta(F)=(\piba )^{-1} (Q_\alpha )$. Let $x\in (\pi^{\beta+1}_\alpha )^{-1}
(Q_\alpha )\setminus F$, so we must have that $x\rest\beta=s_\beta$. Assume $x(\beta)=0$, the case
$x(\beta)=1$ is symmetric. Because $F$ is closed, we can find a basic clopen set $[t]$ in $K_{\beta+1}$
containing $x$ such that $[t]\cap F=\emptyset$. Let $s=t\rest\beta$.
Therefore $s_\beta\in [s]$ holds in $K_\beta$, and by {\bf R7} we can find $y\in {\rm int}(A_\beta
\cap (\piba )^{-1} (Q_\alpha ))\cap [s]$. Using the inductive assumption we conclude
$y\in {\rm int}(A_\beta\cap (\pi^{\beta+1}_\beta ) (F ))\cap [s]$, so there is a basic clopen
set $[v]\subseteq [s]$ in $K_\beta$ such that $y\in [v]$ and $[v]\subseteq A_\beta\cap (\pi^{\beta+1}_\beta ) (F )$.
But then $[v]$ viewed as a clopen set in $K_{\beta+1}$ satisfies $[v]\subseteq [t]$ and yet
$[v]\cap F\neq \emptyset$.
The limit case of the induction is easy by the definition of inverse limits.
$\eop_{\ref{induction}}$
\end{Proof of the Claim}
\begin{Lemma}\label{repeat} Assume the requirements {\bf R1}-{\bf R7}
and let $H$ be a closed subset of $K$. Then
there is an $\alpha <\omega_1$ such that
$\pi^{\omega_1}_\alpha$ is irreducible on $(\pi^{\omega_1}_\alpha )^{-1} (\pi^{\omega_1}_\alpha (H))$.
\end{Lemma}
\begin{Proof of the Lemma}
For each $\gamma < \omega_1$, let $H_\gamma = \pi^{\omega_1}_\gamma (H)$.
Then the $\mu_\gamma (H_\gamma )$ form a non-increasing sequence
of real numbers, so we may fix a $\gamma <\omega_1$
such that for all $\alpha\geq\gamma$, $\mu_\alpha(H_\alpha)=
\mu_\gamma(H_\gamma)$.
Next fix an $\alpha \geq\gamma$ such that $\delta_\alpha = \gamma$ and
$J_\alpha = H_\gamma$.
Then $L_\alpha = C_\alpha = (\pi^\alpha_\gamma)^{-1} (H_\gamma)$.
Hence $H_\alpha$ is a closed
subset of $L_\alpha$ with the same measure as $L_\alpha$,
so $Q_\alpha \subseteq H_\alpha \subseteq L_\alpha$, by the definition of $Q_\alpha$. Recall that by Claim \ref{induction}
we have that $\pi^{\omega_1}_\alpha$ is irreducible on $(\pi^{\omega_1}_\alpha)^{-1}(Q_\alpha)$.
Now we claim that
$\pi^{\omega_1}_\alpha$ is 1-1 on $(\pi^{\omega_1}_\alpha )^{-1}(H_\alpha\setminus Q_\alpha)$. Otherwise, there would be $x\neq y\in
(\pi^{\omega_1}_\alpha )^{-1}(H_\alpha\setminus Q_\alpha)$ with $x\rest\alpha=y\rest\alpha$. Therefore for some
$\beta\ge\alpha$ we have $x\rest\beta=y\rest \beta=s_\beta$, as otherwise $x=(\pi^{\omega_1}_\alpha )^{-1}(\{x\rest\alpha\})$. In
particular $s_\beta\in (\piba)^{-1}(H_\alpha)\subseteq (\piba)^{-1}(L_\alpha)$. On the other hand, if
$s_\beta\in (\piba)^{-1}(Q_\alpha)$ then $\{x,y\}\in (\pi^{\omega_1}_\alpha )^{-1}(Q_\alpha)$- a contradiction- so
$s_\beta\notin (\piba)^{-1}(Q_\alpha)$.
This means
$s_\beta\in (\piba)^{-1}(N_\alpha)$, in contradiction with {\bf R6}.
Thus, $\pi^{\omega_1}_\alpha$ must be irreducible on $(\pi^{\omega_1}_\alpha )^{-1} (H_\alpha )$ as well, and the Lemma is proved.
$\eop_{\ref{repeat}}$
\end{Proof of the Lemma}
Now we comment on how to assure that $K$ is not Rosenthal compact.
A remarkable theorem of Todor\v cevi\'c from \cite{StevoRos}
states that every non-metric Rosenthal compactum contains either
an uncountable discrete subspace or a homeomorphic copy of the
split interval. As our $K$, being HS+HL, cannot have an
uncountable discrete subspace, it will suffice to show that it
does not contain a homeomorphic copy of the split interval.
\begin{Claim}\label{Rosenthal} Suppose that the requirements {\bf R1}-{\bf R7} are met. Then
\begin{description}
\item{(1)} all $\mu$-measure 0 sets in $K$ are second countable and
\item{(2)} $K$ does not contain a homeomorphic copy of the split interval.
\end{description}
\end{Claim}
\begin{Proof of the Claim}
(1) Suppose that $M$ is a $\mu$-measure 0 Borel set in $K$ and let
$N=\pi_\omega^{\omega_1}(M)$, hence $N$ is of measure 0 in
$2^\omega$. Let $\alpha\in[\omega,\omega_1)$ be such that
$\delta_\alpha= \omega$ and $J_\alpha=N$. Then
$C_\alpha=(\pi_\omega^\alpha)^{-1}(N)$ and hence
$\mu_\alpha(C_\alpha)=0$ and so $C_\alpha\subseteq N_\alpha$.
Requirement {\bf R6} implies that for $\beta\ge\alpha$,
$(\pi_\beta^{\omega_1})^{-1}(s_\beta)\cap M= \emptyset$, so the
topology on $M$ is generated by the basic clopen sets of the form
$[s]$ for ${\rm dom}(s)\subseteq\alpha$. So $M$ is 2nd countable.
(2) Suppose that $H\subseteq K$ is homeomorphic to the split interval. Therefore $H$ is compact
and therefore closed in $K$. In particular $\mu(H)$ is defined.
If $\mu(H)=0$ then by (1), $H$ is 2nd countable, a contradiction.
If $\mu(H)>0$ then there is an uncountable set $N\subseteq H$ with
$\mu(N)=0$. Then $N$ is uncountable and 2nd countable,
contradicting the fact that all 2nd countable subspaces of the
split interval are countable. $\eop_{\ref{Rosenthal}}$
\end{Proof of the Claim}
Now we comment on how we assure that any uncountable nice semi-biorthogonal
system in $C(K)$ is $\omega$-determined, i.e. any uncountable semi-bidiscrete sequence in $K$
forms an $\omega$-determined family of pairs of points. For this we make one further
requirement:
\begin{description}
\item{\bf R8}. If $\alpha\,,\beta\in[\omega,\omega_1)$ with $\alpha < \beta$ then $s_\beta\rest\alpha\neq s_\alpha$.
\end{description}
\begin{claim}
\label{nosupernice} Requirements {\bf R1}-{\bf R8} guarantee that
any uncountable semi-bidiscrete sequence in $K$ is
$\omega$-determined.\end{claim}
\begin{Proof of the Claim}
Suppose that $\langle
(x^0_\alpha,x^1_\alpha):\,\alpha<\omega_1\rangle$ forms an
uncountable semi-bidiscrete sequence in $K$ that is not
$\omega$-determined. By the definition of a semi-bidiscrete
sequence, the $ (x^0_\alpha,x^1_\alpha)$'s are distinct pairs of
distinct points. Therefore there must be $s\in 2^\omega$ such that
$A=\{\alpha:\,x^0_\alpha\rest\omega=x^1_\alpha\rest\omega=s\}$ is
uncountable. We have at least one $l < 2$ such that
$\{x^l_\alpha:\,\alpha\in A\}$ is uncountable, so assume, without
loss of generality, that this is true for $l=0$.
Let $\alpha, \beta,\gamma$ be three distinct members of $A$. Then
by Observation \ref{delta} we have $$x^0_\alpha\rest
\Delta(x^0_\alpha, x^0_\beta)=x^0_\beta\rest \Delta(x^0_\alpha,
x^0_\beta) =s_{ \Delta(x^0_\alpha, x^0_\beta)}$$ and similarly
$$x^0_\alpha\rest \Delta(x^0_\alpha, x^0_\gamma)=x^0_\gamma\rest
\Delta(x^0_\alpha, x^0_\gamma)=s_{ \Delta(x^0_\alpha,
x^0_\gamma)}.$$ By {\bf R8} we conclude that $\Delta(x^0_\alpha,
x^0_\beta)$ is the same for all $\beta \in A\setminus\{\alpha\}$
and we denote this common value by $\Delta_\alpha$. Thus for
$\beta \in A\setminus\{\alpha\}$ we have $x^0_\beta\rest
\Delta_\alpha=s_{\Delta_\alpha}$, but applying the same reasoning
to $\beta$ we obtain $x^0_\alpha\rest
\Delta_\beta=s_{\Delta_\beta}$ and hence by {\bf R8} again we have
$\Delta_\alpha=\Delta_\beta$. Let $\delta^\ast$ denote the common
value of $\Delta_\alpha$ for $\alpha \in A$.
Again, taking distinct $\alpha,\beta,\gamma\in A$ we have
$x^0_\alpha\rest\delta^\ast=
x^0_\beta\rest\delta^\ast=x^0_\gamma\rest\delta^\ast$ and that
$x^0_\alpha(\delta^\ast), x^0_\beta(\delta^\ast)$ and
$x^0_\gamma(\delta^\ast)$ are pairwise distinct. This is, however,
impossible as the latter have values in $\{0,1\}$.
$\eop_{\ref{nosupernice}}$
\end{Proof of the Claim}
Finally we show that the space $K$ is a 2-to-1 continuous preimage
of a compact metric space. We simply define
$\varphi:\,K\into2^\omega$ as $\varphi(x)=x\rest\omega$. This is
clearly continuous. To show that it is 2-to-1 we first prove the
following:
\begin{Claim}\label{ultra} In the space $K$ above, for any $\alpha\neq\beta$ we have
$s_\alpha\rest\omega\neq s_\beta\rest\omega$.
\end{Claim}
\begin{Proof of the Claim} Otherwise suppose that $\alpha<\beta$ and yet
$s_\alpha\rest\omega= s_\beta\rest\omega$. By {\bf R8} we have
$s_\alpha\nsubseteq s_\beta$, so
$\omega\le\delta=\Delta(s_\alpha,s_\beta)<\beta$. By Observation
\ref{delta} applied to any $x\supseteq s_\alpha$ and $y\supseteq
s_\beta$ from $K$, we have
$s_\alpha\rest\delta=x\rest\delta=y\rest\delta=s_\beta\rest\delta=s_\delta$.
But this would imply $s_\delta\subseteq s_\beta$, contradicting
{\bf R8}. $\eop_{\ref{ultra}}$
\end{Proof of the Claim}
Now suppose that $\varphi$ is not 2-to-1, that is there are three
elements $x,y,z \in K$ such that $x\rest\omega=y\rest\omega=
z\rest\omega$. Let $\alpha=\delta(x,y)$ and $\beta=\delta(x,z)$,
so $\alpha, \beta \ge\omega$. By Observation \ref{delta} we have
$x\rest\alpha=y\rest\alpha=s_\alpha$,
$x\rest\beta=z\rest\beta=s_\beta$, so by requirement {\bf R8} we
conclude $\alpha=\beta$. Note that then $y(\alpha)=z(\alpha)$ and
so $\delta=\Delta(y,z)>\alpha$ and $y\rest\delta=
s_\delta\supseteq s_\beta$, in contradiction with {\bf R8}.
Therefore $\varphi$ is really 2-to-1.
\subsubsection{Meeting the requirements}
Now we show how to meet all these requirements. It suffices to
show what to do at any successor stage $\alpha+1$ for $\alpha\in
[\omega,\omega_1)$, assuming all the requirements have been met at
previous stages.
First we choose $s_\alpha$. By {\bf R5} for any $\gamma<\alpha$ we
have $\mu_\gamma(\{s_\gamma\})=0$ and
$\mu_\alpha((\pi_\gamma^\alpha)^{-1}(s_\gamma))=0$. Hence the set
of points $s\in K_\alpha$ for which $s\rest \gamma=s_\gamma$ for
some $\gamma<\alpha$ has measure 0, so we simply choose $s_\alpha$
outside of
$\bigcup_{\gamma<\alpha}(\pi^\alpha_\gamma)^{-1}(s_\gamma)$ , as
well as outside of
$\bigcup_{\gamma<\alpha}(\pi^\alpha_\gamma)^{-1}(N_\gamma)$ (to
meet {\bf R6}), which is possible as the $\mu_\alpha$ measure of
the latter set is also 0.
Now we shall use an idea from \cite{DzK}. We fix a strictly
decreasing sequence $\langle V_n:\,n\in\omega\rangle$ of clopen
sets in $K_\alpha$ such that $V_0=K_\alpha$ and
$\bigcap_{n<\omega}V_n=\{s_\alpha\}$. We shall choose a function
$f:\,\omega\to\omega$ such that letting
$$A_\alpha=\bigcup_{n<\omega} (V_{f(2n)}\setminus
V_{f(2n+1)})\cup\{s_\alpha\}$$ and $$B_\alpha=\bigcup_{n<\omega}
(V_{f(2n+1)}\setminus V_{f(2n)})\cup\{s_\alpha\}$$ will meet all
the requirements. Once we have chosen $A_\alpha$ and $B_\alpha$,
we let $$K_{\alpha+1}=A_\alpha\times \{0\}\cup B_\alpha\times
\{1\}.$$ For a basic clopen set $[s]=\{g\in
K_{\alpha+1}:\,g\supseteq s\}$, where $s$ is a finite partial
function from $\alpha+1$ to 2 and $\alpha\in{\rm dom}(s)$, we let
$\mu_{\alpha+1}([s])=1/2\cdot \mu_\alpha([s\rest\alpha])$. We
prove below that this extends uniquely to a Baire measure on
$K_{\alpha+1}$.
The following is basically the same (in fact simpler) argument
which appears in \cite{DzK}. We state and prove it here for the
convenience of the reader.
\begin{Claim}\label{fastfunction} The above choices of $A_\alpha$, $B_\alpha$, and
$\mu_{\alpha+1}$, with the choice of any function $f$ which is
increasing fast enough, will satisfy all the requirements {\bf
R1}-{\bf R8}.
\end{Claim}
\begin{Proof of the Claim} Requirements {\bf R1}-{\bf R4} are clearly met with any choice of $f$.
To see that {\bf R5} is met, let us first prove that
$\mu_{\alpha+1}$ as defined above indeed extends uniquely to a
Baire measure on $K_{\alpha+1}$. We have already defined
$\mu_{\alpha+1}([s])$ for $s$ satisfying $\alpha\in{\rm dom}(s)$. If
$\alpha\notin{\rm dom}(s)$ then we let
$\mu_{\alpha+1}([s])=\mu_{\alpha}(\pi^{\alpha+1}_\alpha [s])$. It
is easily seen that this is a finitely additive measure on the
basic clopen sets, which then extends uniquely to a Baire measure
on $K_{\alpha+1}$. It is also clear that this extension satisfies
{\bf R5}.
Requirements {\bf R6} and {\bf R8} are met by the choice of
$s_\alpha$, so it remains to see that we can meet {\bf R7}. For
each $\gamma\in[\omega,\alpha]$, if $s_\alpha\in
(\pi_\gamma^\alpha)^{-1}(Q_\gamma)$, fix an $\omega$-sequence
$\bar{t}_\gamma$ of distinct points in
$(\pi_\gamma^\alpha)^{-1}(Q_\gamma)$ converging to $s_\alpha$.
Suppose that $\bar{t}_\gamma$ is defined and that both
$A_\alpha\setminus B_\alpha$ and $B_\alpha\setminus A_\alpha$
contain infinitely many points from $\bar{t}_\gamma$. Then we
claim that $A_\alpha\cap (\pi_\gamma^\alpha)^{-1}(Q_\gamma)$ and
$B_\alpha\cap (\pi_\gamma^\alpha)^{-1} (Q_\gamma)$ are
complementary regular closed subsets of $(\pi_\gamma^\alpha )^{-1}
(Q_\gamma)$. Note that we have already observed that $Q_\gamma$
does not have isolated points, so neither does
$(\pi_\gamma^\alpha)^{-1} (Q_\gamma)$. Hence, since
$\{s_\alpha\}\supseteq A_\alpha\cap
(\pi_\gamma^\alpha)^{-1}(Q_\gamma)\cap B_\alpha\cap
(\pi_\gamma^\alpha)^{-1} (Q_\gamma)$, we may conclude that this
intersection is nowhere dense in both $A_\alpha\cap
(\pi_\gamma^\alpha)^{-1}(Q_\gamma)$ and $B_\alpha\cap
(\pi_\gamma^\alpha)^{-1} (Q_\gamma)$. Finally, $A_\alpha\cap
(\pi_\gamma^\alpha)^{-1}(Q_\gamma)$ and $B_\alpha\cap
(\pi_\gamma^\alpha)^{-1} (Q_\gamma)$ are regular closed because we
have assured that $s_\alpha$ is in the closure of both.
Therefore we need to choose $f$ so that for every relevant
$\gamma$, both $A_\alpha\setminus B_\alpha$ and $B_\alpha\setminus
A_\alpha$ contain infinitely many points of $\bar{t}_\gamma$.
Enumerate all the relevant sequences $\bar{t}_\gamma$ as
$\{\bar{z}^k\}_{k<\omega}$. Our aim will be achieved by choosing
$f$ in such a way that, for every $n$, both sets
$V_{f(2n)}\setminus V_{f(2n+1)}$ and $V_{f(2n+1)}\setminus
V_{f(2n+2)}$ contain a point of each $\bar{z}^k$ for $k\le n$.
$\eop_{\ref{fastfunction}}$
\end{Proof of the Claim}
This finishes the proof of the theorem.
$\eop_{\ref{CH}}$
\end{proof}
\section{Bidiscrete systems}\label{bidiscr} The main result of this section is Theorem \ref{lemma2}
below. In the course of proving Theorem 10 in \S7 of
\cite{StevoMM}, Todor\v cevi\'c actually proved that if $K$ is not
hereditarily separable then it has an uncountable bidiscrete
system. Thus his proof yields Theorem \ref{lemma2} for
$d(K)=\aleph_1$ and the same argument can be easily extended to a
full proof of \ref{lemma2}.
Let us first state some general observations about bidiscrete systems.
\begin{observation}
\label{closedsubspace}
Suppose that $K$ is a compact Hausdorff space and $H\subseteq K$ is closed, while
$\{ (x_\alpha^0, x_\alpha^1):\,\alpha<\kappa\}$
is a bidiscrete system in $H$, as exemplified by functions $f_\alpha\,(\alpha<\kappa)$.
Then there are functions $g_\alpha\,(\alpha<\kappa)$ in $C(K)$
such that $f_\alpha\subseteq g_\alpha$ and $g_\alpha\,(\alpha<\kappa)$ exemplify that
$\{ (x_\alpha^0, x_\alpha^1):\,\alpha<\kappa\}$
is a bidiscrete system in $K$.
\end{observation}
\begin{Proof} Since $H$ is closed we can, by Tietze's Extension Theorem,
extend each $f_\alpha$ continuously to
a function $g_\alpha$ on $K$. The conclusion follows
from the definition of a bidiscrete system.
$\eop_{\ref{closedsubspace}}$
\end{Proof}
\begin{Claim}\label{generaldiscrete} Suppose that $K$ is a compact
space and $F_i\subseteq G_i \subseteq K\,$ for $i\in I$ are such
that the $G_i$'s are disjoint open, the $F_i$'s are closed and in
each $F_i$ we have a bidiscrete system $S_i$. Then $\bigcup_{i\in
I}S_i$ is a bidiscrete system in $K$.
\end{Claim}
\begin{proof} For $i\in I$ let the bidiscreteness of $S_i$ be witnessed
by $\{g^i_{\alpha}\,:\,\alpha<\kappa_i\} \subseteq C(F_i)$. We can,
as in Observation \ref{closedsubspace}, extend each $g^i_{\alpha}$
to $h^i_{\alpha} \in C(K)$ which exemplify that $S_i$ is a
bidiscrete system in $K$.
Now we would like to put all these bidscrete systems together, for
which we need to find appropriate witnessing functions. For any
$i\in I$ we can apply Urysohn's Lemma to find functions $f_i \in
C(K)$ such that $f_i$ is 1 on $F_i$ and 0 on the complement of
$G_i$. Let us then put, for any $\alpha$ and $i$,
$f_{\alpha}^i=g^i_{\alpha}\cdot f_i$. Now, it is easy to verify
that the functions $\{f^i_{\alpha}:\,\alpha<\kappa_i, i\in I\}$
witness that $\bigcup_{i\in I}S_i$ is a bidiscrete system in $K$.
$\eop_{\ref{generaldiscrete}}$
\end{proof}
Clearly, Observation \ref{closedsubspace} is the special case of
Claim \ref{generaldiscrete} when $I$ is a singleton and $G_i = K$.
\begin{Claim}\label{splitintervalsplit}
If the compact space $K$ has a discrete subspace of size $\kappa
\ge \omega$ then it has a bidiscrete system of size $\kappa$, as
well.
\end{Claim}
\begin{Proof}
Suppose that $D = \{x_\alpha : \alpha < \kappa\}$ (enumerated in a
one-to-one manner) is discrete in $K$ with open sets $U_\alpha$
witnessing this, i.e. $D \cap U_\alpha = \{x_\alpha\}$ for all
$\alpha < \kappa$. For any $\alpha < \kappa$ we may fix a function
$f_\alpha \in C(K)$ such that $f_\alpha(x_{2\alpha+1}) = 1$ and
$f_\alpha(x) = 0$ for all $x \notin U_{2\alpha+1}$. Obviously,
then $\{f_\alpha : \alpha < \kappa\}$ exemplifies that
$\{(x_{2\alpha},x_{2\alpha+1}) : \alpha < \kappa \}$ is a
bidiscrete system in $K$.
\end{Proof}
The converse of Claim \ref{splitintervalsplit} is false, however
the following is true.
\begin{claim}
\label{discbidisc} Suppose that $B = \{ (x^0_\alpha,
x^1_\alpha):\,\alpha<\kappa\}$ is a bidiscrete system in $K$. Then
$B$ is a discrete subspace of $K^2$.
\end{claim}
\begin{Proof} Assume that the functions $\{f_\alpha:\,\alpha<\kappa\}
\subseteq C(K)$ exemplify the bidiscreteness of $B$. Then
$O_\alpha=f_\alpha^{-1}((-\infty,1/2))\times
f_\alpha^{-1}((1/2,\infty))$ is an open set in $K^2$ containing
$(x^0_\alpha, x^1_\alpha)$. Also, if $\beta\neq\alpha$ then
$(x^0_\beta, x^1_\beta) \notin O_\alpha$, hence $B$ is a discrete
subspace of $K^2$. $\eop_{\ref{discbidisc}}$
\end{Proof}
Now we turn to formulating and proving the main result of this
section.
\begin{Theorem}\label{lemma2} If $K$ is an infinite compact Hausdorff space
then $K$ has a bidiscrete system of size $d(K)$. If $K$ is moreover 0-dimensional
then there is a very nice bidiscrete system in $K$ of size $d(K)$.
\end{Theorem}
\begin{Proof} The proofs of the two parts of the theorem are the same, except that
in the case of a 0-dimensional space every time
that we take functions witnessing bidiscreteness, we need to observe that these functions
can be assumed to take values only in $\{0,1\}$. We leave it to the reader to check that this is indeed
the case.
The case $d(K)=\aleph_0$ is very easy, as it is well known that
every infinite Hausdorff space has an infinite discrete subspace
and so we can apply Claim \ref{splitintervalsplit}. So, from now
on we assume that $d(K)>\aleph_0$.
Recall that a Hausdorff space $(Y,\sigma)$ is said to be {\em
minimal Hausdorff} provided that there does not exist another
Hausdorff topology $\rho$ on $Y$ such that $\rho\subsetneq
\sigma$, i.e. $\rho$ is strictly coarser than $\sigma$. The
following fact is well known and easy to prove, and it will
provide a key part of our argument:
\begin{Fact}\label{coarse} Any compact Hausdorff space is minimal Hausdorff.
\end{Fact}
\begin{Lemma}\label{lemma1} Suppose that $X$ is a compact Hausdorff space with $d(X)\ge\kappa>\aleph_0$
in which every non-empty open (equivalently: regular closed)
subspace has weight $\ge\kappa$.
Then $X$ has a bidiscrete system of size $\kappa$.
\end{Lemma}
\begin{Proof of the Lemma} We shall choose $x_\alpha^0, x_\alpha^1, f_\alpha$ by induction on $\alpha<\kappa$
so that the pairs $(x_\alpha^0, x_\alpha^1)$ form a bidiscrete
system, as exemplified by the functions $f_\alpha$. Suppose that
$x_\beta^0, x_\beta^1, f_\beta$ have been chosen for
$\beta<\alpha<\kappa$.
Let $C_\alpha$ be the closure of the set $\{x_\beta^0,
x_\beta^1:\,\beta<\alpha\}$. Therefore $d(C_\alpha)<\kappa$ and,
in particular, $C_\alpha\neq X$. Let $F_\alpha\subseteq X\setminus
C_\alpha$ be non-empty regular closed, hence
$w(F_\alpha)\ge\kappa$.
Let $\tau_\alpha$ be the topology on $F_\alpha$ generated by the
family
\[
\mathcal{F}_\alpha = \{f_\beta^{-1}(-\infty,q)\cap
F_\alpha\,,\,f_\beta^{-1}(q,\infty)\cap F_\alpha\,
:\,\beta<\alpha,\, q\in {\mathbf Q}\},
\]
where $\mathbf Q$ denotes the set of rational numbers. Then
$|\mathcal{F}_\alpha|<\kappa$ (as $\kappa>\aleph_0$), hence the
weight of $\tau_\alpha$ is less than $\kappa$, consequently
$\tau_\alpha$ is strictly coarser than the subspace topology on
$F_\alpha$. Fact \ref{coarse} implies that $\tau_\alpha$ is not a
Hausdorff topology on $F_\alpha$, hence we can find two distinct
points $x^0_\alpha, x^1_\alpha\in F_\alpha$ which are not
$T_2$-separated by any two disjoint sets in $\tau_\alpha$ and, in
particular, in $\mathcal{F}_\alpha$. This clearly implies that
$f_\beta(x^0_\alpha)=f_\beta(x^1_\alpha) $ for all $\beta<\alpha$.
Now we use the complete regularity of $X$ to find $f_\alpha\in
C(X)$ such that $f_\alpha$ is identically 0 on the closed set
$C_\alpha\cup\{x^0_\alpha\}$ and $f_\alpha(x^1_\alpha)=1$. It is
straight-forward to check that $\{f_\alpha : \alpha < \kappa\}$
indeed witnesses the bidiscreteness of $\{(x_\alpha^0, x_\alpha^1)
: \alpha < \kappa\}$. $\eop_{\ref{lemma1}}$
\end{Proof of the Lemma}
Let us now continue the proof of the theorem. We let $\kappa$
stand for $d(K)$ and let
\[
{\mathcal P}=\{\emptyset\neq O\subseteq K:\,O\mbox{ open such that }[\emptyset\neq U
\mbox{ open}\subseteq O\implies d(U)= d(O)]\}.
\]
We claim that ${\mathcal P}$ is a $\pi$-base for $K$, i.e. that
every non-empty open set includes an element of ${\mathcal P}$.
Indeed, suppose this is not case, as witnessed by a non-empty open
set $U_0$. Then $U_0\notin{\cal P}$, so there is a non-empty open set
$\emptyset\neq U_1\subseteq U_0$ with $d(U_1)<d(U_0)$ (the case
$d(U_1)<d(U_0)$ cannot occur). Then $U_1$ itself is not a member
of ${\cal P}$ and therefore we can find a non-empty open set
$\emptyset\neq U_2\subseteq U_1$ with $d(U_2)<d(U_1)$, etc. In
this way we would obtain an infinite decreasing sequence of
cardinals, a contradiction.
Let now $\mathcal O$ be a maximal disjoint family of members of
${\mathcal P}$. Since ${\mathcal P}$ is a $\pi$-base for $K$ the
union of $\mathcal{O}$ is clearly dense in $K$. This implies that
if we fix any dense subset $D_O$ of $O$ for all $O\in {\mathcal
O}$ then $\bigcup \{ D_O : O \in \mathcal{ O} \}$ is dense in $K$,
as well. This, in turn, implies that $\sum \{d(O) : O \in \mathcal
{O}\} \ge d(K) = \kappa$.
If $|\mathcal{O}| = \kappa$ then we can select a discrete subspace
of $K$ of size $\kappa$ by choosing a point in each $O\in \mathcal
O$, so the conclusion of our theorem follows by Corollary
\ref{splitintervalsplit}.
So now we may assume that $|\mathcal O|<\kappa$. In this case,
since $\kappa>\aleph_0$, letting ${\mathcal O}'=\{O\in
{\mathcal O}:\,d(O)>\aleph_0\}$, we still have $\sum \{d(O) : O \in
\mathcal {O'}\} \ge \kappa$. Next, for each $O\in{\mathcal O}'$ we choose a
non-empty open set $G_O$ such that its closure
$\overline{G}_O\subseteq O$. Then we have, by the definition of
${\cal P}$, that $d(\overline{G}_O)=d(G_O)=d(O)$. By the same token,
every non-empty open subspace of the compact space
$\overline{G}_O$ has density $d(O)$, and hence weight $\ge d(O)$.
Therefore we may apply Lemma \ref{lemma1} to produce a bidiscrete
system $S_O$ of size $d(O)$ in $\overline{G}_O$. But then Claim
\ref{generaldiscrete} enables us to put these systems together to
obtain the bidscrete system $S = \bigcup \{S_O : O \in
\mathcal{O}'\}$ in $K$ of size $\sum \{d(O) : O \in \mathcal
{O'}\} \ge\kappa$. $\eop_{\ref{lemma2}}$
\end{Proof}
It is immediate from Theorem \ref{lemma2} and Observation
\ref{closedsubspace} that if $C$ is a closed subspace of the
compactum $K$ with $d(C) = \kappa$ then $K$ has a bidiscrete
system of size $\kappa$. We recall that the hereditary density
${\rm hd}(X)$ of a space $X$ is defined as the supremum of the
densities of all subspaces of $X$.
\begin{Fact}\label{hd} For any compact Hausdorff space $K$,
${\rm hd}(K)=\sup\{d(C):\,C\mbox{ closed}\subseteq K\}$.
\end{Fact}
From this fact and what we said above we immediately obtain the
following corollary of Theorem \ref{lemma2}.
\begin{Corollary}\label{theorem2} If $K$ is a compact Hausdorff space with
${\rm hd}(K)\ge\lambda^+$ for some $\lambda\ge\omega$, then $K$
has a bidiscrete system of size $\lambda^+$.
\end{Corollary}
\bigskip
We finish by listing some open questions.
\begin{Question} (1) Does every compact space $K$ admit a
bidiscrete system of size ${\rm hd}(K)$?
{\noindent (2)} Define
\[
{\rm bd}(K)=\sup\{|S|\,:\, S \mbox{ is a bidiscrete system in
}K\}.
\]
Is there always a bidiscrete system in $K$ of size ${\rm bd}(K)$?
{\noindent (3)} Suppose that $K$ is a 0-dimensional compact space
which has a bidiscrete system of size $\kappa$. Does then $K$ also
have a very nice bidiscrete system of size $\kappa$ (i.e. such
that the witnessing functions take values only in $\{0,1\}$)? Is
it true that any bidiscrete system in a 0-dimensional compact
space is very nice?
{\noindent (4)} (This is Problem 4 from \cite{JuSz}): Is there a
$ZFC$ example of a compact space $K$ that has no discrete subspace
of size $d(K)$?
{\noindent (5)} If the square $K^2$ of a compact space $K$
contains a discrete subspace of size $\kappa$, does then $K$ admit
a bidiscrete system of size $\kappa$ (or does at least $C(K)$ have
a biorthogonal system of size $\kappa$)? This question is of
especial interest for $\kappa = \omega_1$.
\end{Question}
| 2024-02-18T23:40:06.727Z | 2009-11-03T08:57:06.000Z | algebraic_stack_train_0000 | 1,370 | 8,878 |
|
proofpile-arXiv_065-6756 | \section{Introduction}
The standard model (SM) of particle physics successfully
explains almost of all the experimental results
around the electroweak scale.
Nevertheless, the SM suffers from several problems and
this fact strongly motivates us to explore physics beyond the SM.
One of them is the so-called hierarchy problem
originating from the ultraviolet sensitivity
of the SM Higgs doublet mass, and another one
is the absence of candidates for the dark matter particle.
In this paper we propose an extra-dimensional scenario
which can provide a possible solution to these two problems.
Among many models proposed to solve the hierarchy problem,
we concentrate on the gauge-Higgs unification
scenario~\cite{Manton:1979kb,YH}.
In this scenario, the SM Higgs doublet field is identified
with an extra-dimensional component of the gauge field
in higher-dimensional gauge theories
where the extra spacial dimensions are compactified to
realize four-dimensional effective theory at low energies.
The higher-dimensional gauge symmetry protects the Higgs
doublet mass from ultraviolet divergences~\cite{YH,finiteness},
and hence the hierarchy problem can be solved.
In the context of the gauge-Higgs unification scenario,
many models have been considered
in both the flat~\cite{Csaki}-\cite{GGHU} and
the warped~\cite{RS} background geometries~\cite{GHUinRS}-\cite{pNG}.
However, the latter problem has not been investigated in this scenario,
except for a few literatures~\cite{DMinGHU,Carena:2009yt,hosotani},
and in this paper, we propose a dark matter candidate
which can be naturally incorporated
in the gauge-Higgs unification scenario.
In the next section, we show a simple way to introduce
a candidate for the dark matter particle
in general higher-dimensional models.
In a sharp contrast with the usual Kaluza-Klein (KK) dark matter
in the universal extradimension scenario~\cite{KKDM},
our procedure is independent of the background
space-time metric.
In section 3, we apply this to the gauge-Higgs unification
scenario and show that a dark matter candidate as a
weakly-interacting-massive-particle (WIMP) emerges.
For our explicit analysis, we consider a gauge-Higgs unification model
based on the gauge group SO(5)$\times$U(1)$_X$
in five-dimensional warped background metric
with the fifth dimension compactified on the $S^1/Z_2$ orbifold.
In section 4, we evaluate the relic abundance
of the dark matter particle and its detection rates
in the direct dark matter detection experiments.
Section 5 is devoted to summary.
\section{A new candidate for the dark matter}
\label{Sec:APDM}
A stable and electric charge neutral WIMP is
a suitable candidate for the dark matter.
In general, a certain symmetry (parity) is necessary
to ensure the stability of a dark matter particle.
Such a symmetry can be imposed by hand in some models
or it can be accidentally realized
such as the KK parity~\cite{KKDM}.
The KK parity is actually an interesting possibility
for introducing a dark matter candidate
in higher-dimensional models.
However, we need to elaborate a model in order to
realize the KK parity in general warped
background geometry~\cite{Agashe:2007jb}.
In a simple setup, the KK parity is explicitly broken
by a warped background metric and
the KK dark matter is no longer stable~\cite{OY}.
So, here is an interesting question:
Is it possible in extradimensional models to introduce
a stable particle independently of the background
space-time metric, without imposing any symmetries by hand?
In the following we address our positive answer to this question.
In fact, when we impose the anti-periodic (AP) boundary condition
on bulk fields, the lightest AP field turns out to be stable.
In models with the toroidal compactification, no matter what
further orbifoldings are, the Lagrangian ${\mathcal L}$
should be invariant under a discrete shift of the coordinate
of the compactified direction,
\begin{equation}
{\cal L}(x,y+2\pi R) = {\cal L}(x,y),
\end{equation}
where $x$ and $y$ denote the non-compact four dimensional coordinate
and the compact fifth-dimensional one with a radius $R$, respectively.
When we introduce some fields which have the AP boundary condition as
\begin{equation}
\Phi(x,y+2\pi R) = -\Phi(x,y),
\end{equation}
these fields never appear alone but always do in pairs in the Lagrangian,
since the Lagrangian must be periodic.
Thus, there exists an accidental $Z_2$ parity, under which
the AP (periodic) fields transform as odd (even) fields.
This concludes that the lightest AP field is stable\footnote{
Similarly to the KK parity,
Lagrangian on the boundaries must be restricted
to respect the $Z_2$ parity.
}
and can be a good candidate for the dark matter
if it is colorless and electric-charge neutral.
In this way, a dark matter candidate can be generally incorporated
as the lightest AP field in higher-dimensional models.
However, except for providing the dark matter candidate,
there may be no strong motivation for introducing such AP fields.
In fact, AP fields often plays a crucial role
in the gauge-Higgs unification scenario
to make a model phenomenologically viable,
and therefore a dark matter candidate is simultaneously
introduced in such a model.
\section{Gauge-Higgs Dark Matter}
\label{Sec:GHDM}
We show a model of the gauge-Higgs unification,
which naturally has a dark matter candidate.
The dark matter particle originates from an AP field which is introduced
in a model for a phenomenological reason as will be discussed below.
We know well that, it is difficult,
in simple gauge-Higgs unification models with the flat metric,
to give a realistic top quark mass and a Higgs boson mass
above the current experimental lower bound.
This difficulty originates from the fact that
effective Higgs potential in the gauge-Higgs unification model
results in the Wilson line phase of order one.
When we consider the gauge-Higgs unification scenario
in the warped metric of the extra dimension,
this problem can be solved because of the effect of
the warped metric, although the Wilson line phase
of order one is obtained from effective Higgs potential.
However, as is claimed in Ref.~\cite{EWPMinGHU},
a small Wilson line phase is again required
in order for the scenario to be consistent with
the electroweak precision measurements.
Therefore, it is an important issue
in the gauge-Higgs unification scenario
how to naturally obtain a small Wilson line phase.
A simple way is to introduce AP fermions in a model.
It has been shown in Ref.~\cite{APinGHU} that
a small Wilson line phase is actually obtained
by introducing AP fermions.
This is the motivation we mentioned above\footnote{
In Ref.~\cite{DMinGHU},
with a {\it similar} purpose, a {\it similar} $Z_2$ symmetry
is imposed {\it but}
by hand.}.
An AP fermion, once introduced, not only reduces
unwanted new particle effects to the precisely measured
SM parameters but also provides a dark matter candidate
as its lightest electric-charge neutral component.
We call the dark matter candidate in the AP fermion
``gauge-Higgs dark matter'' in this paper.
The interactions between the dark matter and the Higgs field is
largely controlled by the gauge symmetry,
since the Higgs field is a part of the gauge field
in the gauge-Higgs unification scenario.
This fact leads to a strong predictive power of the model
for the dark matter phenomenology.
\subsection{A model}
\label{Sec:model}
Here we explicitly examine a 5D gauge-Higgs unification model
with a dark matter particle.
The model is based on the gauge symmetry
SO(5)$\times$U(1)$_X$~\cite{Carena:2009yt,hosotani}
compactified on the simplest orbifold $S^1/Z_2$
with the warped metric~\cite{RS}
\begin{eqnarray}
{\rm d} s^2
=
G_{M N} {\rm d} x^M {\rm d} x^N
=
e^{-2\sigma(y)} \eta_{\mu \nu} {\rm d} x^\mu {\rm d} x^\nu
-
{\rm d} y^2,
\end{eqnarray}
where $M=0,1,2,3,5$, $\mu=0,1,2,3$,
$\sigma(y) = k |y|$ at $-\pi R \leq y \leq \pi R$,
$\sigma(y) = \sigma(y + 2 \pi R)$, and
$\eta_{\mu \nu} = {\rm diag}(1, -1, -1, -1)$ is
the 4D flat metric.
We define the warp factor $a= \exp(-\pi k R)$
and as a reference value, we set the curvature $k$ and
the radius $R$ to give the warp factor $a = 10^{-15}$.
The bulk SO(5) gauge symmetry is broken down to
SO(4)$\simeq$SU(2)$_L\times$SU(2)$_R$
by the boundary conditions~\cite{Kawamura}.
Concretely, the gauge field and its 5th component transform
around the two fixed points $y_0=0$ and $y_L=\pi R$ as
\begin{eqnarray}
A_\mu(x,\,y_i-y) &=& P_i A_\mu(x,\,y_i+y) P_i^\dagger, \\
A_5(x,\,y_i-y) &=& -P_i A_5(x,\,y_i+y) P_i^\dagger,
\end{eqnarray}
under the $Z_2$ parity,
where
$P_0=P_L={\rm diag.}(-1,-1,-1,-1,+1)$ for the five-by-five anti-symmetric
matrix representation of the generators acting on the vector
representation, $\bf5$.
As for the remaining SO(4)$\times$U(1)$_X$ gauge symmetry,
the SU(2)$_R\times$U(1)$_X$ is assumed to be broken down to the hypercharge
symmetry U(1)$_Y$
by a VEV of an elementary Higgs field\footnote{
Note that introducing the elementary Higgs field
at the $y=0$ orbifold fixed point has no contradiction
against the motivation of the gauge-Higgs unification scenario
since the mass of the Higgs fields and their VEVs are of
the order of the Planck scale.
In this case, they decouple from TeV scale physics.
}
put on the $y=0$ orbifold fixed point.
Now the remaining gauge symmetry is the same as the SM,
where there exists the zero-mode of $A_5$
which is identified as the SM Higgs doublet
(possessing the right quantum numbers).
When the zero mode of $A_5$ develops a non-trivial VEV,
the SO(4) symmetry is broken down to SO(3)$\simeq$SU(2)$_D$ which
is the diagonal part of SU(2)$_L\times$SU(2)$_R\simeq$SO(4).
Taking the boundary Higgs VEV into account,
the electromagnetic U(1)$_{\rm EM}$ is left with unbroken.
Thanks to the custodial symmetry which is violated
only at the $y=0$ fixed point, that is, a superheavy energy scale,
the correction to the $\rho$-parameter is naturally
suppressed~\cite{EWPMinGHU}.
This allows the KK scale as low as a few TeV
without any contradictions against current experiments.
The components of gauge field are explicitly written as
\begin{equation}
A_M=
\left(
\begin{array}{cccc|c}
0 & A_V^3 & -A_V^2 & A_A^1 & A_H^1 \\
& 0 & A_V^1 & A_A^2 & A_H^2 \\
& & 0 & A_A^3 & A_H^3 \\
& & & 0 & A_H^4 \\ \hline
& & & & 0
\end{array}
\right)_M,
\end{equation}
where
\begin{eqnarray}
A_{\scriptsize\begin{array}{c}V\vspace{-2mm}\\A\end{array}}^i =
\frac1{\sqrt2}(A_L^i\pm A_R^i),\qquad (i=1,\,2,\,3), \\
A_F^\pm = \frac1{\sqrt2}(A_F^1\mp i A_F^2),\qquad (F=V,A,H).
\end{eqnarray}
The zero-modes of $A_5$ exist on $A_H$ and its VEV
can be rotated into only $(A_H^4)_5$ component
by the SO(4) symmetry, by which
the Wilson line phase $\theta_W$ is defined as
\begin{eqnarray}
W \equiv e^{i \theta_W}
=
P \exp\left( {-i g \int^{\pi R}_{-\pi R} {\rm d} y ~G^{55} (A_H^4)_5} \right),
\label{WilsonLinePase}
\end{eqnarray}
where $P$ denotes the path ordered integral.
For vanishing $\theta_W$,
the SM gauge bosons are included
in $A_L$ and $B_X$ (which is
the gauge boson of the U(1)$_X$ symmetry),
while the $A_H$ component is mixed into
the mass eigenstates of weak bosons for non-vanishing $\theta_W$.
We do not specify the fermion sector of the model
but just assume it works well, since this sector
is not strongly limited by the gauge symmetry
and has a lot of model-dependent degrees of freedom.
Thus, in our following analysis we leave the Higgs boson mass $m_h$ and
the Wilson line phase $\theta_W$ as free parameters,
which should be calculated through the loop induced effective
potential~\cite{EffPot}-\cite{EffPotRS}
once the fermion sector of the model is completely fixed.
Let us now consider an AP fermion, $\psi$,
as a ${\bf5}_0$-multiplet under SO(5)$\times$U(1)$_X$,
in which the dark matter particle is contained.
A parity odd bulk mass parameter $c$ of this multiplet
is involved as an additional parameter~\cite{GherghettaPomarol}.
The wave function profile along the compactified direction
is written by the Bessel functions with the index
$\alpha=\left|\gamma_5 c+1/2\right|$~\cite{GherghettaPomarol}
and the localization of the bulk fermion is controlled
by the bulk mass parameter.
We choose the boundary conditions of this multiplet
so that the singlet component
of the SO(4) is lighter than the vector one for small $\theta_W$
with $ c > 0$.
After the electroweak symmetry breaking,
the forth and fifth components are mixed with each other
through the non-vanishing Wilson line phase in $(4,5)$ component,
while the first, second and third ones are not.
The combinations of forth and fifth components make up
two mass eigenstates:
The lighter one is nothing but the dark matter particle, $\psi_{\rm DM}$,
and we denote the heavier state as $\psi_S$.
The first, second and third components denoted
as $\psi_i$\,$(i=1,\,2,\,3)$ have nothing to do
with the electroweak symmetry breaking, and thus
degenerate up to small radiative corrections.
They are heavier than $\psi_S$.
Note that only dark matter particles themselves have no couplings
with the weak gauge bosons,
which are linear combinations of $A_V$, $A_A$ and $A_H$ (and $B_X$),
and the couplings between the dark matter particle and the weak gauge bosons
are always associated with the transition from/to the heavier partners,
$\psi_i$.
On the other hand,
both types of couplings exist among
the dark matter particle and the Higgs boson.
At the energy scale below the 1st KK mode mass,
the effective Lagrangian is expressed as
\begin{eqnarray}
{\cal L}^{\rm 4D}_{\rm DM}&=&
\sum_{i=1,2,3,S,{\rm DM}}
\bar\psi_i (i\partial\hspace{-2.3mm}/ -m_a)\psi_i
+y_{\rm DM} \bar\psi_{\rm DM} H \psi_{\rm DM}
\nonumber\\
&& +\bar\psi_S H
\left(y_S+y_P\gamma_5\right)
\psi_{\rm DM}
+\bar\psi_{\rm DM} H
\left(y_S-y_P\gamma_5\right)
\psi_S
\nonumber\\
&& +\sum_{i=1,2,3}\bar\psi_i
W_i\hspace{-4mm}/\,\,\,
\left(g^V_i+g^A_i\gamma_5\right)
\psi_{\rm DM}
+\bar\psi_{\rm DM}
W_i\hspace{-4mm}/\,\,
\left(g^V_i+g^A_i\gamma_5\right)
\psi_i,
\label{effL}
\end{eqnarray}
where we denote $Z$ as $W^3$, and set $g_1^h=g_2^h$ due to the remaining
U(1)$_{\rm EM}$ symmetry.
Once we fix the free parameters $\theta_W$ and $c$ (also the warp factor),
we can solve the bulk equations of motion for $A_M$ and $\psi$
(see for example Ref.~\cite{sakamura-unitariy})
and obtain the mass spectra of all the states
and effective couplings in Eq.(\ref{effL}) among AP fields,
the gauge bosons and the Higgs boson,
independently of the Higgs boson mass $m_h$
(which is another free parameter of the model as mentioned above).
Using calculated spectra and the effective couplings,
we investigate phenomenology of the gauge-Higgs dark matter
in the next section.
Since we have only three parameters
(or four if we count also the warp factor),
the model has a strong predictive power.
\subsection{Constraints}
\label{Sec:constraints}
Before investigating the gauge-Higgs dark matter phenomenology,
we examine an
experimental constraint on the Wilson line phase $\theta_W$~\footnote{
Constraints in the case with the flat metric is discussed
in Ref.~\cite{ConstraintInFlatGHU}.
}.
In Ref.~\cite{EWPMinGHU},
it is claimed that $\theta_W$ should be smaller than $0.3$
or the KK gauge boson mass larger than 3 TeV
in order to be consistent with the electroweak precision measurements.
Using the relation between $m_W$ and $m_{KK}\equiv\pi k a$
(see for example Ref.~\cite{sakamura-unitariy}),
\begin{equation}
m_W \simeq \frac{\theta_W}{\sqrt{\ln(a^{-1})}}\frac{m_{KK}}\pi,
\end{equation}
and the formula for the first KK gauge boson mass $m_1=0.78m_{KK}$,
the latter constraint is translated as $\theta_W\lesssim0.4$.
According to these bounds, we restrict our analysis
in the range of a small Wilson line phase,
namely, $\theta_W\leq\pi/10$.
We expect that AP fields not only provide the dark matter particle
but also is helpful to realize such small value of $\theta_W$.
\section{Phenomenology of gauge-Higgs dark matter}
Now we are in a position to investigate
the gauge-Higgs dark matter phenomenology.
We first estimate the relic abundance of the dark matter
and identify the allowed region of the model parameter space
which predicts the dark matter relic density consistent with
the current cosmological observations.
Furthermore, we calculate the cross section of the elastic
scattering between the dark matter particle and nucleon
to show implications of the gauge-Higgs dark matter scenario
for the current and future direct dark matter detection experiments.
\subsection{Relic abundance}
In the early universe, the gauge-Higgs dark matter is
in thermal equilibrium through the interactions
with the SM particles.
According to the expansion of the universe,
temperature of the universe goes down and the dark matter eventually
decouples from thermal plasma of the SM particles
in its non-relativistic regime.
The thermal relic abundance of the dark matter can be evaluated
by solving the Boltzmann equation,
\begin{eqnarray}
\frac{d Y}{dx} =
-\frac{s \langle \sigma v \rangle}{x H}
\left( 1- \frac{x}{3 } \frac{d \log g_{*s}}{d x } \right)
\left( Y^2 -Y_{EQ}^2 \right),
\end{eqnarray}
where $x=m_{\rm DM}/T$,
$\langle \sigma v \rangle$
is the thermal averaged product of
the dark matter annihilation cross section ($\sigma $)
and the relative velocity of annihilating dark matter particles ($v$),
$Y(\equiv n/s)$ is the yield defined as the ratio
of the dark matter number density $(n)$ to the entropy density
of the universe $(s)$, and the Hubble parameter $H$
is described as $H= \sqrt{(8 \pi/3) G_N \rho}$
with the Newton's gravitational constant
$G_N=6.708 \times 10^{39}$ GeV$^{-2}$
and the energy density of the universe ($\rho$).
The explicit formulas for the number density of the dark matter particle,
the energy density, and the entropy density are given,
in the Maxwell-Boltzmann approximation, by
\begin{equation}
n = \frac{g_{DM}}{2 \pi^2} \frac{K_2(x)}{x} m^3, \quad
\rho = \frac{\pi^2}{30} g_* T^4, \quad
s = \frac{2 \pi^2}{45} g_{*s} T^3,
\end{equation}
where $K_2$ is the modified Bessel function of the second kind,
$g_{DM}=4$ is the spin degrees of freedom
for the gauge-Higgs dark matter, and
$g_*$ ($g_{*s}$) is the effective massless degrees of freedom
in the energy (entropy) density, respectively.
In non-relativistic limit, the annihilation cross section
can be expanded with respect to a small relative velocity as
\begin{eqnarray}
\sigma v = \sigma_0 + \frac{1}{4} \sigma_1 v^2 +\cdots,
\end{eqnarray}
where $v \simeq 2 \sqrt{1-4 m^2/s}$ in the center-of-mass frame
of annihilating dark matter particles.
The first term corresponds to the dark matter annihilations
via $S$-wave, while the second is contributed by the $S$-
and $P$-wave processes.
In the Maxwell-Boltzmann approximation,
the thermal average of the annihilation cross section
is evaluated as
\begin{eqnarray}
\left< \sigma v \right>
&\equiv&\frac1{8x^4 K_2(x)^2}
\int_{4x^2}^\infty ds\,
\sqrt{s}(s-4x^2)K_1(\sqrt{s})\sigma_{ann} \\
&=& \sigma_0 +\frac32\sigma_1 x^{-1}+\cdots,
\label{left< v}
\end{eqnarray}
where a unit $T=1$ is used in the first line.
There are several dark matter annihilation modes
in both $S$-wave and $P$-wave processes
(see Eq.~(\ref{effL})),
such as
$ \bar{\psi}_{\rm DM} \psi_{\rm DM} \to W^+ W^-, ZZ, HH$
through $\psi_i$, $\psi_S$ and $\psi_{\rm DM}$ exchanges in the $t$-channel
and $ \bar{\psi}_{\rm DM} \psi_{\rm DM} \to f \bar{f}, W^+ W^-, ZZ, HH$
through the Higgs boson exchange in the $s$-channel,
where $f$ stands for quarks and leptons.
Once the model parameters, $\theta_W$, $c$ and $m_h$, are fixed,
magnitudes of $\sigma_0$ and $\sigma_1$ are calculated.
With a given annihilation cross section,
the Boltzmann equation can be numerically solved.
The relic density of the dark matter is obtained as
$\Omega_{\rm DM} h^2 = m_{\rm DM} s_0 Y(\infty)/(\rho_c/h^2)$
with $s_0=2889$ cm$^{-3}$ and
$\rho_c/h^2=1.054\times 10^{-5}$ GeV cm$^{-3}$.
Here, we use an approximate formula~\cite{DMabundanceApp}
for the solution of the Boltzmann equation:
\begin{equation}
\Omega_{\rm DM}h^2 =
8.766\times10^{-11}({\rm GeV}^{-2})
\left(\frac{T_0}{2.75 {\rm K}}\right)^3
\frac{x_f}{\sqrt{g_{*}(T_f)}}
\left(\frac12\sigma_0+\frac38\sigma_1 x_f^{-1}\right)^{-1},
\label{Omegah2App}
\end{equation}
where $x_f=m_{\rm DM}/T_f$ is the freeze-out temperature
normalized by the dark matter mass, and $T_0=2.725$ K is the present
temperature of the universe.
The freeze-out temperature is approximately determined by~\cite{DMabundanceApp}
\begin{eqnarray}
\sqrt{\frac\pi{45G_N}}\frac{45g_{\rm DM}}{8\pi^4}
\frac{\pi^{1/2}e^{-x_f}}{g_{\rm *s}(T_f)x_f^{1/2}
g_{\rm *}(T_f)
m_{\rm DM}
\left(\frac12\sigma_0+\frac34\sigma_1 x_f^{-1}\right)
\delta(\delta+2)=1.
\label{xfApp}
\end{eqnarray}
Here the parameter $\delta$ defines $T_f$ through a relation
between the yield $Y$ and its value in thermal equilibrium,
$Y - Y_{\rm EQ} = \delta Y_{\rm EQ}$,
whose value is chosen so as to keep this approximation good.
We set $\delta=1.5$ according to Ref.~\cite{DMabundanceApp}.
In these approximations, we include the factor $1/2$
due to the Dirac nature of the gauge-Higgs dark matter
(see the discussion below Eq.~(2.16) of Ref.~\cite{DMabundanceApp}).
Let us now compare the resultant dark matter relic density
for various $ \theta_W $ and $c$
with the observed value~\cite{DMabundance}:
\begin{equation}
\Omega_{\rm DM}h^2=0.1143\pm 0.0034.
\end{equation}
\begin{figure}[t]
\begin{center}
\vspace{4mm}
\hspace{-8mm}
\includegraphics[width=0.58\textwidth]{Fig1.eps}
\end{center}
\caption{{\it The relic abundance}:
The relic abundances consistent with the observations are obtained in the two red regions.
In the upper-left corner outside the red region, the relic abundance is predicted to be too little,
while over-abundance of the dark matter relic density is obtained in the other region.
The contours corresponding to fixed dark matter masses are also shown.
Here, the Higgs mass has been taken to be $m_h=120$ GeV.
}
\label{fig:abundance}
\end{figure}
The result is depicted in Figure \ref{fig:abundance},
where the Higgs boson mass is set as $m_h=120$ GeV.
The regions consistent with the observations are indicated by red,
while too little (much) abundances
are obtained in the region above the red line on the upper-left
(in the other regions).
There are two allowed regions:
One is the very narrow region in upper-right,
where the right relic abundance is achieved
by the enhancement of the annihilation cross section
through the $s$-channel Higgs boson resonance,
so that the dark matter mass is $m_{\rm DM} \simeq m_h/2 =60$ GeV there.
The other one appears in upper-left with
the dark matter mass around a few TeV, where dark matter particles
can efficiently annihilate into the weak gauge bosons
and the Higgs bosons through the processes
with heavy fermions in the $t$-channel.
\subsection{Direct detection}
Next we investigate the implication of the gauge-Higgs dark matter
for the direct detection experiments~\cite{TrAnom}.
A variety of experiments are underway to directly detect
dark matter particles through their elastic scatterings off nuclei.
The most stringent limits on the (spin-independent) elastic
scattering cross section have been reported by the recent
XENON10~\cite{XENON10} and CDMS II~\cite{CDMS} experiments:
$\sigma_{el}({\rm cm}^2) \lesssim
7 \times 10^{-44} - 5 \times 10^{-43}$,
for a dark matter mass of
100 GeV$\lesssim m_{\rm DM} \lesssim$ 1 TeV.
Since the gauge-Higgs dark matter particle can scatter off a nucleon
through processes mediated by the Higgs boson
in the $t$-channel, a parameter region of our model
is constrained by this current experimental bound.
The elastic scattering cross section between
the dark matter and nucleon mediated by the Higgs boson is given as
\begin{equation}
\sigma_{el}(DM+N\to DM+N)=\frac{y_{\rm DM}^2 m_N^2 m_{\rm DM}^2}
{\pi v_h^2 m_h^4 (m_{\rm DM}+m_N)^2}
\left|f_N \right|^2,
\end{equation}
where $m_N=0.931{\rm eV}$ is the nucleon mass~\cite{PDG},
and $v_h=246{\rm GeV}$ is the VEV of the Higgs doublet.
The parameter $f_N$ is defined as
\begin{eqnarray}
f_N = \langle N \left| \sigma_q m_q \bar{q}q
- \frac{\alpha_s}{4 \pi} G_{\mu \nu} G^{\mu \nu}
\right| N \rangle
= m_N \left(\frac{2}{9} f_{T_G}+f_{T_u}+f_{T_d}+f_{T_s}\right),
\end{eqnarray}
where $q$ represents light quarks ($u$, $d$, and $s$)
and $G_{\mu \nu}$ is the gluon field strength.
Contributions from the light quarks to the hadron matrix
element is evaluated by lattice QCD simulations~\cite{fbyLattice},
\begin{eqnarray}
f_{T_u}+f_{T_d} \simeq 0.056, \; \; f_{T_s} < 0.038,
\end{eqnarray}
while the contribution by gluon $f_{T_G}$ is determined
from the trace anomaly condition~\cite{TrAnom}:
\begin{equation}
f_{T_G}+f_{T_u}+f_{T_d}+f_{T_s}=1.
\end{equation}
In our analysis, we use the conservative value, $f_{T_s}=0$.
For various values of parameters, $\theta_W$ and $c$,
with $m_h=120$ GeV fixed,
we evaluate the elastic scattering cross sections
between the dark matter particle and nucleon.
\begin{figure}[t]
\begin{center}
\includegraphics[width=0.5\textwidth]{Fig2.eps}
\end{center}
\caption{{\it The direct detection}:
The red regions correspond to parameter sets that predict the right abundance.
The parameter sets with $\theta_W>\pi/10$ are indicated in gray, and those
excluded by the current bound from the direct detection experiments
are in brown.
The expected search limits by future experiments are also shown.}
\label{fig:DD}
\end{figure}
The result is shown in figure \ref{fig:DD}.
The parameter sets in red regions lead to the appropriate
dark matter abundances.
The gray region corresponds to $\theta_W>\pi/10$, which we do not consider
as discussed in section~\ref{Sec:constraints}.
The already excluded region
from XENON10~\cite{XENON10} and CDMS II~\cite{CDMS} experiments
is shown in brown,
by which a part of the red region with $m_{\rm DM} =2-3$ TeV
is excluded.
Here, we naively extrapolate the exclusion limit beyond 1 TeV,
although the experimental bounds shown in the original papers
are depicted in the range $m_{\rm DM} \leq 1$ TeV~\footnote{
We would like to thank Yoshitaka Itow for his advise on the
current experimental bounds for the dark matter mass beyond 1 TeV.
}.
The other three lines indicate expected future limits
by XMASS~\cite{XMASS}, SCDMS~\cite{SCDMS} and , XENON100~\cite{XENON100}
respectively from above to below.
The allowed region with the dark matter mass around TeV is fully
covered by the future experiments.
On the other hand, most of the narrow region consistent
with the observed dark matter abundance
is out side of the reach of the future experiments.
\section{Summary}
In extradimensional theories,
the AP boundary condition for a bulk fermion
can be imposed in general.
We show that the lightest mode of the AP fields
can be stable and hence become a candidate
for the dark matter in the effective 4D theory
due to the remaining accidental discrete symmetry.
This mechanism works even with general non-flat metric,
in contrast to the KK parity which does not work
in a simple warped model.
Although we can introduce the AP fields in various phenomenological
extradimensional models, they are usually not so strongly motivated
except for providing the dark matter particle.
In contrast, it is worth noting that in the
gauge-Higgs unification scenario, AP fields often play
a crucial role to realize a phenomenologically viable model.
Thus, we examine the possibility of the dark matter
in the gauge-Higgs unification scenario.
We find that due to the structure of the gauge-Higgs unification,
the interactions of the dark matter particle
with the SM particles, especially with the Higgs boson,
are largely controlled by the gauge symmetry
and the model has a strong predictive power
for the dark matter phenomenology.
Because of this feature, we call this scenario
as the gauge-Higgs dark matter scenario.
We have investigated this scenario based on
a five-dimensional SO(5)$\times$U(1)$_X$
gauge-Higgs unification model compactified
on the warped metric as an example.
This model is favorable because it contains the bulk
custodial symmetry and thus a few TeV KK scale
can be consistent with the electroweak precision measurement.
We have evaluated the relic abundance of the dark matter particle
and identified the parameter region of the model
to be consistent with the observed dark matter relic density.
We have found two allowed regions:
One is a quite narrow region where the right dark matter relic
density is achieved by the dark matter annihilation
through the Higgs boson resonance,
so that the dark matter mass is close to a half
of the Higgs boson mass.
In the other region, the dark matter annihilation process
is efficient and the dark matter particle with a few TeV mass
is consistent with the observations.
Furthermore, we have calculated the cross section
of the elastic scattering between the dark matter particle
and nucleon and shown the implication of the gauge-Higgs
dark matter scenario for the current and future direct
dark matter detection experiments.
It turns out that the region with a few TeV dark matter mass
is partly excluded by the current experiments
and the whole region can be explored by future experiments.
On the other hand, most of the narrow region is out side of the
experimental reach.
\vspace{1cm}
\leftline{\bf Acknowledgments}
This work is supported in part by a Grant-in-Aid for Science Research from the
Ministry of Education, Culture, Sports, Science and Technology, Japan
(Nos.~16540258 and 17740146 for N.~H., No.~21740174 for S.~M. and
No.~18740170 for N.~O.),
and by the Japan Society for the Promotion of Science (T.Y.).
| 2024-02-18T23:40:06.804Z | 2009-10-20T06:52:49.000Z | algebraic_stack_train_0000 | 1,375 | 5,081 |
|
proofpile-arXiv_065-7033 | \section{Introduction}
Class field theory is a pillar of algebraic number theory; it is mostly concerned with classifying finite abelian extensions of a fixed local or global field $K$. Similarly, abelian covers of a fixed Riemann surface $X$ can be classified in terms of its first homology group $H_1(X,\mathbb{Z})$, or in terms of its Jacobian $J(X)\simeq H_1(X,\mathbb{R}/\mathbb{Z})$. Andr\'e Weil, in a letter to his sister from 1940 (see \cite{Weil_lettertosister}), pointed out the analogy between these two situations, as well as a potential bridge: the theory of abelian extensions of function fields over finite fields, an area that is now known as geometric class field theory (see \cite{Serre_alggroups&classfields}).
More recently, another analogy has entered the mathematical stage: between a Riemann surface $X$ and a metric graph $\Gamma$, or more generally a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$. Many classical geometric constructions for Riemann surfaces, such as the theory of divisors, linear equivalence, Jacobians, theta functions, and moduli spaces, have natural analogues for tropical curves, as beautifully illustrated in \cite{MikhalkinZharkov}.
The success of this analogy is, of course, not a coincidence. A tropical curve naturally arises as the dual graph $\Gamma_X$ of a semistable degeneration $\mathcal{X}$ of an algebraic curve $X$ (with the metric encoding the deformation parameters at the nodes of $\mathcal{X}$). Geometric constructions on $\Gamma_X$ then naturally arise as combinatorial specializations of their classical counterparts on $X$. We refer the reader for example to \cite{BakerJensen} for a survey of this story in the case of linear series.
In this article, we develop a theory of $G$-covers of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$, where $G$ is a finite abelian group. A $G$-cover of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is an unramified harmonic morphism $\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ (such morphisms were studied in \cite{CavalieriMarkwigRanganathan_tropadmissiblecovers} under the name of tropical admissible covers), together with an action of $G$ on $\scalebox{0.8}[1.3]{$\sqsubset$}'$ that preserves and acts transitively on the fibers. We show that such covers are classified by two objects. The first is a {\it dilation stratification} $\mathcal{S}$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$, indexed by the subgroups of $G$, that encodes the local stabilizer subgroups (see Def.~\ref{def:stratificationtropical}). The second is an element of a {\it dilated cohomology group} $H^1(\scalebox{0.8}[1.3]{$\sqsubset$},S)$ associated to $\scalebox{0.8}[1.3]{$\sqsubset$}$ and a dilation stratification $\mathcal{S}$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ (see Def.~\ref{def:dilatedcohomologytropical}). In the spirit of the above analogies, one may think of this work as the starting point for a tropical version of class field theory.
Our principal result is the following (see Thm.~\ref{thm:main5}):
\begin{maintheorem} Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve, let $G$ be a finite abelian group, and let $\mathcal{S}$ be an admissible dilation stratification of $\scalebox{0.8}[1.3]{$\sqsubset$}$. Then there is a natural bijection between the set of unramified $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ having dilation stratification $\mathcal{S}$ and the dilated cohomology group $H^1(\scalebox{0.8}[1.3]{$\sqsubset$},\mathcal{S})$.
\end{maintheorem}
The main technical ingredient in the classification of $G$-covers of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a theory of dilated cohomology groups of a graph marked by subgroups of $G$. This theory generalizes simplicial cohomology with coefficients in $G$ and satisfies a number of natural properties such as functoriality and pullback, and admits a long exact sequence. It seems natural to generalize dilated cohomology to arbitrary simplicial complexes, but this is beyond the scope of our paper. Since our methods are cohomological, they do not readily generalize to non-abelian groups. In a future paper, we plan to treat the non-abelian case by relating dilated cohomology to Bass--Serre theory~\cite{Serre_trees, Bass} and developing a Galois theory for non-abelian unramified covers of tropical curves.
\subsection*{Earlier and related works}
A number of authors study graphs and tropical curves with a group action. The simplest example is the case of tropical hyperelliptic curves, which are $\mathbb{Z}/2\mathbb{Z}$-covers of a tree (\cite{2009BakerNorine}, \cite{2013Chan}, \cite{Caporaso_gonality}, \cite{ABBRII}, \cite{2016Panizzut}, \cite{2017BologneseBrandtChua}, \cite{2017Len}). Brandt and Helminck \cite{2017BrandtHelminck} consider arbitrary cyclic covers of a tree, while Helminck \cite{2017Helminck} looks at the tropicalization of arbitrary abelian covers of algebraic curves from a non-Archimedean perspective. Jensen and Len \cite{2018JensenLen} classify unramified $\mathbb{Z}/2\mathbb{Z}$-covers of arbitrary tropical curves in terms of dilation cycles, which is a special case of our dilation stratification; with this article we aim to generalize this aspect of their work.
While we do not pursue this direction here, $G$-covers of curves may be used to produce interesting loci of special divisors and linear series. For instance, Jensen and Len \cite{2018JensenLen} and Len and Ulirsch \cite{LenUlirsch} develop a theory of tropical Prym varieties associated to $\mathbb{Z}/2\mathbb{Z}$-covers of tropical curves, with applications to algebraic Prym--Brill--Noether theory. In a similar vein, Song \cite{Song_Ginvariantlinearsystems} considers $G$-invariant linear systems with the goal of studying their descent properties to the quotient.
From a moduli-theoretic perspective, studying degenerations of $G$-covers of algebraic curves is equivalent to studying the compactification of the moduli space of $G$-covers in terms of the moduli space of $G$-admissible covers, as constructed in \cite{AbramovichCortiVistoli} and \cite{BertinRomagny}. In \cite[Section 7]{BertinRomagny} the authors have already introduced a graph-theoretic gadget to understand the boundary strata of this moduli space: so-called \emph{modular graphs} with an action of a finite (not necessarily abelian) group $G$.
This idea seems to have appeared independently in other works as well: Chiodo and Farkas \cite{ChiodoFarkas} study the boundary of the moduli space of level curves, which is equivalent to a component of the moduli space of $G$-admissible covers for a cyclic group $G$, and look at cyclic covers of an arbitrary graph. Their work has been extended to an arbitrary finite group $G$ by Galeotti in \cite{2019GaleottiB, 2019GaleottiA}. Finally, in \cite{SchmittvanZelm}, Schmitt and van Zelm apply a graph-theoretic approach to the boundary of the moduli space of $G$-admissible covers (for an arbitrary finite group $G$) to study their pushforward classes in the tautological ring of $\overline{\mathcal{M}}_{g,n}$.
In \cite{CavalieriMarkwigRanganathan_tropadmissiblecovers} Cavalieri, Markwig, and Ranganathan develop a moduli-theoretic approach to the tropicalization of the moduli space of admissible covers (without a fixed group operation). We extend this aspect of their article to the moduli space of $G$-admissible covers in Section \ref{sec:tropicalization} below. In \cite{CaporasoMeloPacini}, Caporaso, Melo, and Pacini study the tropicalization of the moduli space of spin curves, which, in view of the results in \cite{2018JensenLen}, is closely related to to our story in the case $G=\mathbb{Z}/2\mathbb{Z}$.
The problem of classifying covers of a graph with an action of a given group (not necessarily abelian) was studied by Corry in \cite{2011Corry,2012Corry,2015Corry}. However, Corry considered a different category of graph morphisms, allowing edge contraction but not dilation. To the best of our knowledge, no author has considered the problem of classifying all unramified covers of a given graph with an action of a fixed group.
\bigskip
\subsection*{Analogies in topology and algebraic geometry}
It is instructive to recall the theory of abelian covers in two categories, both directly related to tropical geometry: topological covering spaces and algebraic \'etale covers.
\medskip
\subsubsection*{Topological spaces}
Let $X$ be a path-connected, locally path-connected and semi-locally simply connected topological space, let $x_0\in X$ be a base point, and let $G$ be a group. A {\it regular $G$-cover} of $(X,x_0)$ is a based covering space $(Y,y_0)\to (X,x_0)$ together with an $G$-action on $Y$ such that $G$ acts freely and transitively on fibers. Based regular $G$-covers of $(X,x_0)$ are classified by monodromy homomorphisms $\pi_1(X,x_0)\to G$ (the cover is connected if and only if the homomorphism is surjective). If $G$ is a finite abelian group, then we can identify the set of such homomorphisms, canonically and independently of $x_0$, with the cohomology group $H^1(X,G)$. We note that a $G$-cover is rigidified by the $G$-action: for example, if $p$ is a prime number, there is a single connected degree $p$ covering space $S^1\to S^1$, but there are $p-1$ connected $\ZZ/p\ZZ$-covers of $S^1$ corresponding to the non-trivial elements of $H^1(S^1,\ZZ/p\ZZ)\simeq \ZZ/p\ZZ$.
If $X$ is the underlying topological space of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$, then any regular $G$-cover $X'\to X$ can be given the structure of an unramified $G$-cover $\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ of tropical curves by pulling back the genus function from $\scalebox{0.8}[1.3]{$\sqsubset$}$ to $\scalebox{0.8}[1.3]{$\sqsubset$}'$. These $G$-covers, which we call {\it topological}, have the property that $G$-action on the fibers is free (see Ex.~\ref{ex:topologicalcovers} and Ex.~\ref{example:topologicaladmissiblecovers}). The corresponding dilation stratification $\mathcal{S}$ on $\scalebox{0.8}[1.3]{$\sqsubset$}$ is trivial, and the dilated cohomology group $H^1(\scalebox{0.8}[1.3]{$\sqsubset$},\mathcal{S})$ reduces to $H^1(X,G)$.
\subsubsection*{Algebraic varieties}
Let $X$ be an algebraic variety over a field $k$ and $x_0$ a geometric base point of $X$. Like its topological counterpart, the \'etale fundamental group $\pi_1^{\'et}(X,x_0)$ of $X$ classifies finite \'etale covers of $X$. For a finite abelian group $G$ the set of continuous homomorphisms $\Hom(\pi_1^{\'et}(X,x_0),G)$ is equal to the set of Galois coverings of $X$ with Galois group $G$. If $X$ is a smooth projective curve, the abelian coverings of $X$ naturally arise as pullbacks of (always abelian) coverings of its Jacobian $J$ along the Abel-Jacobi map $X\rightarrow J$. In particular, we have an induced isomorphism $\pi_1^{\'et}(X,x_0)^{ab}\simeq\pi_1^{\'et}(J,x_0)$. In Section \ref{sec:tropicalization} we will see how our \emph{a priori} purely combinatorial construction can be thought of a tropical limit of this well-known story.
\subsection*{Organization of the paper}
Our paper is organized as follows. In Sec.~\ref{sec:definitions}, we review the necessary definitions from graph theory and tropical geometry. In Sec.~\ref{sec:dilated}, we introduce $G$-covers, $G$-dilation data, and dilated cohomology groups. We are primarily interested in classifying abelian covers of tropical curves, however, our constructions are purely graph-theoretic in nature and may be of interest to specialists in graph theory and topology. For this reason, we first develop the theory of $G$-covers for unweighted graphs. In Sec.~\ref{sec:classification} we prove our main classification results, and then extend them to weighted graphs, weighted metric graphs, and tropical curves. Finally, in Sec.~\ref{sec:tropicalization} we relate our constructions to the tropicalization of the moduli space of admissible $G$-covers.
\bigskip
\subsection*{Acknowledgments}
The authors would like to thank Matthew Baker, Madeline Brandt, Renzo Cavalieri, Gavril Farkas, Paul Helminck, David Jensen, Andrew Obus, Sam Payne, Matthew Satriano, Johannes Schmitt, and Jason van Zelm for useful discussions.
This project has received funding from the European Union's Horizon 2020 research and innovation programme under the Marie-Sk\l odowska-Curie Grant Agreement No. 793039. \includegraphics[height=1.7ex]{flag_yellow_low.jpeg}
We also acknowledge support from the LOEWE-Schwerpunkt ``Uniformisierte Strukturen in Arithmetik und Geometrie''.
\section{Definitions and notation}\label{sec:definitions}
We develop the theory of $G$-covers of graphs on several levels successively: graphs, weighted graphs, metric graphs, and tropical curves. In this section, we recall the necessary definitions from graph theory.
\subsection{Graphs}
We first consider unweighted graphs without a metric.
\begin{definition} A {\it graph with legs} $\Gamma$, or simply a {\it graph}, consists of the following:
\begin{enumerate}
\item A finite set $X(\Gamma)$.
\item An idempotent {\it root map} $r:X(\Gamma)\to X(\Gamma)$.
\item An involution $\iota:X(\Gamma)\to X(\Gamma)$ whose fixed set contains the image of $r$.
\end{enumerate}
\end{definition}
The image $V(\Gamma)$ of $r$ is the set of {\it vertices} of $\Gamma$, and its complement $H(\Gamma)=X(\Gamma)\backslash V(\Gamma)$ is the set of {\it half-edges} of $\Gamma$. The involution $\iota$ preserves $H(\Gamma)$ and partitions it into orbits of size 1 and 2; we call these respectively the {\it legs} and {\it edges} of $\Gamma$ and denote the corresponding sets by $L(\Gamma)$ and $E(\Gamma)$. The root map assigns one root vertex to each leg and two root vertices to each edge. A {\it loop} is an edge whose root vertices coincide.
We note that, from a graph-theoretic point of view, there is essentially no difference between a leg and an extremal edge. This distinction is important, however, from a tropical viewpoint: legs are the tropicalizations of marked points, while an extremal edge represents a rational tail. Note that, unlike an extremal edge, a leg does not have a vertex at its free end.
The {\it tangent space} $T_v \Gamma$ and {\it valency} $\val (v)$ of a vertex $v\in V(\Gamma)$ are defined by
$$
T_v\Gamma=\big\{h\in H(\Gamma)\big|r(h)=v\big\} \textrm{ and } \val(v)=\#(T_v\Gamma).
$$
\begin{definition} Let $\Gamma$ be a graph. A {\it subgraph} $\Delta$ of $\Gamma$ is a subset of $X(\Gamma)$ closed under the root and involution maps. Given a subgraph $\Delta\subset \Gamma$ and a vertex $v\in V(\Delta)$, we denote $\val_{\Delta}(v)$ the valency of $v$ viewed as a vertex of $\Delta$. A subgraph $\Delta\subset \Gamma$ is called a {\it cycle} if $\val_{\Delta}(v)$ is even for every $v\in V(\Delta)$. A subgraph $\Delta\subset \Gamma$ is called {\it edge-maximal} if every edge $e\in E(\Gamma)$ having both root vertices in $\Gamma$ lies in $\Delta$.
\end{definition}
It is clear that a subgraph of $\Gamma$ is edge-maximal if and only if it is the largest subgraph of $\Gamma$ with a given set of vertices.
\begin{definition} Let $\Gamma$ be a graph. An {\it orientation} on $\Gamma$ is a choice of order $(h,h')$ on each edge $e=\{h,h'\}\in E(\Gamma)$. We call $s(e)=r(h)$ and $t(e)=r(h')$ the {\it source} and {\it target} vertices of $e$.
\end{definition}
\begin{definition} A {\it finite morphism} of graphs $\varphi:\Gamma'\to \Gamma$, or simply a {\it morphism}, is a map of sets $\varphi:X(\Gamma')\to X(\Gamma)$ which commutes with the root and involution maps, such that edges map to edges and legs map to legs.
\end{definition}
An {\it automorphism} $\varphi:\Gamma\to \Gamma$ of a graph $\Gamma$ is a morphism with an inverse. We denote the group of automorphisms of $\Gamma$ by $\Aut (\Gamma)$. We remark that a nontrivial graph automorphism may act trivially on the vertex and edge sets. For example, the graph $\Gamma$ consisting of one vertex $v$ and one loop $e=\{h,h'\}$ has a nontrivial automorphism fixing $v$ and exchanging $h$ and $h'$. To form quotients of graphs by group actions, we need to exclude such automorphisms from consideration.
\begin{definition} Let $\Gamma$ be a graph, and let $G$ be a group. A {\it $G$-action} on $\Gamma$ is a homomorphism of $G$ to the automorphism group $\Aut (\Gamma)$ such that for every $g\in G$, the corresponding automorphism does not flip edges. In other words, for every edge $e=\{h,h'\}\in E(\Gamma)$ either $\varphi(e)\neq e$, or $\varphi(h)=h$ and $\varphi(h')=h'$. Given a $G$-action on $\Gamma$, we define the {\it quotient graph} $\Gamma/G$ by setting $X(\Gamma/G)=X(\Gamma)/G$. The root and involution maps on $\Gamma$ are $G$-invariant and descend to $\Gamma/G$. It is clear that $V(\Gamma/G)=V(\Gamma)/G$ and $H(\Gamma/G)=H(\Gamma)/G$, and the no-flipping assumption implies that the $G$-action does not identify the two half-edges of any edge of $\Gamma$. Therefore $E(\Gamma/G)=E(\Gamma)/G$ and $L(\Gamma/G)=L(\Gamma)/G$, and the quotient map $\pi:\Gamma\to \Gamma/G$ is a finite morphism. \label{def:Gquotient}
\end{definition}
\subsection{Weighted graphs, harmonic morphisms, and ramification}
We now consider graphs with vertex weights. Heuristically, one may think of a vertex of weight $g$ as an infinitesimally small graph with $g$ loops (cf. \cite[Section 5]{AC}).
\begin{definition} A {\it weighted graph} $(\Gamma,g)$ is a pair consisting of a graph $\Gamma$ and a vertex weight function $g:V(\Gamma)\to \mathbb{Z}_{\geq 0}$.
\end{definition}
We will usually suppress $g$ and denote weighted graphs by $\Gamma$. We define the {\it Euler characteristic} $\chi(v)$ of a vertex $v\in V(\Gamma)$ on a weighted graph $\Gamma$ as
$$
\chi(v)=2-2g(v)-\val(v).
$$
The {\it genus} of a connected graph $\Gamma$ is defined to be
$$
g(\Gamma)=\#(E(\Gamma))-\#(V(\Gamma))+1+\sum_{v\in V(\Gamma)}g(v).
$$
We define the {\it Euler characteristic} $\chi(\Gamma)$ of a graph $\Gamma$ by
$$
\chi(\Gamma)=\sum_{v\in V(\Gamma)} \chi(v);
$$
this is not to be confused with the topological Euler characteristic of $\Gamma$. An easy calculation shows that, if $\Gamma$ is connected, then
$$
\chi(\Gamma)=2-2g(\Gamma)-\#(L(V)).
$$
A subgraph $\Delta\subset \Gamma$ of a weighted graph $\Gamma$ is naturally given the structure of a weighted graph by restricting the weight function $g$. In this case, denote $\chi_{\Delta}(v)=2-2g(v)-\val_{\Delta}(v)$ the Euler characteristic of a vertex $v$ of $\Delta$.
We say that a vertex $v\in \Gamma$ of a weighted graph $\Gamma$ is {\it unstable} if $\chi(v)\geq 1$, {\it semistable} if $\chi(v)\leq 0$, and {\it stable} if $\chi(v)\leq -1$. An unstable vertex has genus zero and is either isolated or extremal. A semistable vertex that is not unstable is either an isolated vertex of genus one or a valency two vertex of genus zero, in which case we call it {\it simple}. We say that a graph $\Gamma$ is {\it semistable} if all of its vertices are semistable, and {\it stable} if all of its vertices are stable.
Let $\Gamma$ be a connected weighted graph with $\chi(\Gamma)<0$. Following \cite[Section 8.2]{ACP}, we construct a stable graph $\Gamma_{st}$, called the {\it stabilization} of $\Gamma$, as follows. First, we construct the {\it semistabilization} $\Gamma_{sst}$ of $\Gamma$ by inductively removing all extremal edges ending at an extremal vertex of genus zero (but not the legs). The graph $\Gamma_{sst}$ is a semistable subgraph of $\Gamma$, and it is clear that $\chi(\Gamma_{sst})=\chi(\Gamma)$, and that any vertices of $\Gamma_{sst}$ that are not stable are simple. We then construct $\Gamma_{st}$ by gluing together the two half-edges at each simple vertex $v$ of $\Gamma_{sst}$. Specifically, if $v$ is an endpoint of two edges $e_1$ and $e_2$, we replace $v$, $e_1$, and $e_2$ with a new edge connecting the other endpoints of $e_1$ and $e_2$. If $v$ is an endpoint of an edge $e$ and a leg $l$, we replace $v$, $e$, and $l$ with a new leg rooted at the other endpoint of $e$. The result is a stable graph $\Gamma_{st}$ with $\chi(\Gamma_{st})=\chi(\Gamma_{sst})=\chi(\Gamma)$.
\begin{definition} Let $\Gamma$ and $\Gamma'$ be graphs. A {\it finite harmonic morphism} $\varphi:\Gamma'\to \Gamma$, or simply a {\it harmonic morphism}, consists of a finite morphism $\Gamma'\to \Gamma$ and a map $d_{\varphi}:X(\Gamma')\to \mathbb{Z}_{>0}$, called the {\it degree} of $\varphi$, such that the following properties are satisfied:
\begin{enumerate}
\item If $e'=\{h'_1,h'_2\}\in E(\Gamma')$ is an edge then $d_{\varphi}(h'_1)=d_{\varphi}(h'_2)$. We call this number the {\it degree} of $\varphi$ along $e'$ and denote it $d_{\varphi}(e')$.
\item For every vertex $v'\in V(\Gamma')$ and every tangent direction $h\in T_{\varphi(v)}\Gamma$, we have
$$
d_{\varphi}(v')=\sum_{\substack{h'\in T_{v'}\Gamma', \\ \varphi(h')=h}}d_{\varphi}(h').
$$
In particular, this sum does not depend on the choice of $h$.
\end{enumerate}
\end{definition}
Let $\varphi:\Gamma'\to \Gamma$ be a harmonic morphism of graphs, where $\Gamma$ is connected. The sum
$$
\deg(\varphi)=\sum_{\substack{v'\in V(\Gamma'),\\ \varphi(v')=v}}d_{\varphi}(v')=\sum_{\substack{e'\in E(\Gamma'), \\ \varphi(e')=e}}d_{\varphi}(e')=
\sum_{\substack{l'\in L(\Gamma'),\\ \varphi(l')=l}}d_{\varphi}(l')
$$
does not depend on the choice of $v\in V(\Gamma)$, $e\in E(\Gamma)$ or $l\in L(\Gamma)$ and is called the {\it degree} of $\varphi$ (see Section 2 of \cite{ABBRI}).
\begin{definition} Let $\varphi:\Gamma'\to \Gamma$ be a harmonic morphism of weighted graphs. The {\it ramification degree} $\Ram_{\varphi}(v')$ of $\varphi$ at a vertex $v'\in V(\Gamma')$ is equal to
$$
\Ram_{\varphi}(v')=d_{\varphi}(v')\chi(\varphi(v'))-\chi(v').
$$
We say that $\varphi$ is {\it effective} if
$$
\Ram_{\varphi}(v')\geq 0
$$
for all $v'\in V(\Gamma')$, and {\it unramified} if
\begin{equation}
\Ram_{\varphi}(v')=0
\label{eq:localRH}
\end{equation}
for all $v'\in V(\Gamma')$.
\end{definition}
\begin{remark} Unramified morphisms were studied extensively in~\cite{CavalieriMarkwigRanganathan_tropadmissiblecovers}, where they were called {\it tropical admissible covers}. We partly preserve this terminology: for example, we call a dilation stratification admissible if it corresponds to an unramified cover. A simple calculation shows that our definition of ramification degree agrees with the standard one in the literature (see, for example, Sec.~2.2 in \cite{ABBRII} or Def. 16 in~\cite{CavalieriMarkwigRanganathan_tropadmissiblecovers}):
$$
\Ram_{\varphi}(v')=d_{\varphi}(v')\left(2-2g(\varphi(v'))\right)-\left(2-2g(v')\right)-\sum_{h'\in T_{v'}\Gamma'} \left(d_{\varphi}(h')-1\right).
$$
\end{remark}
For an unramified harmonic morphism $\varphi:\Gamma'\to \Gamma$, we call equation~\eqref{eq:localRH} the {\it local Riemann--Hurwitz condition} at $v'\in V(\Gamma')$. Adding together these conditions at all $v'\in V(\Gamma')$, we obtain the {\it global Riemann--Hurwitz condition}
\begin{equation}
\chi(\Gamma')=\deg(\varphi)\chi(\Gamma).
\label{eq:globalRH}
\end{equation}
\begin{example} Let $\varphi:\Gamma'\to \Gamma$ be an unramified harmonic morphism, and suppose that $d_{\varphi}(v')=1$ for some $v'$. By the harmonicity condition, each $h\in T_{\varphi(v')}\Gamma$ has a unique preimage in $T_{v'}\Gamma'$, hence $\val(v')=\val(\varphi(v'))$. Furthermore, we have $\chi(\varphi(v'))=\chi(v')$, which implies that $g(v')=g(\varphi(v'))$. It follows that $\varphi$ is a local isomorphism of weighted graphs in a neighborhood of $v'$. In particular, an unramified harmonic morphism of degree one is a graph isomorphism, and vice versa.
\end{example}
We observe that if $\varphi:\Gamma'\to \Gamma$ is an effective harmonic morphism and $\Delta\subset \Gamma$ is a subgraph with preimage $\Delta'=\varphi^{-1}(\Delta)$, then the induced map $\varphi|_{\Delta'}:\Delta'\to \Delta$ is also an effective harmonic morphism, since the ramification degree does not decrease when a half-edge and its preimages are removed. However, if $\varphi$ is unramified, then $\varphi|_{\Delta'}$ is not necessarily unramified. We now show that unramified morphisms naturally restrict to stabilizations.
Let $\varphi:\Gamma'\to \Gamma$ be an unramified harmonic morphism of connected graphs, and assume that $\chi(\Gamma)<0$ (or, equivalently by~\eqref{eq:globalRH}, that $\chi(\Gamma')<0$). For any two vertices $v'\in V(\Gamma')$ and $v=\varphi(v')\in V(\Gamma)$, Eq.~\eqref{eq:localRH} implies that $v'$ is unstable if and only if $v$ is unstable, in which case $\chi(v')=\chi(v)=1$ and $d_{\varphi}(v')=1$. Let $v\in V(\Gamma)$ be an extremal vertex of genus 0, let $e\in E(\Gamma)$ be the unique edge rooted at $v$, and let $u\in V(\Gamma)$ be the other root vertex of $v$. By the above, we see that $v\in V(\Gamma)$ has $\deg(\varphi)$ preimages $v'_i$ in $\Gamma'$, each of which is a root vertex of a unique extremal edge $e'_i$ mapping to $e$ with local degree 1. For any $u'\in \varphi^{-1}(u)$, $d_{\varphi}(u')$ of the edges $e'_i$ are rooted at $u'$. Therefore, removing $v'_i$, $e'_i$, $v$, and $e$ increases $\chi(u)$ by 1 and increases each $\chi(u')$ by $d_{\varphi}(u')$, hence does not change the local Riemann--Hurwitz condition at $u'$. Proceeding in this way, we remove all unstable vertices of $\Gamma'$ and $\Gamma$ and obtain an unramified harmonic morphism $\varphi_{sst}:\Gamma'_{sst}\to \Gamma_{sst}$.
Similarly, we see that for two vertices $v'\in V(\Gamma'_{sst})$ and $v=\varphi(v')\in V(\Gamma_{sst})$, Eq.~\eqref{eq:localRH} implies that one is simple if and only if the other is. Furthermore, by the harmonicity condition, the degrees of $\varphi$ at the two half-edges at $v'$ are equal, hence we can remove $v'$ and $v$, glue together the free half-edges, and extend $\varphi$; this does not change the local Riemann--Hurwitz condition at any remaining vertex of $\Gamma'_{sst}$. Proceeding in this way, we obtain an unramified morphism $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$.
\begin{definition} Let $\varphi:\Gamma'\to \Gamma$ be an unramified harmonic morphism of connected weighted graphs, such that $\chi(\Gamma)<0$ (or, equivalently, $\chi(\Gamma')<0$). The unramified morphism $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$ constructed above is called the {\it stabilization} of $\varphi$.
\label{def:coverstabilization}
\end{definition}
Finally, we define the contraction of a graph along a subset of its edges; this can be viewed as a non-finite harmonic morphism of degree one.
\begin{definition} Let $\Gamma$ be a weighted graph, and let $S\subset E(\Gamma)$ be a set of edges of $\Gamma$. We define the {\it weighted edge contraction} $\Gamma/S$ of $\Gamma$ along $S$ as follows. Let $\Delta$ be the minimal subgraph of $\Gamma$ whose edge set contains $S$, and let $\Delta_1,\ldots,\Delta_k$ be the connected components of $\Delta$. We obtain $\Gamma/S$ from $\Gamma$ by contracting each $\Delta_i$ to a vertex $v_i$ of genus $g(\Delta_i)$. \label{def:edgecontraction}
\end{definition}
Given a harmonic morphism $\varphi:\Gamma'\to \Gamma$ of weighted graphs, we can contract a subset of edges $S\subset E(\Gamma)$ of $\Gamma$, and their preimages in $\Gamma'$. Connected components of graphs map to connected components, and degree is constant when restricted to a connected component, so there is a natural harmonic morphism $\varphi_{S}:\Gamma'/\varphi^{-1}(S)\to \Gamma/S$. A simple calculation shows that if $\varphi$ is unramified, then so is $\varphi_S$:
\begin{proposition}[Proposition 19 in~\cite{CavalieriMarkwigRanganathan_tropadmissiblecovers}] Let $\varphi:\Gamma'\to \Gamma$ be an unramified harmonic morphism of unweighted graphs, let $S\subset E(\Gamma)$ be a subset of the edges of $\Gamma$, and let $\Gamma'/\varphi^{-1}(S)$ and $\Gamma/S$ be the weighted edge contractions. Then $\varphi_{S}:\Gamma'/\varphi^{-1}(S)\to \Gamma/S$ is unramified.
\label{prop:edgecontraction}
\end{proposition}
\subsection{Metric graphs and tropical curves}
Finally, we consider weighted graphs with a metric, as well as tropical curves.
\begin{definition} A {\it weighted metric graph} consists of a weighted graph $(\Gamma,g)$ and a function $\ell:E(\Gamma)\to \mathbb{R}_{>0}$. A {\it finite harmonic morphism} of weighted metric graphs $\varphi:(\Gamma',\ell')\to (\Gamma,\ell)$, or simply a {\it harmonic morphism}, is a finite harmonic morphism $\varphi:\Gamma'\to \Gamma$ of the underlying weighted graphs such that for every edge $e'\in E(\Gamma')$ we have
\begin{equation}
\ell(\varphi(e'))=d_{\varphi}(e')\ell'(e').
\label{eq:length}
\end{equation}
In other words, $\varphi$ dilates each edge $e'\in E(\Gamma')$ by a factor of $d_{\varphi}(e')$. A harmonic morphism $\varphi:\Gamma'\to \Gamma$ of weighted metric graphs is called {\it effective} or {\it unramified} if it is so as a map of weighted graphs.
\end{definition}
\begin{remark} Given a finite harmonic morphism $\varphi:\Gamma'\to \Gamma$ of weighted graphs and a length function $\ell$ on $\Gamma$, there is a unique length function $\ell'$ on $\Gamma'$ satisfying the dilation condition \eqref{eq:length}. Similarly, a length function on $\Gamma'$ uniquely induces a length function on $\Gamma$. It follows that the classification of unramified covers of weighted metric graphs, in particular abelian covers, is independent of the choice of metric. For this reason, in this paper we mostly work with graphs and weighted graphs without metrics.
\label{rem:lengths}
\end{remark}
Given a connected weighted metric graph $\Gamma$ with $\chi(\Gamma)<0$, we give $\Gamma_{st}$ the structure of a weighted metric graph in the obvious way, by setting $\ell(e)=\ell(e_1)+\ell(e_2)$ whenever we replace two edges $e_1$ and $e_2$ with a new edge $e$. It is clear that an unramified morphism of weighted metric graphs $\varphi:\Gamma'\to \Gamma$ induces an unramified morphism $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$.
\begin{definition} Let $(\Gamma,\ell)$ be a weighted metric graph. We define a metric space $|\Gamma|$, called the {\it metric realization} of $(\Gamma,l)$, as follows. Consider a closed interval $I_e\subset \mathbb{R}$ of length $\ell(e)$ for each edge $e\in E(\Gamma)$, and a half-open interval $I_l=[0,\infty)$ for each leg $l\in L(\Gamma)$. We obtain $|\Gamma|$ from the $I_e$ and the $I_l$ by treating their endpoints as the root vertices and gluing accordingly. We then give $|\Gamma|$ the path metric. \end{definition}
A harmonic morphism $\varphi:(\Gamma',\ell')\to (\Gamma,\ell)$ of weighted metric graphs naturally induces a continuous map $|\varphi|:|\Gamma'|\to |\Gamma|$ where, for a pair of edges $e=\varphi(e')$, the map is given by dilation by a factor of $d_{\varphi}(e')$, and similarly for a pair of legs $l=\varphi(l')$. The map is piecewise-linear with integer slope with respect to the metric structure.
A basic inconvenience of tropical geometry is that different weighted metric graphs may have the same metric realizations. This motivates the following definition.
\begin{definition} A {\it tropical curve} $(\scalebox{0.8}[1.3]{$\sqsubset$},g)$ is a pair consisting of a metric space $\scalebox{0.8}[1.3]{$\sqsubset$}$ and a weight function $g:\scalebox{0.8}[1.3]{$\sqsubset$}\to \mathbb{Z}_{\geq 0}$ such that there exists a weighted metric graph $(\Gamma,g,\ell)$ and an isometry $m:|\Gamma|\to \scalebox{0.8}[1.3]{$\sqsubset$}$ of its metric realization with $\Gamma$ such that the weight functions agree:
$$
g(x)=\left\{\begin{array}{cc}g(v)& \textrm{ if }x=m(v) \textrm{ for a } v\in V(\Gamma), \\ 0 & \mbox{otherwise}.\end{array}\right.
$$
We call a quadruple $(\Gamma,g,\ell,m)$ satisfying these properties a {\it model} for $\scalebox{0.8}[1.3]{$\sqsubset$}$.
\end{definition}
The {\it genus} of a connected tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ is given by
$$
g(\scalebox{0.8}[1.3]{$\sqsubset$})=b_1(\scalebox{0.8}[1.3]{$\sqsubset$})+\sum_{x\in \scalebox{0.8}[1.3]{$\sqsubset$}} g(x)
$$
and is equal to the genus of any model of $\scalebox{0.8}[1.3]{$\sqsubset$}$.
For a point $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$ on a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ with model $(\Gamma,g,l,m)$, we define its {\it valency} $\val(x)$ to be $\val(v)$ if $x=m(v)$ for some $v\in V(\Gamma)$ and $2$ otherwise. We similarly define the Euler characteristic as $\chi(x)=2-2g(x)-\val(x)$; these numbers do not depend on the choice of model. We define the Euler characteristic of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ to be $\chi(\scalebox{0.8}[1.3]{$\sqsubset$})=\chi(\Gamma)$ for any model $\Gamma$. It is clear that
$$
\chi(\scalebox{0.8}[1.3]{$\sqsubset$})=\sum_{x\in \scalebox{0.8}[1.3]{$\sqsubset$}}\chi(x),
$$
where $\chi(x)=0$ for all but finitely many $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$.
\begin{remark} Our definition differs from Def.~2.14 in \cite{ABBRII}, where a tropical curve was defined as an equivalence class of weighted metric graphs up to tropical modifications.
\end{remark}
Given a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ with model $\Gamma$, we can form another model $\Gamma'$ by splitting any edge or leg of $\Gamma$ at a new vertex. Conversely, any tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ (other than $\mathbb{R}$ and $S^1$) has a unique {\it minimal model} $\Gamma_{min}$ having no simple vertices. We say that a connected tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ is {\it stable} if $\chi(x)\leq 0$ for all $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$, or, equivalently, if its minimal model is a stable graph. We define the {\it stabilization} of a connected tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ with $\chi(\scalebox{0.8}[1.3]{$\sqsubset$})<0$ by removing all trees of edges having no vertices of positive genus, or, equivalently, as the geometric realization of the stabilization of any model of $\scalebox{0.8}[1.3]{$\sqsubset$}$.
Any tropical curve other than the real line has a well-defined set of maximal legs. A morphism of tropical curves is a continuous, piecewise-linear map that sends legs to legs and is eventually linear on each leg.
\begin{definition} A {\it morphism} $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ of tropical curves is a continuous, piecewise-linear map with integer slopes such that for any leg $l'\subset \scalebox{0.8}[1.3]{$\sqsubset$}'$, there exists a leg $l\subset \scalebox{0.8}[1.3]{$\sqsubset$}$ and numbers $a\in \mathbb{Z}_{>0}$ and $b\in \mathbb{R}$ such that, identifying $l'$ and $l$ with $[0,+\infty)$, we have $\tau(x)=ax+b\in l$ for $x\in l'$ sufficiently large. We note that $\tau$ may map a finite section of $l'$ to $\scalebox{0.8}[1.3]{$\sqsubset$}\backslash l$.
Let $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ be a morphism of tropical curves. A {\it model} for $\tau$ is a pair of models $(\Gamma',g',\ell',m')$ and $(\Gamma,g,\ell,m)$ for $\scalebox{0.8}[1.3]{$\sqsubset$}'$ and $\scalebox{0.8}[1.3]{$\sqsubset$}$, respectively, and a morphism $\varphi:\Gamma'\to \Gamma$ of weighted metric graphs such that $m\circ|\varphi|=\tau\circ m'$. We say that $\tau$ is {\it harmonic}, {\it effective} or {\it unramified} if $\varphi$ has the corresponding property.
\end{definition}
Given a morphism $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ of tropical curves, we construct a model $\varphi:\Gamma'\to \Gamma$ by choosing the vertex set $V(\Gamma')$ to contain the finite set of points where $\tau$ changes slope, and then enlarging $V(\Gamma')$ and $V(\Gamma)$ to ensure that the image and the preimage of a vertex is a vertex. We let the degree of $\varphi$ on each edge and leg be the slope of $\tau$. Given a model $\varphi:\Gamma'\to \Gamma$ of $\tau$, we can produce another model by adding more vertices to $\Gamma'$ and $\Gamma$. Conversely, any morphism $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ to a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ with $\chi(\scalebox{0.8}[1.3]{$\sqsubset$})<0$ has a unique {\it minimal model} $\varphi_{min}:\Gamma'_{min}\to \Gamma_{min}$ with the property that every simple vertex $v\in V(\Gamma_{min})$ has at least one preimage that is not simple.
\begin{example} Let $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ be an unramified morphism of tropical curves of local degree one. Then $\varphi$ is a topological covering space of degree $\deg \tau$. Conversely, if $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a tropical curve and $f:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ is a covering space of finite degree, then there is a unique way to give $\scalebox{0.8}[1.3]{$\sqsubset$}'$ the structure of a tropical curve such that $f$ is unramified: we define the genus function on $\scalebox{0.8}[1.3]{$\sqsubset$}'$ as the pullback of the genus function on $\scalebox{0.8}[1.3]{$\sqsubset$}$.
\end{example}
\section{Dilated cohomology} \label{sec:dilated}
In the following two sections, we fix a finite abelian group $G$ and classify the $G$-covers of a given unweighted graph $\Gamma$. These are defined as surjective finite morphisms $\varphi:\Gamma'\to \Gamma$ together with an $G$-action on $\Gamma'$ that preserves and acts transitively on the fibers. We will see that a $G$-cover of $\Gamma$ is uniquely determined by two objects. The first is a {\it $G$-dilation datum} $D$ on $\Gamma$ (equivalently, a {\it $G$-stratification} $\mathcal{S}$ of $\Gamma$), recording the fibers of $\varphi$ in terms of local stabilizer subgroups of $G$. The second is an element of a {\it dilated cohomology group} $H^1(\Gamma,D)$ (or $H^1(\Gamma,\mathcal{S})$), which generalizes the first simplicial cohomology group $H^1(\Gamma,G)$ by taking the local stabilizers into account.
We introduce $G$-covers, $G$-dilation data and $G$-stratifications in Sec.~\ref{sec:dilation}. In Sec.~\ref{sec:cohomology}, we introduce the dilated cohomology groups $H^i(\Gamma,D)$ of a pair $(\Gamma,D)$, where $\Gamma$ is a graph and $D$ is a $G$-dilation datum on $\Gamma$. In Sec.~\ref{sec:relative} we introduce the long exact sequence in dilated cohomology and study the cohomology groups of a subgraph $\Delta\subset \Gamma$. Once all the relevant definitions have been established, we reach Sec.~\ref{sec:classification}, which is mostly dedicated to proving our classification results.
\subsection{$G$-covers, dilation data, and stratifications} \label{sec:dilation}
Throughout this section, we only consider unweighted graphs with legs. We now give the main definition of our paper.
\begin{definition} Let $\Gamma$ be a graph. A {\it $G$-cover} of $\Gamma$ is a finite surjective morphism $\varphi:\Gamma'\to \Gamma$ together with an action of $G$ on $\Gamma'$, such that the following properties are satisfied:
\begin{enumerate}
\item The action is invariant with respect to $\varphi$.
\item For each $x\in X(\Gamma)$, the group $G$ acts transitively on the fiber $\varphi^{-1}(x)$.
\end{enumerate}
\end{definition}
\begin{example} Let $\Gamma$ be a graph with a $G$-action (see Def.~\ref{def:Gquotient}), then the quotient map $\pi:\Gamma\to \Gamma/G$ is a $G$-cover.
\end{example}
\begin{example} Let $\Gamma$ be a graph. Viewing $\Gamma$ as a topological space, an element of $H^1(\Gamma,G)$ determines a covering space $\varphi:\Gamma'\to \Gamma$ with a $G$-action. It is clear that we can equip $\Gamma'$ with the structure of a graph such that $\varphi$ is a $G$-cover of graphs. Such $G$-covers, which we call {\it topological $G$-covers}, are distinguished by the property that $G$ acts freely on each fiber $\varphi^{-1}(x)$. For such covers, the $G$-dilation datum is trivial, while the the dilated cohomology group is $H^1(\Gamma,G)$. An example with $G$ the Klein group is given below in Fig.~\ref{subfig:cover1}.
\label{ex:topologicalcovers}
\end{example}
Our goal is to describe all $G$-covers $\varphi:\Gamma'\to \Gamma$ of a given graph $\Gamma$. We begin our description by considering the local stabilizer subgroups.
\begin{definition} Let $\Gamma$ be a graph. A {\it $G$-dilation datum} $D$ on $\Gamma$ is a choice of a subgroup $D(x)\subset G$ for every $x\in X(\Gamma)$, such that $D(h)\subset D(r(h))$ for every half-edge $h\in H(\Gamma)$, and such that $D(h)=D(h')$ for each edge $e=\{h,h'\}\in E(\Gamma)$. Given $G$-dilation data $D$ and $D'$ on $\Gamma$, we say that $D$ is a {\it refinement} of $D'$ if $D(x)\subset D'(x)$ for all $x\in X(\Gamma)$. A {\it $G$-dilated graph} is a pair $(\Gamma,D)$ consisting of a graph $\Gamma$ and a $G$-dilation datum $D$ on $\Gamma$.
\end{definition}
We call $D(x)$ the {\it dilation group} of $x\in X(\Gamma)$, and for an edge $e=\{h,h'\}\in E(\Gamma)$ we call $D(e)=D(h)=D(h')$ the {\it dilation group} of $e$. If $e$ is an edge with root vertices $u$ and $v$ (which may be the same), then $D(e)\subset D(u)\cap D(v)$. We call $C(e)=D(u)+D(v)$ the {\it vertex dilation group} of the edge $e$.
\begin{remark}\label{rem:graphOfGroups} A $G$-dilation datum on a graph $\Gamma$ is an example of a {\it graph of groups}, as defined by Bass (see Def.~1.4 in \cite{Bass}). In a future paper, we plan to explore the relationship between the cohomology groups $H^i(\Gamma,D)$ and the fundamental group of the graph of groups defined by $D$, with the goal of extending our theory to the non-abelian case.
\end{remark}
\begin{definition} Let $\varphi:\Gamma'\to \Gamma$ be a $G$-cover. We define the {\it $G$-dilation datum} $D_{\varphi}$ of $\varphi$ by setting $D_{\varphi}(x)$ for $x\in X(\Gamma)$ to be the stabilizer group of any $x'\in \varphi^{-1}(x)$.
\end{definition}
The group $G$ is assumed to be abelian, therefore the stabilizer group of $x'\in \varphi^{-1}(x)$ does not depend on the choice of $x'$.
\begin{remark} If $D_{\varphi}$ is the $G$-dilation datum of a $G$-cover $\varphi$ that is the tropicalization of a $G$-cover of algebraic curves, then the dilation subgroup of every half-edge is cyclic (this follows, for instance, from \cite[Lemma 3.1]{SchmittvanZelm}). As a result, many of the covers described throughout this paper are not algebraically realizable, e.g. the cover~\ref{subfig:cover6} below. Our approach is to develop, as far as possible, an independent theory of $G$-covers of graphs, so we do not impose this condition from the start. In any case, as we shall see, the dilation groups of the half-edges play a secondary role in the classification of $G$-covers.
\label{rem:edgecyclic}
\end{remark}
For any $x\in X(\Gamma)$, the fiber $\varphi^{-1}(x)$ of a $G$-cover $\varphi:\Gamma'\to \Gamma$ is a $G/D_{\varphi}(x)$-torsor. If $h\in T_v(\Gamma)$ is a half-edge rooted at $v\in V(\Gamma)$, then the root map $r:\varphi^{-1}(h)\to \varphi^{-1}(v)$ is an equivariant map of transitive $G$-sets, which implies that $D_{\varphi}(h)\subset D_{\varphi}(v)$. Furthermore, it is clear that $D_{\varphi}(h)=D_{\varphi}(h')$ for any edge $e=\{h,h'\}\in E(\Gamma)$. Therefore, $D_{\varphi}$ is a $G$-dilation datum.
The cardinality of each fiber $\varphi^{-1}(x)$ equals the index of $D_{\varphi}(x)$ in $G$:
$$
\#(\varphi^{-1}(x))=[G:D_{\varphi}(x)].
$$
Furthermore, for a half-edge $h\in H(\Gamma)$ rooted at $r(h)=v\in V(\Gamma)$, the $[G:D_{\varphi}(h)]$ half-edges in the fiber $\varphi^{-1}(h)$ are partitioned by their root vertices into $\#(\varphi^{-1}(v))=[G:D_{\varphi}(v)]$ subsets, each containing $[D_{\varphi}(v):D_{\varphi}(h)]$ elements.
\begin{example}[Klein covers] \label{ex:Klein1} We now give several of examples of $G$-covers in the simplest non-cyclic case, when $G=\Zmod{2}\oplus\Zmod{2}$ is the Klein group. The base graph $\Gamma$ consists of two vertices $u$ and $v$ joined by two edges $e$ and $f$.
We use the following notation to describe a $G$-cover $\varphi:\Gamma'\to \Gamma$. We denote the elements of $G$ by $00$, $10$, $01$, and $11$, and denote the subgroups generated by $10$, $01$, and $11$ by respectively $H_1$, $H_2$, and $H_3$. The vertices of $\Gamma'$ lying above $u$ and $v$ are labeled (non-uniquely if the corresponding stabilizer is non-trivial) $u_{ij}$ and $v_{ij}$ for $ij\in G$, and the action of $G$ on $\varphi^{-1}(u)$ and $\varphi^{-1}(v)$ is the natural additive action on the indices. We color the edges $\varphi^{-1}(e)$ and $\varphi^{-1}(f)$ red and blue, respectively, and label them with indices $ij$ in such a way that $e_{ij}$ and $f_{ij}$ are attached to $u_{ij}$. The sizes of the vertices and the thickness of the edges of $\Gamma'$ denote the size of the dilation subgroup. In the caption, we indicate the nontrivial dilation groups. In Ex.~\ref{ex:Klein2}, we will enumerate all Klein covers of $\Gamma$.\end{example}
\begin{figure}
\begin{center}
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\node at (0,-1.5) {\parbox{0.3\linewidth}{\subcaption{$D(u),D(e)=H_3, D(v)=G$}\label{subfig:cover5}}};
\end{scope}
\begin{scope}[shift={(12,-6)}]
\draw [blue,ultra thick] (-1,1.5) .. controls (-1,1) and (1,1) .. (1,1.5);
\draw [red] (-1,1.5) to (1,1.5);
\draw [red] (-1,1.5) .. controls (-1,2) and (1,2) .. (1,1.5);
\draw [red] (-1,1.5) .. controls (-1,2.5) and (1,2.5) .. (1,1.5);
\draw [red] (-1,1.5) .. controls (-1,3) and (1,3) .. (1,1.5);
\draw [fill, white] (0,1.5) circle (1.5mm);
\draw [fill, white] (0,1.87) circle (1.5mm);
\draw [fill, white] (0,2.24) circle (1.5mm);
\draw [fill, white] (0,2.61) circle (1.5mm);
\draw [] (0,1.5) node {\tiny $11$};
\draw [] (0,1.87) node {\tiny $01$};
\draw [] (0,2.24) node {\tiny $10$};
\draw [] (0,2.61) node {\tiny $00$};
\draw [ left] (-1,1.5) node {\small $u_{ij}$};
\draw [ right] (1,1.5) node {\small $v_{ij}$};
\draw[fill] (-1,1.5) circle(1mm);
\draw[fill] (1,1.5) circle(1mm);
\draw [red] (-1,0) .. controls (-1,0.3) and (1,0.3) .. (1,0);
\draw [blue] (-1,0) .. controls (-1,-0.3) and (1,-0.3) .. (1,0);
\draw[fill] (-1,0) circle(.6mm);
\draw[fill] (1,0) circle(.6mm);
\draw [ left] (-1,0) node {\small $u$};
\draw [ right] (1,0) node {\small $v$};
\draw [above] (0,0.25) node {\small $e$};
\draw [below] (0,-0.25) node {\small $f$};
\node at (0,-1.5) {\parbox{0.3\linewidth}{\subcaption{$D(u),D(v),D(f)=G$}\label{subfig:cover6}}};
\end{scope}
\end{tikzpicture}\end{center}
\caption{Klein covers of a genus 1 graph}
\label{fig:Klein}
\end{figure}
We now give an alternative way to record a $G$-dilation datum on $\Gamma$, by means of a stratification of $\Gamma$ indexed by the subgroups of $G$. This description is often easier to visualize, and generalizes more naturally to tropical curves.
\begin{definition} Let $\Gamma$ be a graph. A {\it $G$-stratification} $\mathcal{S}=\big\{\Gamma_H\big\vert H\in S(G)\big\}$ on $\Gamma$ is a collection of subgraphs $\Gamma_H\subset \Gamma$ indexed by the set $S(G)$ of subgroups of $G$, such that
\begin{equation}\label{eq:strat}
\begin{split}
\Gamma_0&=\Gamma,\\
\Gamma_K&\subset \Gamma_H\mbox{ if }H\subset K, \textrm{ and} \\
\Gamma_H\cap \Gamma_K&=\Gamma_{H+K}\mbox{ for all }H,K\in S(G).
\end{split}\end{equation}
We allow the $\Gamma_H$ to be empty or disconnected for $H\neq 0$. The union of the $\Gamma_H$ for $H\neq 0$ is called the {\it dilated subgraph} of $\Gamma$ and is denoted $\Gamma_{dil}$.
\label{def:stratification}
\end{definition}
We can associate a $G$-stratification of $\Gamma$ to a $G$-dilation datum $D$, and vice versa.
\begin{definition} Let $\Gamma$ be a graph, and let $D$ be a $G$-dilation datum on $\Gamma$. We define the {\it $G$-stratification $\mathcal{S}(D)=\{\Gamma_H:H\in S(G)\}$ associated to $D$} as follows:
$$
\Gamma_H=\big\{x\in X(\Gamma)\big\vert H\subset D(x)\big\}.
$$
\end{definition}
We observe that for any half-edge $h\in H(\Gamma)$ we have $D(h)\subset D(r(h))$, therefore each $\Gamma_H$ is indeed a subgraph of $\Gamma$.
\begin{remark} Let $D_{\varphi}$ be the $G$-dilation datum associated to a $G$-cover $\varphi:\Gamma'\to \Gamma$. Then for any $H\in S(G)$, $\Gamma_H$ is the image under $\varphi$ of the subgraph of $\Gamma'$ fixed under the action of $H$.
\end{remark}
A $G$-dilation datum $D$ can be uniquely recovered from a $G$-stratification $\mathcal{S}$ as follows. Condition~\eqref{eq:strat} implies that the set $X(\Gamma)$ is partitioned into disjoint subsets (which are not subgraphs in general)
$$
X(\Gamma)=\coprod_{H\in S(G)}\Gamma_H\backslash \Gamma^0_H,\mbox{ where } \Gamma_{H}^0=\bigcup_{H\subsetneq K} \Gamma_K.
$$
For any $x\in X(\Gamma)$ we set $D(x)=H$, where $H$ is the unique subgroup of $G$ such that $x\in \Gamma_H\backslash \Gamma^0_H$.
We also define a dual stratification associated to a $G$-dilation datum.
\begin{definition} Let $D$ be a $G$-dilation datum on $\Gamma$. The {\it dual stratification} $\mathcal{S}^*(D)=\{\Gamma^H:H\in S(G)\}$ of $\mathcal{S}$ is defined as follows. For $H\in S(G)$, we define $\Gamma^H$ to be the edge-maximal subgraph of $\Gamma$ whose vertex set is
$$
V(\Gamma^H)=\bigcup_{K\subset H}V(\Gamma_K\backslash\Gamma^0_K)=\big\{v\in V(\Gamma)\big\vert D(v)\subset H\big\}.
$$
In other words, a leg of $\Gamma$ with root vertex $v$ lies in $\Gamma^H$ if and only if $D(v)\subset H$, and an edge $e\in E(\Gamma)$ with root vertices $u$ and $v$ lies in $\Gamma^H$ if and only if $C(e)=D(u)+D(v)\subset H$.
The dual stratification satisfies the following properties:
\begin{equation*}\begin{split}
\Gamma^G&=\Gamma,\\
\Gamma^H&\subset \Gamma^K\mbox{ if }H\subset K, \textrm{ and }\\ \Gamma_H\cap\Gamma_K&=\Gamma_{H\cap K}\mbox{ for all }H,K\in S(G).
\end{split}\end{equation*}
\end{definition}
\begin{remark} Unlike $\mathcal{S}(D)$, the dual stratification $\mathcal{S}^*(D)$ of a $G$-dilation datum does not uniquely determine $D$. For a vertex $v\in V(\Gamma)$, we can recover $D(v)$ as the smallest subgroup $H\subset G$ such that $v\in V(\Gamma^H)$, but the dilation groups $D(h)$ of the edges cannot be determined. For example, let $\Gamma$ be the graph consisting of a vertex $v$ and a loop $e$, let $D(v)=H$ be a subgroup of $G$, and let $D(e)$ be any subgroup of $H$. The dual stratification is
$$
\Gamma^K=\left\{\begin{array}{cc} \Gamma & \textrm{ if } \ H\subseteq K,\\ \emptyset & \textrm{ if } \ H\subsetneq K,\end{array}\right.
$$
so we can recover $H$ but not $D(e)$.
\end{remark}
Finally, we define morphisms of $G$-covers of $\Gamma$.
\begin{definition} Let $\varphi_1:\Gamma_1'\to \Gamma$ and $\varphi_2:\Gamma_2'\to \Gamma$ be $G$-covers. A {\it morphism of $G$-covers} from $\varphi_1$ to $\varphi_2$ is a $G$-equivariant morphism $\psi:\Gamma_1'\to \Gamma_2'$ such that $\varphi_1=\varphi_2\circ \psi$.
\label{def:morphismofGcovers}
\end{definition}
We observe that if $\psi:\Gamma_1'\to \Gamma_2'$ is a morphism of $G$-covers from $\varphi_1:\Gamma_1'\to \Gamma$ to $\varphi_2:\Gamma_2'\to \Gamma$, then for any $x\in X(\Gamma)$ the restriction of $\tau$ to the fiber $\varphi_1^{-1}(x)$ is a $G$-equivariant surjective map onto $\varphi_2^{-1}(x)$, which implies that $D_{\varphi_1}(x)\subset D_{\varphi_2}(x)$, in other words $D_{\varphi_1}$ is a refinement of $D_{\varphi_2}$.
\begin{remark} In this paper, we only consider $G$-covers of a fixed base graph $\Gamma$ (except that we do consider restrictions of covers to a subgraph). It is also possible to define morphisms of $G$-covers of graphs that are related by a morphism. For example, given a $G$-cover $\varphi:\Gamma'\to \Gamma$ and a morphism $\psi:\Delta\to \Gamma$, we define the pullback $G$-cover $\varphi':\Delta'\to \Delta$ by taking $\Delta'$ to be the fiber product $\Gamma'\times_{\Gamma} \Delta'$ (defined by $X(\Delta')=X(\Gamma')\times_{X(\Gamma)}X(\Delta')$ with coordinatewise involution and root maps), and letting $G$ act on the first factor. All of the constructions of this chapter are functorial with respect to such operators, so for example the $G$-dilation datum $D_{\varphi'}$ on $\Delta$ is equal to the pullback $G$-dilation datum $\psi^*D_{\varphi}=D_{\varphi}\circ \psi$.
\label{rem:fixedbase}
\end{remark}
\subsection{Cohomology of $G$-data} \label{sec:cohomology}
In this subsection, we define the cohomology groups $H^0(\Gamma,D)$ and $H^1(\Gamma,D)$ of a $G$-dilated graph $(\Gamma,D)$. These groups generalize the simplicial cohomology groups $H^i(\Gamma,G)$ of $\Gamma$ with coefficients in $G$. The groups $H^i(\Gamma,D)$ do not depend on the legs of $\Gamma$, so we assume for simplicity that $\Gamma$ has no legs. The legs of $\Gamma$ will again play a role in Sec.~\ref{sec:admissible}, when we classify unramified $G$-covers of weighted graphs.
Rather than only considering $G$-dilation data on a graph $\Gamma$, we work in a larger category of {\it $G$-data} on $\Gamma$, a $G$-datum being simply a choice of a $G$-group at every vertex and every edge of $\Gamma$ that is consistent with the root maps (see Definition~\ref{rem:graphOfGroups}). A $G$-datum $D_\varphi$ arising from a $G$-cover $\varphi$ is always a $G$-dilation datum. However, cohomology groups of the more general $G$-data appear in the long exact sequence~\eqref{eq:les} that relates the cohomology groups $H^i(\Gamma,D)$ of a $G$-dilation datum $D$ on $\Gamma$ to the cohomology groups $H^i(\Delta,D|_{\Delta})$ of the restriction of $D$ to a subgraph $\Delta\subset \Gamma$.
We begin by recalling the simplicial cohomology groups of a graph $\Gamma$ with coefficients in $G$. Choose an orientation on the edges, and let $s,t:E(\Gamma)\to V(\Gamma)$ be the source and target maps. The simplicial chain complex of $\Gamma$ is
$$
\begin{tikzcd}
0\arrow[r] & \mathbb{Z}^{E(\Gamma)}\arrow[r,"\delta"]& \mathbb{Z}^{V(\Gamma)} \arrow[r]& 0,
\end{tikzcd}
$$
with the boundary map defined on the generators of $\mathbb{Z}^{E(\Gamma)}$ by $\delta(e)=t(e)-s(e)$. Applying the functor $\Hom(-, G)$ and identifying
$$
\Hom(\mathbb{Z}^{V(\Gamma)},G)=G^{V(\Gamma)}\quad \textrm{ and } \quad\Hom(\mathbb{Z}^{E(\Gamma)},G)=G^{E(\Gamma)},
$$
we obtain the simplicial cochain complex of $\Gamma$ with coefficients in $G$:
\begin{equation}
\begin{tikzcd}
0 \arrow[r] & G^{V(\Gamma)} \arrow[r,"\delta^*"] & G^{E(\Gamma)} \arrow[r] & 0.\label{eq:simplicialcomplex}
\end{tikzcd}
\end{equation}
We identify elements of $G^{V(\Gamma)}$ and $G^{E(\Gamma)}$ with functions $\xi:V(\Gamma)\to G$ and $\eta:E(\Gamma)\to G$, respectively. Under this identification, the duals $s^*,t^*:G^{V(\Gamma)}\to G^{E(\Gamma)}$ of the maps $s$ and $t$ are
$$
s^*(\xi)(e)=\xi(s(e))\quad \textrm{ and }\quad t^*(\xi)(e)=\xi(t(e)),
$$
and the coboundary map is equal to
\begin{equation}
\delta^*=t^*-s^*.\label{eq:coboundarygroup}
\end{equation}The \emph{simplicial cohomology groups} of $\Gamma$ with coefficients in $G$ are
$$
H^0(\Gamma,G)=\Ker \delta^* \quad\textrm{ and }\quad H^1(\Gamma,G)= \Coker \delta^*.
$$
We now generalize this construction by replacing every copy of $G$ in the cochain complex~\eqref{eq:simplicialcomplex} with an arbitrary $G$-group. We recall that a {\it $G$-group} is a map of abelian groups $f:G\to H$, and a {\it morphism of $G$-groups} from $f_1:G\to H_1$ to $f_2:G\to H_2$ is a group homomorphism $g:H_1\to H_2$ such that $f_2=g\circ f_1$.
\begin{definition}\label{def:Gdatum} A {\it $G$-datum} $A$ on an oriented graph $\Gamma$ consists of the following:
\begin{enumerate}
\item For every vertex $v\in V(\Gamma)$, a $G$-group $f_v:G\to A(v)$.
\item For every edge $e\in E(\Gamma)$, a $G$-group $f_e:G\to A(e)$ and morphisms of $G$-groups $s_e:A\big(s(e)\big)\to A(e)$ and $t_e:A\big(t(e)\big)\to A(e)$ such that $s_e\circ f_{s(e)}=t_e\circ f_{t(e)}=f_e$, i.e. for which the diagram
$$
\begin{tikzcd}
& G \ar[dl,"f_{s(e)}"'] \ar[d,"f_e"] \ar[dr,"f_{t(e)}"] & \\ A(s(e)) \ar[r,"s_e"'] & A(e) & A(t(e)) \ar[l,"t_e"] \\
\end{tikzcd}
$$
commutes.
\end{enumerate}
\end{definition}
In other words, a $G$-datum on $\Gamma$ is a functor to the category of $G$-groups from the category whose objects are $V(\Gamma)\cup E(\Gamma)$, and whose non-trivial morphisms are the source and target maps. In contrast with Remark \ref{rem:graphOfGroups}, a $G$-datum is not necessarily a graph of groups in the sense of~\cite{Bass}, since the maps $f_{s(e)}$ and $f_{t(e)}$ are not required to be injective.
To verify that $G$-data, in fact, generalize the notion of $G$-dilation data, we associate a $G$-datum $A^D$ to each $G$-dilation datum $D$. First, let $H_1$ and $H_2$ be subgroups of $G$, let $f_i:G\to G/H_i$ be the projections, and let $\iota_i:G/H_i\to G/H_1\oplus G/H_2$ be the embeddings. The coproduct of $f_1$ and $f_2$ is the $G$-group
$$
G/H_1\sqcup_G G/H_2=(G/H_1\oplus G/H_2)/(\Im f_1\oplus -f_2).
$$
The natural map $f_1\sqcup f_2:G\to G/H_1\sqcup_G G/H_2$ is equal to $\pi\circ \iota_1\circ f_1=\pi\circ \iota_2\circ f_2$, where $\pi:G/H_1\oplus G/H_2\to G/H_1\sqcup_G G/H_2$ is the projection. It is clear that $f_1\sqcup f_2$ is surjective and that $\Ker f_1\sqcup f_2=H_1+H_2$, hence the $G$-group $G\to G/H_1\sqcup_G G/H_2$ can be identified with the quotient $G\to G/(H_1+H_2)$.
\begin{definition} Let $\Gamma$ be an oriented graph, and let $D$ be a $G$-dilation datum on $\Gamma$. We define the {\it associated $G$-datum} $A^D$ as follows. For each $v\in V(\Gamma)$, we set $A^D(v)=G/D(v)$, and let $f_v$ be the natural projection map:
$$
f_v:G\to A^D(v)=G/D(v).
$$
For an edge $e\in E(\Gamma)$, we let $f_e=f_{s(e)}\sqcup f_{t(e)}$ be the coproduct. In other words, we let
$$
A^D(e)=[G/D(s(e))\oplus G/D(t(e))]/(\Im f_{s(e)}\oplus -f_{t(e)})\simeq G/C(e),
$$
where $C(e)=D(s(e))+D(t(e))$ is the edge dilation group. We let
$$
f_e:G\to A^D(e)\simeq G/C(e)
$$
be the quotient map, and we let
$$
s_e:A^D(s(e))\to A^D(e) \quad\textrm{ and } \quad t_e:A^D(t(e))\to A^D(e)
$$
be the natural quotient maps $G/D(s(e))\to G/C(e)$ and $G/D(t(e))\to G/C(e)$.
\end{definition}
We now define the cochain complex and cohomology groups of a $G$-datum $A$ on an oriented graph $\Gamma$.
\begin{definition} Let $G$ be an oriented graph, and let $A$ be a $G$-datum on $\Gamma$. We define the {\it cochain groups} of the pair $(\Gamma,A)$ as follows:
$$
C^0(\Gamma,A)=\prod_{v\in V(\Gamma)}A(v)=\Big\{\xi:V(\Gamma)\to \coprod_{v\in V(\Gamma)} A(v):\xi(v)\in A(v)\Big\},
$$
$$
C^1(\Gamma,A)=\prod_{e\in E(\Gamma)}A(e)=\Big\{\eta:E(\Gamma)\to \coprod_{e\in E(\Gamma)} A(e):\eta(e)\in A(e)\Big\}.
$$
We define the morphisms $s^*,t^*:C^0(\Gamma,A)\to C^1(\Gamma,A)$ by
$$
s^*(\xi)(e)=s_e(\xi(s(e)))\quad\textrm{ and } \quad t^*(\xi)(e)=t_e(\xi(t(e))).
$$
We define the {\it cochain complex} of the pair $(\Gamma,A)$ as
$$
\begin{tikzcd}
0 \ar[r] & C^0(\Gamma,A)\ar[r,"\delta_{\Gamma,A}^*"] & C^1(\Gamma,A) \ar[r] & 0,
\end{tikzcd}
$$
where the {\it coboundary map} $\delta_{\Gamma,A}^*$ is
\begin{equation}
\delta_{\Gamma,A}^* =t^*-s^*.
\label{eq:coboundary}
\end{equation}
We define the {\it cohomology groups} of the pair $(\Gamma,A)$ as
$$
H^0(\Gamma,A)=\Ker \delta_{\Gamma,A}^*\quad\textrm{ and } \quad H^1(\Gamma,A)=\Coker \delta_{\Gamma,A}^*.
$$
\end{definition}
Specializing to $G$-dilation data, we obtain the main definition of this section.
\begin{definition} Let $(\Gamma,D)$ be a $G$-dilated graph, and let $A^D$ be the $G$-datum associated to $D$. The {\it cochain complex} of $(\Gamma,D)$ is the cochain complex of the pair $(\Gamma,A^D)$:
$$\begin{tikzcd}
0 \ar[r] & C^0(\Gamma,D)\ar[r,"\delta_{\Gamma,D}^*"] & C^1(\Gamma,D) \ar[r] & 0,
\end{tikzcd}
$$
where
$$
C^i(\Gamma,D)=C^i(\Gamma,A^D) \quad\textrm{ and } \quad
\delta_{\Gamma,D}^*=\delta_{\Gamma,A^D}^*.
$$
The {\it dilated cohomology groups} $H^i(\Gamma,D)$ are the cohomology groups of $(\Gamma,A^D)$:
\begin{equation}
H^0(\Gamma,D)=\Ker \delta_{\Gamma,D}^*=H^0(\Gamma,A^D)\quad \textrm{ and }\quad H^1(\Gamma,D)=\Coker \delta_{\Gamma,D}^*=H^1(\Gamma,A^D).
\end{equation}\label{def:dilatedcohomology}
\end{definition}
For the sake of clarity, and for future use, we give an explicit description of $H^1(\Gamma,D)$ as a quotient. The cochain group $C^1(\Gamma,D)$ is the direct product of $A^D(e)$ over all $e\in E(\Gamma)$, where each $A^D(e)$ is the coproduct $G/C(e)$ of $G\to G/D(s(e))$ and $G\to G/D(t(e))$. In other words, each $\eta\in C^1(\Gamma,D)$ is given by choosing a pair of elements $(\eta_s(e),\eta_t(e))\in G/D(s(e))\oplus G/D(t(e))$ for each $e\in E(\Gamma)$. A tuple $(\eta_s(e),\eta_t(e))_{e\in E(\Gamma)}$ is equivalent to $(\widetilde{\eta}_s(e),\widetilde{\eta}_t(e))_{e\in E(\Gamma)}$ if and only if there exist elements $\omega(e)\in G$ for all $e\in E(\Gamma)$ such that
\begin{equation*}\begin{split}
\eta_s(e)&=\widetilde{\eta}_s(e)+\omega(e)\mod D(s(e))\\ \eta_t(e)&=\widetilde{\eta}_t(e)-\omega(e)\mod D(t(e)).
\end{split}\end{equation*}
Note that, instead of assuming that $\omega(e)\in G$, we may assume that $\omega(e)$ lies in any quotient group between $G$ and $G/(D(s(e))\cap D(t(e)))$, and it is natural to assume that in fact $\omega(e)\in G/D(e)$.
An element of $C^0(\Gamma,D)$ is given by choosing $\xi(v)\in G/D(v)$ for each $v\in V(\Gamma)$. Putting everything together, we see that an element $[\eta]\in H^1(\Gamma,D)$ is given by choosing a pair of elements $(\eta_s(e),\eta_t(e))\in G/D(s(e))\oplus G/D(t(e))$ for each $e\in E(\Gamma)$, and that two choices $(\eta_s(e),\eta_t(e))_{e\in E(\Gamma)}$ and $(\widetilde{\eta}_s(e),\widetilde{\eta}_t(e))_{e\in E(\Gamma)}$ represent the same element of $H^1(\Gamma,D)$ if and only if there exist elements $\omega(e)\in G/D(e)$ for all $e\in E(\Gamma)$ and elements $\xi(v)$ for all $v\in V(\Gamma)$ such that
\begin{equation}\begin{split}
\eta_s(e)&=\widetilde{\eta}_s(e)-\xi(s(e))+\omega(e)\mod D(s(e))\\ \eta_t(e)&=\widetilde{\eta}_t(e)+\xi(t(e))-\omega(e)\mod D(t(e))
\end{split}\label{eq:explicitH1}
\end{equation}
for all $e\in E(\Gamma)$.
\begin{remark} The dilated cochain complex of $(\Gamma,D)$, and hence the cohomology groups $H^i(\Gamma,D)$, depend only on the dilation groups $D(v)$ of the vertices $v\in V(\Gamma)$, and do not depend on the edge groups $D(e)$. Specifically, given a graph $\Gamma$, we can choose the dilation groups $D(v)$ of the vertices $v\in V(\Gamma)$ arbitrarily, and for each edge $e\in E(\Gamma)$ choose $D(e)$ to be any subgroup of $D(s(e))\cap D(t(e))$. The resulting groups $H^0(\Gamma,D)$ and $H^1(\Gamma,D)$ are independent of the choice of the $D(e)$. In other words, the dilated cohomology groups of $(\Gamma,D)$ only depend on the dual stratification $\mathcal{S}^*(D)$.
\end{remark}
For the remainder of the paper, with the exception of Sec.~\ref{sec:relative} below, we restrict our attention to $G$-dilation data and their cohomology groups. Before we proceed, we calculate our first example, showing that we have in fact generalized simplicial cohomology.
\begin{example} Let $\Gamma$ be a graph, and let $A_G$ be the {\it trivial $G$-datum}, namely $A_G(v)=G$ and $A_G(e)=G$ for all $v\in V(\Gamma)$ and all $e\in E(\Gamma)$, with all structure maps being the identity. Alternatively, $A_G$ is the $G$-datum associated to the {\it trivial $G$-dilation datum} $D_0$ given by $D_0(x)=0$ for all $x\in X(\Gamma)$. It is clear that $C^i(\Gamma,A_G)=C^i(\Gamma,G)$, and that the coboundary map $\delta^*_{\Gamma,A_G}$ given by~\eqref{eq:coboundary} is equal to $\delta^*$ given by~\eqref{eq:coboundarygroup}. Hence $H^i(\Gamma,A_G)=H^i(\Gamma,D_0)=H^i(\Gamma,G)$.
\label{ex:trivialdilation}
\end{example}
We now work out several explicit examples of the cohomology groups $H^i(\Gamma,D)$ of $G$-dilated graphs $(\Gamma,D)$. In the previous example, we saw that the cohomology of the trivial $G$-dilation datum on $\Gamma$ is the simplicial cohomology of $\Gamma$ with coefficients in $G$. In particular, $H^1(\Delta,G)$ is trivial for any tree $\Delta$. We now show that $H^1(\Delta,D)=0$ for any $G$-dilation datum $D$ on a tree $\Delta$.
\begin{proposition} Let $\Delta$ be a tree. Then $H^1(\Delta,D)=0$ for any $G$-dilation datum $D$ on $\Delta$.
\label{prop:tree}
\end{proposition}
\begin{proof} Let $\Gamma$ be an arbitrary graph, and suppose that $D$ and $D'$ are two $G$-dilation data on $\Gamma$, such that $D$ is a refinement of $D'$. In this case, we can define natural surjective maps $\pi^i:C^i(\Gamma,D)\to C^i(\Gamma,D')$ by taking coordinatewise quotients. These maps commute with the coboundary maps and induce maps $\pi^i:H^i(\Gamma,D)\to H^i(\Gamma,D')$, and furthermore the map $\pi^1$ is surjective.
Now suppose that $\Delta$ is a tree, and $D$ is a $G$-dilation datum on $\Delta$. Let $D_0$ be the trivial $G$-dilation datum on $\Delta$. Then $D_0$ is a refinement of $D$, so there is a surjective map $H^1(\Delta,D_0)\to H^1(\Delta,D)$. But by Ex.~\ref{ex:trivialdilation} we know that $H^1(\Delta,D_0)=H^1(\Delta,G)=0$, hence $H^1(\Delta,D)=0$.
\end{proof}
We now work out an example of $H^i(\Gamma,D)$ for a topologically non-trivial graph $\Gamma$.
\begin{example} Let $\Gamma$ be the graph consisting of two vertices $v_1$ and $v_2$ joined by $n$ edges $e_1,\ldots,e_n$, oriented such that $s(e_i)=v_1$ and $t(e_i)=v_2$. Let $H_1$ and $H_2$ be two subgroups of $G$, and consider the following $G$-dilation datum on $\Gamma$:
$$
D(v_1)=H_1,\quad D(v_2)=H_2\quad\textrm{ and }\quad D(e_i)\mbox{ are arbitrary subgroups of }H_1\cap H_2.
$$
We see that $C(e_i)=H_1+ H_2$ for all $i$, therefore
$$
C^0(\Gamma,D)=G/H_1\oplus G/H_2\quad\textrm{ and }\quad C^1(\Gamma,D)=[G/(H_1+ H_2)]^n.
$$
The coboundary map $\delta^*_{\Gamma,D}$ is the composition of the projection
$$
\pi:G/H_1\oplus G/H_2\to G/H_1\sqcup G/H_2\simeq G/(H_1+H_2)
$$
and the diagonal map. Therefore
\begin{equation}
H^0(\Gamma,D)=\Ker \pi\simeq G/(H_1\cap H_2)\quad\textrm{ and }\quad H^1(\Gamma,D)\simeq [G/(H_1+ H_2)]^{n-1}.
\end{equation}
\label{ex:nedges}
\end{example}
We also show that cohomology of $G$-dilation data can be used to compute simplicial cohomology of edge-maximal subgraphs of $\Gamma$, with coefficients in any quotient group of $G$.
\begin{example} Let $\Gamma$ be a graph, let $\Delta\subset \Gamma$ be an edge-maximal subgraph, and let $H\subset G$ be a subgroup. Consider the following $G$-dilation datum on $\Gamma$:
$$
D_{\Delta,H}(x)=\left\{\begin{array}{cc} H,& x\in X(\Delta), \\ G, & x\notin X(\Delta).\end{array}\right.
$$
By definition, an edge $e\in E(\Gamma)$ lies in $\Delta$ if and only if both of its root vertices do. It follows that the dilated cochain complex $C^*(\Gamma,D_{\Delta,H})$ is equal to the simplicial cochain complex $C^*(\Delta,G/H)$, and hence
$$
H^i(\Gamma,D_{\Delta,H})=H^i(\Delta,G/H)\mbox{ for }i=0,1.
$$
\label{ex:subgraph}
\end{example}
\begin{remark} We will show in Sec.~\ref{sec:Gcoversofgraphs} that the group $H^1(\Gamma,D)$ classifies $G$-covers of $\Gamma$ with dilation datum $D$. We do not know of a similar geometric interpretation of the group $H^0(\Gamma,D)$. For the trivial dilation datum $D=0$, the group $H^0(\Gamma,D)=H^0(\Gamma,G)$ is equal to $G$ for any connected graph $\Gamma$. In general, the group $H^0(\Gamma,D)$ can be quite large, even on a connected graph. For example, let $\Gamma$ be a chain of $2n$ vertices, let $p$ and $q$ be distinct prime numbers, let $G=\mathbb{Z}/pq\mathbb{Z}$, and let $H_1=\mathbb{Z}/p\mathbb{Z}$ and $H_2=\mathbb{Z}/q\mathbb{Z}$ be the nontrivial subgroups of $G$. Label the vertices of $\Gamma$ by $H_1$ and $H_2$ in an alternating fashion. Then $C(e)=G/(H_1+H_2)=0$ for any edge of $e$, hence $C^1(\Gamma,D)=0$ and therefore $H^0(\Gamma,D)=C^0(\Gamma,D)=H_1^n\oplus H_2^n=G^n$.
\end{remark}
We have already noted (see Rem.~\ref{rem:fixedbase}) that we restrict our attention to a fixed base graph $\Gamma$. It is possible to define morphisms between pairs consisting of a graph and a $G$-datum on it (it is necessary to require that the graph morphism be finite). Such morphisms define natural pullback maps on the cochain and cohomology groups. In the next section, we work out these pullback maps for a single example, namely the relationship (Prop.~\ref{prop:contraction}) between the cohomology groups $H^i(\Gamma,A)$ of a $G$-datum $A$ on $\Gamma$, and the cohomology groups $H^i(\Delta,A|_{\Delta})$ of the restriction of $A$ to a subgraph $\Delta\subset \Gamma$.
\subsection{Relative cohomology and reduced cohomology} \label{sec:relative}
This section is somewhat technical in nature, and deals with a single question: how to relate the cohomology groups $H^i(\Gamma,D)$ of a $G$-dilated graph $(\Gamma,D)$ to the cohomology groups $H^i(\Delta,D|_{\Delta})$ of the restriction of $D$ to a subgraph $\Delta\subset \Gamma$. This question is natural from the point of view of tropical geometry: we often study tropical curves by contracting edges and forming simpler graphs, and hence we may need to understand the classification of $G$-covers of a graph in terms of $G$-covers of its contractions.
We fix a graph $\Gamma$, a subgraph $\Delta\subset \Gamma$, and a $G$-datum $A$ on $\Gamma$. We will see that the cohomology groups $H^i(\Gamma,A)$ and $H^i(\Delta, A|_{\Delta})$ fit into an exact sequence, which is the analogue of the long exact sequence of the cohomology groups of a pair of topological spaces. The relative cohomology groups occurring in this sequence can be computed as reduced cohomology groups of an induced $G$-datum $A_{\Gamma/\Delta}$ on the quotient graph $\Gamma/\Delta$. Unfortunately, the $G$-datum $A_{\Gamma/\Delta}$ is not in general the $G$-datum associated to a $G$-dilation datum on $\Gamma/\Delta$, even when $A$ is associated to a $G$-dilation datum on $\Gamma$.
\begin{definition} Let $\Gamma$ be an oriented graph, let $\Delta\subset \Gamma$ be a subgraph, and let $A$ be a $G$-datum on $\Gamma$. Then $A|_{\Delta}$ is a $G$-datum on $\Delta$. Viewing $C^0(\Gamma,A)$ and $C^0(\Delta,A|_{\Delta})$ as sets of $A(v)$-valued maps from $V(\Gamma)$ and $V(\Delta)$, respectively, we define a surjective map $\iota^0:C^0(\Gamma,A)\to C^0(\Delta,A|_{\Delta})$ by restricting from $V(\Gamma)$ to $V(\Delta)$. We similarly define a surjective restriction map $\iota^1:C^1(\Gamma,A)\to C^1(\Delta,A|_{\Delta})$, and define the {\it relative cochain complex} of the triple $(\Gamma,\Delta,A)$:
$$
\begin{tikzcd}0\arrow[r] & C^0(\Gamma,\Delta,A) \arrow[r,"\delta^*_{\Gamma,\Delta,A}"] & C^1(\Gamma,\Delta,A) \arrow[r] & 0,\end{tikzcd}
$$
by setting $C^i(\Gamma,\Delta,A)=\Ker \iota^i$ for $i=0,1$ and
$\delta^*_{\Gamma,\Delta,A}$ to be the restriction of $\delta^*_{\Gamma,A}$ to $C^0(\Gamma,\Delta,A)$. The {\it relative cohomology groups} of the triple $(\Gamma,\Delta,A)$ are
$$
H^0(\Gamma,\Delta,A)=\Ker \delta^*_{\Gamma,\Delta,A}\quad\textrm{ and } \quad H^1(\Gamma,\Delta,A)=\Coker \delta^*_{\Gamma,\Delta,A}.
$$
\end{definition}
We note that $\delta_{\Gamma,A}^*\circ \iota^0=\iota^1\circ \delta^*_{\Delta,A|_{\Delta}}$, in other words the $\iota^i$ form a cochain map. Hence we have a short exact sequence of cochain complexes:
\begin{equation}
\begin{tikzcd}
& 0 \ar[d]& 0 \ar[d]& \\
0\arrow[r] & C^0(\Gamma,\Delta,A) \arrow[r,"\delta^*_{\Gamma,\Delta,A}"] \arrow[d] & C^1(\Gamma,\Delta,A) \arrow[r] \arrow[d] & 0 \\
0\arrow[r] & C^0(\Gamma,A) \arrow[r,"\delta^*_{\Gamma,A}"] \arrow[d,"\iota^0"] & C^1(\Gamma,A) \arrow[r] \arrow[d,"\iota^1"] & 0 \\
0\arrow[r] & C^0(\Delta,A|_{\Delta}) \arrow[r,"\delta^*_{\Delta,A|_{\Delta}}"] \arrow[d] & C^1(\Delta,A|_{\Delta}) \arrow[r] \arrow[d] & 0 \\
& 0 & 0 &
\end{tikzcd}\label{eq:ses}
\end{equation}
By the snake lemma, the cohomology groups of $(\Gamma,A)$, $(\Delta,A|_{\Delta})$ and the triple $(\Gamma,\Delta,A)$ fit into an exact sequence
\begin{equation}
\begin{tikzcd} 0 \arrow[r]& H^0(\Gamma,\Delta,A)\arrow[r] & H^0(\Gamma,A)\arrow[r] & H^0(\Delta,A|_{\Delta}) \arrow[r] & \, \\
\,\arrow[r] &H^1(\Gamma,\Delta,A) \arrow[r] & H^1(\Gamma,A)\arrow[r] & H^1(\Delta,A|_{\Delta})\arrow[r] &0.\end{tikzcd} \label{eq:les}
\end{equation}
We now show that the relative cohomology groups of the triple $(\Gamma,\Delta,A)$ are equal to the reduced cohomology groups of the contracted graph $\Gamma/\Delta$ with a certain induced $G$-datum. First, we define the reduced cohomology groups of a pair $(\Gamma,A)$.
\begin{definition} Let $\Gamma$ be an oriented graph and let $A$ be a $G$-datum on $\Gamma$. Let $d:G\to C^0(\Gamma,A)$ be the diagonal morphism given by $d(g)=\xi_g\in C^0(\Gamma,A)$, where
$$
\xi_g(v)=f_v(g)\in A(v)\mbox{ for all }v\in V(\Gamma)\mbox{ and all }g\in G.
$$
For any $g\in G$ and any $e\in E(\Gamma)$ we have
$$
\delta_{\Gamma,A}^*(d(g))(e)=t^*(\xi_g)(e)-s^*(\xi_g)(e)=t_e(f_{t(e)}(g))-s_e(f_{s(e)}(g))=f_e(g)-f_e(g)=0,
$$
hence $\Im d\subset \Ker \delta_{\Gamma,A}^*$. Therefore we can define the {\it reduced cochain complex} of the pair $(\Gamma,A)$
$$
\begin{tikzcd}
0 \ar[r] & \widetilde{C}^0(\Gamma,A)\ar[r,"\widetilde{\delta}_{\Gamma,A}^*"] & \widetilde{C}^1(\Gamma,A) \ar[r] & 0
\end{tikzcd}
$$
by
$$ \widetilde{C}^0(\Gamma,A)=C^0(\Gamma,A)/\Im d\quad\textrm{ and } \quad\widetilde{C}^1(\Gamma,A)=C^1(\Gamma,A), $$
and the {\it reduced cohomology groups} of $(\Gamma,A)$:
$$
\widetilde{H}^0(\Gamma,A)=\Ker \widetilde{\delta}_{\Gamma,A}^*=H^0(\Gamma,A)\mod G\quad \textrm{ and }\quad
\widetilde{H}^1(\Gamma,A)=\Coker \widetilde{\delta}_{\Gamma,A}^*=H^1(\Gamma,A).
$$
\end{definition}
We define the quotient of a graph $\Gamma$ by a subgraph $\Delta\subset \Gamma$ by contracting $\Delta$ to a single vertex. Note that this definition comes from topology, and differs from weighted edge contraction (see Def.~\ref{def:edgecontraction}), wherein each connected component of $\Delta$ is contracted to a separate vertex.
\begin{definition} Let $\Gamma$ be an oriented graph and let $\Delta$ be a subgraph. We define the graph $\Gamma/\Delta$ as follows:
$$
V(\Gamma/\Delta)=V(\Gamma)\backslash V(\Delta)\cup \{w\}\quad\textrm{ and }\quad E(\Gamma/\Delta)=E(\Gamma)\backslash E(\Delta),
$$
as well as
$$
s(e)=\left\{\begin{array}{cc} w & \textrm { if }s(e)\in V(\Delta), \\ s(e) & \textrm{ if }s(e)\notin V(\Delta), \end{array}\right.\quad
t(e)=\left\{\begin{array}{cc} w & \textrm{ if } t(e)\in V(\Delta), \\ t(e) & \textrm{ if }t(e)\notin V(\Delta). \end{array}\right.
$$
\end{definition}
Now let $\Gamma$ be an oriented graph, let $\Delta$ be a subgraph, and let $A$ be a $G$-datum on $\Gamma$. We define the $G$-datum $A_{\Gamma/\Delta}$ on $\Gamma/\Delta$ by restricting $A$ to all vertices except $w$ and all edges, and by placing the trivial $G$-datum at $w$. Specifically, the $G$-groups $f'_v:G\to A_{\Gamma/\Delta}(v)$ corresponding to the vertices $v\in V(\Gamma/\Delta)$ are
$$
f'_v:G\to A_{\Gamma/\Delta}(v)=\left\{\begin{array}{ll} f_v:G\to A(v)&\textrm{ if } v\in V(\Gamma)\backslash V(\Delta), \\ \Id:G\to G& \textrm{ if } v=w.\end{array}\right.
$$
The $G$-groups $f'_e:G\to A_{\Gamma/\Delta}(e)$ corresponding to $e\in E(\Gamma/\Delta)$ are the same as $f_e:G\to A(e)$. Finally, the source and target maps $s'_e:A_{\Gamma/\Delta}\big(s(e)\big)\to A_{\Gamma/\Delta}(e)$ and $t'_e:A_{\Gamma/\Delta}\big(t(e)\big)\to A_{\Gamma/\Delta}(e)$ are
\begin{equation*}
s'_e=\left\{\begin{array}{ll} s_e:A\big(s(e)\big)\to A(e) & \textrm{ if } s(e)\neq w, \\
f_e:G\to A(e) & \textrm{ if } s(e)=w,\end{array}\right.
\end{equation*}
and
\begin{equation*}
t'_e=\left\{\begin{array}{ll} t_e:A\big(t(e)\big)\to A(e) & \textrm{ if } t(e)\neq w, \\
f_e:G\to A(e) & \textrm{ if } t(e)=w.\end{array}\right.
\end{equation*}
\begin{remark} If $A=A^D$ is the $G$-datum associated to a $G$-dilation datum $D$, then so is $A|_{\Delta}$, but not, in general, $A_{\Gamma/\Delta}$. Specifically, the edge groups of $A^D$ are the coproducts of the vertex groups, which is no longer the case for $A_{\Gamma/\Delta}$. In other words, the relationship between the dilated cohomology groups $H^i(\Gamma,D)$ and $H^i(\Delta,D|_{\Delta})$ cannot be expressed without using the more general framework of $G$-data and their cohomology. The edge groups of $A_{\Gamma/\Delta}$ retain a record of the dilation groups $D(v)$ of the edges $v\in V(\Delta)$ that are contracted in $\Gamma/\Delta$.
\end{remark}
\begin{proposition} Let $\Gamma$ be an oriented graph, let $\Delta$ be a subgraph, and let $A$ be a $G$-datum on $\Gamma$. The relative cohomology groups of the triple $(\Gamma,\Delta,A)$ are equal to the reduced cohomology groups of $(\Gamma/\Delta,A_{\Gamma/\Delta})$:
\begin{equation}
H^i(\Gamma,\Delta,A)=\widetilde{H}^i(\Gamma/\Delta,A_{\Gamma/\Delta}).
\end{equation} \label{prop:contraction}
\end{proposition}
\begin{proof} The group $G$ acts diagonally on $C^0(\Gamma/\Delta,A_{\Gamma/\Delta})$, and the action is free and transitive on the $w$-coordinate, since by definition $A_{\Gamma/\Delta}(w)=G$. Therefore any element $[\xi]$ in the quotient group $\widetilde{C}^0(\Gamma/\Delta,A_{\Gamma/\Delta})$ has a unique representative $\xi\in C^0(\Gamma/\Delta,A_{\Gamma/\Delta})$ satisfying $\xi(w)=0$. Since $A_{\Gamma/\Delta}(v)=A(v)$ for $v\in V(\Gamma\backslash \Delta)$, we can define an extension by zero map $j^0:\widetilde{C}^0(\Gamma/\Delta,A_{\Gamma/\Delta})\to C^0(\Gamma,A)$ by
$$
j^0([\xi])(v)=\left\{\begin{array}{ll}\xi(v) & \textrm{ if } v\in V(\Gamma)\backslash V(\Delta), \\ 0 & \textrm{ if }v\in V(\Delta).\end{array}\right.
$$
Similarly, since $A_{\Gamma/\Delta}(e)=A(e)$ for all $e\in E(\Gamma/\Delta)$, we can define an extension by zero map $j^1:\widetilde{C}^1(\Gamma/\Delta,A_{\Gamma/\Delta})=C^1(\Gamma/\Delta,A_{\Gamma/\Delta})\to C^1(\Gamma,A)$ by
$$
j^1(\eta)(e)=\left\{\begin{array}{ll}\eta(e) & \textrm{ if }e\in E(\Gamma)\backslash E(\Delta), \\ 0 & \textrm{ if }e\in E(\Delta).\end{array}\right.
$$
We claim that the $j^i$ form a chain map. Denote for simplicity $\delta^*=t^*-s^*=\delta^*_{\Gamma,\Delta}$ and $\widetilde{\delta}^*=\widetilde{\delta}^*_{\Gamma/\Delta,\alpha_{\Gamma/\Delta}}$. For $[\xi]\in \widetilde{C}^0(\Gamma/\Delta,A_{\Gamma/\Delta})$ let $\xi\in C^0(\Gamma/\Delta,A_{\Gamma/\Delta})$ be the representative satisfying $\xi(w)=0$. Let $e\in E(\Gamma)$ be an edge. If $e\in E(\Delta)$ then $(j^1\circ \widetilde{\delta}^*)\big([\xi]\big)(e)=0$. If $e\in E(\Gamma)\backslash E(\Delta)$ has root vertices $u=s(e)$ and $v=t(e)$, then, using
\begin{equation*}
(j^1\circ\widetilde{\delta}^*)\big([\xi]\big)(e)=t'_e\big(\xi(v)\big)-s'_e\big(\xi(u)\big)
\end{equation*}
we find
\begin{equation}
(j^1\circ\widetilde{\delta}^*)\big([\xi]\big)(e)=\left\{\begin{array}{ll} t_e(\xi(v))-s_e(\xi(u)) & \textrm{ if } v\in V(\Gamma)\backslash V(\Delta) \textrm{ and } u\in V(\Gamma)\backslash V(\Delta) , \\ t_e\big(\xi(v)\big) & \textrm{ if } v\in V(\Gamma)\backslash V(\Delta) \textrm{ and } u\in V(\Delta), \\ -s_e\big(\xi(u)\big) & \textrm{ if } v\in V(\Delta) \textrm{ and } u\in V(\Gamma)\backslash V(\Delta), \\ 0 &\textrm{ if } v\in V(\Delta) \textrm{and} u\in V(\Delta),\end{array}\right.
\label{eq:long}
\end{equation}
because $\xi(w)=0$.
On the other hand, $j^0([\xi])$ is the element of $C^0(\Gamma,\Delta)$ obtained by setting $j^0([\xi])(v)=\xi(v)$ for all vertices $v\in V(\Gamma)\backslash V(\Delta)$ and $\xi(v)=0$ for all $v\in V(\Delta)$. It is clear that $(\delta^*\circ j^0)([\xi])(e)$ is given by~\eqref{eq:long} for any $e\in E(\Gamma)\backslash E(\Delta)$, and $(\delta^*\circ j^0)([\xi])(e)=0$ for any $e\in E(\Delta)$, because any such edge has root vertices in $\Delta$ and $\xi$ vanishes at those vertices. It follows that $\delta^*\circ j^0=j^1\circ \widetilde{\delta}^*$, hence the $j^i$ form a chain map.
We now consider the diagram~\eqref{eq:ses}. By definition,
$$
C^0(\Gamma,\Delta,A)=\left\{\xi\in C^0(\Gamma,A):\xi(v)=0\mbox{ for all }v\in V(\Delta)\right\}
$$
and
$$
C^1(\Gamma,\Delta,A)=\left\{\eta\in C^1(\Gamma,A):\chi(e)=0\mbox{ for all }e\in E(\Delta)\right\}.
$$
It is clear that $j^i$ maps $\widetilde{C}^i(\Gamma/\Delta,A_{\Gamma/\Delta})$ bijectively onto $C^i(\Gamma,\Delta,A)$ for $i=0,1$. It follows that $j^i$ is a chain isomorphism from $\widetilde{C}^i(\Gamma/\Delta,A_{\Gamma/\Delta})$ to $C^i(\Gamma,\Delta,A)$, which completes the proof.
\end{proof}
\section{Classification of $G$-covers of graphs and tropical curves} \label{sec:classification}
In this section, we use dilated cohomology groups, defined in the previous section, to classify $G$-covers of graphs and tropical curves. In Sec.~\ref{sec:Gcoversofgraphs} we give the main classification result for unweighted graphs, Thm.~\ref{thm:main}, which identifies the set of $G$-covers of $\Gamma$ with a given dilation datum $D$ with the group $H^1(\Gamma,D)$. Among these covers, we characterize the connected ones in Prop.~\ref{prop:connectedcovers}, and give examples. The case of unramified $G$-covers of a weighted graph is treated in Sec.~\ref{sec:admissible}, the only novelty being a numerical restriction on the dilation datum $D$ imposed by the local Riemann--Hurwitz condition~\eqref{eq:localRH}. Finally, the case of weighted metric graphs and tropical curves is summarized in Sec.~\ref{sec:metric}.
\subsection{$G$-covers of graphs}\label{sec:Gcoversofgraphs}
In this section, we determine all $G$-covers of an unweighed graph $\Gamma$ with a given $G$-dilation datum $D$. Our theorem generalizes the standard result that the set of topological $G$-covers of $\Gamma$ (i.e. with trivial stabilizers) is identified with $H^1(\Gamma,G)$ (see Ex.~\ref{ex:topologicalcovers} and Ex.~\ref{ex:trivialdilation}).
\begin{theorem} \label{thm:main} Let $\Gamma$ be a graph, let $G$ be a finite abelian group, and let $D$ be a $G$-dilation datum on $\Gamma$. Then there is a natural bijection between $H^1(\Gamma,D)$ and the set of $G$-covers having dilation datum $D$.
\end{theorem}
\begin{proof}
We first explain how to associate an element $[\eta_{\varphi}]\in H^1(\Gamma,D)$ to a $G$-cover $\varphi:\Gamma'\to \Gamma$. Pick an orientation on $E(\Gamma)$ and a consistent orientation on $E(\Gamma')$, and denote $s,t:E(\Gamma')\to V(\Gamma')$ and $s,t:E(\Gamma)\to V(\Gamma)$ the source and target maps. For each $x\in X(\Gamma)$, the preimage $\varphi^{-1}(x)$ is a $G/D(x)$-torsor, so pick a $G$-equivariant bijection $f_x:\varphi^{-1}(x)\to G/D(x)$. Namely, for every $x'\in \varphi^{-1}(x)$ and every $g\in G$ we have
$$
f_x(gx')=f_x(x')+g\ \mod D(x).
$$
We require that if $e=(h_1,h_2)\in E(\Gamma)$, then, under the identification $\varphi^{-1}(h_1)=\varphi^{-1}(h_2)=\varphi^{-1}(e)$, the two maps $f_{h_1}$ and $f_{h_2}$ are equal, in which case we denote them by $f_e$.
If $l\in L(\Gamma)$ is a leg, then $D(l)\subset D(r(l))$ and we have the following diagram of $G$-sets:
\begin{center}\begin{tikzcd}
\varphi^{-1}(l)\arrow[r,"r"] \arrow[d,"f_l"] & \varphi^{-1}\big({r(l)}\big) \arrow[d,"f_{r(l)}"]\\
G/D(l)\arrow[r] & G/D\big({r(l)}\big)
\end{tikzcd}\end{center}
The vertical maps are bijections, while the horizontal maps are surjections. Adding a constant to $f_l$ if necessary, we can assume that the lower horizontal map is reduction modulo $D({r(l)})/D(l)$.
Now let $e\in E(\Gamma)$ be an edge, then $D(s(e))$ and $D(t(e))$ are subgroups of $G$ containing $D(e)$. The source and target maps restrict to $G$-equivariant surjections $s:\varphi^{-1}(e)\to \varphi^{-1}(s(e))$ and $t:\varphi^{-1}(e)\to \varphi^{-1}(t(e))$, and we have a commutative diagram of $G$-sets
\begin{center}\begin{tikzcd}
\varphi^{-1}\big(s(e)\big)\arrow[d,"f_{s(e)}"] & \varphi^{-1}(e)\arrow[l,"s"']\arrow[r,"t"] \arrow[d,"f_e"] & \varphi^{-1}\big({t(e)}\big) \arrow[d,"f_{t(e)}"]\\
G/D\big(s(e)\big) & G/D(e) \arrow[l,"+\eta_s(e)"'] \arrow[r,"+\eta_t(e)"] & G/D\big({t(e)}\big)
\end{tikzcd}\end{center}
where the vertical arrows are bijections. The lower horizontal arrows are surjections, and are therefore given by adding certain elements $\eta_t(e)\in G/D({t(e)})$ and $\eta_s(e)\in G/D(s(e))$, and then reducing modulo $D({t(e)})$ and $D(s(e))$, respectively. Hence the cover $\varphi$ determines an element
$$
\eta_{\varphi}=\big(\eta_s(e),-\eta_t(e)\big)_{e\in E(\Gamma)}\in \prod_{e\in E(\Gamma)} G/D\big(s(e)\big)\oplus G/D\big(t(e)\big).
$$
Denote $[\eta_{\varphi}]$ its class in $H^1(\Gamma,D)$.
We need to verify that the association $\varphi\mapsto[\eta_{\varphi}]$ is independent of all choices. Suppose that we chose different bijections $\widetilde{f}_x:\varphi^{-1}(x)\to G/D(x)$. For any leg $l\in L(\Gamma)$ we can assume, as above, that the induced map $G/D(l)\to G/D(r(l))$ is reduction modulo $D(r(l))/D(l)$. Now let $e\in E(\Gamma)$ be an edge. We have a diagram of $G$-sets
\begin{center}\begin{tikzcd}
G/D\big(s(e)\big) & G/D(e) \arrow[r,"+\widetilde{\eta}_t(e)"]\arrow[l,"+\widetilde{\eta}_s(e)"']& G/D\big({t(e)}\big)\\
\varphi^{-1}\big(s(e)\big)\arrow[d,"f_{s(e)}"] \arrow[u,"\widetilde{f}_{s(e)}"']& \varphi^{-1}(e)\arrow[l,"s"']\arrow[r,"t"] \arrow[d,"f_e"] \arrow[u,"\widetilde{f}_e"'] & \varphi^{-1}({t(e)}) \arrow[d,"f_{t(e)}"] \arrow[u,"\widetilde{f}_{t(e)}"']\\
G/D\big(s(e)\big) & G/D(e) \arrow[r,"+\eta_t(e)"]\arrow[l,"+\eta_s(e)"']& G/D\big({t(e)}\big)
\end{tikzcd}\end{center}
The top horizontal maps define an element $\widetilde{\eta}_{\varphi}=(\widetilde{\eta}_s(e), -\widetilde{\eta}_t(e))_{e\in E(\Gamma)}$ and a corresponding class $[\widetilde{\eta}_{\varphi}]$ in $H^1(\Gamma,D)$.
The middle column consists of isomorphisms of $G$-sets, hence the map $\widetilde{f}_e\circ f_e^{-1}:G/D(e)\to G/D(e)$ is the addition of an element $\omega(e)\in G/D(e)$. Similarly, the isomorphisms $\widetilde{f}_{s(e)}\circ f_{s(e)}^{-1}:G/D\big(s(e)\big)\to G/D\big(s(e)\big)$ and $\widetilde{f}_{t(e)}\circ f_{t(e)}^{-1}:G/D\big({t(e)}\big)\to G/D\big({t(e)}\big)$ are given by adding certain elements $\xi\big(s(e)\big)\in G/D\big(s(e)\big)$ and $\xi\big({t(e)}\big)\in G/D\big({t(e)}\big)$. The two vertical rectangles give the following relations on all of these elements:
\begin{equation*}\begin{split}
\eta_s(e)+\xi(s(e))&=\omega(e)+\widetilde{\eta}_s(e)\mod D(s(e))\\
\eta_t(e)+\xi({t(e)})&=\omega(e)+\widetilde{\eta}_t(e) \mod D({t(e)}).
\end{split}\end{equation*}
Comparing this with Eq.~\eqref{eq:explicitH1}, we see that $[\eta_{\varphi}]=[\widetilde{\eta}_{\varphi}]$, so the $G$-cover $\varphi$ determines a well-defined element of $H^1(\Gamma,D)$.
Conversely, let $D$ be a $G$-dilation datum on $\Gamma$, let $[\eta]\in H^1(\Gamma,D)$ be an element, and let $(\eta_t(e),\eta_s(e))_{e\in E(\Gamma)}$ be a lift of $[\eta]$. Running the above construction in reverse, we obtain a $G$-cover $\varphi:\Gamma'\to \Gamma$ with associated $G$-dilation datum $D$. Specifically, let
\begin{itemize}
\item
$V(\Gamma')=\coprod_{v\in V(\Gamma)} G/D(v)$,
\item $E(\Gamma')=\coprod_{e\in E(\Gamma)} G/D(e)$, and
\item $L(\Gamma')=\coprod_{l\in L(\Gamma)} G/D(l)$.
\end{itemize}
We define $\varphi:\Gamma'\to \Gamma$ by sending each $G/D(x)$ to the corresponding $x\in X(\Gamma)$. For a leg $l\in L(\Gamma)$, we define the lifting $r:G/D(l)\to G/D\big(r(l)\big)$ of the root map to $\Gamma'$ as reduction modulo $D(r(l))/D(l)$. Finally, for an edge $e\in L(\Gamma)$, we define the liftings $s:G/D(e)\to G/D(s(e))$ and $t:G/D(e)\to G/D(t(e))$ of the source and target maps as
$$
s(g)=g+\eta_s(e)\mod D(s(e))/D(e) \quad \textrm{ and }\quad t(g)=g-\eta_t(e)\mod D(t(e))/D(e).
$$
\end{proof}
We observe that there is at least one $G$-cover associated to any $G$-dilation datum $D$ on $\Gamma$, namely the {\it trivial $G$-cover with dilation datum} $D$, corresponding to the identity element of $H^1(\Gamma,D)$. Explicitly, the source graph $\Gamma'$ is the union of the sets $G/D(x)$ for all $x\in X(\Gamma)$, and the root maps $G/D(x)\to G/D\big(r(x)\big)$ are the quotient maps corresponding to the injections $D(x)\subset D\big(r(x)\big)$. Note also that the set of $G$-covers with dilation datum $D$ depends only on the vertex groups $D(v)$ for $v\in V(\Gamma)$, or, alternatively, on the dual stratification $\mathcal{S}^*(D)$.
\begin{remark}[Functoriality] The correspondence $\varphi \mapsto \eta_{\varphi}$ between $G$-covers of $\Gamma$ with dilation datum $D$ and elements of $H^1(\Gamma,D)$ given in Thm.~\ref{thm:main} is functorial, in the following sense. Let $\varphi_1:\Gamma'_1\to \Gamma$ and $\varphi_2:\Gamma'_2\to \Gamma$ be $G$-covers, and let $\tau:\Gamma'_1\to \Gamma'_2$ be a morphism of $G$-covers (in the sense of Def.~\ref{def:morphismofGcovers}). Then the dilation datum $D_{\varphi_1}$ is a refinement of $D_{\varphi_2}$, and by the proof of Prop.~\ref{prop:tree} there is a surjective map $\pi:H^1(\Gamma,D_{\varphi_1})\to H^1(\Gamma,D_{\varphi_2})$. It is easy to check that $\pi(\eta_{\varphi_1})=\eta_{\varphi_2}$. More generally, the correspondence $\varphi \mapsto \eta_{\varphi}$ is functorial with respect to pullback maps induced by finite harmonic morphisms $\Delta\to \Gamma$, which, as we have already remarked, are beyond the scope of our paper.
\end{remark}
\begin{remark}[Trivialization along a tree]
We saw in Prop.~\ref{prop:tree} that $H^1(\Delta,D)=0$ for any $G$-dilation datum on a tree $\Delta$. In other words, any $G$-cover of a tree is isomorphic to the trivial $G$-cover associated to some dilation datum $D$. This statement allows us to give a somewhat explicit description of $G$-covers of an arbitrary graph $\Gamma$. Let $\varphi:\Gamma'\to \Gamma$ be a $G$-cover with dilation datum $D$. Pick a spanning tree $\Delta\subset \Gamma$, and let $\{e_1,\ldots,e_n\}=E(\Gamma)\backslash E(\Delta)$ be the remaining edges. The restricted $G$-cover $\varphi|_{\Delta}$ is isomorphic to the trivial $G$-cover of $\Delta$ with dilation datum $D|_{\Delta}$, in other words there is a $G$-equivariant bijection
$$
\tau:\varphi^{-1}(\Delta)\to \coprod_{x\in X(\Delta)} G/D(x),\quad \tau\big(\varphi^{-1}(x)\big)=G/D(x).
$$
The cover $\varphi$ is then completely determined by the way that the fibers $G/D(e_i)$ are attached to the fibers $G/D(s(e_i))$ and $G/D(t(e_i))$. As we saw in the proof above, this attachment datum can be recorded (in general, non-uniquely) by an $n$-tuple of elements of $\eta_i\in A^D(e_i)=G/C(e_i)$. In terms of the dilated cohomology group, we have shown that any element $[\eta]\in H^1(\Gamma,D)$ can be represented by a cochain $\eta\in C^1(\Gamma,D)$ such that $\eta(e)=0$ unless $e=e_i$ for some $i=1,\ldots,n$ (cf. Lemma 2.3.4 in~\cite{LenUlirsch}).
\end{remark}
\subsubsection*{Connected covers}
Given a connected graph $\Gamma$, it is natural to ask which of its $G$-covers constructed above are connected. To answer this question, we first consider the following construction. Let $H$ be a proper subgroup of $G$, and let $D$ be an $H$-dilation datum on a connected graph $\Gamma$. We can then view $D$ as a $G$-dilation datum, which we denote by $D^G$ to prevent confusion. There are natural injective chain maps $\iota^i:C^i(\Gamma,D)\to C^i(\Gamma,D^G)$ that induce maps $\iota^i:H^i(\Gamma,D)\to H^i(\Gamma,D^G)$.
\begin{lemma} The maps $\iota^i:H^i(\Gamma,D)\to H^i(\Gamma,D^G)$ are injective.
\end{lemma}
\begin{proof} The cochain groups $C^0(\Gamma,D)$ and $C^0(\Gamma,D^G)$ are the products of $H/D(v)$ and $G/D(v)$, respectively, over all $v\in V(\Gamma)$. It follows that $\Coker \iota^0$ can be identified with the cochain group $C^0(\Gamma,G/H)$, and similarly $\Coker \iota^1=C^1(\Gamma,G/H)$. By the snake lemma, we have a long exact sequence of cohomology groups:
$$
\begin{tikzcd} 0 \arrow[r]& H^0(\Gamma,D)\arrow[r,"\iota^0"] & H^0(\Gamma,D^G)\arrow[r] & H^0(\Gamma,G/H) \arrow[r] & H^1(\Gamma,D) \arrow[r,"\iota^1"] & H^1(\Gamma,D^G).\end{tikzcd}
$$
Therefore $\iota^0$ is injective. To prove that $\iota^1$ is injective, we show that the map $\pi:H^0(\Gamma,D^G)\to H^0(\Gamma,G/H)$ is surjective. All of our chain complexes split into direct sums over the connected components of $\Gamma$, so we assume that $\Gamma$ is connected. In this case $H^0(\Gamma,G/H)=G/H$, and moreover any $[\xi]\in H^0(\Gamma,G/H)$ is represented by a constant cochain $\xi(v)=\overline{g}$ for some $\overline{g}\in G/H$. Pick $g\in G$ representing $\overline{g}$, then the constant cochain $\xi'(v)=g\mod G/D(v)$ in $C^0(\Gamma,D^G)$ lies in $\Ker \delta^0_{\Gamma,D^G}$, hence represents a class $[\xi']\in H^0(\Gamma,D^G)$, and $\pi\big([\xi']\big)=[\xi]$. Therefore $\pi$ is surjective, so $\iota^0$ is injective.
\end{proof}
There is a natural way to associate a $G$-cover of $\Gamma$ to an $H$-cover of $\Gamma$ that corresponds, under the bijection of Thm.~\ref{thm:main}, to the injective map $\iota^1:H^1(\Gamma,D)\to H^1(\Gamma,D^G)$.
\begin{definition} Let $\Gamma$ be a graph, let $H\subset G$ be abelian groups, and let $\varphi\colon\Gamma'\to \Gamma$ be an $H$-cover with $H$-dilation datum $D$. We define the $G$-cover $\varphi^G:\Gamma'^G\to \Gamma$ with $G$-dilation datum $D^G$, called the {\it extension of $\varphi$ by $G$}, as follows. For each $x\in X(\Gamma)$, pick an identification of $H$-sets, as in the proof of Thm.~\ref{thm:main}, of $\varphi^{-1}(x)$ with $H/D(x)$, and for every edge $e\in E(\Gamma)$ let $\eta_t(e)\in H/D\big(t(e)\big)$ and $\eta_s(e)\in H/D\big(s(e)\big)$ be the elements that determine the root maps $t:\varphi^{-1}(e)\to \varphi^{-1}\big(t(e)\big)$ and $s:\varphi^{-1}(e)\to \varphi^{-1}\big(s(e)\big)$. We define $\varphi^G$ by identifying each fiber $(\varphi^G)^{-1}(x)$ with the $G$-set $G/D(x)$, and rooting $(\varphi^G)^{-1}(e)$ to $(\varphi^G)^{-1}\big(t(e)\big)$ and $(\varphi^G)^{-1}\big(s(e)\big)$ using $\eta_t(e)$ and $\eta_s(e)$, viewed, respectively, as elements of $G/D\big(t(e)\big)$ and $G/D\big(s(e)\big)$.
\end{definition}
Looking at the proof of~\ref{thm:main}, it is clear that $\iota^1(\eta_{\varphi})=\eta_{\varphi^G}$. Furthermore, the cover $\varphi^G$ is disconnected (unless $H=G$), since the root maps $t:G/D(e)\to G/D(t(e))$ and $s:G/D(e)\to G/D(s(e))$ preserve the decomposition into $H$-cosets.
We now show that all disconnected $G$-covers of a connected graph $\Gamma$ arise in this way. Indeed, let $\varphi:\Gamma'\to \Gamma$ be a $G$-cover of a connected graph with $G$-dilation datum $D$, and let $\Gamma'=\Gamma'_1\sqcup \cdots \sqcup \Gamma'_n$ be the connected components of $\Gamma'$. The group $G$ acts on the connected components by permutation. Let $H=\big\{g\in G\big\vert g(\Gamma'_1)=\Gamma'_1\big\}$, then $D(v)\in H$ for all $v\in V(\Gamma)$. We view $D$ as an $H$-dilation datum, which we denote $D_H$. It is clear that the restriction $\varphi_{\Gamma'_1}\colon\Gamma'_1\to \Gamma$ is a connected $H$-cover with $H$-dilation datum $D_H$, and that $\varphi$ is isomorphic to the $G$-extension of $\varphi|_{\Gamma'_1}$ by $G$. In other words, every disconnected $G$-cover of $\Gamma$ is the extension of an $H$-cover, where $H\subset G$ is some proper subgroup. We have proved the following result, which classifies connected $G$-covers of a connected graph $\Gamma$.
\begin{proposition} Let $\Gamma$ be a connected graph, and let $D$ be a $G$-dilation datum on $\Gamma$. If the groups $D(v)$ span $G$, then every $G$-cover with dilation datum $D$ is connected. If not, then the set of disconnected $G$-covers with dilation datum $D$ is the union of the images of the maps $H^1(\Gamma,D_H)\to H^1(\Gamma,D)$ over all proper subgroups $H\subset G$ such that $D(v)\subset H$ for all $v\in V(\Gamma)$, where for each such $H$, $D_H$ denotes $D$ viewed as an $H$-dilation datum. \label{prop:connectedcovers}
\end{proposition}
\begin{example}[Klein covers continued] \label{ex:Klein2} We now apply the results of this section to enumerate all $G$-covers of the graph $\Gamma$ consisting of two vertices $u$ and $v$ joined by two edges $e$ and $f$, when $G=\Zmod{2}\oplus\Zmod{2}$ is the Klein group. In particular, we describe the covers of Ex.~\ref{ex:Klein1} in terms of dilated cohomology. We recall that we denote $00$, $10$, $01$, and $11$ the elements of $G$, and $H_1$, $H_2$, and $H_3$ the subgroups of $G$ generated respectively by $10$, $01$, and $11$. We enumerate the covers in the following way: first, we enumerate the choices for $D(u)$ and $D(v)$, then, for each choice, we consider the possible $D(e), D(f)\subset D(u)\cap D(v)$, and finally $\#H^1(\Gamma,D)$ counts the $G$-covers with such $G$-dilation data (note that the last two steps are independent, since $H^1(\Gamma,D)$ does not depend on the edge dilation groups).
We saw in Ex.~\ref{ex:nedges} that $H^1(\Gamma,D)=G/(D(u)+D(v))$ for any $G$-dilation datum on $\Gamma$. We now make this identification more explicit. Orient $\Gamma$ so that $s(e)=s(f)=u$ and $t(e)=t(f)=v$. An element $[\eta]\in H^1(\Gamma,D)$ is represented by two pairs of elements
$$
\big(\eta_s(e),\eta_t(e)\big), \big(\eta_s(f),\eta_t(f)\big)\in G/D(u)\oplus G/D(v),
$$
modulo the relations~\eqref{eq:explicitH1}. It is clear that for any $[\eta]$ we can pick a representative with $\eta_s(e)=0$, $\eta_s(f)=0$, and $\eta_t(f)=0$ (in other words, we trivialize $[\eta]$ along the spanning tree $\{u,v,f\}$), so we can represent $[\eta]$ with a single element $\eta_t(e)\in G/D(v)$. Furthermore, the class of this $\eta_t(e)$ in $G/(D(u)+D(v))$ is equal to $[\eta]$ under the isomorphism $G\big/\big(D(u)+D(v)\big)=H^1(\Gamma,D)$. Explicitly, the cover corresponding to $[\eta]$ is constructed as follows: define the sets $\{u_{ij}\}=G/D(u)$, $\{v_{ij}\}=G/D(v)$, $\{e_{ij}\}=G/D(e)$, and $\{f_{ij}\}= G/D(f)$ (where the labeling is non-unique for a nontrivial dilation group), attach $f_{ij}$ to $u_{ij}$ and $v_{ij}$, and attach $e_{ij}$ to $u_{ij}$ and $v_{ij+\eta_t(e)}$.
\begin{enumerate} \item $D(u)=D(v)=0$. This is the topological case, with trivial dilation. Here $D(e)=D(f)=0$, $H^1(\Gamma,D)=H^1(\Gamma,G)=G$, and there are four covers, three of them non-trivial. All of these covers are disconnected, since there are no surjective maps $\pi_1(\Gamma)=\mathbb{Z}\to G$. The cover corresponding to $\eta_t(e)=10$ is given in Fig.~\ref{subfig:cover1}.
\item $D(u)=0$, $D(v)=H_i$ for $i=1,2,3$. In this case $D(e)=D(f)=0$, $H^1(\Gamma,D)=G/H_i$, so for each $i$ there is one trivial and one nontrivial cover. For example, Fig.~\ref{subfig:cover2} shows the non-trivial cover with $D(v)=H_1$ and $\eta_t(e)=01$. There are a total of six covers of this type: three trivial disconnected covers and three non-trivial connected covers.
\item $D(u)=H_i$ for $i=1,2,3$, $D(v)=0$. This case is symmetric to the one above, with three connected and three disconnected covers.
\item $D(u)=D(v)=H_i$ for $i=1,2,3$. Each of the groups $D(e)$ and $D(f)$ can be chosen to be $0$ or $H_i$. Since $H^1(\Gamma,D)=G/H_i$, there is one trivial and one non-trivial cover for each choice. For example, when $D(u)=D(v)=D(e)=H_2$ and $D(f)=0$, we obtain the non-trivial cover of Fig.~\ref{subfig:cover3} by choosing $\eta_t(e)=10$, and the trivial cover of Fig.~\ref{subfig:cover4} by choosing $\eta_t(e)=00$. There are a total of 24 such covers, 12 connected and 12 disconnected.
\item $D(u)=H_i$, $D(v)=H_j$, $i\neq j$. The only possibility is $D(e)=D(f)=0$, and $H^1(\Gamma,D)=0$, so for each $i\neq j$ there is a unique trivial cover, for a total of six covers, all connected.
\item If one or both of the groups $D(u)$ and $D(v)$ are equal to $G$, then $H^1(\Gamma,D)=0$. Picking $D(e)$ and $D(f)$ to be arbitrary subgroups of $D(u)\cap D(v)$, we obtain 51 connected trivial covers. Two such covers are given in Figs.~\ref{subfig:cover5} and~\ref{subfig:cover6}. We note that 9 of these covers, including the one on~\ref{subfig:cover6}, have non-cyclic edge dilation groups, and are therefore not algebraically realizable.
\end{enumerate}
In total, there are 97 Klein covers of $\Gamma$, including 75 connected covers.
\end{example}
\subsection{Weighted graphs and unramified $G$-covers} \label{sec:admissible}
We now consider the category of weighted graphs and finite harmonic morphisms between them. Given a weighted graph $\Gamma$ and a $G$-cover $\varphi:\Gamma'\to \Gamma$ (where we view $\Gamma$ as an unweighted graph and $\varphi$ as a morphism), there is a natural way to promote $\varphi$ to a harmonic morphism of degree equal to $\#(G)$. Since the action of $G$ is transitive on the fibers, the genera of all vertices of $\Gamma'$ lying in a single fiber are equal. Therefore a $G$-cover of $\Gamma$ with a given dilation datum $D$ is uniquely specified by an element of $H^1(\Gamma,D)$ and a weight function $g':V(\Gamma)\to \mathbb{Z}_{\geq 0}$ (which we lift to $\Gamma'$). There is a natural way to specify this weight: require $\varphi$ to be unramified. This condition imposes a numerical restriction on the $G$-dilation datum $D$.
\begin{definition} Let $\Gamma$ be a weighted graph, and let $G$ be a finite abelian group. A {\it $G$-cover} of $\Gamma$ is a finite harmonic morphism $\varphi:\Gamma'\to \Gamma$ together with an action of $G$ on $\Gamma'$, such that the following properties are satisfied:
\begin{enumerate}
\item The action is invariant with respect to $\varphi$.
\item For each $x\in X(\Gamma)$, the group $G$ acts transitively on the fiber $\varphi^{-1}(x)$.
\item $\#(G)=\deg \varphi$.
\end{enumerate}
We say that a $G$-cover $\varphi$ is {\it effective} or {\it unramified} if it is so as a harmonic morphism.
\end{definition}
\begin{remark} This definition is similar to Definition 7.1.2 in \cite{BertinRomagny}.
\end{remark}
\begin{example} Let $\Gamma$ be a weighted graph. In Example~\ref{ex:topologicalcovers}, we saw that an element $\eta\in H^1(\Gamma,G)$ determines a topological $G$-cover $\varphi:\Gamma'\to \Gamma$. We now weight $\Gamma'$ by setting $g(v')=g(v)$ for all $v\in V(\Gamma)$ and all $v'\in \varphi^{-1}(v)$. Setting $\deg_{\varphi}(x)=1$ for all $x\in X(\Gamma')$, we see that $\varphi$ is an unramified $G$-cover. Conversely, it is clear that a $G$-cover $\varphi:\Gamma'\to \Gamma$ is a topological $G$-cover if and only if $\deg_{\varphi}(x)=1$ for all $x\in X(\Gamma')$.
\label{example:topologicaladmissiblecovers}
\end{example}
We now classify all $G$-covers and unramified $G$-covers of a given weighted graph $\Gamma$. We first note that there is no difference between studying $G$-covers of a weighted graph and $G$-covers of the underlying unweighted graph. Indeed, let $\varphi:\Gamma'\to \Gamma$ be a $G$-cover of a weighted graph $\Gamma$, and let $D_{\varphi}$ be the associated $G$-dilation datum. An element $g\in G$ determines an automorphism of $\Gamma'$, which in particular is an unramified cover of degree one. Therefore, for any $x\in X(\Gamma)$, the harmonic morphism $\varphi$ has the same degree at $x$ and at $g(x)$. Since $G$ acts transitively on $\varphi^{-1}(x)$, we see that $d_{\varphi}(x')$ is the same for all $x'\in \varphi^{-1}(x)$. Since
$$
\deg \varphi=\sum_{x'\in \varphi^{-1}(x)} d_{\varphi}(x')=d_{\varphi}(x') \#\big(\varphi^{-1}(x)\big)=d_{\varphi}(x')\big[G:D_{\varphi}(x)\big],
$$
we see that
\begin{equation}
d_{\varphi}(x')=\#\big(D_{\varphi}(x)\big)
\label{eq:degreedilation}
\end{equation}
for all $x'\in \varphi^{-1}(x)$. Therefore, the local degrees of $\varphi$ are uniquely defined by the associated dilation datum. Conversely, if $\varphi:\Gamma'\to \Gamma$ is a $G$-cover of $\Gamma$ viewed as an unweighted graph, then Eq.~\eqref{eq:degreedilation} gives the unique way to promote $\varphi$ to a harmonic morphism of degree $\#(G)$. As a result, the classification of $G$-covers of weighted graphs reduces trivially to the unweighted case, except that we need to manually specify the weights on the cover.
\begin{theorem} Let $\Gamma$ be a weighted graph, let $G$ be a finite abelian group, let $D$ be a $G$-dilation datum on $\Gamma$, and let $g':V(\Gamma)\to \mathbb{Z}_{\geq 0}$ be a function. Then the there is a natural bijection between $H^1(\Gamma,D)$ and the set of $G$-covers $\varphi:\Gamma'\to \Gamma$ having dilation datum $D$, such that $g(v')=g'(\varphi(v'))$ for all $v'\in V(\Gamma')$.
\label{thm:main2}
\end{theorem}
\begin{proof} This follows immediately from Thm.~\ref{thm:main}, since $G$ acts transitively on each fiber $\varphi^{-1}(v)$ and therefore the numbers $g(v')$ for $v'\in \varphi^{-1}(v)$ are all equal to some $g'(v)$.
\end{proof}
For the remainder of this section, we restrict our attention to unramified $G$-covers, which are the graph-theoretic analogues of \'etale maps. Given such a cover $\varphi:\Gamma'\to \Gamma$, we consider the Riemann--Hurwitz condition~\eqref{eq:localRH} at all vertices $v'\in V(\Gamma')$. This condition uniquely specifies the genera of the vertices of $\Gamma'$. However, these genera may fail to be non-negative integers, which imposes a numerical constraint on the $G$-dilation data on $\Gamma$ that are associated to unramified $G$-covers.
\begin{definition} Let $(\Gamma,D)$ be a $G$-dilated graph. We define the {\it index function} $a_{\Gamma,D}\colon V(\Gamma)\times S(G)\to \mathbb{Z}_{\geq 0}$ of $(\Gamma,D)$ by
\begin{equation}
a_{\Gamma,D}(v;H)=\#\big\{h\in T_v\Gamma\big\vert D(h)=H\big\}.
\label{eq:dilationindex}
\end{equation}
\label{def:index}
\end{definition}
\begin{proposition} Let $\varphi:\Gamma'\to\Gamma$ be an unramified $G$-cover, let $D_{\varphi}$ be the associated $G$-dilation datum, and let $a_{\Gamma,D}$ be the index function of $(\Gamma,D_{\varphi})$. Let $v\in V(\Gamma)$ be a vertex with dilation group $D(v)$, and let $S(D(v))$ be the set of subgroups of $D(v)$. Then
\begin{equation}
2-2g'(v)-\sum_{K\in S(D(v))} a_{\Gamma,D}(v;K)\big[D(v):K\big]=\#\big(D(v)\big)\Big[2-2g(v)-\sum_{K\in S(D(v))} a_{\Gamma,D}(v;K)\Big]
\label{eq:localRH2}
\end{equation}
where $g'(v)$ is the genus of any vertex $v'\in\varphi^{-1}(v)$. \label{prop:admissibility}
\end{proposition}
\begin{proof} For any half-edge $h\in T_v\Gamma$, the dilation group $D(h)$ is a subgroup of $D(v)$. Hence
$$
\val(v)=\sum_{K\in S(D(v))}a_{\Gamma,D}(v;K).
$$
As noted above, each $h\in T_v\Gamma$ has $[D(v):D(h)]$ preimages in $\Gamma'$ attached to $v'$. Therefore
$$
\val(v')=\sum_{K\in S(D(v))} a_{\Gamma,D}(v;K)\big[D(v):K\big].
$$
Plugging this into~\eqref{eq:localRH}, we obtain~\eqref{eq:localRH2}.
\end{proof}
\begin{definition} Let $\Gamma$ be a weighted graph. A $G$-dilation datum $D$ on $\Gamma$ is called {\it admissible} if for every $v\in V(\Gamma)$ the number
\begin{equation}
g'(v)=\#\big(D(v)\big)\big[g(v)-1\big]+1+\frac{1}{2}\sum_{K\in S(D(v))}a_{\Gamma,D}(v;K)\big(\#(D(v))-[D(v):K]\big)
\label{eq:gtop}
\end{equation}
determined by~\eqref{eq:localRH2} is a non-negative integer. A $G$-stratification $\mathcal{S}$ is called {\it admissible} if the associated $G$-stratification $D$ is admissible.
\label{def:admissibledilation}
\end{definition}
It is clear that the $G$-dilation datum associated to an unramified $G$-cover is admissible. Conversely, if $D$ is an admissible $G$-dilation datum, then~\eqref{eq:gtop} uniquely specifies the weight function on any $G$-cover of $\Gamma$ with dilation datum $D$. Hence we obtain the following result.
\begin{theorem} Let $(\Gamma,D)$ be $G$-dilated weighted graph. If $D$ is admissible, then there is a natural bijection between the set of unramified $G$-covers of $\Gamma$ having dilation datum $D$ and $H^1(\Gamma,D)$. Otherwise, there are no unramified covers of $\Gamma$ having dilation datum $D$.
\label{thm:main3}
\end{theorem}
\begin{proof} The result follows immediately from Thm.~\ref{thm:main2} and Prop.~\ref{prop:admissibility}.
\end{proof}
Condition~\eqref{eq:gtop} imposes two restrictions on a $G$-stratification $\mathcal{S}$ (equivalently, on a $G$-dilation datum $D$): a stability condition ($g'(v)$ is non-negative) and a parity condition ($g'(v)$ is an integer). We make a number of general observations. First, we note that the admissibility condition is trivially satisfied at each undilated vertex $v\in V(\Gamma)\backslash V(\Gamma_{dil})$. Indeed, if $D(v)=0$, then equation~\eqref{eq:gtop} reduces to $g'(v)=g(v)$. We also observe that $g'(v)$ is positive, and hence the stability condition is satisfied, if $g(v)\geq 1$. We also note that $g'(v)$ is an integer if $\#(D(v))$ is odd, so~\eqref{eq:gtop} does not impose a parity condition if the order of $G$ is odd.
\begin{remark} Equation~\eqref{eq:gtop} is the only role that the weight function on $\Gamma$ plays in the classification of unramified $G$-covers of $\Gamma$. Furthermore, when checking the admissibility condition at a vertex $v\in V(\Gamma)$, we only need to know whether $g(v)$ is positive or not, the actual value is not important. Therefore, for example, two weighted graphs having the same underlying unweighted graph, and having the same set of genus zero vertices with respect to the two weight functions, will have the same set of unramified $G$-covers.
\end{remark}
We now show that an admissible $G$-stratification has the following semistability properties.
\begin{proposition} Let $\mathcal{S}$ be an admissible $G$-stratification of a graph $\Gamma$. For every simple vertex $v\in V(\Gamma)$ and for each $H\in S(G)$, either $v\in V(\Gamma_H)$ and $\val_{\Gamma_H}(v)=2$, or $v\notin V(\Gamma_H)$. \label{prop:admissibilitystability}
\end{proposition}
\begin{proof} Suppose that $\mathcal{S}$ is admissible. Let $v\in V(\Gamma)$ be a vertex with $g(v)=0$ and two tangent directions $h_1$ and $h_2$. We write condition~\eqref{eq:gtop} at $v$:
$$
g'(v)=1-\frac{1}{2}\left([D(v):D(h_1)]+[D(v):D(h_2)]\right).
$$
The only way that this number can be a non-negative integer is $D(v)=D(h_1)=D(h_2)$, hence $v\in V(\Gamma_H)$ and $\val_{\Gamma_H}(v)=2$ if $H\subset D(v)$ and $v\notin V(\Gamma_H)$ otherwise.
\end{proof}
\begin{proposition} Let $\mathcal{S}$ be an admissible $G$-stratification of a graph $\Gamma$. Then the dilated subgraph $\Gamma_{dil}\subset \Gamma$ is semistable.
\label{prop:semistable}
\end{proposition}
\begin{proof} We recall that $\Gamma_{dil}$ is the union of the $\Gamma_H$ for all subgroups $H\subset G$ except $H=0$. Let $D$ be the $G$-dilation datum associated to $\mathcal{S}$, and let $v\in V(\Gamma_{dil})$ be a vertex, so that $D(v)\neq 0$, and assume that $g(v)=0$. If $v$ is an isolated vertex of $\Gamma_{dil}$, then $a_{\Gamma,D}(v;K)=0$ for all subgroups $K\subset D(v)$ such that $K\neq 0$. It follows that the sum in the right hand side of~\eqref{eq:gtop} vanishes, hence $g'(v)=-\#\big(D(v)\big)+1<0$. Similarly, suppose that $v$ is an extremal vertex of $\Gamma_{dil}$, so that there exists a unique edge $h\in T_v \Gamma_{dil}$ with $H=D(h)\neq 0$. It follows that $a_{\Gamma,D}(v;H)=1$ and $a_{\Gamma,D}(v;K)=0$ for all $K\neq 0,H$, hence
$$
g'(v)= -\#\big(D(v)\big)+1+\frac{1}{2}\big(\#(D(v))-[D(v):H]\big)=1-\frac{\#\big(D(v)\big)}{2}\left(1+\frac{1}{\#(H)}\right)<0,
$$
since $\#\big(D(v)\big)\geq \#(H)\geq 2$. Therefore $\val_{\Gamma_{dil}}(v)\geq 2$ and $\Gamma_{dil}$ is semistable.
\end{proof}
\subsubsection*{Unramified $G$-covers and stability}
Let $\Gamma$ be a weighted graph, and let $\Gamma_{st}$ be its stabilization. We have seen in Def.~\ref{def:coverstabilization} that any unramified cover $\varphi:\Gamma'\to \Gamma$ descends to an unramified cover $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$. It follows that we can restrict unramified $G$-covers of $\Gamma$ to its stabilization, and vice versa.
\begin{proposition} Let $\Gamma$ be a weighted graph. Then there is a natural bijection between the unramified $G$-covers of $\Gamma$ and the unramified $G$-covers of $\Gamma_{st}$.
\label{prop:coversstability}
\end{proposition}
\begin{proof} Let $\varphi:\Gamma'\to \Gamma$ be an unramified $G$-cover. The $G$-action descends to the subgraph $\Gamma'_{sst}\subset \Gamma'$, hence $\varphi_{sst}:\Gamma'_{sst}\to \Gamma_{sst}$ is a $G$-cover. We note that the supporting arguments for Def.~\ref{def:coverstabilization} show that $\varphi$ is undilated on $\Gamma'\backslash \Gamma_{sst}$; alternatively, this follows from Prop.~\ref{prop:semistable}, since any semistable subgraph of $\Gamma$ is contained in $\Gamma_{sst}$. Therefore, for any vertex $v\in V(\Gamma_{sst})$, any adjacent half-edge $h\in H(\Gamma)\backslash H(\Gamma_{sst})$ has $\deg \varphi$ preimages in $H(\Gamma')$, evenly split among the preimages of $v$. It follows that $\varphi_{sst}$ is an unramified $G$-cover. It is then clear how to descend the $G$-action to $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$: for any $g\in G$ and any simple vertex $v'\in V(\Gamma'_{sst})$ that is replaced by an edge or a leg, $g$ maps that edge or leg to the edge or leg that replaces $g(v)$.
Conversely, let $\Gamma$ be a weighted graph, and let $\varphi_{st}:\Gamma'_{st}\to \Gamma_{st}$ be an unramified $G$-cover, where $\Gamma'_{st}$ is a stable weighted graph. The semistabilization $\Gamma_{sst}$ is obtained from $\Gamma_{st}$ by splitting edges and legs at new vertices of genus 0. Performing the same operation on the preimages of these vertices in $\Gamma'_{st}$, we obtain an unramified $G$-cover $\varphi_{sst}:\Gamma'_{sst}\to \Gamma_{sst}$. The graph $\Gamma$ is obtained from $\Gamma_{sst}$ by attaching trees having no vertices of positive genus. For each such tree $T$ attached at $v\in V(\Gamma_{sst})$, we attach $\#(G)$ copies of $T$ to $\Gamma'_{sst}$ at the fiber $\varphi^{-1}(v)$, and extend the $G$-action in the obvious way. We obtain an unramified $G$-cover $\varphi:\Gamma'\to \Gamma$ whose stabilization is $\varphi_{st}$.
\end{proof}
\subsection{$G$-covers of weighted metric graphs and tropical curves} \label{sec:metric}
In this final section, we reformulate our classification results for weighted metric graphs and tropical curves. There is essentially no new mathematical content obtained by adding metrics to graphs, so this section is essentially a restatement and a summary of the results of the previous sections, and is included for the reader's convenience.
First, we introduce $G$-covers of weighted metric graphs:
\begin{definition} Let $(\Gamma,\ell)$ be a weighted metric graph. A {\it $G$-cover} of $(\Gamma,\ell)$ is a finite harmonic morphism $\varphi:(\Gamma',\ell')\to (\Gamma,\ell)$ together with an action of $G$ on $(\Gamma',\ell')$, such that the following properties are satisfied:
\begin{enumerate}
\item The action is invariant with respect to $\varphi$.
\item For each $x\in X(\Gamma)$, $G$ acts transitively on the fiber $\varphi^{-1}(x)$.
\item $\#(G)=\deg \varphi$.
\end{enumerate}
We say that a $G$-cover $\varphi$ is {\it unramified} if it is an unramified harmonic morphism.
\end{definition}
In other words, a $G$-cover $\varphi:(\Gamma',\ell')\to(\Gamma,\ell)$ is a $G$-cover $\varphi:\Gamma'\to \Gamma$ of the underlying weighted graph $\Gamma$ that satisfies the dilation condition \eqref{eq:length}. Given $\varphi$ and $\ell$, there is a unique way to choose $\ell'$ such that the dilation condition is satisfied (see Rem.~\ref{rem:lengths}). It follows that the classification of $G$-covers of $(\Gamma,\ell)$ is identical to the classification of $G$-covers of $\Gamma$. Specifically, such a cover is uniquely determined by choosing the dilation subgroups, an element of the corresponding dilated cohomology group, and a genus assignment on $\Gamma$ which is then lifted to $\Gamma'$. To obtain unramified $G$-covers, we require the dilation data to be admissible, and pick the genus using Eq.~\eqref{eq:gtop}:
\begin{theorem} Let $(\Gamma,\ell)$ be a weighted metric graph. There is a natural bijection between the set of $G$-covers of $(\Gamma',\ell')$ and the set of triples $(D,\eta,g')$, where
\begin{enumerate}
\item $D$ is a $G$-dilation datum on the underlying weighted graph $\Gamma$,
\item $\eta$ is an element of $H^1(\Gamma,D)$,
\item $g'$ is a map from $V(\Gamma)$ to $\mathbb{Z}_{\geq 0}$.
\end{enumerate}
The set of unramified $G$-covers is obtained by choosing $D$ to be an admissible $G$-dilation datum, and defining $g'$ by Eq.~\eqref{eq:gtop}.
\label{thm:main4}
\end{theorem}
\begin{proof} This follows immediately from Thms.~\ref{thm:main2} and \ref{thm:main3}, and Rem.~\ref{rem:lengths}.
\end{proof}
Finally, we describe $G$-covers of tropical curves.
\begin{definition} Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve. A {\it $G$-cover} of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a finite harmonic morphism $\tau:\scalebox{0.8}[1.3]{$\sqsubset$}'\to \scalebox{0.8}[1.3]{$\sqsubset$}$ together with an action of $G$ on $\scalebox{0.8}[1.3]{$\sqsubset$}'$ such that the following properties are satisfied:
\begin{enumerate}
\item The action is invariant with respect to $\tau$.
\item For each $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$, $G$ acts transitively on the fiber $\tau^{-1}(x)$.
\item $\#(G)=\deg \tau$.
\end{enumerate}
We say that a $G$-cover $\tau$ is {\it unramified} if it is an unramified harmonic morphism.
\end{definition}
To describe $G$-covers of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$, we need to define $G$-dilation data on $\scalebox{0.8}[1.3]{$\sqsubset$}$. We can define this to be a $G$-dilation datum on some model of $\scalebox{0.8}[1.3]{$\sqsubset$}$. It is more convenient to define dilation in terms of the associated stratification, which does not involve choosing a model. The following definition generalizes Defs.~\ref{def:stratification}, \ref{def:index}, and \ref{def:admissibledilation} to the case of tropical curves.
\begin{definition} Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve. A {\it $G$-stratification} $\mathcal{S}=\{\scalebox{0.8}[1.3]{$\sqsubset$}_H:H\in S(G)\}$ on $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a collection of subcurves $\scalebox{0.8}[1.3]{$\sqsubset$}_H\subset \scalebox{0.8}[1.3]{$\sqsubset$}$, indexed by the set $S(G)$ of subgroups of $G$, such that
\begin{itemize}
\item $\scalebox{0.8}[1.3]{$\sqsubset$}_0=\scalebox{0.8}[1.3]{$\sqsubset$}$,
\item $\scalebox{0.8}[1.3]{$\sqsubset$}_K\subset \scalebox{0.8}[1.3]{$\sqsubset$}_H$ if $H\subset K$,
\item $\scalebox{0.8}[1.3]{$\sqsubset$}_H\cap \scalebox{0.8}[1.3]{$\sqsubset$}_K=\scalebox{0.8}[1.3]{$\sqsubset$}_{H+K}$ for all $H,K\in S(G)$.
\end{itemize}
We allow $\scalebox{0.8}[1.3]{$\sqsubset$}_H$ to be empty or disconnected for $H\neq 0$. A $G$-stratification $\mathcal{S}$ partitions $\scalebox{0.8}[1.3]{$\sqsubset$}$ into disjoint subsets
$$
\scalebox{0.8}[1.3]{$\sqsubset$}=\coprod_{H\in S(G)}\scalebox{0.8}[1.3]{$\sqsubset$}_H\backslash \scalebox{0.8}[1.3]{$\sqsubset$}^0_H \quad \textrm{ and }\quad \scalebox{0.8}[1.3]{$\sqsubset$}_{H}^0=\bigcup_{H\subsetneq K} \scalebox{0.8}[1.3]{$\sqsubset$}_K.
$$
For $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$ we define the {\it dilation subgroup} $D(x)$ to be the unique subgroup $H\subset G$ such that $x\in \scalebox{0.8}[1.3]{$\sqsubset$}_H\backslash \scalebox{0.8}[1.3]{$\sqsubset$}^0_H$. We define the {\it index function} $a_{\mathcal{S}}:\scalebox{0.8}[1.3]{$\sqsubset$}\times S(G)\to \mathbb{Z}_{\geq 0}$ of $\mathcal{S}$ by setting $a_{\mathcal{S}}(x,H)$ to be the number of connected components of the intersection of $\scalebox{0.8}[1.3]{$\sqsubset$}_H\backslash \scalebox{0.8}[1.3]{$\sqsubset$}_H^0$ with a sufficiently small punctured neighborhood of $x$. We say that $\mathcal{S}$ is {\it admissible} if for every $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$ the number
\begin{equation}
g'(x)=\#(D(x))\big[g(x)-1\big]+1+\frac{1}{2}\sum_{K\in S(D(x))}a_{\mathcal{S}}(x;K)\big(\#(D(x))-[D(x):K]\big)
\label{eq:genustropical}
\end{equation}
is a non-negative integer.
\label{def:stratificationtropical}
Finally, we define the dual stratification $\mathcal{S}^*$ of a stratification $\mathcal{S}$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ as follows. Choose a model $\Gamma$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ minimal with respect to the property that each element $\scalebox{0.8}[1.3]{$\sqsubset$}_H$ of $\mathcal{S}$ corresponds to a subgraph $\Gamma_H$ of $\Gamma$. Then the $\Gamma_H$ form a $G$-stratification of the weighted metric graph $\Gamma$, and we let $\mathcal{S}^*$ be the dual of this stratification. We note that choosing a larger model $\Gamma'$ will result in a larger dual stratification, which will, however, retract to $\mathcal{S}^*$.
\end{definition}
Similarly, we can define the dilated cohomology groups of a tropical curve with a $G$-stratification:
\begin{definition} Let $\mathcal{S}$ be a stratification of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$. Pick a model $(\Gamma,\ell)$ for $\scalebox{0.8}[1.3]{$\sqsubset$}$ such that each $\scalebox{0.8}[1.3]{$\sqsubset$}_H$ corresponds to a subgraph $\Gamma_H$ of $\Gamma$, then $\mathcal{S}$ is a $G$-stratification of $\Gamma$ and induces a $G$-dilation datum $D$ on $\Gamma$. We define the {\it dilated cohomology group} $H^1(\scalebox{0.8}[1.3]{$\sqsubset$},\mathcal{S})$ as the cohomology group $H^1(\Gamma,D)$; it is clear that this group does not depend on the choice of model.\label{def:dilatedcohomologytropical}
\end{definition}
We can now state our main classification result for $G$-covers of tropical curves, which is simply a restatement of Thm.~\ref{thm:main4} using the equivalent description of dilation by means of a stratification:
\begin{theorem} Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve. There is a natural bijection between the set of $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ and the set of triples $(\mathcal{S},\eta,g')$, where
\begin{enumerate}
\item $\mathcal{S}$ is a $G$-stratification of $\scalebox{0.8}[1.3]{$\sqsubset$}$,
\item $\eta$ is an element of $H^1(\scalebox{0.8}[1.3]{$\sqsubset$},\mathcal{S})$,
\item $g'$ is a function from $\scalebox{0.8}[1.3]{$\sqsubset$}$ to $\mathbb{Z}_{\geq 0}$.
\end{enumerate}
The set of unramified $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is obtained by requiring $\mathcal{S}$ to be an admissible $G$-stratification, and defining $g'$ by~\eqref{eq:genustropical}.
\label{thm:main5}
\end{theorem}
We also restate Prop.~\ref{prop:coversstability} for tropical curves.
\begin{proposition} Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve. Then there is a natural bijection between the unramified $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ and the unramified $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}^{st}$.
\end{proposition}
\begin{remark} Any tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ has infinitely many $G$-covers for any nontrivial group $G$, since we can choose a dilation stratification with arbitrarily many connected components. However, the number of {\it unramified} $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is finite. Indeed, Prop.~\ref{prop:admissibilitystability} shows that if $\mathcal{S}$ is an admissible stratification of $\scalebox{0.8}[1.3]{$\sqsubset$}$, then no $\scalebox{0.8}[1.3]{$\sqsubset$}_H$ can contain any simple point $x\in \scalebox{0.8}[1.3]{$\sqsubset$}$ as an unstable extremal point. Since any tropical curve has only finitely many non-simple points, it follows that the number of admissible stratifications of a tropical curve is finite, and hence so is the number of unramified $G$-covers.
\label{rem:finitelymanycovers}
\end{remark}
\subsubsection*{Cyclic covers of prime order}
We now classify the unramified $G$-covers of a tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ in the case when $G=\ZZ/p\ZZ$, where $p$ is prime. These covers were studied in \cite{2018JensenLen} and \cite{2017BologneseBrandtChua} for $p=2$, and in \cite{2017BrandtHelminck} for arbitrary $p$ in the case when $\scalebox{0.8}[1.3]{$\sqsubset$}$ is a tree.
Let $\scalebox{0.8}[1.3]{$\sqsubset$}$ be a tropical curve, let $p$ be a prime number, and let $G=\mathbb{Z}/p\mathbb{Z}$. A $G$-stratification $\mathcal{S}=\{\scalebox{0.8}[1.3]{$\sqsubset$}_0,\scalebox{0.8}[1.3]{$\sqsubset$}_G\}$ of $\scalebox{0.8}[1.3]{$\sqsubset$}$ has a single nontrivial element $\scalebox{0.8}[1.3]{$\sqsubset$}_G=\scalebox{0.8}[1.3]{$\sqsubset$}_{dil}$, the dilated subcurve. Condition~\eqref{eq:genustropical} is trivially satisfied at any non-dilated point. If $x\in \scalebox{0.8}[1.3]{$\sqsubset$}_G$, then $D(x)=\mathbb{Z}/p\mathbb{Z}$ and $a_{\mathcal{S}}(x;\mathbb{Z}/p\mathbb{Z})=\val_{\scalebox{0.8}[1.3]{$\sqsubset$}_G}(x)$, and condition~\eqref{eq:genustropical} is
$$
g'(x)=\big[g(x)-1\big]p+1+\frac{p-1}{2}\val_{\scalebox{0.8}[1.3]{$\sqsubset$}_G}(x).
$$
We see that $g'(x)$ is non-negative if $g(x)>0$, or if $g(x)=0$ and $\val_{\scalebox{0.8}[1.3]{$\sqsubset$}_G}(x)\geq 2$. Similarly, $g'(x)$ is an integer if $p\geq 3$, or if $p=2$ and $\val_{\scalebox{0.8}[1.3]{$\sqsubset$}_G}(x)$ is even. We therefore have the following result:
\begin{enumerate}
\item For $p\geq 3$, a $\ZZ/p\ZZ$-stratification $\scalebox{0.8}[1.3]{$\sqsubset$}$ is admissible if and only if the dilated subcurve $\scalebox{0.8}[1.3]{$\sqsubset$}_G$ is semistable.
\item For $p=2$, a $\ZZ/p\ZZ$-stratification $\scalebox{0.8}[1.3]{$\sqsubset$}$ is admissible if and only if the dilated subcurve $\scalebox{0.8}[1.3]{$\sqsubset$}_G$ is a semistable cycle. This was observed in \cite{2018JensenLen} (see Corollary~5.5).
\end{enumerate}
If $\mathcal{S}$ is an admissible $\ZZ/p\ZZ$-stratification of $\scalebox{0.8}[1.3]{$\sqsubset$}$, Ex.~\ref{ex:subgraph} and Thm.~\ref{thm:main3} shows that the set of unramified $G$-covers of $\scalebox{0.8}[1.3]{$\sqsubset$}$ having dilation stratification $\mathcal{S}$ is equal to $H^1(\scalebox{0.8}[1.3]{$\sqsubset$}^0,\ZZ/p\ZZ)$, where $\scalebox{0.8}[1.3]{$\sqsubset$}^0$ is the nontrivial element of the dual stratification $\mathcal{S}^*(D)$. Specifically, $\scalebox{0.8}[1.3]{$\sqsubset$}^0$ is obtained from $\scalebox{0.8}[1.3]{$\sqsubset$}$ by removing $\scalebox{0.8}[1.3]{$\sqsubset$}_G$, and then removing any edges or legs that are missing an endpoint. In other words, an unramified $\ZZ/p\ZZ$-cover of $\scalebox{0.8}[1.3]{$\sqsubset$}$ is uniquely specified by choosing a (possibly empty) semistable subcurve $\scalebox{0.8}[1.3]{$\sqsubset$}_G\subset \scalebox{0.8}[1.3]{$\sqsubset$}$, which is required to be a cycle when $p=2$, and an element of $H^1(\scalebox{0.8}[1.3]{$\sqsubset$}^0,\ZZ/p\ZZ)$.
As an example, we count the number of unramified $\ZZ/p\ZZ$-covers of the following genus two tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ with one leg (the edge lengths are arbitrary and irrelevant):
\begin{center}
\begin{tikzpicture}
\draw[ultra thick] (0.5,0) circle(.5);
\draw[ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[ultra thick] (0.5,2.0) circle(.5);
\draw[fill] (0.5,0.5)circle(.10);
\draw[fill] (0.5,1.0)circle(.10);
\draw[fill] (0.5,1.5)circle(.10);
\end{tikzpicture}
\end{center}
This curve has eight admissible $\ZZ/p\ZZ$-stratifications, listed below, with the last one being admissible only when $p$ is odd. We draw the semistable dilated subgraph $\scalebox{0.8}[1.3]{$\sqsubset$}_G$ in blue, and the corresponding element $\scalebox{0.8}[1.3]{$\sqsubset$}^0$ of the dual stratification in red. Below each stratification we list the number $p^{b_1\left(\scalebox{0.8}[1.3]{$\sqsubset$}^0\right)}$ of $\ZZ/p\ZZ$-covers with the given stratification.
\begin{center}
\begin{tabular}{c c c c c c c c}
\begin{tikzpicture}
\draw[red,ultra thick] (0.5,0) circle(.5);
\draw[red,ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[red,ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[red,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[red,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,red] (0.5,0.5)circle(.10);
\draw[fill,red] (0.5,1.0)circle(.10);
\draw[fill,red] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[red,ultra thick] (0.5,0) circle(.5);
\draw[red,ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[red,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[blue,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,red] (0.5,0.5)circle(.10);
\draw[fill,red] (0.5,1.0)circle(.10);
\draw[fill,blue] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[blue,ultra thick] (0.5,0) circle(.5);
\draw[ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[red,ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[red,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[red,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,blue] (0.5,0.5)circle(.10);
\draw[fill,red] (0.5,1.0)circle(.10);
\draw[fill,red] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[red,ultra thick] (0.5,0) circle(.5);
\draw[ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[blue,ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[blue,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[blue,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,red] (0.5,0.5)circle(.10);
\draw[fill,blue] (0.5,1.0)circle(.10);
\draw[fill,blue] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[blue,ultra thick] (0.5,0) circle(.5);
\draw[blue,ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[blue,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[red,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,blue] (0.5,0.5)circle(.10);
\draw[fill,blue] (0.5,1.0)circle(.10);
\draw[fill,red] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[blue,ultra thick] (0.5,0) circle(.5);
\draw[ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[red,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[blue,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,blue] (0.5,0.5)circle(.10);
\draw[fill,red] (0.5,1.0)circle(.10);
\draw[fill,blue] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[blue,ultra thick] (0.5,0) circle(.5);
\draw[blue,ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[blue,ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[blue,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,blue] (0.5,0.5)circle(.10);
\draw[fill,blue] (0.5,1.0)circle(.10);
\draw[fill,blue] (0.5,1.5)circle(.10);
\end{tikzpicture}
&
\begin{tikzpicture}
\draw[blue,ultra thick] (0.5,0) circle(.5);
\draw[blue,ultra thick] (0.5,0.5) -- (0.5,1.0);
\draw[blue,ultra thick] (0.5,1.0) -- (0.5,1.5);
\draw[blue,ultra thick] (0.5,1.0) -- (1.0,1.0);
\draw[blue,ultra thick] (0.5,2.0) circle(.5);
\draw[fill,blue] (0.5,0.5)circle(.10);
\draw[fill,blue] (0.5,1.0)circle(.10);
\draw[fill,blue] (0.5,1.5)circle(.10);
\end{tikzpicture}
\\
$p^2$ & $p$ & $p$ & $p$ & $p$ & 1 & 1 & 1
\end{tabular}
\end{center}
Hence there are a total of $p^2+4p+3$ covers when $p$ is odd, and 14 covers when $p=2$.
\section{Tropicalizing the moduli space of admissible $G$-covers} \label{sec:tropicalization}
In this section we explain how unramified tropical $G$-covers naturally arise as tropicalizations of algebraic $G$-covers from a moduli-theoretic perspective, expanding on~\cite{ACP} and~\cite{CavalieriMarkwigRanganathan_tropadmissiblecovers} (recall that tropical unramified covers are called {\it tropical admissible covers} in~\cite{CavalieriMarkwigRanganathan_tropadmissiblecovers}).
Throughout this section we assume that the genus $g\geq 2$ and we work over an algebraically closed field $k$ endowed with the trivial absolute value. In this section, we do not need to assume that $G$ is abelian.
\subsection{Compactifying the moduli space of $G$-covers}
Let $G$ be a finite group and let $X\rightarrow S$ be a family of smooth projective curves of genus $g$. A $G$-cover of $X$ is a finite unramified Galois morphism $f\colon X'\rightarrow X$ together with an isomorphism $\Aut(X'/X)\simeq G$. Denote by $\mathcal{H}_{g,G}$ the moduli space of connected $G$-covers of smooth curves of genus $g$ (see e.g. \cite{RomagnyWewers} for a construction). There is a good notion of a limit object as $X$ degenerates to a stable curve, as introduced in \cite{AbramovichCortiVistoli}. The definition below generalizes this construction, by allowing a fixed ramification profile along marked points.
\begin{definition}\label{def_admissibleGcover}
Let $G$ be a finite group and let $X\rightarrow S$ be a family of stable curves of genus $g$ with $n$ marked disjoint sections $s_1,\ldots, s_n$. Let $\mu=(r_1,\ldots, r_n)$ be a $n$-tuple of natural numbers that divide $\#(G)$, and denote $k_i=\#(G)/r_i$ for $i=1,\ldots, n$. An \emph{admissible $G$-cover} of $X$ consists of a finite morphism $f\colon X'\rightarrow X$ from a family of stable curves $X'\rightarrow S$ that is Galois and unramified away from the sections of $X$, an action of $G$ on $X'$, and disjoint sections $s'_{ij}$ of $X'$ over $S$ for $i=1,\ldots,n$ and $j=1,\ldots,k_i$,
subject to the following conditions:
\begin{enumerate}[(i)]
\item The map $f:X'\rightarrow X$ is a principal $G$-bundle away from the nodes and sections of $X$.
\item The preimage of the set of nodes in $X$ is precisely the set of nodes of $X'$.
\item The preimage of a section $s_i$ is precisely given by the sections $s'_{i1},\ldots, s'_{ik_i}$.
\item Let $p$ be a node in $X$ and $p'$ a node of $X'$ above $p$. Then \'etale-locally $p'$ is given by $x'y'=t$ for $t\in\mathcal{O}_S$ and $p$ is \'etale-locally given by $xy=t^r$ for some integer $r\geq 1$ with $x'=x^r$ and $y'=y^r$, and the stabilizer of $G$ at $p'$ is cyclic of order $r$ and operates via
\begin{equation*}
(x',y')\longmapsto (\zeta x',\zeta^{-1} y')
\end{equation*}
for an $r$-th root of unity $\zeta\in\mu_r$.
\item \'Etale-locally near the sections $s_i$ and $s'_{ij}$ respectively, the morphism $f$ is given by $\mathcal{O}_S[t_i]\rightarrow \mathcal{O}_S[t_{ij}']$ with $(t_{ij}')^{r_i}=t_i$, and the stabilizer of $G$ along $s_{ij}$ is cyclic or order $r_i$ and operates via $t'_{ij}\mapsto \zeta t$, for an $r_i$-th root of unity $\zeta\in \mu_{r_i}$.
\end{enumerate}
\end{definition}
We emphasize that the $G$-action is part of the data, in particular, an isomorphism between two admissible $G$-covers has to be a $G$-equivariant isomorphism. As explained in \cite{AbramovichCortiVistoli}, the moduli space $\overline{\mathcal{H}}_{g,G}(\mu)$ of $G$-admissible covers of stable $n$-marked curves of genus $g$ is a smooth and proper Deligne-Mumford stack that contains the locus $\mathcal{H}_{g,G}(\mu)$ of $G$-covers of smooth curves of ramification type $\mu$ as an open substack. The complement of $\mathcal{H}_{g,G}(\mu)$ is a normal crossing divisor.
\begin{remark}
Although closely related, the moduli space $\overline{\mathcal{H}}_{g, G}(\mu)$ is not quite the same as the one constructed in \cite{AbramovichCortiVistoli}. The quotient
\begin{equation*}
\big[\overline{\mathcal{H}}_{g, G}(\mu)/S_{k_1}\times \ldots \times S_{k_{n}}\big]
\end{equation*}
which forgets about the order of the marked sections on $s'_{ij}$ of $X'$ over $S$ for $i=1,\ldots,n$ and $j=1,\ldots,k_i$, is equivalent to a connected component of the moduli space of twisted stable maps to $\mathbf{B}G$ in the sense of \cite{AbramovichVistoli, AbramovichCortiVistoli}, indexed by ramification profile and decomposition into connected components. Our variant of this moduli space $\overline{\mathcal{H}}_{g, G}(\mu)$, with ordered sections on $X'$, has also appeared in \cite{SchmittvanZelm} and in \cite{JarvisKaufmannKimura} (the latter permitting admissible covers with possibly disconnected domains).
An object in this stack is technically not a admissible $G$-cover $X'\rightarrow X$ but rather a $G$-cover $X'\rightarrow \mathcal{X}$ of a twisted stable curve $\mathcal{X}$. A \emph{twisted stable curve} $\mathcal{X}\rightarrow S$ is a Deligne-Mumford stack $\mathcal{X}$ with sections $s_1,\ldots, s_n\colon S\rightarrow \mathcal{X}$ whose coarse moduli space $X\rightarrow S$ is a family of stable curves over $S$ with $n$ marked sections (also denoted by $s_1,\ldots, s_n$) such that
\begin{itemize}
\item The smooth locus of $\mathcal{X}$ is representable by a scheme,
\item The singularities are \'etale-locally given by $\big[\{x'y'=t\}/\mu_r\big]$ for $t\in\mathcal{O}_S$, where $\zeta\in\mu_r$ acts by $\zeta\cdot(x',y')=(\zeta x',\zeta^{-1}y')$. In this case the singularity in $X'$ is locally given by $xy=t^{r}$.
\item The stack $\mathcal{X}$ is a root stack $\big[\sqrt[r_i]{s_i/X}\big]$ along the section $s_i$ for all $i=1,\ldots n$.
\end{itemize}
Both notions are naturally equivalent: given a $G$-admissible cover $X'\rightarrow X$ the associated twisted $G$-cover is given by $X'\rightarrow [X'/G]$. Conversely, given a twisted $G$-cover $X'\rightarrow \mathcal{X}$ in the corresponding connected component, the composition $X'\rightarrow\mathcal{X}\rightarrow X$ with the morphism to the coarse moduli space $X$ is a $G$-admissible cover. We refer the interested reader to \cite{BertinRomagny} for an alternative approach to this construction.
\end{remark}
\subsection{The moduli space of unramified tropical $G$-covers}
We now construct a moduli space $H_{g,G}^{trop}(\mu)$ of unramified $G$-covers of stable tropical curves of genus $g$ with $n$ marked points and ramification profile $\mu=(r_1,\ldots,r_n)$, where each $r_i$ divides $\#(G)$. Denote $k_i=\#(G)/r_i$, as well as $k=k_1+\cdots+k_n$, and assume that $n\cdot \#(G)-k$ is even. A point $\big[\varphi,l,l'\big]$ of $H_{g,G}^{trop}(\mu)$ consists of the following data:
\begin{enumerate} \item A $G$-equivariant isomorphism class of an unramified $G$-cover $\varphi\colon \scalebox{0.8}[1.3]{$\sqsubset$}'\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$}$ of a stable tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ of genus $g$ with $n$ legs, by a stable tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}'$ of genus
$$
g'=(g-1)\cdot\#(G)+1+(n\cdot \#(G)-k)/2
$$
with $k$ legs.
\item A marking $l:\{1,\ldots,n\}\simeq L(\scalebox{0.8}[1.3]{$\sqsubset$})$ of the legs of $\scalebox{0.8}[1.3]{$\sqsubset$}$.
\item A marking $l'\colon\big \{(1,\ldots, k_1),\ldots,(1,\ldots,k_n)\big\}\simeq L(\scalebox{0.8}[1.3]{$\sqsubset$}')$ of the legs of $\scalebox{0.8}[1.3]{$\sqsubset$}'$ such that $\varphi(l'_{ij})=l_i$, where we denote $l_i=l(i)$ and $l'_{ij}=l(i,j)$.
\end{enumerate}
\begin{proposition}\label{prop_HgG=genconecomplex}
The moduli space $H_{g,G}^{trop}(\mu)$ naturally carries the structure of a generalized cone complex.
\end{proposition}
\begin{proof}
We need to show that $H_{g,G}^{trop}(\mu)$ is naturally the colimit of a diagram of rational polyhedral cones connected by (not necessarily proper) face morphisms. We first construct an index category $J_{g,G}(\mu)$ as follows:
\begin{itemize}
\item The objects are tuples $\big(\varphi\colon\Gamma'\rightarrow \Gamma,l,l'\big)$, where $\Gamma'$ and $\Gamma$ are stable weighted graphs of genera $g'$ and $g$ having respectively $k$ and $n$ legs, $\varphi$ is an unramified $G$-cover, and $l'$ and $l$ are markings of the legs of $\Gamma'$ and $\Gamma$, respectively, such that $\varphi(l'_{ij})=l_i$.
\item The morphisms are generated by the automorphisms of $\Gamma'\rightarrow\Gamma$ that preserve the markings on both $\Gamma$ and $\Gamma'$, and weighted edge contractions (see Def.~\ref{def:edgecontraction}) of the target graph $\Gamma$. We recall that a weighted edge contraction of the target graph $\Gamma$ induces a weighted edge contraction of the source graph $\Gamma'$ along the preimages of the contracted edges. Moreover, the $G$-action on $\Gamma'$ induces a $G$-action on its weighted edge contraction, which is an unramified $G$-cover by Prop.~\ref{prop:edgecontraction}.
\end{itemize}
We then consider a functor $\Sigma_{g,G}(\mu)\colon J_{g,G}(\mu)\rightarrow \mathbf{RPC}_{face}$ to the category $\mathbf{RPC}_{face}$ of rational polyhedral cones with (not necessarily proper) face morphisms defined as follows:
\begin{itemize}
\item An object $\big(\varphi:\Gamma'\rightarrow \Gamma,l,l'\big)$ is sent to the rational polyhedral cone $\sigma_{\varphi}=\mathbb{R}_{\geq 0}^{E(\Gamma)}$.
\item An automorphism of $\big(\Gamma'\rightarrow \Gamma,l,l'\big)$ induces an automorphism of $\sigma_{\varphi}$ that permutes the entries according to the induced permutation of the edges of $\Gamma$; for a set of edges $S\subset E(\Gamma)$, a weighted edge contraction $\varphi_S:\Gamma'/\varphi^{-1}(S)\to \Gamma/S$ of $\varphi:\Gamma'\to\Gamma$ induces a morphism $\sigma_{\varphi_S}\hookrightarrow \sigma_{\varphi}$ that sends $\sigma_{\varphi_S}$ to the face of $\sigma_{\varphi}$ given by setting all entries of the contracted edges equal to zero.
\end{itemize}
The natural maps $\sigma_{\varphi}\rightarrow H_{g,G}^{trop}(\mu)$ are given by associating to a point $(a_e)_{e\in E(\Gamma)}\in \mathbb{R}_{\geq 0}^{E(\Gamma)}$ an unramified $G$-cover $\big[\scalebox{0.8}[1.3]{$\sqsubset$}'\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$}\big]$ defined as follows: \begin{itemize}
\item In the special case that $a_e\neq 0$ for all $e\in E(\Gamma)$, the tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ is given by the graph $\Gamma$ with the metric $\ell(e)=a_e$. In general, the tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}$ is given by contracting those edges $e\in E(\Gamma)$ for which $a_e=0$ and then by endowing the contracted weighted graph with the induced edge length given by the $a_e\neq 0$.
\item The tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}'$ is defined accordingly: we first contract all edges that map to an edge $e$ with $a_e=0$, and then endow $\scalebox{0.8}[1.3]{$\sqsubset$}'$ with the edge length $\ell'(e')= \ell(\varphi(e'))/d_\varphi(e)$ so that the induced map $\scalebox{0.8}[1.3]{$\sqsubset$}'\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$}$ is an unramified $G$-cover of tropical curves.
\end{itemize}
The maps $\sigma_{\varphi}\rightarrow H_{g,G}^{trop}(\mu)$ naturally commute with the morphisms induced by $J_{g,G}(\mu)$, and therefore descend to a map
\begin{equation*}
\varinjlim_{(\varphi,l,l')\in J_{g,G}(\mu)} \sigma_{\varphi} \simeq H_{g,G}^{trop}(\mu)
\end{equation*}
that is easily checked to be a bijection. This realizes $H_{g,G}^{trop}(\mu)$ as a colimit of a diagram of (not necessarily proper) face morphisms and therefore endows it with the structure of a generalized cone complex.
\end{proof}
There are natural \emph{source} and \emph{target morphisms}
\begin{equation*}
\src_{g,G}^{trop}(\mu)\colon H_{g,G}^{trop}(\mu)\longrightarrow M_{g',k}^{trop}\\
\qquad \textrm{ and } \qquad
\tar_{g,G}^{trop}(\mu)\colon H_{g,G}^{trop}(\mu)\longrightarrow M_{g,n}^{trop}\\
\end{equation*}
that are given by the associations
\begin{equation*}
\big[\scalebox{0.8}[1.3]{$\sqsubset$}'\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$},l,l'\big]\longmapsto \big[\scalebox{0.8}[1.3]{$\sqsubset$}',l'\big] \qquad \textrm{ and } \qquad \big[\scalebox{0.8}[1.3]{$\sqsubset$}'\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$},l,l'\big]\longmapsto \big[\scalebox{0.8}[1.3]{$\sqsubset$},l\big]
\end{equation*}
respectively. By Rem.~\ref{rem:finitelymanycovers}, the map $\tar_{g,G}^{trop}(\mu)$ has finite fibers.
\begin{remark}
The functor $J_{g,G}(\mu)\rightarrow \mathbf{RPC}_{face}$ in the proof of Proposition \ref{prop_HgG=genconecomplex} defines a category fibered in groupoids over $\mathbf{RPC}_{face}$, i.e. a \emph{combinatorial cone stack} in the sense of \cite{CCUW}. So we may think of $H_{g,G}^{trop}(\mu)$ as a "coarse moduli space" of a \emph{cone stack} $\mathcal{H}_{g,G}^{trop}(\mu)$, a geometric stack over the category of rational polyhedral cones (see \cite{CCUW} for details), that parametrizes families of unramified tropical $G$-covers over rational polyhedral cones.
\end{remark}
\subsection{A modular perspective on tropicalization}
Denote by $\mathcal{H}_{g,G}^{an}(\mu)$ the Berkovich analytic space\footnote{We implicitly work with the underlying topological space of the Berkovich analytic stack $\mathcal{H}_{g,G}^{an}(\mu)$, as introduced in \cite[Section 3]{Ulirsch_tropisquot}.} associated to $\mathcal{H}_{g,G}(\mu)$. We define a natural \emph{tropicalization map}
\begin{equation*}\begin{split}
\trop_{g,G}(\mu)\colon \mathcal{H}_{g,G}^{an}(\mu)&\longrightarrow H_{g,G}^{trop}(\mu)\\
[X'\rightarrow X,s_i,s'_{ij}]& \longmapsto \big[\scalebox{0.8}[1.3]{$\sqsubset$}_{X'}\rightarrow\scalebox{0.8}[1.3]{$\sqsubset$}_X,l,l'\big]
\end{split}\end{equation*}
that associates to an admissible $G$-cover $X'\rightarrow X$ of smooth curves over a non-Archimedean extension $K$ of $k$ an unramified tropical $G$-cover $\scalebox{0.8}[1.3]{$\sqsubset$}_{X'}\rightarrow \scalebox{0.8}[1.3]{$\sqsubset$}_X$ of the dual tropical curve $\scalebox{0.8}[1.3]{$\sqsubset$}_X$ of $X$ that is defined in the following way.
Let $X$ be a smooth projective curve of genus $g$ over a non-Archimedean extension $K$ of $k$ with $n$ marked sections $s_1,\ldots, s_n$ over $K$. Let $(X'\rightarrow X,s'_{ij})$ be a $G$-cover of $X$, where $i=1,\ldots,n$ and $j=1,\ldots,k_i$. By the valuative criterion for properness, applied to the stack $\overline{\mathcal{H}}_{g,G}(\mu)$, there is a finite extension $L$ of $K$ such that $X'_L\rightarrow X_L$ extends to a family of admissible $G$-covers $f:\mathcal{X}'\rightarrow\mathcal{X}$ defined over the valuation ring $R$ of $L$ (with marked sections also denoted by $s_i$ and $s'_{ij}$). The \emph{dual tropical curve} $(\scalebox{0.8}[1.3]{$\sqsubset$}_X,l)$ of $\mathcal{X}$ (and similarly $(\scalebox{0.8}[1.3]{$\sqsubset$}_{X'},l')$ of $\mathcal{X}'$) is given by the following data:
\begin{itemize}
\item The dual graph $\Gamma_{\mathcal{X}_0}$ of the special fiber $\mathcal{X}_0$ of $\mathcal{X}$: the components of $\mathcal{X}_0$ correspond to vertices, nodes correspond to edges, and the sections correspond to legs.
\item A vertex weight $V(\Gamma_{\mathcal{X}_0})\rightarrow \mathbb{Z}_{\geq 0}$ that associates to a vertex $v$ the genus of the normalization of the corresponding component of $\mathcal{X}_0$.
\item A marking $l\colon \{1,\ldots n\}\simeq L(\Gamma_{\mathcal{X}_0})$ of the legs of $\Gamma_{\mathcal{X}_0}$ according to the full order of $s_1,\ldots, s_n$.
\item An edge length $\ell\colon E(\Gamma_{\mathcal{X}_0})\rightarrow\mathbb{R}_{>0}$ that associates to an edge $e$ the positive real number $r\cdot\val(t)$, where the corresponding node is \'etale-locally given by an equation $xy=t^r$ for $t\in R$.
\end{itemize}
The map $f:\mathcal{X}'\rightarrow\mathcal{X}$ induces a map $\varphi:\Gamma_{\mathcal{X}'_0}\rightarrow\Gamma_{\mathcal{X}_0}$:
\begin{itemize}
\item Every component $X'_{v'}$ of $\mathcal{X}'_0$ is mapped to exactly one component $X_{v}$ of $\mathcal{X}_0$.
\item Every node $p_{e'}$ of $\mathcal{X}'_0$, over a node $p_e$ of $\mathcal{X}_0$ given by $xy=t^r$ for $t\in R$ on the base, has a local equation $x'y'=t$ which determines the dilation factor $r=d_\varphi(e')$.
\item \'Etale-locally around the sections $s_i$ and $s'_{ij}$ respectively, the morphism $f$ is given by $\mathcal{O}_S[t_i]\rightarrow \mathcal{O}[t'_{ij}]$ with $(t'_{ij})^{r_i}=t_i$, so the dilation factor $d_\varphi(l'_{ij})$ is given by $r_i$.
\end{itemize}
The map $f:\Gamma_{\mathcal{X}'_0}\rightarrow\Gamma_{\mathcal{X}_0}$ is harmonic by \cite[Theorem A]{ABBRI} (identifying both $\scalebox{0.8}[1.3]{$\sqsubset$}_{X'}$ and $\scalebox{0.8}[1.3]{$\sqsubset$}_X$ with the non-Archimedean skeletons of $(X')^{an}$ and $X^{an}$ respectively). Applying the Riemann--Hurwitz formula to $X_{v'}\rightarrow X_v$ shows that it is unramified. The operation of $G$ on $\mathcal{X}'_0$ induces an operation of $G$ on $\Gamma_{\mathcal{X}'_0}$ for which the map $\scalebox{0.8}[1.3]{$\sqsubset$}_{X'}\rightarrow \scalebox{0.8}[1.3]{$\sqsubset$}_X$ is $G$-invariant. The stabilizer at every edge $e'_i$ and of every leg $l'_{ij}$ is a cyclic group of order $r_i$ and $r_{ij}$ respectively by Definition \ref{def_admissibleGcover} (iii) and (iv). Since $\mathcal{X}'_0\rightarrow\mathcal{X}_0$ is a principal $G$-bundle away from the nodes, the operation of $G$ on the fiber over each point in $\scalebox{0.8}[1.3]{$\sqsubset$}_{X}$ is transitive and so $\scalebox{0.8}[1.3]{$\sqsubset$}_{X'}\rightarrow \scalebox{0.8}[1.3]{$\sqsubset$}_X$ is a $G$-cover.
\begin{remark}
In the language of twisted stable curves, the stabilizers of the $G$-operation on the nodes and legs of $\mathcal{X}_0'$ give rise to the dilation datum on the dual graph. So one may think of a dilation datum as a stack-theoretic enhancement of a tropical curve.
\end{remark}
Since the boundary of $\overline{\mathcal{H}}_{g,G}(\mu)$ has normal crossings, the open immersion $\mathcal{H}_{g,G}(\mu)\hookrightarrow \overline{\mathcal{H}}_{g,G}(\mu)$ is a toroidal embedding in the sense of \cite{KKMSD}. Therefore, as explained in \cite{Thuillier_toroidal, ACP}, there is a natural strong deformation retraction $\rho_{g,G}\colon \mathcal{H}_{g,G}^{an}(\mu)\rightarrow \mathcal{H}_{g,G}^{an}(\mu)$ onto a closed subset of $\mathcal{H}_{g,G}^{an}(\mu)$ that carries the structure of a generalized cone complex, the \emph{non-Archimedean skeleton} $\Sigma_{g,G}(\mu)$ of $\mathcal{H}_{g,G}^{an}(\mu)$. Expanding on \cite[Theorem 1 and 4]{CavalieriMarkwigRanganathan_tropadmissiblecovers}, we have:
\begin{theorem}\label{thm_skeletonvstropicalization}
The tropicalization map
$\trop_{g,G}(\mu)\colon\mathcal{H}_{g,G}^{an}(\mu)\longrightarrow H_{g,G}^{trop}(\mu)$
factors through the retraction to the non-Archimedean skeleton $\Sigma_{g,G}(\mu)$ of $\mathcal{H}_{g,G}^{an}(\mu)$, so that the restriction
\begin{equation*}
\trop_{g,G}(\mu)\colon \Sigma_{g,G}(\mu)\longrightarrow H^{trop}_{g,G}(\mu)
\end{equation*}
to the skeleton is a finite strict morphism of generalized cone complexes. Moreover, the diagram
\begin{equation}\label{eq_functorialityoftrop}\begin{tikzcd}
\mathcal{H}_{g,G}^{an}(\mu)\arrow[rr,"\src_{g,G}^{an}(\mu)"] \arrow[dd,"\tar_{g,G}^{an}(\mu)"']\arrow[rd,"\trop_{g,G}(\mu)"]& & \mathcal{M}_{g',k}^{an}\arrow[d,"\trop_{g',k}"]\\
& H_{g,G}^{trop}(\mu) \arrow[r,"\src_{g,G}^{trop}(\mu)"]\arrow[d,"\tar_{g,G}^{trop}(\mu)"']& M_{g',k}^{trop}\\
\mathcal{M}_{g,n}^{an}\arrow[r,"\trop_{g,n}"']& M_{g,n}^{trop} &
\end{tikzcd}\end{equation}
commutes.
\end{theorem}
In other words, the restriction of $\trop_{g,G}(\mu)$ onto a cone in $\Sigma_{g,G}(\mu)$ is an isomorphism onto a cone in $H^{trop}_{g,G}(\mu)$ and every cone in $H_{g,G}^{trop}(\mu)$ has at most finitely many preimages in $\Sigma_{g,G}(\mu)$.
\begin{proof}[Proof of Theorem \ref{thm_skeletonvstropicalization}]
Let $x$ be a closed point in $\overline{\mathcal{H}}_{g,G}(\mu)$, which corresponds to an admissible $G$-cover $X'\rightarrow X$ over $k$. Denote by $\varphi:\Gamma_{X'}\rightarrow \Gamma_X$ the corresponding unramified $G$-cover of the dual graphs. Denote by $\mathfrak{o}_k$ either $k$ when $\characteristic k=0$ or the unique complete local ring with residue field $k$ when $\characteristic k=p>0$ (using Cohen's structure theorem). The complete local ring at $x$ is given by
\begin{equation*}
\widehat{\mathcal{O}}_{\overline{\mathcal{H}}_{g,G}(\mu),x}\simeq \mathfrak{o}_k\big\llbracket t_1,\ldots, t_{3g-3+n}\big\rrbracket
\end{equation*}
where $t_i=0$ for $i=1,\ldots,r$ cuts out the locus where the corresponding node $q_i$ of $X$ remains a node. The retraction to the skeleton is locally given by
\begin{equation}\label{eq_localtrop}\begin{split}
\big(\Spec \widehat{\mathcal{O}}_{\overline{\mathcal{H}}_{g,G}(\mu),x}\big)^\beth &\longrightarrow \overline{\mathbb{R}}_{\geq 0}^r\\
x&\longmapsto \big( -\log\vert t_1\vert_x, \ldots, -\log\vert t_r\vert_x\big) \ ,
\end{split}\end{equation}
where $(.)^\beth$ denotes the generic fiber functor constructed in \cite[Prop./D\'ef. 1.3]{Thuillier_toroidal} and $\overline{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$. We find that under the isomorphism $\sigma_{\varphi}\simeq \mathbb{R}_{\geq 0}^r$ the restriction of \eqref{eq_localtrop} to the preimage of $\mathbb{R}_{\geq 0}^r$ is nothing but the tropicalization map $\trop_{g,G}(\mu)$ defined above.
We observe the following:
\begin{itemize}
\item A degeneration of $X'\rightarrow X$ in $\overline{\mathcal{H}}_{g,G}(\mu)$ to another admissible $G$-cover $X_0'\rightarrow X_0$ with corresponding $\varphi_0:\Gamma_{X'_0}\to \Gamma_{X_0}$ may be described by additional coordinates $t_{r+1}, \ldots, t_{r_0}$ that encode the new nodes $q_{r+1},\ldots, q_{r_0}$ in the degeneration. The induced map $\mathbb{R}_{\geq 0}^r\hookrightarrow\mathbb{R}_{\geq 0}^{r_0}$ describes $\sigma_{\varphi}$ as the face of $\sigma_{\varphi_0}$ that corresponds to letting the edges $e_{r+1},\ldots, e_{r_0}$ have length zero.
\item Denote by $E\subset \overline{\mathcal{H}}_{g,G}(\mu)$ the toroidal stratum containing $x$ and by $\widetilde{E}$ and $\widetilde{x}$ respectively their images in $\overline{\mathcal{M}}_{g,n}$. The operation of the fundamental group $\pi_1(E,x)$ of $E$ on $\mathbb{R}_{\geq 0}^r\simeq\Hom(\Lambda^+_E,\mathbb{R}_{\geq 0})$, where $\Lambda^+_E$ denotes the monoid of effective divisors supported on the closure of $E$, naturally factors through the operation of $\pi_1(\widetilde{E},\widetilde{x})$ on $\mathbb{R}_{\geq 0}^r\simeq\Hom(\Lambda^+_{\widetilde{E}},\mathbb{R}_{\geq 0})$. Analogously, the operation of the automorphisms of $\Gamma_{X'}\rightarrow\Gamma_X$ on $\mathbb{R}_{\geq 0}^r$ naturally factors through the operation of the automorphisms of $\Gamma_X$ on $\mathbb{R}_{\geq 0}^r=\mathbb{R}_{\geq 0}^{E(\Gamma_X)}$. Therefore, by \cite[Proposition 7.2.1]{ACP}, the images of the automorphism groups of both $\pi_1(E,x)$ and $\Aut(\Gamma_{X'}\rightarrow\Gamma_X)$ in the permutation group of the entries of $\mathbb{R}_{\geq 0}^r$ are equal.
\end{itemize}
This shows that the isomorphisms $\mathbb{R}_{\geq 0}^r\simeq\sigma_{\varphi}$ induce a necessarily strict morphism of generalized cone complexes $\Sigma_{g,G}(\mu)\rightarrow H_{g,G}^{trop}(\mu)$ that factors the tropicalization map as $\mathcal{H}_{g,G}^{an}(\mu)\rightarrow \Sigma_{g,G}(\mu)\rightarrow H_{g,G}^{trop}(\mu)$. Its fibers are finite, since above every toroidal stratum of $\overline{\mathcal{M}}_{g,n}$ there are only finitely many toroidal strata of $\overline{\mathcal{H}}_{g,G}(\mu)$.
Finally, the commutativity of \eqref{eq_functorialityoftrop} is an immediate consequence of the definition of $\trop_{g,G}(\mu)$.
\end{proof}
\begin{remark}
In general, not every unramified tropical cover is realizable. We may, for example, consider an unramified cover for which the local ramification profile is not of Hurwitz type (e.g. when $d=4$ and the ramification profile at a vertex is given by $\big\{(3,1),(2,2),(2,2)\big\}$). This explains why the tropicalization map on the moduli space of admissible covers (without the $G$-action), as considered in \cite{CavalieriMarkwigRanganathan_tropadmissiblecovers}, is not surjective. We refer the reader to \cite[Section 2.2]{Caporaso_gonality} for a discussion of this issue in the context of comparing algebraic and tropical gonality and to \cite{PervovaPetronio_HurwitzexistenceI} for a survey of the underlying widely open problem, the so-called \emph{Hurwitz existence problem}.
We do not know whether, for a general finite abelian group $G$, every unramified tropical $G$-cover (with cyclic stablizers at the nodes) is realizable. When $G$ itself is cyclic and there are no marked legs (along which ramification is possible), we do expect every $G$-admissible cover to be realizable, since by \cite[Theorem 3.1]{2018JensenLen} the tropicalization map on the level of $n$-torsion points on Jacobians is surjective. We will return to this topic in its proper setting in the upcoming \cite{LenUlirschZakharov_cyclictropicalcovers}.
\end{remark}
\addcontentsline{toc}{section}{References}
\bibliographystyle{amsalpha}
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proofpile-arXiv_065-7377 | \section{Introduction}
During the past decade, crowd employment has emerged as a new form of short term and flexible employment.
Crowd employment has been defined as a type of employment which ``uses an online platform to enable organisations or individuals to access an indefinite and unknown group of other organisations or individuals to solve specific problems or to provide specific services or products in exchange for payment" \cite{mandl2015new}. While this definition is similar to the concept of crowdsourcing \cite{crowdsourcing_definition}, it explicitly includes only those activities that are performed in exchange for payment.
One type of crowd employment platforms are micro-task platforms such as Amazon Mechanical Turk\footnote{http://www.mturk.com/} (AMT) or CrowdFlower\footnote{http://www.crowdflower.com/}.
On micro-task platforms, crowdworkers are payed on a per-task basis and a single task usually pays only a few cents upon completion.
The micro-tasks offered to workers on these platforms are also called ``human intelligence tasks"
and typically require workers to solve problems that are easy to solve for humans, but hard to solve for computers. This characteristic led Amazon Mechanical Turk to coin the term ``artificial artificial intelligence" for describing this type of work. Typical micro-tasks include classification and tagging of text or images, audio and image transcription, and validating addresses of companies on the Web. Also more complex tasks such as editing text documents \cite{bernstein2015soylent}, ontology alignment \cite{sarasua2012crowdmap} and the evaluation of unsupervised machine learning algorithms (e.g. \cite{chang2009reading, foulds2013stochastic, posch2015polylingual}) have been successfully deployed on micro-task platforms. Anyone, regardless of geographical location or education, can perform micro-tasks -- the only necessary requirement is having access to the Internet.
The emergence of crowd employment and a general trend towards more flexible and shorter term employment has given rise to policy discussions on social protection and working conditions of crowdworkers (e.g. \cite{europarl,euroagenda,codagnone2016future,felstiner2011working}).
One ongoing discussion is whether crowd employment is to be considered ``work” at all, or whether it is mostly considered a spare-time activity by many workers, meaning that payment plays only a minor role for them \cite{europarl}.
Currently, on popular micro-task platforms, the achievable hourly wage is below 2 US\$ (e.g. \cite{ross2010crowdworkers, horton2010labor, khanna2010evaluating}). While this amount is above minimum wage in some countries, in many high-income countries it is far below the wage of any traditional job.
Despite this, the rise of crowd employment is an international phenomenon which does not exclude high-income countries.
Understanding the underlying motivations of the ``indefinite and unknown group" of crowdworkers is therefore crucial in understanding this new form of employment.
This paper develops the groundwork for understanding the motivations of the international crowd workforce by presenting the Multidimensional Crowdworker Motivation Scale (MCMS). The MCMS is theoretically grounded in Self-Determination Theory (SDT) and tailored specifically to the context of crowdsourced micro-labor. Most items in the MCMS are based on items from existing SDT-based motivation scales (such as the Multidimensional Work Motivation Scale \cite{mwms} and the Work Extrinsic Intrinsic Motivation Scale \cite{weims}), which were adapted to the idiosyncrasies of work on micro-task platforms.
The main contributions of this paper are (1) an evaluation of two existing SDT-based work motivation scales with respect to their suitability for micro-task crowdwork, (2) the development of the Multidimensional Crowdworker Motivation Scale (MCMS), (3) a validation of the MCMS in ten countries and three income groups, and (4) an evaluation of the cross-country and cross-group comparability of motivations measured with the MCMS.
To the best of our knowledge, this is the first SDT-based work motivation scale for the crowdworking domain which is validated in multiple countries and groups.
This paper is structured in the following way: Section~\ref{sec:related_work} gives a short overview of the different types of motivation as conceptualized by Self-Determi\-nation Theory and reviews existing SDT-based work motivation scales. Furthermore, it gives an overview of related work on the motivations of crowdworkers on micro-task platforms.
In Section~\ref{sec:existing} we evaluate the suitability of existing SDT-based work motivation scales for the crowdworking domain and show the need for a work motivation scale adapted to the idiosyncrasies of this domain. Section~\ref{sec:development} describes the process of developing the MCMS and Section~\ref{sec:validation} presents a validation of the MCMS in ten countries and three income groups. In Section~\ref{sec:invariance}, we demonstrate the cross-country and cross-income group comparability of motivations measured with the MCMS. Finally, Section~\ref{sec:conclusion} concludes this work and discusses the scale's limitations as well as directions for future research.
\section{Related Work}
\label{sec:related_work}
\textbf{Self-Determination Theory and Work Motivation. }
Self-Determi\-nation Theory (SDT) was developed by Deci and Ryan \cite{deci1980empirical, deci868, deci2000and}.
The theory specifies three general kinds of motivation, \emph{amotivation}, \emph{extrinsic motivation} and \emph{intrinsic motivation}, which lie along a continuum of self-determination. At the one extreme of the continuum lies \emph{amotivation} which completely lacks self-determination; at the other extreme lies \emph{intrinsic motivation} which is completely self-determined \cite{gagne2005self}.
Between these extremes lies \emph{extrinsic motivation}, which is further split up into subtypes with varying degrees of internalisation: \emph{external regulation}, \emph{introjected regulation}, \emph{identified regulation} and \emph{integrated regulation}.
While SDT hypothesizes that the different types of motivation fall along a continuum structure, empirical evidence for this continuum hypothesis is inconsistent (e.g. \cite{chemolli2014evidence, howard2016using}). There is evidence that motivations differ more in \emph{kind} than in \emph{degree} and that SDT-based motivation scales are best represented by multidimensional models \cite{chemolli2014evidence}. Howard and Gagné \cite{howard2016using} found evidence for a continuum structure, but they also highlight the importance of taking into account the quality of motivation.
Figure \ref{fig:sdt} (adopted from \cite{gagne2005self}) shows the types of motivation as specified by SDT.
\emph{Amotivation} is the absence of motivation, a state of acting passively or not intending to act all.
\emph{External regulation} is the least self-determined form of extrinsic motivation. Individuals motivated by external regulation act in order to obtain rewards or avoid punishments \cite{deci2002overview}.
\emph{Introjected regulation} refers to a form of partially internalized extrinsic motivation which aims at the avoidance of guilt or at attaining feelings of worth \cite{deci2002overview}.
\emph{Identified regulation} is a form of extrinsic motivation with a high degree of perceived autonomy, where the action is in alignment with the individual's personal goals.
\emph{Integrated regulation} is the most self-determined form of extrinsic motivation and stems from evaluated identifications that are in alignment with self-endorsed values, goals and needs \cite{deci2002overview}.
The most self-determined form of motivation is \emph{intrinsic motivation}. This form of motivation is non-instrumental and people act freely, driven by interest and enjoyment inherent in the action \cite{ryan2000intrinsic}.
\input{diagram_sdt}
Several work motivation scales have been developed based on SDT.
The first SDT-based work motivation scale was a French scale developed by Blais et al. \cite{blais1993work}. Tremblay et al. \cite{weims} translated this scale into English and conducted an evaluation in different work environments. The resulting Work Extrinsic and Intrinsic Motivation Scale (WEIMS) measures six factors: amotivation, the four external regulation subtypes and intrinsic motivation.
Gagné et al. \cite{gagne2010motivation} created the Motivation at Work Scale (MAWS), a scale which measures the four factors external regulation, introjected regulation, identified regulation and intrinsic motivation. The MAWS was validated in French and in English and was partly based on \cite{blais1993work}.
Later, Gagné et al. \cite{mwms} developed the Multidimensional Work Motivation Scale (MWMS). The MWMS was validated in seven languages and nine countries and does not include any items from the MAWS.
MWMS measures six first-order factors (amotivation, material external regulation, social external regulation, introjected regulation, identified regulation and intrinsic motivation) and one second-order factor (external regulation).
Work motivation scales such as MWMS, MAWS and WEIMS investigate motivations at the domain level of analysis, meaning that they measure the general motivation to perform a job as opposed to specific tasks within a job.
Besides the domain of work, SDT-based scales for measuring motivation also exist for many other domains such as sports \cite{sports,walking}, teaching \cite{teaching} or academics \cite{vallerand1992academic}.
\textbf{Crowdworker Motivation on Micro-Task Platforms.}
Most research investigating the motivations of workers on micro-task platforms has focused on the platform Amazon Mechanical Turk (AMT). Consequently, most studies have focused on American and Indian crowdworkers, which constitute the vast majority of workers on AMT \cite{ipeirotis2010demographics, ross2010crowdworkers}. This country distribution is likely due to the fact that
workers can receive money from AMT in the USA and in India while workers from other countries are paid in Amazon.com gift cards \cite{amtpay}.
One early study on the reasons crowdworkers have for participating on AMT was conducted by Ipeirotis \cite{ipeirotis2010demographics}. He asked the multiple choice question ``Why do you complete tasks in Mechanical Turk?", offering six response possibilities. The study found that more Indians than Americans treat AMT as a primary source of income, and that few Indian workers report the reason ``To kill time."
Hossain \cite{hossain2012users} created a classification of motivation in online platform participation, listing extrinsic and intrinsic motivators and incentives.
Kaufmann et al. \cite{kaufmann2011more} developed an early model for measuring crowdworker motivations on AMT, differentiating between enjoyment based motivation, community based motivation, immediate payoffs, delayed payoffs and social motivation.
They used a sample composed of Indian and US workers on AMT and found that the construct with the highest score was ``immediate payoffs," i.e. payment.
Their study further found that the pastime score correlated positively with household income and negatively with the weekly time spent on AMT, and that workers who spend a lot of time on AMT may be motivated very differently than workers who spend little time on the platform.
Antin and Shaw \cite{antin2012social} used a list experiment for investigating social desirability effects in motivation self-reports of crowdworkers from the USA and India on AMT. Using the four items ``to kill time," ``to make extra money," ``for fun" and "because it gives me a sense of purpose," they found that US workers tend to over-report all four reasons while Indian workers tend to over-report ``sense of purpose" and under-report ``killing time" and ``fun."
For measuring extrinsic motivations of crowdworkers, Naderi et al. \cite{naderi2014development} evaluated a 4-factor model using a subset of WEIMS items on a sample of US workers on AMT. In this model, identified and integrated regulation are merged into one factor, and the intrinsic motivation factor is omitted.
Several qualitative studies on the motivations of crowdworkers have been conducted. For example, Gupta et al. \cite{gupta2014understanding, gupta2014turk} investigated, among other aspects, the motivations of Indian crowdworkers on AMT and Martin et al. \cite{martin2014being} studied the content of a forum for AMT users.
Other research related to the motivations of crowdworkers includes measuring the impact of motivation on performance \cite{rogstadius2011assessment} and manipulating motivations via task framing \cite{chandler2013breaking} or achievement feedback \cite{lee2013experiments}.
\section{Suitability of Existing Work Motivation Scales}
\label{sec:existing}
In order to determine whether existing SDT-based work motivation scales are suitable for application in the crowdworking domain, we conduct an evaluation of the WEIMS and the MWMS with crowdworkers on CrowdFlower.
The reason for choosing CrowdFlower over AMT is that we aim to develop a motivation scale suitable for an international comparison of crowdworker motivations, instead of exclusively focusing on crowdworkers in the USA and in India. We consider CrowdFlower to be better suited for this task as it pays workers via independent partner channels\footnote{http://www.crowdflower.com/labor-channels/} and therefore attracts a more international crowd-workforce.
In order for the scale stems and items to be conceptually applicable to the crowdworking domain, minimal adaptations had to be performed. For WEIMS, we changed the stem ``Why do you do your work?" to ``Why do you do CrowdFlower tasks?" and replaced the word ``it" (referring to ``your work") in the items with ``CrowdFlower tasks."
The stem of MWMS ``Why do you or would you put efforts into your current job?" was changed to ``Why do you or would you put efforts into CrowdFlower tasks?"
and words in the items referring to ``your current job" were replaced with ``CrowdFlower tasks." Additionally, one item in the MWMS was conceptually not applicable to the domain and had to be minimally adapted. There is no concept of ``losing one's job" on micro-task platforms. The closest concept on CrowdFlower is failing many quality control questions, which results in a lower worker account accuracy and consequently in less tasks being offered to the worker. Therefore, the item ``Because I risk losing my job if I don't put enough effort in it." was changed to ``Because I risk not being offered enough tasks if I don't put enough effort into them.".
We choose to use the term ``CrowdFlower tasks" in the stems and items instead of a more general term because workers who are logged into CrowdFlower via the partner channels see that they are doing ``CrowdFlower tasks." We can therefore assume that workers know what CrowdFlower tasks are. In contrast, general terms like ``micro-tasks" or ``human intelligence tasks" are widely used in scientific publications and sometimes in the media but we cannot be sure that workers on CrowdFlower understand this term as it does not appear frequently on channel websites or on the platform itself.
For both of the minimally adapted work motivation scales, we collected 500 responses from crowdworkers residing in the USA. Anonymity was ensured.
After removing spammers (also see Section \ref{sec:validation}),
the sample size was 424 for the WEIMS and 414 for the MWMS. This constitutes a subject to item ratio higher than 20:1 which is a suitable ratio for factor analysis \cite{osborne2009best, froman2001elements}.
\textbf{Confirmatory Factor Analysis.}
We use confirmatory factor analysis (CFA) for evaluating the factor structure of the models.
For all confirmatory factor analyses, we used the R packages lavaan \cite{lavaan} and semtools \cite{semtools}.
As the item distributions were skewed, we used a robust maximum likelihood estimator (as suggested in e.g. \cite{de2013construct}). By specifying a robust maximum likelihood estimator, the model parameters were estimated with robust standard errors and a Satorra-Bentler (S-B) scaled test statistic is reported \cite{lavaan}.
\input{table_existing_scales}
We evaluated the model fit based on the absolute fit measures \emph{root mean squared error of approximation} (RMSEA) and \emph{standardised root mean square residual} (SRMR) as well as the incremental fit measures \emph{comparative fit index} (CFI) and \emph{Tucker-Lewis index} (TLI). For cut-off values of the goodness-of-fit measures, we followed the recommendations in Hooper et al. \cite{hooper2008structural}: A well fitting model should have an RMSEA of less than 0.08, a SRMR of less than 0.05, and ideally a CFI and TLI higher than 0.95 (but at least 0.9).
Furthermore, we report the (S-B scaled) Chi-Square test statistic but do not rely on it for determining model fit as it is very sensitive to sample size (e.g. \cite{bentler1980significance}).
\textbf{Models.}
We test the following models for the adapted WEIMS: (1) The original WEIMS model with 6 factors (\texttt{WEIMS-M1}), (2) an alternative 5 factor WEIMS model with the factors identified regulation and integrated regulation loading onto a single factor (\texttt{WEIMS-M2}), and (3) the 4 factor, 12 item subset of WEIMS items which was used by Naderi et al. \cite{naderi2014development} to measure the extrinsic motivations of workers on Amazon Mechanical Turk (\texttt{WEIMS-M3}).
The rationale for evaluating the alternative model \texttt{WEIMS-M2} is that the integrated regulation factor has been shown to be poorly separable from identified regulation and intrinsic motivation (e.g. \cite{weims, vallerand1992academic}), which is also one of the reasons for why the MWMS does not include an integrated regulation factor \cite{mwms}.
For the MWMS, we test the model originally hypothesized by Gagné et al. \cite{mwms} (\texttt{MWMS-M1}) and the model which had the best fit in \cite{mwms} (\texttt{MWMS-M2}). Furthermore, we test a 6 factor model where material external and social regulation are separate factors, omitting the second-order external regulation factor. This model is tested with a hypothesized correlation of zero between intrinsic motivation and both external regulation factors (\texttt{MWMS-M3}) and without the correlation restriction (\texttt{MWMS-M4}).
\textbf{Evaluation Results.}
None of the evaluated models, with the exception of the 4-factor model \texttt{WEIMS-M3}, which does not measure intrinsic motivation, had an acceptable model fit on our data. Table \ref{table:cfa_existing} shows the goodness-of-fit statistics of the different models.
The WEIMS model with 6 factors (\texttt{WEIMS-M1}) could not be estimated due to the factors identified regulation and integrated regulation not being distinguishable from another, which resulted in the covariance matrix of the factors not being positive definite. This is consistent with the findings of Naderi et al. \cite{naderi2014development}. Also for the alternative 5 factor model \texttt{WEIMS-M2}, the goodness-of-fit measures were not in the acceptable range.
The 4 factor model \texttt{WEIMS-M3} was the only evaluated model with a good fit. However, besides the drawback that the model does not measure intrinsic motivations, it has additional limitations: It includes one item for measuring the amotivation construct which was criticized by Gagné et al. \cite{mwms} for resembling low satisfaction of the need for competence rather than measuring amotivation (``I ask myself this question, I don't seem to be able to manage the important tasks related to this work.").
Furthermore, two of the four factors (external regulation and introjected regulation) are measured by only two items each and external regulation had a low reliability (Crohnbach's alpha of 0.63) in our evaluation. Note, however, that this evaluated subset of WEIMS is not completely identical to the one used by Naderi et al. \cite{naderi2014development} due to the small adaptations made in item phrasing.
Of the evaluated MWMS models, the 6-factor model with separate factors for social external and material external regulation and no correlation restrictions had the best fit. However, the fit was not acceptable, with all goodness-of-fit measures being outside acceptable ranges.
The fit measures for \texttt{MWMS-M2} are not included in Table \ref{table:cfa_existing} due to the covariance matrix of the factors not being positive definite.
Our results show that the evaluated work motivation scales developed for the traditional work context do not work well within the crowdworking context when only minimal adaptations in item phrasing are made. For the development of a reliable motivation scale which measures the motivations of crowdworkers on all dimensions, further adaptations are needed.
\section{Development of the MCMS}
\label{sec:development}
The results of the confirmatory factor analysis conducted on the slightly modified WEIMS and MWMS show the necessity for developing a new scale for measuring the motivations of crowdworkers that is adapted to the idiosyncracies of the crowdwork environment.
To meet this necessity, we developed the Multidimensional Crowdworker Motivation Scale (MCMS) in three steps.
First, we compiled a pool of items conceptually suitable for the characteristics of the crowdworking domain. Most of the items in the pool were adapted from existing SDT-based motivation scales. Second, we selected items from the pool based on exploratory factor analysis on a sample of workers residing in the USA. Third, we further reduced the item pool based on exploratory factor analyses on samples from Spanish\footnote{In this paper, we use country demonyms synonymously with the location of workers for better readability.} and Indian crowdworkers.
For generating the item pool, we first conducted exploratory factor analysis on the data collected with WEIMS and MWMS in order to identify which items of these scales should be included in the MCMS development item pool.
For all exploratory factor analyses, we use oblique rotation (promax), because in line with SDT we expect the latent factors to be correlated.
WEIMS and MWMS items that had very low loadings on the appropriate factor as well as items that had high crossloadings were not included in the pool. Furthermore, we added only those items from WEIMS to the pool which were not deemed problematic by MWMS, i.e. we excluded items that were culturally sensitive or that addressed needs rather than motivations. We included all items from the MWMS which were conceptually suitable for the crowdworking domain. The minimal adaptions to the crowdworking domain described in the previous section were kept.
To extend the pool, we added semantically suitable items from the motivation scales developed in \cite{sports} and \cite{teaching} as well as several new items. The total number of items in the pool was 44. The stem phrasing ``Why do you or would you put efforts into doing CrowdFlower tasks?'' adapted from MWMS was used in order to capture both actual and latent motivations.
For a first selection of items from the pool, we collected answers from 1,000 crowdworkers residing in the USA.
Exploratory factor analysis was conducted on the responses.
Consistent with the findings from Section \ref{sec:existing}, we found that material external regulation and social external regulation did not load on the same factor.
Therefore, we aimed at a 6-factor model with the two separate external regulation factors (social and material).
Items which had very low factor loadings on the appropriate factor ($<$ 0.5), items which loaded on a factor different to the one theorized as well as items with high cross-loadings ($>$ 0.35) were iteratively removed from the initial pool, creating a reduced item pool with 36 items.
We conducted a second round of data collection with this reduced item pool, collecting responses from 1,000 Spanish and 1,200 Indian crowdworkers. The additional 200 responses from the Indian crowdworkers were requested due to the high amount of spam in this response set (also see Section \ref{sec:validation}), with the aim of achieving an item-to-response ratio of close to 1:20.
Again, we iteratively removed items with low loadings (with the higher threshold of $<$ 0.7 if more than three items were left for this construct), wrong loadings, or high cross-loadings (with the threshold of $>$ 0.3). Furthermore, if two items were phrased very similarly and the factor had more than three items remaining, we removed the item with the lower loading.
The final MCMS contains 18 items, with three items loading on each factor. Of the 18 final items, five were adapted from \cite{mwms}, four from \cite{weims}, two from \cite{teaching} and \cite{walking} each, and five items are new (but semantically based on items from existing scales). Like other SDT-based work motivation scales, the MCMS aims to measure motivations at the domain level of analysis (i.e. the job) as opposed to specific tasks within a job. Table \ref{table:mcms_items} shows the scale.
\section{Validation of the MCMS}
\label{sec:validation}
\textbf{Data Collection.}
With the final version of the MCMS, we collected data from 10 countries, with 900 participants from each country for validation.
We selected countries from three World Bank income groups\footnote{http://databank.worldbank.org/data/download/site-content/CLASS.xls}: high income, upper middle income and lower middle income. From each of the three income groups, we selected three countries with high activity on CrowdFlower.
The countries were selected according to the following criteria: The country had to be active on CrowdFlower (either high in the Alexa\footnote{http://www.alexa.com/} ranking or one of the top contributing countries in at least one of the partner channels) and we aimed for a high cultural diversity overall as well as within the income groups.
For the high income group, the selected countries were USA, Germany and Spain. The upper middle income group contains Brazil, Russia and Mexico, and the lower middle income group is comprised of India, Indonesia and the Philippines.
Note that in the rest of this paper, we use the group label ``Middle Income" (MID) for the upper middle income group and ``Low Income" (LOW) for the lower middle income group for better readability.
In addition, we collected responses from Venezuela as it was the most active country on CrowdFlower at the time of data collection, with CrowdFlower receiving 18.5\% of traffic from this country\footnote{Data obtained from http://www.alexa.com/.}. However, we did not include Venezuela in the data grouped by income as we believe it represents a special case: At the time of data collection the US\$ earned on CrowdFlower could be sold on the black market at a rate several orders of magnitude higher than the official exchange rate \cite{blackmarket}.
In order to capture a diverse sample of crowdworkers in each country, the starting times of the survey were divided into three groups: (1) 300 responses were requested during typical working hours (8:00 am to 5:00 pm in the appropriate time zone), (2) 300 responses were requested in the evening (6:00 pm to 11:pm in the appropriate time zone) and finally, (3) 300 responses were requested during weekends. The data was collected in October and November 2016.
\input{table_demographic}
The items in the MCMS were randomly permuted and presented to crowdworkers as a task on CrowdFlower. Besides the MCMS items, the task also included a section with demographic questions and questions about money use, as well as a section where workers were instructed to think of five reasons for why they do tasks on CrowdFlower and asked to write down these reasons. Anonymity was ensured.
\textbf{Spam Detection.}
We expected a significant amount of spam in the responses, such as people not reading the questions and clicking randomly or workers accepting the task despite having insufficient English skills. To counteract a high amount of noise in the dataset, we included three test items in the motivation scale section of the CrowdFlower task, and an additional test question in the demographics section.
The test items in the motivation scale section of the task asked participants to answer with a specific ranking on the Likert scale and the test question in the demographics section consisted of the question ``Are you paying attention to the questions?" with the possible answers ``No," ``Yes" and ``I don't know" selectable from a drop-down list. These questions ensure that less than 0.1\% ($(\tfrac{1}{7})^3*\tfrac{1}{3}$) of spammers pass the test questions.
Table \ref{table:demographic} shows the percentage of spam and the sample size after spam removal in each country and income group. The table also introduces the country and group codes used in the remainder of this paper.
\begin{figure}[t!]
\centering
\scalebox{.8}{\input{diagram_sem}}
\caption{\textbf{Factor structure of the MCMS.} This figure shows the factor structure of the hypothesized MCMS model.}
\label{fig:sem_diagram}
\end{figure}
\textbf{Hypothesized Model.}
Due to the results of the evaluation of existing scales and the results from exploratory factor analysis on the MCMS item pool, we model material external and social external regulation as two separate factors.
Hence, our hypothesized model is a 6-factor model where all factors are first-order factors. Note that the material and social external regulation are not adjacent factors in the continuum hypothesized by SDT but occupy the same spot.
While SDT hypothesizes that intrinsic motivation does not correlate with external regulation \cite{mwms, ryan1989perceived},
Chemolli and Gagné \cite{chemolli2014evidence} found that intrinsic motivation correlates with external regulation significantly for both the MWMS and the Academic Motivation Scale \cite{vallerand1992academic}.
Due to these findings, we decided not to restrict these correlations to zero in our hypothesized model, but we evaluate the model fit of both the correlation-restricted and the unrestricted model. No cross-loadings were hypothesized.
The hypothesized model is depicted in Figure~\ref{fig:sem_diagram}.
\textbf{Descriptive Statistics and Internal Consistency.}
Table~\ref{table:means_sds} summarizes the empirical factor means as well as standard deviations for each of the countries and income groups in our data. Table~\ref{table:factor_correlations} displays the Pearson correlations of the empirical factor means. The observed correlations do not strictly follow the continuum hypothesized by SDT (i.e. adjacent factors correlate stronger than nonadjacent factors). However, we do observe a negative correlation between intrinsic motivation and amotivation, and the strongest correlations are between adjacent factors (intrinsic motivation with identified regulation and introjected regulation with social external regulation).
\input{table_means_sds}
\input{table_factor_correlations}
Cronbach's alpha statistic \cite{cronbach1951coefficient} was used to assess the internal consistency of the MCMS. Table~\ref{table:cronbachs_alpha} displays the values of alpha for each country and income group. As a rule of thumb, values above 0.7 are considered acceptable, values between 0.6 and 0.7 questionable, values between 0.5 and 0.6 poor and values below 0.5 unacceptable \cite{george2003using}. In most countries and groups, alpha exceeds 0.7 for each construct. Exceptions to this are the amotivation factor in Brazil and Venezuela as well as the material external regulation factor in Brazil and Indonesia with values between 0.5 and 0.7. Therefore, when interpreting results from these factors and groups, care should be taken.
\input{table_cronbachs_alpha}
\textbf{Confirmatory Factor Analysis.}
In order to validate the factor structure of our hypothesized model, we conducted confirmatory factor analysis.
The first item of each factor in Table \ref{table:mcms_items} serves as the marker variable (i.e., the loading of this item is fixed to one).
\input{table_cfa_fit}
Table \ref{table:cfa_fit} shows the results of the confirmatory factor analysis of the hypothesized model.
The analysis was conducted on the entire dataset as well as on each group separately. As in Section \ref{sec:existing}, we follow the guidelines of Hooper et al. \cite{hooper2008structural} for evaluating model fit.
The results show that the hypothesized model has adequate fit overall as well as in all groups. The CFI was above 0.95 in all groups except Indonesia and Mexico (with 0.931 and 0.948, respectively). RMSEA was lower than 0.06 in all groups and SRMR was lower than or equal to 0.05 in all groups. We consider the fit measures for Mexico and Indonesia to be borderline acceptable, but some care should be taken when interpreting the results from these countries.
Table \ref{table:model_parameters} shows the item loadings and intercepts estimated by the hypothesized model and Table \ref{table:model_correlations} shows the estimated correlations between the constructs.
All estimated factor correlations are statistically significant at $p < 0.01$ (most at $p < 0.001$).
The alternative model which restricts the correlations of the external regulation factors (material external regulation and social external regulation) with intrinsic motivation to zero did not have an acceptable fit: While CFI was close to 0.95 for most groups (0.948 for the total sample), SRMR was high for the total sample (0.094) as well as for all other groups, ranging between 0.074 in the USA and 0.113 in Russia.
\input{table_model_parameters}
\input{table_model_correlations}
\section{Cross-Group Comparability of MCMS Results}
\label{sec:invariance}
When comparing the results of a measurement instrument across different groups, a key is to ensure that the instrument possesses the same psychometric properties in all groups. This characteristic is referred to as \emph{measurement invariance}. Tests of measurement invariance evaluate ``whether or not, under different conditions of observing and studying phenomena, measurement operations yield measures of the same attribute'' \cite{horn1992practical}. In our case, measurement invariance means that crowdworkers from different countries (or country income groups) assign the same meaning to the items used in the MCMS.
For evaluating measurement invariance of the MCMS, we conduct Multigroup Confirmatory Factor Analysis (MGCFA). In MGCFA, measurement invariance is evaluated via a series of hypothesis tests which test invariance at different levels.
Three levels of measurement invariance are required for comparing latent mean differences across groups: configural, metric and scalar invariance (e.g. \cite{cheung2002evaluating, chen2007sensitivity}).
Configural invariance requires that the items share the same configurations of loadings in all groups.
Metric invariance requires that the loadings of each item on its factor is the same across groups.
Scalar invariance requires that the intercepts of item regressions on the factor are the same across groups.
\input{table_invariance}
We calculate configural, metric and scalar invariance of the MCMS for countries and income groups.
Table \ref{table:invariance} displays the results of the MGCFA.
Configural invariance is indicated by acceptable goodness-of-fit indices in an MGCFA model without any equality constraints \cite{van2012checklist}.
For indications of metric and scalar non-invariance, we follow the guidelines of Chen \cite{chen2007sensitivity}:
For metric and scalar invariance in large samples ($N > 300$), a change of $\geq -0.010$ in CFI supplemented by a change of $\geq 0.015$ in RMSEA indicates non-invariance.
Table \ref{table:invariance} shows the goodness-of-fit indices for the progressively restricted models.
The results show a good fit for the model without equality constraints, indicating that configural invariance is supported.
Full metric invariance was also achieved, indicating that the strength of the relationship between the items and constructs is the same across groups.
Full scalar invariance could not be achieved. However, partial scalar invariance was achieved by releasing one intercept for the income groups (\emph{Am3}) and five intercepts in the countries (\emph{Am3}, \emph{ExMat2}, \emph{ExSoc3}, \emph{Introj2} and \emph{Ident2}).
Achieving partial scalar invariance allows for factor means to be compared. However, Steinmetz' \cite{steinmetz2013analyzing} simulations showed that unequal intercepts may lead to wrong conclusions when comparing observed composite means across groups.
Therefore, as only partial and not full scalar invariance was achieved for the MCMS, a comparison of the scale's latent factor means should rely on the model-implied means instead of observed composite means.
\section{Conclusion}
\label{sec:conclusion}
In this paper, we presented the Multidimensional Crowdworker Motivation Scale (MCMS), a new scale for measuring crowdworker motivations. The scale measures the motivations of crowdworkers on six dimensions, based on the conceptualization of motivation suggested by Self-Determination Theory (SDT).
To the best of our knowledge, the MCMS is the first instrument for measuring motivation in the context of crowdsourced micro-labor which is validated in multiple countries and income groups.
Furthermore, the results of the measurement invariance analyses show that motivations measured with the MCMS are comparable across countries and income groups.
This work constitutes a first step towards understanding the motivations of crowdworkers in an internationally comparable way.
The work presented in this paper has several limitations. First of all, the scale was presented in English to the crowdworkers in all countries, which means that we are only able to capture the motivations of crowdworkers with appropriate English skills.
Regarding the presence of social desirability bias, Blais et al. \cite{blais1993work} found that self-reported work motivations only correlated very weakly with the Marlow-Crowne Social Desirability Scale \cite{crowne1960new} (correlations between -0.18 and 0.20). However, as Antin and Shaw \cite{antin2012social} found evidence for the presence of social desirability bias in self-reported motivations of crowdworkers, further experiments are needed in order to assess the extent to which social desirability bias is present in data collected with the MCMS.
A further limitation of the MCMS is that it does not measure integrated regulation. The lack of an integrated regulation factor in the MCMS is due to problems of statistically distinguishing this factor from identified regulation and intrinsic motivation.
Due to the same problems, this limitation is also present in other work motivation scales such as the MWMS \cite{mwms}.
Another limitation of this work is that the extent to which motivations measured with the MCMS are related to different antecedents and outcomes has not been assessed yet. For investigating the relations between measured motivations and antecedents (e.g. the satisfaction of basic needs) or outcomes (e.g. emotional exhaustion), the scales for measuring the antecedents and outcomes would first have to be validated within the crowdworking domain.
In future work, we plan to conduct a cross-country as well as a cross-income group comparison of crowdworker motivations using the model-implied means. Furthermore, we plan to investigate the relationship between motivations and other demographic factors.
Another direction which we plan to follow in future work is the development of Bayesian models which incorporate not only the responses to the MCMS items but also responses to open-ended survey questions and demographic metadata.
This work is relevant for researchers and practitioners interested in measuring the motivations of crowdworkers.
\section{The Multidimensional Crowdworker Motivation Scale}
\input{table_mcms_items}
\end{document}
| 2024-02-18T23:40:09.745Z | 2017-05-23T02:14:25.000Z | algebraic_stack_train_0000 | 1,508 | 6,166 |
|
proofpile-arXiv_065-7632 | \section{Introduction}
The effectiveness of the Deep Convolutional Neural Networks (DCNNs) has been demonstrated for various computer vision tasks such as image classification \cite{He2015a,Krizhevsky2012,Szegedy2015,Simonyan2015a}, object detection \cite{Girshick2014,Girshick2015,Ren2015}, semantic-segmentation \cite{Long2014,Chen2014a,Noh2015,Pinheiro2014}, human body joint localization \cite{Toshev2014,Tompson2014,Chen2014b}, face recognition \cite{Taigman2014} and so on. Due to the large number of network parameters that need to be trained, DCNNs require a significant number of training samples. For tasks where sufficient number of training samples are not available, a DCNN trained on a large dataset for a different task is tuned to the current task by making necessary modifications to the network and retraining it with the available training data \cite{Yosinski2014a}. This transfer learning technique has been proven to be effective for tasks such as fine-grained recognition \cite{Sharif2014,Gui2015,Azizpour2015}, object detection \cite{Girshick2014,Sermanet2014a}, object classification \cite{Zeiler2014}, attribute detection \cite{Sharif2014,Azizpour2015} and so on.
One of the tasks for which a large number of training samples are not available is the continuous object orientation estimation problem where the goal is to predict the continuous orientation of objects in the range $0^{\circ}$ to $360^{\circ}$. The orientations are important properties of objects such as pedestrians and cars and precise estimation of the orientations allows better understanding of the scenes essential for applications such as autonomous driving and surveillance. Since in general it is difficult to annotate the orientations of objects without a proper equipment, the number of training samples in existing datasets for orientation estimation tasks is limited. Thus, it would be interesting to see if and how it would be possible to achieve good performance by adapting a DCNN trained on a large object recognition dataset \cite{Russakovsky2014} to the orientation estimation task.
The first consideration is the representation used for prediction.
When a DCNN is trained for an image classification task, the layers inside the network gradually transform the raw pixel information to more and more abstract representation suitable for the image classification task. Specifically, to achieve good classification ability, the representations at later layers have more invariance against shift/rotation/scale changes while maintaining a good discriminative power between different object classes. On the other hand, the orientation estimation task requires representation which can capture image differences caused by orientation changes of the objects in the same class. Thus, it is important to thoroughly evaluate the suitability of representations from different layers of a DCNN trained on the image classification task for the object orientation estimation task.
The second consideration is the design of the orientation prediction unit.
For the continuous orientation estimation task, the network has to predict the angular value, which is in a non-Euclidean space, prohibiting the direct use of a typical L2 loss function. To handle this problem, we propose three different approaches. The first approach represents an orientation as a 2D point on a unit circle, then trains the network using the L2 loss. In test time, the network's output, a 2D point not necessarily on a unit circle, is converted back to the angular value by $\mathrm{atan2}$ function. Our second approach also uses a-point-on-a-unit-circle representation, however, instead of the L2 loss, it minimizes a loss defined directly on the angular difference. Our third approach, which is significantly different from the first two approaches, is based on the idea of converting the continuous orientation estimation task into a set of discrete orientation estimation tasks and addressing each discrete orientation estimation task by a standard softmax function. In test time, the discrete orientation outputs are converted back to the continuous orientation using a mean-shift algorithm. The discretized orientations are determined such that all the discretized orientations are uniformly distributed in the output circular space. The mean-shift algorithm for the circular space is carried out to find the most plausible orientation while taking into account the softmax probability for each discrete orientation.
We conduct experiments on car orientation estimation and pedestrian orientation estimation tasks. We observe that the approach based on discretization and mean-shift algorithm outperforms the other two approaches with a large margin. We also find that the final performance significantly varies with the feature map used for orientation estimation. We believe that the findings from the experiments reported here can be beneficial for other object classes as well.
The paper is organized as follows. Section \ref{sec:related_work} discusses related works. Section \ref{sec:methods} presents three proposed approaches. Section \ref{sec:experiments} shows experimental results and conclusions are given in section \ref{sec:conclusions}.
\section{Related Work}\label{sec:related_work}
Object orientation estimation problem has been gaining more and more attention due to its practical importance. Several works treat the continuous orientation estimation problem as a multi-class classification problem by discretizing the orientations. In \cite{Ozuysal2009}, a three-step approach is proposed where a bounding box containing the object is fist predicted, then orientation is estimated based on image features inside the predicted bounding box, and finally a classifier tuned for the predicted orientation is applied to check the existence of the object. \cite{Ghodrati2014} address the orientation estimation task using Fisher encoding and convolutional neural network-based features. \cite{Bakry2014} learns a visual manifold which captures large variations in object appearances and proposes a method which can untangle such a visual manifold into a view-invariant category representation and a category-invariant pose representation.
Some approaches address the task as continuous prediction in order to avoid undesirable approximation error caused by discretization. In \cite{He2015}, a joint object detection and orientation estimation algorithm based on structural SVM is proposed. In order to effectively optimize a nonconvex objective function, a cascaded discrete-continuous inference algorithm is introduced. In \cite{Hara2014,Hara2016}, a regression forest trained with a multi-way node splitting algorithm is proposed. As an image descriptor, HOG features are used. \cite{Teney2014} introduces a representation along with a similarity measure of 2D appearance based on distributions of low-level, fine-grained image features. For continuous prediction, an interpolation based approach is applied. A neural network-based model called Auto-masking Neural Network (ANN) for joint object detection and view-point estimation is introduced in \cite{Yang2014}. The key component of ANN is a mask layer which produces a mask passing only the important part of the image region in order to allow only these regions to be used for the final prediction. Although both our method and ANN are neural network-based methods, the overall network architectures and the focus of the work are significantly different.
Several works consider learning a suitable representation for the orientation estimation task. In \cite{Torki2011}, an embedded representation that reflects the local features and their spatial arrangement as well as enforces supervised manifold constraints on the data is proposed. Then a regression model to estimate the orientation is learned using the proposed representation. Similarly to \cite{Torki2011}, \cite{Fenzi2013,Fenzi2014} learn a representation using spectral clustering and then train a single regression for each cluster while enforcing geometric constraints. \cite{Fenzi2015} formulates the task as a MAP inference task, where the likelihood function is
composed of a generative term based on the prediction error
generated by the ensemble of Fisher regressors as well
as a discriminative term based on SVM classifiers.
\cite{Xiang2014} introduces PASCAL3D+ dataset designed for joint object detection and pose estimation. Continuous annotations of azimuth and elevation for 12 object categories are provided. The average number of instances per category is approximately 3,000. The performance is evaluated based on Average Viewpoint Precision (AVP) which takes into account both the detection accuracy and view-point estimation accuracy. Since the focus of this work is the orientation estimation, we employ the EPFL Multi-view Car Dataset \cite{Ozuysal2009} and the TUD Multiview Pedestrian Dataset \cite{Andriluka2010} specifically designed to evaluate the orientation prediction.
Despite the availability of continuous ground-truth view point information, majority of works \cite{Xiang2014,Tulsiani2015a,Massa2016,Pepik2012,Ghodrati2014,Su2015} using PASCA3D+ dataset predict discrete poses and evaluate the performance based on the discretized poses. \cite{Tulsiani2015a} proposes a method for joint view-point estimation and key point prediction based on CNN. It works by converting the continuous pose estimation task into discrete view point classification task. \cite{Su2015} proposes to augment training data for their CNN model
by synthetic images. The view point prediction is cast as a fine-grained (360 classes for each angle) discretized view point classification problem.
\section{Method}\label{sec:methods}
Throughout this work we assume that a single object, viewed roughly from the side, is at the center of the image and the orientation of the object is represented by a value ranging from $0^{\circ}$ to $360^{\circ}$. Before feeding to the network, we first resize the input image to a canonical size and then subtracted the dataset mean. The network then processes the input image by applying a series of transformations, followed by an orientation prediction unit producing the final estimates. In this section, we present each of the proposed orientation prediction units in details. All the prediction units are trained by back propagation.
\subsection{Orientation Estimation}
\subsubsection{Approach 1}
We first represent orientation angles as points on a unit circle by $v = ( \cos{\theta}, \sin{\theta} )$. We then train a standard regression layer with a Huber loss function, also known as smooth L1 loss function. The Huber loss is used to improve the robustness against outliers, however, a standard L2 or L1 loss can also be used when appropriate. During testing, predicted 2D point coordinate $v = (x,y)$ is converted to the orientation angle by $\theta = \mathrm{atan2}(y,x)$. A potential issue in this approach is that the Huber loss function, as well as L2 or L1 loss functions, consider not only the angular differences but also the radial differences that are not directly related to the orientation.
\subsubsection{Approach 2}
As in approach 1, we represent orientation angles as points on a unit circle and train a regression function, however, we use a loss function which focuses only on the angular differences:
\begin{equation}
L(v_g, v ) = 1-\cos(\theta) = 1 - \frac{ v_g \cdot v }{|v_g||v|} = 1 - \frac{ x_g x + y_g y }{ \sqrt{ x^2 + y^2 } }
\end{equation}
where $v_g = (x_g, y_g )$ is the ground-truth. Note that $|v_g| = 1$ by definition. The derivative of $L$ with respect to $x$ is computed as
\begin{equation}
\frac{\partial L}{\partial x} = \frac{(x_g x + y_g y) \frac{x}{\sqrt{x^2 + y^2}} - x_g \sqrt{x^2 + y^2} }{x^2 + y^2}
\end{equation}
We compute $\frac{\partial L}{\partial y}$ similarly. These derivatives allow us to train the network parameters by back propagation. As in approach 1, during testing, the predicted 2D point coordinates are converted to orientation angles by the atan2 function.
A potential issue in this approach is that the derivatives approaches 0 when angular difference becomes close to $180^{\circ}$, making the optimization more challenging.
\subsubsection{Approach 3}
We propose an approach based on discretization. The network architecture illustrating this approach is presented in Fig.~\ref{fig:network2}. We first discretize the 0-360 range into $N$ unique orientations which are $G=360/N^{\circ}$ degree apart and convert the continuous prediction task into an $N$-class classification task. Each training sample is assigned one of the $N$ class labels based on its orientation's proximity to the discretized orientations. In order to alleviate the loss of information introduced by the discretization, we construct $M$ classification tasks by having a different starting orientation for each discretization. The $M$ starting orientations equally divide $G$ degree. Formally, the discrete orientations for the $m$-th classification task are $\{m \times G/M + k \times G\}_{k=0, \dots, N-1}$, where $m=0, \dots, M-1$. The example discretization with $N=4, M=3$ is depicted in Fig.~\ref{fig:cropped_discretization}. As an orientation estimation unit, we thus have $M$ independent $N$-way softmax classification layers, which are trained jointly.
\begin{figure*}[tbh]
\begin{center}
\includegraphics[width=6.0in]{network2.pdf}
\caption{The network arthitecture for the discretization based approach. \label{fig:network2}}
\end{center}
\end{figure*}
During testing, we compute softmax probabilities for each of the $M$ classification tasks separately. Consequently, we obtain probabilistic votes at all of the $M \times N$ unique orientations. We then define a probability density function using weighted kernel density estimation adopted for the circular space. We use the von-Mises distribution as a kernel function. The von-Mises kernel is defined as
\begin{equation}\label{eq:von_Mises}
k_\nu (\theta) = \frac{1}{2 \pi I_0(\nu)} \exp{( \nu \cdot \cos( \theta ))}
\end{equation}
where $\nu$ is the concentration parameter and $I_0(\nu)$ is the modified Bessel function of order 0.
Formally, the density at the orientation $\theta$ is given by
\begin{equation}
\hat{p}( \theta ; \nu ) \propto \sum_{i=1}^{M \times N} p_i k_\nu ( \theta - \theta_i).
\end{equation}
where $\theta_i$ is the $i$-th discrete orientation and $p_i$ is the corresponding softmax probability.
Then final prediction is made by finding the orientation with the highest density:
\begin{equation}
\hat{\theta} = \operatornamewithlimits{argmax}_{\theta} \hat{p}( \theta ; \nu )
\end{equation}
In order to solve the above maximization problem, we use a mean-shift mode seeking algorithm specialized for a circular space proposed in \cite{Hara2016}
The same level of discretization can be achieved by different combinations of $N$ and $M$. For instance, both $(N,M)=(72,1)$ and $(N,M)=(8,9)$ discretize the orientation into 72 unique orientations, however, we argue that larger $N$ makes the classification task more difficult and confuses the training algorithm since there are smaller differences in appearances among neighboring orientations. Setting $M$ larger than 1 and reducing $N$ could alleviate this problem while maintaining the same level of discretization. This claim is verified through the experiments.
A potential problem of the proposed approach is the loss of information introduced by the discretization step. However, as shown later in the experiment section, the mean-shift algorithm successfully recovers the continuous orientation without the need for further discretization.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=1.7in]{cropped_discretization.pdf}
\caption{The example discretization constructed by $N=4, M=3$. $N$ Orientations depicted in the same color are used for one of the $M$ classification task. \label{fig:cropped_discretization}}
\end{center}
\end{figure}
\section{Experiments}\label{sec:experiments}
We evaluate the effectiveness of the proposed approaches on the EPFL Multi-view Car Dataset \cite{Ozuysal2009} and TUD Multiview Pedestrian Dataset \cite{Andriluka2010}. Both datasets have continuous orientation annotations available.
\subsection{DCNN}
As an underlying DCNN, we employ the Residual Network \cite{He2015a} with 101 layers (ResNet-101) pre-trained on ImageNet image classification challenge \cite{Russakovsky2014} with 1000 object categories. The ResNet-101 won the 1st place on various competitions such as ImageNet classification, ImageNet detection, ImageNet localization, COCO detection, and COCO segmentation. Although ResNet-152, which is deeper than ResNet-101, achieves better performance than the ResNet-101, we employ ResNet-101 due to its smaller memory footprint.
The key component of the ResNet is a residual block, which is designed to make the network training easier. The residual block is trained to output the residual with reference to the input to the block. The residual block can be easily constructed by adding the input to the output from the block. ResNet-101 consists of 33 residual blocks. Each residual block contains three convolution layers, each of which is followed by a Batch Normalization layer, a scale Layer and the ReLU layer.
\subsection{Training details}
Unless otherwise noted, the weights of the existing ResNet-101 layers are fixed to speed up the experiments. The parameters of the orientation prediction unit are trained by Stochastic Gradient Descent (SGD). All the experiments are conducted using the Caffe Deep Learning Framework \cite{Jia2014} on a NVIDIA K40 GPU with 12GB memory. In order to include contextual regions, bounding box annotations are enlarged by a scale factor of 1.2 and 1.1 for EPFL and TUD datasets, respectively. We augment the training data by including the vertically mirrored versions of the samples.
For all experiments, we apply average pooling with size 3 and stride 1 after the last residual block chosen and then attach the orientation prediction unit. The batch size, momentum and weight decay are set to 32, 0.9 and 0.0005, respectively. Weights of the all the orientation prediction layers are initialized by random numbers generated from the zero-mean Gaussian distribution with $\mathrm{std} = 0.0001$. All biases are initialized to 0.
For the approach 3, we set $M$, the number of starting orientations for discretization, to 9 and $N$ to 8 for all the experiments unless otherwise stated.
\subsection{EPFL dataset}
The EPFL dataset contains 20 sequences of images of cars captured at a car show where each sequence contains images of the same instance of car captured with various orientations. Each image is given a ground-truth orientation. We use the first 10 sequences as training data and the remaining 10 sequences for testing. As a result, the number of training samples is 2,358 after data augmentation and that of testing samples is 1,120. We use bounding box information which comes with the dataset to crop out the image region to be fed to the network. The performance of the algorithm is measured by Mean Absolute Error (MeanAE) and Median Absolute Error (MeadianAE) in degree following the practice in the literature. Unless otherwise noted, the number of training iterations is 2000 with 0.000001 as a learning rate, followed by the additional 2,000 iterations with 10 times reduced learning rate.
First we conduct experiments to figure out the most suitable representation for the orientation estimation task by attaching the orientation prediction unit to different residual blocks. For these experiments, we use approach 3.
\begin{figure}[tbh]
\begin{center}
\includegraphics[width=2.8in]{effect_of_feature_map_EPFL.pdf}
\caption{The performance on EPFL dataset obtained by using a different residual block to attach the proposed orientation prediction unit. \label{fig:feature_map_EPFL}}
\end{center}
\end{figure}
In Fig.~\ref{fig:feature_map_EPFL}, we show the MeanAE on the EPFL dataset obtained by using different residual blocks. As can be seen, both the earlier and later residual blocks do not provide a suitable representation for the orientatin estimation task. The 22nd residual block produces the best representation among 33 residual blocks for our task. The following experiments are conducted using the 22nd residual block.
We analyze the effect of $M$, the number of starting orientations for discretization, and $N$, the number of unique orientations in each discretization, of the discretization-based approach. The results are summarized in Table.~\ref{tb:effect_of_M}. It is observed that when $N=8$, increasing $M$ leads to better results, however, no significant improvement is observed after $M=9$. When $N=72$, increasing $M$ from 1 to 5 does not lead to better performance. These results indicate that the larger number of the total orientations leads to better performance upto some point.
When the total number of unique orientations is same, e.g., $(N,M) = (72,1)$ and $(N,M) = (8,9)$, MeanAE is smaller with $(N,M) = (8,9)$ while MedianAE is smaller with $(N,M) = (72,1)$. Since MedianAE is very small with both settings, both of them achieve high accuracy in most of the cases.
\begin{table*}[htb]
\small
\begin{center}
\caption{MeanAE and MedianAE with different values for $N$ and $M$ on the EPFL dataset.\label{tb:effect_of_M}}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
$N$ & \multicolumn{2}{|c|}{72} & \multicolumn{5}{|c|}{8} \\ \hline
$M$ & 1 & 5 & 1 & 3 & 5 & 9 & 15 \\ \hline
MeanAE & 13.6 & 13.7 & 19.0 & 13.3 & 13.2 & 12.6 & 12.6 \\ \hline
MedianAE & 2.7 & 2.8 & 10.8 & 3.0 & 3.0 & 3.0 & 3.1 \\ \hline
\end{tabular}
\end{center}
\end{table*}
Table \ref{tb:other_approaches_EPFL} shows the performance of the other two approaches based on a-point-on-a-unit-circle representation. For the approach 1, we train the model with the same setting used for the approach 3. For the approach 2, we train the model for 40,000 iterations with a learning rate of 0.0000001 as it appears necessary for convergence. As can be seen, the discretization-based approach significantly outperforms the other two approaches.
\begin{table}
\small
\begin{center}
\caption{Comparison among the proposed approaches on the EPFL dataset.\label{tb:other_approaches_EPFL}}
\begin{tabular}{|c|c|c|} \hline
Approach & MeanAE & MedianAE \\ \hline \hline
1 & 22.9 & 11.3 \\ \hline
2 & 26.7 & 10.2 \\ \hline
3 & 12.6 & 3.0 \\ \hline
\end{tabular}
\end{center}
\end{table}
In Table ~\ref{tb:existing_work}, we present results from the literature and the result of our final model ( approach 3 ). For this comparison, instead of fixing the existing ResNet weights, we fine-tune all the network parameters end-to-end, which reduce the MeanAE by 21.7\%. As can be seen, our final model advances the state of the art performance.
The information on whether or not the ground-truth bounding box annotations are used in test time is also included in the table. In methods which do not utilize the ground-truth bounding boxes, an off-the-shelf object detector such as DPM \cite{Felzenszwalb2010} is used to obtain the bounding boxes \cite{Fenzi2015} or the localization and orientation estimation are addressed jointly \cite{He2015,Yang2014,Teney2014,Redondo-cabrera2014,Ozuysal2009}.
\begin{table*}
\small
\begin{center}
\caption{Comparison with the existing works on the EPFL dataset. The performance is measured in Mean Absolute Error (MeanAE) and Median Absolute Error (MedianAE).\label{tb:existing_work}}
\begin{tabular}{|c|c|c|c|} \hline
Methods & MeanAE & MedianAE & Ground-truth Bounding box? \\ \hline \hline
\textbf{Ours} & \textbf{9.86} & \textbf{3.14} & Yes \\ \hline
Fenzi et al. \cite{Fenzi2015} & 13.6 & 3.3 & No \\ \hline
He et al. \cite{He2015} & 15.8 & 6.2 & No \\ \hline
Fenzi and Ostermann \cite{Fenzi2014} & 23.28 & N/A & Yes \\ \hline
Hara and Chellappa \cite{Hara2016} & 23.81 & N/A & Yes \\ \hline
Zhang at al. \cite{Zhang2013} & 24.00 & N/A & Yes \\ \hline
Yang et al. \cite{Yang2014} & 24.1 & 3.3 & No \\ \hline
Hara and Chellappa \cite{Hara2014} & 24.24 & N/A & Yes \\ \hline
Fenzi et al. \cite{Fenzi2013} & 31.27 & N/A & Yes \\ \hline
Torki and Elgammal \cite{Torki2011} & 33.98 & 11.3 & Yes \\ \hline
Teney and Piater \cite{Teney2014} & 34.7 & 5.2 & No \\ \hline
Redondo-Cabrera et al. \cite{Redondo-cabrera2014} & 39.8 & 7 & No \\ \hline
Ozuysal et al. \cite{Ozuysal2009} & 46.5 & N/A & No \\ \hline
\end{tabular}
\end{center}
\end{table*}
Finally, we show representative results in Fig.~\ref{fig:caffe_results_EPFL} with ground-truth bounding boxes overlaid on the images and a ground-truth orientation and predicted orientation indicated in a circle. Note that many of the failure cases are due to the flipping errors ($\approx 180^\circ$) and tend to occur at a specific instance whose front and rear look similar (See the last two examples in the row 4.)
\begin{figure*}[!htbp]
\centering
\begin{tabular}{cccccccccc}
\includegraphics[width=0.52in]{11_1.png} & \includegraphics[width=0.52in]{11_11.png} &
\includegraphics[width=0.52in]{11_21.png} &
\includegraphics[width=0.52in]{11_31.png} &
\includegraphics[width=0.52in]{11_41.png} &
\includegraphics[width=0.52in]{11_51.png} &
\includegraphics[width=0.52in]{11_61.png} &
\includegraphics[width=0.52in]{11_71.png} &
\includegraphics[width=0.52in]{11_81.png} &
\includegraphics[width=0.52in]{11_91.png}
\\
\includegraphics[width=0.52in]{12_1.png} & \includegraphics[width=0.52in]{12_11.png} &
\includegraphics[width=0.52in]{12_21.png} &
\includegraphics[width=0.52in]{12_31.png} &
\includegraphics[width=0.52in]{12_41.png} &
\includegraphics[width=0.52in]{12_51.png} &
\includegraphics[width=0.52in]{12_61.png} &
\includegraphics[width=0.52in]{12_71.png} &
\includegraphics[width=0.52in]{12_81.png} &
\includegraphics[width=0.52in]{12_91.png}
\\
\includegraphics[width=0.52in]{15_1.png} & \includegraphics[width=0.52in]{15_11.png} &
\includegraphics[width=0.52in]{15_21.png} &
\includegraphics[width=0.52in]{15_31.png} &
\includegraphics[width=0.52in]{15_41.png} &
\includegraphics[width=0.52in]{15_51.png} &
\includegraphics[width=0.52in]{15_61.png} &
\includegraphics[width=0.52in]{15_71.png} &
\includegraphics[width=0.52in]{15_81.png} &
\includegraphics[width=0.52in]{15_91.png}
\\
\includegraphics[width=0.52in]{16_1.png} & \includegraphics[width=0.52in]{16_11.png} &
\includegraphics[width=0.52in]{16_21.png} &
\includegraphics[width=0.52in]{16_31.png} &
\includegraphics[width=0.52in]{16_41.png} &
\includegraphics[width=0.52in]{16_51.png} &
\includegraphics[width=0.52in]{16_61.png} &
\includegraphics[width=0.52in]{16_71.png} &
\includegraphics[width=0.52in]{16_81.png} &
\includegraphics[width=0.52in]{16_91.png}
\\
\includegraphics[width=0.52in]{19_1.png} & \includegraphics[width=0.52in]{19_11.png} &
\includegraphics[width=0.52in]{19_21.png} &
\includegraphics[width=0.52in]{19_31.png} &
\includegraphics[width=0.52in]{19_41.png} &
\includegraphics[width=0.52in]{19_51.png} &
\includegraphics[width=0.52in]{19_61.png} &
\includegraphics[width=0.52in]{19_71.png} &
\includegraphics[width=0.52in]{19_81.png} &
\includegraphics[width=0.52in]{19_91.png}
\\
\end{tabular}
\caption{Representative results obtained by the proposed method ( approach 3, $N=8,M=9$). A ground-truth orientation (red) and predicted orientation (blue) are indicated in a circle. Each row contains 10 example results from a testing sequence. From left to right, images are selected with 10 frames apart, starting from the first frame.}\label{fig:caffe_results_EPFL}
\end{figure*}
\subsection{TUD dataset}
The TUD dataset consists of 5,228 images of pedestrians with bounding box annotations. Since the original annotations are discrete orientations, we use continuous annotations provided by \cite{Hara2016}. In total, there are 4,732 images for training, 290 for validation and 309 for testing. Note that the size of the dataset is more than two times larger than that of EPFL Multi-view Car Dataset and unlike the EPFL dataset, images are captured in the wild. Since most of the training images are gray scale images and thus not adequate to feed into the DCNN, we convert all the grey scale images into color images by a recently proposed colorization technique \cite{zhang2016}. The performance of the algorithm is measured by Mean Absolute Error (MeanAE), Accuracy-22.5 and Accuracy-45 as in \cite{Hara2016}. Accuracy-22.5 and Accuracy-45 are defined as the ratio of samples whose
predicted orientation is within $22.5^{\circ}$ and $45^{\circ}$ from the ground truth, respectively. For this dataset, the number of training iterations is 10,000 with 0.00001 as a learning rate.
In Fig.~\ref{fig:feature_map_TUD}, we show the MeanAE obtained by attaching the orientation estimation unit of the approach 3 to different residual blocks. As is the case with the EPFL dataset, the performance varies significantly depending on the residual block used. Furthermore, the use of a proper representation is more critical on this dataset. Interestingly though, as in the EPFL dataset, the 22nd residual block performs well. Following experiments are thus conducted by using the 22nd residual block.
\begin{figure}[htb]
\begin{center}
\includegraphics[width=2.8in]{effect_of_feature_map_TUD.pdf}
\caption{The performance on TUD dataset obtained by using a different residual block to attach the proposed orientation prediction unit. \label{fig:feature_map_TUD}}
\end{center}
\end{figure}
In Table.~\ref{tb:effect_of_M_TUD}, we show the effect of $M$ while keeping $N=8$. As can be seen, in general larger $M$ produces better results, however, no significant improvement is observed after $M=9$. In order to evaluate the effect of having multiple non-overlapping discretization, we compare $N=72, M=1$ setting, whose number of discrete angles is same as $N=8, M=9$ setting. As can be seen in the table, the effect of having multiple discretization is prominent.
\begin{table*}[htb]
\small
\begin{center}
\caption{MeanAE, Accuracy-22.5 and Accuracy-45 with different $M$ on the TUD dataset.\label{tb:effect_of_M_TUD}}
\begin{tabular}{|c||c|c|c|c|c|c|c|} \hline
$N$ & \multicolumn{2}{|c|}{72} & \multicolumn{5}{|c|}{8} \\ \hline
$M$ & 1 & 5 & 1 & 3 & 5 & 9 & 15 \\ \hline
MeanAE & 35.4 & 33.5 & 40.0 & 32.7 & 31.1 & 30.2 & 30.9 \\ \hline
Accuracy-22.5 & 63.1 & 62.5 & 55.0 & 61.5 & 61.5 & 63.1 & 62.5 \\ \hline
Accuracy-45 & 79.6 & 80.6 & 75.4 & 79.6 & 82.2 & 82.8 & 82.5 \\ \hline
\end{tabular}
\end{center}
\end{table*}
Table \ref{tb:other_approaches_TUD} shows the performance of all the proposed approaches. For the approach 2, we increase the training iterations to 70,000 as it appears to take more iterations to converge. It is observed again that the approach 3 performs best.
\begin{table}
\small
\begin{center}
\caption{Comparison among the proposed approaches on the TUD dataset.\label{tb:other_approaches_TUD}}
\begin{tabular}{|c|c|c|c|} \hline
Approach & MeanAE & Accuracy-22.5 & Accuracy-45 \\ \hline \hline
1 & 33.7 & 46.9 & 75.7 \\ \hline
2 & 34.6 & 44.0 & 70.9 \\ \hline
3 & 30.2 & 63.1 & 82.8 \\ \hline
\end{tabular}
\end{center}
\end{table}
Finally, we train our model by fine-tuning all the layer parameters end-to-end. The result is shown in Table \ref{tb:TUD_results} along with the result of prior art. The end-to-end training reduces the MeanAE by 11.9 \%. Our final model outperforms the state-of-the-art with 23.3 \% reduction in MeanAE. The table contains the performance of human which is significantly better than the algorithms, necessitating further algorithm development.
Finally, Fig.~\ref{fig:visualResults_caffe} shows some representative results. The last row includes failure cases.
\begin{table*}[tb]
\small
\begin{center}
\caption{Continuous pedestrian orientation estimation: Mean Absolute Error in degree, Accuracy-$22.5^\circ$ and Accuracy-$45^\circ$ are shown. \label{tb:TUD_results}}
\begin{tabular}{|c|c|c|c| } \hline
Method & MeanAE ($^\circ$) & Accuracy-$22.5^\circ$ & Accuracy-$45^\circ$ \\ \hline
\textbf{Ours} & 26.6 & 70.6 & 86.1 \\ \hline
Hara and Chellappa \cite{Hara2016} & 34.7 & 68.6 & 78.0 \\ \hline \hline
Human & 9.1 & 90.7 & 99.3 \cite{Hara2016} \\ \hline
\end{tabular}
\end{center}
\end{table*}
\begin{figure*}[htb]
\centering
\begin{tabular}{cccccccccc}
\includegraphics[width=0.51in]{test_00001_1.png} & \includegraphics[width=0.51in]{test_00003_1.png} &
\includegraphics[width=0.51in]{test_00005_1.png} &
\includegraphics[width=0.51in]{test_00069_2.png} &
\includegraphics[width=0.51in]{test_00080_1.png} &
\includegraphics[width=0.51in]{test_00085_1.png} &
\includegraphics[width=0.51in]{test_00108_1.png} &
\includegraphics[width=0.51in]{test_00072_1.png} &
\includegraphics[width=0.51in]{test_00111_1.png} &
\includegraphics[width=0.51in]{test_00116_1.png}
\\
\includegraphics[width=0.51in]{test_00001_1_result.png} & \includegraphics[width=0.51in]{test_00003_1_result.png} &
\includegraphics[width=0.51in]{test_00005_1_result.png} &
\includegraphics[width=0.51in]{test_00069_2_result.png} &
\includegraphics[width=0.51in]{test_00080_1_result.png} &
\includegraphics[width=0.51in]{test_00085_1_result.png} &
\includegraphics[width=0.51in]{test_00108_1_result.png} &
\includegraphics[width=0.51in]{test_00072_1_result.png} &
\includegraphics[width=0.51in]{test_00111_1_result.png} &
\includegraphics[width=0.51in]{test_00116_1_result.png}
\\
\includegraphics[width=0.51in]{test_00248_3.png} &
\includegraphics[width=0.51in]{test_00237_1.png} &
\includegraphics[width=0.51in]{test_00200_1.png} &
\includegraphics[width=0.51in]{test_00207_1.png} &
\includegraphics[width=0.51in]{test_00193_2.png} &
\includegraphics[width=0.51in]{test_00160_1.png} &
\includegraphics[width=0.51in]{test_00182_1.png} &
\includegraphics[width=0.51in]{test_00181_1.png} &
\includegraphics[width=0.51in]{test_00093_1.png} &
\includegraphics[width=0.51in]{test_00079_1.png}
\\
\includegraphics[width=0.51in]{test_00248_3_result.png} &
\includegraphics[width=0.51in]{test_00237_1_result.png} &
\includegraphics[width=0.51in]{test_00200_1_result.png} &
\includegraphics[width=0.51in]{test_00207_1_result.png} &
\includegraphics[width=0.51in]{test_00193_2_result.png} &
\includegraphics[width=0.51in]{test_00160_1_result.png} &
\includegraphics[width=0.51in]{test_00182_1_result.png} &
\includegraphics[width=0.51in]{test_00181_1_result.png} &
\includegraphics[width=0.51in]{test_00093_1_result.png} &
\includegraphics[width=0.51in]{test_00079_1_result.png}
\\
\includegraphics[width=0.51in]{test_00009_1.png} &
\includegraphics[width=0.51in]{test_00054_1.png} &
\includegraphics[width=0.51in]{test_00102_1.png} &
\includegraphics[width=0.51in]{test_00154_1.png} &
\includegraphics[width=0.51in]{test_00193_1.png} &
\includegraphics[width=0.51in]{test_00184_4.png} &
\includegraphics[width=0.51in]{test_00227_1.png} &
\includegraphics[width=0.51in]{test_00217_1.png} &
\includegraphics[width=0.51in]{test_00222_1.png} &
\includegraphics[width=0.51in]{test_00221_2.png}
\\
\includegraphics[width=0.51in]{test_00009_1_result.png} &
\includegraphics[width=0.51in]{test_00054_1_result.png} &
\includegraphics[width=0.51in]{test_00102_1_result.png} &
\includegraphics[width=0.51in]{test_00154_1_result.png} &
\includegraphics[width=0.51in]{test_00193_1_result.png} &
\includegraphics[width=0.51in]{test_00184_4_result.png} &
\includegraphics[width=0.51in]{test_00227_1_result.png} &
\includegraphics[width=0.51in]{test_00217_1_result.png} &
\includegraphics[width=0.51in]{test_00222_1_result.png} &
\includegraphics[width=0.51in]{test_00221_2_result.png}
\\
\includegraphics[width=0.51in]{test_00133_1.png} &
\includegraphics[width=0.51in]{test_00173_1.png} &
\includegraphics[width=0.51in]{test_00234_1.png} &
\includegraphics[width=0.51in]{test_00244_1.png} &
\includegraphics[width=0.51in]{test_00244_2.png} &
\includegraphics[width=0.51in]{test_00128_1.png} &
\includegraphics[width=0.51in]{test_00112_1.png} &
\includegraphics[width=0.51in]{test_00096_2.png} &
\includegraphics[width=0.51in]{test_00073_1.png} &
\includegraphics[width=0.51in]{test_00052_1.png}
\\
\includegraphics[width=0.51in]{test_00133_1_result.png} &
\includegraphics[width=0.51in]{test_00173_1_result.png} &
\includegraphics[width=0.51in]{test_00234_1_result.png} &
\includegraphics[width=0.51in]{test_00244_1_result.png} &
\includegraphics[width=0.51in]{test_00244_2_result.png} &
\includegraphics[width=0.51in]{test_00128_1_result.png} &
\includegraphics[width=0.51in]{test_00112_1_result.png} &
\includegraphics[width=0.51in]{test_00096_2_result.png} &
\includegraphics[width=0.51in]{test_00073_1_result.png} &
\includegraphics[width=0.51in]{test_00052_1_result.png}
\end{tabular}
\caption{Representative results obtained by the proposed method ( approach 3, $N=8,M=9$). A ground-truth orientation (red) and predicted orientation (blue) are indicated in a circle. The first three rows show successful cases while the last row shows failure cases.}\label{fig:visualResults_caffe}
\end{figure*}
\section{Conclusion}\label{sec:conclusions}
This work proposed a new approach for a continuous object orientation estimation task based on the DCNNs. Our best working approach works by first converting the continuous orientation estimation task into a set of non-overlapping discrete orientation estimation tasks and converting the discrete prediction to a continuous orientation by a mean-shift algorithm. Through experiments on a car orientation estimation task and a pedestrian orientation estimation task, we demonstrate that the DCNN equipped with the proposed orientation prediction unit works significantly better than the state of the approaches, providing another successful DCNN application. Our experiments also indicate that selecting a suitable representation is critical in transferring DCNNs trained on an image classification task to an orientation prediction task.
\newpage
{\bibliographystyle{ieee}
| 2024-02-18T23:40:10.879Z | 2017-02-07T02:08:55.000Z | algebraic_stack_train_0000 | 1,550 | 5,680 |
|
proofpile-arXiv_065-7658 | \section{Introduction}
Particulate matter suspended in fluids is a common feature of turbulent flows encountered in industrial devices as well as the environment. Typical examples range from combustion processes in diesel engines and gas turbines \cite{Engineering2,lee2002morphological}, ash or aerosols expelled from volcanic eruptions or chemical or nuclear accidents \cite{Engineering1,hidy2012aerosols}, to liquid rain droplets in clouds \cite{FFS,shaw2003particle}. Knowledge about the distribution of these particles can be essential for the working reliability and efficiency of engines \cite{williams1979drop}, climate predictions \cite{Seinfeld,reeks2014transport} and the health of living organisms \cite{Flagan}. Therefore, understanding of particle-turbulence interaction causing heterogeneous particle distributions and identifying potential high-concentration regions is a crucial aspect.
Clustering of inertial particles in homogeneous turbulence conditions has been subject to intensive research for a long time and is thus generally well explained by the phenomenon of small-scale or fractal clustering, see for example \cite{BecCenciniHillerbranddelta,sundaram1997collision,Stefano,JYL,BFF,CenciniBecBiferale,Cencini,monchaux2010preferential,eatonfessler}.
Many flows in the environment or industrial applications are physically bounded by walls, which creates a heterogeneity in the flow. In these non-uniform flows, inertial particles are subject to turbophoretic forces that create a strongly inhomogeneous distribution of particles along the direction of non-uniformity. Turbophoresis is generally associated with a net flux of particles towards regions of lower turbulence diffusivity, thus leading to increased particle concentration in near-wall regions of wall-bounded flows. Since inhomogeneous turbulent flows containing non-passive particles are a common occurrence in nature and industry, turbophoresis is an ubiquitous phenomenon that deserves special attention.
The turbophoretic mechanism was initially described from a theoretical point of view in \cite{caporaloni} and \cite{reeksturbophoresis}. The work of \cite{young} developed a theoretical description of the particle transfer to the wall caused by turbulent sweep and ejection events. Subsequently, several experimental investigations (e.g. \cite{kaftoria,kaftorib,righetti}) have confirmed the accumulation of particles in near-wall regions due to turbophoresis. Increasing computational resources allowed for detailed investigations of turbophoresis based on direct numerical simulations of inhomogeneous turbulent flows. The early work of Eaton \& Fessler \cite{eatonfessler} found heavy particles to be localized preferably in regions close to the wall that feature a low instantaneous velocity.
Further investigations \cite{marchioli,picciotto} indicated that coherent wall structures are very likely responsible for high particle concentrations in low velocity regions in wall-proximity. Furthermore, the spatial development of particle concentrations in a turbulent pipe flow was investigated numerically in \cite{picano}. These investigations mainly focused on the increasing concentration of particles with relatively large inertia in the near-wall region and the possibility to derive information about particle surface deposition.
Preferential concentration arising due to turbophoresis as well as small-scale clustering has been investigated in a channel flow for heavy particles by \cite{sardina}. \cite{de2016clustering} studied the effects of turbophoresis and small-scale clustering in a shear-flow without walls for a wide range of particle inertia.
They showed that turbophoresis is stronger for particles with moderate inertia, whereas small-scale clustering dominates at week inertia.\\
Multifractality was derived originally for spatially uniform turbulence. It is a small-scale phenomenon holding at scales smaller than Kolmogorov scale. Recently, multifractality was generalized to the case of inhomogeneous turbulence \cite{Schmidt}. The derivation assumed that during the characteristic time of formation of fractal structures the motion of small parcels of particles is confined in a not too large region of space where statistics of flow gradients can be considered uniform. It was demonstrated that the pair-correlation function of concentration of particles factorizes in the product of (possibly time-dependent) average local concentrations and a geometrical factor of fractal increase of probability of two particles to be close. Thus we can separate dependencies in the particle distribution. The geometry of the time-dependent multifractal to which the particles are confined is statistically stationary. It is space-dependent because of the inhomogeneity of turbulence. However, the overall number of particles, that distribute over the multifractal locally, is determined by the average local concentration that can be time-dependent. For instance in the case of turbophoresis there is depletion of local average concentration because of the particle flux to the wall.
The present work provides an extensive theoretical and numerical investigation of the clustering degree of weakly inertial particles in a turbulent channel flow. The specific case of interest here is the turbophoretic behavior of weakly inertial particles in flows where the turbulent diffusivity varies in one direction (i.e. wall-normal) but the turbulence is homogenous in wall-parallel planes. The universal framework of weakly compressible flow \cite{FFS,fouxon1,fphl} is therefore modified to describe the combined clustering effects occurring due to the inhomogeneity of the flow and fluctuational clustering present in homogeneous turbulence. The theory is used as basis for the analysis of inertial particles in a direct numerical simulation (DNS) of a turbulent channel flow.
Our main theoretical result in this paper is the formula for the space-dependent rate $\left\langle \sum\lambda_i(y)\right\rangle$ of creation of inhomogeneities of concentration of particles in terms of local statistics of turbulence,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!
\left\langle \sum\lambda_i(y)\right\rangle=\!-\tau^2\int_0^{\infty}\langle\nabla^2p(0)\nabla^2p(t)\rangle_c dt'+\partial_y v_{eff}(y),\ \ \ \ v_{eff}(y)=-\tau\partial_y \langle u_y^2\rangle+\tau\int_0^{\infty}\langle u_y(0) \nabla^2p(t)\rangle_c dt. \label{sum}
\end{eqnarray}
In this formula $y$ is the distance to the wall, $\tau$ is the Stokes time of the particles, $\bm u$ and $p$ are the turbulent flow velocity and pressure, respectively. The angular brackets designate averaging over local statistics of turbulence and brackets with $c$ standing for the dispersion. Besides the average Eulerian velocity of the particles $-\tau\partial_y \langle u_y^2\rangle$, the rest of the terms come from distinction between Lagrangian and Eulerian averages in the formation of particle inhomogeneities. The first term is the local form of the formula known for spatially uniform turbulence that holds in the bulk. Inhomogeneity of turbulence produces an effective velocity term $v_{eff}(y)$ that in contrast with the bulk term is proportional to the first power of $\tau$. Thus clustering of weakly inertial particles can be much stronger because of inhomogeneity effects.
The numerical investigation of this work is based on DNS data obtained from the Johns Hopkins University Turbulence Database (JHTDB). We implement a Lagrangian tracking algorithm for inertial particles based on time-resolved Eulerian DNS results of a turbulent channel flow. This allows to evaluate a wide range of flow quantities along particle trajectories which serves as basis for the quantification of preferential concentration of inertial particles in inhomogeneous turbulence.
This paper commences with a theoretical analysis (Sec.\,\ref{sec:theory}) and the equations used for the analysis of the numerical data. This is followed by a description of the channel flow simulation details, including the implementation of the particle tracking algorithm and a brief characterization of the flow quantities of interest (Sec.\,\ref{sec:dns}). Subsequently, the performed analysis and results based on the direct numerical simulation (Sec.\,\ref{sec:analysis}) are illustrated. A summary of these results and the corresponding conclusions are presented in Sec.\,\ref{sec:discussion}.
\section{Sum of Lyapunov exponents}
\label{sec:theory}
In this Section, we introduce the local rate of production of inhomogeneities of concentration of particles. This rate is the sum of Lyapunov exponents $\sum\lambda_i$. We observe that the local rate of particle density increase is determined by the local divergence of velocity. However that divergence is fluctuating in turbulence. The sum of Lyapunov exponents describes the rate of accumulated growth obtained by proper averaging over the fluctuations and it differs both from fluctuating and average divergences. We will see in the coming Sections that it is the average divergence plus the coherent contribution of the fluctuations.
We consider particles with weak inertia suspended in incompressible turbulent flow in a channel. The strength of inertia of a small spherical particle is quantified by the particle Stokes time $\tau$. This is given by $\tau=2a^2\rho_p/(9\nu\rho)$, with $a$ being the particle radius, $\rho$ and $\rho_p$ as the fluid and particle density respectively and $\nu$ is the fluid viscosity. We consider the case of dense particles with $\rho_p\gg \rho$ so effects such as memory and added mass can be neglected \cite{MaxeyRiley}. Assuming that the Reynolds number $Re_p$ of the flow perturbation caused by the particles is small ($Re_p\ll 1$) we can use the linear law of friction finding the equation of motion,
\begin{equation}
\frac{d\bm x}{dt}=\bm v,\ \ \frac{d\bm v}{dt}=-\frac{\bm v-\bm u[t, \bm x(t)]}{\tau}.\label{lw}
\end{equation}
Here, $\bm x(t)$ is the particle coordinate, $\bm v(t)$ is the velocity and $\bm u(t, \bm x)$ is the turbulent flow velocity. Since we consider particles with small inertia, the velocity may be approximated as \cite{Maxey},
\begin{equation}
\bm v\,=\,\bm u-\tau[\partial \bm u/\partial t + (\bm u\cdot \bm\nabla)\bm u] ,
\label{eq:particlevelocity}
\end{equation} where $\tau[\partial \bm u/\partial t + (\bm u\cdot \bm\nabla)\bm u]$ represents the inertial particle drift.
Despite that the fluid velocity is divergence-free, the inertial drift of the particle velocity results in a 'weakly compressible' particle flow because $\nabla\cdot\bm v\neq 0$. For the case where particles are seeded into a statistically stationary, incompressible channel flow, with two homogeneous $(x,z)$ and one inhomogeneous ($y$) flow direction the mean Eulerian divergence of the particle velocity field reduces to,
\begin{equation}
\langle v_y\rangle=-\tau\langle \nabla \cdot (\bm u u_y)\rangle=-\tau\partial_y \langle u_y^2\rangle\ \ \langle \nabla\cdot\bm v\rangle=\frac{\partial \langle v_y\rangle}{\partial y}=-\tau\frac{\partial^2 \langle u_y^2\rangle}{\partial y^2}.
\label{eq:divv_inh}
\end{equation}
The formula for $\langle v_y\rangle$, obtained by averaging Eq.~(\ref{eq:particlevelocity}), is the well-known result for the average Eulerian particle velocity towards the wall. Since $\langle u_y^2\rangle$ monotonously grows away from the wall reaching a maximum in the log-law region of the channel, then $\langle v_y\rangle$ is directed toward the walls constituting the simplest demonstration of turbophoretic motion of particles towards the regions with smaller intensity of turbulence. In contrast, $\langle \nabla\cdot\bm v\rangle$ is not sign-definite: near the wall where $\langle u_y^2\rangle$ has minimum we have positive $\partial_y^2 \langle u_y^2\rangle$ (right on the wall $\partial_y^2 \langle u_y^2\rangle=0$ but it is positive nearby) and negative $\langle \nabla\cdot\bm v\rangle$, conversely we have $\partial_y^2 \langle u_y^2\rangle<0$ and $\langle \nabla\cdot\bm v\rangle>0$ already in the buffer layer. Thus there is a point $y_*$ where $\langle \nabla\cdot\bm v\rangle=0$ so $\langle \nabla\cdot\bm v\rangle$ is negative for $y<y_*$ and positive otherwise.
Based on Eulerian divergence we could conclude that at $y<y_*$ there are positive correlations in positions of particles separated by distance smaller than the viscous scale $\eta$. These particles are in the same divergence \cite{frisch} that is negative on average and thus approach each other. In contrast, for $y>y_*$ there would seem to be negative correlations of positions. In reality the particles move and correlate in reaction not to Eulerian divergence of the flow but to the divergence of the flow in the frame of reference that moves with the particle. This divergence determines the evolution of infinitesimal volumes $V$ of particles that obey \cite{Batchelor},
\begin{equation}
\frac{d\ln V}{dt}=w(t, \bm x(t)),\ \ w(t, \bm x)=\nabla\cdot\bm v(t, \bm x),
\end{equation}
where $\bm x(t)$ is the trajectory of some particle located inside the considered infinitesimal volume. This equation holds provided the largest linear size of $V$ is much smaller than $\eta$. The solution of this equation for the logarithmic increase rate of $V(t)$ defines the finite-time sum of Lyapunov exponents $\sum\lambda_i(t)$ as,
\begin{equation}
\sum_{i=1}^3\lambda_i(t)=\frac{1}{t}\ln \left(\frac{V(t)}{V(0)}\right)=\frac{1}{t}\int_0^t w(t', \bm x(t'))dt'. \label{finite}
\end{equation}
The RHS of this equation has the form that occurs in the ergodic theorem \cite{Sinai}. Indeed, in spatially uniform turbulence ergodicity implies that the RHS converges to the deterministic limit in the limit of $t\to\infty$ (we remark that the convergence holds for almost every point with respect to the stationary measure but in this context it holds for almost every point in space that is with possible exception of points with zero total volume). This limit is independent of the initial position of $V(t)$. Thus the limit equals to the average over the initial position defining the sum of Lyapunov exponents of spatially uniform turbulence \cite{review,ff},
\begin{equation}
\left(\sum \lambda_i\right)_{bulk}=\lim_{t\to\infty}\sum\lambda_i(t)=\lim_{t\to\infty}\frac{1}{t}\int_0^t \langle w(t', \bm x(t'))\rangle dt'=\lim_{t\to\infty}\langle w(t, \bm x(t))\rangle,
\end{equation}
where angular brackets stand for averaging over $\bm x(0)$. Here the subscript signifies that this formula holds in the bulk of the flow, far from the boundaries, where the spatially uniform turbulence framework applies. It was demonstrated in \cite{ff}, and will be proved below somewhat differently, that this can be written as,
\begin{equation}
\left(\sum \lambda_i\right)_{bulk}=-\int_0^\infty \langle w(0)w(t)\rangle dt, \label{fluct}
\end{equation}
which clarifies that $\sum\lambda_i<0$ see details in \cite{ff}. Taking the divergence of the Navier-Stokes equations obeyed by $\bm u$ we find that $w=\tau\nabla^2 p$ where $p$ is the turbulent pressure. Thus we can write
\begin{equation}
\left(\sum \lambda_i\right)_{bulk}=-\tau^2 \int_0^\infty \langle\nabla^2p(0)\nabla^2p(t)\rangle dt\propto St^2, \label{blk}
\end{equation}
where we defined the Stokes number $St=\tau\sqrt{\epsilon/\nu}$. Here $\epsilon$ is the average rate of energy dissipation per unit mass.
In practice, the infinite time limit must be understood as near convergence beyond a certain convergence time $t_c$ when for most of the trajectories $\bm x(t)$, the LHS of Eq.~(\ref{finite}) becomes constant at $t\sim t_{c}$. This time is strongly different for $\sum\lambda_i$ and other combinations of Lyapunov exponents. These are defined very similarly where $\lambda_1$ and $\lambda_1+\lambda_2$ are logarithmic increase rates of infinitesimal line and surface elements, respectively \cite{review}. For instance
\begin{equation}
\lambda_1(t)+\lambda_2(t)=\frac{1}{t}\ln \left(\frac{S(t)}{S(0)}\right), \label{area}
\end{equation}
where $S(t)$ is the infinitesimal area of fluid particles \cite{review,Schmidt}. For $\lambda_1$ and $\lambda_2$ the convergence time $t_c$ is a few Kolmogorov times $\tau_{\eta}$. Here $\tau_{\eta}=\sqrt{\nu/\epsilon}$, which is usually defined as typical time-scale of turbulent eddies at the viscous scale \cite{frisch}, is also the correlation time of flow gradients (and thus $w$) in the fluid particle frame \cite{review,ff}. In contrast, for $\sum\lambda_i$ the convergence time is much longer as seen considering the dispersion the of $\sum\lambda_i(t)$,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\left\langle \left(\sum\lambda_i(t)\right)^2\right\rangle\!-\!\left\langle \sum\lambda_i(t)\right\rangle^2\!=\!\frac{1}{t^2}\int_0^t \!\!dt_1dt_2 \langle w(t_1, \bm x(t_1))w(t_2, \bm x(t_2))\rangle_c\!\sim \!\frac{1}{t}\int_0^\infty \!\!\langle w(0)w(t)\rangle dt\sim \frac{|\sum\lambda_i|}{t},\ \ t\gg \tau_{\eta}, \label{sps}
\end{eqnarray}
where $c$ stands for cumulant (dispersion) and we used that the correlation time $\tau_{\eta}$ of $w(t, \bm x(t))$ is much smaller than $t$. We observe that the normalized rms deviation of $\sum\lambda_i(t)$ obeys,
\begin{equation} \frac{\sqrt{\left\langle \left(\sum\lambda_i(t)\right)^2\right\rangle-\left\langle \sum\lambda_i(t)\right\rangle^2}}{|\sum\lambda_i(t)|}\sim \frac{1}{\sqrt{|\sum\lambda_i| t}}\sim \frac{1}{St \sqrt{t/\tau_{\eta}}}.\label{rms}
\end{equation}
Thus fluctuations of $\sum \lambda_i(t)$ are small only at quite large times $t\gg \tau_{\eta}/St^2$. This is because non-zero $\sum\lambda_i$ is due to fluctuations, as evident from Eq.~(\ref{fluct}), so dispersion of $\sum \lambda_i(t)$ has the same order in $St$ as the average. Thus the convergence time for the sum of Lyapunov exponents obeys $t_c\sim \tau_{\eta}/St^2$. This slow convergence of long-time limit $\sum\lambda_i(t)$ at small $St$, seems to be unobserved previously.
Clearly the long-time convergence of $t^{-1}\ln V(t)/V(0)$ to a deterministic limit makes it a very useful quantity because it allows a deterministic prediction despite the randomness of turbulence. We determine $\sum\lambda_i$ once and then we can predict it for any arbitrary trajectory. The sum of the Lyapunov exponents, made dimensionless by dividing by a factor of order of $\sqrt{\epsilon/\nu}$ determines the strength of the clustering by giving fractal dimensions of the random attractor formed by particles in space, see e. g. \cite{fouxon1} and Section \ref{pc}.
Fortunately, we can extend the notion of the sum of Lyapunov exponents to the inhomogeneous case with which we can describe clustering closer to the walls. We consider the situation when $\bm x(t_c)$ does not deviate from $\bm x(0)$ by the characteristic scale of inhomogeneity of turbulent statistics which is the distance to the wall $y$. This demands that the $y-$dependent inhomogeneity time $t_{in}(y)=y/\langle u_y^2\rangle^{1/2}$, where $\langle u_y^2\rangle^{1/2}$ is the typical transversal velocity, is much larger than $t_c$. Here $t_{in}(y)$ is the time during which the trajectory passes a distance comparable with $y$ over which the statistics changes. If $t_c\ll t_{in}(y)$ holds then during the convergence time $\bm x(t)$ in Eq.~(\ref{finite}) stays in the region where the statistics is roughly uniform. Thus we can use the results for spatially uniform turbulence locally. We could then guess that,
\begin{equation}
\left(\sum \lambda_i\right)_{bulk}(y)\sim -\tau^2 \int_0^\infty \langle\nabla^2p(0, y)\nabla^2p(t)\rangle dt, \label{sps2}
\end{equation}
where the RHS depends on the $y$ coordinate. Here the local correlation function can be defined with the help of temporal averaging. In the case of the channel instead of time averaging we can use averaging over the symmetry plane of the statistics,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\langle\nabla^2p(0, y)\nabla^2p(t)\rangle=\!\!\lim_{S\to\infty}\!\int \!\!\frac{dx dz}{S} \nabla^2p(0, x, y, z))\nabla^2p[t, \bm q(t, x, y, z)],\end{eqnarray} where $S$ is the area of the $x-z$ plane and we introduced 'Lagrangian' trajectories of particles labeled by their initial positions at $t=t_0$,
\begin{eqnarray}&&
\partial_t \bm q(t| t_0, \bm x)=\bm v[t, \bm q(t| t_0, \bm x)],\ \ \bm q(t| t_0, \bm x)=\bm x,\ \ \bm q(t, \bm x)=\bm q(t| t_0=0, \bm x). \label{bas2}
\end{eqnarray}
The temporal and "planar" averages are identical because the temporal average \\$\langle\nabla^2p(0, y)\nabla^2p(t)\rangle_t$ is independent of $x$ and $z$ so that,
\begin{equation}
\langle\nabla^2p(0, y)\nabla^2p(t)\rangle_t=\lim_{S\to\infty}\!\int \!\!\frac{dx dz}{S} \langle\nabla^2p(0)\nabla^2p(t)\rangle_t
\end{equation} where interchanging the order of averages in the last term proves that time and plane averages coincide (time average of the plane average is the plane average). Below we will use planar averaging designating it by angular brackets and demonstrate that the guess given by Eq.~(\ref{sps2}) is incomplete.
\section{Identity for Lagrangian averages}
In this Section we derive an identity for Lagrangian average $\langle f(t, y)\rangle$ of an arbitrary stationary random function $f(t, \bm x)$ in the frame of particles released at the same distance $y$ from the wall,
\begin{equation}
\langle f(t, y)\rangle=\int\frac{dxdz}{S} f(t, \bm q(t, x, y, z)).
\end{equation}
Here this quantity is of interest in the case of $f=w(t, \bm x)$ when it provides the average sum of Lyapunov exponents studied in the next Section. However other cases of this quantity can be of interest in future studies so we keep arbitrariness of $f$. The derivation is a changed line of thought that appeared in \cite{ff}. We observe that $\bm q(t| t_0, \bm x)$ as a function of the initial time $t_0$ obeys,
\begin{eqnarray}&&
\partial_{t_0} \bm q(t| t_0, \bm x)+[\bm v(t_0, \bm x)\cdot\nabla]\bm q(t| t_0, \bm x)=0.\label{initime}
\end{eqnarray}
This expresses that changing initial time and position so that the initial position stays on the same trajectory does not change that trajectory:
$\bm q(t| t_0+\epsilon, \bm q(t_0+\epsilon|t_0, \bm r))=\bm q(t|t_0, \bm r)$ (the trajectory that passes through $\bm r$ at time $t_0$ is the same trajectory that passes through $\bm q(t_0+\epsilon|t_0, \bm r)$ at time $t_0+\epsilon$). Differentiating over $\epsilon$ and setting $\epsilon=0$ one finds the equation above.
We introduce the two-time version of average of $w$,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle f(t, t_0, y)\rangle=\lim_{S\to\infty}\int \frac{dx dz}{S} f[t, \bm q(t| t_0, x, y, z)].\label{def}
\end{eqnarray}
Because of stationarity $\langle f(t, t_0, y)\rangle$ depends on $t $, $t_0$ only through the difference of the time arguments $t-t_0$. We consider the time derivative of Eq.~(\ref{def}) over $t_0$ using Eq.~(\ref{initime}),
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{\partial \langle f\rangle}{\partial t_0 }\!=\!-\!\lim_{S\to\infty}\int \!\!\frac{dx dz}{S} v_i(t_0, x, y, z)\nabla_i f[t, \bm q(t| t_0, x, y, z)]\nonumber\\&& \!\!\!\!\!\!\!\!\!\!\!\!\!=\!-\!\lim_{S\to\infty}\int \!\!\frac{dx dz}{S} \nabla_i \left(v_i(t_0, x, y, z)f[t, \bm q(t| t_0, x, y, z)]
\right)
\nonumber\\&& \!\!\!\!\!\!\!\!\!\!\!\!\!+\langle w(0)f(t-t_0)\rangle,
\label{der1}
\end{eqnarray}
where we used stationarity and defined the correlation function of $w$,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle w(0)f(t)\rangle\!=\!\!\lim_{S\to\infty}\!\int \!\!\frac{dx dz}{S} w(0, x, y, z))f[t, \bm q(t, x, y, z)].
\end{eqnarray}
Finally, observing that derivatives over $x$ and $z$ in Eq.~(\ref{der1}) give zero as integrals of complete derivative and taking the $y-$derivative outside the integration we find
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\frac{\partial \langle f(t, t_0, y)\rangle}{\partial t_0}\!=\!\langle w(0)f(t-t_0)\rangle\!-\!\partial_y\langle v_y(0)f(t-t_0)\rangle,\label{der}
\end{eqnarray}
where we defined the correlation function,
\begin{eqnarray}&& \!\!\!\!\!\!\!\!\!\!\!\!\!
\langle v_y(0)f(t)\rangle\!=\!\!\lim_{S\to\infty}\!\int \!\!\frac{dx dz}{S} v_y(0, x, y, z))f[t, \bm q(t, x, y, z)].
\end{eqnarray}
We find integrating Eq.~(\ref{der}) over $t_0$ from $t_0=0$ up to $t_0=t$ and using $\langle f(t, y)\rangle=\langle f(t, t_0=0, y)\rangle$ that
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle f(t, y)\rangle\!-\langle f(y)\rangle=\!\int_0^t \left[\partial_y\langle v_y(0)f(t')\rangle\!-\!\langle w(0)f(t')\rangle\right]dt'.\label{integral}
\end{eqnarray}
Here $\langle f(y)\rangle=\langle f(t=0, y)\rangle$ is the Eulerian average of $f(\bm x)$ over the horizontal plane. The RHS describes the difference between Lagrangian and Eulerian averages that holds because of preferential concentration of particles and thus quantifies the strength of the clustering. This integrals in Eq.~(\ref{integral}) do not necessarily converge in the long-time limit. We separate the possibly divergent term introducing cumulants (dispersion)
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle v_y(0)f(t')\rangle=\langle v_y(0)f(t')\rangle_c+\langle v_y(0)\rangle\langle f(t')\rangle,\nonumber\\&& \!\!\!\!\!\!\!\!\!\!\!\!\!
\langle w(0)f(t')\rangle=\langle w(0)f(t')\rangle_c+\langle w(0)\rangle\langle f(t')\rangle,
\label{eq:cumulants}
\end{eqnarray}
where the angular brackets stand for average over $x-z$ coordinates. We find using $\partial_y \langle v_y(0)\rangle=\langle w(0)\rangle$ that
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle f(t, y)\rangle\!=\langle f(y)\rangle+\int_0^t \left[\partial_y\langle v_y(0)f(t')\rangle_c-\langle w(0)f(t')\rangle_c\right]dt'-\tau(\partial_y \langle u_y^2\rangle)\partial_y \int_0^t \langle f(t')\rangle dt'.\label{identity}
\end{eqnarray}
where we used $\langle v_y(0)\rangle$ provided by Eq.~(\ref{eq:divv_inh}). The last term does not necessarily converge in $t\to\infty$ limit. For instance if $\langle f(t)\rangle$ has a finite long-time limit then this term grows linearly with time. We consider this identity in the case of our interest.
\section{Space-dependent sum of Lyapunov exponents for channel turbulence}
\label{sec:lyapunov}
In this Section we study the average sum of Lyapunov exponents. That describes the average logarithmic rate of growth of infinitesimal volumes that start at the same distance from the wall $y$. Performing averaging over initial $x-z$ coordinates of the volume we find from Eq.~(\ref{finite}),
\begin{equation}
\left\langle \sum\lambda_i(t, y)\right\rangle=\frac{1}{t}\int_0^t \langle w(t', y)\rangle dt'=\frac{1}{t}\int_0^t dt'\int\frac{dxdz}{S} w(t', \bm q(t', x, y, z)). \label{dfnt}
\end{equation}
The identity given by Eq.~(\ref{identity}) gives in the leading order in $St$ for $f=w$,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!
\langle w(t, y)\rangle\!=\tau\partial_y\int_0^t \langle u_y(0)\nabla^2 p(t')\rangle_c dt'-\tau^2\int_0^t \langle \nabla^2 p(0)\nabla^2 p(t')\rangle_c dt'-\tau^2(\partial_y \langle u_y^2\rangle)\partial_y \int_0^t \langle \nabla^2 p(t')\rangle dt'-\tau\partial_y^2 \langle u_y^2\rangle.\label{complete}
\end{eqnarray}
where we used $\langle w\rangle$ given by Eq.~(\ref{eq:divv_inh}). In different time correlation functions in this formula we can use trajectories of the tracers in the leading order in $St$. The first term in the integrand is of order $St$ while the second is of order $(St)^2$. However both terms have to be kept because the former vanishes in the bulk due to spatial uniformity. In contrast, the third term in the integrand which is of order $(St)^2$ is smaller than the first term and can be neglected. The remaining integrals have finite $t\to\infty$ limit. We make the plausible assumption that the integrals converge over the local Kolmogorov time-scale $\tau_{\eta}(y)=\sqrt{\nu/\epsilon(y)}$. Then we find that,
\begin{eqnarray}&&\!\!\!\!\!\!\!\! \langle w(t, y)\rangle=\!-\tau^2\int_0^{\infty}\langle\nabla^2p(0)\nabla^2p(t)\rangle_c dt'+\partial_y v_{eff}(y),\ \ v_{eff}(y)=\langle v(y)\rangle+\int_0^{\infty}\langle u_y(0) w(t)\rangle_c dt,\ \ t\gg \tau_{\eta}(y).\label{eq:29}\end{eqnarray}
Finally, we find Eq.~(\ref{sum}) using Eq.~(\ref{dfnt}). This formula is valid at not too large times because the third term in Eq.~(\ref{complete}) grows with time with $\tau\langle \nabla^2 p\rangle\approx \sum\lambda_i=const$ in the integrand. This term becomes non-negligible at times $t$ obeying $\tau t\partial_y \langle u_y^2\rangle \partial_y \ln |\sum\lambda_i(y)|\gtrsim 1$. Since the scale of variations of the involved quantities is $y$ then this gives equality $\tau t \langle (u_y/y)^2\rangle\gtrsim 1$. For spatially uniform turbulence, applicable in the bulk, using Kolmogorov theory we find that $\langle (u_y/y)^2\rangle\sim \epsilon^{2/3}y^{-4/3}$ is smaller than $\langle (\nabla u)^2\rangle$ by a factor of the Reynolds number $Re$. This would give $t\gtrsim (Re/St)\tau_{\eta}$. If we assume that this time is much larger than $t_c$ which demands $Re\gg St^{-1}$ then Eq.~(\ref{sum}) describes $\sum\lambda_i(t)$ for one trajectory so we can use $\sum\lambda_i(t)\approx \langle\sum\lambda_i(t)\rangle$. A similar consideration can be made in the near wall region of small $y$ where $\langle u_y^2\rangle\propto y^4$. Below we assume that the correction term is negligible in regions of interest (as confirmed by results of a DNS in section \ref{sec:analysis}).
The formula given by Eq.~(\ref{sum}) has reductions in the bulk and in the turbulent boundary layer. In the bulk the statistics is uniform and we recover Eq.~(\ref{blk}) where there is no dispersion sign in the average because $\langle \nabla^2p \rangle=0$ in the bulk. In spatially uniform turbulence the sum of Lyapunov exponents is of order $St^2$. In contrast, the inhomogeneous terms are proportional to $St$ and dominate regions of strong inhomogeneity,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!
\left(\left\langle\sum \lambda_i\right\rangle\right)_{inhom}\approx \partial_y v_{eff}\label{eq:30}\propto St.
\end{eqnarray}
We see that the RHS has the form of the divergence of an effective velocity which is the average Eulerian velocity of particles plus the correction. We can interpret $v_{eff}$ as the average velocity of turbophoretic particles to the wall which differs from the average Eulerian velocity $\langle v_y\rangle$ because of the difference between Lagrangian and Eulerian averages. The reason $\left(\left\langle\sum \lambda_i\right\rangle\right)_{inhom}$ scales linearly with $St$ is that in the bulk non-zero $\sum\lambda_i$ appears because of fluctuations and thus is proportional to $St^2$ but in the boundary layer there is an average effect proportional to $St$.
The complete equation (\ref{complete}) is an identity that holds for arbitrary $y$ including those close to the wall. In the passages between this identity and Eq.~(\ref{sum}) we introduced the assumption that the third term in Eq.~(\ref{complete}) can be neglected and the integrals converge over the local Kolmogorov time-scale $\tau_{\eta}(y)$. In the case of inhomogeneous statistics the integrands can have non-trivial time-dependence because the trajectory samples regions with statistics different from that at the initial point. Nevertheless the convergence seems reasonable.
We consider the question of how well $\left\langle \sum\lambda_i(y)\right\rangle$ approximates the fluctuating finite-time Lyapunov exponent $\sum \lambda_i(t)=t^{-1}\ln V(t)/V(0)$ for volumes whose initial vertical position is $y$ and horizontal position is arbitrary. Proceeding as we did in studying the similar question in the spatially uniform situation, see Eq.~(\ref{sps2}), we consider the dispersion at $t\gg \tau_{\eta}$,
\begin{eqnarray}&&\!\!\!\!\!\!\!\!\!\!\!\!\!\!
\left\langle \left(\sum\lambda_i(t)\right)^2\right\rangle\!-\!\left\langle \sum\lambda_i(t)\right\rangle^2\!=\!\frac{1}{t^2}\int_0^t \!\!dt_1dt_2 \langle w(t_1, \bm x(t_1))w(t_2, \bm x(t_2))\rangle_c\!\sim \!\frac{1}{t}\int_0^\infty \!\!\langle w(0)w(t)\rangle_c,\end{eqnarray}
where we used Eq.~(\ref{finite}) and consider $t\ll t_{in}(y)$ so the statistics of $w(t)$ does not change over the considered time interval. The difference from the spatially uniform case is that the last integral is no longer necessarily $\sum\lambda_i$, see Eq.~(\ref{sum}). For $y$ which are not too far from the bulk so we have $\tau^2\int_0^{\infty}\langle\nabla^2p(0)\nabla^2p(t)\rangle_c dt'\gtrsim |\partial_y v_{eff}(y)|$, see Eq.~(\ref{sum}), we can use Eq.~(\ref{rms}). Thus for these $y$ we can use for one trajectory $\sum\lambda_i(t)\approx \langle\sum\lambda_i(t)\rangle$ for $t\gtrsim \tau_{\eta}(y)/St^2(y)$ where $St(y)=\tau/\tau_{\eta}(y)$. In contrast, when $y$ is closer to the wall region where $\tau^2\int_0^{\infty}\langle\nabla^2p(0)\nabla^2p(t)\rangle_c dt'\ll |\partial_y v_{eff}(y)|$ we have in that region,
\begin{equation} \frac{\sqrt{\left\langle \left(\sum\lambda_i(t)\right)^2\right\rangle-\left\langle \sum\lambda_i(t)\right\rangle^2}}{|\sum\lambda_i(t)|}\sim \frac{\sqrt{\int_0^\infty \!\!\langle w(0)w(t)\rangle_c}}{|\sum\lambda_i|\sqrt{ t}},
\end{equation}
cf. Eq.~(\ref{rms}). In this case convergence is faster.
\section{Preferential concentration and distinction between Kaplan-Yorke and correlation dimensions}\label{pc}
The theory described in previous Sections provided the local rate of production of inhomogeneities of inertial particles in channel turbulence. This rate provides the growth of concentration of particles $n(t, \bm x)$. Solving the continuity equation at $t>0$, we have
\begin{eqnarray}&&
\frac{1}{t}\ln \frac{n(t, \bm q(t, \bm x))}{n(0, \bm x)}=-\frac{1}{t}\int_0^t w(t', \bm q(t', \bm x))dt',\label{slv}
\end{eqnarray}
where the RHS is $\sum\lambda_i$ up to the sign. This formula is implied by mass conservation in the particle's frame, $n(t)V(t)=const$. We see from Eq.~(\ref{slv}) that concentration grows at large times as $\exp\left[|\langle\sum\lambda_i\rangle|t\right]$. We consider the history of creation of fluctuation of concentration at scale $r\ll \eta$, see \cite{FFS,fphl}. This starts from compression of the volume of particles whose initial size is the correlation length $\eta$ of $w$. Over the initial volume the concentration is effectively uniform \cite{fouxon1}. The smallest dimension of the compressed volume decreases with time $t$ as $\eta\exp[-|\lambda_3|t]$ where $\lambda_3$ is the third Lyapunov exponent \cite{Schmidt,review}. Since $\lambda_3$ is non-zero for tracers then, in the leading order in Stokes number, we can use $\lambda_3$ of the fluid particles. However for fluid particles volumes are conserved. Thus the smallest dimension decreases so that its product with the growing area, giving the volume, stays constant. This gives $|\lambda_3|=\lambda_1+\lambda_2$ where $\lambda_1+\lambda_2$ is the growth exponent of areas defined in Eq.~(\ref{area}). The fluctuation of concentration grows until the time $t_*=|\lambda_3|^{-1}\ln(\eta/r)$ when the smallest dimension becomes equal to $r$. Beyond this time there is no growth of correlated fluctuations of concentration \cite{fphl}. We find that the factor of increase of concentration is $\exp[|\sum\lambda_i|t_*]=(\eta/r)^{D_{KY}}$ where,
\begin{eqnarray}&&
D_{KY}(y)=\frac{|\langle \sum \lambda_i(y)\rangle|}{|\langle \lambda_3(y)\rangle|}.
\label{eq:dky}
\end{eqnarray}
We observe that $D_{KY}$ has the structure of the reduced formula for the Kaplan-Yorke fractal codimension in the case of weak compressibility \cite{ky,fouxon1}. We defined
\begin{eqnarray}&&
\langle \lambda_3(y)\rangle=\frac{1}{t}\int\frac{dxdz}{S} \ln\left(\frac{S(t, x, y, z)}{S(0, x, y, z)}\right),\ \ \tau_{\eta}\ll t\ll t_{in}(y),
\end{eqnarray}
where $S(t, x, y, z)$ is an arbitrarily oriented infinitesimal area element located near $(x, y, z)$ initially. This average is independent of $t$ in the considered time interval. Further, the orientation of the surface, that can be defined by the normal, is irrelevant for the long-time limit despite the anisotropy of the statistics of turbulence. This is because orientation reaches (anisotropic) steady state quite fast \cite{review} (this is not the case for sedimenting inertial particles where relaxation of orientation is long \cite{fphl}). There is a significant difference of convergence time for $\sum\lambda_i$ and $\lambda_3$ as remarked previously: we have $\lambda_3(t, y)\approx \langle \lambda_3(y)\rangle$ at $t\gg\tau_{\eta}(y)$, cf. spatially uniform case \cite{BFF,review}.
In the case of spatially uniform turbulence $D_{KY}$ describes all the fractal codimensions \cite{fouxon1}. For instance the pair-correlation function of concentration scales as $(\eta/r)^{\Delta}$ where $\Delta=2D_{KY}$ is the correlation codimension \cite{BFF,fouxon1}. However in the case of inhomogeneous turbulence this is no longer true. This can be seen in the simplest context by considering the exponential growth of moments of concentration which in spatially uniform case is determined by $\sum\lambda_i$ completely. We have,
\begin{eqnarray}&&
\left\langle \left[\frac{n(t, \bm q(t, \bm x))}{n(0, \bm x)}\right]^k\right\rangle=\left\langle \exp\left[-k\int_0^t w(t', \bm q(t', \bm x))dt'\right]\right\rangle.
\end{eqnarray}
We can use the formula $\langle \exp[X]\rangle=\exp[\langle X\rangle+\langle X^2\rangle_c/2]$ for averaging of Gaussian random variable $X$ doing averaging of the last term. This can be proved using cumulant expansion and smallness of compressibility \cite{fouxon1}. We find at $t\gg \tau_{\eta}(y)$,
\begin{eqnarray}&&
\left\langle \left[\frac{n(t, \bm q(t, \bm x))}{n(0, \bm x)}\right]^k\right\rangle=\exp\left[-kt\left\langle\sum\lambda_i\right\rangle+k^2t\int_0^{\infty} \langle w(0)w(t)\rangle_c dt\right].
\end{eqnarray}
In the spatially uniform case described in Section \ref{sec:theory} $|\sum\lambda_i|=\int_0^{\infty} \langle w(0)w(t)\rangle_c dt$ so the growth exponents are $k(k+1)|\langle\sum\lambda_i\rangle|$. The exponent is zero at $k=-1$ because of the conservation of the number of particles \cite{BFF}. The exponents are determined by $\sum\lambda_i$ completely. In contrast, for inhomogeneous turbulence factors near $k$ and $k^2$ in the formula above become independent. The average concentration can increase or decrease locally without contradicting the global conservation of the number of particles as in turbophoresis. The growth exponents of the moments of concentration are no longer determined uniquely by $\sum\lambda_i$. The formulas derived above hold for concentration in the particle's frame. Similar formulas can be written for the growth of moments of concentration at a fixed spatial point \cite{fouxonnlin}.
We consider statistics of particle distribution in space after transients. If we seed particles in the channel then, after transients that at scale $r$ have typical time-scale $t_*$, they distribute over a multifractal structure in space \cite{BFF,FFS,fouxon1}. The statistics of the distribution can be obtained by averaging over the $x-z$ plane as in the previous Sections. It was demonstrated in \cite{Schmidt} that pair-correlation function of concentration of particles $n(t, \bm x)$ factorizes in product of (possibly time-dependent) average local concentrations and geometrical factor of fractal increase of probability of two particles to be close. The obvious change of the formula for $w$ in \cite{Schmidt} gives,
\begin{eqnarray}&&
\langle n(t, \bm x)n(t, \bm x+\bm r)\rangle=\langle n(t, \bm x)\rangle \langle n(t, \bm x+\bm r)\rangle\left(\frac{\eta}{r}\right)^{\Delta(y)},\ \ \ \ \frac{1}{|\lambda_3(y)|}\ln\left(\frac{\eta}{r}\right)\ll t_{in},\\&&
\Delta(y)=\frac{\tau^2}{|\lambda_3(y)|}\int_{-\infty}^{\infty} \langle \nabla^2 p(0, y)\nabla^2 p(t)\rangle_c dt=2D_{KY}(y)+\frac{\partial_y v_{eff}(y)}{|\lambda_3(y)|}.
\label{eq:delta}
\end{eqnarray}
We see that the correlation codimension is not $2D_{KY}$. It scales proportionally with $St^2$ when $D_{KY}$ has both the term that scales linearly and the terms that scales quadratically with $St$. The reason why the $\partial_y v_{eff}(y)$ term in $\sum\lambda_i$ drops from $\Delta$ is that this term originates in the average velocity that affects equally the average $n$ and its fluctuations disappearing from the ratio $n/\langle n\rangle$. The form of $\langle n(t, \bm x)\rangle$ in Eq.~(\ref{eq:delta}) is determined by initial and boundary conditions on the concentration and is problem-dependent in contrast with the power-law factor. Further the correlations do not depend on the direction of $\bm r$ despite anisotropy of the statistics of turbulence. This is restoration of isotropy that originates in independence of Lyapunov exponents on the initial orientations.
Finally we consider the coarse-grained concentration $n_l(\bm x)$,
\begin{eqnarray}&&
n_l(\bm x)=\frac{m_l(\bm x)}{4\pi l^3/3},\ \ m_l(\bm x)=\int_{|\bm x'-\bm x|<l}n(0, \bm x') d\bm x',
\end{eqnarray}
which is mass $m_l(\bm x)$ in small volume of radius $l\ll \eta$ divided by the volume. We can find $m_l(\bm x)$ using the consideration of \cite{fouxon1,Schmidt} by tracking the ball of the particles back in time to time $t=-t_*$ where $t_*=-|\lambda_3(y)|^{-1}\ln(\eta/l)$. Since there are no fluctuations of concentration at scale $\eta$ then the mass of the ball at that time is volume $(4\pi l^3/3)\exp[-\int_{-t_*}^0 w(t', \bm q(t', \bm x))dt']$ times the average concentration $\langle n(-t_*, \bm q(-t_*, \bm x))\rangle$ see details in \cite{fouxon1}. Comparing the resulting formula for $n_l(\bm x)$ with the formula for $\langle n(\bm x, t)\rangle$ from \cite{Schmidt} we find
\begin{eqnarray}&&
\frac{n_l(\bm x)}{\langle n(\bm x, t)\rangle}=\frac{\exp[-\int_{-t_*}^0 w(t', \bm q(t', \bm x))dt']}{\left\langle\exp[-\int_{-t_*}^0 w(t', \bm q(t', \bm x))dt']\right\rangle}.
\end{eqnarray}
This formula was provided (with a typo) in \cite{Schmidt} where the result of the averaging was presented,
\begin{eqnarray}&&
\frac{\langle n_l^k(\bm x)\rangle}{\langle n(\bm x, t)\rangle^k}=\left(\frac{\eta}{l}\right)^{\Delta(y)k(k-1)/2},
\end{eqnarray}
which for $k=2$ corresponds with the previously derived formula for the pair-correlation of concentration. Thus if we use the scaling of $n_l/\langle n(\bm x, t)\rangle$ for defining fractal dimensions \cite{fouxon1} then none of the dimensions is $D_{KY}$.
\section{Numerical Simulation}\label{sec:dns}
We use the direct numerical simulation (DNS) of a turbulent channel flow provided by the JHTDB. A large variety of time and space dependent, Eulerian simulation results are stored on a cluster of databases, which is made accessible to the public. The functionality of this database systems and details on the simulations available, as well as confirmations of their validity, are described in \cite{perlman2007data,li2008public,yu2012studying,graham2016web} and other references therein. All details on the DNS computation, specifically numerical schemes, discretization methods and further simulation details of the turbulent channel flow are extensively described in \cite{graham2016web}.
The turbulent channel flow with a friction Reynolds number $Re_{\tau}\approx1000$, considered in this work is a wall bounded flow with no-slip conditions at the top and bottom walls ($y/h=\pm1$, where $h$ corresponds to half of the channel height) and periodic boundary conditions in the longitudinal and transverse directions. In this channel flow DNS, the streamwise direction $x$ and the transverse direction $z$ can be considered homogeneous, whereas the wall-normal direction $y$ serves as inhomogeneous direction for the purpose of generating turbophoretic drift of inertial particles. The domain spans over the three directions as follows: $L_x \times L_y \times L_z = 8\pi h\times 2h\times 3\pi h$, where $h=1$ in dimensionless units. Quantities normalized by the friction velocity $u_{\tau}$, the viscous length $\nu/u_{\tau}$ ($\nu=$ viscosity) or the viscous time scale $\nu/u_{\tau}^2$ are presented with the superscript $+$. The wall of the channel is located at $y^+=0$, the center of the channel is at $y^+=1000$.
An overview over the main simulation, flow and grid parameters is given in Table \ref{table:parametes}.
\begin{table}
\begin{tabular}{K{2.6cm} K{2.8cm} K{1.2cm} K{1.2cm} K{1cm} K{1cm} K{1.1cm} K{1cm} K{1.1cm} K{1.1cm} K{1cm} K{1cm}}
\hline\hline \noalign{\vskip 2mm}
$L_x\times L_y\times L_z$ &$N_x\times N_y\times N_z$ & $\delta t$ & $\nu$ & $U_c$ & $u_{\tau}$ & $Re_{\tau}$ &$St^+$ & $\Delta x^+$ & $\Delta y_1^+$ & $\Delta y_c^+$ & $\Delta z^+$ \\ \hline \noalign{\vskip 2mm}
$8h\pi\times 2h\times 3h\pi $ & $2048\times 512\times 1536$ & $0.0065$ & $5\times10^{-5}$ & $1.13$ & $0.0499$ & $999.4$ & $1$ & $12.26 $ & $0.0165$ & $6.16$ & $6.13$\\
[0.25cm] \hline\hline \noalign{\vskip 2mm}
\end{tabular}
\caption{\label{table:parametes} Simulation, flow and grid parameters - $L_{i}$: domain size in all directions $i=x,y,z$, $N_{i}$: number of grid points in all directions, $\delta t$: time step, $\nu$: viscosity, $U_c$: centerline velocity, $u_{\tau}$: friction velocity, $Re_{\tau}$: friction Reynolds number, $St^+$: viscous Stokes number, $\Delta x^+$: grid spacing streamwise direction, $\Delta y_1^+$: grid spacing wall-normal direction (first point), $\Delta y_c^+$: grid spacing wall-normal direction (central point), $\Delta z^+$: grid spacing spanwise direction}
\end{table}
The time step, $\delta t$, at which the Eulerian flow data can be extracted from the database is $0.0065$ and the total available flow time is approximately $26$ (non-dimensional time units), which corresponds to approximately one flow through time. The total duration of the simulation is thus $t_{end}^+=2.6\times10^4$.
We have used the Eulerian results of the channel flow DNS provided by the JHTDB to perform Lagrangian tracking of inertial particles in the channel flow. At total number of, $4\times10^6$ point-particles are randomly seeded across the entire channel domain. The JHTDB allows to extract velocity, velocity gradients and the Hessian of pressure at any arbitrary particle position. We use Eq.\,(\ref{eq:particlevelocity}) to determine the inertial particle velocity. We compute the second term on the RHS of Eq.\,(\ref{eq:particlevelocity}) $(\partial_t\bm u+(\bm u\cdot\nabla)\bm u)$ based on the material derivative $D\bm u/Dt$ along a tracer particle trajectory. This is done by applying a simple finite difference scheme using the tracer particle velocity at the tracer particle position of two consecutive time steps. The inertial particles are advected with time step $\delta t$ applying a second order Adams-Bashforth method for temporal integration.
\begin{figure}[h]
\centering
\includegraphics[scale=0.37]{Figure1-eps-converted-to.pdf}
\caption{\label{fig:Ch_restress_full}(a) - Components of the Reynolds stress tensor and turbulent kinetic energy $k$ normalized by $u_{\tau}^2$ plotted along non-dimensional channel height. The turbulent kinetic energy (red solid line) peaks at $y^+=10-11$. The square of the wall-normal velocity fluctuations $\langle u'_yu'_y\rangle/u_{\tau}^2$ (red dashed line) has its maximum at $y^+=100$. (b) - Variation of the the Kolmogorov time-scale $\tau_{\eta}$ (solid line) and the inhomogeneity time $t_{in}=y/\langle u_y^2\rangle^{1/2}$ (dashed line) along the channel height.}
\end{figure}
A verification of the channel flow DNS, including a comparison with previous works, has been done in \cite{graham2016web}. We show the Reynolds stresses $u'_iu'_j$ as well as the turbulent kinetic energy, normalized by the friction velocity $u_{\tau}$ versus the viscous wall distance $y^+$ in Fig.\,\ref{fig:Ch_restress_full}(a) and plot the wall-distance logarithmically in order to focus the visualization on the near-wall behavior of the flow. The dashed red line, indicating the Reynolds stresses based on the wall-normal velocity $u'_y$, shows a maximum at around $y^+=100$ dropping quite steeply towards the wall. We highlight this term in Fig.\,\ref{fig:Ch_restress_full}, since its second derivative is responsible for the turbophoretic migration of the particles as shown previously in Eq.\,(\ref{eq:divv_inh}).
The ratio between the particle response time $\tau$ and the Kolmogorov time scale $\tau_{\eta}$, defines the particle Stokes number $St$. Due to the strong dependence of the turbulent kinetic energy on the wall-normal direction (red solid line in Fig.\,\ref{fig:Ch_restress_full}(a)), also the dissipation of the turbulent kinetic energy and thus the Kolmogorov microscales, depend on $y$. Fig.\,\ref{fig:Ch_restress_full}(b) shows $\tau_{\eta}$ versus the inhomogeneous spatial direction $y^+$. For the presented theory to be valid we choose a rather weak particle inertia by setting the Stokes number averaged over the whole channel to $\langle St\rangle=0.1$, based on the averaged Kolmogorov time $\langle\tau_{\eta}\rangle=0.2133$. This yields $\tau=0.0213$ and thus a viscous Stokes number of $St^+\approx1$. The viscous Stokes number is defined in terms of the friction velocity $u_{\tau}$ as $St^+=\tau u_{\tau}^2/\nu$. Due to the variation of $\tau_{\eta}$ along wall-normal direction (Fig.\,\ref{fig:Ch_restress_full}) also the local $St$ depends strongly on $y$, reaching the largest values in the vicinity of the wall.
\section{Results and Discussion}\label{sec:analysis}
\subsection{Results}
Before going into the analysis of cluster formation, we start with discussing the effect of turbophoresis. In a channel flow inertial particle migration due to turbophoresis changes the initially random distribution of particles by driving particles towards the wall. A snapshot of the distribution of tracer and inertial particles at $t^+=2.5\times 10^4$ in the $x-y$ plane near the wall (Fig.\,\ref{fig:particledistribution}) shows the increased inertial particle concentration near the wall qualitatively. The plane chosen here is located at $z=1.5\pi$ and covers the full extension of the channel in streamwise direction but only the near wall region between $y^+=0$ and $y^+=15$. Particles are plotted as black points on top of the time-averaged $\partial_y^2\langle u_y^2\rangle$ field. It is visible that the tracer particles shown in Fig.\,\ref{fig:particledistribution}(a) distribute randomly in space and appear uniformly distributed while inertial particles (Fig.\,\ref{fig:particledistribution}(b)) accumulate near the wall.
\begin{figure}[h]
\centering
\includegraphics[scale=0.5]{Figure2_new.pdf}
\caption{\label{fig:particledistribution} Comparison of the instantaneous particle distribution of tracer particles ($St^+=0$ - plot a) and inertial particles ($St^+=1$ - plot b) at time $t^+=2.5\times10^4$ in the near-wall region between $y+=0-15$ (dimensions are not to scale). The particles are plotted on top of the time-averaged $\partial_y^2\langle u_y^2\rangle$ field.}
\end{figure}
The temporal evolution of wall-normal particle concentration profiles are depicted in Fig.\,\ref{fig:particleconcentration}, which compares the particle concentration $n$ at the initial time step $t^+=0$ (circles), as well as $t^+=10^4$ (triangle), $t^+=2\times10^4$ (triangle upside down) and $t^+=2.5\times10^4$ (squares), normalized by the initial, random particle concentration $n_0$. It is visible that with time more particles accumulate in the vicinity of the wall. In the region below $y^+=10$ a significant increase of the particle concentration is observable. In the vicinity of the wall, the ratio $n/n_0$ rises almost up to factor three in the considered time span, despite the relatively weak inertia of the particles. Above $y^+=10$ the particle concentration decreases below the initial concentration where a minimum of $n/n_0=0.95$ is reached at $y^+=20$. This leads to a specific interest of the degree of clustering in both, the region where the number of particles is high (below $y^+=10$) and the region where the number of particles is low (between $y^+=10-70$).
\begin{figure}[h]
\centering
\includegraphics[scale=0.37]{Figure3-eps-converted-to.pdf}
\caption{\label{fig:particleconcentration} Particle concentration $n$ normalized by the initial, random, particle concentration $n_0$ versus the wall-normal distance $y^+$. Different symbols indicate the temporal evolution from $t^+=0$ to $t^+=2.3\times10^4$. The region below $y^+=10$ shows an increase in particle concentration, whereas the particle concentration drops below the initial concentration between $y^+=10$ and $y^+=60$.}
\end{figure}
The degree of clustering is not only determined by the particle concentration itself but instead rather by a high probability of finding particles with very small inter-particle distance. As explained in section \ref{sec:theory}, particles approach each other and thus form clusters if $\nabla\cdot\bm v<0$ along particle trajectories. This effect is quantified by the previously introduced sum of Lyapunov exponents, $\sum\lambda_i$.
Negative values of $\sum\lambda_i$ correspond to a compression of infinitesimal volumes formed by particles, whereas positive values of $\sum\lambda_i$ correspond to diverging volumes. For a precise quantification of the degree of clustering arising from the combined action of inhomogeneous and homogeneous clustering effects, it is necessary to compute the finite-time Lyapunov exponents. We do this in the following by the computation of each individual term of the RHS in Eq.\,(\ref{sum}) or Eq.\,(\ref{complete}), respectively
Since all these terms depend on time, it is important to look at the convergence time of the individual terms. Theoretically, convergence within a few Kolmogorov time scales of the first term ($\tau\partial_y\int_0^t\langle u_y(0)\nabla^2p(t')\rangle_c dt'$) as well as the second term ($-\tau^2\int_0^t\langle\nabla^2p(0)\nabla^2p(t')\rangle_c dt'$) on the RHS of Eq.(\ref{complete}) is expected. These two integrals use the cumulant terms as described in Eq.\,(\ref{eq:cumulants}). As discussed in section \ref{sec:lyapunov} the third integral, which is $\int_0^t\langle\nabla^2p\rangle$, does not converge. However, at the time where the other terms have converged the full term $\tau^2(\partial_y\langle u_y^2\rangle)\partial_y\int_0^t\langle\nabla^2p(t')\rangle dt'$ (including this integral) remains small and can therefore be excluded from the computation of $\sum\lambda_i$ as shown below. For the approximations to be valid the inhomogeneity time $t_{in}$ (Fig.\,\ref{fig:Ch_restress_full})(b) has to be larger than the convergence time of these integrals.
The temporal evolution of the three integrals of Eq.\,(\ref{sum}) in Fig.\,\ref{fig:integrals}. The color shading is darker for increasing $y^+$, i.e. light curves refer to regions near the wall and dark curves in the bulk of the channel, respectively. The integral of the first term on the RHS of Eq.\,(\ref{sum}) reaches relatively low values in the viscous sublayer and in the bulk, whereas it becomes more significant in the intermediate (log-layer) regions where convergence takes longer. In the bulk as well as in the viscous sublayer the curves converge to relatively low values.
The second integral of Eq.\,(\ref{sum}) is shown in Fig.\,\ref{fig:integrals}(b), where it is seen that all curves converge fast. The largest values are found in the regions between $y^+=6-25$, where turbulence intensity peaks (see Fig.\,\ref{fig:Ch_restress_full}(a)). Figure \,\ref{fig:integrals}(c) shows $\int_0^t\langle\nabla^2p(t')\rangle dt'$ that as expected does not converge. Generally, the convergence time is smaller than $t_{in}$. However, as one can see from Fig.\,\ref{fig:integrals}(a) in the buffer layer region the convergence of the first term is rather slow and an upper limit can be estimated at about $100\langle\tau_{\eta}\rangle$, which is the time we choose to evaluate the mean Lyapunov exponents.
\begin{figure}[h]
\centering
\includegraphics[scale=0.45]{Figure4-eps-converted-to.pdf}
\caption{\label{fig:integrals} The three integrals of Eq.\,(\ref{sum}) are plotted versus time normalized by the average Kolmogorov time for all wall distances used for the computation. Darker curves present the regions in the center of the channel and lighter curves the near-wall region. (a): $\tau\partial_y\int_0^t\langle u_y(0)\nabla^2p(t')\rangle_c dt'$ (b): $-\tau^2\int_0^t\langle\nabla^2p(0)\nabla^2p(t')\rangle_c dt'$ (c): $\int_0^t\langle\nabla^2p(t')\rangle dt'$ }
\end{figure}
Now we evaluate the individual terms on the RHS of Eq.\,(\ref{complete}) in Fig.\,\ref{fig:3termcorrelation}(a). Circles show the wall-normal profile of the first term which is the sole term causing small-scale clustering of inertial particles in homogeneous turbulence.
This correlation is zero at the wall but its absolute value increases further away from the wall reaching the largest negative values in the range between $y^+=5-20$. Beyond $y^+=20$ the curve approaches small magnitudes as turbulence becomes more homogeneous but will not vanish.
The second term on the RHS of Eq.\,(\ref{complete}) ($\tau\nabla_y\int_0^T\langle u_y(0)\nabla^2p(t)\rangle$ - x symbols) is negative and contributes to clustering in a part of the buffer layer, whereas it is positive (and counteracting clustering) outside that region.
The third term of Eq.\,(\ref{complete}) ($-\tau\nabla_y\langle u_y^2\rangle$), shown as $+$ symbols, accounting for the turbophoretic effect solely is negative below $y^+=15$. This indicates a compression of infinitesimal volumes and contribution towards clustering (this location has been defined as $y*$ in section \ref{sec:theory}). The curve changes sign contributing against clustering above $y^+=15$. Above $y^+=15$ the term becomes slightly positive before converging towards $0$ at $y^+=100$.
The fourth term of Eq.\,(\ref{complete}) shown as filled black points in Fig.\,\ref{fig:integrals}(a) is rather small at the time where the other terms have reached convergence as predicted and will be neglected in the remaining analysis below.
\begin{figure}[h]
\centering
\includegraphics[scale=0.49]{Figure5-eps-converted-to.pdf}
\caption{\label{fig:3termcorrelation}:(a) All four terms of the RHS of Eq.\,(\ref{complete}) versus channel height; $-\tau^2\int_0^T\langle \nabla^2p(0)\nabla^2p(t)\rangle dt$ - circles; $-\tau\nabla_z^2\langle u_z^2\rangle$ - crosses; $\tau\nabla_z \int_0^T\langle u_z(0)\nabla^2p(t)\rangle dt$ - stars; $\tau^2(\partial_y\langle u_y^2\rangle)\partial_y\int_0^t\langle\nabla^2p(t')\rangle dt'$ - black points. (b): We show $\langle\sum\lambda_i\rangle$ separately for the homogeneous (circles) and the inhomogeneous contribution (squares), consisting of the sum of $-\tau\nabla_z^2\langle u_z^2\rangle$ and $\tau\nabla_z \int_0^T\langle u_z(0)\nabla^2p(t)\rangle dt$ but not including the last term of Eq.\,(\ref{complete}) The sum of the two components $\langle\sum\lambda_i\rangle_{tot}$, according to Eq.\,(\ref{eq:29}) is presented as blue diamonds.}
\end{figure}
In Fig.\,\ref{fig:3termcorrelation}(b) the terms of Eq.\,(\ref{eq:29}) are displayed. We divide the total $\sum\lambda_i$ (diamond symbols) in a homogeneous or bulk component (circles) and an inhomogeneous component (squares). Despite the linear dependence of the inhomogeneous component on $St$ the clustering due to the homogenous contribution generally exceeds the inhomogeneous component. Both components add up and reach the maximal negative $\sum\lambda_i\approx -1$ at $y^+=6$
Since we aim to quantify the preferential concentration via $D_{KY}$ and the correlation codimension $\Delta$ according to Eq.\,(\ref{eq:dky}) and (\ref{eq:delta}) respectively, in dependence on the inhomogeneous flow direction, the third Lyapunov exponent $\lambda_3$ has to be computed. The calculation of $\lambda_3$ is performed via the finite-time Lyapunov exponents that provide a measure of the cumulative deformation of the particles \cite{meneveau2016analysis}. This requires an estimation of the instantaneous deformation rate of the particle trajectory, which is done via the instantaneous Lyapunov exponents $\lambda'_{ii}$. These instantaneous Lyapunov exponents can be computed by the alignment of the eigenvectors of the Cauchy-Green tensor $C_{ij}$ of a particle trajectory and the velocity gradient tensor. More precisely the instantaneous Lyapunov exponents at each particle location can be found, using
\begin{equation}
\lambda'_{ii}=cos^2(\theta_{ij})P_j ,
\label{eq:FLTLE}
\end{equation}
where $P_j$ are the eigenvalues of the strain rate tensor and $\theta_{ij}$ is the angle between eigenvector $i$ of the Cauchy-Green tensor and eigenvector $j$ of the strain rate tensor. Averaging those instantaneous Lyapunov exponents along a Lagrangian path enables us to determine $\lambda_3$ depending on the wall distance $y$ of the channel. The corresponding result is presented in the inset of Fig.\,\ref{fig:lambda_3}(a). The maximum $|\lambda_3|$ is reached at $y^+=10$, the curve drops quite fast in both directions. As apposed to spatially uniform turbulence the estimate that $\lambda_3\approx\tau_{\eta}^{-1}$ is not true along the entire channel. Below $y^+=100$ the product of $\tau_{\eta}\lambda_3$ drops significantly, whereas it is constant above $y^+=100$.
The clustering introduced by $(\langle\sum\lambda_i\rangle)_{bulk}$ is quantified using $(D_{KY})_{hom}$. The resulting values are shown as blue circles in Fig.\,\ref{fig:lambda_3}(a). With $(D_{KY})_{hom}=0.1$ the maximum level is moderately high but interestingly it stays almost constant across a large region ranging from $y^+=3$ to $y^+=30$.
The values obtained for $\Delta_{hom+inhom}$, combining homogeneous and inhomgeneous clustering, are relatively high in regions below $y^+=10$, see Fig.\,\ref{fig:lambda_3}(a). At the wall $\Delta$ is $0$ but starts increasing rapidly. The curve peaks around $y^+=4$ with $\Delta_{hom+inhom}\approx 0.3$. It then decreases slowly to almost $0$ at $y^+=200$. In contrast to $(D_{KY})_{hom}$ the correlation codimension takes on very large values in a much narrower range from $y^+=2-10$.
\begin{figure}[h]
\centering
\includegraphics[scale=0.49]{Figure6-eps-converted-to.pdf}
\caption{\label{fig:lambda_3} The blue circles show $(D_{KY})_{hom}$, obtained from Eq.\,(\ref{eq:dky}) over the wall-normal distance showing a strong increase in the near-wall region and the largest values in the region between $y^+=2-8$. The diamonds indicate the correlation co-dimension $\Delta_{hom+inhom}$, computed according to Eq.\,(\ref{eq:delta}), varying over wall-distance. Inset: Dependence of $\lambda_3$, computed according to the procedure described above, on the viscous wall distance $y^+$. The largest value can be observed at $y^+=10$ (b): Plot of $\left|\frac{\langle\sum\lambda_i\rangle_inhom}{\langle\sum\lambda_i\rangle_{hom}}\right|$ form $y^+=1$ to $y^+=100$ for three different Stokes numbers - $St^+=0.1$ (filled squares); $St^+=1$ (empty circles); $St^+=10$ (filled triangles). Note that the strong outliers in a few positions result from a change of sign of the inhomogeneous term.}
\end{figure}
One key difference between the clustering in homogeneous and inhomogeneous turbulence is that for weakly inertial particles there is a linear dependence on St (for the first term on the RHS of Eq.\,(\ref{complete})). The ratio $\left|\frac{\langle\sum\lambda_i\rangle_inhom}{\langle\sum\lambda_i\rangle_{hom}}\right|$ changes not only significantly throughout the channel height but also for different Stokes numbers. One would expect this term to dominate in inhomogeneous regions of the flow. In the case examined in this study so far, this is not the case. The homogeneous term is generally higher or at least of the same magnitude as the inhomogeneous term. Therefore, we want to extend this study to other $St$. In Fig.\,\ref{fig:lambda_3}(a) we show the ratio between the homogeneous and the inhomogeneous contributions to clustering additionally for the case where $\langle St\rangle=1$ or $St^+=1$ (empty circles) and $St^+=0.1$ (filled squares) and $St^+=10$ (filled triangles).
We find that the Stokes number has a strong impact on which term dominates the clustering. However, predicting the behavior is not trivial due to the variations of the different terms along the $y-$ direction. As mentioned before, for $St^+=1$ the ratio of $\left|\frac{\langle\sum\lambda_i\rangle_inhom}{\langle\sum\lambda_i\rangle_{hom}}\right|$ stays below $1$ throughout the entire channel. Interestingly, for particles with small inertia the inhomogeneous term will dominate, as can be seen for the case of $St^+=0.1$ in Fig.\,\ref{fig:lambda_3}(b). For particles with larger inertia the behavior becomes more complex. The inhomogeneous terms will dominate or be of the same order as the homogeneous terms right at the wall but become less important throughout the rest of the channel. The homogeneous terms dominate even more strongly than in the case of $St^+=1$ further away from the wall.
\subsection{Discussion}
The results presented in the previous section indicate a strong dependence of particle concentration and clustering on the wall-normal direction. Turbophoresis in a turbulent channel flow drives particles towards the wall. After $t^+=2.5\times10^4$ the initial concentration at the wall is exceeded by almost factor 3 within the considered time span. The turbophoretic particle migration is driven by $-\tau\nabla_z^2\langle u_z^2\rangle$ shown in Fig.\,\ref{fig:3termcorrelation}(a). The change of sign in this term at $y^+\approx 15$ determines the wall-distance below which particles start accumulating and above which the particle concentration decreases.
We analyze the space-dependent rate of creation of inhomogeneous particle concentration to quantify inertial particle clustering in inhomogeneous turbulence. All three terms investigated according to Eq.\,(\ref{sum}) and Eq.\,(\ref{complete}) determining $\langle\sum\lambda_i\rangle$, depend differently on the wall distance. The term $\tau^2(\partial_y\langle u_y^2\rangle)\partial_y\int_0^{\infty}\langle\nabla^2p(t')\rangle dt'$ was small enough to be neglected for times within the convergence time of the other terms. Particle clustering for the case of $St^+=1$ is dominated by the homogeneous fractal clustering, represented by $-\tau^2\int_0^t\langle\nabla^2p(0)\nabla^2p(t')\rangle_c dt'$ even close to the wall where inhomogeneity is strongest. Its dependence on $y$ (Fig.\,\ref{fig:3termcorrelation}) is similar to the one of the turbulent kinetic energy, shown in Fig.\,\ref{fig:Ch_restress_full}(a). In regions of strong turbulence, $\tau\nabla^2p$ takes on large values and causes stronger clustering. Despite the linear dependence on $\tau$ the inhomogeneous contribution $(\langle\sum\lambda_i\rangle)_{inhom}$ to the overall clustering degree is smaller than $(\langle\sum\lambda_i\rangle)_{bulk}$ in the case of $St^+1$. Though $(\langle\sum\lambda_i\rangle)_{inhom}$ behaves similarly to $(\langle\sum\lambda_i\rangle)_{bulk}$ its absolute value peaks already at $y^+=4-5$ and is below the peak of $(\langle\sum\lambda_i\rangle)_{bulk}$. As can be seen in Fig.\,\ref{fig:lambda_3} the significance of the inhomogeneous terms decreases with larger wall-distance.
We use the correlation codimension of the multifractal formed by inertial particles to quantify the strength of particle clustering. The values found for the the correlation codimension ($\Delta$) for the case under investigation here are relatively large for particles with such small inertia. In the region between $y^+=2$ to $y^+=10$, $\Delta$ reaches up to $0.3$. In the region between $y^+=10-70$ --- where the particle concentration drops below the initial concentration --- the correlation codimension decreases rapidly. Whereas, the local homogeneous turbulence contribution to clustering $(D_{KY})_{hom}$ stays almost constant at $(D_{KY})_{hom}=0.1$ from $y^+=3-30$. Studies with particles of similar inertia find lower values of $\Delta$ (between $10^{-2}$ and $10^{-1}$) in homogeneous isotropic turbulence, e.g. \cite{saw1,Collinstwo}. This shows that turbulent inhomogeneity enhances the clustering degree particularly in the viscous sublayer and the onset of the buffer layer for $St^+=1$. The difference between $\Delta$ and $D_{KY}$ shows also that the inhomogeneous terms enhance clustering but also affect the regions where clustering occurs by the peak of $\Delta$ towards the wall. This can be explained by the maximal absolute values of $(\langle\sum\lambda_i\rangle)_{inhom}$ and $(\langle\sum\lambda_i\rangle)_{bulk}$ in Fig.\,\ref{fig:3termcorrelation}(b). Therefore, a complete investigation, unifying the clustering effects of both mechanism is essential for a precise quantification of the preferential concentration in inhomogeneous turbulent flows.
Due to the linear dependence of $(\langle\sum\lambda_i\rangle)_{inhom}$ on $\tau$ and the quadratic dependence of $(\langle\sum\lambda_i\rangle)_{bulk}$ on $\tau$, the situation changes for different particle inertia. In Fig.\,\ref{fig:lambda_3}(a) we saw that for particles with even smaller inertia $St^+=0.1$ the inhomogeneous terms dominate clustering throughout the entire channel. Instead for particles with large inertia ($St^+=10$) the inhomogeneous terms dominate near the wall but then become outweight by the homogeneous clustering term. The complementary effects of inhomogeneous and homogeneous clustering cause a different degree of clustering depending on inertia and the wall-normal direction.
\section{Conclusion}\label{sec:discussion}
In this study the combined effects of turbophoresis and small-scale fractal clustering of weakly inertial particles have been analyzed theoretically as well as numerically. A novel theoretically approach allows to describe the preferential concentration resulting from homogeneous and inhomogeneous particle clustering in inhomogeneous turbulence. We determine a space-dependent rate ($\langle\sum\lambda_i(y)\rangle$) that create inhomogeneities of concentration of particles. We find that $\sum\lambda_i$ depends linearly on the particle Stokes number, as opposed to homogeneous turbulence where $\sum\lambda_i\propto St^2$. The theoretical predictions for the creation of particle inhomogeneities have been investigated by direct numerical simulations of a turbulent channel flow using the JHTDB. The results reveal a strong turbophoretic migration of particles, despite the relatively small inertia. We find that the clustering degree depends strongly on the wall-normal direction. The strongest effects of preferential concentration can be observed in the transition of the viscous sublayer to the buffer layer at $y^+=2-10$, where local homogeneous terms contribute to clustering and clustering due to the inhomogeneity of the flow is strong. The correlation codimension rises up to $0.3$ which is remarkably high for particles with relatively weak inertia of $St^+=1$ or $\langle St\rangle=0.1$. The values found for $\Delta$ in the near-wall region are larger compared to what particles with the same inertia would show in homogeneous turbulence. The contributions to clustering from homogeneous and inhomogeneous terms varies strongly with the wall-distance but as we show also with particle inertia. For particles with very small inertia the inhomogeneous terms outweigh the homogeneous contributions significantly. However, for particles with large inertia homogeneous turbulence will mainly determine clustering except in the vicinity of the wall, where the contribution of inhomogeneous turbulence might be stronger.
The findings of this work allow for a precise quantification of inhomogeneous preferential concentration of weakly inertial particles in non-uniform turbulent flows. The case of a turbulent channel flow investigated here, serves as general example for all inhomogeneous turbulent flows. Thus, the presented results can be easily transferred to investigate preferential concentration of weakly inertial particles in other frequently occurring turbulent flows, e.g. pipe or free shear flows.
\section{Acknowledgments}
L.S. would like to thank Stephen S. Hamilton for his support regarding the work with the database provided by the Johns Hopkins University.
Financial support from the Swiss National Science Foundation (SNSF) under Grant No. 144645 is gratefully acknowledged.
| 2024-02-18T23:40:10.978Z | 2017-02-07T02:07:23.000Z | algebraic_stack_train_0000 | 1,557 | 11,771 |
|
proofpile-arXiv_065-7790 | \section{Introduction}
The Atiyah-Singer (AS) index theorem \cite{Atiyah:1963zz} on a closed manifold without boundary $X$,
\begin{align}
{\rm Ind} (D)=
\frac{1}{32\pi^2}\int_{X} d^4x \epsilon_{\mu\nu\rho\sigma}{\rm tr}[F^{\mu\nu}F^{\rho\sigma}],
\end{align}
is well-known and appreciated in physics.
The number of chiral zero modes $n_\pm$ with the $\pm$ chirality of the Dirac operator $D$
defines the index ${\rm Ind}(D) = n_+-n_-$, which is related to
the winding number or topological charge of the gauge fields.
The index is essential in understanding
the tunneling effect among different vacuua in quantum-chromo dynamics (QCD).
This text-book-level formula can be confirmed by only one-loop computations,
and therefore, physicist-friendly.
However, the Atiyah-Patodi-Singer (APS) index theorem \cite{Atiyah:1975jf}, which is
an extension of the AS theorem to a manifold $X$ with boundary $Y$,
\begin{align}
\label{eq:APS}
{\rm Ind}(D_{\rm APS}) =
\frac{1}{32\pi^2}\int_{X} d^4x \epsilon_{\mu\nu\rho\sigma}{\rm tr}[F^{\mu\nu}F^{\rho\sigma}] \textcolor{black}{-\frac{\eta(iD_Y)}{2}},
\end{align}
is less known.
Here the second term is called the eta invariant, which is given by a regularized difference
between the number of positive and negative eigenvalues of the boundary Dirac operator $iD_Y$.
This is not surprising since we have been not very interested in manifolds with boundary in physics until very recently.
Recently the theorem became important in condensed matter physics,
as it was pointed out in \cite{Witten:2015aba}
that the APS index is a key to understand bulk-edge correspondence \cite{Hatsugai}
of symmetry protected topological insulator, which is a gapped material in the bulk
but a good conductivity is seen on the edge.
The carrier of the charge, the so-called edge mode,
is described as a massless $2+1$-dimensional Dirac fermion,
whose partition function suffers from an anomaly of the time-reversal symmetry.
This anomaly is not a problem in the total system since
it is precisely canceled by the bulk fermion determinant.
Each piece of the APS index in Eq.~(\ref{eq:APS}) corresponds
to each phase of the fermion determinants. Namely, the APS theorem
is a mathematical guarantee that the time reversal symmetry
is protected in the total system.
However, the left-hand side of the theorem defined
by a massless Dirac operator with nonlocal boundary condition
is physicist-unfriendly.
The difficulty is in the fact that if we impose local and Lorentz symmetric boundary condition,
the reflected particle at the boundary flips its momentum but keeps the
angular momentum unchanged. This means that the chirality is not conserved
and the $\pm$ chirality sectors do not decouple anymore: $n_\pm$ and the index do not make sense.
The Atiyah-Patodi-Singer boundary condition \cite{Atiyah:1975jf} gives up the locality
and rotational symmetry to respect the chirality.
In the vicinity of the boundary, the Dirac operator has a form
\begin{align}
D = \gamma_4\left(\partial_4 + H \right),
\end{align}
where we take $x_4$ in the normal direction to the boundary,
the $A_4=0$ gauge is chosen,
and $H = \gamma_4\sum_i \gamma_i D^i $ is a Hermitian operator.
The APS boundary condition is given by this $H$ operator,
such that its positive eigenmode's components are forced to zero at the boundary: $(H+|H|)\psi=0$.
This is a non-local boundary condition requiring
information of eigenfunction extended on the whole boundary manifold.
Note that $H$ commutes with $\gamma_5$ and chirality is conserved.
Then we can define the index by the chiral zero-modes.
This outcome is mathematically beautiful but physicist-unfriendly,
since locality (or causality) is much more important than chirality for physicists,
otherwise, we may allow information to propagate faster than speed of light.
In physics, we should not accept the non-local APS boundary condition no matter how mathematically beautiful it is
\footnote{
In Refs.\cite{Witten:2019bou, Kobayashi:2021jbn}, it was shown that
the non-local feature of the APS boundary condition has no problem
when it is Wick-rotated to a ``state'' at a time-slice.
}.
Instead, we need to give up the chirality and consider left-right mixing of massive fermion.
The question is if we can make a fermionic integer even within massive systems.
Our answer is ``Yes we can''.
Here is our reference list. We proposed a new formulation of the APS index using the domain-wall fermion
\cite{Jackiw:1975fn,Callan:1984sa,Kaplan:1992bt}
in 2017 \cite{Fukaya:2017tsq} with Onogi and Yamaguchi.
One year later, three mathematicians Furuta, Matuo and Yamashita joined
and we succeeded in a mathematical proof that our proposal is mathematically correct \cite{Fukaya:2019qlf}.
The reformulation is so physicist-friendly that application to lattice gauge theory
is straightforward \cite{Fukaya:2019myi} (see \cite{Onogi:2021slv} for a relation to the Berry phase).
In this conference, we had two parallel talks on an application to a curved domain-wall
fermion by Aoki \cite{AokiS} and that to the mod-two APS index \cite{Fukaya:2020tjk, Matsuki} by Matsuki.
We also refer the readers to our review paper \cite{Fukaya:2021sea} on the whole project.
\section{Massive Dirac operator index without boundary}
The key point of this work is if we can reformulate the index of the Dirac operator
in terms of massive fermions,
where either notion of chiral or zero mode is lost.
Let us start with an easier case without boundary.
We consider a Dirac fermion with a negative mass, compared to that of the regulator.
Here we choose the Pauli-Villars regularization.
For simplicity, we couple the $SU(N)$ gauge fields to the fermion
on an even-dimensional flat Euclidean space.
The fermion determinant is expressed as
\begin{align}
\frac{\det(D-m)}{\det(D+M)}.
\end{align}
In the large mass limit,
or simply taking the physical mass $m$ and Pauli-Villars mass $M$ the same,
let us perform an axial $U(1)$ rotation with angle $\pi$
to flip the sign of the mass.
Apparently the fermion looks decoupling from the theory,
but taking the anomaly \cite{Adler:1969gk,Bell:1969ts}
into account, we have a shift in the $\theta$ term by $\pi$
and the sign is controlled by the Atiyah-Singer index $I_{\rm AS}(D)$:
\begin{align}
\frac{\det(D-M)}{\det(D+M)} = \frac{\det(D\textcolor{black}{+M})}{\det(D+M)}
\times\exp\left(-i\textcolor{black}{\pi}\underbrace{\frac{1}{32\pi^2}\int d^4x FF}_{=I_{\rm AS}(D)}\right) = (-1)^{-I_{\rm AS}(D)}.
\end{align}
Even though we consider this massive fermion case,
which does not have chiral symmetry or zero modes, the index still appears.
Our proposal is then to use the massive Dirac operator to “define” the index.
We use anomaly rather than symmetry.
Specifically, multiplying $i\gamma_5$ to the numerator and denominator in the determinant
to make the operator anti-Hermitian,
the determinant can be expressed by the products of pure imaginary eigenvalues
of massive operators:
\begin{align}
\frac{\det(D-M)}{\det(D+M)} &=\frac{\det i\gamma_5 (D-M)}{\det i\gamma_5 (D+M)} = \frac{\prod_{\lambda_{-M}}i\lambda_{-M}}{\prod_{\lambda_{+M}}i\lambda_{+M}}
= \exp\left[\frac{i\pi}{2} \left(\sum_{\lambda_{-M}}{\rm sgn}\lambda_{-M}-\sum_{\lambda_{+M}}{\rm sgn}\lambda_{+M}\right)\right].
\end{align}
Now we have shown the equality
\begin{align}
I_{\rm AS}(D) = \frac{1}{2}\eta(\gamma_5(D-M))^{reg.} = -\frac{1}{2}\left[\eta(\gamma_5(D-M))-\eta(\gamma_5(D+M))\right].
\end{align}
On the right-hand side, chiral symmetry is no more essential
but it is not written by the zero modes only but also by the non-zero modes.
\section{New formulation of the index with boundary}
Next we consider the nontrivial case with boundary.
Before going into the details, let us discuss
what a more physical set-up should be.
In physics, any boundary has its ``outside''.
Topological insulator is nontrivial because its outside
is surrounded by normal insulators.
It is more natural to regard the surface of
material as a wall between domains of different physical
profiles, rather than a boundary of a manifold.
This domain-wall should keep the
angular momentum in its normal direction,
rather than helicity of the reflecting particles.
Namely, we should consider a massive fermion system.
The boundary condition should not be put by hand
but should be automatically chosen by nature.
The edge-localized modes should play a key role.
Is there any candidate?
Yes, we have the domain-wall fermion \cite{Jackiw:1975fn,Callan:1984sa,Kaplan:1992bt}.
Let us consider a four-dimensional closed manifold,
which is extended from the original boundary,
and a massive Dirac fermion operator on it,
\begin{align}
D+\varepsilon M,
\end{align}
where the sign function takes $\varepsilon=-1$ in our target domain (say $x_4>0$),
and $\varepsilon=+1$, otherwise\footnote{
In \cite{Kanno:2021bze}, more general position-dependent mass is discussed.
}.
Here we do not assume any boundary condition
on the domain-wall, expecting it dynamically given.
Unlike the standard domain-wall fermion employed
in lattice QCD simulations,
our domain-wall fermion lives in
four dimensions and the surface modes
are localized in the three dimensional wall.
Our proposal for the new expression of the index is a natural extension
of the Atiyah-Singer index in the previous section.
We find that the $\eta$ invariant
of the domain-wall Dirac operator with kink structure,
coincides with the APS index,
\begin{align}
\label{eq:APSDW}
I_{\rm APS}(D) = \frac{1}{2}\eta(\gamma_5(D+\varepsilon M))^{reg.} = \frac{1}{2}\left[\eta(\gamma_5(D+\varepsilon M))-\eta(\gamma_5(D+M))\right].
\end{align}
This equality can be shown by Fujikawa method, which consists of three steps.
1) choosing regularization: we employ the Pauli-Villars subtraction,
2) choosing the complete set to evaluate the trace: we take
the eigenmode set of free domain-wall Dirac operator squared,
and 3) perturbative evaluation.
See \cite{Fukaya:2017tsq} for the details of the computation.
Here we give two remarks about the evaluation.
First, the eigenmodes of the free domain-wall fermion,
or the solutions to
\begin{align}
\{\gamma_5(D^{\rm free}+M\varepsilon(x_4))\}^2 \phi = \left[-\partial_\mu^2 + M^2 \textcolor{black}{-2M\gamma_4 \delta(x_4)}\right]\phi = \lambda^2 \phi
\end{align}
are given by a direct product $\phi =\varphi_{\pm,e/o}^{\omega/{\rm edge}}(x_4) \otimes e^{i\bm{p}\cdot \bm{x}}$,
where $\bm{p}$ denotes the momentum vector in the three dimensions.
The bulk wave functions in the $x_4$ direction
\begin{align}
\varphi^\omega_{\pm, o}(x_4)&=\frac{e^{i\omega x_4}-e^{-i\omega x_4}}{\sqrt{2\pi}},\;\;\;
\varphi^{\omega}_{\pm,e}(x_4)=
\frac{(i\omega\pm M)e^{i\omega |x_4|}+(i\omega\mp M)e^{-i\omega |x_4|}}{\sqrt{2\pi(\omega^2+M^2)}},
\end{align}
have the eigenvalues $\lambda^2=\bm{p}^2+\omega^2+M^2$ and
the $\pm$ eigenvalue of $\gamma_4$ indicated by the subscripts $\pm$.
Another subscript $e/o$ denotes the even/odd parity under the
reflection $x_4\to -x_4$.
In the complete set, we have the edge localized solutions with
\begin{align}
\varphi^{\rm edge}_{-, e}(x_4)&=\sqrt{M}e^{-M|x_4|},
\end{align}
with the eigenvalue $\lambda^2=\bm{p}^2$.
These modes are chiral and massless.
The second remark is that we did not give any boundary condition by hand
but the following non-trivial boundary condition
\begin{align}
\left[\partial_4 \mp M\varepsilon\right]\varphi^{\omega/{\rm edge}}_{\pm,e}(x_4)|_{x_4=0} = 0,\;\;\;
\varphi^{\omega}_{\pm,o}(x_4=0)=0,
\end{align}
is automatically satisfied due to the domain-wall.
More importantly, this boundary condition preserves
the angular momentum in the $x_4$ direction but does not keep helicity.
Nature chooses the rotational symmetry, rather than chirality.
Now we can compute the $\eta$ invariant in a simple perturbative expansion.
From the bulk mode part, we obtain the curvature term but with the sign flipping:
\begin{align}
\label{eq:bulk}
\frac{1}{2}\eta(H_{DW})^{\rm bulk}&= \frac{1}{2} \sum_{\rm bulk}(\phi^{\rm bulk})^\dagger {\rm sgn}(H_{DW})\phi^{\rm bulk}
=
\frac{1}{64\pi^2}\int d^4x \textcolor{black}{\epsilon(x_4)}\epsilon_{\mu\nu\rho\sigma}{\rm tr}_cF^{\mu\nu}F^{\rho\sigma}(x) + O(1/M),
\end{align}
where we have denoted $H_{DW}=\gamma_5(D+\varepsilon M)$.
From the edge modes, we obtain the $\eta$ invariant of the three-dimensional massless Dirac operator
on the domain-wall,
\begin{align}
\label{eq:edge}
\frac{1}{2}\eta(H_{DW})^{\rm edge} &= \frac{1}{2}\sum_{\rm edge}\phi^{\rm edge}(x)^\dagger {\rm sgn}(H_{DW})\phi^{\rm edge}(x)
= -\frac{1}{2}\eta(iD^{\rm 3D})|_{x_4=0}.
\end{align}
Together with the Pauli-Villars contribution with $H_{PV}=\gamma_5(D+M)$,
\begin{align}
-\frac{1}{2}\eta(H_{PV}) &=\frac{1}{64\pi^2}\int d^4x\; \epsilon_{\mu\nu\rho\sigma}{\rm tr}_cF^{\mu\nu}F^{\rho\sigma}(x) + O(1/M),
\end{align}
we finally obtain,
\begin{align}
\frac{1}{2}\eta(\gamma_5(D+\varepsilon M))^{reg.}=
\frac{1}{32\pi^2}\int_{x_4>0} d^4x \epsilon_{\mu\nu\rho\sigma}{\rm tr}[F^{\mu\nu}F^{\rho\sigma}] \textcolor{black}{-\frac{\eta(iD^{\rm 3D})}{2}},
\end{align}
which is equal to the APS index.
The above evaluation of the bulk and edge modes separately
makes the roles of them in the anomaly inflow manifest:
the time-reversal ($T$) symmetry breaking of the edge (\ref{eq:edge})
(the $\eta$ invariant is odd in $T$) is precisely
canceled by that of the bulk modes (\ref{eq:bulk})
when they are exponentiated in the fermion partition function:
$Z_{\rm edge}Z_{\rm bulk}\propto (-1)^{-\frac{1}{2}\eta(\gamma_5(D+\varepsilon M))^{reg.}}$.
\section{Mathematical justification}
In the previous section, we have perturbatively shown on a flat Euclidean space
that the $\eta$ invariant of the domain-wall Dirac operator equals to the APS index
of a domain with negative mass, where the APS boundary condition is imposed on the
location of the domain-wall, in spite of the fact that we did not impose any
boundary condition in the former set up.
As the two formulations are given on different manifolds,
the reader may wonder if the equivalence is just a coincidence.
Since the problem is nontrivial not only in physics but also in mathematics,
our collaboration invited three mathematicians and
the interdisciplinary collaboration went successful
in giving a general proof for the equivalence.
In \cite{Fukaya:2019qlf} we gave a proof that for any APS index of a Dirac operator
on a manifold $X_+$ with boundary $Y$, there exists
a massive domain-wall fermion Dirac operator on a closed manifold $X$
extended from $X_+$, and its $\eta$ invariant equals to the original index.
Here we just give a sketch of the proof.
We introduce an extra dimension to consider $X \times \mathbb{R}$, and
the following Dirac operator on it,
\begin{align}
D^{\rm 5D} = \left(\begin{array}{cc}
0 & \partial_5 + \gamma_5 (D^{\rm 4D} + m(x_4,x_5))\\
-\partial_5 + \gamma_5 (D^{\rm 4D} + m(x_4,x_5)) & 0
\end{array}\right),
\end{align}
where the mass term is given negative in a positive region of $x_4$ and $x_5$ and positive otherwise:
\begin{align}
m(x_4,x_5) = \left\{
\begin{array}{cc}
-M & \mbox{for}\; x_4>0\; \&\; x_5>0\\
0 & \mbox{for}\; x_4=0\; \&\; x_5=0\\
M_2 & \mbox{otherwise}\\
\end{array}\right.
\end{align}
Here the gauge fields are copied in the $x_5$ direction
and we set $A_5=0$.
Then we evaluate the index of this Dirac operator
in two different ways.
With the so-called localization technique \cite{Witten:1982im,FurutaIndex}, which focuses on
the edge-localized modes on the domain-wall,
we can show that the index is equal to the original APS index,
together with an equality between the index and
that with an half-infinite cylinder.
The index can be also computed by counting
the zero-crossings of the eigenvalues of
the Dirac operator in the 4 dimensions,
which leads to our new proposal, the
$\eta$ invariant of domain-wall Dirac operators.
The equality (\ref{eq:APSDW}) always holds since the left and right-hand sides
are just two different evaluations of the same quantity.
\section{APS index on a lattice}
Finally let us discuss the lattice version of the APS index.
In the standard lattice gauge theory on a square hyper-cubic
and periodic lattices, it is well-known that
the overlap fermion action \cite{Neuberger:1997fp}
$S = \sum_x \bar{q}(x) D_{ov}q(x)$ is invariant under
a modified chiral transformation \cite{Ginsparg:1981bj, Luscher:1998pqa}:
\begin{align}
q \to e^{i\alpha\gamma_5(1-aD_{ov})}q,\;\;\;\bar{q} \to \bar{q}e^{i\alpha\gamma_5}.
\end{align}
But the fermion measure transforms as
\begin{align}
Dq\bar{q} \to \exp\left[2i\alpha {\rm Tr}(\gamma_5+\gamma_5(1-aD_{ov}))/2\right]D q\bar{q},
\end{align}
which reproduces the axial $U(1)$ anomaly.
Moreover, the AS index can be defined as
\begin{align}
\label{eq:latticeAS}
I_{\rm AS}(D_{ov})={\rm Tr}\gamma_5\left(1-\frac{aD_{ov}}{2}\right),
\end{align}
and it can reach the continuum value even at a finite lattice spacing \cite{Hasenfratz:1993sp},
when the gauge link variables are smooth enough.
But how to formulate the APS index has not been known.
In continuum theory, the APS boundary condition is imposed by separating the
normal and tangent parts of the Dirac operator to the boundary
and requiring the positive components of the tangent Dirac operator
of fermion fields to vanish at the boundary.
On the lattice, however, such a separation is difficult
as is clear from the explicit form of the overlap Dirac operator,
\begin{align}
\label{eq:Dov}
D_{ov} = \frac{1}{a}\left(1+\gamma_5\frac{H_W}{\sqrt{H_W^2}}\right),
\end{align}
where $H_W=\gamma_5(D_W-1/a)$ is the Hermitian Wilson Dirac operator.
Even if we managed to impose a lattice version of the
APS boundary condition, the Ginsparg-Wilson relation or
the definition of the index would no longer been guaranteed.
In fact, an alternative direction is hidden in the index of the overlap Dirac operator.
If we substitute the explicit form (\ref{eq:Dov}) into the
definition of the index (\ref{eq:latticeAS}), we end up with
the $\eta$ invariant of a negatively very massive Wilson Dirac operator,
\begin{align}
I_{\rm AS}(D_{ov})= - \frac{1}{2}{\rm Tr}\frac{H_W}{\sqrt{H_W^2}} = - \frac{1}{2}\eta(\gamma_5(D_W-1/a)).
\end{align}
Interestingly, the lattice index ``knew''
1) the AS index can be written by the $\eta$ invariant of a massive Dirac operator\footnote{
This fact was known, for example, in \cite{Itoh:1987iy, Adams:1998eg}
but its mathematical importance was not discussed, as far as we know.
},
and 2) chiral symmetry is not important: the Wilson Dirac operator is enough to define it\footnote{
The issue was recently revisited by some of mathematicians \cite{Yamashita:2020nkf,Kubota:2020tpr}.
}.
The situation can be summarized in two tables.
With the conventional massless Dirac operator, we need a significant effort to formulate the index theorems.
For the APS index we need the unphysical boundary condition.
For the lattice Atiyah-Singer index, we need the overlap fermion.
The lattice APS is even not known yet.
However, the $\eta$ invariant with massive Dirac operator on a closed manifold
gives a united treatment of index theorems which is easy to handle.
The APS in continuum theory is given by a kink structure in the mass term
and the lattice AS is given by Wilson Dirac operator.
Now we can easily speculate that the lattice APS index
must be given by the eta-invariant of the Wilson Dirac operator with a domain-wall mass
$-\frac{1}{2}\eta(\gamma_5(D_W-\varepsilon M))$.
\begin{table}[hbt]
\caption{The standard formulation of the index with massless Dirac operator}
\label{tab:massless}
\centering
\begin{tabular}{|c|c|c|}
\hline
& continuum & lattice\\\hline
AS & ${\rm Tr}\gamma_5e^{D^2/M^2}$ & ${\rm Tr}\gamma_5(1-aD_{ov}/2)$\\\hline
APS & ${\rm Tr}\gamma_5e^{D^2/M^2}$ w/ APS b.c. & not known.\\\hline
\end{tabular}
\vspace{1cm}
\caption{The $\eta$ invariant of massive Dirac operator}
\centering
\begin{tabular}{|c|c|c|}
\hline
& continuum & lattice\\\hline
AS & $-\frac{1}{2}\eta(\gamma_5(D- M))^{reg.}$ & $-\frac{1}{2}\eta(\gamma_5(D_W-M))$\\\hline
APS & $-\frac{1}{2}\eta(\gamma_5(D-\varepsilon M))^{reg.}$ & $-\frac{1}{2}\eta(\gamma_5(D_W-\varepsilon M))$ \\\hline
\end{tabular}
\label{tab:massive}
\end{table}
In \cite{Fukaya:2019myi}, we have perturbatively shown on a four-dimensional Euclidean flat lattice that
$-\frac{1}{2}\eta(\gamma_5(D_W-\varepsilon M))$ in the classical continuum limit is
\begin{align}
\label{eq:latAPS}
-\frac{1}{2}\eta(\gamma_5(D_W-\varepsilon M))=& \displaystyle{\frac{1}{32\pi^2} \int_{0<x_4<L_4} d^4x \epsilon^{\mu\nu\rho\sigma} {\rm tr} F_{\mu\nu}F_{\rho\sigma}}
-\frac{1}{2}\eta(iD^{3\mathrm{D}})|_{x_4=0}+\frac{1}{2}\eta(iD^{3\mathrm{D}})|_{x_4=L_4},
\end{align}
which coincides with the APS index on $T^3\times I$ with $I=[0,L_4]$.
Here we have put two domain-walls at $x_4=a/2$ and $x_4=L_4-a/2$ and set $M=1/a$.
Since the left-hand side of (\ref{eq:latAPS}) is always an integer,
we expect that this definition is non-perturbatively valid.
\section{Summary}
In this work, we have shown that the massive domain-wall fermion
is physicist-friendly: the APS index can be formulated (even on a lattice)
without any unphysical boundary conditions.
Moreover, it is mathematically rich:
the $\eta$ invariant of the massive Dirac operator
on a closed manifold unifies the index theorems.
The author thanks M.~Furuta, N.~Kawai, S.~Matsuo, Y.~Matsuki, M.~Mori, K.~Nakayama,
T.~Onogi, S.~Yamaguchi and M.~Yamashita for the fascinating collaborations.
This work was supported in part by JSPS KAKENHI Grant Number
JP18H01216 and JP18H04484.
| 2024-02-18T23:40:11.458Z | 2021-12-22T02:09:09.000Z | algebraic_stack_train_0000 | 1,586 | 3,714 |
|
proofpile-arXiv_065-8036 | \section{\@startsection{section}{1}{\z@}%
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\title{Boundary Deformation Theory \\[2mm] and \\[2mm]
Moduli Spaces of D-Branes\\[2mm] \phantom{b}}
\author{Andreas Recknagel \rnote{$\;$1$*$} \ \ \ and \ \
Volker Schomerus \rnote{$\;$2}
\\[7mm]
\note{1} Max-Planck-Institut f\"ur Mathematik in den Naturwissenschaften
\\ Inselstra\ss e 22-26, D--04103 Leipzig, Germany
\\[3mm]
\note{2} II. Institut f\"ur Theoretische Physik, Universit\"at Hamburg,
\\ Luruper Chaussee 149, D--22761 Hamburg, Germany}
\date{}
\begin{document}
\begin{titlepage} \maketitle \thispagestyle{empty}
\begin{abstract}
\noindent
Boundary conformal field theory is the suitable framework
for a microscopic treatment of D-branes in arbitrary CFT backgrounds.
In this work, we develop boundary deformation theory in order to
study the changes of boundary conditions generated by marginal
boundary fields. The deformation
parameters may be regarded as continuous moduli of D-branes.
We identify a large class of boundary fields which are shown
to be truly marginal, and we derive closed formulas describing
the associated deformations to all orders in perturbation theory.
This allows us to study the global topology properties of the
moduli space rather than local aspects only.
As an example, we analyse in detail the moduli space of $c=1$
theories, which displays various stringy phenomena.
\end{abstract}
\vspace*{-18cm}
{\tt {DESY 98-185 \hfill MIS-preprint\ No.\ 60}}\\
{\tt {hep-th/9811237 \hfill}}
\vfill
\noindent\phantom{wwwx}{\small e-mail addresses: }
{\small\tt [email protected], [email protected]}
\smallskip
\noindent\phantom{wwwx}{\small ${}^*\;$Address after March 1, 1999:
Max-Planck-Institut f\"ur Gravitationsphysik,}
\newline
\noindent\phantom{wwwx{\small ${}^*\;$}
{\small Albert-Einstein-Institut, Schlaatzweg 1, D--14473 Potsdam, Germany}
\end{titlepage}
\thispagestyle{empty}
\setcounter{page}{0}
\phantom{b}
\newpage
\section{Introduction}
Since Polchinski's discovery that D-branes \cite{DLP} provide
a string realization of supergravity solitonic $p$-branes
in \cite{Pol1,Pol2,Pol3}, non-perturbative
effects have become accessible within string theory. This has changed
the perspective of both string theory and gauge theories drastically.
In particular, a net of dualities has emerged relating different field or
string theories in the unified picture of $M$-theory \cite{Wit1}; see e.g.\
\cite{Ler,GiKu,Bac1,Sen5} for reviews and further references.
More recently, this has led to conjectures of rather direct equivalences
between string and supergravity theories on one side and
gauge theories on the other \cite{Mal2,GKP,Wit3}.
D-branes are the most important new objects in this development. They have
mainly been investigated from a target geometry and classical field
theory point of view, where they appear as ``defects'' of various dimensions
to which closed strings can couple and which support gauge theories.
In the flat background case, there exists a well-known alternative
world-sheet approach using the boundary state formalism \cite{CLNY,PoCa}; it
provides an effective handle on explicit string calculations but also allows
to reproduce the classical behaviour of D-branes in the low-energy limit,
see e.g.\ \cite{Bac,Li,CaKl,GrGu2,BG1,BCV,VFPSLR}. This formulation was
extended somewhat beyond the flat case e.g.\ in \cite{HINS2}, see also
\cite{OOY,Stan}, but to give a fully general formulation of D-branes
in arbitrary CFT backgrounds \cite{ReSc,FuSc2} with no a priori classical
counterpart requires more refined techniques. Those are provided by
conformal field theory on surfaces with boundaries as developed mainly
by Cardy \cite{Car1,Car2,Car3,CaLe} and first introduced into string
theory by Sagnotti \cite{Sag1,BiSa1,Sag2}.
Techniques from conformal field theory are particularly well
developed for rational models in which the state space
decomposes into a finite number of sectors of some chiral
symmetry algebra. This general remark applies to boundary
theories in particular and means that boundary conditions
with a large symmetry are the easiest to construct.
In fact, for a certain class of rational models,
Cardy managed to write down universal solutions \cite{Car3}.
A variant of Cardy's ideas was used in \cite{ReSc} to obtain
boundary conditions that describe D-branes in Gepner models.
The set of such rational boundary theories is typically
discrete.
Continuous moduli, therefore, are an important feature of strings
and branes that is rather difficult to handle with the algebraic
techniques of CFT. Here, geometry and gauge theory undoubtedly
are more efficient in producing quick results. Still, there are
reasons to try and investigate moduli spaces within the
CFT approach: First of all, it is one of the fundamental ideas
of string theory to treat space-time as a derived concept, not as
part of the input data. Moreover, when starting a discussion of
string or brane moduli spaces from geometrical notions, one runs the
risk of missing some of the non-classical features of the moduli space
and of the dynamics of massless fields. Finally, the efficiency of
geometric approaches to moduli very much depends on the background and on
space-time supersymmetry; CFT methods, on the other hand, not only are
background independent but also more robust when
the amount of supersymmetry is reduced.
Within the CFT setting, moduli are the parameters of deformations
generated by marginal operators -- more specifically, of marginal
{\em boundary} perturbations if one is interested in D-brane
moduli. Up to now, there does not seem to exist
a systematic treatment of marginal deformations of boundary CFTs
in the literature. There are, however, interesting case studies partly
motivated by open string theory \cite{BiSa2,BPS,GrGu1}, partly by
dissipative quantum mechanics \cite{CFF,CaKle,CKLM,CKMY,PoTh0,PoTh}.
The present paper aims at closing this gap and at presenting a
general treatment of marginal perturbations of conformal boundary
conditions.
A careful analysis of the properties of marginal operators
reveals that there is a large class of deformations which
can be treated to all orders in perturbation theory.
For deformations of CFTs on the plane, this is possible
only for very few cases so that, usually, only local properties
of the closed string moduli space are accessible from CFT. In contrast,
the closed formulas we obtain for marginal boundary deformations
allow us to recover global topological aspects of the D-brane moduli
space from CFT.
{}From the $\s$-model interpretation one expects that continuous
brane moduli should reveal some information about the underlying
{\em target} space itself, the simplest geometric moduli being the position
coordinates of D-branes in the target. And indeed, we shall see
target geometry -- ``blurred'' and enriched by stringy effects --
emerging from our CFT analysis even though our
starting point is purely algebraic with no initial reference to a
classical $\sigma$-model description.
The simplest class of deformations we consider are the so-called
{\em chiral deformations}. Roughly speaking, branes obtained from
each other by chiral deformations are related through continuous
symmetries of the target space. {\em Non-chiral} deformations,
however, are capable of moving branes between inequivalent positions
not related by any continuous symmetry. In particular,
they can push the brane into some singularity of the underlying
target space (e.g.\ a fixed point of some orbifold group). The
geometric singularity becomes manifest within the CFT description
through a breakdown of certain sewing relations
resp.\ the {\em cluster property} to be discussed below.
In addition, we shall encounter some non-chiral deformations
without an immediate target interpretation.
\bigskip
The paper is organized as follows:
Section 2 introduces some tools from boundary conformal field
theory needed throughout the text. It is also designed so as
to make the presentation self-contained.
In the end, we will explain the cluster property mentioned above
and introduce the
notion of a ``self-local'' boundary field that will become
a crucial ingredient in our discussion of D-brane moduli spaces.
In Section 3, we will give a detailed general discussion
of marginal boundary deformations. We will show that whenever
a marginal boundary operator is self-local it is truly
marginal to all orders in the perturbation parameter (Subsection 3.2).
Moreover, we present formulas which allow to compute structure constants
of the deformed theory to all orders in perturbation theory.
For reasons to become clear later, deformations generated by
self-local boundary fields will also be called ``analytic''.
Currents from the chiral symmetry algebra are special cases
of self-local marginal fields; they generate group manifold pieces within
the moduli space, and the corresponding deformed models can be described
through simple closed formulas (Subsections 3.3-4).
Subsection 3.5 contains further observations on the effect of (non-chiral)
analytic deformations on Ward identities and spectrum of boundary
excitations. In particular, we shall see which symmetries remain
unbroken and which part of the brane partition function
is independent of the strength of the perturbation. This
explains and generalizes observations made for deformations
of free bosonic boundary theories in \cite{CKLM}.
Section 4 contains a more or less complete analysis of truly marginal
boundary deformations of $c=1$ theories, which provide explicit examples
for all elements of our general construction. During the discussion,
which subsumes the material of \cite{CKLM,PoTh} and leads to new
results on orbifold models, we shall see that quantum field theoretical
``subtleties'' like the cluster property are crucial in determining
the topology of the moduli space of boundary conditions.
A summarizing description of this $c=1$ brane moduli space is given in
Section 5, with emphasis put on its non-classical features. Some of
these are familiar effects from stringy geometry, while the interpretation
of others remains to be found. We
conclude the paper with a brief outlook on possible extensions and on
applications of our framework to the investigation of D-brane moduli
spaces in arbitrary backgrounds.
\medskip
We hope that our methods will also be useful for condensed
matter problems, which represent the second important field of
application of boundary CFT. We have already mentioned investigations
of boundary perturbations in connection with dissipative quantum
mechanics. The influence of dissipation on a particle in an infinite
periodic potential is described by the boundary sine-Gordon model
which at the same time appears to be closely related to the Kondo problem
\cite{FLS}. The latter deals with the marginally relevant perturbation
induced by an impurity spin in a magnetic alloy, see e.g.
\cite{AfLu1,Aff,Lud} and references therein.
\section{Boundary conditions in conformal field theory}
In this section we present a brief survey of boundary conformal
field theory and fix the notations
used throughout the paper. It is explained in some detail
how boundary theories are parameterized by the choice of {\em
gluing maps} $\Omega$ and the {\em structure constants} $A^\a_\varphi$
appearing in the 1-point functions for bulk fields of the theory.
The last subsection is devoted to boundary fields. In particular,
we introduce a notion of {\em locality} that will become crucial for
the deformation theory to be developed below.
\subsection{The bulk conformal field theory.}
All constructions of boundary conformal field theories start
from a usual conformal field theory on the complex plane,
which we shall refer to as {\em bulk theory}. It consists
of a space $\cH^{(P)}$ of states equipped with the action of a
Hamiltonian $H^{(P)}$ and of field operators $\varphi(z,\bar z)$,
which can be assigned uniquely to elements in the state space $\cH^{(P)}$
via the state-field correspondence, i.e.\
\begin{equation}} \def\ee{ \end{equation} \varphi(z,\bar z) \ = \ \Phi^{(P)}(|\varphi\rangle ;z,\bar z) \ \ \
\mbox{ for all } \ \ \ |\varphi\rangle \in \cH^{(P)}\ \ .
\label{stfldcor}
\ee
The reverse relation is given by $\varphi(0,0) | 0\rangle
= |\varphi \rangle$ where $|0\rangle$ denotes the vacuum
state in $\cH^{(P)}$.
\newline
The CFT is completely determined once we know all possible 3-point
functions, or, equivalently, the coefficients of the operator
product expansions (OPEs) for all fields in the theory.
This task is often tractable since fields and states
can be organized into irreducible representations of the
observable algebra generated by the energy-momentum tensor and other
chiral fields \cite{BPZ}.
\medskip
Chiral fields depend on only one of the coordinates $z$ or $\bar z$
so that they are either holomorphic, $W = W(z)$, or anti-holomorphic,
$\bW = \bW(\bar z)$. The (anti-)\-holomorphic fields of a given
bulk theory, or their Laurent modes $W_n$ and $\bW_n$ defined through
\begin{equation}} \def\ee{ \end{equation} W(z) \ = \ \sum \, W_n \ z^{-n-h} \ \ \ , \ \ \
\bW(\bar z) \ = \ \sum \, \bW_n \ \bar z^{-n-\bar h} \ \ ,
\label{modexp}
\ee
generate two commuting {\em chiral algebras}, $\cW$ and ${\overline {\cal W}}} \def\bJ{{\overline J}$.
The Virasoro fields $T$ and $\overline T$ with modes $L_n$ and
$\overline{L}_n$ are among the chiral fields of a CFT and, above,
$h$ and $\bar h$ are
the (half-) integer conformal weights of $W$ and $\bW$ wrt.\ $L_0$ and
$\overline{L}_0$. {}From now on we shall assume the two chiral
algebras $\cW$ and ${\overline {\cal W}}} \def\bJ{{\overline J}$ to be isomorphic.
The state space of a CFT on the plane admits a
decomposition $\cH^{(P)} = \bigoplus_{i,j} {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i \otimes} \def\bz{\bar z {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^j$
into irreducible representations of the two commuting chiral algebras.
${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^0$ refers to the vacuum representation --
which is mapped to $\cW$ via the state-field correspondence $\Phi^{(P)}$.
\newline
The irreducible representations ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i$ of $\cW$
acquire a (half-)integer grading under the action of $L_0$
so that they may be decomposed as ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i = \bigoplus_{n\geq 0} V^i_n$.
We assume that the $V^i_n$ are finite-diemensional.
Let $V^i_0 \subset {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i$
be the eigenspace of $L_0$ with lowest eigenvalue. It carries an
irreducible action of all the zero modes $W_0$. We will denote
the corresponding linear maps by $X^i_W$,
\begin{equation}} \def\ee{ \end{equation} X^i_W \ := \ W_0\ |_{V^i_0} : V^i_0 \ \longrightarrow \
V^i_0 \ \ \ \mbox{ for all chiral fields $W$}
\ \ .
\label{Xdef}
\ee
The whole irreducible representation ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i$ may be recovered
from the elements of the finite-dimen\-sional subspace $V^i_0$
by acting with $W_n,\ n < 0$.
Using the state-field correspondence $\Phi^{(P)}$, we can assign
fields to all states in $V^i_0 \otimes} \def\bz{\bar z V^j_0$. We shall assemble them
into a single object which one can regard as a matrix of fields
after choosing some basis in the subspaces $V^i_0$ and $V^j_0$,
\begin{equation}} \def\ee{ \end{equation} \varphi_{ij}(z,\bar z) \ : = \ \Phi^{(P)}(V^i_0 \otimes} \def\bz{\bar z V^j_0; z,
\bar z)\,:\ V^j_0 \otimes} \def\bz{\bar z \cH^{(P)} \ \longrightarrow \ V^i_0 \otimes} \def\bz{\bar z
\cH^{(P)} \ \ .
\label{phiij}
\ee
In case the ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i$ are $W$-algebra highest weight representations,
$\varphi_{ij}(z,\bar z)$ are simply all the (Virasoro primary) fields
which arise from a $\cW$-primary through the action of $W$-algebra
zero modes.
\subsection{Boundary theories and the gluing map.}
With some basic notations for the (``parent'') bulk theory set up, we can
begin our analysis of {\em associated} boundary theories (``descendants'').
These are conformal field theories on the upper half-plane ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z \geq 0$
which, in the interior ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z>0$, are locally equivalent to the
given bulk theory: The state space $\cH^{(H)}$ of
the boundary CFT is equipped with the action of a Hamiltonian
$H^{(H)}$ and of bulk fields $\varphi(z,\bar z)$
-- still well-defined for ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z >0$ -- assigned to the
{\em same} elements $\varphi$ that were used to label fields in the
bulk theory. Accordingly, we demand that all the OPEs of bulk fields
coincide with the OPEs of the bulk theory.
\newpag
Note that, in general, the boundary theory contains a lot more bulk
fields than it has states. We will see shortly which fields are
in one-to-one correspondence to the states in $\cH^{(H)}$.
\medskip
Considering all possible conformal boundary theories associated to a bulk
theory whose chiral algebra is a true extension of the Virasoro
algebra is, at present, too difficult a problem to be addressed
seriously. For the moment, we restrict our considerations
to that class of boundary conditions which leave the whole symmetry
algebra $\cW$ unbroken. More precisely, we assume that all chiral
fields $W(z), \bW(\bz)$ can be extended analytically to the
real line and that there exists a local automorphism $\Omega$
-- called the {\em gluing map} -- of the chiral algebra $\cW$
such that \cite{ReSc}
\begin{equation}} \def\ee{ \end{equation} T(z) \ = \ {\overline T}} \def\bW{{\overline W} (\bar z) \ \ \ \mbox{ and }\ \ \
W(z) \ = \ \Omega(\bW) (\bar z) \ \ \mbox{ for } \ \ z = \bar z\ \ .
\label{gluecond}
\ee
The first condition simply forbids an energy flow across the
boundary; it is included in the second equation if we require
$\Omega$ to act trivially on the Virasoro field.
Note also that $\Omega$ induces an automorphism $\omega$
of the fusion rule algebra.
\medskip
Our assumption on the existence of the gluing map $\Omega$ has
the powerful consequence that it gives rise to an action of
one chiral algebra $\cW$ on the state space $\cH\equiv \cH^{(H)}$
of the boundary theory. To see this, we combine the chiral
fields $W(z)$ and $\Omega \bW(\bar z)$ into a single object
$\sW (z)$ defined on the whole complex plane such that
$$ \sW(z) \ :=\ \left\{ \begin{array}{ll}
W(z) \ \ &\mbox{ for } \ \ {\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z \geq 0 \\[2mm]
\Omega \bW(\bar z) \ \ &\mbox{ for } \ \ {\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z < 0
\end{array} \right. \ \ .
$$
Because of the gluing condition along the boundary, this field
is analytic and we can expand it in a Laurent series $\sW(z) =
\sum_n \, W^{(H)}_n z^{-n-h}$, thereby introducing the modes
$W_n\equiv W_n^{(H)}$.
These operators on the state space $\cH$ are easily seen to
obey the defining relations of the chiral algebra $\cW$.
Note that there is just one such action of $\cW$ constructed
out of the two chiral fields $W(z)$ and $\Omega \bW(\bar z)$.
\medskip
In the usual way, the representation of $\cW$ on $\cH$ leads
to Ward identities for correlation functions of the boundary
theory. They follow directly from the singular parts of the
operator product expansions of the field $\sW$ with the bulk
fields $\varphi(z,\bar z)$ which are fixed by our
requirement of local equivalence between the bulk theory and
the bulk of the boundary theory. To make this more precise,
we introduce the notation $\sW_>(z) = \sum_{n > -h} W_n
z^{-n-h}$ for the singular part of the field $\sW$. The
singular part of the OPE is then given by
\newpage
\begin{eqnarray}} \def\ea{ \end{eqnarray}
{\lefteqn{\sW(w) \, \varphi(z,\bar z) \sim
[\, \sW_>(w)\, ,\, \Phi(\varphi;z,\bar z)\, ]}} \label{WOPE}\\[2mm]
& = & \sum_{n > -h} \left( \frac{1}{(w-z)^{n+h}} \, \Phi\bigl(W^{(P)}_n
\varphi; z, \bar z\bigr) + \frac{1}{(w-\bar z)^{n+h}} \,
\Phi\bigl(\Omega\bW{}^{(P)}_n \varphi; z,\bar z\bigr)\right) \ \ . \nonumber} \def\ssstyle{\scriptscriptstyle
\ea
Here, the symbol $\sim$ means that the right hand side gives only
the singular part of the operator product expansion, and we have
placed a superscript $(P)$ on the modes $W_n, \bW_n$ to
display clearly that they act on the elements $\varphi \in
\cH^{(P)}$ labeling the bulk fields in the theory (superscripts
$(H)$, on the other hand, will be dropped).
The sum on the right hand side of eq.\ (\ref{WOPE}) is always
finite because $\varphi$ is annihilated by all modes
$W{}^{(P)}_m, \bW{}^{(P)}_m$ with sufficiently large $m$. For ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} w >0$
only the first terms involving $W{}^{(P)}_n$ can become singular
and the singularities agree with the singular part of the
OPE between $W(w)$ and $\varphi(z,\bar z)$ in the bulk theory.
Similarly, the singular part of the OPE between $\Omega \bW(w)$ and
$\varphi(z,\bar z)$ in the bulk theory is reproduced by the
terms which contain $\bW{}^{(P)}_n$, if ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} w < 0$.
\smallskip
As it stands, the previous formula is rather abstract. So,
let us spell out at least one more concrete example in which
the chiral field $\sW$ has dimension $h=1$ (we shall denote
such chiral currents by the letter $J$ from now on) and in
which the field $\varphi$ is replaced by the matrix $\varphi_{ij}$
of fields that were assigned to states $\varphi \in V^i_0 \otimes} \def\bz{\bar z V^j_0$
through eq.\ (\ref{phiij}). Since the latter are annihilated by all
the modes $J_n, \bJ_n$ with $n > 0$, equation (\ref{WOPE})
reduces to
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\sJ(w) \, \varphi_{ij}(z,\bar z) \ \sim \
\frac{ X_J^i }{w-z} \, \varphi_{ij}(z,\bar z) \ - \
\varphi_{ij}(z,\bar z) \, \frac{X_{\Omega \bJ}^j}{w-\bar z} \ \ .
\label{JOPE}
\ea
The linear maps $X_W^i$ and $X_{\Omega \bJ}^j$ were introduced
in eq.\ (\ref{Xdef}) above; they act on $\varphi_{ij}: V^j_0
\otimes} \def\bz{\bar z \cH \rightarrow V^i_0 \otimes} \def\bz{\bar z \cH$ by contraction in the first
tensor component $V^i_0$ resp.\ $V^j_0$.
\smallskip
Ward identities for arbitrary $n$-point functions of fields
$\varphi_{ij}$ follow directly from eq.\ (\ref{WOPE}). They have
the same form as those for chiral conformal blocks in a bulk CFT
with $2n$ insertions of chiral vertex operators with charges
$i_1,\dots,i_n,\omega(j_1),..,\omega(j_n)$, see e.g.\
\cite{Car1,Car2,ReSc,FuSc2}. In many concrete examples,
one has rather explicit expressions for such chiral blocks.
So we see that objects familiar from the construction of
bulk CFT can be used as building blocks of correlators in
the boundary theory (``doubling trick''). Note, however, that
the Ward identities depend on the gluing map $\Omega$.
\subsection{One-point functions.}
Using the Ward identities described in the previous subsection
together with the OPE in the bulk, we can
reduce the computation of correlators involving $n$ bulk fields
to the evaluation of $1$-point functions $\langle \varphi_{ij}
\rangle_\a$. They need not vanish in a boundary CFT because translation
invariance along the imaginary axis is broken, and they may depend on the
possible boundary conditions $\a$ along the real line.
To control the remaining freedom, we notice that the transformation
properties of $\varphi_{ij}$ with respect to $L_n,\ n= 0, \pm 1,$
and the zero modes $W_0$,
\begin{eqnarray}} \def\ea{ \end{eqnarray} [\, W_0\, , \, \varphi_{ij}(z,\bar z)\, ] & = &
X_W^i\ \varphi_{ij}(z,\bar z) \ - \ \varphi_{ij}(z,\bar z)
\ X_{\Omega \bW}^j \ \ , \ \ \nonumber} \def\ssstyle{\scriptscriptstyle \\[3mm]
[\, L_n\, , \, \varphi_{ij}(z,\bar z)\, ] & = &
z^n\, ( \, z \partial + h_i(n+1)\, ) \varphi_{ij}(z,\bar z)
\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm] & & \hspace*{5mm} +
\bar z^n\, ( \, \bar z {\overline \partial} + \bar h_i(n+1)\, )
\varphi_{ij} (z,\bar z) \ \ \nonumber} \def\ssstyle{\scriptscriptstyle
\ea
determine the 1-point functions up to scalar factors. Indeed,
an elementary computation reveals that $\langle \varphi_{ij}
\rangle_\a$ must be of the form
\begin{equation}} \def\ee{ \end{equation}
\langle \varphi_{ij} (z,\bar z) \rangle_\alpha =
\frac{ A^\alpha_{ij}}{(z - \bar z)^{h_i + h_j}}\ \
\label{1ptfct}
\ee
where
$$
A^\a_{ij}: V^j_0 \rightarrow V^i_0 \quad\quad \mbox{ obeys } \quad
\ \ \ X_W^i \ A^\a_{ij} \ = \ A^\a_{ij} \ X_{\Omega \bW}^j \ \ .
$$
The intertwining relation in the second line implies $j = \omega^{-1}(i^+)$
as a necessary condition for a non-vanishing 1-point function ($i^+$ denotes
the representation conjugate to $i$), and
thus we can put $h_i + h_j= 2 h_i $ in the exponent in eq.\ (\ref{1ptfct})
because the gluing map acts trivially on the Virasoro field.
Irreducibility of the zero mode representations on the subspaces $V^i_0$
and Schur's lemma imply that each matrix
$A^\a_{ij}$ is determined up to one
scalar factor. Hence, if there exist several boundary conditions
associated with the same bulk theory and the same gluing map
$\Omega$, they can differ only by these scalar parameters in
the 1-point functions. Once we know their values, we have
specified the boundary theory. In particular, one can express
the partition function $Z_{(\Omega, \a)}(q)$ of the
theory in terms of the coefficients $A^\a_{ij}$ (see eqs.\
(\ref{partfct},\ref{AeqBB}) below for precise formulas).
\begin{figure}
{\epsfbox{nto1.ps}} \vspace*{4mm}
\caption{ \small With the help of operator product expansions in
the bulk, the computation of $n$-point functions in a
boundary theory can be reduced to computing 1-point
functions on the half-plane. Consequently, the latter
must contain all information about the boundary
condition.}
\end{figure}
\medskip
The parameters in the 1-point functions are not completely
free. In fact, there exist strong {\em sewing constraints}
on them that have been worked out by several authors
\cite{CaLe,Lew,PSS1,PSS,BPPZ}. The basic relation can be derived from the
following {\em cluster property} of correlation functions:
\begin{eqnarray}} \def\ea{ \end{eqnarray} \lefteqn{\hspace*{-10mm} \lim_{a \ra \infty} \ \langle
\varphi_1(z_1,\bz_1) \dots
\varphi_{P-1}(z_{P-1},\bz_{P-1}) \varphi_{P} (z_{P}+a,
\bz_P+a) \dots \varphi_N(z_N+a,\bz_N+a)\rangle} \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \langle \varphi_1(z_1,\bz_1) \dots
\varphi_{P-1}(z_{P-1},\bz_{P-1})\rangle \ \langle
\varphi_{P} (z_{P},\bz_P) \dots \varphi_N(z_N,\bz_N)\rangle
\ \ . \label{cluster}\ea
Here, $a$ is a real parameter, and the fields
$\varphi_\nu =\varphi_{i_\nu,\bar\imath_\nu}$ on the
right hand side can be placed at $(z_\nu,\bz_\nu)$ since the
whole theory is invariant under translations parallel to the
boundary. If the cluster property is combined with the Ward
identities to evaluate 2-point functions of bulk fields,
one obtains a constraint of the form
\begin{equation}} \def\ee{ \end{equation} A^\a_i \, A^\a_j \ = \ \sum_k \ \Xi^{k}_{ij} \
A^\alpha_0 \ A^\a_k
\quad \ \ \mbox{ with } \ \ \ A^\a_l \ =
\ A^\a_{l \omega(\bar l)} \ \ . \label{class} \ee
It holds whenever the vacuum representation ``0'' occurs in the
fusion product of $i$ with $\omega(\bar\imath)$ and of $j$ with
$\omega(\bar\jmath)$.
The coefficient $\Xi^k_{ij}$ can be expressed as a combination
of the coefficients in the bulk OPE and of the fusing matrix.
In some cases, this combination has been shown to agree
with the fusion multiplicities or some generalizations thereof
(see e.g.\ \cite{PSS,FuSc1,FuSc3,BPPZ}). The importance of eq.\
(\ref{class}) for a classification of boundary conformal field
theories has been stressed in a number of publications recently
\cite{FuSc1,BPPZ,FuSh4} and is further supported by their
close relationship with algebraic structures that entered
the classification of bulk conformal field theories already
some time ago (see e.g.\ \cite{Pas,ZuPe1,ZuPe2}).
Let us remark that, from the string theory point of view,
the 1-point functions give the couplings of closed string modes
to a D-brane, i.e.\ generalized tensions and RR-charges.
Eq.\ (\ref{class}) provides an example of non-linear constraints
imposed on these couplings.
\smallskip
In our discussion of boundary perturbations, we shall always
depart from a set of correlation functions satisfying relation
(\ref{cluster}). Anticipating a more detailed discussion
below, we stress that boundary perturbations do not preserve
this property in general. One can often interpret the breakdown of
rel.\ (\ref{cluster}) as a signal for the theory to develop a
mixture of different ``pure'' (i.e.\ clustering) boundary
conditions. Such phenomena are certainly expected to occur
upon boundary perturbation and we will present some concrete
examples later on.
\subsection{Boundary fields.}
The action of $\cW$ on the state space of the boundary theory
induces a decomposition $\cH= \bigoplus_i {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^i$ (possibly
with multiplicities) into irreducibles of $\cW$. It also implies
that the partition function may be expressed as a sum of
characters $\chi_i(q)$ of the chiral algebra,
$$ Z_{(\Omega,\a)}(q) \ := \ \tr_\cH (q^{L_0-c/24}) \ = \
\sum_i n^{\Omega\a}_i \,
\chi_i(q) \ \ \mbox{ where } \ \ n^{\Omega\a}_i \ \in \
\mathbb{N}\ \ .
$$
There exists a one-to-one state-field correspondence
$\Phi\equiv\Phi^{(H)}$ between states $\psi \in \cH$
and so-called boundary fields $\psi(x)$ which are
defined (at least) for $x$ on the real line \cite{ReSc}. The conformal
dimension of a boundary field $\psi(x)$ can be read off from
the $L_0$-eigenvalue of the corresponding state $\psi \in \cH$.
The boundary fields assigned to elements in the vacuum sector ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^0$
coincide with the chiral fields in the theory, i.e.\
$\sW(x) = \Phi(w;x)$ for some $w \in {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{0}$ and ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} x = 0$.
These fields can always be extended beyond the real line
and coincide with either $W$ or $\Omega \bW$ in the bulk.
If other boundary fields admit such an extension, this
suggests an enlargement of the chiral algebra in the bulk
theory.
\medskip
Following the standard reasoning in CFT, it is easy to conclude that
the bulk fields $\varphi_{ij}(z,\bar z)$ give singular contributions
to the correlation functions whenever $z$ approaches the real line.
This can be seen from the fact that the Ward identities describe a
mirror pair of chiral charges $i$ and $\omega(j)$ placed on both sides
of the boundary.
Therefore, the singularities in an expansion of $\varphi(z,\bar z)
\equiv \varphi_{ij}(z,\bar z)$ around $x = {\Re} z$ are given
by primary fields which are localized at t $x$ on the
the real line, i.e.\ the boundary fields $\psi(x)$. In other words,
the observed singular behaviour of bulk fields $\varphi(z,\bar z)$ near
the boundary may be expressed in terms of a bulk-boundary OPE
\cite{CaLe}
\begin{equation}} \def\ee{ \end{equation} \varphi(z,\bar z) \ = \ \sum_k \ (2 y)^{h_k - h - \bar h}
\ C^{\,\alpha}_{\varphi\; k}\ \psi_k(x) \ \ .
\label{bbOPE}
\ee
Here, $\psi_k(x)$ are primary fields of conformal weight $h_k$,
and $z= x+iy$. Which $\psi_k$ can possibly appear on the rhs.\
of (\ref{bbOPE}) is determined by the chiral fusion of $i$ and $\omega(j)$,
but some of the coefficients $C^{\,\alpha}_{\varphi\, k}$ may vanish for
some $\alpha$. One can show
that $ C^{\,\alpha}_{\varphi\; {0}} = A^\alpha_\varphi /A^\alpha_0$;
moreover, the $C^{\,\alpha}_{\varphi\; k}$ are related to the 1-point
functions by generalizations of the constraints (\ref{class}), see
e.g.\ \cite{Lew,PSS}.
In boundary conformal field theory one also considers boundary
fields which induce transitions (``jumps'') between different boundary
conditions $\a,\b$, see e.g.\ \cite{Car3}.
These ``boundary condition changing operators'' are
associated with vectors in a state space $\cH_{\a\b}$ depending
on both boundary conditions, and they cannot be obtained
from bulk fields through a bulk-boundary OPE. Even though
we shall only consider homogeneous perturbations of boundary
conditions which are constant all along the boundary, we
will meet some boundary condition changing operators eventually:
At certain values of the deformation parameter $\lambda$, it
may happen that a perturbed theory describes a mixture (or
superposition) of different clustering boundary conditions.
In such cases, no jump is visible along the real axis,
but there exist boundary fields which induce transitions
between the various ``pure'' boundary conditions.
\medskip
Having introduced the boundary fields $\psi(x)$, it is natural
to extend the set of correlation functions and to consider
correlators in which a number of boundary fields
$\psi_\nu(x_\nu) = \Phi(\psi_\nu,x_\nu)$ are inserted along with
bulk fields:
\begin{equation}} \def\ee{ \end{equation}
\langle \psi_1(x_1) \dots \psi_M(x_M) \, \varphi_1(z_1,\bz_1) \dots
\varphi_N(z_N,\bz_N) \rangle_\a \ \ \ \mbox{ for } \ \ \
x_\nu < x_{\nu+1}\ \ \ .
\label{bcorr} \ee
These functions are analytic in the variables $z_\nu$ throughout the
whole upper half-plane ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z_\nu > 0$. For the variables $x_\nu$,
the domain of analyticity is restricted to the interval $x_\nu \in \;
]x_{\nu-1}, x_{\nu+1}[$ on the boundary. In most cases, there exists
no {\em unique} analytic continuation of $\psi_\nu(x_\nu)$ to other
points on the real axis which lie beyond the insertion points
of the neighbouring boundary fields. In fact, if we continue
analytically along curves like the one shown in Figure 2, the result
will typically depend on whether we move the field $\psi_\nu$
around $\psi_{\nu+1}$ in clockwise or anti-clockwise direction.
There are certainly exceptions: Chiral fields $\sW(x)$, for
example, do possess a unique analytic continuation to
$x \in \R \setminus\{ x_\nu\}$.
\begin{figure}
{\epsfbox{pic2.ps}}
\caption{ \small The curve $\gamma^\nu_{\nu+1}$ along which
correlation functions are analytically continued to exchange
the position of two neighboring boundary fields. In most cases
the result depends on the orientation of the curve.}
\end{figure}
Based on this discussion on analyticity, we want to introduce
a notion of locality that will become important later on: Two
boundary fields $\psi_1 (x_1)= \Phi(\psi_1;x_1)$ and $ \psi_2(x_1)
= \Phi(\psi_2;x_2)$ are said to be {\em mutually local} if
\begin{equation}} \def\ee{ \end{equation}
\Phi(\psi_1;x_1)\ \Phi(\psi_2;x_2) \ = \ \Phi(\psi_2;x_2)\
\Phi(\psi_1;x_1) \ \ \ \mbox{ with } \ \ x_1 < x_2\ \ .
\label{local}
\ee
The equation is supposed to hold after insertion into arbitrary
correlation functions, and for the right hand side to make sense
it is required that there exists a unique analytic continuation
from $x_1< x_2 $ to $x_1 > x_2$.
\newline
A boundary field $\psi(x)
= \Phi(\psi;x)$ will be called {\em self-local} or {\em analytic} in
the following if it is mutually {\em local with respect to itself}.
(The second expression is chosen in view of the properties
its correlation functions and of perturbations with self-local
marginal operators.)
\newline
Let us note that the OPE of two mutually
local fields contains only pole singularities. In particular,
the OPE of a self-local boundary field $\psi$ with conformal
dimension $h_\psi = 1$ is determined up to a constant $K$
to be
\begin{equation}} \def\ee{ \end{equation} \label{locmargOPE}
\psi(x_1) \, \psi(x_2) \ = \ \frac{K}{(x_1 - x_2)^2}\ +\ {\rm reg} \ \
\ \ \mbox{ if } \ \ h_\Psi \, = \, 1 \ \ .
\ee
Boundary fields $\sW(x)$ from the chiral algebra are the simplest
examples of analytic fields. They are not only local with respect to
themselves but to all other boundary fields in the theory.
It is crucial for our analysis of D-brane moduli to observe
that further (non-chiral) analytic boundary fields $\psi$ can
exist depending on the boundary condition under consideration.
Unless they belong to some extended chiral symmetry (which
means that the original chiral algebra $\cW$ was not chosen
to be the maximal chiral symmetry), these self-local boundary
fields $\psi$ will not possess a unique analytic continuation
into the full upper half-plane. In fact, ``moving'' the boundary
field $\psi$ around the insertion point of a bulk field $\varphi$
(by analytic continuation)
can lead to a non-trivial monodromy in general. Whenever this
happens, $\psi$ has no chance to be local wrt.\ all the
boundary fields that appear in the bulk-boundary OPE of the
bulk field $\varphi$. Consequently, a non-chiral analytic boundary
field $\psi$ is only expected to be local wrt.\ a subset of boundary
fields. The latter includes at least the chiral boundary fields
$\sW$ in addition to the field $\psi$ itself.
The existence of non-chiral self-local fields is signaled by a
partition function $Z_{\Omega\a}(q)$ that contains the vacuum
character of a $W$-algebra $\cW_{\Omega\a}$ extending the chiral
algebra $\cW \subset \cW_{\Omega\a}$ of the model. We shall see
several examples in Section 4.
\section{Marginal perturbations of boundary conditions}
Our aim is to study {\em perturbations} (or {\em
deformations}) of a boundary condition which are generated by
{\em marginal} boundary fields. After some general remarks, we
describe a class of perturbations -- which we call {\em
analytic deformations} -- that are truly marginal to all
orders in the perturbation expansion. They are induced by
self-local boundary fields of dimension one. Therefore,
deformations generated by chiral currents are among them
and may serve us to illustrate the more general construction
we propose. In the last subsection
we investigate arbitrary non-chiral analytic deformations and derive
some of their properties which hold to all orders
in perturbation theory.
\subsection{The general prescription.}
\def\lambda{\lambda}
Let us start from some boundary conformal field theory
with state space $\cH = \cH^{(H)}_{(\Omega,\alpha)}$, where
$(\Omega,\alpha)$ denotes the boundary condition along the
real line. Boundary operators $\psi(x) \in
\Phi(\cH)$ may be used to define a new perturbed theory whose
correlation functions are constructed from the unperturbed
ones by the formal prescription
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\lefteqn{\langle\, \varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle_{\alpha;\ \lambda\psi}
= Z^{-1} \cdot \langle\, I_{\lambda\psi} \, \varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N) \,\rangle_{\alpha} }
\label{defcorra} \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& := & Z^{-1} \sum_n \lambda^n \int \cdots \int_{x_i < x_{i+1}}
\frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi}
\ \langle\, \psi(x_1) \cdots \psi(x_n) \ \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha} \nonumber} \def\ssstyle{\scriptscriptstyle\\[2mm]
& = & Z^{-1} \sum_n \frac{\lambda^n}{n!} \sum_{\s \in S_n}
\int \cdots \int_{\raisebox{-3mm}{$ \scriptstyle
x_{\s(i)}\, <\, x_{\s(i+1)}$}}
\hspace*{-15mm} \frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi}
\langle\, \psi(x_{\s(1)}) \cdots \psi(x_{\s(n)} )
\ \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha}
\nonumber} \def\ssstyle{\scriptscriptstyle \ea
where $\lambda$ is a real parameter. The second sum in the lowest line
runs over all elements in the permutation group $S_n$.
Since all the $n!$ summands are identical, the last equality is
obvious. It shows, however, that the symbol $I_{\lambda \psi}$
in the first line should be understood as a path ordered exponential
of the perturbing operator,
\begin{equation}} \def\ee{ \end{equation} I_{\lambda\psi} \ = \ P \exp \bigl\lbrace\, \lambda
S_\psi\,\bigr\rbrace \ :=\ P \exp \bigl\lbrace\,
\lambda \int_{- \infty}^\infty \psi(x) \,
\frac {dx} {2\pi}\,\bigr\rbrace\ \ .
\label{defcorrb}\ee
The normalization $Z$ is defined as the expectation value
$Z = (A^\a_0)^{-1}\,\langle I_{\lambda \psi}\rangle_\a$. These expressions
deal with deformations of bulk correlators only. If there are extra
boundary fields present in the correlation function, the formulas
need to be modified in an obvious fashion so that these boundary
fields are included in the ``path ordering''. A particularly simple
example of this type will be discussed shortly, but we refrain
from spelling out the general formula here.
To make sense of the above expressions (beyond the formal level), it is
certainly necessary to regularize the integrals (introducing UV- and
IR-cutoffs) and to renormalize couplings and fields (see e.g.\
\cite{CarL} for a discussion of bulk perturbations in 2D conformal
field theory). IR divergences are usually cured by putting the
system into a ``finite box'', i.e., in our case, by studying perturbations
of finite temperature correlators; but this will not play
any role below. On the other hand, we have to deal with UV
divergences. So, let us introduce a UV cutoff $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$ such that
the integrations are restricted to the region $|x_i - x_j|>\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$.
Thereby, all integrals become UV-finite before we perform the
limit $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle \rightarrow 0$.
\medskip
In the following, we consider marginal boundary deformations
where the conformal dimension $h$ of the perturbing operator
$\psi(x)$ is $h=1$ so that there is a chance to stay at the
conformal point for arbitrary values of the real coupling $\lambda$
(we choose $\psi(x)$ to be anti-selfadjoint).
If $h \neq 1$, the perturbation will automatically introduce a
length scale and one has to follow the renormalization group
(RG)-flow to come back to a boundary conformal field theory. For
$h >1$, the perturbations are irrelevant so that one ends up with
the original boundary theory. For $h < 1$, the perturbation is
relevant and it is usually quite difficult to say precisely
which conformal fix-point one reaches with a given relevant
perturbation. Nevertheless, several non-trivial examples have been
studied in the literature, see \cite{FSW,War2,Chim,DoPTWa,LeSa1,LeSaSi}
and references therein, partially with the help of the thermodynamic
Bethe ansatz.
All these cases, however, share the common feature that (at the
RG fix-point) the new conformal boundary theory is associated to
the same bulk CFT -- since the local properties in the bulk are
not affected by the ``condensate'' along the boundary. Thus, boundary
perturbations can only induce changes of the boundary conditions.
\smallskip
To begin our discussion of marginal perturbations with a
boundary field $\psi(x)$, let us
investigate the change of the two-point function $\langle \psi(x_1)
\psi(x_2) \rangle_\a $ of the perturbing field $\psi$ itself
under the deformation. Obviously, the first order contribution involves
the following sum of integrals:
$$ \int_{-\infty}^{x_1-\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle} \hspace*{-5mm} dx \,\langle \psi(x)
\psi(x_1) \psi(x_2) \rangle_\a
+ \int_{x_1+\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle}^{x_2-\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle} \hspace*{-5mm} dx \,\langle \psi(x_1)
\psi(x) \psi(x_2) \rangle_\a
+ \int_{x_2-\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle}^{\infty} \hspace*{-5mm} dx \,\langle \psi(x_1)
\psi(x_2) \psi(x) \rangle_\a
. $$
{}From the general form of the three point function
(with $x_1 < x_2 < x_3$)
$$ \langle \psi(x_1) \psi(x_2) \psi(x_3)\rangle_\a =
\frac{C^\alpha_{\psi \psi \psi}}
{(x_1-x_2)(x_1-x_3)(x_2-x_3)} \ ,
$$
it is easy to see
that the first order contribution to the perturbation expansion
is logarithmically divergent unless the structure constant
$C^\a_{\psi\psi\psi}$ from the OPE of boundary fields
vanishes. The divergence would
force the conformal weight of the field $\psi$ away from the
initial value $h_\psi=1$ as we turn on the perturbation, i.e.\
the marginal field $\psi$ is not truly marginal unless
$C^\a_{\psi\psi\psi} = 0$. If there are
several marginal boundary fields in the theory, degenerate
perturbation theory gives a somewhat stronger condition: A
marginal field $\psi$ is {\em truly marginal only if $C^\a_{\psi\psi
\psi'} = 0 $} for all marginal boundary fields $\psi'$ in the
theory -- cf.\ e.g.\ \cite{DVV} for the bulk case.
Eq.\ (\ref{locmargOPE}) shows that self-local marginal boundary
operators do satisfy this first order condition.
One should stress, however, that this is merely a
necessary condition. Since it was derived within first order
perturbation theory it is by no means sufficient to guarantee
true marginality in higher orders of the perturbation series.
\subsection{Truly marginal operators.}
The first order condition is how far general investigations of marginal
{\em bulk} perturbations go.
Our main aim here is to prove that every {\em self-local marginal boundary
operator} is indeed {\em truly marginal} to all orders and therefore
generates a deformation of a boundary CFT.
To this end, let us assume that
the perturbing marginal field $\psi(x)$ is self-local in the sense discussed
at the end of the previous section. Then the above expression for the
deformed bulk correlation functions can be rewritten as
\begin{eqnarray}} \def\ea{ \end{eqnarray} \label{bcfirst}
\lefteqn{\langle\, \varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle^{\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle}_{\alpha;\ \lambda\psi}} \\[2mm]
& = & Z^{-1} \sum_n \frac{\lambda^n}{n!}
\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty
\frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi}
\langle\, \psi(x_1) \cdots \psi(x_n )
\ \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha}
\nonumber} \def\ssstyle{\scriptscriptstyle \ea
where all integrals are taken over the real line with the
regions $|x_i - x_j| < \varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$ removed as before. Based on the
OPE (\ref{locmargOPE}) of $\psi$, it is not difficult to see that
the divergences in $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$ from the numerator cancel those from the
denominator so that the limit $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle \to 0$ of the deformed bulk field
correlator can be taken. Moreover, as we are dealing with a self-local
marginal operator, this limit can be
written as
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\lefteqn{
\langle\, \varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle_{\alpha;\ \lambda\psi
= \lim_{\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle\to 0} \; \langle\, \varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle^{\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle}_{\alpha;\ \lambda\psi}}
\label{analdef} \\[2mm]
& \phantom{xxxxx} = & \sum_n \frac{\lambda^n}{n!}
\int_{\gamma_1} \cdots \int_{\gamma_n}
\frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi}
\langle\, \psi(x_1) \cdots \psi(x_n )
\ \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha}
\nonumber} \def\ssstyle{\scriptscriptstyle \ea
where $\gamma_p$ is the straight line parallel to the real
axis with ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} \gamma_p = i \varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle/p $, and it can be computed
through contour integration. The expression on the
right hand side is manifestly finite, and it is independent
of $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$ as long as $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle <$ {\rm min\/}$({\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z_i)$ where $z_i$
denote the insertion points of bulk fields. Thus, the above
formula allows us to construct the perturbed bulk correlators
to all orders in perturbation theory. In particular, it
determines the deformation of bulk 1-point functions and
hence the deformation of the structure constants
$A^\a_{\varphi}$ which parameterize the possible boundary
theories along with the gluing map.
\smallskip
The extension of these ideas to the deformation of
boundary correlators meets some obstacles. In fact,
formula (\ref{bcfirst}) admits for the obvious
generalization
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\lefteqn{\langle\, \psi_1(u_1) \cdots \psi_M(u_M)\
\varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle_{\alpha;\ \lambda\psi}}
\label{bbcfirstdef}\\[2mm]
& = & Z^{-1} \sum_n \frac{\lambda^n}{n!}
\int_{-\infty}^\infty \cdots \int_{-\infty}^\infty
\frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi} \;
\langle\, \psi(x_1) \cdots \psi(x_n ) \;\psi_1
\cdots \psi_M
\; \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha}
\nonumber} \def\ssstyle{\scriptscriptstyle
\ea
{\em if and only if} the boundary fields $\psi_1,\ldots,\psi_M$ are
{\em local} with respect to the perturbing field $\psi$. As we have
argued in the previous section, this is usually a strong
constraint on boundary fields. The integrals on the rhs.\ of
eq.\ (\ref{bbcfirstdef}) diverge as $\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle \rightarrow 0$ whenever
the iterated OPE of the perturbing field $\psi$ with one of
the boundary fields $\psi_i$ contains poles of even order. The
(renormalized) correlation functions are again obtained
through contour integration,
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\lefteqn{\langle\, \psi_1(u_1) \cdots \psi_M(u_M)\
\varphi_1(z_1,\bar z_1) \cdots
\varphi_N(z_N, \bar z_N)\,\rangle_{\alpha;\ \lambda\psi}}
\label{banaldef}\\[2mm]
& = & \sum_n \frac{\lambda^n}{n!}
\int_{\gamma_1} \cdots \int_{\gamma_n}
\frac{dx_1}{2\pi} \cdots \frac{dx_n}{2\pi} \;
\langle\, \psi(x_1) \cdots \psi(x_n ) \; \tilde \psi_1
\cdots \tilde \psi_M
\; \varphi_1 \cdots
\varphi_N \,\rangle_{\alpha}
\nonumber} \def\ssstyle{\scriptscriptstyle
\ea
where the fields $\tilde \psi_i$ in the correlator on the
right hand side are given by
$$ \tilde \psi_i\ = \ \Bigl[ e^{\frac12 \lambda \psi}\, \psi_i
\Bigr](u_i) := \sum_{n=0}^{\infty} {\lambda^n\over 2^n n!}\;
\oint_{C_1} \!{dx_1\over 2\pi} \cdots \oint_{C_n}
\!{dx_n\over 2\pi} \; \psi_i(u_i)\psi(x_n) \cdots \psi(x_1)
\ \ ,
$$
and $C_\nu$ are small circles around the insertion point
of $\psi_i$. Since the contour integrals on the rhs.\ pick
out simple poles, the fields $\psi_i$ and $\tilde \psi_i$
have the same conformal dimension -- $\tilde \psi_i$ can be
regarded as the image of $\psi_i$ under a ``rotation'' generated
by the perturbing field $\psi$.
With the help of eq.\ (\ref{banaldef}) we are able to study
the deformation of $n$-point functions of the (self-local) perturbing
field $\psi$ itself. Notice that the OPE (\ref{locmargOPE})
contains no first order poles so that the fields $\tilde \psi$
and $\psi$ coincide; in fact, all the contour integrals in eq.\
(\ref{banaldef}) are zero if there is no bulk field inserted
in the upper half-plane. Hence, any perturbative correction to the
$n$-point function of $\psi$ vanishes -- which implies that self-local
marginal field are truly marginal.
\subsection{Chiral marginal boundary perturbations.}
For the time being, let us restrict to perturbations with local
boundary fields $\sJ$ taken from the chiral algebra, i.e.\ we
shall analyze perturbations generated by fields assigned to
elements in the subspace ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^0_1 \subset \cH$. Such fields
are local wrt.\ all bulk and boundary fields, so that eq.\
(\ref{banaldef}) may be applied to correlators involving
arbitrary bulk and boundary fields. Consequently, a complete
non-perturbative picture of the deformation can be given,
including a proof of the invariance of the partition
function.
\subsubsection{Deformation of the gluing map.}
\def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle}
Our first goal is to describe the effect a marginal
perturbation with the boundary current $\sJ$ has on
the {\em gluing map} $\Omega$. To this end, we phrase the
content of the gluing condition (\ref{gluecond}) as follows: Suppose we insert
the field $W(z+2i\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) - \Omega\bW(\bar z - 2i\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon)$ with $z =
\bar z$ into an arbitrary correlation function of the
unperturbed theory. Then, by taking the limit $\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \rightarrow
0^+$, we move $W$ and $\Omega\bW$ to the boundary until the
correlator vanishes at $\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon = 0$. Now we want to understand
how the presence of the perturbation $P \exp ( \lambda S_J)$
influences this situation. In more formal terms, we need to
evaluate the expression
\begin{eqnarray}} \def\ea{ \end{eqnarray} \lefteqn{ 0 = \lim_{\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \ra 0} \ P e^{ \lambda S_J} \ \ \left( W(z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) -
\Omega\bW (\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \right)} \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \lim_{\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \ra 0}\
\sum_{n=0}^{\infty} \frac{\lambda^n}{n!} \int_{\gamma_1}
\cdots \int_{\gamma_n} \!\frac{dx_1}{2\pi}\cdots\frac{dx_n}{2\pi} \;
\sJ(x_1) \cdots \sJ(x_n) \, \left( W(z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) -
\Omega\bW (\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \right)\nonumber} \def\ssstyle{\scriptscriptstyle
\ea
where we have used $z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon = z+2 i\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon$, $\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon =z-2 i \delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon$ and inserted the
definition of the operator $ P\exp (\lambda S_J)$ underlying formula
(\ref{analdef}). Our next step involves closing
the integration contours $\gamma_i$ either in the upper or in the
lower half-plane. Let us choose the upper half-plane ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z >0$ for all
contours (the final result is certainly independent of this choice).
If there are other bulk fields in the correlator, we split the closed
contour into a small circle $C$ around $z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon$ and
a part surrounding the location of all other fields.
The latter correlation function vanishes
separately for $\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \ra 0$ due to the ``old'' gluing conditions, whereas
the former part yields the equation
\begin{eqnarray}} \def\ea{ \end{eqnarray} 0 & = & \lim_{\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \ra 0}\
\sum_{n=0}^{\infty} \frac{\lambda^n}{n!} \int_C
\cdots \int_C \!\frac{dx_1}{2\pi} \cdots\frac{dx_n}{2\pi}\;
\sJ(x_1) \cdots \sJ(x_n) \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& & \hspace*{3cm} \left( \Phi(w\otimes} \def\bz{\bar z \vac;z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) -
\Phi(\vac \otimes} \def\bz{\bar z \Omega w; z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \right)\ \ .
\nonumber} \def\ssstyle{\scriptscriptstyle \ea
Here, we have described the fields $W$ and $\Omega\bW$ in terms of
the corresponding states $w, \Omega w \in {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^0$.
Now we insert the formula (\ref{WOPE}) for
the operator product expansion between $\sJ$ and the chiral fields.
Only the residues survive the contour integration so that we get
$$ \int_C \frac{dx}{2\pi}\; \sJ(x) \bigl( \Phi(w \otimes} \def\bz{\bar z \vac;z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon)
+ \Phi(\vac \otimes} \def\bz{\bar z \Omega w; z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \bigr)
\ = \ i\,\Phi(J_0 w \otimes} \def\bz{\bar z \vac;z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \ \ . $$
The second term associated with $\Omega \bW$ cannot contribute
since it is holomorphic in the upper half-plane. Iteration leads to
\begin{eqnarray}} \def\ea{ \end{eqnarray} 0 & = & \lim_{\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon \ra 0}\
\sum_{n=0}^{\infty} \frac{(i \lambda)^n}{n !}
\left( \Phi(J_0^n w \otimes} \def\bz{\bar z \vac; z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon,\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon ) -
\Phi(\vac \otimes} \def\bz{\bar z \Omega w;\bar z_\delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon) \right) \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \sum_n\; \bigl(\Phi(\exp(i \lambda J_0) w; z\bigr)
- \Phi(\Omega w; z) \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \ e^{i \lambda J_0}
\, W(z) e^{-i \lambda J_0} - \Omega \bW(\bar z) \ \ .
\nonumber} \def\ssstyle{\scriptscriptstyle
\ea
Our last step follows from $J_0 \vac = 0$ and the state field
correspondence for boundary fields. Conjugation with $\exp(i
\lambda J_0)$ induces an inner automorphism $\gamma_J$ of the chiral
algebra $\cW$, defined by
$$ \gamma_J (W) \ := \ \exp(- i \lambda J_0)\,
W\, \exp( i \lambda J_0)\ \ \ \mbox{ for all } \ \ \ W \in \cW\ \ .
$$
Replacing $W$ by $\gamma_J (W)$ in the last line of our short
computation, the final result for the change of the gluing
conditions under chiral marginal deformations becomes
\begin{equation}} \def\ee{ \end{equation}
W(z) \ = \ \Omega \circ \gamma_{\bJ} (\bW) (\bar z) \ \ \
\mbox{ for } \ \ \ z = \bar z \ \ .
\label{form1}
\ee
\newline
Observe that $\gamma_J$ acts trivially on the Virasoro field
because a current zero mode $J_0$ commutes with all the modes $L_n$.
Hence, the gluing condition $T = {\overline T}} \def\bW{{\overline W}$ and those
of all other generators $W \in \cW$ that commute with $J_0$
remain unchanged under the deformation with $\sJ$. These fields
then generate the same Ward identities as before the perturbation.
\subsubsection{Deformation of the 1-point functions.}
We will now analyze the change of 1-point functions under
the deformation induced by $\sJ$. Our aim is to derive an
exact formula for the perturbed 1-point function.
To this end, we evaluate the terms in eq.\ (\ref{analdef}) order
by order in $\lambda$ using the operator product expansion
(\ref{JOPE}) between the field $\varphi_{ij}$
and our current $\sJ(x)$. Thereby, the calculation of the
perturbed 1-point function is essentially reduced to the
following simple computation:
\begin{eqnarray}} \def\ea{ \end{eqnarray} \lefteqn{ \langle \varphi_{ij}(z,\bar z) S_J \rangle_\a =
\int_{-\infty}^{\infty} \frac{dx}{2\pi} \, \left(\ \frac{X^i_J}{x-z} \,
\langle \varphi_{ij}(z,\bar z)\rangle_\a - \langle
\varphi_{ij}(z,\bar z)\rangle_\a \,
\frac{X_{\Omega \bJ}^j}{x-\bar z } \right)} \phantom{XXXXX}\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \int_{-\infty}^{\infty} \frac{dx}{2\pi} \, \left(\frac{X^i_J}{x-z}
\frac{A^\a_{ij}}{(z-\bar z)^{2h_i}}
- \frac{A^\a_{ij}}{(z-\bar z)^{2h_i}}
\frac{X_{\Omega \bJ}^j}{x-\bar z} \right)\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \frac{X_J^i A^\a_{ij}}{(z-\bar z)^{2h_i}}
\int_{-\infty}^{\infty} \frac{dx}{2\pi} \left(
\frac{1}{x-z} - \frac{1}{x-\bar z}\right)
\ = \ \frac{i\,X_J^i A^\a_{ij}}{(z-\bar z)^{2h_i}} \ \ . \nonumber} \def\ssstyle{\scriptscriptstyle
\ea
\newline
It involves the same kind of arguments as in the previous
subsection and, in addition, the intertwining relation after
eq.\ (\ref{1ptfct}). The higher order terms can be computed
in the same way and give
\begin{equation}} \def\ee{ \end{equation} \langle \varphi_{ij} (z,\bar z) \rangle_{\a:\,\lambda J}
\ = \ \frac{e^{i \lambda X_J^i} A^\a_{ij}}{(z-\bar z)^{2h_i}}\ \ .
\label{form2}
\ee
Consequently, the effect of the perturbation is to ``rotate'' the matrix
$A^\a_{ij}$ with $\exp(i \lambda X_J^i)$. This behavior
is consistent with the change of the gluing automorphism and the
intertwining relation of the linear map
$A^\a_{ij}$.
\subsubsection{Partition function and cluster property.}
We have argued in the first subsection that correlation functions
involving boundary fields can be deformed by the simple prescription
(\ref{banaldef}) if all boundary fields in the correlator are local
with respect to the perturbing field. In the case of a chiral
marginal perturbation, all boundary fields have this property
so that formula (\ref{banaldef}) can be used without any
restriction on the fields $\psi_i$. For correlation functions
without insertions of bulk fields, there are no singularities
in the upper half-plane. Consequently, the effect of the
deformation on pure boundary correlators is trivial. In
particular, the conformal weight of all boundary fields is
unaffected by the perturbation. Hence, {\em the partition function
$Z_{\Omega,\a)}(q)$ is invariant under chiral deformations}.
\medskip
Let us also briefly discuss the fate of the cluster property
(\ref{cluster}) under chiral deformations. Without loss of
generality, we can restrict ourselves to the investigation
of 2-point functions.
The basic idea is simple: After expanding the perturbing
operator $P \exp(S_{\lambda J})$, we deform the integration
contours (which originally are parallel to the real axis)
so that they surround the two insertion points $z_1$ and
$z_2$ in the upper half-plane (see Figure 3). Thereby we
rewrite the deformed correlation function in each order of
the perturbation expansion as a sum of unperturbed 2-point
functions involving descendants of the original bulk fields.
These functions can be split into products of perturbed
1-point functions by the cluster property of the undeformed
theory. This last step involves a standard re-summation,
and the details are left to the reader.
\begin{figure}
{\epsfbox{pic3.ps}} \vspace*{3mm}
\caption{ \small For chiral deformations the original curves
$\gamma_p$ in the contour integrals (\ref{analdef}) can be deformed
into small circles surrounding the insertion points of two
bulk fields. The result is expressible through descendants
of the original bulk fields. }
\end{figure}
We will see in the next subsection that our assertions
on chiral deformations can be derived rather easily in the
boundary state formalism. Here we have chosen an alternative
(and certainly more cumbersome) route because it allows for
a first illustration of the prescriptions that underlie
analytic marginal perturbations. We shall return to more
general cases in the last subsection after a brief interlude
on the boundary state formalism, which is very effective for
chiral marginal deformations but difficult to adapt to other cases.
\subsection{The boundary state formalism.}
Most aspects of CFTs on the upper half-plane can be studied equally
well by introducing boundary states into the ``parent'' CFT on the full
plane -- more precisely, on the annulus or on the complement of the
unit disk. Boundary states can be viewed as special linear combinations
of generalized coherent states (the so-called Ishibashi states), which
are placed at the boundary of the annulus resp.\ disk complement and which
provide sources for the bulk fields. This leads to a generalized notion of
D-branes coupling to closed string modes \cite{ReSc}.
An abstract characterization of a boundary state can be given in
terms of bulk field correlation functions: Let us use
$z, \bar z$ as coordinates on the upper half-plane as before
and $\xi, \bar\xi$ with
\begin{equation}} \def\ee{ \end{equation}
\xi \ =\ e^{{{2 \pi i}\over {\beta_0}} \ln z } \ \ \hbox{\rm and}
\ \ \
\bar \xi \ =\ e^{-{{ 2 \pi i}\over{\beta_0}} \ln \bar z }
\label{coortrsf}\ee
to denote coordinates on the annulus within the full complex plane;
$\beta_0$ is an inverse ``temperature'', i.e.\ we have identified
the semi-circles $|z|=t_0$
and $|z|=t_0 \exp \beta_0$ with some positive imaginary time $t_0$.
\footnote{The transformation is easier visualized when split up into
the two consecutive maps $z \longmapsto w := \ln z$ from the upper
half-plane to the strip -- hereby the boundary is broken up into
two components -- and $w \longmapsto \xi$ from the strip to the annulus.}
\smallskip
The boundary state $|\alpha\rangle$ which implements a boundary condition
$\alpha$ of the boundary CFT (with Hamiltonian $H^{(H)}\,$)
into the plane theory (with Hamiltonian $H^{(P)}\,$) is
defined by demanding the relation \cite{ReSc}
\begin{eqnarray}} \def\ea{ \end{eqnarray}
&\; {\rm Tr}_{{\cal H}_\alpha}\bigl( e^{-\beta_0 H^{(H)}}
\varphi^{(H)}_1(z_1,\bar z_1) \cdots
\varphi^{(H)}_N(z_N,\bar z_N)\bigr)
\phantom{XXXXXXXXXXXXXX} \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
&=\ {\cal J}(z,\bar z; \xi, \bar\xi) \cdot
\langle \Theta \alpha |\,e^{- {{2\pi^2} \over {\beta_0}} \, H^{(P)}}
\varphi^{(P)}_1(\xi_1,\bar \xi_1) \cdots
\varphi^{(P)}_N(\xi_N, \bar\xi_N)\, | \alpha
\rangle\label{corrrel}
\ea
for arbitrary bulk fields $\varphi^{(H)}_i(z_i,\bar z_i) = \varphi_i
(z_i,\bar z_i)$ of the half-plane theory. The Jacobians that appear
due to the conformal transformation from $(z_i,\bz_i)$ to $(\xi_i,
\bar \xi_i)$ are collectively denoted by ${\cal J}(z,\bar z; \xi,
\bar\xi)$; $\Theta$ is the CPT-operator.
The above definition may be extended so as to allow for two different
boundary states $|\alpha\rangle$, $|\beta\rangle$ at
the boundaries of the annulus, corresponding to a strip with two
different boundary conditions $\a,\b$, or to a jump in the
boundary condition along the real line.
There exists an alternative way to introduce boundary
states, namely by equating zero-temperature correlators on the half-plane and
on the complement of the unit disk in the plane.
Since this is useful to compute the variation of 1-point functions under
chiral marginal deformations, we present the formulas. With $z, \bar z$ as
before, we introduce coordinates $\zeta, \bar\zeta$ on the complement of the
unit disk by
\begin{equation}} \def\ee{ \end{equation}
\zeta = {1-iz \over 1+i z}\quad\quad {\rm and} \quad\quad
\bar\zeta = {1+i\bar z \over 1-i \bar z}\ ;
\label{coortrsfzeta}\ee
if $\,|0\rangle$ denotes the vacuum of the bulk CFT, then the requirement
\begin{eqnarray}} \def\ea{ \end{eqnarray}
&\; \langle\,
\varphi^{(H)}_1(z_1,\bar z_1) \cdots \varphi^{(H)}_N(z_N,\bar z_N)\,
\rangle_{\alpha}
\phantom{XXXXxxXXXXXXXxxxxxxX}\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
&=\ {\cal J}(z,\bar z; \zeta, \bar\zeta) \cdot
\langle 0|\, \varphi^{(P)}_1(\zeta_1,\bar\zeta_1) \cdots
\varphi^{(P)}_N(\zeta_N, \bar\zeta_N)\, | \alpha
\rangle\label{corrrelzeta}
\ea
defines the same boundary states as before; see e.g.\ \cite{CaLe,ReSc}.
The concrete construction of boundary states
proceeds in two steps: Given a gluing automorphism $\Omega$ of the chiral
algebra $\cW$, one first associates Ishibashi states
$|i\rangle\!\rangle_{\Omega}$ to each pair $(i,\omega^{-1}(i^+))$ of irreducibles
that occur in the bulk Hilbert space \cite{Ish1};
$|i\rangle\!\rangle_{\Omega}$ is unique
up to a scalar factor (fixed by relation (\ref{ishchar}) below)
and implements the gluing map in the sense that
\begin{equation}} \def\ee{ \end{equation}
\bigl[\,W_n-(-1)^{h_W}\Omega\bW_{-n}\,\bigr]\,|i\rangle\!\rangle_{\Omega}=0\ .
\label{gluconplane}\ee
Full boundary states $|\alpha\rangle_{\Omega} \equiv |(\Omega,\alpha)\rangle$
are given as certain linear combinations of Ishibashi states,
$$
|\alpha\rangle_{\Omega} = \sum_i B^i_{\alpha}\, |i\rangle\!\rangle_{\Omega}\ .
$$
The complex coefficients $B^i_{\alpha}$ are subject
to various consistency conditions, most notably to ``Cardy's conditions''
arising from world-sheet duality -- see \cite{Car3} for details: The
partition function of the boundary theory on a strip can be calculated on the
annulus as a transition amplitude between two boundary states,
\begin{equation}} \def\ee{ \end{equation}
Z_{\alpha \beta}(q) \equiv
{\rm tr}_{{\cal H}^{(H)}}\bigl( q^{L_0^{(H)}-\frac c {24}} \bigr)
= \langle \Theta\beta|\, \tilde q^{L_0^{(P)}-\frac c {24}} \,|\alpha\rangle\ .
\label{partfct}
\ee
This is the two-boundary-state generalization of (\ref{corrrel})
without bulk insertions, and $q=\exp(2\pi i\tau) = \exp(-\beta_0)$,
$\tilde q = \exp(-2\pi i /\tau)$. The rhs.\ of eq.\ (\ref{partfct}) can
be calculated with the help of
\begin{equation}} \def\ee{ \end{equation}
{}_{\Omega}\langle\!\langle j |\, \tilde q^{L_0^{(P)}-\frac{c}{24}}
\,|i\rangle\!\rangle_{\Omega} \ = \ \delta_{i,j}\, \chi^{\cal W}_i(\tilde q)
\label{ishchar}\ee
and, on general grounds, the lhs.\ in the expression (\ref{partfct}) must
be a sum of $\cW$-characters with (positive) {\em integer} coefficients.
After a modular transformation, this implies Cardy's
non-linear constraints \cite{Car3} on the coefficients $ B^i_{\alpha}$. In
particular, the boundary ``states'' should be regarded
as labels for sectors, not as elements in some vector space.
With the help of (\ref{corrrelzeta}), one can show \cite{CaLe,ReSc} that
there is a simple relation to the 1-point functions and structure constants
of the bulk-boundary OPE -- which are subject to further non-linear sewing
constraints like (\ref{class}) --
namely
\begin{equation}} \def\ee{ \end{equation}
A_{i,\omega^{-1}(i^+)}^{\alpha} = B^{i^+}_{\alpha} \quad\ \ \mbox{and}\ \quad
C^{\,\alpha}_{\varphi\; 0}
\ =\ \frac{B^{i^+}_{\alpha}}{B^0_{\alpha}} \ .
\label{AeqBB}\ee
The decomposition of a boundary state into Ishibashi states contains
the same information as the set of 1-point functions and therefore
specifies the ``descendant'' boundary CFT of a given bulk CFT completely.
\medskip
Now let us exploit the boundary state formalism
for the discussion of marginal boundary perturbations by
$W$-algebra currents $J(x)$. To this end, we use eqs.\ (\ref{corrrel})
or (\ref{corrrelzeta}) to transport the perturbation from the
boundary of the upper half-plane to the boundary of the annulus resp.\
unit disk. This is possible since $J$ is a local field of the bulk theory
so that its image under the conformal transformation acts on the state
space of the bulk theory. With (\ref{coortrsfzeta}) and $h_{J}=1$, we obtain
\begin{equation}} \def\ee{ \end{equation}
\int_{\R} \frac{dx}{2\pi}\, J^{(H)}(x)
= \int_{|\zeta|=1}\! \frac{d\zeta}{2\pi}\,J^{(P)}(\zeta) = i J^{(P)}_0\ .
\label{pertint}\ee
An analogous formula
results from the map (\ref{coortrsf}) to the annulus, this time the rhs.\
consists of one integral for each boundary component
at $|\xi|=1$ resp.\ $|\xi|= \frac{2\pi^2}{\beta_0}\,$.
Since chiral currents are analytic, we need not worry about possible
divergences, as they can be avoided by deforming the integration contour.
It is the last equality in (\ref{pertint}) that makes it easy to treat
perturbations by chiral currents in the boundary state formalism: We
could {\em not} conclude that $\int_{\R}\!dx\, J^{(H)}(x) = J^{(H)}_0$
on the half-plane because of the different integration contour in the
definition of half-plane modes, see \cite{Car1,Car2,ReSc}.
Using boundary states, however, the
effect of marginal boundary perturbations on a half-plane theory
reduces to the action of current zero modes -- as long as the
perturbing fields are taken from the chiral algebra.
\medskip
The boundary states which describe the boundary conditions before
and after the chiral perturbation are related by a simple ``rotation''.
Correlators of the deformed boundary CFT can be obtained upon
replacing $|\alpha\rangle$ in the correspondences (\ref{corrrel})
or (\ref{corrrelzeta}) by
\begin{equation}} \def\ee{ \end{equation}
|(\Omega,\alpha)\rangle_{\lambda\,J} \equiv
|(\Omega,\alpha)\,;\,{\lambda\,J}\rangle \ \;
= \ \,e^{i\lambda J_0}\, |(\Omega,\alpha)\rangle\
\label{lambdst}\ee
where $J_0 \equiv J^{(P)}_0$ is the zero mode of the left-moving current
on the plane.
We have made the unperturbed gluing map $\Omega$ explicit in (\ref{lambdst}).
Indeed, from this formula, we can immediately re-derive the change
(\ref{form1}) of the gluing conditions under the marginal deformation by
$J(x)$: Using that left- and right-movers commute, as well as the simple
relation $J_n\,|(\Omega,\alpha)\rangle = -\,\Omega\bJ_{-n}\,|(\Omega,
\alpha)\rangle$, eq.\ (\ref{gluconplane}) gets replaced by \vspace*{2mm}
\begin{equation}} \def\ee{ \end{equation}
\bigl[\,W_n-(-1)^{h_W}\,\Omega \circ \gamma_{\bJ}\,\bigl(\bW_{-n}\bigr)
\,\bigr]\, |(\Omega,\alpha)\rangle_{\lambda\,J} \ = \ 0\
\label{defgluconplane} \vspace*{2mm}
\ee
with $\gamma_{\bJ}(\bW) := \exp(-i\lambda \bJ_0)\, \bW\, \exp(i\lambda
\bJ_0)$; see also \cite{GrGu1} for special cases of (\ref{defgluconplane}).
Likewise, eq.\ (\ref{form2}) for the change of the 1-point functions
follows from (\ref{corrrelzeta}), (\ref{AeqBB}) and (\ref{lambdst}).
\medskip
Finally, let us use the boundary state formalism to verify -- without
resorting to perturbative arguments -- that the partition
function $Z_{\alpha}(q)\equiv Z_{\alpha\alpha}(q)$ of a boundary CFT
with boundary condition $\alpha$ along the real line stays invariant under
marginal perturbations by a chiral boundary field $J(x)$: We have to compute
the transition amplitude between $|\alpha;\, \lambda\,J \rangle$ and
$\langle\Theta\,(\alpha;\, \lambda\,J) \,|$.
But this equals the unperturbed amplitude $Z_{\alpha\alpha}(q)$ because
$\langle\Theta\, \exp(i\lambda J_0)\,\alpha|=
\langle\Theta\alpha|\, \exp(-i\lambda J_0)$ and because
$\exp(i\lambda J_0)$ commutes with $L^{(P)}_0$.
The spectrum of the boundary theory does not change.
\smallskip
Up to now, we have always started from a boundary CFT with a constant
boundary condition $\alpha$ along the real line and considered
boundary perturbations involving marginal fields that were integrated
over the whole boundary -- which corresponds to simultaneous deformation
of one and the same boundary state $|\alpha\rangle$ on both ends of the
annulus. Generalizations of this would involve jumps in the boundary
conditions along the real line and different boundary operators
integrated over the segments of constant
boundary condition.
The boundary state formalism allows to discuss the basic case
with one such jump, using different perturbations for (possibly different)
in- and out-boundary states in equations (\ref{corrrel},\ref{lambdst}).
Generically, the partition functions $Z_{(\alpha;\lambda_1J^1),(\beta;
\lambda_2J^2)}(q)$ for such systems will show a different spectrum than
in the unperturbed situation, and they will involve ``twisted characters''
of the symmetry algebra -- more precisely, characters of representations
twisted by inner automorphisms Ad$_U$ with $U=\exp\{i(\lambda_1J^1_0-
\lambda_2J^2_0)\}$. We shall take advantage of this fact in Section 4.
\subsection{Non-chiral analytic perturbations.}
Let us now turn towards deformations generated by marginal boundary
fields $\psi(x)$ that are self-local (in the sense of
Section 2.4) but {\em not} taken from the chiral algebra. We have
seen already that these fields are truly marginal to all orders in
$\lambda$, so we can ask how gluing conditions and 1-point functions
behave under finite perturbations. We will settle the former issue
completely in 3.5.1 and make some general statements on 1-point-functions
and on the spectrum in Subsection 3.5.2.
\subsubsection{Change of the gluing map.}
As in the case of chiral boundary perturbations, we would like to study
the effect of non-chiral marginal deformations on the gluing conditions
$W = \Omega (\bW)$ for the generators of the observable algebra
${\cal W}$. We start the discussion by showing that $T = {\overline T}} \def\bW{{\overline W}$ is not
changed under analytic deformations to all orders of $\lambda$.
This follows essentially from the OPE between $\sT(z)$ and $\psi(x)$:
For a field $\psi$ of conformal dimension $h=1$, the singular part of
the OPE is a total derivative,
$$ \sT(z) \, \psi(x) \ = \ \frac{1}{(z-x)^2} \, \psi(x)
+ \frac{1}{z-x} \, \partial_x \psi(x)\ +\;{\rm reg} \ = \
\partial_x\ \left( \frac{1}{z-x} \, \psi(x) \right)\ +\;{\rm reg} \ \ .
\ \
$$
We can test the gluing condition for the Virasoro field by inserting
$T(z)$ into the correlation function (\ref{analdef}) such that ${\mbox{\it Im\/}}} \def\Re{{\mbox{\it Re\/}} z >
\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle$, and then moving $T(z)$ down towards the real axis, where it can
be compared to ${\overline T}} \def\bW{{\overline W}(\bz)$. While passing through one of the contours
$\gamma_i$, we pick up a term
$$ \int_C \frac{dx}{2\pi}\ \sT(z) \, \psi(x)\ = \ \int_C \frac{dx}{2\pi}\
\partial_x\, \left( \frac{1}{z-x} \,
\psi(x) \right) \ = \ 0
$$
where $C$ is a small circle surrounding the insertion point of the
Virasoro field. The contour integral along $C$ vanishes, which means
that the Virasoro field $T$ cannot feel the presence of the
perturbation and hence the gluing condition stays intact.
\medskip
The previous argument can be generalized to the following
simple criterion:
\newline
{\em Under analytic deformation with a self-local perturbing field
$\psi(x)$, a prescribed gluing condition for a chiral
field $W(z)$ stays invariant to all orders in $\lambda$
if the singular part of the OPE $\;W(z)\,\psi(x)\,$
is a total derivative with respect to $x$}.
\newline
We will encounter several examples later in the text. Let us
remark that the same criterion is at least necessary for
other (non-analytic) marginal perturbations to preserve a
given gluing condition.
Perturbations with currents $\sJ$ from the chiral algebra often
lead to a non-trivial deformation (\ref{defgluconplane},\ref{form1})
of the gluing condition of a symmetry generator $W(z)$,
without destroying the associated Ward identity.
We will see that this is impossible for non-chiral analytic
deformations: In Subsection 3.3.1, the change in the
$W(z)$-gluing condition was obtained by moving the chiral
field $W(z)$ through the stack of integration contours.
After a bit of combinatorics, the same procedure for non-chiral
analytic deformations results in
$$
W(z) \, e^{\lambda\! \int\!{dx\over2\pi}\,\psi(x)}
= e^{\lambda\! \int\!{dx\over2\pi}\,\psi(x)}\,
\Bigl[ e^{\lambda \psi}\,W\Bigr](z)
$$
(to be understood in the limit $z \ra \bar z$) with
\begin{equation}} \def\ee{ \end{equation}
\Bigl[ e^{\lambda \psi}\,W\Bigr](z) :=
\sum_{n=0}^{\infty} {\lambda^n\over n!}\;
\oint_{C_1} \!{dx_1\over 2\pi} \cdots \oint_{C_n} \!{dx_n\over 2\pi} \;
W(z)\,\psi(x_1) \cdots \psi(x_n) \ \ .
\label{exppsiop}\ee
The curves $C_i$ encircle the point $z$ in the upper half-plane
as in Figure 3. To each order $n$, the integrals will pick some
term $\psi^{(n)}$ from the OPE of $W(z)$ with the product of
perturbing fields.
At least part of the $\psi^{(n)}$ are true boundary fields which
are not defined away from the boundary, thus they do not belong
to the chiral algebra and the above gluing does not produce
Ward identities for $W(z)$ in the deformed boundary CFT.
This shows that a non-chiral analytic perturbation either
breaks or leaves invariant the Ward identity associated to
a given generator of $\cW$. In general, this leads to a new
conformally invariant boundary theory with Ward identities
governed by a subalgebra $\cU$ of the original chiral
algebra $\cW$.
\medskip
Let us add a few comments on deformations of boundary
conditions for $N=2$ superconformal CFTs because they
constitute an important motivation for the present work
and because they nicely illustrate the criterion given
above.
In such theories, one considers two types (A,B) of gluing
conditions for the chiral fields $G^\pm,J,T$,
\def\bG{\overline G}
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\mbox{A-type:} & & J(z) \ = \ - \bJ (\bz) \ \ , \ \
G^\pm(z) \ = \eta \, \bG{}^\mp(\bz) \label{Atype}
\\[2mm]
\mbox{B-type:} & & J(z) \ = \ \ \bJ (\bz)\ \ , \ \
G^\pm(z) \ = \ \eta \, \bG{}^\pm(\bz) \label{Btype}
\ea
supplemented by $T = {\overline T}} \def\bW{{\overline W}$ in both cases. The
parameter $\eta$ is restricted in order to have a
supersymmetric ``space-time'' theory.
More precisely, one requires that an $N=1$ subalgebra
with generating supercurrent $G(z) := G^+(z)
+ G^-(z)$ or $G'(z) := i(G^+(z) - G^-(z))$ is preserved
by any boundary condition. This leaves us with the choice
$\eta = \pm 1$. The gluing conditions (\ref{Atype},\ref{Btype})
were first introduced in \cite{OOY}, where the connection
with supersymmetric cycles in Calabi-Yau manifolds was investigated.
A quite non-trivial realization in CFTs associated with homogeneous
spaces was constructed in \cite{Stan}.
\smallskip
It is natural to try and deform an $N=2$ superconformal boundary
CFT with the chiral U(1) current. According to our general
formulas, such deformations lead to
$$ \mbox{A type:} \ \ \ G^\pm(z) \ = \ e^{-i\lambda} \eta \,\bG^\mp(\bz)
\ , \ \quad \
\mbox{B type:} \ \ \ G^\pm(z) \ = \ e^{i \lambda} \eta \, \bG^\pm(\bz)
\ .
$$
This however, spoils the condition $\eta = \pm 1$ and hence the
``space-time'' supersymmetry unless $\lambda$ is a multiple of
$\pi$. Thus there is no family of supersymmetric boundary CFTs
generated by perturbing an $N=2$ model by the U(1) boundary
current $J$.
\medskip
On the other hand, marginal deformations associated with chiral
or anti-chiral primaries can exist and preserve $N=2$ supersymmetry.
A state $|\psi_{c,a}\rangle$ (or the corresponding conformal field)
in an $N=2$ superconformal field theory is called {\em chiral} resp.\
{\em anti-chiral} primary if it satisfies
$$
G^+_{-{1\over2}}\, |\psi_c\rangle = 0 \quad\quad {\rm resp.} \ \ \quad
G^-_{-{1\over2}}\, |\psi_a\rangle = 0\ .
$$
It follows that $|\psi_{c,a}\rangle$ are $N=2$ highest weight
states with charge and dimension related as $q = 2h$ resp.\ $q = -2h$,
see \cite{LVW}.
\smallskip
Suppose there is a chiral primary boundary field $\psi_c(x)$ of
conformal dimension $1/2$ in an $N=2$ boundary CFT, and set
$\psi(x) := G^-_{-1/2} \psi_c(x) - G^+_{-1/2} \psi_a(x)$ where
$\psi_a(x) = \bigl(\psi_c(x)\bigr)^*$ is the anti-chiral conjugate
of $\psi_c(x)$. Then $\psi(x)$ is anti-selfadjoint, uncharged and
marginal, and we can study the deformations it induces.
\smallskip
\def\sG{{\sf G}} \def\sbG{{\sf \bG}}
Typically, there will be other boundary and bulk operators that are
non-local wrt.\ $\psi(x)$, so we have to rely on the methods developed
for non-chiral perturbations. The gluing condition for the Virasoro
field is preserved because of $h_{\psi}=1$ (see above). Since $\psi(x)$
carries no charge, the singular contribution to the operator product
of the current $\sJ(w)$ with $\psi(x)$ vanishes so that the gluing
condition for the current $J$ is untouched. As for the supercurrents
$\sG^{\pm}(z)$, we use the state-field correspondence and the $N=2$
relations to find
$$
\sG^-(z) \, \Bigl( G^-_{-{1\over2}} \psi_c\Bigr) (x) \ \sim \ 0\ \ , \ \ \
\sG^+(z) \, \Bigl( G^-_{-{1\over2}} \psi_c\Bigr) (x) \ \sim \
\partial_x\,\Bigl( {2 \, \psi_c(x)\over z-x} \Bigr)\
$$
together with the analogous relations for the anti-chiral contribution
$G^+_{-1/2} \psi_a(x)$ to the perturbing field $\psi$. The first equation
already holds when $\psi_c(x)$ is any $N=2$ primary, whereas in the
second it is crucial that $\psi_c(x)$ is chiral. Our general criterion
shows that deformations with $\psi(x)$ do not affect the prescribed
$N=2$ gluing conditions -- whether they are of A-type or of B-type --
to first order in the perturbation parameter; hence they are invariant
to all orders if $\psi(x)$ is a self-local marginal field.
\medskip
Deformations induced by chiral primaries as above could serve as
a starting point to define topological $N=2$ boundary CFTs. In the
bulk case \cite{DVVtop,War1}, topological field theories yield families
of commutative associative rings, parameterized by the perturbation
parameter, which often can be interpreted as quantum cohomology
rings of complex manifolds. It would be interesting to see which
new structures arise from topological boundary correlators. Since
the topology of the ``supporting space'', i.e.\ of the world-sheet
boundary, does not allow to continuously interchange arguments in
correlation functions, one may expect that non-commutative rings
appear quite naturally.
\subsubsection{One-point-functions, spectrum, and the cluster property.}
A boundary conformal field theory is determined by the gluing conditions
and the 1-point functions. We have discussed the change of gluing conditions
under non-chiral analytic deformations, but it is difficult to obtain general
statements on the deformed 1-point functions, in particular because they
are to be computed for all primary fields of the smaller (``unbroken'')
subalgebra $\cU \subset \cW$ associated to the reduced set of Ward
identities that may survive after turning on the perturbation.
Nevertheless, as we will see later on, there are examples of
non-trivial analytic deformations for which the deformed 1-point
functions can be constructed to all orders.
At the moment, we limit ourselves to a simple first order criterion
for the {\em invariance} of a 1-point function. Let $\varphi(z,\bar z)$
be an arbitrary quasi-primary bulk field, e.g.\ a primary field
for the reduced chiral algebra $\cU \subset \cW$. Conformal
transformation properties fix the 2-point function of $\varphi(z,
\bar z)$ with the perturbing field $\psi(x)$ up to
a constant,
$$ \langle \varphi(z,\bar z) \psi(x) \rangle_\a \ = \
\frac{C^\a_{\varphi\psi}}{(z-\bar z)^{2h-1} (z-x)(\bar z -x)}
\ \ . $$
Here, $h = \bar h$ is the conformal weight of the field $\varphi$,
and the bulk-boundary OPE coefficient $C^\a_{\varphi \psi}$ depends
on the original boundary condition $\a$. By the residue theorem, we
get the following first order correction for the perturbed 1-point
function
\begin{equation}} \def\ee{ \end{equation}
\langle \varphi(z,\bar z) \rangle_{\a;\,\lambda \psi} \ = \
\langle \varphi(z,\bar z) \rangle_\a + i\,\lambda\,
\frac{C^\a_{\varphi\psi}} {(z-\bar z)^{2h}} + O(\lambda^2) \ \ .
\ee
To leading order, a 1-point function $\langle
\varphi(z,\bar z) \rangle_\a$ is invariant under a
perturbation with $\psi$ if and only if $C^\a_{\varphi\psi} = 0$.
Again this is a necessary condition for the invariance of a
given 1-point function under any truly marginal perturbation,
but it is certainly not sufficient.
\medskip
A full computation of the partition function requires complete
knowledge of all 1-point functions and hence it is at best
accessible through a case by case study. On the other hand,
there are some general statements we can make about the behaviour
of $Z_\a (q)$ under analytic deformations. We have argued above
that the formula (\ref{banaldef}) can be used to construct perturbed
correlators of boundary fields $\psi_i$ which are local with
respect to the perturbing field $\psi$. By the same arguments
as in chiral deformation theory, we conclude that the conformal
weights of such fields $\psi_i$ are invariant under the
deformation. While this criterion does not protect the
full spectrum of boundary conformal weights (as in the
case of chiral deformations where all boundary fields are
local with respect to $\psi$), it shows that {\em part of
the partition function stays intact}. In particular, all
chiral fields $\sW$ are local with respect to $\psi$ so
that the partition function will always contain the
vacuum character of the original chiral algebra $\cW$ {\em even
if} gluing conditions and Ward identities are broken down to
a subalgebra $\cU \subset \cW$. Furthermore, while the
``gluing'' (\ref{exppsiop}) of a chiral field $W(z)$ to boundary
operators destroys the Ward-identity for $W(z)$, it still leads
to a (possibly twisted) action of the full chiral algebra $\cW$
on the state space $\cH$. This effect can be read off from
the partition function of the deformed theory which still
decomposes into characters of (twisted) representations
of $\cW$, see the examples below.
\medskip
The cluster property is somewhat more difficult to attack.
Note that the argument at the end of Subsection 3.3.3 cannot
be used in this simple form because the deformed correlators
are not expressible through correlators of descendants of
the original bulk fields. There exists a variant of the
previous reasoning which takes into account the specific
analyticity properties of correlators with insertions
of self-local non-chiral boundary fields and bulk fields. Its
convergence behaviour in the limit $n \rightarrow
\infty$, however, is not easy to control. It is likely
that the cluster property is preserved for an open
neighbourhood of $\lambda = 0$ but is bound to break
down at certain finite values of the perturbation parameter
$\lambda$ whenever we deform with some non-chiral boundary
fields. This agrees with the examples we analyse below.
Often, the breakdown of the cluster property has
an interesting physical or geometric interpretation.
\section{Example: Boundary deformations for {\fatma c}$\;$=$\;$1 theories}
The results of the previous section hold for arbitrary boundary CFTs.
We will now illustrate them in a simple example, namely the free bosonic
field. To begin with, we present the uncompactified theory with Neumann
and Dirichlet boundary conditions and study their deformations. Then the
same analysis is made for the compactified boson. In the third subsection,
we investigate boundary perturbations of $c=1$ orbifold theories.
Although the models under consideration are simple enough, we will
encounter rich patterns in the brane moduli space, including some
unexpected phenomena.
\subsection{The uncompactified theory.}
The dynamical degrees of freedom of the bulk theory
are obtained from a single field $X(z,\bar z)$ which obeys the
usual equation of motion $ \partial \bar \partial X(z,\bar z) = 0$.
The modes of the left- and right-moving chiral currents
$J(z) = 2 i\, \partial X(z,\bar z) = \sum \, a_n\, z^{-n-1}$
and $\bJ(\bar z) = 2i\, \bar \partial X(z,\bar z)=
\sum {\bar a}_n \,\bar z^{-n-1}$
generate a U(1)$\,\times\,$U(1) algebra with canonical
commutation relations
$$
[\, a_n\, ,\, a_m\, ] \ = \ n\, \delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon_{n,-m} \ \ , \ \
[\, {\bar a}_m \, ,\, {\bar a}_m\, ] \ = \ n\, \delta } \def\e{\epsilon} \def\varepsilon} \def\ra{\rightarrow} \def\vac{|0\rangle{\varepsilon_{n,-m}\ \ .
$$
The Virasoro fields are obtained from $J,\bJ$ by
normal-ordering, $ T(z) = \frac12\, {\bf :}\, J\,J\,{\bf :}\,(z)$
and ${\overline T}} \def\bW{{\overline W}(\bar z) = \frac12\, {\bf :}\, \bJ\, \bJ\,{\bf :}\,(\bar z)\;$.
\newline
The abelian current algebra has irreducible representations ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g$
labeled by real numbers $g$, the U(1) charge. ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g$ is
generated from a ground state $|g\rangle$ with the
properties
$$ a_n\;|g\rangle \ = \ 0 \ \ \mbox{ for all } \ \ n > 0 \ \
\ \mbox{ and } \ \ a_0 \; |g\rangle \ = \ g\; |g \rangle \ \ .$$
The lowest-energy subspace $V^g_0$ of ${\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g$ is one-dimensional
and spanned by $|g\rangle$, the element $a_0$ acts on $V^g_0$ by
$X^g_J = g$.
Putting things together, one can realize
the bosonic field $X$ on the state space $\cH^{(P)} = \bigoplus_g
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g \otimes} \def\bz{\bar z {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g$ which is a diagonal sum with equal
U(1) charges for both chiralities. In the explicit
formula
$$ X(z,\bar z) \ = \ x - \frac{i}{4}\,p\, \ln (z\bar z) +
\frac{i}{2} \sum_{n \neq 0} \, \left( \frac{a_n}{n} \, z^{-n} +
\frac{{\bar a}_n}{n} \, \bar z^{-n} \right)\ , $$
one new element $x$ appears which acts as differentiation
$x = i \partial_g$ on the state space. We have also introduced
the operator $p = a_0 + {\bar a}_0 $ which has the usual
Heisenberg commutation relation with $x$. Bulk fields
$\varphi_{g_1g_2}(z, \bar z)$ exist only for $g_1 = g_2$ so that
we will omit one index in the following. $\varphi_g =
\varphi_{gg}$ is obtained from the bosonic field by
$$ \varphi_g(z,\bar z) \ = \ {\bf:}\,\exp(2 i g X(z,\bar z))\,{\bf:} \ \ ,
$$
and with proper normal-ordering these fields can be shown to
obey the operator product expansions
$$ \varphi_{g_1}(z_1,\bar z_1) \ \varphi_{g_2}(z_2, \bar z_2) \
\sim \ |z_1 - z_2|^{2 g_1 g_2} \varphi_{g_1 +g_2}
(z_2,\bar z_2) + \dots \ \ . $$
The conformal weights $h_g = \bar h_g$ of $\varphi_g(z,\bar z)$
are given by
$h_g = \frac12 g^2$.
\bigskip
We will look for boundary conditions that preserve the chiral
symmetry algebra generated by the U(1) current $J$. There are
two possibilities for the gluing map which we can use:
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\mbox{Neumann boundary condition:} \ \ \ \ \ \
J(z) & = & \Omega_N\bJ(\bar z) \ \equiv \ \ \bJ(\bar z) \
\label{N} \\[2mm]
\mbox{Dirichlet \ boundary condition:} \ \ \ \ \ \
J(z) & = & \Omega_D\bJ(\bar z) \ \equiv \ - \bJ(\bar z) \
\label{D}
\ea
The Neumann type boundary conditions are realized by a bosonic
field
$$ X(z,\bar z) \ = \ x - \frac{i}{4}\,p\, \ln (z\bar z) +
\frac{i}{2} \sum_{n\neq 0}
\, \frac{a_n}{n}\ \left( \, z^{-n} + \bar z^{-n}\, \right) $$
acting on a state space $\cH = \bigoplus_g {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^g$. Here,
$x = i \partial_g$ as before, and $a_n$, $p= 2 a_0$ are the modes of the
generator $\sJ$ of the U(1) symmetry in the boundary Hilbert space.
The computation of the 1-point
function of $\varphi_g(z,\bar z) =\; {\bf :}\,\exp(2 i g X(z,\bar z))
\,{\bf :}$ is a straightforward exercise, leading to
$$ \langle \varphi_g(z,\bar z) \rangle_N \ = \ \delta_{g,0} \ \ .$$
Note that there appears no free parameter in these 1-point
functions, i.e.\ there is only one boundary theory with Neumann
boundary conditions for an uncompactified free boson.
\smallskip
For Dirichlet boundary conditions, we build the bosonic
field according to
$$
X(z,\bar z) \ = \ x_0 + \frac{i}{2} \sum_{n\neq 0}
\, \frac{a_n}{n}\ \left( z^{-n} - \bar z^{-n} \right)
\ \ .
$$
Here, $x_0$ is a free real parameter describing the
value of the bosonic field along the boundary, i.e.\
$X(z,\bar z) \equiv X(z)+X(\bar z) = x_0$ for $z = \bar z$. The field
$X$ acts on a state space $\cH = {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^0$ consisting only of
the vacuum representation. This time, calculation of 1-point
functions for $\varphi_g(z,\bar z)$ results in
the formula
$$ \langle \varphi_g(z,\bar z) \rangle_{D\, x_0}
\ = \ \frac{e^{2 i g x_0}}{(z-\bar z)^{2h_g}} \ \
$$
which depends on $x_0$, parameterizing different possible
boundary theories with Dirichlet boundary conditions.
Note that, for the free boson theory, the structure constants
$\Xi$ in the sewing constraint (\ref{class}) are given by
$\Xi^{g'}_{g_1,g_2} =
\delta_{g', g_1+g_2}$. The numbers $A^{x_0}_g
= \exp(2 i g x_0)$ solve eq.\ (\ref{class}) and hence the
theory has the cluster property (\ref{cluster}); at the same time,
this means that superpositions (``mixtures'') of ``pure'' Dirichlet
boundary conditions do not cluster.
\medskip
\subsubsection{Chiral deformations.}
Let us now study marginal deformations and start with
the chiral current $\sJ$ which is the only field
of weight $h=1$ in the chiral algebra. Since the
zero mode $a_0$ commutes with all other elements
in $\cW$, it generates the trivial inner automorphism
$\gamma_J = id$ on the chiral algebra. It follows then
from eq.\ (\ref{form1}) that the gluing conditions
are invariant under the
deformation, i.e.\ they are given by the formulas (\ref{N},\ref{D})
for all values of the perturbation parameter
$\lambda$. Consequently, the only possible
effect of the perturbation on the boundary theories
is due to changes of the 1-point functions.
For Neumann boundary conditions we have $A^N_{g} =
\delta_{g,0}$ so that, according to our formula
(\ref{form2}), this coefficient -- and therefore
the Neumann boundary theory -- stays invariant under
deformations with $\sJ(x)$. For Dirichlet boundary
conditions, things are a bit more interesting. The coefficients
$A^{x_o}_{g} = e^{2 i g x_0}$ of the Dirichlet boundary
condition behave as
$$ A^{x_0}_{g} \ \longrightarrow \ e^{i \lambda g}
\, e^{2 i g x_0}
\ = \ e^{2 i g (x_0+\frac{\lambda}{2})} \ \ $$
when we turn on the perturbation. As a result of
the deformation, the parameter $x_0$ gets shifted by
$\lambda/2$ -- i.e.\ the D-brane is displaced.
\subsubsection{Non-chiral deformations.}
In the case of Neumann boundary conditions, there are two other
boundary fields of conformal dimension $h=1$. We will consider
perturbations by the combinations
$$
\psi^1(x) \ := \ \sqrt{2} \,\cos \{2 \sqrt{2} X(x)\} \ \ \ \ \
\mbox{ and }\\ \ \ \
\psi^2(x) \ :=\ \sqrt{2} \,\sin \{2 \sqrt{2} X(x)\}\ ,
$$
which will be seen to break the chiral symmetry down to
the Virasoro algebra by inducing a periodic ``potential''
along the boundary. This has been studied in some detail in
\cite{CaKle,CKLM,PoTh}.
The boundary fields $\psi^a(x)$, $a=1,2$, are local with respect
to themselves. By our general considerations of Section 3.2
on analytic perturbations, $\psi^a(x)$ are truly marginal (to all
orders in the ``coupling'' $\lambda$). At the same time, we
expect the spectrum of boundary fields to change when the boundary
potential is turned on, because the boundary Hilbert space $\cH =
\bigoplus_g {\cal V}^g$ of the Neumann theory contains fields which
are non-local wrt.\ $\psi^a(x)$ -- in fact, only the scaling
dimensions of operators with charges in $\sqrt{2}\Z$ are protected.
\medskip
Let us first see how the U(1) gluing conditions
behave under perturbations with e.g.\ $\psi^1(x)$. The criterion for
invariance of $\Omega$ given in Section 3.5 required that the
singular part of the OPE between a chiral symmetry generator $W(z)$
and $\psi^1(x)$ is a total $\partial_x$-derivative. This is true for
the Virasoro field $W(z) = T(z)$, but not for the current $W(z)=J(z)$.
So we have to determine the effect of pushing $J(z)$ through the
$x_i$-integration contour when moving the field towards the real
line, in order to evaluate (\ref{exppsiop}) describing the change
of $J(z) = \overline{J}(\bar z)$. The OPE of $J(z)$ with $\psi^1(x)$
is given by
\begin{equation}} \def\ee{ \end{equation}
J(z) \, \psi^1(x) \ = \ { i \sqrt{2} \over z-x} \,\psi^2(z) \ +\ {\rm reg}
\label{JjOPE}\ee
so that we pick up a term $i \sqrt{2} \psi^2(z)$ whenever $J(z)$
passes one of the contours. The effect of moving the field $\psi^2(z)$
towards the real axis is determined by the OPE
\begin{equation}} \def\ee{ \end{equation}
\psi^2(z)\, \psi^1(x) \ = \ {-i \sqrt{2} \over z-x}\, J(z) +\ {\rm reg}
\ \ . \label{jjOPE}
\ee
We can now apply our general formula (\ref{exppsiop}) to derive
the following closed expression for the $\psi^1$-deformed gluing
conditions:
\begin{equation}} \def\ee{ \end{equation}
J(z) \ = \ \sin(\sqrt{2} \lambda)\, \psi^2(x) + \cos(\sqrt{2} \lambda)\,
\overline{J}(\bar z)
\label{newglcond}\ee
for $z=\bar z = x$. By the same reasoning, one can determine the
effect of perturbations with $\psi^2$ on the Neumann gluing condition:
\begin{equation}} \def\ee{ \end{equation}
J(z) \ = \ - \sin(\sqrt{2} \lambda)\, \psi^1(x) + \cos(\sqrt{2} \lambda) \,
\overline{J}(\bar z)\ .
\label{newglcondtwo}\ee
These equations, which were also found in \cite{CKLM}, mean that
the boundary reflects left-moving into right-moving currents
only at the expense of marginal boundary fields. As a consequence,
the correlation functions of the perturbed boundary CFT no longer
obey Ward identities for the U(1) current $\sJ$. Exceptions occur
whenever $\lambda = n\,{\pi\over\sqrt{2}}$ for some integer $n$:
Then $\psi^a(x)$ disappear from (\ref{newglcond},\ref{newglcondtwo})
and, if $n$ is odd, the original Neumann conditions for $J(z)$
are turned into Dirichlet conditions. We will refer to the latter
values of the perturbation parameter as {\em Dirichlet-like}
points.
Broken U(1) symmetry
complicates computations considerably. Since it is only the
Virasoro algebra that remains at our disposal, we have to
characterize the deformed theory through the 1-point functions
of all Virasoro primary fields. The decomposition of irreducible
U(1) modules into $c=1$ Virasoro modules is well known.
Both coincide as long as the
conformal dimension of the primary field is not given by
$h = {m^2}$ for any $m \in \frac12 \Z$, but for those cases,
the U(1) modules decompose into a sequence of irreducible
Virasoro representations,
\begin{equation}} \def\ee{ \end{equation}
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{{\sqrt{2} m}} \ = \ \bigoplus_{l=0}^{\infty}
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm Vir}_{{(|m|+l)^2}}
\label{UVirdecomp}\ee
-- the subscript on U(1) modules is the charge, the one on
Virasoro modules the conformal dimension. There is
a corresponding identity for the characters,
\begin{equation}} \def\ee{ \end{equation}\chi^{\rm Vir}_{{m^2}}(q) \ = \ \chi^{\rm
U(1)}_{{\sqrt{2}m }}(q) -\chi^{\rm U(1)}_{ \sqrt{2}{(|m|+1)}}(q)\ .
\label{excepVirchars}\ee
It means that, for the values $h={m^2},\ m \in \frac12 \Z$, the
$c=1$ Virasoro Verma modules contain a singular vector at level
$2|m|+1$. \smallskip
Coming back to our problem of describing the deformed boundary
theories, we first remark that the theory has a rather
useful ``hidden'' SU(2) symmetry which also governs the deformed
theories. In fact, this symmetry is obvious from the OPEs of $J,
\psi^1, \psi^2$ which, while not forming an algebra of true local
currents for the full boundary CFT, still lead to the same
algebraic structure for various quantities of interest, in
particular for 1-point functions of bulk fields. This SU(2)
symmetry is also visible in the structure of the decomposition
(\ref{UVirdecomp}). Indeed, the Virasoro highest weight vectors at
energy $h = j^2$ in the state space $\cH$ of the Neumann theory
span an SU(2) multiplet of length $2j+1$ so that
$$ \cH \ = \ \int_{g \neq \sqrt{2}m} {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_g \ \oplus \
\bigoplus_{m \in \frac12 \Z} {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{\sqrt{2} m }
\ = \ \int_{g \neq \sqrt{2}m } {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm Vir}_{g^2 \over 2}
\ \oplus \ \bigoplus_{j\in\oh \Z_+}^{\infty} \Bigl({\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm Vir}_{{j^2}}
\Bigr)^{\oplus\,2 j +1}\ .
$$
A similar structure is observed for the state space $\cH^{(P)}$
of the bulk theory,
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\cH^{(P)} & = & \int_{g \neq \sqrt{2}m } {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_g \otimes} \def\bz{\bar z \
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_g \ \oplus \ \bigoplus_{m \in \frac12 \Z}
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{{\sqrt{2}m}}\otimes{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{{\sqrt{2}m}}
\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
& = & \int_{g \neq \sqrt{2}m} {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{Vir}_{g^2 \over 2} \otimes} \def\bz{\bar z
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{Vir}_{g^2 \over 2} \ \oplus \
\bigoplus_{j\in{\oh}\Z_+} \Bigl({\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm Vir}_{{j^2}}
\otimes {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm Vir}_{{j^2}} \Bigr)^{\oplus\,2 j +1} \
\oplus \ \ldots
\label{flatbulkVir}
\ea
where the dots denote terms with $h\neq \bar h$, which are of
no concern to us since they cannot couple to a conformal
boundary state. {}From these formulas we conclude that spin-less
(i.e.\ $h = \bar h$) Virasoro primary bulk fields come in two
families:
$$ (1) \ \ \ \varphi_{g,g} (z,\bar z) \ \ \ \mbox{ with } \ \ \
g \not \in{\textstyle \frac{1}{\sqrt{2}}}\Z \ \ \ \ \ \ \ \ \
(2) \ \ \ \varphi^j_{m,m} (z,\bar z) \ \ \mbox{ with }
j \in {\textstyle \frac12} \Z_+
$$
and $m = -j, -j+1, \dots , j-1,j$. The fields of the second
family have U(1) charges $g = \bar g = \sqrt{2} m \in
{\textstyle\frac{1}{\sqrt{2}}} \Z$ with respect to $J_0$.
\smallskip
Since the perturbing fields span the charge lattice
$\sqrt{2} \Z$, U(1) charge conservation implies that the
1-point functions of fields $\varphi_{g,g}$ in the first
family are not perturbed, i.e.\
\begin{equation}} \def\ee{ \end{equation}
\langle \varphi_{g,g}(z,\bar z)\rangle_{N;\lambda \psi^a}
\ = \ 0 \ \ \ \mbox{ for } \ \ \
g \not \in {\textstyle\frac{1}{\sqrt{2}}} \Z \ \ .
\label{unchonept}\ee
For the fields $\varphi^j_{m,m}$, results get more interesting.
In the evaluation of the deformed correlators we continue
the perturbing field analytically into the upper half-plane
and compute the usual contour integrals. This leads to an action
of the SU(2) generators $J_0, \psi^a_0 := \int\!\frac{dx}{2\pi}\,\psi^a(x)$
on the left index of the fields, i.e.\
\begin{equation}} \def\ee{ \end{equation}
\langle \varphi^{\,j}_{m,m}(z,\bar z)
\rangle_{N;\,\lambda \psi^a} \ =\
\sum_{m'=-j}^j D^{\,j}_{m,m'} (\Gamma^a_{\lambda}) \;
\langle \varphi^j_{m',m}(z,\bar z) \rangle_{N}
\label{pertonepoint}\ee
where $\Gamma^a_{\lambda} = \exp(i\lambda \psi^a_0)$ is regarded as an
SU(2)-element, and $D^{\,j}_{m,m'} (\Gamma^a_{\lambda})$ are the
entries of its spin $j$ representation matrix expressed in a
spin$_z$ eigenbasis. Finally, the correlator on the rhs.\ of
(\ref{pertonepoint}) stands for the function
$$
\langle \varphi^{\,j}_{m',m}(z,\bar z) \rangle_{N} \ = \
\delta_{m', -m}\, {1\over (z-\bar z)^{m^2} }\ ,
$$
even if $\varphi^{\,j}_{-m,m}(z,\bar z)$ does not occur in the
uncompactified free boson theory.
We can also encode the outcome of this computation in the
following formula for the $\psi^a$-perturbed flat Neumann
boundary state:
\begin{equation}} \def\ee{ \end{equation}
|N;\,\lambda \psi^a\rangle \ = \ \sum_{j\in{\oh}\Z_+}
\sum_{m=-j}^j \, D^{\,j}_{m,-m} (\Gamma^a_{\lambda}) \,|j,m,m \rangle\!
\rangle
\label{flatpertbdst}
\ee
where $ |j,m,m \rangle\!\rangle$ are Virasoro Ishibashi states associated
to the primaries $ \varphi^{\,j}_{m,m}(z,\bar z)$.
\medskip
While (\ref{flatpertbdst}) in principle gives complete information on the
perturbed boundary theory, it looks essentially hopeless to compute
the perturbed partition function $Z_{\alpha}(q)$ directly via a
modular transformation of $\sum_{j,m} | D^{\,j}_{m,-m}
(\Gamma^a_{\lambda}) |^2 \chi^{{\rm Vir}}_{j^2}(\tilde q)$ -- simply
because the matrix elements $D^{\,j}_{m,n} (\Gamma)$ are given by the
rather cumbersome formula
\begin{eqnarray}} \def\ea{ \end{eqnarray}
D^{\,j}_{m,n} (\Gamma) & = & \sum_{\mu={\rm max}(0,n-m)}^{{\rm min}
(j-m,j+n)}\ {[(j+m)!(j-m)![(j+n)!(j-n)!]^{\oh} \over
(j-m-\mu)!(j+n-\mu)! \mu! (m-n+\mu)! } \, \phantom{xxxxxxxxxx}
\nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm] & & \phantom{xxxxxxxxxxxxxxxx}\times\;
a^{j+n-\mu}\, (a^*)^{j-m-\mu} \, b^{\mu}\, (-b^*)^{m-n+\mu} \label{Djmn}
\ea
in which the group element $ \Gamma \in$ SU(2) was parameterized
by $\Gamma = { a\ b \choose -b^* a^* }\,$; see e.g.\ \cite{Ham}. At
the Dirichlet-like points $\lambda = \,{2k+1\over\sqrt{2}}\pi,\ k\in\Z$,
however, the formula simplifies considerably, and modular
transformation yields
\begin{equation}} \def\ee{ \end{equation}
Z_{\alpha_D}(q)\ \sim \ \sum_{n\in\Z} \; {q^{n^2} \over \eta(q) }
\label{Zdirlike}\ee
for $\alpha_D =$ Dirichlet-like boundary conditions: The initially
continuous Neumann spectrum is reduced to a discrete one (which
furthermore is the same as the one of a boundary CFT of a free boson
compactified at the self-dual radius). The boundary condition can be
viewed as a superposition of flat D-branes located at the sites of an
infinite lattice. The boundary fields with non-zero U(1) charge should
be attributed to ``solitons'' interpolating between different
minima of the boundary potential.
\smallskip
In \cite{PoTh}, the partition function for an arbitrary perturbation was
computed along a different route, namely by passing to a free fermion
representation and by explicitly diagonalizing the Hamiltonian consisting
of a free part and the boundary interaction. Since the technical details
are not very illuminating, we merely state their result: The spectrum of
the perturbed Neumann boundary states $|\alpha\rangle_\lambda :=
|N;\,\lambda \psi^1\rangle$ is given by
\begin{equation}} \def\ee{ \end{equation}
Z_{\alpha_\lambda}(q)
\ = \ \eta(q)^{-1}\;\sum_{m\in\Z} \int_{0}^2 \!d\zeta \;
q^{(2m +f_{\lambda}(\zeta))^2}
\label{ZPolTh}\ee
with
\begin{equation}} \def\ee{ \end{equation}
f_{\lambda}(\zeta) \ = \ {1\over \pi}
\arcsin[\, \cos{{\textstyle \frac{\lambda}{\sqrt{2}}}} \,\cdot\, \sin\pi \zeta \,]\ ;
\label{arcsinfunc}
\ee
the arcsin-branch is to be chosen such that
$\lim_{\lambda\to0} f_\lambda(\zeta) = \zeta$; the integral is over the
half-open interval, which becomes important in the discrete variants to be
discussed below. (\ref{ZPolTh}) displays a band structure of the
spectrum which is typical for a theory of electrons moving
in a crystal. As soon as an infinitesimal periodic potential is
turned on, the continuous spectrum rips apart at the values
$h= \frac{n^2}4$, and the gaps open up as the strength $\lambda$ of the
potential grows. The bands are reduced to points at the Dirichlet-like value
$\lambda={\pi\over\sqrt{2}}$, where only primaries with dimension
$h=n^2$ for $n \in \Z$ remain (tight-binding limit).
Naively one would expect this to occur at $\lambda = \infty$
but, loosely speaking, the period of the potential introduces a
``scale'' $r_{\rm s.d.} = \frac{1}{\sqrt{2}}$ into the problem
so that special effects are bound to appear whenever $\lambda$ is
in resonance with $r_{\rm s.d.}$.
\smallskip
The structure of the spectrum is in line with our general expectation.
In fact, it does decompose into U(1) characters and all states of
U(1) charge in the lattice $\sqrt{2} \Z$ -- which correspond to
fields that are local with respect to the perturbing fields --
do remain in the boundary theory.
\medskip
The physical interpretation of the periodic boundary potential with
Dirichlet-like coupling strength as generating a mixture
of elementary Dirichlet conditions is rather compelling,
but suggests that the perturbed boundary theory violates the
cluster property. Indeed, at $\lambda={2k+1\over\sqrt{2}}\pi$,
the cluster relation together with the Dirichlet Ward identities
for the U(1) current would imply the sewing constraint
\begin{equation}} \def\ee{ \end{equation}
A^{\alpha_{D}}_{g_1} \, A^{\alpha_{D}}_{g_2}
\ \ {\buildrel ? \over =} \ \ A^{\alpha_{D}}_{g_1+g_2}\ .
\label{clcondflat}\ee
Choosing $g_1,g_2$ such that $g_i \notin \frac{1}{\sqrt{2}} \Z$
but $g_1+g_2 = \frac{1}{\sqrt{2}}$, the structure constants
$A^{\a_D}_{g_i}$ vanish as in the original Neumann boundary theory,
cf.\ (\ref{unchonept}), while $A^{\a_D}_{g_1+g_2}\neq 0$.
In order to test clustering for arbitrary values of
$\lambda$, we would need a lot of information on fusion and chiral blocks
of $c=1$ Virasoro modules, about which virtually nothing is known.
We expect, however, that the boundary states (\ref{flatpertbdst})
obey the cluster condition as long as $|\lambda| < \frac{\pi}{\sqrt{2}}$,
for the following reasons: Our study of orbi\-fold models will show that
this is true for the $r_{\rm circ}= \sqrt{2}$ circle model --
which possesses analogous deformations with exactly the same algebraic
properties as the $r=\infty$ theory. Furthermore, the general argument in
favour of clustering which was sketched in Subsections 3.3.3 and 3.5.3
indicates that finite domains of convergence could spoil clustering at
finite perturbation strength.
\subsection{The compactified theory.}
If we take a circle of radius $r$ as the target space for
the free boson, we can again impose Dirichlet and Neumann
boundary conditions, but now there are continuous parameters
in both cases. Compared to the uncompactified case, the mode
expansion of the bulk field $X(z,\bar z)= X_L(z) + X_R(\bar z)$
additionally involves a winding number operator $w$ as well as two
independent zero mode operators $x_{L,R}$:
\begin{eqnarray}} \def\ea{ \end{eqnarray}
X_L(z) & = & x_L - \frac i 4\ p\; \ln z - \frac i 2\ r\,w \;
\ln z + \frac i 2 \ \sum_{n \neq 0} \, \frac{a_n}{n} \, z^{-n}
\ \ , \ \ \nonumber} \def\ssstyle{\scriptscriptstyle \\[2mm]
X_R(\bar z) & = & x_R - \frac i 4\ p\; \ln \bar z + \frac i 2\ r\,w \;
\ln \bar z + \frac i 2 \ \sum_{n \neq 0} \, \frac{{\bar a}_n}{n} \, \bar z^{-n}
\ \ . \nonumber} \def\ssstyle{\scriptscriptstyle
\ea
The normalizations are chosen so as to preserve the canonical commutation
relations from the uncompactified case: The winding operator $w$ commutes
with $x:=x_L+x_R$, $p$ and all oscillators, all other relations follow
from the exchanges $p \leftrightarrow 2rw\,,\ x \leftrightarrow \tilde
x:= x_L-x_R$. For later convenience, we introduce the zero modes $a_0
:= p/2 + r w$ and $\overline{a}_0 := p/2 - r w$. Chiral currents
$J(z)$ and $\overline{J}(\bar z)$ are obtained from the modes $a_n,
{\bar a}_n$ as before.
Because of the new degree of freedom, primary fields
$\varphi_{g,\bar g}(z,\bar z) = e^{2ig X_L(z)} e^{2i\bar gX_R(\bar z)}$
can carry different left- and right-moving charges wrt.\ $a_0$ and
$\overline{a}_0$, namely $g=k/2r + rw$ and $\bar g =k/2r - rw$, where
$k :=rp$ and $w$ take integer values. Again one can easily solve the
Dirichlet and Neumann boundary conditions eqs.\ (\ref{D}) and (\ref{N})
in terms of the bosonic field with values on the circle to arrive
at
$$
\langle \varphi_{g,\bar g}(z,\bar z) \rangle_{D\, x_0}
\ = \ \delta_{g,\bar g}\; \frac 1 {\sqrt{2r}} \; \frac {e^{i k x_0/r}}
{(z-\bar z)^{ \frac{k^2}{4r^2}} } \ \
$$
for the Dirichlet case -- the real parameter $x_0 \in \R\; {\rm mod}\; 2\pi r$
can again be interpreted as the location of the brane, i.e.\ $X(z) = x_0 -
X(\bar z)$ for $z= \bar z$. The Neumann case is obtained via $T$-duality;
here the 1-point functions are
$$
\langle \varphi_{g,\bar g}(z,\bar z) \rangle_{N\, \tilde x_0}
\ = \ \delta_{g,-\bar g}\; \sqrt{r}\;\frac { e^{2ir w \tilde x_0}}
{(z-\bar z)^{r^2w^2} } \ \ ;
$$
now we have $X(z) = \tilde x_0 + X(\bar z)$ for $z= \bar z$,
where $\tilde x_0 \in \R\; {\rm mod}\; \frac{\pi}{r}$ parametrizes
representations of the fundamental group $\pi_1(S^1)$ (`Wilson lines').
The $r$-dependent normalization arises from the non-trivial one-point
function $\langle {\bf 1} \rangle_\a$.
Passing to boundary states and applying a modular transformation as in
(\ref{partfct}), we obtain the following formulas for the partition
functions of the theories with boundary conditions $D\,x_0$
respectively $N\,\tilde x_0$ along the real line,
\begin{equation}} \def\ee{ \end{equation}
Z_{D\,x_0}(q) \ = \ \frac 1 {\eta(q) }\; \sum_{k\in\Z} q^{2r^2k^2}
\ \ , \quad\quad
Z_{N\,\tilde x_0}(q)\ =\ \frac 1 {\eta(q) }\; \sum_{w\in\Z}
q^{\frac{w^2}{2r^2}}\ \ ;
\label{freepartfcts}
\ee
they depend on the compactification radius (i.e., on the bulk modulus),
but not on $x_0$ or $\tilde x_0$.
\subsubsection{Chiral deformations.}
Marginal deformations with the chiral current $\sJ(x)$ can be treated
in close analogy to the uncompactified case: First observe that the
gluing conditions are invariant under $\gamma_J$ and that only
the coefficients $A_{ij}^{\alpha}$ can be effected by the perturbation.
As before, the matrix $X^i_{J}$ acts on $\varphi_{g,\bar g}(z, \bar z)$
through multiplication by $g$, therefore eq.\ (\ref{form2}) leads to
\begin{eqnarray}} \def\ea{ \end{eqnarray}
\langle \varphi_{g,\bar g}(z,\bar z) \rangle_{D\, x_0;\ \lambda J}
\ & = &\ \delta_{g,\bar g}\; \frac 1 {\sqrt{2r}} \;
\frac {e^{i k (x_0+\frac{\lambda}2)/r}}
{(z-\bar z)^{ \frac{k^2}{4r^2}} } \ \ , \nonumber} \def\ssstyle{\scriptscriptstyle\\[2mm]
\langle \varphi_{g,\bar g}(z,\bar z) \rangle_{N\, \tilde x_0;\ \lambda J}
\ & = &\ \delta_{g,-\bar g}\; \sqrt{r}\;
\frac { e^{2ir w (\tilde x_0+\frac{\lambda}2)}}
{(z-\bar z)^{r^2w^2} } \ \ :\nonumber} \def\ssstyle{\scriptscriptstyle
\ea
The marginal perturbations with the current $\sJ(x)$ induce translations
in the Dirichlet resp.\ Neumann parameters $x_0$ and $\tilde x_0$ --
periodic in $\lambda$ with period $4\pi r$ resp.\ $2\pi/r$.
For the ``rational radii'' $r= \sqrt{M/N}$ with positive coprime integers
$M,N$, additional chiral (local) fields
$$ W^\pm_{g_{{\rm loc}}}(z) \ = \ :\,e^{\pm i 2 g_{\rm loc} X_L(z)}\,:
\ \ \ \mbox{ and }\ \ \
\bW{}^\pm_{g_{{\rm loc}}}(\bz) \ = \ :\,e^{\pm i 2 g_{\rm loc} X_R(\bz)}\,:
$$
(along with products) appear in the bulk theory; the charge $g_{{\rm loc}}$ is
$2\sqrt{MN}$ if $N$ is odd and $\sqrt{MN}$ if $N$ is even. These extended
chiral algebras in the bulk theories are a well-known feature of the
rational Gaussian models, see \cite{DVVV} and references therein.
\newline
We may ask whether this additional symmetry is preserved by the boundary
conditions and how the gluing conditions, if they exist, behave under
marginal perturbations with the chiral current $\sJ(x)$. It is easily seen
from the bosonization formula for $W_{g_{{\rm loc}}}^{\pm}(z)$ and
$\bW{}^{\pm}_{g_{{\rm loc}}}(\bz)$ that all Dirichlet boundary theories
respect the enhanced symmetry with gluing conditions
\begin{equation}} \def\ee{ \end{equation}
W^\pm_{g_{{\rm loc}}}(z) \ = \ \Omega_D\bigl[
\overline{W}{}^\pm_{g_{{\rm loc}}}\bigr]
(\bar z) \ := \ e^{\pm 2 i g_{{\rm loc}} x_0}\,
\overline{W}{}^\mp_{g_{{\rm loc}}}
(\bar z)
\label{locfiegluconddir}
\ee
for $z = \bar z$. If the free boson satisfies Neumann boundary
conditions, $X_L(x) = X_R(x) + \tilde x_0$ leads to
\begin{equation}} \def\ee{ \end{equation}
W^\pm_{g_{{\rm loc}}}(z) \ = \ \Omega_N\bigl[\overline{W}{}^\pm_{g_{{\rm
loc}}}\bigr](\bar z) \ := \ e^{\pm 2 i g_{{\rm loc}} \tilde x_0}\,
\overline{W}{}^\pm_{g_{{\rm loc}}}(\bar z)
\label{locfieglucondneu}
\ee
along the boundary. Consequently, under marginal boundary perturbations
with $\sJ(x)$, these enhanced gluing conditions are no longer invariant,
instead they behave according to eq.\ (\ref{form1}) simply because
$W^\pm_{g_{{\rm loc}}}$ are charged wrt.\ $\sJ$.
\medskip
Something special occurs at the ``self-dual point'' $r=1/\sqrt{2}$:
Here, the local chiral fields $J^{\pm}(z) := W^\pm_{\sqrt{2}}(z)$
and $\bJ{}^\pm (\bz) := \bW{}^\pm_{\sqrt{2}}(\bz)$ have conformal
dimension $h_\pm = 1$. This means that there are new marginal
operators {\em within} the enlarged
chiral algebra -- which is simply the non-abelian current algebra
${\rm SU}(2)_1$. We have seen that $J^{\pm}(z), \bJ{}^\pm(\bz)$
automatically obey the gluing conditions $\Omega$ from equation
(\ref{locfiegluconddir}) or (\ref{locfieglucondneu}) for all
Dirichlet or Neumann boundary theories. Therefore, SU(2)$_1$ is
preserved at the boundary.
\smallskip
The general results of Subsections 3.3 and 3.4 show that an arbitrary real
linear combination of $\sJ^1 = \frac{1}{\sqrt{2}} (\sJ^+ + \sJ^-),
\sJ^2 = \frac{1}{\sqrt{2}i} (\sJ^+ - \sJ^-)$ and $ \sJ^3 = \sJ $
can be used to deform the free bosonic boundary theories at
$r_{\rm s.d.}$. The new models satisfy all sewing constraints and
can be described by the boundary states
\begin{equation}} \def\ee{ \end{equation}
|\Gamma \rangle_{\rm s.d.} \ = \ \Gamma \, |N(0)\rangle_{\rm s.d.}
\quad\quad
{\rm with} \ \ \Gamma \,= \, e^{i \sum \lambda_a J^a_0}\
\label{su2fam}\ee
This family contains
the boundary states $|N(\tilde x_0)\rangle$, but also other cases where
$\sJ^3(z)$ does not obey simple Neumann gluing conditions.
Naively, one might expect to obtain a second component of the moduli
space of boundary theories by SU(2)-deformations of the Dirichlet
boundary state $|D(0)\rangle_{\rm s.d.}$. However, the Dirichlet
boundary states are already included
in the set (\ref{su2fam}): By means of the SU(2)-deformations
at the self-dual radius, we can rotate Neumann conditions for $J =
J ^3$ into Dirichlet conditions; a perturbation with $\lambda\,\sJ^1$
changes the gluing condition $J^3(z) = \pm \bJ{}^3(\bar z)$ to
$$
J^3(z) \ = \ \pm (\cos \sqrt{2}\lambda)\, \bJ{}^3(\bar z)
\pm (\sin \sqrt{2}\lambda) \,\bJ{}^2(\bar z) \ ,
$$
cf.\ the general formula (\ref{form1}) and also (\ref{newglcond}) for
the non-chiral deformation $\psi^1$. When approaching $\lambda
= \pi/\sqrt{2}$, Neumann conditions for $J^3$ turn into Dirichlet
conditions -- by a continuous deformation. More precisely, we can write
\begin{equation}} \def\ee{ \end{equation}
|D(0)\rangle_{\rm s.d.} \ = \ e^{i \frac{\pi}{\sqrt2} J^1_0}\
|N(0)\rangle_{\rm s.d.} \ ,
\label{DirNeuRsd}\ee
showing that there is one connected SU(2)-family of boundary
conditions for $r_{\rm s.d.}$.
\smallskip
Let us compare the structure of
boundary theories at the self-dual radius to the known boundary
states of the $SU(2)_1$ WZW-model \cite{Ish1,Car3}. The possibility
of non-standard gluing conditions for the currents was not realized in
these works, but with the help of the general formalism explained in
Section 2 it is straightforward to extend Cardy's classification of
boundary
states to arbitrary gluing maps $\Omega$ in the SU(2) current algebra.
Per fixed gluing condition $\Omega$, one finds two boundary states
$$ |i\rangle_{\Omega} \ = \ 2^{1/4}\, |0\rangle\!\rangle_{\Omega}
+ (-1)^i\, 2^{1/4}\, |1\rangle\!\rangle_{\Omega}
$$
where $i=0,1$ labels the two irreducible heighest weight
representations of SU(2)$_1$. To re-discover those in the family
(\ref{su2fam}), observe that formula (\ref{locfieglucondneu})
with $g_{\rm loc}= \sqrt{2}$ is
invariant under the shift $x_0 \longmapsto x_0 + \pi/\sqrt{2}$,
while the marginal perturbation $\exp(-i \sqrt{2}\pi J^3_0)$
implementing this shift acts non-trivially on the full boundary
state $|N(\tilde x_0)\rangle$, producing precisely the sign in
front of the spin $1/2$ Ishibashi state $|1\rangle\!\rangle_{N}$.
Thus, $|N(0)\rangle$ and $|N(\pi \sqrt{2})\rangle$ -- sitting
at opposite points of the $\tilde x_0$-circle -- coincide with
Cardy's rational SU(2)$_1$ boundary states. Analogous results
hold for other gluing conditions, which
are parameterized by SO(3) since central elements of SU(2) yield
trivial $\gamma_J$ in eq.\ (\ref{form1}). But there are two
different boundary theories sitting over each point of this
SO(3) which resolve the full SU(2) moduli space we found before.
Cardy's boundary states for the SU(2)$_1$ WZW-model are simply
assigned to elements in the centre of SU(2).
\medskip
All the time, we have implicitly assumed that the boundary
conditions in (\ref{su2fam}) are pairwise inequivalent -- which
is not clear a priori. In the self-dual bulk theory, e.g., all
operators $J'(z)\overline{J}{}'(\bar z)$ with $J'=\sum \lambda_a J^a$
are marginal
and we can move away from $r=1/\sqrt{2}$ along an $S^3$ of different
directions. But all these deformations result in equivalent bulk CFTs
because of SU(2)$\,\times\,$SU(2) symmetry, leaving only the ordinary
change-of-radius deformation.
For boundary CFTs, we arrive at a similar scenario if we declare
boundary conditions $(\Omega_i, \alpha_i)$, $i = 1,2$, equivalent as
soon as there is an automorphism (a ``gauge transformation'') of the
bulk CFT which intertwines the gluing conditions
$\Omega_1$ and $\Omega_2$ and maps the set of 1-point-functions
$A^{\a_1}$ to $A^{\a_2}$. This criterion, however, would even identify
all possible Dirichlet conditions $(D, x_0)$ for a free boson, simply
because of translational invariance.
We can formulate a sharper criterion by composing new
systems from two different boundary conditions, e.g.\ by putting
the CFT on the strip with boundary conditions $\alpha$ on one end
and $\beta$ on the other. Then, additional data like the partition
function $Z_{\a\b}(q)$ discussed in Section 3.4 are available, and
we can certainly conclude
that $\alpha \not\simeq \beta$ if $Z_{\a\a}(q) \neq Z_{\a\b}(q)$.
In this way, not only can we resolve all the free boson boundary conditions
at generic radii, but also the family (\ref{su2fam}) at the self-dual
point. Since every SU(2)$_1$ boundary state
can obtained from one out of Cardy's list by the action of an SU(2)
element $g = \exp(i J'_0)$, we have to
compute $Z_{\a\b}(q)$ for some $|\alpha\rangle$ (which obeys, say, standard
gluing conditions) and arbitrary $|\b\rangle := g|\alpha\rangle$.
First note that $Z_{h\a, h\b}(q) = Z_{\a,\b}(q)$ and
$Z_{\a, hgh^{-1}\b}(q) = Z_{\a,g\b}(q)$ for all
$g,h\in\, $SU(2) -- this follows from $h\,|\alpha\rangle =
\bar h^{-1} |\alpha\rangle$ and $\Theta h = h \Theta$. Therefore,
$Z_{\a,g\a}(q)$ depends only on the conjugacy class of $g$, and we
can in particular choose an $h$ such that $h g h^{-1} = t \equiv
\exp{(i\lambda J^3_0)}$ is in a given torus of SU(2).
Partition functions $Z_{\a\b}(q)$ where one of the boundary states
has been twisted by a current in a maximal abelian subgroup can be
computed with standard modular transformation rules. If $\alpha$ is
one of Cardy's boundary conditions, we find the expression
$$
Z_{\a,t\a}(q) = \langle \Theta \a|\, \tilde q^{L_0 - \frac {c}{24}}
e^{i\lambda J^3_0}\, |\alpha\rangle =
\sum_i N_{\a, \a^+}^i \, {\rm tr}_{{\cal H}_i}
q^{L_0 + \frac{\lambda}{2\pi} J^3_0 + \frac{\lambda^2}{8\pi^2} -
\frac {c}{24}} \ ,
$$
which involves twisted SU(2)-characters that depend on $\lambda$. \newline
Finally, if $g \neq g'$ are conjugate to the same torus element, one
can show that there is a boundary condition $\a'$ such that
$Z_{\a', g\a}(q) \neq Z_{\a', g'\a}(q)$ -- yielding a complete
resolution of the SU(2)-family (\ref{su2fam}) as desired.
\subsubsection{Non-chiral deformations.} At special values of the
compactification radius, there are extra non-chiral marginal deformations
similar to the ones present for the uncompactified
free boson with Neumann boundary condition.
The partition functions (\ref{freepartfcts}) show that these radii are
$r = N/ \sqrt{2}$ for integer $N$ in the Neumann and $r = 1/(\sqrt{2} N)$
in the Dirichlet case -- in accordance with an interpretation of
the perturbation as a periodic boundary potential with period $1/\sqrt{2}$.
\newline
The two self-local and mutually local primaries $\psi^{a}(x)$,
$a= 1,2$, appearing there lead to similar effects as the non-chiral
marginal operators in the uncompactified theory with Neumann boundary
conditions. For $r = N/ \sqrt{2}$, the decomposition of the bulk Hilbert
space into Virasoro modules results in a formula analogous to
(\ref{flatbulkVir}); we write it in the form
$$
\cH^{(P)} \ = \ \bigoplus_{k,w \in \Z}
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{{k\over \sqrt{2} N} + {wN\over\sqrt{2}}}
\otimes {\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{\rm U(1)}_{{k\over \sqrt{2} N} - {wN\over\sqrt{2}}}
\ \oplus \ldots \ =\ \bigoplus_{j\in{\oh}\Z_+} \;
\bigoplus_{k',w \in \Z}
{\cal V}} \def\cH{{\cal H}} \def\cU{{\cal U}^{{\rm Vir} \times {\rm Vir}}_{{j^2, {k'+Nw\over2},{k'-Nw\over2} }} \
\oplus \ \ldots
$$
where again the dots indicate terms that do not couple to the
boundary, either because $h\neq \bar h$ or because the charge
condition $g + \bar g \in \sqrt{2} \Z$ is not met. We have
indicated the SU(2) quantum numbers explicitly, adopting the
convention that a module ${\cal V}_{j, m,n}$ is empty unless
$m$ and $n$ are in the range $-j, \ldots, j$. Precisely the
same Virasoro primaries contribute if we consider the
perturbation of a boundary CFT with Dirichlet conditions at
radius $r = \frac{1}{\sqrt{2}N}$.
We can apply the methods used in the uncompactified case
to determine the deformed boundary states, and we find
\begin{equation}} \def\ee{ \end{equation}
|N(\tilde x_0);\,\lambda \psi^a\rangle \ =\
\sum_{j\in{\oh}\Z_+} \sum_{w, k' \in \Z}
D^{\ j}_{{k'+Nw\over2}, {-k'+Nw\over2}} (\Gamma^a_{\tilde x_0,\lambda}) \;
|j, {\scriptstyle {k'+Nw\over2}}, { \scriptstyle {k'-Nw\over2}}
\rangle\!\rangle\ .
\label{comppertbdst}\ee
Now, the SU(2)-element $\Gamma^a_{\tilde x_0,\lambda} =
e^{i \lambda \psi^a_0} e^{2i \tilde x_0 J_0}$ contains the
perturbation parameter $\lambda$ along with $\tilde x_0$ specifying
the original Neumann condition. The latter is recovered for $\lambda=0$,
where only the terms with $k'=0$ contribute (the $N$-dependence encodes
the information on the radius).
\smallskip
Again, the modular transformation to obtain the spectrum from the boundary
states is not manageable except for the Dirichlet-like points
$\lambda={2k+1\over \sqrt{2}}\pi$ (they are ``Neumann-like'' points if
we start from Dirichlet conditions at the dual radius). There, the
prefactors of the Virasoro Ishibashi states are given by the phases
$$
D^{\;j}_{{k'+Nw\over2},{-k' + Nw\over2}}
(\Gamma^1_{\tilde x_0,\lambda_{{\rm Dir}}})
\ = \ \delta_{w,0} (-1)^j e^{-\sqrt{2} i \tilde x_0 k'}\ \ , $$
which lead to the same perturbed partition function (\ref{Zdirlike})
as in the uncompactified case. In particular, the parameter $\tilde x_0$
does not appear in $Z_{\alpha_D}(q)$, and in the boundary state itself
it shows up with a different periodicity: The information about the
original radius $r={N\over\sqrt{2}}$ has been lost during the
perturbation.
\medskip
The alternative method of \cite{PoTh} applies again, and it leads
to a formula for the partition function similar to eq.\ (\ref{ZPolTh}),
only that the $\zeta$-integral is to be replaced by a sum since the
spectrum is discrete from the start.
For later purposes, let us give the explicit formula for the case
$r = \sqrt{2}$:
With $|\alpha\rangle_\lambda := |N(\tilde x_0);\,\lambda \psi^a\rangle$,
restriction of the $\zeta$-integral in (\ref{ZPolTh}) to the sum over
$0, {\scriptstyle {\oh}}, 1, {\scriptstyle {3\over2}}$ yields
\begin{equation}} \def\ee{ \end{equation}
Z_{\alpha_\lambda}(q)
= \eta(q)^{-1}\;\sum_{m\in\Z} \, \Bigl(\, q^{m^2} +
q^{(m + \oh + {\lambda \over \sqrt{2}\pi} )^2} \Bigr) \ .
\label{compZPolTh}\ee
Generally, the charges $g\neq n \sqrt{2}$
follow the flow prescribed by the function (\ref{arcsinfunc}),
the corresponding fields being those which are non-local
wrt.\ to the perturbing field. Finally, only charges $g = n
\sqrt{2}$ are left at $\lambda = {\pi\over \sqrt{2}}$. It is
once more easy to show that the cluster property is broken
at the Dirichlet-like point, but we have no direct handle on
clustering for intermediate $\lambda$. Employing the higher symmetry
algebras present at rational radii does not seem to yield additional
insight into the clustering properties, either. Surprisingly, however,
the study of orbifold models will provide further information.
\medskip
\subsection{The {\fatma c}$\;$=$\;$1 orbifold theories.}
The moduli space of $c=1$ theories on the plane has another branch which
parameterizes orbifolds of the circle theories. This family is constructed
by ``dividing out'' the left-right symmetric $\Z_2$-action $X \longmapsto
-X$ on the compactified free boson theories -- see e.g.\ \cite{DVVV,Gin1} and
references therein. The chiral fields are the invariant elements of
the U(1)$\,\times\,$U(1) current algebra, the bulk Hilbert space consists of
an untwisted sector containing all $\Z_2$-invariant states of the free
boson Hilbert space and of two twisted sectors $\cH^{\rm tw}_{0}$ and
$\cH^{\rm tw}_{\pi r}$ built up over twist fields of left and right
conformal dimension $h^{\rm tw}_{0,\pi r} = 1/16 $. The subscripts
refer to the endpoints of the interval $[0,\pi r]$ which can be
regarded as the target space of the orbifold model at radius $r$.
For $r=r_{\rm s.d.}$, there are three further orbifold models that
arise from dividing out finite subgroups of SO(3), see \cite{Gin2,DVVV},
but we will not discuss these cases here.
We give the description of the associated boundary orbifold models in
terms of boundary states, which can e.g.\ be found in \cite{OsAff}.
Consider the untwisted sector first. The free boson Ishibashi states
are given as $\Z_2$-invariant exponentials of $\sum a_{-n}\bar a_{-n}$
acting on U(1) ground states; therefore one merely has to symmetrize
in the latter to obtain ``untwisted'' orbifold boundary states from
ordinary free boson Dirichlet or Neumann boundary states,
\begin{eqnarray}} \def\ea{ \end{eqnarray}
|D(x_0)\rangle^{\rm orb} & := & \frac1{\sqrt2}\,\Bigl(\;
|D(x_0)\rangle^{\rm circ} + |D(-x_0)\rangle^{\rm circ} \;\Bigr)\ ,
\label{orbDir}\\[2mm]
|N(\tilde x_0)\rangle^{\rm orb} & := & \frac1{\sqrt2}\,\Bigl(\;
|N(\tilde x_0)\rangle^{\rm circ} + |N(-\tilde x_0)\rangle^{\rm circ}
\;\Bigr)\ ; \label{orbNeu}
\ea
The parameters range over the intervals $0< x_0 < \pi r$ and
$0< \tilde x_0 < \frac {\pi} {2 r}$. In terms of 1-point functions,
(\ref{orbDir}) e.g.\ means that
$$
\langle \, \cos \bigl({\textstyle \frac{k}{r}} X(z,\bar z)\bigr)\,
\rangle^{\rm orb}_{D\, x_0} \ = \ \frac 1 {\sqrt{2r}} \;
\frac{\cos \frac{kx_0}{r}}{(z-\bz)^{k^2/4r^2}}
$$
and that no twist fields couple to the
identity on the boundary. A similar formula holds for Neumann boundary
conditions of the orbifold theory.
\smallskip
To each fixed point of the $\Z_2$-action on $S^1$, one assigns two
{\em twisted} Dirichlet and two {\em twisted} Neumann boundary states
made up from the corresponding circle boundary states and the
(appropriately symmetrized) Dirichlet or Neumann Ishibashi states of
$\cH^{\rm tw}_{0,\pi r}$, see \cite{OsAff} for more details.
With $\xi = 0, \pi r$ and $\tilde \xi = 0,
\frac{\pi}{2r}$, we write
\begin{eqnarray}} \def\ea{ \end{eqnarray}
|D(\xi), \pm \rangle^{\rm orb} & := & 2^{-\frac1 2}\,
|D(\xi)\rangle^{\rm circ} \pm 2^{-\frac1 4}\, |D(\xi)\rangle^{\rm tw} \ ,
\label{orbtwDir} \\[2mm]
|N(\tilde\xi), \pm \rangle^{\rm orb} & := & 2^{-\frac1 2}\,
|N(\tilde\xi)\rangle^{\rm circ} \pm 2^{-\frac1 4}\,
|N(\tilde\xi)\rangle^{\rm tw} \ .
\label{orbtwNeu}
\ea
The prefactors ensure proper normalization of all partition functions
$Z_{\a\b}(q)$ for $\a,\b$ taken from the two sets (\ref{orbDir}-\ref{orbtwNeu}).
For our purposes,
the cases with $\a=\b$ are most important since they provide
the number of marginal boundary operators induced by the boundary
condition $\alpha$. In the case of Dirichlet gluing conditions, one
obtains
\begin{eqnarray}} \def\ea{ \end{eqnarray}
Z_{\a}(q) & = & \sum_{k\in\Z}\; \frac {q^{2r^2k^2}}{\eta(q)} +
\sum_{k\in\Z}\; \frac {q^{2(rk+\frac{x_0}{\pi})^2}}{\eta(q)}\quad
\ \ \mbox{\rm for}\ \ |\a\rangle \ = \ |D(x_0)\rangle^{\rm orb} \ ,
\label{Zuntw}\\[2mm]
Z_{\b}(q) & = & \sum_{k=1}^{\infty}\; \frac {q^{2r^2k^2}}{\eta(q)}
+ \sum_{n=0}^{\infty} \ \chi^{\rm Vir}_{4n^2}(q)\quad \ \ \
\mbox{\rm for}\ \ \ |\b\rangle \ =\ | D(\xi), \pm \rangle^{\rm orb}\ ;
\label{Ztw}
\ea
the Neumann partition functions follow when $r$ is replaced with $1/2r$.
The Virasoro characters $\chi^{\rm Vir}_h(q)$ were introduced in Subsection
4.1.2. They coincide with $\eta^{-1} q^h$ if $h\neq
m^2$ for any $m\in \frac12 \Z$, and are given by the difference
(\ref{excepVirchars}) of U(1) characters otherwise.
\smallskip
Since the U(1) current algebra is reduced by the orbifold procedure,
the occurrence of Virasoro characters for twisted boundary states
is not surprising. Indeed, (\ref{Ztw}) is precisely the $\Z_2$-projection
of the circle Dirichlet partition function, the second sum being
the vacuum character of the $\Z_2$-invariant subalgebra of U(1).
On the other hand, the partition functions for untwisted Dirichlet boundary
conditions (\ref{orbDir}) are sums of U(1) characters; the state
space of the corresponding boundary theories is {\em not} $\Z_2$-invariant,
and (\ref{Zuntw}) should be interpreted as the total excitation spectrum of
a superposition of two branes in the circle theory. (Nevertheless,
the boundary states above obey the cluster property with respect to
the reduced set of bulk fields present in the orbifold theory.) The first
sum in (\ref{Zuntw}) describes strings starting and ending on the same
brane, whereas the $x_0$-dependent
characters are associated with excitations of open strings stretching
between the Dirichlet brane at $x_0$ to the one at $-x_0$, up to
identification of strings running in opposite directions. The corresponding
boundary fields are induced by the bulk-boundary OPE of the twist fields
$\sigma_{0,\pi r}(z,\bz)$ in the bulk \cite{OsAff}.
\smallskip
The marginal boundary operator content of the orbifold models, too,
depends on $r$ and $x_0$. Let us look at untwisted Dirichlet boundary
conditions first (always, the Neumann cases follow upon $T$-dualizing
the radius): For arbitrary radius $r$, one marginal operator $J(x)$
occurs in the parameter-independent part of the partition function
for arbitrary radius $r$, in the vacuum U(1) character. This field
is the boundary value of the original bulk current of the circle theory
which was removed by the orbifolding procedure, and it appears through
the bulk-boundary OPE of the bulk fields $\cos(\frac{k}{r}X)$ with
$k\neq 0$,
\begin{equation}} \def\ee{ \end{equation} \cos \bigl({\textstyle \frac{k}{r}} X(z,\bar z)\bigr) \ = \
\frac{\cos \frac{kx_0}{r}}{(z-\bz)^{\frac{k^2}{4r^2}}}\ {\bf 1} \ - \
\frac{i\, \frac{k}{2r}\, \sin \frac{kx_0}{r}}{(z-\bz)^{\frac{k^2}{4r^2}
- 1 }}\ J(x) \ + \ \dots \label{bbOPEcos} \ee
$J(x)$ is local with respect to all other boundary fields from the
$x_0$-independent part of the spectrum, but non-local wrt.\
those fields which have an $x_0$-dependent conformal dimension,
since the latter arise through the bulk-boundary OPE of twist
fields. Consequently, the second part of the boundary spectrum
(\ref{Zuntw}) is not protected against changes under a perturbation
with $J$. This is perfectly consistent with our findings below that
$J$ simply moves the position $x_0$ of the brane.
For the special radii $r=\frac 1 {\sqrt{2}N}$, two additional
states $\psi^{a}(x),\ a=1,2$, of dimension 1 show up in the
$x_0$-independent part of the partition function (\ref{Zuntw}).
They are self-local and give rise to the familiar periodic boundary
potentials.
The parameter-dependent part of $Z_{\a}(q)$ can contain further
marginal operators if the distance of the two branes -- the
length of the stretched open string --
is appropriately
adjusted: If $r= 1 /(\sqrt{2} N)$ with $N\in \Z$, this fine-tuning
cannot be achieved, but for all other radii there is one marginal
field $\psi'(x)$ whenever $x_0 = 1/ \sqrt2 - k_0 r\;$\ or\ $\;x_0 =
- 1 / \sqrt2 + (k_0+1)\, r$, where $k_0$ is the positive integer
satisfying $\sqrt{2} r k_0 < 1 < \sqrt{2} r (k_0+1)$. Since
these massless excitations originate from the
bulk-boundary OPE of a twist field in the bulk, they will
have non-trivial monodromy wrt.\ the twist field and wrt.\ themselves,
hence they are non-local and do not give rise to analytic deformations.
The picture is simpler for twisted boundary conditions:
There is no field of dimension $h=1$ in the $\Z_2$-invariant
subalgebra of the U(1) current algebra, and the first sum in
(\ref{Ztw}) contributes one marginal operator iff $r=1/(\sqrt{2}N)$;
this is just the boundary field $\sqrt{2}\,\cos(2\sqrt{2} X(x))$,
leading to similar effects as $\psi^a(x)$.
\medskip
When constructing the deformed boundary theories, one encounters
the same general phenomenon as for the unorbifolded models: Some
of the boundary conditions listed above are connected by boundary
perturbations and, at special values of the bulk parameters, new
boundary states are generated that would have been hard to discover
directly without using marginal deformations.
\newline
Let us first focus on the perturbation of the untwisted boundary
states generated by the self-local marginal field $J(x)$. This
deformation does not change the
Dirichlet or Neumann gluing conditions of the orbifold theory.
Furthermore, since $J(x)$ was defined through the bulk-boundary
OPE (\ref{bbOPEcos}) of a bulk field from the untwisted sector,
we conclude that the 1-point functions of bulk twist
fields continue to vanish in the $J$-deformed theory.
To calculate the effect on the 1-point functions of untwisted fields,
we use (\ref{orbDir}) to pass to the underlying circle theory,
where the deformation by a current is easy to handle. However,
observe that the coefficient of $J(x)$ in the bulk-boundary
OPE (\ref{bbOPEcos}) is antisymmetric upon replacing $x_0$ by
$-x_0$, so the definition of the current $J$ picks up an extra
minus sign when acting on the second term in the boundary
state (\ref{orbDir}). The result is that, as long as $0 <
x_0+{\textstyle\frac{\lambda}2} < \pi r$,
\begin{equation}} \def\ee{ \end{equation}
|D(x_0)\rangle^{\rm orb}_{\lambda J} \ = {\textstyle \frac 1{\sqrt{2}}}\,
e^{i \lambda J_0}
\ |D(x_0)\rangle^{\rm circ} + {\textstyle \frac 1{\sqrt{2}}} \,
e^{-i\lambda J_0} \ |D(-x_0)\rangle^{\rm circ} \ = \
|D(x_0+{\textstyle\frac{\lambda}2})\rangle^{\rm orb}\ .
\label{orbDirdef}
\ee
The marginal operator $J(x)$ moves the untwisted orbifold brane along
the interval $]0,\pi r[$. Continuation into the end-points $\xi$ leads to
the boundary states $|D(\xi),+\rangle^{\rm orb}+|D(\xi),-\rangle^{\rm orb}$,
which are inconsistent in the sense that
they violate the sewing relation (\ref{class}) for the twist fields.
In the interior of the interval, however, the deformed theory has the cluster
property in spite of being generated by a non-chiral deformation, and
the spectrum behaves as expected.
The perturbations with $\psi^a(x)$ from untwisted or with the marginal
operator from twisted boundary conditions have to be treated in analogy to
the unorbifolded case, and the technical details were provided in
Subsections 4.1.2 and 4.2.2. Let us, however, have a closer look at
the radius $r = 1/\sqrt{2}$, which is again exceptional. Among the bulk
fields, there is one chiral current, $J^1_{\rm orb}(z) :=\sqrt{2}\,
\cos 2\sqrt{2} X(z)$, and it is easy to see that some of the boundary
conditions (\ref{orbDir}-\ref{orbtwNeu}) preserve this extended symmetry:
\newline
$J^1_{\rm orb}(z)$ satisfies Dirichlet gluing conditions
for $|D(x_0)\rangle^{\rm orb}$ or $|N(\tilde x_0)\rangle^{\rm orb}$
if $x_0=\tilde x_0 = \frac{\pi}{2\sqrt{2}}$. The eight twisted boundary
states (\ref{orbtwDir},\ref{orbtwNeu}) induce Neumann boundary conditions
on $J^1_{\rm orb}(z)$.
\newline
In those cases, $J^1_{\rm orb}$ is a chiral
local field of the full boundary CFT, and it follows from the general theory
developed in Subsections 3.3, 3.4 that the boundary value $J^1_{\rm orb}(x)$
generates deformations which neither change the spectrum nor violate clustering
conditions. We obtain two continuous U(1) families of deformed boundary states,
containing the two untwisted resp.\ the eight twisted boundary states from
above which exist at generic radii. The first family is further enlarged by
$J_{\rm orb}$, see also below.
In the bulk, the $r=1/\sqrt2$ orbifold model is equivalent to
the $r=\sqrt2$ circle theory, see e.g.\ \cite{Gin1}: The identification of the
two models starts from the $r = 1/\sqrt{2}$ circle theory, where the two
different orbifoldings $X \sim -X$ (i.e.\ $J^3 \sim -J^3$) and
$X \sim X + \frac{2\pi r}{2}$ (i.e.\ $J^1 \sim -J^1$) are equivalent
by SU(2)-symmetry; the second procedure leads to a circle model at $r=1/\sqrt{8}$,
which in turn is $T$-dual to the $r=\sqrt{2}$ theory.
It is quite instructive to investigate how this equivalence relates
boundary conditions for the bulk theories, so we give an outline.
The chain of isomorphisms sketched above implies that Dirichlet resp.\
Neumann gluing conditions for $J^3_{\rm circ}$ in the $r_{\rm circ}= \sqrt{2}$
model correspond to Neumann resp.\
Dirichlet conditions for $J^1_{\rm orb}$ in the $r_{\rm orb} = 1/\sqrt{2}$
theory. We have already singled out the latter orbifold boundary states,
and the following partition functions indeed coincide:
\begin{eqnarray}} \def\ea{ \end{eqnarray}
Z_{\a}^{\sqrt{2}} (q) &=& Z_{\b}^{1/\sqrt{2}} (q)
\quad\ \ {\rm for}\ \ |\a\rangle = |N(\tilde x_0)\rangle^{\rm circ}\ , \ \
|\b\rangle = |G({\textstyle \frac{\pi}{2\sqrt{2}}})\rangle^{\rm orb}\ ,
\phantom{xxx}\label{Zciruntw} \\
Z_{\a}^{\sqrt{2}} (q) &=& Z_{\b}^{1/\sqrt{2}} (q)
\quad\ \ {\rm for}\ \ |\a\rangle = |D(x_0)\rangle^{\rm circ}\ , \ \
|\b\rangle = |G(\xi),\pm\rangle^{\rm orb}\ ; \phantom{xxx}
\label{Zcirtw}\ea
the gluing conditions ``$G$'' in the orbifold theory can be both $N$ or $D$,
and the circle parameters take values $x_0 \in [\,0,2\sqrt{2}\pi]$ and
$\tilde x_0 \in [\,0, \pi / \sqrt{2}]$ as usual.
It is possible to pin down the one-to-one equivalence of boundary states by
comparing the 1-point functions of corresponding bulk fields from
circle and orbifold model; e.g., the twist fields $\sigma_0$ and
$\sigma_{\pi r}$ are to be identified with $\sin(\frac1{\sqrt{2}} X)$
and $\cos(\frac 1{\sqrt{2}} X)$ in the $r_{\rm circ}= \sqrt{2}$ theory
on dimensional grounds. We restrict ourselves to some general observations:
As $\tilde x_0$ in (\ref{Zciruntw}) is varied by the deformation
$\lambda J^3_{\rm circ}$,
the corresponding operator $\lambda J^1_{\rm orb}$ generates the
U(1) family of orbifold boundary states mentioned above, with Dirichlet
and Neumann gluing conditions for $J^3_{\rm orb}$ showing up at the opposite
points $\lambda = 0$ and $\lambda = \pi / \sqrt{2}$ (compare the
discussion of the self-dual circle model).
The twisted boundary
states in (\ref{Zcirtw}), too, are members of a family generated by
$J^1_{\rm orb}$. The identification of twist fields with vertex operators
of the circle theory shows that under this deformation -- resp.\ under
the $J^3_{\rm circ}$-perturbation -- the 1-point functions of $\sigma_0$ and
$\sigma_{\pi r}$ can be turned on and off smoothly. We may say that
$J^1_{\rm orb}$ induces a {\em tunneling} of the twisted D-brane states
between the two $\Z_2$-fixed points.
Let us try to match the ``missing'' orbifold boundary states, namely
(\ref{orbDir},\ref{orbNeu}) with $x_0, \tilde x_0 \neq \frac{\pi}{2\sqrt{2}}$,
to boundary conditions of the circle model.
The isomorphism from the $r_{\rm orb} = 1/\sqrt{2}$ to the $r_{\rm circ} =
\sqrt{2}$ theory not only maps $J^1_{\rm orb}(z)$ to $J^3_{\rm circ}(z)$,
but also allows us to identify the non-chiral boundary field
$J^3_{\rm orb}(x) := J_{\rm orb}(x)$ with $\psi^2_{\rm circ}(x)$ and
$J^2_{\rm orb}(x) := \sqrt{2}\sin(2\sqrt 2 X(x))$ with $\psi^1_{\rm circ}(x)$.
The orbifold boundary states in question are
generated by $J_{\rm orb}(x)$ and do not preserve the $J^1_{\rm orb}$-symmetry.
Likewise, the $\psi^2_{\rm circ}(x)$-deformed boundary states
$|N(\tilde x_0);\, \lambda \psi^2\,\rangle^{\rm circ}$ of the
circle model break the $J^3_{\rm circ}$ gluing conditions.
Furthermore, eqs.\ (\ref{Zuntw}) and (\ref{compZPolTh}) show that the
following partition functions coincide,
\begin{equation}} \def\ee{ \end{equation}
Z_{\a}^{\sqrt{2}} (q) = Z_{\b}^{1/\sqrt{2}} (q) \ \ \quad {\rm for}\ \
|\a\rangle = |N(\tilde x_0);\, \lambda \psi^a\,\rangle^{\rm circ} \ , \quad
|\b\rangle = |G(x_0')\rangle^{\rm orb}
\label{partfctmatch}\ee
if $x_0' = \frac{\pi}{2\sqrt{2}} + \frac{\lambda}2\,$.
All this tells us that the family of orbifold boundary states
generated from $|N(\frac{\pi}{2\sqrt{2}})\rangle^{\rm orb}$
by $J^a_{\rm orb}$, $a=1,2,3$,
corresponds to the family of circle boundary states generated from
$|N(0)\rangle^{\rm circ}$ by $J^3_{\rm circ}$ and $\psi^a_{\rm circ}$.
Because of the degeneracy in the partition functions, (\ref{partfctmatch})
does not quite allow us to match individual members of the families, and
a direct comparison of 1-point-functions is virtually impossible because
of the complicated matrix elements $D^{\,j}_{mn}(\Gamma)$
in eq.\ (\ref{comppertbdst}).
Still, we can now draw general conclusions on the $\psi^a_{\rm circ}$-deformed
boundary conditions of the $r_{\rm circ} =\sqrt{2}$ circle model
that were inaccessible before:
Perturbations by $\lambda \psi^a_{\rm circ}$ do preserve the cluster
property for $|\lambda| < \frac{\pi}{\sqrt{2}}$ since the corresponding
orbifold boundary conditions do. It follows that the subfamilies
of boundary conditions generated by $\psi^1$ or $\psi^2$ form
open intervals. Altogether, $J^3_{\rm circ}$ and $\psi^a_{\rm circ}$
generate a patch of moduli space with the topology of the interior
of a solid 2-torus (of a ``bagel''), which can be seen as follows:
As long as we ignore clustering issues, these marginal operators lead to
an SU(2) $\simeq S^3$ of boundary conditions when applied to
$|N(0)\rangle^{\rm circ}$. We have to remove all points
where clustering is violated -- which are characterized by Dirichlet gluing
conditions for $J^3_{\rm circ}$. The latter are broken by any infinitesimal
perturbation with $\psi^a_{\rm circ}$, but $J^3_{\rm circ}$ itself maps
the Dirichlet-like points into each other. Therefore, the
remaining space of clustering boundary conditions is the bagel $S^3 \setminus
S^1$.
A direct isomorphism between orbifold and circle model can be exploited
only for $r_{\rm circ} =\sqrt{2}$. Nevertheless, we expect the same
topology to arise from the non-chiral perturbations at other radii
$r_{\rm circ} =N / \sqrt{2} $, and a similar one in the uncompactified
case (see below). As we have argued before, the
breakdown of cluster properties at finite perturbation strength
$\lambda = \pi / \sqrt{2} $ in $\lambda \psi^a(x)$ should be
due to a finite domain of convergence in the proof of clustering
mentioned in Subsection 3.5.3.
\section{The {\fatma c} =1 brane moduli space, string geometry, and open problems}
Putting together the pieces found in the last section, we can give
a global description of the moduli
space of $c=1$ conformal boundary conditions. This is possible
because we could analyse marginal deformations
to all orders in the perturbation parameter; first order results would
have allowed for a local picture only.
\newline
The (brane) moduli space $\cM_B$ can be viewed as a fibration
over the (closed string) moduli space $\cM_S$ of bulk
CFTs, $ \cM_B = \bigcup_{m\in \cM_S} \bigl(\cM_B\bigr)_m\,$. We focus
on the connected part $\cM_S = \cM_S^{\rm circ} \cup \cM_S^{\rm orb}$
and ignore the three exceptional orbifold points. Both
branches of $\cM_S$ are parameterized as half-lines $\R_{\geq 1/\sqrt{2}}$,
since radii below the self-dual one lead to equivalent theories upon
$T$-duality $r \leftrightarrow 1/2r$ and exchange of Dirichlet and Neumann
boundary conditions.
The topological
type of the fiber $\bigl(\cM_B\bigr)_m$ depends on $m$:
For $m = r_{\rm circ} \in \cM_S^{\rm circ}$, we found
\def\cup\hskip-8.5pt\cdot\hskip8.5pt{\cup\hskip-8.5pt\cdot\hskip8.5pt}
\begin{equation}} \def\ee{ \end{equation}
\bigl(\cM_B\bigr)_{r_{\rm circ}} \ = \ \left\{ \begin{array}{ll}
S^1_r \cup\hskip-8.5pt\cdot\hskip8.5pt S^1_{1/{2r}} &\ \ r_{\rm circ} \neq \frac N{\sqrt{2}} \\[2mm]
S^1_r \cup\hskip-8.5pt\cdot\hskip8.5pt B_{1/2r} &\ \ r_{\rm circ} = \frac N{\sqrt{2}}\,,\ N \geq 2 \\[2mm]
S^3 &\ \ r_{\rm circ} = \frac 1{\sqrt{2}} \\[2mm]
\R \cup\hskip-8.5pt\cdot\hskip8.5pt \widetilde{B} &\ \ r_{\rm circ} = \infty \ \; .
\end{array} \right.
\label{modspacecirc}
\ee
Points $x_0$ in $S^1_r$ label positions of Dirichlet branes, while the
Neumann parameter $\tilde x_0 \in S^1_{1/{2r}}$ distinguishes Wilson lines.
\newline
The spaces
$B_{1/2r} \simeq \mbox{{\it \r{D}}}\,{}^2_{\pi/\sqrt{2}} \times S^1_{1/2r}$
have the topology of the soft interior of a bagel before baking,
cf.\ the end of Subsection 4.3. The boundary of the 2-disk
$D^2_{\pi/\sqrt{2}}$ corresponds to Dirichlet-like mixtures of pure
boundary conditions, which violate the cluster property.
\newline
The uncompactified case emerges in the $N \ra \infty$ limit of the
second line in (\ref{modspacecirc}): The component $\R$ indicates
that the brane can be placed anywhere in the flat target. The second
component $\widetilde{B} \simeq
B_{1/2r}\,/(0 \times S^1_{1/2r}) $ has the topology of an open solid torus with
the central circle shrunk to a point. This can be seen from the matrix
elements $D^{\,j}_{m,-m} (\Gamma^a_{\tilde x_0, \lambda})$ which, for
$\lambda = 0$, become independent of the parameter $\tilde x_0$ --
in agreement with the fact that $\R$ is simply connected. Switching
on a periodic boundary potential, however, lifts the $\tilde x_0$-degeneracy.
Note that the radii indicated as subscripts in (\ref{modspacecirc})
reflect our normalization conventions for the perturbing fields:
Those for $J(x)$ -- dictating the radii of Dirichlet and
Neumann circles -- are fixed by the choices in the bulk -- i.e.\ by
$r_{\rm circ}$ --, and we have put the constant $K$ in (\ref{locmargOPE})
to 1 for the non-chiral deformations.
The fibers over the bulk moduli space of orbifold models have the
following form:
\begin{equation}} \def\ee{ \end{equation}
\bigl(\cM_B\bigr)_{r_{\rm orb}}\ = \ \left\{ \begin{array}{ll}
\widehat{I}_r \cup\hskip-8.5pt\cdot\hskip8.5pt \widehat{I}_{1/2r} &\ \ r_{\rm orb} \neq \frac N{\sqrt{2}} \\[2mm]
\hat I_r \cup\hskip-8.5pt\cdot\hskip8.5pt \widehat{C}_{1/2r} &\ \ r_{\rm orb}= \frac N{\sqrt{2}}\,,\
N \geq 2 \\[2mm]
S^1_{\sqrt{2}} \cup\hskip-8.5pt\cdot\hskip8.5pt B_{1/\sqrt{8}} &\ \ r_{\rm orb} = \frac 1{\sqrt{2}}
\end{array} \right.
\label{modspaceorb}\ee
$\widehat{I}_r$ denotes the disjoint union of the open interval
$\mbox{{\it \r{I}}}\,{
= ]0,\pi r[$
with four extra points for the twisted boundary states.
The spaces $\widehat{C}_r$ arise from the non-chiral orbifold
deformations we did not discuss in detail above. $\widehat{C}_r$
consists of five disjoint parts; one has the topology of an open ball
$D^3_r \simeq \mbox{{\it \r{D}}}\,{
^2_{\pi/\sqrt{2}} \times \mbox{{\it \r{I}}}$
(from the action of $\psi^a_{\rm orb}$ and $J_{\rm orb}$
on the untwisted Neumann boundary states), the four remaining components
are open intervals (from the action of $\sqrt{2}\cos(2\sqrt{2}X)$ on
the twisted Neumann boundary states). These four intervals would form
a single circle (and in fact do at $r_{\rm orb} = 1/\sqrt{2}$)
were it not for the four Dirichlet-like points at which clustering
is violated.
Some of the identifications above are as yet conjectural: Only for
$r_{\rm circ} = \sqrt{2}$ was it possible to give precise arguments
for the ``bagel topology'' in $\bigl(\cM_B\bigr)_m$, but it is highly
plausible that the same picture emerges at the other exceptional radii
$r_{\rm circ}$. The same proviso applies to the pieces $\widehat{C}$
in (\ref{modspaceorb}). Also, we cannot exclude the possibility that
there are further conformal boundary conditions at $c=1$ which
are not continuously connected to Dirichlet or Neumann conditions
for the current.
Except for the jumps in the fiber types occurring at multiples
of $r=1/\sqrt{2}$, the whole space $\cM_B$ is continuous. We have indicated
in Subsection 4.3 how to identify the fibers $S^1 \times B$ over
$r_{\rm circ} = \sqrt{2}$ and $r_{\rm orb} = 1 / \sqrt{2}$, where
$\cM_S^{\rm circ}$ and $\cM_S^{\rm orb}$ intersect. Over the circle
branch, the cones describing
Dirichlet and Neumann conditions for $r_{\rm circ} > 1 / \sqrt{2}$
are glued smoothly into the $S^3$ at the self-dual point. There, we
can continuously ``change the sheet'' from Dirichlet to Neumann conditions
for the free boson.
\newline
This has consequences for generic radii, too. Suppose that Dirichlet
conditions are given for a boson compactified on an arbitrary
radius $r_{\rm circ}$.
Combining bulk and boundary perturbations, we can continuously deform this
situation to Neumann conditions: We first apply a marginal bulk deformation
by $J(z)\bJ(\bar z)$ until we reach the self-dual radius. There, additional
marginal boundary fields are at our disposal to rotate the Dirichlet to Neumann
gluing conditions on the U(1) current $J(z)$. Afterwards, $J(z)\bJ(\bar z)$ may
shift us back to the original radius, where now Neumann conditions hold.
The whole process never leads out the space of conformal field theories, unlike
the D-N-transition by relevant perturbations suggested e.g.\ in \cite{FSW}.
It shows that the dimension of a D-brane may not only change under discrete
transformations like $T$-duality, but is not even a ``homotopy invariant''
for a family of boundary CFTs.
Obviously, the moduli space of boundary conditions or
of D-branes is much richer than that of bulk theories. In view
of the findings of \cite{DKPS} and others that D-branes probe
smaller distance scales in the target than strings with their
soft scattering behaviour can do, we could say that
``space-time'' looks richer at shorter scales.
\newline
Let us try to explore the relation between ``space-time'' or target
geometry and the D-brane moduli space (\ref{modspacecirc},\ref{modspaceorb})
in more detail; after all, the study of marginal deformations should
allow us to {\em derive} geometrical features from CFT, even when starting
from a purely algebraic formulation of the latter.
The $c=1$ models can be written as $\sigma$-models with $S^1$ or $S^1/\Z_2$
as the classical targets. The bulk moduli space $\cM_S$ only discloses that
there are radii $r$ parameterizing the targets, but not their actual shape.
It does tell us, on the other hand, that string effects induce equivalences
between geometrically different targets: By $T$-duality, the CFT-description
of the $\sigma$-models on $S^1_r$ and $S^1_{1/2r}$ are isomorphic, and the
same holds for $S^1_{\sqrt{2}}$ and $S^1_{1/\sqrt{2}}/\Z_2$.
\newline
The fibers of the brane moduli space show much more of the target
geometry -- but still they do not simply coincide with it. Instead,
each fiber $\bigl(\cM_B\bigr)_m$ has more connected components or even
a higher dimension than the target corresponding to $m$. This hints at
``non-geometric'' moduli.
Certainly, space-time supersymmetry can eliminate the corresponding
deformations, by restricting to marginal operators which leave
the (e.g., Dirichlet) gluing automorphism for the current intact.
This would reinstate the standard folklore that ``the moduli space of
BPS D0-branes is just the target of the underlying $\sigma$-model'',
but at the cost of sweeping string-theoretic phenomena under the carpet,
as we will see shortly. {}From a pure string world-sheet point of view,
there is no reason anyway to discard marginal deformations that change
$\Omega_D$. As a consequence, the very notion of world-volume dimension
of a brane becomes ``blurred'' through (open) string effects.
Even without invoking supersymmetry as a selection principle,
our investigation of the $c=1$ examples suggests an interpretation
of marginal boundary deformations that should hold in general:
Only the operators
present for generic values of the bulk moduli correspond to classical
geometric moduli. At generic radius, the Dirichlet-Neumann doubling of
the target circle or interval remains, but this is due to the discrete
string equivalence between $T$-dual radii (note that the same
$\cW$-automorphism governs $T$-duality and the flip of gluing conditions).
One is inclined then to interpret any non-generic marginal perturbation
as a signal for additional ``external'' structures like periodic tachyon
backgrounds, which disappear as soon as an infinitesimal change in the bulk
moduli is introduced. Sometimes, however, different such deformations
are available, and it depends on the direction of the bulk perturbation
which marginal boundary operators survive as ``geometric'' moduli. This
happens at points with an ambiguous classical target interpretation of
the bulk theory, like the meeting point of $\cM_S^{\rm circ}$ and
$\cM_S^{\rm orb}$: While the interval swept out by the
$\psi^2_{\rm circ}$-deformation looks non-geometric from the circle
point of view, it is perfectly ``classical'' within the orbifold
interpretation.
\newline
Had we restricted ourselves to deformations which preserve the gluing
conditions for the currents, we would have discarded $\psi^2_{\rm circ}$
from the start and would have seen no trace of the string-geometric
identification of $S^1_{\sqrt{2}}$ and $S^1_{1/\sqrt{2}}/\Z_2$ in
(\ref{modspacecirc},\ref{modspaceorb}). Likewise, the ``minimal
resolution point'' $r_{\rm circ}= 1/\sqrt{2}$ would have lost all its
significance. If we want the D-brane moduli space to display string
rather than classical geometry, we have to allow for seemingly
non-geometric, gluing condition changing marginal perturbations.
There is a finer hierarchy among the ``generic'' marginal
operators, which reflects the global symmetries of the classical
target. Over the orbifold branch, the D-brane motion is generated by
the ``generic'' non-chiral marginal field $J_{\rm orb}$. These deformations
explore the underlying target even though there is no continuous
target-symmetry left after the orbifold projection from $S^1$ to
$S^1/ \Z_2$ -- but this lack
of symmetry becomes manifest in the partition function: The branes
related by $J_{\rm orb}$ possess open
string spectra which depend on the brane's distance to the
orbifold fixed points.
On the other hand, the periodic brane motion generated by
$J_{\rm circ}$ (or by $J^1_{\rm orb}$ at $r_{\rm orb} = q/\sqrt{2}$)
corresponds to a continuous target symmetry, and
the open string spectrum is indeed invariant under the deformation.
Note that this is just the simplest example of the (abelian or
non-abelian) Lie group structure generally associated with chiral marginal
perturbations. For free bosons in a torus of dimension $d\geq 1$ e.g., we
would find subvarieties of the bulk moduli space over which the brane moduli
space is enlarged from U(1)$^d$ to (products of)
ADE groups of (total) rank $d$ -- see also \cite{GrGu1}.
\smallskip
The status and the interpretation of the higher-dimensional fibers over
exceptional points of the bulk moduli space should certainly be studied
in more detail:
Their topology is classical, but not group-like; they are obtained
via an SU(2)-operation, but the matrix elements in
(\ref{flatpertbdst},\ref{comppertbdst}) are truncated like in ``fuzzy''
spaces of non-commutative geometry \cite{Con} -- see also \cite{FG,FGR}
for relations of NCG to QFT and string theory.
\newline
Non-commutativity in brane moduli spaces was first uncovered in \cite{Wit2};
see also \cite{CDS,DHu}. It should be a general phenomenon occurring
for higher central charge, connected with the interplay between marginal
deformations and continuous parameters in the gluing conditions.
To resolve such additional structures of the moduli space, and also
in order to determine properties of the moduli space like
2-body brane potentials and its metric, finer tools
as in the exemplary treatment of \cite{Bac} will be necessary.
In this way, it should also be possible to make contact to geometry
and gauge theory inspired investigations of brane moduli spaces like
e.g.\ in \cite{DGM}.
It should not be difficult to incorporate perturbations
by boundary condition changing operators into our analysis. The
most prominent example where such operators occur is the condensate
of D1-D5-strings in the D-brane derivation of the Bekenstein-Hawking
entropy, see \cite{StVa} and also \cite{Hor,Mal1} for further details
and references.
\newline
Relevant boundary perturbations are important in string theory since
they trigger the formation of D-brane bound states, see e.g.\
\cite{GNS,Sen1}. The CFT approach allows to study
non-BPS bound states, too, and it was used in \cite{Sen3,Sen2,BG}
to identify an $S$-dual pair of such states. It remains to be seen whether
such results can be recovered directly from relevant perturbation theory,
by studying properties of RG-fixed points. On the other hand, very
interesting recent work by Sen \cite{Sen4,Sen6,Sen7} shows that marginal
boundary perturbations can often be employed as an efficient tool even
for the study of D-brane bound states. Therefore, our general
investigations should have applications to the $K$-theory classification
of branes proposed in \cite{Wit4}.
As a more immediate task, the general constructions discussed in this
paper should be applied to the
supersymmetric case. One of the original motivations behind this work
was to prepare the ground for a geometric interpretation of the
Gepner model boundary states constructed in \cite{ReSc} by purely
algebraic methods. Some promising results in this direction have been
obtained in \cite{GuSa}, where it was also shown how the ``algebraic''
boundary states of \cite{ReSc} can be used to explicitly determine
geometric quantities connected with non-perturbative D-instanton
corrections to the moduli space geometry \cite{BBS,OOY}. We
hope that the present methods are also
useful in establishing further links to supersymmetric cycles
in Calabi-Yau manifolds.
\bigskip
{\bf Acknowledgements:} We would like to thank C.\ Bachas, I.\ Brunner,
R. Dijkgraaf, M.\ Douglas, J.\ Fr\"ohlich, K.\ Gawedzki, O.\ Grandjean,
M.\ Gutperle, N.\ Hambli, W.\ Nahm, B.\ Pioline, A.\ Sagnotti, Y.\ Satoh,
A.\ Schwarz, A.\ Sen, Y.\ Stanev, J.-B.\ Zuber and in particular G.\ Watts
for very useful and stimulating comments. We also benefitted from
conversations with the participants of the DESY-workshop on {\em Conformal
field theory of D-branes} \footnote{for further information see
{\tt http://www.desy.de/$\sim$jfuchs/CftD.html}}
which provided a unique opportunity to discuss various topics
related to this work.
\newcommand{\sbibitem}[1]{\vspace*{-1.5ex} \bibitem{#1}}
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|
proofpile-arXiv_065-8054 | \section{#1}}
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proofpile-arXiv_065-8106 | \section{Introduction}
{\it Introduction}.---Despite much experimental evidence of
pseudogap phenomena
in the underdoped cuprates, their microscopic mechanism is not
understood.\cite{randeria97}
However, a pairing precursor as the origin of the pseudogap is one
prominent possibility.\cite{exp}
An active endeavor has been to incorporate
strong pairing fluctuations to account for the pseudogap phenomena,
especially above the superconducting critical temperature
$T_c$.\cite{maly,varlamov}
In this report, we take a different tack
and study the order parameter fluctuation effects in the superconducting
state.
From a phenomenological standpoint, the pseudogap state can be considered
as a superconductor whose phase coherence is destroyed by strong phase
fluctuations whereas the gap is robust.\cite{Emery}
Therefore, it will bare similarity
to a superconducting state with strong order parameter phase and
amplitude fluctuations.
Here we do not attempt to reproduce pseudogap phenomena since we
study fluctuations in the weak-coupling BCS theory below $T_c$ and do not
consider vortex
pair unbinding transition, but
much of the qualitative trend is expected to pertain to the pseudogap state.
Using the effective low-energy theory approach,
we may describe the
problem with relatively few physical parameters and separate
the effect of order parameter phase and amplitude fluctuation
effects.\cite{ours,xo}
We address the following two issues concerning the fluctuation effects.
First, we examine the effect of the order parameter fluctuations on the
size of the spectral gap below $T_c$.
It has been shown that the fluctuations reduce the magnitude of the
order parameter and the critical temperature.\cite{varlamov,varlamov1,smith}
In this paper we also show the reduction in the
spectral gap in a wide temperature range. For simplicity we do not the
include Coulomb interaction
or disorder although they would alter the form of the order parameter
fluctuations and further modify our result if properly
included.\cite{varlamov,varlamov1}
Secondly, we examine the angular variation of the fluctuation effect.
Angle-resolved photoemission spectroscopy (ARPES) data on underdoped cuprates
show that above
or below $T_c$ the shape of the gap near the node significantly deviates
from the simple $d$-wave shape.\cite{norman,mesot}
We show that this
may be due to the fluctuation of the phase rather than the amplitude
of the order parameter, and moreover that the amplitude fluctuation effect is
the strongest near the antinode. Also we discuss the angular variation
of the quasiparticle lifetime.
{\it Low-energy effective theory}.---We consider the weak-coupling mean-field
BCS theory
in which the pairing potential gives
a $d$-wave order parameter which is effective only within the
momentum thickness $2\Lambda \ll k_F$ around the Fermi surface.
Then the effective fermion Hilbert space is a thin momentum shell of a
characteristic thickness $2\Lambda $ around the Fermi surface where
$\Lambda \ll k_F$.
For convenience, we coarse-grain the momentum shell into small boxes
and label them with an angular variable $\phi$.
The effective action of the fermions and the order
parameter is
\begin{eqnarray}
S_{\rm eff} &=&\int d^2x \int_0^\beta d\tau \left[\sum_{\phi,\sigma}
c^{\dag}_\sigma(\phi ;{\bf x},\tau)
\left(\partial_\tau +{\nabla^2 \over 2m} -\mu\right)
c_\sigma(\phi ;{\bf x},\tau)\right. \nonumber \\
& & + \sum_{\phi} \Psi^*({\bf x},\tau)
w({\phi})
c_\downarrow (\phi ;{\bf x},\tau)
c_\uparrow (\phi+\pi ;{\bf x},\tau) +{\rm h.c.} \nonumber \\
& & \left. -{1\over g}\Psi^*({\bf x},\tau)\Psi({\bf x},\tau)
\right]~,
\end{eqnarray}
where $\Psi$ is the superconducting order parameter introduced via
Hubbard-Stratonovich transformation to decouple the pairing interaction.
In the above we assume a pairing potential of the form
$V(\phi ,\phi^\prime )=g~w(\phi)w(\phi^\prime)$
where $w(\phi) = \cos 2\phi $ and $g<0$ which produces a $d$-wave
order parameter. It is understood that in writing
$c_\sigma(\phi ;{\bf k},\tau)$, the momentum $\bf k$ lives only inside
the small box labeled by the angular variable $\phi $ near the Fermi
surface.
In order to explicitly separate the order parameter phase and amplitude
degrees of freedom, we re-express $\Psi({\bf x},\tau)=\Delta({\bf x},\tau)
e^{i\theta({\bf x},\tau)}$ where $\Delta({\bf x},\tau)$ takes a real value.
In the mean-field approximation, we replace $\Delta({\bf x},\tau)$
with $\Delta_0$ and obtain a d-wave gap $\Delta(\phi)=\Delta_0
\cos 2\phi $ using the following self-consistent gap equation:
\begin{equation}
{1\over |g|} = T\sum_{\omega}\sum_{\phi ,{\bf k}}
{w^2(\phi) \over \omega ^2 +\xi_{\bf k}^2 +\Delta^2(\phi)}~,
\label{gapeq}
\end{equation}
where $\xi_{\bf k}=k^2/2m -\mu $. The momentum summation above is
constrained by the condition $|\xi_{\bf k}| < v_F \Lambda $.
Here we consider the fluctuation around the mean-field
value and re-express $\Delta({\bf x},\tau) = \Delta_0+d({\bf x},\tau)$.
Then we perform a gauge transformation
$ \psi_\sigma({\bf x},\tau) = c_\sigma({\bf x},\tau)
e^{-i\theta ({\bf x},\tau)/2}$, to couple the phase fields to the
fermions explicitly.
The resulting effective action is expressed in terms of the Nambu
spinor notation,
$\hat{\psi} = (\psi_{\uparrow}, \psi^{\dag}_{\downarrow}) $, as
$S_{\rm eff}= S_0 + S_I$,
with
\begin{eqnarray}
S_0&=& T\sum_{\omega}
\sum_{\phi ,{\bf k}} \hat{\psi}^{\dag} \hat{G}_0^{-1} \hat{\psi}
\\ \nonumber
&& +\int_0^\beta d\tau
\int d^2x \big{\{} \, {n_f \over 8m} \,[\nabla \theta ({\bf x},\tau )]^2
+{1\over g} [d({\bf x},\tau)]^2 \big{\}}
\label{Act0}
\end{eqnarray}
and
\begin{eqnarray}
S_I &=& T\sum_\omega T\sum_\nu \sum_{\phi, \bf k,q}
\hat{\psi}^{\dag}(\phi ; {\bf k},\omega)\Big{\{}
{1\over 2}[-\nu +i{\bf v}_F(\phi)\cdot {\bf q}
]\theta({\bf q},\nu) \nonumber \\
&& +w(\phi)
\hat{d}({\bf q},\nu)\Big{\}}\hat{\psi}(\phi ; {\bf k-q},\omega -\nu)
~.
\label{ActI}
\end{eqnarray}
Here
\begin{equation}
\hat{G}_0^{-1} = \left(
\begin{array}{cc}
i\omega - \xi_{\bf k} & \Delta_0 w(\phi) \\
\Delta_0 w(\phi) & i\omega + \xi_{\bf k} \\
\end{array}
\right),
\end{equation}
and $ \hat{d}(\nu , {\bf q}) =\hat{\sigma}_x~{d}(\nu , {\bf q})$,
with $\hat{G}_0$ the bare Green's function for the neutral fermions.
In the above, we approximate $\xi_{\bf k}=k^2/2m -\mu $ as
$\xi_{\bf k} \approx v_F( |{\bf k}| -k_F)$ and ${\bf v}_F(\phi)$
is the Fermi velocity in the $\phi$ direction.
In building this effective theory
we have not considered vortex pair unbinding which
leads to the Kosterlitz-Thouless transition.
This is justified well outside the fluctuation regime. We also assume that
we are in the temperature range where the BCS mean-field theory is justified,
namely, that $\delta T/T_c \gg \Delta_0/E_F $
in a two dimensional clean superconductor.\cite{AL}
One thing we observe from the form of the effective theory is that
the strength of the coupling between fermions and the amplitude
fluctuations has an
angle-dependence; the amplitude fluctuation effect
is suppressed near
the gap nodes, as is evident in Eq. (\ref{selfE0}) where the
self-energy correction is multiplied by a factor of $w^2(\phi)$.
The strength of the coupling to the phase fluctuations is not suppressed
at the node, however.
In studying the finite temperature superconducting to normal state
transition, it should be sufficient to consider only the static
fluctuations of the phase, and we may suppress the time-dependence
in $\theta $ and $d$ and retain only the spatial fluctuations.
Eq. (\ref{ActI}) is then modified as
\begin{eqnarray}
S_I &=& T\sum_\omega \sum_{\phi, \bf k,q}
\hat{\psi}^{\dag}(\phi ;{\bf k},\omega)\Big{\{}
{1\over 2}i{\bf v}_F(\phi)\cdot {\bf q}
\theta({\bf q}) \nonumber \\
&& +w(\phi)
\hat{d}({\bf q})\Big{\}}\hat{\psi}(\phi ; {\bf k-q},\omega)
~.
\label{ActIm}
\end{eqnarray}
The simplified form above, however, does not produce reliable results
near zero temperature.
{\it Fermion single-particle properties}.---In evaluating the quasiparticle
self-energy by perturbative expansions,
we take advantage
of the fact that the effective theory resides in the thin shell around
the large Fermi surface and select the diagrams which are of leading
order in
$\Lambda /k_F$,
which amounts to summing over the ring diagrams
in calculating the order parameter correlation function.
The fermion self-energy can be obtained self-consistently from the
Dyson equation.
We first evaluate the correlation functions of the amplitude
fluctuations:
\begin{eqnarray}
\langle d({\bf q})~d({-\bf q})\rangle _{\rm ring} &=&
{g~T\over 1 + {g}\Pi_{dd}({\bf q},0)} ~,
\label{ddcor}
\end{eqnarray}
where
\begin{eqnarray}
\Pi_{dd}({\bf q},\nu) &=&
{1\over 2}T\sum_{\omega }\sum_{\phi , \bf k}
w^2(\phi) {\rm Tr} \left[ \hat{G}_0(\phi ;{\bf k},\omega)
\hat{\sigma}_x \right. \nonumber \\
&& \left. \times \hat{G}_0(\phi ;{\bf k+q},\omega+\nu)\hat{\sigma}_x
\right] ~.
\label{Pdd}
\end{eqnarray}
From Eq. (\ref{ddcor}) we obtain
$\langle d({\bf q})~d({-\bf q})\rangle _{\rm ring}={T/( a+b~q^2)}$
where
the temperature-dependent coefficients $a$ and $b$ can be evaluated
by carefully expanding $\Pi_{dd}$ in $\bf q$ from Eq. (\ref{Pdd}).
If we only consider the spatial fluctuations in the order parameter,
the $\langle d~\theta\rangle $ terms are zero
in the Gaussian approximation.
Also the phase fluctuation has the following well-known correlation
function\cite{ours}
\begin{equation}
\langle \theta({\bf q})\theta({\bf -q})\rangle _{\rm ring}=
{4mT \over n_s(T)~q^2}~,
\end{equation}
where $n_s(T)$ is the superfluid density at temperature $T$.
Now we can determine the quasiparticle self-energy correction using
the self-consistent Dyson equation, neglecting the vertex corrections:
\begin{eqnarray}
\hat{\Sigma}(\phi ;{\bf k}, \omega) &\approx &
\sum_{\bf q} \left\{ {1\over 4} \left[ {\bf v}_F(\phi)\cdot {\bf q} \right] ^2
\langle \theta({\bf q}) \theta({\bf - q})\rangle _{\rm ring}
\right. \\ \nonumber
&& + w^2(\phi) \langle d({\bf q}) d({\bf - q}) \rangle _{\rm ring}
\bigg{\}} \hat{G}(\phi ;{\bf k-q}, \omega -\nu)~,
\label{selfE0}
\end{eqnarray}
where $\hat{G}$ is the full fermion Green's function, given
self-consistently by $\hat{G}^{-1} = \hat{G}^{-1}_0 - \hat{\Sigma}$.
In general the self-energy has both a momentum and frequency dependence,
but we focus on the behavior of the self-energy near the Fermi
surface, assuming that it varies smoothly near the Fermi surface. Therefore
we neglect the $\xi_{\bf k}$-dependence so that the only
momentum dependence is through the angle $\phi$ on the
Fermi surface.
Then we can approximately obtain the self-energy:
\begin{eqnarray}
\hat{\Sigma}(\phi ,\omega) &\approx & \left\{
{4mT\over n_s(T)}\,{1\over 16\pi} \ln \left[
{\Lambda ^2 + \tilde{\Delta}^2(\phi)+\tilde{\omega} ^2 \over
\tilde{\Delta}^2(\phi)+\tilde{\omega} ^2} \right] \right. \nonumber \\
&& +\left. \sum_{\bf q} {T~w^2(\phi)\over a+b~q^2}~{1\over
\tilde{\omega} ^2 +({\bf v}_F(\phi)\cdot {\bf q})^2 +\tilde{\Delta}^2(\phi)}
\right\} \nonumber \\
&& \times \left(
\begin{array}{cc}
-i\tilde{\omega} & \tilde{\Delta}(\phi) \\
\tilde{\Delta}(\phi) & -i\tilde{\omega} \\
\end{array} \right),
\label{selfE}
\end{eqnarray}
where $\tilde{\omega}$ and $\tilde{\Delta}$ can be calculated
self-consistently by
$\hat{G}^{-1}(\phi ;{\bf k},\omega) = \hat{G}_0^{-1}(\phi ;{\bf k},\omega)
-\hat{\Sigma}(\phi, \omega)$.
From the self-energy obtained in Eq. (\ref{selfE}), by analytically
continuing
the frequency $i\omega \rightarrow
\omega +i\eta $,
we can calculate various single-particle
properties such as density of states (DOS), spectral functions,
and single-particle
scattering rates. In this paper we focus on DOS since
it is gauge-invariant and measurable via the tunneling spectroscopy\cite{sts}
or the momentum-integrated ARPES data\cite{photo}.
We are especially interested in the
angle-resolved DOS:
\begin{equation}
N(\phi ,\omega) = -{1\over \pi} {\rm Im}\int d\xi _{\bf k}~ {\rm Tr}~
\hat{G}(\phi ;{\bf k},\omega)~,
\end{equation}
as it gives information about the angular variation of the fluctuation
effect.
Throughout this paper, we set the relevant energy scales
$\Lambda \approx 10\Delta _0(T=0)/v_F $ and $E_F \approx 5 \Lambda v_F$ so that
we are well in the BCS weak-coupling regime; these relative energy scales
give much
stronger pairing strength than ordinary superconductors but significantly
weaker than the cuprates.
With the energy scales so chosen, we may estimate the regime of the
validity of the mean-field theory. If we apply the Ginzburg criterion,
namely, $|\Delta_0(T)|^2 \gg \langle d({\bf x})~d({\bf x})\rangle $,
which may be estimated from Eq. (\ref{ddcor}),
the mean-field theory breaks down only near $T/T_c \sim 0.98 $.
Therefore, the BCS framework is reliable in most of the temperature
range that we consider.
In Fig. \ref{TDOS} we show the total DOS. As the temperature
increases, the DOS peak is widely smeared. Very close to
$T_c$, the DOS peak has almost disappeared and the spectral gap is only
manifested by the depletion in the DOS around $\omega =0$ as compared to
the normal state DOS.
Figure \ref{peak} shows the DOS peak position as a function of temperature;
in most of the temperature range, we can interpret the peak position roughly
as the spectral gap.
This figure shows that
the spectral gap is reduced in a wide temperature range due to the order
parameter fluctuations.
Near $T/T_c\approx 0.98$, the DOS peak structure has almost disappeared,
and the DOS maxima do not have a meaning as the spectral gap. Therefore,
we need to
estimate the size of the spectral gap from the width of the DOS depletion
in this case.
It is difficult to study the evolution of the spectral gap
through $T_c$ in this framework because the mean-field theory breaks down
sufficiently close to $T_c$ as discussed above.
Also the result near zero temperature is not reliable
due to the negligence of the time-dependence of the fluctuations.
The angle-dependent DOS peak near $T_c$ is shown in Fig.
\ref{dp45}.
At $T\ll T_c$, the DOS peak contour follows the $d$-wave shape.
As the temperature approaches $T_c$, we observe that the DOS
is widely smeared to low-energy states especially near the antinode
($\phi=0$).
Figure \ref{dp45} shows that
the shape of the angle-resolved gap (DOS peak curve) is deformed
from the original $d$-wave shape near the node. We argue that the
downward bending of the DOS peak curve near the node ($\phi = \pi/4$)
is due to the phase fluctuations since the amplitude fluctuations alone
do not cause the downward bending as illustrated in the same figure.
We find that the angle-dependence of the spectral gap near the node is
\begin{eqnarray}
|\tilde{\Delta}(\phi)| &\approx & \Delta_0(T) |\cos(2\phi)| \\
&&~\times \left\{ 1-{mT\over2\pi n_s(T)}
\ln \left[{\Lambda\over \Delta_0(T)|\cos(2\phi)|}
\right] \right\}~, \nonumber
\end{eqnarray}
and the slope of the gap near the node is reduced.
It has similarity to the shape of the gap obtained by ARPES on
underdoped $\rm Bi_2Sr_2CaCu_2O_{8+\delta}$ (Bi2212) in the superconducting
state,\cite{mesot} although its microscopic origin is not well understood.
Figure \ref{rate} shows angular variation of the scattering rate due to the
order parameter fluctuation.
We observe that the maximum scattering rate occurs near the antinode
and also that the rate decreases as one approaches the node.
This variation is due to the angular dependence of the order parameter
magnitude fluctuations.
This feature may contribute to the anisotropy of the quasiparticle
scattering rate in cuprate superconductors.\cite{photo}
{\it Discussions and Conclusions}.---Here we discuss the qualitative
effects of amplitude and phase fluctuations.
In Fig. 3, we find that if we omit the phase fluctuation effect,
the apparent gap is enhanced. This is because the amplitude
fluctuations tend to increase the gap magnitude. This can be understood
from the Fermi liquid reference frame as following:
The fermion self-energy correction due to the amplitude fluctuations
can be roughly estimated as
\begin{eqnarray}
\Sigma ({\bf p}, \omega) & \approx & -\int d \nu d^2 q
G({\bf p -q},\omega -\nu )~\langle \Delta ({\bf q},\nu)\Delta ({-\bf q},-\nu)
\rangle \nonumber \\
&\approx & {1\over i\omega + \xi_p} \langle \Delta (x) \Delta (x)\rangle ~,
\end{eqnarray}
and therefore the effective spectral gap is enhanced by the fluctuations as
$|\Delta_{\rm eff}|^2 = \langle \Delta (x) \Delta (x)\rangle
=|\Delta_0|^2+\langle \delta\Delta (x) \delta\Delta (x)\rangle$.
The effect of phase fluctuation can be considered as a Doppler shift in
the fermionic spectrum by ${\bf k}_F\cdot {\bf v}_s$ where ${\bf v}_s \sim
\nabla \theta /m$. Due to thermally fluctuating superfluid velocity,
the DOS near the gap
is now shifted
since the energy levels at the gap nodes are enhanced.
As a result, more states would be occupied near the nodes, and hence
decrease in the slope of the gap nodes as shown in Fig. 3.
On including the vortex pair unbinding transition of the BKT type,
which gives stronger phase fluctuation effects, we can obtain a
Fermi arc-like phenomenon.\cite{ours,franz98}
In order to obtain the correct effect on the size of the spectral gap,
both the phase and amplitude fluctuations have to be self-consistently
taken into account. The total effect is reduction in the spectral gap as
shown in Fig. 2 and 3.
However, the spectral gap
may not be equal to the order parameter magnitude especially if the
fluctuation is strong,\cite{kosztin} and hence more careful
study is needed to separate these two quantities.
The results presented in this report are equally well applicable to any
unconventional superconducting symmetry. In principle, any superconductor
would have a window of temperatures near $T_c$ where such fluctuations are
visible, depending on the pairing strength and the superfluid density.
In case of underdoped cuprates,
however, it would be essential to include the effect of vortex pair
unbinding in the pseudogap state, due to the small superfluid density.
The result of this report may nevertheless pertain to its superconducting
state. Indeed, a recent
observation of the deformed gap shape
in Bi2212,\cite{mesot} which is only observed in
underdoped regime,
may be related to the order parameter fluctuation effects,
considering that the phase fluctuations are more important in underdoped
cuprates due to reduced superfluid density. Since the
microscopic origin of this deformation is not understood, further experimental
investigation on the temperature variation of the gap anisotrpy would
be desirable.
Some of the above features are shown to be shared by other
theoretical results in the normal state counterpart.
For instance,
the form of the density of states obtained above is similar to that above $T_c$
when the Gaussian pairing fluctuations are incorporated.\cite{old}
Also a similar but much more pronounced deformation of the $d$-wave spectral
gap was obtained above $T_c$ using a self-consistent conserving
approximation.\cite{jan}
The author thanks Alan Dorsey and Rob Wickham for helpful discussions and
comments.
This work was supported by the
National High Magnetic Field Laboratory and by NSF grant DMR 96-28926.
| 2024-02-18T23:40:12.572Z | 1999-01-18T21:52:25.000Z | algebraic_stack_train_0000 | 1,658 | 3,193 |
|
proofpile-arXiv_065-8147 | \section*{Acknowledgments}
This work was supported in part by the National Science Foundation
under grant PHY/94-07195 and NASA under grant NAGW-931, and GF
acknowledges discussions at the Institute for Theoretical Physics
Program on Astrophysical Turbulence, April-June, 2000.
| 2024-02-18T23:40:12.690Z | 2000-10-09T05:07:11.000Z | algebraic_stack_train_0000 | 1,664 | 43 |
|
proofpile-arXiv_065-8220 | \section{Introduction}
There has recently been considerable interest in the Galactic
evolution of the abundances of the light elements Li, Be and B
(Feltzing \& Gustafsson 1994; Reeves 1994; Cass\'e et al. 1995;
Fields et al. 1994,1995; Ramaty et al. 1996,1997; Vangioni-Flam et
al. 1998). The Be abundance is particularly interesting because this
element is thought to be produced exclusively by spallation reactions
involving collisions between nuclei of the CNO group of elements and
protons or alpha particles at energies greater than about $30\,\rm
MeV$ per nucleon (MeV/n). Thus the evolution of the Be abundance
contains information about the particle acceleration and cosmic ray
history of the Galaxy.
The evolution of the Be abundance, and indeed the evolution of all
elemental abundances, has to be deduced from observations of the
fossil abundances preserved in the oldest halo stars. Advances in
spectroscopy over the last decade have greatly improved the quality of
the data available (Duncan et al. 1992,1997; Edvardsson et al. 1994;
Gilmore et al. 1992; Kiselman \& Carlsson 1996; Molaro et al. 1997;
Ryan et al. 1994) and the main result is easily summarized~: in old
halo stars of low metallicity, the ratio of the Be abundance to the
Iron (Fe) abundance appears constant, that is to say the Be abundance
rises {\em linearly} with the Fe abundance.
This has been a surprising result. Naively one had expected that,
because Be is a secondary product produced from the primary CNO
nuclei, its abundance should vary {\em quadratically} as a function of
the primary abundances at low metallicities. Indeed, considering that
the cosmic rays (CRs) responsible for the Be production are somehow
related to the explosion of supernovae (SNe) in the Galaxy, it is
natural to assume that their flux is proportional to the SN rate, $\mathrm{d}
N_{\mathrm{SN}}/\mathrm{d} t$. Now since the number of CNO nuclei present in
the Galaxy at time $t$ is proportional to the total number of SN
having already exploded, $N_{\mathrm{SN}}(t)$, the Be production rate
has to be proportional to $N_{\mathrm{SN}}\mathrm{d} N_{\mathrm{SN}}/\mathrm{d} t$.
Therefore, the integrated amount of Be grows as $N_{\mathrm{SN}}^{2}$,
that is quadratically with respect to the ambient metallicity (C,N,O
or Fe, assumed to be more or less proportional to one another).
The above reasoning, however, relies on two basic assumptions that
need not be fulfilled~: i) the CRs recently accelerated interact with
all the CNO nuclei already produced and dispersed in the entire Galaxy
and ii) the CRs are made of the ambient material, dominated by H and
He nuclei. Instead, it might be i) that the proton rich CRs recently
accelerated interact predominantly with the freshly synthesized CNO
nuclei near the explosion site and ii) that a significant fraction of
the CNO rich SN ejecta are also accelerated. In both cases, a linear
growth of the Be abundance with respect to Fe or O would arise very
naturally, since the number of Be-producing spallation reactions
induced by each individual supernova would be directly linked to its
local, individual CNO supply, independently of the accumulated amount
of CNO in the Galaxy.
In fact, as emphasised by Ramaty et al. (1997), the simplest
explanation of the observational data is to assume that each
core-collapse supernova produces on average 0.1~\hbox{$M_{\odot}$}~of Fe, one to
few \hbox{$M_{\odot}$}~of the CNO elements and $2.8\times10^{-8}\,\hbox{$M_{\odot}$}$ of Be,
{\em with no metallicity dependence}. Clearly if this is the case and
the production of Be is directly linked to that of the main primary
elements, the observed linear relation between Be and Fe will be
reproduced whatever the complications of infall, mixing and outflow
required by the Galactic evolution models. On the other hand,
although the simplest explanation of the data is clearly to suppose a
primary behaviour for the Be production, it is possible that this
could be an artifact of the evolutionary models (as argued, e.g., by
Casuso \& Beckman 1997).
Most work in this area has attempted to deduce information about
cosmic ray (or other accelerated particle populations) in the early
galaxy by working backwards from the abundance observations. While
perfectly legitimate, our feeling is that the observational errors and
the uncertainties relating to Galactic evolution in general make this
a very difficult task. We have chosen to approach the problem from
the other direction and ask what currently favoured models for
particle acceleration in supernova remnants (SNRs) imply for light
element production. This is in the general spirit of recent
calculations of the $\pi^0$-decay gamma-ray luminosity of SNRs (Drury
et~al. 1994) and the detailed chemical composition of SNR shock
accelerated particles (Ellison, Drury \& Meyer, 1997) where we look
for potentially observable consequences of theoretical models for
cosmic ray production in SNRs.
Interestingly enough, the study of particle acceleration in SNRs
suggests that both alternatives to the naive scenario mentioned above
do occur in practice, as demonstrated qualitatively in
Sect.~\ref{PartAccInSNRs}. The first of these alternatives, namely
the local interaction of newly accelerated cosmic rays in the vicinity
of SN explosion sites, has already been called upon by Feltzing \&
Gustaffson (1994), as well as the second, the acceleration of enriched
ejecta through a SN reverse shock, by Ramaty et al. (1997). However,
no careful calculations have yet been done, taking the dynamics of the
process into account, notably the dilution of SN ejecta and the
adiabatic losses. Yet we show below that they have a crucial
influence on the total amount of Be produced, and that a
time-dependent treatment is required. Indeed, the evolution of a SNR
is essentially a dynamical problem in which the acceleration rate as
well as the chemical composition inside the remnant are functions of
time. The results of the full calculation of both processes and the
discussion of their implications for the chemical evolution of the
Galaxy will be found in an associated paper (Parizot \&~Drury 1999).
Here we present simple analytical calculations which provide an
accurate understanding of the dynamics of light element production in
SNRs and elucidates the role and influence of the different
parameters, notably the ambient density.
Although Li and B are also produced in the processes under study, we
shall choose here Be as our `typical' light element, because nuclear
spallation of CNO is thought to be its only production mechanism,
while Li is also (and actually mainly) produced through $\alpha +
\alpha$ reactions, $^{7}$Li may be produced partly in AGB stars (Abia
et~al. 1993), and $^{11}$B neutrino spallation may be important as
seems to be required by chemical evolution analysis (Vangioni-Flam
et~al. 1996). In order to compare our results with the observations,
we simply note that, as emphasized in Ramaty et al. (1997), the data
relating to the Galactic Be evolution as a function of [Fe/H] indicate
that $\sim 1.6\,10^{-6}$ nuclei of Be must be produced in the early
Galaxy for each Fe nucleus. Therefore, if Be production is indeed
induced, directly or indirectly, by SNe explosions, and since the
average SN yield in Fe is thought to be $\sim 0.11\,\hbox{$M_{\odot}$}$, each
supernova must lead to an average production of $\sim 3.8\,10^{48}$
nuclei (or $\sim 2.8\,10^{-8}\,\hbox{$M_{\odot}$}$) of Be, with an uncertainty of
about a factor of~2 (Ramaty et al. 1997). We adopt this value as the
`standard needed number' of Be per supernova explosion. To state this
again in a different way, for an average SN yield in CNO of, say,
$\sim 1\,\hbox{$M_{\odot}$}$, the required spallation rate per CNO atom is $\sim
3\,10^{-8}$.
\section{Particle acceleration in SNRs}
\label{PartAccInSNRs}
It is generally believed that cosmic ray production in SNRs occurs
through the process of diffusive shock acceleration operating at the
strong shock waves generated by the interaction between the ejecta
from the supernova explosion and the surrounding medium. Significant
effort has been put into developing dynamical models of SNR evolution
which incorporate, at varying levels of detail, this basic
acceleration and injection process (one of the major advantages of
shock acceleration is that it does not require a separate injection
process). Qualitatively the main features can be crudely summarised as
follows.
In a core collapse SN the collapse releases roughly the gravitational
binding energy of a neutron star, some $10^{53}\,\rm erg$, but most of
this is radiated away in neutrinos. About $\hbox{${E_{\rm SN}}$} = 10^{51}\,\rm erg$ is
transferred, by processes which are still somewhat obscure, to the
outer layers of the progenitor star which are then ejected at
velocities of a few percent of the speed of light. Initially the
explosion energy is almost entirely in the form of kinetic energy of
these fast-moving ejecta. As the ejecta interact with the surrounding
circumstellar and interstellar material they drive a strong shock
ahead into the surrounding medium. The region of very hot high
pressure shocked material behind this forward shock also drives a
weaker shock backwards into the ejecta giving rise to a characteristic
forward reverse shock pair separated by a rather unstable contact
discontinuity.
This initial phase of the remnant evolution lasts until the amount of
ambient matter swept up by the remnant is roughly equal to the
original ejecta mass. At this so-called sweep-up time, \hbox{${t_{\rm SW}}$}, the
energy flux through the shocks is at its highest, the expansion of the
remnant begins to slow down, and a significant part of the explosion
energy has been converted from kinetic energy associated with the bulk
expansion to thermal (and non-thermal) energy associated with
microscopic degrees of freedom of the system. The remnant now enters
the second, and main, phase of its evolution in which there is rough
equipartition between the microscopic and macroscopic energy
densities. The evolution in this phase is approximately self-similar
and resembles the exact solution obtained by Sedov for a strong point
explosion in a cold gas.
It is important to realise that the approximate equality of the energy
associated with the macroscopic and microscopic degrees of freedom in
the Sedov-like phase is not a static equilibrium but is generated
dynamically by two competing processes. As long as the remnant is
compact the energy density, and thus pressure, of the microscopic
degrees of freedom is very much greater than that of the external
medium. This strong pressure gradient drives an expansion of the
remnant which adiabatically reduces the microscopic degrees of freedom
of the medium inside the remnant and converts the energy back into
bulk kinetic energy of expansion. At the same time the strong shock
which marks the boundary of the remnant converts this macroscopic
kinetic energy of expansion back into microscopic internal form. Thus
there is a continuous recycling of the original explosion energy
between the micro and macro scales. This continues until either the
external pressure is no longer negligible compared to the internal, or
the time-scales become so long that radiative cooling becomes
important. The time scales for the conversion of kinetic energy to
internal energy and vice versa are roughly equal and of order the
dynamical time scale of the remnant which is of order the age of the
remnant, hence the approximately self-similar evolution.
In terms of particle acceleration the theory assumes that strong
collisionless shocks in a tenuous plasma automatically and
inevitably generate an approximately power law distribution of
accelerated particles which connects smoothly to the shock-heated
particle distribution at `the\-rmal' energies and extends up to a
maximum energy constrained by the shock size, speed, age and magnetic
field. The acceleration mechanism is a variant of Fermi acceleration
based on scattering from magnetic field structures on both sides of
the shock. A key point is that these scattering structures are not
those responsible for general scattering on the ISM, but strongly
amplified local structures generated in a non-linear bootstrap process
by the accelerated particles themselves. As long as the shock is
strong it will be associated with strong magnetic turbulence which
drives the effective local diffusion coefficient down to values close
to the Bohm value. As pointed out by Achterberg et al. (1994) the
extreme sharpness of the radio rims of some shell type SNRs can be
interpreted as observational evidence for this type of effect. The
source of free energy for the wave excitation is of course the strong
gradient in the energetic particle distribution at the edge. Thus in
the interior of the remnant, where the gradients are absent or much
weaker, we do not expect such low values of the diffusion coefficient.
The net effect is that the edge of the remnant, as far as the
accelerated particles are concerned, is both a self-generated diffusion
barrier and a source of freshly accelerated particles. Except at the
very highest energies the particles produced at the shock are
convected with the post-shock flow into the interior of the remnant
and effectively trapped there until the shock weakens to the point
where the self-generated wave field around the shock can no longer be
sustained. At this point the diffusion barrier collapses and the
trapped particle population is free to diffuse out into the general
ISM.
In terms of bulk energetics, the total energy of the accelerated
particle population is low during the first ballistic phase of the
expansion (because little of the explosion energy has been processed
through the shocks) but rises rapidly as $t\approx\hbox{${t_{\rm SW}}$}$. During the
sedov-like phase the total energy in accelerated particles is roughly
constant at a significant fraction of the explosion energy (0.1 to 0.5
typically). However, this is because of the dynamic recycling
described above. Any individual particle is subject to adiabatic
losses on the dynamical time-scale of the remnant, while the
energy lost this way goes into driving the shock and thus generating
new particles, distributed over the whole energy spectrum.
\section{Spallation reactions within SNRs}
\subsection{Qualitative overview}
We now turn to the production of Li, Be and B (LiBeB) by spallation
reactions within a SNR. As emphasized above, there are two obvious
mechanisms. One is the irradiation of the CNO ejecta by accelerated
protons and alphas. It is clear that the fresh CNO nuclei produced by
the SN will, for the lifetime of the SNR, be exposed to a flux of
energetic particles (EPs) very much higher than the average
interstellar flux, and this must lead to some spallation production of
light elements. This process starts at about \hbox{${t_{\rm SW}}$}~with a very intense
radiation field and continues with an intensity decreasing roughly as
$R^{-3}\propto t^{-6/5}$ (where $R$ is the radius of the SNR) until
the remnant dies.
The second process is that some of the CNO nuclei from the ejecta are
accelerated, either by the reverse shock in its brief powerful phase
at $t\approx\hbox{${t_{\rm SW}}$}$ or by some of this material managing to get ahead of
the forward shock. This later possibility is not impossible, but
seems unlikely to be as important as acceleration by the reverse
shock. Calculations of the Raleigh-Taylor instability of the contact
discontinuity do suggest that some fast-moving blobs of ejecta can
punch through the forward shock at about \hbox{${t_{\rm SW}}$} (Jun \& Norman 1996),
and in addition Ramaty and coworkers have suggested that fast moving
dust grains could condense in the ejecta at $t<\hbox{${t_{\rm SW}}$}$ and then
penetrate through into the region ahead of the main shock. In all
these pictures acceleration of CNO nuclei takes place only at about
\hbox{${t_{\rm SW}}$}~and the energy deposited in these accelerated particles is
certainly less than the explosion energy \hbox{${E_{\rm SN}}$}, although it might
optimistically reach some significant fraction of that value (say $\la
10\%$). Crucially the accelerated
CNO nuclei are then confined to the interior of the SNR and will thus
be adiabatically cooled on a rather rapid time-scale, initially of
order \hbox{${t_{\rm SW}}$}.
\subsection{Evaluation of the first process (forward shock)}
From the above arguments, it is clear that SNe do induce some Be
production. Now the question is~: how much? Let us first consider
the irradiation of the ejecta by particles (H and He nuclei)
accelerated at the forward shock during the Sedov-like phase --
process~1. We have already indicated that detailed studies of
acceleration in SNRs show that the fraction of the explosion energy
given to the EPs is roughly constant during the Sedov-like phase and
of order 0.1 to 0.5 or so. Let $\theta_{1}$ be that fraction. Since
the EPs are distributed more or less uniformly throughout the interior
of the remnant, the energy density can be estimated as
\begin{equation}
\hbox{${\cal E}_{\rm CR}$} \approx \frac{3\theta_{1}\hbox{${E_{\rm SN}}$}}{4\pi R^{3}}
\label{EdCR}
\end{equation}
where $R$ is the radius of the remnant and \hbox{${E_{\rm SN}}$}~is the explosion
energy.
To derive a spallation rate from this we need to assume some form for
the spectrum of the accelerated particles. Shock acceleration
suggests that the distribution function should be close to the
test-particle form $f(p)\propto p^{-4}$ and extend from an injection
momentum close to `thermal' values to a cut-off momentum at about
$p_{\mathrm{max}} = 10^{5}\,{\rm GeV}/c$. The spallation rate per
target CNO atom to produce a Be atom is then obtained by integrating
the cross sections
\begin{equation}
\nu_{\rm spall} = \int_{\hbox{${p_{\rm th}}$}}^{p_{\mathrm{max}}}
\sigma v f(p) \mathrm{d} p,
\end{equation}
with the normalisation $\int E(p)f(p)4\pi p^{2}\mathrm{d} p =
\hbox{${\cal E}_{\rm CR}$}$. Looking at graphs of the spallation cross-sections for Be (as
given, e.g., in Ramaty et al. 1997), it is clear that these
cross-sections can be well approximated as zero below a threshold at
about 30--40~MeV/n and a constant value $\sigma_0\simeq 5\times
10^{-27}\,\mathrm{cm}^{2}$ above it. One then obtains roughly~:
\begin{equation}
\nu_{\rm spall} \approx \frac{\sigma_{0}}{mc} \hbox{${\cal E}_{\rm CR}$}
\frac{1-\ln(\hbox{${p_{\rm th}}$}/mc)}{1 + \ln(p_{\rm max}/mc)}
\simeq 0.2 \sigma_0 c \frac{\hbox{${\cal E}_{\rm CR}$}}{m c^2},
\end{equation}
where $\hbox{${p_{\rm th}}$} \simeq mc/5$ is the momentum corresponding to the
spallation threshold and $m$ refers to the proton mass. Fortunately,
for this form of the spectrum the upper cut-off and the spallation
threshold only enter logarithmically. A softer spectrum would lead to
higher spallation yields and a stronger dependence on the spallation
threshold.
Using Eq.~(\ref{EdCR}) and the adiabatic expansion law for the forward
shock radius, $R = R_{\mathrm{SW}}(t/\hbox{${t_{\rm SW}}$})^{2/5}$, we can now
estimate the total fraction of the CNO nuclei which will be converted
to Be during the Sedov-like phase as~:
\begin{eqnarray}
\phi_{1} &=& \int_{\hbox{${t_{\rm SW}}$}}^{\hbox{${t_{\rm end}}$}} 0.2 \sigma_0 c
\frac{\theta_{1}\hbox{${E_{\rm SN}}$}}{mc^{2}} \frac{3}{4\pi R^3} \mathrm{d} t\\
&=& 0.2 \sigma_0 c \frac{\theta_{1}\hbox{${E_{\rm SN}}$}}{mc^{2}} \frac{3}{4\pi
\hbox{${R_{\rm SW}}$}^3} \int_{\hbox{${t_{\rm SW}}$}}^{\hbox{${t_{\rm end}}$}} \left(\frac{t}{\hbox{${t_{\rm SW}}$}}\right)^{-6/5} dt,
\label{phi1Interm}
\end{eqnarray}
or
\begin{equation}
\phi_{1} = \sigma_0 c \frac{\theta_{1}\hbox{${E_{\rm SN}}$}}{mc^{2}}
\frac{\rho_{0}}{\hbox{${M_{\rm ej}}$}} \hbox{${t_{\rm SW}}$}
\left[1 - \left(\frac{\hbox{${t_{\rm SW}}$}}{\hbox{${t_{\rm end}}$}}\right)^{1/5}\right]
\label{phi1}
\end{equation}
where as usual $\rho_{0}$ denotes the density of the ambient medium
into which the SNR is expanding and \hbox{${M_{\rm ej}}$}~is the total mass of the SNR
ejecta. We now recall that the sweep-up time is given in terms of the
SN parameters and the ambient number density, $n_{0} \approx
\rho_{0}/m$, as
\begin{equation}
\hbox{${t_{\rm SW}}$} = \frac{n_{0}^{-1/3}}{v_{\mathrm{ej}}}
\left(\frac{3}{4\pi}\frac{M_{\mathrm{ej}}}{m}\right)^{1/3},
\end{equation}
where $v_{\mathrm{ej}}\approx (2\hbox{${E_{\rm SN}}$}/\hbox{${M_{\rm ej}}$})^{1/2}$ is the velocity of
the ejecta, or numerically~:
\begin{equation}
\hbox{${t_{\rm SW}}$} = (1.4\,10^{3}\,\mathrm{yr})
\left(\frac{M_{\mathrm{ej}}}{10\hbox{$M_{\odot}$}}\right)^{\hspace{-2pt}\frac{5}{6}}
\hspace{-4pt}
\left(\frac{E_{\mathrm{SN}}}{10^{51}\mathrm{erg}}\right)^{\hspace{-2pt}-\frac{1}{2}}
\hspace{-4pt}
\left(\frac{n_{0}}{1\mathrm{cm}^{-3}}\right)^{\hspace{-2pt}-\frac{1}{3}}.
\label{SweepUpTime}
\end{equation}
Replacing in Eq.~(\ref{SweepUpTime}) and using canonical values of $\hbox{${E_{\rm SN}}$} =
10^{51}\,\rm erg$ and $\hbox{${M_{\rm ej}}$} = 10\,\hbox{$M_{\odot}$}$, we finally get~:
\begin{equation}
\phi_{1} \simeq 4\times10^{-10}\,\theta_{1}
\left(\frac{n_{0}}{1\,\mathrm{cm}^{-3}}\right)^{2/3} \left[1 -
\left(\frac{\hbox{${t_{\rm SW}}$}}{\hbox{${t_{\rm end}}$}}\right)^{1/5}\right].
\label{phi1Num}
\end{equation}
Clearly this falls short of the value of order $10^{-8}$ required to
explain the observations, even for values of $\theta_{1}$ as high as
$0.5$. It might seem from Eq.~(\ref{phi1Num}) that very high ambient
densities could help to make the spallation yields closer to the
needed value. This is however not the case. First, the above
estimate does not take energy losses into account, while both
ionisation and adiabatic losses act to lower the genuine production
rates. Second, and more significantly, the ratio $\hbox{${t_{\rm SW}}$}/\hbox{${t_{\rm end}}$}$ (and
\emph{a fortiori} its fifth root) becomes very close to 1 in dense
environments, lowering $\phi_{1}$ quite notably (see
Fig.~\ref{BeYield}). In fact, it turns out that there is no
Sedov-like phase at all in media with densities of order
$10^{4}\,\mathrm{cm}^{-3}$, the physical reason being that the
radiative losses then act on a much shorter time-scale, eventually
shorter than the sweep-up time.
\subsection{Evaluation of the second process (reverse shock)}
\label{SecondProcess}
Let us now turn to the second process, namely the spallation of
energetic CNO nuclei accelerated at the reverse shock from the SN
ejecta and interacting within the SNR with swept-up ambient material.
We have argued above that this reverse shock acceleration is only
plausible at times around \hbox{${t_{\rm SW}}$}~and certainly the amount of energy
transferred to CNO nuclei cannot be more than a fraction of \hbox{${E_{\rm SN}}$}. Let
$\theta_2$ be the fraction of the explosion energy that goes into
accelerating the ejecta at or around \hbox{${t_{\rm SW}}$}, and $\theta_{\mathrm{CNO}}$
the fraction of that energy that is indeed transferred to CNO nuclei.
These particles are then confined to the interior of the remnant where
they undergo spallation reactions as well as adiabatic losses. Let us
again assume that the spectrum is of the form $f(p)\propto p^{-4}$.
Then the production rate of Be atoms per unit volume is approximately
\begin{equation}
0.2 {n_{0} \sigma_0\over m c} {\hbox{${\cal E}_{\rm CR}$}\over 14}
\label{ProdRate2}
\end{equation}
where \hbox{${\cal E}_{\rm CR}$}~now refers to the accelerated CNO nuclei, the factor~14
comes from the mean number of nucleons per CNO nucleus and the factor
0.2, as before, from the $f(p)\propto p^{-4}$ spectral shape (assuming
the same upper cut-off position, but this only enters
logarithmically). Integrating over the remnant volume, we obtain the
spallation rate at \hbox{${t_{\rm SW}}$}~:
\begin{equation}
\frac{\mathrm{d}\mathcal{N}_{\mathrm{Be}}}{\mathrm{d} t} \approx 0.2 \sigma_{0}c
\frac{\theta_{\mathrm{CNO}}\theta_{2}E_{\mathrm{SN}}}{14 m c^{2}}.
\label{ProdRate2bis}
\end{equation}
Now the adiabatic losses need to be evaluated rather carefully. It is
generally argued that they act so that the momentum of the particles
scales as the inverse of the linear dimensions of the volume occupied.
Accordingly, in the expanding spherical SNR the EPs should lose
momentum at a rate $\dot{p}/p = - \dot{R}/R$, reminiscent,
incidentally, of the way photons behave in the expanding universe. In
our case, however, the situation is complicated by the fact that the
EPs do not push directly against the `walls' limiting the volume of
confinement, which move at the expansion velocity, $V = \dot{R}$, but
are reflected off the diffusion barrier consisting of magnetic waves
and turbulence at rest with respect to the downstream flow, and thus
expanding at velocity $\frac{3}{4}\dot{R}$.
To see how this influences the actual adiabatic loss rate, it is safer
to go back to basic physical laws. Adiabatic losses must arise
because the EPs are more or less isotropised within the SNR and
therefore participate to the pressure. Now this pressure, $P$, works
positively while the remnant expands, implying an energy loss rate
equal to the power contributed, given by~:
\begin{equation}
\frac{\mathrm{d} U}{\mathrm{d} t} = - \int\hspace{-4pt}\int_{\mathcal{S}}\vec{F}\cdot\vec{v}
= - \int\hspace{-4pt}\int_{\mathcal{S}}P\mathrm{d} S\times\frac{3}{4}\dot{R}
= - 3\pi R^{2}\dot{R}P,
\label{KinEnTh}
\end{equation}
where $U = \frac{4}{3}\pi R^{3}\epsilon$ is the total kinetic energy
of the particles. Considering that $P = \frac{2}{3}\epsilon$ in the
non-relativistic limit (NR) and $P = \frac{1}{3}\epsilon$ in the
ultra-relativistic limit (UR), Eq.~(\ref{KinEnTh}) can be re-writen
as~:
\begin{equation}
\begin{split}
\frac{\mathrm{d}\epsilon}{\mathrm{d} t} = -\frac{9}{4}P\frac{\dot{R}}{R}
&= - \frac{3}{2}\epsilon\frac{\dot{R}}{R}\quad(\mathrm{NR})\\
&= - \frac{3}{4}\epsilon\frac{\dot{R}}{R}\quad(\mathrm{UR}).
\end{split}
\end{equation}
Finally, dividing both sides by the space density of the EPs and
noting that $E = p^{2}/2m$ in the NR limit, and $E = pc$ in the UR
limit, we obtain the momentum loss rate for individual particles,
valid in any velocity range~:
\begin{equation}
\frac{\dot{p}}{p} = -\frac{3}{4}\frac{\dot{R}}{R}.
\label{AdiabLossRate}
\end{equation}
From this one deduces that at the time when the remnant has expanded
to radius $R$, only those particles whose {\em initial} momenta at
\hbox{${t_{\rm SW}}$}~were more than $(R/\hbox{${R_{\rm SW}}$})^{3/4}\hbox{${p_{\rm th}}$}$ are still above the
spallation threshold. For a $p^{-4}$ distribution function the
integral number spectrum decreases as $p^{-1}$ and thus the number of
accelerated nuclei still capable of spallation reactions decreases as
$R^{-3/4}\propto t^{-3/10}$. For a softer accelerated spectrum the
effect would be even stronger because there are proportionally fewer
particles at high initial momenta.
This being established, we can integrate Eq.~(\ref{ProdRate2bis}) over
time, to obtain the total production of Be atoms~:
\begin{equation}
\mathcal{N}_{\mathrm{Be}} = 0.2 n \sigma_{0} c
\frac{\theta_{\mathrm{CNO}}\theta_{2}\hbox{${E_{\rm SN}}$}}{14 mc^{2}}
\int_{\hbox{${t_{\rm SW}}$}}^{\hbox{${t_{\rm end}}$}} \left(\frac{t}{\hbox{${t_{\rm SW}}$}}\right)^{-3/10}\,\mathrm{d} t
\label{phi2Interm}
\end{equation}
that is~:
\begin{equation}
\mathcal{N}_{\mathrm{Be}} = \frac{2}{7} n \sigma_{0} c
\frac{\theta_{\mathrm{CNO}}\theta_{2}\hbox{${E_{\rm SN}}$}}{14 mc^{2}} \hbox{${t_{\rm SW}}$}
\left[\left(\frac{\hbox{${t_{\rm end}}$}}{\hbox{${t_{\rm SW}}$}}\right)^{7/10} - 1\right].
\end{equation}
Dividing by the total number of CNO nuclei in the ejecta,
$N_{\mathrm{ej,CNO}} = (\theta_{\mathrm{CNO}}/14) N_{\mathrm{ej,tot}}
\simeq (\theta_{\mathrm{CNO}} M_{\mathrm{ej}}/14 m)$, we get the final
result~:
\begin{equation}
\phi_{2} = \sigma_{0}c \frac{\theta_{2}E_{\mathrm{SN}}}{m
c^{2}}\frac{\rho_{0}}{M_{\mathrm{ej}}}\hbox{${t_{\rm SW}}$}
\frac{2}{7}\left(\frac{\hbox{${t_{\rm end}}$}}{\hbox{${t_{\rm SW}}$}}\right)^{7/10}
\left[1 - \left(\frac{\hbox{${t_{\rm SW}}$}}{\hbox{${t_{\rm end}}$}}\right)^{7/10}\right].
\label{phi2}
\end{equation}
Note that we assumed that the mass fraction of CNO in the ejecta is
the same as the energy fraction of CNO in the EPs (which was the
original meaning of $\theta_{\mathrm{CNO}}$). Considering that all
nuclear species have the same spectrum in MeV/n, and thus a total
energy proportional to their mass number, this simply means that the
acceleration process is not chemically selective, in the sense that
the composition of the EPs is just the same as that of the material
passing through the shock.
Numerically, again with $\hbox{${E_{\rm SN}}$}=10^{51}\,\rm erg$ and $\hbox{${M_{\rm ej}}$} =
10\,\hbox{$M_{\odot}$}$, we finally obtain~:
\begin{equation}
\begin{split}
\phi_2 \simeq 1\times 10^{-10}&\theta_{2}
\left(n_0\over 1\,\rm cm^{-3}\right)^{2/3}\\
&\times\left(\frac{\hbox{${t_{\rm end}}$}}{\hbox{${t_{\rm SW}}$}}\right)^{7/10}
\left[1 - \left(\frac{\hbox{${t_{\rm SW}}$}}{\hbox{${t_{\rm end}}$}}\right)^{7/10}\right].
\label{phi2Num}
\end{split}
\end{equation}
\subsection{Relative contribution of the two processes}
\label{RelativeContribution}
It is worth emphasizing the similarity between expressions
(\ref{phi1}) and (\ref{phi2}) that we obtained for the spallation
rates per CNO nuclei by the two processes considered here. This
formal analogy allows us to write down their relative contributions
straightforwardly~:
\begin{equation}
\frac{\phi_{2}}{\phi_{1}} = \frac{\theta_{2}}{\theta_{1}}
\times \frac{2}{7}\left(\hbox{${t_{\rm end}}$}/\hbox{${t_{\rm SW}}$}\right)^{7/10}
\frac{\left[1 - \left(\hbox{${t_{\rm SW}}$}/\hbox{${t_{\rm end}}$}\right)^{7/10}\right]}
{\left[1 - \left(\hbox{${t_{\rm SW}}$}/\hbox{${t_{\rm end}}$}\right)^{1/5}\right]}.
\label{phi2/phi1}
\end{equation}
As is often the case, this similarity is not fortuitous and has a
physical meaning. The two processes may indeed be regarded as `dual'
processes, the first consisting of the irradiation of the SN ejecta by
the ambient medium, and the second of the ambient medium by the SN
ejecta. The `symmetry' is only broken by the dynamical aspect of the
processes. First, of course, the energy imparted to the EPs in both
cases needs not be the same, for it depends on the acceleration
efficiency as well as the total energy of the shock involved (forward
or reverse). This is expressed by the expected ratio
$\theta_{2}/\theta_{1}$. And secondly, in the first process one has
to fight against the dilution of the ejecta -- integration of
$(t/\hbox{${t_{\rm SW}}$})^{-6/5}$, see Eq.~(\ref{phi1Interm}) -- while in the second
process one fights against the adiabatic losses -- integration of
$(t/\hbox{${t_{\rm SW}}$})^{-3/10}$, see Eq.~(\ref{phi2Interm}). This is expressed by
the last factor in Eq.~(\ref{phi2/phi1}).
Clearly the latter decrease of the production rates is the least
dramatic, and the reverse shock process must dominate the LiBeB
production in supernova remnants. However, this conclusion still
depends on the genuine efficiency of reverse shock acceleration, and
once the relative acceleration efficiency $\theta_{2}/\theta_{1}$ is
given, the weight of the first process relative to the second still
depends on the total duration of the Sedov-like phase, appearing
numerically in Eq.~(\ref{phi2/phi1}) through the ratio \hbox{${t_{\rm end}}$}/\hbox{${t_{\rm SW}}$},
which in turn depends on the ambient density, $n_{0}$. The expression
of \hbox{${t_{\rm SW}}$}~as a function of the parameters has been given in
Eq.~(\ref{SweepUpTime}), so we are left with the evaluation of the time,
\hbox{${t_{\rm end}}$}, when the magnetic turbulence collapses and the EPs leave the
SNR, putting an end to Be production. We argued above that
\hbox{${t_{\rm end}}$}~should correspond to the end the Sedov-like phase, when the
shock induced by the SN explosion becomes radiative, that is when the
cooling time of the post-shock gas becomes of the same order as the
dynamical time.
In principle, the cooling rate can be derived from the so-called
cooling function, $\Lambda(T) (\mathrm{erg~cm}^{3}\mathrm{s}^{-1}$),
which depends on the physical properties of the post-shock material,
notably on its temperature, $T$, and metallicity, $Z$~:
\begin{equation}
\tau_{\mathrm{cool}}\approx\frac{\frac{3}{2}k_{\mathrm{B}}T}{n\Lambda(T)},
\label{tauCool}
\end{equation}
where $n$ is the post-shock density, equal to $4n_{0}$ if the
compression ratio is that of an ideal strong shock (nonlinear effects
probably act to increase the compression ratio to values larger
than~4). As for the dynamical time, we simply write
\begin{equation}
\tau_{\mathrm{dyn}} \approx \frac{\dot{R}}{R} \approx \frac{5}{2}\,t.
\label{tauDyn}
\end{equation}
To obtain \hbox{${t_{\rm end}}$}, we then need to solve the following equation in the
variable $t$, obtained by equating $\tau_{\mathrm{cool}}$ and
$\tau_{\mathrm{dyn}}$ given above~:
\begin{equation}
t \approx \frac{3k_{\mathrm{B}}T}{20n_{0}\Lambda(T)},
\label{tEndEq}
\end{equation}
where it should be clear that the right hand side also depends on time
through the temperature, $T$, and thus indirectly through the cooling
function too. In the non-radiative SNR expansion phase, the function
$T(t)$ is obtained directly from the hydrodynamical jump conditions at
the shock discontinuity~:
\begin{equation}
T \approx \frac{3m}{8k_{\mathrm{B}}}V^{2},
\end{equation}
or numerically~:
\begin{equation}
T \approx (2\times 10^{5}\,\mathrm{K})
\left(\frac{E_{\mathrm{SN}}}{10^{51}\mathrm{erg}}\right)^{\hspace{-2pt}\frac{2}{5}}
\hspace{-4pt}
\left(\frac{n_{0}}{1\mathrm{cm}^{-3}}\right)^{\hspace{-2pt}-\frac{2}{5}}
\hspace{-4pt}
\left(\frac{t}{10^{5}yr}\right)^{\hspace{-2pt}-\frac{6}{5}}.
\label{Temperature}
\end{equation}
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig1.eps}}
\caption{Cooling function (bold line) as a function of the temperature
for a medium with metallicity lower than $\sim 10^{-2}Z_{\odot}$. The
dashed lines illustrate the graphical determination of \hbox{${t_{\rm end}}$}, for an
ambient density $n_{0} = 10\,\mathrm{cm}^{-3}$ (see text).}
\label{CoolingFunction}
\end{figure}
To solve Eq.~(\ref{tEndEq}), we still need to know the cooling
function $\Lambda(T)$. In the range of temperatures corresponding to
the end of the Sedov-like phase, $10^{5}\,\mathrm{K}\la T\la
10^{7}\,\mathrm{K}$, it happens to depend significantly on
metallicity, with differences up to two orders of magnitude for
metallicities from $Z = 0$ to $Z = 2 Z_{\odot}$ (B\"oh\-rin\-ger \&
Hens\-ler 1989). Because we focus on Be production in the early
Galaxy, we adopt the cooling function corresponding to zero
metallicity, represented in Fig.~\ref{CoolingFunction} (adapted from
B\"oh\-rin\-ger \& Hens\-ler 1989), which holds for values of $Z$ up
to $\sim 10^{-2}Z_{\odot}$.
For high enough ambient densities, the shock will become radiative
early in the SNR evolution, when the temperature is still very high,
say above $T\ga 2\,10^{6}$~K. In this case, the cooling function is
dominated by Bremsstrahlung emission and can be written analytically
as~:
\begin{equation}
\Lambda_{\mathrm{Br}}(T) \approx
(2.4\,10^{-23}\,\mathrm{erg~cm}^{3}\mathrm{s}^{-1})
\left(\frac{T}{10^{8}\,\mathrm{K}}\right)^{1/2}.
\label{LambdaBr}
\end{equation}
Substituting from (\ref{Temperature}) and (\ref{LambdaBr}) in
Eq.~(\ref{tEndEq}) and solving for $t$, we find~:
\begin{equation}
\hbox{${t_{\rm end}}$} = (1.1\,10^{5}\,\mathrm{yr})
\left(\frac{E_{\mathrm{SN}}}{10^{51}\mathrm{erg}}\right)^{1/8}
\left(\frac{n_{0}}{1\mathrm{cm}^{-3}}\right)^{-3/4}.
\label{tEnd}
\end{equation}
To check the consistency of our assumption $\Lambda \approx
\Lambda_{\mathrm{Br}}$ (i.e. $T\ga 2\,10^{6}$~K), let us now report
Eq.~(\ref{tEnd}) in (\ref{Temperature}) and write down the temperature
$T_{\mathrm{End}}$ at the end of the Sedov-like phase~:
\begin{equation}
T_{\mathrm{End}} \approx (2\times 10^{5}\,\mathrm{K})
\left(\frac{E_{\mathrm{SN}}}{10^{51}\mathrm{erg}}\right)^{1/4}
\left(\frac{n_{0}}{1\mathrm{cm}^{-3}}\right)^{1/2},
\end{equation}
which means that the above analytical treatment is valid only for
ambient densities greater than about $100\,\mathrm{cm}^{-3}$. For
lower densities, we must solve Eq.~(\ref{tEndEq}) graphically. First,
we invert Eq.~(\ref{Temperature}) to express $t$ as a function of
temperature, then we plot the function $f(T)\equiv
3k_{\mathrm{B}}T/(20n_{0}t)$ on the same graph as $\Lambda$ (see
Fig.~\ref{CoolingFunction} for an example), find the value of $T$ at
intersection, and finally convert this value into the sought time
\hbox{${t_{\rm end}}$}~making use again of Eq.~(\ref{Temperature}). The results,
showing \hbox{${t_{\rm end}}$}~as a function of the ambient density, are shown in
Fig.~\ref{TimeScales}.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig2.eps}}
\caption{Comparison of \hbox{${t_{\rm end}}$}~and \hbox{${t_{\rm SW}}$}~as a function of the ambient
density, $n_{0}$, for two different values of the mass ejected by the
supernova (10 and 30~\hbox{$M_{\odot}$}). The dashed line shows the asymptotic
analytic estimate of Eq.~(\ref{tEnd}).}
\label{TimeScales}
\end{figure}
We now have all the ingredients to plot the efficiency ratio of the
two processes calculated above. Figure~\ref{Phi2/Phi1} shows the
ratio $\phi_{2}/\phi_{1}$ given in Eq.~(\ref{phi2/phi1}) as a function
of the ambient density, assuming that $\theta_{1} = \theta_{2}$. Two
different values of the ejected mass have been used, corresponding to
different progenitor masses ($\sim 10-40\,\hbox{$M_{\odot}$}$). It can be seen
that low densities are more favourable to the reverse shock
acceleration process. This is due to $\hbox{${t_{\rm end}}$}/\hbox{${t_{\rm SW}}$}$ being larger,
implying a larger dilution of the ejecta (process~1 less efficient)
and smaller adiabatic losses, which indeed decrease as $t^{-1}$
(process~2 more efficient). The part of the plot corresponding to
$\phi_{2}/\phi_{1}\le 1$ is not physical, because it requires
$\hbox{${t_{\rm end}}$}\le\hbox{${t_{\rm SW}}$}$, which simply means that the Sedov-like phase no longer
exists and the whole calculation becomes groundless. Note however
that in Fig.~\ref{Phi2/Phi1} the energy imparted to the EPs has been
assumed equal for both processes, which is most certainly not the
case. Actually, if $\theta_{2}/\theta_{1} = 0.1$ (e.g. $\theta_{1} =
10\%$ and $\theta_{1} = 1\%$), then process~1 is found to dominate Be
production during the Sedov-like phase, regardless of the ambient
density.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig3.eps}}
\caption{Comparison of the Be (and B) production efficiency through
the forward and reverse shock acceleration processes, for two values
of the ejected mass (10~and 30~\hbox{$M_{\odot}$}). The ratio $\phi_{2}/\phi_{1}$
(see text) is plotted as a function of the ambient density, assuming
that both processes impart the same total energy to the EPs
($\theta_{1} = \theta_{2}$).}
\label{Phi2/Phi1}
\end{figure}
\section{Spallation reactions after the Sedov-like phase}
At the end of the Sedov-like phase, the EPs are no longer confined and
leave the SNR to diffuse across the Galaxy. At the stage of chemical
evolution we are considering here, there are no or few metals in the
interstellar medium (ISM), so that energetic protons and $\alpha$
particles accelerated at the forward shock will not produce any
significant amount of Be after \hbox{${t_{\rm end}}$}~(although Li production will
still be going on through $\alpha + \alpha$ reactions). In the case
of the second process, however, the EPs contain CNO nuclei which just
cannot avoid being spalled while interacting with the ambient H and He
nuclei at rest in the Galaxy. This may be regarded as a third process
for Be production, which lasts until either the EPs are slowed down by
Coulombian interactions to subnuclear energies (i.e. below the
spallation thresholds) or they simply diffuse out of the Galaxy.
Since the confinement time of cosmic rays in the early Galaxy is
virtually unknown, we shall assume here that the Galaxy acts as a
thick target for the EPs leaving the SNR, an assumption which actually
provides us with an upper limit on the spallation yields.
Unlike the first two processes evaluated above, this third process is
essentially independent of dynamics. Thus, time-dependent
calculations are no longer needed and, from this stage on, the
calculations made by Ramaty et~al. (1997) or any steady-state
calculation is perfectly valid. In particular, the ambient density
has no influence on light element production, since a greater number
of reactions per second, as would result from a greater density,
implies an equal increase of both the spallation rates and the energy
loss rate. Once integrated over time, both effects cancel out
exactly, and in fact, given the energy spectrum of the EPs, the
efficiency of Be production (and Li, and B), expressed as the number
of nuclei produced per erg injected in the form of EPs, depends only
on their chemical composition.
Results are shown in Fig.~\ref{Be/erg} for different values of the
source abondance ratios, $\mathrm{H}/\mathrm{He}$ and
$(\mathrm{H}+\mathrm{He})/(\mathrm{C}+\mathrm{O})$, allowing one to
derive the spallation efficiency for any composition. Two-steps
processes (such as $^{12}\mathrm{C} + ^{1}\mathrm{H} \longrightarrow
^{10}\mathrm{B}$ followed by $^{10}\mathrm{B} + ^{1}\mathrm{H}
\longrightarrow ^{9}\mathrm{Be}$) have been taken into account. Test
runs show good agreement with the results of Ramaty et~al. (1997).
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig4.eps}}
\caption{Production efficiency of Be, as a function of the EP
composition (all abundances are by number). The ordinate is the
number of Be nuclei produced by spallation reactions per erg injected
in the form of EPs. A thick target has been assumed, with zero
metallicity. Carbon and Oxygen abundances were set equal in the EP
composition.}
\label{Be/erg}
\end{figure}
As can be seen on Fig.~\ref{Be/erg}, pure Carbon and Oxygen have a
production efficiency of about $0.22\,\mathrm{nuclei}/\mathrm{erg}$,
while this efficiency decreases by at least a factor of~10 for
compositions with hundred times more H and He than metals (or about
ten times more by mass). According to models of explosions for SN
with low metallicity progenitors, the average
$(\mathrm{H}+\mathrm{He})/(\mathrm{C}+\mathrm{O})$ ratio among the EPs
should indeed be expected to be $\ga 200$, unless selective
acceleration occurs to enhance the abundance of the metals. As a
consequence, efficiencies greater than $\sim
10^{-2}\,\mathrm{nuclei}/\mathrm{erg}$ should not be expected, so that
a production of $\sim 4\,10^{48}$~atoms of Be requires an energy of
$\sim 4\,10^{50}$~erg to be imparted to the EPs. This seems very
unlikely considering that the total energy available in the reverse
shock (the source of the EPs) should be of order one tenth of the SN
explosion energy, not to mention the acceleration efficiency.
Moreover, a significant fraction of the energy originally imparted to
the EPs has been lost during the Sedov-like phase of the SNR evolution
through adiabatic losses.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig5.eps}}
\caption{Fraction of the energy imparted to the EPs at time \hbox{${t_{\rm SW}}$}~which
is still available at \hbox{${t_{\rm end}}$}, after the Sedov-like phase, as a function
of othe ambient density.}
\label{EnergyReduction}
\end{figure}
To evaluate the `surviving' fraction of energy, it sufficies to go
back to Eq.~(\ref{AdiabLossRate}), which indicates that when the
radius of the shock is multiplied by a factor $\eta$, the momentum $p$
of all the particles is multiplied by a factor $\eta^{-3/4}$. It is
worthwhile noting that, because of their specific momentum dependence,
adiabatic losses do not modify the shape of the EP energy spectrum.
In our case, $f(p)\propto p^{-4}$, so that when all momenta $p$ are
divided by a factor $\zeta$, the distribution function $f(p)$ is
divided by the same factor $\zeta$. To see that, the easiest way is
to work out the number of particules between momenta $p$ and $p + \mathrm{d}
p$ after the momentum scaling. This number writes $\mathrm{d} N^{\prime} =
f^{\prime}(p)4\pi p^{2}\mathrm{d} p$, where $f^{\prime}(p)$ is the new
distribution function. Now $\mathrm{d} N^{\prime}$ must be equal to the
number of particles that had momentum between $\zeta p$ and $\zeta(p +
\mathrm{d} p)$, which is, by definition, $\mathrm{d} N = f(\zeta p)4\pi(\zeta
p)^{2}\zeta\mathrm{d} p$. Equating $\mathrm{d} N$ and $\mathrm{d} N^{\prime}$ yields the
result $f^{\prime}(p) = \zeta^{-1}f(p)$.
Putting all pieces together, we find that when the shock radius $R$ is
multiplied by a factor $\eta$, the distribution function and, thus,
the total energy of the EPs are multiplied by $\eta^{-3/4}$. Now
considering that $R$ increases as $t^{2/5}$ during the Sedov-like
phase, we find that the total energy of the EPs decreases as
$t^{-3/10}$. Note that this is nothing but an other way to work out
the decrease of the spallation rates for our second process during the
Sedov-like phase (cf. Sect.~\ref{SecondProcess}). Finally, we find
that a fraction $(\hbox{${t_{\rm end}}$}/\hbox{${t_{\rm SW}}$})^{-3/10}$ of the initial energy imparted
to the EPs is still available for spallation at the end of the
Sedov-like phase. This factor is plotted on
Fig.~\ref{EnergyReduction}, as a function of the ambient density. It
can be seen that for $n_{0} = 1\,\mathrm{cm}^{-3}$, the energy
available to power our third process of light element production has
been reduced by adiabatic losses to not more than one third of its
initial value, and less than one half for densities up to
$100\,\mathrm{cm}^{-3}$. Clearly, high densities are favoured
(energetically) because they tend to shorten the Sedov-like phase, and
therefore merely avoid the adiabatic losses.
\section{Discussion}
Since light element production in the interstellar medium obviously
requires a lot of energy in the form of supernuclear particles (i.e.
with energies above the nuclear thresholds) as well as metals
(especially C and O), it is quite natural to consider SNe as possible
sources of the LiBeB observed in halo stars. We have analysed in
detail the spallation nucleosynthesis induced by a SN explosion on the
basis of known physics and theoretical results relating to particle
shock acceleration. Two major processes can be identified, depending
on whether the ISM or the ejecta are accelerated, respectively at the
forward and reverse shocks. In the first case, the EPs consist mostly
of protons and alpha particles and must therefore interact with C and
O nuclei, which are much more numerous within the SNR than in the
surrounding medium (especially at early stages of Galactic evolution).
The process will thus last as long as the EPs stay confined in the
SNR, i.e. approximately during the Sedov-like phase, but not more.
In the second case, freshly synthesized CNO nuclei are accelerated,
and Be production occurs through interaction with ambient H and He
nuclei. The process is then divided into two, one stretching over the
Sedov-like phase, with the particles suffering adiabatic losses, and
the other one occuring outside the remnant, with only Coulombian
losses playing a role.
\begin{figure}
\resizebox{\hsize}{!}{\includegraphics{fig6.eps}}
\caption{Spallation efficiency of CNO during the Sedov-like phase, as
a function of the ambient density. The fraction of freshly
synthesized CNO nuclei being spalled to Be by processes~1 ($\phi_{1}$)
and~2 ($\phi_{2}$) is obtained from Eqs.~(\ref{phi1Num})
and~(\ref{phi2Num}) and the values of \hbox{${t_{\rm end}}$}~derived in
Sect.~\ref{RelativeContribution}, for two values of the ejected mass
(10~and 30~\hbox{$M_{\odot}$}).}
\label{BeYield}
\end{figure}
We have calculated the total Be production in these three processes,
taking the dynamics of the SNR evolution into account (dilution of the
ejecta by metal-poor material and adiabatic losses). The results are
shown in Fig.~\ref{BeYield} for processes 1 and~2 (from
Eqs.~(\ref{phi1Num}) and~(\ref{phi2Num})). We find that with
canonical values of $\theta_{1} = 0.1$, $\theta_{2} = 0.01$,
$M_{\mathrm{ej}} = 10\,\hbox{$M_{\odot}$}$ and a mean ambient density $n_{0} =
10\,\mathrm{cm}^{-3}$, the fraction of freshly synthesized CNO nuclei
spalled into Be in these processes is $\phi_{1} \sim 8\,10^{-11}$ and
$\phi_{2} \sim 3\,10^{-11}$, respectively, which is very much less
than the value `required' by the observations, discussed in the
introduction ($\phi_{\mathrm{obs}}\sim 3\,10^{-8}$). Even allowing
for unreasonably high values of the acceleration efficiency,
$\theta_{1}\sim \theta_{2}\la 1$, the total Be production by processes
1 and~2 would still be more than one order of magnitude below the
observed value.
As suggested by Fig.~\ref{BeYield} and our analytical study, higher
densities improve the situation. However, even with $n_{0} =
10^{3}\,\mathrm{cm}^{-3}$ and acceleration efficiencies equal to~1,
the Be yield is still unsufficient. Moreover, it should be noted that
our calculations did not consider Coulombian energy losses (because
they are negligeable as compared to adiabatic losses for usual
densities), which become important as the density increases and
therefore make the Be yield smaller. Finally, since we are trying to
account for the mean abundance of Be in halo stars, as compared to Fe,
we have to evaluate the Be production for an ambient density
corresponding to the mean density encountered around explosion sites
in the early Galaxy, which is very unlikely to be as high as
$10^{3}\,\mathrm{cm}^{-3}$. It could even be argued that although the
gas density might have been higher in the past than it is now (hence
our `canonical value' $n_{0} = 10\,\mathrm{cm}^{-3}$), the actual mean
density about SN explosion sites could be lower than
$1\,\mathrm{cm}^{-3}$, because most SNe may explode within superbubble
interiors, where the density is much less than in the mean ISM.
Thus, our conclusion is that processes 1 and~2 both fail in accounting
for the Be observed in metal-poor stars in the halo of our Galaxy.
Concerning the third process, adopting canonical values for the
parameters again leads to unsufficient Be production, as noted in the
previous section. While higher densities improve the situation by
avoiding the adiabatic energy losses, one should nevertheless expect
at least half of the EP energy to be lost in this way, for any
reasonable density. This means that even if 10\%
of the explosion energy is imparted to EPs accelerated at the reverse
shock, which is certainly a generous upper limit, the required number
of $\sim 4\,10^{48}$ nuclei of Be per SN implies a spallation
efficiency of $\sim 0.1$~nucleus/erg. Now Fig.~\ref{Be/erg} shows
that this requires an EP composition in which at least one particle
out of ten is a CNO nucleus. In other words, the ejected mass of CNO
must be of the order of that of H and He together. None of the SN
explosion models published so far can reproduce such a requirement,
and so there is clearly a problem with Be production in the early
Galaxy.
The results presented here are in fact interesting in many regards.
First, they show that it is definitely very difficult to account for
the amount of Be found in halo stars. Consequently, we feel that the
main problem to be addressed in this field of research is probably not
the chemical evolution of Be (and Li and B) in the Galaxy, as given by
the ensemble of the data points in the abundance vs metallicity
diagrams (e.g. whether Be is proportional to Fe or to its square)
but, to begin with, the position of any of these points. Are we able
to describe in some detail one process which could explain the amount
of Be (relative to Fe) present in any of the stars in which it is
observed? The answer, we are afraid, seems to be no at this stage.
It is however instructive to ask why the processes investigated here
have failed. Concerning process~1 (acceleration of ISM, interaction
with fresh CNO within the SNR), the main reason is that the CNO rich
ejecta are `too much diluted' by the swept-up material as the SNR
expands, so that the spallation efficiency is too low (or the
available energy is too small). However, it seems rather hard to
think of any region in the Galaxy where the concentration in CNO is
higher than inside a SNR during the Sedov-like phase (especially in
the first stages of chemical evolution)! So the conclusion that
process~1 cannot work, even with a 100\% acceleration
efficiency, seems to rule out any other process based on the
acceleration of the ISM, initially devoided of metals.
The other solution is then of course to accelerate CNO nuclei
themselves, which provides the maximum possible spallation efficiency,
independently of the ambient metallicity. Every energetic CNO will
lead to the production of as much Be as possible given the spallation
cross sections and the energy loss rates. The latter cannot
physically be smaller than the Coulombian loss rate in a neutral
medium, and this leads to the efficiency plotted in Fig.~\ref{Be/erg}.
Unfortunately, a significant amount of the CNO rich ejecta of an
isolated SN can only be accelerated at the reverse shock at a time
around the sweep-up time, \hbox{${t_{\rm SW}}$}. This means that i) the total amount of
energy available is smaller than the explosion energy (probably of
order 10\%, i.e. $\sim 10^{50}$~erg), and ii) the accelerated nuclei
will suffer adiabatic losses during the Sedov-like phase, reducing
their energy by a factor of 2 or~3. As shown above, this makes
process~2-3 incapable of producing enough Be, as long as the EPs have
a composition reflecting that of the SN ejecta.
This suggest that a solution to the problem could be that the reverse
shock accelerates preferentially CNO nuclei rather than H and He. For
example, recent calculations have shown that such a selective
acceleration arises naturally if the metals are mostly condensed in
grains (Ellison et~al. 1997). The proposition by Ramaty et~al.
(1997) that grains condense in the ejecta before being accelerated
could then help to increase the abundance of CNO in the EPs. However,
we have to keep in mind that any selective process called upon must be
very efficient indeed, since as we indicated above, the data require
that the EP composition be as rich as one CNO nuclei out of ten EPs,
which means that CNO nuclei must be accelerated at least ten times
more efficiently than H and He. This would have to be increased by
another factor of ten if the energy initially imparted to the EPs by
the acceleration process were only a factor 2 or~3 lower (i.e. $\sim
3\,10^{49}$~erg, which is more reasonable from the point of view of
particle acceleration theory). Clearly, more work is needed in this
field before one can safely invoke a solution in terms of selective
acceleration.
As can be seen, playing with the composition to increase the
spallation efficiency has its own limits, and in any case,
Fig.~\ref{Be/erg} gives an unescapable upper limit, obtained with pure
Carbon and Oxygen (at least for the canonical spectrum considered here
- other spectra were also investigated, as in Ramaty et al. (1997),
leaving the main conclusions unchanged). This would then suggest that
another source of energy should be sought. However, the constraint
that it should be more energetic than SNe is rather strong.
Another interesting line of investigations could be the study of the
collective effects of SNe. Most of the massive stars and SN
progenitors are believed to be born (and indeed observed, Melnik
\&~Efremov 1995) in associations, and their joint explosions lead to
the formation of superbubbles which may provide a very favourable
environment for particle acceleration (Bykov \&~Fleishman 1992).
Parizot et~al. (1998) have proposed that these superbubbles could be
the source of most of the CNO-rich EPs, and Parizot \&~Knoedlseder
(1998) further investigated the gamma-ray lines induced by such an
energetic component. The most interesting features of a scenario in
which Be-producing EPs are accelerated in superbubbles is that i) when
a new SN explodes, the CNO nuclei ejected by the previous SNe are
accelerated at the \emph{forward shock}, instead of the reverse shock
in the case of an isolated SN, which implies a greater energy, and ii)
no significant adiabatic losses occur, because of the dimensions and
low expansion velocity of the superbubble. This makes the superbubble
scenario very appealing, and it will be investigated in detail in a
forthcoming paper.
However that may be, we should also keep in mind that when we say that
a process does not produce enough Be, it always means that it does not
produce enough Be \emph{as compared to Fe}. Now it could also be that
SN explosion models actually produce too much Fe. The point is that
Be is compared to Fe in the observations, while it has no direct
physical link with it. Indeed, Be is not made out of Fe, but of C
and~O. So to be really conclusive, the studies of spallative
nucleosynthesis should compare theoretical Be/O yields to the
corresponding abundance ratio in metal-poor stars. Unfortunately, the
data are much more patchy for Be as a function of [O/H] than as a
function of [Fe/H], especially in very low metallicity stars. The
usually assumed proportionality between O and Fe could turn out to be
only approximate, as recent observational works possibly indicate
(Israelian et~al. 1998; Boesgaard et~al. 1998; these observations,
however, still ?? need to be confirmed by an independent method, all
the more that they come into conflict with several theoretical and
observational results; cf. Vangioni-Flam et~al. 1998b). We shall
address this question in greater detail in the attending paper
(Parizot and Drury, 1999, Paper~II).
Finally, we wish to stress that the calculations presented in this
paper rely on a careful account of the dynamics of the problem. More
generally, time-dependent calculations are required to properly
evaluate the spallation processes in environments where compositions
and energy densities are evolving. In particular, as argued in
Parizot (1998), no variation with density can be obtained with a
stationary model, since an increase in the density induces an
equivalent and cancelling increase in the spallation rates and the
energy loss rates. By contrast, we have shown that all three of the
processes considered here are more efficient at higher density -- a
result which could not have been found otherwise. Detailed, numerical
time-dependent calculations will be presented in paper~II, with
conclusions similar to those demonstrated here.
\begin{acknowledgements}
This work was supported by the TMR programme of the European Union
under contract FMRX-CT98-0168. It was initiated during a visit by LD
to the Service d'Astrophysique, CEA Saclay, whose hospitality is
gratefully acknowledged. We wish to thank M. Cass\'e and Elisabeth
Vangioni-Flam for stimulating discussions of these and related topics.
\end{acknowledgements}
| 2024-02-18T23:40:12.910Z | 1998-11-30T19:50:33.000Z | algebraic_stack_train_0000 | 1,677 | 10,035 |
|
proofpile-arXiv_065-8222 | \section{Introduction}
\qquad The gravitational effects of quantum fields in black hole spacetimes
has long been studied. Since Hawking's discovery that black holes radiate
\cite{ref:hawkevap} much interesting work has been done in this area.
Quantities of interest include the expectation value $\langle\phi^{2}\rangle$, which describe
vacuum polarization effects, and $\langle T_{\mu \nu}\rangle$, the expectation value of the
stress-energy tensor of the field. This latter quantity may then be used
in the Einstein Field equations:
\begin{equation}
R_{\mu\nu}-\frac{1}{2}Rg_{\mu\nu}+\Lambda g_{\mu\nu} = 8\pi\langle
T_{\mu\nu} \rangle \label{eq:einst}
\end{equation}
to determine the back-reaction of the field on the original spacetime.
The effects of the back-reaction may also include the
removal of singularities (\cite{ref:rem1}-\cite{ref:hiscock}). This is the
main motivation for the work presented here and the answer would have
consequences to many fundamental questions including the information loss
problem.
\qquad Hiscock et.al \cite{ref:hiscock} have done an extensive
study of
these effects on the Schwarzschild interior and have found cases where
curvature is initially slowed in the interior as well as cases where the
curvature is initially strengthened (such as the case of the massless
conformally coupled scalar field). They have also studied the effects on
black hole anisotropy. It is interesting to ask whether or
not the results are a product of the symmetry chosen or are general. It
is also interesting to ask whether the presence of a cosmological
constant will alter the situation. The study here attempts to address both
issues by studying a black cosmic string which is asymptotically
anti-deSitter. The field is in the Hartle-Hawking vacuum state
\footnote{Due to the fact that the black string has positive specific
heat, the Hartle-Hawking vacuum state is particularly applicable here.
For a discussion of black string thermodynamic properties see
\cite{ref:brillthermo}, \cite{ref:mythermo} and \cite{ref:pecathermo}.}
\cite{ref:harthawk} and the
stress-energy tensor is found using the
approximation of Page \cite{ref:pageappx} which is particularly useful
here since the spacetime is an Einstein spacetime (in an Einstein
spacetime the relation $R_{\mu\nu}=\Lambda g_{\mu\nu}$ holds). Black
string
solutions are of relevance to cosmic strings. It has also been shown how
such black holes may form by gravitational collapse \cite{ref:smithmann}
\cite{ref:lemoscollapse}. This type of collapse has astrophysical
relevance as the collapse of a finite spindle can behave as an infinite
cylinder near its central region \cite{ref:spindle}.
\qquad It may be thought that, since no external observer can view the
interior without falling into the black hole, that a study of the
interior is not physically meaningful. However, as pointed out in
\cite{ref:hiscock} black hole evaporation reveals more and more of the
black hole interior as time progresses and therefore the interior has
relevance to exterior observers in this way. Also, the issue of whether or
not spacetime singularities actually exist has been one of intense
interest ever since Oppenheimer and Snyder's \cite{ref:opsnyd} original
collapse calculation.
\section{Black String Spacetime}
\qquad The black hole studied here is the cylindrical black hole
spacetime developed by Lemos and Zanchin \cite{lemos} and also studied by
Kaloper \cite{kaloper}. If charge and angular momentum are not present
the metric has the form
\begin{equation}
ds^{2}=-(\alpha^{2}\rho^{2}-\frac{4M}{\alpha\rho}) dt^{2} +
\frac{d\rho^{2}}{(\alpha^{2}\rho^{2}-\frac{4M}{\alpha\rho})} + \rho^{2}
\,d\varphi^{2} + \alpha^{2}\rho^{2}\, dz^{2}. \label{eq:metric}
\end{equation}
where $M$ is the mass per unit length, $\alpha^{2}=-\frac{1}{3}\Lambda$ and
the coordinates take on the following ranges:
\begin{eqnarray}
-\infty < t < \infty , \nonumber \\
0 \leq \rho < \infty , \nonumber \\
0\leq \varphi <2\pi , \nonumber \\
-\infty < z < \infty. \nonumber
\end{eqnarray}
An event horizon exists at $\rho=\rho_{H}\equiv
\frac{(4M)^{1/3}}{\alpha}$ and the cosmological constant
(which is negative and necessary for cylindrical black hole solutions),
$\Lambda$, dominates in the limit $\rho\rightarrow\infty$ giving the
spacetime its asymptotically anti-deSitter behaviour.
\qquad The apparently singular behaviour of the spacetime at
$\rho=\rho_{H}$ is a coordinate effect and not a true singularity. On
calculating the Kretschmann scalar one obtains
\begin{equation}
K\equiv
R_{\delta\lambda\mu\nu}R^{\delta\lambda\mu\nu} =
24\alpha^4\left(1+\frac{8M^{2}}{\alpha^{6}\rho^{6}}\right) \label{eq:origK}
\end{equation}
from which it can be seen that the only true singularity is a polynomial
singularity at $\rho=0$. Thus, this solution violates the hoop conjecture
but not the cosmic censorship conjecture. The hoop conjecture is
therefore not valid in spacetimes with a cosmological constant.
As calculations will be extended to the interior, it is convenient to
re-write the metric using the following coordinate redefinitions:
\begin{eqnarray}
t \rightarrow R , \nonumber \\
\rho \rightarrow T. \nonumber
\end{eqnarray}
Where $T$ is timelike in the interior and $R$ is spacelike. The ``interior"
metric now has the form \begin{equation}
ds^{2}_{interior}=-\frac{dT^{2}}{\left(\frac{4M}{\alpha T} -
\alpha^{2}T^{2}\right)} + \left(\frac{4M}{\alpha T}
-\alpha^{2}T^{2}\right) dR^{2}+ T^2\,d\varphi^{2} + \alpha^{2}T^{2} \,
dz^{2} \label{eq:intmet}
\end{equation}
where the interior region corresponds to $0 \leq T \leq \rho_{H}$.
\section{Stress-Energy Tensor}
\qquad In this section the stress energy tensor is calculated which will
eventually be used in (\ref{eq:einst}) to calculate back-reaction
effects. The expectation value of stress-energy tensors have been
calculated in exterior Schwarzschild spacetime by Howard and Candelas
\cite{ref:handctmn} and Page \cite{ref:pageappx} as well as by Anderson
et. al \cite{ref:ahl} who
studied the stability in the extreme Reissner-Nordstr\"{o}m black hole.
Anderson, Hiscock and Samuel \cite{ref:ahs} have
developed an approximation for both massive and massless fields in arbitrary
spherically symmetric spacetimes and have used this approximation to
calculate $\langle T_{\mu \nu}\rangle$\ in the exterior Reissner-Nordstr\"{o}m geometry. The
Kerr and Kerr-Newman spacetimes have also been studied in \cite{ref:frolov},
\cite{ref:frovandthorne} and \cite{ref:frovandzel}. Quantum effects in
lower dimensional black hole exteriors may be found
in \cite{ref:btz1}-\cite{ref:btz10}. For a calculation of
$\langle\phi^{2}\rangle$\ in the spacetime studied here see \cite{ref:myphi}.
\qquad Various works on back-reaction effects of quantum fields have also
been produced. Hiscock and Weems \cite{ref:handw}, Bardeen \cite{ref:B1},
Balbinot \cite{ref:B2} and York \cite{ref:york} have studied
effects in Schwazschild and Reissner-Nordstr\"{o}m exteriors. Few
calculations, however, have been performed on the interiors of black
holes. One such study has been done by Hiscock, Larson
and Anderson \cite{ref:hiscock} where they have extended their analysis to
the Schwarzschild interior and calculated back-reaction effects on curvature
invariants.
\subsection{Stress-Energy Tensor for the Conformally Coupled Scalar Field}
\qquad The calculation of the stress energy tensor will be done using
the Eucldeanized metric. This is obtained by making the transformation
($t\rightarrow -i\tau$) in (\ref{eq:metric}) giving the metric positive
definite signature so that
\begin{equation}
ds^{2}_{Euclidean}=(\alpha^{2}\rho^{2}-\frac{4M}{\alpha\rho}) d\tau^{2} +
\frac{d\rho^{2}}{(\alpha^{2}\rho^{2}-\frac{4M}{\alpha\rho})} + \rho^{2}
\,d\varphi^{2} + \alpha^{2}\rho^{2}\, dz^{2}. \label{eq:eucmetric}
\end{equation}
To calculate $\langle T_{\mu \nu}\rangle$\ exactly is an extremely
difficult task
which normally involves acting on $\langle\phi^{2}\rangle$\ with a complicated differential
operator. It is useful therefore to use an approximation which will give
an analytic result from which information on back-reaction effects may be
calculated. The approximation used here is the approximation of Page for
thermal stress-energy tensors in static spacetimes\cite{ref:pageappx}. This
approximation is especially good if the spacetime under consideration is
an Einstein spacetime such as the one considered here and contains no
ambiguities in the case of scalar fields. The
Bekenstein-Parker \cite{ref:bekpark} Gaussian path integral approximation
is utilized from which the thermal propagator is constructed. This
construction is done in an (Euclideanized) ultrastatic spacetime
($g_{00}=k,\,\, k$
is a constant chosen to be $1$ in this work) which is related to the
physical spacetime by
\begin{equation}
g_{\mu\nu}=|g_{00\,\,(p)}|^{-1}g_{\mu\nu\,\,(p)}.
\end{equation}
The subscript $p$ will be used to indicate quantities calculated using the
physical metric (all other tensors in this section are obtained using the
ultrastatic metric).
This approximation gives, for the stress-energy tensor in the physical
spacetime:
\begin{eqnarray}
T^{\mu}_{\nu\,\, (p)}=|g_{00\,\, (p)}|^{-2} \lbrace T^{\mu}_{\nu} +
[8\lambda |g_{00\,\,(p)}|^{-1}(|g_{00\,\,(p)}|^{1/2})_{;\alpha}
(|g_{00\,\,(p)}|^{1/2})^{;\beta} \nonumber \\
-4(\lambda +\beta) R^{\beta}_{\alpha}]C^{\alpha\mu}_{\beta\nu}+
2\beta [H^{\mu}_{\nu}+3\alpha^{4}|g_{00\,\,(p)}|^{2}\delta^{\mu}_{\nu}]+
\frac{1}{6}\gamma I^{\mu}_{\nu}\rbrace , \label{eq:tmn1}
\end{eqnarray}
where $C^{\alpha\mu}_{\beta\nu}$ is the Weyl tensor
and the coefficients $\lambda,\,\beta$ and $\gamma$ are given as follows:
\begin{equation} \lambda=\frac{12h(0)}{2^{9}45\pi^{2}},
\,\,\beta=\frac{-4h(0)}{2^{9}45\pi^{2}},
\,\,\gamma=\frac{8h(0)}{2^{9}45\pi^{2}}. \end{equation}
The number of helicity states, $h(0)$, simply counts the number of scalar
fields present.
$T^{\mu}_{\nu}$ is the stress-energy tensor in the ultrastatic metric,
\begin{equation}
T^{\mu}_{\nu}=\frac{\pi^{2}}{90}T^{4}(\delta^{\mu}_{\nu}-
4\delta^{\mu}_{0}\delta^{0}_{\nu}),
\end{equation}
were $T$ is the temperature of the black string which can be found by
demanding
that the Euclidean extension of (\ref{eq:metric}) be regular on the
horizon;
\begin{equation}
T=\frac{3\alpha}{4\pi}(4M)^{1/3}. \label{eq:temp}
\end{equation}
The quantities $H^{\mu}_{\nu}$ and $I^{\mu}_{\nu}$ are given by:
\begin{eqnarray}
H^{\mu}_{\nu}=-R^{\alpha\mu}R_{\alpha\nu}+ \frac{2}{3}RR^{\mu}_{\nu} +
(\frac{1}{2}R^{\alpha}_{\beta}R^{\beta}_{\alpha}-
\frac{1}{4}R^{2})\delta^{\mu}_{\nu}, \nonumber \\
I^{\mu}_{\nu}=2R^{;\mu}_{;\nu}-2RR^{\mu}_{\nu}+(\frac{1}{2}R^2-
2R^{;\alpha}_{;\alpha})\delta^{\mu}_{\nu}. \label{eq:handI}
\end{eqnarray}
\qquad The calculation of $\langle T_{\mu \nu}\rangle$\ is carried out on the exterior of the
black hole. However, since the result is finite at the horizon, it is
easily extended to the interior where the field equations will be solved.
For the spacetime considered here, the stress-energy tensor is calculated
to be
\begin{eqnarray}
T^{\mu}_{\nu\,\,(p)}=-\lbrace
1920\pi^{2}\epsilon
[\alpha^{4}\rho^{6}(\alpha^{3}\rho^{3}-4M)^{2}]\rbrace^{-1}
[-27\,2^{2/3}\,M^{4/3}\alpha^{8}\rho^{8}(\delta^{\mu}_{\nu}-
4\delta^{\mu}_{0}\delta^{0}_{\nu}) \nonumber \\
-16M\alpha^{9}\rho^{9}
(3\delta^{\mu}_{0}\delta^{0}_{\nu}+\delta^{\mu}_{1}\delta^{1}_{\nu})
-128\alpha^{3}\rho^{3}M^{3}(\delta^{\mu}_{\nu} -12\delta^{\mu}_{0}
\delta^{0}_{\nu}) \nonumber \\
+192M^{4}(\delta^{\mu}_{\nu}- 12\delta^{\mu}_{0}
\delta^{0}_{\nu}- \frac{8}{3}\delta^{\mu}_{1}\delta^{1}_{\nu}) \nonumber \\
+96M^{2}\alpha^{6}\rho^{6}(\delta^{\mu}_{\nu} -5\delta^{\mu}_{0}
\delta^{0}_{\nu} +\delta^{\mu}_{1}\delta^{1}_{\nu})
+2\alpha^{12}\rho^{12}\delta^{\mu}_{\nu}], \label{eq:setensor}
\end{eqnarray}
where $\epsilon= \hbar\alpha^{2}$.
This function remains unchanged when analytically continued to the
Lorentzian sector by the transformation $\tau \rightarrow it$ and
has trace consistent with anomaly calculations.
Far from the black string, (\ref{eq:setensor}) takes on its pure
anti-deSitter value of
$-\frac{\alpha^{4}}{960\pi^{2}}\delta^{\mu}_{\nu}$ \cite{ref:pbo} whereas
at the horizon (\ref{eq:setensor}) is also well defined and given by
\begin{equation}
T^{\mu}_{\nu\,\,(p)}(\rho_H)=
\frac{\alpha^{4}}{\pi^{2}}\left(\begin{array}{cccc}
\frac{1}{640}& 0 & 0 & 0 \\
0 & \frac{1}{640}& 0 & 0 \\
0 & 0 & -\frac{1}{640} & 0 \\
0 & 0 & 0 & -\frac{1}{640} \end{array} \right). \label{eq:tmnhorz}
\end{equation}
Inspection of (\ref{eq:tmnhorz}) immediately shows that the weak energy
condition (WEC) is violated. The qualitative behaviour of the energy density
($\varepsilon=-T^{0}_{0}$) is shown in figure 1 where it can be seen that
the WEC is violated throughout the interior of the black hole ($\rho<
\sim 1.6$). However, it is unknown how relevant the classical energy
conditions are in the case of quantum matter where violations are common
(for example in the case of the Casimir effect) and are in fact required
for a self consistent picture of Hawking evaporation.
\begin{figure}[ht]
\label{fig1}
\includegraphics[bb=70 286 507 720, width=0.8\textwidth,clip]{edenspic.eps}
\caption{{\small Energy density of the quantum scalar field in the
cylindrical
black hole spacetime. Weak energy condition violation can be seen
throughout the interior ($\rho<\sim 1.6$) and part of the exterior. The
interior energy density is given by $-T^{T}_{T}$ whereas on the exterior
it is given by $-T^{t}_{t}$.}}
\end{figure}
\section{Gravitational Back-Reaction}
\qquad In this section the gravitational effects of the quantum field on
the background spacetime will be calculated using the perturbed metric
\begin{eqnarray}
ds^2=-\frac{dT^2}{\frac{4M}{\alpha T}-\alpha^2
T^2}\left(1+\epsilon\eta (T)\right) &+& \left(\frac{4M}{\alpha T}-\alpha^2
T^2\right)\left(1+\epsilon\sigma (T)\right)\,dR^{2} \nonumber \\
&+& T^2\,d\varphi^{2}+\alpha^{2}T^{2}\left(1+\epsilon\psi
(T)\right)\,dz^{2}. \label{eq:pertmet}
\end{eqnarray}
The functions $\eta(T)$, $\sigma(T)$ and $\psi(T)$ are to be solved for and
the coupling constant, $\epsilon$ is assumed to
be small. The Einstein field equations, to first order in $\epsilon$ yield
\begin{subequations}
\begin{align}
& \sigma (T)'(\alpha^{3}T^{3}-4M)+\psi (T)'(\alpha^{3} T^{3}+M) =
\left(\frac{8\pi
\langle T^{T}_{T} \rangle}{\epsilon} +3\alpha^{2} \eta (T)\right)\alpha
T^{2}, \label{eq:E1} \\
& \frac{d}{dT}\left[\eta (T)(4M-\alpha^{3} T^{3}) + \psi
(T)'\left(\frac{\alpha^{3}T^{4}}{2}-2TM\right)\right]-3M\psi (T)'
\nonumber \\
& = \frac{8\pi \langle T^{R}_{R} \rangle \alpha T^{2}}{\epsilon} ,
\label{eq:E2} \\
& \frac{d}{dT}\left[(\sigma
(T)'+\psi (t)')\left(\frac{\alpha^{3}T^{4}}{2}-2TM\right)
\right]+3M\sigma (T)'+\eta (T)'(M-\alpha^{3}T^{3}) \nonumber \\
& = \left(\frac{ 8\pi\langle T^{\varphi}_{\varphi}\rangle}{\epsilon}
+ 3 \alpha^{2}\eta (T)\right) \alpha T^{2},
\label{eq:E3} \\
& \frac{d}{dT}\left[\sigma
(T)'\left(\frac{\alpha^{3}T^{4}}{2}-2TM\right)\right] +3M\sigma (T)' +
\eta (T)'(M-\alpha^{3}T^{3}) \nonumber \\
&= \left(\frac{ 8\pi \langle T^{z}_{z} \rangle}{\epsilon} + 3
\alpha^{2}\eta (T)\right)\alpha T^2, \label{eq:E4}
\end{align}
\end{subequations}
where primes denote ordinary differentiation with respect to $T$.
\qquad Since solving (\ref{eq:E1}-\ref{eq:E4}) using (\ref{eq:setensor})
and calculating the resulting relevant quantities such as the Riemann
curvature tensor and Kretschmann scalar would be an enourmous task, some
simplifying assumptions are first made. It is noted from
(\ref{eq:setensor}) that equations (\ref{eq:E3}) and (\ref{eq:E4}) must
be equal. It is therefore assumed that the function $\psi(T)$ is equal to
a constant and therefore does not appear in the field equations.
Secondly, the stress energy tensor will be approximated by its value near
the event horizon. This should not introduce too large of an error in the
calculations as the perturbative scheme here is only valid in regions
where the spacetime curvature is not large (such as near the horizon in
the small $\alpha$ limit).
\qquad Utilizing the above, the following solutions are obtained for the
perturbations:
\begin{eqnarray}
\eta (T)&=&\frac{1}{\pi(4M-\alpha^{3}T^{3})} \left[ \frac{13}{240}
\alpha^{3}T^{3}-\frac{3}{160}\alpha^{4}T^{4}M^{-1/3}-\frac{M}{15}\right]
\label{eq:eta} \\
\sigma(T)&=&-\frac{1}{\pi(4M-\alpha^{3}T^{3})} \left[\frac{13}{240}
\alpha^{3}T^{3}-\frac{3}{160}\alpha^{4}T^{4}M^{-1/3}- \frac{M}{15}\right]
\nonumber \\
&+& \frac{1}{30}\frac{\alpha T 2^{1/3}}{\pi
M^{1/3}}
+ \frac{1}{45\pi}\ln\left(\frac{(4M)^{1/3}-\alpha T}{4M-\alpha^3 T^3}\right)
\nonumber \\
&-& \frac{1}{90\pi}\ln (\alpha^{2}T^{2}+\alpha
T(4M)^{1/3}+(4M)^{2/3}) \nonumber \\
&-&\frac{\sqrt{3}}{45\pi}\arctan\left[\frac{1}{\sqrt{3}}\left( \frac{\alpha T
2^{1/3}}{M^{1/3}}+1\right)\right]+k_{0}.
\label{eq:sigma}
\end{eqnarray}
The integration constant $k_{0}$ may be left arbitrary as it does not
enter subsequent calculations. Both solutions are well behaved in the
domain of validity as $\eta(\rho_H)=\frac{1}{240\pi}$ and
$\sigma(\rho_H)=\frac{153-72\ln(3)-96\ln(2)+48\ln(M)+16\sqrt{3}\pi}{2160\pi}$.
\qquad Attention is now turned to the effects of the quantum perturbation
on the black hole spacetime. It has long been thought that quantum
effects may remove the singular behaviour of physical spacetimes.
Although the perturbative scheme can not determine whether or not the
actual singularity is removed, it can give information on the growth of
curvature scalars on the interior spacetime. Using the perturbed metric,
the Kretschmann scalar can now be written as:
\begin{equation}
K=K_{orig} + \epsilon\delta K,
\end{equation}
where $K_{orig}$ is the unperturbed value given by (\ref{eq:origK}) and
$\delta K$ is the first order correction term.
Whether or not curvature is strengthened depends on the sign of $\delta
K$. If it is positive, the initial curvature growth is
strengthened. If it is negative, it is weakened. The analysis will be
limited to the region near the horizon.
\qquad The correction term is calculated to be
\begin{eqnarray}
\delta K &=&\frac{-1}{15}\left[25\,\alpha^{17}T^{17}
2^{2/3}+54\,\alpha^{16}T^{16}(2M)^{1/3}+116\,\alpha^{15}T^{15}M^{2/3}
\right. \nonumber \\
&-&286\,\alpha^{14}T^{14} 2^{2/3}M- 624\,\alpha^{13}T^{13}2^{1/3}M^{4/3}-
1352\,\alpha^{12}T^{12}M^{5/3}\nonumber \\
&+& 1104\,\alpha^{11}T^{11}2^{2/3}M^2+
2256\,\alpha^{10}T^{10}2^{1/3}M^{7/3}
4608\,\alpha^{9}T^{9}M^{8/3} \nonumber \\
&-& 2656\,\alpha^{8}T^{8} 2^{2/3}M^{3}-
3456\,\alpha^{7}T^{7}2^{1/3}M^{10/3}-
3200\,\alpha^{6}T^{6}M^{11/3} \nonumber \\
&+& 9472\,\alpha^{5}T^{5}2^{2/3}M^{4}+ 9984\,\alpha^{4}T^{4}2^{1/3}M^{13/3}
+2048\,\alpha^{3}T^{3}M^{14/3}\nonumber \\
&-& \left.18432\, \alpha^{2}T^{2}2^{2/3}M^{5}-
24576\,\alpha T 2^{1/3}M^{16/3}-24576M^{17/3}\right] \nonumber \\
&\times&\left[\alpha^{2}T^{6}M^{1/3}\pi (\alpha^{2}T^{2}+\alpha T (4M)^{1/3}
+ (4M)^{2/3})^{2} \right. \nonumber \\
&\times& \left. (4M^{2/3}+\alpha^{2} T^{2} 2^{2/3} + 2\alpha T
(2M)^{1/3})^{2} (\alpha T- (4M)^{1/3})^{2}\right]^{-1}
\end{eqnarray}
Although this result is slightly complicated, a few relevant properties
may be obtained. The value of
$\delta K$ near the horizon behaves as
\begin{equation} \delta K \approx \frac{3}{10}\frac{\alpha^{4}}{\pi}-
2\frac{\alpha^{5}}{\pi}\left(\frac{2}{M}\right)^{1/3} \left(T-
\frac{(4M)^{1/3}}{\alpha}\right), \label{eq:expdK}
\end{equation}
so that the limiting value at $T = \rho_{H}$ is given by
$\delta K(\rho_{H})= \frac{3}{10}\frac{\alpha^{4}}{\pi}$.
from which it can be seen that curvature growth is {\em strengthened} at the
horizon. The function (\ref{eq:expdK}) increases as one passes
through the
horizon towards the singularity and the perturbation eventually becomes
positive. Near the singularity the curvature diverges as $1/T^{16}$
although in this regime the approximation breaks down and the expression
has no physical meaning.
\qquad For the case of Schwarzschild geometry the curvature perturbation
diverges as $1/T^{9}$ near the singularity for massless conformally coupled
scalars \cite{ref:hiscock} and in the regime where the perturbation is
valid, curvature invariants are always strengthened. At the event
horizon
of a Schwarzschild black hole, for example, the perturbation is $\delta
K= \frac{1965}{2M^{4} 45\times 2^{13}\pi}$.
\section{Conclusion}
The stress energy tensor for a conformally coupled quantum scalar field
has been calculated in the black string spacetime and it is found that,
as is common with quantum
fields in curved spacetime, there exist regions where the weak energy
condition is violated. The
violation occurs on the interior and near the horizon on the exterior of
the black hole. From the stress energy tensor, the back-reaction has been
calculated in the form of the perturbed metric and Kretschmann scalar.
Similar to the case of spherical symmetry without cosmological constant, it
is found that curvature is strengthened on the interior indicating
that quantum effects may be geometry and cosmological constant independent.
\section{Acknowledgments}
The author would like to thank Dr. K.S. Viswanathan for helpful advice
during the production of this work. Discussions with Dr. A. Das on
relativity were also helpful.
\newpage
\bibliographystyle{unsrt}
| 2024-02-18T23:40:12.915Z | 1998-11-18T22:24:36.000Z | algebraic_stack_train_0000 | 1,679 | 3,717 |
|
proofpile-arXiv_065-8226 | \section{Introduction}
At the Rutherford Appleton Laboratory (RAL) the KARMEN collaboration
is studying neutrino-nuclear reactions, induced from the decay
products of positive pions, which are produced and stopped in the
proton beam dump. In 1995 KARMEN for the first time
reported~\cite{armbruster:1995} an anomaly in the time distribution of
single prong events concerning the time interval corresponding to muon
decay. Even with a much improved active detector shielding the anomaly
has persisted in new KARMEN data~\cite{zeitnitz:1998}.
This anomaly has been suggested to originate from the observation of a
hitherto unknown weakly interacting neutral and massive fermion,
called \ensuremath{\mathrm{x}}, from a rare pion decay process $\ensuremath{\mathrm{\pi^+}}\to\ensuremath{\mathrm{\mu^+}}\ensuremath{\mathrm{x}}$. After a
mean flight path of $\rm 17.5\,m$ \ensuremath{\mathrm{x}}\ is registered in the KARMEN
calorimeter after $\rm t_{TOF}=(3.60\pm0.25)\,\mu{}s$ beam on target
by its decay resulting in visible energies of typically $\rm
T_{vis}=11-35\,MeV$. The observed velocity and the two-body kinematics
of the assumed pion decay branch lead to a mass $\rm
m_\ensuremath{\mathrm{x}}=33.9\,MeV/c^2$, extremely close to the kinematical limit.
The hypothetical decay $\ensuremath{\mathrm{\pi^+}}\to\ensuremath{\mathrm{\mu^+}}\ensuremath{\mathrm{x}}$ has been searched for at PSI
in a series of experiments using magnetic spectrometers by studying
muons from pion decay in flight~\cite{bilger:1995:plb, daum:1995,
daum:1998}, the latest measurement resulting in an upper limit for
the branching ratio of $\rm BR(\ensuremath{\mathrm{\pi^+}}\to\ensuremath{\mathrm{\mu^+}}\ensuremath{\mathrm{x}})<1.2\cdot10^{-8}$
(95\% C.L.) \cite{daum:1998}. Combined with theoretical constraints
which assume no new weak interaction~\cite{barger:1995:plb:352} this
result rules out the existence of this rare pion decay branch if \ensuremath{\mathrm{x}}\
is an isodoublet neutrino. However, if \ensuremath{\mathrm{x}}\ is mainly isosinglet
(sterile), the branching ratio can be considerably
lower~\cite{barger:1995:plb:356}. From the number of observed \ensuremath{\mathrm{x}}\
events in comparison with the total number of \ensuremath{\mathrm{\pi^+}}\ decays the KARMEN
collaboration gives a lower limit for the branching ratio of
$10^{-16}$.
Very recently Gninenko and Krasnikov have
proposed~\cite{gninenko:1998} that the observed time anomaly can also
be explained by an exotic \ensuremath{\mathrm{\mu}}\ decay branch $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$
resulting in the production of a new, weakly interacting neutral boson
with mass $\rm m_\ensuremath{\mathrm{X}}=103.9\,MeV/c^2$. They show that a second
exponential in the KARMEN time distribution with time constant equal
to the muon lifetime and shifted by the flight time of the
\ensuremath{\mathrm{X}}-particle $\rm t_{TOF}=3.60\,\mu{}s$ gives an acceptable fit to the
KARMEN data. Considering three possible \ensuremath{\mathrm{X}}-boson phenomenologies,
they predict branching ratios for $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$ in the order of
$10^{-2}$, if \ensuremath{\mathrm{X}}\ is a scalar particle; $10^{-5}$, if \ensuremath{\mathrm{X}}\ decays via
a hypothetical virtual charged lepton; and $10^{-13}$, if \ensuremath{\mathrm{X}}\ decays
via two additional hypothetical neutral scalar bosons.
In this paper we present a direct experimental search for the \ensuremath{\mathrm{X}}\
particle by studying the low energy part of the Michel spectrum
looking for a peak from mono-energetic positrons with energy $\rm
T_\ensuremath{\mathrm{e}}=(m_\ensuremath{\mathrm{\mu}}^2+m_\ensuremath{\mathrm{e}}^2-m_\ensuremath{\mathrm{X}}^2)/(2m_\ensuremath{\mathrm{\mu}})-m_\ensuremath{\mathrm{e}}=1.23\,MeV$
resulting from the two-body decay $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$.
In the past, searches for exotic two-body \ensuremath{\mathrm{\mu}}\ decay modes have
already been performed~\cite{bryman:1986:prl} motivated by predictions
about the existence of light, weakly interacting bosons like axions,
majorons, Higgs particles, familons and Goldstone bosons resulting in
upper limits for the branching ratio of approximatley $3\cdot10^{-4}$
(90\% C.L.). However, these searches are not sensitive to the
suggested \ensuremath{\mathrm{X}}\ boson with $\rm m_\ensuremath{\mathrm{X}}=103.9\,MeV/c^2$ since the lowest
positron energy region studied was between 1.6 and 6.8\,MeV,
corresponding to the \ensuremath{\mathrm{X}}\ mass region 103.5 to $\rm 98.3\,MeV/c^2$.
\section{The Experiment}
The basic idea is to stop a $\mu^+$ beam inside a germanium detector.
The low energy decay positrons of interest also deposit their entire
kinetic energy in the detector volume. For a sizeable fraction of
events the subsequent annihilation radiation does not interact with
the detector thus preserving the positron energy information.
This experiment has been performed at the $\rm \mu{}E4$ channel at PSI
(see Fig.~\ref{fig:setup}). The beam line is optimized for intense
polarized muon beams in the momentum range between 30 and 100\,MeV/c
with very low pion and positron contamination. Pions from the
production target are collected at an angle of $90^\circ$ relative to
the primary proton beam and are injected into a long 5\,T
superconducting solenoid in which they can decay. The last part of the
beam line is the muon extraction section which allows the selection of
a central muon momentum different from that of the injected pions.
The detector setup consists of a large ($\rm 120\times200\,mm^2$)
2\,mm thick plastic scintillator counter S1 followed by a 35\,mm
diameter hole in a 10\,cm thick lead shielding wall and a small ($\rm
20\times20\,mm^2$) 1\,mm thick plastic scintillator counter S2
directly in front of a 9\,mm thick planar high purity germanium (HPGe)
detector with an area of $\rm 1900\,mm^2$. In addition, we have placed
a 127\,mm (5 inch) diameter, 127\,mm thick NaI detector shielded
against the \ensuremath{\mathrm{\mu}}-flux adjacent to the HPGe for detecting 511\,keV
$\gamma$ rays from positron annihilation.
\begin{figure}
\centerline{\epsfig{file=mue4_kanal_detector.eps,width=0.80 \textwidth}}
\caption[]{Schematical layout of the experimental setup. The $\rm
\ensuremath{\mathrm{\mu}}{}E4$ low energy \ensuremath{\mathrm{\mu}}\ channel at PSI is shown in the left
part of the figure together with a sketch of the detector setup,
which is shown in more detail in the right part of the figure.}
\label{fig:setup}
\end{figure}
The coincidence $\rm S1\times{}S2\times{}HPGe$ was used as a trigger
which generated --- in addition to a prompt gate --- a delayed gate
$\rm 2.2-7.2\,\mu{}s$ after the prompt muon signal for the expected
decay events. During the time period for the delayed gate, S1 was
used as a veto detector to discriminate against further beam
particles. Timing and energy information from the dectetors utilizing
several different methods for signal discrimination, amplification,
shaping and digitization were recorded for both prompt and delayed
signals using the MIDAS data acquisition system~\cite{midas:1998}.
For the energy calibration of signals occuring during the prompt gate,
\ensuremath{\mathrm{\gamma}}\ rays from $\rm ^{22}Na$ and $\rm ^{60}Co$ sources were used. In
order to derive the energy information from the HPGe detector signal,
both spectroscopy amplifiers and peak-sensitive ADCs as well as a
timing filter amplifier (TFA) connected to a charge sensitive QDC were
employed. In addition, sample signals from the HPGe detector, both
before and after amplification, were recorded and stored with a
digital oscilloscope. It turned out that every spectroscopy amplifier
available during the course of the experiment showed a significantly
varying baseline shift for a few microseconds following a prompt
signal. The variations of the baseline level just after the prompt
signal were due to fluctuations in time for the onset of the baseline
restoration circuitry. Thus, for spectroscopy amplifiers, a
sufficiently accurate energy calibration for the delayed signal was
not possible.
The TFA branch did not have such baseline problems, however the energy
resolution for the delayed signal in this branch is 100\,keV FWHM
only. A short shaping time of $\rm 0.25\,\mu$s and low amplification
to avoid saturation from the high-amplitude prompt signal had to be
used to be ready in time for the delayed pulse.
During 12 hours of data taking $1.3\cdot10^7$ events were recorded on
tape. Saturation of the HPGe pre-amplifier at a singles rate of
$5-6\cdot10^3\,\rm s^{-1}$ was limiting the event rate.
\section{Results}
The energy deposition of the stopped muons in the HPGe detector is
$\rm 11.3\!\pm\!0.7\,MeV$ (see
Fig.~\ref{fig:muex_hpge_delayed_tdc_prompt_adc}). The cut on the
energy of the prompt signal is $9.9 - 12.7$\,MeV. The delayed signal
has to occur within the time interval of $\rm 3.4-7.2\,\mu{}s$ after
the prompt signal. The time distribution (see
Fig.~\ref{fig:muex_hpge_delayed_tdc_prompt_adc}) nicely shows the
expected exponential shape with $\rm \tau=2.21\pm0.02\,\mu{}s$. For
shorter times the tail of the prompt signal still causes a varying
effective discriminator threshold thus the TDC spectrum deviates from
an exponential shape. The information from the NaI detector is used to
check the consistency of the analysis, but is not used for the
determination of the branching ratio.
\begin{figure}
\begin{center}
\epsfig{file=muex_hpge_prompt_signal_energy_2.ps,width=0.45 \textwidth,bbllx=32,bblly=160,bburx=560,bbury=648}
\epsfig{file=muex_hpge_delayed_tdc_2.ps,width=0.45 \textwidth,bbllx=0,bblly=160,bburx=528,bbury=648}
\end{center}
\caption[]{Left: Energy spectrum of prompt signals resulting from
muons stopping in the HPGe detector with an additional constraint
requiring the presence of an afterpulse arriving during the
delayed gate. Right: Spectrum for the time difference between
delayed and prompt signals. The time constant $\tau=2.21\pm0.02\mu
s$ of the exponential shape is in very good agreement with the
muon life time.}
\label{fig:muex_hpge_delayed_tdc_prompt_adc}
\end{figure}
After energy and time cuts $1.32\cdot10^6$ events remain.
Accounting for high energy positrons from muon decay causing a signal
in the veto counter S1, a 3\% correction results in $1.36\cdot10^6$
good muon decays for normalization.
GEANT \cite{geant:1998} based Monte Carlo studies have provided an
understanding of the shape of the delayed signal energy spectrum (see
inset in Fig.~\ref{fig:muex_branching}). The two peaks are due to an
asymmetric \ensuremath{\mathrm{\mu}}\ stop distribution with respect to the symmetry plane
perpendicular to the beam axis of the cylindrical HPGe detector
resulting in different energy distributions for Michel positrons
emitted in the backward and forward hemispheres of the detector,
respectively.
The interaction of the annihilation \ensuremath{\mathrm{\gamma}}\ rays with the detector
has also been studied. For positrons in the considered energy range
the double escape probability is 40-44\% (no 511\,keV \ensuremath{\mathrm{\gamma}}\ rays
interacting in the HPGe), the single escape probability being a factor
4 lower. The search for $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$ events as described below
concentrates on double escape events.
Assuming a smooth and gently varying background as confirmed by the
Monte Carlo studies, the search for a peak structure in the delayed
signal energy spectrum (see Fig.~\ref{fig:muex_branching}) has been
done for energies from $0.3$ to $2.2$\,MeV. The lower energy limit is
given by the effective discriminator threshold, the upper energy limit
from the positron zero transmission range in germanium. Since the beam
muons are stopped after $\rm 2-3\,mm$ (2$\sigma$) in the HPGe
detector, and since the 2.2\,MeV electrons have a zero transmission
range of 2\,mm, this is the highest energy for which all positrons
remain within the detector volume thus completely depositing their
kinetic energy.
\begin{figure}
\centerline{\epsfig{file=muex_branching_2.eps,width=0.80 \textwidth}}
\caption[]{Plots showing the energy deposition during the delayed
gate in the HPGe detector (top) and fit results leading to upper
limits for the branching ratio for the decay $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$.
For the abscissa two corresponding scales, which are the same for
all graphs (except for the inset at the top, which shows the full
\ensuremath{\mathrm{e}^+}\ energy range recorded), are drawn, one is the positron
kinetic energy $\rm T_\ensuremath{\mathrm{e}}$, the other the \ensuremath{\mathrm{X}}\ boson mass $\rm
m_\ensuremath{\mathrm{X}}$. In the graph at the top the Gaussian centered at 1.23\,MeV
gives the expected detector response if $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$ would
contribute with a branching ratio of $5\cdot10^{-3}$. The second
graph shows the reduced $\chi^2$, dashed line for a
polynomial-only fit, solid line for a combined polynomial and
Gaussian fit. The third graph, with ordinate units already
converted into branching ratio, shows the contents (solid line)
and the error (dashed line) of the Gaussian from this fit. The
graph at the bottom gives the upper limit for a $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$
decay branch at 90\% confidence level by applying the Bayesian
method to the fit results.}
\label{fig:muex_branching}
\end{figure}
For all positron energies between 0.3 and 2.2\,MeV a typically
1.2\,MeV wide energy interval is chosen and a polynomial fitted to
this part of the spectrum. For a polynomial of low order the fit has
an unrealistically high $\chi^2$. Increasing the order of the
polynomial the resulting $\rm \chi^2/D.F.$ first decreases and then
remains roughly constant with values around one.
A polynomial of order seven was chosen as the lowest order to have a
suitable reduced $\chi^2$ (second graph in
Fig.~\ref{fig:muex_branching}). Then a simultaneous fit of a Gaussian
(position and width fixed) and a polynomial provides the area and
error for a possible peak. In the third graph of
Fig.~\ref{fig:muex_branching} these results have already been
converted in branching ratio (BR) units. With a Bayesian
approach~\cite{barnett:1996} one can derive from these results an
upper limit with a given confidence level. Shown on the bottom of
Fig.~\ref{fig:muex_branching} is the 90\% C.L.\ upper limit.
For the positron energy $\rm T_\ensuremath{\mathrm{e}}=1.23\,MeV$ corresponding to an \ensuremath{\mathrm{X}}\
particle with mass $\rm m_\ensuremath{\mathrm{X}}=103.9\,MeV/c^2$ as suggested by Gninenko
and Krasnikov~\cite{gninenko:1998} the 90\% C.L.\ upper limit for the
branching ratio in the decay $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$ is $\rm
BR=4.9\cdot10^{-4}$.
\section{Summary and Outlook}
Following the proposition that a new, weakly interacting boson \ensuremath{\mathrm{X}}\
with mass $\rm m_\ensuremath{\mathrm{X}}=103.9\,MeV/c^2$ produced in $\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}}$
might be the reason for the observed anomaly in the KARMEN data, we
have searched for this two-body \ensuremath{\mathrm{\mu}}\ decay branch by inspection of
the low energy end of the Michel spectrum. Utilizing a clean \ensuremath{\mathrm{\mu}}\
beam from the $\mu{}E4$ channel at PSI and stopping the muons in a
planar HPGe detector this work is the first direct search for such an
exotic \ensuremath{\mathrm{\mu}}\ decay process for \ensuremath{\mathrm{X}}\ boson masses $\rm
103\,MeV/c^2<m_\ensuremath{\mathrm{X}}<105\,MeV/c^2$ corresponding to positron energies
$\rm 0.3\,MeV<T_e<2.2\,MeV$. Our first results give branching ratios
$\rm BR(\ensuremath{\mathrm{\mu^+}}\to\ensuremath{\mathrm{e}^+}\ensuremath{\mathrm{X}})<5.7\cdot10^{-4}$ (90\% C.L.) over most of the
accessible region, such excluding the simplest scenario for the \ensuremath{\mathrm{X}}\
boson phenomenology suggested in Ref.~\cite{gninenko:1998}. By
refining the experimental method used in this experiment it will be
feasible to improve on this result.
\bigskip
\noindent We gratefully acknowledge valuable support from and
discussions with D.~Branford, M.~Daum, T.~Davinson, F.~Foroughi,
C.~Petitjean, D.~Renker, U.~Rohrer, and A.C.~Shotter. We also would
like to thank the Paul Scherrer Institut for assistance in setting up
this experiment in a very short time.
| 2024-02-18T23:40:12.926Z | 1998-11-15T14:47:32.000Z | algebraic_stack_train_0000 | 1,681 | 2,717 |
|
proofpile-arXiv_065-8264 | \section{Introduction}
Since its very origins, the exact renormalization group (ERG)
\cite{KW,SW,WH,JP} has proved to be a powerful tool for studies of
non-perturbative effects in quantum field theory (see recent
reviews in \cite{YK,TM1}).
A particularly interesting case is that of an effective scalar
field theory in two dimensions. As it was first conjectured by
Zamolodchikov \cite{AZ}, for a $Z_2$ symmetric theory there
should exist an infinite set of non-perturbative fixed
points corresponding to the unitary minimal series of $(p,p+1)$
conformal field theories, where $p=3,4,\ldots,\infty$
\cite{BPZ}. Morris \cite{TM} showed numerically that such
points do exist.
The calculation was performed with a reparametrization
invariant version of the Legendre ERG equation \cite{TM2}
expanded in powers of derivatives. It was also pointed out
there that to the level of the local potential
approximation only the continuum limits described by periodic
solutions and corresponding to critical sine-Gordon models
could be obtained. To find the expected set of fixed points the
calculations had to be taken to the next order in the derivative
expansion. This constituted a manifestation of the non-perturbative
nature of the phenomena, and remarkably the Legendre ERG equation was
powerful enough to locate and describe with good accuracy
the expected set of 2D field theories.
In this work we study the same $Z_2$ symmetric scalar
field theories in two dimensions but now with the
Polchinski ERG equation \cite{JP}. We present preliminary results
which complement the results obtained with the Legendre ERG
equation. In Sect. 2 we follow the article by Ball
et al. \cite{BHLM} to present the basic equations
of the formalism. This will allow us to set up notation for
Sect. 3 where we analyse the equations to second order in
the derivative expansion. In Sect. 4 we discuss the results and
present our conclusions.
\section{Polchinski equation and derivative expansion}
The Polchinski equation \cite{JP} for a scalar theory can be
written as follows \cite{BT}
\begin{eqnarray}
& & {\partial\over{\partial t}}\hat{S}={\int_{\hat{p}}}
K'({\hat{p}^2})\left[{{\delta \hat{S}}\over{\delta{\hat{\varphi}_p}}}
{{\delta \hat{S}}\over{\delta{\hat{\varphi}_{-p}}}}-
{{{\delta^2}\hat{S}}\over{\delta{\hat{\varphi}_p}
\delta{\hat{\varphi}_{-p}}}}\right]+dS+ \nonumber \\
& & {\int_{\hat{p}}}\left[1-{d\over{2}}-{\eta(t)
\over{2}}-2{\hat{p}^2}{{K'({\hat{p}^2})}\over{K({\hat{p}^2})}}\right]
{\hat{\varphi}_p}{{\delta \hat{S}}\over{\delta{\hat{\varphi}_p}}} -
{\int_{\hat{p}}}{\hat{\varphi}_p}\hat{p}\cdot
{\partial'\over{\partial\hat{p}}}{{\delta \hat{S}}
\over{\delta{\hat{\varphi}_p}}} \label{cv1}.
\end{eqnarray}
Here $\hat{S}$ is a general Wilsonian action which can be written
in terms of dimensionless variables as follows
\begin{eqnarray}
\hat{S}[\hat{\varphi};t] & = & \frac{1}{2} \int_{p} \hat{\phi}_{p}
p^{2} \left( K(\hat{p}^{2}) \right)^{-1} \hat{\phi}_{-p} +
\hat{S}_{int}[\hat{\varphi};t], \label{action1} \\
\hat{S}_{int}[\hat{\varphi};t] & = & \int dy \left[ v(\hat{\phi}(y),t)
+ z(\hat{\phi}(y),t) \left( \frac{\partial \hat{\phi}}{\partial y^{\mu}}
\right)^{2} + \ldots \right].
\label{action2}
\end{eqnarray}
In Eq.\ (\ref{cv1}) the partial derivative on $\hat{S}$ means it
only acts on the explicit
$t=\ln({\Lambda_0}/\Lambda)$ dependence
of the couplings and the prime in the momentum derivative means
it does not act on the delta function of the energy-momentum
conservation, and ${\int_{\hat{p}}}\equiv\int{{{d^d}\hat{p}}
\over{{(2\pi)}^d}}$. $K(\hat{p}^2)$ is a (smooth)
regulating function which damps the high energy modes satisfying
the normalization condition $K(0)=1$.
The renormalized field $\hat{\varphi}_p$ changes with
scale according to
\begin{equation}
\Lambda \frac{\partial}{\partial \Lambda}\hat{\varphi}_{p}=
\left[ 1+{d\over{2}}-{1\over{2}}\eta(t) \right] \hat{\varphi}_{p},
\end{equation}
where $\eta(t)$ is the anomalous scaling dimension.
To the second order in the derivative expansion we
consider the two terms which are written explicitly in
Eq.\ (\ref{action2}). Within this approximation the Polchinski ERG
equation reduces to the following system \cite{BHLM}
\begin{eqnarray}
\dot{f} & = & f''+2Az'-2ff'+{\Delta^+}f+{\Delta^-}xf',
\label{eq1} \\
\dot{z} & = & z''+B{{f'}^2}-4zf'-2z'f+{\Delta^-}xz'-
\eta z-\eta/2, \label{eq2}
\end{eqnarray}
where ${\Delta^\pm}=1\pm d/2-\eta/2$, $f(x)=v'(x)$, $x\equiv\hat{\varphi}$
and the potentials $v(x,t)$ and $z(x,t)$ are defined in
Eq.\ (\ref{action2}). The dots and primes denote the partial derivatives
with respect to $t$ and $x$ respectively. The parameters $A$ and $B$ reflect
the scheme dependence of the equations and are equal to
$A=({I_1}{K_0})/{I_0}$, $B={K_1}/{K_0^2}$.
Here $K_n$ and $I_n$, $n=0,1,\cdots$ parametrize the regulating
function in Eq.\ (\ref{action1}) and are defined by
\begin{eqnarray}
{K_n} & = & {{(-1)}^{n+1}}{K^{(n+1)}}(0), \nonumber \\
{I_n} & = & -{\int_{\hat{p}}}{{({\hat{p}^2})}^n}K'
({\hat{p}^2})=-{\Omega_d}{\int_0^\infty}dz {z^{d/2-1-n}}K'(z),
\nonumber
\end{eqnarray}
where $K^{(n)}$ stands for the $n$-th derivative of $K$ and
${\Omega_d}=2/(\Gamma(d/2){{(4\pi)}^d})$.
In the next section we search for fixed-point solutions, i.e.
for functions $f(x)$ and $z(x)$ which are independent of $t$
and satisfy the system
\begin{eqnarray}
& & f''+2Az'-2ff'+{\Delta^+}f+{\Delta^-}xf'=0, \label{eqfp1} \\
& & z''+B{{f'}^2}-4zf'-2z'f+{\Delta^-}xz'-
\eta z-\eta/2=0. \label{eqfp2}
\end{eqnarray}
We will choose the initial conditions (according to the terminology
adopted in the literature on the ERG equations) set by the
$Z_2$ symmetry: $f(0)=0$ and $z'(0)=0$ and by the normalization
condition: $z(0)=0$. For the value of the first derivative of $f(x)$
at the origin we will take the condition $f'(0)=\gamma$, where
$\gamma$ is a free parameter. The anomalous dimension $\eta(t)$ at a
fixed point becomes the critical exponent $\eta_{*}$.
\section{Fixed points and critical exponents}
To solve Eqs. (\ref{eqfp1}), (\ref{eqfp2}) for $d=2$ we consider
the recursive numerical method already tested for $d=3$
\cite{BHLM}. The physical fixed point solutions
$f_{*}(x)$, $z_{*}(x)$ at the fixed point value $\eta=\eta_{*}$
are regular for $x > 0$ and have a certain asymptotic
behavior as $x\rightarrow +\infty$. Thus
the natural method for finding the correct numerical
solution is to select those which can be integrated as
far as possible in $x$. A generic solution will end at a
sharp singularity for a finite value of $x$.
The difficulty lies in the nonlinear and stiff nature of
the equations and the need to fine tune $\eta$ and $\gamma$.
This makes the direct integration of the system too hard.
One way out is to solve it recursively.
Unlike the case $d=3$ studied in a number of articles \cite{BHLM,D3},
one faces an additional difficulty in two dimensions.
It is not possible to start the iterative procedure by setting
in Eq.\ (\ref{eqfp1}) $z=0$ and $\eta=0$ as it is prescribed by
the consistency of the leading approximation. For $d=2$ the
Polchinski equation in the leading order has only
periodic or singular solutions for all values of $\gamma$.
To overcome this difficulty one has to consider
$\eta\not=0$ as the initial value to start the iterations.
Consequently, an analysis of the leading order Polchinski equation
\begin{equation}
f''-2ff'+{\Delta^+}f+{\Delta^-}xf'=0, \label{eqfp01}
\end{equation}
with the initial conditions $f(0)=0$, $f'(0)=\gamma$
for non-zero $\eta$ is required.
We studied Eq.\ (\ref{eqfp01}) for $d=2$ numerically for
a wide range of values of $\eta$ and $\gamma$. Our results
show that for each $0<\eta\leq 1$ we can fine tune $\gamma$
in such a way as to obtain a non-trivial regular
fixed point solution. The set of such values $(\eta, \gamma)$
form a discrete series of continuous lines $\eta(\gamma)$
(see Fig. 1).
In fact simple arguments can be presented which explain the
appearance of the lines in the parameter space corresponding
to regular fixed-point solutions $f_{*}(x)$. Let
$x_{0}(\eta,\gamma)$ denote position of the pole of a generic
solution of Eq.\ (\ref{eqfp01}). Suppose that for some values
$(\eta',\gamma')$ the solution is regular, i.e.
$x_{0}(\eta',\gamma')=+\infty$. Let us take another value
$\eta''$ sufficiently close to $\eta'$. Assuming that the
function $x_{0}(\eta,\gamma)$ is continuous, it is clear that
there should exist the value $\gamma''$ such that again
$x_{0}(\eta'',\gamma'')=+\infty$. Hence there is a line of
the "constant value" $x_{0}(\eta,\gamma)=+\infty$ in the parameter
space. When we move along a fixed line the solutions $f_{*}(x)$
do not change their shape significantly. Moreover, their shape
follows a regular pattern when passing from one curve to the
other similar to solutions obtained by Morris \cite{TM}.
This can be considered as a sign for the existence of the
infinite discrete set of fixed points corresponding to the
minimal unitary series of conformal models. We also would
like to note that for $-2 < \eta < 6$ and $-1 < \gamma < 1$
there are no other non-trivial fixed-point solutions besides
the ones corresponding to the lines discussed here.
\vspace{-1cm}
\begin{figure}[ht]
\epsfxsize=1.0\hsize
\epsfbox{plotd32.eps}
\vspace{-1.2cm}
\caption{Non-trivial fixed-point lines for $d=2$ and $d=3$.
For clarity, only the first 7 lines are shown.}
\end{figure}
\vspace{-0.1cm}
We would like to note that a similar picture takes place in
other dimensions. For $d=3$ we found the same discrete
set of lines in the $(\gamma, \eta)$-plane corresponding
to regular solutions of Eq.\ (\ref{eqfp01}), but in this
case they are situated in the interval $-1<\eta\leq 1/2$
(see Fig. 1). As one can see there is a line
(upper line in Fig. 1) which crosses the $\gamma$-axis at
$\gamma_{*}=-0.229\ldots$. This is the value of the parameter
$\gamma$ for which a non-trivial fixed point solution of the
Polchinski equation was found in the leading order (local
potential approximation) \cite{BHLM}. The important
observation is that there is only one line in the
$(\gamma, \eta)$-plane with positive values of $\eta$. Since
according to general arguments at physical fixed points
$\eta_{*}>0$, this suggests that for $d=3$ there is only one
non-trivial fixed point. This is the Wilson-Fischer fixed point
found in numerous previous studies \cite{D3}.
Recall that for $d=2$ all the lines are
situated in the $\eta > 0$ half-plane, hence one can expect
an infinite number of non-trivial fixed points.
One more remark is relevant here. By a
simple scaling analysis of Eq.\ (\ref{eqfp01}) it can be shown
that there is a certain mapping between the lines $\eta(\gamma)$
in different dimensions. When we pass from one dimension to another
the line experiences a vertical shift and scaling transformation.
More details about this mapping will be presented elsewhere.
We now pass to the study of the system (\ref{eqfp1}), (\ref{eqfp2}).
We have seen that there are families of solutions of
Eq.\ (\ref{eqfp01}) corresponding to a given fixed point. It
turns out that when the second equation of the system is taken
into account, this degeneracy disappears. We solved the system
using the following iteration procedure developed by Ball et al.
\cite{BHLM}. First we set $z(x)=0$, choose some initial
value $\eta=\eta_{0}$ and fine tune $\gamma$ to the value
$\gamma={\gamma_0}$ corresponding to the regular solution
$f_{0}(x)$ of the first equation (\ref{eqfp1}) (or (\ref{eqfp01})).
Of course, the point $(\eta_{0},\gamma_{0})$ lies on one of the
lines described above (see Fig. 1)). As the next step we insert the
function $f_{0}(x)$ into the second equation (\ref{eqfp2}) for
a fixed $B$ and fine tune $\eta$ to the value $\eta={\eta_1}$
for which a regular solution $z_{1}(x)$ exists. Then we substitute
$z_{1}(x)$ and $\eta_{1}$ into the first equation of the system
and find a regular solution for a fixed value of $A$ thus
obtaining a new value $\gamma={\gamma_1}$ and a new function
$f_{1}(x)$. We repeat this process keeping $A$ and $B$ fixed.
As a result a sequence of functions $f_0 (x)$, $z_1 (x)$,
$f_1 (x)$, $z_2 (x), \ldots$, and a sequence of numbers
$\gamma_0$, $\eta_1$, $\gamma_1$, $\eta_2, \ldots$,
are obtained, and we test them for convergence.
For $d=3$ and for the relevant line, associated
with the Wilson-Fischer fixed point, we confirmed the
results by Ball et al. \cite{BHLM}. The new feature in
our calculations is that we took $\eta_{0} \neq 0$ as the
initial value of the iterating procedure, whereas in \cite{BHLM}
only $\eta_{0}=0$ was considered.
We conclude that the numerical method converges and that
the rate of convergence is controlled by $A$ for fixed
$\eta_0$. The best $A$ was shown to correspond to
the inflexion point where $\Delta\eta=
{\eta_2}-{\eta_1}$ changes sign. The important observation is
that the final values $\eta_{*}$ and $\gamma_{*}$ to which the
iterations converge (i.e. the fixed-point values) do not
depend on the initial value $\eta_{0}$. When $\eta_0$ is closer
to the fixed-point value the rate of convergence is of course
faster. The final value $\eta_{*}$ depends on $B$ linearly.
For $A=0$ the two equations decouple
and there is no need for iterations to find a solution
of the system. We just have to adjust $B$ such that
${\eta_1}={\eta_0}$. For the Wilson-Fischer fixed
point ${\eta_0}={\eta_1}=0.04$ we found $B=0.666768\ldots$ and
$\gamma_{*}={\gamma_0}=-0.197435\ldots$.
\vspace{-1cm}
\begin{figure}[ht]
\epsfxsize=1.0\hsize
\epsfbox{fzplpi2.eps}
\vspace{-1.5cm}
\caption{The solutions $f(x)$ and $z(x)$ for $p=3,4,5,6$.}
\end{figure}
\vspace{-0.1cm}
For $d=2$ the situation is totally different.
For any line $\eta (\gamma)$ we start with and
the initial value $(\eta_0, \gamma_{0})$
the iterative procedure turns out to be divergent if
$A \not=0$. Only for $A=0$ we have been able to find a
solution to the now decoupled system (\ref{eqfp1}), (\ref{eqfp2})
by adjusting $B$. Similar to the case $d=3$ we have not found any
natural criteria to select the value of $B$ since for each line $B$
depends monotonically on $\eta$ decreasing as $\eta\rightarrow 0$.
To determine the fixed-point solutions, corresponding to the
minimal unitary series of conformal field theories, we have fixed
the value of $B$ by a fit to the series of exact values for
the anomalous scaling dimension $\eta_{*}$. In this way we found
$B \approx 0.25$. The fixed-point solutions for $f_{*}(x)$ display
regular behaviour and are reminiscent of those obtained by Morris
\cite{TM}. In particular they have $p-2$ extrema,
$p=3,4,\ldots$ (see Fig. 2). The fixed
point solutions for $z_{*}(x)$ have the same pattern of extrema,
though their profiles are different from those of Morris.
Next for $B=1/4$ and and corresponding values $\eta=\eta_{*}$ we
have calculated the critical exponent $\nu$. For this we considered
perturbations of the functions $f(x)$ and $z(x)$ around the
fixed-point solutions,
\[
f(x)={f_*}(x)+{\sum_{n=1}^\infty}{g_n}(x){e^{{\lambda_n}t}},
\quad
z(x)={z_*}(x)+{\sum_{n=1}^\infty}{h_n}(x){e^{{\lambda_n}t}},
\]
and substituted them into Eqs.\ (\ref{eq1}), (\ref{eq2}). After
linearization we obtained the system
\vspace{-0.3cm}
\begin{eqnarray}
{g_n}''-2{g_n}{f_*}'-2{f_*}{g_n}'+{\Delta_{*}^{+}}{g_n}+
{\Delta_{*}^{-}}x{g_n}'&=& {\lambda_n}{g_n}, \nonumber \\
{h_n}''-2{f_*}{h_n}'-2{g_n}{z_*}'-4{f_*}'{h_n}-4{g_n}'{z_*}
\qquad&& \nonumber \\
+{\Delta_{*}^{-}}x{h_n}'-{\eta_*}{h_n}+ 2B{f_*}'{g_n}'&=&{\lambda_n}{h_n},
\nonumber
\end{eqnarray}
where $\Delta_{*}^{+}$ and $\Delta_{*}^{-}$ are calculated for
$\eta=\eta_{*}$.
For $Z_2$ perturbations the initial conditions are
${g_n}(0)=0$, ${h_n}'(0)=0$. We also imposed the
normalization condition ${g_n}'(0)=1$. Away from the
fixed point (but sufficiently close to it) we relaxed ${h_n}$
to be different from zero, ${h_n}=\delta$.
Then $\lambda_n$ and $\delta$ were fine tuned so that
polynomially growing eigenfunctions were obtained.
For $d=2$, $A=0$ and $B=1/4$ we have calculated the critical exponent
$\nu=1/{\lambda_1}$. The results for $\eta_{*}$ and $\nu$
are given in Table 1. We conclude that the Polchinski ERG
equation gives the values for the critical exponent which
match quite well the exact conformal field theory
values \cite{AZ} and the results by Morris \cite{TM}, also
included in the table.
We note that the best fit value for $B$ that we have found
actually corresponds to a well defined subset of the regulating
functions. An explicit example is $K(x)=(1+ax+b{x^2}){e^{-x}}$,
where $a+2b+1=0$ to ensure ${I_1}=0$ and ${I_0}={\Omega_d}$,
and $a=-5\pm 2\sqrt{6}$ to give $B=1/4$.
\vspace{0.5cm}
\leftline{
\begin{tabular*}{11cm}{|r|@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c|
@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c|
@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c|
@{\extracolsep{\fill}}c|@{\extracolsep{\fill}}c@{\extracolsep{\fill}}|}
\hline
$p$ & 3 & 4 & 5 & 6 & 7 & 8 & 9 & 10 & 11 & 12 \\
\hline
$\eta_{*}$ & .204 & .183 & .0865 & .0708 & .0496 & .0413 &.0328
& .0281 & .0234 & .0206 \\
\hline
$\eta_{\hbox{\tiny{CFT}}}$ & .25 & .15 & .10 & .0714 & .0536 & .0417
& .0333 & .0273 & .0227 & ..0192 \\
\hline
$\eta'$ & .309 & .200 & .131 & .0920 & .0679 &
.0521 & .0412 & .0334 & 0277 & .0233 \\
\hline
$\nu$ & 1.41 & .560 & .527 & .521 & .515 & .512 & .509 & .508 & .507
& .506 \\
\hline
$\nu_{\hbox{\tiny{CFT}}}$ & 1 & .556 & .536 & .525 & .519 & .514
& .511 & .509 & .508 & .506 \\
\hline
$\nu'$ & .863 & .566 & .545 & .531 & .523 & .517 &
.514 & .511 & .509 & .508 \\
\hline
\end{tabular*}
}
\vspace{0.5cm}
\leftline{Table 1: The critical exponents $\eta_{*}$ and $\nu$
obtained, as compared to the }
\hspace{0.65cm} values obtained by Morris \cite{TM} ($\eta'$, $\nu'$)
and the exact results of
\hspace{0.65cm} conformal field theory ($\eta_{\hbox{\tiny{CFT}}}$, $\nu_{\hbox{\tiny{CFT}}}$).
\section{Discussion and conclusions}
In this work we have studied the solutions of the
Polchinski ERG equation for an effective $Z_2$-symmetric scalar
field theory in the two-dimensional space $R^{2}$.
We have seen that this equation provides a reliable
non-perturbative evidence for the existence of the fixed-point
solutions corresponding to the minimal unitary series of
conformal field theories and allows to calculate the anomalous
dimension and the critical exponents with good accuracy.
This constitutes another positive test of the power
of the ERG approach. At the same time our studies are complementary
to similar calculations within the ERG approach based on
the equation for the Legendre action \cite{TM}.
As mentioned above, in the leading order of the derivative expansion
(local potential approximation) the consistent value for $\eta$ is 0
and only periodic sine-Gordon type fixed-point solutions can be
obtained. However, we have found that there are continuous
families of fixed-point solutions corresponding to a series of
lines $\eta(\gamma)$ in the $(\gamma, \eta)$ plane which have a
part with positive values $\eta>0$. It was also argued
that these lines correspond to the multicritical fixed points
of the theory. It is by taking into account the second order
in the derivative expansion that we find isolated fixed-point solutions.
We have found the first 10 points out of the infinite series and
calculated the critical exponent $\nu$ for them.
The results depend on the choice of the regulating function.
The value of $B$, for which a regular solution exists, depends linearly
on $\eta$, so the criterium of minimal sensitivity cannot be applied to
fix $B$. The best fit to the conformal field
theory values of $\eta$ gives us $A=0$, $B \approx 0.25$. No other
regulating functions have been found to work. For $d=2$ whenever
$A\not=0$, the iterative procedure is not seen to converge.
In fact, for $A=0$ there is no need for iterations since the
two equations of the Polchinski approach decouple.
This is in sharp contrast to the case of $d=3$ where convergence
was checked for $A\not=0$ \cite{BHLM}.
For the values $A=0$, $B=1/4$, giving the best fit, our results
are comparable in accuracy and sometimes
better (the accuracy also increases with multicriticality) than
those of Morris \cite{TM}. We would like to note that fixing
$B$ by the best fit to exact results for the anomalous dimensions
$\eta$ is reminiscent of fixing the renormalization scheme
dependence in the perturbative renormalization group. It is also
similar to fixing the regulator by the condition of the
reparametrization
invariance for the ERG equation for the Legendre action
(that corresponds to the limiting case of $A=0$, $B =
\infty$) \cite{TM}.
Another important point we would like to mention is
that in our analysis we have not found
non-trivial fixed points other than those corresponding
to the minimal models. This is what one expects from
Zamolodchikov's $c$-theorem~\cite{AZ1}. The conclusion
is already clear from the analysis of the leading order Polchinski
equation for $\eta\not=0$ when the lines in the
$(\gamma,\eta)$-plane corresponding to non-trivial fixed points
are plotted.
As final comments we would like to mention that to obtain an
estimate of the error of our numerical results in Table 1
one needs to carry out the calculations to the next order
of the derivative expansion.
It would be also interesting to expand the analysis
for higher dimension operators.
\vspace{0.3cm}
\leftline{\bf Acknowledgements}
\vspace{0.15cm}
We would like to thank Tim Morris and Jos\'e Latorre
for some fruitful discussions during the Workshop.
We also acknowledge financial support by Funda\c {c}\~ao para a
Ci\^encia e a Tecnologia under grant number CERN/S/FAE/ \\
1177/97.
Yu.K.\ acknowledges financial support from
fellowship PRAXIS XXI \\
/BCC/4802/95.
R.N.\ acknowledges financial support from
fellowship PRAXIS XXI/BPD/14137/97.
\vspace{0.5cm}
\leftline{\bf References}
| 2024-02-18T23:40:13.053Z | 1998-11-16T17:26:23.000Z | algebraic_stack_train_0000 | 1,693 | 3,725 |
|
proofpile-arXiv_065-8346 | \section{Introduction}
\paragraph*{}
The motivation for the investigation of the domain
walls$^{\cite{um}}$ evolution stems for inflationary
universes$^{\cite{dois}}$. The physical motivation to investigate
charged domain walls$^{\cite{tres}}$ and domain walls with spinning
particles$^{\cite{quatro}\cite{cinco}}$ were provided mainly through
the works of C.A. Lopez$^{\cite{tres,cinco}}$ and \O . Gr\o
n$^{\cite{quatro}}$ in their study of repulsive gravitational
fields$^{\cite{seis,sete}}$.
Their physical motivation was basic to build electron models
extending on Lorentz stress to General Relativity (GR).
Since Hehl et al$^{\cite{oito}}$ have shown that Cartan torsion is
the geometrical interpretation of spin of the particles this provides a
natural motivation for the study of spinning particles in charged
domain walls. Therefore here I shall be concerned with an exact
static solution of ECM-field equations representing a charged planar
domain wall endowed with particles with spin. The Einstein
cosmological constant although it is one of the responsibles for the
repulsive gravitation$^{\cite{nove}\cite{sete}}$ is absent here. In
fact the spin energy density plays the role of a ``cosmological
constant'' on a sort of torsion vacuum.
Here we use the fact discovered recently by myself and
Lemos$^{\cite{dez}}$ that in EC-gravity space-time defects can be
dealt with simply by substitution of the energy density and pressures
by effective quantities with the contribution of spin energy densities.
This allow us to reach a series of interesting physical conclusions
concerning the gravitationally repulsive character$^{\cite{onze}}$ of
domain walls as well as the horizon position in relation to
the planar wall. This paper in a certain extent completes and
generalizes my previous works on torsion walls$^{\cite{doze,treze}}$.
In section II we review ECM theory and apply in
the case of static Taub$^{\cite{quinze}}$ wall. It is important to
note that Taub wall in the context of General Relativity (GR) is the
well known no wall since $(\sigma =p=0)$ where $\sigma$ and $p$ are
respectively the energy density and the longitudinal pressure. This
situation is not trivial in EC-gravity since $\sigma^{eff}\equiv
(\sigma -2\pi S^2)$ and $p^{eff}\equiv p-2\pi S^2$
vanishing do not imply that $\sigma =0=p$. This is an important
different between our paper and the others dealing with domain walls
in (GR). In section III some consequences of Section II as the
investigation of repulsive gravitation in
(EC)-gravity are found.
\section{Einstein-Cartan-Maxwell Gravitational Fields and
Electrostatic Domain Walls}
\paragraph*{}
Let us consider the formulation of ECM-field equations as given in
Tiwari and Ray$^{\cite{quinze}}$. The main difference is that here we
use (TR)-equations with plane
symmetry while in the TR-paper their solution is spherically
symmetric. In general the ECM-equations in the quasi-Einsteinian form
maybe written as
\begin{equation}
G^b_a(\{\})=-8\pi\!\!\stackrel{eff}{T}{}^{\!\!b}_{\!\!a}
\end{equation}
which corresponds to
\begin{equation}
G^0_0(\{ \})=-2\mu_{zz}-3\mu^2_z+2\mu_z\nu_z=8\pi
[\sigma+\sigma_E-2\pi S^2]\delta (z)e^{2\nu}\equiv 8\pi
\sigma^{eff}\delta (z)
\end{equation}
\begin{equation}
G^2_2(\{ \})=G^3_3(\{ \})=-\mu_{zz}-\mu^2_{zz}-\nu_{zz}=-
8\pi [\sigma_E-2\pi S^2]e^{2\nu}\delta (z)=-8\pi p^{eff}_{||}e^{2\nu}\delta (z)
\end{equation}
\begin{equation}
G^1_1(\{ \})=-(\mu_z\nu_z+\mu^2_z)=8\pi[p^{eff}_{||}]\delta (z)e^{2\nu}=
8\pi [p_{\bot}](\sigma_E+2\pi S^3)]
\end{equation}
Here $\{ \}$ represents the Riemannian connection $\delta (z)$ is
the Dirac $\delta$-distribution and where we have taken the
Taub$^{\cite{dezesseis}}$ plane symmetric space-time as
\begin{equation}
ds^2=e^{2\nu }(-dt^2+dz^2)+e^{2\mu}(dx^2+dy^2)
\end{equation}
and we have also considered $S^0_{23}\equiv u^0S_{23}$ as the only
nonvanishing component of the spin density tensor and $u^0$ is the
zero-velocity of the four-velocity. Here $S_{23}$ is the spin angular
momentum tensor. In (GR) the orthogonal pressure $p_{\bot}$ vanishes.
But here what vanishes is the effective orthogonal pressure in (4)
\begin{equation}
p_{\bot}^{eff}\equiv 0\rightarrow p_{\bot}=(\sigma_E+2\pi S^2)
\end{equation}
where here $\sigma_E\equiv E^2/8\pi$ is the electrostatic surface
energy density. Since torsion does not couple with the electrostatic
field$^{\cite{dezesete}}$ the Maxwell equations are the same as in
(GR). In our case we take the electric and torsion fields as
constants at the charged domain wall. Thus a simply first integration
of the system of (ECM)-differential system reads
\begin{equation}
\mu^+_z=-\frac{1}{2}\ \nu^+_z
\end{equation}
\begin{equation}
-4\mu^+_z= 8\pi^{eff}e^{2\nu}
\end{equation}
\begin{equation}
2\mu^+_z-2\nu^+_z=-8\pi p^{eff}e^{2\nu}
\end{equation}
Now the Taub$^{\cite{dezesseis}}$ wal is defined in EC-gravity as
\begin{equation}
\sigma^{eff}=p_{||}^{eff}=0
\end{equation}
which from (6) yields
\begin{equation}
\sigma_E=+2\pi S^2
\end{equation}
or $E^2=16\pi^2S^2$ which implies
\begin{equation}
E=\pm 4\pi S
\end{equation}
Where the plus sign denotes the upper half Minkowski space-time. The
non-Riemannian space-time deffect here is obtained by gluing together
two half Minkowski spaces across spin-torsion-charge function.
Where the result (11) comes
from classical electrostatics$^{\cite{dezoito}}$ is the electrostatic
field at the plane.
Therefore the spin density squared $S\equiv \sqrt{S^2}$ and $E=\pm 4\pi
\stackrel{0}{\sigma}_E$. Of course the plus and minus signs corresponds
to the upper and lower sides of the plane.
The Taub solution is the same as in (GR) and it
is$^{\cite{vinte}\cite{vum}}$
\begin{equation}
ds^2=-\frac{dt^2+dz^2}{\sqrt{1+Kz}}+(1+Kz)(dx^2+dy^2)
\end{equation}
Let us now solve eqns. (7-9) for the Vilenkin$^{\cite{um}}$ wall
where $\sigma^{eff}=-4p_{||}^{eff}$ which from (2) and (3) reads
\begin{equation}
\sigma +\sigma_E-2\pi S^2=-4(\sigma_E-2\pi S^2)
\end{equation}
or
\begin{equation}
\sigma =-5(\sigma_E+2\pi S^2)
\end{equation}
for the surface energy density of the wall. By solving the eqns.
(7-9) one obtains
\begin{equation}
e^{2\nu}=-e^{-\mu}=(1-4\pi \sigma^{eff}|z|)^{-1/2}
\end{equation}
by substitution of the RHS of (2) into (16) one obtains the final
``Vilenkin'' charged domain wall with spin and torsion
\begin{equation}
e^{2\nu}=e^{-\mu}\left(1-[4\pi \sigma +E^2+8\pi S^2]|z|\right)
\end{equation}
where $E^+=4\pi \stackrel{0}{\sigma}_E$ is the electrostatic field on
upper half of the planar wall. Notice that the gravitational
potential $g_{00}$ in (17) is written in terms of the electrostatic
potential when the weak field limit is taken. We may write the whole
energy density as $\sigma^{eff}=\sigma +2\sigma_E+2\pi S^2$.
\section{Gravitational Repulsion of the Charged Domain Wall With Spin and
Torsion}
\paragraph*{}
In this Section I shall review the Riemannian Gauss-Coddazzi
equations given in Ipser and Skivie$^{\cite{onze}}$. Since the only
modification of charged spinning particles domain wall is the
introduction of spin and electric surface charge densities into the
effective energy densities, the modification of physical conclusions
of (ECM)-domain walls is basically due to the changes in the
densities. Before the examination of the Gauss-Codazzi eqns. let us
consider the first important physical change due to the introduction
of spin, torsion and the electrostatic field into the domain wall.
The first change is obvious from the metric (17) and is given by the
``new horizon'' singularity
\begin{equation}
g_{00}=\left(1-\left[4\pi \sigma
+E^2+8\pi^2S^2\right]|z|\right)^{-1/2}\rightarrow \infty
\end{equation}
where
\begin{equation}
\sigma^{eff}=\sigma +2\sigma_E+2\pi S^2
\end{equation}
yielding
\begin{equation}
|z|=\frac{1}{4\pi [\sigma^{eff}]}
\end{equation}
>From (19) one sees that $2\pi S^2$ term gives an extra spin
contribution to the wall surface density.
Expression (20) simply tell us that the position of the singularity
becomes closer to the wall due to the influences of charge and spin.
A similar situation appears in the case of the cosmological constant
in the case of plane symmetric exact solution of Einstein equations.
Since the first term on the RHS of (21) is positive if $E>0$, $\sigma
<0$ which reminds us of the wormwhole geometry$^{\cite{dezenove}}$.
Let us now turn to the Gauss-Coddazzi equations
\begin{equation}
^3R+\pi_{ij}\pi^{ij}-\pi^2=-2G_{ij}\xi^i\xi^j
\end{equation}
\begin{equation}
h_{ij}D_K\pi^{jK}-D_i\pi =G_{jK}h^j_i\xi^K
\end{equation}
where $^3R$ is the Ricci scalar curvature of the 3-dimensional
geometry $h_{ij}$ of the surface $S,$ $\pi \equiv \pi^i_i$ where.
\begin{equation}
\pi_{ij}\equiv D_i\xi_j=\pi_{ji}
\end{equation}
is the extrinsic curvature and
\begin{equation}
D_i\equiv h^j_i\nabla_j
\end{equation}
is the covariant derivative projected onto the surface $S$ and
$h_{ij}\equiv g_{ij}-\xi_i\xi_j$ is the 3-dimensional geometric metric
tensor. From the above eqns, Ipser and Skivie$^{\cite{onze}}$ were
able to deduce the equation for the acceleration of a test particle
{\it off} the wall
\begin{equation}
\xi_iu^j\nabla_j\xi^i|_+=-\xi_iu^j\nabla_j\xi^i|_-=2\pi
G_N(\sigma^{eff}-2p^{eff})
\end{equation}
where $G_N$ is the Newtonian gravitational constant. By substitution
of $\sigma^{eff}$ and $p^{eff}_{||}$ into the eq. (26).
\begin{eqnarray}
\xi_iu^j\nabla_j\xi^i|_+ &=& 2\pi G_N(\sigma +\sigma_E-2\pi S^2-
2\sigma_E+4\pi S^2)\nonumber \\
&=& 2\pi G_N(\sigma -\sigma_E+2\pi S^2)
\end{eqnarray}
Since one knows that an observer who wishes to remain stationary next
to the wall must accelerate away from the wall if
$(\sigma^{eff}-2p^{eff}_{||})>0$ and towards to the wall if
$(\sigma^{eff}-2p^{eff}_{||})<0$ one notes from (27) that repulsive
domain wall would have
\begin{equation}
\sigma <\sigma_E-2\pi S^2
\end{equation}
Since in most of the physical systems in nature the spin energy
density $2\pi S^2$ is never higher than the surface electrostatic
energy density $\sigma_E$, $\sigma$ is not necessarily negative.
Nevertheless for attractive domain walls
\begin{equation}
\sigma >\sigma_E-2\pi S^2>0
\end{equation}
and the stress-energy surface domain wall density should not violate
the weak energy condition, then the wormhole geometry is not
possible in this case.
Other applications of charged domain walls with spin and torsion maybe
found elsewhere. In particular a detailed account of the relation
between domain walls in EC-gravity and traversable wormholes maybe
found in recent paper by myself and Lemos$^{\cite{dez}}$.
Also recently other type of topological defect (cosmic strings) have
been investigated by Cl\'ement$^{\cite{vdois}}$ as a source of
flat wormholes.
Finally it is important to mention that the matching conditions
in EC-theory were not
necessary to deal with the non-Riemannian domain
walls$^{\cite{vtres}}$ discussed here. Distributional curvature of
cosmic strings has been recently investigated by
Wilson$^{\cite{vquatro}}$.
\section*{Acknowledgements}
\paragraph*{}
I am very much indebt to P.S. Letelier, Prof. C.A. Lopez and A. Wang
for helpful discussions on this subject of this paper. Some
suggestions of an unknown referee are gratefully acknowledged. Grants
from CNPq (Ministry of Science of Brazilian Government) and
Universidade do Estado do Rio de Janeiro (UERJ) are acknowledged.
| 2024-02-18T23:40:13.279Z | 1998-11-05T13:01:24.000Z | algebraic_stack_train_0000 | 1,711 | 1,962 |
|
proofpile-arXiv_065-8424 |
\section{Introduction}
Since the discovery of the Jones polynomial \cite{Jones:85}, several
new invariants of knots, links and tangles have become available due to
the development of sophisticated mathematical techniques. Among these,
the quantum algebras as defined by Drinfeld \cite{Drinfeld:87} and
Jimbo \cite{Jimbo:85}, being examples of quasi-triangular Hopf
algebras, provide a systematic means of solving the Yang--Baxter
equation and in turn may be employed to construct representations of
the braid group. From each of these representations, a prescription
exists to compute invariants of oriented knots and links
\cite{Reshetikhin:87,Turaev:88,ZhangGouldBracken:91b}, from which the
Jones polynomial is recoverable using the simplest quantum algebra
$U_q\[sl(2)\]$ in its minimal (2-dimensional) representation.
\vfill
\pagebreak
From such a large class of available invariants, it is natural to ask
if generalisations exist, with the view to gaining a classification.
One possibility is to look to multiparametric extensions in order
to see which invariants occur as special cases. A notable example is
the HOMFLY invariant \cite{FreydYetterHosteLickorishMilletOcneanu:85}
which includes both the Jones and Alexander--Conway invariants
\cite{Alexander:23,Conway:70} as particular cases as well as the
invariants arising from minimal representations of $U_q\[sl(n)\]$
\cite{Turaev:88}. Another is the Kauffman polynomial which includes the
Jones invariant as well as those obtained from the quantum algebras
$U_q\[o(n)\]$ and $U_q\[sp(2n)\]$ in the $q$-deformations of the
defining representations \cite{Turaev:88}.
The work of Turaev and Reshetikhin \cite{ReshetikhinTuraev:90} shows
that the algebraic properties of quantum algebras are such that an
extension of this method to produce invariants of oriented tangles is
permissible. A tangle diagram is a link diagram with free ends. An
associated invariant takes the form of a tensor operator acting on a
product of vector spaces. Zhang \cite{Zhang:95} has extended this
formalism to the case of quantum superalgebras which are
$\mathbb{Z}_2$-graded generalisations of quantum algebras.
Since quantum superalgebras give rise to nontrivial one-parameter
families of irreducible representations, it is possible to utilise them
for the construction of two variable invariants. This was first shown
by Links and Gould \cite{LinksGould:92b} for the simplest case using the
family of four dimensional representations of $U_q\[gl(2|1)\]$. It was
also made known that a one variable reduction of this invariant
coincides with a one variable reduction of the Kauffman polynomial by
the use of the Birman--Wenzl--Murakami algebra. Extensions to more
general representations of quantum superalgebras are discussed in
\cite{GouldLinksZhang:96b}.
Thus far, little has been investigated with regard to the Links--Gould
invariant. Here we report on some properties and behaviour. The method
of evaluating the invariant involves a prior construction of the
quantum $R$-matrix associated with a family of four dimensional
representations. Having obtained this matrix, the construction of the
invariant follows from properties of ribbon Hopf (super)algebras and
their representations. Here we consider the invariants of $(1,1)$
tangles for the following reason: for invariants derived from
representations of quantum superalgebras with zero $q$-superdimension,
the corresponding invariant is also zero. If the representation is
irreducible, the quantum superalgebra symmetry of the procedure ensures
that the invariant of $(1,1)$ tangles takes the form of some scalar
multiple of the identity matrix. (See \cite{ReshetikhinTuraev:90} for
a discussion of this symmetry.) We take this scalar to be the
invariant.
In this paper, we prove that the Links--Gould invariant is not able to
distinguish between mutant links (\secref{KTMutants}), nor is it able
to distinguish a knot from its inverse (i.e. from the knot obtained by
reversing the orientation) (\propref{inversion}). However it is good at
distinguishing some knots and links from their mirror images (see
Propositions \ref{prp:chiralmeanspalindromic} and
\ref{prp:Rhadinocentrus}), and it is distinguished from the HOMFLY and
Jones polynomials by this behaviour (see
\secref{NineFortyTwoandTenFortyEight} for specific examples). We
provide many examples and a complete description of the state model for
the invariant in abstract tensor form. This description of the
invariant directly facilitates the construction of a computer program
in \textsc{Mathematica} for calculation of the invariant.
\section{Construction of the $ R $ Matrix}
\seclabel{Rmatrixdef}
We consider the family of four dimensional representations of the
quantum superalgebra $ U_q \[ gl \( 2 | 1 \) \] $, which depend on a
free complex parameter $ \alpha $.
This superalgebra has $ 7 $ simple generators
$
\{
{E^1}_1, {E^2}_2, {E^3}_3,
{E^1}_2, {E^2}_1,
{E^2}_3, {E^3}_2
\}
$
on which we define a $ \BZ_2 $ grading in
terms of the natural grading on the indices $ \[ 1 \] = \[ 2 \] = 0 $,
$ \[ 3 \] = 1 $ by:
\be
\[ {E^i}_j \] = \[ i \] + \[ j \]
\qquad \( \mod~2 \).
\ee
The $ U_q \[ gl \( 2 | 1 \) \] $ generators satisfy the commutation
relations:
\be
\[ {E^1}_2, {E^2}_1 \]
\eq
{\[ {E^1}_1 - {E^2}_2 \]}_q
\\
\{ {E^2}_3, {E^3}_2 \}
\eq
{\[ {E^2}_2 + {E^3}_3 \]}_q
\\
\[ {E^i}_i, {E^j}_k \]
\eq
{\delta^j}_i {E^i}_k
-
{\delta^i}_k {E^j}_i,
\qquad
i, j, k = 1, 2, 3,
\ee
where $ \[~,~\] $ and $ \{~,~\} $ denote the usual commutator and
anticommutator, respectively and
we have employed the \emph{$ q $ bracket}, defined for a wide
class of objects $ x $ by:
\be
{[x]}_q
\defeq
\frac{q^x - q^{-x}}{q - q^{-1}}.
\ee
Let $ \{ \ket{i} \}_{i = 1}^4 $ denote a basis for
the four dimensional $ U_q \[ gl \( 2 | 1 \) \] $ module
$ V $. Consistent with the $ \mathbb{Z}_2 $ grading
on $ U_q \[ gl \( 2 | 1 \) \] $, we grade the basis states by:
\be
\[ \ket{1} \] = \[ \ket{4} \] = 0,
\qquad
\[ \ket{2} \] = \[ \ket{3} \] = 1.
\ee
We define a dual basis $ \{ \bra{i} \}_{i = 1}^4 $; in component form,
these are represented by the transpose complex conjugates
of the original basis:
$ \bra{i} = \overline{\ket{i}}^t \equiv \ket{i}^\dagger $. Then:
$ \bra{i} \ket{j} \equiv \langle i | j \rangle = \delta_{i j} $.
In terms of these dual bases, we define a representation $ \pi $ of the
$ U_q \[ gl \( 2 | 1 \) \] $ generators; their action on the basis
vectors is given by:
\be
\pi \( {E^1}_1 \)
\eq
- \ket{2} \bra{2} - \ket{4} \bra{4}
\\
\pi \( {E^2}_2 \)
\eq
- \ket{3} \bra{3} - \ket{4} \bra{4}
\\
\pi \( {E^3}_3 \)
\eq
\alpha \ket{1} \bra{1}
+
(\alpha+1) \( \ket{2} \bra{2} + \ket{3} \bra{3} \)
+
(\alpha+2) \ket{4} \bra{4}
\\
\pi \( {E^1}_2 \)
\eq
-
\ket{3} \bra{2}
\\
\pi \( {E^2}_1 \)
\eq
-
\ket{2} \bra{3}
\\
\pi \( {E^2}_3 \)
\eq
{[\alpha]}_q^{1/2} \ket{1} \bra{3}
-
{[\alpha+1]}_q^{1/2} \ket{2} \bra{4}
\\
\pi \( {E^3}_2 \)
\eq
[\alpha]_q^{1/2} \ket{3} \bra{1}
-
{[\alpha +1 ]}_q^{1/2} \ket{4} \bra{2}.
\ee
\pagebreak
Associated with $U_q\[gl(2|1)\]$ there is a co-product structure
($\mathbb{Z}_2$-graded algebra homomorphism)
$
\Delta : U_q\[gl(2|1)\] \to U_q\[gl(2|1)\] \otimes U_q\[gl(2|1)\]
$
given by:
\be
\Delta ( {E^i}_i)
\eq
I\otimes {E^i}_i + {E^i}_i \otimes I,
\qquad
i = 1, 2, 3,
\\
\Delta ( {E^1}_2 )
\eq
{E^1}_2 \otimes q^{-\frac{1}{2} ( {E^1}_1 - {E^2}_2 )}
+
q^{\frac{1}{2} ( {E^1}_1 - {E^2}_2 )} \otimes {E^1}_2
\\
\Delta ( {E^2}_1 )
\eq
{E^2}_1 \otimes q^{-\frac{1}{2} ( {E^1}_1 - {E^2}_2)}
+
q^{\frac{1}{2} ( {E^1}_1 - {E^2}_2 )} \otimes {E^2}_1
\\
\Delta ( {E^2}_3 )
\eq
{E^2}_3 \otimes q^{-\frac{1}{2} ( {E^2}_2 + {E^3}_3 )}
+
q^{\frac{1}{2}( {E^2}_2 + {E^3}_3 )} \otimes {E^2}_3
\\
\Delta ( {E^3}_2 )
\eq
{E^3}_2 \otimes q^{-\frac{1}{2}( {E^2}_2 + {E^3}_3 )}
+
q^{\frac{1}{2}( {E^2}_2 + {E^3}_3 )} \otimes {E^3}_2.
\ee
There exists another possible co-product structure:
$ \overline{\Delta} $, defined by
$ \overline{\Delta} = T \cdot \Delta $, where
$
T :
U_q \[ gl \( 2 | 1 \) \] \otimes U_q \[ gl \( 2 | 1 \) \]
\to
U_q \[ gl \( 2 | 1 \) \] \otimes U_q \[ gl \( 2 | 1 \) \]
$
is the twist map, defined for homogeneous elements
$ a, b \in U_q \[ gl \( 2 | 1 \) \] $:
\be
T \( a \otimes b \)
=
{\( - \)}^{\[ a \] \[ b \]}
\( b \otimes a \).
\ee
The tensor product module has the following decomposition with respect
to the co-product for generic values of $ \alpha $:
\bse
V \otimes V
=
V_1 \oplus V_2 \oplus V_3.
\eqlabel{TPdecomp}
\ese
We construct symmetry adapted bases
$ \{ \ket{\Psi^k_1} \}_{k=1}^4 $ and $ \{ \ket{\Psi^k_3} \}_{k=1}^4 $,
for the spaces $ V_1 $ and $ V_3 $ respectively in terms of the basis
elements of $ V $:
\be
\ket{\Psi^1_1}
\eq
\ket{1} \otimes \ket{1}
\\
\ket{\Psi^1_2}
\eq
{(q^{\alpha}+q^{-\alpha})}^{-\frac{1}{2}}
\(
q^{\alpha/2} \ket{1} \otimes \ket{2}
+
q^{-\alpha/2} \ket{2} \otimes \ket{1}
\)
\\
\ket{\Psi^1_3}
\eq
(q^{\alpha}+q^{-\alpha})^{-\frac{1}{2}}
\(
q^{\alpha/2} \ket{1} \otimes \ket{3}
+
q^{-\alpha/2} \ket{3} \otimes \ket{1}
\)
\\
\ket{\Psi^1_4}
\eq
{\( q^{\alpha} + q^{-\alpha} \)}^{-\frac{1}{2}}
{[2\alpha+1]}_q^{-\frac{1}{2}}
\times
\\
& &
\! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \!
\[
{[\alpha+1]}_q^{\frac{1}{2}}
\(
q^{\alpha} \ket{1} \otimes \ket{4}
+
q^{-\alpha} \ket{4} \otimes \ket{1}
\)
+
{[\alpha]}_q^{\frac{1}{2}}
\(
q^{\frac{1}{2}} \ket{2} \otimes \ket{3}
-
q^{-\frac{1}{2}} \ket{3} \otimes \ket{2}
\)
\]
\\
\ket{\Psi^3_1}
\eq
{(q^{\alpha+1}+q^{-\alpha-1})}^{-\frac{1}{2}}
{[2\alpha+1]}_q^{-\frac{1}{2}}
\times
\\
& &
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\[
{[\alpha]}_q^{\frac{1}{2}}
\(
q^{\alpha+1} \ket{4} \otimes \ket{1}
+
q^{-\alpha-1} \ket{1} \otimes \ket{4}
\)
+
[\alpha+1]_q^{\frac{1}{2}}
\(
q^{-\frac{1}{2}} \ket{3} \otimes \ket{2}
-
q^{\frac{1}{2}} \ket{2} \otimes \ket{3}
\)
\]
\\
\ket{\Psi^3_2}
\eq
{(q^{\alpha+1}+q^{-\alpha-1})}^{-\frac{1}{2}}
\(
q^{(\alpha+1)/2} \ket{4} \otimes \ket{2}
+
q^{-(\alpha+1)/2} \ket{2} \otimes \ket{4}
\)
\\
\ket{\Psi^3_3}
\eq
{(q^{\alpha+1}+q^{-\alpha-1})}^{-\frac{1}{2}}
\(
q^{(\alpha+1)/2} \ket{4} \otimes \ket{3}
+
q^{-(\alpha+1)/2} \ket{3} \otimes \ket{4}
\)
\\
\ket{\Psi^3_4}
\eq
\ket{4} \otimes \ket{4}.
\ee
Dual bases
$ \{ \bra{\Psi^k_1} \}_{k=1}^4 $ and $ \{ \bra{\Psi^k_3} \}_{k=1}^4 $,
are found from the definitions:
\bne
\bra{\Psi^k_j}
\eq
\ket{\Psi^k_j}^{\dagger},
\qquad \qquad \qquad
k = 1, 3,
\qquad
j = 1, \dots, 4,
\eqlabel{dualrule1}
\\
{( \ket{i} \otimes \ket{j} )}^{\dagger}
\eq
{\( - \)}^{\[ \ket{i} \] \[ \ket{j} \]}
\( \bra{i} \otimes \bra{j} \),
\qquad
i, j = 1, \dots, 4.
\eqlabel{dualrule2}
\ene
Now, the general form of the basis vectors $ \ket{\Psi^k_j} $ is:
\be
\ket{\Psi^k_j}
=
\sum_{m}
\theta^{k j}_{m}
\( \ket{x^{k j}_{m}} \otimes \ket{y^{k j}_{m}} \),
\ee
where the $ \theta^{k j}_{m} $ are in general complex scalar
functions of $ q $ and $ \alpha $. From \eqref{dualrule1} and
\eqref{dualrule2}, and choosing the parameters $ q $ and $ \alpha $ to
be real and positive, the duals of these vectors are given by:
\be
\bra{\Psi^k_j}
=
\sum_{m}
{\( - \)}^{ \[ \ket{x^{k j}_{m}} \] \[ \ket{y^{k j}_{m}} \]}
\theta^{k j}_{m}
\( \bra{x^{k j}_{m}} \otimes \bra{y^{k j}_{m}} \).
\ee
As the $ R $ matrix is unique,
analytic continuation makes our final results valid for any
complex $ q $ and $ \alpha $. For the duals, we obtain:
\be
\bra{\Psi^1_1}
\eq
\bra{1} \otimes \bra{1}
\\
\bra{\Psi^1_2}
\eq
(q^{\alpha}+q^{-\alpha})^{-\frac{1}{2}}
\(
q^{\frac{1}{2}\alpha} \bra{1} \otimes \bra{2}
+
q^{-\frac{1}{2}\alpha} \bra{2} \otimes \bra{1}
\)
\\
\bra{\Psi^1_3}
\eq
(q^{\alpha}+q^{-\alpha})^{-\frac{1}{2}}
\(
q^{\frac{1}{2}\alpha} \bra{1} \otimes \bra{3}
+
q^{-\frac{1}{2}\alpha} \bra{3} \otimes \bra{1}
\)
\\
\bra{\Psi^1_4}
\eq
{\( q^{\alpha} + q^{-\alpha} \)}^{-\frac{1}{2}}
[2\alpha+1]_q^{-\frac{1}{2}}
\times
\\
& &
\! \! \! \!
\! \! \! \!
\! \! \! \!
\[
[\alpha+1]_q^{\frac{1}{2}}
\(
q^{\alpha} \bra{1} \otimes \bra{4}
+
q^{-\alpha} \bra{4} \otimes \bra{1}
\)
-
[\alpha]_q^{\frac{1}{2}}
\(
q^{\frac{1}{2}} \bra{2} \otimes \bra{3}
-
q^{-\frac{1}{2}} \bra{3} \otimes \bra{2}
\)
\]
\\
\bra{\Psi^3_1}
\eq
(q^{\alpha+1}+q^{-\alpha-1})^{-\frac{1}{2}}
[2\alpha+1]_q^{-\frac{1}{2}}
\times
\\
& &
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\! \! \! \!
\[
[\alpha]_q^{\frac{1}{2}}
\(
q^{\alpha+1} \bra{4} \otimes \bra{1}
+
q^{-\alpha-1} \bra{1} \otimes \bra{4}
\)
-
[\alpha+1]_q^{\frac{1}{2}}
\(
q^{-\frac{1}{2}} \bra{3} \otimes \bra{2}
-
q^{\frac{1}{2}} \bra{2} \otimes \bra{3}
\)
\]
\\
\bra{\Psi^3_2}
\eq
{\( q^{\alpha+1}+q^{-\alpha-1} \)}^{-\frac{1}{2}}
\(
q^{\frac{1}{2}(\alpha+1)} \bra{4} \otimes \bra{2}
+
q^{-\frac{1}{2}(\alpha+1)} \bra{2} \otimes \bra{4}
\)
\\
\bra{\Psi^3_3}
\eq
(q^{\alpha+1}+q^{-\alpha-1})^{-\frac{1}{2}}
\(
q^{\frac{1}{2}(\alpha+1)} \bra{4} \otimes \bra{3}
+
q^{-\frac{1}{2}(\alpha+1)} \bra{3} \otimes \bra{4}
\)
\\
\bra{\Psi^3_4}
\eq
\bra{4} \otimes \bra{4}.
\ee
From the basis vectors $ \ket{\Psi^k_j} $ and their duals
$ \bra{\Psi^k_j} $ for $ V_1 $ and $ V_3 $, we
construct projectors $ P_1 $ and $ P_3 $, defined by:
\be
P_1
=
\sum_{k=1}^4
\ket{\Psi^k_1} \bra{\Psi^k_1},
\qquad
P_3
=
\sum_{k=1}^4
\ket{\Psi^k_3} \bra{\Psi^k_3}.
\ee
Note that the multiplication operation on the graded space
$ V \otimes V $ is given by:
\bse
\( \ket{i} \otimes \ket{j} \) \( \bra{k} \otimes \bra{l} \)
=
{\( - \)}^{\[ \ket{j} \] \[ \bra{k} \]}
\( \ket{i} \bra{k} \otimes \ket{j} \bra{l} \),
\qquad
i, j, k, l = 1, 2, 3, 4.
\eqlabel{Callistemon}
\ese
Now let $ I $ be the identity operator on $ V \otimes V $, viz:
$
I
=
\sum_{i j=1}^4
{e^i}_i \otimes {e^j}_j
$,
where $ {e^k}_l = \ket{k} \bra{l} $ is an elementary rank $ 2 $ tensor.
As we have $ P_1 + P_2 + P_3 = I $, we thus do not need to explicitly
construct $ P_2 $ (or even a basis for $ V_2 $); we simply set:
\bse
P_2 = I - P_1 - P_3.
\eqlabel{P2def}
\ese
Where $g$ is a classical Lie superalgebra, the corresponding quantum
superalgebra $U_q[g]$ admits a universal $R$ matrix
$R\in U_q[g]\otimes U_q[g]$ satisfying (among other relations):
\bne
R \Delta \( a \)
\eq
\overline{\Delta} \( a \) R,
\qquad \quad \;
\forall a \in U_q \[ g \],
\nonumber
\\
R_{12} R_{13} R_{23}
\eq
R_{23} R_{13} R_{12},
\qquad
\mathrm{in}
\;
U_q \[ g \] \otimes U_q \[ g \] \otimes U_q \[ g \],
\eqlabel{YB}
\ene
where the subscripts refer to the embedding of $ R $ acting on the
triple tensor product space. From any representation of
$ U_q \[ g \] $, one may obtain a tensor solution of \eqref{YB} by
replacing the superalgebra elements with their matrix representatives.
Similarly to \eqref{Callistemon}, multiplication of tensor products of
matrices $ a, b, c, d $ is governed by:
\be
\( a \otimes b \)
\( c \otimes d \)
=
{\( - \)}^{\[ b \] \[ c \]}
\( a c \otimes b d \),
\qquad
\textrm{homogeneous} \;
b, c.
\ee
We introduce the \emph{graded permutation operator $ P $} on the
tensor product space $ V \otimes V $,
defined for graded basis vectors
$ v^k, v^l \in V $ by:
\be
P ( v^k \otimes v^l )
=
{\( - \)}^{\[ k \] \[ l \]}
( v^l \otimes v^k ),
\ee
and extended by linearity.
(We use the shorthand $ \[ v^k \] \equiv \[ k \] $.)
With this, we define:
\be
\sigma = P R,
\ee
which can be shown to satisfy the equation:
\bse
\( \sigma \otimes I \)
\( I \otimes \sigma \)
\( \sigma \otimes I \)
=
\( I \otimes \sigma \)
\( \sigma \otimes I \)
\( I \otimes \sigma \).
\eqlabel{Isigma}
\ese
From \cite{LinksGould:92b}, we have (with a slight change of notation
and a convenient choice of normalisation):
\be
\sigma
=
q^{-2 \alpha} P_1
-
P_2
+
q^{2 \alpha + 2} P_3.
\ee
Using \eqref{P2def}, this simplifies to:
\be
\sigma
=
\( 1 + q^{-2 \alpha} \) P_1
+
\( 1 + q^{2 \alpha + 2} \) P_3
-
I.
\ee
From the above form of $ \sigma $, it is straightforward to deduce that
$ \sigma $ satisfies the polynomial identity:
\be
q^{-1} \sigma^3
+
\( q^{-1} - q^{- 2 \alpha - 1} - q^{2 \alpha + 1} \) \sigma^2
+
\( q - q^{- 2 \alpha - 1} - q^{2 \alpha + 1} \) \sigma
+
q I
=
0.
\ee
The above skein relation may be used to evaluate the invariant in some
cases, but not all since it is of third order. The invariant may also
be directly evaluated for a class of links using quantum superalgebra
theoretic results \cite{GouldLinksZhang:96b}.
\pagebreak
We will represent rank $ 2 $ tensors as matrices, that is, the
elementary rank $ 2 $ tensor $ {e^i}_k $ is represented by the
elementary ($ 4 \times 4 $) matrix $ e_{i,k} $.
We adopt the (standard) convention that the elementary rank $ 4 $
tensor $ {e^{i j}}_{k l} = {e^i}_k \otimes {e^j}_l $ is constructed by
insertion of a copy of the elementary rank $ 2 $ tensor $ {e^j}_l $ at
each location of $ {e^i}_k $ (i.e.
each element of $ {e^i}_k $ is multiplied by the whole of $ {e^j}_l $).
This means that
$ {e^{i j}}_{k l} $ is represented by the elementary ($ 16 \times 16 $)
matrix $ e_{4(i-1)+j,4(k-1)+l} $.
Let $ A $ be an arbitrary graded rank $ 4 $ tensor acting on
$ V \otimes V $, then for scalar coefficients $ {A^{i j}}_{k l} $:
\be
A
=
\sum_{i j k l}
{A^{i j}}_{k l}
\( {e^i}_k \otimes {e^j}_l \).
\ee
Our convention then tells us that the coefficient $ {A^{i j}}_{k l} $ is
the $ \( 4(i-1)+j, 4(k-1)+l \) $ entry of
$ A $, written explicitly:
\be
{A^{i j}}_{k l}
\mapsto
A_{4(i-1)+j,4(k-1)+l}.
\ee
We wish to remove the grading on $ V $, and convert the matrix
representing $ \sigma $ to its ungraded counterpart.
Recall that basis vectors $ v^k $ satisfy
$ {e^i}_j v^k = {\delta^k}_j v^i $, hence the action of $ A $ on
$ V \otimes V $ is:
\be
A ( v^k \otimes v^l )
\eq
\sum_{i j m n}
{A^{i j}}_{m n}
\( {e^i}_m \otimes {e^j}_n \)
( v^k \otimes v^l )
\\
\eq
\sum_{i j m n}
{A^{i j}}_{m n}
{\( - \)}^{\[ k \] \( \[ j \] + \[ n \] \)}
\( {e^i}_m v^k \otimes {e^j}_n v^l \)
\\
\eq
\sum_{i j m n}
{A^{i j}}_{m n}
{\( - \)}^{\[ k \] \( \[ j \] + \[ n \] \)}
\( {\delta^k}_m v^i \otimes {\delta^l}_n v^j \)
\\
\eq
\sum_{i j m n}
{A^{i j}}_{m n}
{\( - \)}^{\[ k \] \( \[ j \] + \[ n \] \)}
{\delta^k}_m {\delta^l}_n
\( v^i \otimes v^j \)
\\
\eq
\sum_{i j}
{A^{i j}}_{k l}
{\( - \)}^{\[ k \] \( \[ j \] + \[ l \] \)}
\( v^i \otimes v^j \).
\ee
Now, in this sum, the parity factor is constructed from the degrees
of vectors; in the ungraded case, there would be no such factor, indeed
we would have:
\be
\overline{A} ( v^k \otimes v^l )
\eq
\sum_{i j}
{\overline{A}^{i j}}_{k l}
\( v^i \otimes v^j \).
\ee
This motivates us to set:
\be
{\overline{A}^{i j}}_{k l}
=
{\( - \)}^{\[ k \] \( \[ j \] + \[ l \] \)}
{A^{i j}}_{k l}.
\ee
Under these conventions, the explicit form of $ \sigma $ is presented
(as a matrix!) in \secref{lglink}.
\section{Knot Theory}
\subsection{Link Examples}
In \tabref{linknamesandwrithes}, we list the links to be studied. (By
the term `knot', we intend a link of one component.) We use the
well-known notation of Alexander and Briggs (1926)
\cite{AlexanderBriggs:26}, the data being abstracted from
\cite{Adams:94}, itself citing \cite{Rolfsen:76} and
\cite{DollHoste:91} (beware that the tables in this latter article are
presented in microfiche form only).
\defNo{No}
\defYes{Yes}
\def\scriptstyle{\scriptstyle}
\begin{table}[ht]
\centering
\begin{tabular}{||l|c|p{32mm}|p{32mm}||}
\hline\hline
& & & \\[-3mm]
\multicolumn{1}{||c|}{$ K $} &
$ w \( K \) $ &
\multicolumn{1}{c|}{Chiral?} &
\multicolumn{1}{c||}{Invertible?} \\
& & & \\[-3mm]
\hline\hline
& & & \\[-2mm]
$ 0_1 $ (Unknot) &
$ 0 $ & $ No $ (trivial) & $ Yes $ (trivial) \\[2mm]
\hline
& & & \\[-2mm]
$ 2^2_1 $ (Hopf Link) &
$ 2 $ & $ No $ (trivial) & $ Yes $ (trivial) \\[2mm]
\hline
& & & \\[-2mm]
$ 3_1 $ (Trefoil) &
$ 3 $ & $ Yes $ \cite[p~176]{Adams:94} & $ Yes $ (trivial)
\\[2mm]
\hline
& & & \\[-2mm]
$ 4_1 $ (Figure Eight) &
$ 0 $ &
$ No $
(\cite[p~14]{Adams:94};
see \cite[p~198]{Kauffman:88} for
an elegant graphical proof) &
$ Yes $ ({\raggedright as $ 8_{17} $ is \\}
the smallest noninvertible knot)
\\[2mm]
\hline
& & & \\[-2mm]
$ 5^2_1 $ (Whitehead Link) &
$ 1 $ & $ Yes $ \cite[pp~49-50]{Kauffman:87a} & $ Yes $
\\[2mm]
\hline
& & & \\[-2mm]
$ 8_{17} $ &
$ 0 $ & $ No $ \cite[p~455]{Kauffman:87a} &
$ No $ \cite[p~162]{Kauffman:97a}
\\[2mm]
\hline
& & & \\[-2mm]
$ 9_{42} $ &
$ 1 $ & $ Yes $ \cite[p~218]{Kauffman:93} & $ Yes $ \\[2mm]
\hline
& & & \\[-2mm]
$ 10_{48} $ &
$ 0 $ & $ Yes $ \cite[p~218]{Kauffman:93} & $ Yes $ \\[2mm]
\hline\hline
\end{tabular}
\caption{%
Data for the links to be investigated, including their
Alexander--Briggs (and common) names,
their writhes $ w \( K \) $, and whether they are chiral and
invertible. Diagrams of the links are presented in Figures
\ref{fig:HopfLinkTrefoil} to \ref{fig:TenFortyEight}.
}
\tablabel{linknamesandwrithes}
\end{table}
\subsection{Reflection and Inversion -- Chirality and Invertibility}
Throughout, we shall write ``='' to denote \emph{ambient isotopy} of
link diagrams, meaning that they are equivalent under the Reidemeister
moves (original: \cite{Reidemeister:48}, but see, e.g.
\cite{Kauffman:93}). We shall use the following definitions, but the
reader must be aware that conflicting terminology appears in the
literature.
\pagebreak
\begin{description}
\item[Reflection:]
We shall denote by $ K^* $ the mirror image (or reflection) of a knot
$ K $. A knot is \emph{chiral} if it is distinct from its mirror
image; i.e. there are actually two distinct knots with the same name,
$ K^* \neq K $, e.g. the trefoil knot is chiral: $ {\( 3_1 \)}^*
\neq 3_1 $. Note that this definition doesn't require an
orientation. A knot is \emph{amphichiral} if it is ambient isotopic
to its mirror image, i.e. $ K^* = K $.
The HOMFLY%
\footnote{
The HOMFLY polynomial is named by the conjunction of the initials
of six of its discoverers
\cite{FreydYetterHosteLickorishMilletOcneanu:85}, omitting those
(``P'' and ``T'') of
two independent discoverers \cite{PrzytyckiTraczyk:87}.
Przytycki, the omitted ``P'', has furthered the entymological
spirit with the suggestion ``FLYPMOTH'' \cite[p~256]{Przytycki:89},
which includes all the discoverers and has a muted reference to the
``flyping'' operation of the Tait, Kirkwood and Little -- the
original compilers of knot tables.
(Another possibility is the letter sequence ``HOMFLYPT''.)
Bar-Natan (Prasolov and Sossinsky \cite[p~36]{PrasolovSossinsky:96}
cite Bar-Natan \cite{BarNatan:95}, who cites ``L Rudulph'')
goes further, adding a ``U'' for good measure, to account for any
unknown discoverers, yielding the unpalatable ``LYMPHTOFU''!
}
(and hence the Jones) polynomial and the
Kauffman polynomial can distinguish many (but not all) knots from
their reflections.
The first chiral knot that neither the HOMFLY nor the Kauffman
polynomial can distinguish is $ 9_{42} $,
i.e. $ 9_{42}^* \neq 9_{42} $, but the polynomials are equal.
Similarly, the knot $ 10_{48} $ is chiral, but the HOMFLY polynomial
fails to detect this, although the Kauffman does detect it
\cite[p~218]{Kauffman:93} (wrongly labeled $10_{79}$).
\item[Inversion:]
Assign an orientation to a knot. Denote the \emph{inverse}
of a knot $ K $ by $ K^{-1} $, obtained by reversing
the orientation. Whilst this is a simple concept for a knot, there
are of course many possibilities for the reversal of only some
components of oriented, multi-component links; we shall not go into
these here.
Commonly, $ K = K^{-1} $, and we say that $ K $ is
\emph{invertible}. For example, the trefoil knot is invertible
$ {\( 3_1 \)}^{-1} = 3_1 $. Less commonly, $ K \neq K^{-1} $, and we
say that $ K $ is \emph{noninvertible}. The first example of a
noninvertible (prime) knot is $ 8_{17} $.
\end{description}
Both the reflection and the inverse are automorphisms of order two, i.e.
$ {( K^* )}^* = K $ and $ {( K^{-1} )}^{-1} = K $.
The notions may of course be combined, we obtain:
$ {( K^* )}^{-1} = {( K^{-1} )}^* $.
To illustrate, using the Trefoil Knot $ 3_1 $
(see \figref{HopfLinkTrefoil}). We have two equivalence classes:
$ 3_1 = {\( 3_1 \)}^{-1} $ and
$ {\( 3_1 \)}^* = {( {\( 3_1 \)}^{-1} )}^* = {( \( 3_1 \) ^* )}^{-1} $.
\subsection{Abstract Tensor Conventions}
By a `positive oriented' or `right-handed' crossing, we shall intend a
crossing such that if the thumb of the right hand points in the
direction of one of the arrows, the fingers of the right hand will
point in the direction of the other arrow. The opposite situation is
naturally called a `negative oriented' or `left-handed' crossing.
If the two outward-pointing arrows of a positive oriented crossing are
pointed upwards, then we shall label the components of the crossing
with indices $ a $ in top left, $ b $ in bottom left, $ c $ in top
right, and $ d $ in bottom right, and associate with the crossing the
(rank $ 4 $) tensor $ {\sigma}^{a~c}_{b~d} $, where the position of the
indices in the tensor corresponds with the positioning of the labels in
the crossing. The inverse of $ \sigma $ will represent a negative
oriented crossing, with the convention on the indices being the same as
that of $ \sigma $. A diagram of $ \sigma $ and $ \sigma^{-1} $ is
provided in \figref{RandS}.
\begin{figure}[htbp]
\begin{center}
\input{RandS.pstex_t}
\caption{
Definition of the tensors $ \sigma $ and $ \sigma^{-1} $
representing positive oriented and negative oriented crossings
with upward pointing arrows, respectively.
}
\figlabel{RandS}
\end{center}
\end{figure}
We shall also require four (rank $ 2 $) tensors (i.e. genuine matrices)
to represent all possible horizontally-oriented half-loops. We shall
call these `cap' and `cup' matrices, and label them with the suggestive
$ \Omega^\pm $ and $ \mho^\pm $, e.g. $ \Omega^+ $ is the upper loop
with arrow pointing right. A diagram is provided in
\figref{CapsandCups}.
\begin{figure}[htbp]
\begin{center}
\input{CapsandCups.pstex_t}
\caption{
Definition of the tensors (matrices)
$ \Omega^\pm $ and $ \mho^\pm $, representing all possibilities
of horizontally-aligned half-loops.
}
\figlabel{CapsandCups}
\end{center}
\end{figure}
With these basic tensors $ \sigma $, $ \sigma^{-1} $, $ \Omega^\pm $
and $ \mho^\pm $, we may evaluate an invariant for any particular
link. However, this direct procedure tends to be computationally
expensive, and parts of the computation are often repeated, so in
practice, we define auxiliary symbols. We shall use the notation
$ X $ to represent a rank $ 4 $ tensor such as $ \sigma $ or
$ \sigma^{-1} $ with parallel pointing arrows
(i.e. a `channel' crossing in the terminology of Kauffman
\cite[p~76]{Kauffman:93}.)
The primary auxiliary tensors used are listed below; secondary ones
will be mentioned where necessary. The Einstein summation convention
is used throughout.
\begin{itemize}
\item
The first auxiliary symbols are those of crossings that have been
`twisted' relative to $ \sigma $ and $ \sigma^{-1} $. The left,
right, and upside-down-twisted versions of $ X $ will be called
$ X_l $, $ X_r $ and $ X_d $ respectively. They are defined in the
following manner:
\bne
{\( X_l \)}^{a~c}_{b~d}
& \defeq &
{X}^{e~a}_{d~h}
\cdot
{\( \Omega^- \)}_{b~e}
\cdot
{\( \mho^- \)}^{h~c}
\nonumber
\\
{\( X_r \)}^{a~c}_{b~d}
& \defeq &
{X}^{c~g}_{f~b}
\cdot
{\( \mho^+ \)}^{a~f}
\cdot
{\( \Omega^+ \)}_{g~d}
\eqlabel{XRdef}
\\
{\( X_d \)}^{a~c}_{b~d}
& \defeq &
{X}^{e~g}_{f~h}
\cdot
{\( \mho^+ \)}^{a~h}
\cdot
{\( \Omega^+ \)}_{g~b}
\cdot
{\( \mho^+ \)}^{c~f}
\cdot
{\( \Omega^+ \)}_{e~d}.
\nonumber
\ene
Observe that $ X_d $ is a `channel' crossing, whilst $ X_l $ and
$X_r$ are `cross-channel' crossings. Diagrams are found in Figures
\ref{fig:XLandXR} and \ref{fig:XD}.
\begin{figure}[htbp]
\begin{center}
\input{XLandXR.pstex_t}
\caption{The primary auxiliary tensors
$ X_l $ and $ X_r $, where
$ X $ is one of $ \sigma $ or $ \sigma^{-1} $.
}
\figlabel{XLandXR}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{XD.pstex_t}
\caption{The primary auxiliary tensor
$ X_d $, where
$ X $ is one of $ \sigma $ or $ \sigma^{-1} $.
}
\figlabel{XD}
\end{center}
\end{figure}
\item
The next set of auxiliary symbols represent $ p $ copies of the
\emph{same} crossing $ X $ (for any channel crossing $ X $) atop one
another (see \figref{Xp}). They are defined recursively in the
following manner:
\be
{\( X^{p+1} \)}^{a~c}_{b~d}
\defeq
X^{a~c}_{e~f}
\cdot
{\( X^p \)}^{e~f}_{b~d},
\qquad
p = 1, 2, \dots.
\ee
\begin{figure}[htbp]
\begin{center}
\input{Xp.pstex_t}
\caption{The primary auxiliary tensors
$ X^{p+1} $ in terms of $ X $ and $ X^p $;
$ X $ is one of $ \sigma $ or $ \sigma^{-1} $. If all
arrows are reversed, then the definition also holds for $ X $
being $ \sigma_d $ or $ \sigma^{-1}_d $; that is, any channel
crossing.
}
\figlabel{Xp}
\end{center}
\end{figure}
\item
The third set of frequently-encountered patterns are where a crossing
$ X $ is to the left or right of its own `upside-downness' $ X_d $.
That is, fix an $ X $ as either $ \sigma $ or $ \sigma^{-1} $, and
examine the patterns formed from juxtaposing $ X $ and $ X_d $. They
are defined in the following manner:
\be
{\( X_d X \)}^{a~c}_{b~d}
& \defeq &
{\( X_d \)}^{a~e}_{b~f}
\cdot
{X}^{g~c}_{h~d}
\cdot
{\( \Omega^- \)}_{e~g}
\cdot
{\( \mho^+ \)}^{f~h}
\\
{\( X X_d \)}^{a~c}_{b~d}
& \defeq &
{X}^{a~e}_{b~f}
\cdot
{\( X_d \)}^{g~c}_{h~d}
\cdot
{\( \Omega^+ \)}_{e~g}
\cdot
{\( \mho^- \)}^{f~h}.
\ee
A diagram is found in \figref{XXDandXDX}.
\begin{figure}[htbp]
\begin{center}
\input{XXDandXDX.pstex_t}
\caption{The primary auxiliary tensors
$ X_d X $ and $ X X_d $;
$ X $ is one of $ \sigma $ or $ \sigma^{-1} $.
}
\figlabel{XXDandXDX}
\end{center}
\end{figure}
\item
The final set of frequently-encountered patterns are where a crossing
$ X_l $ is placed atop above a crossing $ X_r $ (or vice-versa). We
obtain:
\be
{\( X_l X_r \)}^{a~c}_{b~d}
& \defeq &
{\( X_l \)}^{a~c}_{e~f}
\cdot
{\( X_r \)}^{e~f}_{b~d}
\\
{\( X_r X_l \)}^{a~c}_{b~d}
& \defeq &
{\( X_r \)}^{a~c}_{e~f}
\cdot
{\( X_l \)}^{e~f}_{b~d}.
\ee
A diagram is found in \figref{XLXRandXRXL}. A moment's thought
demonstrates that the diagram for $ X_l X_r $ is a right rotation of
the diagram for $ X_d X $. In fact, we have the identity:
\be
{\( X_l X_r \)}^{a~c}_{b~d}
=
{\( X_d X \)}^{e~a}_{d~h}
\cdot
{\( \mho^+ \)}^{h~c}
\cdot
{\( \Omega^+ \)}_{b~e},
\ee
although in practice we shall not use it. (A diagram parallel to
\figref{XLandXR} would demonstrate this.)
\begin{figure}[htbp]
\begin{center}
\input{XLXRandXRXL.pstex_t}
\caption{The primary auxiliary tensor
$ X_l X_r $;
$ X $ is one of $ \sigma $ or $ \sigma^{-1} $.
$ X_r X_l $ is obtained by swapping every $ r $ and $ l $
in this diagram.
}
\figlabel{XLXRandXRXL}
\end{center}
\end{figure}
\end{itemize}
\subsection{The Effects of Reflection and Inversion on the Tensors}
\begin{description}
\item[Reflection:]
Let $ K^* $ be the reflection of a tangle $ K $; and say that we have
constructed a tensor representing $ K $. Every positive (respectively
negative) crossing in $ K $ will have been replaced by the equivalent
negative (respectively positive) crossing in $ K^* $. Thus, the
tensor corresponding to $ K^* $ will be that of $ K $ with every
$ \sigma $ replaced by $ \sigma^{-1} $, and every $ \sigma^{-1} $
replaced by $ \sigma $. This carries through to the auxiliary
tensors; i.e. $ \sigma_d \sigma $ will be replaced with
$ \sigma^{-1}_d \sigma^{-1} $, etc. The caps $ \Omega^\pm $ and cups
$ \mho^\pm $ will remain unchanged.
From the uniqueness \cite{KhoroshkinTolstoy:91} of the universal
$ R $ matrix for any quantum superalgebra the following relation
holds (for appropriate normalisation):
\be
R^{-1} ( q )
=
R ( q^{-1} ),
\ee
which in turn leads to the relation
\be
\sigma^{-1} ( q )
=
P \sigma ( q^{-1} ) P.
\ee
Thus, up to a basis transformation, $ \sigma $ and $ \sigma^{-1} $ are
interchangeable by the change of variable $ q \mapsto q^{-1} $. It
then follows that the invariant for $ K^* $ is obtainable from that
of $ K $ by the same change of variable, which leads to the
following:
\begin{proposition}
If $ K $ is amphichiral then the invariant ${LG}_K$ is palindromic.%
\footnote{%
We intend ``palindromic'' to mean that the polynomial is
invariant under the mapping $ q \mapsto q^{-1} $.
}
\proplabel{chiralmeanspalindromic}
\end{proposition}
\vfill
\item[Inversion:]
Again, if $ K^{-1} $ is the inverse of $ K $, then every arrow in
$ K $ will have been replaced with an arrow in the opposite
direction. The tensor corresponding to $ K^{-1} $ will thus have the
following changes: For the crossings,
where $ X $ is either $ \sigma $ or $ \sigma^{-1} $, interchange
$ X \iff X_d $ and $ X_l \iff X_r $;
and for the caps and cups, interchange only the signs, i.e.
$ \Omega^\pm \iff \Omega^\mp $
and
$ \mho^\pm \iff \mho^\mp $.
This has the effect that the tensor representing $ K $ is replaced by
the dual tensor acting on the dual space \cite{ReshetikhinTuraev:90}.
Recalling that the tensors representing $ \( 1, 1 \) $ tangles act as
scalar multiples of the identity on $ V $, then the dual tensor has
exactly the same form, from which we conclude:
\begin{proposition}
A knot invariant derived from an irreducible representation of a
quantum (super)algebra is unable to detect inversion.
\proplabel{inversion}
\end{proposition}
\end{description}
\subsection{Abstract Tensor Expressions for the Example Links}
We list the abstract tensors $ {\( T_K \)}^y_x $ that represent the
$ \( 1, 1 \) $-tangle (open diagram) forms of the example links. In
each case, the indices $ x $ and $ y $ are the lower and upper loose
ends of the tangle in question.
The Links--Gould invariant is then formed by setting
$ x $ and $ y $ to be the same, i.e.
\be
{LG}_K \( q, p = q^\alpha \)
\defeq
{\( T_K \)}^i_i
\ee
(no sum on $ i $), for any allowable index $ i $. We typically choose
$ i = 1 $.
Our invariant does not need to be \emph{writhe-normalised}, due to the
choice of normalisation of $ \sigma $ and the cap and cup matrices
$ \Omega^\pm $ and $ \mho^\pm $. \figref{Loop} depicts removal of a
loop from a diagram.
\begin{figure}[htbp]
\begin{center}
\input{Loop.pstex_t}
\caption{Removal of a Single, Positive Loop.}
\figlabel{Loop}
\end{center}
\end{figure}
Braid presentations for the example knots are taken
from \cite[pp~109-110]{Jones:85} and \cite[pp~381-386]{Jones:87}.
\begin{description}
\item[$ \mathbf{0_1} $ (Unknot):]
A braid presentation is the trivial $ e \in B_1 $.
As the unclosed tangle representing the Unknot is rather meaningless,
we use simply $ {\( T_{0_1} \)}^y_x \defeq \delta^y_x $.
\item[$ \mathbf{2^2_1} $ (Hopf Link):]
A braid presentation is $ {\sigma_1}^2 \in B_2 $.
Diagrams pertaining to the Hopf Link and Trefoil are found in
\figref{HopfLinkTrefoil}.
\be
{( T_{2^2_1} )}^y_x
\defeq
{\( \sigma^2 \)}^{y~a}_{x~b}
\cdot
{\( \Omega^+ \)}_{a~c}
\cdot
{\( \mho^- \)}^{b~c}.
\ee
\item[$ \mathbf{3_1} $ (Trefoil):]
A braid presentation is $ {\sigma_1}^3 \in B_2 $.
This knot has also been
called the \emph{overhand knot} (as that is how it is tied) and the
\emph{cloverleaf knot} \cite[pp~3-4]{CrowellFox:77}.
\be
{\( T_{3_1} \)}^y_x
\defeq
{\( \sigma^3 \)}^{y~a}_{x~b}
\cdot
{\( \Omega^+ \)}_{a~c}
\cdot
{\( \mho^- \)}^{b~c}.
\ee
\begin{figure}[htbp]
\begin{center}
\input{HopfLinkTrefoil.pstex_t}
\caption{Tangle form of $ 2^2_1 $ (the Hopf Link) and $ 3_1 $
(the positive Trefoil).
}
\figlabel{HopfLinkTrefoil}
\end{center}
\end{figure}
\pagebreak
\item[$ \mathbf{4_1} $ (Figure Eight):]
A braid presentation is
$ {\( \sigma_1 \sigma_2^{-1} \)}^2 \in B_3 $, and
a diagram is found in \figref{FigureEight}. This knot has also been
called the \emph{Four-Knot} (as it is the only $ 4 $ crossing knot)
and \emph{Listing's Knot} \cite[p~4]{CrowellFox:77}.
\be
{\( T_{4_1} \)}^y_x
\defeq
{\( \sigma^{-1}_l \sigma^{-1}_r \)}^{y~b}_{a~c}
\cdot
{\( \sigma_r \)}^{c~e}_{d~f}
\cdot
{\sigma}^{a~d}_{x~g}
\cdot
{\( \Omega^- \)}_{b~e}
\cdot
{\( \mho^- \)}^{g~f}.
\ee
\begin{figure}[ht]
\begin{center}
\input{FigureEight.pstex_t}
\caption{Tangle form of $ 4_1 $ (the Figure Eight Knot).
}
\figlabel{FigureEight}
\end{center}
\end{figure}
\item[$ \mathbf{5^2_1} $ (Whitehead Link):]
A braid presentation is
$ {\( \sigma_1 \sigma_2^{-1} \)}^2 \sigma_2^{-1} \in B_3 $ and a
diagram is found in \figref{WhiteheadLink}. (This link is named after
the topologist J H C Whitehead, not the logician Alfred North
Whitehead \cite[p~200]{Kauffman:88}.)
Firstly, we define a temporary tensor to reduce computation:
\be
{\( W \)}^{c~i}_{x~d}
\defeq
{\( \sigma^{-2} \)}^{c~e}_{x~f}
\cdot
{( \sigma_d^2 )}^{g~i}_{h~d}
\cdot
{\( \Omega^+ \)}_{e~g}
\cdot
{\( \mho^- \)}^{f~h},
\ee
where we have written $ \sigma^{-2} \defeq {\( \sigma^{-1} \)}^2 $.
With this, we have:
\be
{( T_{5^2_1} )}^y_x
\defeq
{\( W \)}^{c~i}_{x~d}
\cdot
{\( \sigma_r \sigma_l \)}^{a~y}_{i~b}
\cdot
{\( \Omega^+ \)}_{c~a}
\cdot
{\( \mho^+ \)}^{d~b}.
\ee
\begin{figure}[htbp]
\begin{center}
\input{WhiteheadLink.pstex_t}
\caption{Tangle form of $ 5^2_1 $ (the Whitehead Link).
}
\figlabel{WhiteheadLink}
\end{center}
\end{figure}
\item[$ \mathbf{8_{17}} $:]
A braid presentation is
$
{\( \sigma_1^{-1} \sigma_2 \)}^2 \sigma_2^2 \sigma_1^{-2} \sigma_2
\in B_3
$, and a diagram is found in \figref{EightSeventeen}.
Again, we define some temporary tensors to reduce computation:
\be
{\( EA \)}^{y~c~e}_{b~d~f}
& \defeq &
{\( \sigma^{-2} \)}^{c~e}_{g~f}
\cdot
{\( \sigma^2 \)}^{y~g}_{b~d}
\\
{\( EB \)}^{b~d~f}_{x~i~j}
& \defeq &
{{\( \sigma^{-1} \)}}^{d~f}_{k~l}
\cdot
{\sigma}^{b~k}_{m~n}
\cdot
{{\( \sigma^{-1} \)}}^{n~l}_{o~j}
\cdot
{\sigma}^{m~o}_{x~i}.
\ee
With these, we have:
\be
{\( T_{8_{17}} \)}^y_x
\defeq
{\( EA \)}^{y~c~e}_{b~d~f}
\cdot
{\( EB \)}^{b~d~f}_{x~i~j}
\cdot
{\( \Omega^+ \)}_{c~r}
\cdot
{\( \mho^- \)}^{i~r}
\cdot
{\( \Omega^+ \)}_{e~q}
\cdot
{\( \mho^- \)}^{j~q}.
\ee
To reduce computation, we may define even more auxiliary tensors:
\be
{\( EB \)}^{b~d~f}_{x~i~j}
=
{\( EC \)}^{b~d~f}_{m~n~l}
\cdot
{\( ED \)}^{m~n~l}_{x~i~j},
\ee
where:
\be
{\( EC \)}^{b~d~f}_{m~n~l}
& \defeq &
{{\( \sigma^{-1} \)}}^{d~f}_{k~l}
\cdot
{\sigma}^{b~k}_{m~n}
\\
{\( ED \)}^{m~n~l}_{x~i~j}
& \defeq &
{{\( \sigma^{-1} \)}}^{n~l}_{o~j}
\cdot
{\sigma}^{m~o}_{x~i}.
\ee
\begin{figure}[htbp]
\begin{center}
\input{EightSeventeen.pstex_t}
\caption{Tangle form of $ 8_{17} $.}
\figlabel{EightSeventeen}
\end{center}
\end{figure}
\pagebreak
\item[$ \mathbf{9_{42}} $:]
A braid presentation is
$
\sigma_1^3 \sigma_3
\sigma_2^{-1} \sigma_3
\sigma_1^{-2} \sigma_2^{-1}
\in B_4
$,
and a diagram is found in \figref{NineFortytwo}.
Again, we define a temporary tensor to reduce computation:
\be
{\( N \)}^{a~y}_{b~h}
& \defeq &
{( \sigma_d^2 )}^{a~c}_{b~d}
\cdot
{\( \sigma^{-3} \)}^{e~y}_{f~h}
\cdot
{\( \Omega^- \)}_{c~e}
\cdot
{\( \mho^+ \)}^{d~f}.
\\
{\( T_{9_{42}} \)}^y_x
& \defeq &
{\( N \)}^{a~y}_{b~h}
\cdot
{\( \sigma^{-1}_d \sigma^{-1} \)}^{b~h}_{i~j}
\cdot
{\( \sigma \sigma_d \)}^{k~i}_{x~m}
\cdot
{\( \mho^+ \)}^{m~j}
\cdot
{\( \Omega^+ \)}_{k~a}.
\ee
\item[$ \mathbf{10_{48}} $:]
A braid presentation is
$
\sigma_1^{-2} \sigma_2^{4} \sigma_1^{-3} \sigma_2
\in B_3
$,
and a diagram is found in \figref{TenFortyEight}.
\be
{\( T_{10_{48}} \)}^y_x
\defeq
{{\( \sigma^{-2} \)}}^{a~y}_{b~f}
\cdot
{\( \sigma^{4} \)}^{f~g}_{d~h}
\cdot
{\( \sigma^{-3} \)}^{b~d}_{c~e}
\cdot
{\( \sigma \)}^{e~h}_{x~i}
\cdot
\\
\qquad \qquad
{\( \Omega^- \)}_{j~a}
\cdot
{\( \mho^+ \)}^{j~c}
\cdot
{\( \Omega^+ \)}_{g~k}
\cdot
{\( \mho^- \)}^{i~k}.
\ee
\begin{figure}[htbp]
\begin{center}
\input{TenFortyEight.pstex_t}
\caption{Tangle form of $10_{48}$.}
\figlabel{TenFortyEight}
\end{center}
\end{figure}
\begin{figure}[ht]
\begin{center}
\input{NineFortytwo.pstex_t}
\caption{Tangle form of $ 9_{42} $.}
\figlabel{NineFortytwo}
\end{center}
\end{figure}
\end{description}
\clearpage
\section{The Links--Gould Tangle Invariant}
\seclabel{lglink}
\subsection{Crossing Matrices $ \sigma $ and $ \sigma^{-1} $}
From the results of \secref{Rmatrixdef}, we have the matrices
$ \sigma $ and $ \sigma^{-1} $,
using the substitution $ p \defeq q^\alpha $:
\be
\sigma
=
\m{[}{@{}*{3}{*{3}{c@{}@{}}c|}*{3}{c@{}@{}}c@{}}
{\scriptstyle p^{-2}} &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &{\scriptstyle p^{-1}} &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &. &. &. &. &{\scriptstyle p^{-1}} &. &. &. &. &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &. &{\scs1} &. &. &. \\
\hline
. &{\scriptstyle p^{-1}} &. &. &{\scriptstyle p^{-2}-1} &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &. &{\scriptstyle -1} &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &. &. &{\scriptstyle q^2-1} &. &. &{\scriptstyle -q} &. &. &{\scriptstyle - q Y} &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p q} &. &. \\
\hline
. &. &{\scriptstyle p^{-1}} &. &. &. &. &. &{\scriptstyle p^{-2}-1} &. &. &. &. &. &. &. \\
. &. &. &. &. &. &{\scriptstyle -q} &. &. &. &. &. &{\scriptstyle Y} &. &. &. \\
. &. &. &. &. &. &. &. &. &. &{\scriptstyle -1} &. &. &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p q} &. \\
\hline
. &. &. &{\scriptstyle 1} &. &. &{\scriptstyle - q Y} &. &. &{\scriptstyle Y} &. &. &{\scriptstyle Y^2} &. &. &. \\
. &. &. &. &. &. &. &{\scriptstyle p q} &. &. &. &. &. &{\scriptstyle p^2 q^2 -1} &. &. \\
. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p q} &. &. &{\scriptstyle p^2 q^2 -1} &. \\
. &. &. &. &. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p^2 q^2}
\me,
\ee
\be
\! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \!
\sigma^{-1}
=
\m{[}{@{}*{3}{*{3}{c@{}@{}}c|}*{3}{c@{}@{}}c@{}}
{\scriptstyle p^2} &. &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\
. &{\scriptstyle p^2 -1} &. &. &{\scriptstyle p} &. &. &. &. &. &. &. &. &. &. &. \\
. &. &{\scriptstyle p^2 -1} &. &. &. &. &. &{\scriptstyle p} &. &. &. &. &. &. &. \\
. &. &. &{\scriptstyle Y^2 q^{-2}} &. &. &{\scriptstyle Y q^{-1}} &. &. &{\scriptstyle -Y q^{-2}} &. &. & {\scriptstyle 1} &. &. &. \\
\hline
. &{\scriptstyle p} &. &. &. &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &. &{\scriptstyle -1} &. &. &. &. &. &. &. &. &. &. \\
. &. &. &{\scriptstyle Y q^{-1}} &. &. &. &. &. &{\scriptstyle -q^{-1}} &. &. &. &. &. &. \\
. &. &. &. &. &. &. &{\scriptstyle p^{-2} q^{-2} - 1} &. &. &. &. &. &{\scriptstyle p^{-1} q^{-1}} &. & . \\
\hline
. &. &{\scriptstyle p} &. &. &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &{\scriptstyle -Y q^{-2}} &. &. &{\scriptstyle -q^{-1}} &. &. &{\scriptstyle q^{-2} - 1} &. &. &. &. &. & . \\
. &. &. &. &. &. &. &. &. &. &{\scriptstyle -1} &. &. &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p^{-2} q^{-2} - 1} &. &. &{\scriptstyle p^{-1} q^{-1}} & . \\
\hline
. &. &. &{\scriptstyle 1} &. &. &. &. &. &. &. &. &. &. &. &. \\
. &. &. &. &. &. &. &{\scriptstyle p^{-1} q^{-1}} &. &. &. &. &. &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p^{-1} q^{-1}} &. &. &. &. \\
. &. &. &. &. &. &. &. &. &. &. &. &. &. &. &{\scriptstyle p^{-2} q^{-2}}
\me,
\ee
where $ Y = {\( p^{-2} - q^2 + p^2 q^2 - 1 \)}^{1/2} $.
\subsection{Caps and Cups $ \Omega^\pm $ and $ \mho^\pm $}
Where $ \mho^\pm = {\( \Omega^\pm \)}^{-1} $, we will use:
\be
\Omega^-
\eq
\m{[}{c@{}c@{}c@{}c}
{\scriptstyle q^{- 2 \alpha}} & . & . & . \\
. & {\scriptstyle - q^{- 2 \( \alpha + 1 \)}} & . & . \\
. & . & {\scriptstyle - q^{- 2 \alpha}} & . \\
. & . & . & {\scriptstyle q^{- 2 \( \alpha + 1 \)}}
\me
\equiv
\m{[}{cccc}
{\scriptstyle p^{-2}} & . & . & . \\
. & {\scriptstyle - p^{-2} q^{-2}} & . & . \\
. & . & {\scriptstyle - p^{-2}} & . \\
. & . & . & {\scriptstyle p^{-2} q^{-2}}
\me,
\\
\qquad
\mho^-
\eq
\m{[}{c@{}c@{}c@{}c}
{\scriptstyle q^{2 \alpha}} & . & . & . \\
. & {\scriptstyle - q^{2 \( \alpha + 1 \)}} & . & . \\
. & . & {\scriptstyle - q^{2 \alpha}} & . \\
. & . & . & {\scriptstyle q^{2 \( \alpha + 1 \)}}
\me
\equiv
\m{[}{cccc}
{\scriptstyle p^2} & . & . & . \\
. & {\scriptstyle - p^2 q^2} & . & . \\
. & . & {\scriptstyle - p^2} & . \\
. & . & . & {\scriptstyle p^2 q^2}
\me,
\\
\Omega^+
\eq
\mho^+
=
I_4.
\ee
The choices for $ \Omega^\pm $ and $ \mho^\pm $ are not unique.
\begin{itemize}
\item
$ \Omega^+ $ and $ \mho^+ $ may be chosen from
consistency considerations in \figref{XLandXR}. The simple choices:
\be
{\( \Omega^+ \)}_{a~b}
=
{\( \mho^+ \)}^{a~b}
=
\delta_{a~b}
\ee
(i.e. $ \Omega^+ = \mho^+ = I_4 $),
ensure that the definition \eqref{XRdef}, i.e.
\be
{\( X_r \)}^{a~c}_{b~d}
\defeq
{X}^{c~g}_{f~b}
\cdot
{\( \mho^+ \)}^{a~f}
\cdot
{\( \Omega^+ \)}_{g~d}
\ee
(where $ X $ is either $ \sigma $ or $ \sigma^{-1} $),
simplifies to the elegant form:
\be
{\( X_r \)}^{a~c}_{b~d}
=
{X}^{c~d}_{a~b}.
\ee
\item
For the choice of $ \Omega^- $ and $ \mho^- $,
we invoke the following result from
\cite[Lemma~2,~p~354]{LinksGouldZhang:93}
(see also \cite{LinksGould:92b}):
\be
\( I \otimes \mathrm{str} \)
[ ( I \otimes q^{-2 h_\rho} ) \sigma ]
=
k I,
\ee
for some constant $ k $ depending on the normalisation of $ \sigma $.
Note that $ \mathrm{str} $ denotes the supertrace, and that in this
case:
\be
\pi ( q^{-2 h_\rho} )
=
\m{[}{cccc}
q^{- 2 \alpha} & . & . & . \\
. & q^{- 2 \alpha - 2} & . & . \\
. & . & q^{- 2 \alpha} & . \\
. & . & . & q^{- 2 \alpha - 2}
\me.
\ee
\pagebreak
From \figref{Loop}, we require:
\be
\sigma^{y a}_{x b}
\cdot
{\( \Omega^+ \)}_{a c}
\cdot
{\( \mho^- \)}^{b c}
=
\delta^y_x,
\ee
which, along with the condition:
\be
{\( \Omega^- \)}_{a b}
\cdot
{\( \mho^- \)}^{b c}
=
\delta^c_a,
\ee
imposes the choice:
\be
{\( \mho^- \)}^{b c}
\eq
{\( - \)}^{\[ b \]}
\pi {( q^{-2 h_\rho} )}^b_c,
\\
{\( \Omega^- \)}_{b c}
\eq
{\( - \)}^{\[ b \]}
\pi {( q^{2 h_\rho} )}^b_c.
\ee
\end{itemize}
For other references on the construction of the cap and cup matrices,
see the papers by Reshetikhin and Turaev \cite{ReshetikhinTuraev:90},
and particularly Zhang \cite{Zhang:95} for the superalgebra case.
\subsection{Results}
Some evaluations of the invariant are presented in
\tabref{gl21oriented}. This $ U_q \[ gl \( 2 | 1 \) \] $ oriented
invariant \emph{is} an invariant of ambient isotopy.
\begin{table}[ht]
\centering
\begin{tabular}{||c|l||}
\hline\hline
& \\[-3mm]
$ K $ &
\multicolumn{1}{c||}{$ {LG}_K \( q, p \) $}
\\[1mm]
\hline\hline
& \\[-2mm]
$ 0_1 $ &
$ 1 $
\\[2mm]
\hline
& \\[-2mm]
$ 2^2_1 $ &
$ -1 + p^{-2} - q^2 + p^2 q^2 $
\\[2mm]
\hline
& \\[-2mm]
$ 3_1 $ &
$
1 + p^{-4} - p^{-2} + 2 q^2 - p^{-2} q^2 - p^2 q^2
- p^2 q^4 + p^4 q^4
$
\\[2mm]
\hline
& \\[-2mm]
$ 4_1 $ &
$
7
+ \( p^{-4} q^{-2} + p^4 q^2 \)
- 3 \( p^{-2} + p^2 \)
- 3 \( p^{-2} q^{-2} + p^2 q^2 \)
+ 2 \( q^{-2} + q^2 \)
$
\\[2mm]
\hline
& \\[-2mm]
$ 5^2_1 $ &
$
\begin{array}{l}
- 10
+ p^{-6} q^{-2}
- 3 p^{-4}
- 3 p^{-4} q^{-2}
+ 4 p^{-2} q^{-2}
+ 9 p^{-2}
- 2 q^{-2}
\\
\quad
- 8 q^2
+ 2 p^{-2} q^2
+ 9 p^2 q^2
+ 4 p^2
+ 2 p^2 q^4
- 3 p^4 q^2
- 3 p^4 q^4
+ p^6 q^4
\end{array}
$
\\[2mm]
\hline\hline
& \\[-2mm]
$ 8_{17} $ &
see \secref{EightSeventeen}
\\[2mm]
\hline
& \\[-2mm]
$ 9_{42}, 10_{48} $ &
see \secref{NineFortyTwoandTenFortyEight}
\\[2mm]
\hline\hline
\end{tabular}
\caption{%
The Links--Gould $ U_q \[ gl \( 2 | 1 \) \] $ oriented
polynomial invariant $ {LG}_K \( q, p \) $,
evaluated using the open diagram form of various links $ K $.}
\tablabel{gl21oriented}
\end{table}
\subsection{Behaviour of the Invariant}
Fix a knot $ K $, and denote by $ K^* $ the reflection of $ K $ and by
$ K^{-1} $ the inverse of $ K $. From the polynomial for $ K $, we may
immediately write down the polynomials for $ K^* $ and $ K^{-1} $. For
the reflection, we have:
\bse
{LG}_{K^*} \( q, p \)
=
{LG}_K \( q^{-1}, p^{-1} \).
\eqlabel{Utricularia.1}
\ese
For the inverse, we have:
\be
{LG}_{K^{-1}} \( q, p \)
=
{LG}_K \( q, q^{-1} p^{-1} \).
\ee
(this follows from $ \alpha \mapsto - \( \alpha + 1 \) $).
\begin{description}
\item[Chirality:]
As we have:
\be
K = K^*
\quad \to \quad
{LG}_K \( q, p \)
=
{LG}_{K^*} \( q, p \),
\ee
then we have, conversely, that:
\bse
{LG}_K \( q, p \)
\neq
{LG}_{K^*} \( q, p \)
\quad \to \quad
K \neq K^*,
\eqlabel{Utricularia.3}
\ese
i.e. if the polynomials corresponding to $ K $ and $ K^* $ are
distinct, then $ K $ must be chiral. Using the identity
\eqref{Utricularia.1}, the test of \eqref{Utricularia.3} becomes:
\be
{LG}_K \( q, p \)
\neq
{LG}_K \( q^{-1}, p^{-1} \)
\quad \to \quad
K \neq K^*,
\ee
i.e. if $ {LG}_K \( q, p \) $
is \emph{not} palindromic, then $ K $ is chiral.
\item[Invertibility:]
We make the observation that the representation of
$ U_q \[ gl \( 2 | 1 \) \] $ acting on the dual module $ V^* $
is given by the replacement $ \alpha \mapsto - \( \alpha + 1 \) $
(with an appropriate redefinition of the Cartan elements). Thus
for a given $ \( 1, 1 \) $ tangle $ K $, with invariant
$ {LG}_K \( q, p \) $, the tangle invariant
$ {LG}_{K^{-1}} $ of its inverse $ K^{-1} $ is obtained as
$ {LG}_{K^{-1}} \( q, p \) = {LG}_K \( q, q^{-1} p^{-1} \) $.
However, in view of \propref{inversion}, such an invariant is unable
to detect inversion.
\end{description}
We summarise these results in a proposition:
\begin{proposition}
Let $ {LG}_K \( q, p \) $ be the Links--Gould polynomial invariant for
the knot $ K $.
\begin{itemize}
\item
If $ {LG}_K \( q, p \) $ is \emph{not}
invariant under the transformation $ q \mapsto q^{-1} $
(which implies $ p \mapsto p^{-1} $), then
$ LG $ detects the chirality of $ K $.
\item
$ {LG}_K \( q, p \) $ enjoys the symmetry property:
\bse
{LG}_K \( q, p \) = {LG}_K \( q, q^{-1} p^{-1} \).
\eqlabel{Melanotaenia}
\ese
\end{itemize}
\proplabel{Rhadinocentrus}
\end{proposition}
\subsection{The Chirality of $9_{42}$ and $10_{48}$}
\seclabel{NineFortyTwoandTenFortyEight}
The polynomials for $9_{42}$ and $10_{48}$ are:
\begin{eqnarray*}
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \!
{LG}_{9_{42}} \( q, p \)
=
\\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\begin{array}{l}
3
+ p^{-8} q^{-6}
- 2 p^{-6} q^{-6}
- 2 p^{-6} q^{-4}
+ p^{-4} q^{-6}
+ 3 p^{-4} q^{-4}
+ p^{-4} q^{-2}
+ p^{-4}
- p^{-2} q^{-4}
\\
- p^{-2} q^{-2}
- 3 p^{-2}
- 3 p^{-2} q^2
+ 6 q^2
+ 2 q^4
- p^2 q^{-2}
- p^2
- 3 p^2 q^2
- 3 p^2 q^4
+ p^4 q^{-2}
\\
+ 3 p^4
+ p^4 q^2
+ p^4 q^4
- 2 p^6
- 2 p^6 q^2
+ p^8 q^2
\end{array}
\\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \!
{LG}_{10_{48}} \( q, p \)
=
\\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\begin{array}{l}
165
+ 5 p^{-8}
- 25 p^{-6}
+ 68 p^{-4}
- 129 p^{-2}
- 132 p^{2}
+ 67 p^{4}
- 22 p^{6}
+ 4 p^{8}
+ p^{-16} q^{-8}
\\
- 3 p^{-14} q^{-8}
+ 4 p^{-12} q^{-8}
- 4 p^{-10} q^{-8}
+ 4 p^{-8} q^{-8}
- 2 p^{-6} q^{-8}
- 3 p^{-14} q^{-6}
+ 12 p^{-12} q^{-6}
\\
- 21 p^{-10} q^{-6}
+ 24 p^{-8} q^{-6}
- 22 p^{-6} q^{-6}
+ 13 p^{-4} q^{-6}
- 3 p^{-2} q^{-6}
+ 16 q^{-4}
+ 5 p^{-12} q^{-4}
\\
- 23 p^{-10} q^{-4}
+ 50 p^{-8} q^{-4}
- 69 p^{-6} q^{-4}
+ 67 p^{-4} q^{-4}
- 43 p^{-2} q^{-4}
- 3 p^{2} q^{-4}
+ 94 q^{-2}
\\
- 6 p^{-10} q^{-2}
+ 29 p^{-8} q^{-2}
- 72 p^{-6} q^{-2}
+ 119 p^{-4} q^{-2}
- 132 p^{-2} q^{-2}
- 43 p^{2} q^{-2}
+ 13 p^{4} q^{-2}
\\
- 2 p^{6} q^{-2}
+ 88 q^{2}
- 2 p^{-6} q^{2}
+ 12 p^{-4} q^{2}
- 39 p^{-2} q^{2}
- 129 p^{2} q^{2}
+ 119 p^{4} q^{2}
- 69 p^{6} q^{2}
\\
+ 24 p^{8} q^{2}
- 4 p^{10} q^{2}
+ 12 q^{4}
- 2 p^{-2} q^{4}
- 39 p^{2} q^{4}
+ 68 p^{4} q^{4}
- 72 p^{6} q^{4}
+ 50 p^{8} q^{4}
- 21 p^{10} q^{4}
\\
+ 4 p^{12} q^{4}
- 2 p^{2} q^{6}
+ 12 p^{4} q^{6}
- 25 p^{6} q^{6}
+ 29 p^{8} q^{6}
- 23 p^{10} q^{6}
+ 12 p^{12} q^{6}
- 3 p^{14} q^{6}
- 2 p^{6} q^{8}
\\
+ 5 p^{8} q^{8}
- 6 p^{10} q^{8}
+ 5 p^{12} q^{8}
- 3 p^{14} q^{8}
+ p^{16} q^{8}.
\end{array}
\end{eqnarray*}
Neither of these polynomials are palindromic, hence $LG$ \emph{does}
distinguish the chirality of these knots.
\subsection{The Noninvertibility of $ 8_{17} $ is not Detected}
\seclabel{EightSeventeen}
Recall that $ 8_{17} $ is the smallest noninvertible knot.
We find its polynomial invariant to be given by:
\be
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \!
{LG}_{8_{17}} \( q, p \)
=
\\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\begin{array}{l}
139
+ \( p^{-12} q^{-6} + p^{12} q^6 \)
- 4 \( p^{-10} q^{-6} + p^{10} q^6 \)
- 4 \( p^{-10} q^{-4} + p^{10} q^4 \)
\\
+ 7 \( p^{-8} q^{-6} + p^8 q^6 \)
+ 18 \( p^{-8} q^{-4} + p^8 q^4 \)
+ 7 \( p^{-8} q^{-2} + p^8 q^2 \)
- 7 \( p^{-6} q^{-6} + p^6 q^6 \)
\\
- 36 \( p^{-6} q^{-4} + p^6 q^4 \)
- 36 \( p^{-6} q^{-2} + p^6 q^2 \)
- 7 \( p^{-6} + p^6 \)
+ 3 \( p^{-4} q^{-6} + p^4 q^6 \)
\\
+ 40 \( p^{-4} q^{-4} + p^4 q^4 \)
+ 82 \( p^{-4} q^{-2} + p^4 q^2 \)
+ 40 \( p^{-4} + p^4 \)
+ 3 \( p^{-4} q^2 + p^4 q^{-2} \)
\\
- 22 \( p^{-2} q^{-4} + p^2 q^4 \)
- 102 \( p^{-2} q^{-2} + p^2 q^2 \)
- 102 \( p^{-2} + p^2 \)
- 22 \( p^{-2} q^2 + p^2 q^{-2} \)
\\
+ 4 \( q^{-4} + q^4 \)
+ 68 \( q^{-2} + q^2 \).
\end{array}
\ee
As $ 8_{17} $ is amphichiral, the polynomial invariant is palindromic,
as predicted by \propref{Rhadinocentrus}.
Furthermore, we may observe the invariance:
\be
{LG}_{8_{17}} \( q, p \)
=
{LG}_{8_{17}} \( q, q^{-1} p^{-1} \),
\ee
which is consistent with our assertion that our polynomial invariant
cannot detect the noninvertibility of \emph{any} knot. More
experiments to illustrate this claim are supplied in
\secref{trotterpretzel}.
\subsection{A Class of Noninvertible Pretzel Knots}
\seclabel{trotterpretzel}
A class of noninvertible knots has been presented by Trotter
\cite{Trotter:64}; this class provides an easily-programmable
set of examples to see if a knot invariant detects noninvertibility.
Trotter is of the opinion that the knots are chiral. These pretzel
knots were in fact the \emph{first} noninvertible knots to be described
\cite[p~25]{Livingston:93}.
The structure of the knots $ \( p, q, r \) $ in this family is depicted
by its simplest example in \figref{TrotterPretzel}. Note that $ p, q $,
and $ r $ must all be distinct, odd, and greater than $ 1 $. In
\figref{TrotterPretzel}, the notation $ X_{rlr}^N $ refers to the
$ 2 $-braid of $ N $ crossings formed by the placing of $ X_r $ atop
$ X_l $, with $ X_r $ as the top and bottom crossings, for $ X $ being
either $ \sigma $ or $ \sigma^{-1} $. The recursive definition for such
knots is provided in \figref{XRLR}.
\begin{figure}[htbp]
\begin{center}
\input{XRLR.pstex_t}
\caption{%
Recursive definition of the towers $ X_{rlr} $ used in the
evaluation of the Links--Gould link invariant for the Trotter
pretzel knots; $X$ is either $ \sigma $ or $ \sigma^{-1} $.
The minimum is the case $ N = 1 $, which
corresponds to $ X_r $, i.e. $ X_{rlr}^1 \defeq X_r $.
A parallel definition of $ X_{lrl} $ might be given.
}
\figlabel{XRLR}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{TrotterPretzel.pstex_t}
\caption{The (noninvertible) pretzel knots of
Trotter, in tangle form. This illustration is of the smallest
possible one, with $ p = 7 $, $ q = 3 $, $ r = 5 $.
}
\figlabel{TrotterPretzel}
\end{center}
\end{figure}
The tensor associated with the pretzel is:
\be
{T_{TP \( p, q, r \)}}^y_x
& \defeq &
{\( \sigma^{-p}_{rlr} \)}^{a~c}_{x~d}
\cdot
{\( \sigma^{-q}_{rlr} \)}^{e~g}_{f~h}
\cdot
{\( \sigma^{-r}_{rlr} \)}^{i~k}_{j~l}
\cdot
\\
& & \qquad
{\( \Omega^- \)}_{a~k}
\cdot
{\( \Omega^+ \)}_{c~e}
\cdot
{\( \Omega^+ \)}_{g~i}
\cdot
{\( \mho^+ \)}^{d~f}
\cdot
{\( \mho^+ \)}^{h~j}
\cdot
{\( \mho^+ \)}^{l~y}.
\ee
Experiments show that the Links--Gould invariant for this class of
noninvertible knots always displays the symmetry of
\eqref{Melanotaenia}, for all $ p, q, r \leqslant 67 $. This amounts to
$5456$ knots, the smallest being the $(3,5,7)$ pretzel, a knot of
$3+5+7=15$ crossings, and the largest being the $(63,65,67)$ pretzel, a
knot of $63+65+67=195$ crossings. Incidentally, we find that the
invariant demonstrates that all those pretzels are chiral.
\subsection{The Kinoshita--Terasaka Pair of Mutant Knots}
\seclabel{KTMutants}
The Kinoshita--Terasaka pair is an example of a pair of $ 11 $ crossing
mutant knots that are known to be distinct. To be precise, more
commonly, the first of the pair is usually known as the
``Kinoshita--Terasaka Knot'', and the second has been called the
``Conway Knot''. In the original source by Kinoshita and Terasaka
\cite[p~151]{KinoshitaTerasaka:57}, the knot involved is the one
labelled $ \kappa \( 2, 2 \) $ (reproduced in
\cite[p~53]{Livingston:93}). They had constructed this knot as an
example of a nontrivial $ 11 $ crossing knot with Alexander polynomial
equal to $ 1 $. The source used to draw our example is from
\cite[p~174]{Adams:94}; note that these diagrams have $ 12 $ crossings,
so they are not minimal.
A number of proofs of their distinctness are at hand:
\begin{itemize}
\item
Adams \cite[p~106]{Adams:94} states that Francis Bonahon and Lawrence
Siebenmann first showed this in 1981.
Adams \cite[p~174]{Adams:94} goes on to state that
in 1986 David Gabai \cite{Gabai:86} showed that
their \emph{minimal genus Seifert surfaces} have different
\emph{genera}%
\footnote{%
\textsl{
If $ L $ is an oriented link in $ S^3 $
}
(i.e. the $ 3 $-sphere),
\textsl{
then a Seifert surface for $ L $ is an oriented surface $ R $
embedded in $ S^3 $ such that $ \partial R = L $ and no component
is closed.}
\cite[p~677]{Gabai:86}.
That is, a Seifert surface is a $ 2 $-manifold with boundary
being the link in question; the genus of such a surface being a
topological classifying label \cite[p~95-106]{Adams:94}. The
original reference for the Seifert algorithm is contained in
\cite{Seifert:34}.
}.
\item
More recently, Morton and Cromwell \cite{MortonCromwell:96} have
constructed a Vassiliev invariant of \emph{type}%
\footnote{%
A Vassiliev invariant is defined \cite[p~229]{MortonCromwell:96} to
be of \emph{type} $d$ if it is zero on any link diagram of
$d+1$ nodes, and to be of \emph{degree} $d$ if it is of type $d$
but not of type $d-1$.
}
$11$ which distinguishes them. This Vassiliev invariant is based on
the HOMFLY polynomial for framed links, and the authors compare it
with another invariant, itself coming from $ SU_q \( 3 \) $, which
does \emph{not} distinguish them.
More specifically, they show that the `$ SU_q \( N \) $ invariant'
for the module with Young diagram
$ \stackrel{\textstyle \Box \! \Box}{\Box \; \;} $ will distinguish
at least \emph{some} mutant pairs (in particular the KT pair), for
all $ N \geqslant 4 $, but will definitely \emph{not} distinguish any
for $ N = 2, 3 $.
\end{itemize}
More generally, it is known that neither the HOMFLY nor the Kauffman
polynomial can distinguish \emph{any} pair of mutants
\cite[p~174]{Adams:94}. In fact Lickorish \cite{Lickorish:87} used
skein theoretical arguments to show this; and furthermore, Lickorish
and Lipson \cite{LickorishLipson:87} and, independently Przytycki
\cite{Przytycki:89} again used skein theoretical arguments to show that
two equally twisted \emph{$ 2 $-cables}
(definition in \cite[p~118]{Adams:94})
of a mutant pair would have the same HOMFLY polynomial. Perhaps the
strongest statement that can be made in this direction was provided in
1994 by Chmutov, Duzhin and Lando \cite{ChmutovDuzhinLando:94}, who
proved that \emph{all} Vassiliev invariants of type less than $ 9 $
will agree on \emph{any} pair of mutants.
The question of whether the Links--Gould invariant is able to
distinguish mutants is immediately answerable in the negative. Theorem
5 of \cite{MortonCromwell:96} states that if the modules occurring in
the decomposition of $ V \otimes V $ each have unit multiplicity, as
indeed \eqref{TPdecomp} shows in our case, then the invariant is unable
to detect mutations. Whilst this was proved in
\cite{MortonCromwell:96} for the case of quantum algebras, the
extension to the case of quantum superalgebras is quite
straightforward. As an example, we have explicitly evaluated the
Links--Gould invariant for the aforementioned pair of mutants.
\pagebreak
We illustrate $ KT $, the first of Kinoshita--Terasaka pair, in
\figref{KT}, where the tensors $ KTA $ and $ KTB $ are
defined below, in Figures \ref{fig:KTA} and \ref{fig:KTC}. From $ KT $,
we may build the mutant $ KT' $ by replacing the component $ KTA $ with
$ KTA' $ (depicted in \figref{KTAprime}), which is formed by reflection
of $ KTA $ about a horizontal line.
\begin{figure}[htbp]
\begin{center}
\input{KT.pstex_t}
\caption{%
$ KT $, the first of the Kinoshita--Terasaka pair of mutant knots,
where the subdiagrams $ KTA $, $ KTA' $ and $ KTC $ are found in
Figures \ref{fig:KTA}, \ref{fig:KTAprime} and \ref{fig:KTC}
respectively. (The mutant $ KT' $ of $ KT $ is obtained by
exchanging $ KTA $ with $ KTA' $.)
}
\figlabel{KT}
\end{center}
\end{figure}
The tensors associated with $ KT $ and $ KT' $ are:
\be
{\( T_{KT} \)}^y_x
& \defeq &
{\( KTA \)}^{a~b}_{x~c}
\cdot
{\( KTC \)}^{d~j}_{e~k}
\cdot
{\( \Omega^- \)}_{b~d}
\cdot
{\( \mho^+ \)}^{c~e}
\cdot
{\( \Omega^+ \)}_{a~j}
\cdot
{\( \mho^+ \)}^{k~y},
\\
{\( T_{KT'} \)}^y_x
& \defeq &
{\( KTA' \)}^{a~b}_{x~c}
\cdot
{\( KTC \)}^{d~j}_{e~k}
\cdot
{\( \Omega^- \)}_{b~d}
\cdot
{\( \mho^+ \)}^{c~e}
\cdot
{\( \Omega^+ \)}_{a~j}
\cdot
{\( \mho^+ \)}^{k~y},
\ee
where
\be
{\( KTA \)}^{a~b}_{q~c}
& \defeq &
{\( \sigma \sigma_d \)}^{a~b}_{d~e}
\cdot
{\( \sigma^{-2} \)}^{d~f}_{q~g}
\cdot
{\( \sigma^{-1}_d \)}^{h~e}_{i~c}
\cdot
{\( \Omega^+ \)}_{f~h}
\cdot
{\( \mho^- \)}^{g~i}
\\
{\( KTA' \)}^{a~b}_{q~c}
& \defeq &
{\( \sigma^{-2} \)}^{a~f}_{d~g}
\cdot
{\( \sigma^{-1}_d \)}^{h~b}_{i~e}
\cdot
{\( \sigma \sigma_d \)}^{d~e}_{q~c}
\cdot
{\( \Omega^+ \)}_{f~h}
\cdot
{\( \mho^- \)}^{g~i}
\\
{\( KTC \)}^{d~j}_{e~k}
& \defeq &
{\( KTB \)}^{d~f}_{e~g}
\cdot
{\( \sigma^{-1}_l \sigma^{-1}_r \)}^{h~j}_{i~k}
\cdot
{\( \Omega^- \)}_{f~h}
\cdot
{\( \mho^+ \)}^{g~i}
\\
{\( KTB \)}^{d~f}_{e~g}
& \defeq &
\sigma^{d~b}_{a~c}
\cdot
{( \sigma_d^2 )}^{l~f}_{m~n}
\cdot
{\( \sigma^{-1} \sigma^{-1}_d \)}^{a~n}_{e~g}
\cdot
{\( \Omega^+ \)}_{b~l}
\cdot
{\( \mho^- \)}^{c~m}.
\ee
\begin{figure}[htbp]
\begin{center}
\input{KTC.pstex_t}
\caption{%
The component $ KTC $ of the Kinoshita--Terasaka pair
of mutant knots $ KT $ and $ KT' $.
}
\figlabel{KTC}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{KTA.pstex_t}
\caption{%
The component $ KTA $ of $ KT $, the first of the
Kinoshita--Terasaka pair.
}
\figlabel{KTA}
\end{center}
\end{figure}
\begin{figure}[htbp]
\begin{center}
\input{KTAprime.pstex_t}
\caption{%
The component $ KTA' $ of $ KT' $, the second of the
Kinoshita--Terasaka pair.
}
\figlabel{KTAprime}
\end{center}
\end{figure}
\pagebreak
\subsection{Links--Gould Polynomials of the K--T Mutants}
We find the Links--Gould polynomials of both mutants to be:
\begin{eqnarray*}
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\! \! \! \! \! \! \! \!
{LG}_{KT} \( q, p \)
=
\\
& &
\! \! \! \! \! \! \! \! \! \! \! \! \! \! \! \!
\begin{array}{l}
- 23
- p^{-6} q^{-8}
- p^{-6} q^{-6}
+ 2 p^{-6} q^{-4}
+ p^{-6} q^{-2}
- p^{-6}
+ p^{-4} q^{-8}
+ 6 p^{-4} q^{-6}
- 3 p^{-4} q^{-4}
\\
- 9 p^{-4} q^{-2}
+ 2 p^{-4}
+ 3 p^{-4} q^2
- 7 p^{-2} q^{-6}
- 7 p^{-2} q^{-4}
+ 18 p^{-2} q^{-2}
+ 9 p^{-2}
- 11 p^{-2} q^2
\\
- 2 p^{-2} q^4
+ 2 q^{-6}
+ 14 q^{-4}
- 8 q^{-2}
+ 6 q^2
+ 10 q^4
- 7 p^2 q^{-4}
- 7 p^2 q^{-2}
+ 18 p^2
+ 9 p^2 q^2
\\
- 11 p^2 q^4
- 2 p^2 q^6
+ p^4 q^{-4}
+ 6 p^4 q^{-2}
- 3 p^4
- 9 p^4 q^2
+ 2 p^4 q^4
+ 3 p^4 q^6
- p^6 q^{-2}
- p^6
\\
+ 2 p^6 q^2
+ p^6 q^4
- p^6 q^6,
\end{array}
\end{eqnarray*}
hence the Links--Gould link invariant \emph{does not} distinguish
between these mutants.
As predicted by the theorem of \cite{MortonCromwell:96}, the tensors
$KTA$ and $KTA'$ are in fact \emph{identical}, which explains why the
pair of mutants yield the same invariant.
\pagebreak
\section*{Acknowledgements}
Louis Kauffman thanks the National Science Foundation for support of
this research under grant number DMS-9205277 and the NSA for partial
support under grant number MSPF-96G-179.
Jon Links is supported by an Australian Postdoctoral Research
Fellowship and a University of Queensland New Staff Research Grant.
The authors wish to thank Mark Gould for his enthusiastic support of
this research.
\input{bibabb}
\bibliographystyle{plain}
| 2024-02-18T23:40:13.519Z | 1998-11-25T08:08:50.000Z | algebraic_stack_train_0000 | 1,732 | 10,519 |
|
proofpile-arXiv_065-8462 | \section{Introduction}
According to the standard scenarios of cosmology, the hydrogen
becomes neutral at $z\simeq 10^{3}$.
The absence of Ly$\alpha$ trough in quasar spectra, however, severely
constrains the neutral fraction of the uniformly distributed intergalactic
medium at $z<5$.
This fact indicates that the hydrogen has to be reionized by energy
injection from some objects in an intervening period.
It is suggested that ultraviolet radiation from the known quasar
population is insufficient to photoionize the universe (Shapiro \&
Giroux 1987).
Recently, the role of the first stars in reionizing the universe has
been emphasized by many authors (e.g., Fukugita \& Kawasaki
\markcite{FK94}1994; Ostriker \& Gnedin \markcite{OG96}1996; Haiman \&
Loeb \markcite{HL97}1997).
However, star formation in the galaxy formation epoch is one of
the most poorly understood processes involved in galaxy formation.
Consequently, theories on the stellar reionization are undermined by many
uncertainties, for example, the initial mass function,
star formation rate and efficiency of the first stars.
In order to resolve these uncertainties, detailed knowledge of the formation
processes of a first star from primordial gas seems to be required in
advance.
Primordial star formation can be viewed as the successive fragmentation and
contraction processes of collapsed cosmological objects.
In this paper we discuss the latter process, namely the contraction of
primordial protostellar clouds \footnote{We use the term
``protostellar cloud'' to identify a gravitationally bound
contracting gas cloud that, in part or as a whole, becomes a star as it
evolves. On the other hand, the term ``protostar'' means an accreting
core in hydrostatic equilibrium formed at the center of a protostellar
cloud} into stars.
This problem has been investigated with a one-zone approximation by e.g.,
Carlberg (1981); Palla, Salpeter, \& Stahler (1983; see also Matsuda,
Sato, \& Takeda 1969; Yoneyama 1972).
On the other hand, Stahler, Palla, \& Salpeter \markcite{SPS86}(1986)
studied the main accretion phase of a primordial protostar assuming
stationary accretion of ambient material onto a central hydrostatic
core, which is a primordial counterpart of the present-day stellar core
(Larson\markcite{Lar69}1969).
In that paper, they assumed Shu-like collapse (Shu
\markcite{Shu77}1977), i.e., the cloud initially in an unstable hydrostatic
equilibrium is overrun by an expansion wave that propagates outward
from the center, leaving behind a growing central point mass and a
free-fall velocity field.
Further, the mass accretion rate is constant with time and its rate is \.{\it
M}$ = 0.975 c_{\rm s}^{3}/G $, where $c_{\rm s}$ is the sound
velocity in the cloud.
Higher temperatures, $\sim 10^3$ K, in primordial clouds results in
a higher mass accretion rate $(4.41\times 10^{-3} M_{\sun}
{\rm yr}^{-1})$ than the present-day value ($10^{-5} M_{\sun} {\rm
yr}^{-1}$,
e.g. Stahler, Shu, \& Taam \markcite{SST80}1980).
However, it remains unclear how the primordial protostellar cloud
reaches the main accretion phase.
Villere \& Bodenheimer \markcite{VB87}(1987) investigated this problem
by using hydrodynamical methods.
They studied the spherically symmetric collapse of a primordial cloud of
$5 \times 10^{5} M_{\sun}$ into a protostar and pointed out the
importance of H$^{-}$ cooling for the determination of the minimum mass
of primordial stars.
However, they did not treat the radiative transfer of H$_{2}$ lines, and
their efficiency of cooling in optically thick H$_{2}$ lines was
different from that of Palla et al. (1983).
Therefore, in order to follow the collapse of primordial protostellar
clouds accurately, we need to employ the hydrodynamical method together
with solving the radiative transfer of H$_{2}$ lines.
In this paper we study the evolution of primordial protostellar clouds by
performing hydrodynamical calculations, with radiative transfer of
H$_{2}$ lines and continuous radiation for spherically symmetric clouds
up to the main accretion phase.
The outline of this paper is as follows.
In \S 2, we describe the method of our calculation.
In \S 3, results of calculations are presented.
We summarize our work and discuss its implication in \S 4.
In the Appendix, our treatment of radiative transfer is described.
\section{Method of Calculation}
\subsection{Hydrodynamics}
We assume the cloud is spherically symmetric and neglect rotation, magnetic
field, and external radiation field for simplicity.
The deviation from spherical symmetry and the effects of rotation may be
important.
However, these effects make the problem two or three-dimensional and
intractable so far.
In primordial star formation, magnetic fields are considered to play a
less important role than in present-day star formation.
Since star formation takes place in the deep interior of a protogalactic
cloud, and, moreover, in such a region the electron fraction is tiny, the
effects of cosmic background radiation are not significant.
The hydrodynamic code we use is a usual spherical symmetry,
explicit Lagrangian finite-difference scheme with
von Neumann-Richtmyer artificial viscosity (e.g., Mihalas \& Mihalas
1984; Thoul \& Weinberg 1995).
The Lagrangian equations of hydrodynamics describing the collapse of a
spherically symmetric cloud are
the continuity equation
\begin{equation}
\frac{\partial m}{\partial r}=4 \pi r^2 \rho,
\end{equation}
the momentum equation
\begin{equation}
\frac{Dv}{Dt}=-4 \pi r^2 \frac{\partial p}{\partial m}-\frac{Gm}{r^2},
\end{equation}
the energy equation
\begin{equation}
\frac{D \epsilon}{Dt}=-p\frac{D}{Dt}(\frac{1}{\rho})-\frac{\Lambda}{\rho},
\end{equation}
and the equation of state
\begin{equation}
p=(\gamma_{ad}-1)\rho \epsilon.
\end{equation}
In the above equations, the quantities $m,\rho,v,p,\varepsilon$, and $\Lambda$
are the mass within radius $r$, the density, velocity,
pressure, thermal energy per unit mass, and net cooling rate per unit
volume, respectively.
The adiabatic exponent $\gamma_{\rm ad}$ is given by
\begin{equation}
\gamma_{\rm ad}=1+ \sum_{i=1}^{8} y_{i} /\sum_{i=1}^{8}
\frac{y_{i}}{\gamma_{i}-1},
\end{equation}
where the summation is over all chemical species, which include $i={\rm
H^{0}}$,${\rm H^{+}}$,${\rm H^{-}}$,${\rm H_{2}}$,${\rm He^{0}}$,${\rm
He^{+}}$,${\rm He^{++}}$, and $e^{-}$.
The adiabatic exponent for H$_{2}$ is given by
\begin{equation}
\label{eq;gamH2}
\frac{1}{\gamma_{\rm H_{2}}-1}
=\frac{1}{2}[5+2x^{2}\frac{e^{x}}{(e^{x}-1)^{2}}]
, ~~x\equiv \frac{6100{\rm K}}{T}
\end{equation}
for H$_{2}$, where the second term in equation (\ref{eq;gamH2}) counts
the vibrational degrees of freedom and by $1/(\gamma_{i}-1)$=1.5 for
the other species.
The concentration of the $i$ th species $y_{i}$ is defined by
\begin{equation}
y_{i}=\frac{n(i)}{n_{\rm H}},
\end{equation}
where $n(i)$ is the number density of the $i$ th species, and $n_{\rm H}=n({\rm
H^{0}})+n({\rm H^{+}})+n({\rm H^{-}})+2n({\rm H_{2}})$ is
the number density of hydrogen nuclei.
We do not include the radiation force term in the momentum equation (2),
since it is always negligible compared with the gas pressure in our runs.
The net cooling rate $\Lambda$ consists of two parts, that is, the
radiative cooling $\Lambda_{\rm{rad}}$ and the chemical cooling
$\Lambda_{\rm{chem}}$.
The former can be written by using the luminosity $L(m)$ as
\begin{equation}
\Lambda_{\rm{rad}}(m)=\rho \frac{\partial L(m)}{\partial m}.
\end{equation}
The luminosity $L(m)$ is obtained by solving radiative transfer as
described in \S 2.3 and the Appendix.
We consider H$_{2}$ lines and continuum components that consist of
H$_{2}$ collision-induced absorption (CIA), H$^{-}$ bound-free absorption,
Lyman continuum absorption, etc., as sources of opacity.
The net chemical cooling rate $\Lambda_{\rm{chem}}$ is given by
\begin{equation}
\Lambda_{\rm{chem}}(m)=-\rho \frac{\partial \epsilon_{\rm
chem}}{\partial t},
\end{equation}
where $\epsilon_{\rm chem}$ is the chemical binding energy per unit
mass.
We keep the external pressure constant with time as the boundary condition.
The timestep is the smaller of that determined by the Courant
condition (Courant number=0.2) and $10^{-3}$ of the local cooling time.
\subsection{Chemical Reactions}
We consider the following eight species: ${\rm
H^{0}}$,${\rm H^{+}}$,${\rm H^{-}}$,${\rm H_{2}}$,${\rm He^{0}}$,${\rm
He^{+}}$,${\rm He^{++}}$, and $e^{-}$.
The abundance of helium atoms is assumed to be 28 \% by mass.
At low density ($n_{\rm H}< 10^{15} {\rm cm^{-3}}$), reaction equations
between ${\rm H^{0},H^{+},H^{-},H_{2}}$, and ${e^{-}}$ are
solved using an implicit difference scheme.
Considered reactions are the same as Palla et
al. (1983), while some reaction coefficients are
altered (Table 1).
The H$^{-}$ fraction is assumed to be the equilibrium value.
Helium atoms are considered to be all neutral, because in our case this low
density range corresponds to low temperature, namely $T<2100{\rm K}$.
Before the number density exceeds $10^{15} {\rm cm^{-3}}$, the H$_{2}$
fraction easily reaches the equilibrium value.
At $n_{\rm H}>10^{15} {\rm cm^{-3}}$ we switch the computation to the
equilibrium chemistry.
Although the $e^{-}$ fraction has not reached equilibrium yet, this
does not cause any significant error, since $e^{-}$ plays no role in H$_{2}$
formation/dissociation at that time.
The equilibrium chemistry ($n_{\rm H}>10^{15} {\rm cm^{-3}}$) is
computed by solving coupled Saha equations between
${\rm H^{0},H^{+},H_{2},He^{0},He^{+},He^{++}}$, and ${e^{-}}$.
\subsection{Radiative Transfer}
We consider H$_{2}$ rovibrational lines and continuum components as
sources of opacity and treat their transfer separately.
H$_{2}$ line transfer is dealt with as a frequency
dependent transfer problem as is described in detail in the Appendix.
We consider the first three vibrational
states with rotational levels up to $J=20$ in each vibrational state
following Palla et al. (1983).
The population of each level is determined by assuming local
thermodynamic equilibrium (LTE), since we consider the density range
well above the
critical number density $n_{\rm{cr}}\sim 10^{4} {\rm cm^{-3}}$, where
radiative and collisional deexicitation rates become equal for the
molecular hydrogen.
We consider both thermal and kinematic Doppler shifts as a line broadening
mechanism.
Because of the lack of dust grains, primordial gas has only very weak
continuum opacity.
Continuum transfer is treated as a gray problem using
mean opacity.
When the cloud becomes substantially optically thick, it becomes
difficult to keep numerical accuracy in solving the gray transfer
equation.
In such a case, we separate the cloud
into two regions, namely, the opaque interior and transparent envelope.
We solve the transfer equation only in the outer region, while the
diffusion approximation is employed in the interior.
As the mean opacity, we take the continuum-Planck mean opacity for
metal free gas at 1000K$<T<$7000K from Lenzuni, Chernoff \& Salpeter
(1991), which includes all the
important continuum processes, specifically, bound-free absorption by
${\rm H^{0}}$ and ${\rm H^{-}}$, free-free absorption by ${\rm H^{0}}$,
${\rm H^{-}}$, ${\rm H_{2}}$, ${\rm H_{2}^{-}}$, ${\rm H_{2}^{+}}$,
${\rm H_{3}}$, ${\rm He^{0}}$, and ${\rm He^{-}}$, photodissociation of
${\rm H_{2}}$, and ${\rm H_{2}^{+}}$, Rayleigh scattering by ${\rm H^{0}}$,
${\rm H_{2}}$, and ${\rm He^{0}}$, Thomson scattering by $e^{-}$, and
collision-induced absorption by ${\rm H_{2}}$ due to collisions with
${\rm H_{2}}$, ${\rm He^{0}}$, and ${\rm H^{0}}$.
They omit all lines, for example, Ly$\alpha$ or ${\rm H_{2}}$
rovibrational (quadrupole) transitions in their opacity.
We treat the ${\rm H_{2}}$ lines as was described above.
The Ly$\alpha$ lines are not so important in cooling in our case, because at
such high temperatures where they are excited, the density is so high and
their optical depth is so huge that these photons are always absorbed
before escaping the cloud.
At $T<1000{\rm K}$, we set the continuum opacity to be zero.
At $T>7000{\rm K}$, it is supplemented by using the Rosseland mean
opacity for metal free gas from Rogers \& Iglesias \markcite{RI92}(1992).
Our adoption of the Planck mean opacity at $T<7000{\rm K}$ and the
Rosseland mean opacity at higher temperature may deserve some comments.
At low temperature, and then at low density,
protostellar clouds are transparent to continuum emission, so that Planck
mean opacity is appropriate.
While the continuum optical depth exceeds unity $(T>2000{\rm K})$,
both mean opacities have close values within a factor of 2 in our
relevant range (Lenzuni et al. \markcite{LCS91}1991).
Actually, they match smoothly to each other at the interface.
We solve H$_{2}$ line transfer and continuum transfer
independently, i.e., In solving H$_{2}$
line transfer, we neglect continuum opacity.
However, errors caused by this are not significant.
The reason is as follows:
once the clouds become opaque to continuum opacity in a small
central region where all the evolution occurs essentially, H$_{2}$ line
cooling is negligible compared with continuum and chemical cooling there.
There still exists the outer region where H$_{2}$ line radiation is the
dominant cooling agent.
However, in such a region the evolutionary timescale is so long in
comparison with the central one that this region remains practically
unchanged up to the formation of the hydrostatic core at the center (i.e.,
the end of our calculations).
After the continuum optical depth exceeds unity, we set the cooling rate
due to H$_{2}$ lines equal to zero.
\section{Results}
\subsection{Initial Conditions}
We consider primordial protostellar clouds which are fragments of larger
ones.
Fragmentation of a collapsing cosmological object into primordial
protostellar clouds can be viewed as follows.
It is known that a spherical cloud in pressure-free collapse is unstable
with respect to non-spherical perturbations (Lin, Mestel, \& Shu 1965;
Hutchins 1976; Susa, Uehara, \& Nishi 1996).
As a consequence, the collapsing cosmological object first becomes a
pancaking disk, which is gravitationally unstable and fragments
more easily into filamentary clouds rather than into spherical clouds
(Miyama, Narita, \& Hayashi 1987a, 1987b).
Such a filamentary cloud is also gravitationally unstable and subject to
fragmentation but does not fragment immediately, as long as the
filamentary cloud collapses so fast that the density perturbation has
not sufficient time to grow (Inutsuka \& Miyama 1992, 1997).
Because the virial temperature of a filamentary cloud is
constant with time, the free-fall time $t_{\rm ff}$ decreases faster
than the cooling time $t_{\rm cool}$ as contraction proceeds.
Eventually, $t_{\rm ff} \simeq t_{\rm cool}$, i.e., the collapse becomes
slow, then the filamentary cloud fragments.
At the time of fragmentation, fragments cannot be highly
gravitationally unstable.
Hence, we take hydrostatic equilibrium
clouds with adiabatic stratification as the initial conditions for
simplicity.
Namely, the density distribution is represented by $\rho=\rho_{c}
(\theta_{n})^{n}$, where $\rho_{c}$ is the central density,
$n=1/(\gamma_{ad}-1)$ is the polytropic index, and $\theta_{n}$ is the
Emden function of index $n$.
The pressure distribution is given by $p=K \rho^{\gamma_{ad}}$,
where $K$ is a free parameter fixed by giving the number density at
half mass radius $n_{h}$ and the total mass of the cloud $M$.
These clouds are cut off at the radius $r_{s}$ where the density falls
off $10^{-3}$ from the central value.
Then parameters needed to specify an initial state are the mass of the
cloud $M$, the number density at the half mass radius $n_{h}$, and the
concentrations of chemical species.
The initial parameters for several models are given in Table 2.
Here, $N$ is the number of radial grids.
As shown in Table 2, the initial width of the radial
zones is increased by constant ratio for each zone as distance from the
center is increased.
This ratio is determined so as to resolve the smallest structure found in test
runs with a sufficient number of radial grids.
The mass and initial parameters of run A are close to those of a
primordial cylindrical cloud at the fragmentation epoch (Uehara 1998).
Hereafter, we call run A the fiducial run.
The parameters of run B are the same as those of run A, except the
number of radial grids is reduced to 100 in order to check the accuracy of our
calcualtion.
The match of the evolutionary paths between run A and run B is fairly good.
Therefore, radial grids of 100 cells are enough to follow the collapse as
long as the resolution around the center is sufficient.
Runs C and D correspond to smaller and denser fragments than the fiducial
run, which result from the later fragmentation of cylindrical clouds
(Uehara et al. 1996).
The evolution in these runs becomes almost the same as that of the
corresponding inner mass regions of the fiducial run.
If the H$_{2}$ fraction is too small at the begining, heat generated by
H$_{2}$ formation exceeds the radiative loss by H$_{2}$ lines and the cloud
expands until a sufficient amount of H$_{2}$ is produced.
However, this expansion is clearly a result of inappropriate initial
conditions.
We put as much H$_{2}$ fraction as necessary in these runs to avoid such
expansion.
In the above runs, we set the H$_{2}$ concentration $f_{\rm H_{2}}={\rm
const.}$ with radius initially, while in reality the $f_{\rm H_{2}}$
distribution has a gradient, that is, higher in the interior and lower in
the envelope.
To see its effect, we set $f_{\rm H_{2}}=0$ initially in run E.
In this case the cloud does not collapse dynamically until a sufficient
amount of H$_{2}$ is formed.
It is found that, however, the evolution thereafter is the same as
in the fiducial case.
As was mentioned above, the structure of collapsing primordial clouds becomes
almost the same after initial transient phases that merely reflect
inappropriate initial conditions.
Therefore, we describe the evolution in the fiducial run in detail hereafter.
\subsection{The collapse of a primordial protostellar cloud}
Initially the cloud is in hydrostatic equilibrium, and the cooling time
is somewhat longer than the free-fall time.
For the fiducial run the cooling time $t_{\rm cool}\equiv \rho \epsilon /
\Lambda=5.1 \times 10^{5}$ yr and the free-fall time $t_{\rm
ff}\equiv (3 \pi /32G \rho)^{1/2}=3.2 \times 10^{5}$ yr at the
center of the cloud initially.
After some quasi-static contraction, the cooling time at the
center becomes shorter than the free-fall time.
At this time, the cloud begins to collapse dynamically in a free-fall
time scale.
In reality, the evolution should only be physically
relevant thereafter, since a fragment is in the state where $t_{\rm cool}\simeq
t_{\rm ff}$ at the fragmentation epoch (Low \& Lynden-Bell 1976; Rees
1976; Silk 1977; Uehara et al. 1996).
The evolutionary sequences of number density, temperature, velocity, and
H$_{2}$ concentration distributions are illustrated for the fiducial
run in Figure 1 and for run C, in Figure 3, as a function of
the radial distance.
As we can see from Figures 1 a, the collapse proceeds
almost self-similarly and is very reminiscent of Larson-Penston's
similarity solution (Penston 1969; Larson 1969).
Since the free-fall time depends inversely on the density, the collapse
proceeds most rapidly in the center where the density is the highest,
leaving the outer part practically unchanged.
The dash in the figure indicates the slope of density
gradient for the isothermal case, namely $\rho \propto r^{-2}$
(Bodenheimer \& Sweigart 1968).
The slope in our case is slightly steeper, and its value is about
$-2.2$.
For the Larson-Penston similarity solution in the case of a polytropic
equation of state, i.e., $p=K \rho ^{\gamma}$, the density gradient may
be written as (Larson 1969)
\begin{equation}
\frac{\partial {\rm ln} \rho}{\partial {\rm ln} r} = \frac{-2}{2- \gamma}.
\end{equation}
Therefore $\gamma \simeq 1.1$ in our case (see \S 3.3).
The evolution of the central temperature and the ratio of specific heats
$\Gamma=\frac{\partial {\rm ln} p/ \partial t}{\partial {\rm ln} \rho /
\partial t}$ at the center are illustrated in Figure 3 a and 3 b.
According to Figure 3 b, the central value of $\Gamma$ is about 1.1 in
a large range of density, which coincides with the value expected from
the density gradient.
An initial low value of $\Gamma$ at the center does not mean that the
central part of the cloud cools efficiently at that time.
The reason is the following.
In quasi-static contraction, although $\Gamma$ becomes nearly equal to the
critical value 4/3 on average over a whole cloud, this needs not be
$\Gamma \simeq 4/3$ at the center.
Since the inner region cools faster than the outer part, a specific
entropy gradient (i.e., $\frac{\partial s}{\partial r}>0$) is
established in the course of contraction.
In the case that the cloud contracts from the initial adiabatic
stratification (i.e., $\frac{\partial s}{\partial r}=0$ ) to such a
state, $\Gamma$ tends to become lower in the inner region, while larger than
4/3 in the outer region in quasi-static contraction (see Omukai et
al. 1998).
Our initial low value of $\Gamma$ at the center merely
reflects this fact.
After the cloud starts dynamical collapse, efficient H$_{2}$-line cooling
makes the ratio of specific heats $\Gamma$ continue to decrease toward unity
(corresponding to the isothermal case) on average, until the cloud
becomes optically thick to several lines when the central number density
$n_{\rm c}\sim 10^{11} {\rm cm^{-3}}$.
Cooling due to molecular hydrogen formed by efficient three-body
processes, which become the dominant mode of H$_{2}$ production for
number density $n >10^{8} {\rm cm^{-3}}$, makes the temperature in the
central region even lower than in the outer part, as we can see in Figure 1
(b).
These processes completely transform hydrogen in the central part of
about $1M_{\sun}$ into molecular form before the temperature becomes so
high that efficient H$_{2}$ dissociation begins.
However, the outer region, i.e., $m>{\rm several}~M_{\sun}$ in mass
coordinate, remains totally atomic to the end (Figs. 1 and 2 d).
By the time the central number density reaches about $10^{11} {\rm
cm^{-3}}$ and the central temperature $T_{c}\simeq 900{\rm K}$, more
than 60 \% of hydrogen atoms have been converted into molecular form in a
small region around the center, and some of the H$_{2}$ lines begin to
become optically thick.
Eventually the cooling rate per unit mass around the
center becomes lower than that in the outer region.
Although this raises the ratio of specific heats $\Gamma$
steadily, it does not exceed the critical value 4/3, as we can see in
Figure 3 (b).
The reason is that there are always sufficient lines that are efficient in
cooling, namely, those whose optical depth is nearly unity.
When the central number density reaches about $3\times 10^{13} {\rm cm^{-3}}$
and the central temperature about $1600 {\rm K}$, the hydrogen molecules
begin to dissociate gradually.
However, this is only temporary.
At the same time, H$_{2}$ CIA-continuum cooling comes into play.
Continuum cooling is so strong as to stop and even to reverse
dissociation, until it becomes optically thick and radiative cooling
is no longer efficient.
After a small central part of the cloud becomes opaque ($ 3\times 10^{16}
{\rm cm^{-3}}, 2000 {\rm K} $), the full-scale dissociation begins
subsequently.
In the case of the star formation in present-day molecular clouds,
after the central region becomes opaque to dust continuum, a transient
core in hydrostatic equilibrium is formed with central density and
temperature $n_{\rm c}=10^{14}{\rm cm^{-3}}$, and $T_{\rm c}=170 {\rm K}$
immediately after the formation (Larson 1969).
In the primordial case, no transient core is formed when the central
part of the cloud becomes optically thick to continuous radiation, since
H$_{2}$ dissociation follows successively.
The density and temperature at the center at the begining of the full-scale
dissociation are nearly the same as in the present-day case.
Then the evolution also becomes nearly the same.
When most of the hydrogen molecules are dissociated, the ratio of
specific heats $\Gamma$ rises above the critical value 4/3 (Fig. 3 b).
The partial ionization of the hydrogen atoms reduces $\Gamma$
temporarily but not
below 4/3, since the density is already sufficiently high for
compressional work to dominate cooling owing to ionization.
After the central part of the cloud contracts almost adiabatically to
some extent, the hydrostatic
core, whose mass is about $5 \times 10^{-3} {\rm M_{\sun}}$, forms at the
center when the central number density reaches $\sim 10^{22} {\rm
cm^{-3}}$ and the central temperature $3 \times 10^{4} {\rm K}$.
These physical dimensions are almost the same as those of the
stellar core found in the calculation of present-day star
formation.
The formation of the core can be seen more clearly in the velocity
distributions (Figs. 1 c and 2 c).
When the central hydrostatic core forms, a shock front develops at the
surface.
The core grows in mass by accretion of envelope material and
eventually will become an ordinary star.
\subsection{The Mass Accretion Rate}
Because of our explicit method of hydrodynamics, the Courant condition makes
time steps extremely short and it becomes numerically very costly to
pursue the evolution after the core formation.
Moreover our artificial viscosity method smears the accretion shock front,
thus producing a small temperature gradient, which results in a low
luminosity and low radiative cooling when we use the diffusion approximation
(Winkler \& Newman \markcite{WN80}1980).
We then stop our calculation at some arbitrary time after the core formation.
Therefore the evolution thereafter cannot be known exactly.
However, we found that the Larson-Penston-type similarity solution for
the equation of state $p=K \rho ^{\gamma}$ (Yahil \markcite{Yahil83}1983;
Suto \& Silk \markcite{SS88}1988), where $K=4.2\times 10^{11}$ (in
cgs) and $\gamma=1.09$ reproduces well our results of the evolution
in the fiducial run before the core formation (see Figure 4).
Here we assume that the evolution of envelope after the central core formation
is also described by the same similarity solution.
Larson-Penston type similarity solutions can be extended naturally even
after the central density becomes infinite (i.e., core formation)
(Hunter \markcite{Hun77}1977; Yahil \markcite{Yahil83}1983).
The solutions after the central density becomes infinite have the finite
point mass at the center (i.e., protostar), which grows in mass with
time as (Yahil \markcite{Yahil83}1983; Suto \& Silk \markcite{SS88}1988)
\begin{eqnarray}
M_{\ast}&=&K^{3/2} G^{(1-3 \gamma)/2} t^{4-3 \gamma} m_{0} \nonumber \\
&=&0.11 M_{\sun} (\frac{t}{1 {\rm yr}})^{0.73},
\label{eq:mass}
\end{eqnarray}
where $M_{\ast}$ is the mass of the central protostar, $t$ is the time
elapsed since the core formation, and $m_{0}$ is a
non-dimensional constant whose value is 20 for $\gamma=1.09$
Larson-Penston type solution.
We used $K=4.2\times 10^{11}$ and $\gamma=1.09$ in the second expression
of equation (\ref{eq:mass}).
Thus the mass accretion rate to the central protostar is given by
\begin{eqnarray}
{\dot M_{\ast}}&=&(4-3 \gamma) K^{3/2} G^{(1-3 \gamma)/2}
t^{-3(\gamma-1)} m_{0} \nonumber \\
&=&8.3 \times 10^{-2} M_{\sun} {\rm yr^{-1}} (\frac{t}{1
{\rm yr}})^{-0.27}.
\label{eq:acc}
\end{eqnarray}
Using the relation in equation (\ref{eq:mass}), we can write the mass accretion rate
as a function of an instantaneous mass of the protostar as
\begin{equation}
{\dot M_{\ast}}=3.7 \times 10^{-2} M_{\sun} {\rm yr^{-1}}
(\frac{M_{\ast}}{M_{\sun}})^{-0.37}.
\label{eq:acc2}
\end{equation}
According to equation (\ref{eq:acc}) or (\ref{eq:acc2}), the mass
accretion rate is huge and is diminishing with time.
From the first expression of equation (\ref{eq:acc}), we can see that
there are three parameters that determine the mass accretion rate, i.e.,
$\gamma,K$ and $m_{0}$.
The decrease of the accretion rate with time is the common nature of
self-similar collapse with $\gamma>1$ as can be seen by the first
expression of equation (\ref{eq:acc}).
The parameter $K$ is related to the temperature of protostellar clouds, and
the parameter $m_{0}$ concerns the type of the similarity solution
(i.e., the Larson-Penston-type or the Shu-type solution, etc.) as
well as $\gamma$.
Stahler et al. \markcite{SPS86}(1986) pointed out the higher mass accretion
rate $4.41 \times 10^{-3} M_{\sun} {\rm yr}^{-1}$ of primordial
protostars than present-day value because of higher temperature (then
the higher $K$ value) in primordial protostellar clouds, although they
thought of Shu-like collapse.
In our analysis, however, the collapse proceeds like the Larson-Penston-type
solution, whose parameter $m_{0}$ is about an order of magnitude higher
than that of the Shu-type solution.
For example, in the isothermal ($\gamma=1$) Larson-Penston collapse the
density is 4.4 times higher than the hydrostatic equilibrium value and
the fluid velocity is 3.3 times than the sound speed at the time of
central core formation.
This results in the value of $m_{0}$ and the mass accretion rate 48
times higher than that in isothermal Shu collapse of the same
temperature (Hunter \markcite{Hun77}1977).
\section{Conclusion}
In this paper we have investigated the collapse of primordial clouds
into protostars using a spherically symmetric
Lagrangian hydrodynamic code combined with radiative transfer of H$_{2}$
lines/continuum and chemical reactions.
The collapse proceeds self-similarly like the Larson-Penston
similarity solution until the central region reaches stellar
density.
During the collapse, the central part of about $1M_{\sun}$ becomes
fully molecular before the dissociation, while the outer part ($m> {\rm
several}~M_{\sun}$ in mass coordinate) remains almost fully atomic.
When the small central part of the cloud reaches stellar densities ($\sim
10^{-2} {\rm g~cm^{-3}}$), the hydrostatic core, whose physical
dimensions are roughly the same as that in present-day star formation,
is formed at the center of the cloud.
The mass accretion rate is typically $\sim
10^{-2} M_{\sun} {\rm yr^{-1}}$, which is about 3 orders of magnitude higher
than in the present-day case, and it declines with time.
Our results suggest a stellar core of
$5 \times 10^{-3}M_{\sun}$ at its formation.
On the other hand, Palla et al. (1983) argued the minimum Jeans mass in
the primordial gas clouds to be $0.05M_{\sun}$ at H ionization.
This mass can be identified with the mass of the stellar core at the time of
formation (Carlberg 1981).
This descrepancy seems to come not only from Palla et al. (1983)'s
one-zone treatment, but also from the fact that Palla et al. (1983) did
not take into account H$_{2}$ CIA continuum.
\acknowledgements
We are grateful to H. Masunaga, H. Susa, and H. Uehara for fruitful
discussions, to H. Sato for his continuous encouragement,
to T. Chiba and N. Sugiyama for reading our manuscript carefully,
to S. Hayward for checking the English, and to the referee, F. Palla,
for improving this manuscript.
We also would like to thank participants in the workshop ``Molecular
Hydrogen in the Early Universe'' (Firenze, 1997 December 4-6) for invaluable
comments.
We owe thanks to the YITP computer system for the numerical analyses.
This work is supported in part by a Grant-in-Aid of Scientific Research
from the Ministry of Education, Science, Sports and Culture
No. 09740174(RN).
\newpage
| 2024-02-18T23:40:13.693Z | 1998-11-19T16:36:52.000Z | algebraic_stack_train_0000 | 1,737 | 5,314 |
|
proofpile-arXiv_065-8463 | \section{Introduction}
A number of the experiments being performed at the Thomas Jefferson
National Accelerator Facility (TJNAF) involve the elastic and
inelastic scattering of electrons off the deuteron at space-like
momentum transfers of the order of the nucleon mass. In building
theoretical models of these processes, relativistic kinematics and
dynamics would seem to be called for. Much theoretical effort has
been spent constructing relativistic formalisms for the two-nucleon
bound state that are based on an effective quantum field theory
lagrangian. If the usual hadronic degrees of freedom appear in the
lagrangian then this strategy is essentially a logical extension of
the standard nonrelativistic treatment of the two-nucleon system.
Furthermore, regardless of the momentum transfer involved, it is
crucial that a description of the deuteron be used which incorporates
the consequences of electromagnetic gauge invariance. Minimally this
means that the electromagnetic current constructed for the deuteron
must be conserved.
Of course, the two-nucleon bound state can be calculated and a
corresponding conserved deuteron current constructed using
non-relativistic $NN$ potentials which are fit to the $NN$ scattering
data. This approach has met with considerable success. (For some
examples of this program see Refs.~\cite{Ad93,Wi95}.) Our goal here is
to imitate such calculations---and, we hope, their success!---in a
relativistic framework. To do this we construct an $NN$ interaction,
place it in a relativistic scattering equation, and then fit the
parameters of our interaction to the $NN$ scattering data. We then
calculate the electromagnetic form factors of the deuteron predicted
by this $NN$ model. By proceeding in this way we hope to gain
understanding of the deuteron electromagnetic form factors in a model
in which relativistic effects, such as relativistic kinematics,
negative-energy states, boost effects, and relativistic pieces of the
electromagnetic current, are explicitly included at all stages of the
calculation.
This program could be pursued using a four-dimensional formalism based
on the Bethe-Salpeter equation. Indeed, pioneering calculations of
electron-deuteron scattering using Bethe-Salpeter amplitudes were
performed by Zuilhof and Tjon almost twenty years
ago~\cite{ZT80,ZT81}. However, despite increases in computer power
since this early work the four-dimensional problem is still a
difficult one to solve. Since the $NN$ interaction is somewhat
phenomenological ultimately it is not clear that one gains greatly in
either dynamics or understanding by treating the problem
four-dimensionally. Therefore, instead we will employ a
three-dimensional formalism that incorporates what we believe are the
important dynamical effects due to relativity at the momentum
transfers of interest.
We will use a three-dimensional (3D) formalism that, in principle, is
equivalent to the four-dimensional Bethe-Salpeter formalism. This
approach has been developed and applied in
Refs.~\cite{PW96,PW97,PW98}. In this paper we will focus on the
calculation of elastic electron-deuteron scattering. Here we review
the formalism for relativistic bound states and show how to construct
the corresponding electromagnetic current. Calculations of elastic
electron-deuteron scattering are performed both in the impulse
approximation and with some meson-exchange currents included. The
results for the observables $A$, $B$ and $T_{20}$ are presented.
Many other 3D relativistic treatments of the deuteron dynamics which
are similar in spirit to that pursued here exist (see for instance
Refs.~\cite{vO95,Ar80}). Of these, our work is closest to that of
Hummel and Tjon~\cite{HT89,HT90,HT94}. However, in that work
approximations were employed for ingredients of the analysis, such as
the use of wave functions based on the 3D quasipotential propagator of
Blankenbecler-Sugar~\cite{BbS66} and Logunov-Tavkhelidze~\cite{LT63},
approximate boost operators, and an electromagnetic current which only
approximately satisfies current conservation. Calculations of elastic
electron-deuteron scattering also were performed by Devine and Wallace
using a similar approach to that pursued here~\cite{WD94}. Here we
extend these previous analyses by use of our systematic 3D formalism.
In this way we can incorporate retardations into the interaction and
also use a deuteron electromagnetic current that is specifically
constructed to maintain the Ward-Takahashi identites.
The paper is organized as follows. In Section~\ref{sec-Section2} we
explain our reduction from four to three dimensions. In
Section~\ref{sec-Section3} we present a four-dimensional equation
which is a modified version of the ladder Bethe-Salpeter equation.
This modified equation has the virtue that it, unlike the ladder BSE,
incorporates the correct one-body limit. By applying our
three-dimensional reduction technique to this four-dimensional
equation we produce an equation which has the correct one-body limit and
contains the correct physics of negative-energy states. In
Section~\ref{sec-Section4} we explain the various potentials that are
used in calculations of deuteron wave functions. These can be divided
into two classes: instant potentials, and potentials that include
meson retardation. Within either of these classes versions of the
potentials are constructed that do and do not include the effects of
negative-energy states, in order to display the role played by such
components of the deuteron wave function. Section~\ref{sec-Section5}
discusses our 3D reduction of the electromagnetic current that
maintains current conservation. This completes the laying out of a
consistent formalism that includes the effects of relativity
systematically, has the correct one-body limits, and maintains current
conservation. In Section~\ref{sec-Section6} we apply this machinery to
the calculation of electron-deuteron scattering both in the impulse
approximation and when corrections due to some meson-exchange currents
are included. Finally, discussion and conclusions are presented in
Section~\ref{sec-Section7}.
\section{The reduction to three dimensions}
\label{sec-Section2}
The Bethe-Salpeter equation,
\begin{equation}
T=K + K G_0 T,
\label{eq:BSE}
\end{equation}
for the four-dimensional $NN$ amplitude $T$ provides a theoretical
description of the deuteron which incorporates relativity.
Here $K$ is the Bethe-Salpeter kernel, and $G_0$ is the
free two-nucleon propagator. In a strict quantum-field-theory
treatment, the kernel $K$ includes the infinite set of two-particle
irreducible $NN \rightarrow NN$ Feynman graphs.
For the two-nucleon system an application of the full effective
quantum field theory of nucleons and mesons is impractical and
perhaps, since hadronic degrees of freedom are not fundamental,
inappropriate. In other words, the Bethe-Salpeter formalism may serve
as a theoretical framework within which some relativistic effective
interaction may be developed. But, if the $NN$ interaction is only an
effective one, then it would seem to be equally appropriate to develop
the relativistic effective interaction within an equivalent
three-dimensional formalism which is obtained from the
four-dimensional Bethe-Salpeter formalism via some systematic
reduction technique.
One straightforward way to reduce the Bethe-Salpeter equation to three
dimensions is to approximate the kernel $K$ by an instantaneous
interaction $K_{\rm inst}$. For example, if $q=(q_0,\bf q)$ is the
relative four-momentum of the two nucleons then
\begin{equation}
K(q)=\frac{1}{q^2 - \mu^2} \qquad \rightarrow \qquad
K({\bf q})=-\frac{1}{{\bf q}^2 + \mu^2}.
\end{equation}
This, admittedly uncontrolled, approximation, yields from the Bethe-Salpeter
equation the Salpeter equation:
\begin{equation}
T_{\rm inst}=K_{\rm inst} + K_{\rm inst} \langle G_0 \rangle T_{\rm inst},
\label{eq:Salpeter}
\end{equation}
where the three-dimensional Salpeter propagator $\langle G_0 \rangle$ is
obtained by integrating over the time-component of relative momentum,
\begin{equation}
\langle G_0 \rangle=\int \frac{dp_0}{2 \pi} G_0(p;P).
\end{equation}
Throughout this paper we denote the integration over the zeroth
component of relative momenta, which is equivalent to consideration of
an equal-time Green's function, by angled brackets. We shall consider
only spin-half particles, and so
\begin{equation}
\langle G_0 \rangle=\frac{\Lambda_1^+ \Lambda_2^+}{E - \epsilon_1
- \epsilon_2} - \frac{\Lambda_1^- \Lambda_2^-}{E + \epsilon_1
+ \epsilon_2};
\label{eq:aveG0}
\end{equation}
where $\Lambda^{\pm}$ are related to projection operators onto
positive and negative-energy states of the Dirac equation, $E$ is the
total energy, and $\epsilon_i=({\bf p}_i^2 + m_i^2)^{1/2}$. Note that
for spin-half particles, this propagator $\langle G_0 \rangle$ is not
invertible.
In order to systematize this kind of 3D reduction one must split the
4D kernel $K$ into two parts. One of these, $K_1$, is to be understood
as a three-dimensional interaction in the sense that it does not
depend on the zeroth component of relative four momentum~\footnote {Of
course, this is not a covariant reduction, but covariance can be
maintained by a suitable generalization of this idea~\cite{PW97}.}.
We then seek to choose this $K_1$ such that the 3D amplitude $T_1$
defined by
\begin{equation}
T_1=K_1 + K_1 \langle G_0 \rangle T_1,
\label{eq:3Dscatt}
\end{equation}
has the property that
\begin{equation}
\langle G_0 \rangle T_1 \langle G_0 \rangle=\langle G_0 T G_0 \rangle.
\end{equation}
It is straightforward to demonstrate that such a $K_1$ is defined by
the coupled equations:
\begin{equation}
K_1 = \langle G_0 \rangle ^{-1} \langle G_0 K {\cal G} \rangle
\langle G_0 \rangle ^{-1} ,
\label{eq:K1}
\end{equation}
which is three-dimensional, and
\begin{equation}
{\cal G} = G_0 + G_0 (K - K_1) {\cal G},
\label{eq:<G0>}
\end{equation}
which is four dimensional. The $K_1$ of Eq.~(\ref{eq:K1}) does this by
ensuring that
\begin{equation}
\langle {\cal G} \rangle=\langle G_0 \rangle.
\label{eq:calGeq}
\end{equation}
The formalism is systematic in the sense that, given a perturbative
expansion for the 4D kernel, $K$, a perturbative expansion for the 3D
kernel, $K_1$, can be developed. At second order in the coupling this gives:
\begin{equation}
K_1^{(2)}=\langle G_0 \rangle^{-1} \langle G_0 K^{(2)} G_0 \rangle
\langle G_0 \rangle^{-1}.
\label{eq:K12}
\end{equation}
In $++ \rightarrow ++$ states this is just the usual energy-dependent
one-particle-exchange interaction of time-ordered perturbation theory,
but with relativistic kinematics, i.e. ignoring spin and isospin:
\begin{equation}
K_1^{(2)}=\frac{g^2}{2 \omega}\left[\frac{1}{E^+ - \epsilon_1 -
\epsilon_2' - \omega} + (1 \leftrightarrow 2)\right],
\end{equation}
where $\omega$ is the on-shell energy of the exchanged particle. Note
that $\langle G_0 \rangle$ must be invertible in order for the 3D
reduction to be consistent. (Similar connections between three and
four-dimensional approaches are discussed in
Refs.~\cite{LT63,Kl53,KL74,BK93B,LA97}.)
Equation~(\ref{eq:3Dscatt}) leads to an equation for the bound-state
vertex function:
\begin{equation}
\Gamma_1 = K_1 \langle G_0 \rangle \Gamma_1,
\label{eq:3Deqn}
\end{equation}
where $\Gamma_1$ is the vertex function in the three-dimensional
theory. The 4D vertex function, $\Gamma$, and the
corresponding 3D one, $\Gamma_1$, are related via
\begin{equation}
G_0 \Gamma = {\cal G}\Gamma_1.
\label{eq:GcalGamma1}
\end{equation}
\section{The one-body limit}
\label{sec-Section3}
As mentioned above, and discussed many years ago by Klein~\cite{Kl53},
the propagator $\langle G_0 \rangle$ is not invertible and therefore
the above reduction is not consistent. We shall show in this section
that this difficulty is connected to the behavior of the
three-dimensional equation in the one-body limit. In this limit we
allow one particle's mass to tend to infinity. We expect that the
amplitude $T_1$ then reduces to that given by the Dirac equation for a
light particle moving in the static field of the heavy particle. In
fact, this does not happen unless we include an infinite number of
graphs in the kernel of the integral equation Eq.~(\ref{eq:Salpeter}).
In fact, if a scattering equation with a kernel which contains only a
finite number of graphs is to possess the correct one-body limit, two
distinct criteria must be satisfied. First the 3D propagator should
limit to the one-body propagator for one particle (the Dirac
propagator in this case) as the other particle's mass tends to
infinity. Second, as either particle's mass tends to infinity, the
equation should become equivalent to one in which the interaction,
$K_1$, is static.
Equation~(\ref{eq:Salpeter})'s lack of either of these properties
stems from Eq.~(\ref{eq:BSE}) not having the correct one-body limit if
any kernel which does not include the infinite set of crossed-ladder
graphs is chosen~\cite{Gr82}. Solution of Eq.~(\ref{eq:BSE}) with such
a kernel is impractical in the $NN$ system. Nevertheless, the
contributions of crossed-ladder graphs to the kernel may be included
in an integral equation for $T$ by using a 4D integral equation for
$K$, the kernel of Eq.~(\ref{eq:BSE})
\begin{equation}
K = U + U G_C K.
\label{eq:U}
\end{equation}
Once $G_C$ is defined this equation defines a reduced kernel $U$ in
terms of the original kernel $K$. The propagator $G_C$ is chosen so
as to separate the parts of the kernel $K$ that are necessary to
obtain the one-body limit from the parts that are not. $U$ may then
be truncated at any desired order without losing the one-body limits.
The following 4D equation for the t-matrix is thus equivalent to
Eqs.~(\ref{eq:BSE}) and (\ref{eq:U}),
\begin{equation}
T = U + U (G_0 + G_C) T.
\label{eq:4DETampl}
\end{equation}
We can now remedy the defects of our previous 3D reduction. Applying
the same 3D reduction used above to Eq.~(\ref{eq:4DETampl}) gives:
\begin{equation}
T_1 = U_1 + U_1 \langle G_0 + G_C \rangle T_1,
\label{eq:3DETampl}
\end{equation}
where the 3D propagator is
\begin{eqnarray}
\langle G_0 + G_C \rangle&=&\frac{ \Lambda_1^+ \Lambda_2^+}{{P^0}^+
- \epsilon_1 - \epsilon_2} - \frac{ \Lambda_1^+ \Lambda_2^-}{2
\kappa_2^0 - {P^0}^+ + \epsilon_1 + \epsilon_2} \nonumber\\
&-& \frac{ \Lambda_1^- \Lambda_2^+}{{P^0}^- - 2 \kappa_2^0 + \epsilon_1
+ \epsilon_2} - \frac{ \Lambda_1^- \Lambda_2^-}{{P^0}^- +
\epsilon_1 + \epsilon_2},
\label{eq:aveG0GCgeneral}
\end{eqnarray}
and $\kappa_2^0$ is a parameter that enters through the construction
of $G_C$. This three-dimensional propagator was derived by Mandelzweig
and Wallace with the choice $\kappa_2^0 = P^0/2 - (m_1^2 -
m_2^2)/(2P^0)$~\cite{MW87,WM89}. With $\kappa_2^0$ chosen in this way
$\langle G_0 + G_C \rangle$ has the correct one-body limits as either
particle's mass tends to infinity and has an invertible form. The
kernel $U_1$ is defined by Eqs.~(\ref{eq:K1}) and (\ref{eq:calGeq})
with the replacements $G_0 \rightarrow G_0 + G_C$, $K \rightarrow U$,
and $K_1 \rightarrow U_1$.
Here we are interested in the scattering of particles of equal mass
and so we make a different choice for $\kappa_2^0$. Specifically,
\begin{equation}
\kappa_2^0 = \frac{P^0 - \epsilon_1 + \epsilon_2}{2}.
\label{eq:kappachoice}
\end{equation}
This form avoids the appearance of unphysical singularities when
electron-deuteron scattering is calculated~\cite{PW98}. It yields a
two-body propagator:
\begin{equation}
\langle G_0 + G_C \rangle=\frac{ \Lambda_1^+ \Lambda_2^+}{{P^0}^+
- \epsilon_1 - \epsilon_2} - \frac{ \Lambda_1^+ \Lambda_2^-}{2
\epsilon_2} - \frac{ \Lambda_1^- \Lambda_2^+}{2 \epsilon_1} -
\frac{ \Lambda_1^- \Lambda_2^-}{{P^0}^- + \epsilon_1 + \epsilon_2},
\label{eq:aveG0GC}
\end{equation}
which is consistent with that required by low-energy theorems for
Dirac particles in scalar and vector fields~\cite{Ph97}. Another way
of saying this is to realize that if we compare the the $++
\rightarrow ++$ piece of the amplitude
\begin{equation}
V_1 \langle G_0 + G_C \rangle V_1
\end{equation}
to the amplitude obtained at fourth order in the full 4D field theory
then the contribution of negative-energy states agrees at leading
order in $1/M$~\cite{PW98}.
For bound states the argument of the previous section leads to the 3D
equation:
\begin{equation}
\Gamma_1=U_1 \langle G_0 + G_C \rangle \Gamma_1.
\label{eq:3DET}
\end{equation}
Equation (\ref{eq:3DET}) is a bound-state equation which incorporates
relativistic effects and the physics of negative-energy states.
For instance, fig.~\ref{fig-Zgraph} is one example of a
graph which is included if Eq.~(\ref{eq:3DET}), even if only the
lowest-order kernel $U_1^{(2)}$ is used, because of our careful treatment
of the one-body limit.
\begin{figure}[h]
\centerline{\BoxedEPSF{fig1.eps scaled 350}}
\caption{One example of a Z-graph which is included in our
3D equation (\ref{eq:3DET}).}
\label{fig-Zgraph}
\end{figure}
\section{Results for the deuteron}
\label{sec-Section4}
To calculate observables in the deuteron we now consider two types of
kernels $U_1$, both of which are calculated within the framework
of a one-boson exchange model for the $NN$ interaction:
\begin{enumerate}
\item $U_1=U_{\rm inst}$, the instantaneous interaction.
\item A kernel $U_1^{(2)}$ which is a retarded interaction. This is
obtained from Eq.~(\ref{eq:K12}) by the substitutions $K_1^{(2)}
\rightarrow U_1^{(2)}$ and $G_0 \rightarrow G_0 + G_C$.
\end{enumerate}
These interactions are used in a two-body equation with the full ET
Green's function given by Eq.~(\ref{eq:3DET}), and also in an equation
in which only the $++$ sector is retained. For the instant
interaction, we follow the practice of Devine and Wallace~\cite{WD94}
and switch off couplings between the $++$ and $--$ sectors, and
between the $+-$ and $-+$ sectors. A partial justification of this
rule follows from an analysis of the static limit of our 3D retarded
interaction.
The mesons in our one-boson exchange model are the $\pi(138)$, the
$\sigma(550)$, the $\eta(549)$, the $\rho(769)$, the $\omega(782)$,
and the $\delta(983)$. All the parameters of the model, except for
the $\sigma $ coupling, are taken directly from the Bonn-B fit to the
$NN$ phase shifts~\cite{Ma89}---which is a fit performed using a
relativistic wave equation and relativistic propagators for the
mesons. The $\sigma$ coupling is varied so as to achieve the correct
deuteron binding energy for each interaction considered. Of course, we
should refit the parameters of our $NN$ interaction using our
different scattering equations. However, for a first estimate of the
importance of negative-energy states and retardation we adopt this
simpler approach to constructing the interaction. Work on improving
the $NN$ interaction model is in progress~\cite{PW98B}.
Once a particular interaction is chosen, the integral equation
(\ref{eq:3DET}) is solved for the bound-state energy. In each
calculation, the $\sigma$ coupling is adjusted to get the correct
deuteron binding energy, producing the results (accurate to three
significant figures) given in Table~\ref{table-sigmacoupling}. The
value given for the instant calculation with positive-energy states
alone is that found in the original Bonn-B fit. In all other cases the
$\sigma$ coupling must be adjusted to compensate for the inclusion of
retardation, the effects of negative-energy states, etc. We believe
that this adjustment of the scalar coupling strength is sufficient to
get a reasonable deuteron wave function. The static properties of this
deuteron are very similar to those of a deuteron calculated with the
usual Bonn-B interaction.
With the bound-state wave function in the center-of-mass frame has
been determined in this fashion, it is a simple matter to solve the
integral equation (\ref{eq:3DET}) in any other frame. We choose to
calculate electron-deuteron scattering in the Breit frame. The
interaction is recalculated in the Breit frame for a given $Q^2$, and
then the integral equation is solved with this new interaction.
Because the formalism we use for reducing the four-dimensional
integral equation to three dimensions is {\it not} Lorentz invariant
there is a violation of Lorentz invariance in this calculation.
Estimations of the degree to which Lorentz invariance is violated are
displayed in Ref.~\cite{PW98}.
\begin{table}[htbp]
\caption{Sigma coupling required to produce the correct deuteron binding
energy in the four different models under consideration here.}
\label{table-sigmacoupling}
\begin{center}
\begin{tabular}{|c|c|c|}
\hline
{\bf Interaction} & {\bf States included} & {\bf $g_\sigma^2/4 \pi$} \\
\hline
Instant & ++ & 8.08 \\ \hline
Retarded & ++ & 8.39 \\ \hline
Instant & All & 8.55 \\ \hline
Retarded & All & 8.44 \\ \hline
\end{tabular}
\end{center}
\end{table}
\section{Current conservation}
\label{sec-Section5}
\subsection{Currents in the three-dimensional formalism}
\label{sec-3Dcurrents}
As discussed in the Introduction, we now want to compare the predictions
of this formalism with experimental data gained in electron scattering
experiments. In calculating the interaction of the electron
with the hadronic bound state it is crucial to derive a
3D reduction of the electromagnetic current which is consistent with
the reduction of the scattering equation we have chosen to use here.
The current in the full four-dimensional formalism is obtained by
coupling photons everywhere on the right-hand side of
Eq.~(\ref{eq:BSE}). This produces the following gauge-invariant
result for the photon's interaction with the bound state:
\begin{eqnarray}
{\cal A}_\mu&=&\bar{\Gamma}(P') G_0(P') J_\mu G_0 (P) \Gamma(P) \nonumber\\
&+& \bar{\Gamma}(P') G_0(P') K_\mu^\gamma G_0 (P) \Gamma(P),
\label{eq:gi4damp}
\end{eqnarray}
where $P$ and $P'$ are the initial and final total four-momenta of the
deuteron bound state. Here $J_\mu$ contains the usual one-body
currents and $K_\mu^\gamma$ represents two-body contributions which
are necessary for maintaining the Ward-Takahashi identities. All
integrals implicitly are four-dimensional. The connection to the
three-dimensional amplitude, $\Gamma_1$, obtained from
Eq.~(\ref{eq:3DET}) is made by inserting Eq.~(\ref{eq:GcalGamma1})
into Eq.~(\ref{eq:gi4damp}), giving
\begin{equation}
{\cal A}_\mu=\bar{\Gamma}_1(P') \langle {\cal G}(P')
\left[J_\mu + K_\mu^\gamma \right] {\cal G}(P) \rangle \Gamma_1(P).
\label{eq:Amu}
\end{equation}
Once the effective operator $\langle {\cal G}(P') \left[J_\mu +
K_\mu^\gamma \right] {\cal G}(P) \rangle$ is calculated the
expression (\ref{eq:Amu}) involves only three-dimensional integrals.
Since ${\cal G}$ is an infinite series in $K-K_1$ this result would
not be much help on its own. But, given a result for $\Gamma_1$
obtained by systematic expansion of $K_1$, the amplitude ${\cal
A}_\mu$ can be analogously expanded in a way that maintains current
conservation. $K_1$ as defined by Eq.~(\ref{eq:K1}) is an infinite
series and the condition (\ref{eq:calGeq}) is imposed order-by-order
in the expansion in $K-K_1$ defines $K_1$ to some finite order. The
question is: Does a corresponding 3D approximation for the current
matrix element (\ref{eq:Amu}) exist that maintains the Ward-Takahashi
identities of the theory? {\it It turns out that the current matrix
element (\ref{eq:Amu}) is conserved if ${\cal G} (J_\mu +
K^\gamma_\mu) {\cal G}$ on the right-hand side of Eq.~(\ref{eq:Amu})
is expanded to a given order in the coupling constant and the kernel
$K_1$ used to define $\Gamma_1$ is obtained from
Eq.~(\ref{eq:calGeq}) by truncation at the same order in the
coupling constant.}
This is done by splitting the right-hand side of Eq.~(\ref{eq:Amu})
into two pieces, one due to the one-body current $J_\mu$, and one due
to the two-body current $K^\gamma_\mu$. If $K_1$ has been truncated at
lowest order---i.e., $K_1=K_1^{(2)}$---then, in the $J_\mu$ piece, we
expand the $\cal G$s and retain terms up to the same order in
$K^{(2)}-K_1^{(2)}$. A piece from the two-body current, in which we
write ${\cal G}=G_0$, is added to this. That is, we define our
second-order approximation to ${\cal A}_\mu$, ${\cal A}_\mu^{(2)}$, by
\begin{eqnarray}
{\cal A}^{(2)}_\mu&=&\bar{\Gamma}_1(P') \langle G^\gamma_{0 \mu}
\rangle \Gamma_1(P) \nonumber \\ &+& \bar{\Gamma}_1(P') \langle G_0(P') (K^{(2)}(P')-K^{(2)}_1(P'))
G^\gamma_{0 \mu} \rangle \Gamma_1(P)
\nonumber\\
&+& \bar{\Gamma}_1(P') \langle G^{\gamma}_{0 \mu} (K^{(2)}(P)-K^{(2)}_1(P)) G_0(P) \rangle
\Gamma_1(P)
\nonumber\\
&+& \bar{\Gamma}_1(P') \langle G_0(P') K^{\gamma (2)}_\mu G_0(P) \rangle
\Gamma_1(P),
\label{eq:A2mu}
\end{eqnarray}
where ${G_0^\gamma}_\mu=G_0(P') J_\mu G_0(P)$. It can now be shown
that if Eq.~(\ref{eq:calGeq}) expanded to second order defines
$K_1^{(2)}$, the corresponding amplitude for electromagnetic
interactions of the bound state, as defined by Eq.~(\ref{eq:A2mu}),
exactly obeys
\begin{equation}
Q^\mu {\cal A}^{(2)}_\mu=0.
\label{eq:A1WTI}
\end{equation}
It is straightforward to check that the same result holds if
Eq.~(\ref{eq:calGeq}) for $K_1$ is truncated at fourth order, while
the one-body and two-body current pieces are expanded to fourth order.
The amplitude ${\cal A}_\mu^{(2)}$ includes contributions from
diagrams where the photon couples to particles one and two while
exchanged quanta are ``in-flight''. These contributions are of two
kinds. Firstly, if the four-dimensional kernel $K$ is dependent on
the total momentum, or if it involves the exchange of charged
particles, then the WTIs in the 4D theory require that $K_\mu^\gamma$
contain terms involving the coupling of the photon to internal lines
in $K$. Secondly, even if such terms are not present, terms arise in
the three-dimensional formalism where the photon couples to particles
one and two while an exchanged meson is ``in-flight''. These must be
included if our 3D approach is to lead to a conserved current. (See
Fig.~\ref{fig-inflight} for one such mechanism.)
\begin{figure}[h]
\centerline{\BoxedEPSF{fig2.eps scaled 350}}
\caption{One example of a two-body current that is required in our formalism
in order to maintain current conservation.}
\label{fig-inflight}
\end{figure}
A special case of the above results occurs when retardation effects
are omitted, i.e., the kernel $K_1=K_{\rm inst}$, is chosen, and the
bound-state equation (\ref{eq:3Deqn}) is solved to get the vertex
function $\Gamma_1=\Gamma_{\rm inst}$. Then a simple conserved current
is found:
\begin{equation}
{\cal A}_{{\rm inst},\mu}=\bar{\Gamma}_{\rm inst}(P') \langle
G_{0 \mu}^\gamma \rangle \Gamma_{\rm inst}(P) + \bar{\Gamma}_{\rm
inst}(P') \langle G_0(P') \rangle {K^\gamma_{\rm inst}}_\mu
\langle G_0(P) \rangle \Gamma_{\rm inst}(P),
\label{eq:instantme}
\end{equation}
where we have also replaced the meson-exchange current kernel
$K^\gamma_\mu$ by the instant approximation to it.
\subsection{Current conservation in the 4D formalism with $G_C$}
In Ref.~\cite{PW98} we showed how to construct a conserved current
consistent with the 4D equation
\begin{equation}
\Gamma=U (G_0 + G_C) \Gamma.
\label{eq:4DET}
\end{equation}
This turns out to be a moderately complicated exercise, because the
propagator $G_C$ depends on the three-momenta of particles one and
two, not only in the usual way, but also through the choice
(\ref{eq:kappachoice}) made for $\kappa_2^0$ above. However, a 4D
current ${\cal G}_{0,\mu}^\gamma = {G_0^\gamma}_\mu +
{G_C^\gamma}_\mu$ corresponding to the free Green's function $G_0 +
G_C$ can be constructed. Its form is displayed in Ref.~\cite{PW98} and
is not really germane to our purposes here, for, as we shall see
hereafter, only certain pieces of the current ${\cal
G}_{0,\mu}^\gamma$ are actually used in our calculations.
\subsection{Reduction to 3D and the ET current}
Having constructed a 4D current for the formalism involving $G_C$ that
obeys the required Ward-Takahashi identity, we can apply the reduction
formalism of Section~\ref{sec-3Dcurrents} to obtain the currents
corresponding to the 3D reduction of this 4D theory. The result is:
\begin{eqnarray}
{\cal A}^{(2)}_\mu&=&\bar{\Gamma}_{1,{\rm ET}}(P') \langle {\cal
G}^\gamma_{0,\mu} \rangle \Gamma_{1,{\rm ET}}(P) \nonumber\\
&+& \bar{\Gamma}_{1,{\rm ET}}(P') \langle
(G_0 + G_C)(P') (K^{(2)}(P')-U_1^{(2)}(P')) {\cal G}^\gamma_{0,\mu}
\rangle \Gamma_{1,{\rm ET}}(P)
\nonumber\\
&+& \bar{\Gamma}_{1,{\rm ET}}(P') \langle {\cal G}^{\gamma}_{0,\mu}
(K^{(2)}(P)-U_1^{(2)}(P)) (G_0 + G_C)(P) \rangle \Gamma_{1,{\rm ET}}(P)
\nonumber\\
&+& \bar{\Gamma}_{1,{\rm ET}}(P') \langle (G_0 + G_C)(P') K^{\gamma (2)}_\mu
(G_0 + G_C)(P) \rangle \Gamma_{1,{\rm ET}}(P),
\label{eq:general3DETcurrent(2)}
\end{eqnarray}
where $\Gamma_{1,{\rm ET}}$ is the solution of Eq.~(\ref{eq:3DET}) with
$U_1=U_1^{(2)}$.
This current obeys the appropriate Ward-Takahashi identity. In fact in
one-boson exchange models the only contributions to $K^{\gamma
(2)}_\mu$ give rise to isovector structures, and so their
contribution to electromagnetic scattering off the deuteron is zero.
\subsection{Impulse-approximation current based on the instant
approximation to ET formalism}
Just as in the case of the Bethe-Salpeter equation, if the instant
approximation is used to obtain a bound-state equation with an instant
interaction from Eq.~(\ref{eq:4DET}) then a corresponding simple
conserved impulse current can be constructed:
\begin{equation}
{\cal A}_{{\rm inst},\mu}=\bar{\Gamma}_{\rm inst}
\langle {\cal G}^\gamma_{0,\mu} \rangle
\Gamma_{\rm inst}.
\end{equation}
Now we note that the full result for ${\cal G}^\gamma_{0,\mu}$ was
constructed in order to obey Ward-Takahashi identities in the full
four-dimensional theory. It is not necessary to use this result if we
are only concerned with maintaining WTIs at the three-dimensional
level in the instant approximation. Therefore we may construct the
corresponding current
\begin{eqnarray}
&& {\cal G}_{{\rm inst},\mu}^\gamma({\bf p}_1,{\bf p}_2;P,Q)=i \langle d_1(p_1)
d_2(p_2+Q) j^{(2)}_\mu d_2(p_2) \nonumber\\
&& \qquad + d_1(p_1) d_2^{\tilde{c}}(p_2+Q) j^{(2)}_{c,\mu} d_2^c(p_2) \rangle
+ (1 \leftrightarrow 2).
\end{eqnarray}
Here $d_i$ is the Dirac propagator for particle $i$, and $j_\mu=q
\gamma_\mu$ is the usual one-body current, with $q$ is the charge of
the particle in question. Meanwhile $d_i^c$ is a one-body Dirac
propagator used in $G_C(P)$ to construct the approximation to the
crossed-ladder graphs. Correspondingly, $d_i^{\tilde{c}}$ appears in
$G_C(P+Q)$, which does {\it not} equal $d_i^c$, even if particle $i$
is not the nucleon struck by the photon. Finally,
\begin{equation}
j^{(2)}_{c,\mu}= q_2 \gamma_\mu - \tilde{j}^{(2)}_\mu,
\end{equation}
where
\begin{equation}
\tilde{j}^{(2)}_\mu=q_2 \frac{\hat{p}_{2 \mu}' + \hat{p}_{2
\mu}}{\epsilon_2' + \epsilon_2} {\gamma_{2}}_0,
\label{eq:tildej2}
\end{equation}
with $\hat{p}_2=(\epsilon({\bf p}_2),{\bf p}_2)$. (For further
explanation of these quantities and the necessity of their appearance
here the reader is referred to Ref.~\cite{PW98}.)
If a vertex function $\Gamma_{{\rm inst}}$ is constructed to be a
solution to Eq.~(\ref{eq:3DET}) with an instant interaction then the
three-dimensional hadronic current:
\begin{equation}
{\cal A}_{{\rm inst},\mu}= \bar{\Gamma}_{\rm inst} {\cal G}_{{\rm inst},\mu}^\gamma \Gamma_{\rm inst}
\label{eq:instAmu}
\end{equation}
is conserved. This current is simpler than the full ET current and
omits only effects stemming from retardation in the current. Our
present calculations are designed to provide an assessment of the role
of negative-energy states and retardation effects in the vertex
functions. Therefore we use the simple current (\ref{eq:instAmu}) in
{\it all} of our calculations here---even the ones where $\Gamma_1$ is
calculated using a retarded two-body interaction. The effects stemming
from retardation in the current are expected to be minor, and so we
expect this to be a good approximation to the full current in the
three-dimensional theory. Future calculations should be performed to
check the role of meson retardation in that current.
\section{Results for electron-deuteron scattering}
\label{sec-Section6}
\subsection{Impulse approximation}
We are now ready to calculate the experimentally observed deuteron
electromagnetic form factors $A$ and $B$, and the tensor polarization
$T_{20}$. These are straightforwardly related to the charge,
quadrupole, and magnetic form factors of the deuteron, $F_C$, $F_Q$,
and $F_M$. These form factors in turn are related to the Breit frame
matrix elements of the current ${\cal A}_\mu$ discussed in the
previous section,
\begin{eqnarray}
F_C&=&\frac{1}{3\sqrt{1 + \eta}e} (\langle 0|{\cal A}^0| 0 \rangle
+ 2 \langle +1|{\cal A}^0|+1 \rangle),\\
F_Q&=&\frac{1}{2 \eta \sqrt{1 + \eta} e} (\langle 0|{\cal A}^0| 0 \rangle
- \langle +1|{\cal A}^0|+1 \rangle),\\
F_M&=& \frac{-1}{\sqrt{2 \eta (1 + \eta)}e} \langle +1|{\cal A}_+|0\rangle,
\end{eqnarray}
where $|+1 \rangle$, $|0 \rangle$ and $|-1 \rangle$ are the
three different spin states of the deuteron.
We take the wave functions constructed for the four different
interactions of Section~\ref{sec-Section4} and insert them into the
expression (\ref{eq:instAmu}). In using any of the interactions
obtained with only positive-energy state propagation we drop all
pieces of the operator ${\cal G}^\gamma_{{\rm inst},\mu}$ in
negative-energy sectors.
The single-nucleon current used in these calculations is the usual one
for extended nucleons. We choose to parametrize the single-nucleon
form factors $F_1$ and $F_2$ via the 1976 Hohler fits~\cite{Ho76}.
Choosing different single-nucleon form factors does not affect our
qualitative conclusions, although it has some impact on our
quantitative results for $A$, $B$, and $T_{20}$.
Using this one-body current we then calculate the current matrix
elements via Eq.~(\ref{eq:Amu}). This is a conserved current if the
vertex function $\Gamma_1$ is calculated from an instant potential.
However, if a potential including meson retardation is used it
violates the Ward-Takahashi identities by omission of pieces that are
required because of the inclusion of retardation effects in the
calculation. Work is in progress to estimate the size of these
effects.
\begin{figure}[htbp]
\centerline{\BoxedEPSF{IA.eps scaled 500}}
\caption{The form factors $A(Q^2)$ and $B(Q^2)$ and the tensor
polarization $T_{20}$ for the deuteron calculated in impulse
approximation. The dash-dotted line represents a calculation using a
vertex function generated using the instant interaction. Meanwhile
the solid line is the result obtained with the retarded vertex
function. The dotted and long dashed lines are obtained by
performing a calculations with instant and retarded interactions in which
no negative-energy states are included.}
\label{fig-IA}
\end{figure}
The results for the impulse approximation calculation of the
experimental observables $A$, $B$, and $T_{20}$ are displayed in
Fig.~\ref{fig-IA}. We also show experimental data from
Refs.~\cite{El69,Ar75,Si81,Cr85,Pl90} for $A$, from
Refs.~\cite{Si81,Cr85,Au85,Bo90} for $B$ and from
Ref.~\cite{Sc84} for $T_{20}$. A number of
two-body effects must be added to our calculations before they can be
reliably compared to experimental data. However, even here we see the
close similarity of the results for these observables in all four
calculations. The only really noticeable difference occurs at the
minimum in $B$. There, including the negative-energy states in the
calculation shifts the minimum to somewhat larger $Q^2$. A similar
effect was observed by van Orden {\it et al.}~\cite{vO95} in
calculations of electron-deuteron scattering using the spectator
formalism. However, note that here, in contradistinction to the
results of Ref.~\cite{vO95}, the inclusion of negative-energy states
does {\it not} bring the impulse approximation calculation into
agreement with the data.
The fact that negative-energy states seem to have a smaller effect on
observables in the ET analysis than in the spectator analysis of van
Orden et al.~\cite{vO95} is somewhat surprising since our ``ET''
propagator has twice the negative-energy state propagation amplitude
of the spectator propagator. Thus, other differences between the ET and
spectator models, not just differences in the role of negative-energy
states in the two approaches, appear to be responsible for Ref.~\cite{vO95}'s
success in reproducing the minimum in $B$.
For the tensor polarization $T_{20}$ the different models produce
results which are very similar. This suggests that this observable is
fairly insensitive to dynamical details of the deuteron model, at
least up to $Q^2=4 \,\, \rm{GeV}^2$.
\subsection{Meson-exchange currents}
As $Q^2$ increases the cross-section due to the impulse approximation
diagrams drops precipitously. Thus we expect that in some regime other
interactions may become competitive with the impulse mechanism. One
such possibility is that the photon will couple to a meson while that
meson is in flight. Because of the deuteron's isoscalar nature and
the conservation of G-parity, the lowest mass state which can
contribute in such meson-exchange current (MEC) diagrams is one where
the photon induces a transition from a $\pi$ to a $\rho$.
This $\rho \pi \gamma$ MEC is a conserved current whose structure can
be found in Refs.~\cite{HT89,De94}. The couplings and form factors for
the meson-nucleon-nucleon vertices are all taken to be consistent with
those used in our one-boson-exchange interaction. Meanwhile, the $\rho
\pi \gamma$ coupling is set to the value $g_{\rho \pi \gamma}=0.56$,
and a vector meson dominance form factor is employed at the $\rho \pi
\gamma$ vertex: $F_{\rho \pi \gamma}(q)=1/(q^2 - m_\omega^2)$. The
value of this MEC is added to the impulse contribution calculated
above and $A$, $B$, and $T_{20}$ are calculated. This is done with the
vertex function obtained from an instant interaction, and consequently
the electromagnetic current is exactly conserved. The results of this
calculation are displayed in Fig.~\ref{fig-RPG}. We see that at $Q^2$
of order 2 ${\rm GeV}^2$ the $\rho \pi \gamma$ MEC makes a significant
contribution to all three observables. However, far from improving the
agreement of the position of the minimum in the $B$ form factor with
the experimental data, this particular MEC moves the theoretical result
{\it away} from the data---as noted by Hummel and Tjon~\cite{HT89},
and seen within a simplified version of the formalism presented here
by Devine~\cite{De94}. Thus, it would seem that some physics beyond
the impulse approximation other than the $\rho \pi \gamma$ MEC plays a
significant role in determining the position of the minimum in
$B(Q^2)$.
\begin{figure}[htbp]
\centerline{\BoxedEPSF{RPG.eps scaled 500}}
\caption{The form factors $A(Q^2)$ and $B(Q^2)$ together with the
tensor polarization for the deuteron. The long dashed line is an
impulse approximation calculation with an instant interaction. The
solid line includes the effect of the $\rho \pi \gamma$
MEC.}
\label{fig-RPG}
\end{figure}
\section{Conclusion}
\label{sec-Section7}
A systematic theory of the electromagnetic interactions of
relativistic bound states is available in three dimensions. In this
formalism integrations are performed over the zeroth component of the
relative momentum of the two particles, leading to the construction of
``equal-time'' (ET) Green's functions. If the formalism is to
incorporate the Z-graphs that are expected in a quantum field theory,
then the propagator must include terms coming from crossed Feynman
graphs. Here we have displayed a three-dimensional propagator that
includes these effects correctly to leading order in $1/M$.
Given a suitable choice for the ET propagator, the electromagnetic and
interaction currents which should be used with it can be calculated.
If these are truncated in a fashion consistent with the truncation of
the $NN$ interaction in the hadronic field theory then the
Ward-Takahashi identities are maintained in the three-dimensional
theory. A full accounting of the dynamical role played by
negative-energy states and of retardations in electromagnetic
interactions of the deuteron is thereby obtained.
Calculations have been performed for both the impulse approximation
and when the $\rho \pi \gamma$ MEC is included. In our MEC
calculations we use an instant approximation for the electromagnetic
current. This current satisfies current conservation when used with
deuteron vertex functions that are calculated with instant interactions.
We also have used this simpler current with vertex functions which
are calculated with the retarded interactions obtained within the
ET formalism.
Comparing impulse approximation calculations with and without
negative-energy states indicates that the role played by
negative-energy state components of the deuteron vertex function is
small. This corroborates the results of Hummel and Tjon and is in
contrast to those obtained in Ref.~\cite{vO95}. Because the ET
formalism incorporates the relevant Z-graphs in a preferable way, we
are confident that these Z-graphs really do play only a minor role in
calculations that are based upon standard boson-exchange models of the
$NN$ interaction.
The results for impulse approximation calculations of the
electromagnetic observables are relatively insensitive to the
distinction between a vertex calculated with retardations included and
one calculated in the instantaneous approximation. The results of both
calculations fall systematically below experimental data for the form
factors $A$ and $B$ for $Q$ of order 1 GeV. This deficiency at higher
$Q$ suggests that mechanisms other than the impulse approximation
graph should be significant. Indeed, when the $\rho \pi \gamma$ MEC
graph is included in our calculation it somewhat remedies the result
for $A(Q^2)$. However, it fails to narrow the gap between our result
for $B(Q^2)$ and the existing experimental data. The significant gap
that remains between our theoretical result for $B(Q^2)$ and the data
indicates that it is an interesting observable in which to look for
physics of the deuteron other than the simple impulse mechanism or the
standard $\rho \pi \gamma$ MEC. Finally, the existing tensor
polarization data are reasonably well described. This is consistent
with previous analyses which have shown $T_{20}$ to be less sensitive
to non-impulse mechanisms.
\section*{Acknowledgments}
It is a pleasure to thank Steve Wallace for a fruitful and enjoyable
collaboration on this topic, and for his comments on this manuscript.
I am also very grateful to Neal Devine for giving us the original
version of the computer code to calculate these reactions, and to
Betsy Beise for useful information on the experimental situation.
Finally, I want to thank the organizers of this workshop for a
wonderful week of physics in Elba! This work was supported by the U.~S.
Department of Energy under grant no. DE-FG02-93ER-40762.
| 2024-02-18T23:40:13.696Z | 1998-11-03T05:53:17.000Z | algebraic_stack_train_0000 | 1,738 | 7,018 |
|
proofpile-arXiv_065-8506 | \section{Introduction}
Superstring theories are powerful candidates for the unification
theory of all forces including gravity.
The supergravity theory (SUGRA) is effectively constructed from
4-dimensional
(4D) string model using several methods \cite{ST-SG,OrbSG,OrbSG2}.
The structure of SUGRA is constrained
by gauge symmetries including an anomalous $U(1)$ symmetry
($U(1)_A$) \cite{ST-FI} and stringy symmetries
such as duality \cite{duality}.
4D string models have several open questions and
two of them are pointed out here.
The first one is what the origin of supersymmetry (SUSY) breaking is.
Although intersting scenarios
such as SUSY breaking mechanism due to gaugino condensation \cite{gaugino}
and Scherk-Schwarz mechanism \cite{SS}
have been proposed, realistic one has not been identified yet.
The second one is how the vacuum expectation value (VEV) of dilaton
field $S$ is stabilized.
It is difficult to realize the stabilization with a realistic VEV of $S$
using a K\"ahler potential at the tree level alone
without any conspiracy among several terms which
appear in the superpotential \cite{S-Stab}.
A K\"ahler potential generally receives radiative
corrections as well as non-perturbative ones.
Such corrections may be sizable for the part related to
$S$ \cite{corr1,corr2}.
It is important to solve these enigmas
in order not only to understand the structure of more fundamental theory
at a high energy scale but also to know the complete SUSY particle spectrum
at the weak scale, but it is not an easy task
because of ignorance of the explicit forms of
fully corrected total K\"ahler potential.
At present, it would be meaningful to get any information on SUSY
particle spectrum model-independently.\footnote{
The stability of $S$ and soft SUSY breaking parameters are discussed
in the dilaton SUSY breaking scenario in Ref.\cite{Casas}.}
In this paper, we study the magnitudes of soft SUSY breaking parameters
in heterotic string models with $U(1)_A$ and derive model-independent
predictions for them
without specifying SUSY breaking mechanism
and the dilaton VEV fixing mechanism.
The idea is based on that in the work by Ref.\cite{ST-soft}.
The soft SUSY breaking terms have been derived from
$\lq\lq$standard string model" and analyzed
under the assumption that SUSY is broken by $F$-term condensations
of the dilaton field and/or moduli fields $M^i$.
We relax this assumption such that
SUSY is broken by $F$-term condensation of $S$, $M^i$ and/or
matter fields with non-vanishing $U(1)_A$ charge
since the scenario based on $U(1)_A$ as a mediator
of SUSY breaking is also possible \cite{DP}.
In particular, we make a comparison of magnitudes between
$D$-term contribution to scalar masses and $F$-term ones and
a comparison of magnitudes among scalar masses, gaugino masses
and $A$-parameters.
The features of our analysis are as follows.
The study is carried out in the framework of SUGRA
model-independently,\footnote{The model-dependent analyses are carried out
in Ref.\cite{DP,model-dep1,model-dep2}.}
i.e., we do not specify SUSY breaking mechanism, extra matter contents,
the structure of superpotential and
the form of K\"ahler potential related to $S$.
We treat all fields including $S$ and $M^i$ as dynamical fields.
The paper is organized as follows.
In the next section, we explain the general structure of SUGRA briefly
with some basic assumptions of SUSY breaking.
We study the magnitudes of soft SUSY breaking parameters
in heterotic string models with $U(1)_A$ model-independently in section 3.
Section 4 is devoted to conclusions and some comments.
\section{General structure of SUGRA}
We begin by reviewing the scalar potential in SUGRA \cite{SUGRA,JKY}.
It is specified by two functions, the total K\"ahler
potential $G(\phi, \bar \phi)$ and the gauge kinetic function
$f_{\alpha \beta}(\phi)$ with $\alpha$, $\beta$ being indices of
the adjoint representation of the gauge group.
The former is a sum of the K\"ahler potential $K(\phi, \bar \phi)$
and (the logarithm of) the superpotential $W(\phi)$
\begin{equation}
G(\phi, \bar \phi)=K(\phi, \bar \phi) +M^{2}\ln |W (\phi) /M^{3}|^2
\label{total-Kahler}
\end{equation}
where $M=M_{Pl}/\sqrt{8\pi}$ with $M_{Pl}$ being the Planck mass, and is
referred to as the gravitational scale.
We have denoted scalar fields in the chiral multiplets by $\phi^I$
and their complex conjugate by $\bar \phi_J$.
The scalar potential is given by
\begin{eqnarray}
V &=& M^{2}e^{G/M^{2}} (G_I (G^{-1})^I_J G^J-3M^{2})
+ \frac{1}{2} (Re f^{-1})_{\alpha \beta} \hat D^{\alpha} \hat D^{\beta}
\label{scalar-potential}
\end{eqnarray}
where
\begin{eqnarray}
\hat D^\alpha = G_I ( T^\alpha \phi)^I = (\bar \phi T^\alpha)_J G^J.
\label{hatD}
\end{eqnarray}
Here $G_I=\partial G/\partial \phi^I$,
$G^J=\partial G/\partial \bar \phi_J$ etc, and
$T^\alpha$ are gauge transformation generators.
Also in the above, $(Re f^{-1})_{\alpha \beta}$ and
$(G^{-1})^I_J$ are the inverse matrices of $Re f_{\alpha \beta}$ and
$G^I_J$, respectively, and a summation over $\alpha$,... and $I$,... is
understood.
The last equality in Eq.(\ref{hatD}) comes from the
gauge invariance of the total K\"ahler potential.
The $F$-auxiliary fields of the chiral multiplets are given by
\begin{equation}
F^I =Me^{G/2M^{2}} (G^{-1})^I_J G^J.
\label{F}
\end{equation}
The $D$-auxiliary fields of the vector multiplets are given by
\begin{equation}
D^{\alpha} = (Re f^{-1})_{\alpha\beta} \hat{D}^{\beta}.
\label{D}
\end{equation}
Using $F^I$ and $D^\alpha$, the scalar potential is rewritten down by
\begin{eqnarray}
V &=& V_F + V_D , \nonumber \\
V_F &\equiv& F_I K^I_J F^J - 3M^{4} e^{G/M^{2}} ,
\label{VF}\\
V_D &\equiv& \frac{1}{2} Re f_{\alpha \beta} D^{\alpha} D^{\beta}.
\label{scalar-potential 2}
\end{eqnarray}
Let us next summarize our assumptions on SUSY breaking.
The gravitino mass $m_{3/2}$ is given by
\begin{equation}
m_{3/2}= \langle Me^{G/2M^{2}} \rangle
\label{gravitino}
\end{equation}
where $\langle \cdots \rangle$ denotes the VEV.
As a phase convention, it is taken to be real.
We identify the gravitino mass with the weak scale in most cases.
It is assumed that SUSY is spontaneously broken by some $F$-term condensations
($\langle F \rangle \neq 0$) for singlet fields under
the standard model gauge group
and/or some $D$-term condensations ($\langle D \rangle \neq 0$)
for broken gauge symmetries.
We require that the VEVs of $F^I$ and $D^{\alpha}$ should satisfy
\begin{eqnarray}
&~& \langle (F_I K^I_J F^J)^{1/2} \rangle \leq O(m_{3/2}M) ,
\label{VEV-F} \\
&~& \langle D^\alpha \rangle \leq O(m_{3/2}M)
\label{VEV-D}
\end{eqnarray}
for each pair $(I,J)$ in Eq.(\ref{VEV-F}).
Note that we allow the non-zero vacuum energy $\langle V \rangle$
of order $m_{3/2}^2 M^2$ at this level, which could be canceled
by quantum corrections.
In order to discuss the magnitudes of several quantities, it is
necessary to see consequences of the stationary condition
$\langle \partial V /\partial \phi^I \rangle =0$. From
Eq.(\ref{scalar-potential}), we find
\begin{eqnarray}
\partial V /\partial \phi^I &=& G_I ( {V_F \over M^2}
+ M^{2}e^{G/M^{2}} ) + M e^{G/2M^{2}} G_{IJ} F^J \nonumber \\
&~& - F_{I'} G^{I'}_{J'I} F^{J'}
- \frac{1}{2} (Re f_{\alpha \beta}),_I D^\alpha D^\beta
\nonumber \\
&~& + D^\alpha (\bar \phi T^\alpha )_J G^J_I .
\label{VI}
\end{eqnarray}
Taking its VEV and using the stationary condition, we derive the formula
\begin{eqnarray}
m_{3/2} \langle G_{IJ} \rangle \langle F^J \rangle &=&
- \langle G_I \rangle ( {\langle V_F \rangle \over M^2} + m_{3/2}^2 )
+ \langle F_{I'} \rangle \langle G^{I'}_{J'I} \rangle
\langle F^{J'} \rangle
\nonumber \\
&~& + \frac{1}{2} \langle (Re f_{\alpha \beta}),_I \rangle
\langle D^\alpha \rangle \langle D^\beta \rangle
- \langle D^\alpha \rangle \langle (\bar \phi T^\alpha )_J \rangle
\langle G^J_I \rangle .
\label{<VI>}
\end{eqnarray}
We can estimate the magnitude of SUSY mass parameter
$\mu_{IJ} \equiv m_{3/2} (\langle G_{IJ} \rangle + \langle G_{I} \rangle
\langle G_{J} \rangle/M^2 - \langle G_{I'} (G^{-1})^{I'}_{J'}
G^{J'}_{IJ} \rangle)$ using Eq.(\ref{<VI>}).
By multiplying $(T^\alpha \phi)^I$ to Eq.(\ref{VI}),
a heavy-real direction is projected on.
Using the identities derived from the gauge
invariance of the total K\"ahler potential
\begin{eqnarray}
& & G_{IJ}(T^\alpha \phi)^J + G_J (T^\alpha )_I^J
- K^J_I(\bar \phi T^\alpha)_J = 0,
\\
& & K_{IJ'}^J (T^\alpha \phi)^{J'} + K^J_{J'} (T^\alpha)^{J'}_I
-[G^{J'} (\bar \phi T^\alpha )_{J'}]_I^J = 0,
\end{eqnarray}
we obtain
\begin{eqnarray}
{\partial V \over \partial \phi^I} (T^\alpha \phi)^I &=&
( {V_F \over M^2} + 2 M^2 e^{G/M^2} ) \hat D^\alpha
- F_I F^J (\hat{D}^{\alpha})^I_J
\nonumber \\
&~& - \frac{1}{2} (Re f_{\beta \gamma}),_I (T^\alpha \phi)^I
D^\beta D^\gamma
+ (\bar \phi T^\beta)_J G^J_I (T^\alpha \phi)^I D^\beta .
\label{VI2}
\end{eqnarray}
Taking its VEV and using the stationary condition,
we derive the formula
\begin{eqnarray}
&~& \{ {(M_V^2)^{\alpha\beta} \over 2g_\alpha g_\beta}
+ ({\langle V_F \rangle \over M^2} + 2 m_{3/2}^2)
\langle Re f_{\alpha\beta} \rangle \} \langle D^\beta \rangle
= \langle F_I \rangle \langle F^J \rangle
\langle (\hat{D}^{\alpha})^I_J \rangle \nonumber \\
&~& ~~~~~~~~~~~~~~~~~~~~~~~~~~~~
+ \frac{1}{2} \langle (Re f_{\beta \gamma}),_I \rangle
\langle (T^\alpha \phi)^I \rangle
\langle D^\beta \rangle \langle D^\gamma \rangle
\label{<VI2>}
\end{eqnarray}
where $(M_{V}^{2})^{\alpha \beta}=
2g_\alpha g_\beta \langle (\bar \phi T^\beta)_J K^J_I
(T^\alpha \phi)^I \rangle$
is the mass matrix of the gauge bosons and
$g_\alpha$ and $g_\beta$ are the gauge coupling constants.
Using Eq.(\ref{<VI2>}), we can estimate the magnitude of
$D$-term condensations $\langle D^\beta \rangle$.
Using the scalar potential and gauge kinetic terms,
we can obtain formulae of soft
SUSY breaking scalar masses $(m^2)_I^J$, soft SUSY breaking
gaugino masses $M_{\alpha}$ and $A$-parameters $A_{IJK}$ \cite{KMY2,Kawa1},
\begin{eqnarray}
(m^2)^J_I &=& (m^2_F)^J_I + (m^2_D)^J_I , \\
(m^2_F)^J_I &\equiv& (m_{3/2}^2 + {\langle V_F \rangle \over M^2})
\langle K^J_I \rangle \nonumber \\
& & + \langle {F}^{I'} \rangle
\langle ({K}_{I'I}^{I"} (K^{-1})^{J"}_{I"}
{K}^{JJ'}_{J"} - {K}_{I'I}^{J'J}) \rangle \langle {F}_{J'} \rangle + \cdots
\label{mF} ,\\
(m^2_D)^J_I &\equiv& \sum_{\hat{\alpha}} q^{\hat{\alpha}}_{I}
\langle D^{\hat{\alpha}} \rangle \langle K^J_I \rangle ,
\label{mD} \\
M_\alpha &=& \langle F^I \rangle \langle (Re f_\alpha)^{-1} \rangle
\langle f_\alpha,_I \rangle
\label{Ma} ,\\
A_{IJK} &=& \langle F^{I'} \rangle (\langle f_{IJK},_{I'} \rangle
+ {\langle K_{I'} \rangle \over M^2} \langle f_{IJK} \rangle
\nonumber \\
&~& - \langle K_{(II'}^{J'} \rangle \langle (K^{-1})_{J'}^{J"} \rangle
\langle f_{J"JK)} \rangle )
\label{A}
\end{eqnarray}
where the index $\hat{\alpha}$ runs over broken gauge generators,
$Re f_\alpha \equiv Re f_{\alpha\alpha}$ and $f_{IJK}$'s are Yukawa
couplings some of which are moduli-dependent.
The $(I \cdots JK)$ in Eq.(\ref{A}) stands for a cyclic permutation
among $I$, $J$ and $K$.
The ellipsis in $(m^2_F)^J_I$ stands for extra $F$-term contributions
and so forth.
The $(m^2_D)^J_I$ is a $D$-term contribution
to scalar masses.
\section{Heterotic string model with anomalous $U(1)$}
Effective SUGRA is derived from 4D string models taking a field
theory limit.
In this section, we study soft SUSY breaking parameters
in SUGRA from heterotic string model with $U(1)_A$.\footnote{
Based on the assumption that SUSY is broken by $F$-components of
$S$ and/or a moduli field, properties of
soft SUSY breaking scalar masses have been studied in Ref.\cite{N,KK}.}
Let us explain our starting point and assumptions first.
The gauge group $G=G_{SM} \times U(1)_A$ originates from the
breakdown of $E_8 \times E_8'$ gauge group.
Here $G_{SM}$ is a standard model gauge group
$SU(3)_C \times SU(2)_L \times U(1)_Y$ and $U(1)_A$ is an anomalous $U(1)$
symmetry.
The anomaly is canceled by the Green-Schwarz mechanism \cite{GS}.
Chiral multiplets are classified into two categories.
One is a set of $G_{SM}$ singlet fields which the dilaton field
$S$, the moduli fields $M^i$ and
some of matter fields $\phi^m$ belong to.
The other one is a set of $G_{SM}$ non-singlet
fields $\phi^k$.
We denote two types of matter multiplet as
$\phi^\lambda = \{\phi^m, \phi^k\}$ .
The dilaton field $S$ transforms as
$S \rightarrow S-i\delta_{GS}^{A} M \theta(x)$ under $U(1)_A$
with a space-time dependent parameter $\theta(x)$.
Here $\delta_{GS}^{A}$ is so-called Green-Schwarz coefficient
of $U(1)_A$ and is given by
\begin{eqnarray}
\delta_{GS}^{A} &=& {1 \over 96\pi^2}Tr Q^A
= {1 \over 96\pi^2} \sum_{\lambda} q^A_{\lambda} ,
\label{delta_GS}
\end{eqnarray}
where $Q^A$ is a $U(1)_A$ charge operator,
$q^A_{\lambda}$ is a $U(1)_A$ charge of $\phi^{\lambda}$
and the Kac-Moody level of $U(1)_A$ is rescaled as $k_A=1$.
We find $|\delta_{GS}^{A}/q^A_m| = O(10^{-1}) \sim O(10^{-2})$
in explicit models \cite{KN,KKK}.
The requirement of $U(1)_A$ gauge invariance yields the form
of K\"ahler potential $K$ as,
\begin{eqnarray}
K &=& K(S + {\bar S} + \delta_{GS}^{A} V_A, M^i, {\bar M}^i,
{\bar \phi}_\mu e^{q^A_\mu V_A}, \phi^\lambda)
\label{K-st}
\end{eqnarray}
up to the dependence on $G_{SM}$ vector multiplets.
We assume that derivatives of the K\"ahler potential $K$ with respect to
fields including
moduli fields or matter fields are at most of order unity
in the units where $M$ is taken to be unity.
However we do not specify the magnitude of
derivatives of $K$ by $S$ alone.
The VEVs of $S$ and $M^i$ are supposed to be
fixed non-vanishing values by some non-perturbative effects.
It is expected that the stabilization of $S$ is
due to the physics at the gravitational scale $M$
or at the lower scale than $M$.
Moreover we assume that the VEV is much bigger than
the weak scale, i.e., $O(m_{3/2}) \ll \langle K_S \rangle$.
The non-trivial transformation property of $S$
under $U(1)_A$ implies that $U(1)_A$ is broken down at some high energy
scale $M_I$.
Hereafter we consider only the case with overall modulus
field $T$ for simplicity.
It is straightforward to apply our method to more complicated
situations with multi-moduli fields.
The K\"ahler potential is, in general, written by
\begin{eqnarray}
K &=& K^{(S)}(S + {\bar S} + \delta_{GS}^{A} V_A) +
K^{(T)}(T + {\bar T}) + K^{(S, T)}
\nonumber\\
&~& + \sum_{\lambda, \mu} ( s_{\lambda}^{\mu}(S + {\bar S}
+ \delta_{GS}^{A} V_A)
+ t_{\lambda}^{\mu}(T + {\bar T}) + u_{\lambda}^{{\mu}(S,T)} )
\phi^\lambda {\bar \phi}_\mu + \cdots
\label{K-st2}
\end{eqnarray}
where $K^{(S, T)}$ and $u_{\lambda}^{{\mu}(S,T)}$ are mixing terms between
$S$ and $T$.
The magnitudes of $\langle K^{(S, T)} \rangle$,
$\langle s_{\lambda}^{\mu} \rangle$ and
$\langle u_{\lambda}^{{\mu}(S,T)} \rangle$
are assumed to be $O(\epsilon_1 M^2)$, $O(\epsilon_2)$ and $(\epsilon_3)$
where $\epsilon_n$'s ($n=1,2,3$) are model-dependent parameters
whose orders are expected not to be more than one.\footnote{
The existence of $s_{\lambda}^{\mu} \phi^\lambda {\bar \phi}_\mu$
term in $K$
and its contribution to soft scalar masses are discussed in
4D effective theory derived through the standard embedding
from heterotic M-theory \cite{het/M}.}
We estimate the VEV of derivatives of $K$
in the form including $\epsilon_n$.
For example, $\langle K^\mu_{\lambda S} \rangle \leq O(\epsilon_p/M)$
($p=2,3$).
Our consideration is applicable to models in which some of $\phi^\lambda$
are composite fields made of original matter multiplets in string
models if the K\"ahler potential meets the above requirements.
Using the K\"ahler potential (\ref{K-st2}), $\hat{D}^A$ is given by
\begin{eqnarray}
\hat{D}^A &=& -K_S \delta_{GS}^{A} M
+ \sum_{\lambda, \mu} K_\lambda^\mu {\bar \phi}_\mu (q^A \phi)^\lambda
+ \cdots .
\label{D-st}
\end{eqnarray}
The breaking scale of $U(1)_A$ defined by
$M_I \equiv |\langle \phi^m \rangle|$ is estimated as
$M_I = O((\langle K_S \rangle \delta^A_{GS} M/q^A_m)^{1/2})$
from the requirement $\langle D^A \rangle \leq O(m_{3/2} M)$.
We require that $M_I$ should be equal to or be less than $M$,
and then we find that the VEV of $K_S$ has an upper bound such as
$\langle K_S \rangle \leq O(q^A_m M/\delta^A_{GS})$.
The $U(1)_A$ gauge boson mass squared $(M_{V}^{2})^A$ is given by
\begin{eqnarray}
(M_{V}^{2})^A = 2 g_A^2 \{ \langle K^S_S \rangle (\delta_{GS}^A M)^2
+ \sum_{m, n} q^A_m q^A_n
\langle K^{n}_{m} \rangle
\langle \phi^m \rangle \langle {\bar \phi}_n \rangle \}
\label{MV2A}
\end{eqnarray}
where $g_A$ is a $U(1)_A$ gauge coupling constant.
The magnitude of $(M_{V}^{2})^A/g_A^2$ is estimated as
$Max(O(\langle K^S_S \rangle (\delta^A_{GS} M)^2), O(q^{A2}_m M_I^2))$.
We assume that the magnitude of $(M_{V}^{2})^A/g_A^2$ is
$O(q^{A2}_m M_I^2)$.
It leads to the inequality
$\langle K^S_S \rangle \leq O((q^{A}_m M_I/\delta^{A}_{GS} M)^2)$.
The formula of soft SUSY breaking scalar masses on $G_{SM}$ non-singlet
fields is given by \cite{KK}
\begin{eqnarray}
(m^2)^k_l &=& (m_{3/2}^2 + {\langle V_F \rangle \over M^2})
\langle K^k_l \rangle
+ \langle {F}^{I} \rangle \langle {F}_{J} \rangle
(\langle R_{Il}^{Jk} \rangle + \langle X_{Il}^{Jk} \rangle) , \\
\langle R_{Il}^{Jk} \rangle &\equiv&
\langle ({K}_{Il}^{I'} (K^{-1})^{J'}_{I'}
{K}^{kJ}_{J'} - {K}_{Il}^{Jk}) \rangle , \\
\langle X_{Il}^{Jk} \rangle &\equiv&
q^A_k ((M^2_V)^A)^{-1} \langle ({\hat D}^A)_I^J \rangle
\langle K^k_l \rangle.
\end{eqnarray}
Here we neglect extra $F$-term contributions and so forth
since they are model-dependent.
The neglect of extra $F$-term contributions is justified
if Yukawa couplings between heavy and light fields
are small enough and the $R$-parity violation is also tiny enough.
We have used Eq.(\ref{<VI2>}) to derive the part related to
$D$-term contribution.
Note that the last term in r.h.s. of Eq.(\ref{<VI2>}) is negligible
when $(M_{V}^{2})^A/g_A^2$ is much bigger than $m_{3/2}^2$.
Using the above mass formula, the magnitudes of
$\langle R_{Il}^{Jk} \rangle$ and
$\langle X_{Il}^{Jk} \rangle$ are estimated and given in Table 1.
Here we assume $q^A_k/q^A_m = O(1)$.
\begin{table}
\caption{The magnitudes of $\langle R_{Il}^{Jk} \rangle$ and
$\langle X_{Il}^{Jk} \rangle$}
\begin{center}
\begin{tabular}{|c|l|l|}
\hline
$(I,J)$ & $\langle R_{Il}^{Jk} \rangle$ & $\langle X_{Il}^{Jk} \rangle$ \\
\hline\hline
$(S,S)$ & $O(\epsilon_p/M^2)$ &
$Max(O(\langle K^S_{SS} \rangle/\langle K_{S} \rangle),
O(\epsilon_p/M^2))$ \\ \hline
$(T,T)$ & $O(1/M^2)$
& $Max(O(\epsilon_1/(\langle K_{S} \rangle M)),
O(1/M^2))$ \\ \hline
$(m,m)$ & $O(1/M^2)$ & $O(1/M_I^2)$ \\ \hline
$(S,T)$ & $O(\epsilon_p/M^2)$
& $Max(O(\epsilon_1/(\langle K_{S} \rangle M)),
O(\epsilon_3/M^2))$ \\ \hline
$(S,m)$ & $O(\epsilon_p M_I/M^3)$
& $Max(O(\epsilon_p/(\langle K_{S} \rangle M)),
O(\epsilon_p/(M M_I))$ \\ \hline
$(T,m)$ & $O(M_I/M^3)$
& $Max(O(\epsilon_3/(\langle K_{S} \rangle M)),
O(1/(M M_I))$ \\ \hline
\end{tabular}
\end{center}
\end{table}
Now we obtain the following generic features on $(m^2)^k_l$.\\
(1) The order of magnitude of $\langle X_{Il}^{Jk} \rangle$ is
equal to or bigger than that of $\langle R_{Il}^{Jk} \rangle$
except for an off-diagonal part $(I,J)=(S,T)$.
Hence {\it the magnitude of $D$-term contribution is comparable to
or bigger than that of $F$-term contribution except for the universal part}
$(m_{3/2}^2 + \langle V_F \rangle/M^2)\langle K_l^k \rangle$.\\
(2) In case where the magnitude of $\langle F_m \rangle$ is bigger than
$O(m_{3/2} M_I)$ and $M > M_I$,
we get the inequality $(m^2_D)_k > O(m_{3/2}^2)$ since the magnitude of
$\langle \hat{D}^A \rangle$ is bigger than $O(m_{3/2}^2)$.\\
(3) In order to get the inequality $O((m^2_F)_k) > O((m^2_D)_k)$, the
following conditions must be satisfied simultaneously,
\begin{eqnarray}
&~& \langle F_T \rangle, \langle F_m \rangle \ll O(m_{3/2} M) ,~~
\langle F_S \rangle = O({m_{3/2} M \over \langle K^S_S \rangle^{1/2}})
\nonumber \\
&~& {M^2 \langle K^S_{SS} \rangle \over \langle K_S \rangle
\langle K^S_S \rangle} < O(1) ,~~
{\epsilon_p \over \langle K^S_S \rangle} < O(1) ,~~(p=2,3)
\label{conditions}
\end{eqnarray}
unless an accidental cancellation among
terms in $\langle \hat{D}^A \rangle$ happens.
To fulfill the condition $\langle F_{T,m} \rangle \ll O(m_{3/2} M)$,
a cancellation among various terms including $\langle K_{I} \rangle$ and
$\langle M^2 W_{I} / W \rangle$ is required.
Note that the magnitudes of
$\langle K_{T} \rangle$ and $\langle K_{m} \rangle$
are estimated as $O(M)$ and $O(M_I)$, respectively.
The gauge kinetic function is given by
\begin{eqnarray}
f_{\alpha\beta} = k_\alpha {S \over M}\delta_{\alpha\beta}
+ \epsilon_\alpha {T \over M}\delta_{\alpha\beta}
+ f_{\alpha\beta}^{(m)}(\phi^\lambda)
\label{f}
\end{eqnarray}
where $k_{\alpha}$'s are Kac-Moody levels
and $\epsilon_\alpha$ is a model-dependent parameter \cite{epsilon}.
The gauge coupling constants $g_\alpha$'s are related to the
real part of gauge kinetic functions such that
$g_\alpha^{-2} = \langle Re f_{\alpha\alpha} \rangle$.
The magnitudes of gaugino masses and $A$-parameters
in MSSM particles are estimated using the formulae
\begin{eqnarray}
M_a &=& \langle F^I \rangle \langle h_{aI} \rangle
\label{Ma2} ,\\
\langle h_{aI} \rangle &\equiv& \langle Re f_a \rangle^{-1}
\langle f_a,_I \rangle \\
A_{kll'} &=& \langle F^I \rangle \langle a_{kll'I} \rangle
\label{A2}, \\
\langle a_{kll'I} \rangle &\equiv& \langle f_{kll'},_I \rangle
+ {\langle K_I \rangle \over M^2} \langle f_{kll'} \rangle
- \langle K_{(kI}^{I'} \rangle \langle (K^{-1})_{I'}^J \rangle
\langle f_{Jll')} \rangle .
\end{eqnarray}
The result is given in Table 2.
Here we assume that $g_\alpha^{-2} = O(1)$.
\begin{table}
\caption{The magnitudes of $\langle h_{aI} \rangle$
and $\langle a_{kll'I} \rangle$}
\begin{center}
\begin{tabular}{|c|l|l|}
\hline
$I$ & $\langle h_{aI} \rangle$ & $\langle a_{kll'I} \rangle$
\\ \hline\hline
$S$ & $O(1/M)$ &
$Max(O(\langle K_{S} \rangle/M^2), O(\epsilon_p/M))$ \\ \hline
$T$ & $O(\epsilon_\alpha/M)$ & $O(1/M)$ \\ \hline
$m$ & $O(M_I/M^2)$ & $O(M_I/M^2)$ \\ \hline
\end{tabular}
\end{center}
\end{table}
In case that SUSY is broken by the mixture of $S$, $T$ and matter
$F$-components such that
$\langle (K^S_S)^{1/2} F_S \rangle$, $\langle F_T \rangle$,
$\langle F_m \rangle= O(m_{3/2} M)$ ,
we get the following relations among soft SUSY breaking parameters
\begin{eqnarray}
(m^2)_k &\geq& (m^2_D)_k = O(m_{3/2}^2 {M^2 \over M_I^2}) \geq
(A_{kll'})^2 = O(m_{3/2}^2) ,
\label{Mix-rel1}\\
(M_a)^2 &=& O(m_{3/2}^2) \cdot Max(O({\langle K^S_S \rangle}^{-1}),
O(\epsilon_\alpha^2), O({M_I^2 \over M^2})) .
\label{Mix-rel2}
\end{eqnarray}
Finally we discuss the three special cases of SUSY breaking scenario.
\begin{enumerate}
\item In the dilaton dominant SUSY breaking scenario
\begin{eqnarray}
\langle (K^S_S)^{1/2} F_S \rangle = O(m_{3/2} M) \gg
\langle F_T \rangle, \langle F_m \rangle ,
\label{S-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2) \cdot Max(O(1), O({M^2 \langle K^S_{SS} \rangle
\over \langle K^S_S \rangle \langle K_S \rangle}),
O({\epsilon_p \over \langle K^S_S \rangle})) , \nonumber \\
M_a &=& O({m_{3/2} \over \langle K^S_S \rangle^{1/2}}) , ~~~
A_{kll'} = O(m_{3/2}) \cdot Max(O({\langle K_{S} \rangle \over M}),
O(\epsilon_p)) . \nonumber
\end{eqnarray}
Hence we have a relation such that $O((m^2)_k) \geq O((A_{kll'})^2)$.
As discussed in Ref.\cite{model-dep1}, gauginos can be heavier than
scalar fields if $\langle K^S_S \rangle$ is small enough and
$O(M^2 \langle K^S_{SS} \rangle) < O(\langle K_S \rangle)$.
In this case, dangerous flavor changing neutral current
(FCNC) effects from squark mass non-degeneracy are avoided
because the radiative correction
due to gauginos dominates in scalar masses at the weak scale.
On the other hand, in Ref.\cite{model-dep2},
it is shown that gauginos are much lighter than
scalar fields from the requirement of
the condition of vanishing vacuum energy
in the SUGRA version of model proposed in Ref.\cite{BD}.
In appendix, we discuss the relations among the magnitudes of
$\langle K_S \rangle$, $\langle K^S_S \rangle$ and
$\langle K^S_{SS} \rangle$ under some assumptions.
\item In the moduli dominant SUSY breaking scenario
\begin{eqnarray}
\langle F_T \rangle = O(m_{3/2} M) \gg
\langle (K^S_S)^{1/2} F_S \rangle, \langle F_m \rangle ,
\label{T-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2) \cdot Max(O(1), O({\epsilon_1 M \over
\langle K_S \rangle})) ,\nonumber \\
M_a &=& O(\epsilon_\alpha m_{3/2}) , ~~~
A_{klm} = O(m_{3/2}) . \nonumber
\end{eqnarray}
Hence we have a relation such that
$O((m^2)_k) \geq O((A_{kll'})^2) \geq O((M_a)^2)$.
The magnitude of $\mu_{TT}$ is estimated as $\mu_{TT} = O(m_{3/2})$.
\item In the matter dominant SUSY breaking scenario
\begin{eqnarray}
\langle F_m \rangle = O(m_{3/2} M) \gg
\langle (K^S_S)^{1/2} F_S \rangle, \langle F_T \rangle ,
\label{matter-dom}
\end{eqnarray}
the magnitudes of soft SUSY breaking parameters are estimated as
\begin{eqnarray}
(m^2)_k &=& O(m_{3/2}^2 {M^2 \over M_I^2}) , ~~~
M_a, A_{kll'} = O(m_{3/2} {M_I \over M}) . \nonumber
\end{eqnarray}
The relation $(m^2)_k \gg O((M_a)^2) = O((A_{kll'})^2)$
is derived when $M \gg M_I$.
The magnitude of $\mu_{mn}$ is estimated as
$\mu_{mn} = O(m_{3/2} M/M_I)$.
This value is consistent with that in Ref.\cite{DP}.
\end{enumerate}
\section{Conclusions}
We have studied the magnitudes of soft SUSY breaking parameters
in heterotic string models with $G_{SM} \times U(1)_A$,
which originates from the breakdown of $E_8 \times E'_8$, and derive
model-independent predictions for them
without specifying SUSY breaking mechanism
and the dilaton VEV fixing mechanism.
In particular, we have made a comparison of magnitudes between
$D$-term contribution to scalar masses and $F$-term ones and
a comparison of magnitudes among scalar masses, gaugino masses
and $A$-parameters
under the condition that $O(m_{3/2}) \ll \langle K_S \rangle \leq
O(q^A_m M/\delta^A_{GS})$, $(M_V^2)^A/g_A^2 = O(q^{A2}_m M_I^2)$
and $\langle V \rangle \leq O(m_{3/2}^2 M^2)$.
The order of magnitude of $D$-term contribution of $U(1)_A$
to scalar masses is comparable to
or bigger than that of $F$-term contribution $\langle F^I \rangle
\langle F_J \rangle \langle R_{Il}^{Jk} \rangle$
except for the universal part
$(m_{3/2}^2 + \langle V_F \rangle/M^2)\langle K_l^k \rangle$.
If the magnitude of $F$-term condensation of matter fields
$\langle F_m \rangle$ is bigger than
$O(m_{3/2} M_I)$, the magnitude of $D$-term contribution $(m_D^2)_k$
is bigger than $O(m_{3/2}^2)$.
In general, it is difficult to realize the inequality
$O((m^2_D)_k) < O((m^2_F)_k)$ unless conditions such as
Eq.(\ref{conditions}) are fulfilled.
We have also discussed relations among soft SUSY breaking
parameters in three special scenarios on SUSY breaking,
i.e., dilaton dominant SUSY breaking scenario,
moduli dominant SUSY breaking scenario
and matter dominant SUSY breaking scenario.
The $D$-term contribution to scalar masses with different broken
charges as well as the $F$-term contribution from the difference among
modular weights can destroy universality among scalar masses.
The non-degeneracy among squark masses of first and second families
endangers the discussion of the suppression
of FCNC process.
On the other hand, the difference among broken charges
is crucial for the scenario of fermion mass hierarchy
generation \cite{texture}.
It seems to be difficult to make two discussions compatible.
There are several way outs.
The first one is to construct a model that the fermion mass hierarchy is
generated due to non-anomalous $U(1)$ symmetries.
In the model, $D$-term contributions of non-anomalous $U(1)$ symmetries
vanish in the dilaton dominant SUSY breaking case
and it is supposed that anomalies from contributions of
the MSSM matter fields are canceled out
by an addition of extra matter fields.
The second one is to use ``stringy'' symmetries
for fermion mass generation in the situation with degenerate
soft scalar masses \cite{texture2}.
The third one is to use a parameter region that
the radiative correction due to gauginos, which is flavor independent,
dominates in scalar masses
at the weak scale.
It can be realized when $\langle K^S_S \rangle$ is small enough and
$O(M^2 \langle K^S_{SS} \rangle) < O(\langle K_S \rangle)$.
Finally we give a comment on moduli problem \cite{cosmo}.
If the masses of dilaton or moduli fields are of order of the weak scale,
the standard nucleosynthesis should be modified
because of a huge amount of entropy production.
The dilaton field does not cause dangerous contributions
in the case with $\langle (K^S_S)^{1/2} F_S \rangle = O(m_{3/2} M)$
if the magnitude of $\langle K_S^S \rangle$
is small enough.\footnote{
This possibility has been pointed out
in the last reference in \cite{corr2}.}
Because the magnitudes of $(m_F^2)_S$ is
given by $O(m_{3/2}^2/\langle K_S^S \rangle^2)$.
\section*{Acknowledgements}
The author is grateful to T.~Kobayashi, H.~Nakano, H.P.~Nilles and
M.~Yamaguchi for useful discussions.
| 2024-02-18T23:40:13.831Z | 1998-11-19T06:07:10.000Z | algebraic_stack_train_0000 | 1,746 | 5,241 |
|
proofpile-arXiv_065-8719 | \section{Introduction}
The production mechanism\footnotetext{Talk at the Fourth Workshop
on Quantum Field Theory Under the Influence of External Conditions,
Leipzig, 14--18 September, 1998} of the intense flashes of light
which occur at the end of
bubble collapse in sonoluminescence remains mysterious.\cite{review}
A particularly intriguing possibility, put forth by Schwinger,
was that the Casimir effect in some dynamical manifestation was
responsible.\cite{js1,js2,js3,js4,js5} This idea was extended first
by Eberlein,\cite{eberlein} and later by Carlson, Liberati, and
others.\cite{carlson,visser}
Let us start by reviewing the relevant numbers for sonoluminscent
light emission. Typically, a bubble of air in water is held in the
node of an acoustic standing wave with overpressure of about 1
atmosphere,
of a frequency 20 kHz. The bubble goes from a maximum radius of $\sim
4\times
10^{-3}$ cm to a minimum radius of $\sim4\times 10^{-4}$ cm with a
time
scale $\tau_c$ of $10^{-5}$ s. The flash of light, which occurs near
minimum
radius, has a time scale $\tau_f$ of less than $10^{-11}$ s, and is
characterized by the emission of $10^6$ optical photons, so about
$10$ MeV of
light energy is emitted per flash.
It seems likely that the adiabatic approximation should be valid. If
the
flash scale is not orders of magnitude less than $10^{-11}$ s, that
scale is
long compared to the optical time scale, $\tau_o\sim10^{-15}$ s. In
that case,
we can immediately test the Casimir idea. The Casimir energy of a
dielectric
ball in vacuum is equivalent to that of a bubble in a dielectric
medium, and
has recently been definitively evaluated.\cite{brevik,barton} The
Casimir
energy of a dilute ball, of dielectric constant $\epsilon$,
$|\epsilon-1|\ll1$, of radius $R$
is
\begin{equation}
E={23\over1536\pi R}(\epsilon-1)^2,
\label{casball}
\end{equation}
which may be alternatively calculated by summing the van der Waals
energies
between the molecules that make up the medium.\cite{milton}
This value is 10 orders of magnitude too small to be relevant, as
well
as of the wrong sign. This is hardly a surprising result, since the
magnitude
of the effect is what one would expect from dimensional
considerations.
However, others have come to an opposite conclusion. In particular,
Schwinger,
\cite{js2} without relying on detailed calculations, asserted that
the
`dielectric energy, relative to that of the vacuum' was
\begin{equation}
E_c=-\int{(d{\bf r})(d{\bf k})\over(2\pi)^3}{1\over2}
k\left(1-{1\over
\epsilon({\bf r})^{1/2}}\right).
\label{casbulk}
\end{equation}
Although he argued this was true for slow variation in the dielectric
constant, he applied it to a hole of radius $a$ with a dielectric
medium,
therefore with a discontinuous boundary:
\begin{equation}
E_c={R^3\over12\pi}K^4\left(1-{1\over\epsilon^{1/2}}\right).
\label{casbulk2}
\end{equation}
Here $K$ represents an ultraviolet cutoff, which
Schwinger took to be $K\sim 2\times 10^5$ cm$^{-1}$, which gives a
sufficient energy, $E_c\sim 6$ MeV, to be relevant.
This conclusion is supported by the work of Carlson et
al.,\cite{carlson}
who obtain the identical result.
Why is there a discrepancy of the conclusion of these authors
with the result given in Eq.~(\ref{casball})?
The answer is simple. The term that Schwinger \cite{js2} and Carlson
et
al.\cite{carlson} keep is indeed present as a quartically divergent
term if one simply sums normal modes. But this is a intrinsic
contribution
to the self-energy of the dielectric medium. It was quite properly
subtracted off at the outset in the first paper on the Casimir energy
of
a dielectric ball,\cite{me} as it was in Schwinger's own detailed
papers
on the Casimir effect.\cite{jscas} A detailed analysis of this issue
is given
in Ref.~\cite{milton}. As Barton has noted, such divergent
volume and surface terms
`would be combined with other contributions to the bulk and to the
surface
energies of the material, and play no further role if one uses the
measured values.' \cite{barton} In other words, they serve to
renormalize
the phenomenological parameters of the model.
Further support for the irrelevance of the bulk energy comes from the
above-noted identity between the dilute Casimir energy and the van
der
Waals energy.\cite{brevik,barton,milton}
This would seem {\it prima facie\/} evidence that the
finite remainder is unambiguously determined. Note that the summed
van der
Waals energy must go like $(\epsilon-1)^2$, not the $\epsilon-1$
behavior
that Eq.~(\ref{casbulk}) displays.
\section{Acceleration and Temperature}
It seems plausible that the dynamical Casimir effect is closely
allied
with the so-called Unruh effect, \cite{unruh} wherein an accelerated
observer, with acceleration $a$, sees a bath of photons with
temperature $T$,
\begin{equation}
T={a\over2\pi}.
\label{unform}
\end{equation}
Indeed, the observed radiation in sonoluminescence is consistent with
the tail of a blackbody spectrum, with temperature $\sim$20,000
K.\footnote{The
temperature may be even higher. If so, $\tau_f$ is correspondingly
reduced.}
That is,
$kT$ is about 1 eV. Let us, rather naively, apply this to the
collapsing
bubble, where $a=d^2 R/dt^2\sim R/\tau_f^2$, where $\tau_f$ is some
relevant
time scale for the flash. We then have
\begin{equation}
kT\sim{R\over(c\tau_f)^2}\hbar c,
\end{equation}
or
\begin{equation}
1 \,\mbox{eV}\sim {10^{-3} \mbox{cm}\, 2\times 10^{-5}
\mbox{eV-cm}\over
\tau_f^2(3\times
10^{10}\mbox{cm\,s}^{-1})^2}\sim{10^{-29}\mbox{eV}\over\tau_f^2
(\mbox{s}^2)}.
\end{equation}
That is, $\tau_f\sim10^{-15}$ s, which seems implausibly short; it
implies a
characteristic velocity $R/\tau_f\sim10^{12}$ cm/s $\gg c$. It is
far shorter
than the upper limit to the flash duration, $10^{-11}$ s. Indeed, if
we
use the latter in the Unruh formula (\ref{unform}) we get a
temperature
about 1 milli Kelvin! This conclusion seems consistent with that of
Eberlein,\cite{eberlein} who indeed stressed the connection with
the Unruh effect, but whose numbers required superluminal velocities.
However, we must remain open to the possibility that discontinuities,
as in a shock, could allow changes on such short time scales without
requiring superluminal speeds. Indeed, Liberati et al.,\cite{visser}
following Schwinger's earlier suggestion,\cite{js1,js3} indeed assume
an extremely short time scale, so that rather than the adiabatic
approximation
discussed above being valid, a sudden approximation is more
appropriate.
We therefore turn to an analysis of that situation.
\section{Instantaneous collapse and photon production}
The picture offered by Liberati et al.\cite{visser} is that of the
abrupt
disappearance of the bubble at $t=0$, as shown in Fig.~\ref{fig1}.
\begin{figure}
\centering
\begin{picture}(200,100)
\thicklines
\put(100,0){\line(0,1){100}}
\thinlines
\put(50,50){\circle{20}}
\put(110,0){\line(1,1){90}}
\put(130,0){\line(1,1){70}}
\put(150,0){\line(1,1){50}}
\put(170,0){\line(1,1){30}}
\put(190,0){\line(1,1){10}}
\put(110,20){\line(1,1){80}}
\put(110,40){\line(1,1){60}}
\put(110,60){\line(1,1){40}}
\put(110,80){\line(1,1){20}}
\put(70,0){\line(1,1){20}}
\put(50,0){\line(1,1){40}}
\put(30,0){\line(1,1){60}}
\put(10,0){\line(1,1){40}}
\put(60,50){\line(1,1){30}}
\put(0,10){\line(1,1){40}}
\put(50,60){\line(1,1){40}}
\put(0,30){\line(1,1){70}}
\put(0,50){\line(1,1){50}}
\put(0,70){\line(1,1){30}}
\put(0,90){\line(1,1){10}}
\put(50,-10){\makebox(0,0){$t=0-$}}
\put(150,-10){\makebox(0,0){$t=0+$}}
\end{picture}
\caption{The sudden collapse of an otherwise static bubble.}
\label{fig1}
\end{figure}
On the face of it, this picture seems preposterous---the bubble
simply
disappears
and water is created out of nothing. It is no surprise that a large
energy
release would occur in such a case. Further, the static Casimir
effect
calculations employed in Ref.~\cite{visser} are invalid in this
instantaneously
changing model. Therefore, rather than computing Bogoliubov
coefficients
from the overlap of states belonging to two static configurations,
let us
follow the original methodology of Schwinger,\cite{js1,js3}
which is essentially equivalent.
As in Schwinger's papers, let us confine our attention to the
electric (TM)
modes. They are governed by the time-dependent Green's function
satisfying
\begin{equation}
(\partial_0\epsilon(x)\partial_0-\nabla^2)G(x,x')=\delta(x-x').
\end{equation}
The photon production is given by the effective two-photon source
\begin{equation}
\delta(JJ)=i\delta G^{-1}=i\partial_0\delta\epsilon(x)\partial_0.
\label{jj}
\end{equation}
The effectiveness for producing a photon in the momentum element
centered
about $\bf k$ is
\begin{equation}
J_k=\sqrt{{(d{\bf k})\over(2\pi)^3}{1\over2\omega}}\int(dx)
e^{-i({\bf k\cdot
r}-i\omega t)}J(x),\quad \omega=|{\bf k}|.
\label{jk}
\end{equation}
Let us follow Schwinger and consider one complete cycle of
disappearance
and re-appearance of the bubble, which we assume disappears for a
time $\tau_c$:
For a bubble centered at the origin, the dielectric constant as a
function
of time within the volume of the bubble is then taken to be
\begin{equation}
r<R:\quad \epsilon(r)=1+(\epsilon'-1)\eta(\tau_c/2-|t|).
\end{equation}
Here $\epsilon'$ is the dielectric constant of all space when the
bubble is gone. The dielectric constant of the region outside the
volume occupied by the bubble is
\begin{equation}
r>R:\quad
\epsilon(r)=\epsilon+(\epsilon'-\epsilon)\eta(\tau_c/2-|t|).
\end{equation}
Here $\epsilon$ is the dielectric constant outside the bubble when
the
bubble is present.
Occurring here is the unit step function,
\begin{equation}
\eta(x)=\left\{\begin{array}{cc}
1,&x>0,\\
0,&x<0.\end{array}\right.
\end{equation}
Clearly, this model is based on the assumption
that the disappearance time is short compared to the complete cycle
time of bubble collapse and re-expansion.
In the spirit of a first approximation, let us suppose all the
dielectric
constants are nearly unity, that is, that we are dealing with dilute
media. Let us further assume, appropriate to the instantaneous
approximation,
that the medium is a gas, which is capable of instantaneously filling
the
bubble. Then because the deviation of the dielectric constant from
unity is proportional to the matter number density $N$,
\begin{equation}
\epsilon-1=4\pi N\alpha,
\end{equation}
where $\alpha$ is the {\it constant\/} molecular polarizability,
matter conservation implies
\begin{equation}
(\epsilon'-1)V=(\epsilon-1)(V-v),
\end{equation}
where $V$ is the volume of all space, and $v$ is the volume of the
bubble.
Thus the change of the dielectric constant inside the bubble, and
outside,
respectively, is
\begin{eqnarray}
\delta\epsilon_{\rm in}&=&(\epsilon'-1)\eta(\tau_c/2-|t|),\nonumber\\
\delta\epsilon_{\rm out}&=&(\epsilon'-\epsilon)\eta(\tau_c/2-|t|)
=-(\epsilon'-1){v\over V-v}\eta(\tau_c/2-|t|).
\end{eqnarray}
The latter term here appears to be very small, and was therefore
disregarded
in Ref.~\cite{js1,js3,visser}. However, we will see that the
inclusion of
this term could be significant.
{}From Eqs.~(\ref{jj}) and (\ref{jk}), the two-photon production
amplitude is
proportional to ($v\ll V$)
\begin{eqnarray}
J_kJ_{k'}&=&\sqrt{{(d{\bf k})\over(2\pi)^3}{(d{\bf k'})\over(2\pi)^3}
{1\over2\omega2\omega'}}\int(d{\bf r'})\int_{-\tau/2}^{\tau/2}
dt \,e^{-i({\bf k+k')\cdot
r}+i(\omega+\omega')t}(-i\omega\omega')\nonumber\\
&&\times(\epsilon'-1)\left[\eta(a-r)-{v\over
V}\eta(r-a)\right]\nonumber\\
&\propto&(\epsilon'-1)\int_{-\tau/2}^{\tau/2}
dt\,e^{i(\omega+\omega')t}
(-i\omega\omega')\bigg[\int_{\rm in}(d{\bf r})e^{-i({\bf k+k')\cdot
r}}
\nonumber\\
&&\qquad\mbox{}-{v\over V}
\int_{\rm out}(d{\bf r})e^{-i({\bf k+k')\cdot r}}\bigg].
\label{twophoton}
\end{eqnarray}
The probability of emitting two photons is proportional to the
square
of this amplitude. For sufficiently short wavelengths, $\lambda\ll
R$,
the square of the quantity in square brackets in
Eq.~(\ref{twophoton})
is the product of $(2\pi)^3\delta({\bf k+k'})$ and $v$, that is, if
the
exterior contribution is negligible,
\begin{equation}
|J_kJ_{k'}|^2\propto(\epsilon'-1)^2\omega^2\sin^2\omega\tau_c\,
\delta({\bf k+k'})v.
\label{prob}
\end{equation}
This is the same result found by Schwinger,\cite{js1,js3} and by
Liberati
et al.\cite{visser} However, if as is plausible, the effective
exterior
volume $V$ is not much bigger that the volume of the bubble $v$, a
larger
contribution results. Indeed, a careful discretized version of the
momentum
integrals in Eq.~(\ref{twophoton}) gives in general for the factor
multiplying the delta function in Eq.~(\ref{prob})
$v(1+v/V)^2$. The interference is {\em
constructive}, not destructive as I erroneously
claimed in my Leipzig talk, and negligible
as $V\to\infty$. Taking the latter limit (but remembering that there
might
be up to a factor of 4 enhancement), and, appropriate for
$\tau_c/\tau_o
\gg1$, replacing $\sin^2\omega\tau_c\to1/2$, we obtain the
probability
of emitting a pair with momenta $\bf k$ and $\bf -k$ just as given by
Schwinger \cite{js3} (this now includes the equal contribution from
the
magnetic modes):
\begin{equation}
P_{\gamma\gamma}=v{(d{\bf
k})\over(2\pi)^3}\left(\epsilon-1\over4\right)^2,
\quad |\epsilon-1|\ll1.
\end{equation}
[For $|\epsilon-1|$ not small, Schwinger \cite{js3} generalized this
to
\begin{equation}
P_{\gamma\gamma}=2v{(d{\bf
k})\over(2\pi)^3}\ln{\epsilon^{1/4}+\epsilon^{-1/4}
\over2}.
\end{equation}
The numerical effect of this correction is not significant for a
first
estimate.]
The total number of photon pairs emitted is then, if dispersion is
ignored,
\begin{equation}
N=\left(4\pi\over3\right)^2\left(R\over\Lambda\right)^3
\left(\epsilon-1\over4
\right)^2,
\label{nophoton}
\end{equation}
where the cutoff wavelength is given by $K=2\pi/\Lambda$. Such a
divergent
result should be regarded as suspect.\footnote{Although it is not
clear how
this is to be related to the divergent energy (\ref{casbulk2}),
Schwinger obtained both in Ref.~\cite{js3} as the imaginary and real
parts,
respectively, of a complex action.}
It was Eberlein's laudable goal
\cite{eberlein} to put this type of argument on a sounder footing.
Nevertheless, if we put in plausible numbers, $\sqrt{\epsilon}=4/3$,
$R=4\times
10^{-3}$ cm, and, as in Schwinger's earlier estimate,
$\Lambda=3\times
10^{-5}$ cm, we obtain the required $N\sim 10^6$ photons per flash.
The problem with this estimate is one of time and length scales---for
the
instantaneous approximation to be valid, the flash time $\tau_f$ must
be
much less than the period of optical photons, $\tau_o\sim10^{-15}$ s.
This is consistent with the discussion in \S2, and acknowledged by
Liberati et al.\cite{visser} On the other hand, the collapse time
$\tau_c\sim
10^{-5}$ s is vastly longer than $\tau_f$, and is therefore totally
irrelevant to the photon production mechanism. The flash occurs near
minimum
radius, and thus the appropriate value of $R$ in Eq.~(\ref{nophoton})
would seem to be at least an order of magnitude smaller, $R\sim
10^{-4}$ em.
This would lead to $N<10^3$ photon pairs, totally insufficient.
\section{Conclusions}
We conclude by stating that the Casimir model fo sonoluminescence
remains
`unproven.' The static Casimir effect can be applied only in the
adiabatic
approximation, where it seems clearly irrelevant. The instantaneous
approximation grafted onto static configurations seems logically
deficient,
and again numerically irrelevant unless implausible parameters are
adopted. What is still needed is a dynamical calculation of the
Casimir
effect. The burden of proof is on the proponents of this mechanism.
\section*{Acknowledgments}
I thank Iver Brevik, Gabriel Barton, and Michael Bordag for useful
conversations. This work was supported in part by a grant from the
U.S. Department of Energy.
\section*{References}
| 2024-02-18T23:40:14.572Z | 1998-11-19T08:48:25.000Z | algebraic_stack_train_0000 | 1,789 | 2,738 |
|
proofpile-arXiv_065-8790 | \section{Introduction}
$\quad$In considering problems of mathematical physics with definite spatial
and/or dynamical symmetries, one commonly uses various decompositions
of vector fields over scalar potentials. These decompositions supplement
the famous Helmholtz theorem and reduce its ``gauge freedoms''.
Let $\exists \ \mbox{\bf V}: \ \mbox{\bf r} \in R^3 \rightarrow \mbox{\bf V}(\mbox{\bf r})
\in R^3 $ with all ``good'' properties. We may represent the given field
$\mbox{\bf V}$ in different ways
$$\mbox{\bf V} \rightarrow \{ V_x, V_y, V_z\} \rightarrow \{ \varphi ,
\mbox{\bf A} \}_{\mbox{div} \mbox{\bf A}=0} \rightarrow \{\varphi , \psi , \chi\}.$$
In any case, it is necessary to constrain superfluous components, if they
take place, by introducing conditions similar to $\mbox{div} \mbox{\bf A}=0.$
The last variant, diffeomorphic scalarization of a vector field, is the most economical and
convenient approach to vector boundary-value problems of mathematical physics.
But to use this approach, we have to be able to invert
decomposition formulas, i.e. to deduce the integral representations of scalar
potentials through the original vector field.
{\large Approach.}
To obtain a vector function from a (pseudo)scalar function set
$\varphi(\mbox{\bf r}), \psi(\mbox{\bf r})$ and $\chi(\mbox{\bf r})$, one must
act on them by some vector
operator $\hat F(\mbox{\bf r}, \mbox{\boldmath $\nabla$} ).$ Besides $\mbox{\bf r} \ \mbox {and} \
\mbox{\boldmath $\nabla$} $ themselves, we may from them construct three simplest operators:
$\ \mbox{\bf L}:=-\mbox{\bf r} \times \mbox{\boldmath $\nabla$}, \
\mbox{\bf N}:= \mbox{\boldmath $\nabla$} \times
\mbox{\bf L} \ \mbox {and} \ \mbox{\bf M}:=-\mbox{\bf r} \times \mbox{\bf L}.$ One can see their
symplectic nature because of their correspondence to the frames of reference
in a phase space: $\mbox{\bf r},\ \mbox{\bf k},\ \mbox{\bf r} \times \mbox{\bf k}, $
$\mbox{\bf r}\times (\mbox{\bf r}\times \mbox{\bf k})$
and $\mbox{\bf k} \times (\mbox{\bf k} \times \mbox{\bf r}).$
The {\it trio} of vectors $\mbox{\bf k},\ \mbox{\bf L},\ \mbox{\bf N}$ and $\mbox{\bf r},\ \mbox{\bf L},\ \mbox{\bf N}$, being
immersed into the spaces $R_{\mbox{\bf r}}^{3}$ and $R^{3}_{\mbox{\bf p}}$, respectively, form orthogonal
bases in them, which are important for different applications.
One may verify the
following projection and commutation properties of $\hat F$:\\
$[\mbox{\bf L}, r^2] = [\mbox{\bf L}, p^2] = 0$
$[L^2, r] = i [\mbox{\bf r} \times \mbox{\bf L}] - i[\mbox{\bf L} \times \mbox{\bf r}]$
$[L^2, k] = i [\mbox{\bf k} \times \mbox{\bf L}] - i[\mbox{\bf L} \times \mbox{\bf k}]$
$\mbox{\bf r} \cdot \mbox{\bf L} = \mbox{\boldmath $\nabla$}
\mbox{\bf L} = \mbox{\bf r} \cdot \mbox{\bf N} =
\mbox{\bf L} \cdot \mbox{\bf N} = \mbox{\bf L} \cdot \mbox{\bf M} = \mbox{\bf M} \cdot \mbox{\bf L} = 0,
\mbox{curl} \mbox{\bf N} = -\mbox{\bf L} \triangle; $
$[\mbox{\bf L}, \triangle ]=0,\ [\mbox{\bf N}, \triangle ]=0,\
[\mbox{\bf M}, \triangle ]=-6 \mbox{\boldmath $\nabla$},$ etc.
$[r_i, \nabla_k]=-\delta_{ik},\ \ [r_i,L_k]=-\epsilon_{ikj}r_j, \ \
[\nabla_i,L_k]=-\epsilon_{ikj} \nabla_j,$
$[\nabla_i,N_k]=\nabla_i\nabla_k-\triangle\delta_{ik}, \ \
[r_i,M_k]=r_ir_k-r^2\delta_{ik}$
$[L_i,M_k]=\epsilon_{ikj}r_j - r^2\epsilon_{ikj}\nabla_j.$
$..................................................................$
It is taken into account that
\begin{eqnarray}
\mbox{\bf N}=-\mbox{\bf r}\triangle + \mbox{\boldmath $\nabla$}
(\mbox{\bf r}\mbox{\boldmath $\nabla$}) + \mbox{\boldmath $\nabla$},
\nonumber \\ \mbox{\bf r}\times \mbox{\bf N} = -(1+\mbox{\bf r}
\mbox{\boldmath $\nabla$})\mbox{\bf L}=-\mbox{\bf L}(1+\mbox{\bf r}
\mbox{\boldmath $\nabla$}), \nonumber \\ \mbox{curl} \mbox{\bf N}=-\mbox{\bf L}
\triangle, \ \mbox{etc.} \nonumber
\end{eqnarray}
Note that in the space $R_{\mbox{\bf r}}^{3}$, $\mbox{\bf L}$ and $\mbox{\bf N}$ form the following
algebra of differential operators
$$[L_i, L_j]=\epsilon _{ijk}L_k,\ [L_i,N_j]=\epsilon _{ijk}N_k,\ [N_i,N_j]=
-\epsilon _{ijk}L_k \triangle .$$
After the rescaling of $\mbox{\bf N} \rightarrow \mbox{\bf N}/ \sqrt{\triangle}$ over a
supporting function space, we acquire the $O(3,1)$ Lie algebra representation
in terms of the $\hat F(\mbox{\bf r}, \mbox{\boldmath $\nabla$} )$ operators in $R_{\mbox{\bf r}}^{3}$. They act in the
$O(3)\times O(3)$ function space suitable for arranging the multipole
phenomenology in electromagnetic theory (see e.g. \cite{Bl}, \cite{DbCh},
\cite{DT}).
The operators $\mbox{\bf L}$ and $\mbox{\bf M}$ obey the same algebra in the space of wave
vectors $R_{\mbox{\bf k}}^{3}$ that is Fourier-conjugate to $R_{\mbox{\bf r}}^{3}$.
In the preceding paper \cite{DM} the emphasis has been made on the inversion of
different formulas for decomposition of vector fields in the mathematical
aspect.
In the physical aspect, two vector-potential formulations of the theory
of continuous media with taking
into account both magnetic and electric toroid polarizations \cite{DTT},
(see also \cite{DubTug}, part II) are published for the first time.
Here we consider the mathematical underlying reason of uniqueness of
division of the transverse electric distribution density
$E_{lm}(\mbox{\bf k}^{2}, \, t)$ into two independent multipole specimen~\cite{DbCh}:
\begin{equation}
E_{lm}(\mbox{\bf k}^{2}, \, t) = \dot{Q}_{lm}(0, \, t) + \mbox{\bf k}^2 T_{lm} (\mbox{\bf k}^{2}, \, t),
\label{N1}
\end{equation}
where $\dot{Q}_{lm}(0, \, t)$ are the time-derivative of the
Coulomb (charge) multipole $l-$moment and $T_{lm} (\mbox{\bf k}^{2}, \, t)$ are
the toroid multipole form factor of $l-$order. Note that a secularized
relation when neglecting toroid contributions and known as the Siegert theorem
may be correct for low-symmetric electromagnetic systems only.
The mathematical question of condition responsible for the possibility of
identification of multipole moments (the leading ones for each given $l$)
in the transverse and longitudinal
parts of current raised in \cite{French} will be tried to clear in this paper.
From our representation theorem (see also remark
II) it follows that definitions of multipole moments are unique.
Recall that the procedure of multipole
decomposition of the current and field densities in electrodynamics in fact
corresponds to the description of properties of a system by a set of numerical
characteristics which are assigned to a point, ``center'' of the density
distribution of the system considered.
In this case, for the poloidal and potential parts of
the current an additional connection arises between its longitudinal and
transverse components \cite{DbCh} due to degeneracy of boundary conditions
of longitudinality and transversality in $\mbox{\bf r}$-space at the self-similar
shrinkage of the definition domain of current to the chosen center.
To prove this statement, we could probably use the transfer technique of
boundary conditions \cite{Vlad}, \cite{Fedor}.
It might have lead to the separation of multipole moments
called in \cite{DbCh} the toroid ones~\footnote{In fact, their exact
title should be the poloidal moments, see~\cite{DT}, \cite{DubTug}.}.
But ways of this kind is very difficult in the general framework of
the distribution theory. Here we use the simple concrete approach.
The main feature of the multipole expansion procedure is a
special choice of basis functions, which in actual practice ensues the
rapid convergence of multipole series.
This circumstance forces us to weaken requirements of the usual
Helmholtz theorem and, respectively, to take account of a
gauge freedom extension.
In sections 1 and 2, we discuss gauge freedoms in the Helmholtz
and Neumann-Debye decomposition. In section 3, we turn to their realizations
within the multipole expansion of the electromagnetic current.
\section{The enlarged Helmholtz theorem}
$\quad$We begin our consideration with the Helmholtz Decomposition:\\
$\triangleleft \qquad \forall \mbox{\bf V}$ with properties of single-valuedness,
continuity, boundedness or convergence
$$|\mbox{\bf V}|<\frac{k}{r^{2+\epsilon}},\ \epsilon>0 \ \mbox{at}\ r\rightarrow
{\infty}$$
in the space may be represented in the form [see e.g. \cite{Lee}]:
$$\mbox{\bf V}=\mbox{\boldmath $\nabla$} \varphi +
\mbox{curl}\mbox{\bf A} \qquad
\mbox{with} \qquad \mbox{div}\mbox{\bf A}=0.$$
Indeed, the theorem requirements are proved to be sufficient in order to
reexpress tautologically the given vector field $\mbox{\bf V}$ through its
divergence and vorticity. The explicit realization of the theorem could be
attained due to the following operations
\begin{eqnarray}
\mbox{div} \mbox{\bf V}^{\|}&=&\triangle \varphi,
\varphi=\triangle ^{-1}\mbox{div}\mbox{\bf V}, \nonumber \\
\mbox{where} \quad
\triangle^{-1}&:=&-\int_{\Omega \subseteq R^3/\{0\}}{\frac{d^3r'}{4\pi |\mbox{\bf r}-
\mbox{\bf r}'|}}, \nonumber\\
\mbox{curl} \mbox{\bf V}^{\bot}&=&\mbox{curl}\ \mbox{curl}\mbox{\bf A}=-\triangle \mbox{\bf A}, \quad
\mbox{\bf A}=-\triangle^{-1}\mbox{curl}\mbox{\bf V}.\nonumber
\end{eqnarray}
So we have
\begin{equation}
\mbox{\bf V}\equiv \mbox{\bf V}^{\|}+\mbox{\bf V}^{\bot}=\mbox{\boldmath $\nabla$}
\triangle^{-1}\mbox{div} \mbox{\bf V}
-\mbox{curl}\triangle^{-1}\mbox{curl}\mbox{\bf V}. \qquad \triangleright
\label{1}
\end{equation}
Form (\ref{N1}) often produces misunderstanding
(e.g. \cite{Bosko}) that the representation of $\mbox{div} \mbox{\bf V}$ and
$\mbox{curl} \mbox{\bf V}$ is equivalent to the representation of $\mbox{\bf V}$ itself.
We discuss here to what extent single-valued is this representation.
In fact
$(1)$ contains the evident ``gauge freedom'':
$$\varphi = \varphi+ \omega, \quad (\triangle \omega=0), \qquad \mbox{\bf A}=\mbox{\bf A}+
\mbox{\boldmath $\nabla$}
w.$$
If we remove the demands of topological triviality $\Omega$ or/and
the boundedness of functions
$\omega$ and $w$,
then as an example important for physical application, we may represent the
gauge freedoms in the form of special additional functions to $\mbox{\bf V}$:
$$0\neq\mbox{\bf V}_{\mbox{\bf N}}=\mbox{curl}\mbox{\bf L} {{r^l}\choose {1/r^{l+1}}} Y_{lm}
\equiv
{{-(l+1)\mbox{\boldmath $\nabla$}
r^l Y_{lm}}\choose {l\mbox{\boldmath $\nabla$} r^{-l-1}Y_{lm}}}, $$
which have a nonzero finite value all over the space except $r \to \infty$ and
$r \to 0$, respectively, as far as
$$ \mbox{div}\mbox{\bf V}_{\mbox{\bf N}}= \mbox{curl} \mbox{\bf V}_{\mbox{\bf N}}\equiv 0, \quad
\mbox{in} \quad R^3 / S_{r \to \infty}^{2} \quad \mbox{and}\quad
R^3/ \{0\}.$$
Therefore these functions~\footnote{Note that similar functions may be
generated using the common relation
$$\mbox{curl}(\mbox{\bf r}\times \mbox{\boldmath
$\nabla$}) r^{\kappa}Y_{lm}=
-(\kappa +1)\mbox{\boldmath
$\nabla$} r^{\kappa}Y_{lm}+(\kappa -l)(\kappa+l+1)\mbox{\bf r} r^
{\kappa -2}Y_{lm},$$
and taking into account that
$$\kappa =l \ \ \mbox{and} \ \ \kappa=-l-1$$
(see \cite{Af}, Appendix A).}
cannot be represented by the usual Helmholtz
decomposition, and manifest the gauge freedom of its enlarged formulation.
{\bf\small Remark I.} {\small Note that functions $\mbox{\bf V}_{N}$ are
longitudinal and transverse simultaneously, since they represent the
vector solutions of the Laplace equation ($\triangle \equiv
\mbox{grad} \, \mbox{div} - \mbox{curl} \, \mbox{curl}$).
Moreover, in our context it is important to emphsize that
they are topologically equivalent to the poloidal (meridional) harmonics on
the toruslike surface covering the whole space of $R^3$
except one deleted axis. }
Thus, under this gauge freedom the Helmholtz decomposition takes the
following alternative forms
$$ \mbox{\bf V}\equiv \mbox{\bf V}^{\|}+\mbox{\bf V}^{\bot}= $$
\begin{eqnarray}
= \mbox{\boldmath $\nabla$} \left( \triangle^{-1}\mbox{div} \mbox{\bf V} - (l+1) \sum_{lm} \bigl[
C_{lm} r^l + C_{lm}^{\prime} r^{-l-1}\bigr] Y_{lm} \right)
+\mbox{curl} \triangle^{-1}\mbox{curl}\mbox{\bf V}=
\label{N3} \\
= \mbox{\boldmath $\nabla$} \triangle^{-1}\mbox{div} \mbox{\bf V}+ \mbox{curl} \left( \triangle^{-1}\mbox{curl}\mbox{\bf V}+ \mbox{\bf L} \sum_{lm} \bigl[
C_{lm} r^l + C_{lm}^{\prime} r^{-l-1}\bigr] Y_{lm} \right).
\nonumber
\end{eqnarray}
\section{The Neumann-Debye decomposition}
$\quad$The well-known mathematical physicist W.M.Elsasser
in \cite{El} has already observed that every vector field of the
form $\mbox{\boldmath $\nabla$}\times \mbox{\boldmath $\nabla$}\times \mbox{\bf r}\chi +\mbox{\boldmath $\nabla$}\times
\mbox{\bf r} \psi$, where $\psi$ and $\chi$ are any scalars, is solenoidal. In
paper \cite{Bs} it has been shown that if $\mbox{div} \mbox{\bf V} =0$ in $R^3$, then for every
choice of the origin there exist unique scalars $\psi$ and $\chi$ such that
$\mbox{\bf V}=\mbox{\bf L} \psi +\mbox{\bf N} \chi$ while $\psi$ and $\chi$ average to zero
on every spherical surface concentric with the origin. The complete theorem
of the possibility of decomposition of $\mbox{\bf V}(\mbox{\bf r})$ in terms of scalar
functions reads as follows:
{\large Representation Theorem.}
Given a region $\Omega \subseteq R^3\backslash \{0\},$ with a regular boundary
and $R^3-$vector field, $\mbox{\bf V}:\ \mbox{\bf r}\in \Omega \rightarrow \mbox{\bf V}(\mbox{\bf r})
\in R^3.$ Then, there exist three scalar functions $\varphi (\mbox{\bf r}),
\psi (\mbox{\bf r}) \ \mbox{and} \ \chi (\mbox{\bf r}) \ \mbox{on} \ \Omega$ which
define this $\mbox{\bf V}$ \cite{Mul}.
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The most used decomposition, which we call the Neumann-Debye one, has the
form
\begin{equation}
\triangleleft \qquad \mbox{\bf V}(\mbox{\bf r}):=
\mbox{\boldmath $\nabla$}
\varphi(\cdot)+\mbox{curl}\mbox{\bf r}\psi(\cdot)+
\mbox{curl} \ \mbox{curl}\mbox{\bf r}\chi(\cdot) \equiv
\mbox{\boldmath $\nabla$} \varphi +\mbox{\bf L}\psi +
\mbox{\bf N}\chi .
\end{equation}
Here $\psi$ and $\chi$ are the so-called Debye potentials and $\varphi$
is the usual (electric) scalar one.
We found
fundamental solutions of the inversion problem of (1) in the form~\cite{DM}
\begin{eqnarray}
\mbox{div} \mbox{\bf V}=\triangle \varphi \qquad \rightarrow \qquad
\varphi&=&\triangle^{-1}\mbox{div} \mbox{\bf V}, \nonumber\\
\mbox{\bf L}\mbox{\bf V} = L^2\psi \qquad \rightarrow \qquad
\psi&=& - L^{-2}\mbox{\bf L}\mbox{\bf V}\equiv \mbox{\bf L}^{-2}\mbox{\bf r}\ \mbox{curl}\mbox{\bf V}, \label{4} \\
\mbox{\bf r}\mbox{\bf V}=(\mbox{\bf r}\mbox{\boldmath $\nabla$} )
\varphi +L^2\chi \qquad \rightarrow \qquad
\chi&=& L^{-2}(\mbox{\bf r}\mbox{\boldmath $\nabla$})
\triangle^{-1} \mbox{div}\mbox{\bf V} -
L^{-2}(\mbox{\bf r}\mbox{\bf V}), \nonumber
\end{eqnarray}
where \cite{Cou}
\begin{displaymath}
\mbox{\bf L}^{-2}:=\int_{\sigma}{{\frac{d\omega'}{4\pi}}ln(1-\hat r\cdot
\hat r')}. \quad \triangleright
\end{displaymath}
{\bf\small Remark II.} {\small Eigenfunctions of the square of the
operator angular momentum $iL:\ (iL)^2=-L^2$ are usual spherical
functions satisfying the equation
\begin{eqnarray}
L^2\ Y_{lm}(\hat r)=-l(l+1)Y_{lm}(\hat r) \nonumber
\end{eqnarray}
The corresponding Green function for this equation can be found with the
help of the known Mercer theorem (\cite{Cou}, v.1) which in the given case
yields}
\begin{eqnarray}
\frac{1}{4\pi}\sum_{l,m}\frac{Y_{lm}^*(\hat r)Y_{lm}(r')}{-l(l+1)}=
-\sum_{l}\frac{2l+1}{l(l+1)}P_l(\hat r\cdot \hat r')=1-ln2+ln(1-\hat r\cdot
\hat r'). \quad
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\nonumber
\end{eqnarray}
{\bf\small Remark III.} {\small It is well-known that the gauge freedom of
(\ref{4}) is the following:
$$\varphi \rightarrow \varphi +C, \quad \psi \rightarrow \psi +\mu (r),
\quad \chi \rightarrow \chi +\nu (r).$$
Requiring $\varphi$ to vanish on the boundary, and $\psi, \ \chi$ not to
contain spherically symmetric components
$$\int_{S^2}{dw \ \psi}\ = \ \int_{S^2}{dw\ \chi} \ =\ 0; $$
we put these functions in one-to-one correspondence to $\mbox{\bf V}$.}
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{\large Uniqueness Theorem.} If a vector field $\mbox{\bf V}$ (with the
properties determined in the theorem (I) ) is defined on every $S_r^2$ in
some range $r_0<r<r_1$ and in that range $V_r=0$ while $V_{\theta}(r,\theta ,
\varphi)$
and $V_{\varphi}(r,\theta,\varphi)$ are bounded for each fixed $r$ and are
continuously differentiable except possibly at $\theta=0$ and $\theta=\pi$
and further $\mbox{div} \mbox{\bf V}=0$ and $\mbox{curl} \mbox{\bf V}=0$, then $\mbox{\bf V}\equiv 0$.
Our inversion formulas (3) demonstrate that immediately
(cp. with \cite{Bs}, p.383, where the condition $\mbox{\bf L}\mbox{\bf V}=0$ has been
used instead of our $\mbox{curl} \mbox{\bf V}=0$).
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Now we have to reconstruct the representation of $\chi$ such that
it depends on $\mbox{curl} \mbox{\bf V}$ and $\mbox{div} \mbox{\bf V}$ only. Really, the
latter quantities have the physical meaning but not the radial component of
$\mbox{\bf V}$. Moreover, we may expect $\chi$ not to depend on
$\mbox{div} \mbox{\bf V}$ generally because this potential defines the transverse
part of the vector field $\mbox{\bf V}$. Nevertheless, because of gauge freedom,
the situation is not so simple as it seems to be.
Indeed, we may substitute the Helmholtz
decomposition (the last expression in (\ref{N3})) into the term with
$\mbox{\bf r} \mbox{\bf V}$ and see that $\chi$ takes the form
\begin{eqnarray}
\chi &=& L^{-2} \mbox{\bf L} \triangle^{-1} \mbox{curl} \mbox{\bf V}
\sum_{lm} L^{-2} \bigl[
C_{lm} \nabla r^l + C_{lm}^{\prime} \nabla r^{-l-1}\bigr] Y_{lm}.
\end{eqnarray}
Further our vector field corresponding to the gauge freedom may be
transformed as
\begin{eqnarray}
\mbox{curl} \mbox{\bf L} L^{-2} (r \nabla) r^{l}Y_{lm}=
(l+1) \mbox{curl} r^{l} \mbox{\bf L} L^{-2} Y_{lm}= \nonumber \\
= (l+1) \mbox{curl} r^{l} \mbox{\bf L} L^{-2} \frac{L^2}{l(l+1)}Y_{lm}=
\frac{1}{l}\mbox{curl} \mbox{\bf L} r^{l} Y_{lm}=
-\frac{l+1}{l} \mbox{\boldmath $\nabla$} r^{l}Y_{lm}.
\label{N7}
\end{eqnarray}
So, we found fundamental solutions of the inversion problem of (\ref{4})
in the form~\cite{DM}
\begin{eqnarray}
\mbox{div} \mbox{\bf V}=\triangle \varphi \qquad \rightarrow \qquad
\varphi&=&\triangle^{-1}\mbox{div} \mbox{\bf V}, \nonumber\\
\mbox{\bf L}\mbox{\bf V} = L^2\psi \qquad \rightarrow \qquad
\psi&=& - L^{-2}\mbox{\bf L}\mbox{\bf V}\equiv \mbox{\bf L}^{-2}\mbox{\bf r}\ \mbox{curl}\mbox{\bf V}, \label{N6} \\
\mbox{\bf r}\mbox{\bf V}=(\mbox{\bf r}\mbox{\boldmath $\nabla$}
)\varphi +L^2\chi \qquad \rightarrow \qquad
\chi&=& L^{-2}(\mbox{\bf r}\mbox{\boldmath $\nabla$})\triangle^{-1} \mbox{div}\mbox{\bf V} -
L^{-2}(\mbox{\bf r}\mbox{\bf V}), \nonumber \\
&=& L^{-2} \mbox{\bf L} \triangle^{-1} \mbox{curl} \mbox{\bf V} \nonumber \\
&+& \sum_{lm} L^{-2} \bigl[
C_{lm} \nabla r^l + C_{lm}^{\prime} \nabla r^{-l-1}\bigr] Y_{lm} \nonumber
\end{eqnarray}
with taking into account the gauge freedom in (\ref{N3}). $\quad $
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\section{The multipole expansion of the \\
electromagnetic current}
$\quad$Now we compare our abstract exercises and the
procedure of
multipole expansion of electromagnetic current
on the basis of the vector Helmholtz equation solutions constructed through
the Neumann-Debye decomposition (\ref{N6}).
The multipole representation of current $\mbox{\bf J}(\mbox{\bf r})$ by (\ref{4}) may be
obtained by the standard
expansion of three scalar densities into series:
\begin{eqnarray}
\phi \sim \sum_{lmk} j_{l}(kr) Y_{lm} \dot{Q}_{lm} (k^2, t);
\nonumber \\
\psi \sim \sum_{lmk} j_{l}(kr) Y_{lm} M_{lm} (k^2, t);
\label{N9} \\
\chi \sim \sum_{lmk} j_{l}(kr) Y_{lm} E_{lm} (k^2, t).
\nonumber
\end{eqnarray}
Therefore the multipole representations of the transverse
part of current ($\mbox{div} \, \mbox{\bf J} =0$) are determined by the magnetic
form factors $M_{lm}(k^2, \, t)$, and the transverse electric contributions
$E_{lm}(k^2, \, t)$, and the scalar part of current ($\mbox{curl} \, \mbox{\bf J} =0$)
are expressed due to the $4-$current conservation law ($\mbox{div} \, \mbox{\bf J} =
- \dot{\rho}$) through the Coulomb (charge) multipole moments $Q_{lm}(0 \, t)$
and their mean $2n-$power radii
$$ Q_{lm}(k^2, \, t) = Q_{lm} (0, \, t) + \sum_{n=1}^{\infty}
\frac{k^{2n}}{n!} Q_{lm}^{(2n)} (0, \, t).$$
As a result, $\mbox{\bf J}$ may be represented as \cite{DbCh}
\begin{eqnarray}
\mbox{\bf J}(\mbox{\bf r}, \, t)& &= (2\pi)^{-3} \sum_{l,m} \int_{0}^{\infty} dk
(-ik)^{l} \frac{\sqrt{4 \pi (2l+1)}}{l(2l+1)!!}
\{ - i k \, \mbox{\bf L} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}})
M_{lm}(k^2, \, t) \nonumber \\
& & -ik \mbox{curl} \mbox{\bf L}
f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}})
E_{lm}(k^2, \, t) +
l \, \mbox{\boldmath $\nabla$} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}}) \dot{Q}_{lm}(k^2, \, t) \}.
\label{N10}
\end{eqnarray}
We repeat here the procedure used firstly in \cite{DbCh} for the
ascertainment of the exact structure of transverse electric
contributions $E_{lm}(k^2, \, t)$. To this end, it one should rewrite
explicitly our basis functions in terms of the vector harmonics
\begin{eqnarray}
\mbox{curl} \mbox{\bf L} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}}) = (2l+1)^{-1/2} \{ f_{l-1}
(kr) \sqrt{l+1} \mbox{\bf Y}_{ll-1m}(\hat{\mbox{\bf r}}) +
\nonumber \\
+ f_{l+1}(kr) \sqrt{l} \mbox{\bf Y}_{ll+1m}(\hat{\mbox{\bf r}}) \};
\nonumber
\end{eqnarray}
\begin{eqnarray}
\mbox{\boldmath $\nabla$} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}}) = (2l+1)^{-1/2} \{ f_{l-1}
(kr) \sqrt{l} \mbox{\bf Y}_{ll-1m}(\hat{\mbox{\bf r}}) -
\nonumber \\
- f_{l+1}(kr) \sqrt{l+1} \mbox{\bf Y}_{ll+1m}(\hat{\mbox{\bf r}}) \},
\nonumber
\end{eqnarray}
where $\mbox{\bf Y}_{ll-1m}(\hat{\mbox{\bf r}})$ is a harmonic polynomial function defined as
$$ r^{l-1}\mbox{\bf Y}_{ll-1m}(\hat{\mbox{\bf r}}) = \frac{1}{\sqrt{l(2l+1)}}
\mbox{\boldmath $\nabla$
} r^{l} Y_{lm}.$$
As is obvious, in the wave-length approximation, the leading contributions in the
latter expressions are delivered by the vector harmonic functions
$f_{l-1}Y_{ll-1m}$ from which it follows that
\begin{eqnarray}
\mbox{curl} \mbox{\bf L} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}}) \approx_{k \rightarrow 0}
\sqrt{(l+1)/l} \mbox{\boldmath $\nabla$} f_{l}(kr)Y_{lm}(\hat{\mbox{\bf r}})=
\nonumber \\
=\frac{4\pi (ikr)^{l-1}}{(2l+1)!!} \sqrt{\frac{l+1}{l}}
\mbox{\boldmath $\nabla$} r^{l} Y_{lm}(\hat{\mbox{\bf r}})+ O[(kr)^{l+1}]
\label{N11}
\end{eqnarray}
It is the relation that permits us to identify the leading term in
$E_{lm}(k^2, \, t)$ with time-derivatives of $Q_{lm}(0, \, t)$.
However all functually independent contributions in $E_{lm}(k^2, \, t)$
give the so-called toroid moments and their $2n-$power radii
$$ T_{lm}^{(2n)}(0, \, t) = - \frac{\sqrt{\pi l}}{2l+1} \int r^{l+2n+1}
\left[ \mbox{\bf Y}_{ll-1m}^{*} (\mbox{\bf r}) + \frac{2\sqrt{l/(l+1)}}{2l+3} \mbox{\bf Y}_{ll+1m}^{*}(\mbox{\bf r})
\right] \mbox{\bf J} (r, t) d^3 r.$$
As the toroid moments have a distinct geometrical meaning
(diverse details and representations are given in \cite{DbCh}, \cite{Vlad},
\cite{DubTug}, \cite{Shab} and see also \cite{Saha}), the rejection of
$T_{lm}(t)$ is generally invalid like it was done in the Siegert theorem
$E_{lm} (k^2, \, t) \rightarrow_{k \rightarrow 0} \dot{Q}_{lm} (0, \, t)$.
Neglect of $T_{lm}(t)$ in comparison with $Q_{lm}(t)$ is analogous to the
neglect of a higher multipole moment (contributions of highest symmetries
of a given system) in comparison with the lower ones, which is of course
permissible only when the lower moments (low symmetries) of this system
do exist. So, the strict theorem determining the electric part structure
has the form (\ref{N1})
$E_{lm}(\mbox{\bf k}, \, t) = \dot{Q}_{lm}(0, \, t) + \mbox{\bf k}^2 T_{lm} (\mbox{\bf k}, \, t),$
and its validity and uniqueness rely on the
gauge freedom which has been obtained for the enlarged Helmholtz theorem and
transferred to
Debye potentials (compare (\ref{N6}) and (\ref{N11})).
Moreover, by using the exact relation (\ref{N1}),
it is not so hard to find an
expression for the complete parametrization of the current in
terms of generalized functions~\footnote{The contribution of $l=0, \, n=0$ to the last term is
forbidden by total charge conservation, whereas other terms contain
no contribution of $l=0$, formally, owing to
$(\mbox{\bf r} \times \mbox{\boldmath $\nabla$}) \delta(\mbox{\bf r}) \equiv 0$.} \cite{DubTug}
\begin{eqnarray}
\label{N12}
\mbox{\bf J} (\mbox{\bf r}, \, t) = \sum_{l=1}^{\infty} \sum_{m=-l}^{m=l} \sum_{n=0}^{\infty}
\frac{(2l+1)!!}{2^n n! l(2l+2n+1)!!} \sqrt{\frac{4 \pi}{2l+1}}
\{ M_{lm}^{(2n)} (t) (\mbox{\bf r} \times \mbox{\boldmath $\nabla$}) \Delta^{n} \delta_{lm}(\mbox{\bf r}) \nonumber\\
\\
+ \bigl[ \dot{Q}_{lm} (t) \delta_{n,0} \Delta^{-1} - T_{lm}^{(2n)} (t)
\bigr] \mbox{curl} (\mbox{\bf r} \times \mbox{\boldmath $\nabla$}) \Delta^{n} \delta_{lm} (\mbox{\bf r}) -
l \, \dot{Q}_{lm}^{(2n)} (t) \mbox{\boldmath $\nabla$} \Delta^{n-1} \delta_{lm}(\mbox{\bf r})\}.\nonumber
\end{eqnarray}
{\bf\small Remark IV.} {\small However, the expansion via of spherical
harmonics emerges rapidly convergent. We remind (see e.g. \cite{Bl}, p.806)
that if the field decreases for large distances very slowly, slower than
$r^{-2}$, the divergence and curl of the vector field considered are assigned
arbitrary independent values. Conversely, if we know that $\mbox{\bf J}$ vanishes
identically outside some source radius $R$, $\mbox{\boldmath $\nabla$}
\mbox{\bf J}$ and $\mbox{\boldmath $\nabla$} \times \mbox{\bf J}$
are no longer independent of each other. As far as the expansion via
spherical harmonics is rapidly convergent, it is realized in the latter
representation (\ref{N12}) immediately.}
Thus, we strictly demonstrated that the gauge freedom in division
of the electromagnetic current into the transverse and longitudinal parts
leads to the fact that the multipole contributions to the
transverse part of current $E_{lm} (k^2, t)$ are represented in the form
(\ref{N1}) and its leading terms
may be identified with $\dot{Q}_{lm} (0 \, t)$ from the
longitudinal part of current for all $l$. Note that, since the coefficient
$C_l$ does not depend on the wave number $k$, we can use hereafter the
Lorentz gauge condition in the calculation of the vector potential.
\section{ Conclusion}
$\quad$The representation of $\chi$ in the Neumann-Debye
scalarization already assumes that the prohibition of the electric type of
radiation imposes some conditions both on ${\rm curl} \mbox{\bf J}$ and ${\rm div} \mbox{\bf J}$.
But we could not reveal the ones due to their non-division in the
scalarization mentioned.
Exploitation of the enlarged Helmholtz theorem for this operation has
inserted the extended gauge freedom (let us recall that the Neumann-Debye
representation the gauge freedom reduce to functions of the scalar
argument $|\mbox{\bf r}|$ only). It is just this freedom, consideration of which made it
possible to identify the coefficients of leading order of the expansion
of transverse and longitudinal electric parts of the current!
The form of the expression (\ref{N1}) shows the possibility of compensation
of the electric type radiation if the toroid and charge moments are switched on
as ``anti-phase'' ones~\cite{DbCh} (see also \cite{Afan}).
| 2024-02-18T23:40:14.788Z | 1998-11-13T13:15:18.000Z | algebraic_stack_train_0000 | 1,806 | 4,733 |
|
proofpile-arXiv_065-8824 | \section{Introduction}
Nonzero ground state entropy, $S_0 \ne 0$, is an important subject in
statistical mechanics. One physical example is provided by ice, for which
$S_0 = 0.82 \pm 0.05$ cal/(K-mole), i.e., $S_0/R = 0.41 \pm 0.03$, where
$R=N_{Avog.}k_B$ \cite{ice,lp,liebwu}. A particularly simple model exhibiting
ground state entropy without the complication of frustration is the $q$-state
Potts antiferromagnet (AF) \cite{potts,fk,wurev} on a graph $G$
\cite{graphdef}
(which may or may not be a regular lattice) for $q \ge \chi(G)$, where
$\chi(\Lambda)$ denotes the chromatic number of $G$, i.e., the minimum
number of colors necessary to color the vertices of the lattice such that no
two adjacent vertices have the same color. As is evident, there is a deep
connection with graph theory here, since the zero-temperature partition
function of the above-mentioned $q$-state Potts antiferromagnet on a graph
$G$ satisfies
\begin{equation}
Z(G,q,T=0)_{PAF}=P(G,q)
\label{zp}
\end{equation}
where $P(G,q)$ is the chromatic
polynomial expressing the number of ways of coloring the vertices of the graph
$G$ with $q$ colors such that no two adjacent vertices (connected by a bond of
the graph) have the same color \cite{birk}--\cite{biggsbook}.
Since the strict mathematical definition of a graph $G$ requires the number of
vertices $n=v(G)$ to be finite, we shall denote
\begin{equation}
\lim_{n \to \infty}G = \{G\}
\label{ginfinite}
\end{equation}
In this limit, the ground state entropy per vertex (site) is given by
\begin{equation}
S_0 = k_B \ln W(\{G\},q)
\label{s0}
\end{equation}
where $W(\{G\},q)$, the ground state degeneracy per vertex, is
\begin{equation}
W(\{G\},q) = \lim_{n \to \infty} P(G,q)^{1/n}
\label{w}
\end{equation}
As noted, in the limit $n \to \infty$, nonzero ground state entropy,
$S_0(\{G\},q) > 0$, or equivalently, ground state degeneracy $W(\{G\},q) > 1$
generically occurs in the $q$-state Potts antiferromagnet for sufficiently
large $q$ on a given lattice $G$; equivalently, the number of ways of coloring
the graph subject to the constraint that no two adjacent vertices have the
same color grows exponentially rapidly with the number of vertices on the
graph.
Given the fact that $P(G,q)$ is a polynomial, there is a natural
generalization, which we assume here, of the variable $q$ from integer to
complex values. $W(\{G\},q)$ is an analytic function in the $q$ plane except
along a continuous locus of points (and at possible isolated points which will
not be relevant here).
Following the terminology in our earlier papers \cite{p3afhc}--\cite{strip}, we
denote this continuous locus as ${\cal B}$. In the limit as $n \to \infty$, the
locus ${\cal B}$ forms by means of a coalescence of a subset of the zeros of
$P(G,q)$ (called chromatic zeros of $G$) \cite{early}. Because $P(G,q)$ has
real (indeed, integer) coefficients, ${\cal B}$ has the basic property of
remaining invariant under the replacement $q \to q^*$:
\begin{equation}
{\cal B}(q) = {\cal B}(q^*)
\label{binvariance}
\end{equation}
In a
series of papers we have calculated and analyzed the loci ${\cal B}$ for a
variety of families of graphs \cite{p3afhc}--\cite{strip} and investigated the
connections of these loci with the behavior of $W(\{G\},q)$ for the physical
values $q \in {\mathbb Z}_+$, which we have also studied by means of rigorous
upper and lower bounds and large--$q$ series \cite{ww,wn} and Monte Carlo
measurements \cite{p3afhc,w,ww}. In many cases the loci ${\cal B}$ enclose
and form the boundaries of two or more regions in the complex $q$ plane.
Since this is the case for the families of graphs studied here, we shall use
the abbreviated nomenclature ``$W$ boundary'' for ${\cal B}$. We note that in
other families of graphs, such as the infinitely long, finite-width
homogeneous strip graphs of various lattices (with open boundary conditions)
studied in Ref. \cite{strip}, the loci ${\cal B}$ form arcs that do
not enclose any region. The study of these loci for various families of
graphs also makes a very interesting connection with algebraic geometry.
With regard to the properties of the ground state entropy of the Potts
antiferromagnet on a given ($n \to \infty$ limit of a graph) $\{G\}$, two of
the most important properties of
$W(\{G\},q)$ and the associated nonanalytic locus ${\cal B}$ include (i)
the maximal finite value of $q$ at which ${\cal B}$ crosses the real axis in
the $q$ plane, which we denote $q_c$, as in our previous work; and (ii) the
question of whether ${\cal B}(q)$ is bounded or unbounded (noncompact).
Property (i) is important because in cases where ${\cal B}$ encloses regions,
the behavior of the ground state degeneracy $W(\{G\},q)$ changes qualitatively
(nonanalytically) as $q$ (taken to be real) decreases through
$q_c$. \cite{exception}.
Concerning the importance of property (ii), we first recall that
since an obvious upper bound on $P(G,q)$
describing the coloring of an $n$-vertex graph with $q$ colors is
$P(G,q) \le q^n$, and hence $W(\{G\},q) \le q$, it is natural to define a
reduced function that is bounded as $q \to \infty$:
\begin{equation}
W_{red.}(\{G\},q) = q^{-1}W(\{G\},q)
\label{wr}
\end{equation}
(This function was denoted $W_r(\{G\},q)$ in our Refs. \cite{w,wa}.)
If and only if $W_{red.}(\{G\},q)$ is analytic at $z=1/q=0$, there exists a
Taylor
series expansion of this function around this point. Such large--$q$ series
expansions are a useful means for obtaining approximate values of the
ground state degeneracy of the $q$-state Potts antiferromagnet for finite $q$
on regular lattices \cite{nagle,baker,kewser,ww,wn}. In turn,
$W_{red.}(\{G\},q)$
is nonanalytic at $1/q=0$ if and only if the locus ${\cal B}$
is unbounded in the $q$ plane, i.e., passes through the origin of the $1/q$
plane. Thus, families of graphs with unbounded ${\cal B}(q)$ do not have
large--$q$ Taylor series expansions for $W_{red.}(\{G\},q)$.
In Ref. \cite{w} we drew attention to this for bipyramid graphs
\cite{read91} and in Ref. \cite{wa}
we presented a general method for constructing families of graphs with
unbounded, noncompact loci ${\cal B}$. We also presented a general (necessary
and sufficient) algebraic condition such that ${\cal B}$ is unbounded.
Clearly it is important to understand better the differences between
the families of graphs that yield $W_{red.}(\{G\},q)$ functions analytic at
$1/q=0$ and those that produce $W_{red.}(\{G\},q)$ functions that are
nonanalytic at $1/q=0$. Since ${\cal B}$ forms by merging of
chromatic zeros of $G$ as $v(G) \to \infty$, a necessary condition for
${\cal B}$ to be noncompact in the $q$ plane, extending infinitely far from the
origin, is that as $v(G) \to \infty$, the magnitudes $|q|$ of some chromatic
zeros must increase without bound. That this is not a sufficient condition is
illustrated by the chromatic zeros of the complete graph $K_p$
\cite{complete}: these occur at $q=0,1,..,p-1$ so that, as $p \to \infty$, the
magnitude of the largest chromatic zero grows without bound, but the locus
${\cal B}$ for this family is not only not noncompact, it is trivial,
consisting of the empty set.
In the present paper, continuing our earlier studies
\cite{w,wa}, we shall address this problem. We shall present several
additional methods for constructing families of graphs with unbounded
${\cal B}$. We shall use these to obtain exact
calculations of the corresponding chromatic polynomials, $W$ functions, and
boundaries ${\cal B}$. In particular, we shall generate an infinite number of
such families via the method of homeomorphic expansion. Homeomorphic classes
of graphs have been of continuing interest in graph theory \cite{biggsbook};
for some recent theorems in a somewhat different direction from
the present work, see Ref. \cite{rw}.
The study of these families is valuable because it
enables one gain further insight into the origin of the nonanalyticity of
$W_r(\{G\},q)$ at $1/q=0$ and the consequent nonexistence of large--$q$ series
expansions for the ground state degeneracy of the Potts antiferromagnet on such
graphs. Here we shall give detailed analyses of families with loci ${\cal B}$
that have a simple structure in the vicinity of $1/q=0$. In a companion
paper, we shall present results for families of
graphs with loci ${\cal B}$ that are more complicated in the vicinity of
$1/q=0$.
Before proceeding, we recall two subtleties in the definition of
$W(\{G\},q)$ when $q$ is not a positive integer \cite{w}. First,
for certain ranges of real $q$, $P(G,q)$ can be negative, and, of course,
when $q$ is complex, so is $P(G,q)$ in general; in these cases it is not
obvious, {\it a priori}, which of the $n$ roots
\begin{equation}
P(G,q)^{1/n} = \{ |P(G,q)|^{1/n}e^{i \arg(P(G,q))/n}e^{2\pi i r/n} \} \ ,
\quad r=0,1,...,n-1
\label{pphase}
\end{equation}
to choose in eq. (\ref{w}).
Consider the function $W(\{G\},q)$ defined via eq. (\ref{w})
starting with $q$ on the positive real axis where $P(G,q) > 0$, and consider
the maximal region in the complex $q$ plane that can be reached by analytic
continuation of this function. We denote this region as $R_1$. Clearly, the
phase choice in (\ref{pphase}) for $q \in R_1$ is that given by $r=0$.
For families of graphs $\{G\}$ where there are regions $R_i$ of analyticity of
$W(\{G\},q)$ that are not analytically connected with $R_1$, there is no
canonical choice of phase in eq. (\ref{pphase}) and hence it is only
possible to determine the magnitude $|W(\{G\},q)|$ unambiguously.
A second subtlety in the definition of $W(\{G\},q)$ concerns the fact that
at certain special points $q_s$, the following limits do not commute \cite{w}
(for any choice of $r$ in eq. (\ref{pphase})):
\begin{equation}
\lim_{n \to \infty} \lim_{q \to q_s} P(G,q)^{1/n} \ne
\lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}
\label{wnoncomm}
\end{equation}
The set $q_s$ includes points where $P(G,q)$ has zeros whose multiplicity
does not scale like the number of vertices \cite{w} (typically these are simple
zeros). Following our earlier work \cite{w}, we define
\begin{equation}
W(\{G\},q_s) \equiv W(\{G\},q_s)_{D_{qn}} \equiv
\lim_{q \to q_s} \lim_{n \to \infty} P(G,q)^{1/n}
\label{wdefqn}
\end{equation}
This definition has the advantage of maintaining the analyticity of
$W(\{G\},q)$ at the special points $q_s$.
A general form for the chromatic polynomial of an $n$-vertex graph $G$ is
\begin{equation}
P(G_n,q) = c_0(q) + \sum_{j=1}^{N_a} c_j(q)a_j(q)^{t(n)}
\label{pgsum}
\end{equation}
where
\begin{equation}
t(n) = t_1 n + t_0
\label{tn}
\end{equation}
and $c_j(q)$ and $a_j(q)$ are certain functions of $q$. Here the $a_j(q)$ and
$c_{j \ne 0}(q)$ are independent of $n$, while $c_0(q)$ may contain
$n$-dependent terms, such as $(-1)^n$, but does not grow with $n$ like
$(const.)^n$.
Obviously, the reality of $P(G,q)$ for real $q$ implies that $c_j(q)$ and
$a_j(q)$ are real for real $q$. The condition that ${\cal B}$ does not
extend to infinite distance from the origin in the $q$ plane is
equivalent to the condition that for sufficiently large $|q|$, there is one
leading term $a_j(q)$ in eq. (\ref{pgsum}). Here we recall that ``leading
term $a_\ell(q)$ at a point $q$'' was defined in Ref. \cite{w} as a term
satisfying $|a_\ell(q)| \ge 1$ and $|a_\ell(q)| > |a_j(q)|$ for $j \ne \ell$.
(If the $c_0$ term is absent and $N_a=1$, then the sole $a_1(q)$ may be
considered to be leading even if $|a_j(q)| < 1$.) In the limit as $n \to
\infty$ (i.e., for fixed $p$, $r \to \infty$), the leading term $a_\ell$ in a
given region determines the limiting function $W$, with $|W|=|a_\ell|^{t_1}$
and the boundary ${\cal B}$ occurs where there is a nonanalytic change in $W$
as it switches between being determined by different leading terms $a_\ell$
in eq. (\ref{pgsum}).
Since for the families of graphs studied here, the boundary ${\cal B}$
is noncompact in the $q$ plane, it is often more convenient to describe the
boundary in the complex $z$ or $y$ planes, where
\begin{equation}
z \equiv \frac{1}{q}
\label{z}
\end{equation}
and
\begin{equation}
y \equiv \frac{1}{q-1}
\label{y}
\end{equation}
The variable $y$ is commonly used in large--$q$ series expansions. Some useful
relations between these variables are
\begin{equation}
q = 1 + \frac{1}{y} \ , \qquad z = \frac{y}{1+y} \ , \qquad y = \frac{z}{1-z}
\label{zyrel}
\end{equation}
We define polar coordinates as
\begin{equation}
z = \zeta e^{i\theta}
\label{zpolar}
\end{equation}
and
\begin{equation}
y = \rho e^{i\beta}
\label{ypolar}
\end{equation}
We shall
use some standard notation from graph theory and combinatorics:
$v(G)$ and $e(G)$ denote the numbers of vertices and edges in the graph $G$,
and the symbol $q^{(n)}$ is given by
\begin{equation}
q^{(p)} = p! {q \choose q-p} = \prod_{j=0}^{p-1}(q-s)
\label{ff}
\end{equation}
The terms ``edge'' and ``bond'' will be used synonymously.
For any partition of $n$, $n=p+u$,
\begin{equation}
q^{(p)}(q-p)^{(u)} = q^{(u)}(q-u)^{(p)} = q^{(p+u)}
\label{ffidentity}
\end{equation}
For our results below, it will also be convenient to define the polynomial
\begin{equation}
D_k(q) = \frac{P(C_k,q)}{q(q-1)} = a^{k-2}\sum_{j=0}^{k-2}(-a)^{-j} =
\sum_{s=0}^{k-2}(-1)^s {{k-1}\choose {s}} q^{k-2-s}
\label{dk}
\end{equation}
where
\begin{equation}
a=q-1
\label{a}
\end{equation}
and $P(C_k,q)$ is the chromatic polynomial for the circuit graph,
\begin{equation}
P(C_k,q) = a^k + (-1)^ka
\label{pck}
\end{equation}
Some useful properties of $D_k$ are listed in an appendix.
The organization of the paper is as follows. In Section II we review some
methods and results from our previous work on families of the form
$(K_p)_b + G_r$ that will be necessary for our current discussion (where $K_p$
is the complete graph on $p$ vertices \cite{complete} and,
following standard notation in the mathematical literature on graph theory, we
use the symbol $G+H$ for the ``join'' of these graphs; see below for
definitions). Section III contains another method of constructing
families of graphs with noncompact $W$
boundaries ${\cal B}$ involving the removal of bonds not just from one vertex
of the $K_p$ subgraph in $K_p + G_r$. In Section IV we present and discuss a
number of ways to construct families of graphs with noncompact $W$ boundaries
based on homeomorphic expansion of starting sets of graphs. In Section V we
analyze two such methods involving homeomorphic expansion of the $K_p$ subgraph
in a larger graph. Sections VI-VIII are detailed analyses of the respective
chromatic polynomials, $W$ functions, and their boundaries of regions of
analyticity ${\cal B}$ for families of graphs obtained by specific
homeomorphic expansions. In Section IX we include some remarks on general
geometrical features of the families of graphs with noncompact $W$ boundaries
${\cal B}$. Finally, Section X contains our conclusions.
\section{Families of the Form
$(K_{\lowercase{p}})_{\lowercase{b}}+G_{\lowercase{r}}$ }
Since we will construct and analyze a number of new families of graphs with
noncompact boundaries ${\cal B}$ as homeomorphic expansions of families
studied in Ref. \cite{wa}
it is first necessary to review briefly the method that we formulated and
used in Ref. \cite{wa} to generate such families. Consider a family of
graphs $G$ and its $v(G) \to \infty$ limit, $\{G\}$ .
If this family already has the
property that the limiting function $W(\{G\},q)$ has a region boundary
${\cal B}$ that extends to complex infinity in the $q$ plane, then we have no
work to do to get such a boundary. So assume that this family is such that
$W(\{G\},q)$ has a region boundary ${\cal B}$ that does not extend to complex
infinity in the $q$ plane. Specifically, we start with an
$r$-vertex graph $G_r$ and then form the graph
$K_p + G_r$, where, as above, $K_p$ denotes the complete graph on $p$
vertices \cite{complete}, and we adopt the notation $G + H$ that is commonly
used in graph theory to indicate that every vertex of $G$ is connected by a
bond to every vertex of $H$; this is called the ``join'' of $G$ and $H$
in the mathematical literature. We then remove $b$ bonds
from one vertex in the $K_p$ subgraph; the resultant family is denoted
\begin{equation}
(K_p)_b + G_r
\label{kpbgr}
\end{equation}
(In Ref. \cite{wa}
we used the notation $G \times H$ for what is called $G+H$ here,
and the notation $(K_p \times G_n)_{rb}$ for the family denoted
$(K_p)_b + G_r$ above, where the subscript $rb$ signified ``removed bonds''.)
Since each such vertex has
degree $\Delta = p-1$ (i.e., has $p-1$ bonds connecting to it), it follows that
\begin{equation}
1 \le b \le p-1
\label{bcondition}
\end{equation}
It was proved in Ref. \cite{wa} that in the
limit $r \to \infty$, the family $(K_p)_b + G_r$ has a noncompact $W$ region
boundary ${\cal B}(q)$, extending infinitely
far from the origin of the $q$ plane. A number of families of this type were
constructed and the chromatic polynomials, resultant $W$ functions, and
boundaries ${\cal B}$ calculated \cite{wa}. Specifically, by means of the
relation
\begin{equation}
P(K_p + G_r,q) = q^{(p)}P(G_r,q-p) \ ,
\label{pkpg}
\end{equation}
the chromatic polynomial of $(K_p)_b + G_r$ was calculated to be \cite{wa}
\begin{eqnarray}
P((K_p)_b + G_r,q) =
P(K_p + G_r,q) + b P(K_{p-1} + G_r,q) \qquad \qquad \qquad \qquad
\qquad \qquad \cr\cr
= q^{(p-1)}\Bigl [ (q-(p-1))P(G_r,q-p) +b P(G_r,q-(p-1)) \Bigr ]
\label{pkpcutb}
\end{eqnarray}
Substituting the expression (\ref{pgsum}) yields
\begin{eqnarray}
P((K_p)_b + G_r,q) = q^{(p-1)}\biggl [(q-(p-1)) \Big \{ c_0(q-p) +
\sum_{j=1}^{N_a} c_j(q-p)a_j(q-p)^{t(r)} \Big \} \cr
+ b \Big \{ c_0(q-(p-1)) +
\sum_{j=1}^{N_a} c_j(q-(p-1))a_j(q-(p-1))^{t(r)} \Big \} \biggr ]
\label{pkpcutbfull}
\end{eqnarray}
For a given $q$, the boundary ${\cal B}$ as $r \to \infty$ (with $p$ fixed)
is determined by the degeneracy in
magnitude of the leading terms in eq. (\ref{pkpcutbfull}) at this value of $q$,
viz.,
\begin{equation}
|a_\ell(q-p)| = |a_\ell(q-(p-1))|
\label{mageq}
\end{equation}
This equation, and hence also ${\cal B}$ are independent of $b$ for the
interval (\ref{bcondition}), $1 \le b \le p-1$. In passing, we note that since
the total number of vertices of $(K_p)_b + G_r$ is $v((K_p)_b + G_r)=p+r$,
another way to take $n \to \infty$ is to let $p \to \infty$ with $r$ fixed,
rather than letting $r \to \infty$ with $p$ fixed. However, from the viewpoint
of either statistical mechanics or graph theory, this is not as interesting a
limit, since for any given graph $G_r$ and for any given (finite) value of
$q \in {\mathbb Z}_+$, as $p$ becomes sufficiently large, one will not be able
to color the graph $(K_p)_b + G_r$ and the chromatic polynomial will vanish.
We shall therefore restrict ourselves to the other way of letting the number of
vertices go to infinity, viz., $r \to \infty$ with $p$ fixed.
Writing
\begin{equation}
a_\ell = \sum_{s=0}^{s_{max}}\alpha_{\ell,s}q^s \ ,
\label{aellsum}
\end{equation}
recalling that the basic theorem that the coefficient of the
highest-order term, $q^n$, in the chromatic polynomial $P(G_n,q)$ of any
$n$-vertex graph $G_n$ is unity implies that
\begin{equation}
\alpha_{s_{max}}=1 \ ,
\label{alphasmax}
\end{equation}
next dividing eq. (\ref{mageq}) by $|q^{s_{max}}|$, and finally
reexpressing the degeneracy equation in the more convenient variable $z$
given in eq. (\ref{z}), we obtain
\begin{eqnarray}
& & |(1-pz)^{s_{max}} +
\sum_{s=0}^{s_{max}-1}\alpha_{\ell,s}(1-pz)^sz^{s_{max}-s}| \cr
& = &
|(1-(p-1)z)^{s_{max}} +
\sum_{s=0}^{s_{max}-1}\alpha_{\ell,s}(1-(p-1)z)^sz^{s_{max}-s}|
\label{mageqz}
\end{eqnarray}
This equation is clearly satisfied for $z=0$, which shows that the boundary
${\cal B}$ for the $r \to \infty$ limit of the family of graphs
$(K_p)_b + G_r$ is noncompact in the $q$ plane, passing through $z=1/q=0$.
As noted above, it is thus more convenient to analyze ${\cal B}$ in the $z$
plane. We next show that ${\cal B}$ passes vertically
through the origin of the $z$ plane for the $r \to \infty$ limit of all of the
families of the type $(K_p)_b + G_r$. Writing eq. (\ref{mageqz}) in polar
coordinates, using (\ref{zpolar}), we have, for $|z|=\zeta \to 0$,
\begin{equation}
\zeta s_{max} \cos \theta + O(\zeta^2) = 0
\label{rsmax}
\end{equation}
Since $s_{max} \ne 0$, it follows that
\begin{equation}
\cos \theta \to \pm \frac{\pi}{2} \quad {\rm as} \quad \zeta \to 0
\label{vertical}
\end{equation}
which proves the above assertion, that ${\cal B}$ passes vertically through the
point $z=0$.
The simplest families of graphs of the form $(K_p)_b + G_r$ are those for which
$P(G_r,q)$ has, in the notation of eq. (\ref{pgsum}), $c_0=0$, $N_a=1$, and
$a_1(q)$ a linear function of $q$,
\begin{equation}
a_1(q) = q + \alpha_0
\label{a1}
\end{equation}
For such families, in the $q$ plane, ${\cal B}$ is a vertical line with
$q_{_I}$ arbitrary and
\begin{equation}
q_{_R} = q_c = p - (\frac{1}{2} + \alpha_0) \quad {\rm for} \quad s_{max}=1
\label{qrsmax1}
\end{equation}
where $Re(q)=q_{_R}$ and $Im(q)=q_{_I}$.
In the $z$ plane, ${\cal B}$ is the circle defined by
\begin{equation}
|z-\frac{z_c}{2}| = \frac{z_c}{2}
\label{circlemag}
\end{equation}
with
\begin{equation}
z_c = \frac{1}{q_c} = \frac{1}{p - (\frac{1}{2} + \alpha_0)}
\label{zcsmax1}
\end{equation}
Equivalently, ${\cal B}$ is the circle $|y-\frac{y_c}{2}| = \frac{y_c}{2}$ in
the $y$ plane, where $y_c=1/(p-(\frac{3}{2}+\alpha_0))$.
Thus, ${\cal B}$ divides the $z$ (equivalently $q$ or $y$) plane into two
regions: (i) $R_1$, as defined above to include the region on the positive
$z$ axis contiguous with the origin, and (ii) its complement, denoted $R_2$.
We find that
\begin{equation}
W(\lim_{r \to \infty}[(K_p)_b + G_r],q) =
q-p+1+\alpha_0 \quad {\rm for} \quad q \in R_1
\label{wkpgrsmax1region1}
\end{equation}
\begin{equation}
|W(\lim_{r \to \infty}[(K_p)_b + G_r],q)| =
|q-p+\alpha_0| \quad {\rm for} \quad q \in R_2
\label{wkpgrsmaxregion2}
\end{equation}
These families include the cases $G_r=$ (i) edgeless graphs
$\overline K_r$; (ii) tree graphs $T_r$; and (iii) chains of triangles with
each pair of adjacent triangles intersecting on a mutual edge, i.e.
$(Ch)_{3,r}$ in the notation of Ref. \cite{wa} (The general chain graph of
$n_p$ $k$-sided polygons with adjacent $k$-gons intersecting on a mutual
edge is denoted $(Ch)_{k,r}$; the number of vertices, $r$, is given by
$r=(k-2)n_p+2$.) The chromatic polynomials for these families are
\begin{equation}
P((K_p)_b + \overline K_r ,q) = q^{(p)}\Bigl [ (q-p)^r + b(q-(p-1))^{r-1}
\Bigr ]
\label{pkper}
\end{equation}
\begin{equation}
P((K_p)_b + T_r,q) = q^{(p+1)}\Bigl [ (q-(p+1))^{r-1} + b(q-p)^{r-2}
\Bigr ]
\label{pkptr}
\end{equation}
\begin{equation}
P((K_p)_b + (Ch)_{k,r},q) =
q^{(p+1)}\Bigl [ (q-(p+1))D_k(q-p)^{n_p} + bD_k(q-(p-1))^{n_p} \Bigr ]
\label{pkpchainkr}
\end{equation}
Thus
\begin{equation}
\alpha_0 = 0, \ -1, \ -2 \quad {\rm for} \quad G_r = \overline K_r, \ T_r, \
(Ch)_{3,r}
\label{alpha0values}
\end{equation}
so that
\begin{equation}
q_c = p-\frac{1}{2} \ , \ p+\frac{1}{2} \ , \ p+\frac{3}{2} \ \quad {\rm for}
\quad G_r = \overline K_r \ , \ T_r \ , (Ch)_{3,r}
\label{qcthreecases}
\end{equation}
The boundaries ${\cal B}$ for other families have the same behavior near the
origin, $z=0$, as shown by eq. (\ref{rsmax}), but have different features where
they cross the positive real $z$ axis. For example, consider $G_r=C_r$,
which has, in the notation of eq. (\ref{pgsum}), $N_a=1$ and a linear
$a_1(q)=q-1$ but also a nonzero $c_0$ term, viz., $c_0=(-1)^r(q-1)$.
For $\lim_{r \to \infty}[(K_p)_b + C_r]$, the portion
of ${\cal B}$ near to $z=0$ forms part of the circle (\ref{circlemag}), but to
the right of the points $z_{int.}=1/q_{int.}$ and $z_{int.}^*$, where
\begin{equation}
q_{int.} = p + \frac{1}{2} + \frac{i\sqrt{3}}{2}
\label{qint}
\end{equation}
the boundary bifurcates into two arcs, which cross the positive $z$ axis at
\begin{equation}
z_c = \frac{1}{q_c} = \frac{1}{p+1} \quad {\rm for} \quad G_r = C_r
\label{qckpcr}
\end{equation}
and at $z=1/p$. Since we shall compare this in detail with homeomorphic
expansions of this family, we show this boundary in Fig. \ref{bipzplane}. For
this and similar plots given later we also show chromatic zeros calculated for
a reasonably large finite value of $r$ (in this case, $r=27$. The comparison
of these chromatic zeros with the $r \to \infty$ locus ${\cal B}$ shows
quantitatively how ${\cal B}$ forms via the merging of the zeros. Of course,
since for finite $r$, the chromatic polynomial is of finite degree (equal to
the number of vertices on the graph), its zeros (i.e., the chromatic zeros of
the graph) are of bounded magnitude in $q$ and hence do not track the
asymptotic curve ${\cal B}$ all the way in to the origin of the $z=1/q$ plane.
The same comment applies to the chromatic zeros that we will show for other
families below.
\begin{figure}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{bipzplane.ps}
\end{center}
\vspace{-1cm}
\caption{\footnotesize{Boundary ${\cal B}$ in the $z=1/q$ plane for the $r \to
\infty$ limit of the family of graphs $(K_2)_{b=1} + C_r = \overline K_2 +
C_r$. Chromatic zeros for $r=27$ are shown for comparison.}}
\label{bipzplane}
\end{figure}
Similar comments apply for other families such as
$(K_p)_b + L_r$ and for the $k \ge 4$ members of the family
$(K_p)_b + (Ch)_{k,r}$ \cite{wa}, where $L_r$ denotes
a cyclic square strip of width one square, i.e. cyclic ladder graph, and
$(Ch)_{k,r}$ was defined above in our discussion of the special case $k=3$.
Of the families of graphs that we have discussed,
with noncompact boundaries ${\cal B}$ of regions
of analyticity in $W$, most have chromatic polynomials that depend on three
parameters, $p$, $r$, and $b$, and the resultant ${\cal B}$ for the limit
$r \to \infty$ depends on one
parameter, $p$ (and is independent of $b$). The family
$(K_p)_b + (Ch)_{k,r}$ illustrates a case where the chromatic
polynomial depends on four parameters -- \ $p$, $k$, $r$, and $b$ -- and
${\cal B}$ thus depends on two parameters: $p$ and $k$.
Here we present results on another such family,
\begin{equation}
G_{p,b,s,u} = (K_p)_b + sK_u
\label{gpbsu}
\end{equation}
where $sK_u$ denotes the disjoint union of
$s$ copies of the $K_u$ graph (where the vertices of each of the $s$ $K_u$
subgraphs are not connected to each other by bonds).
That is, one adjoins $K_p$ to $G_r = sK_u$ and then removes $b$ bonds from
one vertex of the $K_p$ subgraph. An illustration of the special case
$p=2$, $b=1$, $s=3$, and $u=2$ is shown in Fig. \ref{vhegraph1}(d).
\begin{figure}
\centering
\leavevmode
\epsfxsize=3.5in
\epsffile{vhegraph1.ps}
\caption{\footnotesize{Illustrations of families of graphs considered in the
text: (a) $T_{k,r} = HEG_{k-2}(\overline K_2 + T_r)$;
(b) $C_{k,r} = HEG_{k-2} (\overline K_2 + C_r)$;
(c) $HEG_{k-2}(\overline K_3 + T_r)$; all for $k=3$ and $r=4$;
(d) $\overline K_2 + sK_u$ for $s=3$ and $u=2$. }}
\label{vhegraph1}
\end{figure}
\vspace{4mm}
Clearly,
the number of vertices in the disjoint union $sK_u$ is $r=v(sK_u)=su$, so that
\begin{equation}
n=v((K_p)_b + sK_u)=p+su
\label{vkpsku}
\end{equation}
For $s=1$ and $b=0$, one has the simplification $K_p + K_u = K_{p+u}$.
Henceforth, we shall assume that $b$ satisfies the restriction
(\ref{bcondition}). For general $s$ and $u=1$,
\begin{equation}
(K_p)_b + sK_1 = (K_p)_b + \overline K_s
\label{rk1er}
\end{equation}
so that our analysis for the latter family applies. For general $u$ and $s$,
from eq. (\ref{pkpcutb}) we have
\begin{equation}
P((K_p)_b + sK_u,q) = q^{(p-1)}\Bigl \{ (q-(p-1))[(q-p)^{(u)}]^s
+ b [(q-(p-1))^{(u)}]^s \Bigr \}
\label{pkprksff}
\end{equation}
This chromatic polynomial has the form
\begin{equation}
P((K_p)_b + sK_u,q) = q^{(p)}\Bigl [ (q-p)^{(u-1)} \Bigr ]^s P_{b,s}
\label{pkprkub}
\end{equation}
where $P_{b,s}$ is a polynomial in $q$ of degree $s$.
Now consider the limit $s \to \infty$ with $p$ and $u$ fixed. Equation
(\ref{pkprksff}) has the form of eq. (\ref{pgsum}), so that
the degeneracy equation is
\begin{equation}
|(q-p)^{(u)}| = |(q-(p-1))^{(u)}|
\label{degeqrks}
\end{equation}
Taking into account that $q=0$ is never a solution of eq. (\ref{degeqrks}) and
dividing by $|q|^u$, we obtain the corresponding degeneracy equation of leading
terms in the $z$ plane:
\begin{equation}
|\prod_{j=0}^{u-1}(1-(p+j)z)| = |\prod_{j'=0}^{u-1}(1-(p-1+j')z)|
\label{degeqrksz}
\end{equation}
By the same reasoning as before, it follows that the resultant locus
${\cal B}$ which is the solution to the degeneracy equation is noncompact
in the $q$ plane, passing through $z=1/q=0$.
The general solution to the degeneracy equation (\ref{degeqrks}) in the $q$
plane is the vertical line given by
\begin{equation}
Re(q) = q_c = p -1 + \frac{u}{2}
\label{bkpku}
\end{equation}
or equivalently, in the $z$ plane, the circle
\begin{equation}
|z - \frac{z_c}{2}| = \frac{z_c}{2} \ , \quad z_c =
\frac{1}{p-1+\frac{u}{2}}
\label{bkpkuz}
\end{equation}
and similarly, in the $y$ plane, the circle
$|y-\frac{y_c}{2}|=\frac{y_c}{2}$ where $y_c = 1/(p-2+\frac{u}{2})$.
The $q$ (or $z$ or $y$) plane is divided into two regions by the boundary
${\cal B}$, namely, $R_1$ with $Re(q) > p-1+\frac{u}{2}$, and $R_2$, with
$Re(q) < p-1+\frac{u}{2}$.
The dependence of $q_c$ on $u$ is easy to understand: clearly, just as
increasing $p$ reduces the number of colors available to color the rest of the
graph and hence increases the value of $q_c$ (serving as the demarcation on the
real axis between the region of $q$ values with $W > 1$ and nonzero ground
state entropy $S_0 = k_B \ln W$, from the region where the behavior of $W$ is
different), so also increasing $u$ for fixed $p$ has the same effect.
We find that
\begin{equation}
W(\lim_{s \to \infty}[(K_p)_b + sK_u],q) = [(q-(p-1))^{(u)}]^{1/u} \quad
{\rm for} \quad q \in R_1
\label{wkpkubregion1}
\end{equation}
where we recall the definition (\ref{ff}). Further,
\begin{equation}
|W(\lim_{s \to \infty}[(K_p)_b + sK_u],q)| = |(q-p)^{(u)}|^{1/u}
\quad {\rm for} \quad q \in R_2
\label{wkpkubregion2}
\end{equation}
It follows that
\begin{equation}
W(\lim_{s \to \infty}[(K_p)_b + sK_u],q=q_c) = 0 \quad {\rm if}
\quad q_c\in {\Bbb Z}_+ \quad {\rm i.e. \ \ if} \quad u \quad {\rm is \ even}
\label{wboundaryqcz}
\end{equation}
Note that the zeros in $W$ extend into region $R_1$, to the right of $q_c$, if
$p+u-2 > q_c$, i.e, if $u \ge 3$. For example, for the case $u=3$, for which
$q_c = p + \frac{1}{2}$, the expression (\ref{wkpkubregion1}) has formal
zeros at $q=p-1$, $q=p$, and $q=p+1$. The first two of these are not
relevant since they lie to the left of $q_c$ in the region $R_2$
where the expression (\ref{wkpkubregion1}) does not apply; however, the third
zero, at $q=p+1$, is in region $R_1$ and is a true zero of $W$. For this
illustrative value $u=3$, the expression (\ref{wkpkubregion2}) has
formal zeros at $q=p$, $q=p+1$, and $q=p+2$. The last two of these are not
relevant since they lie to the right of $q_c$ in region $R_1$ where the
expression (\ref{wkpkubregion2})
does not apply, while the first one, at $q=p$, lies in region $R_2$
where this expression does apply and hence is a true
zero of $W$ in this region.
It is interesting to comment on the comparative values of $q_c$ defined in the
limit as $r \to \infty$ and the chromatic number $\chi$ for specific members of
a family of graphs parametrized by $r$. Before doing this, one should note
that there are cases where ${\cal B}$ is trivial or does not cross the real
axis, so that no $q_c$ is defined; an example is given by tree graphs $T_r$,
for which ${\cal B}=\emptyset$ and $\chi=2$. For circuit graphs $C_r$,
$\chi=2$ if $r$ is even and $\chi=3$ if $r$ is odd. In the limit $r \to
\infty$, ${\cal B}$ is the circle $|q-1|=1$, so that $q_c=2$; hence this family
illustrates the possibilities $q_c=\chi$ and $q_c < \chi$. For the square
lattice with free boundary conditions or (periodic or toroidal) boundary
conditions that preserve the bipartite nature of the lattice, $\chi=2$ while in
all cases $q_c=3$ \cite{w}, so that this illustrates the possibility $q_c >
\chi$. For the infinitely long strip of the square lattice with width
$L_y=3$ vertices ($w=L_y-1$ squares), $q_c=2$, equal to $\chi$ for this
family; for the
infinitely long strip of the triangular lattice with $L_y=3$, $q_c \simeq
2.57$, less than the chromatic number $\chi=3$ for this family, while for the
infinitely long triangular strip with width $L_y=4$, ${\cal B}$ does not cross
the real $q$ axis, so no $q_c$ is defined \cite{w,strip}. Clearly,
there is a large variety of behavior as regards the question of whether $q_c$
for the infinite-vertex limit of a family of graphs is smaller than, equal to,
or larger than the chromatic number $\chi$ for specific members of the family.
As an illustration of the families discussed here,
$\overline K_2 + \overline K_r$ has $\chi=2$ and $q_c=3/2 < \chi$;
$\overline K_2 + T_r$ has $\chi=3$ and $q_c=5/2 < \chi$, and
$\overline K_2 + (Ch)_{3,r}$ has $\chi=4$ and $q_c=7/2 < \chi$. We shall
discuss the situation for homeomorphic expansions of graphs in these families
below.
\vspace{10mm}
\section{Families of the Form $(K_{\lowercase{p}})_{ \{ \lowercase{b} \}} +
G_{\lowercase{r}}$}
Another way to construct families with noncompact $W$ boundaries ${\cal B}$ is
to remove bonds not just from one vertex of the $K_p$ subgraph but also from
one or more other vertices of this subgraph that are not adjacent to the first
vertex, i.e., are not connected to this first vertex by any bonds. We
symbolize this family by
\begin{equation}
(K_p)_{ \{b \}} + G_r
\label{kpgmultipleb}
\end{equation}
where the subscript $\{b\}$ refers to the removal of multiple bonds from
non-adjacent vertices of the $K_p$ subgraph.
For this family, the maximum number of bonds
that can be removed from the $K_p$ subgraph is the number of bonds in
this subgraph, namely,
\begin{equation}
e(K_p) = {p \choose 2} = \frac{p(p-1)}{2}
\label{ekp}
\end{equation}
In general, the chromatic polynomial for such families is not quite
as easily derived as that for the type of families where all bonds removed are
from one vertex, where we obtained the simple formula (\ref{pkpcutb})
\cite{wa}.
A particularly simple family of graphs
is obtained when one removes all of the
$p \choose 2$ bonds in the $K_p$ subgraph, thereby yielding the subgraph
$\overline K_p$ and the resultant family
\begin{equation}
\overline K_p + G_r
\label{epg}
\end{equation}
Clearly, for $p=2$,
\begin{equation}
(K_2)_{b=1} + G_r = \overline K_2 + G_r
\label{e2grrel}
\end{equation}
For the general family $\overline K_p + G_r$, one can use the
addition-contraction \cite{thm}
theorem to reexpress $P(\overline K_p + G_r,q)$ as the linear combination
\begin{eqnarray}
P(\overline K_p + G_r,q) & = & \sum_{j=1}^p {\cal S}_p^{(j)}P(K_j+G_r,q) \cr
& = & \sum_{j=1}^p {\cal S}_p^{(j)} q^{(j)}P(G_r,q-j)
\label{pepk}
\end{eqnarray}
where the coefficients ${\cal S}_p^{(j)}$ are the Stirling numbers of the
second kind, defined by the equation (cf. eq. (\ref{ff}))
\begin{equation}
q^p = \sum_{j=1}^p {\cal S}_p^{(j)} q^{(j)}
\label{qpff}
\end{equation}
A closed-form expression is \cite{combin}
\begin{equation}
{\cal S}_p^{(j)} = \frac{1}{j!} \sum_{k=0}^j (-1)^{j-k}{j \choose k} k^p
\label{stirlings2}
\end{equation}
Given that $P(G,q)$
has the form (\ref{pgsum}), it follows, by the same argument as
was used for the $(K_p)_b + G_r$ families with removal of $b$ bonds connecting
to a given vertex, that ${\cal B}$ is noncompact in the $q$ plane,
passing through $z=1/q=0$.
It is worthwhile to consider some special cases for $G_r$. For
$G_r= \overline K_r$, we calculate
\begin{equation}
P(\overline K_p + \overline K_r,q) =
\sum_{j=1}^p {\cal S}_p^{(j)}q^{(j)}(q-j)^r
\label{peper}
\end{equation}
Hence,
in the limit $r \to \infty$, ${\cal B}$ is determined by the degeneracy
equation
\begin{equation}
|q-1|=|q-p| \ , \quad i.e., \quad |1-z|=|1-pz|
\label{degeqeper}
\end{equation}
Clearly, $z=0$ is a solution, and ${\cal B}$ is, in the $q$ plane, the vertical
line with
\begin{equation}
Re(q) = q_c = \frac{p+1}{2}
\label{qceper}
\end{equation}
i.e., in the $z$ plane the circle
\begin{equation}
|z- \frac{z_c}{2}| = \frac{z_c}{2} \ , \quad z_c = \frac{2}{p+1}
\label{bzeper}
\end{equation}
Let us compare the values of $q_c$ for the respective $r \to \infty$ limits of
$\overline K_p + \overline K_r$ and $(K_p)_b + \overline K_r$ (where it is
understood in the latter case that $b$ bonds are removed from a
vertex in the $K_p$ subgraph).
For $p=2$, the value of $q_c$ in eq. (\ref{qceper}) is equal to that
for $G_r = \overline K_r$ in eq. (\ref{qcthreecases}),
as follows from eq. (\ref{e2grrel}). For $p \ge 3$, the
value of $q_c$ in eq. (\ref{qceper}) is smaller than that for
$G_r = \overline K_r$ in eq. (\ref{qcthreecases}). This is a consequence of
the fact that the graph $(K_p)_b + \overline K_r$ can be obtained from
$\overline K_p + \overline K_r$ by addition of bonds and hence, by the
theorem in section VII of Ref. \cite{wn},
the coloring of the former graph with $q$ colors is
more constrained than the coloring of the latter graph; the same theorem
implies that for $q \in {\mathbb Z}_+$,
\begin{equation}
W(\lim_{r \to \infty}[(K_p)_b + \overline K_r],q) \le
W(\lim_{r \to \infty}[\overline K_p + \overline K_r],q)
\label{wkpkrrel}
\end{equation}
In region $R_1$ to the right of the vertical line (\ref{qceper}) in the $q$
plane,
\begin{equation}
W(\lim_{r \to \infty}(\overline K_p + \overline K_r),q) = q-1
\quad {\rm for} \quad q \in R_1
\label{weperregion1}
\end{equation}
while in the complement, $R_2$,
\begin{equation}
|W(\lim_{r \to \infty}(\overline K_p + \overline K_r),q)| = |q-p|
\quad {\rm for} \quad q \in R_2
\label{weperregion2}
\end{equation}
A second case is $G_r=T_r$, i.e., the family $\overline K_p + T_r$. We find
\begin{equation}
P(\overline K_p + T_r,q) = \sum_{j=1}^p {\cal S}_p^{(j)}q^{(j)}
(q-j)(q-j-1)^{r-1}
\label{peptr}
\end{equation}
Hence, in the limit $r \to \infty$, ${\cal B}$ is determined by the degeneracy
equation
\begin{equation}
|q-2|=|q-p-1| \ , \quad i.e., \quad |1-2z|=|1-(p+1)z|
\label{degeqeptr}
\end{equation}
In the $q$ plane the locus of solutions to this equation is the vertical line
with
\begin{equation}
Re(q) = q_c=\frac{p+3}{2}
\label{qceptr}
\end{equation}
which again is a circle in the $z$ plane,
\begin{equation}
|z- \frac{z_c}{2}| = \frac{z_c}{2} \ , \quad z_c = \frac{2}{p+3}
\label{bzeptr}
\end{equation}
For $p=2$, the value of $q_c$ in eq. (\ref{qceptr}) is equal to that
for $G_r = T_r$ in eq. (\ref{qcthreecases}),
as follows from eq. (\ref{e2grrel}). For $p \ge 3$, $q_c$
is larger for $\lim_{r \to \infty}[(K_p)_b + T_r]$ with $b$ bonds removed from
one vertex in $K_p$ than for $\lim_{r \to \infty}[\overline K_p + T_r]$, for
the reason given above. For region $R_1$ to the right of the vertical line
(\ref{qceptr}) in the $q$ plane,
\begin{equation}
W(\lim_{r \to \infty}[\overline K_p + T_r],q) = q-2
\quad {\rm for} \quad q \in R_1
\label{weptrregion1}
\end{equation}
while in the complement, $R_2$,
\begin{equation}
|W(\lim_{r \to \infty}[\overline K_p + T_r],q)| = |q-p-1|
\quad {\rm for} \quad q \in R_2
\label{weptrregion2}
\end{equation}
\vspace{10mm}
\section{Construction of Families of Graphs with Noncompact ${\cal B}$ via
Homeomorphic Expansion}
A major result of the present paper is the construction of families of graphs
with noncompact $W$ boundaries by means of homeomorphic expansion ($HE$) (also
called inflation) of a beginning set of families of graphs. We recall the
definition that two graphs $G$ and $H$ are homeomorphic to each other if $H$,
say, can be obtained from $G$ by successive insertions of degree-2 vertices
on bonds of $G$ \cite{biggsbook}.
Each such insertion subdivides an existing edge of $G$ into two,
connected by the inserted degree-2 vertex. This process is called
homeomorphic expansion and its inverse is called
homeomorphic reduction, i.e. the successive removal of vertices of degree 2
from a graph $H$. Clearly, homeomorphic expansion of a graph always yields
another graph. The inverse is not necessarily true; i.e., homeomorphic
reduction of a graph can produce a multigraph or pseudograph instead of a
(proper) graph \cite{graphdef}. Here, a multigraph is a finite set of
vertices and bonds that, like a graph, has no bonds that loop around from a
given vertex back to itself but, in contrast to a (proper) graph,
may have more than one bond connecting two vertices. A pseudograph is a finite
set of vertices and bonds that may have multiple bonds connecting two
vertices and may also have looping bonds. For example, consider homeomorphic
reduction of a circuit graph $C_r$: removing one of the vertices (all of
which have degree 2), one goes from $C_r$ to $C_{r-1}$, and so forth, until one
gets to $C_3$. During this sequence of homeomorphic reductions, one remains
within the category of graphs. However, the next homeomorphic reduction takes
$C_3$ to $C_2$, which is a multigraph, not a proper graph. Removing one of the
two vertices in $C_2$ produces a pseudograph consisting of a single vertex and
a bond that goes out and loops back to this vertex. Thus homeomorphism is an
equivalence relation on pseudographs. This complication will not
be relevant here because we shall only use homeomorphic expansions, not
reductions, of graphs, and the homeomorphic expansion of a proper graph always
yields another proper graph. For our subsequent discussion, we shall
denote the homeomorphic expansion involving the insertion of $k-2$ additional
vertices (where $k \ge 3$)
on a specific bond $b$ of a graph $G$ as $HEG_{k-2;b}(G)$.
Most of our studies will be of graphs in which the homeomorphic expansion of
$G$ is performed systematically on each bond of $G$; in these cases, we shall
denote the resultant graph as $HEG_{k-2}(G)$.
The homeomorphic expansion of a family of graphs with a compact (empty or
nontrivial) locus ${\cal B}$ can produce a family of graphs with either a
compact or noncompact ${\cal B}$; this depends on the nature of the original
family and of the homeomorphic expansion.
For example, if one starts with the tree
graph $T_r$ and inserts $k-2$ degree-2 vertice on each of the bonds, one
obtains another tree graph, $T_{r'}$, where $r'=r+(r-1)(k-2)$. In the limit
$r \to \infty$, both $T_r$ and its homeomorphic expansion, $T_{r'}$,
have a trivial ${\cal B}=\emptyset$.
If one starts with the circuit graph $C_r$ and inserts $k-2$ degree-2 vertices
on each bond of this graph, one obtains another circuit graph, $C_{r'}$, where
$r'=r+r(k-2)$. In the limit $r \to \infty$, both $C_r$ and its homeomorphic
expansion $C_{r'}$ have a compact ${\cal B}$ given by the unit circle centered
at $q=1$, $|q-1|=1$. We next proceed to discuss the cases of main interest
here, where the homeomorphic expansion (i) leads from a family with a compact
${\cal B}$ to one with a noncompact ${\cal B}$ or (ii) takes a family that
already has a noncompact ${\cal B}$ to another that again has a noncompact
${\cal B}$. We comment on the effect of a homeomorphic expansion of a graph
$G$ on its girth $\gamma(G)$, defined as the length of (= number of vertices
on) a minimal-length closed circuit in this graph. Clearly, a homeomorphic
expansion applied to one or more bonds of a graph increases the girth of the
graph if and only if these bonds lie on the minimal-length circuits of the
original graph.
We shall consider four main types of homeomorphic expansions,
as well as combinations thereof:
\vspace{10mm}
\begin{flushleft}
1. \ $HEK0$
\vspace{6mm}
Start with the family $K_p + G_r$ (where no bonds have been removed from any
vertex of the $K_p$ subgraph). If in the limit $r \to \infty$, ${\cal B}$ for
the $G_r$ family itself is compact (bounded) in the $q$ plane, then the same
holds for the family $K_p + G_r$. Now insert degree-2 vertices on one or more
bonds of the $K_p$. We shall denote this generically as an $HEK0$ homeomorphic
expansion, meaning that the homeomorphic expansion
acts on the $K_p$ subgraph and that originally
there were $b=0$ bonds removed from this $K_p$. Specifically, we
denote the graph obtained by successive insertion of $k-2$ degree-2 vertices on
a single bond $b_j$ of the $K_p$ as
\begin{equation}
HEK_{k-2;b_j}(K_p + G_r)
\label{hekb0}
\end{equation}
where $k \ge 3$,
and so forth for similar homeomorphic insertions on other bonds of the $K_p$.
We will show below that this homeomorphic expansion leads from the family
$K_p + G_r$ with compact (trivial or nontrivial) ${\cal B}$ to the family
(\ref{hekb0}) with a locus ${\cal B}$ that is noncompact in the $q$ plane.
The labelling convention is chosen so that as one moves along the expanded set
of bonds linking a vertex of the original $K_p$ to what was originally an
adjacent vertex, one traverses a total of $k$ vertices, including the original
pair, i.e., $k-2$ inserted vertices.
\vspace{8mm}
Next, we consider cases (nos. 2-5 below)
where we begin with a family of graphs that already has
a locus ${\cal B}$ that is unbounded in the $q$ plane. For these the
homeomorphic expansion produces another family of graphs again with an
unbounded ${\cal B}$:
\vspace{8mm}
2. \ $HEK$
\vspace{6mm}
Start with the family $(K_p)_b + G_r$ where $b$ bonds have been removed from
one vertex of the $K_p$ subgraph and insert degree-2 vertices on one or more
bonds of the $K_p$. We shall denote this generically as an $HEK$ homeomorphic
expansion, meaning that the homeomorphic expansion
acts on the $K_p$ subgraph. Specifically, we
denote the graph obtained by successive insertion of $k-2$ degree-2 vertices on
a single bond of the $K_p$ as
\begin{equation}
HEK_{k-2;b_j}[(K_p)_b + G_r]
\label{hekb}
\end{equation}
where again $k \ge 3$,
and so forth for similar insertions on other bonds of the $K_p$ subgraph.
\vspace{8mm}
3. \ $HEG$
\vspace{6mm}
Start with either $(K_p)_b + G_r$ or $(K_p)_{\{b\}} + G_r$
and add vertices to bonds in the $G_r$ subgraph.
We label this type of homeomorphic expansion generically as $HEG$.
Analogously to the previous category, we denote the respective graphs
obtained by successive insertion of $k-2$ degree-2 vertices on
a single bond of the $G_r$ subgraph as
\begin{equation}
HEG_{k-2;b_j}[(K_p)_b + G_r] \ , \quad
HEG_{k-2;b_j}[(K_p)_{\{b\} } + G_r]
\label{hegb}
\end{equation}
and the graphs obtained by successive insertions of $k-2$ degree-2 vertices on
all of the bonds of $G_r$ as
\begin{equation}
HEG_{k-2}[(K_p)_b + G_r] \ , \quad
HEG_{k-2}[(K_p)_{\{ b \} } + G_r]
\label{heg}
\end{equation}
\vspace{8mm}
4. \ $HEC$
\vspace{6mm}
Start with either $(K_p)_b + G_r$ or $(K_p)_{\{b\}} + G_r$
and add vertices to the bonds connecting vertices in
the $K_p$ subgraph to vertices in the $G_r$ subgraph. We label this type of
homeomorphic expansion generically as $HEC$, where the ``C'' refers to the fact
that the homeomorphic expansion is performed on the above-mentioned connecting
bonds. We denote the respective graphs
obtained by successive insertion of $k-2$ degree-2 vertices on
a single bond $b_{ij}$ connecting a vertex $v_i \in K_p$ to a vertex
$v_j \in G_r$ as
\begin{equation}
HEC_{k-2; b_{ij}}[(K_p)_b + G_r] \ , \quad
HEC_{k-2; b_{ij}}[(K_p)_{\{ b \} } + G_r]
\label{hecij}
\end{equation}
and the graphs obtained by successive insertions of $k-2$ degree-2 vertices on
all of the bonds connecting vertices of $K_p$ to vertices of $G_r$ as
\begin{equation}
HEC_{k-2}[(K_p)_b + G_r] \ , \quad
HEC_{k-2}[(K_p)_{\{ b \} } + G_r ]
\label{hec}
\end{equation}
\vspace{8mm}
5. \ Combinations
\vspace{6mm}
Clearly, one can combine several types of homeomorphic expansion. For example,
starting with $(K_p)_b + G_r$, one can add vertices both to bonds in the
subgraph $K_p$, to bonds that connect $K_p$ to $G_r$, and to bonds in the $G_r$
subgraph.
\end{flushleft}
In the present paper we shall concentrate on homeomorphic expansions of types
(1)-(3). Our results for homeomorphic expansions of type (4)
involve somewhat more complicated boundaries ${\cal B}$ than those discussed
here and will be dealt with in a separate paper. Composite homeomorphic
expansions of type (5) can be studied by similar means. We now proceed to
consider the various homeomorphic classes in more detail.
Given that the number of vertices in the original graph is a linear function of
the two (positive integer) parameters $p$ and $r$, the number $n$ of vertices
of the homeomorphic expansion is a linear function of $p$, $r$, and $k$. There
are therefore three basic ways of producing the limit $n \to \infty$, namely
($L$ denotes limit)
\begin{equation}
L_p: \ p \to \infty \quad {\rm with} \quad r \quad {\rm and} \quad k \quad
{\rm fixed}
\label{pinf}
\end{equation}
\begin{equation}
L_r: \ r \to \infty \quad {\rm with} \quad p \quad {\rm and} \quad k \quad
{\rm fixed}
\label{rinf}
\end{equation}
\begin{equation}
L_k: \ k \to \infty \quad {\rm with} \quad p \quad {\rm and} \quad r \quad
{\rm fixed}
\label{kinf}
\end{equation}
We have explained above (after eq. (\ref{mageq})) why the limit $L_p$ is not
very interesting. From the viewpoint of the present work on boundaries
${\cal B}$ that are noncompact in the $q$ plane, the limit $L_k$ is also not
of primary interest, since it generically yields a compact boundary
${\cal B}$, as we shall illustrate below. Hence we shall concentrate on the
limit $L_r$ here.
\section{$HEK0$ and $HEK$ Homeomorphic Classes}
Let us consider first the homeomorphic expansions of type (1), namely,
$HEK_{k-2,b_j}(K_p+G_r)$.
By use of the deletion-contraction theorem and eq. (\ref{pkpg}), we
find that, for arbitrary $G_r$,
\begin{eqnarray}
P \Bigl ( HEK_{k-2,b_j}(K_p+G_r),q \Bigr ) & = & D_k P(K_p + G_r,q) +
[D_k + (-1)^{k-1}]P(K_{p-1} + G_r,q) \cr
& = & D_k q^{(p)} P(G_r,q-p) + [D_k + (-1)^{k-1}] q^{(p-1)}P(G_r,q-(p-1))
\label{phkkpg}
\end{eqnarray}
It follows by the same argument as that given with eq. (\ref{pkpcutb}) that as
$r \to \infty$, this family $HEK_{k-2 \ge 1,b_j}(K_p+G_r)$ has a $W$ boundary
${\cal B}$ that is noncompact in the $q$ plane.
This is true independent of whether the family $G_r$ has
this property. Indeed, as is evident from a comparison of eq. (\ref{phkkpg})
with eq. (\ref{pkpcutb}), for a given $G_r$,
\begin{equation}
{\cal B}\Bigl ( \lim_{r \to \infty} HEK_{k-2,b_j}(K_p+G_r) \Bigr ) =
{\cal B}\Bigl ( \lim_{r \to \infty} [(K_p)_b + G_r] \Bigr )
\label{bsame}
\end{equation}
That is, if we start with the family $K_p + G_r$, remove $b$ bonds from a
given vertex in the $K_p$ subgraph, and take $r \to \infty$, the resultant
boundary ${\cal B}$ is the same as if we had instead homeomorphically added
some number $k-2 \ge 1$ degree-2 vertices to a bond in the $K_p$ subgraph. One
can, of course, continue this process with homeomorphic expansions of other
bonds of the $K_p$. Note that if one uses the limit $L_k$ in eq.
(\ref{kinf}) to get $n \to \infty$ for this class of homeomorphic expansions,
then, for arbitrary
(finite) $G_r$, the resultant boundary is the compact locus comprised by the
unit circle $|a|=1$ (where $a$ was defined in eq. (\ref{a}), i.e.,
\begin{equation}
{\cal B}_{L_k} = q \quad {\rm such \ \ that} \quad |q-1|=1
\label{blk}
\end{equation}
\vspace{6mm}
We next consider the homeomorphic expansions of type (2), namely,
$HEK_{k-2;b_j}[(K_p)_b + G_r]$. Here, the boundary ${\cal B}$ for the
$r \to \infty$ of the beginning family $(K_p)_b + G_r$ is
already noncompact in the $q$ plane, and the families generated by the
homeomorphic expansion maintain this property. As an illustration, consider
the family $(K_3)_1 + G_r$ and homeomorphically expand one of the two remaining
bonds, denoted $b_j$, in the $(K_3)_1 = T_3$ subgraph.
We find for the resultant chromatic
polynomial the result
\begin{eqnarray}
P(HEK_{k-2;b_j}[(K_3)_1 + G_r],q) & = & D_k q^{(3)}P(G_r,q-3) \cr\cr
& & + [2D_k + (-1)^{k-1}]q^{(2)}P(G_r,q-2)
\label{hekexample}
\end{eqnarray}
Again, using the same reasoning that we employed before with eq.
(\ref{pkpcutb}), we deduce that as $r \to \infty$, this family
$HEK_{k-2;b_j}[(K_3)_1 + G_r]$ has a noncompact locus ${\cal B}$ in the $q$
plane. As was the case for the $HEK0$ class, for the limit $L_k$, the
resultant boundary is the unit circle $|q-1|=1$ given in eq. (\ref{blk}).
\section{The Family $T_{\lowercase{k,r}} =
HEG_{\lowercase{k}-2}(\overline K_2 +T_{\lowercase{r}})$}
We next proceed to analyze in detail several homeomorphic classes of graph
families of the form $HEG$. One
interesting family of graphs with noncompact ${\cal B}(q)$ is obtained by
homeomorphic expansion starting with the family $(K_p)_b + T_r$ for
$r \ge 2$ and adding vertices on each of the bonds in the $T_r$ subgraph.
We let $k$ be the number of vertices on $T_r$ between each pair of vertices
that connect with the $K_p$, including this pair (which were originally
adjacent on $T_r$ before the homeomorphic expansion).
We denote this family as
\begin{equation}
T_{p,b,k,r} = HEG_{k-2}[(K_p)_b + T_r]
\label{tpbkr}
\end{equation}
where as above, $r \ge 2$ and $1 \le b \le p-1$.
The number of vertices is given by
\begin{equation}
v(T_{p,b,k,r}) = (r-1)(k-1)+p+1
\label{etpbkr}
\end{equation}
For the lowest value $k=2$,
\begin{equation}
T_{p,b,k=2,r} = (K_p)_b + T_r
\label{tpbkeq2r}
\end{equation}
which we studied previously \cite{wa}.
We shall thus concentrate on the cases $k \ge 3$ here.
It suffices for our present purposes to consider the
simplest nontrivial case $p=2$ and hence $b=1$, for which
\begin{equation}
(K_2)_{b=1} + T_r = \overline K_2 + T_r
\label{k2bartr}
\end{equation}
An illustration of a graph of this type is given in Fig. \ref{vhegraph1}(a).
For brevity, we define
\begin{equation}
T_{k,r} = T_{p=2,b=1,k,r} = HEG_{k-2}(\overline K_2 + T_r)
\label{tkrdef}
\end{equation}
We observe that the chromatic number is
\begin{eqnarray}
\chi(T_{k,r}) & = & 2 \quad {\rm for} \quad k \ \ {\rm odd} \cr
& = & 3 \quad {\rm for} \quad k \ \ {\rm even}
\label{chitkr}
\end{eqnarray}
The girth, i.e., the length of a minimal-length circuit on this graph, is
\begin{equation}
\gamma(T_{k,r}) = k+1
\label{girthtkr}
\end{equation}
By the deletion-contraction theorem, we find the recursion relation
\begin{equation}
P(T_{k,r},q) = \Bigl [ D_3D_k + (-1)^{k-1} \Bigr ] P(T_{k,r-1},q) +
q(q-1)D_k (D_{k+1})^{r-2}
\label{tkrrecursion}
\end{equation}
Solving this, we obtain the chromatic polynomial
\begin{equation}
P(T_{k,r},q) = A_k \Bigl [ D_3D_k + (-1)^{k-1} \Bigr ]^{r-2}
+ q(q-1)(D_{k+1})^{r-1}
\label{ptkr}
\end{equation}
where
\begin{equation}
A_k = (q-3)P(C_{k+1},q) + (q-1)P(C_{k-1},q)
\label{ak}
\end{equation}
$P(T_{k,r},q)$ has the general factors $q(q-1)$ for $k$ odd and
$q(q-1)(q-2)$ for $k$ even. As is evident from eq. (\ref{ptkr}),
$P(T_{k,r},q)$ has the form of eq. (\ref{pgsum}) with
\begin{equation}
a_1 = D_{k+1}
\label{a1tkr}
\end{equation}
and
\begin{equation}
a_2 = D_3D_k + (-1)^{k-1}
\label{a2tkr}
\end{equation}
(recall that $D_3=q-2$). In the limit $r \to \infty$ (with $k$ fixed
\cite{lknote})
the locus ${\cal B}$ is determined by the degeneracy of magnitudes
\begin{equation}
|a_1|=|a_2|
\label{degeneqtkr}
\end{equation}
This equation can be simply expressed in terms of the variable $a=q-1$ which
was defined above in eq. (\ref{a}) (and should not be confused with the
variables $a_1(q)$ and $a_2(q)$). To do this, we multiply both sides of eq.
(\ref{degeneqtkr}) by $|q(q-1)|=|(a+1)a|$ (the spurious solutions at $q=0$
and $q=1$ thereby introduced are discarded) and use eq. (\ref{pck}) to get
\begin{equation}
|a(a^k - a^{k-1} + 2(-1)^{k+1})| = |a(a^k + (-1)^{k+1})|
\label{aeqtkr}
\end{equation}
i.e., after dividing by $|a|^{k+1}$,
\begin{equation}
|1-y+2(-1)^{k+1}y^k| = |1+(-1)^{k+1}y^k|
\label{yeqtkr}
\end{equation}
Since $y=0$ is a solution of this equation, ${\cal B}$ is noncompact in the $q$
plane, passing through $y=z=0$. In polar coordinates, with $y=\rho
e^{i\beta}$, eq. (\ref{yeqtkr}) yields
\begin{equation}
\rho\Bigl [3\rho^{2k-1} + \rho - 2\cos \beta + 2(-1)^{k-1}\rho^{k-1}
\biggl \{ \cos(k\beta) - 2\rho \cos((k-1)\beta) \biggr \} \Bigr ] = 0
\label{yeqtkrpolar}
\end{equation}
As $\rho \to 0$, it follows that $\cos\beta=0$, i.e. $\beta = \pm \pi/2$, so
that ${\cal B}$ approaches $y=z=0$ vertically.
To calculate the point at which the boundary ${\cal B}$ crosses the real $q$,
or equivalently, $y$ or $z$ axes, we set $\beta=0$ in
eq. (\ref{yeqtkrpolar}); for $\rho \ne 0$ this gives
\begin{equation}
3\rho^{2k-1} + \rho - 2 + 2(-1)^{k-1}\rho^{k-1}(1-2\rho)=0
\label{yeqtkrpolarpositivey}
\end{equation}
(Note that $Re(q) > 0 \Longleftrightarrow
Re(z) > 0$ and $Re(q) > 1 \Longleftrightarrow Re(y) > 0$ so that setting
$\beta=0$, i.e. making $y$ real and positive, implies that $q$ is real and $q >
1$; we comment later on the interval $0 < q < 1$.) Since $\rho=|y|$ is real
and positive, we are only interested in roots of eq.
(\ref{yeqtkrpolarpositivey})
that have this property. For both even and odd $k$, there is only one such
root, which is thus $y_c$ (equivalently, $z_c$), the minimum nonzero value of
$y$ ($z$) respectively, at which the boundary ${\cal B}$ crosses the
positive real axis in the $y$ ($z$) plane, corresponding to the maximal finite
real point $q_c$ at which the pre-image of this boundary ${\cal B}$ crosses the
positive real axis in the $q$ plane. For $k$ odd, this single root is
$\rho=y=1$ (the expression on the left-hand side contains a factor
$(\rho-1)$) so that $z_c=1/2$, and
\begin{equation}
q_c = 2 = \chi \quad {\rm for} \quad k \quad {\rm odd}
\label{qce2tkr_kodd}
\end{equation}
For $k$ even, the single such root of eq. (\ref{yeqtkrpolarpositivey})
has a value that increases monotonically from $\rho=y_c=2/3$ (equiv.
$z_c=2/5$) for $k=2$ toward the limit $\rho=y_c=1$ (equiv.
$z_c=1/2$) as $k \to \infty$ through even integers. Correspondingly, for $k$
even, $q_c$ decreases monotonically from 5/2 for $k=2$ toward 2 in the same
limit; some values are
$q_c=2.2564, \ 2.1736, \ 2.1315$ for $k=4, \ 6, \ 8$). Hence,
\begin{equation}
2 < q_c < \frac{5}{2} < \chi = 3 \quad {\rm for} \quad k \quad {\rm even}
\label{qce2tkr_keven}
\end{equation}
To show that ${\cal B}$ does not cross the positive real $q$ axis in the
interval $0 \le q < 1$, we take $\beta=\pi$, i.e., $y$ real and negative, in
eq. (\ref{yeqtkrpolar}), which, for $\rho \ne 0$, yields the equation
\begin{equation}
3\rho^{2k-1}+\rho+2-2\rho^{k-1}-4\rho^k=0
\label{betapieq}
\end{equation}
This range $-\infty < y \le 0$
includes both the interval $0 \le q < 1$ and the interval $-\infty < q < 0$.
We find that in this range of $y$, eq. (\ref{betapieq}) has no
real positive roots for $\rho$ except for $\rho=1$, i.e., $y=-1$ or
equivalently, $q=0$, but this is just a spurious solution introduced when we
multiplied both sides of eq. (\ref{degeneqtkr}) by $|q(q-1)|$ to express it as
eq. (\ref{aeqtkr}). Hence, ${\cal B}$ crosses
the positive $z$ or equivalently $q$ axis only once for this family of graphs.
We proceed to discuss the boundary ${\cal B}$ for specific values of $k$
further. For $k=2$ \cite{wa}
${\cal B}$ is the vertical line with $Re(q)=5/2$ in the $q$
plane, or equivalently, the circles $|z-z_c/2|=z_c/2$, $|y-y_c/2|=y_c/2$ in the
$z$ and $y$ planes, where $z_c=2/5$, $y_c=2/3$ as given above.
\pagebreak
\begin{figure}
\vspace{-4cm}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{e2hvk3tr16z.ps}
\end{center}
\vspace{-2cm}
\begin{center}
\leavevmode
\epsfxsize=3.0in
\epsffile{e2hvk4tr16z.ps}
\end{center}
\vspace{-2cm}
\caption{\footnotesize{Boundary ${\cal B}$ in the $z=1/q$ plane for the $r \to
\infty$ limit of the family of graphs $HEG_{k-2}(\overline K_2 + T_r)$
with $k=$ (a) 3 (b) 4. Chromatic zeros for $r=16$ are shown for
comparison.}}
\label{boundarye2hvktr1}
\end{figure}
\pagebreak
\begin{figure}
\vspace{-4cm}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{e2hvk5tr10z.ps}
\end{center}
\vspace{-2cm}
\begin{center}
\leavevmode
\epsfxsize=3.0in
\epsffile{e2hvk6tr8z.ps}
\end{center}
\vspace{-2cm}
\caption{\footnotesize{As in Fig. \ref{boundarye2hvktr1} for $k=$ (a) 5 (b) 6.
Chromatic zeros for $r=$ (a) 10 (b) 8 are shown for comparison.}}
\label{boundarye2hvktr2}
\end{figure}
The boundary ${\cal B}$ is shown in the $z$ plane for $3 \le k \le 6$ in
Figs. \ref{boundarye2hvktr1} and \ref{boundarye2hvktr2} \cite{thesis}.
In region $R_1$
containing the portion of the positive $z$ axis analytically connected to the
point $z=0$,
\begin{equation}
W([\lim_{r \to \infty} T_{k,r}],q) = (a_1)^{1/(k-1)} = (D_{k+1})^{1/(k-1)}
\quad {\rm for} \quad q \in R_1
\label{wtkr_r1}
\end{equation}
In region $R_2$ occupying the rest of the $z$ plane for $2 \le k \le 4$, and
the rest of the $z$ plane except for the interiors of the other closed loops
for $k \ge 5$,
\begin{equation}
|W([\lim_{r \to \infty} T_{k,r}],q)| = |a_2|^{1/(k-1)} = |D_3D_k +
(-1)^{k-1}|^{1/(k-1)} \quad {\rm for} \quad q \in R_2
\label{wtkr_r2}
\end{equation}
For odd $k$, as $q$ decreases toward 2 one has, using eqs. (\ref{wtkr_r1})
and (\ref{dkoddq2}),
\begin{equation}
\lim_{q \searrow 2}W([\lim_{r \to \infty} T_{k,r}],q) = 1
\label{tkrwq2region1}
\end{equation}
and, as $q$ increases toward 2, one has, using eq. (\ref{wtkr_r2}),
\begin{equation}
\lim_{q \nearrow 2}||W([\lim_{r \to \infty} T_{k,r}],q)| =1
\label{tkrwq2region2}
\end{equation}
The equality of the right-hand sides of eqs. (\ref{tkrwq2region1}) and
(\ref{tkrwq2region2}) follows from the fact that $q=2$ is on the boundary
${\cal B}$ for odd $k$.
In each of the additional regions enclosed by closed curves,
\begin{equation}
|W([\lim_{r \to \infty} T_{k,r}],q)| = |a_1|^{1/(k-1)} = |D_{k+1}|^{1/(k-1)}
\label{wtkr_r3}
\end{equation}
For comparison, we show chromatic zeros for
reasonably large finite values $r$ in each of these figures.
We find several general features of the boundary ${\cal B}$ for the $r \to
\infty$ limit of this family of graphs. The first feature is that
${\cal B}$ has support only for non-negative $Re(z)$ or equivalently,
non-negative $Re(q)$:
\begin{equation}
z \in {\cal B} \Rightarrow Re(z) \ge 0 \quad {i.e.,} \quad Re(q) \ge 0
\label{bnonnegative}
\end{equation}
Indeed, the only place on the boundary ${\cal B}$ where $Re(z)$ vanishes is at
the origin $z=0$ itself:
\begin{equation}
(z \in {\cal B} \quad {\rm and} \quad Re(z) = 0) \Rightarrow z = 0
\label{only0}
\end{equation}
Thus, in the vicinity of the point $z=0$, the curve comprising ${\cal B}$ bends
to the right as one increases $Im(z)$ above zero or decreases it below zero; we
have shown above that this curve crosses the point $z=0$ vertically.
The second feature is that ${\cal B}$ has no multiple points \cite{mp}.
The third feature concerns the
number of different connected components $N_{comp.}$ comprising ${\cal B}$.
In the present case, this is simply related to the number of regions $N_{reg.}$
by the equation $N_{reg.}=N_{comp.}+1$. One might hope
that there would be a general mathematical theorem that would state the number
of different connected components $N_{comp.}$
of the solution set of a given polynomial equation
without requiring an explicit solution. However this question, which is
related to the sixteenth Hilbert Problem, still remains unanswered
\cite{hilbert16,amstran}. One can make use of an upper bound on $N_{comp.}$
contained in the Harnack theorem, which is $N_{comp} \le g+1$,
where $g$ denotes the genus of the algebraic curve \cite{alg}.
For plane algebraic curves of the type relevant here,
\begin{equation}
\sum_{m,n} c_{m,n} y_{_R}^m y_{_I}^n = 0
\label{yeqgeneral}
\end{equation}
with maximal degree $d$, where $y_{_R}=Re(y)$, $y_{_I} = Im(y)$,
the genus is $g=(d-1)(d-2)/2$. Thus, for a given case, one
writes the degeneracy equation (\ref{yeqtkr}) or its equivalent in the $z$
plane, out into cartesian components.
The case $k=2$ has already been discussed and leads to the equation
$(y_{_R}- 1/3)^2 + y_{_I}^2 = (1/3)^2$, or equivalently,
$(z_{_R}- 1/5)^2 + z_{_I}^2 = (1/5)^2$ where $z_{_R}=Re(z)$ and
$z_{_I}=Im(z)$. These equations have homogeneous degree \cite{alg}
$d=2$, hence genus $g=0$, so Harnack's theorem yields the bound $N_{comp.}
\le 1$ which, together with the fact that $N_{comp.} \ge 1$ implies that
$N_{comp.}=1$, in agreement with the elementary solution of the explicit
equation above. However, for higher values of $k$, the Harnack upper bound is
not very restrictive. For $k \ge 3$, eq. (\ref{yeqtkr}), when written out in
cartesian components, is
\begin{equation}
\Bigl [ (y_{_R}+1)^2 + y_{_I}^2 \Bigr ]F(y_{_R}, y_{_I})=0
\label{yeqk3}
\end{equation}
where $F(y_{_R}, y_{_I})=3(y_{_R}^{2k-2} + y_{_I}^{2k-2})+$ lower order terms.
Because the first factor is positive definite, eq. (\ref{yeqtkr}) thus reduces
to $F(y_{_R}, y_{_I})=0$, of homogeneous degree $d=2k-2$ and hence genus
$g=(2k-3)(k-2)$. The Harnack theorem then yields the upper bound
$N_{comp.} \le 2k^2-7k+7$. As an example, for $k=3,4,5,6$, this bound has
the respective values 4,11,22,37, while the actual values are
$N_{comp.}=1,1,3,3$, and so forth for higher $k$.
For both of the cases $k=3,4$, as was
true for $k=2$, ${\cal B}$ divides the $z$ plane into two regions: (i)
the region $R_1$ including the vicinity of the positive real axis contiguous
with the origin, $z=0$; and (ii) the region $R_2$ occupying the rest of the $z$
plane outside of $R_1$. In the $z$ plane, the chromatic zeros tend to cluster
on the regions of the curve ${\cal B}$ in the ``northeast'' and ``southeast''
directions. Indeed, as $k$ increases from 3 to 4, ${\cal B}(z)$ develops
protuberances in these ``northeast'' and ``southeast'' directions. Since for a
given $r$ and $k$ the chromatic zeros are bounded, they avoid the origin of the
$z$ plane, as can be seen in the figures. Between
$k=4$ and $k=5$ there occurs a qualitative change in ${\cal B}$, namely, that,
whereas it consisted of a single component for $2 \le k \le 4$, it consists of
three disconnected components for $5 \le k \le 7$, comprised of a
self-conjugate closed curve together with a complex-conjugate pair of closed
curves. Hence, while ${\cal B}$ divides the $z$ plane into two regions for
$2 \le k \le 4$, it divides this plane into four regions for $5 \le k \le 7$.
In a figurative language, one can think of the northeast and southeast
bulges that are evident in the $k=4$ case as breaking off to form
the two separate closed curves that first appear for $k=5$. We also observe
that the additional closed curves and associated disconnected regions appear
to line up approximately along the vertical line with $Re(z)=1/2$. It is
interesting to note that this line maps to unit circles in the $y$ and $q$
planes:
\begin{equation}
z: \ Re(z)=\frac{1}{2}, \ Im(z) \ {\rm arbitrary} \Longleftrightarrow
|y|=1 \quad \Longleftrightarrow \quad |q-1|=1
\label{lineimages}
\end{equation}
Thus, the disconnected phases that appear in the $z$ plane are clustered
around the unit circle $|y|=1$ in the $y$ plane. In the $q$ plane, portions of
the boundary ${\cal B}$ are also clustered around the circle $|q-1|=1$ while
one portion extends infinitely far from the origin.
\section{The Family $C_{\lowercase{k,r}} =
HEG_{\lowercase{k}-2}(\overline K_2 + C_{\lowercase{r}})$}
We obtain a further infinite family of graphs with noncompact ${\cal B}(q)$
by the same steps as in the previous section, but replacing the subgraph
$T_r$ by the circuit subgraph $C_r$. That is, we homeomorphically expand the
family $(K_p)_b + C_r$ for $r \ge 2$ by adding vertices on each of the
bonds in the $C_r$ subgraph. As before, we let $k$ be the number of vertices
on $C_r$ between each pair of vertices that connect with the $K_p$ subgraph,
including this pair. We denote this family as
\begin{equation}
C_{p,b,k,r} = HEG_{k-2}[(K_p)_b + C_r]
\label{cpbkr}
\end{equation}
where as above, $r \ge 2$ and $1 \le b \le p-1$.
For a given set of parameters $p$, $k$, and $r$, the number of vertices in this
family is
\begin{equation}
v(C_{p,b,k,r}) = r(k-1)+p
\label{vcpbkr}
\end{equation}
For the lowest value, $k=2$,
\begin{equation}
C_{p,b,k=2,r} = (K_p)_b + C_r
\label{cpbkeq2r}
\end{equation}
which we studied previously \cite{wa}; accordingly, we concentrate here
on the cases $k \ge 3$. It suffices for our present purposes to consider the
simplest nontrivial case $p=2$ and hence $b=1$, for which
\begin{equation}
(K_2)_{b=1} + C_r = \overline K_2 + C_r
\label{c2bartr}
\end{equation}
As before, a short notation is useful:
\begin{equation}
C_{k,r} = C_{p=2,b=1,k,r} = HEG_{k-2}(\overline K_2 + C_r)
\label{ckrdef}
\end{equation}
An illustration of a graph of this type is given in Fig. \ref{vhegraph1}(b).
The chromatic number and girth are the same as for $T_{k,r}$, given by the
right-hand sides of eqs. (\ref{chitkr}) and (\ref{girthtkr}).
By the deletion-contraction theorem, we find the recursion relation
\begin{equation}
P(C_{k,r},q) = D_k P(T_{k,r},q) + (-1)^{k-1}P(C_{k,r-1},q)
\label{ckrrecursion}
\end{equation}
For the lowest value of $r$, namely, $r=2$, we have
\begin{equation}
P(C_{k,2},q) = \Bigl [ D_k + (-1)^{k-1} \Bigr ] P(T_{k,2},q) +
(-1)^k (D_3)^2 P(C_k,q)
\label{ckr2}
\end{equation}
Solving the recursion relation (\ref{ckrrecursion}) with (\ref{ckr2}), we
calculate the chromatic polynomial. As before, we express this in the form of
eq. (\ref{pgsum}). (In order to render the polynomial property manifest, one
divides through by the factors in the denominator, thereby generating a series,
using the identity $(x^m-1)/(x-1)=\sum_{j=0}^{m-1}x^j$.) We get
\begin{eqnarray}
P(C_{k,r},q) & = &
D_k \Biggl [ A_k a_2 \Bigl ( a_2 + (-1)^{k-1}\Bigr )
\biggl [ \frac{a_2^{r-2}-(-1)^{(k-1)r}}{a_2^2-1} \biggr ] + \cr\cr\cr
& & q(q-1)a_1^2\Bigl (a_1 + (-1)^{k-1} \Bigr )
\biggl [ \frac{a_1^{r-2}-(-1)^{(k-1)r}}{a_1^2-1} \biggr ] \Biggr ] +
(-1)^{(k-1)r}P(C_{k,2},q)
\label{pckr}
\end{eqnarray}
where $A_k$ was defined in eq. (\ref{ak}).
$P(C_{k,r},q)$ has the general factors $q(q-1)$ for $k$ odd and
$q(q-1)(q-2)$ for $k$ even. $P(C_{k,r},q)$ has the form of
eq. (\ref{pgsum}) with $a_1$ and $a_2$ as given in eqs. (\ref{a1tkr}) and
(\ref{a2tkr}), together with
\begin{equation}
a_3 = 1
\label{a3ckr}
\end{equation}
For large $|q|$ (small $|z|$),
the terms $a_1$ and $a_2$ have larger magnitudes than $a_3$, so that in the
limit $r \to \infty$ (with $k$ fixed \cite{lknote})
the boundary ${\cal B}$ is determined by the degeneracy equation
$|a_1|=|a_2|$, and the same analysis goes through as before for
the family $HEG_{k-2}(\overline K_2 + T_r)$, with the same conclusions that
${\cal B}$ is noncompact in the $q$ plane, passing through the point $z=y=0$
vertically. For the lowest case $k=2$, i.e., $\overline K_2 + C_r$,
${\cal B}$ consists of three regions, as shown in Fig. \ref{bipzplane}):
$R_1$ including the real axis in the interval $0 \le z \le z_c$ with
$z_c=1/3$ as given by eq. (\ref{qckpcr}); $R_3$ centered around $z=2/5$ and
occupying the interval $z_c \le z \le 1/2$ on the real axis; and $R_2$
occupying the complement of the $z$ plane extending outward to $|z| \to
\infty$. This region diagram may be contrasted with
the simpler one for $\overline K_2 + T_r$, which was just a circle, separating
the $z$ plane into two regions.
In Figs. \ref{boundarye2hvkcr1} and \ref{boundarye2hvkcr2} we
show the boundary ${\cal B}$ in the $z$ plane for $3 \le k \le 6$
\cite{thesis}.
Again, for comparison, we show chromatic zeros for reasonably large finite
values $r$ in each of these figures. The property that ${\cal B}$ passes
through $z=0$ vertically is evident in these figures, as are the properties
(\ref{bnonnegative}) and (\ref{only0}). For each value of $k$,
the region $R_1$ is the one occupying the non-negative interval of the $z$ axis
adjacent to the origin. The region $R_2$ occupies
the rest of the $z$ plane for $2 \le k \le 4$, and
the rest of the $z$ plane except for the interiors of the other closed loops
for $k \ge 5$. For odd $k$, there are two regions on the real $z$ axis:
$R_1$ for $0 \le z \le 1/2$ and $R_2$ for $z < 0$ and $z > 1/2$, while for even
$k \ge 4$ there are these phases and, in addition, a third ``pocket'' phase
$R_3$ whose right-hand boundary with $R_2$ occurs at $z=1/2$ and whose
left-hand boundary with $R_1$ occurs at the point $z=z_c$ slightly less than
1/2 (see further below).
As was the case for
$\lim_{r \to \infty}[HEG_{k-2}(\overline K_2 + T_r)]$, we find
\begin{equation}
W([\lim_{r \to \infty} C_{k,r}],q) = (a_1)^{1/(k-1)} = (D_{k+1})^{1/(k-1)}
\quad {\rm for} \quad q \in R_1
\label{wckr_r1}
\end{equation}
and
\begin{equation}
|W([\lim_{r \to \infty} C_{k,r}],q)| = |a_2|^{1/(k-1)} = |D_3D_k +
(-1)^{k-1}|^{1/(k-1)} \quad {\rm for} \quad q \in R_2
\label{wckr_r2}
\end{equation}
Hence for $k=3$ the equalities (\ref{tkrwq2region1}) and
(\ref{tkrwq2region2}) hold, with $T_{k,r}$ replaced by $C_{k,r}$.
For $k \ge 3$, additional regions appear as ``outgrowths'' on the right-hand
boundary of $R_1$ in the $z$ plane, bounded on the left by $R_1$ and on the
right by $R_2$. We label these regions generically as $R_{1-2;j}$, where $j
\ge 3$ indexes the region. For example, there are two such regions for
$k=3$, which are complex-conjugates of each other, and there are three such
regions for $k=4$, consisting of a complex-conjugate pair and the region $R_3$
that includes an interval of the real $z$ axis between $z=0.42495$ and
$z=1/2$. For
arbitrary $k \ge 3$, in these outgrowth regions lying between $R_1$ and $R_2$,
the leading term
in the limit $r \to \infty$ is the last one in eq. (\ref{pckr}),
viz., $(-1)^{(k-1)r}P(C_{k,2},q)$, so that
\begin{equation}
|W([\lim_{r \to \infty} C_{k,r}],q)| = 1
\quad {\rm for} \quad q \in R_{1-2,j}
\label{wckr_rj}
\end{equation}
For $k \ge 5$, additional sets of regions occur that are not connected with
$R_1$; these consist of complex-conjugate pairs, lying near to the vertical
line in the $z$ plane with $Re(z) = 1/2$.
For the cases that we have studied, each of these disconnected sets of regions
actually consists of two: for the ones with $Im(z) > 0$, one part occupying
a ``northeast'' position and the other a ``southwest'' position, as is evident
in the figures. We label these regions disconnected from $R_1$ as
$R_{disc,NE;j}$, $R_{disc,SW;j}$, and their complex-conjugates as
$R_{disc,NE;j}^*$, $R_{disc,SW;j}^*$
. We find that
\begin{equation}
|W([\lim_{r \to \infty} C_{k,r}],q)| = |a_1|^{1/(k-1)} = |D_{k+1}|^{1/(k-1)}
\quad {\rm for} \quad q \in R_{disc,SW;j}, R_{disc,SW;j}^*
\label{wckr_rdisc_sw}
\end{equation}
\begin{equation}
|W([\lim_{r \to \infty} C_{k,r}],q)| = 1
\quad {\rm for} \quad q \in R_{disc,NE;j}, R_{disc,NE;j}^*
\label{wckr_rdisc_ne}
\end{equation}
As was noted before in the case of the $r \to \infty$ limit of the family
$T_{k,r}$, this clustering of the disconnected phases along the
vertical line $Re(z)=1/2$ is equivalent to their clustering around the unit
circle in the $y$ plane.
For odd $k$, ${\cal B}$ crosses the real $z$ axis away from $z=0$
(equivalently the real $q$ axis) once, at $z=1/2$ (i.e., $q=2$).
For even $k$, ${\cal B}$ crosses the real $z$ axis at two points away from
the origin $z=0$: (i) at $z=1/2$, and (ii) at a value of $z$ that increases
monotonically from $z=1/3$ for $k=2$, approaching $z=1/2$ as
$k \to \infty$ through even integers. Hence, taking into account that
\begin{eqnarray}
\chi(C_{k,r}) & = & 2 \quad {\rm for} \quad k \quad {\rm odd} \cr
& = & 3 \quad {\rm for} \quad k \quad {\rm even}
\label{chickr}
\end{eqnarray}
we have
\begin{equation}
q_c = 2 = \chi \quad {\rm for \ odd} \quad k
\label{qce2ckr_kodd}
\end{equation}
while
\begin{equation}
2 < q_c \le 3 = \chi \quad {\rm for \ even} \quad k
\label{qce2ckr_keven}
\end{equation}
Equations (\ref{qce2ckr_kodd}) and (\ref{qce2ckr_keven}) may be compared with
the analogous eqs. (\ref{qce2tkr_kodd}) and (\ref{qce2tkr_keven}) for
the family $T_{k,r}= HEG_{k-2}(\overline K_2 + T_r)$.
To derive the above results, we observe first that for
the case of odd $k$, this follows from the same analysis as applied to the
$r \to \infty$ limit of the family $ HEG_{k-2}(\overline K_2 + T_r)$ in the
previous section, since the crossing is determined by the degeneracy condition
$|a_1|=|a_2|$. For even $k$, the crossing point at the smaller value of $z$ is
determined by the degeneracy condition of leading terms
\begin{equation}
{\cal B}(R_1,R_3) \ : \quad |a_1|=|a_3|=1
\label{br1r3}
\end{equation}
To put this into a form that can be used for the analysis of our region
diagrams in the $z$ plane, Figs.
\ref{boundarye2hvkcr1} and \ref{boundarye2hvkcr2}, we
multiply both sides of eq. (\ref{br1r3}) by $|q(q-1)|=|(a+1)a|$
(the spurious solutions at $q=0$ and $q=1$ thereby introduced are understood
to be discarded). The resulting equation is most
simply expressed in the $y$ plane, as
$|1-y^k|=|y^{k-1}(y+1)|$, i.e.,
\begin{equation}
{\cal B}(R_1,R_3): \quad
\rho^{2k-2} + 2\rho^k \Bigl (\rho^{k-1}\cos\beta + \cos(k\beta)\Bigr ) -1 = 0
\label{br1r3polar}
\end{equation}
For $\beta=0$, this equation has a single acceptable (real positive) root for
$\rho=y_c$ that increases monotonically from 1/2 (i.e., $z_c=1/3$) for $k=2$
through 0.738984 ($z=0.424951$) for $k=4$, toward $\rho=y_c=1$ ($z_c=1/2$)
as $k$ goes to infinity through even values;
equivalently, $q_c$ decreases from
3 at $k=2$ toward 2 as $k \to \infty$ through even values. Some explicit
values are $q_c=2.35321$, 2.21486, 2.15442 for $k=4,6,8$.
The crossing point at the larger value of $z$ (for even $k$),
viz., $z=1/2$, is determined by the degeneracy condition of leading terms
$|a_2|=|a_3|$. Multiplying by $|q(q-1)|$ and reexpressing this in terms of $y$
as before, we obtain $|1-y+2(-1)^{k-1}y^k|=|(1+y)y^{k-1}|$, i.e.,
\begin{eqnarray}
{\cal B}(R_2,R_3): \quad & &
3\rho^{2k}-\rho^{2k-2}+\rho^2+1 - 2\rho(\rho^{2k-2}+1)\cos\beta \cr\cr
& & - 4\rho^k\Bigl [ \cos(k\beta) - \rho\cos((k-1)\beta) \Bigr ] = 0
\label{br2r3polar}
\end{eqnarray}
For $\beta=0$, this equation always has a root at $\rho=y=1$, (i.e., $z=1/2$,
$q=2$). (It also has a real positive root at a value of $\rho < 1$
(i.e., $z < 1/2$), but $a_2$ is not a leading term in this region, so that
this root is irrelevant.)
The number of multiple points on ${\cal B}$ is
\begin{equation}
N_{m.p.}=2(k-1) \ , \quad {\rm for \ even} \quad k
\label{nmpe2hvcr_evenk}
\end{equation}
comprised of $k-1$ complex-conjugate pairs, and
\begin{equation}
N_{m.p.}=2k-3
\label{nmpe2hvcr_oddk} \ , \quad {\rm for \ odd} \quad k
\end{equation}
consisting of a real one at $z=1/2$, i.e.,
$q=2$, together with $k-2$ complex-conjugate pairs.
It is interesting to compare the boundaries ${\cal B}$ and associated region
diagrams for the $r \to \infty$ limits of the two families of graphs
$HEG_{k-2}(\overline K_2 + T_r)$ and $HEG_{k-2}(\overline K_2 + C_r)$ for a
given value of $k$. We have done this above for $k=2$. For $k \ge 3$, one
sees that the diagrams for $HEG_{k-2}(\overline K_2 + C_r)$ look somewhat
similar to those for $HEG_{k-2}(\overline K_2 + T_r)$ with the differences that
(i) there are additional ``outgrowth'' regions contiguous to the right-hand
side of $R_1$; (ii) the additional disconnected regions that appear for $k
\ge 5$ are themselves composed of additional regions; (iii) ${\cal B}$
contains multiple points; and (iv) for even $k$ there are three, rather than
just two regions along the real $z$ axis. Owing to the presence of the
multiple points on ${\cal B}$ (i.e. singular points, in the sense of algebraic
geometry), the analysis of bounds on the number of disconnected components,
$N_{comp.}$ is more difficult than for algebraic curves without singular
points, such as the curves ${\cal B}$ for $\lim_{r \to \infty}T_{k,r}$.
\pagebreak
\begin{figure}
\vspace{-4cm}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{e2hvk3cr12z.ps}
\end{center}
\vspace{-2cm}
\begin{center}
\leavevmode
\epsfxsize=3.0in
\epsffile{e2hvk4cr12z.ps}
\end{center}
\vspace{-2cm}
\caption{\footnotesize{Boundary ${\cal B}$ in the $z=1/q$ plane for the $r \to
\infty$ limit of the family of graphs $HEG_{k-2}(\overline K_2 + C_r)$
with $k=$ (a) 3 (b) 4. Chromatic zeros for $r=12$ are shown for
comparison.}}
\label{boundarye2hvkcr1}
\end{figure}
\pagebreak
\begin{figure}
\vspace{-4cm}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{e2hvk5cr10z.ps}
\end{center}
\vspace{-2cm}
\begin{center}
\leavevmode
\epsfxsize=3.0in
\epsffile{e2hvk6cr8z.ps}
\end{center}
\vspace{-2cm}
\caption{\footnotesize{As in Fig. \ref{boundarye2hvkcr1} for $k=$ (a) 5 (b) 6.
Chromatic zeros for $r=$ (a) 10 (b) 8 are shown for comparison.}}
\label{boundarye2hvkcr2}
\end{figure}
\section{The Family $S_{\lowercase{k,r}} =
HEG_{\lowercase{k}-2}(\overline K_3 + T_{\lowercase{r}})$}
To illustrate one infinite family of graphs with noncompact ${\cal B}$ of the
type $HEG_{k-2}(\overline K_p + G_r)$ with $p>2$, we discuss an infinite
family constructed by homeomorphic expansion of the bonds of the subgraph
$G_r=T_r$ in the graph $(\overline K_3 + T_r)$. The beginning family,
$\overline K_3 + G_r$, is of type (\ref{epg}) (and can be obtained by
complete removal of all three of the bonds in the $K_3$ subgraph in
$K_3 + G_r$). We denote this family as
\begin{equation}
S_{\lowercase{k,r}} = HEG_{k-2}(\overline K_3 + T_r)
\label{skr}
\end{equation}
The number of vertices is given by
\begin{equation}
v(S_{k,r})=(r-1)(k-1)+4
\label{vskr}
\end{equation}
An example of a graph of this type is shown in Fig. \ref{vhegraph1}(c).
The chromatic number and girth are the same as those of the graph $T_{k,r}$,
given in eqs. (\ref{chitkr}) and (\ref{girthtkr}).
By the deletion-contraction theorem, we find the recursion relation
\begin{equation}
P(S_{k,r},q) = \Bigl [ (q-3)D_k + (-1)^{k-1} \Bigr ] P(S_{k,r-1},q) +
3D_k P(T_{k,r-1},q) - q(q-1)D_k (D_{k+1})^{r-2}
\label{skrrecursion}
\end{equation}
Solving this, we obtain the chromatic polynomial
\begin{eqnarray}
P(S_{k,r},q) & = & 3 \biggl [ q(q-1)D_{k+1}-P(T_{k,2},q) \biggr ]
\biggl [ a_3^{r-2}-a_2^{r-2} \biggr ]
+q(q-1)a_1 \biggl [a_1^{r-2}-a_3^{r-2} \biggr ] \cr\cr
& & + a_3^{r-2}P(S_{k,2},q)
\label{pskr}
\end{eqnarray}
where $a_1$ and $a_2$ were given in eqs. (\ref{a1tkr}) and (\ref{a2tkr}),
\begin{equation}
a_3=(q-3) D_k + (-1)^{k-1} \ ,
\label{a3skr}
\end{equation}
and
\begin{equation}
P(S_{k,2},q)=(q-2)P(T_{k,2},q)+q(q-1)^3D_{k-1}
\label{pskr2}
\end{equation}
In the limit $r \to \infty$ with $k$ fixed \cite{lknote}, the locus
${\cal B}$ is determined by the degeneracy of magnitudes of leading
terms
\begin{equation}
|a_1|=|a_3|
\label{degeneqskr}
\end{equation}
($a_2$ is never a leading term in this case). The degeneracy equation in the
$y$ variable takes the form
\begin{equation}
|1-2y+3(-1)^{k+1}y^k| = |1+(-1)^{k+1}y^k|
\label{yeqskr}
\end{equation}
Since $y=0$ is a solution of this equation, ${\cal B}$ is noncompact in the
$q$ plane, passing through $y=z=0$. In polar coordinates, with
$y=\rho e^{i\beta}$, eq. (\ref{yeqskr}) yields
\begin{equation}
\rho\Bigl [2\rho^{2k-1} + \rho - \cos \beta + (-1)^{k-1}\rho^{k-1}
\biggl \{ \cos(k\beta) - 3\rho \cos((k-1)\beta) \biggr \} \Bigr ] = 0
\label{yeqskrpolar}
\end{equation}
As $\rho \to 0$, it follows that $\cos\beta=0$, i.e. $\beta = \pm \pi/2$, so
that ${\cal B}$ approaches $y=z=0$ vertically.
To calculate the point at which the boundary ${\cal B}$ crosses the real $q$,
or equivalently, $y$ or $z$ axes, we set $\beta=0$ in
eq. (\ref{yeqskrpolar}); for $\rho \ne 0$ this gives
\begin{equation}
2\rho^{2k-1} + \rho - 1 + (-1)^{k-1}\rho^{k-1}(1-3\rho)=0
\label{yeqskrpolarpositivey}
\end{equation}
For both even and odd $k$, eq. (\ref{yeqskrpolarpositivey}) has only one
acceptable (real positive) root for $\rho$, which is thus $y_c$.
For odd $k$ this root is $y_c=1$
(equivalently, $z_c=1/2$, $q_c=2$). For even $k$, the real positive crossing
point increases monotonically from $y_c=0.647799$ ($z_c=0.39313$) for $k=4$
to $y_c=1$ ($z_c=1/2$) as $k$ goes to infinity through even values; i.e.,
$q_c$ decreases monotonically from 2.54369 to 2 over the same range.
In Fig. \ref{boundarye3hvktr1} we show the
respective boundaries ${\cal B}$ in the $z$ plane for $k=3,4$ \cite{thesis}.
Note that for $k=3$, ${\cal B}$ is close to being, but is not, a circle.
Again, for comparison, we
show chromatic zeros for reasonably large finite values of $r$ in each of these
figures. As is evident from Fig. \ref{boundarye3hvktr1} and from the figures
for higher values of $k$ \cite{thesis},
${\cal B}$ for $\lim_{r \to \infty} S_{k,r}$ displays some similarities with
that for $\lim_{r \to \infty} T_{k,r}$. Hence we do not show these figures for
higher values of $k$ here. We remark, however, that there are some
differences; for example, the number of connected components, $N_{comp.}$ is
equal to the respective values 1,1,3,3,3,5 for $3 \le k \le 8$ for
$\lim_{r \to \infty} T_{k,r}$ but is equal to 1,1,3,3,5,5 for the present case,
$\lim_{r \to \infty} S_{k,r}$ with $3 \le k \le 8$. As was true with
$N_{comp.}$ for $\lim_{r \to \infty}T_{k,r}$, for $k \ge 4$,
the values of $N_{comp.}$ are
only weakly bounded by the Harnack theorem.
For each value of $k$,
the region $R_1$ is the one occupying the non-negative interval of the $z$ axis
adjacent to the origin. The region $R_2$ occupies
the rest of the $z$ plane for $2 \le k \le 4$, and
the rest of the $z$ plane except for the interiors of the other closed loops
for $k \ge 5$. We find
\begin{equation}
W(\lim_{r \to \infty} S_{k,r},q) = (a_1)^{1/(k-1)} = (D_{k+1})^{1/(k-1)}
\quad {\rm for} \quad q \in R_1
\label{wskr_r1}
\end{equation}
and
\begin{equation}
|W([\lim_{r \to \infty} S_{k,r}],q)| = |a_3|^{1/(k-1)} = |(q-3)D_k +
(-1)^{k-1}|^{1/(k-1)} \quad {\rm for} \quad q \in R_2
\label{wskr_r2}
\end{equation}
For $k=5,6$ ($k=7,8$) there is one pair (are two pairs) of complex-conjugate
closed loops disconnected from $R_1$, which we denote $R_{disc,j}$ and
$R_{disc,j}^*$. In these regions we find
\begin{equation}
|W([\lim_{r \to \infty} S_{k,r}],q)| = (a_1)^{1/(k-1)} = (D_{k+1})^{1/(k-1)}
\quad {\rm for} \quad q \in R_{disc,j}, R_{disc,j}^*
\end{equation}
The feature observed in the $r \to \infty$ limits of the families
$T_{k,r}$ and $C_{k,r}$ that the
disconnected regions $R_{disc,j}$ and $R_{disc,j}^*$ tend to lie along the
line $z=1/2$ is also present in this case.
\pagebreak
\begin{figure}
\vspace{-4cm}
\centering
\leavevmode
\epsfxsize=3.0in
\begin{center}
\leavevmode
\epsffile{e3hvk3tr14z.ps}
\end{center}
\vspace{-2cm}
\begin{center}
\leavevmode
\epsfxsize=3.0in
\epsffile{e3hvk4tr14z.ps}
\end{center}
\vspace{-2cm}
\caption{\footnotesize{Boundary ${\cal B}$ in the $z=1/q$ plane for the $r \to
\infty$ limit of the family of graphs $HEG_{k-2}(\overline K_3 + T_r)$
with $k=$ (a) 3 (b) 4. Chromatic zeros for $r=14$ are shown for
comparison.}}
\label{boundarye3hvktr1}
\end{figure}
\section{General Condition for Noncompact ${\cal B}$}
The key ingredient
to construct families of graphs with noncompact $W$ boundaries ${\cal B}$ in
the $q$ plane and resultant reduced functions $W_{red.}(\{G \},q)$ that are
nonanalytic at $z=1/q=0$ is to produce a chromatic polynomial with
the feature that there are two leading terms with a degeneracy condition
(\ref{mageq}) that has a solution at $z=0$ (cf. eq. (\ref{mageqz})).
A general statement of this condition was given as the theorem of Section IV of
Ref. \cite{wa}. From our studies in the present paper, we can add some
geometrical comments to this algebraic condition.
An important property of our families of graphs with noncompact $W$ boundaries
is that none of these families is a regular lattice graph. This is in accord
with the derivation of the large--$q$ expansion for regular lattices
\cite{nagle,baker}. (Of course, the property that $\{G\}$ is a regular
lattice is not a necessary condition for the associated boundary ${\cal B}$ to
be compact in the $q$ plane and hence $W_{red.}(\{G\},q)$ to be
analytic at $z=0$.) A basic feature of a regular lattice graph is that (except
for boundary vertices, which yield a negligible effect in the thermodynamic
limit and are absent if one uses periodic boundary conditions) all vertices
have the same coordination number (= degree $\Delta$ in usual graph theory
terminology). A generalization of this is encountered in duals of
Archimedean lattices, such as the $[4 \cdot 8^2]$ (union-jack) lattice, where
the vertices fall into a finite number of sets with different coordination
numbers (e.g. $\Delta=4$ and $\Delta=8$ for the $[4 \cdot 8^2]$ lattice). In
both of these classes of lattices, there are no vertices with infinite degree.
In contrast, a property of the families of graphs that we have studied with
noncompact loci ${\cal B}(q)$ in the limit $r \to \infty$ is that in this limit
they all contain an infinite number of different, non-overlapping
(and non-self-intersecting) circuits,
each of which passes through at least two fixed, nonadjacent vertices. This
immediately implies that these aforementioned nonadjacent vertices have
degrees $\Delta$ that go to infinity in this limit.
In the families that we constructed earlier \cite{wa}
and some of the additional families discussed in sections II and
III of the present work, this property was produced by starting with a
family of the form $K_p + G_r$ and removing one or more bonds from the $K_p$
subgraph, thereby rendering two or more vertices that were originally adjacent
no longer adjacent. As we showed in Section IV, it can also be produced by
homeomorphically adding one or more vertices to a bond in the $K_p$ subgraph,
thereby again rendering two vertices that were originally adjacent no longer
so. It should be mentioned that the condition of having two or more vertices
with $\lim_{r \to \infty} \Delta(r) = \infty$ is not, by itself, sufficient to
produce a noncompact ${\cal B}(q)$. For example,
consider the family of ``$p$-wheels'',
\begin{equation}
(Wh)_n^{(p)} = K_p + C_r
\label{wheel}
\end{equation}
(where $n=p+r$ is the number of vertices) that we constructed in Ref.
\cite{wc}. The degree of each of the $p$ vertices in the
$K_p$ subgraph of the graph $K_p + C_r$ is $\Delta = p-1+r$, so that in the
limit $r \to \infty$ (with $p$ fixed), this degree $\Delta \to \infty$.
However, the corresponding boundary ${\cal B}$ is compact; specifically, we
showed that it is the unit circle $|q-(p+1)|=1$ \cite{wc}.
Furthermore, we observe that the presence of non-adjacent vertices with
degrees $\Delta$ such that $\lim_{r \to \infty}\Delta = \infty$ is also
not, by itself sufficient to guarantee that ${\cal B}(q)$ is noncompact.
This is easily seen by considering, for example, two $p=1$ $p$-wheel graphs,
$(Wh)_n^{(1)}$ whose central vertices are connected by a tree graph
containing at least two bonds. As $n \to \infty$, the degrees $\Delta$ of the
central vertices go to infinity, but again the resultant $W$ boundary
${\cal B}$ is compact; indeed, it is the same as for a single $p=1$ $p$-wheel,
viz., $|q-2|=1$. Thus, a necessary feature is observed to be the presence, in
the limit as the number of vertices of the graph goes to infinity, of an
infinite number of different, non-overlapping (non-self-intersecting)
circuits, each of which pass
through the two or more nonadjacent vertices.
\section{Conclusions}
In this paper we have further explored a fundamental problem in statistical
mechanics -- nonzero ground state entropy -- using as a theoretical laboratory
the $q$-state Potts antiferromagnet. We have presented a number of exact
calculations of the zero-temperature partition function $Z(G,q,T=0)$ or
equivalently the chromatic polynomial $P(G,q)$, and the corresponding limiting
function representing the ground-state degeneracy, $W(\{G\},q)$, for this model
on various families of graphs $G$ for which the boundary ${\cal B}$ of
regions of analyticity of $W$ in the complex $q$ plane is noncompact, passing
through $z=1/q=0$. The study of these graphs thus gives insight into the
conditions for the validity of the large--$q$ Taylor series expansions of the
reduced function $W_{red.}(\{G\},q)$. In addition to families obtained by
removal of bonds from nonadjacent vertices in the $K_p$ subgraph of
$K_p + G_r$, we have constructed and investigated a number of families of
graphs by the powerful method of homeomorphic expansion from respective
starting families. We have shown how the families thus obtained have, in the
limit of an infinite number of vertices, noncompact boundaries in the $q$
plane which pass vertically through the origin of the $z=1/q$ plane and have
support for $Re(q) \ge 0$.
This research was supported in part by the NSF grant PHY-97-22101.
\vspace{20mm}
\section{Appendix 1}
In this Appendix we gather together some convenient formulas concerning the
function $D_k$, defined in eq. (\ref{dk}) in the text. One has
\begin{equation}
D_k(q=0) = (-1)^k (k-1)
\label{dk0}
\end{equation}
and
\begin{equation}
D_k(q=1)=(-1)^k
\label{dk1}
\end{equation}
Since for $k$ even, the circuit graph $C_k$ is bipartite, which is equivalent
to the fact that the chromatic number $\chi(C_{k \ even})=2$, and thus
$P(C_{k \ even},q=2)=2$, it follows that
\begin{equation}
D_{k \ even}(q=2)=1
\label{dkevenq2}
\end{equation}
Since for $k$ odd, $\chi(C_{k \ odd})=3$ and $P(C_{k \ odd},q=2)=0$, we have
\begin{equation}
D_{k \ odd}(q=2)=0
\label{dkoddq2}
\end{equation}
This zero results from a linear factor, i.e.,
\begin{equation}
D_{k \ odd}(q=2)=(q-2)Pol(q)
\label{dkkoddfactor}
\end{equation}
where $Pol(q)$ is a polynomial of degree $k-3$ in $q$ with $Pol(q=2) \ne 0$.
Two identities that we have derived and used for our calculations
are listed below:
\begin{equation}
D_k-aD_{k-1}=(-1)^k
\label{dkdkm1rel}
\end{equation}
\begin{equation}
D_{k+1} - D_k = D_3D_k + (-1)^{k+1}
\label{dkpdkrel}
\end{equation}
The proofs follow immediately from the definition of $D_k$.
\pagebreak
| 2024-02-18T23:40:14.887Z | 1998-11-30T00:58:36.000Z | algebraic_stack_train_0000 | 1,815 | 16,995 |
|
proofpile-arXiv_065-8849 |
\section{Introduction}
In his provoking classic, {\em Chance and Necessity\/}
\cite{monod:71}, Jacques Monod expressed the belief that a
``universal theory'', despite anticipating the appearance of certain
{\em classes\/} of objects (such as galaxies, planetary systems,
molecules, atoms, and the like), would not be able to account for the
biosphere. The biosphere, Monod says, does not contain a predictable
class of objects. Essential unpredictability from first principles
does not imply that the biosphere is not explicable through these
principles: biological objects have, in Monod's words, ``no obligation
to exist, but they have the right to'' (\cite{monod:71}, p. 44). From
a more rigorous point of view, the problem of predicting the class of
objects that are outcomes of biological evolution, that is, species of
organisms, is ill-posed as long as we lack a formal specification of
this class \cite{fontana:94}. A judgement on Monod's position must,
therefore, remain open for the time being.
While the selection pressures of natural evolution arise endogenously,
artificial evolution allows to specify desired outcomes implicitly by
fixing those pressures exogenously. For example, when RNA molecules
are being intentionally evolved in the test tube to perform certain
functions or to bind certain targets \cite{ellington:94a}, an outcome
is being implicitly specified in advance. The evolutionary problem
then reduces to producing an actual RNA sequence folding into a shape
implementing some prespecified properties. This is an adaptative
process {\em within\/} a class of objects rather than the evolution
{\em of\/} a class of objects (the process which Monod believed to be
intrinsically unpredictable). Issues concerning the interplay of
chance and necessity become smaller in scope but also better defined
in the context of evolutionary adaptation. For example, is it
predictable whether a desired outcome can be attained? And how
predictable are adaptive trajectories given some level of abstraction?
When considering the production of antibodies during an immune
response, Monod -- like many -- was puzzled by the effectiveness of
adaptation as a search engine driven by selection and mutation alone.
The puzzle about the effectiveness of adaptation is, as we shall
argue, only apparent. It results from a misframing of the problem not
unlike in the so-called Levinthal paradox \cite{levinthal:69} of
protein folding, where the puzzle is (or rather, was) about how
proteins can fold into their native structure despite a
combinatorially large space of possible configurations. The Levinthal
paradox has come to be recognized as resulting from a false dichotomy
\cite{dill:97}: either the protein has to make an extensive search
through its conformation space to find the lowest energy state (but
then it should not fold within observable times), or it folds ``down''
some energy path (but then it should get stuck in a suboptimal
trap). However, like protein folding, both the limitations and the
opportunties for evolutionary adaptation derive from specific features
of the ``landscape'' on which it occurs. The term landscape denotes
here a space of appropriately weighted pertinent configurations -
self-reproducing units and their fitness in the context of
evolutionary adaptation, or molecular conformations and their (free)
energy in the context of protein folding. Paths on such landscapes are
not equiprobable. The stochastic dynamics of both the adaptive process
(selection) and the folding process (least action) is guided by the
landscape structure and explores only a tiny fraction of the possible
configurations. This raises the issue about what exactly a ``folding
pathway'' or an ``evolutionary trajectory'' consist in, and how
well-defined they are. The ability to address the problem of
predictability in adaptation depends on such a characterization.
Exploring the structure of landscapes is a very active area of
empirical and theoretical research \cite{frauenfelder:97}. Despite the
similarity of certain questions, and despite the fact that both
processes crucially depend on special features of their landscapes,
the analogies between folding of an individual molecule and
evolutionary adaptation are limited. Landscapes underlying
evolutionary processes differ fundamentally from energy landscapes,
because biological entities are organized by a genotype-phenotype map.
Replication and mutation occur at the level of the genotype, but the
selective amplification of a genotype depends on the performance of
its phenotype. This dichotomy wouldn't be fundamental, if it were not
for the fact that the genotype-phenotype map is characteristically
many-to-one. The fact that many genotypes form the same phenotype
enables (as it turns out) evolutionary adaptation to be successful.
However, it also constitutes a major source of uncertainty in
evolutionary trajectories.
The split into genotype and phenotype implies two different notions of
``innovation'', and, hence, mandates care when speaking about
``trajectories''. Should an evolutionary trajectory refer to a
temporal succession of genotypes or of phenotypes? And how are the
two related? It is important to realize here that genotypic and
phenotypic innovation are not on an equal footing. A phenotype cannot
be modified directly, but only indirectly through variation of its
genotype. This indirection means that phenotypic innovation is
mediated and, hence, biased by the genotype-phenotype map. Genetic
mutations are random, but their consequences are far from random, as
they depend on the context in which they are expressed. Assume, for
example, that all one-error mutants of a gene are equally
likely. Despite the absence of a bias at the genetic level, the
resultant protein shapes would, however, occur with biased
probabilities. Innovation is locally isotropic for genotypes, but the
genotype-phenotype map channels phenotypic change along specific and
much fewer directions. We should, therefore, expect regularities in
the genotype-phenotype map to reduce the ``phenotypic uncertainty'' of
evolutionary trajectories as compared to their ``genotypic
uncertainty''.
Imagine a game that is played on two boards, the g(enotype)-board and
the p(henotype)-board. The game allows the player to make local moves
on the g-board only, while the actual pay-off is determined by moves
on the p-board. Some unknown machinery transduces the g-board-moves
into p-board-moves. Even if the g-board-moves were entirely random, an
observer of the p-board would pick up statistical regularities
reflecting the transducing mechanism. Suppose further that an
``invisible hand'' holds the player's stone on the p-board fixed. The
transducing machine then acts back to confine the player to a subset
of moves on the g-board which are compatible with the stone's fixed
position on the p-board. We call these moves neutral, as they don't
change pay-off. The set of squares on the g-board that the player can
access by moving under this constraint we call a neutral
network. Imagine now the invisible hand softening its grip to allow
moves on the p-board that do not decrease pay-off (such as
selection). If none of the neighboring squares of the player's
position on the g-board yields a pay-off increase, the player won't be
stuck, as moves to neighboring neutral squares are still an
option. {\sl Their} neighbors might trigger pay-off increasing moves
on the p-board. As long as the stone on the p-board remains in a fixed
position, an observer of the p-board may conclude that nothing is
happening, and an observer of the g-board notes only ``more of the
same'' as the pay-off stays constant. The point, however, is that by
moving neutrally something {\em does\/} vary: the potential for
change. Without neutrality the game would quickly become stuck in a
suboptimal trap, and the ``evolutionary Levinthal paradox'' would be a
real one. However, with a sufficient degree of neutrality the very
notion of a ``trap'' looses its relevance, as ``lateral''
(fitness-neutral) moves change opportunities at no cost.
In this contribution we review work on the most realistic molecular
model for the genotype-phenotype dichotomy available to date. RNA
unites both the genotypic and the phenotypic level in a single
molecular object. The genotype is the sequence which can be
replicated in the test tube by suitable enzymes, and the phenotype is
the structure which can be subject to selection. We will summarize the
salient properties of the folding relation between sequences and
structures. This sets the stage for discussing some central aspects
of evolutionary dynamics in models of evolving populations of RNA
molecules. We shall focus in particular on an emerging notion of
``evolutionary trajectory'' which is far from being fully understood
but which invites deeper investigation. This will prompt us to think
about what is and what is not predictable.
The main take-home should be an appreciation for the central and
apparently paradoxical role played by neutrality. On the one hand,
neutrality (or redundancy, for that matter) is conservative, because
it buffers against inevitable mutations ensuring that ``nothing
happens'', that once attained success is not too easily lost. Yet at
the same time neutrality is deeply enabling, because it permits
evolutionary change to occur by allowing the sequence context to vary
until a mutation can become phenotypically consequential. In the
dynamics picture neutrality sets the stage for adaptation in jumps,
that is, intrinsic punctuation. But neutrality does more: it endows
the set of phenotypes with a topological structure. A many-to-one map
induces equivalence classes of genotypes labelled by
phenotypes. Nearness (or accessibility) between phenotypes, then,
should be defined to reflect nearness between equivalence classes of
genotypes. Once we possess a topology we can start thinking about
whether and when evolutionary trajectories are
``discontinuous''. While chance events will not allow to predict when
a discontinuity will happen, the topology enables the prediction of
what class of changes will be involved. We conclude by placing the
lessons learned from RNA into the larger perspective of evolution.
\section{RNA: an experimental and theoretical model system}
\label{RNAdef}
Understanding how notions like ``evolutionary path'' and
``continuity'' (or ``discontinuity'') are shaped by adaptive
landscapes, requires a model system that captures essential properties
of the relationship between genotype and phenotype. Since we don't
know these properties {\em a priori}, any theoretical model must have
a firm empirical grounding to begin with. In addition to being
computationally tractable, the system should also be a laboratory
model of evolutionary adaptation. This excludes multicellular organisms,
since their complexity does not permit at the present state of knowledge
an empirical and theoretical tracking of how the genotype-phenotype
relation (i.e., development) plays out during an evolution experiment
in the laboratory. Prokaryotic organisms don't undergo development
but the cellular metabolism is still too complex to be included
in a genotype-phenotype map. Settling for less than organismal complexity,
we arrive at RNA as the simplest non-trivial molecular system fulfilling
``ideal model'' requirements.
What makes RNA unique is the simultaneous presence of both levels,
genotype and phenotype, in a single molecule. RNA molecules are
heteropolymers of (predominantly) four units called ribonucleotides.
Ribonucleotides have the ribose phosphate in common, but differ in
the base attached to the sugar. RNA molecules are represented as
sequences over a four letter alphabet, with each letter standing for a
particular base - {\bf A} for adenine, {\bf U} for uracil, {\bf C} for
cytosine, and {\bf G} for guanine. Interactions mediated by hydrogen
bond patterns give rise to a stereoselective recognition between
specific pairs of bases - {\bf A}$\cdot${\bf U} and {\bf G}$\cdot${\bf
C}. This specific base pairing enables an RNA sequence to be copied
{\it via\/} a complementary negative, and hence to function as a
genotype. At the same time it enables segments of a sequence to pair
with other segments within the same sequence, causing the sequence to
fold back on itself into a structure. (In the formation of intramolecular
structure {\bf G}$\cdot${\bf U} pairs are possible as well.) This
structure mediates the chemical interactions of the sequence, and
hence constitutes its phenotype.
``Structure'' can be seen at many levels of resolution, extending from
atomic coordinates to the mean radius of gyration. This raises the
question of what constitutes a relevant level of structure resolution
for understanding adaptation. We take a pragmatic stance here, but we
shall return to this issue in conclusion. On the theoretical side we
require computability from the sequence, and on the empirical side we
require usefulness in interpreting molecular function and evolutionary
data. This leaves us with an empirically well established level of
resolution known as secondary structure. The secondary structure of
an RNA molecule refers to a topology of binary contacts arising from
specific base pairing, rather than a geometry cast in terms of
coordinates and distances (Figure \ref{secondar}). The driving force
behind secondary structure formation is the stacking between
contiguous base pairs. The formation of an energetically favorable
paired (or double-stranded) region implies, however, the formation of
an energetically unfavorable loop. This ``frustrated'' energetics
leads to a vast combinatorics of stack and loop arrangements spanning
the structural repertoire of an individual RNA sequence.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/secondar.ps,width=8cm}}
\caption[]{\label{secondar}{\small {\bf An RNA secondary structure graph.}
Unpaired positions not enclosed by base pairs, such as free ends or
links between independent structure modules, are called ``external''.
Here they are marked by ticks.}}
\end{figure}
A secondary structure can be conveniently discretized as a graph
representing the pattern of base pair contacts (Figure
\ref{secondar}). The nodes of the graph represent bases at positions
$i=1,\ldots,n$ along a sequence of length $n$. The set of edges
consists of two parts. One is common to all secondary structure
graphs, and represents the covalent backbone connecting node $i$ with
node $i+1$, $i=1,\ldots,n-1$. The other part is the secondary
structure proper, and consists of a set $P$ of edges $i\cdot j$,
$P=\{\;i\cdot j\;|\; i\ne j \text{ and } j\ne i+1\;\}$, representing
admissible hydrogen bonds between the bases at positions $i$ and
$j$. The set $P$ has to satisfy two conditions: (i) every edge in $P$
connects a node to at most one other node, and (ii) if both $i\cdot j$
and $k\cdot l$ are in $P$, then $i<k<j$ implies that $i<l<j$. Failure
to meet condition (ii) results in interactions (pseudoknots)
considered as belonging to the next - the tertiary - level of
structure. Both conditions distinguish RNA structure from protein
structure, in particular condition (i) which builds RNA secondary
structure exclusively from binary interactions. We use a picture of
the graph as the visually most immediate representation of a secondary
structure. It proves convenient to also use a line oriented
representation, such as ``{\tt ((((.(((...))).(((...))).))))}'', where
a dot stands for an unpaired position, and a pair of matching
parentheses indicates positions that are paired with one another.
Observe that the building blocks of a secondary structure are classes
of loops (Figure \ref{elements}): a hairpin loop is delimited by one
base pair which encloses a number of unpaired positions, a stack is
delimited by two base pairs and has no unpaired positions, while an
internal loop is delimited by two base pairs that enclose unpaired
positions. An internal loop is called a bulge, if either side has no
unpaired positions. Finally, the class of multiloops consists of loops
delimited by more than two base pairs. A position that does not belong
to any loop class is called external, such as free ends or joints
between loops (see Figure \ref{secondar}). The importance of these
loop classes derives from the reasonable assumption that the overall
energy of a secondary structure is the sum of its loop energies, and
from the fact that their free energies have been measured and
tabulated \cite{freier:86,turner:88,jaeger:89,he:91} as a function of
loop size and the nature of the delimiting base pairs.
\begin{figure}
\centerline{\psfig{figure=figures/elements.ps,width=10cm}}
\caption[]{\label{elements} {\small {\bf Secondary structure elements.}}}
\end{figure}
Secondary structure graphs are formal combinatorial objects which can
be subject to mathematical treatment (they can be counted, for
instance). Of particular interest are secondary structures possessing
some extremal property with respect to a given sequence, such as
minimizing the free energy. The theoretical importance of RNA as a
model system for sequence-structure relations in biopolymers lies in
the fact that structures of this kind can be computed by dynamic
programming
\cite{nussinov:78,waterman:78b,nussinov:80,zuker:81,zuker:84}. This
method produces a single structure that minimizes free energy.
Following an idea by Waterman \cite{waterman:85} we have recently
extended the standard RNA folding algorithm to compute all structures
within some energy range above the minimum free energy
\cite{wuchty:98}. However, for the present discussion the minimum free
energy structure will suffice, and we shall consider it to be the
phenotype of an RNA sequence.
The secondary structure is not just an utter abstraction, but it
provides both geometrically and thermodynamically a scaffold for the
tertiary structure. Its free energy accounts for a large share of the
overall free energy of the full structure. This linkage puts the
secondary structure in correspondence with functional properties of
the tertiary structure. Consequently, selection pressures (and hence
functional elements) become observable at the secondary structure
level in terms of conserved base pairs.
The extremely reduced amplification time and the minimal complexity of
the phenotype make RNA a tractable laboratory model. RNA molecules
can be evolved in the test tube using a variety of techniques for
amplification, variation and selection. In fact, such experiments have
shown that evolutionary optimization of RNA properties in the test
tube occurs readily and effectively. Examples are the optimization of
replicative efficiency \cite{mills:67,spiegelman:71}, the production
of RNA molecules binding optimally to prespecified target molecules by
means of the SELEX technique \cite{ellington:90,tuerk:90},
evolutionary induced changes in the activity and specificity of
catalytic RNA molecules \cite{beaudry:92} (so-called ribozymes), and
the evolutionary design of ribozymes with new functions
\cite{bartel:93,ekland:95}. Often the interpretation of such
experiments occurs by computing the secondary structure and placing it
in relation to molecular performance.
Until now the bulk of empirical interest has been directed at the
outcomes of evolutionary test tube experiments. However, we shall take
here outcomes for granted, and focus instead on dynamics and paths
toward outcomes. These aspects hold the key for understanding why (and
which) desired outcomes can be achieved at all. An emerging
evolutionary technology will depend on a comprehensive dynamical
theory of molecular adaptation, much like chemical engineering depends
on chemical kinetics.
\section{Characteristic features of the RNA folding map}
The first computer experiments on the adaptive dynamics of replicating
and mutating RNA populations \cite{fontana:87} were very encouraging,
but they also made it clear that to understand adaptive dynamics, we
must first understand the features of the landscape induced by RNA
folding. The term ``landscape'' emphasizes the folding map as a
function from a {\it space\/} of sequences into a {\it space\/} of
structures (with possibly a further assignment of numerical values to
each structure). The set of RNA sequences of length $n$ comprises
$4^n$ possible variations that are organized into a metric space by a
natural distance measure reflecting the allowed physical
interconversions of sequences. The Hamming distance is a natural
distance measure, if only point mutations are allowed. It counts the
number of positions in which two sequences differ, that is, the
minimum number of point mutations required to convert one sequence
into the other. In the case of a two-letter alphabet this space is
the well-known $n$-dimensional hypercube. The set of secondary
structures can also be made into a metric space. Structure distance
functions are typically based on some notion of minimum edit cost for
transforming one structure into another, such as the Hamming distance
or the base pair distance \cite{hofacker:94} defined on the
dot/parentheses representation of secondary structures, or the
tree-edit-distance \cite{fontana:93b}. If the structure space is
metric, it becomes possible to assess the ``ruggedness'' of the
landscape by autocorrelation functions \cite{fontana:93a}. However, it
is not always necessary or useful to think of the set of secondary
structures as a metric space. In section \ref{RNAtopo} we shall think
of it as a topological space.
Our interest is thus primarily aimed at extracting robust statistical
properties pertaining to the mapping as a whole. Although we use state
of the art algorithms \cite{zuker:81,hofacker:94,package} that are
routinely applied to predict the secondary structure of sequences
occurring naturally or having evolved in the laboratory, our main
focus is not the accurate prediction of a structure from a particular
sequence. (This would anyway require an integration with data from
phylogenetic comparison.) We seek qualitative and generic features of
adaptive dynamics in RNA, and these should not depend on whether the
employed algorithms reproduce the fine details of the actual secondary
structure in a particular instance. Rather, we rely on the fact that
the employed algorithms are sufficiently mature that they correctly
capture the logic and basic energetics of constrained base-pair
optimization inherent in RNA folding. In fact, the generic features of
the folding map summarized below have been found to be insensitive to
the choice of structure formation criteria, such as minimizing free
energy, maximizing base pairing, or kinetic folding. They also are
numerically robust to variations in the set of empirical energy
parameters or the thermodynamic level of description (one minimum free
energy structure versus the Boltzmann ensemble for a given sequence)
\cite{tacker:96}. It is encouraging that similar properties have also
been recently discovered in lattice models of protein folding
\cite{goldstein:96,goldstein:97,li:96}. For the remainder of this
paper we shall - for the sake of brevity - refer to secondary
structures simply as ``shapes''.
A systematic study of the mapping from RNA sequences to shapes was
based on the statistics of appropriately chosen samples
\cite{fontana:93b,fontana:93a,schuster:94a} as well as on the
exhaustive folding of all sequences of a given length
\cite{gruener:96a,gruener:96b}. The regularities found depend on
two simple and fundamental facts. First, both the sequence and the
shape space are very high dimensional spaces (forget three-dimensional
caricatures), and, second, the sequence space is substantially larger
than the shape space. An upper bound along the lines of
\cite{stein:78} yields only $S_{n}=1.48 \times n^{-3/2}(1.85)^{n}$
shapes vis \`a vis $4^n$ sequences \cite{hofacker:99}. It is clear
that the mapping from sequences to shapes is significantly
many-to-one, even if the alphabet were binary. These, then, are the
major generic properties that were found:
Property 1 ({\it ``typical shapes''\/}) states that some shapes are
supported by significantly larger equivalence classes of sequences
(i.e., occur more frequently) than others. These relatively few
typical shapes are set apart from many rare shapes which can hardly
play a role in evolution. The property of being ``typical'' is made
more precise by the observation that in the limit of long chains the
fraction of such shapes tends to zero (their number grows nevertheless
exponentially), while the fraction of sequences folding into them
tends to one.\footnote{This asymptotic condition for ``typical'' is
fulfilled by a whole class of definitions. A simple and
straightforward one is that of a so-called ``common shape'' which
refers to a shape formed by more sequences than the average given by
the number of sequences divided by the number of realized shapes
\cite{pks:234}.} A numerical example may help: The space of
{\bf GC}-only sequences of length $n=30$ contains $1.07\times10^{9}$
sequences folding into $218,820$ structures of which $22,718$
($=10.4\%$) classify as typical (in the sense of ``common'', see footnote
1). In this case $93.4\%$ of all sequences fold into these $10.4\%$
shapes. Property 1 also implies that any statistical statement we
make about the folding map, and {\it a fortiori\/} about adaptive
dynamics, can only hold for the set of typical shapes.
Property 2 ({\it ``neutral networks''\/}) is a statement about the
connectedness in sequence space of sequences folding into the same
shape. Typical shapes are characterized by a high degree of
neutrality expressed as the average fraction of nearest neighbors of a
sequence possessing a typical shape that retain that shape. A large
enough degree of neutrality, expressed as the mean fraction $\lambda$
of neutral neighbors with Hamming distance one, leads to percolation
in sequence space, that is, to the existence of extended neutral
networks connecting sequences with the same shape by one or at most
two point mutations (Figure \ref{neutnets}).
\begin{figure}[ht]
\centerline{\psfig{figure=figures/neutnets.eps,width=8cm}}
\caption[]{\label{neutnets}{\small {\bf Neutral networks in
sequence space.} The lower half exhibits a typical structure with a
connected network of sequences folding into it. The network reaches
through sequence space. In contrast, the upper half shows an
``untypical'' or rare structure whose neutral sequences are far less
in number. They are organized into a so-called giant component and many
small pockets. Connectivity of neutral networks depends on the mean
fraction of neutral neighbors ($\lambda$).}}
\vspace*{0.3cm}
\end{figure}
Property 3 ({\it ``shape space covering''\/}) is a statement about the
mutual entanglement of neutral networks belonging to different
shapes. All typical shapes are realized within a small neighborhood
(compared to sequence length) of any arbitrarily chosen sequence
(Figure \ref{covering}). For example, 15 mutations are sufficient on
average for a chain of length $n=100$ to find at least one instance of
every typical shape.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/covering.eps,width=11cm}}
\caption[]{\label{covering}{\small {\bf RNA shape space covering.}
To find a sequence folding into a typical structure, only a relatively
small hyperspherical region around any random sequence needs to be
searched. The covering radius can be computed from properly chosen
samples of structures.}}
\vspace*{0.3cm}
\end{figure}
These statistical properties, in particular neutral networks, have led
to a mathematical model based on percolation in random graphs
\cite{reidys:97}. There is no doubt that neutrality is an essential
feature of adaptive topographies. It is, therefore, crucial that
models of adaptive landscapes take neutrality into account, see for
example
\cite{derrida:91,gavrilets:97,newman:98,nimw98a,nimw98b,pks:248}. We
shall explore some of its effects in the following two sections.
\section{Error propagation in genotype and phenotype populations}
\label{RNAdyn}
The evolutionary dynamics of molecules based on replication, mutation
and selection induced by a constant population size in a flow reactor
has been analyzed in terms of chemical reaction kinetics by Manfred
Eigen \cite{eigen:71}, and was further developed in subsequent studies
\cite{pks:78,eigen:89}. Faithful copying and mutation
are treated on an equal footing by viewing the replication of an RNA
sequence as a branching reaction with many channels. In principle
every sequence can be obtained as a mutant from every sequence
(although the probabilities vary dramatically). The materials consumed
by RNA synthesis are replenished by a continuous flow in a reactor
resembling a chemostat for bacterial cultures (Figure
\ref{chemostat}). The character of the dynamics of a sequence
population depends critically on two factors, the accuracy of
replication which governs the ``width'' of individual reaction
channels and the degree of neutrality - a property of the
genotype-phenotype map - which governs the decoupling of the
propagation of a given phenotype from the propagation of its
underlying genotypes. The presence of neutrality mandates a
distinction between a genotypic and a phenotypic error threshold
\cite{huynen:96a}. The point of this section is to caricature the
essence of this distinction in a very simple way. This will set the
stage for discussing trajectories as influenced by the RNA
genotype-phenotype map in section \ref{RNAtopo}.
\begin{figure}[ht]
\vspace*{0.3cm}
\centerline{\psfig{figure=figures/flowreact.eps,width=10cm}}
\caption[]{\label{chemostat}{\small {\bf Flow reactor.} The flow
reactor (continuously stirred tank reactor, CSTR) is a device for
recording chemical reactions with instantaneous replenishment of
consumed materials in continuous time. Like a chemostat for bacterial
cultures, it is used here to study the evolution of populations under
replication and mutation. The influx of stock solution, containing the
materials required for replication, is compensated by an unspecific
outflux of the reaction mixture which is kept homogeneous by
mechanical stirring. The flow is adjusted to yield an average of
$N\pm\sqrt N$ RNA molecules in the tank. In the computer experiments
described here the chain length $n$ of the molecules was kept constant
by restricting variation to point mutations. Parameters are the
population size $N$, the chain length $n$, and the mutation rate per
nucleotide and replication, $p$.}}
\vspace*{0.3cm}
\end{figure}
Selection dynamics in sequence populations can be described by
ordinary differential equations. The variable $ x_i, 0\le x_i\le1$,
denotes the relative frequency of a sequence type (i.e., genotype)
I$_i$ of length $n$. For the sake of brevity we shall overload the
word ``sequence'' to mean an individual instance or a type, depending
on context. Since frequencies are normalized, we have $\sum x_i=1$.
We further denote the population size with $N$, and the number of
possible sequences with $M=4^n$. Let the rate constants for replication
and degradation of sequence I$_i$ be $a_i$ and $d_i$,
respectively. The time dependence of the sequence distribution is then
described by the kinetic equations
\begin{equation}
\dot x_i\ =\ \Bigl(a_i Q_{ii}-d_i-\bar E(t)\Bigr) x_i\,+\,\sum_{j\neq
i} a_j Q_{ji} x_j\ ,\quad i,j\,=\,1,\hdots,M\ .
\end{equation}
It has been assumed for simplicity that replication is direct, rather
than proceeding through a complementary negative, as the base pairing
rules would require (see \cite{eigen:71,eigen:89} for details). The
width of the reaction channel from sequence I$_i$ to sequence I$_j$ is
given by the mutation matrix $Q\doteq\{Q_{ij};\,i,j=1,\ldots
M\}$. $Q_{ij}$ denotes the likelihood that a replication of sequence
I$_i$ yields sequence I$_j$, and the diagonal element $Q_{ii}$ is the
fraction of correct replicas synthesized on template I$_i$. To fulfill
$\sum \dot x_i = 0$, the reactor outflow $\bar E(t)$ matches exactly
the average productivity, $\bar E(t)=\sum_{i=1}
(a_i-d_i)\, x_i(t)$. When degradation is negligible, as
in the test tube evolution of RNA molecules or when all degradation
rates are essentially the same, $d_1\approx d_2\approx
\ldots\approx d_m\approx d$, degradation has no influence on
the selection dynamics and can be neglected. In this case the
quantities which determine the selection dynamics are given by the
so-called value matrix: $W\doteq\{w_{ij}=a_i Q_{ij}\,;\,i,j=1,\ldots
M\}$, whose diagonal elements $w_{ii}$ were called selective values.
The selective value of sequence I$_i$ amounts to its fitness in the
case of vanishing mutational backflow, $\sum_{j\neq i} a_j Q_{ji}
x_j\,<<\,a_i Q_{ii}\, x_i\,=\,w_{ii}\, x_i\,\, , \forall\,
i=1,\ldots,M$. Under these conditions the sequence I$_m$ with the
maximal selective value
\begin{equation}
w_{mm}\ =\ \max\,\{w_{ii}\,|\,i=1,\ldots,M\}\ ,
\end{equation}
dominates a population in selection equilibrium, and is called the
master sequence.
The term quasispecies was introduced for the stationary sequence
distribution, whose values $\bar x_i$ are computed as the solutions
of $\dot x_i=0;\,i=1,\ldots,M$, from the eigenvalue problem
\cite{thompson:74,jones:75}
\begin{equation}
\left(W\,-\,\omega\,I\right)\,\bar x\ =\ 0\ , \label{eigen}
\end{equation}
where $I$ denotes the unit matrix, $\omega$ an eigenvalue and
$\bar x=(\bar x_1,\bar x_2,\ldots,\bar x_M)$ the corresponding
eigenvector.
\subsection{Genotypic error threshold}
To understand the difference between genotypic and phenotypic error
thresholds, we consider the so-called single peak landscape which
assigns a higher replication rate to the master and identical values
to all others, $a_m=\sigma_m\cdot a$ and $a_i=a\, ,\forall\,i\neq m$,
where $\sigma_m$ is the superiority of the master sequence. The
assumption behind the single peak landscape is to lump together all
mutants into a cloud with average fitness. The relative population
frequency of the cloud is simply $ x_c\,=\,\sum_{j\neq i}
x_j\,=\, 1- x_m$, and the replication-mutation problem boils down to
an exercise in one variable, $ x_m$, the frequency of the master. The
approach has something in common with the mean field approximation
often used in physics, since the mutant cloud can be characterized by
``mean except the master'' properties. For example, the ``mean except
the master'' replication rate constant $\bar a\,=\,\sum_{j\neq m}
a_j x_j \big/ (1- x_m)$. The superiority then reads: $\sigma_m = a_m
\big/\bar a$.
Neglecting muational backflow we can readily compute the stationary
frequency of the master sequence in a ``zeroth-order'' approximation:
\begin{equation}
\bar x_m\ =\ \frac{a_m Q_{mm}\,-\bar a}{a_m\,-\,\bar a}\ =\
\frac{\sigma_m Q_{mm}\,-\,1}{\sigma_m\,-\,1}\ .
\end{equation}
In this expression the master sequence vanishes at some finite
replication accuracy,
$Q_{mm}\bigm|_{\bar x_m=0}\,=\,Q_{min}\,=\,\sigma_m^{\ -1}$. A
non-zero stationary frequency of the master, that is, its indefinite
propagation, thus requires $Q_{mm}>Q_{min}$. This is the so-called
error threshold condition. It is illuminating to introduce a simple
model for the elements of the mutation matrix, called uniform error
rate approximation \cite{pks:78}. It assumes the existence of a per
nucleotide mutation probability $p$ that is independent of the nature
of a nucleotide and its position in the sequence. In terms of the
single digit accuracy $q=1-p$ (the mean fraction of correctly
incorporated nucleotides) the elements of the mutation matrix for a
sequence of length $n$ take the form:
\begin{equation}
Q_{ij}\ =\ q^n\,\left(\frac{1-q}{q}\right)^{d_{ij}}\ ,
\end{equation}
with $d_{ij}$ being the Hamming distance (i.e., the number of
mismatches) between two sequences I$_i$ and I$_j$. The critical
condition occurs at $q_{min}=1-p_{max}=\root n\of{\mathstrut
Q_{min}}=\sigma_m^{\ -1/n}$. The replication accuracy of RNA viruses
was indeed found to occur near this critical value
\cite{domingo:96b,domingo:97a}.
To study stochastic features of the population dynamics around the
critical replication accuracy, the replication-mutation system has
been modeled by a multitype branching process \cite{pks:152}. The main
result of this study was the derivation of an expression for the
probability of survival to infinite time for the master sequence and
its mutants. In the regime of sufficiently accurate replication the
survival probability is non-zero and decreases with increasing error
rate. At the critical accuracy $q_{min}$ this probability becomes
zero, which implies that all molecular species currently in the
population will die out in finite time and new variants will replace
them. This scenario is tantamount to migration of the population
through sequence space.
Further details of the population structure require consideration of
the mutational backflow (and thus a solution to the eigenvalue
problem, equation \ref{eigen}). The two eigenvectors associated with
the two largest eigenvalues at $q>q_{cr}$ represent the quasispecies
and the uniform distribution,
$\bar x_1=\bar x_2=\ldots=\bar x_M=1/M$, respectively. These
eigenvectors show avoided crossing at the critical accuracy $q=q_{cr}$
\cite{pks:178}. Since the off-diagonal elements are small and decrease
further with increasing chain length $n$, the zeroth order
approximation to the frequency of the master is fairly good
($q_{min}\approx q_{cr}$), and the transition from the quasispecies to
the uniform distribution is sharp
\cite{eigen:89,pks:126}. Obviously, the deterministic approach
becomes meaningless at accuracies below threshold, when the uniform
distribution, $1/M=1/{4^n}$, implies less than a single molecule for
each variant. This is already the case for fairly small chain lengths
of $n>24$, when the population size required for the deterministic
approach ($N=4^n$) exceeds the feasible sizes realizable in test tube
evolution experiments ($N\approx10^{15}$).
\subsection{Phenotypic error threshold}
In the case of neutrality the superiority of the master becomes
$\sigma_m=1$, which pushes the error threshold towards absolute
replication accuracy, $q_{min}=1$, and the deterministic model must
fail. Any sequence on a neutral network is inevitably lost at non-zero
mutation rates, but the phenotype associated with that network may
nevertheless persist. Neutrality, thus, shifts the focus from
genotypes to phenotypes, suggesting a treatment where phenotypes
become the relevant units by lumping together sequences with equal
phenotype. Given $L$ possible phenotypes, we define new aggregate
variables, $\eta_{\alpha}$ ($\alpha=1,\ldots,L$), by collecting the
set of sequences with given phenotype $\alpha$ and replication rate
$a_{\alpha}$ (latin letters refer to sequences and greek ones to
phenotypes):
\begin{equation}
\eta_{\alpha}\ =\negthickspace\negthickspace
\sum_{\substack{\text{I$_i$ with}\\
\text{phenotype $\alpha$}}}
\negthickspace\negthickspace x_i\ ,
\end{equation}
with $\sum_{\alpha=1}^L\eta_{\alpha}=\sum_{i=1}^M x_i=1$. In
analogy to the previous treatment we call the phenotype $\mu$ with
maximal fitness $a_{\mu}$ the master phenoptype. Since we are heading
again for a zeroth-order approximation, we only consider the master
phenotype. Without loss of generality we index the sequence types
possessing the master phenotype from $1$ to $k$, so that
$\eta_{\mu}=\sum_{i=1}^k x_i$. This yields the kinetic equations for
the {\it set\/} of sequences with master phenotype:
\begin{equation}
\dot \eta_{\mu}\,=\,\sum_{i=1}^k \dot x_i\,=\,\eta_{\mu}\,\bigl(
a_{\mu} Q_{kk} - \bar E\bigr)\,+\,
\sum_{i=1}^k \sum_{j\neq i} a_j Q_{ji} x_j\ . \label{pheno-kin}
\end{equation}
The mean excess productivity of the population is, of course, independent
of the choice of variables:
\begin{equation}
\bar E\,=\,\sum_{\alpha=1}^L a_{\alpha} \eta_{\alpha}\,=\,\sum_{i=1}^M
a_i x_i\ .
\end{equation}
To derive a suitable expression for the phenotypic error threshold, we
split the mutational backflow into two contributions, (i) a mutational
backflow within the neutral network of the master phenotype and (ii) a
mutational backflow on to the master network from sequences not
belonging to it:
\begin{equation}
\sum_{i=1}^k \sum_{j\neq i} a_j Q_{ji} x_j\,=\,\left\{
a_{\mu} \sum_{i=1}^k \sum_{j=1,j\neq i}^k Q_{ji} x_j\right\} +
\left\{\sum_{i=1}^k \sum_{j=k+1}^M a_j Q_{ji} x_j\right\}\ .
\end{equation}
We approximate the within-network backflow by assuming that a sequence
on the network has a constant fraction $\lambda_{\mu}$ of neutral
neighbors. We further assume equal mutation rates
($Q_{ji}=\bar{Q}_j;i,j=1,\ldots,k;i\neq j$) on the master network and
find:
\begin{eqnarray}
\sum_{i=1}^k \sum_{j=1,j\neq i}^k Q_{ji} x_j\,&\approx&\,
\frac{\lambda_{\mu}(1-Q_{kk})}{k-1} \sum_{i=1}^k
\sum_{j=1,j\neq i}^k x_j\,=\\
&=&\,\frac{\lambda_{\mu}(1-Q_{kk})}{k-1}
\sum_{j=1,j\neq i}^k \sum_{i=1}^k x_j\,=\,\lambda_{\mu}
(1-Q_{kk})\,\eta_{\mu}\ .
\end{eqnarray}
To keep things comparable, we make the same approximation as in the
genotypic error threshold and neglect mutational backflow from other
networks ($\eta_{\alpha},\alpha\neq\mu$) on to the master network. The
kinetic equation (\ref{pheno-kin}) for the master phenotype can now be
written as:
\begin{equation}
\dot \eta_{\mu}\,=\,\Bigl(a_{\mu}\tilde Q_{\mu\mu} - \bar E\Bigr)\eta_{\mu},
\end{equation}
where $\tilde Q_{\mu\mu}$ expresses an effective replication accuracy of the
master network as such:
\begin{equation}
\tilde Q_{\mu\mu}\,=\,Q_{kk}+\lambda_{\mu}(1-Q_{kk})\ .
\end{equation}
Proceeding in complete analogy with the derivation of the genotypic
error threshold, we find
\begin{equation}
\tilde Q_{min}\,=\,Q_{kk}+\lambda_{\mu}(1-Q_{kk})\,=\,\sigma_{\mu}^{-1}
\end{equation}
where $\sigma_{\mu}$ is the superiority of the master phenotype. The
uniform error rate model yields for the stationary frequency of the
master phenotype:
\begin{equation}
\bar\eta_{\mu}(p)\ =\ \frac{\sigma_{\mu}\,Q_{\mu\mu}(p)-1}
{\sigma_{\mu}-1}\ =\ \frac{(1-p)^n\,\sigma_{\mu}\,(1-\lambda_{\mu})\,+\,
\sigma_{\mu}\lambda_{\mu}\,-\,1}{\sigma_{\mu}-1}\ .
\end{equation}
The ``zeroth-order'' approximation for the phenotypic error threshold
($\bar\eta_{\mu}=0$) now becomes:
\begin{equation}
\tilde q_{min}\,=\,(1-\tilde p_{max})\,=\,
\left(\frac{1-\lambda_{\mu}\sigma_{\mu}}
{(1-\lambda_{\mu})\,\sigma_{\mu}}\right)^{1/n}\ .
\end{equation}
The function $q=\tilde q_{min}(n,\lambda_{\mu},\sigma_{\mu})$ is
illustrated in Figure \ref{phenothresh}. In the limit of no neutrality,
$\lambda_{\mu}\to 0$, both phenotypic and genotypic error threshold
are the same, $\tilde q_{min}=\sigma_{\mu}^{-1/n}=\sigma_m^{-1/n}$. In the
limit of ``extensive'' neutrality,
$\lambda_{\mu}\to\sigma_{\mu}^{-1}$, the minimal replication accuracy
$\tilde q_{\min}$ approaches zero. This means that when the degree of
neutrality exceeds the reciprocal superiority, the master phenotype is
never lost from the population, no matter what the mutation rate is.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/phe-thre.eps,width=11cm}}
\caption[]{\label{phenothresh}{\small {\bf Phenotypic error threshold.}
The error threshold, $\tilde p_{\max}$, is shown as a function of the
error rate $p$ and the mean degree of neutrality, $\lambda$. The line
separates the domains of a stationary distriution of phenotypes and
migrating populations. More erros can be tolerated at higher degrees
of neutrality.
}}
\vspace*{0.3cm}
\end{figure}
This section's goal was to show that neutrality leads to populations
whose sequences drift randomly on the neutral network of the master
phenotype, or, for that matter, any temporary most fit phenotype
\cite{huynen:96a}. Even if the sequence distribution drifts, the
phenotype is maintained. However, if the replication accuracy falls
below a critical minimum value - the phenotypic error threshold - the
population drifts through both sequence and shape space.
Interestingly, if the neutrality associated with a phenotype exceeds a
certain level, the phenotypic error threshold disappears, and that
phenotype cannot be lost from the population at any mutation rate.
Neutrality suggests a reformulation of Eigen's original kinetic
description and its stochastic versions in terms of neutral networks
as the units of aggregation. The switch in variables is complicated by
the fact that the sequence mutation matrix $Q_{ji}$ must be translated
into a ``mutation'' matrix $\tilde Q_{\alpha\beta}$ which refers to
transitions between neutral networks, i.e. phenotypes. This is where
we heavily glossed over in this section, since it requires an
understanding of the sense in which networks are ``adjacent'' to one
another. One quite fruitful approach is to define prototype landscapes
\cite{reidys:97,nimw98a,nimw98b,nimw97a}, another is to analyze this
adjacency relation in the concrete case of RNA
\cite{fontana:98b}, and to extract regularities which may guide the
design of model landscapes. We shall turn to this in the next
section.
\section{Adaptive paths and intrinsic punctuation}
\label{RNAtopo}
In this section we lay some groundwork for framing adaptive
trajectories in evolving RNA populations. To this end we must first
understand the organization of RNA shape space.
\subsection{The topology of RNA shape space}
We have seen in section \ref{RNAdyn} that neutrality loosens the
requirements on the replication accuracy for selection to remain
effective. This increases the uncertainity of adaptive trajectories
through sequence space, as populations with tiny sizes compared to the
number of all possible sequences undergo neutral diffusion. At the
same time, however, adaptive trajectories are subject to quite
specific constraints. To see this informally, consider that for
phenotypes to adapt, transitions between neutral genotype networks
(that is, phenotypes) must occur. Some transitions are easy, others
not. If the replication accuracy is sufficiently high, so that most
mutants are one-error mutants, the degree to which a transition from
one network to another is easy depends on how much of their boundary
both have in common. This suggests defining a nearness relation
between phenotypes (a topology) based on the extent to which their
corresponding neutral networks are adjacent in genotype space
\cite{fontana:98b,fontana:98a}. Notice that a so defined nearness
relation is independent of a notion of similarity between phenotypes.
Transitions between neutral networks sharing a small fraction of their
boundary will, then, act as bottlenecks, and the uncertainty of
adaptive trajectories will be reduced to the extent that they must
pass such bottlenecks. If we can characterize the set of possible
bottlenecks, we should be able to predict a {\it class\/} of phenotype
transformations that any adaptive trajectory must go through, although
we may not be able to predict the exact phenotypes involved or their
temporal succession. We shall call this class a class of ``way
points''.
In RNA, as in many biological situations, there are two mappings
involved which need to be kept distinct: the mapping from genotype to
phenotype and the mapping from phenotype to fitness. Both are
typically many-to-one and induce neutrality. However, the map from
phenotypes to fitness depends on the (exogenous or endogenous)
selection criteria, while at least in the case of RNA folding the
genotype to phenotype map does not. Because the class of phenotype
transformations that we are seeking is an intrinsic property of a
given genotype-phenotype relation, it constrains adaptive trajectories
in a fashion that holds under any (non-trivial) fitness assignment.
We shall now make these intuitions precise, and characterize the class
of way points for RNA secondary structures based on a statistical
analysis of the folding map. We then juxtapose these results with
adaptive trajectories obtained from simulating a population of RNA
sequences that replicate and mutate in a flow reactor under selection.
A particular adaptive path consists of a temporal succession of
sequences and their associated shapes. We refer to the temporal series
of shapes as the phenotypic trace. Whether a shape $\beta$ succeeds a
shape $\alpha$ will be strongly influenced by fitness. Yet for this to
be an issue at all $\beta$ must first occur somehow, that is, $\beta$
must be {\it accessible\/} from $\alpha$ by a mutation of $\alpha$'s
sequence. If $\beta$ is very likely to be accessible from $\alpha$,
we shall call $\beta$ ``near'' $\alpha$. In the case of neutrality a
shape $\alpha$ is realized by a large set of sequences, and a robust
notion of accessibility then comes to mean that $\beta$ must arise
from $\alpha$ with a high probability {\it when averaged over all
sequences folding into $\alpha$}. Only then are the shapes in the
phenotypic neighborhood of $\alpha$ a robust property of $\alpha$
itself, independent of a particular sequence.
This notion of neighborhood is illustrated by considering a tRNA-like
shape of length 76 (inset of Figure \ref{zipf}). A sample of the many
sequences folding into this shape was obtained by an inverse folding
procedure \cite{hofacker:94} available with the {\it Vienna RNA
package\/} \cite{package}. For every sequence in the sample we compute
all shapes realized in its sequence space neighborhood, consisting of
all $228$ one-error mutants. From this data we determine the fraction
of sequence neighborhoods in which a particular mutant shape appeared
at least once. The totality of these mutant shapes, irrespective of
how often they occurred, is termed the shape space {\it boundary\/} of
the tRNA shape.
When rank-ordering the boundary shapes with decreasing frequency, we
obtain Figure \ref{zipf}. The most salient feature is a marked change
in the scaling exponent, suggesting a natural cut-off point for the
definition of neighborhood. In the present case, the high frequency
range comprises some 20 shapes, which we define to be near the tRNA
shape \cite{fontana:98b}. These shapes constitute the characteristic
set of the tRNA, that is, its most specific neighborhood. The topmost
12 shapes are shown in Figure \ref{top12}, and exhibit two properties
we found to hold for all shapes whose neighborhoods we studied.
First, most shapes in the characteristic set of a shape $\alpha$ are
highly similar to $\alpha$, typically differing in a stack size by
single base pairs. Second, some shapes, such as tRNA$_8$ (the shape
ranked 8th in Figure \ref{top12}), differ by the loss of an entire
stack. The latter finding illustrates that nearness of a shape to
another does not imply their similarity. More importantly, it
illustrates that nearness is not a symmetric relation. In fact, the
tRNA shape was not found in the characteristic set of the tRNA$_8$,
and it did not even occur in its boundary sample. Not surprisingly,
the destruction of a structural element (in a random sequence bearing
it) through a single point mutation is easier than its creation. While
the high frequency of the event is surprising, it is ultimately a
consequence of the average base pair composition of stacks and the
markedly different stacking energies of AU and GC base pairs
\cite{fontana:98b}.
\begin{figure}[ht]
\vspace*{0.3cm}
\centerline{\psfig{figure=figures/zipf.eps,width=10cm,angle=-90}}
\caption[]{\label{zipf}{\small {\bf Rank-ordered frequency distribution
of shapes in the tRNA boundary.} A sample of 2,199 sequences whose
minimum free energy secondary structure is a tRNA clover-leaf (inset)
was generated. All their one-error mutants (501,372 sequences) were
folded. 28\% of the mutants retained the original structure (i.e.~were
neutral). The remaining 358,525 sequences realized 141,907 distinct
shapes. The frequency $f(\alpha)$ is the number of one-error
neighborhoods in which $\alpha$ appeared at least once, divided by the
number of sequences in the sample. The log-log plot shows the rank of
$\alpha$ versus $f(\alpha)$. Rank $n$ means the $n$th most frequent
shape. The dotted line indicates a change in the slope which we take
to naturally delimit the high frequency domain (to the left) whose
shapes form the characteristic set of the tRNA.}}
\vspace*{0.3cm}
\end{figure}
\begin{figure}[ht]
\centerline{\psfig{figure=figures/top12.eps,width=10cm}}
\caption[]{\label{top12}{\small {\bf Shapes near the tRNA shape.}
The figure shows the 12 highest ranked shapes (left to right, top to
bottom) in the characteristic set. See also Figure \ref{zipf}.}}
\vspace*{0.3cm}
\end{figure}
The nearness relation developed here defines a topology on the set of
RNA shapes. We call a transition from shape $\alpha$ to a near shape
$\beta$ {\it continuous\/} (in the spirit of topology), if that
transition is caused by a single point mutation (since in that case
the sequence of $\beta$ is in the obvious sense near the sequence of
$\alpha$).
Generalizing from this example, we can characterize continuous
transformations as those structural rearrangements which fine tune a
shape architecture in a sequential fashion by lengthening or
shortening stacks, or which destroy a stack element and the loop
implied by it (Figure \ref{trans}). This leaves the {\it
discontinuous\/} transformations characterized by the two remaining
possible structural changes: (i) the creation of a long stack in a
single step, and (ii) generalized shifts (see Figure
\ref{trans}). As an example of a shift consider one strand
of a stacked region sliding past the other by a few positions (simple
shift). Notice here that structural similarity does not imply
nearness. Both types of discontinuous transformations require the
synchronous participation of several bases (or base pairs) in a
fashion that {\it cannot be sequentialized on thermodynamic grounds}.
An example for a discontinuous transition of type (i) is the formation
of a multiloop (a loop issuing more than two stacking
regions). Generally, the free energy gain upon formation of a stack
must offset the free energy loss from the loop caused by it. A stack
closing a multiloop must, therefore, come into existence with some
minimum length (typically more than 5 bp) in a single step. Likewise,
the discontinuity of generalized shifts (type ii) has thermodynamic
and structural origins. Shifting a stack by shifting its base pairs in
random order would cause energetically unfavorable internal loops as
well as severe sterical conflicts, besides violating the formal
no-knot condition (section \ref{RNAdef}). As a consequence, the
shifting of a stack requires that all base pairs move synchronously.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/transit.ps,width=8.5cm}}
\caption[]{\label{trans}{\small {\bf Continuous and discontinuous
RNA shape transformations.} The figure illustrates transformations
between RNA secondary structure parts. Solid (dashed) arrows indicate
continuous (discontinuous) transformations in our topology. Three
groups of transformations are shown. Top: the loss and formation of a
base pair adjacent to a stack are both continuous. Middle: the opening
of a constrained stack (e.g. closing a multiloop) is continuous, while
its creation is discontinous. This reflects the fact that the
formation of a long helix between two unpaired random segments upon
mutation of a single position is a highly improbable event, whereas
the unzipping of a random helix is likely to occur as soon as a
mutation blocks one of its base pairs. Bottom: generalized shifts are
discontinuous transformations in which one or both strands of a helix
shift ending up with or without an overlap. Accordingly, we partition
generalized shifts into the four classes shown. The intersecting
disks are a schematic representation of continuous and discontinuous
transitions between two shapes $\alpha$ and $\beta$. The disk with
center $\alpha$ ($\beta$) stands for the set of shapes that are near
$\alpha$ ($\beta$). If $\beta$ is a member of $\alpha$'s disk
(neighborhood), a transition from $\alpha$ to $\beta$ is continuous
(solid arrow). A discontinuous transition leaves the neighborhood of
$\alpha$ (dashed arrow). Note that even if $\alpha$ and $\beta$ are
highly dissimilar, $\alpha$ might nontheless be transformed
continuously into $\beta$ through intermediate shapes whose
neighborhoods have sufficient overlap.}}
\end{figure}
\afterpage{\clearpage}
We may visualize the neighborhood structure on the set of all shapes
(the topology) as a directed graph. Each shape is represented by a
node. Directed edges fan out from a node $\alpha$ to the nodes in its
characteristic set. We can think of a continuous transformation of
shape $\alpha$ into a shape $\beta$ that is not nearby $\alpha$ as a
path from $\alpha$ to $\beta$ following the direction of the edges.
Discontinuous transformations are transitions between disconnected
components of the graph.
\subsection{Adaptive dynamics in RNA}
We next discuss how the topology of RNA shape space shapes adaptive
histories. Our simulation of replicating and mutating RNA populations
is cast in terms of stochastic chemical kinetics
\cite{gillespie:76,gillespie:77}, and represents a continuous time model
of Spiegelman's classic serial transfer experiments
\cite{spiegelman:71,kramer:74}. It's implementation is described elsewhere
\cite{fontana:87,fontana:89}. In the laboratory a goal might be to
find an RNA aptamer binding to some molecule
\cite{gold:90}, and the evolutionary end product is typically
unknown. In principle one can think of the end product as some shape
that is specified implicitly by the imposed selection criterion.
Since our intent is to study evolutionary trajectories rather than end
products, we short cut by simply defining a target shape in advance.
We then assume the replication rate constant of a sequence to be a
function of the distance between its shape and the target. Given a
distance measure $d$, a shape replicates faster, the more it resembles
the target. In all simulations reported here, the replication rate
$r_i$ of a sequence I$_i$ of length $n$ with shape $\alpha$ at
distance $d(\alpha,\tau)$ from a target shape $\tau$ is given by
$r_i=(0.01 + d(\alpha,\tau)/n)^{-1}$. Using an exponential or a
linear function did not make any difference with regard to the issues
we are interested in. The error rate was $p=0.001$ per nucleotide. At
this rate, the difference between parent and a mutant offspring is
mostly one point mutation, and the topology described above applies.
In the examples reported here the target shape is a tRNA clover leaf,
and the distance between shapes is measured as the Hamming distance
between their line oriented representations cast in terms of dots and
parentheses (see section \ref{RNAdef}).
Figure \ref{evolve1} and Figure \ref{evolve2} both show the approach
toward the target shape as indicated by the average Hamming distance
in the population (inversely related to fitness, black curve). Aside
from a short initial phase, the entire history is dominated by steps,
that is, flat periods of no apparent adaptive progress, interrupted by
sudden improvements toward the target. The initial relaxation period
is understood by considering that many modifications of a random
initial shape increase its similarity to any randomly chosen
target. Beyond this phase, however, adaptation becomes much harder,
and the character of adaptive dynamics changes.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/traject1.eps,width=12cm,angle=-90}}
\caption[]{\label{evolve1}{\small {\bf Simulation of an evolving
RNA population.} An RNA population evolves towards a tRNA target
shape (inset of Figure \ref{zipf}) in a flow reactor logistically
constrained to a capacity of 1,000 sequences on average. The
replication accuracy per position is $0.999$. The initial population
consisted of 1,000 identical sequences folding into a random shape.
The target was reached after approximately $11\times 10^6$
replications. The black trace shows the average structure distance of
the shapes in the population to the target. The relay series (see
text) linking the initial shape to the target comprises 43 shapes. To
each of these corresponds one horizontal level placed above the black
curve. The topmost (bottom) level belongs to the initial (target)
shape. For these levels only the time axis has a meaning. At each
level the series of presence intervals for the corresponding shape is
shown. The step curve indicates the transitions between relay shapes,
and hence the time spent by each relay shape on the adaptive
trajectory. Each transition was caused by a single point mutation in
the underlying sequences. The vertical dotted lines and the labels
mark transitions referred to in the text.}}
\vspace*{0.3cm}
\end{figure}
\begin{figure}[ht]
\centerline{\psfig{figure=figures/traject2.eps,width=12cm}}
\caption[]{\label{evolve2}{\small {\bf Simulation of an evolving
RNA population.} Same situation as in Figure \ref{evolve1}, except
for a different initial random shape. The labels refer to shape
transformations shown in Figure \ref{relay2}}}
\vspace*{0.3cm}
\end{figure}
By adaptive trajectory we mean something like a path taken by the
population as a whole (caveats below), rather than a single lineage.
We refer to its projection on shape space as its phenotypic trace. In
a simulation of this kind one has to cope with a huge amount of data,
and one possibility of obtaining an approximation to the phenotypic
trace of an adaptive trajectory is to record only data about mutation
events that generate an ``innovation'', that is, a shape which is new
in the population at the time $t$ of its appearance. This does not
neccesarily imply that the shape hasn't been in the population in the
past; it could have become extinct at some earlier time, being
``rediscovered'' at time $t$. For each shape innovation $\alpha$ we
record entry times, $l_i^{\alpha}$, and exit (extinction) times
$h_i^{\alpha}$. This yields for each shape ever seen during the
adaptive process a set of ``presence intervals''
${L}_{\alpha}=\{[l_i^{\alpha}, h_i^{\alpha}],\;l_i^{\alpha} <
h_i^{\alpha} < l_{i+1}^{\alpha}\}$, marking the presence of shape
$\alpha$ in the system's history. After the target has been found (or
the simulation has been stopped), we trace back through the history
data in the following way. Each presence interval $[l_i^{\alpha},
h_i^{\alpha}]$ of $\alpha$ has a unique ancestor with shape $\beta$
which spawned that interval at time $l_i^{\alpha}$, meaning that a
sequence folding into $\beta$ produced at time $l_i^{\alpha}$ a mutant
which folded into $\alpha$, and $\alpha$ was not in the population at
that time. Let $\omega$ be the target shape, and $l_i^{\omega}$ the
time of its first appearance (the beginning of its presence interval).
Let the shape from which $\omega$ was derived at time $l_i^{\omega}$
be $\omega_{-1}$. In the set ${L}_{\omega_{-1}}$ there is a unique
presence interval $[l_j^{\omega_{-1}}, h_j^{\omega_{-1}}]$ containing
the time instant $l^{\omega}_i$, and we proceed searching for the
unique ancestor of $[l_j^{\omega_{-1}}, h_j^{\omega_{-1}}]$. Upon
repeating this procedure we eventually end up at one of the initial
shapes (see Figure \ref{relay}). At this point we have reconstructed a
succession of shapes
$\alpha\equiv\omega_{-n}\;\omega_{-n+1}\;\cdots\;\omega_{-i}\;
\cdots\;\omega_{-1}\;\omega_{0}\equiv\beta$ connecting an initially
present shape $\alpha$ with the target (or final) shape $\beta$. This
chain is uninterrupted in time, in the sense that for every $n\ge i\ge
1$, $\omega_{-i}$ is ancestor of $\omega_{-i+1}$ and there exists a
pair $[l_r^{\omega_{-i}}, h_r^{\omega_{-i}}]\; [l_s^{\omega_{-i+1}},
h_s^{\omega_{-i+1}}]$ with $l^{\omega_{-i}}_r < l^{\omega_{-i+1}}_s <
h_r^{\omega_{-i}}$. The chain depends on the presence interval of the
final shape $\beta$ from where the trace starts, but it is unique for
that interval. Typically we are interested in the chain that
originates in the first instance of the target, and call it the
``relay series''.
\vspace*{0.3cm}
\begin{figure}[ht]
\centerline{\psfig{figure=figures/relay.eps,width=8cm}}
\caption[]{\label{relay}{\small {\bf Relay series concept.}
The figure shows schematically the presence intervals of 6 shapes, and
the reconstruction of the uninterrupted chain of innovations called
the relay-series. See text for details.}}
\vspace*{0.3cm}
\end{figure}
Figures \ref{relay1} and \ref{relay2} show the complete relay series
for the computer simulations reported in Figures \ref{evolve1} and
\ref{evolve2}, respectively. While the discontinuities in the fitness
trace of Figures \ref{evolve1} and \ref{evolve2} are apparent, it is
their comparison with the shape discontinuities along the relay series
which yields insight. The result is that fitness triggered selection
events do not always line up with shape transformations that are
discontinuous in the sense of the previously defined topology. In
Figure \ref{evolve1}, events (a) and (b) are rapid successions of
continuous transitions shortening and elongating stacks by single base
pairs. This shows that sudden changes in fitness do not imply
discontinuous phenotypic transformations. The reverse isn't true
either, as shown by the discontinuous shift event (c) which is silent
in terms of fitness. All remaining fitness changes do, however,
coincide with discontinuous shape transformations. These are the
simple shift events (e), (g), (h), (i), the double flip (d), and the
flip (f). Similar observations hold for Figure \ref{evolve2}. Here we
have an initial phase of rapid improvments (a), four simple shift
events (b), (f), (g), (h), a flip (c), a double flip (e) as well as
a ``silent shift'' (d) being a neutral discontinuous transitions,
that is, a shift which does not change fitness.
\begin{figure}[ht]
\centerline{\psfig{figure=figures/relay-1.eps,width=10cm}}
\caption[]{\label{relay1}{\small {\bf Relay series of Figure \ref{evolve1}.}
The full series of 45 relay shapes is shown. Different gray levels indicate
different classes of neighboring shapes which are accessible from the
precursor class by a discontinuous transition only. The stretch 19 to
37 (corresponding to the long plateau e-f in figure \ref{evolve1})
shows a sequence of continuous transformations between closely related
shapes some of them occurring more than once in the relay series:
$\alpha\to\beta\to\gamma\to\beta\to\gamma\to\beta\to\alpha\to\delta\to\alpha$.
}}
\vspace*{0.3cm}
\end{figure}
\begin{figure}[ht]
\centerline{\psfig{figure=figures/relay-2.eps,width=10cm}}
\caption[]{\label{relay2}{\small {\bf Relay series of Figure \ref{evolve2}.}
This series involves 42 relay shapes. Discontinuous transitions
classified as in Figure \ref{trans} are marked by letters referring
to Figure \ref{evolve2}. Again, shapes occur repeatedly in the series
as indicated, for example, by the white shapes on gray background.}}
\vspace*{0.3cm}
\end{figure}
To trigger a generalized shift or a multiloop closure by a single
point mutation puts constraints on the required sequence context,
making such sequences rare. When a shape $\alpha$ is under strong
selection, neutral drift is the only means for eventually producing
such a sequence \cite{huynen:96a,huynen:96c}. This is why
discontinuous transitions are preceded by extended periods of neutral
drift in Figures \ref{evolve1} and \ref{evolve2}.
It is important to realize that the present reconstruction of an
adaptive trajectory is not identical to the succession of shapes
associated with the actual lineage of sequences that led to the
target. The relay series reports for all $n$ from start to target
that (some sequence folding into) shape $\omega_n$ was sufficient for
giving rise to a new shape $\omega_{n+1}$. In the actual successful
lineage $\omega_{n+1}$ may not have arisen from the first sequence
with shape $\omega_n$; for that matter, it may not even have arisen
from any sequence with shape $\omega_n$. (Sequences with other shapes
may also have produced $\omega_{n+1}$, while $\omega_{n+1}$ was
already present in the population.) In sum, the relay series is a
{\it fictitious\/} path. The point is that it is a {\it possible\/}
path and a very useful representative of the {\it ensemble\/} of paths
shown to be possible by a population in a particular adaptive history.
Let us explain. At any given time $t$ a population contains a number
of individuals (instances) for each of the sequence types
present. Each individual has its unique lineage of ancestors all the
way back to the initial individuals. At the level of individuals there
would obviously be only one successful lineage or causal path to the
first appearance of the target. However, it makes sense not to
distinguish among identical individuals, and to think in terms of
sequence types (or molecular ``species'') instead. The consequence is
that a given sequence type can be produced by more than one other
sequence type, thus giving rise to a network. At this level of
resolution there is a combinatorics of paths relating the first
sequence type possessing target shape to the sequence types present
intially. Although we have lost strict causality, each of these paths
could have been a causal one if the dice had rolled differently. We
hence refer to this set as the ``ensemble of paths shown to be
possible by a population in a particular history''. To each path in
this ensemble corresponds a phenotypic trace, and the relay series is
one of those. Now comes the salient point. All these different paths
must coincide at least at the discontinuous transitions revealed by
the relay series. These transitions are seen, after the fact, as those
key innovations that enabled the population to reach the target in a
particular experiment. If such a transition is associated with a
fitness advantage, then we have a typical founder effect, where one
(or a few related) sequences conveying this advantage are amplified by
selection, giving rise to all future lineages, while the other
lineages up to that epoch are terminated. If a discontinuous
transition is not associated with a fitness advantage we have what we
might call a silent founder effect. All lineages other than those
emanating from the silent founder are doomed at some later point,
since only a lineage from the silent founder enables the next major
transition. In other words, discontinuities in the relay series
indicate way points in shape space where ensembles of different paths
coincide as they zero in (even in the absence of fitness advantage) on
the target. The few paths reaching such a way point diversify
afterwards on shape neutral as well as fitness neutral networks, until
they are reduced again at the next way point. To be sure, any fitness
improving transition has this reducing effect, whether it is
associated with a discontinuous transformation or not. The important
point here is that fitness improving transitions can be anything
depending on the selection criterion, and hence cannot be
characterized in general. In contrast, the characterizable class of
discontinuous shape transformations is an intrinsic property of RNA
folding, and constitutes a set of potential way points for {\it any\/}
selection criterion.
Preliminary results from simulations with different population sizes
(same initial conditions, same target) show roughly a constant number
of discontinuous transitions for different adaptive trajectories,
while consistently reducing the number of continuous transitions.
To summarize, the emerging picture of an adaptive trajectory is not
one of a single path, but rather of an ensemble of diverse paths
coming together at way points characterizable as discontinuous shape
transformations. (This ``ensemble view'' contains imagery borrowed
from \cite{dill:97,nimw97a}.) Given a particular initial condition
and a target, different simulations yield different adaptive
trajectories. However, all involve the same class of way points.
Future work will have to track several individual lineages throughout
the adaptive process and ``align'' them to corroborate this picture. A
hint in this direction is provided by the pattern of presence
intervals of relay shapes during the history of a simulation (see
Figures \ref{evolve1} and \ref{evolve2}). These patterns nicely
visualize the nearness relation between shapes and the way point
concept. When a shape $\alpha$ is succeeded by a shape $\beta$ that is
near $\alpha$, $\beta$ is present intermittently in the population
well prior to becoming part of the relay path. That is, once $\alpha$
is present, $\beta$ is unavoidable, and a transition to $\beta$ is
continuous. Conversely, at a discontinuous transition, when $\alpha$
is succeeded by a shape $\beta$ that is not near $\alpha$, $\beta$ has
almost always its first ever appearance just prior to that
transition. Seen together, the presence intervals of successive shapes
on the relay path form blocks of continuous (within-neighborhood)
transitions, separated by discontinuous transitions (neighborhood
escapes). During within-neighborhood transitions many relay shapes
coexist and the ensemble of possible paths is distributed over these
shapes.
\section{Conclusions}
We have reviewed our theoretical understanding of evolutionary
adaptation in model populations of RNA sequences subject to selection
at the level of their secondary structures. The focus was on the
evolutionary consequences of neutrality. The issue of neutrality in
estimating the rate of evolution was brought to the fore by the
Japanese population geneticist Motoo Kimura in the late sixties
\cite{kimura:68,kimura:83}, and was triggered by the observation of
high aminoacid substitution rates in proteins. Neutrality was recently
emphasized again by Sergey Gavrilets in the context of model
landscapes \cite{gavrilets:97}, so-called ``holey adaptive
landscapes'' consisting of neutral networks with a ``swiss
cheese''-like structure. The extreme case of stochastic dynamics on a
flat landscape was approached by Peliti and Derrida
\cite{derrida:91}, while recent progress focussed on stochastic
adaptive dynamics in piece-wise neutral prototype landscapes, such as
landscapes of a wedding-cake structure \cite{nimw98a,nimw98b,nimw96a},
``neutralized'' NK-type (Kauffman) landscapes
\cite{barnett97,newman:98}, or landscapes based on random graphs
\cite{pks:248}. The term neutral network was coined in
\cite{schuster:94a} where the phenomenon was found in the context of
the RNA folding map.
The interest in RNA secondary structure folding derives from providing
a compromise between landscapes designed for analytic tractability and
landscapes grounded in molecular reality that are also suited as
laboratory models \cite{biebricher:97,joyce:93}. Our findings on RNA
folding provide a microfoundation for Kimura's phenomenological approach,
and led to insights which are hard to obtain without a mechanistic model.
We briefly summarize these insights.
First, the important fact about neutrality in RNA is that sets of
shape neutral sequences are connected by single point mutations, and,
hence, are not just sets but networks. This alone means that adaptive
populations can tolerate higher mutation rates for transmitting a
master phenotype over generations, than if a particular genotype had
to be transmitted.
Second, while neutrality means robustness against mutations, it also
means an increased {\it ability to adapt\/}. This is less paradoxical
than it may look at first sight. The effect of a mutation depends on
the context in which it is expressed. By permitting the sequence
context to vary while preserving a shape, neutrality is a prerequisite
for making certain point mutations consequential, and hence enables
phenotypic innovation. Phrased in terms of landscape vocabulary,
neutral networks delocalize ``local traps'', and change the way a
``trap'' works. In a barrier-like trap nothing happens for a long
time, while the system is waiting for an improbable
macromutation event (such as several point mutations at once) to
occur. In contrast, a neutral network is a diffusion-like trap in
which the population spends time drifting in sequence space (as well
as in fitness neutral parts of shape space) until it hits some rare
region of the network where an adaptive transition to some other
network becomes possible. Intuitively, converting a barrier-like trap
into a (comparable) diffusion-like trap increases the likelihood of a
transition, but analytic calculations are required to spell out the
exact conditions for this to be the case \cite{nimw:pers}. (The
terminology of a barrier-like and diffusion-like trap is derived from
the similar notions of an energy barrier and an entropic barrier in
entropic spin-glasses \cite{newman:pers}.) The importance of extended
neutral networks is nicely illustrated by our repeated failure to
evolve in computer simulations a tRNA shape with {\bf GC}-only
sequences. {\bf GC}-only sequences with tRNA shape do exist, since we
could obtain thousands of them by inverse folding. Yet, each simulated
adaptive process starting from a random initial condition would get
stuck far away from any tRNA shape. Structure landscapes based on
binary {\bf GC}-only sequences are very rugged \cite{fontana:93a}, and
do not have sufficient neutrality to remove the barrier-like nature of
local traps
\cite{schuster:94a}. We had the same failure in the case of {\bf
AU}-only sequences. However, we were unable to find any {\bf AU}-only
sequence with tRNA shape by inverse folding, and the failure to evolve
one might well be due to its non-existence.
Third, neutral networks suggest a topology for RNA shapes reflecting
their mutual ``accessibility''. Accessibilty (by point mutation) of one
shape from another depends on the fraction of boundary shared by their
corresponding neutral networks. This is not a symmetric relation. A
random step out of Monaco has a very high probability of ending in
France, but the reverse is not true. In this topology a transition
from shape $\alpha$ to $\beta$ might be rare along the direct route,
but feasible in an incremental fashion by chaining together frequent
transitions. However, we found that there exist rare transitions whose
likelihood cannot be increased by any indirect route, and, hence, they
are irreducibly rare. These transitions are discontinuous in a
topological sense, and we did characterize that set as those structure
transformations which require the coordinated change of several parts
of the molecule at once. This characterization may apply to systems
other than RNA as well. The fact that for RNA that set also has a
thermodynamic raison d'\^etre, nicely illustrates how the physics of
biopolymer structure constrains adaptive trajectories quite
independently of external fitness criteria.
Fourth, neutrality increases the uncertainty in the genotypic trace of
adaptive trajectories. However, their phenotypic trace must go through
discontinuous transitions, no matter what the target shape is made to
be. This again reduces to some extent the uncertainty of adaptive
trajectories at the phenotypic level.
The same punctuated dynamics that we have observed in RNA, was also
found with evolving bacteria under precisely controlled constant
conditions in chemostats \cite{elena:96} stressing once more its
intrinsic nature. Punctuation has been reported as well in a quite
different context where cellular automata are
evolved to perform specific computational tasks (yet another kind of
``genotype-phenotype'' map) \cite{crutchfield:94a,crut98c}. This
raises the possibility that neutral networks giving rise to a
punctuation dynamics are a quite general phenomenon, not limited to
materials or structures related to strictly biological systems.
It is thought provoking to consider neutrality as a function of the
resolution at which we conceptualize ``shape'' (or ``structure'' in
general). As we decrease the resolution, formely distinct neutral
networks will merge, and diversity is lost until it makes no sense to
speak of an adaptive process, since the features captured by that
level of resolution have all the same adaptive value, that is, none.
On the other hand, as we increase the level of resolution, a formerly
single neutral network splits into smaller networks. In view of the
importance we ascribe to neutrality for adaptability, it becomes
meaningful to ask at which level of resolution networks become so
confined as to destroy the adaptability of a system. It is doubtful,
for example, whether RNA could be evolvable at all, if the full set of
atomic coordinates of an RNA shape were to matter for its function.
In that case no two sequences were structurally the same. This train
of thought suggests that the first appearance of percolating
neutrality on the structure resolution scale defines
the evolutionary relevant notion of ``structure''.
| 2024-02-18T23:40:14.965Z | 1998-11-19T00:14:38.000Z | algebraic_stack_train_0000 | 1,824 | 12,865 |
|
proofpile-arXiv_065-8869 | \section{Introduction}
Traditional shell-model studies
have recently received a renewed
interest through large scale shell model calculations
in both the $1p0f$ shell and the $2s1d0g_{7/2}$ shells with
the inclusion of the $0h_{11/2}$ intruder state as well.
It is now therefore fully possible to perform large-scale
shell-model investigations
and study the excitation spectra
for systems with
some 10 million basis states. With recent advances in
Monte Carlo methods one is also
able to enlarge the dimensionality
of the systems under study considerably,
and important information on e.g., ground state properties
has thereby been obtained.
An important feature of such large scale calculations
is that it allows one to probe the underlying many-body
physics in a hitherto unprecedented way.
The crucial starting point in all such shell-model
calculations is
the derivation of an effective interaction, be it
either an approach based on a microscopic theory
starting from the free nucleon-nucleon ($NN$) interaction or a more
phenomenologically determined interaction.
In shell-model studies of e.g., the Sn isotopes, one may have
up to 31 valence particles or holes interacting via e.g.,
an effective two-body interaction. The results of such
calculations can therefore yield, when compared with
the availiable body of experimental data, critical
inputs to the underlying theory of the effective interaction.
Until very recently, realistic shell-model effective interactions have
mainly been applied to nuclei with two or a few valence particles beyond
closed shells, such as the oxygen and calcium isotopes. Thus, by going to the
tin isotopes, in which the major neutron shell between neutron numbers 50 and
82 is being filled beyond the $^{100}$Sn closed shell core, we have the opportunity
of testing the potential of large-scale
shell-model calculations as well as the reliability of
realistic effective interactions in systems with many valence particles.
Clearly, although the $NN$ interaction is of short
but finite range, with typical interparticle
distances of the order of $1\sim 2$ fm, there are
indications from both studies of few-body systems such as the triton and
infinite nuclear matter, that at least three-body
interactions, both real and effective ones, may be of
importance.
Thus, with many valence nucleons present, such
large-scale shell-model calculations may
tell us how well e.g., an effective interaction
which only includes two-body terms does in
reproducing properties such as excitation spectra and
binding energies.
This work deals therefore with various ways of
deriving the effective
interaction or effective operator needed
in shell-model calculations, starting from the
free $NN$ interaction.
Normally, the problem of deriving such effective operators and interactions are solved
in a limited space, the so-called model space, which is a subspace of the
full Hilbert space. The effective operator and interaction
theory is then introduced in order to
systematically take into account contributions from the complement
(the excluded space) of the chosen model space. Several formulations
for such expansions of effective operators and interactions exit in the literature, following
time-dependent or time-independent
perturbation theory \cite{so95,brandow67,ko90,hko95,lm85,so84}.
Formulations like the coupled-cluster
method or exponential ansatz
\cite{lm85,arponen97,lk72a,lk72b,zabolitzky74,klz78,ticcm,hm98},
the summation of the Parquet class of diagrams
\cite{dm64,nozieres,babu,jls82,br86,scalapino,ym96,dya97}, or
the so-called $\hat{Q}$-box method with the folded-diagram
formulation of Kuo and co-workers \cite{ko90,hko95} have been extensively
applied to systems in nuclear, atomic, molecular and solid-state physics.
Here we will focus on the above-mentioned $\hat{Q}$-box approach
and the summation of the so-called Parquet diagrams.
For the description of other many-body methods such as the Hypernetted-chain or the
correlated basis function \cite{adelchi98} methods, monte-carlo related
methods
\cite{david97,vijay}, Unitary-correlation operator method \cite{so84} etc.\
see other contributions in this volume.
The
$\hat{Q}$-box has been introduced in Rayleigh-Schr\"odinger perturbation
theory as the definition of all non-folded diagrams to a given order in
the expansion parameter, in nuclear physics the so-called $G$-matrix.
The $G$-matrix renders the free $NN$ interaction $V$, which
is repulsive at small internucleon distances, tractable
for a perturbative analysis through the summation of ladders diagrams
to infinite order. Stated differently, the $G$-matrix, through the
solution of the Bethe-Brueckner-Goldstone equation, accounts for the
short-range correlations involving high-lying states.
Folded diagrams are a class of diagrams which arise
due to the removal of the dependence of the exact model-space energy
in the Brillouin-Wigner perturbation expansion. Through the $\hat{Q}$-box
formulation and its derivatives, this set of diagrams can easily be summed
up.
In addition to the evaluation of folded diagrams and the inclusion
of ladder diagrams to infinite order included in the $G$-matrix, there
are other classes of diagrams which can be summed up.
These take into account
the effect of long-range correlations involving low-energy excitations.
A frequently applied formalism is
the Tamm-Dancoff (TDA) or the random-phase
(RPA) approximations. In their traditional formulation one allows
for the summation of all particle-hole excitations, both forward-going
and backward going insertions.
This set of diagrams, as formulated by Kirson \cite{kirson74} and reviewed
in Ref.\ \cite{eo77}, should account for correlations arising
from collective particle-hole correlations. Another possibility,
is to include any number of particle-particle and hole-hole
correlations in the screening of particle-hole correlations.
The inclusion of these kind of correlations is conventionally labelled
particle-particle (pp) RPA. It has been used both in nuclear matter
studies \cite{angels88,rpd89,yhk86,syk87}
and in evaluations of ground state properties of closed-shell
nuclei \cite{hmtk87,emm91,hmm95}.
Recently, Ellis, Mavromatis and M\"uther \cite{emm91,hmm95} have
extended the pp RPA to include the particle-hole (ph) RPA, though
screening of two-particle-one-hole (2p1h) and two-hole-one-particle
(2h1p) vertices was not included.
The latter works can be viewed as a step towards the full summation of the
Parquet class of diagrams.
The summation of the Parquet diagrams entails a self-consistent
summation of both particle-particle and hole-hole ladder diagrams
and particle-hole diagrams. Practical solutions to this many-body
scheme for finite nuclei will be discussed here.
This work falls in six sections.
In the next section we present various definitions pertinent
to the determination of effective interactions, with an emphasis
on perturbative methods.
The resummation of the ladder type of
diagrams is then presented in section \ref{sec:sec3}.
In that section we also discuss the summation of so-called
folded diagrams which arise in the evaluation of
valence space effective interactions. Further perturbative
corrections are also discussed and selected results
for light nuclei in the $1s0d$ and $1p0f$ shells and for
Sn isotopes are presented.
The summation of the TDA and RPA class of diagrams is discussed in
section \ref{sec:sec4}. Other screening corrections are also
discussed in that section. The self-consistent approach to the
summation of both ladder type diagrams and screening terms
through the solution
of equations for the Parquet class of diagrams will be presented
in section \ref{sec:sec5}.
Concluding remarks are given in section \ref{sec:sec6}.
\section{Perturbative methods}
\label{sec:sec2}
In order to derive a microscopic approach to the effective interaction and/or operator
within the framework of perturbation theory, we need to introduce various
notations and definitions pertinent to the methods exposed.
In this section we review how to calculate an effective
operator within the framework of
degenerate Rayleigh-Schr\"{o}dinger
(RS) perturbation theory \cite{ko90,lm85}.
It is common practice in perturbation theory to reduce the infinitely
many degrees of freedom of the Hilbert space to those represented
by a physically motivated subspace, the model space.
In such truncations of the Hilbert space, the notions of a projection
operator $P$ onto the model space and its complement $Q$ are
introduced. The projection operators defining the model and excluded
spaces are defined by
\begin{equation}
P=\sum_{i=1}^{D} \left|\Phi_i\right\rangle
\left\langle\Phi_i\right |,
\end{equation}
and
\begin{equation}
Q=\sum_{i=D+1}^{\infty} \left|\Phi_i\right\rangle
\left\langle\Phi_i\right |,
\end{equation}
with $D$ being the dimension of the model space, and $PQ=0$, $P^2 =p$,
$Q^2 =Q$ and $P+Q=I$. The wave functions $\left|\Phi_i\right\rangle$
are eigenfunctions
of the unperturbed hamiltonian $H_0 = T+U$, where $T$ is the kinetic
energy and $U$ and appropriately chosen one-body potential, that of the
harmonic oscillator (h.o.) in most calculations. The full hamiltonian
is then rewritten as $H=H_0 +H_1$ with $H_1=V-U$, $V$ being e.g.\ the
$NN$ interaction. The eigenvalues
and eigenfunctions of the full hamiltonian are denoted by
$\left|\Psi_{\alpha}\right\rangle$
and $E_{\alpha}$,
\begin{equation}
H\left|\Psi_{\alpha}\right\rangle=
E_{\alpha}\left|\Psi_{\alpha}\right\rangle.
\end{equation}
Rather than solving the full Schr\"{o}dinger equation above, we define
an effective hamiltonian acting within the model space such
that
\begin{equation}
PH_{\mathrm{eff}}P\left|\Psi_{\alpha}\right\rangle=
E_{\alpha}P\left|\Psi_{\alpha}\right\rangle=
E_{\alpha}\left|\Phi_{\alpha}\right\rangle
\end{equation}
where $\left|\Phi_{\alpha}\right\rangle=P\left|\Psi_{\alpha}\right\rangle$
is the projection of the full wave function
onto the model space, the model space wave function.
In RS perturbation theory, the effective interaction
$H_{\mathrm{eff}}$ can be written out order by order in the
interaction $H_1$ as
\begin{equation}
PH_{\mathrm{eff}}P=PH_1P +PH_1\frac{Q}{e}H_1 P+
PH_1\frac{Q}{e}H_1 \frac{Q}{e}H_1 P+\dots,
\label{eq:effint}
\end{equation}
where terms of third and higher order also
include the aforementioned folded diagrams.
Further, $e=\omega -H_0$,
where $\omega$ is the so-called starting energy, defined as the unperturbed
energy of the interacting particles..
Similarly,
the exact wave
function $\left|\Psi_{\alpha}\right\rangle$
can now be written in terms of the model space wave function as
\begin{equation}
\left|\Psi_{\alpha}\right\rangle=
\left|\Phi_{\alpha}\right\rangle+
\frac{Q}{e}H_1\left|\Phi_{\alpha}\right\rangle
+\frac{Q}{e}H_1\frac{Q}{e}H_1\left|\Phi_{\alpha}\right\rangle
+\dots
\label{eq:wavef}
\end{equation}
The wave operator is often expressed as
\begin{equation}
\Omega = 1 +\chi,
\end{equation}
where $\chi$ is known as the correlation operator. The correlation
operator generates the component of the wave function in the $Q$-space
and must therefore contain at least one perturbation. Observing
that $P\Omega P = P$, we see that the correlation operator $\chi$
has the properties
\begin{equation}
P\chi P = 0, \hspace{1cm} Q\Omega P =
Q\chi P =\chi P. \label{eq:chi1}
\end{equation}
Since $\left|\Psi_i\right\rangle=\Omega\left|\Psi_i^{M}\right\rangle$
determines the wave operator
only when it operates to the right on the model space, i.e., only the
$\Omega P$ part is defined, the term $\Omega Q$
never appears in the theory,
and we could therefore add the conditions $Q\chi Q =0$ and $P\chi Q =0$
to Eq.\ (\ref{eq:chi1}). This leads to the following choice for $\chi$
\begin{equation}
\chi = Q\chi P. \label{eq:chi2}
\end{equation}
This has been the traditional choice in perturbation theory \cite{so95,lm85}.
The wave operator $\Omega$ can then be ordered in terms of the number
of interactions with the perturbation $H_1$
\begin{equation}
\Omega = 1 +\Omega^{(1)} + \Omega^{(2)}+\dots ,
\end{equation}
where $\Omega^{(n)}$ means that we have $n$ $H_1$ terms.
Explicitely, the above
equation reads
\begin{eqnarray}
\Omega\left|\Phi_i\right\rangle=
&{\displaystyle\left|\Phi_i\right\rangle
+\sum_{\alpha}\frac{\left|\alpha\right\rangle
\left\langle\alpha\right|
V\left|\Phi_i\right\rangle}{\varepsilon_i -\varepsilon_{\alpha}}
+\sum_{\alpha\beta}\frac{\left|\alpha\right\rangle
\left\langle\alpha\right| V
\left|\beta\right\rangle\left\langle\beta\right| V
\left|\Phi_i\right\rangle }
{(\varepsilon_i-\varepsilon_{\alpha})
(\varepsilon_i-\varepsilon_{\beta})} }\\ \label{eq:wavefu}\nonumber
& {\displaystyle -\sum_{\alpha j}\frac{\left|\alpha\right\rangle
\left\langle\alpha\right|
V\left|\Phi_j\right\rangle
\left\langle\Phi_j\right| V\left|\Phi_i\right\rangle}
{(\varepsilon_i-\varepsilon_{\alpha})
(\varepsilon_i-\varepsilon_{j})} }
+\dots ,
\end{eqnarray}
where $\varepsilon$ are the unperturbed energies of the $P$-space
and $Q$-space states defined by $H_0$.
The greek letters refer to
$Q$-space states, whereas a latin letter refers to model-space
states. The second term
in the above equation corresponds to $\Omega^{(1)}$ while the third
and fourth define $\Omega^{(2)}$.
Note that the fourth term diverges
in case we have a degenerate or nearly degenerate model space. It is
actually divergencies like these which are to be removed by the folded
diagram procedure to be discussed in the next section. Terms like these
arise due to the introduction of an energy independent perturbative
expansion. Conventionally, the various contributions to the
perturbative expansion are represented by Feynman-Goldstone diagrams.
In Fig.\ \ref{fig:wavef1} we display the topologically distinct
contributions to first order in the
interaction $V$ to
the wave operator Eq.\ (\ref{eq:wavefu}). These diagrams all
define the correlation operator $\chi$ to first order in $V$.
Diagrams with Hartree-Fock contributions
and single-particle potential terms
are not included. The possible
renormalizations of these diagrams will be discussed
in the next three sections.
The reader should note that with respect to
the nomenclature in Eq.\ (\ref{eq:wavefu}), we will hereafter
employ the following notation in our discussion of various diagrams
and vertex renormalizations:
\begin{itemize}
\item roman letters $p,q,r,s,t,\dots$ refer to particle single-particle
states, either
within the model-space or from the excluded space. An arrow pointing
upwards represents such a particle state.
\item greek letters $\alpha ,\beta ,\gamma ,\delta \dots$ refer to hole
single-particle states. An arrow pointing downward is a hole state.
\end{itemize}
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=8cm \epsfbox{vertex.eps}}
\end{picture}
\caption{The various vertices to first order in the interaction
$V$ which contribute to the wave operator
$\Omega =1+\chi$. Hartree-Fock
terms are not included. Possible hermitian conjugate
diagrams are also not shown. Indicated are also possible
angular momentun coupling orders.}
\label{fig:wavef1}
\end{figure}
\subsection{Expressions for the wave operator}
We end this section with the equations for the diagrams
in Fig.\ \ref{fig:wavef1} representing $\chi$ to first order in $V$.
Moreover, in order to introduce the various channels needed to sum the
Parquet class of diagrams, we will find it convenient here to
classify these channels in terms of angular momentum
recouplings. Later on, we will also introduce the pertinent
definitions of energy and momentum variables in the various channels.
The nomenclature we will follow in our labelling is that
of Blaizot and Ripka, see Ref.\ \cite{br86} chapter 15.
All matrix elements in the definitions below
are antisymmetrized and unnormalized.
The first channel is the $[12]$ channel, or the $s$-channel
in field theory, and its angular momentum coupling order
is depicted in Fig.\ \ref{fig:channelsdef}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,40)
\put(35,0){\epsfxsize=7cm \epsfbox{recouple.eps}}
\end{picture}
\caption{Coupling order for the $[12]$ (a), $[13]$ (b) and
$[14]$ (c) channels.}
\label{fig:channelsdef}
\end{figure}
In this figure
we do not distinguish between particles and holes, all single-particle
labels $1,2,3,4$ can represent either a hole or particle
single-particle state. It is the coupling order which is
of interest here.
The matrix element $V^{[12]}$ in the $[12]$ channel is then
\begin{equation}
V_{1234J}^{[12]}
=\left\langle (12)J\right | V
\left | (34)J\right\rangle,
\label{eq:12channel}
\end{equation}
meaning that the single-particle state $1(3)$ couples to the state
$2(4)$ to yield a total angular momentum $J$.
This channel is commonly denoted as the particle-particle (pp)
or particle-particle-hole-hole (pphh) channel, meaning that when
we will sum classes of diagrams to infinite order in this channel, the only
intermediate states which are allowed are those of a pphh character,
coupled to a final $J$ in the above order.
In the next section we will explicitely discuss ways to evaluate
the equations for the $[12]$ channel.
This coupling order is also the standard way of representing
effective interactions for shell-model
calculations.
If we now specialize to particles and holes (these matrix
elements were shown in Fig.\ \ref{fig:wavef1}) we obtain for the case
with particle states only, diagram (a),
\begin{equation}
V_{\mathrm{2p}}=V_{pqrs J}^{[12]}=
\left\langle (pq)J\right | V\left | (rs)J\right\rangle.
\label{eq:2pv}
\end{equation}
The corresponding one for holes only, diagran (b), is
\begin{equation}
V_{\mathrm{2h}}=V_{\alpha\beta\gamma\delta J}^{[12]}=
\left\langle (\alpha\beta)J\right | V
\left | (\gamma\delta)J\right\rangle.
\label{eq:2hv}
\end{equation}
Thus, in the forthcoming discussion, we will always employ as our
basic notation for a matrix element that of the $[12]$ channel,
meaning that matrix elements of the other two channels
can always be rewritten in terms of those in $[12]$ channel
We see this immediately by looking at the expression for the
matrix element in the $[13]$ channel, the $t$-channel in field
theory, see Fig.\
\ref{fig:channelsdef}(b). Here the single-particle state $3(4)$
couples to the single-particle state $1(2)$\footnote{In a Goldstone-Feynman
diagram in an angular momentum representation, the coupling direction
will always be from incoming single-particle states to outgoing
single-particle states.}.
Through simple angular momentum algebra we have
\begin{equation}
V_{1234J}^{[13]}=
{\displaystyle \sum_{J'}}(-)^{j_1+j_4+J+J'}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_3&j_1&J\\j_2&j_4&J'
\end{array}
\right\}V_{1234J}^{[12]},
\label{eq:13channel}
\end{equation}
where the symbol with curly brackets represents a $6j$-symbol and
$\hat{J'}=\sqrt{2J'+1}$.
In a similar way we can also express the matrix
element in the $[14]$ channel, the $u$-channel in field theory,
through
\begin{equation}
V_{1234J}^{[14]}=
{\displaystyle \sum_{J'}}(-)^{j_1+j_4+J+2j_3}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_4&j_1&J\\j_2&j_3&J'
\end{array}
\right\}
V_{1234J}^{[12]}.
\label{eq:14channel}
\end{equation}
It is also possible to have the inverse relations or to express
e.g., the $[14]$ channel through the $[13]$ channel as
\begin{equation}
V_{1234J}^{[14]}=
{\displaystyle \sum_{J'}}(-)^{2j_1+2j_2+2j_3}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_4&j_1&J\\j_3&j_2&J'
\end{array}
\right\}
V_{1234J}^{[13]}.
\label{eq:1413channel}
\end{equation}
The matrix elements defined in Eqs.\
(\ref{eq:12channel})-(\ref{eq:1413channel}) and the inverse relations
are the
starting points for various resummation of diagrams.
In the next section we will detail ways of solving equations
in the $[12]$ channel, whereas various approximations for the
$[13]$ channel and $[14]$ channel such as the TDA and RPA
and vertex and propagator renormalization schemes
will be discussed in section \ref{sec:sec4}. Finally, how to
merge self-consistently all three channels will be discussed
in section \ref{sec:sec5}.
We end this section by giving the expressions in an angular momentum
basis\footnote{Note that we only include angular momentum factors,
other
factors coming from the diagram rules\cite{kstop81},
like number of hole lines,
number of closed loops etc.\ are omitted here.} for the remaining
diagrams of Fig.\ \ref{fig:wavef1}. The coupling order is indicated
in the same figure.
Thus, the 2p1h vertex $V_{\mathrm{2p1h}}$,
diagram (c) in Fig.\ \ref{fig:wavef1},
is coupled
following the prescription of the $[13]$ channel and reads
\begin{equation}
V_{\mathrm{2p1h}}=V_{pqr\alpha J}^{[13]}=
{\displaystyle \sum_{J'}}(-)^{j_{\alpha}+j_p+J+J'}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_r&j_p&J\\j_q&j_{\alpha}&J'
\end{array}
\right\}
V_{pqr\alpha J'}^{[12]}.
\label{eq:2p1hv}
\end{equation}
The 2p2h ground-state correlation $V_{\mathrm{2p2h}}$, diagram (d),
which will enter
in the RPA summation discussed in section \ref{sec:sec4} is given by,
the coupling order is that of the $[13]$ channel,
\begin{equation}
V_{\mathrm{2p2h}}=V_{pq\alpha\beta J}^{[13]}=
{\displaystyle \sum_{J'}}(-)^{j_{\beta}+j_p+J+J'}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_{\alpha}&j_p&J\\j_q&j_{\beta}&J'
\end{array}
\right\}
V_{pq\alpha\beta J'}^{[12]}.
\label{eq:2p2hv}
\end{equation}
The 2h1p vertex $V_{\mathrm{2h1p}}$, diagram (e),
still in the representation of
the $[13]$ channel, is defined as
\begin{equation}
V_{\mathrm{2h1p}}=V_{\alpha\beta\gamma p J}^{[13]}=
{\displaystyle \sum_{J'}}(-)^{j_{\alpha}+j_p+J+J'}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_{\gamma}&j_{\alpha}&J\\j_{\beta}&j_p&J'
\end{array}
\right\}
V_{\alpha\beta\gamma p J'}^{[12]}.
\label{eq:2h1pv}
\end{equation}
Note well that the vertices of Eqs.\ (\ref{eq:2p1hv})-(\ref{eq:2h1pv})
and their respective
hermitian conjugates can all be expressed in the $[14]$ channel
or $[12]$ channel as well.
However, it is important to note that the expressions in the various
channels are different, and when solving the equations for the various
channels, the renormalizations will be different. As an example, consider
the two particle-hole vertices $V_{\mathrm{ph}}$
of Fig.\ \ref{fig:wavef1}, i.e., diagrams (f) and (g).
Diagram (g) is just the exchange diagram of (f) when seen in the
$[12]$ channel. However, if (f) is coupled as in the $[13]$ channel,
recoupling this diagram to the $[14]$ channel will not give
two particle-hole
two-body states coupled to a final $J$ but rather
a particle-particle two-body state and a hole-hole two-body state.
But why bother at all about such petty details? The problem arises when we
are to sum diagrams in the $[13]$ channel and $[14]$ channel.
In the $[12]$ channel we allow only particle-particle and hole-hole
intermediate states, whereas in the $[13]$ channel and $[14]$ channel
we allow only particle-hole intermediate states, else we may risk
to double-count various contributions.
If we therefore recouple
diagram (f) to the $[14]$ representation, this contribution
does not yield an intermediate particle-hole state
in the $[14]$ channel.
Thus, diagram (f), whose expression is
\begin{equation}
V_{\mathrm{ph}}=V_{p\beta \alpha q J}^{[13]}=
{\displaystyle \sum_{J'}}(-)^{j_p+j_q+J+J'}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_{\alpha}&j_p&J\\j_{\beta}&j_q&J'
\end{array}
\right\}
V_{p\beta \alpha q J'}^{[12]},
\label{eq:ph13}
\end{equation}
yields a particle-hole contribution only in the $[13]$ channel,
whereas the exchange diagram (g), which reads
\begin{equation}
V_{\mathrm{ph}}=V_{p\beta q\alpha J}^{[14]}=
{\displaystyle \sum_{J'}}(-)^{2j_q+j_{\alpha}+j_p+J}\hat{J'}^2
\left\{
\begin{array}{ccc}
j_{\alpha}&j_p&J\\j_{\beta}&j_{q}&J'
\end{array}
\right\}
V_{p\beta q\alpha J'}^{[12]},
\label{eq:ph14}
\end{equation}
results in the corresponding particle-hole contribution in the
$[14]$ channel.
In electron gas theory, the latter expression
is often identified as the starting point for the self-screening
of the exchange term. In the discussion of the TDA series in
section \ref{sec:sec4} we will give the expressions for the screening
corrections based on Eqs.\ (\ref{eq:ph13}) and (\ref{eq:ph14}).
An important aspect to notice in connection with the latter
equations and the discussions in sections \ref{sec:sec4}
and \ref{sec:sec5} is that
\begin{equation}
V_{p\beta q\alpha J}^{[14]}=-V_{p\beta \alpha q J}^{[13]},
\end{equation}
i.e., just the exchange diagram, as it should be.
This is however important to keep in mind, since we later
on will sum explicitely sets of diagrams in the
$[13]$ channel and the $[14]$ channel, implying thereby that
we will obtain screening and vertex corrections
for both direct and exchange diagrams.
\section{Summation of diagrams in the $[12]$ channel}
\label{sec:sec3}
In order to write down the equation for the renormalized
interaction $\Gamma^{[12]}$ in the
$[12]$ channel we need first to present some further definitions.
We will also assume that the reader has some familiarity with the theory
of Green's function. Thorough discussions of such topics
can be found in the recent reviews of Dickoff and M\"uther \cite{dm92}
and Kuo and Tzeng \cite{kt94}. In our presentation below we will
borrow from these works and the monograph of Blaizot and Ripka \cite{br86}.
The vertex $\Gamma^{[12]}$ is in lowest order identical with the
interaction $V^{[12]}$ and obeys also the same symmetry relations
as $V$, i.e.,
\begin{equation}
\Gamma^{[12]}_{1234J}=\Gamma^{[12]}_{2143J}=-\Gamma^{[12]}_{2134J}=
\Gamma^{[12]}_{1243J}.
\label{eq:symproperties}
\end{equation}
We also need to define energy variables. Since we are going to
replace the interaction $V$ with the $G$-matrix, or certain
approximations to it, defined below in all
of our practical calculations, the momentum variables are already
accounted for in $G$. The basis will be that of harmonic oscillator
wave functions, and the labels $1234$ will hence refer to oscillator
quantum numbers, which in turn can be related to the momentum
variables. The labels $1234$, in addition to representing
single-particle quantum numbers, define also the energy of the single-particle
states. With a harmonic oscillator basis, the starting point for the
single-particle energies $\varepsilon_{1,2,3,4}$ are the unperturbed
oscillator energies. When iterating the equations for $\Gamma^{[12]}$,
self-consistent single-particle energies can be introduced.
The total energy in the $[12]$ channel $s$ is
\begin{equation}
s=\varepsilon_1+\varepsilon_2=\varepsilon_3+\varepsilon_4.
\label{eq:energy12}
\end{equation}
The equation for the vertex $\Gamma^{[12]}$ is,
in a compact matrix notation, given by \cite{br86}
\begin{equation}
\Gamma^{[12]}=V^{[12]}+V^{[12]}(gg)\Gamma^{[12]},
\label{eq:schematic12}
\end{equation}
where $g$ is the one-body Green's function representing
the intermediate states.
The diagrammatic expression for this equation is
given in Fig.\ \ref{fig:selfcons12}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=7cm \epsfbox{sigmagamma.eps}}
\end{picture}
\caption{(a) represents the two-body vertex $\Gamma$ function while (b)
represents the self-energy $\Sigma$.}
\label{fig:selfcons12}
\end{figure}
The expression of Eq.\ (\ref{eq:schematic12}) is known as the Feynman-Galitskii
equation. This equation is normally solved iteratively.
In the first iteration
the irreducible
vertex $V^{[12]}$ is then often chosen as the bare $NN$
interaction. This interaction is then typically assumed to be energy
independent and we can drop the $s$ dependence of $V^{[12]}$. Moreover,
the single-particle propagators are chosen as the
unperturbed ones. The first iteration of
Eq.\ (\ref{eq:schematic12}) can then be rewritten as
\begin{equation}
\Gamma^{[12]}_{1234J}(s) =
V^{[12]}_{1234J}+\frac{1}{2}
\sum_{56}
V^{[12]}_{1256J}\hat{{\cal G}}^{[12]}
\Gamma^{[12]}_{5634J}(s),
\label{eq:first12}
\end{equation}
with the unperturbed particle-particle and hole-hole propagators
\begin{equation}
\hat{{\cal G}}^{[12]}=
\frac{Q^{[12]}_{\mathrm{pp}}}{s-\varepsilon_5-\varepsilon_6+\imath \eta}-
\frac{Q^{[12]}_{\mathrm{hh}}}{s-\varepsilon_5-\varepsilon_6-\imath \eta},
\label{eq:paulioperator12}
\end{equation}
which results from the integration over the energy variable
in the product of the two single-particle
propagators in Eq.\ (\ref{eq:schematic12}).
The factor $1/2$ follows from
one of the standard Goldstone-Feynman diagram
rules \cite{kstop81}, which state
that a factor $1/2$ should be associated with each pair of lines
which starts at the same interaction vertex and ends at the same
interaction vertex.
The reader should note that the intermediate states $56$
can represent a two-particle state or a two-hole state.
In Eq.\ (\ref{eq:paulioperator12}) we have assumed unperturbed single-particle
energies.
In our iterations we will approximate the single-particle energies
with their real part only. Thus,
the two-particle propagator
$\hat{{\cal G}}^{[12]}$
with renormalized single-particle energies has the same
form as the unperturbed one.
The operators $Q^{[12]}_{\mathrm{pp}}$ and $Q^{[12]}_{\mathrm{hh}}$
ensure that the intermediate states are of two-particle
or two-hole character.
In order to obtain a self-consistent scheme, Eq.\ (\ref{eq:first12})
has also to be accompanied with the
equation for the single-particle propagators $g$
given by Dyson's equation
\begin{equation}
g=g_0-g_0\Sigma g,
\label{eq:dyson12}
\end{equation}
with $g_0$ being the unperturbed single-particle
propagator and $\Sigma$ the self-energy. We will however defer a discussion
of these quantities to section \ref{sec:sec5}. Here it will suffice to state
that the self-energy is related to the vertex
$\Gamma^{[12]}$ as
\begin{equation}
\Sigma \sim g\Gamma.
\label{eq:sigma12}
\end{equation}
The similarity sign is meant to indicate that, although being formally
correct, great care has to be exercised in order not to double-count
contributions to the self-energy \cite{jls82}.
The set of equations for the vertex function and the self-energy
is shown pictorially in Fig.\ \ref{fig:selfcons12}.
Assume now that we have performed the first iteration. The question which now
arises is whether the obtained vertex $\Gamma^{[12]}$ from the solution of
Eq.\ (\ref{eq:first12}) should replace the bare vertex $V^{[12]}$
in the next iteration. Before answering this question, let us give some examples
of diagrams which can be generated from the first iteration.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,70)
\put(35,0){\epsfxsize=8cm \epsfbox{channel12.eps}}
\end{picture}
\caption{Diagrams (a)-(d) give examples of
diagrams which are summed up by
the use of Eq.\ (\protect{\ref{eq:schematic12}}).
Diagrams (e) and (f) are examples of core-polarization
terms which are not generated by the $[12]$ channel.}
\label{fig:gamma12}
\end{figure}
These contributions
are given by e.g., diagrams (a)-(d) in Fig.\ \ref{fig:gamma12}.
Diagrams (a) and (b) are examples of contributions to second order
in perturbation theory, while diagrams (c) and (d) are higher
order terms. Diagrams (e) and (f) are higher-order core-polarization terms,
which can e.g., be generated through the solution of the equations for
the $[13]$ and $[14]$ channels discussed in the next section.
If we were to include diagrams (a)-(d) in the definition of the bare
vertex in our next iteration,
we are prone to double-count, since such contributions are generated
once again. Diagrams which contain particle-hole intermediate state are however
not generated by the solution of Eq.\ (\ref{eq:first12}).
We need therefore to define the vertex $V^{[12]}$ used in every iteration
to be the sum of diagrams which are irreducible in the $[12]$ channel.
With irreducible we will mean all diagrams which cannot be reduced to
a piece containing the particle states $12$ entering or leaving
the same interaction vertex
and another part containing the states $34$ at the same interaction
vertex by cutting two internal lines.
Clearly, if we cut diagrams (a) and (b) we are just left with two bare
interaction vertices. Similarly, cutting two lines of an intermediate
state in diagrams (c) and (d) leaves us with two second-order terms
of the type (a) and (b) and one bare interaction.
Diagrams (e) and (f) are however examples of diagrams which are
irreducible in the $[12]$ channel. Diagram (e) is
irreducible in the $[13]$ channel, but not in the $[14]$ channel.
Similarly, diagram (g) is reducible in the $[13]$ channel and
irreducible in the $[14]$ channel.
This means that, unless we solve equations similar to Eq.\ (\ref{eq:first12})
in the $[13]$ channel and $[14]$ channels as well, changes
from further iterations
of Eq.\ (\ref{eq:first12}) will only come from
the single-particle terms defined
by Dyson's equation in Eq.\ (\ref{eq:dyson12}).
In the remaining part of this section, we will try to delineate ways
of solving the above equations, and discuss possible approximations,
their merits and faults. First of all, we will reduce
the propagator of Eq.\ (\ref{eq:paulioperator12}) to only include
particle-particle intermediate states. This will lead us to the familiar
$G$-matrix in nuclear many-body theory.
Based on the $G$-matrix, we will construct effective interactions
through perturbative summations. Applications of such
effective interacions to selected nuclei will then be discussed.
Thereafter, we will try to
account for hole-hole contributions and end this section with a
discussion on self-consistent determinations of the single-particle
energies through the solution of Dyson's equation.
\subsection{The $G$-matrix}
In nuclear structure and nuclear matter calculations one has to
face the problem that any realistic $NN$ interaction $V$
exhibits a strong short-range repulsion, which in turn makes
a perturbative treatment of the nuclear many-body problem
prohibitive. If the interaction has a so-called hard core,
the matrix elements of such an interaction $\bra{\psi}V\ket{\psi}$
evaluated
for an uncorrelated two-body wave function $\psi (r)$ diverge,
since the uncorrelated wave function is different from zero also for
relative distances $r$ smaller than the hard-core radius. Similarly,
even if one uses interactions with softer cores, the matrix elements of the
interaction become very large at short distances.
The above problem was however overcome by
introducing the reaction matrix $G^{[12]}$ (displayed by the summation
of ladder type of diagrams in Fig.\ \ref{fig:gamma12}
with particle-particle intermediate states only),
accounting thereby for short-range two-nucleon correlations.
The $G^{[12]}$-matrix represents just a subset to the solution
of the equations for the interaction $\Gamma^{[12]}$ in the $[12]$ channel,
we have clearly neglected the possibility of having intermediate states
which are of the hole-hole type.
The matrix elements of the
interaction $V^{[12]}$ then become
\begin{equation}
\bra{\psi}G^{[12]}\ket{\psi} =\bra{\psi}V^{[12]}\ket{\Psi}
\end{equation}
where $\Psi$ is now the correlated wave function
containing selected correlation from the excluded space.
By accounting for these
correlations in the two-body wave functon $\Psi$, the matrix elements of
the interaction become finite, even for a hard-core interaction $V$. Moreover,
as will be discussed below, compared with the uncorrelated
wave function, the correlated wave function enhances the
matrix elements of $V$ at distances for which the interaction is
attractive.
The type of correlations which typically are included in the evaluation
of the $G^{[12]}$-matrix are those of the two-particle type.
If we label the operator $Q$ in this case by $Q^{[12]}_{\mathrm{pp}}$, we can write
the integral equation for the $G$-matrix as
\begin{equation}
G^{[12]}(s)=V^{[12]}+V^{[12]}\frac{Q^{[12]}_{\mathrm{pp}}}
{s -H_0+\imath \eta}G^{[12]}(s),
\label{eq:g1}
\end{equation}
implicitely assuming that $\lim \eta \rightarrow \infty$.
The variable $s$ represents normally the unperturbed
energy of the incoming two-particle state. We will suppress $\imath \eta$ in the
following equations. Moreover, since one is often interested only in the
$G^{[12]}$-matrix
for negative starting energies, the $G^{[12]}$-matrix commonly used in studies
of effective interactions has no divergencies.
We can also write
\begin{equation}
G^{[12]}(s )=V^{[12]}+V^{[12]}Q^{[12]}_{\mathrm{pp}}
\frac{1}{s -Q^{[12]}_{\mathrm{pp}}H_0Q^{[12]}_{\mathrm{pp}}}
Q^{[12]}_{\mathrm{pp}}
G^{[12]}(s ).
\label{eq:g2}
\end{equation}
The former equation applies if the Pauli operator $Q^{[12]}_{\mathrm{pp}}$ commutes
with the unperturbed hamiltonian $H_0$, whereas the latter is
needed if $[H_0,Q^{[12]}_{\mathrm{pp}}]\neq 0$.
Similarly, the correlated wave function $\Psi$
is given as
\begin{equation}
\ket{\Psi}=\ket{\psi}+
\frac{Q^{[12]}_{\mathrm{pp}}}{s - H_0}G^{[12]}\ket{\psi},
\label{eq:wave}
\end{equation}
or
\begin{equation}
\ket{\Psi}=\ket{\psi}+Q^{[12]}_{\mathrm{pp}}\frac{1}
{s - Q^{[12]}_{\mathrm{pp}}H_0Q^{[12]}_{\mathrm{pp}}}
Q^{[12]}_{\mathrm{pp}}G^{[12]}\ket{\psi}.
\end{equation}
In order to evaluate the $G^{[12]}$-matrix for finite nuclei,
we define first a useful identity following Bethe, Brandow and
Petschek \cite{bbp63}. Suppose we have two
different $G$-matrices\footnote{For notational economy,
we drop the superscript $^{[12]}$. Furthermore,
in the subsequent discussion in
this subsection it is understood that all operators $Q$
refer to particle-particle intermediate states only. The subscript
$\mathrm{pp}$ is also dropped.}, defined by
\begin{equation}
G_1=V_1+V_1\frac{Q_1}{e_1}G_1,
\end{equation}
and
\begin{equation}
G_2=V_2+V_2\frac{Q_2}{e_2}G_2,
\end{equation}
where $Q_1/e_1$ and $Q_2/e_2$ are the propagators of
either Eq.\ (\ref{eq:g1}) or Eq.\ (\ref{eq:g2}). $G_1$ and $G_2$
are two different $G$-matrices having two different interactions
and/or different propagators. We aim at an identity
which will enable us to calculate $G_1$ in terms of $G_2$,
or vice versa.
Defining the wave operators
\begin{equation}
\Omega_1=1+\frac{Q_1}{e_1}G_1,
\end{equation}
and
\begin{equation}
\Omega_2=1+\frac{Q_2}{e_2}G_2,
\end{equation}
we can rewrite the above $G$-matrices as
\begin{equation}
G_1=V_1\Omega_1,
\label{eq:omega1}
\end{equation}
and
\begin{equation}
G_2=V_2\Omega_2.
\label{eq:omega2}
\end{equation}
Using these relations, we rewrite $G_1$ as
\begin{eqnarray}
G_1=&G_1 -{\displaystyle
G_2^{\dagger}\left(\Omega_1-1-\frac{Q_1}{e_1}G_1\right)
+\left(\Omega_2^{\dagger}-1-G_2^{\dagger}\frac{Q_2}{e_2}\right)G_1}
\nonumber \\
=&{\displaystyle G_2^{\dagger} +G_2^{\dagger}\left(\frac{Q_1}{e_1}-
\frac{Q_2}{e_2}\right)G_1
+\Omega_2^{\dagger}G_1 -G_2^{\dagger}\Omega_1},
\end{eqnarray}
and using Eqs.\ (\ref{eq:omega1}) and (\ref{eq:omega2}) we obtain
the identity
\begin{equation}
G_1=G_2^{\dagger} +G_2^{\dagger}
\left(\frac{Q_1}{e_1}-\frac{Q_2}{e_2}\right)G_1
+\Omega_2^{\dagger}(V_1-V_2)\Omega_1.
\label{eq:gidentity}
\end{equation}
The second term on the rhs.\ is called the propagator-correction term;
it vanishes if $G_1$ and $G_2$ have the same propagators. The third term
is often referred to as the potential-correction term, and it disappears
if $G_1$ and $G_2$ have the same potentials.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=8cm \epsfbox{ho.eps}}
\end{picture}
\caption{Classification of harmonic oscillator single-particle
orbits.}
\label{fig:orbits}
\end{figure}
The reader may now ask what is the advantage of the above identity. If
we assume that by some physical reasoning we are able to calculate
$G_2$ and that the expression for $G_2$ can be calculated
easily, and further that $G_2$ is a good approximation
to the original $G$-matrix, then we can use the above identity to
perform a perturbative calculation of $G_1$ in terms of $G_2$.
Before we proceed in detailing the calculation of the $G$-matrix
appropriate for finite nuclei, certain approximations need be explained.
As discussed above, the philosophy behind perturbation theory is
to reduce the intractable full Hilbert space problem to one which
can be solved within a physically motivated model space, defined by the
operator $P$. The excluded degrees of freedom are represented by the
projection operator $Q$. The definition of these operators is connected
with the nuclear system and the perturbative expansions discussed
in section \ref{sec:sec2}. Consider the evaluation of the effective interaction
needed in calculations of the low-lying states of $^{18}$O. From
experimental data and theoretical calculations the belief is that
several properties of this nucleus can be described by a model
space consisting of a closed $^{16}$O core (consisting of the filled
$0s$- and $0p$-shells) and two valence neutrons
in the $1s0d$-shell. In Fig.\ \ref{fig:orbits} we exhibit this division
in terms of h.o.~sp orbits.
The active sp states in the $1s0d$-shell are then given by the $0d_{5/2}$,
$0d_{3/2}$ and $1s_{1/2}$ orbits, labels $4-6$ in Fig.\ \ref{fig:orbits}.
The remaining states enter the definition of
$Q$. Once we have defined $P$ and $Q$ we proceed in constructing the $G$-matrix
and the corresponding perturbative expansion in terms of the $G$-matrix.
There are however several ways of choosing $Q$. A common procedure is to
specify the boundaries of $Q$ by three numbers, $n_1$, $n_2$ and $n_3$, explained
in Fig.\ \ref{fig:qoperat}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=7cm \epsfbox{paulifin.eps}}
\end{picture}
\caption{Definition of the $P$ (shaded area) and $Q$ operators
appropriate for the definition of the $G$-matrix and the effective
interaction. Outside the shaded area limited by the boundaries $n_1$,
$n_2$ and $n_3$ $P=0$ and $Q=1$.}
\label{fig:qoperat}
\end{figure}
For $^{18}$O we would choose $(n_1=3,n_2=6,n_3=\infty)$.
Our choice of
$P$-space implies that the single-particle states outside the model space
start from the
$1p0f$-shell (numbers 7--10 in Fig.\ \ref{fig:orbits}), and orbits 1, 2
and 3 are hole states. Stated differently, this means that $Q$
is constructed so as to prevent scattering into intermediate
two-particle states
with one particle in the $0s$- or $0p$-shells or both particles
in the $1s0d$-shell. This definition of the $Q$-space influences the determination
of the effective shell-model interaction. Consider the diagrams displayed
in Fig.\ \ref{fig:qboxexam1}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,50)
\put(35,5){\epsfxsize=8cm \epsfbox{qdiags.eps}}
\end{picture}
\caption{Examples of diagrams which may define the effective valence space
interaction. The wavy line is the $G$-matrix.}
\label{fig:qboxexam1}
\end{figure}
Diagram (a) of this figure is just the $G$-matrix and is allowed in the definition
of the $\hat{Q}$-box. With our choice $(n_1=3,n_2=6,n_3=\infty)$, diagram (b) is not
allowed since the intermediate state consists of passive particle
states and is already included in the evaluation of the $G$-matrix. Similarly,
diagram (c) is also not allowed whereas diagram (d) is allowed. Now an important
subtlety arises. If we evaluate the $G$-matrix with the boundaries
$(n_1=3,n_2=10,n_3=\infty)$, and define the $P$-space of {\em
the effective interaction}
by including orbits 4 to 6 only, then diagrams (b) and (c)
are allowed if $7\leq p_1 , p_2 \leq 10$
In this way we allow for
intermediate two-particle states as well with orbits outside the
model-space of the effective interaction. The reader should notice the above
differences, i.e.\ that the $Q$-space defining the $G$-matrix and
$H_{\mathrm{eff}}$
may differ.
In order to calculate the $G$-matrix we will henceforth employ a
so-called double-partitioned scheme.
Let us be more specific and detail this double-partitioned procedure.
We define first a reference $G$-matrix $\tilde{G}$
in terms of plane wave intermediate states only, meaning that $H_0$ is
replaced by the kinetic energy operator $T$ only
while $G$ has harmonic oscillator intermediate states (this is one
possible choice for $U$). We divide the exclusion operator
into two parts, one which represents the low-lying states $Q_l$ and
one which accounts for high-lying states $Q_h$, viz.\
\[
Q=Q_l+Q_h=Q_l+\tilde{Q}.
\]
If we consider $^{18}$O as our pilot nucleus, we may define $Q_l$ to consist
of the sp orbits of the $1p0f$-shell, orbits $7-10$ in Fig.\ \ref{fig:orbits},
described by h.o.\ states. $Q_h$ represents then the remaining orthogonalized
intermediate states.
Using the identity of Bethe, Brandow and Petschek \cite{bbp63} of
Eq.\ (\ref{eq:gidentity}) we
can first set up $\tilde{G}$ as
\begin{equation}
\tilde{G}=V+V\frac{\tilde{Q}}{s -T}\tilde{G},
\label{eq:gfinite}
\end{equation}
and express $G$ in terms of $\tilde{G}$ as
\begin{equation}
G=\tilde{G} +\tilde{G}
\left(\frac{Q_l}{s -H_0}\right)G,
\label{eq:gidfinite}
\end{equation}
and we have assumed that $\tilde{G}$ is hermitian and that $[Q_l,H_0]=0$.
Thus, we first calculate
a ``reference'' $G$-matrix ($\tilde{G}$ in our case), and then insert this
in the expression for the full $G$-matrix. The novelty here is that
we are able to calculate $\tilde{G}$ exactly through operator relations
to be discussed below. In passing we note that $G$ depends significantly
on the choice of $H_0$, though the low-lying intermediate states
are believed to be fairly well represented by h.o.\ states.
Also, the authors of ref.\ \cite{kkko76} demonstrate that low-lying
intermediate states are not so important in $G$-matrix calculations,
being consistent with the short-range nature of the $NN$ interaction.
Since we let $Q_l$ to be defined by the orbits of the $1p0f$-shell,
and the energy difference between two particles in the
$sd$-shell and $pf$ shell is of the order $-14$ MeV, we can treat
$G$ as a perturbation expansion in $\tilde{G}$.
Eq.\ (\ref{eq:gidfinite}) can then be written as
\begin{equation}
G=\tilde{G} +\tilde{G}
\left(\frac{Q_l}{s -H_0}\right)\tilde{G}
+\tilde{G}
\left(\frac{Q_l}{s -H_0}\right)\tilde{G}
\left(\frac{Q_l}{s -H_0}\right)\tilde{G} +\dots
\end{equation}
The only intermediate states are those defined by the $1p0f$-shell.
The second term on the rhs.\ is nothing but the second-order
particle-particle ladder. The third term is then the third-order ladder
diagram in terms of
$\tilde{G}$. As shown by the authors of ref.\ \cite{kkko76}, the inclusion
of the second-order particle-particle diagram in the evaluation
of the $\hat{Q}$-box, represents a good approximation.
The unsettled problem is however how to define
the boundary between
$Q_l$ and $Q_h$.
Now we will discuss how to compute $\tilde{G}$.
One can solve the equation for the $G$-matrix
for finite nuclei by employing
a formally
exact technique for handling $\tilde{Q}$
discussed in e.g., Ref.\ \cite{kkko76}.
Using the matrix identity
\begin{equation}
\tilde{Q}\frac{1}{\tilde{Q}A\tilde{Q}}
\tilde{Q}=\frac{1}{A}-
\frac{1}{A}\tilde{P}\frac{1}{\tilde{P}A^{-1}\tilde{P}}\tilde{P}\frac{1}{A},
\label{eq:matrix_relation_q}
\end{equation}
with $A=s -T$, to rewrite Eq.\ (\ref{eq:gfinite}) as\footnote{We will omit the
label $\tilde{G}$ for the $G$-matrix for finite nuclei, however it is
understood that the $G$-matrix for finite nuclei is calculated according
to Eq.\ (\ref{eq:gfinite}) This means that we have to
include the particle-particle ladder diagrams in the
$\hat{Q}$-box. }
\begin{equation}
G = G_{F} +\Delta G,\label{eq:gmod}
\end{equation}
where $G_{F}$ is the free $G$-matrix defined as
\begin{equation}
G_{F}=V+V\frac{1}{s - T}G_{F}. \label{eq:freeg}
\end{equation}
The term $\Delta G$ is a correction term defined entirely within the
model space $\tilde{P}$ and given by
\begin{equation}
\Delta G =-V\frac{1}{A}\tilde{P}
\frac{1}{\tilde{P}A^{-1}\tilde{P}}\tilde{P}\frac{1}{A}V.
\end{equation}
Employing the definition for the free $G$-matrix of Eq.\ (\ref{eq:freeg}),
one can rewrite the latter equation as
\begin{equation}
\Delta G =-G_{F}\frac{1}{e}\tilde{P}
\frac{1}{\tilde{P}(e^{-1}+e^{-1}G_{F}e^{-1})
\tilde{P}}\tilde{P}\frac{1}{e}G_F,
\end{equation}
with $e=s -T$.
We see then that the $G$-matrix for finite nuclei
is expressed as the sum of two
terms; the first term is the free $G$-matrix with no Pauli corrections
included, while the second term accounts for medium modifications
due to the Pauli principle. The second term can easily
be obtained by some simple matrix operations involving
the model-space matrix $\tilde{P}$ only.
However, the second term is a function of the variable
$n_3$. The convergence in terms of $n_3$ was discussed ad extenso
in Ref.\ \cite{hko95} and we refer the reader to that work.
The equation for the free matrix $G_F$ is solved in momentum space in the
relative and centre of mass system and thereafter transformed to the
relevant expression in terms of harmonic ocillator single-particle
wavefunctions in the laboratory system. This yields final
matrix elements of the type
\begin{equation}
\bra{(ab)J}G\ket{(cd)J}
\end{equation}
where $G$ is the given by the sum $G = G_{F} +\Delta G$.
The label $a$ represents here all the single particle quantum numbers
$n_{a}l_{a}j_{a}$.
\subsection{Folded diagrams and the effective valence space interaction}
Here we discuss further classes of diagrams
which can be included in the evaluation of effective interactions
for the shell model.
Especially, we will focus on the summations of so-called folded
diagrams.
One way of obtaining the wave operator
$\Omega$ is through the generalized Bloch
equation given by Lindgren and Morrison \cite{lm85}
\begin{equation}
[\Omega, H_0]P=QH_1\Omega P-\chi PH_1\Omega P,
\label{eq:lind}
\end{equation}
which offers a suitable way of generating the RS perturbation expansion.
Writing Eq.\ (\ref{eq:lind}) in terms of $\Omega^{(n)}$ we have
\begin{equation}
[\Omega^{(1)}, H_0]P=QH_1P,
\end{equation}
\begin{equation}
[\Omega^{(2)}, H_0]P=QH_1\Omega^{(1)} P- \Omega^{(1)} PH_1P,
\end{equation}
and so forth, which can be generalized to
\begin{equation}
[\Omega^{(n)}, H_0]P=QH_1\Omega^{(n-1)} P- \sum_{m=1}^{n-1}
\Omega^{(n-m)} PH_1\Omega^{(m-1)}P.
\end{equation}
The effective interaction to a given order can then be obtained from
$\Omega^{(n)}$, see \cite{lm85}.
Another possibility is obvioulsy the coupled-cluster method discussed
elewhere in this volume.
Here we will assume that we can start with a given
approximation to $\Omega$, and through an iterative scheme generate
higher order terms. Such schemes will in general differ from the
order-by-order scheme of Eq.\ (\ref{eq:lind}). Two such iterative
schemes were derived by Lee and Suzuki \cite{ls80}. We will focus
on the folded diagram method of Kuo and co-workers \cite{ko90}.
Having defined the wave operator $\Omega = 1 +\chi$ (note that $
\Omega^{-1}=1-\chi$) with
$\chi$ given by Eq.\ (\ref{eq:chi2}) we can obtain
\begin{equation}
QHP-\chi HP +QH\chi - \chi H\chi = 0. \label{eq:basic}
\end{equation}
This is the basic equation to which a solution to $\chi$ is
to be sought.
If we choose to work with a degenerate model space we define
\[
PH_0 P = s P,
\]
where $s$ is the unperturbed model space eigenvalue
(or starting energy) in the degenerate case,
such that Eq.\ (\ref{eq:basic}) reads in a slightly modified form
($H=H_0 + H_1$)
\[
(s -QH_0 Q -QH_1 Q)\chi = QH_1 P -\chi PH_1 P -\chi PH_1 Q\chi,
\]
which yields the following equation for $\chi$
\begin{equation}
\chi = \frac{1}{s - QHQ}QH_1 P -\frac{1}{s -QHQ}\chi\left(PH_1 P +
PH_1 Q\chi P\right).\label{eq:chi3}
\end{equation}
Observing that the $P$-space effective hamiltonian is given as
\[
H_{\mathrm{eff}}= PHP+PH\chi=PH_0 P + V_{\mathrm{eff}}(\chi),
\]
with $V_{\mathrm{eff}}(\chi)= PH_1 P + PH_1Q\chi P$, Eq. (\ref{eq:chi3}) becomes
\begin{equation}
\chi = \frac{1}{s - QHQ}QH_1 P -\frac{1}{s -QHQ}
\chi V_{\mathrm{eff}}(\chi ).
\label{eq:chi4}
\end{equation}
Now we find it convenient to introduce the so-called $\hat{Q}$-box,
defined as
\begin{equation}
\hat{Q}(s)=PH_1 P + PH_1 Q\frac{1}{s - QHQ}
QH_1 P.\label{eq:qbox}
\end{equation}
The $\hat{Q}$-box is made up of non-folded diagrams which are irreducible
and valence linked. A diagram is said to be irreducible if between each pair
of vertices there is at least one hole state or a particle state outside
the model space. In a valence-linked diagram the interactions are linked
(via fermion lines) to at least one valence line. Note that a valence-linked
diagram can be either connected (consisting of a single piece) or
disconnected. In the final expansion including folded diagrams as well, the
disconnected diagrams are found to cancel out \cite{ko90}.
This corresponds to the cancellation of unlinked diagrams
of the Goldstone expansion.
We illustrate
these definitions by the diagrams shown in Fig.\
\ref{fig:diagsexam}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,40)
\put(35,0){\epsfxsize=8cm \epsfbox{diagsexam.eps}}
\end{picture}
\caption{Different types of valence-linked diagrams. Diagram (a)
is irreducible and connected, (b) is reducible, while (c) is irreducible
and disconnected.}
\label{fig:diagsexam}
\end{figure}
Diagram (a) is irreducible, valence linked and connected,
while (b) is reducible since
the intermediate particle states belong to the model space.
Diagram (c) is irreducible, valence linked and disconnected.
It is worth noting that general form of the $\hat{Q}$-box
is the same as that of the $G$-matrix, or the equations
of the $[12]$ channel or those of the $[13]$ and $[14]$
channels to be discussed in section \ref{sec:sec4}.
In Ref.\ \cite{hko95}, the $\hat{Q}$-box was defined
to be the sum all diagrams to third order in the $G$-matrix.
Multiplying both sides of Eq.\ (\ref{eq:chi4}) with $PH_1$ and
adding $PH_1 P$ to both sides we get
\[
PH_1 P + PH_1 \chi =
PH_1 P + PH_1 Q\frac{1}{s - QHQ}QH_1 P -
PH_1 \frac{1}{s -QHQ}\chi V_{\mathrm{eff}}(\chi ),
\]
which gives
\begin{equation}
V_{\mathrm{eff}}(\chi )=\hat{Q}(s)-
PH_1 \frac{1}{s -QHQ}\chi V_{\mathrm{eff}}(\chi ).
\label{eq:veff}
\end{equation}
There are several ways to solve Eq.\ (\ref{eq:veff}). The idea is
to set up an iteration scheme where we determine $\chi_n$ and
thus $V_{\mathrm{eff}}(\chi_n )$ from
$\chi_{n-1}$ and $V_{\mathrm{eff}}(\chi_{n-1})$.
For the mere sake of simplicity we write
$V_{\mathrm{eff}}^{(n)}=V_{\mathrm{eff}}(\chi_{n})$.
Let us write Eq.\ (\ref{eq:veff}) as
\[
V_{\mathrm{eff}}^{(n)}=\hat{Q}(s)-
PH_1 \frac{1}{s -QHQ}\chi_n V_{\mathrm{eff}}^{(n-1)}.
\]
The solution to this equation can be shown to be \cite{ls80}
\begin{equation}
V_{\mathrm{eff}}^{(n)}=\hat{Q}+{\displaystyle\sum_{m=1}^{\infty}}
\frac{1}{m!}\frac{d^m\hat{Q}}{ds^m}\left\{
V_{\mathrm{eff}}^{(n-1)}\right\}^m .
\label{eq:fd}
\end{equation}
Observe also that the
effective interaction is $V_{\mathrm{eff}}^{(n)}$
is evaluated at a given model space energy
$s$. If
$V_{\mathrm{eff}}^{(n)}=V_{\mathrm{eff}}^{(n-1)}$, the iteration is said to
converge. In the limiting case $n\rightarrow \infty$, the
solution $V_{\mathrm{eff}}^{(\infty)}$ agrees with the formal solution of
Brandow
\cite{brandow67} and Des Cloizeaux \cite{des}
\begin{equation}
V_{\mathrm{eff}}^{(\infty)}=\sum_{m=0}^{\infty}\frac{1}{m!}
\frac{d^{m}\hat{Q}}{ds^{m}}\left\{
V_{\mathrm{eff}}^{(\infty)}\right\}^{m}.\label{eq:pert}
\end{equation}
Alternatively, we can generate the contribution
from $n$ folds the following way. In
an $n$-folded $\hat{Q}$-box there are of course $n+1$ $\hat{Q}$-boxes. The
general expression for an $n$-folded $\hat{Q}$-box is then
\begin{equation}
\hat{Q} -\hat{Q}\int\hat{Q}
+\hat{Q}\int\hat{Q}\int\hat{Q} -\dots=
{\displaystyle\sum_{m_1m_2\dots m_n}}
\frac{1}{m_1!}\frac{d^{m_1}\hat{Q}}{ds^{m_1}}P
\frac{1}{m_2!}\frac{d^{m_2}\hat{Q}}{ds^{m_2}}P
\dots
\frac{1}{m_n!}\frac{d^{m_n}\hat{Q}}{ds^{m_n}}P\hat{Q},
\label{eq:fdfinal}
\end{equation}
where we have the constraints
\[
m_1+m_2+\dots m_n=n,
\]
\[
m_1\geq 1,
\]
\[
m_2, m_3, \dots m_n \geq 0,
\]
and
\[
m_k \leq n-k+1.
\]
The last restriction follows from the fact that there are only
$n-k+1$ $\hat{Q}$-boxes to the right of $k^{\mathrm{th}}$ $\hat{Q}$-box.
Thus, it can at most be differentiated $n-k+1$ times.
We have inserted the model-space projection operator in the above
expression, in order to emphasize that folded diagrams have as intermediate
states between successive $\hat{Q}$-boxes
only model-space states. Therefore, the sum in Eq.\ (\ref{eq:fdfinal})
includes a sum over all model-space states with the same quantum
numbers such as isospin and total angular momentum. It is understood
that the $\hat{Q}$-box and its derivatives are evaluated at the
same starting energy, which should correspond to the unperturbed
energy of the model-space state.
It is then straightforward to recast Eq.\ (\ref{eq:fdfinal})
into the form of Eq.\ (\ref{eq:fd}).
Note that although $\hat{Q}$ and its derivatives contain disconnected
diagrams, such diagrams cancel exactly in each order \cite{ko90}, thus
yielding a fully connected expansion in Eq.\ (\ref{eq:fd}).
However, in order to achieve this, disconnected diagrams have
to be included in the definition of the $\hat{Q}$-box. An example
is given by diagram (c) Fig.\ \ref{fig:diagsexam}.
Such a diagram will generate a contribution to
the first fold $\frac{d\hat{Q}}{ds}\hat{Q}=-\hat{Q}\int\hat{Q} $
which cancels exactly diagram (c) when all time-ordered
contributions to this diagram are accounted for, see Ref.\ \cite{ko90}
for more details.
It is moreover important to note in connection with the above
expansion, that a term like
$F_1= \hat{Q}_1 \hat{Q}$ actually means $P\hat{Q}_1 P\hat{Q}P$ since
the $\hat{Q}$-box is defined in the model space only. Here we have defined
$\hat{Q}_{m}=\frac{1}{m!}\frac{d^{m}\hat{Q}}
{ds^{m}}$.
Due to this structure, only so-called folded diagrams
contain $P$-space intermediate states.
The folded diagram expansion discussed above yields however
a non-hermitian effective interaction. This may happen even at the
level of the $G$-matrix, or any of the effective interactions
we will derive in this work, from Parquet theory to
the order-by-order perturbation expansion.
We will therefore end this section with a way to cure
this non-hermiticity.
A hermitian effective interaction has recently been derived by
Ellis, Kuo, Suzuki and co-workers \cite{so95,so84,kehlsok93} through the
following steps\footnote{The reader who wishes more details can consult
Refs.\ \cite{so95,kehlsok93}.}.
To obtain a hermitian effective interaction,
let us define
a model-space eigenstate
$\left | b_{\lambda}\right\rangle$ with eigenvalue $\lambda$ as
\begin{equation}
\left | b_{\lambda}\right\rangle=\sum_{\alpha =1}^{D}
b_{\alpha}^{(\lambda )}\left | \psi_{\alpha}\right\rangle
\end{equation}
and the biorthogonal wave function
\begin{equation}
\left | \overline{b}_{\lambda}\right\rangle=\sum_{\alpha=1}^{D}
\overline{b}_{\alpha}^{(\lambda )}
\left | \overline{\psi}_{\alpha}\right\rangle,
\end{equation}
such that
\begin{equation}
{\left\langle \overline{b}_{\lambda} | b_{\mu} \right\rangle}=
\delta_{\lambda\mu}.
\end{equation}
The model-space eigenvalue problem can be written
in terms of the above non-hermitian effective interaction unperturbed
wave functions
\begin{equation}
{\displaystyle
\sum_{\gamma =1}^{D}b_l^{(\lambda )}\left\langle \psi_{\sigma}\right |
H_0+V+VQ\chi\left | \psi_{\gamma}\right\rangle} =
E_{\lambda}b_{\sigma}^{(\lambda )}.
\end{equation}
The exact wave function expressed in terms of the correlation
operator is
\begin{equation}
\left | \Psi_{\lambda}\right\rangle=
({\bf 1}+\chi)\left | \psi_{\lambda}\right\rangle.
\end{equation}
The part $\chi\left | \psi_{\lambda}\right\rangle$ can be expressed in terms
of the time-development operator or
using the time-independent formalism as
\begin{equation}
\chi\left | \psi_{\lambda}\right\rangle=
\frac{Q}{E_{\lambda}-QHQ}QVP\left | \psi_{\lambda}\right\rangle,
\label{eq:newchi}
\end{equation}
where $Q$ is the exclusion operator. Note that this equation is given
in terms of the Brillouin-Wigner perturbation expansion, since
we have the exact energy $E_{\lambda}$ in the denominator.
Using the normalization condition for the true wave function
we obtain
\begin{equation}
\left\langle \Psi_{\gamma} |\Psi_{\lambda}\right\rangle
N_{\lambda}\delta_{\lambda\gamma}=
\left\langle \psi_{\gamma}\right |({\bf 1} +
\chi^{\dagger}\chi\left | \psi_{\lambda}\right\rangle,
\label{eq:newnorm}
\end{equation}
where we have used the fact that
$\left\langle \psi_{\gamma}\right |
\chi\left | \psi_{\lambda}\right\rangle=0$. Recalling that the
time-development operator is hermitian we have that $\chi^{\dagger}\chi$ is
also hermitian. We can then define an orthogonal basis $d$ whose eigenvalue
relation is
\begin{equation}
\sum_{\alpha}\left\langle \psi_{\beta}\right | \chi^{\dagger}
\chi\left | \psi_{\alpha}
\right\rangle d_{\alpha}^{\lambda}=\mu^{2}_{\lambda}
d_{\beta}^{\lambda},
\label{eq:newbasis1}
\end{equation}
with eigenvalues greater than $0$.
Using the definition in Eq.\ (\ref{eq:newchi}), we note
that the diagonal element of
\begin{equation}
\left\langle \psi_{\lambda}\right | \chi^{\dagger}
\chi\left | \psi_{\lambda}\right\rangle=
\left\langle \psi_{\lambda}\right | PVQ\frac{1}{(E_{\lambda}-QHQ)^2}QVP
\left | \psi_{\lambda}\right\rangle,
\label{eq:chichi}
\end{equation}
which is nothing but the derivative of the $\hat{Q}$-box, with an additional
minus sign. Thus, noting that if $\gamma\neq\lambda$
\begin{equation}
\left\langle \Psi_{\gamma} |\Psi_{\lambda}\right\rangle=0=
\left\langle \psi_{\gamma} |\psi_{\lambda}\right\rangle
+\left\langle \psi_{\gamma}\right |
\chi^{\dagger}\chi\left | \psi_{\lambda}\right\rangle,
\end{equation}
we can write $\chi^{\dagger}\chi$
in operator form as
\begin{equation}
\chi^{\dagger}\chi =-\sum_{\alpha}\left |
\overline{\psi}_{\alpha}\right\rangle
\left\langle \psi_{\alpha}\right |
\hat{Q}_1(E_{\alpha})\left | \psi_{\alpha}\right\rangle
\left\langle \overline{\psi}_{\alpha} \right |
-\sum_{\alpha\neq\beta}\left | \overline{\psi}_{\alpha}\right\rangle
\left\langle \psi_{\alpha} |\psi_{\beta}\right\rangle
\left | \overline{\psi}_{\beta}\right\rangle.
\end{equation}
Using the new basis in Eq.\ (\ref{eq:newbasis1}), we see that
Eq.\ (\ref{eq:newnorm}) allows us to define another
orthogonal basis $h$
\begin{equation}
h_{\alpha}^{\lambda}=\sqrt{\mu_{\alpha}^2+1}\sum_{\beta}
d_{\beta}^{\alpha}b_{\beta}^{\lambda}\frac{1}{\sqrt{N_{\lambda}}}
=\frac{1}{\sqrt{\mu_{\alpha}^2+1}}\sum_{\beta}
d_{\beta}^{\alpha}\overline{b}_{\beta}^{\lambda}\sqrt{N_{\lambda}},
\label{eq:hbasis}
\end{equation}
where we have used the orthogonality properties of the vectors
involved. The vector $h$ was used by the authors of Ref.\ \cite{kehlsok93}
to obtain a hermitian effective interaction as
\begin{equation}
\left\langle \psi_{\alpha}\right |
V_{\mathrm{eff}}^{\mathrm{(her)}}\left | \psi_{\beta}\right\rangle=
\frac{\sqrt{\mu_{\alpha}^2+1}\left\langle \psi_{\alpha} \right |
V_{\mathrm{eff}}^{\mathrm{(nher)}}\left | \psi_{\beta}\right\rangle
+\sqrt{\mu_{\beta}^2+1}\left\langle \psi_{\alpha}\right |
V_{\mathrm{eff}}^{\dagger\mathrm{(nher)}}\left |
\psi_{\beta} \right\rangle}
{\sqrt{\mu_{\alpha}^2+1}+\sqrt{\mu_{\beta}^2+1} },
\label{eq:hermitian}
\end{equation}
where (her) and (nher) stand for hermitian and non-hermitian
respectively.
This equation is rather simple to compute, since
we can use the folded-diagram method
to obtain the non-hermitian part.
To obtain the total hermitian effective interaction, we have to
add the $H_0$ term. The above equation is manifestly hermitian.
Other discussion of the hermiticity problem can be found
in Refs.\ \cite{lm85,arponen97}.
The remaining question is how to evaluate the $\hat{Q}$-box.
Obviously, we are not in the position where we can evaluate it
exactly, i.e., to include all possible many-body terms.
Rather, we have to truncate somewhere.
Several possible approaches exist, but all have in common that
there is no clear way which tells us where to stop.
However, as argued by the authors of Ref.\ \cite{jls82}, there is a
minimal class of diagrams which need to be included
in order to fulfull necessary conditions. This class
of diagrams includes both diagrams which account for short-range
correlations such as the $G$-matrix and long-range
correlations such as those accounted for by various
core-polarization terms.
The importance of such diagrams has been extensively documented
in the literature and examples can be found
in Refs.\ \cite{hko95,eo77}. In Ref.\ \cite{hko95} we included
all core-polarization contributions to third-order
in the $G$-matrix, in addition to including other diagrams
which account for short-range correlations as well.
In the next subsection we present results from recent
large scale shell-model Monte Carlo calculations \cite{drhklz99}
for light nuclei in the $1s0d$ and $1p0f$ shells based
on an effective interaction for the two shells.
The effective interaction was derived following the methods
outlined hitherto in this section, taking into account
the non-hermiticity which arises when dealing
with more than one major shell.
The closed shell core is $^{16}$O and all diagrams through
second-order where employed in the definition of the
$\hat{Q}$-box. Folded diagrams were calculated using the abovementioned
folded-diagrams method and the interaction was made explicitely
hermitian through Eq.\ (\ref{eq:hermitian}). A discussion of results
for Sn isotopes will also be given.
\subsection{Selected applications}
\subsubsection{Neutron-rich nuclei in the $1s0d$-$1p0f$ shells}
Studies of extremely neutron-rich nuclei have revealed a number of
intriguing new phenomena. Two sets of these nuclei that have received
particular attention are those with neutron number $N$ in the vicinity
of the $1s0d$ and $0f_{7/2}$ shell closures ($N \approx 20$ and $N
\approx 28$). Experimental studies of neutron-rich Mg and Na isotopes
indicate the onset of deformation, as well as the modification of the
$N = 20$ shell gap for $^{32}$Mg and nearby nuclei \cite{r:motobayashi}.
Inspired by the rich set of phenomena occurring near the $N = 20$
shell closure when $N \gg Z$, attention has been directed to nuclei
near the $N = 28$ (sub)shell closure for a number of S and Ar isotopes
\cite{r:brown1,r:brown2} where similar, but less dramatic, effects
have been seen as well.
\begin{table}[hbtp]
\begin{center}
\caption{The computed and measured values of $B(E2)$ for
the nuclei in this study using $e_p=1.5$ and $e_n=0.5$.
}
\begin{tabular}{|cccc|}\hline
& $B(E2; 0^+_{gs} \rightarrow 2^+_1)_{Expt}$ & $B(E2, total)_{SMMC}$ &
$B(E2; 0^+_{gs} \rightarrow 2^+_1)$ \\\hline
$^{22}$Mg & $458 \pm 183$ & $334 \pm 27 $
& \\
$^{30}$Ne & & $303 \pm 32$
& 342 \cite{r:fukunishi},171 \cite{r:poves2} \\
$^{32}$Mg & $454 \pm 78$ \cite{r:motobayashi} & $494 \pm 44 $
& 448 \cite{r:fukunishi},205 \cite{r:poves2} \\
$^{36}$Ar & $296.56 \pm 28.3$ \cite{r:ensdf} & $174 \pm 48$
& \\
$^{40}$S & $334 \pm 36$ \cite{r:brown1} & $270 \pm 66$
& 398 \cite{r:brown2},390 \cite{r:retamosa} \\
$^{42}$S & $397 \pm 63$ \cite{r:brown1} & $194 \pm 64$
& 372 \cite{r:brown2},465 \cite{r:retamosa} \\
$^{42}$Si & & $445 \pm 62$
& 260 \cite{r:retamosa} \\
$^{44}$S & $314 \pm 88$ \cite{r:brown2} & $274 \pm 68$
& 271 \cite{r:brown2},390 \cite{r:retamosa} \\
$^{44}$Ti & $610 \pm 150$ \cite{r:raman} & $692 \pm 63$
& \\
$^{46}$Ar & $196 \pm 39$ \cite{r:brown1} & $369 \pm 77 $
& 460 \cite{r:brown1},455 \cite{r:retamosa} \\\hline
\end{tabular}
\end{center}
\label{t:tab1}
\end{table}
In parallel with the experimental efforts, there have been several
theoretical studies seeking to understand and, in some cases, predict
properties of these unstable nuclei. Both mean-field
\cite{r:werner,r:campi} and shell-model calculations
\cite{r:brown1,r:brown2,r:wbmb,r:poves1,r:fukunishi,r:retamosa,r:caurier}
have been proposed. The latter require a severe truncation to
achieve tractable model spaces, since the successful description of these
nuclei involves active nucleons in both the $1s0d$- and the $1p0f$-shells.
The natural basis for the problem is therefore the full $1s0d$-$1p0f$
space, which puts it out of reach of exact diagonalization on current
hardware\footnote{For a treatment of the c.m. problem, see Ref.\
\cite{drhklz99}.}.
Shell-Model Monte Carlo (SMMC) methods
\cite{r:david97,r:smmc_ar,r:lang} offer an alternative to direct
diagonalization when the bases become very large. Though SMMC provides
limited detailed spectroscopic information, it can predict, with good
accuracy, overall nuclear properties such as masses, total strengths,
strength distributions, and deformation, precisely those quantities
probed by the recent experiments.
There is limited experimental information about the highly unstable,
neutron-rich nuclei under consideration. In many cases only the mass,
excitation energy of the first excited state, the $B(E2)$ to that state,
and the $\beta$-decay rate is known, and not even all of this
information is available in some cases. From the
measured $B(E2)$, an estimate of the nuclear deformation parameter,
$\beta_2$, has been obtained via the usual relation
\begin{equation}
\beta_2 = 4 \pi \sqrt{B(E2; 0^+_{gs} \rightarrow 2^+_1)}/3 Z R_0^2 e
\end{equation}
with $R_0 = 1.2 A^{1/3}$ fm and $B(E2)$ given in $e^2$fm$^4$.
Much of the interest in the region stems from the unexpectedly large
values of the deduced $\beta_2$, results which suggest the onset of
deformation and have led to speculations about the vanishing of the $N
= 20$ and $N = 28$ shell gaps. The lowering in energy of the 2$^+_1$
state supports this interpretation. The most thoroughly studied case,
and the one which most convincingly demonstrates these phenomena, is
$^{32}$Mg with its extremely large $B(E2) = 454 \pm 78 \, e^2$fm$^4$ and
corresponding $\beta_2 = 0.513$ \cite{r:motobayashi}; however, a word of
caution is necessary when deciding on the basis of this
limited information that we are in the presence of well-deformed
rotors: for $^{22}$Mg, we would obtain $\beta_2 = 0.67$, even more
spectacular, and for $^{12}$C, $\beta_2 = 0.8$, well above the
superdeformed bands.
Most of the measured observables can be calculated within the SMMC
framework. It is well known that in {\it deformed} nuclei the total
$B(E2)$ strength is almost saturated by the $0^+_{gs} \rightarrow
2_1^+$ transition (typically 80\% to 90\% of the strength lies in this
transition).
Thus the total strength calculated by SMMC should only
slightly overestimate the strength of the measured transition. In
Table 1 the SMMC computed values of $B(E2, total)$ are
compared both to the experimental $B(E2; 0^+_{gs} \rightarrow 2^+_1)$
values and to the values found in various truncated shell-model
calculations. Reasonable agreement with experimental data across the
space is obtained when one chooses effective charges of $e_p=1.5$ and
$e_n=0.5$.
All of the theoretical
calculations require excitations to the $1p0f$-shell before reasonable
values can be obtained. We note a general agreement among all
calculations of the $B(E2)$ for $^{46}$Ar, although they are typically
larger than experimental data would suggest. We also note a somewhat
lower value of the $B(E2)$ in this calculation as compared to
experiment and other theoretical calculations in the case of $^{42}$S.
Table 2 gives selected occupation numbers for the nuclei
considered. We first note a difficulty in extrapolating some of the
occupations where the number of particles is nearly zero. This leads
to a systematic error bar that we estimate at $\pm 0.2$ for all
occupations shown, while the statistical error bar is quoted in the
table. The extrapolations for occupation numbers were principally
linear. Table 2 shows that $^{22}$Mg remains as an almost
pure $sd$-shell nucleus, as expected. We also see that the protons in
$^{30}$Ne, $^{32}$Mg, and $^{42}$Si are almost entirely confined to the
$sd$ shell. This latter is a pleasing result in at least two regards.
First, it shows that the interaction does not mix the two shells to an
unrealistically large extent. Second, if spurious c.m.\ contamination
were a severe problem, we would expect to see a larger proton
$0f_{7/2}$ population for these nuclei due to the $0d_{5/2}$-$0f_{7/2}$
``transition'' mediated by the center-of-mass creation operator. The fact that
there is little proton $f_{7/2}$ occupation for these nuclei confirms
that the c.m.\ contamination is under reasonable control.
See Ref.\ \cite{drhklz99} for further details.
\begin{table}[hbtp]
\begin{center}
\caption{The calculated SMMC neutron and
proton occupation numbers for the $sd$ shell, the
$0f_{7/2}$ sub-shell, and the remaining orbitals of the
$pf$ shell. The statistical errors are given for linear
extrapolations. A systematic error of $\pm 0.2$ should also
be included. The first row represents neutron results, while the
second row represents protons.}
\begin{tabular}{|ccccc|}\hline
& $N,Z$ & $1s0d$ & $0f_{7/2}$ & $1p0f_{5/2}$\\ \hline
$^{22}$Mg & 10,12 & $3.93 \pm 0.02$ & $0.1 \pm 0.02$ &
$-0.05 \pm 0.01$ \\ && $2.04 \pm 0.02$ & $0.00 \pm 0.01$ &
$-0.05 \pm 0.01$ \\
$^{30}$Ne & 20,10 & $9.95 \pm 0.03$ & $2.32 \pm 0.03$ &
$-0.26 \pm 0.02$ \\ && $2.03 \pm 0.02$ & $-0.01 \pm 0.01$ &
$-0.02 \pm 0.01$ \\
$^{32}$Mg & 20,12 & $9.84 \pm 0.03$ & $ 2.37 \pm 0.03$ &
$-0.21 \pm 0.02$ \\ && $3.99 \pm 0.03$ & $0.05 \pm 0.02$ &
$-0.05 \pm 0.01$ \\
$^{36}$Ar & 18,18 & $9.07 \pm 0.03$ & $1.08 \pm 0.02$ &
$-0.15 \pm 0.02$ \\ && $9.07 \pm 0.03$ & $1.08 \pm 0.02$ &
$-0.15 \pm 0.02$ \\
$^{40}$S & 24,16 & $11.00 \pm 0.03$ & $ 5.00 \pm 0.03 $ &
$-0.01\pm 0.02$ \\ && $7.57 \pm 0.04$ & $0.54 \pm 0.02$ &
$-0.12 \pm 0.02$ \\
$^{42}$Si & 28,14 & $11.77 \pm 0.02$ & $7.34 \pm 0.02$ &
$0.90 \pm 0.03$ \\ && $5.79 \pm 0.03$ & $0.25 \pm 0.02$ &
$-0.07 \pm 0.01$ \\
$^{42}$S & 26,16 & $11.41 \pm 0.02$ & $6.33 \pm 0.02$ &
$0.25 \pm 0.03$ \\ && $7.49 \pm 0.03$ & $0.58 \pm 0.02$ &
$-0.09 \pm 0.02$ \\
$^{44}$S & 28,16 & $11.74 \pm 0.02$ & $7.18 \pm 0.02$ &
$1.06 \pm 0.03$ \\ && $7.54 \pm 0.03$ & $0.56 \pm 0.02$ &
$-0.12 \pm 0.02$ \\
$^{44}$Ti & 22,22 & $10.42 \pm 0.03$ & $3.58 \pm 0.02$ &
$0.00 \pm 0.02$\\ & & $10.42 \pm 0.03$ & $3.58 \pm 0.02$ &
$0.00 \pm 0.02$ \\
$^{46}$Ar & 28,18 & $11.64 \pm 0.02$ & $7.13 \pm 0.02$ &
$1.23 \pm 0.03$ \\ && $8.74 \pm 0.03$ & $1.34 \pm 0.02$ &
$-0.08 \pm 0.02$ \\\hline
\end{tabular}
\end{center}
\label{t:tab3}
\end{table}
An interesting feature of Table 2 lies in the neutron
occupations of the $N = 20$ nuclei ($^{30}$Ne and $^{32}$Mg) and the $N =
28$ nuclei ($^{42}$Si, $^{44}$S, and $^{46}$Ar). The neutron
occupations of the two $N = 20$ nuclei are quite similar, confirming
the finding of Fukunishi {\it et al.} \cite{r:fukunishi} and Poves and
Retamosa \cite{r:poves1} that the $N= 20$ shell gap is modified. In
fact, the neutron $0f_{7/2}$ orbital contains approximately two
particles before the $N=20$ closure, thus behaving like an intruder
single-particle state. Furthermore, we see that 2p-2h excitations
dominate although higher excitations also play some role. We also see
that the neutrons occupying the $1p0f$-shell in $N=20$ systems are
principally confined to the $0f_{7/2}$ sub-shell.
The conclusions that follow from looking at nuclei with $N > 20$,
particularly those with $N = 28$, are that the $N = 20$ shell is nearly
completely closed at this point, and that the $N=28$ closure shell is
reasonably robust, although approximately one neutron occupies the upper
part of the $1p0f$ shell. Coupling of the protons with the low-lying
neutron excitations probably accounts for the relatively large
$B(E2)$, without the need of invoking rotational behavior.
\begin{table}[hbtp]
\begin{center}
\caption{The calculated total Gamow-Teller strength, $GT^-$,
from this study. The results of other studies, when
available, are presented for comparison.
}
\begin{tabular}{|ccc|}\hline
Nucleus & SMMC & Other \\
\hline
$^{22}$Mg & $0.578 \pm 0.06$ & \\
$^{30}$Ne & $29.41 \pm 0.25$ & \\
$^{32}$Mg & $24.00 \pm 0.34$ & \\
$^{36}$Ar & $2.13 \pm 0.61$ & \\
$^{40}$S & $22.19 \pm 0.44$ & 22.87 \cite{r:retamosa} \\
$^{42}$S & $28.13 \pm 0.42$ & 28.89 \cite{r:retamosa} \\
$^{42}$Si & $40.61 \pm 0.34$ & \\
$^{44}$S & $34.59 \pm 0.39$ & 34.93 \cite{r:retamosa} \\
$^{44}$Ti & $4.64 \pm 0.66$ & \\
$^{46}$Ar & $29.07 \pm 0.44$ & 28.84 \cite{r:retamosa} \\\hline
\end{tabular}
\end{center}
\label{t:tab4}
\end{table}
In Table 3 we show the SMMC total Gamow-Teller (GT$^-$)
strength. We compare our results to those of previous truncated
calculations, where available. In all cases, our results are slightly
smaller than, but in good accord with, other calculations. Since we
do not calculate the strength function, we do not compute
$\beta$-decay lifetimes.
\subsubsection{Heavy Sn isotopes}
Effective two-hole matrix elements are calculated based on
a $Z = 50, \quad N = 82$ asymmetric core and
with the active $P$-space for holes
based on the $2s_{1/2}$, $1d_{5/2}$, $1d_{3/2}$, $0g_{7/2}$ and $0h_{11/2}$
hole orbits. The CD-Bonn model for the NN interaction was employed
\cite{cdbonn} and all diagrams through third-order in the $G$-matrix
were included in the evaluation of the $\hat{Q}$-box. Folded diagrams
were again included through the method exposed above, see
Refs.\ \cite{hko95,ehho98} for further details.
The corresponding single-hole energies
$\varepsilon(d_{3/2}^{+}) = 0.00$~MeV,
$\varepsilon(h_{11/2}^{-}) = 0.242$~MeV,
$\varepsilon(s_{1/2}^{+}) = 0.332$~MeV,
$\varepsilon(d_{5/2}^{+}) = 1.655$~MeV and
$\varepsilon(g_{7/2}^{+}) = 2.434$~MeV
are taken from \mbox{Ref. \cite{jan}}
and the shell model calculation amounts to studying
valence neutron holes outside this core.
The shell model problem requires the solution of a real symmetric
$n \times n$ matrix eigenvalue equation
\begin{equation}
\widetilde{H}\left | \Psi_k\right\rangle =
E_k \left | \Psi_k\right\rangle .
\label{eq:shell_model}
\end{equation}
where for the present cases the dimension of
the $P$-space reaches $n \approx 2 \times 10^{7}$.
At present our basic approach in finding
solutions to Eq.\ (\ref{eq:shell_model})
is the Lanczos algorithm; an iterative method
which gives the solution of the lowest eigenstates. This method was
already applied to nuclear physics problems by Whitehead et al.
in 1977. The technique is described in detail in Ref.\ \cite{whit77},
see also Ref.\ \cite{ehho95}.
The results of the shell model calculation are presented in Table 4.
All experimental information in the present analysis is taken
from the data base of the
National Nuclear Data Center at \mbox{Brookhaven \cite{brook}}.
\begin{table}[t]
\caption{Exitation spectra for the heavy Sn isotopes. \label{tab:2}}
\vspace{0.2cm}
\begin{center}
\footnotesize
\begin{tabular}{|cccc|cccc|}
\hline
&&&&&&&\\[-5pt]
\multicolumn{4}{|c|}{$^{130}$Sn}&\multicolumn{4}{|c|}{$^{128}$Sn}\\
$J^{\pi}$&Exp.&$J^{\pi}$&Theory&$J^{\pi}$&Exp.&$J^{\pi}$&Theory\\
\hline
&&&&&&&\\[-3pt]
$(2^{+})$ & $1.22$ & $2^{+}$ & $1.46$ &$(2^{+})$ & $1.17$ & $2^{+}$ & $1.28$\\
$(4^{+})$ & $2.00$ & $4^{+}$ & $2.39$ &$(4^{+})$ & $2.00$ & $4^{+}$ & $2.18$\\
$(6^{+})$ & $2.26$ & $6^{+}$ & $2.64$ &$(6^{+})$ & $2.38$ & $6^{+}$ & $2.53$\\[3pt]
\hline \hline
&&&&&&&\\[-5pt]
\multicolumn{4}{|c|}{$^{126}$Sn}&\multicolumn{4}{|c|}{$^{124}$Sn}\\
$J^{\pi}$&Exp.&$J^{\pi}$&Theory&$J^{\pi}$&Exp.&$J^{\pi}$&Theory\\
\hline
&&&&&&&\\[-3pt]
$2^{+}$ & $1.14$ & $2^{+}$ & $1.21$ &$2^{+}$ & $1.13$ & $2^{+}$ & $1.17$\\
$4^{+}$ & $2.05$ & $4^{+}$ & $2.21$ &$4^{+}$ & $2.10$ & $4^{+}$ & $2.26$\\
$ $ & & $6^{+}$ & $2.61$ & & & $6^{+}$ & $2.70$\\[3pt]
\hline \hline
&&&&&&&\\[-5pt]
\multicolumn{4}{|c|}{$^{122}$Sn}&\multicolumn{4}{|c|}{$^{120}$Sn}\\
$J^{\pi}$&Exp.&$J^{\pi}$&Theory&$J^{\pi}$&Exp.&$J^{\pi}$&Theory\\
\hline
&&&&&&&\\[-3pt]
$2^{+}$ & $1.14$ & $2^{+}$ & $1.15$ & $2^{+}$ & $1.17$ & $2^{+}$ & $1.14$\\
$4^{+}$ & $2.14$ & $4^{+}$ & $2.30$ & $4^{+}$ & $2.19$ & $4^{+}$ & $2.30$\\
$6^{+}$ & $2.56$ & $6^{+}$ & $2.78$ & & & $6^{+}$ & $2.86$\\[3pt]
\hline \hline
&&&&&&&\\[-5pt]
\multicolumn{4}{|c|}{$^{118}$Sn}&\multicolumn{4}{|c|}{$^{116}$Sn}\\
$J^{\pi}$&Exp.&$J^{\pi}$&Theory&$J^{\pi}$&Exp.&$J^{\pi}$&Theory\\
\hline
&&&&&&&\\[-3pt]
$2^{+}$ & $1.22$ & $2^{+}$ & $1.15$ & $2^{+}$ & $1.30$ & $2^{+}$ & $1.17$\\[3pt]\hline
\end{tabular}
\end{center}
\end{table}
The isotopes above $^{116}$Sn (heavy Sn) are treated based
on the asymmetric $Z = 50, N = 82$ core.
This simplifies the shell model calculation,
but in addition it is of interest to see how successful a hole-hole effective
interaction calculated with respect to $^{132}$Sn is.
Only some selected states are displayed. First of all, the well-known
near constant $0^{+} - 2^{+}$ spacing is well reproduced.
all the way down to
$^{116}$Sn.
Also the additional calculated states are in very good agreement
with experiment. However more detailed analysis of
the results close to $^{116}$Sn
indicates that our effective two-particle interaction
has difficulties in reproducing
the shell closure which is believed to occur in this region.
The increase of the
the $0^{+} - 2^{+}$ splitting is not as sharp as found
experimentally, even if the
phenomenon is rather weak in the case of Sn.
We have observed a similar feature around $^{48}$Ca \cite{hko95}
which is generally agreed
to be a good closed shell nucleus. There the deviation
between theory and experiment
is severe. Preliminary analysis indicates that our effective interaction
may be slightly too actractive when the two particles
occupy different single-particle orbits.
This may be related to the radial wave functions which in our calculation are
chosen to be harmonic oscillator functions.
\subsection{Inclusion of hole-hole contributions and
single-particle propagators}
With the $G$-matrix defined according
to the double-partitioned scheme we can easily solve Eq.\ (\ref{eq:first12})
through matrix inversion. The number of hole-hole and particle-particle
configurations is then rather small, typically smaller than
$\sim 100$, and a matrix inversion
is then rather trivial.
Before we discuss the solution of Eq.\ (\ref{eq:first12}), it is always
instructive to consider the contributions to second order in
perturbation theory, i.e., diagrams (a) and (b) of Fig.\
\ref{fig:gamma12}. The external legs can be
particle states or hole states. Diagram (a) reads
\begin{equation}
(a)=\frac{1}{2}\sum_{pq}V^{[12]}_{12pq J}
\frac{1}{s-\varepsilon_p-
\varepsilon_q} V^{[12]}_{pq34 J},
\label{eq:secondg}
\end{equation}
and
\begin{equation}
(b)=\frac{1}{2}\sum_{\alpha\beta}V^{[12]}_{12\alpha\beta J}
\frac{1}{-s+\varepsilon_{\alpha}+
\varepsilon_{\beta}} V^{[12]}_{\alpha\beta 34 J}.
\end{equation}
We note here the minus sign in the energy denominator,
since in the latter expression we are using
the hole-hole term of the propagator of
Eq.\ (\ref{eq:paulioperator12}).
If we use a double-partitioned $G$-matrix for say $^{16}$O and
are interested in an effective valence space interaction
for the $1s0d$-shell\footnote{This means that the labels $1234$ will
refer to particle states in the $1s0d$-shell.}, then typically
the single-particle orbits of the intermediate states will be represented
by states in the $1p0f$ major shell. Hole states are then defined
by single-particle states in the $0s$ and $0p$ shells.
Clearly, the number of two-body intermediate states is rather limited.
To third order we have diagrams like (c) and (d) of Fig.\
\ref{fig:gamma12}. Diagram (c) is just the third-order equivalent of
Eq.\ (\ref{eq:secondg}) and reads
\begin{equation}
(c)=\frac{1}{4}\sum_{pqrw}V^{[12]}_{12pq J}
\frac{1}{s-\varepsilon_p-
\varepsilon_q}
V^{[12]}_{pqrw J}
\frac{1}{s-\varepsilon_r-
\varepsilon_w}
V^{[12]}_{rw34 J},
\label{eq:thirdg}
\end{equation}
while diagram (d) contains both a two-particle and a two-hole
intermediate state and reads
\begin{equation}
(d)=\frac{1}{4}\sum_{\alpha\beta pq}V^{[12]}_{12pq J}
\frac{1}{s-\varepsilon_p-
\varepsilon_q+\varepsilon_{\alpha}+
\varepsilon_{\beta}}
V^{[12]}_{pq\alpha\beta J}
\frac{1}{s-\varepsilon_p-
\varepsilon_q}
V^{[12]}_{\alpha\beta 34 J}.
\label{eq:thirdg2h}
\end{equation}
Thus, solving Eq.\ (\ref{eq:first12}) will then
yield contributions to the effective interaction
such as the above expressions.
Here we have also tacitly assumed that the energy denominators
do not diverge, i.e., we have chosen an energy $s$
so that we avoid the poles. This has always been the standard
approach in calculations of shell-model
effective interactions. To give an example, consider
now diagram (b) and suppose that we are using harmonic
oscillator wave functions. Let us also assume that the two
hole states are from the $0p$-shell and that the valence
particles are in the $1s0d$-shell. If we rescale the
energies of the valence space to zero, then the two-hole
state would yield $-28$ MeV with an oscillator parameter
$b=1.72$ fm. If $s=-28$, the denominator diverges.
In this case it is rather easy to obtain the imaginary
part, and even if we were to chose $s$ different
from $-28$ MeV, the imaginary part will influence the real
part of the effective interaction through dispersion
relations, see e.g., Refs.\ \cite{angels88,rpd89,ms92}.
It is therefore at best just a first approximation
to neglect the imaginary term.
Moreover, if we solve Dyson's equation for the
self-energy, the single-particle energies may contain
an imaginary part.
Technically it is however not difficult to deal
with imaginary contributions, one needs to
invert a complex matrix rather than a real one.
However, care must be exercised in localizing poles,
see e.g., Ref.\ \cite{landau97} for a
computational approach to this problem. In our actual
calculations we will also follow Ref.\
\cite{landau97}. These technicalities
will however be described elsewhere \cite{mhj99}.
Using the double-partitioned $G$-matrix, we can then rewrite
Eq.\ (\ref{eq:first12}) as
\begin{equation}
\Gamma^{[12]}_{1234J}(s) =
G^{[12]}_{1234J}+\frac{1}{2}
\sum_{56}
G^{[12]}_{1256J}\hat{{\cal G}}^{[12]}
\Gamma^{[12]}_{5634J}(s),
\label{eq:newfirst12}
\end{equation}
where $G^{[12]}$ is just the double-partitioned $G$-matrix
discussed above. It is also energy dependent,
in contrast to $V$.
In case we were to employ this equation for effective
interactions in the $1s0d$-shell, the intermediate two-particle
states would then come from just e.g., the $1p0f$-shell.
This equation, which now is solved within a much smaller space
than the original one spanned by the total $Q_{pp}$, allows
clearly for computationally amenable solutions. It corresponds
to the so-called {\em model-space approach} to the solution
of the Feynman-Galitskii equations as advocated by
Kuo and co-workers, see e.g., Ref.\ \cite{kt94}
for more details. Thus, a possible approach would consist
of the following steps
\begin{enumerate}
\item Solve the $G$-matrix equation from Eq.\ (\ref{eq:gmod})
using the double-partitioning scheme.
\item The next step is then to solve Eq.\ (\ref{eq:first12})
and Dyson's equation for the self-energy.
\item This scheme is iterated till self-consistency is achieved,
see the discussion below.
\end{enumerate}
We will however not employ this {\em model-space} scheme in our
actual calculations. There are several reasons for not doing so.
Let us first assume that we omit the $[13]$ and $[14]$ channels in our
iterative scheme for Eq.\ (\ref{eq:newfirst12}).
The next iteration of Eq.\ (\ref{eq:newfirst12})
would then look like
\begin{equation}
\Gamma^{[12]}_{(1)} =
\Gamma_{(0)}^{[12]}+
\Gamma_{(0)}^{[12]}
\hat{{\cal G}}^{[12]}\Gamma^{[12]}_{(1)},
\label{eq:second12}
\end{equation}
where the vertex function $\Gamma_{(0)}^{[12]}$ is the solution
of Eq.\ (\ref{eq:newfirst12}). However, we cannot define
the ``bare'' vertex $\Gamma_{(0)}^{[12]}$ to be the
solution of Eq.\ (\ref{eq:newfirst12}) simply because then we
would be double-counting contributions.
Thus, $\Gamma_{(0)}^{[12]}$ has to equal the $G$-matrix.
The only change in Eq.\ (\ref{eq:second12})
arises from the solution of Dyson's equation
and thereby new single-particle energies.
Let us then for the sake of simplicity assume that the
single-particle energies are just the Hartree-Fock
solutions. The problem we are aiming at arises at the
Hartree-Fock level.
In order to obtain Hartree-Fock solutions which
are independent of the chosen harmonic oscillator
parameter $b$, we typically need to include single-particle
orbits from quite many major shells.
Typical constraints we have found when we do so-called
Brueckner-Hartree-Fock (BHF) calculations for finite nuclei is that
we need at least $2n+l \leq 20$ in order to obtain
a result which is independent of the chosen
$b$\footnote{Throughout this work our unperturbed single-particle
basis par excellence will always be that of the harmonic
oscillator.} value. The way we solve the
BHF equations is to expand the new single-particle
wave functions $\psi_{\lambda}$, with $\lambda$
representing the quantum numbers $nlj$,
in terms of harmonic oscillator wave functions,
i.e.,
\begin{equation}
\left | \psi_{\lambda}\right\rangle=
\sum_{\alpha =1}^{2n+l\leq 20}
C_{\alpha}^{(\lambda )}\left | \phi_{\alpha}\right\rangle
\label{eq:selfconstbasis}
\end{equation}
where $\phi_{\alpha}$ are the harmonic oscillator wave functions
with quantum numbers $\alpha=nlj$ and $C$ are the coefficients
to be varied in the Hartree-Fock calculations.
The single-particle energies at the
Hartree-Fock level are just
\begin{equation}
\varepsilon_{\alpha}=t_{\alpha}+
\sum_h \bra{\alpha h} G(\varepsilon_{\alpha}+
\varepsilon_h)
\ket{\alpha h},
\end{equation}
where the single-particle states are just those
of the harmonic oscillator. The $G$-matrix used in the
first iteration in the BHF calculation is the one given
by the solution of Eq.\ (\ref{eq:gmod}).
The coefficients
$C_{\alpha}$ can then be obtained by diagonalizing
a matrix of dimension $N\times N$, where $N$ is the number
of single-particle orbits with the same $lj$ values.
As an example, suppose that we are considering the
$s_{1/2}$ state with $l=0$ and $j=1/2$. With the above
requirement $2n+l\leq 20$, we may have
that $N_{\mathrm{max}}=10$, the dimensionality being equal to
the quantum number $n$.
The way to proceed in a BHF calculation is to calculate
the reference $G$-matrix $G_F$ in Eq.\ (\ref{eq:freeg})
once and for all\footnote{This matrix is typically
set up in the relative and c.m. system and calculated
only once and for all
for $b=1$ fm. Other $b$-values involve simply
a multiplication with a constant. The matrix $G_F$ can therefore
be used for other mass areas as well.} .
Thereafter, the change in single-particle wave functions
is introduced in the calculations of Eq.\ (\ref{eq:gmod}).
We see then that if we choose to do
the Hartree-Fock self-consistency
employing the double-partitioned $G$-matrix and summing
the pphh diagrams as in Eq.\ (\ref{eq:second12}),
our single-particle basis will just be
defined by the $0s$, $0p$, $1s0d$ and $1p0f$ shells in the case
of an effective interaction in the $1s0d$-shell.
{\em This is simply
not enough in order to obtain a stable Brueckner-Hartree-Fock
result.}
The reader could infer that why do we not perform first
a BHF calculation for the $G$-matrix in Eq.\ (\ref{eq:gmod}),
and then solve Eq.\ (\ref{eq:first12}) and stop there.
This would be in line with the abovementioned
{\em model-space approach} of Kuo and co-workers \cite{kt94}.
However, if one calculates the self-energy by only
including particle-particle intermediate states
in the vertex function $\Gamma$, which is the case
if we do a standard BHF calculation, one may seriously violate
various sum rules, as demonstrated in Refs.\
\cite{ms92,mahaux85}.
Thus, to respect sum rules such as the conservation
of number of particles constrains severely the way we
solve Eqs.\ (\ref{eq:first12}) and (\ref{eq:dyson12}).
The vertex function needs both particle-particle and hole-hole
intermediate states in order to eventually
satisfy e.g., the conservation
of the number of particles.
Moreover, our interest lies in solving
the Parquet equations. This
entails simply that we perform the above
self-consistency.
It is therefore not only a matter of many-body aestethics for
embarking on the solution of the Parquet equations.
We are also able to satisfy various sum rules.
In section \ref{sec:sec4} we will also show that
at every level of approximation, the solution
of the equations in the $[13]$ and $[14]$ channels will
also result in an antisymmetric vertex function in these
channels.
In section \ref{sec:sec5} we will come back to the
technical solution of Eq.\ (\ref{eq:first12}) and its
iterations.
\section{Screening corrections and vertex renormalization, the equations
for the $[13]$ and $[14]$ channels}
\label{sec:sec4}
We start as in the previous section with the definition of the interaction
vertices in the $[13]$ and $[14]$ channels and the corresponding
integral equations. Thereafter, we discuss various approximations
to these equations such as the summation of TDA and RPA diagrams.
Eventually, the aim is to merge the discussion in this section and
the preceeding one into equations for a self-consistent scheme which combines
all three channels, namely the so-called set of Parquet equations to
be discussed in section \ref{sec:sec5}.
The equations for the renormalized vertex in the $[13]$ and $[14]$
channels have the same form as Eq.\ (\ref{eq:schematic12}), namely
\begin{equation}
\Gamma^{[13]}=V^{[13]}+V^{[13]}(gg)\Gamma^{[13]},
\end{equation}
and
\begin{equation}
\Gamma^{[14]}=V^{[14]}+V^{[14]}(gg)\Gamma^{[14]}.
\end{equation}
The matrix elements which enter are however
defined differently and the irreducible
diagrams of $V^{[13]}$ and $V^{[14]}$ can obviously not be the same.
With irreducible in the $[13]$ channel we will mean a diagram, which by
cutting an internal particle-hole pair, cannot be separated into a piece
containing the external legs $1,3$ and another piece containing
$2,4$ as external legs. The definition for the irreducible vertex in the
$[14]$ channel is similar and we illustrate these differences in Fig.\
\ref{fig:1314channel}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,20)
\put(35,0){\epsfxsize=7cm \epsfbox{diagramsph.eps}}
\end{picture}
\caption{Examples of irreducible and reducible diagrams in the
$[13]$ and $[14]$ channels. See text for further details.}
\label{fig:1314channel}
\end{figure}
Diagram (a) is just the lowest-order
interaction in the $[13]$ channel and is therefore irreducible.
Diagram (b) is an irreducible diagram in the $[13]$ channel,
whereas it is reducible in the $[14]$ channel. Diagram (c) is in turn
irreducible in the $[14]$ channel and reducible in the $[13]$ channel.
Diagram (d) is an example of a diagram which is irreducible in both
channels. This diagram stems from the $[12]$ channel.
The energy variables in these channels are, following
Fig.\ \ref{fig:channelsdef} and Eqs.\ (\ref{eq:13channel}) and
(\ref{eq:14channel}),
\begin{equation}
t=\varepsilon_3-\varepsilon_1=\varepsilon_2-\varepsilon_4,
\end{equation}
for the $[13]$ channel and
\begin{equation}
u=\varepsilon_1-\varepsilon_4=\varepsilon_3-\varepsilon_2,
\end{equation}
for the $[14]$ channel.
Defining the unperturbed particle-hole propagators
in the energy representation as \cite{kt94}
\begin{equation}
\hat{{\cal G}}^{[13]}=
\frac{Q^{[13]}_{\mathrm{ph}}}{t-\varepsilon_p+\varepsilon_h+\imath \eta}
-\frac{Q^{[13]}_{\mathrm{hp}}}{t+\varepsilon_p-\varepsilon_h-\imath \eta},
\label{eq:paulioperator13}
\end{equation}
and
\begin{equation}
\hat{{\cal G}}^{[14]}=
\frac{Q^{[14]}_{\mathrm{ph}}}{u-\varepsilon_p+\varepsilon_h+\imath \eta}
-\frac{Q^{[14]}_{\mathrm{hp}}}{u+\varepsilon_p-\varepsilon_h-\imath \eta}
\label{eq:paulioperator14}
\end{equation}
we arrive at the following equations
for the interaction vertex in these two channels
\begin{equation}
\Gamma^{[13]}_{1234J}(t) =
V^{[13]}_{1234J}+
\sum_{ph}
V^{[13]}_{12phJ}\hat{{\cal G}}^{[13]}
\Gamma^{[13]}_{ph34J}(t),
\label{eq:first13}
\end{equation}
and
\begin{equation}
\Gamma^{[14]}_{1234J}(u) =
V^{[14]}_{1234J}-
\sum_{ph}
V^{[14]}_{12phJ}\hat{{\cal G}}^{[14]}
\Gamma^{[14]}_{ph34J}(u).
\label{eq:first14}
\end{equation}
These equations, together with Eq.\ (\ref{eq:first12}),
can then form the basis for the first iteration in a self-consistent
scheme for renormalization corrections of the Parquet type.
The origin of the minus sign in Eq.\ (\ref{eq:first14}) follows from
the diagram rules \cite{kstop81} and will be examplified below.
A graphical view of these equations is given in Fig.\
\ref{fig:figs1314}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,70)
\put(35,0){\epsfxsize=7cm \epsfbox{ch1314.eps}}
\end{picture}
\caption{(a) shows the structure of the integral equation for
the interaction vertex in the $[13]$ channel. (b) represents
the integral channel for the $[14]$ channel.
The coupling order is displayed as well.}
\label{fig:figs1314}
\end{figure}
The reader should also keep in mind the two contributions to the particle
propagators of Eqs.\ (\ref{eq:paulioperator13}) and (\ref{eq:paulioperator14}).
See Ref.\ \cite{kt94} for a physical interpretation.
In this section we will omit a discussion of the self-energy corrections
which arise from these channels. This
will be relegated to the next section. In the nuclear
case, due to the strongly repulsive short-range character
of the interaction, we will have
to replace in actual calculations
the bare interaction in the irreducible vertices $[13]$ and $[14]$
with the $G$-matrix discussed in the previous section.
Doing this entails already a first step towards the Parquet set of equations,
in the sense that we are including short-range correlations from the
$[12]$ channel.
Here we will however limit the discussion to expressions
in terms of the interaction $V$.
The aim is to try
to recover from Eqs.\ (\ref{eq:first13})
and (\ref{eq:first14}) the familiar TDA and RPA equations and the
so-called self-consistent coupled equations of Kirson \cite{kirson74}.
The hope is that these intermediate steps can bridge the
gap between the familiar
TDA, RPA and $G$-matrix equations and the Parquet set of equations
in section \ref{sec:sec5}.
\subsection{Screened ph and 2p2h interactions}
Here we study the screening of the particle-hole
and the 2p2h interactions given in Fig.\ \ref{fig:wavef1},
indicated by $V_{ph}$ and $V_{2p2h}$, respectively.
Before we list the final expression, it is however instructive
to consider the corrections to second order in the interaction $V$ to the ph
and 2p2h vertices.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,40)
\put(35,0){\epsfxsize=7cm \epsfbox{secph.eps}}
\end{picture}
\caption{Second-order perturbation theory corrections to the ph
interaction vertex.}
\label{fig:phvertex}
\end{figure}
In Fig.\ \ref{fig:phvertex} we display the second-order corrections to
the ph diagrams of Fig.\ \ref{fig:wavef1}.
Diagram (a) is the core-polarization correction term to
the particle-hole
interaction, and corresponds to a contribution from the
$[14]$ channel, as indicated by the coupling order.
The term labeled (b) corresponds to the exchange
term of (a) and is coupled in the $[13]$-order, see also
the discussion in connection with Eqs.\ (\ref{eq:ph13})
and (\ref{eq:ph14}).
The other corrections, like (c) and (d) include
particle-particle and hole-hole intermediate states, respectively.
They are irreducible in both the $[13]$ channel and the $[14]$ channel,
and can therefore enter the irreducible vertices of these two channels in
later iterations.
They are however not generated by various iterations of Eqs.\
(\ref{eq:first13})
and (\ref{eq:first14}). In fact, if we replace $V$ by $G$ in Eqs.\
(\ref{eq:first13})
and (\ref{eq:first14}), diagram (c) is already accounted for by the
$G$-matrix. It may however be included if the double-partitioned $G$-matrix
of the previous section is used.
Let us now look at the analytical expressions
in an angular momentum coupled basis for diagrams (a) and (b)
of Fig.\ \ref{fig:phvertex}. Here we just include the first term of
the propagators of Eqs.\ (\ref{eq:paulioperator13}) and
(\ref{eq:paulioperator14}). The second terms will give rise to the
2p2h contributions discussed below. In the following discussion we will
also assume that the interaction $V$ does not depend on the energy,
although
it is rather easy to generalize to an energy dependent
interaction.
Diagram (a) reads
\begin{equation}
(a)=-\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}(-)^{2j_{\gamma}}
V^{[14]}_{p\gamma r\alpha J}
\frac{1}{u+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[14]}_{r\beta q\gamma J},
\label{eq:secordph}
\end{equation}
The factor $(-)^{2j_{\gamma}}$ stems from the opening up
and recoupling of an internal particle-hole pair \cite{kstop81}
and the phase $(-)^{j_r+j_{\gamma}-J}$ is needed in order
to rewrite the matrix elements in the coupling order of
Eq.\ (\ref{eq:13channel}).
The general structure of Eq.\ (\ref{eq:secordph}) is
just of the form $-V_{\mathrm{ph}}^{[14]}Q^{[14]}_{\mathrm{ph}}/\epsilon^{[14]}V_{\mathrm{ph}}^{[14]}$, with
$ \epsilon^{[14]}=\varepsilon_{q}+\varepsilon_{\gamma}-\varepsilon_{\beta}-
\varepsilon_{r}=u+\varepsilon_{\gamma}-\varepsilon_{r}$
and we have defined
\begin{equation}
u=\varepsilon_{q}-\varepsilon_{\beta}=\varepsilon_{p}-\varepsilon_{\alpha},
\end{equation}
for the on-shell energy case.
This is the equivalent of the energy variable of Eq.\ (\ref{eq:energy12}) in
the $[12]$ channel.
Diagram (b) is in turn given by
\begin{equation}
(b)=\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}(-)^{2j_{\gamma}}
V^{[13]}_{\gamma pr\alpha J}
\frac{1}{\epsilon^{[13]}}V^{[13]}_{\beta rq\gamma J} ,
\label{eq:secordphdirect}
\end{equation}
and we note that the contributions are clearly different.
The minus sign in Eq.\ (\ref{eq:secordph}) stems from the standard
diagram rules \cite{kstop81}.
In our use of the diagram rules below, we will omit the use
of the rule for the number of external valence hole lines.
In our case then, as can also be deduced from inspection of
Fig.\ \ref{fig:phvertex}, diagram (a) has zero closed loops and
three hole lines,
giving thereby rise to a minus sign.
Diagram (b) has an additional closed
loop and thereby yielding the plus sign.
The energy denominator is in this case
\begin{equation}
\epsilon^{[13]}=t+\varepsilon_{\gamma}-\varepsilon_{r},
\end{equation}
with
\begin{equation}
t=\varepsilon_{q}-\varepsilon_{\beta}=\varepsilon_{p}-\varepsilon_{\alpha}.
\end{equation}
We notice, using the relations discussed
in Eqs.\ (\ref{eq:ph13})
and (\ref{eq:ph14}), that diagram (a) is simply the
exchange diagram of (b). We need however to include
both diagrams in order to obtain an
antisymmetric equation for the particle-hole channels
which exhibits the same properties as the $[12]$ channel
shown in Eq.\ (\ref{eq:symproperties}). {\em This is actually crucial
in solving the Parquet equations. We wish namely that
every iteration, with a given approximation to the
vertex function $V$, preserves the antisymmetry property.}
This point cannot be emphasized enough. Let us now see what
happens to third order in the interaction.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,40)
\put(35,0){\epsfxsize=7cm \epsfbox{thirdph.eps}}
\end{picture}
\caption{Corrections beyond second order in the interaction $V$
to the ph
interaction vertex. (a) is in the $[14]$ channel and (b) is in
$[13]$ channel.}
\label{fig:phhigher}
\end{figure}
Third order corrections to the ph vertices (a) and (b)
involving only ph intermediate states are
shown in (a) and (b) of Fig.\ \ref{fig:phhigher}, respectively.
The analytical expression for the third-order contribution (a) is given by
\begin{equation}
(a)=\sum_{rs\gamma\delta}f
V^{[14]}_{p\gamma r\alpha J}
\frac{1}{u+\varepsilon_{\gamma}
-\varepsilon_{r} } V^{[14]}_{r\delta s\gamma J}
\frac{1}{u+\varepsilon_{\delta}
-\varepsilon_{s} } V^{[14]}_{s\beta q\delta J},
\label{eq:thirdpha}
\end{equation}
with $f=(-)^{j_r+j_s+j_{\gamma}+j_{\delta}-2J}
(-)^{2j_{\gamma}+2j_{\delta}}$. This
equation has the general structure
\[
V_{\mathrm{ph}}^{[14]}
\frac{Q_{\mathrm{ph}}^{[14]}}{\epsilon^{[14]}}
V_{\mathrm{ph}}^{[14]}
\frac{Q_{\mathrm{ph}}^{[14]}}{\epsilon^{[14]}}
V_{\mathrm{ph}}^{[14]}.
\]
A similar expression applies to diagram (b), whose expression
is
\begin{equation}
(b)=\sum_{rs\gamma\delta}f
V^{[13]}_{\gamma pr\alpha J}
\frac{1}{t+\varepsilon_{\gamma}
-\varepsilon_{r} } V^{[13]}_{\gamma rs\delta J}
\frac{1}{t+\varepsilon_{\delta}
-\varepsilon_{s} } V^{[13]}_{\beta s q\delta J}.
\label{eq:thirdphb}
\end{equation}
It has the general structure
\[
V_{\mathrm{ph}}^{[13]}
\frac{Q_{\mathrm{ph}}^{[13]}}{\epsilon^{[13]}}
V_{\mathrm{ph}}^{[13]}
\frac{Q_{\mathrm{ph}}^{[13]}}{\epsilon^{[13]}}
V_{\mathrm{ph}}^{[13]}.
\]
But these expressions have the same sign! Diagram (a) counts
now 4 hole lines, and (b) counts also 4 hole lines and 2 closed
loops. However, {\em there are three interaction terms $V$},
and taking
the exchange term of each of these in diagram (a) leads to
the desired results, namely $(a)=-(b)$, as it should.
Thus, to third order we keep the antisymmetry property of
$\Gamma$ in the $[13]$ and $[14]$ channels.
It is easy to see that in the $[14]$ channel we will always
have an alternating sign in front of each contribution,
since every new order in perturbation theory brings a new hole
line and no closed loop, and thus a new minus sign.
In the $[13]$ channel we have always one new hole line and
one new closed loop for every new vertex.
If we consider only the screening of the ph vertex, we can then set up
a perturbative expansion in terms of the ph vertex for the
vertex functions $\Gamma^{[13]}$ and $\Gamma^{[14]}$.
For notational economy we will skip the Pauli operators
$Q_{\mathrm{ph,hp}}^{[ij]}$ in the discussions below.
It will always be understood that the intermediate states
are two-body particle-hole states, $\left| \mathrm{ph}\right\rangle$ or
$\left| \mathrm{hp}\right\rangle$.
Consider e.g.,
$\Gamma^{[14]}$
\begin{equation}
\Gamma^{[14]}=V^{[14]}_{\mathrm{ph}}-
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{ph}}^{[14]}+
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V^{[14]}_{\mathrm{ph}}-+\dots,
\end{equation}
which can be summed up to yield
\begin{equation}
\Gamma^{[14]}=V^{[14]}_{\mathrm{ph}}-
V^{[14]}_{\mathrm{ph}}
\frac{1}
{\epsilon^{[14]}-V^{[14]}_{\mathrm{ph}}}
V^{[14]}_{\mathrm{ph}}=
V^{[14]}_{\mathrm{ph}}-
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}\Gamma^{[14]},
\label{eq:screening1}
\end{equation}
which is the standard TDA expression for the ph term.
The corresponding expression in the $[13]$ channel
results in
\begin{equation}
\Gamma^{[13]}=V^{[13]}_{\mathrm{ph}}+
V^{[13]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[13]}}\Gamma^{[13]}.
\end{equation}
The signs agree with the
expressions of Blaizot and Ripka \cite{br86}, see chapter 15 and
Eq.\ (15.50).
The summations in both channels ensures that the final vertex
is antisymmetric and the combination of
the latter two equations results in the familiar
TDA equations, see e.g., Ref.\ \cite{kt94} for a matrix equation
version.
We next look at the $2p2h$ matrix element and show the
corresponding corrections to second order in perturbation
theory in Fig.\ \ref{fig:pphhvertex}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=8cm \epsfbox{ph22.eps}}
\end{picture}
\caption{Corrections to second order in $V$ of the 2p2h
vertex.}
\label{fig:pphhvertex}
\end{figure}
If we omit diagrams (e) and (f) which contain 2p and 2h
intermediate states generated by the solutions
in $[12]$ channel, we have
for diagram (a)
\begin{equation}
(a)=-\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[14]}_{\gamma\beta qr J}
\frac{1}{-u+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[14]}_{pr\gamma\alpha J},
\label{eq:2p2ha}
\end{equation}
with the general structure
\begin{equation}
-V_{\mathrm{2p2h}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p2h}}^{[14]}.
\end{equation}
Note well the minus sign in front of $u$. The contribution from the
propagator can in this case be retraced to the second
term of the propagator of Eq.\ (\ref{eq:paulioperator14}).
Diagram (a) follows the coupling order of the $[14]$ channel.
It is also easy to see that diagram (b) is given by
\begin{equation}
(b)=-\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[14]}_{\gamma q \beta r J}
\frac{1}{-u+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[14]}_{pr\gamma\alpha J},
\label{eq:2p2hb}
\end{equation}
and has the structure
\begin{equation}
-V_{\mathrm{ph}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p2h}}^{[14]}.
\end{equation}
Similarly, if we now move to the $[13]$ channel we have
the following expressions
\begin{equation}
(c)=\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[13]}_{\beta\gamma qr J}
\frac{1}{-t+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[13]}_{qp\gamma\alpha J},
\label{eq:2p2hc}
\end{equation}
and
\begin{equation}
(d)=\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[13]}_{\gamma q r\alpha J}
\frac{1}{-t+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[13]}_{pr\beta\gamma J},
\label{eq:2p2hd}
\end{equation}
with the general structure
\begin{equation}
V_{\mathrm{2p2h}}^{[13]}
\frac{1}{\epsilon^{[13]}}
V_{\mathrm{2p2h}}^{[13]},
\end{equation}
and
\begin{equation}
V_{\mathrm{ph}}^{[13]}
\frac{1}{\epsilon^{[13]}}
V_{\mathrm{2p2h}}^{[13]},
\end{equation}
respectively. Diagram (c) is just the exchange of (a)
and includes two 2p2h vertices, while diagram (d) is the exchange
of diagram (b) and includes a ph vertex multiplied with
a 2p2h vertex. We note again that the antisymmetry is
ensured at a given order in the interaction only
if we include the corrections at the same level in both
channels.
One can then easily sum up higher-order corrections
to the 2p2h diagrams as well in both channels.
The inclusion of the backward going particle-hole
pair in the propagators of Eqs.\ (\ref{eq:paulioperator13})
and (\ref{eq:paulioperator14}) ensures thus that we will
also sum to infinite order 2p2h corrections.
This leads ultimately to the familiar RPA equations,
see e.g., Refs.\ \cite{eo77,kt94}.
A closer inspection of Eqs.\ (\ref{eq:2p2hb}) and
(\ref{eq:2p2hd}) shows that if we only include ph vertices,
we could resum these corrections to infinite
order for the 2p2h vertex by observing that the
structure of such diagrams would be of the form
(e.g., in the $[14]$ channel )
\begin{equation}
\Gamma_{\mathrm{2p2h}}^{[14]}=V^{[14]}_{\mathrm{2p2h}}-
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p2h}}^{[14]}+
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V^{[14]}_{\mathrm{ph}}
\frac{1}{\epsilon^{[14]}}
V^{[14]}_{\mathrm{2p2h}}.
+\dots,
\end{equation}
which can be summed up to yield
\begin{equation}
\Gamma^{[14]}_{\mathrm{2p2h}}=V^{[14]}_{\mathrm{2p2h}}-
V^{[14]}_{\mathrm{ph}}
\frac{1}
{\epsilon^{[14]}-V^{[14]}_{\mathrm{ph}}}
V^{[14]}_{\mathrm{2p2h}},
\label{eq:screening2}
\end{equation}
and similarly for the $[13]$ channel, but with a plus sign.
The modification discussed in Eqs.\ (\ref{eq:screening1})
and (\ref{eq:screening2})
serve to modify the propagation of a particle-hole pair
and have normally been termed for propagator
renormalizations, as can easily be seen from
Eqs.\ (\ref{eq:screening1})
and (\ref{eq:screening2}) where the propagation of
a free particle-hole pair is modified by the presence
of the interaction $V$ in the energy denominator.
Other important processes which can affect e.g.,
various polarization terms are those
represented by so-called vertex renormalizations,
a term originally introduced by Kirson and Zamick \cite{kz70}.
These authors studied the renormalizations of the
2p1h and 2h1p vertices as well, see also Kirson \cite{kirson74}
and Ellis and Osnes \cite{eo77} for further discussions.
We will therefore end the discussion in this section
by looking at such renormalizations.
\subsection{Further renormalizations}
In the previous subsection we dealt mainly with what has
conventionally been labelled for propagator
renormalizations. We will therefore extend the standard TDA and RPA
scheme by looking at further ways of renormalizing
interaction vertices.
The approach discussed here follows
Kirson \cite{kirson74}. Extensions were made later
by Ellis and Goodin \cite {eg80} and
Ellis, Mavrommatis and M\"uther \cite{emm91}.
We will limit the discussion here to the scheme of Kirson.
We start therefore with the contributions to second
order to the 2p1h vertex\footnote{The discussion here applies to the
other interaction vertices discussed in Fig.\ \ref{fig:wavef1}.}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,60)
\put(35,0){\epsfxsize=8cm \epsfbox{ph21.eps}}
\end{picture}
\caption{The corrections to second order in $V$ of the 2p1h
vertex.}
\label{fig:2p1hvertex}
\end{figure}
These contributions are shown in Fig.\ \ref{fig:2p1hvertex}.
Diagram (a) consists of a 2p1h vertex multiplied with a
ph vertex whereas (b) stems from the multiplication
of a 2p2h vertex with a 2p1h vertex. They both contain
a particle-hole pair as an intermediate state and
follow the coupling order of the $[14]$ channel.
The exchange diagram of (b) is given by
(c), while that of (a) ig diagram (d). Diagrams (e) and (f)
represent contributions from the $[12]$ channel
and are irreducible in both particle-hole channels.
Before we sketch the
general structure of the renormalization
procedure of Kirson, it is instructive to consider
again the equations to second order in perturbation theory,
as the general expressions can be deduced from inspection
of these diagrams.
Diagram (a) is
\begin{equation}
(a)=\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[14]}_{p\gamma r\alpha J}
\frac{1}{u+\varepsilon_{\gamma}-
\varepsilon_{r}} V^{[14]}_{rsq\gamma J},
\label{eq:2p1ha}
\end{equation}
and the plus sign stems
from the diagram rules \cite{kstop81},
i.e., we
have two hole lines and no closed loop.
The propagator is that arising
from the first term in Eq.\ (\ref{eq:paulioperator14}).
The general structure is
\begin{equation}
V_{\mathrm{ph}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p1h}}^{[14]}.
\label{eq:2p1hseca}
\end{equation}
Diagram (b) reads
\begin{equation}
(b)=\sum_{r\gamma}(-)^{j_r+j_{\gamma}-J}
(-)^{2j_{\gamma}}
V^{[14]}_{pr\gamma\alpha J}
\frac{1}{-u+\varepsilon_{\gamma}-
\varepsilon_r} V^{[14]}_{\gamma sqr J},
\label{eq:2p1hb}
\end{equation}
with the following structure
\begin{equation}
V_{\mathrm{2p1h}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p2h}}^{[14]}.
\label{eq:2p1hsecb}
\end{equation}
In this case the propagator stems from the second term
in Eq.\ (\ref{eq:paulioperator14}).
Similar equations arise for e.g., the 2h1p
vertices of Fig.\ \ref{fig:wavef1}.
Before we write down the self-consistent equations
of Kirson \cite{kirson74}, let us assume that we
can approximate the 2p1h vertex in the $[14]$ channel
by the first order term and diagrams (a) and (b).
This renormalized vertex, which we here label
$\tilde{V}_{2p1h}$, is then given by
\begin{equation}
\tilde{V}_{2p1h}\approx {V}_{2p1h}^{[14]}
+V_{\mathrm{ph}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p1h}}^{[14]}+
V_{\mathrm{2p1h}}^{[14]}
\frac{1}{\epsilon^{[14]}}
V_{\mathrm{2p2h}}^{[14]}.
\label{eq:2p1hsecondorder}
\end{equation}
If we now allow for the screening to infinite order
of the ph vertex given by Eq.\ (\ref{eq:screening1})
and replace the 2p2h vertex in the above equation
with Eq.\ (\ref{eq:screening2}) we obtain the following
renormalization of the 2p1h vertex
\begin{equation}
\tilde{V}_{2p1h}= {V}_{2p1h}^{[14]}
+V_{\mathrm{ph}}^{[14]}
\frac{1}
{\epsilon^{[14]}-V_{\mathrm{ph}}^{[14]}}
V_{\mathrm{2p1h}}^{[14]}+
V_{\mathrm{2p1h}}^{[14]}
\frac{1}
{\epsilon^{[14]}-V_{\mathrm{ph}}^{[14]}}
V_{\mathrm{2p2h}}^{[14]}.
\label{eq:screening3}
\end{equation}
A similar equation applies to the 2h1p vertex of
Fig.\ \ref{fig:wavef1} and
for the $[13]$ channel. Eqs.\ (\ref{eq:screening1}),
(\ref{eq:screening2}) and (\ref{eq:screening3}) form then
the starting point for the approach of Kirson \cite{kirson74}.
Examples of diagrams which can be obtained through the
iterative solution of Eqs.\ (\ref{eq:screening1}),
(\ref{eq:screening2}) and (\ref{eq:screening3})
are given in Fig.\ \ref{fig:kirsoniterate}.
\begin{figure}[hbtp]
\setlength{\unitlength}{1mm}
\begin{picture}(100,50)
\put(35,0){\epsfxsize=8cm \epsfbox{kirson.eps}}
\end{picture}
\caption{Examples of diagrams which can arise from Kirson's
self-consistent equations.}
\label{fig:kirsoniterate}
\end{figure}
The question now is however how to relate
Eqs.\ (\ref{eq:screening1}),
(\ref{eq:screening2}) and (\ref{eq:screening3}) with those
from Eqs.\ (\ref{eq:first13}) and
(\ref{eq:first14}).
This is rather trivial if we recall that the labels
$1234$ can, as was also discussed in section \ref{sec:sec2},
represent whatever single-particle states, either holes
or particles. Thus, $V_{1234}$ can represent
a 2p1h, 2h1p, 2p2h, 2p, 2h or a ph vertex.
This means that, due to the choice of propagators
in Eqs.\ (\ref{eq:paulioperator13}) and
(\ref{eq:paulioperator14}) equations like Eq.\
(\ref{eq:screening3}) are already inherent in Eqs.\
(\ref{eq:first13}) and
(\ref{eq:first14}). If we e.g., approximate Eq.\ (\ref{eq:first14})
to second order in the interaction $V$ and let the
single-particle labels $1234$ represent a 2p1h interaction
vertex, we immediately reobtain Eq.\
(\ref{eq:2p1hsecondorder}). If we let $1234$ represent
a 2p2h vertex, we find to second order diagrams
(a)-(d) of Fig.\ \ref{fig:pphhvertex}.
Till now we have however refrained from discussing the contributions
from the $[12]$ channel, examples were only shown in (c) and (d) of
Fig.\ \ref{fig:phvertex}, (e) and (f) of Figs.\ \ref{fig:pphhvertex}
and \ref{fig:2p1hvertex}.
These diagrams cannot be generated by simply iterating
the equations for the $[13]$ and $[14]$ channels, but they could enter
as contributions to the irreducible vertex
in the $[13]$ and $[14]$ channels from the first iteration
in the $[12]$-channel.
We see thus the emerging contour of an iterative scheme.
The crucial point is however how to perform the next iteration of say
Eqs.\ (\ref{eq:screening1}),
(\ref{eq:screening2}) and (\ref{eq:screening3}) or
Eqs.\ (\ref{eq:first13}),
(\ref{eq:first14}) and (\ref{eq:first12}) from the
$[12]$ channel
The question is how do we include the results
from the first iteration into the next one, i.e.,
how to modify the bare vertices $V^{[13]}$ and $V^{[14]}$
in e.g., Eqs.\ (\ref{eq:first13}) and
(\ref{eq:first14}) in order to obtain
an effective interaction for the shell model.
We have also not addressed how to deal
with the solution of Dyson's equation for the one-body
Green's function.
We mention also that Ellis and Goodin \cite{eg80}
included pp correlations, i.e., terms from the $[12]$ channel
such as diagram (e) of Fig.\ \ref{fig:2p1hvertex},
as well when they considered the screening of the 2p1h and 2h1p vertices.
Furthermore, as already mentioned in the introduction
the authors of Refs.\ \cite{emm91,hmm95}
extended the pp RPA to include the particle-hole (ph) RPA, though
screening of the 2p1h and 2h1p vertices was not included. In Ref.\
\cite{hmm95} however, a study with
self-consistent single-particle energies was also performed.
These works represent thus a first serious step towards the solution
of the Parquet equations, i.e., a many-body scheme which solves
self-consistently the equations in the $[12]$, $[13]$ and $[14]$
channels, with the addition of the self-consistent evaluation
of the self-energy. It ought also to be mentioned
that one of the really
first applications for nuclear systems was performed
in a series of papers by Dickhoff and M\"uther and co-workers
\cite{nuclearmatter} for nuclear matter. These authors
actually performed the first iteration of the three
channels.
Such a synthesis of the equations for the
three channels discussed here
will be made in the next section.
\section{Effective interactions for finite nuclei
from Parquet diagrams}
\label{sec:sec5}
The equations we discussed in the two previous sections can
be generalized
in matrix form as
\begin{equation}
\Gamma= \Gamma^{[ij]}+\Gamma^{[ij]}\hat{{\cal G}}^{[ij]}\Gamma,
\label{eq:generalchannel}
\end{equation}
where obviously $[ij]$ represents a given channel,
$\hat{{\cal G}}^{[ij]}$ is the particle-particle, or hole-hole or particle-hole
propagator.
The propagator is a product of two single-particle propagators $g$
which we do not specify any further here. They are
defined by the solution of Dyson's equation in Eq.\ (\ref{eq:dyson12}).
The irreducible vertices must appear in the solution of
the self-energy, and conversely, the self-energy must appear in all
single-particle propagators within the expressions for the three
channels $[12]$, $[13]$ and $[14]$.
For all of our practical purposes, the irreducible vertex used in all
channels is the so-called free $G$-matrix defined in Eq.\
(\ref{eq:freeg}). We will explain more why this is our preferred choice
below.
Let us now define the contribution from the $[12]$, $[13]$ and
$[14]$ channels following Ref.\ \cite{jls82}.
Eq.\ (\ref{eq:first12}) is then rewritten as
\begin{equation}
L=\Gamma^{[12]}-G_F,
\end{equation}
where obviously $L$ stands for ladder and if we neglect hole-hole
terms in Eq.\ (\ref{eq:first12}) we obtain
\begin{equation}
L=\Delta G =G-G_F,
\end{equation}
that is the ladder corrections beyond first order in our
irreducible vertex function $G_F$, as discussed in section \ref{sec:sec3}.
The ladder term can then be rewritten as
\begin{equation}
L=G_F\hat{{\cal G}}^{[12]}G_F+G_F\hat{{\cal G}}^{[12]}L.
\label{eq:ladder}
\end{equation}
In a similar way we can define the diagrams beyond
first order in the particle-hole channel as
\begin{equation}
R^{[13]}=\Gamma^{[13]}-G_F,
\end{equation}
and
\begin{equation}
R^{[14]}=\Gamma^{[14]}-G_F,
\end{equation}
where $G_F$ now is coupled in either the $[13]$ or $[14]$ way.
Rewriting $R^{[ij]}$, where $R$ refers to ring diagrams,
we obtain
\begin{equation}
R^{[ij]}=G_F\hat{{\cal G}}^{[ij]}G_F+G_F\hat{{\cal G}}^{[ij]}R^{[ij]},
\label{eq:ring}
\end{equation}
where $ij$ stands for either $[13]$ or $[14]$.
The equation for the vertex function $\Gamma$ in Eq.\
(\ref{eq:generalchannel}) becomes then
\begin{equation}
\Gamma=G_F+L+R^{[13]}+R^{[14]}.
\label{eq:gammap}
\end{equation}
{\em Note here that the vertex $\Gamma$ can be represented in the coupling
order of any of the above channels}. Our convention is that of the
$[12]$ channel.
If Eqs.\ (\ref{eq:ladder}) and (\ref{eq:ring}) define the first iteration,
it should be fairly obvious to see that the next iteration would be
\begin{equation}
L=\left(G_F+R^{[13]}+R^{[14]}\right)\hat{{\cal G}}^{[12]}\left(G_F+R^{[13]}+R^{[14]}\right)
+\left(G_F+R^{[13]}+R^{[14]}\right)\hat{{\cal G}}^{[12]}L,
\label{eq:laddernext}
\end{equation}
where the contributions $R^{[13]}+R^{[14]} $ are recoupled according to the coupling order
of the $12$-channel. These contributions are irreducible in the $[12]$ channel.
Similarly, for the rings we have
\begin{equation}
R^{[13]}=\left(G_F+L+R^{[14]}\right)\hat{{\cal G}}^{[13]}
\left(G_F+L+R^{[14]}\right) +
\left(G_F+L+R^{[14]}\right)\hat{{\cal G}}^{[13]}R^{[14]},
\label{eq:ring13next}
\end{equation}
and
\begin{equation}
R^{[14]}=\left(G_F+L+R^{[13]}\right)\hat{{\cal G}}^{[14]}
\left(G_F+L+R^{[13]}\right) +
\left(G_F+L+R^{[13]}\right)\hat{{\cal G}}^{[14]}R^{[14]}.
\label{eq:ring14next}
\end{equation}
Our scheme for calculating $\Gamma$ will be an iterative one based on
Eqs.\ (\ref{eq:gammap})-(\ref{eq:ring14next})
and the solution of Dyson's equation
for the single-particle propagator. This set of equations will then yield
the two-body Parquet diagrams.
Relating the above equations to the discussions of sections \ref{sec:sec3}
and \ref{sec:sec4}, it is rather easy to see
that the $G$-matrix, TDA, RPA and Kirson's
screening scheme are contained in Eqs.\
(\ref{eq:gammap})-(\ref{eq:ring14next}) .
\subsection{Petit Parquet}
The aim here is to present a numerically viable approach
to the Parquet equations.
Here
we will limit ourself to just sketch
the structure of the solution, more technical details
will be represented elsewhere \cite{mhj99}.
The iterative scheme starts with the solution of Eq.\ (\ref{eq:first12}).
The bare vertex which is irreducible in all three channels
is the so-called free $G$-matrix $G_F$ defined in Eq.\ (\ref{eq:freeg}).
Let us use the identity from Ref.\ \cite{bbp63} and rewrite the
vertex function $\Gamma^{[12]}$ as
\begin{equation}
\Gamma^{[12]}=G_F+G_F\left(\frac{Q^{[12]}}{s-H_0}
-\frac{1}{s-H_0}\right)\Gamma^{[12]},
\end{equation}
where
\[
G_F = V+V\frac{1}{s-H_0}G_F.
\]
We define then $Q^{[12]}=Q_{pp}+Q_{hh}$ and use
Eq.\ (\ref{eq:matrix_relation_q}) to rewrite $Q_{pp}$ and obtain
\begin{equation}
\Gamma^{[12]}_{(0)}=G_F-G_F\left(\frac{Q_{hh}}{s-H_0}+
\frac{1}{s-H_0}\tilde{P}\frac{1}{\tilde{P}(s-H_0)^{-1}
\tilde{P}}\tilde{P}\frac{1}{s-H_0}\right)\Gamma^{[12]}_{(0)}.
\label{eq:approx12channel}
\end{equation}
The subscript $(0)$ means that this is just the first iteration.
The single-particle energies are the unperturbed
harmonic oscillator energies.
Since we will always deal with real single-particle energies,
the two-body propagators of Eqs.\ (\ref{eq:paulioperator12}),
(\ref{eq:paulioperator13}) and (\ref{eq:paulioperator14})
will have the same expressions except for the fact that
the single-particle energies get renormalized after each
iteration. This means that
the denominators
in $G_F$, $\Gamma^{[12]}_{(0)}$ and subsequent iterations
and the expressions for the ring diagrams from the $[13]$ and
$[14]$ channels,
can be rewritten via the simple relation
\begin{equation}
\frac{1}{s\pm \imath\eta}=P\frac{1}{s}\mp\imath\pi\delta(s).
\label{eq:principalvalue}
\end{equation}
With this caveat, we can in turn obtain the real and imaginary parts
of all matrices involved.
The first step in our calculations is to evaluate $G_F$. It is
calculated in momentum space for a series of starting
energies (typically $\sim 30$).
The principal value integral from Eq.\ (\ref{eq:principalvalue})
is solved using Kowalski's
\cite{kowalski67}
method. This method ensures a numerical
stable treatment with large numbers of
mesh points in momentum space. From the principal value integral we
can in turn define the real and imaginary part of $G_F$, see
e.g., Ref.\ \cite{rpd89} for technical details.
The matrix $G_F$ is solved only once and transformed
to a harmonic oscillator basis in the lab from the
relative and center of mass system, see Ref.\ \cite{hko95}
for details.
The second step is to solve Eq.\ (\ref{eq:approx12channel}),
which is now a complex equation.
Since the number of two-hole states is rather limited,
typically $\le 100$
for a given $J$ even for heavy nuclei like Pb, the major
problem in the matrix inversion of Eq.\ (\ref{eq:approx12channel})
resides in the dimensionality of $\tilde{P}$ discussed in section
\ref{sec:sec3}. What conditions our truncation of $\tilde{P}$
and the number $n_3$, is dictated by the
Brueckner-Hartree-Fock (BHF) independence on the chosen oscillator parameter
$b$. This choice leads to the inclusion of more than ten major shells
in the computation of the $G$-matrix.
For a given $J$ value, the total number of two-body states
needed can then be of the order $ \sim 10^4$.
{\em Our first approximation} is therefore to truncate the available
space of single-particle states to be within $\sim 10-20$ major shells.
The choice being conditioned by the BHF criteria and the possibility
to store these large matrices in the RAM of available computing
facilities.
If we choose to use a double-partitioned model space, we need
to sum up further diagrams with particle-particle intermediate states.
These two steps lead then to the first iteration of
the ladders, i.e.,
\begin{equation}
L_{(0)}=\Gamma^{[12]}_{(0)}-G_F.
\end{equation}
It contains both hole-hole and particle-particle intermediate
states and is a complex matrix. The external single-particle
legs can be particles or holes. Only unperturbed single-particle
energies enter the definition of the two-body propagators.
The third step is to calculate the first iteration for the
rings, namely
\begin{equation}
R_{(0)}^{[13]}=\left(G_F+L_{(0)}\right)\hat{{\cal G}}^{[13]}
\left(G_F+L_{(0)}\right) +
\left(G_F+L_{(0)}\right)\hat{{\cal G}}^{[13]}R_{(0)}^{[13]},
\label{eq:ring13first}
\end{equation}
and
\begin{equation}
R_{(0)}^{[14]}=\left(G_F+L_{(0)}\right)\hat{{\cal G}}^{[14]}
\left(G_F+L_{(0)}\right) +
\left(G_F+L_{(0)}\right)\hat{{\cal G}}^{[14]}R_{(0)}^{[14]}.
\label{eq:ring14first}
\end{equation}
The equations for $L$ and $R$ are all defined within a
truncated Hilbert space. They can therefore be recast into
matrix equations of finite dimensionality.
Recall also that we need to recouple the contribution
from the $[12]$ into the relevant ones for the $[13]$ and $[14]$
channels. This is done employing Eqs.\ (\ref{eq:13channel}) and
(\ref{eq:14channel}).
With these contributions, we can now obtain the vertex function
$\Gamma$ after the first interaction
\begin{equation}
\Gamma_{(0)}=G_F+L_{(0)}+R_{(0)}^{[13]}+R_{(0)}^{[14]}.
\end{equation}
The fourth step is to compute the self-energy and thereby obtain
new single-particle energies. In so doing, care has to be exercised
in order to avoid double-counting problems. A thourough discussion
of this topic can be found in Ref.\ \cite{jls82}. More details
will also be presented in Ref.\ \cite{mhj99}.
Through dispersion relations \cite{rpd89} we can in turn obtain
the real part of the self-energy and our single-particle energies
will be approximated by
\begin{equation}
\varepsilon_{\alpha}=t_{\alpha}+\mathrm{Re}\Sigma_{\alpha}.
\end{equation}
This is {\em our second approximation}.
The new single-particle wave functions of
Eq.\ (\ref{eq:selfconstbasis}) are obtained
by diagonalizing a matrix of dimension
$n_{\alpha}\times n_{\alpha}$, $n_{\alpha}$ the quantum
number $n$ of the single-particle state $\alpha$.
The fifth step is to repeat steps 1-4 with the new single-particle
energies till a predetermined self-consistency is obtained.
But now the rings have to be included in all equations, i.e.,
we solve Eqs.\ (\ref{eq:gammap})-(\ref{eq:ring14next}).
The final vertex $\Gamma$ can then be used to define a
new effective interaction to be applied in shell model studies,
where many more diagrams are considered than in present
state of the art calculations, see e.g., Fig.\ 8 of Ref.\ \cite{jls82}
for a list of diagrams to sixth order entering the definition
of the irreducible vertex $\Gamma$.
\section{Perspectives}
\label{sec:sec6}
The reader should always
keep in mind that the many-body scheme we have focussed on
is only one of several
possible ways of calculating effective interactions. Other methods
are also discussed in this volume.
The aim of this work has however been to show how one can
practically implement
the Parquet equations in order to obtain effective interactions
for the nuclear shell model. The emphasis here has been to connect
these equations
with state of the art approaches to
effective interactions. Thus,
how to recover the standard $G$-matrix equation and
the TDA and RPA equations from the Parquet equations has been
outlined. Applications of this many-body scheme will be presented
elsewhere \cite{mhj99}, although in depth discussions and applications
of both
the $G$-matrix, folded diagrams and other perturbative
resummations can be found in e.g., Refs.\ \cite{hko95,eo77}.
There are also subtle technical details which deal with
double-counting problems in the outlined iterative Parquet scheme
that we have left out, due to space limits,
in the discussion. We take the liberty here to
refer to e.g., Refs.\ \cite{jls82,scalapino}. They will also be discussed
in Ref.\ \cite{mhj99}.
In the scheme we have sketched, there are obviously other important
many-body contributions at the two-body level
which cannot be generated by the two-body Parquet equations.
We have also omitted any discussion of three-body terms.
Such terms could be generated if
we were to solve the three-body Parquet equations, see Lande and
Smith \cite{jls82}.
An implementation of the three-body Parquet equations is in progress.
Three-body terms, as also mentioned in the introduction, are believed
to be important, it should just suffice
to mention studies of the Triton \cite{nogga97}.
With two-body forces only one is also not able
to reproduce properly the nuclear matter saturation point
\cite{apr98}.
In connection with three-body contributions one has to carefully
distinguish between three-body forces and effective three-body
interactions. To understand this point consider the following arguments
from studies of infinite nuclear matter and finite nuclei.
The last three years have seen quite some advances in
many-body studies of dense infinite matter, see e.g., Refs.\
\cite{apr98,engvik97,baldo98} for recent surveys.
These results can be summarized as follows.
Firstly, new $NN$ interactions such as
the CD-Bonn potential \cite{cdbonn}, different Nijmegen interactions
\cite{nijmegen94} and the recent Argonne $V_{18}$ interaction
\cite{argv18}
all fit
the set of scattering data of the
Nijmegen group
with a $\chi^2$ per datum close to 1. All these interactions,
when applied to calculations of the equation of state (EoS),
yield essentially similar equations of state up to
densities of $3-4$ nuclear matter saturation density
for both pure neutron matter
and $\beta$-stable matter when
the non-relativistic lowest-order Brueckner theory (LOB) is used.
Other properties like the
symmetry energy and proton fractions do also show
a similar quantitative agreement, see Ref.\ \cite{engvik97} for more
details.
Secondly, the inclusion of more complicated
many-body terms at the two-body \cite{apr98}
level does not alter this picture and even the recent
summation of three-hole line diagrams of Baldo and co-workers
\cite{baldo98} results in an EoS which is
close to LOB when a continuous
choice is used for the single-particle energies in matter
\cite{baldo98}. The latter are examples of effective
three-body contributions. These findings
are also in line with
recent works on the energy of pure neutron
drops, where three-body clusters are included \cite{ndrops97},
and large-scale shell-model calculations of Sn
isotopes including
effective three-body interactions \cite{eho99}.
Differences do however occur
when one introduces real three-body forces.
These are necessary
in order to reproduce the saturation properties of nuclear matter
\cite{apr98} and the binding energy of light nuclei \cite{vijay}.
The reader may then eventually ask why do we bother at all to solve
the two-body set of Parquet equations when most likely these new
effective interactions will not cure any of the problems
seen in shell model studies or nuclear matter. One needs at least to
include some realistic three-body force and thereby to solve
the three-body Parquet equations.
The problem however with real three-body forces is that presently
we have no serious candidates which exhibit the same level
of quality and sophistication
as the $NN$ interactions mentioned above.
The introduction of such forces lead therefore to strong model
dependencies. Moreover, to assess properly many-body terms at the two-body
level is an important, and not yet solved, problem per se.
The Parquet equations allow one also to include in a self-consistent way
several many-body terms, fulfill crossing symmetries and
satisfy certain sum rules. In addition, the Green's function formalism
can be taylored for finite temperatures.
This is clearly of interest for studies of infinite matter at finite
temperature, such as e.g., dense matter occuring in a newly born
neutron star.
The application of the Parquet equation in nuclear and neutron matter will also
allow for a consistent treatment of screening effects relevant for
the pairing problem. Such work is in progress.
\subsection*{Acknowledgements}
I have greatly benefitted from many discussions on Parquet theory with
Andy Jackson. Moreover, many interactions with David Dean, Paul Ellis,
Torgeir Engeland, Tom Kuo,
Herbert M\"uther, Artur Polls, Eivind Osnes and Andres Zuker
have hopefully matured
some of the ideas exposed, although any eventual flaw(s) is(are) obviously
to be retraced to the author.
| 2024-02-18T23:40:15.030Z | 1998-11-30T09:11:30.000Z | algebraic_stack_train_0000 | 1,829 | 22,743 |
|
proofpile-arXiv_065-9056 | \section{Introduction}
It is widely accepted that the disk instability model (DIM) provides
the correct description of dwarf nova outbursts (e.g Bath \& Pringle
1982, Smak 1984, Lin, Papaloizou \& Faulkner 1985, Cannizzo 1993a) and
probably of soft X-ray transient events (van Paradijs \& Verbunt 1984,
Mineshige \& Wheeler 1989, Cannizzo 1998a). In this model, the
variability of a steadily fed thin accretion disc is due to the
occurrence of a thermal-viscous instability in the disc. For a certain
range of mass transfer rates from the secondary, $\dot{M}_T$, the
effective temperature of a disc annulus lies within the unstable range
$5000-8000$ K, which corresponds to hydrogen recombination inside the
disc. The unstable annulus then experiences a limit cycle during which
matter is processed at rates larger or smaller than the mean rate
$\dot{M}_T$ (see Cannizzo 1993b or Osaki 1996 for reviews).
Transition fronts are essential to the DIM because they are the link
between the local instability of an annulus and the global evolution
of the disc. The time required by a heating front to cross the disc is
related (depending on wavelength) to the rise time
of an outburst. Similarly, the time required for a cooling front to
cross the disc defines the decay time of an outburst in the DIM (if
the entire disc is brought to the hot state, the decay time is
increased by a ``viscous plateau'' phase, see Cannizzo 1993a).
Observations of the rise, decay and recurrence times therefore allow a
direct test of the predictions of the DIM.
Several attempts to determine the structure and properties of
transition fronts exist in the literature. Analytical studies lead
only to qualitative results or to complex results that require
numerical simulations for calibration (Meyer 1984, Lin et al. 1985,
Vishniac \& Wheeler 1996, Vishniac 1997). On the other hand, numerical
studies have difficulties in correctly handling transition fronts
because the fronts are very narrow and their propagation involves time
and length scales which vary by several orders of magnitude in a
single outburst. As discussed by Hameury et al. (1998, hereafter
HMDLH), heating fronts are unresolved in most numerical simulations
while cooling fronts are often barely resolved, in particular close to
the disc inner edge (see e.g. Cannizzo, Chen \& Livio 1995). The poor
resolution of transition fronts not only affects localized regions of
the disc but it has also consequences for the global evolution of the
disc (Cannizzo 1993a, HMDLH).
HMDLH have recently developed an implicit time-dependent numerical
code which solves the disc equations on an adaptive grid. The code
allows the resolution of very narrow structures in the disc at a
relatively low computational cost. Here we use this code to study the
detailed structure and the propagation velocities of heating and cooling
fronts. We show that the width of any transition front is proportional
to the disc pressure scale height $H$, and not to $(HR)^{1/2}$ as
proposed by Cannizzo et al. \shortcite{ccl95}. We confirm the
existence of a self-similar regime of the hot inner disc during the
propagation of a cooling front, and we discuss the possible
reflections of cooling (resp. heating) fronts into heating
(resp. cooling) fronts.
\section{Disc instability model and transition fronts}
\subsection{Disc equations}
The time evolution of an accretion disc is driven by thermal-viscous
processes. {For a Keplerian disc t}he viscous equation, describing mass and
angular momentum conservation, can be written as
\cite{l74,p81}:
\begin{equation}
\frac{\partial \Sigma}{\partial t}=\frac{3}{R}
\frac{\partial}{\partial R} \left[ R^{1/2} \frac{\partial}{\partial R}
\left( \nu \Sigma R^{1/2} \right) \right],
\end{equation}
where $\nu$ is the kinematic viscosity,
$\Sigma$ is the surface density of the disc, and $R$ is the
distance from the central object. The viscosity is parameterized
according to $\nu = \alpha c_{\rm s} H$, where $c_{\rm s}$ is the
sound speed in the disc midplane, $H$ is the vertical scale height of the
disc and the parameter $\alpha$ ($\leq 1$) describes our ignorance of
the viscous processes. Shakura and Sunyaev (1973) first proposed this
prescription with $\alpha$ constant, but it was later shown by Smak
(1984) that $\alpha$ must be larger for the hot ionized disc than for
the cold neutral disc in order to obtain large amplitude outbursts in
the DIM.
In the following, we use two parameterizations often adopted in the
literature. First, we adopt an $\alpha_{\rm hot}-\alpha_{\rm cold}$
prescription and an arbitrary functional dependence of $\alpha$ with
the disc central temperature $T_c$ which is similar to that used in
HMDLH, i.e.
\begin{eqnarray}
\log (\alpha)=\log(\alpha_{\rm cold})& +& \left[ \log(\alpha_{\rm hot})-
\log( \alpha_{\rm cold} ) \right] \nonumber \\ & &
\times \left[1+ \left( \frac{2.5 \times 10^4 \; \rm K}{T_{\rm c}} \right)^8
\right]^{-1},
\label{eq:alpha}
\end{eqnarray}
where $\alpha_{\rm hot}$ and $\alpha_{\rm cold}$ are the two constant
values of $\alpha$ on the hot and the cold stable branches of the thermal
equilibrium curves (or S-curves), respectively.
The second parameterization we consider is
\begin{equation}
\alpha = \alpha_0 (H/R)^{1.5},
\end{equation}
where $\alpha_0$ is a constant ($\alpha_0=50$ here). Such a dependence
has been strongly advocated by several authors to explain the observed
exponential decays of soft X-ray transients \cite{ccl95,vw96}. This
interpretation may face some difficulties, however (\S 4.6). Note
that the viscosity parameter $\alpha \propto R^{3/4}$ in this case
(e.g. Mineshige \& Wheeler 1989).
The importance of allowing the disc outer edge, $R_{\rm out}$, to vary
with time in order to reproduce correct outburst cycles was emphasized
by HMDLH and before by Smak (1984) and Ichikawa and Osaki (1992).
Transition fronts are, however, localized structures and should not be
influenced by the physical conditions at the disc outer edge. This is
confirmed by a comparison between models differing only in whether or
not $R_{\rm out}$ varies with time, which shows that the structure and
the properties of the fronts are identical in the two classes of models
(except when a front reaches $R_{\rm out}$). In the following, we
assume that $R_{\rm out}$ is fixed.
The thermal equation (describing energy conservation; Faulkner, Lin \&
Papaloizou 1983, HMDLH) is given by:
\begin{equation}
{\partial T_{\rm c} \over \partial t} = { 2 (Q^ + -Q^- + J) \over C_P
\Sigma} - {\Re T_{\rm c} \over \mu C_P} {1 \over R} {\partial (R
V_{\rm r}) \over \partial R} - V_{\rm r} {\partial T_{\rm c} \over
\partial R},
\label{eq:heat}
\end{equation}
where $Q^+$ and $Q^-$ are the vertically-averaged rates of viscous
heating and radiative cooling, respectively, $\Re$ is the perfect gas constant,
$C_P$ is the specific heat at constant pressure of the gas, $\mu$ its
mean molecular weight and $V_{\rm r}$ its radial velocity (positive
outward). The second term on the r.h.s. of Eq.~(\ref{eq:heat})
represents the contribution from pressure work and will be referred
to as $Q_{\rm pdv}$. The third term ($Q_{\rm adv}$)
represents the contribution from energy advection. The term $J$ (or
equivalently $Q_j$) accounts for the radial transport of energy
(viscous or radiative). The (viscous) flux carried by turbulent eddies
with a characteristic velocity $v_{\rm e}$ and size $l_{\rm e}$ is
obtained dimensionally in the framework of the $\alpha$-prescription:
\begin{equation}
F_{\rm e} = C_P \Sigma v_{\rm e} {\partial T_{\rm c} \over \partial R}
l_{\rm e} = {3 \over 2} \nu C_P \Sigma {\partial T_{\rm c} \over
\partial R},
\label{eq:fturb}
\end{equation}
and $J$ is then given by:
\begin{equation}
J = \frac{1}{R} \frac{\partial}{\partial R} \left( R F_{\rm e}
\right).
\label{eq:j}
\end{equation}
A similar expression is obtained for the radiative transport of energy
\cite{lpf85}. Other prescriptions were proposed for $J$ but they give
results which are similar, in terms of outburst cycles, to those
obtained with Eqs.~(\ref{eq:fturb}) and~(\ref{eq:j}) (HMDLH; see also
\S 6).
Simulations show that most of the disc remains close to thermal
equilibrium during its time evolution. In that case, the thermal
equation basically reduces to $Q^+ = Q^-$. Transition fronts
correspond, however, to regions far from thermal equilibrium in which
the other terms in Eq.~(\ref{eq:heat}) are no longer negligible.
\subsection{Numerical models}
We use the numerical code developed by HMDLH with 800 grid
points. This corresponds to about 100 points in the transition fronts,
and is more than sufficient to avoid resolution-limited results.
All the models predict strictly periodic sequences of outbursts with
(eventually) various amplitudes (see e.g. Fig.~8 of HMDLH). The
sequences differ from one model to another (in number and amplitude of
the outbursts). In the following, we refer to such a sequence as the
outburst cycle of a model.
The disc outer radius is fixed to $R_{\rm out}=4 \times 10^{10}$ cm.
The disc inner radius, $R_{\rm in}=5 \times 10^{8}$ cm, is equal to
the white dwarf radius of mass $M_1=1.2~M_{\odot}$. A comparison of
models with various mass transfer rates ($\dot{M}_T$) shows that,
although the outburst cycles differ from one model to another, the
intrinsic structure and properties of transition fronts do not depend
on $\dot{M}_T$. A value $\dot{M}_T=10^{-9}~M_{\odot}~{\rm yr}^{-1} =
6.66 \times 10^{16}$ g s$^{-1}$ is used in all models presented here.
We have considered four models to investigate the influence of the
viscosity prescription on the structure and the properties of
transition fronts. In three of them, we use an $\alpha_{\rm
hot}-\alpha_{\rm cold}$ prescription: $\alpha_{\rm hot}=0.1$ and
$\alpha_{\rm cold}=0.02$ (model {\it h0.1.c0.02}), $\alpha_{\rm
hot}=0.2$ and $\alpha_{\rm cold}=0.02$ (model {\it h0.2.c0.02}) and
finally $\alpha_{\rm hot}=0.1$ and $\alpha_{\rm cold}=0.01$ (model
{\it h0.1.c0.01}). The fourth model uses $\alpha=50(H/R)^{1.5}$ (model
{\it 50hr1.5}).
Global stability considerations (e.g. Cannizzo 1993b) and numerical
simulations show that cooling fronts invariably appear in the outer
disc. On the contrary, outbursts may be triggered anywhere in the
disc, and both inside-out and outside-in heating fronts are observed
in simulations (Smak 1984, HMDLH). All outbursts are of the
inside-out type in the models presented here, except the largest
amplitude outbursts of model {\it h0.1.c0.01} and all the outburst of
model {\it 50hr1.5} which are of the outside-in type.
\subsection{Transition fronts}
We define the location of a front, $R_{\rm front}$, as the radius at
which the central temperature $T_c = 1.8 \times 10^4$~K in the
disc. This value of $T_c$ is a signature of the presence of a
transition front because it lies in the range of central temperatures
for which an annulus is thermally and viscously unstable. The speed of
a front, $V_{\rm front}$, is obtained by numerical differentiation of
$R_{\rm front}$.
In the following, the structure of transition fronts is shown each
time for a specific value of $R_{\rm front}$. Simulations show that
the structures of transition fronts are qualitatively independent of
their location in the disc. (In addition, cooling fronts evolve in a
nearly self-similar way.) The general properties of the fronts do not
depend on the viscosity prescription either.
From now on, the various terms $Q/Q^+$ in Eq.~(\ref{eq:heat}) used to
describe the thermal structure of a transition front are defined as
positive if they correspond to a local heating and negative if they
correspond to a local cooling in the disc (i.e. contrary to the
positive convention for $Q^-$ in Eq.~(\ref{eq:heat})).
\section{Heating fronts}
\subsection{Structure of inside-out heating fronts}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{inoutheat2.epsi}
\end{displaymath}
\caption{Structure of an inside-out heating front located at $R_{\rm
front} \sim 3 \times 10^9~{\rm cm}$ (model {\it h0.1.c0.02}). The
structure is dominated by a strong outflow of gas
and by viscous heating, except in a narrow
precursor region where radial diffusion and advection of energy
dominate.}
\label{fig:inoutheat}
\end{figure}
Figure~\ref{fig:inoutheat} shows the structure of an inside-out
heating front located at $R_{\rm front} \sim 3 \times 10^9~{\rm cm}$
in model {\it h0.1.c0.02}. It is characterized by a strong outflow of
gas which is coincident with a sharp spike in the $\Sigma$ profile and
a steep gradient of $T_c$ (or equivalently viscosity). The outflow and
the formation of the spike in $\Sigma$ were explained by Lin et
al. (1985); they appear because the gas which carries
excess angular momentum from the inner disc reaches colder regions
where transport of angular momentum is reduced.
The dominant term in the thermal equation inside an inside-out heating
front is, by far, the viscous heating term $Q^+$.
Figure~\ref{fig:inoutheat} shows, however, the existence of a
substructure in the front: a precursor region (narrower than the front
itself) in which the energetics of the gas is dominated by advection
and radial transport of energy. The contribution from pressure work is
small but not negligible in the precursor region. The thermal
structure of the front can be understood as follows. The hot gas which
penetrates in colder regions of the disc carries a substantial amount of
heat ($Q_{\rm adv}$). The gas is slowed down where the transport of
angular momentum is reduced and it contracts ($Q_{\rm pdv}$). The
precursor region is the region of strongest gradient of $T_c$ (hence
the large value of $Q_j$).
Note that the region where heating ($Q^+$) dominates over cooling is
broader than the spike in $\Sigma$ and the precursor region
corresponds to the rising part of the spike of $\Sigma$.
\subsection{Structure of outside-in heating fronts}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{outinheat2.epsi}
\end{displaymath}
\caption{Structure of an outside-in heating front located at $R_{\rm
front} \sim 1.53 \times 10^{9}~{\rm cm}$ (model {\it 50hr1.5}). The
structure is dominated by a strong inflow of gas and by viscous
heating, except in a very narrow precursor region where radial
diffusion and advection of heat dominate.}
\label{fig:outinheat}
\end{figure}
Figure~\ref{fig:outinheat} shows the structure of an outside-in
heating front located at $R_{\rm front} \sim 1.5 \times 10^9~{\rm cm}$
in model {\it 50hr1.5}. The structure is expected to be different from
that of an inside-out heating front since the matter crossing the
transition region no longer carries excess angular momentum from the
inner disc. Instead, this matter now corresponds to the bulk of
accretion in the disc, while excess angular momentum is being freely
carried away outward \cite{lpf85}.
The accretion of hot gas is revealed by the strong inflow and the
spike of $\Sigma$ where the gas enters colder regions. Note the very
steep gradient of $T_c$ at the interface between the hot and cold
regions of the disc and the presence of a substructure in the profiles
of $\Sigma$, $V_r$ and $\dot{M}$ (corresponding to the precursor
region).
The thermal structure of an outside-in heating front is similar to
that of an inside-out heating front, despite their opposite direction
of propagation. Viscous heating ($Q^+$) is dominant, except in the
precursor region where advection and radial diffusion of energy
dominate the energetics of the gas. Once again, the pressure work term
($Q_{\rm pdv}$) is small but not negligible in the precursor
region. The precursor region is, however, much narrower (as compared
to the total width of the front) and its structure more complex than
in an inside-out heating front.
\subsection{Heating front velocity}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{heatspeedmult.eps}
\end{displaymath}
\caption{The four panels show the speed of successive heating fronts
in our four models with various $\alpha$-prescriptions
(dots). Outside-in (model {\it 50hr1.5}) and inside-out (three other
models) heating fronts slow down during their propagation in the
disc. The speed of heating fronts significantly depends on the value
of $\alpha_{\rm hot}$.}
\label{fig:heatspeedmult}
\end{figure}
Figure~\ref{fig:heatspeedmult} shows the speed of heating fronts for
the four models with various $\alpha$-prescriptions. Each dot
represents the speed of the front, $V_{\rm front}$, at a specific
location $R_{\rm front}$ in the disc. Successive outbursts are
triggered at nearly the same ``ignition'' radius during an outburst
cycle in all the models, except in model {\it h0.1.c0.01} for which
the outburst cycle is made of one outside-in (not shown here for
clarity) and several inside-out outbursts. The simulations show that
the front velocities strongly depend on the surface density profile at
the onset of the thermal-viscous instability. Since mass piles up and
diffuses from the outer regions of the disc during an outburst cycle,
the front velocities vary from one outburst to another during the
cycle, as seen from Fig.~\ref{fig:heatspeedmult}. For comparison, the
sound speed is $\sim 15$ km s$^{-1}$ for the characteristic
temperature $T_c= 18,000$ K in the transition front.
The velocity profiles of outside-in heating fronts (model {\it
50hr1.5}) are very different from those of inside-out heating fronts
(three other models). (This is not just due to the different
$\alpha$-prescriptions.) A common characteristic of the two types of
fronts is, however, a deceleration as the front propagates in
the disc (a characteristic of cooling fronts as well; see \S 4.2).
It is not possible to compare quantitatively the speed of inside-out
and outside-in heating fronts from Fig.~\ref{fig:heatspeedmult}
because a different $\alpha$-prescription is used in the models where
the two types of fronts appear. Fortunately, we obtain both outside-in
and inside-out outbursts in our model {\it h0.1.c0.01}. The maximum
speeds of the two types of fronts in this model are similar ($\sim
3~{\rm km~s^{-1}}$) but outside-in heating fronts cross the disc
faster than inside-out heating fronts because their speed is high over
a larger range of radii. This is in agreement with the fact that
outside-in outbursts are known to produce more asymmetric lightcurves
(i.e. shorter rise times) than inside-out outbursts (e.g. Smak 1984,
HMDLH).
The speed of heating fronts slightly increases when $\alpha_{\rm cold}$
decreases. Increasing the value of $\alpha_{\rm hot}$ results in a
substantial, approximately linear, increase of $V_{\rm front}$ which is
of order of $\alpha_{\rm hot} c_{\rm s}$ as in an ignition front
(Meyer 1984). We find no significant change in the speed of heating
fronts at a given $R_{\rm front}$ when the mass of the central white
dwarf is decreased from $M_1=1.2~M_{\odot}$ to $0.6~M_{\odot}$ in model
{\it h0.1.c0.02}.
It was argued by Lin et al. (1985) that the velocity of a transition
front must be close to the speed of gas in the front. We compared the
speed of heating fronts with the maximum speed of gas inside the
fronts, $V_{\rm r,max}$ (see Figs.~\ref{fig:inoutheat}
and~\ref{fig:outinheat}). The velocity of inside-out and outside-in
heating fronts appears, indeed, close to $V_{\rm r,max}$ in the
fronts. For instance, $V_{\rm front} \sim 3/2~V_{\rm r,max}$ in models
{\it h0.1.c0.02} and {\it 50hr1.5}. Note that the speed of the cold
gas into which a heating front penetrates is negligibly small compared
to the speed of the front.
\subsection{Width of heating fronts}
The width of a transition front was defined in HMDLH as the region of
the disc over which the variation of the viscosity parameter $\alpha=
\alpha(T_c)$ is 90 \% of the total variation between
$\alpha_{\rm cold}$ and $\alpha_{\rm hot}$. This definition, although
appropriate for an $\alpha_{\rm hot}-\alpha_{\rm cold}$ prescription,
cannot be used when $\alpha$ is a smooth function of $H/R$ (in
particular model {\it 50hr1.5}).
We instead define the transition region as a zone with large thermal
imbalance where the temperature changes on short timescales. More
specifically, if the local timescale of change in temperature is
\begin{equation}
t_{\rm T} (R) \equiv \left| \frac{\partial \ln T_c}{\partial t} (R)
\right| ^{-1},
\end{equation}
we define the transition region as the region of the disc where
$t_{\rm T} (R) < 3 t_{\rm T} (R_{\rm front})$. $R_{\rm front}$ is the
location of the front, as defined previously ($T_c(R_{\rm front})
\equiv 18,000$ K). The factor 3 is arbitrary but it has been chosen to
give results close to what an eye estimate would give. We confirmed
that the above definition leads to widths which are comparable to
those found in HMDLH but also comparable to those found with a
definition of the transition region based on a deviation of the ratio
$| Q^-/Q^+|$ from unity. (This last definition should be equivalent to
the definition used by, e.g., Cannizzo 1994; Cannizzo et al. 1995.)
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{dwmult1.2.eps}
\end{displaymath}
\caption{The four panels show the fractional width ($\delta w/R_{\rm
front}$) of cooling fronts and heating fronts in our four models with
various $\alpha$-prescriptions (dots). In each panel, the upper line
of dots corresponds to the width of successive cooling fronts and the
lower line of dots corresponds to the width of successive heating
fronts. The solid line shows the fractional scale height of the disc
($H/R_{\rm front}$). Note that heating fronts in model {\it 50hr1.5}
are of the outside-in type.}
\label{fig:dwmult1.2}
\end{figure}
Figure~\ref{fig:dwmult1.2} shows the fractional width ($\delta
w/R_{\rm front}$) of heating and cooling fronts (dots) as a function
of the front location in the four models with various
$\alpha$-prescriptions. In each panel, the upper line of dots
corresponds to cooling fronts and the lower line of dots corresponds
to heating fronts. The width of cooling fronts is further discussed in
\S 4.3.
The width of a heating front is roughly proportional to the local
vertical scale height $H$ (note that $H=c_s/\Omega_K$ is defined at a
fixed value $T_c=1.8 \times 10^4$ K here). This is not unexpected
since, except for the narrow precursor region, the thermal structure
of the front is dominated by the heating term, $Q^+$. Consequently,
assuming that $V_{\rm front}$ varies much less rapidly than $R_{\rm
front}^{3/2}$ (in agreement with our results, \S 3.3), $\delta w$ is
the distance traveled by the heating front in a few thermal
timescales $\tau_{\rm th} \equiv (\alpha \Omega_{\rm K})^{-1}$
(e.g. Frank, King \& Raine 1992), i.e.
\begin{equation}
\delta w \propto V_{\rm front} \tau_{\rm th} \propto (\alpha
c_s/\alpha \Omega_K) \propto H \propto R_{\rm front}^{3/2}.
\label{eq:deltaw}
\end{equation}
These results are consistent with the conclusions of Meyer (1984,
1986) and Papaloizou \& Pringle (1985).
A comparison of the various panels in Fig.~\ref{fig:dwmult1.2} shows
the influence of $\alpha$ on the width of a heating front. Increasing
$\alpha_{\rm hot}$ or decreasing $\alpha_{\rm cold}$ results in
narrower fronts. It is not possible to interpret this result simply
from Eq.~(\ref{eq:deltaw}), however, because the speed of heating
fronts is also affected in a non-trivial way by changes in the
viscosity parameter $\alpha$ (cf \S 3.3).
In model {\it 50hr1.5}, transition fronts have profiles of $\delta
w/R_{\rm front}$ which are slightly flatter than the solid line
($H/R_{\rm front}$). According to the three other models, reduced
widths are expected for larger values of $\alpha_{\rm hot}$. The
flatter profiles of $\delta w$ are therefore consistent, at least
qualitatively, with the additional variation of $\alpha$ with radius in
this particular model ($\alpha \propto R^{3/4}$).
We also investigated the effect of varying the mass of the central
object on the width of transition fronts. We decreased $M_1$ from
1.2~M$_{\odot}$ to 0.6~M$_{\odot}$ in model {\it h0.1.c0.02}, which
results in a reduced vertical gravity and therefore an increased
vertical scale height $H$ of the disc. At a given $R_{\rm front}$, the
front widths are increased, but the ratio of heating front (or
equivalently cooling front) width to $H$ is reasonably constant. This
is additional evidence for a proportionality between $\delta w$ and
the local thermal timescale in the disc, which depends on $M_1$ via
$\Omega_K$.
\subsection{Pairs of heating fronts}
When the thermal-viscous instability is triggered at a given ignition
radius, two heating fronts appear which propagate in opposite
directions in the disc. Because the ignition radius is often close to
one of the disc edges, one of the two fronts is actually short-lived
(i.e. quickly reaches the disc edge). The other heating front is long
lived, propagates over a large range of radii and determines the
global evolution of the disc.
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{twinfronts2.epsi}
\end{displaymath}
\caption{Radial profiles of $\Sigma$ and $T_c$ in a situation of
coexistence of an inside-out and an outside-in heating front in the
disc (model {\it 50hr1.5}). Both fronts will significantly contribute
to the global evolution of the disc.}
\label{fig:twinfronts}
\end{figure}
It is possible, however, to obtain models in which a pair of heating
fronts appears somewhere in the middle of the disc.
Figure~\ref{fig:twinfronts} shows such a situation (model {\it
50hr1.5}), where an inside-out and an outside heating fronts appear
around $R \sim 1.5 \times 10^{10}$ cm. In that case, the two fronts
contribute significantly to the global evolution of the disc. We
referred to this type of intermediate outburst as an outside-in
outburst.
More generally, simulations show that the value of the ignition radius
in a specific model depends on the mass transfer accretion rate
$\dot{M}_T$ (HMDLH). This implies that a continuum of lightcurve
shapes is possible, from quasi-symmetric (firm inside-out outburst) to
strongly asymmetric (firm outside-in outburst). Consequently, the
classification in two types of fronts initially proposed by Smak
(1984) must be used with caution.
A careful look at Figure~3 of HMDLH reveals a sudden increase and then
decrease of the inner accretion rate, $\dot{M}_{\rm acc}$, which
precedes the bulk of accretion in inside-out outbursts; this spike is
very short lived, contains almost no energy, and is probably not
observable. This is a signature of the arrival of an in-propagating
heating front at the disc inner edge. This front is the short-lived
companion of the long lived inside-out heating front which propagates
over a large range of radii in the disc. Outside-in outbursts
(e.g. Fig.~4 of HMDLH) do not show this burst of $\dot{M}_{\rm acc}$
because the outside-in heating front which reaches the disc inner edge
is responsible for the bulk of the outburst accretion in that case.
\subsection{Reflection of an inside-out heating front}
The surface density after the passage of a heating front must be
larger than the local value of $\Sigma_{\rm min}$ (the minimum density
on the hot branch of the S-curve) to allow the front to propagate
further in the disc. If not, a cooling front appears where $\Sigma <
\Sigma_{\rm min}$ in the disc (just behind the $\Sigma$ spike). Once
the cooling front develops, the transport of excess angular momentum
from the inner disc is strongly reduced and the propagation of the
heating front is stopped. The spike in $\Sigma$ then diffuses under
the action of viscous processes. This process is usually referred to
as a reflection, although it corresponds more exactly to the
appearance of a new cooling.
In the case of outside-in fronts, matter flows from the outer parts of
the disc, which contain most of the disc mass, so that $\Sigma$
steadily increases at any given radius, and the heating front is
always able to reach the disc inner edge without being reflected in
our models.
The situation is quite different for inside-out heating fronts, for
which mass is transferred from the inner parts to the outer parts of
the disc. The comparatively small amount of mass in the inner disc
leads to several reflections of inside-out heating fronts during one
outburst cycle. After these multiple reflections, an inside-out
heating front can reach $R_{\rm out}$ without being reflected because
enough mass has accumulated in the outer disc and $\Sigma$ behind the
front remains above $\Sigma_{\rm min}$ during the entire propagation.
The reflections also occur in mini-outbursts (as compared to
``normal'' outbursts in which most of the disc -- if not all --
undergo a transition to the hot state) which are found in many
numerical models (see e.g. Cannizzo 1993a, HMDLH). The significance of
these mini-outbursts is unclear because they have probably not been
observed.
\section{Cooling fronts}
\subsection{Structure of cooling fronts}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{cool2.epsi}
\end{displaymath}
\caption{Structure of a cooling front located at $R_{\rm front} \sim 3
\times 10^9~{\rm cm}$ (model {\it h0.1.c0.02}). The structure is
dominated by an outflow of gas and by radiative cooling.}
\label{fig:cool}
\end{figure}
Figure~\ref{fig:cool} shows the structure of a cooling front located
at $R_{\rm front} \sim 3 \times 10^9~{\rm cm}$ in model {\it
h0.1.c0.02}. The innermost, hot, regions of the disc accrete at a much
higher rate than the outer cold regions (see e.g. Fig.~9 of Cannizzo
et al. 1995). A substantial amount of angular momentum flows outward
in the inner regions of the disk. The gas which carries this excess
angular momentum is stopped when it reaches colder regions of the
disc, where the transport of angular momentum is reduced. This results
in a large outflow and an abrupt increase of $\Sigma$ where rapid
cooling sets in, as seen from Fig.~\ref{fig:cool}.
As noted by Vishniac \& Wheeler (1996) (see also Vishniac 1997), the
inner disc is in quasi-steady state during the propagation of a
cooling front; the accretion rate is almost independent of radius and
it slowly decreases with time. The profiles of $\Sigma$ and $T_c$ in
these regions (not shown in Fig.~\ref{fig:cool}, but see, e.g.,
Fig.~5a of HMDLH) are close to those of the steady solution of Shakura
\& Sunyaev (1973). Between the inner quasi-steady disc and the front
itself, there is a relatively broad region (called precursor region by
Vishniac \& Wheeler 1996) where the profiles of $\Sigma$ and $T_c$
deviate substantially from the Shakura-Sunyaev profiles.
Figure~\ref{fig:cool} shows that the thermal structure of a cooling
front is dominated, although not strongly, by radiative cooling
($Q^-$): in first approximation, gas freely cools inside a cooling
front (Vishniac \& Wheeler 1996). The three non-local terms in the
energy equation, namely $Q_{\rm adv}$, $Q_{\rm j}$ and $Q_{\rm pdv}$,
are small but not negligible. The radial transport of energy ($Q_{\rm
j}$) occurs in the region of strong gradient of $T_c$. Advection of
energy and adiabatic compression of the gas occur mainly in a region
corresponding to the outflow of gas, i.e. where the gas which carries
a substantial amount of heat ($Q_{\rm adv}$) is compressed as it
enters colder regions of the disc ($Q_{\rm pdv}$).
\subsection{Cooling front velocity}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{coolspeedmult.eps}
\end{displaymath}
\caption{The four panels show the speed of successive cooling fronts
in our four models with various $\alpha$-prescriptions (dots). Cooling
fronts appearing at various radii in the outer disc quickly converge
to the same asymptotic speed (in a given model) as they propagate
inward. (Note that all cooling fronts appear at the same location in
model {\it 50hr1.5}.) The asymptotic speed of a cooling front slightly
depends on the value of $\alpha_{\rm hot}.$}
\label{fig:coolspeedmult}
\end{figure}
Figure~\ref{fig:coolspeedmult} shows the speed of cooling fronts in
the four models with various $\alpha$-prescriptions. Successive
cooling fronts in an outburst cycle appear at various locations in the
disc in models using an $\alpha_{\rm
hot}-\alpha_{\rm cold}$ prescription. All
the cooling fronts appear at the same location, close to the
disc's outer edge, in model {\it 50hr1.5}. The initial speed of all
cooling fronts is relatively high but soon, as they propagate
inward, they relax to a reduced and common asymptotic speed which
depends only on the current location of the front in the disc, $R_{\rm
front}$, in agreement with early calculations by Cannizzo (1994).
The effect of varying $\alpha_{\rm cold}$ on the (asymptotic) speed of
a cooling front is small (at most). Increasing the value of $\alpha_{\rm
hot}$ results, however, in higher asymptotic speeds.
In the asymptotic regime, $V_{\rm front}$ has a weak dependence on
$R_{\rm front}$. This is especially true in models with an
$\alpha_{\rm hot}-\alpha_{\rm cold}$ prescription (see also Cannizzo
1994). The different variation of $V_{\rm front}$ with $R_{\rm front}$
in model {\it 50hr1.5} can be accounted for by the variation of
$\alpha$ with radius in this specific model, according to the scaling
of $V_{\rm front}$ versus $\alpha_{\rm hot}$ in the three other
models. Note that the sharp increase of $V_{\rm front}$ close to the
disc inner edge observed in all models is a boundary condition effect:
the surface density $\Sigma$ goes to zero at $R_{\rm in}$ and this
causes the front to propagate faster as $\Sigma$ is closer to
$\Sigma_{\rm min}$ in the innermost regions of the disc.
The speed of a cooling front, at a given radius $R_{\rm front}$, is
slightly sensitive to the mass of the central object. We obtained an
increase of $V_{\rm front} \sim 30 \%$ in the asymptotic regime when
the mass $M_1$ was changed from $1.2~M_{\odot}$ to $0.6~M_{\odot}$.
$V_{\rm front}$ in the asymptotic regime is not much
larger than the speed of the hot gas into which the front penetrates;
both are of order of a fraction of $1~{\rm km~s^{-1}}$. This is
consistent with the idea that an evolution of the inner hot disc on a
viscous timescale is typically required to allow the propagation of
the cooling front (Lin et al. 1985, Vishniac \& Wheeler 1996). But
contrary to the case of heating fronts, we find no good correlation
between the value of $V_{\rm r,max}$ inside the transition region and
the speed of cooling fronts.
Recently, Bobinger et al. (1997) directly determined the speed of a
cooling front with the eclipse mapping technique. They observed the
evolution of the brightness temperature profile in the disc of the
dwarf nova IP Peg during four successive days of its decay from
outburst. They infer a mean value of $V_{\rm front} \sim 0.8~{\rm
km~s^{-1}}$ over the four days, which is consistent with the speeds
shown in Fig.~\ref{fig:coolspeedmult}. It will be difficult, however,
to determine the viscosity parameter $\alpha$ from such observations,
even if variations of $V_{\rm front}$ with $R_{\rm front}$ were
detected. For instance, models {\it h0.1.0c0.02} and {\it 50hr1.5}
produce very different outburst cycles which could be distinguished
from observed lightcurves; yet the variation of $V_{\rm front}$ with
$R_{\rm front}$ is not drastically different. A strong prediction of
the DIM (which probably holds for any reasonable
$\alpha$-prescription) is, however, the rapid decrease of $V_{\rm
front}$ with $R_{\rm front}$ shortly after the appearance of a cooling
front. The detection (or non-detection) of this characteristic would
be strong evidence (or disproof) that the DIM -- as we understand it
-- operates in accretion discs.
Vishniac \& Wheeler (1996) proposed an analytical model of cooling
fronts which uses the numerical results of Cannizzo et al. (1995) as
calibration. In the derivation of Vishniac \& Wheeler (1996), the speed
of a cooling front does not depend on the physical conditions in the
cold gas behind the front. This is in agreement with the independence
of $V_{\rm front}$ with $\alpha_{\rm cold}$ observed in our
simulations.
Vishniac \& Wheeler (1996) predict a speed for the cooling front
\begin{equation}
V_{\rm front} = \alpha_F c_F \left( \frac{H}{R} \right)^{0.54}
\label{eq:vfrontvw1}
\end{equation}
if $\alpha=50 (H/R)^{1.5}$ (and assuming a Kramer's law opacity is
valid for the hot gas), and a speed
\begin{equation}
V_{\rm front} = \alpha_{\rm hot} c_F \left( \frac{H}{R} \right)^{7/10}
\label{eq:vfrontvw2}
\end{equation}
in the case of an $\alpha_{\rm hot}-\alpha_{\rm cold}$ prescription. Here
$\alpha_F$ and $c_F$ are the viscosity parameter and the sound speed
at the cooling front (i.e. where rapid cooling sets in).
Equation~(\ref{eq:vfrontvw1}) predicts a $V_{\rm front}$ which is
slightly smaller ($20 \%$ typically) than what we find in the
asymptotic regime of model {\it 50hr1.5} but the dependence with
$R_{\rm front}$ appears good. Equation~(\ref{eq:vfrontvw2}) predicts,
however, a $V_{\rm front}$ which is typically twice as small as what
is observed in the asymptotic regime of model {\it h0.1.c0.02}.
Vishniac \& Wheeler (1996) assume that the gas velocity at the cooling
front, $V_F$, is $\sim 1/6~\alpha_F c_F$ and that $V_F$ is much larger
than the cooling front velocity. In our simulations, $V_F \sim
1/7~\alpha_F c_F$, but the cooling front velocity is not much smaller
than $V_F$. The ratio of the gas velocity at the cooling front to the
cooling front velocity is typically of order 2 (see
Figs.~\ref{fig:cool} and \ref{fig:coolspeedmult}). Neglecting $V_{\rm
front}$ is probably one of the reason for the discrepancy between the
predictions of Vishniac \& Wheeler (1996) and our results.
\subsection{Width of cooling fronts}
Figure~\ref{fig:dwmult1.2} shows the width of cooling fronts in the
four models with various $\alpha$-prescriptions. Cooling fronts are
much broader than heating fronts but their width $\delta w$ remains
proportional to $H$. As for heating fronts, we interpret the width of
a cooling front as the distance traveled by the front during a few
thermal timescales (the thermal structure of the front being dominated
by radiative cooling): $\delta w$ is proportional to $R_{\rm
front}^{3/2} \propto H$ (Eq.~(\ref{eq:deltaw})).
\footnote{Naively, one would expect from Eq.~(\ref{eq:deltaw}) smaller
$\delta w$ for cooling fronts than for heating fronts (because of
smaller $V_{\rm front}$). Equation~(\ref{eq:deltaw}) does not take
into account, however, several complications in the problem. The
thermal timescale $\tau_{\rm th} \equiv (\alpha \Omega_K)^{-1}$, which
is very different for annuli in the hot and the cold regions of the
disc, is probably increased because of the ionization energy in
transition regions (Papaloizou \& Pringle 1985). The precursor region
drives out of thermal equilibrium the disc annuli entering the
transition region in the case of heating fronts (see
Fig.~\ref{fig:scurve}), while there is no equivalent for cooling
fronts. Finally, radiative cooling clearly does not dominate the
thermal structure of a cooling front as much as viscous heating does
in a heating front, which probably leads to comparatively longer
thermal timescales for cooling fronts. A key feature that
Eq.~(\ref{eq:deltaw}) includes, however, is the strong variation of
the thermal timescale with radius.}
This conclusion is in contradiction with the results of Cannizzo et
al. (1995) who find $\delta w = \sqrt{H R}$ in a numerical model using
$\alpha=50 (H/R)^{1.5}$. The profiles of $\delta w/R_{\rm front}$ as a
function of $R_{\rm front}$ shown in Fig.~\ref{fig:dwmult1.2} for
model {\it 50hr1.5} are consistent with those of Cannizzo et
al. (1995). Cannizzo et al. (1995) claim that $\delta w = \sqrt{H R}$
is also true in models using an $\alpha_{\rm hot}-\alpha_{\rm cold}$
prescription. Our three models with $\alpha_{\rm hot}-\alpha_{\rm
cold}$ prescriptions rule out this possibility and show that $\delta w
\propto H$. As mentioned previously, we interpret the flatter profiles
of $\delta w$ in model {\it 50hr1.5} as due to the variation of
$\alpha$ with radius in this particular model. Models using $\alpha=
\alpha_0 (H/R)^{n}$ with various $n$ produce various slopes for the
profiles of $\delta w$ as a function of $R_{\rm front}$.
A comparison of the various panels in Fig.~\ref{fig:dwmult1.2} shows
that the effect of varying $\alpha$ on the width of a cooling front is
small.
\subsection{Self-similar solutions}
Vishniac (1997) proposed a self-similar solution for the structure of
the hot inner disc during the propagation of a cooling front. We
confirm the existence of a self-similar regime, although the
self-similarity described here is somewhat different from the solution
of Vishniac (1997) because $\Sigma$ is found to scale naturally with
$\Sigma_{\rm min}$. We leave a detailed analysis of the
self-similar solution for a future paper, but we give here our main
qualitative results.
\begin{figure}
\epsfysize=10cm
\epsfxsize=8cm
\begin{displaymath}
\epsfbox{selfsim.epsi}
\end{displaymath}
\caption{Successive normalized profiles of $\Sigma$ and local mass
flow rate $\dot M$ as a function of the dimensionless radius
$(R-R_{\rm in})/(R_{\rm front}-R_{\rm in})$ during the propagation
of a cooling front (model {\it h0.1.c0.02}). The superposition of
profiles reveals the nearly self-similar evolution of the inner hot
disc.}
\label{fig:selfsim}
\end{figure}
Figure~\ref{fig:selfsim} shows successive profiles of $\Sigma$ and
local mass transfer rate $\dot{M}$ during the propagation of a cooling
front in model {\it h0.1.c0.02}, when the asymptotic regime is
reached. The profiles of $\Sigma$ have been normalized to the minimum
surface density in the cooling front, which is extremely close to
$\Sigma_{\rm min}$ at $R_{\rm front}$. This is not unexpected, since
the transition from the hot to the cold state occurs precisely when
$\Sigma$ reaches $\Sigma_{\rm min}$. The mass flow rate $\dot{M}$ has
been normalized, following Vishniac \shortcite{v97}, to the mass
transfer rate at the disc inner edge, $\dot{M}_{\rm acc}$. The
horizontal coordinate chosen here is $(R-R_{\rm in})/(R_{\rm
front}-R_{\rm in})$ in order to minimize the effect of the inner
boundary condition on the profiles. The value of $R_{\rm front}$
varies by more than an order of magnitude for the successive profiles
shown in Fig.~\ref{fig:selfsim} and yet the normalized $\Sigma$ and
$\dot{M}$ remain remarkably similar: except for the effect of boundary
conditions, the evolution of the inner hot disc is close to
self-similar.
The mass accretion rate onto the compact object, $\dot{M}_{\rm acc}$,
is proportional to $\dot{M}_{\rm front}$ at the transition radius. But
\begin{equation}
\dot{M}_{\rm front} = 2 \pi R_{\rm front} \Sigma(R_{\rm front})
V_r(R_{\rm front}),
\end{equation}
$\Sigma(R_{\rm front}) = \Sigma_{\rm min}(R_{\rm front}) \propto
R_{\rm front}^{1.1}$ (e.g. HMDLH), and $V_r(R_{\rm front})$ is
comparable to the front velocity. If $\alpha$ in the hot inner disc
is constant, one expects a weak dependence of $V_r(R_{\rm front})$
with $R_{\rm front}$ and $\dot{M}_{\rm acc} \propto R_{\rm front}^n$,
with $n$ slightly larger than 2. Indeed, numerical simulations show
that $n \simeq 2.2$ for an $\alpha_{\rm hot}-\alpha_{\rm cold}$
prescription, and $n \simeq 2.4$ when $\alpha \propto (H/R)^{1.5}$.
As discussed by Vishniac \shortcite{v97}, the hot inner disc evolves
on a viscous time during the propagation of a cooling front, and is
therefore very close to thermal equilibrium. The inner disc empties as
a result of accretion onto the compact object, but also because an
even larger amount of mass is transferred into the cold outer regions
of the disc. These parts of the disc are essentially frozen on the
time scale of propagation of the cooling front. Consequently,
$\Sigma(R)$ behind the front remains constant in time and is equal to
$K \Sigma_{\rm min}(R)$, where $K$ is a self-similarity constant that
depends on the mass of the primary and the viscosity parameter
$\alpha$ both in the hot and the cold regions of the disc. For the
parameters of Fig.~\ref{fig:selfsim}, $K \sim 4$. The constant $K$
increases with the mass of the primary (for example, $K \sim 6-7$ for
a 7 M$_\odot$ primary typical of a black hole SXT).
The sudden increase of $\Sigma$ just after the passage of a cooling
front is actually smaller in the pre-self-similar regime, so that
$\Sigma / \Sigma_{\rm min}$ increases as the front propagates toward
the asymptotic regime. This accounts for the density profile in the
outer disc which is relatively flat close to the outer edge, steepens
at smaller radii and finally becomes proportional to $\Sigma_{\rm min}
\propto r^{1.1}$ in the self-similar regime.
\subsection{Reflection of a cooling front}
The sudden increase of $\Sigma$ just after the passage of a cooling
front cannot be arbitrarily large (Vishniac, 1997). If $\Sigma >
\Sigma_{\rm max}$ behind the cooling front, an inside-out heating
front develops and propagates over some distance in the disc because
$\Sigma$ is substantially larger than $\Sigma_{\rm min}$ in the outer
disc. The initial cooling front soon disappears as hot gas from the
outer regions of the disc accretes inward.
This situation corresponds to the reflection of a cooling front into a
heating front and it occurs whenever $\Sigma_{\rm max}/\Sigma_{\rm
min}$ is too small. The ratio $\Sigma_{\rm max}/\Sigma_{\rm min}$ has
a weak radial dependence ($R^{0.02 - 0.03}$; e.g. HMDLH). If the
self-similarity constant $K$ is too large, the cooling front will
experience a reflection typically when it enters the self-similar
regime: the gradual increase of $\Sigma / \Sigma_{\rm min}$ preceding
the self-similar regime will drive $\Sigma$ behind the front above
$\Sigma_{\rm max}$ and lead to the reflection.
The multiple reflections of cooling fronts are well known if
$\alpha_{\rm cold} = \alpha_{\rm hot}$ \cite{sma84}. In this case, the
inner disc can never be brought entirely into quiescence, and the
reflected transition fronts propagate back and forth in a restricted
region of the disc, causing small amplitude quasi-oscillations of the
disc luminosity.
Note that because $K$ increases with the mass of the primary, the
reflections of cooling fronts are more likely to occur in BH SXTs than
in neutron star SXTs or dwarf novae.
\subsection{Exponential decays}
We reexamine, in light of our results on cooling fronts, the claim by
Cannizzo et al. (1995; see also Vishniac \& Wheeler 1996) that
$\alpha=\alpha_0 (H/R)^{1.5}$ is necessary to account for exponential
decays of soft X-ray transients (SXTs; but see also the similarity
between SXTs and dwarf novae pointed out by Kuulkers, Howell \& van
Paradijs 1996)
Our simulations show that the inner accretion rate $\dot M_{\rm acc}
\propto R_{\rm front}^n$ in the self-similar regime of propagation of
a cooling front ($n = 2 - 2.5$ depending on the
$\alpha$-prescription).
Since X-rays are produced close to the central object in an SXT, the
exponentially decaying lightcurves imply an exponential decay of
$\dot M_{\rm acc}$ as well, i.e.
\begin{equation}
\frac{d \dot M_{\rm acc}}{dt} \propto \dot M_{\rm acc}.
\end{equation}
If we assume that the decay phase corresponds to the propagation of a
cooling front in the disc and that our results apply (no irradiation
effect is included in our calculations), then
\begin{equation}
\frac{d \dot M_{\rm acc}}{dt} \propto R_{\rm front}^{n-1} V_{\rm front} \propto \dot M_{\rm acc} \frac{V_{\rm front}}{R_{\rm front}},
\end{equation}
and an exponential decay is possible only if $V_{\rm front}/R_{\rm front}$
remains constant, meaning that $R_{\rm front}$ must also vary exponentially
with time:
\begin{equation}
V_{\rm front} = \frac{d R_{\rm front}}{dt} \propto R_{\rm front}.
\label{eq:exp}
\end{equation}
Note that this requirement was already deduced,
although differently, by Cannizzo et al. (1995) and Vishniac \&
Wheeler (1996).
Figure~\ref{fig:coolspeedmult} shows that $V_{\rm front}$ is primarily
independent of $R_{\rm front}$ (in the asymptotic regime) in the three
models with $\alpha_{\rm hot}-\alpha_{\rm cold}$ prescriptions. On the
contrary, model {\it 50hr1.5} shows a variation of $V_{\rm front}$
with $R_{\rm front}$ which is linear in first approximation. Our
calculations therefore confirm that a dependence of the viscosity
parameter $\alpha$ with radius (close to $\alpha \propto R^{3/4}$ like
in model {\it 50hr1.5}) is required to obtain exponential decays of
$\dot M_{\rm acc}$ in the DIM.
A major concern however is the relevance of interpreting exponentially
decaying lightcurves as due to the propagation of a cooling front in
the discs of SXTs. In particular, Shahbaz et al. (1998) argue that
both linear and exponential decays are observed in SXTs (see also the
compilation of lightcurves by Chen et al. 1997). Since there is no
obvious physical reason why $\alpha$ would vary markedly from one
system to another, the possibility that both linear and exponential
decays are observed in SXTs argues against an interpretation of the
exponential decays as due to a specific functional form of $\alpha$.
An alternative and perhaps more promising explanation for the
exponential decays has been proposed by King \& Ritter (1998). The
authors argue that strong irradiation of the disc may prolongate the
phase during which the entire disc is in the hot state and lead to an
evolution of the disc on a (long) viscous time (see however
Dubus et al. 1998).
\subsection{Cooling fronts and the periodicity of outburst cycles}
The asymptotic regime of cooling front propagation has a profound
influence on the outburst cycles experienced by the disc. Since the inner hot
disc evolves in a self-similar way, the profiles of $\Sigma$ and $T_c$ inside
$R_{\rm front}$ are uniquely determined by the current value of $R_{\rm
front}$.
A cooling front following a heating front that reached the outermost
regions of the disc therefore propagates in the exact same way a
previous cooling front starting from $R_{\rm out}$ did. This type of
cooling front erases the history of the disc and is responsible for
the periodicity of the outburst cycles in the simulations. An outburst
cycle ends and a new cycle begins when a heating front reaches $R_{\rm
out}$ (also corresponding to the largest amplitude outbursts).
Observed outburst cycles in dwarf novae are usually regular but
not periodic (e.g. Warner 1995, Cannizzo \& Mattei 1992). This may hint
that a piece of physics is missing in the standard DIM. A variation of
the mass transfer accretion rate $\dot{M}_T$ appears as a natural
candidate for explaining the irregularity of observed cycles.
\section{Evolution around S-curves}
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{scurve.eps}
\end{displaymath}
\caption{Limit cycles experienced by disc annuli located at $R=10^{9}$
and $10^{10}$ cm in the $\Sigma-T_{\rm eff}$ plane. The left panels
correspond to model {\it h0.1.c0.02} and the right ones to model {\it
50hr1.5}. The solid lines are the thermal equilibrium curves, where
$Q^+ = Q^-$ (standard S-curves). The dashed lines trace the successive
states of a disc annulus, while the density of circles is
representative of the amount of time spent by an annulus in a given
state. The evolution of the disc annulus shown in the lower left panel
is complex because this annulus is located in a region of front
reflections.}
\label{fig:scurve}
\end{figure}
Figure~\ref{fig:scurve} shows limit cycles experienced by two disc annuli in
models {\it h0.1.c0.02} and {\it 50hr1.5} in the $\Sigma-T_{\rm eff}$ plane,
where $T_{\rm eff}$ is the effective temperature of the disc.
Global effects (i.e influence by neighboring annuli) appear when an annulus
no longer lies on the S-curve. The passage of a cooling front corresponds to
a jump from the upper hot branch of the S-curve to the lower cold branch.
Inversely, the passage of a heating front corresponds to the jump from the
lower branch to the upper branch. Note that the maximum value of $\Sigma$
reached by a disc annulus on the lower cold branch of the S-curve during a
limit cycle is indicative of the proximity of this annulus to the ignition
radius in the disc.
Figure~\ref{fig:scurve} shows the evolution of disc annuli during the
passage of a transition front: a sudden increase of $\Sigma$ is as
good a signature of the transition as a change of $T_{\rm eff}$ is.
For a cooling front, the sudden change of $\Sigma$ shortly precedes
the phase of rapid cooling of the annulus. For a heating front, the
sudden change of $\Sigma$ corresponds to the passage of the spike of
$\Sigma$. This shows that inside a transition front, the viscous time
is not much larger but comparable to the thermal time since surface
density and effective temperature vary on similar timescales.
The complexity of the limit cycles experienced by the annulus at
$R=10^{10}$ cm in model {\it h0.1.c0.02} is due to the many
reflections of inside-out heating fronts into cooling fronts around
$10^{10}$ cm in this model. Note that the track followed by the
annulus states during the successive jumps from the upper branch to
the lower branch do not cover each other in that case because cooling
fronts have not yet reached the asymptotic regime at $10^{10}$ cm in
model {\it h0.1.c0.02}.
Note that the evolutions around S-curves shown here differ
significantly from the results of Ludwig, Meyer-Hofmeister \& Ritter
(1994) using the theory of infinitely thin transition fronts (Meyer
1984). Some additional limitations of the theory were pointed out by
Ludwig \& Meyer (1998).
\section{Validity of the disc equations for the transition fronts}
It is standard in the theory of thin accretion discs to assume that
the vertical structure of the disc is decoupled from its radial
structure because the former evolves on much shorter timescales than
the latter (e.g. Frank et al. 1992). Because $\Sigma$ and $T_{\rm
eff}$ vary on similar timescales, this assumption is no longer valid
inside transition fronts: the vertical structure is coupled to the
radial structure of the disc. In particular, radial transport of
energy becomes important in the transition regions, while the
transport occurs only vertically in steady thin discs (Frank et
al. 1992).
A correct treatment of the coupling between the radial and vertical
structures of the disc would probably involve 2D simulations. For
simplicity, however, the radial transport of energy is usually
parameterized in a simple manner in 1D disc equations (see also the
approximations of decoupling used for the determination of the cooling
rate $Q^-$, HMDLH). The fact that various prescriptions exist for $J$
(Eq.~(\ref{eq:heat})) shows that the radial transport of energy in the
disc is not well known, and surely not within a factor of a few.
\begin{figure}
\epsfysize=8.5cm
\epsfxsize=8.5cm
\begin{displaymath}
\epsfbox{rad3speed.eps}
\end{displaymath}
\caption{An illustration of the sensitivity of the propagation
velocity of heating fronts to the poorly known efficiency of radial
transport of energy (term J in Eq.~(2)), in model {\it h0.1.c0.02}.}
\label{fig:rad3speed}
\end{figure}
Figure~\ref{fig:rad3speed} shows the effect on the speed of heating
fronts of increasing or decreasing by a factor $3$ the intensity of
the term $J$. The large uncertainty in $J$ translates into a
significant uncertainty on the speed of heating fronts. (Note that the
speed of cooling fronts is not affected because the radial transport
of energy has a minor influence on their structure; cf
Fig.~\ref{fig:cool}).
An additional concern about the validity of the disc equations inside
transition fronts comes from the narrowness of the fronts. Lin et
al. (1985) pointed out that having a front width $\delta w \sim H$,
although it does not affect much the Keplerian character of the flow,
could lead to a Rayleigh-unstable situation in the disc. A comparison
between the criterion derived by Lin et al. (1985) and the structure
of the narrow heating fronts found here suggests that this type of fronts
may indeed experience the development of a Rayleigh
instability. This could result in an enhanced radial transport (Lin et
al. 1985) and an additional uncertainty in $J$.
The equations used in this study are only valid in the thin disc
approximation, i.e. when radial pressure gradients can be neglected
and both $\Omega$ and $d\Omega /dR$ are equal to the Keplerian values.
The effects of deviations from Keplerianity and radial pressure
gradients were shown, for instance, by Ludwig \& Meyer (1998) using
full hydrodynamical disc equations and by Kley \& Lin (1996) in
boundary layer computations. Although these effects become
non-negligible in narrow heating fronts, we do not expect our main
results on the structure and properties of transition fronts to be
affected if they were fully taken into account. This could be checked
by taking the full equations, which is beyond the scope of the present
paper.
Finally, we note that the use of a local $\alpha$-prescription in
narrow transition regions ($\delta w \sim H$) is also subject to
caution because the approach of scale separation becomes much less
rigorous on scales so close to the size of the ``turbulent eddies''.
\section{Discussion}
Our simulations show that transition fronts systematically
experience a deceleration during their propagation in the disc,
whatever their direction of propagation. Yet, the front widths
increase with radius in all the cases. This shows that the speed of a
front is not directly related to its width (see also a related claim
by Vishniac \& Wheeler 1996 for cooling fronts). A description
according to which the width of a transition front, which defines the
gradients and the fluxes in the front, also determines its speed
thus appears oversimplified.
Our simulations reveal a qualitative
agreement with one of the predictions of the theory of infinitely thin
transition fronts (Meyer 1984, 1986; see also Papaloizou \& Pringle
1985): we clearly observe that the speed of transition fronts is
related to the proximity of $\Sigma$ in the disc to the critical
values $\Sigma_{\rm min}$ (for cooling fronts) and $\Sigma_{\rm max}$
(for heating fronts).
Although Ludwig et al. (1994) pointed out that this theory may not be
applicable to relatively broad cooling fronts, it could provide an
explanation for the rapid deceleration of cooling fronts: the values
of $\Sigma$ around the region where the cooling front appears are
close to $\Sigma_{\rm min}$, so that the front speed is initially
high; soon, however, the front enters regions with larger and larger
values of $\Sigma$ as compared to $\Sigma_{\rm min}$
(quasi-Shakura-Sunyaev profile), which results in a gradual decrease
of the speed; later, the front must wait for the inner hot disc to
evolve on a viscous timescale, typically, before $\Sigma$ approaches
$\Sigma_{\rm min}$ in the disc, which explains the small speeds in the
asymptotic regime. On the contrary, the speed of a heating front is
able to remain large during the propagation, possibly because the
profile of $\Sigma$ encountered in the disc is typically in between
$\Sigma_{\rm min}$ and $\Sigma_{\rm max}$ (Cannizzo 1993b), i.e. not
too far from $\Sigma_{\rm max}$ everywhere in the disc. If this is
true, the deceleration of heating fronts is due to the (necessary)
proximity of $\Sigma$ to $\Sigma_{\rm max}$ close to the ignition
radius but not elsewhere in the disc.
We have not considered the structure and properties of transition
fronts close to $R_{\rm out}$ in this paper. Indeed, additional
simulations show that in a small region close to the disc outer edge,
transition fronts are affected in a complex way if $R_{\rm out}$ is
allowed to vary with time. The behavior of the fronts in this region
of the disc may also depend sensitively on the precise choice of the
outer boundary condition and how mass is deposited in the disc.
\section{Conclusion}
In this paper, we have investigated the structure and properties of
transition fronts in thin accretion discs with detailed numerical
calculations.
We showed that heating fronts are very narrow and have complex
structures. They propagate at a speed which depends on the profile of
surface density in the disc, the radial transport term in the energy
equation and the value of the viscosity parameter $\alpha$, but
is typically of order a few ${\rm km~s^{-1}}$.
Cooling fronts are broader, have a simpler structure and have smaller
speeds (of order a fraction of a km s$^{-1}$) than heating fronts. We
found that their width is not equal to $\sqrt{HR}$, but is rather
$\propto H$, the local disc scale height, as for heating fronts.
We confirmed that the structure of the inner hot disc is well
described by a self-similar solution during the propagation of a
cooling front. We proposed such a solution in which the surface
density does not scale arbitrarily but with the value of the critical
surface density $\Sigma_{\rm min}$. The self-similarity of the disc
appears responsible for the periodicity of the outburst cycles in the
simulations.
Since all our models show a deceleration of transition fronts during
their propagation in the disc, the observation of such a deceleration
would constitute a nice confirmation that the thermal-viscous disc
instability, as we understand it, is responsible for the large
amplitude variability of discs around white dwarfs, neutron stars and
black holes.
\section*{Acknowledgments}
We are grateful to Charles Gammie, Jean-Pierre Lasota, Ramesh Narayan,
John Raymond and Ethan Vishniac for useful discussions. We also thank
Guillaume Dubus and Eliot Quataert for comments on the manuscript.
This work was supported in part by NASA grant NAG 5-2837. KM was
supported by a SAO Predoctoral Fellowship and a French Higher
Education Ministry grant. RS is supported by a PPARC Rolling Grant for
theoretical astrophysics to the Astronomy Group at Leicester.
| 2024-02-18T23:40:15.647Z | 1998-11-12T02:06:05.000Z | algebraic_stack_train_0000 | 1,850 | 10,544 |
|
proofpile-arXiv_065-9111 | \section{Analytic results}
The Dirac-Kogut-Susskind operator of QCD at finite chemical potential can be
written as
\begin{equation}\label{1}
2 \Delta = 2mI + e^\mu G + e^{-\mu} G^\dagger + V
\end{equation}
\noindent
where $G$ ($G^+$) contains all forward (backward) temporal links and V all
space-like links.
The determinant of $\Delta$ in the integration
measure can be replaced, at large fermion masses $m$, by
\begin{equation}\label{2}
\det \Delta = m^{3V_sL_t}\det \left( I + \frac{e^\mu}{2m} G \right)
\end{equation}
If the fugacity $e^\mu$ is much smaller than $2m$, the second factor of (2)
can be replaced by 1 and the theory is independent of the chemical potential.
Therefore, in order to get a non trivial $\mu$ dependence, we need to go
to a region of large chemical potential in which the fugacity is of the
order of $2m$ \cite{TOUS}.
Since all space-like links have disappeared in equation (2), the determinant
of $\Delta$ factorizes as a product of $V_s$ determinants for the single
temporal chains. A straightforward calculation allow us to write
\begin{equation}\label{3}
\det \Delta = e^{3V_sL_t\mu} \prod_{i=1}^{V_s} \det (c + L_i )
\end{equation}
\noindent
with $c=({2m\over{e^\mu}})^{L_t}$, $L_t$ is the lattice
temporal extent and $L_i$
the SU(3) variable representing the forward straight Polyakov loop starting
from the spatial site $i$ and circling once the lattice in the temporal
direction. The determinants in (3) are gauge invariant quantities which can
therefore be written as functions of the trace and the determinant of $L_i$.
Since the gauge group is a unitary group, $\det(L_i)=1$
and therefore the only contributions depending on the gauge configuration
will be functions of $Tr(L_i)$. In fact simple algebra allows to write
\begin{equation}\label{4}
\det (c + L_i ) = c^3 + c^2 Tr (L_i) + c Tr (L_i^*) + 1
\end{equation}
In the infinite gauge coupling limit, the integration over the gauge group
is trivial since we get factorization \cite{LAT98}.
The final result for the partition
function at $\beta=0$ is
\begin{equation}\label{5}
{\cal Z} =
V_G e^{3V_sL_t\mu}
\left( \left(\frac{2m}{e^\mu}\right)^{3L_t} +1 \right)^{V_s}
\end{equation}
\noindent
where $V_G$ is a constant irrelevant factor diverging exponentially
with the lattice
volume which accounts for the gauge group volume. Equation (5)
gives for the free energy density $f={1\over{3V_sL_t}}\log{\cal Z}$
\begin{equation}\label{6}
f = \mu +
\frac{1}{3L_t} \log \left( \left(\frac{2m}{e^\mu}\right)^{3L_t} +1 \right)
\end{equation}
The first contribution in (6) is an analytical function of $\mu$. The second
contribution has, in the limit of infinite temporal lattice extent,
a non analyticity at $\mu_c=\log(2m)$ which induces in the number density
a step jump, indication of a saturation transition of first order at the
value of $\mu_c$ previously given.
This is an expected result on physical grounds. In fact in the infinite
fermion mass limit baryons are point-like particles, and
pion exchange interaction vanishes, since pions are also very heavy.
Therefore we are dealing with a system of very heavy free
fermions (baryons) and by increasing the
baryon density in such a system we expect an onset at
$\mu_c={1\over3}m_b$, i.e., $\mu_c=\log(2m)$ since $3\log(2m)$ is the baryon
mass at $\beta=0$ for large $m$ \cite{SACLAY}.
Let us now discuss the relevance of the phase of the
fermion determinant at $\beta=0$. The standard wisdom based on random matrix
model results is that the phase of the fermion determinant plays a fundamental
role in the thermodynamics of $QCD$ at finite baryon density \cite{RMT}
and that if
the theory is simulated by replacing the determinant by its absolute value,
one neglects a contribution to the free energy density which could be
fundamental in order to understand the critical behavior of this model.
We are going to show now that, contrary to this wisdom, the phase of the
determinant can be neglected in the large $m$ limit at $T=0$.
Equations (3) and (4) imply that an upper bound for the absolute
value of the fermion determinant is given by the determinant of the free
gauge configuration. Therefore the mean value of the phase factor in the
theory defined taking the absolute value of the determinant
\begin{equation}\label{7}
\left\langle e^{i\phi} \right\rangle_\| =
\frac{\int [dU] e^{-\beta S_G(U)}\det\Delta}
{\int [dU] e^{-\beta S_G(U)} | \det\Delta |}
\end{equation}
\noindent
is, at $\beta=0$, bounded from below by the ratio
\begin{equation}\label{8}
\left( \frac
{\left( \frac{2m}{e^\mu}\right)^{3L_t} + 1 }
{\left( \left( \frac{2m}{e^\mu}\right)^{L_t} + 1 \right)^3 }
\right)^{V_s}
\end{equation}
At zero temperature $(L_t=L, V_s=L^3)$, and letting $L\rightarrow\infty$,
it is straightforward to verify that the ratio (8)
goes to 1 except at $\mu_c=\log(2m)$
(at $\mu=\mu_c$ the ratio
goes to zero but it is bounded from below by $(1/4)^{V_s}$).
Therefore the mean value of
the cosine of the phase in the theory where the fermion determinant
is replaced by its absolute value gives zero contribution.
At $T \neq 0$, i.e. taking the infinite $V_s$ limit by keeping
fixed $L_t$, the lower bound
(8) for the mean value of the phase factor (7) goes to zero exponentially
with the spatial lattice volume $V_s$. This suggests that the
phase will contribute in finite temperature $QCD$.
In fact, it
is easy to convince oneself that expression (7), at $\beta=0$, vanishes
also exponentially with the lattice spatial volume at finite temperature
(see fig. 1).
The contribution of the phase is therefore non zero (in the limit considered
here) in simulations of $QCD$ at finite temperature.
The free energy density at finite temperature (equation (6)) is an analytic
function of the fermion mass and chemical potential. It develops a
singularity only in the limit of zero temperature $(T={1\over{L_t}})$.
Therefore $QCD$ at large $m$ and finite temperature does not show
phase transition in the
chemical potential but a crossover at $\mu=\log(2m)$ which becomes a true
first order phase transition at $T=0$.
The standard way to define the theory at zero temperature is to consider
symmetric lattices.
However a more natural way to define the theory at $T=0$
is to take the limit of finite temperature $QCD$ when the physical
temperature $T\rightarrow 0$.
In other words, we should take first the infinite spatial
volume limit and then the infinite temporal extent limit.
We will show here that, as expected, physical results are independent of
the procedure choosen.
The free energy density of the
model can be written as the sum of two contributions $f=f_1+f_2$. The first
contribution $f_1$ is the free energy density of the theory where the fermion
determinant in the integration measure is replaced by its absolute value.
The second contribution $f_2$, which comes from the phase of the fermion
determinant, can be written as
\begin{equation}\label{9}
f_2 = {1\over{V_sL_t}}\log\left\langle e^{i\phi} \right\rangle_\|.
\end{equation}
\noindent
Since the mean value of the phase factor (7) is less or equal than 1,
$f_2$ is bounded from above by zero and from below by
\begin{equation}\label{10}
{1\over{L_t}}\log{\left( \frac
{\left( \frac{2m}{e^\mu}\right)^{3L_t} + 1 }
{\left( \left( \frac{2m}{e^\mu}\right)^{L_t} + 1 \right)^3 }
\right)}
\end{equation}
When $L_t$ goes to infinity, expression (10) goes to zero for all the
values of $\mu$ and therefore the
only contribution to the free energy density which survives in the zero
temperature limit is $f_1$.
Again, we conclude that zero temperature QCD in the strong coupling
limit at finite chemical potential
and for large fermion masses is well described by the theory obtained by
replacing the fermion determinant by its absolute value.
These results are not surprising as follows from the fact that at $\beta=0$
and for large $m$ the system factorizes as a product of $V_s$ noninteracting
$0+1$ dimensional $QCD's$ and from the relevance (irrelevance) of the phase
of the fermion determinant in $0+1$ QCD at finite (zero) "temperature"
\cite{LAT97}.
More surprising maybe is that, as we will see in the following,
some of these results do not change when we put a finite gauge coupling.
The inclusion of a non trivial pure gauge Boltzmann factor in the integration
measure of the partition function breaks the factorization property. The
effect of a finite gauge coupling is to induce correlations between the
different temporal chains of the determinant of the Dirac operator. The
partition function is given by
\begin{equation}\label{11}
{\cal Z} = \int [dU] e^{-\beta S_G(U)} \prod_{i=1}^{V_s}
(c^3 + 1 + c Tr (L_i^*) + c^2 Tr (L_i) )
\end{equation}
\noindent
and can be written as
\begin{equation}
{\cal Z}(\beta,\mu) = {\cal Z}_{pg}\cdot {\cal Z}(\beta=0,\mu)\cdot
R(\beta,\mu)
\end{equation}
\noindent
where ${\cal Z}_{pg}$ is the pure gauge partition function,
${\cal Z}(\beta=0,\mu)$ the strong coupling partition function
(equation (5)) and $R(\beta,\mu)$ is given by
\begin{equation}\label{12}
R(\beta,\mu) = \frac
{\int [dU] e^{-\beta S_G(U)} \prod_{i=1}^{V_s} \left(
1 + \frac{c Tr (L_i) + c^2 Tr (L_i^*)}{c^3 + 1} \right)}
{\int [dU] e^{-\beta S_G(U)}}
\end{equation}
In the zero temperature limit ($L_t=L, L_s=L^{3}, L\rightarrow\infty$) the
productory in the numerator of (13) goes to 1 independently of the gauge
configuration. In fact each single factor has an absolute value equal to
1 up to corrections which vanish exponentially with the lattice size $L$
and a phase which vanishes also exponentially with $L$. Since the total number
of factors is $L^3$, the productory goes to 1 and therefore $R=1$ in the
zero temperature limit.
The contribution of $R$ to the free energy
density vanishes therefore in the infinite volume limit at zero temperature.
In such a case, the free energy density is the sum of the free energy density
of the pure gauge $SU(3)$ theory plus the free energy density of the model at
$\beta=0$ (equation (6)). The first order phase transition found at $\beta=0$
is also present at any $\beta$ and its location and properties do not
depend on $\beta$ since all $\beta$ dependence in the partition function
factorizes in the pure gauge contribution.
Again at finite gauge coupling
the phase of the fermion determinant is irrelevant at zero temperature.
At finite temperature and finite gauge coupling the first order phase
transition induced by the contribution (6) to the free energy density at
zero temperature disappears and becomes a crossover.
Furthermore expression (13) gives also a non vanishing contribution
to the free energy density if $L_t$ is finite.
The common physical interpretation for
the theory with the absolute value of the fermion determinant
is that it possesses
quarks in the {\bf 3} and {\bf 3}$^*$ representations of SU(3),
having baryonic states made up of two quarks which would give account for the
physical differences respect to real QCD. We have proven analytically (at
$\beta=0$) that the relation between modulus and real QCD is
temperature dependent, $i.e.$ they are different only at $T \ne 0$,
a feature that does not support the above interpretation.
\section{Numerical results}
From the point of view of simulations, work has been done by
several groups mainly to develop numerical algorithms capable to overcome
the non positivity of the fermionic determinant.
The most promising of these algorithms \cite{BAR}, \cite{NOI1}
are based on the GCPF formalism and try to calculate extensive quantities
(the canonical partition functions at fixed baryon number).
Usually they measure quantities that, with actual statistics, do not
converge.
In a previous paper \cite{NOI2} we have given arguments to conclude that,
if the phase is relevant, a statistics exponentially increasing
with the system volume
is necessary to appreciate its contribution to the observables (see also
\cite{BAR2} ).
What happens if we consider a case where the phase is not relevant
($i.e.$ the large mass limit of QCD at zero temperature, as discussed
in the previous section)?
To answer this question we have reformulated the GCPF formalism by
writing the partition function as a polynomial in
$c$ and studied the convergence properties of the coefficients at $\beta=0$
using an ensemble of (several thousands) random configurations.
This has been done as in standard numerical simulations ({\it i.e.}
without using the factorization property) for lattices $4^4$ (fig. 2a),
$4^3\times 20$ (fig. 2b), $10^3\times 4$ (fig. 2c) \cite{LAT98} and the
results compared with the analytical predictions (\ref{5}) (solid lines
in the figures).
From these plots we can see that, unless we consider a large lattice temporal
extension, our averaged coefficients in the infinite coupling limit still
suffer from sign ambiguities $i.e.$ not all of them are positive.
For large $L_t$ the {\it sign problem}
tends to disappear because the determinant of the one dimensional system
(\ref{4}) becomes an almost real quantity for each gauge configuration and
obviously the same happens to the determinant of the Dirac operator
(\ref{3}) in the four dimensional lattice.
It is also interesting to note that the sign of the averaged
coefficients is very stable and a different set of random configurations
produces almost the same shape.
However, the sign of the determinant is not the only problem: in fact,
as one can read from fig. 2, even considering the modulus of
the averaged coefficients we do not get the correct result.
We used the same configurations to calculate the average of the modulus of
the coefficients. We expect this quantity to be larger
than the analityc results reported in fig. 2.
The data, however, contrast with this scenario:
the averages of the modulus are always smaller (on a logarithmic scale)
than the analityc results from formula (\ref{5}).
In fact these averages are indistinguishable from the absolute values of
the numerical results reported in fig. 2.
In conclusion, even if the phase of the fermion
determinant is irrelevant in QCD at finite density ($T=0$ and heavy quarks)
the numerical evaluation of the Grand Canonical Partition Function
still suffers from sampling problems.
A last interesting feature which can be discussed on the light of our results
concerns the validity of the quenched approximation in finite density $QCD$.
An important amount of experience in this field \cite{KOGUT} suggests that
contrary to what happens in $QCD$ at finite and zero temperature, the quenched
approximation does not give correct results in $QCD$ at finite chemical
potential. Even if
the zero flavour limit of the theory with the absolute value of the
fermion determinant and of actual $QCD$ are the same (quenched approximation),
the failure of this approximation has been assigned in the past \cite{RMT}
to the fact that it corresponds to the zero flavour limit of the theory
with $n$ quarks in the fundamental and $n$ quarks in the complex
representation of
the gauge group. In fig. 3 we have plotted the number density at $\beta=0$ and
for heavy quarks in three interesting cases: actual $QCD$, the theory
with the absolute value of the fermion determinant and quenched $QCD$.
It is obvious that the quenched approximation produces results far from those
of actual $QCD$ but also far from those of $QCD$ with the modulus of the
determinant of the Dirac operator. The former results are furthermore very
near to those of actual $QCD$. In other words, even if the phase is relevant
at finite temperature, its contribution to the number density is almost
negligible.
It seems unplausible on the light of these results to assign the failure of
the quenched approximation to the feature previously discussed \cite{RMT}.
It seems more natural to speculate that it fails because does not
incorporate correctly in the path integral the Fermi-Dirac statistics and
we do expect that Pauli exclusion principle play, by far, a more relevant
role in finite density $QCD$ than in finite temperature $QCD$.
\vskip 0.3truecm
\noindent
{\bf Acknowledgements}
\vskip 0.3truecm
This work has been partially supported by CICYT and INFN.
\vskip 1 truecm
| 2024-02-18T23:40:15.841Z | 1998-11-26T11:23:30.000Z | algebraic_stack_train_0000 | 1,858 | 2,731 |
|
proofpile-arXiv_065-9117 |
\section{Introduction}
Feynman \cite{Fey82} pointed out that a Turing machine
cannot simulate a quantum mechanical process efficiently and
suggested that a computing machine
based on quantum mechanics might be more powerful than Turing machines.
Deutsch introduced quantum Turing machines \cite{Deu85} and
quantum circuits \cite{Deu89} for establishing the notion
of quantum algorithm exploiting ``quantum parallelism''.
A different approach to quantum Turing machines was taken
earlier by Benioff \cite{Beni80} based on the Hamiltonian description
of Turing machines.
Bernstein and Vazirani \cite{BV97} instituted quantum complexity
theory based on quantum Turing machines and constructed an
efficient universal quantum Turing machine.
Yao \cite{Yao93} reformulated the quantum circuit models
by singling out the acyclic ones and
showed that a computation by a quantum Turing
machine can be simulated by a polynomial size quantum circuit.
The search for an efficient quantum algorithm for a well-studied
but presumably intractable problem was achieved strikingly by
Shor \cite{Sho94}, who found bounded error probability quantum
polynomial time algorithms for the factoring problem
and the discrete logarithm problem.
In this paper, foundations of the concept of quantum Turing machines
are examined.
In Deutsch's formulation \cite{Deu85},
a quantum Turing machine is defined to be a quantum system
consisting of a processor, a moving head, and a tape, obeying a unitary
time evolution determined by local interactions between its components.
The machine is then allowed to be in a superposition of
computational configurations.
Deutsch \cite{Deu85} pointed out that the global transition function
between computational configurations should be determined by a local
transition function which depends only on local configurations.
Bernstein and Vazirani \cite{BV97} found a simple
characterization of the local transition functions for the restricted
class of quantum Turing machines in which the head must move either
to the right or to the left at each step.
Since the above characterization constitutes an alternative
definition of quantum Turing machines more tractable in the field of
theoretical computer science, it is an interesting problem to find
a general characterization valid even when the head is not required
to move or more generally when the machine has more than one tape.
The purpose of this paper is to solve this problem,
while for this and foundational purposes we also provide a completely
formal treatment of the theory of quantum Turing machines.
Extending the Bernstein-Vazirani theory \cite{BV97},
the computational complexity theory for general quantum Turing machines
defined by the conditions given in this paper
will be published in our forthcoming paper \cite{NO99}.
The paper is organized as follows.
In Section 2, quantum Turing machines are introduced along with
Deutsch's original formulation.
We extend Deutsch's formulation to the case where the head is not
required to move every step.
In Section 3, the local transition functions of quantum Turing
machines are introduced along with Deutsch's requirement of
operations by finite means and the problem of the characterization
of local transition functions is formulated.
In Section 4, quantum Turing machines are formulated as mathematical
structures and we prove a characterization theorem of the local
transition functions of quantum Turing machines.
We adopt here the column vector approach, where the characterization
is obtained from the requirement that the column vectors of the
transition matrix are orthonormal.
In Section 5, we prove an alternative characterization theorem of
the local transition functions along with the row vector approach.
In Section 6, the characterization is extended to multi-tape
quantum Turing machines.
\section{Quantum Turing machine as a physical system}
A {\em quantum Turing machine} ${\cal Q}$ is a quantum system consisting of
a {\em processor}, a bilateral infinite {\em tape},
and a {\em head} to read and write a symbol on the tape.
Its configuration is determined by the {\em processor configuration} $q$
from a finite set $Q$ of symbols,
the {\em tape configuration} $T$ represented by an infinite string from a
finite set $\Sigma$ of symbols,
and the discretized {\em head position} $\xi$ taking values in the set ${\bf Z}$
of integers.
The tape consists of {\em cells} numbered by the integers.
The head position $\xi\in{\bf Z}$ stands for the place of the cell numbered
by $\xi$.
We assume that
$\Sigma$ contains the symbol $B$ representing
the blank cell in the tape.
For any integer $m$ the symbol at the cell $m$ on
the tape is denoted by $T(m)$.
We assume that the possible tape configurations are such that
$T(m)=B$ except for finitely many cells $m$.
The set of all the possible tape configurations is denoted by
$\Sigma^{\#}$. The set $\Sigma^{\#}$ is a countable set.
Thus, any configuration $C$ of ${\cal Q}$ is represented by a triple
$C=(q,T,\xi)$ in the configuration space
${\cal C}(Q,\Sigma)=Q\times\Sigma^{\#}\times{\bf Z}$.
The quantum state of ${\cal Q}$ is represented by
a unit vector in the Hilbert
space ${\cal H}(Q,\Sigma)$ generated by the configuration space
${\cal C}(Q,\Sigma)$ so that the vectors in ${\cal H}(Q,\Sigma)$
can be identified with the square summable complex-valued functions
defined on $Q\times\Sigma^{\#}\times{\bf Z}$.
The complete orthonormal basis canonically in one-to-one correspondence
with the configuration space is called the {\em computational basis}.
Thus, the computational basis is represented by
$\ket{C}=\ket{q,T,\xi}$
for any configuration $C=(q,T,\xi)\in {\cal C}(Q,\Sigma)$.
In classical physics, physical quantities are represented by
real-valued functions defined on the phase space
coordinated by the configuration and the generalized momentum.
In quantum mechanics, they are called observables and
represented by self-adjoint operators
on the Hilbert space of quantum states.
The procedure to define the observables from the classical
description of the system is usually called the quantization.
In order to define the observables quantizing the configurations,
we assume the numbering of the sets $Q$ and $\Sigma$ such that
$Q=\{q_{0},\ldots,q_{|Q|-1}\}$ and
$\Sigma=\{\sigma_{0},\ldots,\sigma_{|\Sigma|-1}\}$,
where we denote by $|X|$ the number of the elements of a set $X$.
We define observables $\hat{q}$, $\hat{T}(m)$ for $m\in{\bf Z}$,
and $\hat{\xi}$ representing the processor configuration,
the symbol at the cell $m$, and the head position, respectively,
as follows.
$$
\hat{q}=\sum_{n=0}^{|Q|-1}n\ket{q_{n}}\bra{q_{n}},\quad
\hat{T}(m)=\sum_{n=0}^{|\Sigma|-1}n\ket{\sigma_{n}}\bra{\sigma_{n}},\quad
\hat{\xi}=\sum_{\xi\in{\bf Z}}\xi\ket{\xi}\bra{\xi}.
$$
The computation begins at $t=0$ and
proceeds in steps of a fixed unit duration $\tau$.
The dynamics of ${\cal Q}$ are described by a unitary operator
$U$ on ${\cal H}(Q,\Sigma)$ which specifies
the evolution of the system during a single {\em computational step}
so that we have
$$
U^{\dagger}U=UU^{\dagger}=I,\quad\ket{\psi(n\tau)}=U^{n}\ket{\psi(0)}
$$
for all positive integers $n$.
\section{Local transition functions}
Deutsch \cite{Deu85} required that the quantum Turing machine operates
finitely, i.e., (i) only a finite system is in motion during any one
step, (ii) the motion depends only on the quantum state of a local subsystem,
and (iii) the rule that specifies the motion can be given finitely in
the mathematical sense.
To satisfy the above requirements, the matrix elements of $U$ are
required to take the following form\footnotemark:
\footnotetext{This condition is a natural extension of Deutsch's
condition \cite{Deu85} to the case where the head is not
required to move.
}
\begin{equation}\label{eq:31}
\bra{q',T',\xi'}U\ket{q,T,\xi}=\left\{
\begin{array}{ll}
\delta(q,T(\xi),q',T'(\xi),1)& \mbox{if}\ \xi'=\xi+1\\
\delta(q,T(\xi),q',T'(\xi),0)& \mbox{if}\ \xi'=\xi\\
\delta(q,T(\xi),q',T'(\xi),-1)& \mbox{if}\ \xi'=\xi-1
\end{array}
\right.
\end{equation}
whenever $T'(m)=T(m)$ for all $m\neq \xi$,
and $\bra{q',T',\xi'}U\ket{q,T,\xi}=0$ otherwise,
for any configurations $(q,T,\xi)$ and $(q',T',\xi')$.
The above condition ensures that the tape is changed
only at the head position $\xi$ at the beginning of each computational step,
and that during each step the head position cannot change
by more than one unit. The function $\delta(q,T(\xi),q',T'(\xi),d)$,
where $q,q'\in Q$, $T(\xi),T'(\xi)\in \Sigma$,
and $d\in\{-1,0,1\}$, represents a dynamical motion depending only on
the local observables $\hat{q}$ and $\hat{T}(\xi)$.
It follows that the relation $\delta(q,\sigma,q',\tau,d)=c$ can be interpreted
as the following operation of ${\cal Q}$:
if the processor is in the configuration $q$ and
if the head reads the symbol $\sigma$,
then it follows with the amplitude $c$ that
the processor configuration turns to $q'$,
the head writes the symbol $\tau$,
and that the head moves one cell to the right if $d=1$,
to the left if $d=-1$, or does not move if $d=0$.
We call $\delta$ the {\em local transition function}
of the quantum Turing machine ${\cal Q}$.
The local transition function $\delta$ can be arbitrarily given except
for the requirement that $U$ be unitary.
Each choice defines a different quantum Turing machine ${\cal Q}[\delta]$
with the same configuration space ${\cal C}(Q,\Sigma)$.
Thus, if we have an intrinsic characterization of the local transition
function $\delta$, quantum Turing machines can be defined formally without
referring to the unitary operator $U$ as a primitive notion.
From Eq.\ (\ref{eq:31}) the time evolution operator $U$ is determined
conversely from the local transition function $\delta$ by
\begin{equation}\label{eq:32}
U\ket{q,T,\xi}
=\sum_{p,\tau,d}\delta(q,T(\xi),p,\tau,d)
\ket{p,T^{\tau}_{\xi},\xi+d}
\end{equation}
for any configuration $(q,T,\xi)$, where $T^{\tau}_{\xi}$ is the tape
configuration defined by
$$
T^{\tau}_{\xi}(m)=\left\{
\begin{array}{ll}
\tau& \mbox{if}\ m=\xi,\\
T(m)& \mbox{if}\ m\not=\xi.
\end{array}
\right.
$$
Now we can formulate the characterization problem of local transition
functions of quantum Turing machines:
{\em Let $\delta$ be a complex-valued function on
$Q\times\Sigma\times Q\times\Sigma\times\{-1,0,1\}$
and let $U$ be the operator on ${\cal H}(Q,\Sigma)$ defined by
Eq.\ (\ref{eq:32}).
Then, what conditions ensure that the operator $U$ is unitary?}
This problem is answered by the following statement:
{\em The operator $U$ is unitary if and only if $\delta$ satisfies the
following conditions.
{\rm (a)} For any $(q,\sigma)\in Q\times\Sigma$,
$$
\sum_{p,\tau,d}|\delta(q,\sigma,p,\tau,d)|^{2}=1.
$$
{\rm (b)} For any $(q,\sigma), (q',\sigma')\in Q\times\Sigma$
with $(q,\sigma)\ne (q',\sigma')$,
$$
\sum_{p,\tau,d}
\delta(q',\sigma',p,\tau,d)^{*}\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (c)} For any $(q,\sigma,\tau),(q',\sigma',\tau')
\in Q\times\Sigma^{2}$, we have
$$
\sum_{p\in Q,d=0,1}\delta(q',\sigma',p,\tau',d-1)^{*}
\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (d)} For any $(q,\sigma,\tau),(q',\sigma',\tau')
\in Q\times\Sigma^{2}$, we have
$$
\sum_{p\in Q}\delta(q',\sigma',p,\tau',-1)^{*}\delta(q,\sigma,p,\tau,1)=0.
$$
}
The proof will be given in the next section.
If it is assumed that the head must move either to the right or to
the left at each step (two-way quantum Turing machines),
the condition (c) is automatically satisfied.
In this case, the above statement is reduced to the result due to
Bernstein and Vazirani \cite{BV97}.
In Section 5, we will also characterize the local transition functions
of multi-tape quantum Turing machines.
In order to maintain the Church-Turing thesis, we need to require
that the unitary operator $U$ is constructive, or that the
range of the local transition function $\delta$ is in the computable
complex numbers.
From the complexity theoretical point of view, we need also to
require that the matrix elements of $U$ are polynomially computable
complex numbers, or that the range of the transition function $\delta$ is
in the polynomially computable complex numbers.
\section{Quantum Turing machine as a mathematical structure}
In order to formulate the notion of a quantum Turing machine as a
formal mathematical structure rather than a well-described
physical system, we shall introduce the following
mathematical definitions.
A {\em Turing frame\/} is a pair $(Q,\Sigma)$ of a finite set $Q$
and a finite set $\Sigma$ with a specific element denoted by $B$.
In what follows, let $(Q,\Sigma)$ be a Turing frame.
Let $\Sigma^{\#}$ be the set of functions $T$ from the set ${\bf Z}$ of integers
to $\Sigma$ such that $T(m)=B$ except for finitely many $m\in{\bf Z}$.
The {\em configuration space} of $(Q,\Sigma)$
is the product set ${\cal C}(Q,\Sigma)=Q\times\Sigma^{\#}\times{\bf Z}$.
For any $(p,\tau,d)\in Q\times\Sigma\times\{-1,0,1\}$, denote by
${\cal C}(p,\tau,d)$ the set of configurations
$(p,T,\xi)\in{\cal C}(Q,\Sigma)$ such that
$T(\xi-d)=\tau$.
Let $(p,\tau,d)\in Q\times\Sigma\times \{-1,0,1\}$.
We define the transformation $\alpha(p,\tau,d)$ from ${\cal C}(Q,\Sigma)$ to
${\cal C}(p,\tau,d)$ by
\begin{equation}\label{eq:41}
\alpha(p,\tau,d)(q,T,\xi)=(p,T_{\xi}^{\tau},\xi+d)
\end{equation}
for all $(q,T,\xi)\in{\cal C}(Q,\Sigma)$.
It is easy to see that $\alpha(p,\tau,d)$ represents the operation such
that the processor configuration turns to $p$,
the head writes the symbol $\tau$, and then moves with $|d|$ step to the
direction $d$.
We define the transformation $\beta(p,\tau,d)$ from ${\cal C}(Q,\Sigma)$
to ${\cal C}(p,\tau,0)$ by
\begin{equation}\label{eq:42}
\beta(p,\tau,d)(q,T,\xi)=(p,T^{\tau}_{\xi-d},\xi-d)
\end{equation}
for any $(q,T,\xi)\in{\cal C}(Q,\Sigma)$.
It is easy to see that $\beta(p,\tau,d)$ represents the operation such
that the processor configuration turns to $p$,
the head moves with $|d|$ step to the direction $-d$
and then writes the symbol $\tau$.
The following proposition can be checked by straightforward
verifications.
\begin{Proposition}\label{th:622a}
{\rm (i)} Let $d\in \{-1,0,1\}$.
If $(q,\sigma)\not=(q',\sigma')\in Q\times\Sigma$ then
${\cal C}(q,\sigma,d)\cap{\cal C}(q',\sigma',d)=\emptyset$ and
$$
{\cal C}(Q,\Sigma)=\bigcup_{(q,\sigma)\in Q\times\Sigma}{\cal C}(q,\sigma,d).
$$
{\rm (ii)} Let $(q,\sigma,p,\tau,d)\in Q\times\Sigma\times
Q\times\Sigma\times\{-1,0,1\}$.
We have
$$
\beta(q,\sigma,d)\alpha(p,\tau,d)C=C
$$
for all $C\in{\cal C}(q,\sigma,0)$ and
$$
\alpha(p,\tau,d)\beta(q,\sigma,d)C'=C'
$$
for all $C'\in{\cal C}(p,\tau,d)$.
{\rm (iii)} The mapping $\alpha(p,\tau,d)$
restricted to ${\cal C}(q,\sigma,0)$
has the inverse mapping $\beta(q,\sigma,d)$
restricted to ${\cal C}(p,\tau,d)$, i.e.,
$$
{\cal C}(q,\sigma,0)
\begin{array}{c}
\stackrel{\alpha(p,\tau,d)}{\longrightarrow}\\
\stackrel{\beta(q,\sigma,d)}{\longleftarrow}
\end{array}
{\cal C}(p,\tau,d).
$$
\end{Proposition}
\sloppy
A configuration $(q,T,\xi)$ is said to {\em precede\/} a configuration
$(q',T',\xi')$, in symbols $(q,T,\xi)\prec (q',T',\xi')$,
if $T'(m)=T(m)$ for all $m\not=\xi$
and $|\xi'-\xi|\le 1$.
\begin{Proposition}\label{th:825a}
For any $C,C'\in{\cal C}(Q,\Sigma)$, the following
conditions are equivalent.
{\rm (i)} $C\prec C'$.
{\rm (ii)} There is some $(p,\tau,d)\in Q\times\Sigma\times\{-1,0,1\}$
such that $C'=\alpha(p,\tau,d)C$.
{\rm (iii)} There is some $(q,\sigma,d)\in Q\times\Sigma\times\{-1,0,1\}$
such that $C=\beta(q,\sigma,d)C'$.
\end{Proposition}
\begin{Proof}
Let $C=(q,T,\xi)$ and $C'=(q',T',\xi')$.
(i)$\Rightarrow$(ii): If (i) holds,
we have $C'=\alpha(q',T'(\xi),\xi'-\xi)C$
so that (ii) holds.
(ii)$\Rightarrow$(iii): Suppose that (ii) holds.
Since $C\in{\cal C}(q,T(\xi),0)$, by Proposition \ref{th:622a} (ii)
we have
$$
\beta(q,T(\xi),d)C'=
\beta(q,T(\xi),d)\alpha(p,\tau,d)C=C.
$$
(iii)$\Rightarrow$(i): If (iii) holds, we have
$C=(p,T'{}_{\xi'-d}^{\sigma},\xi'-d)$ and hence $\xi'-\xi=d$
and $T(m)=T'{}_{\xi'-d}^{\sigma}(m)=T'(m)$ for $m\not=\xi'-d=\xi$
so that (i) holds.
\end{Proof}
The {\em quantum state space} of the Turing frame $(Q,\Sigma)$ is
the Hilbert space ${\cal H}(Q,\Sigma)$ spanned by ${\cal C}(Q,\Sigma)$ with
the canonical basis $\{\ket{C}|\ C\in{\cal C}(Q,\Sigma)\}$ called the
{\em computational basis}.
A {\em local transition function} for
$(Q,\Sigma)$ is a function from
$Q\times\Sigma\times Q\times\Sigma\times \{-1,0,1\}$
into the complex number field ${\bf C}$.
In what follows, let $\delta$ be a local transition function for $(Q,\Sigma)$.
The {\em evolution operator} of $\delta$ is a linear operator $M_{\delta}$ on
${\cal H}(Q,\Sigma)$ such that
\begin{equation}\label{eq:43}
M_{\delta}\ket{q,T,\xi}=
\sum_{p,\tau,d}\delta(q,T(\xi),p,\tau,d)\ket{p,T^{\tau}_{\xi},\xi+d}
\end{equation}
for all $(q,T,\xi)\in{\cal C}(Q,\Sigma)$; the summation $\sum_{p,\tau,d}$
is taken over all $(p,\tau,d)\in Q\times\Sigma\times \{-1,0,1\}$ above and
in the rest of this section unless stated otherwise.
Eq.\ (\ref{eq:43}) uniquely defines the bounded operator $M_{\delta}$
on the space ${\cal H}(Q,\Sigma)$ as shown in Appendix A.
Let $(q,T,\xi),\ (q',T',\xi')\in{\cal C}(Q,\Sigma)$.
The following formula can be verified from Eq.\ (\ref{eq:43})
by straightforward calculation.
\begin{equation}\label{eq:44}
\bra{q',T',\xi'}M_{\delta}\ket{q,T,\xi}
=
\left\{
\begin{array}{ll}
\delta(q,T(\xi),q',T'(\xi),\xi'-\xi)&\quad
\mbox{if $(q,T,\xi)\prec (q',T',\xi')$,}\\
0 &\quad\mbox{otherwise.}
\end{array}
\right.
\end{equation}
A configuration $(q,T,\xi)$ is said to be {\em locally like\/}
a configuration $(q',T',\xi')$ if $q=q'$ and
$T(\xi+d)=T'(\xi'+d)$ for all $d\in\{-1,0,1\}$.
\begin{Lemma}\label{th:825b}
For any $C_{1},C_{2}\in{\cal C}(Q,\Sigma)$,
if they are locally like each other,
we have
$$
\bra{C_{1}}M_{\delta}M_{\delta}^{\dagger}\ket{C_{1}}=
\bra{C_{2}}M_{\delta}M_{\delta}^{\dagger}\ket{C_{2}}.
$$
\end{Lemma}\label{th:622b}
\begin{Proof}
Let $\tau_{-1},\tau_{0},\tau_{1}\in \Sigma$.
Suppose that a configuration $C'=(p,T',\xi')$ is such that
$T'(\xi'-d)=\tau_{d}$ for all $d\in\{-1,0,1\}$.
Since every configuration locally like $C'$ also satisfies the above
condition, it suffices to show that
$\bra{C'}M_{\delta}M_{\delta}^{\dagger}\ket{C'}$
depends only on $p,\tau_{-1},\tau_{0},\tau_{1}$.
By Proposition \ref{th:825a} and Eq.\ (\ref{eq:44}) we have
\begin{eqnarray*}
\bra{C'}M_{\delta}M_{\delta}^{\dagger}\ket{C'}
&=&
\sum_{C\in{\cal C}(Q,\Sigma)}|\bra{C'}M_{\delta}\ket{C}|^{2}\\
&=&
\sum_{C\prec C'}|\bra{C'}M_{\delta}\ket{C}|^{2}\\
&=&
\sum_{q,\sigma,d}
|\bra{C'}M_{\delta}\ket{\beta(q,\sigma,d)C'}|^{2}\\
&=&
\sum_{q,\sigma,d}
|\bra{p,T',\xi'}M_{\delta}\ket{q,{T'}^{\sigma}_{\xi'-d},\xi'-d}|^{2}\\
&=&
\sum_{q,\sigma,d}
|\delta(q,{T'}^{\sigma}_{\xi'-d}(\xi'-d),p,T'(\xi'-d),d)|^{2}\\
&=&
\sum_{q,\sigma,d}
|\delta(q,\sigma,p,\tau_{d},d)|^{2}.
\end{eqnarray*}
The first equality above follows from Parseval's identity.
Thus, $\bra{C'}M_{\delta}M_{\delta}^{\dagger}\ket{C'}$
depends only on $p,\tau_{-1},\tau_{0},\tau_{1}$ and the proof is
completed.
\end{Proof}
For the case where the head is required to move,
a proof of the following lemma appeared first in \cite{BV97}.
The following proof not only covers the general case but also
simplifies the argument given in \cite{BV97}.
\begin{Lemma}\label{th:518a}
The evolution operator $M_{\delta}$ of a local transition function $\delta$
is unitary if it is an isometry.
\end{Lemma}
\begin{Proof}
Suppose that $M_{\delta}$ is an isometry,
i.e., $M_{\delta}^{\dagger}M_{\delta}=1$.
Obviously, $M_{\delta}M_{\delta}^{\dagger}$ is a projection.
If $\bra{C}M_{\delta}M_{\delta}^{\dagger}\ket{C}=1$ for every
$C\in{\cal C}(Q,\Sigma)$, the computational basis
is included in the range of $M_{\delta}M_{\delta}^{\dagger}$
and then, since the range of any projection is a closed linear subspace,
we have $M_{\delta}M_{\delta}^{\dagger}=1$ so that $M_{\delta}$ is unitary.
Thus, it suffices to show that
$\bra{C}M_{\delta}M_{\delta}^{\dagger}\ket{C}=1$
for every $C\in{\cal C}(Q,\Sigma)$.
To show this, suppose that there is a configuration
$C_{0}\in{\cal C}(Q,\Sigma)$
such that $\bra{C_{0}}M_{\delta}M_{\delta}^{\dagger}\ket{C_{0}}
=1-\epsilon$ with $\epsilon>0$.
For any $n>2$ and $d\in\{-1,0,1\}$,
let $S(n,d)$ be the set of configurations such that
\begin{eqnarray*}
S(n,d)
=\{(q,T,\xi)\in{\cal C}(Q,\Sigma)|\ T(m)=B\!\!\!
&\mbox{for}&\!\!\!\!\mbox{all}\ \ m\not\in\{1,\ldots,n\}\\
&\mbox{and}&\!\!\!\xi\in\{1-d,\ldots,n+d\} \}.
\end{eqnarray*}
Let
\begin{equation}\label{eq:45}
A=\sum_{(C,C')\in S(n,0)\times S(n,1)}|\bra{C'}M_{\delta}\ket{C}|^{2}
\end{equation}
and we shall consider evaluations of $A$ in terms of the numbers
of elements of the sets $S(n,0)$ and $S(n,1)$.
It is easy to see that if $C\in S(n,0)$ and $C\prec C'$ then
$C'\in S(n,1)$.
It follows from Eq.\ (\ref{eq:44})
that $\bra{C'}M_{\delta}\ket{C}=0$ for any pair
$(C,C')$ with $C\in S(n,0)$ and $C'\not\in S(n,1)$ so that
the summation over $(C,C')\in S(n,0)\times S(n,1)$ in Eq.\ (\ref{eq:45})
can be replaced by the summation over
$(C,C')\in S(n,0)\times {\cal C}(Q,\Sigma)$.
By Parseval's identity, we have
$$
A=\sum_{(C,C')\in S(n,0)\times {\cal C}(Q,\Sigma)}
|\bra{C'}M_{\delta}\ket{C}|^{2}
=\sum_{C\in S(n,0)}\bra{C}M_{\delta}^{\dagger}M_{\delta}{\ket{C}}.
$$
Since $M_{\delta}$ is an isometry, we have
$$
A=|S(n,0)|.
$$
Let $S(C_{0})$ be the set of all configurations in $S(n,-1)$
locally like $C_{0}$. Then, $S(C_{0})\subseteq S(n,1)$.
By Lemma \ref{th:825b},
$\bra{C'}M_{\delta}M_{\delta}^{\dagger}\ket{C'}=1-\epsilon$
for all $C'\in S(C_{0})$.
Thus, we have
\begin{eqnarray*}
A
&\le&\sum_{(C,C')\in {\cal C}(Q,\Sigma)\times S(n,1)}
|\bra{C'}M_{\delta}\ket{C}|^{2}\\
&=&\sum_{C'\in S(n,1)}\bra{C'}M_{\delta}M_{\delta}^{\dagger}\ket{C'}\\
&\le& (1-\epsilon)|S(C_{0})|+|S(n,1)|-|S(C_{0})|\\
&=&|S(n,1)|-\epsilon|S(C_{0})|.
\end{eqnarray*}
The cardinalities of $S(n,d)$ and $S(C_{0})$ are given by
$|S(n,d)|=(n+2d)|Q|\,|\Sigma|^{n}$ and
$|S(C_{0})|=(n-2)|\Sigma|^{n-3}$.
Therefore, we have
$$
|\Sigma|^{n-3}(2|Q||\Sigma|^{3}-\epsilon(n-2))
=|S(n,1)|-\epsilon|S(C_{0})|-|S(n,0)|
\ge 0
$$
for all $n>2$.
But, for $n>2+2\epsilon^{-1}|Q||\Sigma|^{3}$,
this yields an obvious contradiction and the proof is completed.
\end{Proof}
According to discussions in Section 3, a quantum Turing machine
can be defined as a mathematical structure $(Q,\Sigma,\delta)$ consisting of
a Turing frame $(Q,\Sigma)$ and a local transition function $\delta$
such that the evolution operator $M_{\delta}$ is unitary.
The following theorem characterizes intrinsically the local transition
functions that give rise to quantum Turing machines.
\begin{Theorem}\label{th:518b}
The evolution operator $M_{\delta}$ of a local transition function $\delta$
for the Turing frame $(Q,\Sigma)$ is unitary if and only if $\delta$
satisfies the following conditions.
{\rm (a)} For any $(q,\sigma)\in Q\times\Sigma$,
$$
\sum_{p,\tau,d}|\delta(q,\sigma,p,\tau,d)|^{2}=1.
$$
{\rm (b)} For any $(q,\sigma), (q',\sigma')\in Q\times\Sigma$
with $(q,\sigma)\ne (q',\sigma')$,
$$
\sum_{p,\tau,d}
\delta(q',\sigma',p,\tau,d)^{*}\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (c)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times\Sigma^{2}$,
we have
$$
\sum_{p\in Q,d=0,1}\delta(q',\sigma',p,\tau',d-1)^{*}
\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (d)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times\Sigma^{2}$,
we have
$$
\sum_{p\in Q}\delta(q',\sigma',p,\tau',-1)^{*}\delta(q,\sigma,p,\tau,1)=0.
$$
\end{Theorem}
\begin{Proof}
Let $\delta$ be a local transition function for a Turing frame $(Q,\Sigma)$.
Let $C=(q,T,\xi)\in {\cal C}(Q,\Sigma)$.
From Eq.\ (\ref{eq:43}) we have
\begin{eqnarray*}
\lefteqn{\bra{C}M_{\delta}^{\dagger}M_{\delta}\ket{C}}\quad\\
&=&
\sum_{p,\tau,d}\sum_{p',\tau',d'}
\delta(q,T(\xi),p',\tau',d')^{*}\delta(q,T(\xi),p,\tau,d)
\braket{p',T^{\tau'}_{\xi},\xi+d'|p,T^{\tau}_{\xi},\xi+d}\\
&=&
\sum_{p,\tau,d}|\delta(q,T(\xi),p,\tau,d)|^{2}.
\end{eqnarray*}
Since for any $\sigma\in\Sigma$
there are some $T\in\Sigma^{\#}$ and $\xi\in{\bf Z}$
such that $T(\xi)=\sigma$, condition (a) holds
if and only if $\bra{C}M_{\delta}^{\dagger}M_{\delta}\ket{C}=1$ for any
$C\in{\cal C}(Q,\Sigma)$.
Let $C=(q,T,\xi)\in{\cal C}(Q,\Sigma)$ and
$C'=(q',T',\xi')\in{\cal C}(Q,\Sigma)$.
From Eq.\ (\ref{eq:43}) we have
\begin{eqnarray*}
\lefteqn{\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}}\quad\\
&=&\!\!\!\!\!
\sum_{p,\tau,d}\sum_{p',\tau',d'}
\delta(q',T'(\xi'),p',\tau',d')^{*}\delta(q,T(\xi),p,\tau,d)
\braket{p',{T'}^{\tau'}_{\xi'},\xi'+d'|p,T^{\tau}_{\xi},\xi+d}\\
&=&\!\!\!\!\!
\sum{}^{*}
\delta(q',T'(\xi'),p,\tau',d')^{*}\delta(q,T(\xi),p,\tau,d),
\end{eqnarray*}
where the summation $\sum^{*}$ is taken over all $p\in Q$,
$\tau,\tau'\in\Sigma$, and $d,d'\in\{-1,0,1\}$ such that
$T^{\tau}_{\xi}={T'}^{\tau'}_{\xi'}$ and $\xi+d=\xi'+d'$.
For any $k\in{\bf Z}$,
let ${\cal C}(k)$ be a subset of ${\cal C}(Q,\Sigma)^{2}$ consisting of all
pairs $C=(q,T,\xi)$ and $C'=(q',T',\xi')$ with $C\not=C'$
such that $T(m)=T'(m)$ for all $m\not\in\{\xi,\xi'\}$ and that
$\xi'-\xi=k$.
It is easy to see that if $C\neq C'$ and
$$
(C,C')\not\in\bigcup_{k\in\{0,\pm 1,\pm 2\}}{\cal C}(k)
$$
then $\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}=0$.
We shall show that condition (b), (c), or (d) holds if and only if
$\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}=0$ holds
for all $(C,C')\in{\cal C}(0)$,
$(C,C')\in{\cal C}(1)$, or $(C,C')\in{\cal C}(2)$, respectively.
For any $(C,C')\in{\cal C}(0)$ with $C=(q,T,\xi)$ and $C'=(q',T',\xi')$,
we have $T^{\tau}_{\xi}={T'}^{\ \tau'}_{\xi'}$ and $\xi+d=\xi'+d'$
if and only if $\tau=\tau'$ and $d=d'$,
so that we have
\begin{eqnarray*}
\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}
&=&\sum_{p,\tau,d}\delta(q',T'(\xi'),p,\tau,d)^{*}\delta(q,T(\xi),p,\tau,d).
\end{eqnarray*}
Since for any $(q,\sigma),(q',\sigma')\in Q\times \Sigma$ with
$(q,\sigma)\not=(q',\sigma')$
there are configurations
$C=(q,T,\xi)$ and $C'=(q',T',\xi')$ such that
$(C,C')\in {\cal C}(0)$, $T(\xi)=\sigma$ and $T'(\xi')=\sigma'$,
condition (b) holds
if and only if $\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}=0$ for all
$(C,C')\in{\cal C}(0)$.
For any $(C,C')\in{\cal C}(1)$ with $C=(q,T,\xi)$ and $C'=(q',T',\xi')$,
we have $T^{\tau}_{\xi}={T'}^{\tau'}_{\xi'}$ and
$\xi+d=\xi'+d'$
if and only if $\tau=T'(\xi)$, $\tau'=T(\xi')$,
and $(d,d')\in\{(0,-1),(1,0)\}$, so that we have
\begin{eqnarray*}
\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}
&=&\sum_{p\in Q,d=0,1}\delta(q',T'(\xi'),p,T(\xi'),d-1)^{*}
\delta(q,T(\xi),p,T'(\xi),d).
\end{eqnarray*}
Since for any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^{2}$
there are configurations $C=(q,T,\xi)$ and $C'=(q',T',\xi')$ such that
$C,C'\in{\cal C}(1)$,
$(T(\xi),T'(\xi))=(\sigma,\tau)$, and
$(T'(\xi'),T(\xi'))=(\sigma',\tau')$,
condition (c) holds if and only if
$\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}=0$ for any
$(C,C')\in{\cal C}(1)$.
For any $(C,C')\in{\cal C}(2)$ with $C=(q,T,\xi)$ and $C'=(q',T',\xi')$,
we have $T^{\tau}_{\xi}={T'}^{\tau'}_{\xi'}$ and
$\xi+d=\xi'+d'$
if and only if $\tau=T'(\xi)$, $\tau'=T(\xi')$, $d=1$,
and $d'=-1$, so that we have
\begin{eqnarray*}
\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}
&=&\sum_{p\in Q}\delta(q',T'(\xi'),p,T(\xi'),-1)^{*}
\delta(q,T(\xi),p,T'(\xi),1).
\end{eqnarray*}
Thus,
condition (d) holds if and only if
$\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}=0$ for all
$(C,C')\in{\cal C}(2)$.
Since $M_{\delta}^{\dagger}M_{\delta}$ is self-adjoint,
$M_{\delta}$ is an isometry if and only if $
\bra{C'}M_{\delta}^{\dagger}M_{\delta}\ket{C}$ $=$ $\langle C'|C\rangle$
for any $C=(q,T,\xi),\ C'=(q',T',\xi')\in{\cal C}(Q,\Sigma)$
with $\xi\le \xi'$.
Therefore, we have proved that conditions (a)--(d) hold if and
only if $M_{\delta}$ is an isometry.
Now, Lemma \ref{th:518a} concludes the assertion.
\end{Proof}
A quantum Turing machine $M=(Q,\Sigma,\delta)$ is called {\em unidirectional},
if we have $d=d'$
whenever $\delta(q,\sigma,p,\tau,d)\delta(q',\sigma',p,\tau',d')
\not=0$ for any $q,q'\in Q$, $\sigma,\sigma', \tau,\tau'\in\Sigma$,
and $d,d'\in \{-1,0,1\}$.
It is easy to see that conditions (c) and (d) are automatically satisfied
by every unidirectional quantum Turing machine.
Thus, if every quantum Turing machine can be efficiently simulated
by a unidirectional one without error, complexity theoretical
consideration on quantum Turing machines can be done much easier.
For two-way quantum Turing machines, this was shown by
Bernstein and Vazirani \cite{BV97}. For general quantum Turing
machines defined by the above conditions, the positive answer
will be given in our forthcoming paper \cite{NO99},
including extension to multi-tape quantum Turing machines
defined by the conditions of Theorem 6.2.
\section{Alternative approaches to the characterization of
local transition functions}
Hirvensalo \cite{Hir97} gave
the following set of conditions for a local transition function
$\delta$ to give the unitary evolution operator (see also \cite{Gru99}):
{\rm (H-a)} For any $(q,\sigma)\in Q\times\Sigma$,
$$
\sum_{p,\tau,d}|\delta(q,\sigma,p,\tau,d)|^{2}=1.
$$
{\rm (H-b)} For any $(q,\sigma), (q',\sigma')\in Q\times\Sigma$
with $(q,\sigma)\ne (q',\sigma')$,
$$
\sum_{p,\tau,d}
\delta(q',\sigma',p,\tau,d)^{*}\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (H-c)} For any $(p,\tau,d),(p',\tau',d')\in Q\times\Sigma\times\{-1,0,1\}$
with $(p,\tau,d)\ne (p',\tau',d')$, we have
$$
\sum_{(q,\sigma)\in Q\times\Sigma}\delta(q,\sigma,p,\tau,d)^{*}
\delta(q,\sigma,p',\tau',d')=0.
$$
{\rm (H-d)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times\Sigma^{2}$
and $d\neq d'\in\{-1,0,1\}$, we have
$$
\sum_{p\in Q}\delta(q,\sigma,p,\tau,d)^{*}\delta(q',\sigma',p,\tau',d')=0.
$$
However, the above set of conditions consists of only a sufficient
condition, not a necessary one. To show this, let $Q=\{0,1\}$,
$\Sigma=\{B\}$, and define a local transition function $\delta$ as follows.
$$
\begin{array}{lll}
\delta(0,B,0,B,-1)=0,& \delta(0,B,0,B,0)=1/2,& \delta(0,B,0,B,1)=-1/2,\\
\delta(0,B,1,B,-1)=1/2,& \delta(0,B,1,B,0)=1/2,& \delta(0,B,1,B,1)=0,\\
\delta(1,B,0,B,-1)=0,& \delta(1,B,0,B,0)=1/2,& \delta(1,B,0,B,1)=1/2,\\
\delta(1,B,1,B,-1)=1/2,& \delta(1,B,1,B,0)=-1/2,& \delta(1,B,1,B,1)=0.
\end{array}
$$
Then, $\delta$ satisfies conditions (a)--(d) of Theorem \ref{th:518b}
and hence gives the unitary evolution operator, but does not satisfy
Hirvensalo's conditions. In fact, $\delta$ does not satisfy condition (H-c),
since
$$
\delta(0,B,0,B,0)^{*}\delta(0,B,1,B,-1)
+\delta(1,B,0,B,0)^{*}\delta(1,B,1,B,-1)
=1/2,
$$
and $\delta$ does not satisfy condition (H-d), since
$$
\delta(0,B,0,B,0)^{*}\delta(1,B,0,B,1)
+\delta(0,B,1,B,0)^{*}\delta(1,B,1,B,1)=1/4.
$$
Thus, conditions (H-d) and (H-c) are not necessary.
The conditions in Theorem \ref{th:518a} are obtained from the requirement
that the column vectors of the evolution operator are orthonormal in
the matrix representation in the computational basis.
Hirvensalo's conditions mix requirements for column vectors and
for row vectors. In the rest of this section,
from the sole requirement that the row vectors are orthonormal,
we shall obtain a set of necessary and
sufficient conditions for the unitarity of the evolution operator.
The proof of the following lemma is similar to that of Lemma \ref{th:518a}.
\begin{Lemma}\label{th:4}
The evolution operator $M_\delta$ of a local transition function $\delta$
is unitary if $M_\delta M_\delta^{\dagger}=1$.
\end{Lemma}
Now we give another characterization of the local transition functions
that give rise to quantum Turing machines.
\begin{Theorem}\label{th:5}
The evolution operator $M_\delta$ of a local transition function $\delta$
for the Turing frame $(Q,\Sigma)$ is unitary
if and only if $\delta$ satisfies the following conditions.
{\rm (a)} For any $p\in Q$ and $\tau_{-1},\tau_0,\tau_1\in\Sigma$,
$$
\sum_{q\in Q,\sigma\in\Sigma,d\in\{-1,0,1\}}|\delta(q,\sigma,p,\tau_d,d)|^2=1.
$$
{\rm (b)} For any $p,p'\in Q$ with $p\neq p'$
and any $\tau_{-1},\tau_0,\tau_1\in\Sigma$,
$$
\sum_{q\in Q,\sigma\in\Sigma,d\in\{-1,0,1\}}\delta(q,\sigma,p',\tau_d,d)^*
\delta(q,\sigma,p,\tau_d,d)=0.
$$
{\rm (c)} For any $p,p'\in Q$ and $\tau_0,\tau_1\in\Sigma$,
$$
\sum_{q\in Q,\sigma\in\Sigma,d=0,1}\delta(q,\sigma,p',\tau_d,d-1)^*
\delta(q,\sigma,p,\tau_d,d)=0.
$$
{\rm (d)} For any $(p,\tau),(p',\tau')\in Q\times \Sigma$
with $\tau\neq\tau'$ and any $d\in\{-1,0,1\}$, we have
$$
\sum_{q\in Q,\sigma\in\Sigma}
\delta(q,\sigma,p',\tau',d)^{*}\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (e)} For any $(p,\tau),(p',\tau')\in Q\times \Sigma$
with $\tau\neq\tau'$ and any $d=0,1$, we have
$$
\sum_{q\in Q,\sigma\in\Sigma}
\delta(q,\sigma,p',\tau',d-1)^{*}\delta(q,\sigma,p,\tau,d)=0.
$$
{\rm (f)} For any $(p,\tau),(p',\tau')\in Q\times\Sigma$, we have
$$
\sum_{q\in Q,\sigma\in\Sigma}
\delta(q,\sigma,p',\tau',-1)^*\delta(q,\sigma,p,\tau,1)=0.
$$
\end{Theorem}
\begin{Proof} Let $\delta$ be a local transition function for a Turing frame
$(Q,\Sigma)$. Let $C=(p,T,\xi)\in{\cal C}(Q,\Sigma)$.
From Proposition \ref{th:825a} and Eq.\ (\ref{eq:44}), we have
\begin{eqnarray}\label{eq:51}
M_\delta^{\dagger}|p,T,\xi\rangle &=&
\sum_{C'\in{\cal C}(Q,\Sigma)}|C'\rangle
\langle C'|M_\delta^{\dagger}|C\rangle\nonumber\\
&=& \sum_{C':C'\prec C}{\langle C|M_\delta|C'\rangle}^*|C'\rangle\nonumber\\
&=& \sum_{q,\sigma,d}{\langle C|M_\delta|\beta(q,\sigma,d)C\rangle}^*
\ket{\beta(q,\sigma,d)C}\nonumber\\
&=& \sum_{q,\sigma,d}
{\langle p,T,\xi|M_\delta|q,T_{\xi-d}^{\sigma},\xi-d\rangle}^*
\ket{q,T_{\xi-d}^{\sigma},\xi-d}\nonumber\\
&=& \sum_{q,\sigma,d}\delta(q,T_{\xi-d}^\sigma(\xi-d),p,T(\xi-d),d)^*
\ket{q,T_{\xi-d}^{\sigma},\xi-d}\nonumber\\
&=& \sum_{q,\sigma,d}\delta(q,\sigma,p,T(\xi-d),d)^*
\ket{q,T_{\xi-d}^\sigma,\xi-d}.
\end{eqnarray}
From Eq.\ (\ref{eq:51}) we have
\begin{eqnarray*}
\langle C|M_\delta M_\delta^{\dagger}|C\rangle
&=& \sum_{q,\sigma,d}\sum_{q',\sigma',d'}
\delta(q,\sigma,p,T(\xi-d),d)^*
\delta(q',\sigma',p,T(\xi-d'),d')\\
& & \times \langle q',T_{\xi-d'}^{\sigma'},\xi-d'
\ket{q,{T}_{\xi-d}^{\sigma},\xi-d}
\\
&=& \sum_{q,\sigma,d}|\delta(q,\sigma,p,T(\xi-d),d)|^2.
\end{eqnarray*}
Since for any $\tau_{-1},\tau_0,\tau_1\in\Sigma$ there are
some $T\in\Sigma^\#$ and $\xi\in{\bf Z}$ such that $T(\xi-d)=\tau_d$,
condition (a) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C\rangle=1$
for any $C\in{\cal C}(Q,\Sigma)$.
Let $C=(p,T,\xi)\in{\cal C}(Q,\Sigma)$ and
$C'=(p',T',\xi')\in{\cal C}(Q,\Sigma)$.
From Eq.\ (\ref{eq:51}) we have
\begin{eqnarray*}
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle
&=& \sum_{q,\sigma,d}\sum_{q',\sigma',d'}
\delta(q',\sigma',p',T'(\xi'-d'),d')^*
\delta(q,\sigma,p,T(\xi-d),d)\\
& & \times
\langle q,T_{\xi-d}^{\sigma},\xi-d
|q',{T'}_{\xi'-d'}^{\sigma'},\xi'-d'\rangle
\\
&=& \sum{}^{**}\delta(q,\sigma,p',T'(\xi'-d'),d')^*
\delta(q,\sigma,p,T(\xi-d),d),
\end{eqnarray*}
where the summation $\sum^{**}$ is taken over all $q\in Q$, $\sigma\in\Sigma$,
and $d,d'\in\{-1,0,1\}$
such that $T_{\xi-d}^{\sigma}={T'}_{\xi'-d'}^{\sigma}$ and $\xi-d=\xi'-d'$.
For any $k\in{\bf Z}$ and $d\in\{-1,0,1\}$, let
$A(k)$ be a subset of ${\cal C}(Q,\Sigma)^2$
consisting of all pairs $C=(p,T,\xi)$
and $C'=(p',T',\xi')$ with $C\neq C'$, $T=T'$ and $\xi-\xi'=k$, and
$B(k,d)$ be a subset of
${\cal C}(Q,\Sigma)^2$ consisting of all pairs $C=(p,T,\xi)$ and
$C'=(p',T',\xi')$ with $T\neq T'$ and $\xi-\xi'=k$
such that $T(m)=T'(m)$ for all $m\neq\xi-d$.
It is easy to see that if $C\neq C'$ and
$$
(C,C')\not\in\left(\bigcup_{k\in\{0,\pm 1,\pm 2\}}A(k)\right)\cup\left(
\bigcup_{(k,d):|k-d|\le 1}B(k,d)\right)
$$
then $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$.
Let $B(0)=B(0,1)\cup B(0,0)\cup B(0,-1)$ and $B(1)=B(1,1)\cup B(1,0)$.
We shall show that condition (b), (c), (d), (e) or (f) holds
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in A(0)$,\ $(C,C')\in A(1)$,\ $(C,C')\in B(0)$,\
$(C,C')\in B(1)$, or $(C,C')\in A(2)\cup B(2,1)$,
respectively.
For any $(C,C')\in A(0)$ with $C=(p,T,\xi)$ and $C'=(p',T,\xi)$,
we have $T_{\xi-d}^\sigma={T}_{\xi-d'}^{\sigma}$ and $\xi-d=\xi-d'$
if and only if $d=d'$, so that we have
$$
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=\sum_{q,\sigma,d}
\delta(q,\sigma,p',T(\xi-d),d)^*\delta(q,\sigma,p,T(\xi-d),d).
$$
Since for any $p,p'\in Q$ with $p\neq p'$ and
any $\tau_{-1},\tau_0,\tau_1\in\Sigma$
there are configurations $C=(p,T,\xi)$ and $C'=(p',T,\xi)$
such that $(C,C')\in A(0)$ and $T(\xi-d)=\tau_d$ for all $d\in\{-1,0,1\}$,
condition (b) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in A(0)$.
For any $(C,C')\in A(1)$ with $C=(p,T,\xi)$ and $C'=(p',T,\xi')$,
we have $T_{\xi-d}^{\sigma}={T}_{\xi'-d'}^{\sigma}$ and $\xi-d=\xi'-d'$
if and only if $(d,d')\in\{(1,0),(0,-1)\}$, so that we have
$$
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=
\sum_{q\in Q,\sigma\in\Sigma,d=0,1}\delta(q,\sigma,p',T(\xi-d),d-1)^*
\delta(q,\sigma,p,T(\xi-d),d).
$$
Since for any $p,p'\in Q$ and $\tau_0,\tau_1\in\Sigma$
there are configurations $C=(p,T,\xi)$ and $C'=(p',T,\xi')$
such that $(C,C')\in A(1)$ and $T(\xi-d)=\tau_d$ for all $d\in\{0,1\}$,
condition (c) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in A(1)$.
For any $(C,C')\in B(0,1)$ with $C=(p,T,\xi)$ and $C'=(p',T',\xi)$,
we have $T_{\xi-d}^{\sigma}={T'}_{\xi-d'}^{\sigma}$ and $\xi-d=\xi-d'$
if and only if $d=d'=1$, because $T(\xi-1)\neq T'(\xi-1)$
and $T_{\xi-d}^\sigma(\xi-1)={T'}_{\xi-d'}^{\sigma}(\xi-1)$. Thus we have
$$
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=
\sum_{q\in Q,\sigma\in\Sigma}\delta(q,\sigma,p',T'(\xi-1),1)^*
\delta(q,\sigma,p,T(\xi-1),1).
$$
Since for any $(p,\tau),(p',\tau')\in Q\times\Sigma$ with $\tau\neq\tau'$
there are configurations $C=(p,T,\xi)$ and $C'=(p',T',\xi')$
such that $(C,C')\in B(0,1),T(\xi-1)=\tau$, and $T'(\xi-1)=\tau'$,
the case $d=1$ of condition (d) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(0,1)$.
Similarly we can show the case $d=0$ or $d=-1$ of condition (d) holds
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(0,0)$ or $B(0,-1)$.
Thus condition (d) holds
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(0)$.
For any $(C,C')\in B(1,1)$ with $C=(p,T,\xi)$ and $C'=(p',T',\xi')$,
we have $T_{\xi-d}^{\sigma}={T'}_{\xi'-d'}^{\sigma}$ and $\xi-d=\xi'-d'$
if and only if $d=1$ and $d'=0$,
because $T(\xi-1)\neq T'(\xi-1)$ and $T_{\xi-d}^\sigma(\xi-1)
={T'}_{\xi'-d'}^{\sigma}(\xi-1)$. Thus we have
$$
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=
\sum_{q\in Q,\sigma\in\Sigma}\delta(q,\sigma,p',T'(\xi-1),0)^*
\delta(q,\sigma,p,T(\xi-1),1).
$$
Since for any $(p,\tau),(p',\tau')\in Q\times\Sigma$ with $\tau\neq\tau'$
there are configurations $C=(p,T,\xi)$ and $C'=(p',T',\xi')$
such that $(C,C')\in B(1,1),T(\xi-1)=\tau$, and $T'(\xi-1)=\tau'$,
the case $d=1$ of condition (e) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(1,1)$.
Similarly we can show the case $d=0$ of condition (e) holds
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(1,0)$. Thus condition (e) holds
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in B(1)$.
For any $(C,C')\in A(2)\cup B(2,1)$ with $C=(p,T,\xi)$ and $C'=(p',T',\xi')$,
we have $T_{\xi-d}^{\sigma}={T'}_{\xi'-d'}^{\sigma}$ and $\xi-d=\xi'-d'$
if and only if $d=1$ and $d'=-1$, so that we have
$$
\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=
\sum_{q\in Q,\sigma\in\Sigma}\delta(q,\sigma,p',T'(\xi-1),-1)^*
\delta(q,\sigma,p,T(\xi-1),1).
$$
Since for any $(p,\tau),(p',\tau')\in Q\times\Sigma$
there are configurations $C=(p,T,\xi)$ and $C'=(p',T',\xi')$
such that $(C,C')\in A(2)\cup B(2,1),T(\xi-1)=\tau$, and $T'(\xi-1)=\tau'$,
condition (f) holds if and only if
$\langle C|M_\delta M_\delta^{\dagger}|C'\rangle=0$ holds
for all $(C,C')\in A(2)\cup B(2,1)$.
Since $M_\delta M_\delta^{\dagger}$ is self-adjoint,
$M_\delta M_\delta^{\dagger}=1$
if and only if $\langle C|M_\delta M_\delta^{\dagger}|C'\rangle$
$=$ $\langle C|C'\rangle$
for any $C=(p,T,\xi),\ C'=(p',T',\xi')\in{\cal C}(Q,\Sigma)$
with $\xi'\le\xi$.
Therefore, we have proved that conditions (a)--(f) hold if and only if
$M_\delta M_\delta^{\dagger}=1$.
Now, Lemma \ref{th:4} concludes the assertion.
\end{Proof}
\section{Multi-tape quantum Turing machines}
In the preceding sections, we have discussed solely single tape
quantum Turing machines, but our arguments can be adapted easily to
multi-tape quantum Turing machines, which are quantum analogues of
multi-tape deterministic Turing machines.
First, we explain how to adapt our arguments to
multi-tape quantum Turing machines
by considering two-tape quantum Turing machines.
A two-tape quantum Turing machine is a quantum system
consisting of a processor, two bilateral infinite tapes with
heads to read and write symbols on their tapes.
In order to discuss local transition functions,
we adapt the formal definitions as follows.
Let $(Q,\Sigma_{1},\Sigma_{2})$ be a triple, called a
{\em {two-tape} Turing frame},
consisting of a finite sets $Q$, $\Sigma_{1}$, and $\Sigma_{2}$
with specific elements $B_{1}\in\Sigma_{1}$ and $B_{2}\in\Sigma_{2}$.
The {\em configuration space} of $(Q,\Sigma_{1},\Sigma_{2})$ is the product
set
${\cal C}(Q,\Sigma_{1},\Sigma_{2})
=Q\times\Sigma_{1}^{\#}\times\Sigma_{2}^{\#}\times{\bf Z}^{2}$.
Thus, the configuration of a two-tape quantum Turing machine ${\cal Q}$ with
the frame $(Q,\Sigma_{1},\Sigma_{2})$ is determined by the processor
configuration $q\in Q$,
the first and second tape configurations $T_{1}\in\Sigma_{1}^{\#}$,
$T_{2}\in\Sigma_{2}^{\#}$,
and the head positions $\xi_{1}\in{\bf Z}$, $\xi_{2}\in{\bf Z}$ in
the first and second tapes.
The {\em quantum state space} of $(Q,\Sigma_{1},\Sigma_{2})$ is the
Hilbert space ${\cal H}(Q,\Sigma_{1},\Sigma_{2})$ generated by
${\cal C}(Q,\Sigma_{1},\Sigma_{2})$.
A {\em local transition function} for $(Q,\Sigma_{1},\Sigma_{2})$ is
defined to be a complex-valued function on
$Q\times\Sigma\times Q\times\Sigma\times\{-1,0,1\}^{2}$,
where $\Sigma=\Sigma_{1}\times\Sigma_{2}$.
The relation
$\delta(q,(\sigma_1,\sigma_2),p,(\tau_1,\tau_2),(d_1,d_2))=c$
can be interpreted as the following operation of ${\cal Q}$:
if the processor is in the configuration $q$
and if the head of the $i$-th tape
($i=1,2$) reads the symbol $\sigma_i$, then it follows with the amplitude $c$
that the processor configuration turns to $p$, the head of the $i$-th tape
writes the symbol $\tau_i$ and moves one cell to the right if
$d_i=1$, to the left if $d_i=-1$, or does not move if $d_i=0$.
The {\em evolution operator} of $\delta$ is a linear operator
$M_\delta$ on ${\cal H}(Q,\Sigma_{1},\Sigma_{2})$ such that
\begin{eqnarray*}
\lefteqn{M_\delta|q,(T_1,T_2),(\xi_1,\xi_2)\rangle}\quad\\
&=&\!\!\!\!\!\su
\delta(q,(T_1(\xi_1),T_2(\xi_2)),p,(\tau_1,\tau_2),(d_1,d_2)
|p,({T_1}_{\xi_1}^{\tau_1},{T_2}_{\xi_2}^{\tau_2}),(\xi_1+d_1,\xi_2+d_2)\rangle
\end{eqnarray*}
for all $(q,(T_1,T_2),(\xi_1,\xi_2))\in {\cal C}(Q,\Sigma_{1},\Sigma_{2})$,
where the summation is taken over all
$(p,(\tau_1,\tau_2),(d_1,d_2))\in Q\times\Sigma\times\{-1,0,1\}^2$.
Then, local transition functions of two-tape quantum Turing
machines are characterized as follows.
\begin{Theorem}\label{th:1112a}
The evolution operator $M_\delta$ of a local transition function
$\delta$ for the two-tape Turing frame $(Q,\Sigma_{1},\Sigma_{2})$ is unitary
if and only if $\delta$ satisfies the following conditions.
{\rm (1)} For any $(q,\sigma)\in Q\times\Sigma$,
$$
\sum_{p\in Q,\tau\in\Sigma,d_1,d_2\in \{-1,0,1\}}
|\delta(q,\sigma,p,\tau,(d_1,d_2))|^2=1.
$$
{\rm (2)} For any $(q,\sigma),(q',\sigma')\in Q\times \Sigma$
with $(q,\sigma)\neq(q',\sigma')$,
$$
\sum_{p\in Q,\tau\in\Sigma,d_1,d_2\in \{-1,0,1\}}
\delta(q',\sigma',p,\tau,(d_1,d_2))^{*}
\delta(q,\sigma,p,\tau,(d_1,d_2))=0.
$$
{\rm (3)} For any $(q,\sigma,\tau_2),(q',\sigma',\tau'_2)
\in Q\times\Sigma\times\Sigma_2$,
$$
\sum_{
\begin{array}{c}
{\scriptstyle p\in Q,\tau_1\in\Sigma_1}\\
{\scriptstyle d_1\in\{-1,0,1\},d_2=0,1}
\end{array}
}
\delta(q',\sigma',p,(\tau_1,\tau'_2),(d_1,d_2-1))^*
\delta(q,\sigma,p,(\tau_1,\tau_2),(d_1,d_2))=0.
$$
{\rm (4)} For any $(q,\sigma,\tau_2),(q',\sigma',\tau'_2)
\in Q\times\Sigma\times\Sigma_2$,
$$
\sum_{p\in Q,\tau_1\in\Sigma_1,d_1\in\{-1,0,1\}}
\delta(q',\sigma',p,(\tau_1,\tau'_2),(d_1,-1))^*
\delta(q,\sigma,p,(\tau_1,\tau'_2),(d_1,1))=0.
$$
{\rm (5)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q,d_1=0,1}\delta(q',\sigma',p,\tau',(d_1-1,1))^*
\delta(q,\sigma,p,\tau,(d_1,-1))= 0.
$$
{\rm (6)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q,d_1=0,1,d_2=0,1}
\delta(q',\sigma',p,\tau',(d_1-1,d_2))^*\delta(q,\sigma,p,\tau,(d_1,d_2-1))=0.
$$
{\rm (7)} For any $(q,\sigma,\tau_1),(q',\sigma',\tau'_1)
\in Q\times\Sigma\times\Sigma_1$,
$$
\sum_{
\begin{array}{c}
{\scriptstyle p\in Q,\tau_1\in\Sigma_1}\\
{\scriptstyle d_1=0,1,d_2\in\{-1,0,1\}}
\end{array}
}
\delta(q',\sigma',p,(\tau'_1,\tau_2),(d_1-1,d_2))^*
\delta(q,\sigma,p,(\tau_1,\tau_2),(d_1,d_2))=0.
$$
{\rm (8)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q, d_1=0,1,d_2=0,1}
\delta(q',\sigma',p,\tau',(d_1-1,d_2-1))^*\delta(q,\sigma,p,\tau,(d_1,d_2))=0.
$$
{\rm (9)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q,d_1=0,1}\delta(q',\sigma',p,\tau',(d_1-1,-1))^*
\delta(q,\sigma,p,\tau,(d_1,1))=0.
$$
{\rm (10)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q}\delta(q',\sigma',p,\tau',(-1,1))^*
\delta(q,\sigma,p,\tau,(1,-1))=0.
$$
{\rm (11)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q,d_2=0,1}\delta(q',\sigma',p,\tau',(-1,d_2))^*
\delta(q,\sigma,p,\tau,(1,d_2-1))=0.
$$
{\rm (12)} For any $(q,\sigma,\tau_1),(q',\sigma',\tau'_1)
\in Q\times\Sigma\times\Sigma_1$,
$$
\sum_{p\in Q,\tau_2\in\Sigma_2,d_2\in\{-1,0,1\}}
\delta(q',\sigma',p,(\tau'_1,\tau_2),(-1,d_2))^*
\delta(q,\sigma,p,(\tau_1,\tau_2),(1,d_2))=0.
$$
{\rm (13)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q,d_2=0,1}\delta(q',\sigma',p,\tau',(-1,d_2-1))^*
\delta(q,\sigma,p,\tau,(1,d_2))= 0.
$$
{\rm (14)} For any $(q,\sigma,\tau),(q',\sigma',\tau')\in Q\times \Sigma^2$,
$$
\sum_{p\in Q}\delta(q',\sigma',p,\tau',(-1,-1))^*
\delta(q,\sigma,p,\tau,(1,1))= 0.
$$
\end{Theorem}
If each head is required to move either to the right
or to the left at each step, conditions (3),(5)--(9),(11), and (13)
are automatically satisfied.
It is also easy to see that conditions (3)--(14) are automatically
satisfied by unidirectional two-tape quantum Turing machines,
for which $(d_{1},d_{2})$ is uniquely determined by $p$ in the
non-zero amplitude $\delta(q,\sigma,p,\tau,(d_{1},d_{2}))$.
The proof of Theorem \ref{th:1112a} is analogous to the proof of
Theorem \ref{th:518b}.
Let ${\cal C}(k_1,k_2)$ be a subset of ${\cal C}(Q,\Sigma_{1},\Sigma_{2})^2$
consisting of all pairs
$C=(q,(T_1,T_2),(\xi_1,\xi_2))$ and $C'=(q',(T'_1,T'_2),(\xi'_1,\xi'_2))$
with $C\neq C'$ such that $T_{i}(m_i)=T'_{i}(m_i)$
for $m_{i}\not\in\{\xi_{i},\xi'_{i}\}$
and that $\xi'_i-\xi_i=k_i$ for $i=1,2$.
This plays a role similar to ${\cal C}(k)$
in the proof of Theorem \ref{th:518b}.
In the proof of Theorem \ref{th:518b},
we showed that condition (b),(c), or (d) holds
if and only if $\langle C'|M_\delta^{\dagger}M_\delta|C\rangle=0$ holds
for all $(C,C')\in{\cal C}(0),(C,C')\in{\cal C}(1),$
or $(C,C')\in{\cal C}(2)$, respectively.
In the case of Theorem \ref{th:1112a}, we can show similarly that
for $(k_1,k_2)\in(\{0\}\times\{0,1,2\})
\cup(\{1,2\}\times\{0,\pm 1,\pm 2\})$,
condition $(5k_{1}+k_{2}+2)$ holds if and only if
$\langle C'|M_\delta^{\dagger}M_\delta|C\rangle=0$ holds for all
$(C,C')\in{\cal C}(k_1,k_2)$.
For example, condition (2) holds if and only if
$\langle C'|M_\delta^{\dagger}M_\delta|C\rangle=0$ holds for all
$(C,C')\in{\cal C}(0,0)$ (This is the case of $k_1=k_2=0$).
Moreover, it is trivial that
condition (1) holds if and only if
$\langle C|M_\delta^{\dagger}M_\delta|C\rangle=1$ holds
for all $C\in{\cal C}(Q,\Sigma_{1},\Sigma_{2})$,
and that if $C\ne C'$ and
$$
(C,C')\not\in \bigcup_{(k_1,k_2)\in\{0,\pm 1,\pm 2\}^2}{\cal C}(k_1,k_2)
$$
then $\langle C'|M_\delta^{\dagger}M_\delta|C\rangle=0$.
Since $M_\delta^{\dagger}M_\delta$ is self-adjoint, $M_\delta$ is an isometry
if and only if $\langle C'|M_\delta^{\dagger}M_\delta|C\rangle
=\langle C'|C\rangle$
for any $C=(q,(T_1,T_2),(\xi_1,\xi_2))$, $C'=(q,(T'_1,T'_2),(\xi'_1,\xi'_2))
\in {\cal C}(Q,\Sigma_{1},\Sigma_{2})$ with $\xi_1<\xi'_1$
or with $\xi_1=\xi'_1$ and
$\xi_2\le\xi'_2$. Therefore, we can show that
conditions (1)--(14) hold if and only if $M_\delta$ is an isometry.
We can also show that $M_\delta$ is unitary if it is an isometry
by a similar argument with the proof of Lemma \ref{th:518a}.
Thus we can prove Theorem \ref{th:1112a}.
We now consider $k$-tape quantum Turing machines.
In what follows, $\vec{a}$ abbreviates $(a_1,\ldots,a_k)$.
For $j\in\{0,\ldots,k-1\}$, let $\vec{a}_{\le j}=(a_1,\ldots,a_j)$
and $\vec{a}_{> j}=(a_{j+1},\ldots,a_k)$.
For any set $S=\{i_1,\ldots,i_m\}\subseteq \{1,\ldots,k\}$, let
$\vec{a}[S]=(a_{i_1},\ldots,a_{i_m})$ and
$\bar{S}=\{1,\ldots,k\}\backslash S$. Moreover,
for any tuple $(a_{i_1},\ldots,a_{i_m})$,
the symbol $(a_{i_1},\ldots,a_{i_m})^t$
denotes $(a_{I(1)},\ldots,a_{I(m)})$,
where $\{I(1),\ldots,I(m)\}=\{i_1,\ldots,i_m\}$
and $I(1)<\cdots<I(m)$.
Extending the arguments for the two-tape quantum Turing machines,
the local transition functions of $k$-tape quantum Turing machines
can be characterized as follows.
\begin{Theorem}\label{th:009}
The evolution operator $M_\delta$ of a local transition function
$\delta$ for the $k$-tape Turing frame
$(Q,\Sigma_{1},\Sigma_{2},\ldots,\Sigma_{k})$ is unitary
if and only if $\delta$ satisfies the following conditions.
{\rm (1)} For any $(q,\vec{\sigma})\in Q\times\Sigma$,
$$
\sum_{p\in Q,\vec{\tau}\in\Sigma,\vec{d}\in \{-1,0,1\}^k}
|\delta(q,\vec{\sigma},p,\vec{\tau},\vec{d})|^2=1.
$$
{\rm (2)} For any $(q,\vec{\sigma}),(q',\vec{\sigma'})\in Q\times \Sigma$
with $(q,\vec{\sigma})\neq(q',\vec{\sigma'})$,
$$
\sum_{p\in Q,\vec{\tau}\in\Sigma,\vec{d}\in \{-1,0,1\}^k}
\delta(q',\vec{\sigma'},p,\vec{\tau},\vec{d})^*
\delta(q,\vec{\sigma},p,\vec{\tau},\vec{d})=0.
$$
{\rm (3)} For each $j\in\{1,\ldots,k\}$ and
$\vec{D}_{>k-j}=(D_{k-j+1},\ldots,D_k)
\in\{1,2\}\times\{0,\pm 1,\pm 2\}^{j-1}$,
the following condition holds.
For any $(q,\vec{\sigma},\vec{\tau}[S(\vec{D}_{>k-j})]),
(q',\vec{\sigma'},\vec{\tau'}[S(\vec{D}_{>k-j})])\in
Q\times\Sigma\times \prod_{i\in S(\vec{D}_{>k-j})}\Sigma_i$ we have
\begin{eqnarray*}
\sum& &\!\!\!\!\!\delta(q',\vec{\sigma'},p,
(\vec{\tau}[\bar{S}(\vec{D}_{>k-j})],
\vec{\tau'}[S(\vec{D}_{>k-j})])^t,
(\vec{d}_{\le k-j},\vec{d'}_{>k-j}))^*\\
& &\times\delta(q,\vec{\sigma},p,
(\vec{\tau}[\bar{S}(\vec{D}_{>k-j})],
\vec{\tau}[S(\vec{D}_{>k-j})])^t,
(\vec{d}_{\le k-j},\vec{d}_{>k-j}))=0,
\end{eqnarray*}
where the summantion is taken over $p\in Q$,
$\vec{\tau}[\bar{S}(\vec{D}_{>k-j})]
\in\prod_{i\in \bar{S}(\vec{D}_{>k-j})}\Sigma_i$,
$\vec{d}_{\le k-j}\in\{-1,0,1\}^{k-j}$,
and $\vec{d'}_{>k-j},\vec{d}_{>k-j}\in\{-1,0,1\}^j$
such that $\vec{d}_{>k-j}-\vec{d'}_{>k-j}=\vec{D}_{>k-j}$.
Here, $S(\vec{D}_{>k-j})=\{i\in\{k-j+1,\ldots,k\}|\ D_i\neq 0\}$.
\end{Theorem}
Note that condition (3) of Theorem \ref{th:009}
contains $2\times\sum_{j=0}^{k-1}5^j$ conditions
(the number of different pairs $(j,\vec{D}_{>k-j})$).
Thus, the local transition functions of
$k$-tape quantum Turing machines can be characterized by
$$
1+1+2\times\sum_{j=0}^{k-1}5^j=1+(1/2)(5^k+1)
$$
conditions.
Multi-tape Turing machines are often used for theoretical
consideration in complexity theory \cite{Pap94}
because it is often easier to construct
a multi-tape machine than a single tape machine
in order to realize a given algorithm.
Hence, multi-tape quantum Turing machines can be expected as useful
tools for quantum complexity theory.
In such applications, it appears to be a tedious task to check that a
constructed local transition function satisfies the unitarity
conditions. However, it should be noted that
restricted classes of multi-tape machines
are characterized much more simply; the unidirectional
multi-tape machines are characterized by only two conditions,
conditions (1) and (2) in Theorem \ref{th:009}.
| 2024-02-18T23:40:15.862Z | 2000-12-26T16:43:10.000Z | algebraic_stack_train_0000 | 1,860 | 10,216 |
|
proofpile-arXiv_065-9171 | \section{Introduction}
The Cosmic Lens All-Sky Survey (CLASS; \cite{class}) is a large-scale
survey for gravitational lenses among flat-spectrum radio sources.
The primary goals of CLASS are to find lenses which may be suitable
for determinations of the Hubble Constant, $H_0$ (e.g., \cite{refsdal};
\cite{bn}), and to study the lensing rate in a large, homogeneous
sample in order to place limits on the cosmological constant,
$\Lambda$ (e.g., \cite{tog}; \cite{turner}; \cite{ffk}; \cite{ft}).
Over 12,000 sources have been observed with the VLA in three sessions
-- the first in the spring of 1994 (CLASS~1; $\sim$3300 sources), the
second in the summer of 1995 (CLASS~2; $\sim$4500 sources) and the
third in the spring of 1998 (CLASS~3; $\sim$5000 sources). The vast
majority of flat-spectrum radio sources are dominated by emission from
a single compact core; all CLASS sources containing multiple compact
components are selected as gravitational lens candidates, amounting to
50 -- 100 candidates in each phase of the survey. These are followed
up with high resolution radio imaging using MERLIN; the best surviving
candidates are then imaged using the VLBA. During this procedure, the
majority of VLA candidates are rejected based on surface brightness
and morphology criteria. The candidates that survive the radio
filters are then investigated further with optical and/or infrared imaging,
spectroscopy and more radio imaging. By these means we have
discovered 11 new lenses in the survey, in addition to the one
reported here. We are investigating an
additional $\sim$25 promising lens candidates.
We use $H_0 = 100 h\ {\rm km\ s}^{-1}\ {\rm Mpc}^{-1}$ and,
except where noted, assume $q_0 = 0.5$\ throughout this paper.
\section{VLA Observations}
The lens system B2045+265 (GB6 J2047+2643) was observed on 1995
September 02 as part of CLASS~2. The observations were made at
8.4~GHz with the VLA in A configuration, giving a resolution of
$\sim$0\farcs25. Observation and data reduction techniques for the
survey will be discussed in Myers et al.\ (1998). The survey image of
B2045+265 shows four components, with a possible detection of a fifth,
weak component. The system has a standard lens geometry
(e.g., \cite{bn}) and is similar in appearance to B1422+231
(\cite{p1422}).
The source was re-observed with the VLA on 1996 December 31.
Observations were made in A configuration at 1.4, 4.9, 8.5 and
14.9~GHz. Details of the observations are given in
Table~{\ref{tab_af311obs}}. The source 3C\,286 was used as a flux
calibrator. Phase calibrators were selected from the VLA
calibrator list. The data were calibrated using standard AIPS
routines, and maps were made using DIFMAP (\cite{difmap}). At the
frequencies above 1.4~GHz, the source can be characterized as five
point components, with no significant extended structure. Hence, the
maps were made by fitting five point components to the data and then
repeating a cycle of model-fitting and phase self-calibration. The
self-calibration time scale was set to the length of the individual
scans at each frequency. Both the positions and the flux densities were
allowed to vary in the model fitting. At 1.4~GHz the beam size is
large compared to the component separation, so the model fitting was
done by fixing the component positions at their 8.5~GHz values and
varying only the component flux densities. In addition, there were
confusing sources in the 1.4~GHz map that had to be included as
components in the model.
\begin{deluxetable}{llrrl}
\tablewidth{0pt}
\scriptsize
\tablecaption{Radio Observations\label{tab_af311obs}}
\tablehead{
\colhead{}
& \colhead{}
& \colhead{$\nu$}
& \colhead{$t_{tot}$}
& \colhead{Angular}
\\
\colhead{Array}
& \colhead{Date}
& \colhead{(GHz)}
& \colhead{(min)}
& \colhead{Resolution}
}
\startdata
VLA & 1995 Sep 02 & 8.5 & 0.5 & 0\farcs25 \\
VLA & 1995 Sep 11 & 14.9 & 7 & 0\farcs14 \\
VLBA & 1995 Nov 12 & 5.0 & 35 & 0\farcs0025 \\
VLA & 1996 Dec 31 & 1.4 & 12 & 1\farcs5 \\
VLA & 1996 Dec 31 & 4.9 & 12 & 0\farcs43 \\
VLA & 1996 Dec 31 & 8.5 & 192 & 0\farcs25 \\
VLA & 1996 Dec 31 & 14.9 & 20 & 0\farcs14 \\
MERLIN & 1997 Nov 30 & 5.0 & 750 & 0\farcs50 \\
\enddata
\end{deluxetable}
The final maps are shown in Figures \ref{fig_vla_x} --
\ref{fig_vla_l}. In all cases, the morphology seen in the discovery
map is duplicated, with five distinct components clearly present in
the high-dynamic-range 5 and 8.5~GHz maps. Component positions and
flux densities were obtained using the model fitting procedures in
DIFMAP. The brightest component (A) has an 8.5~GHz flux density of
16.55~mJy and is located at 20:47:20.29, +26:44:02.7 (J2000); the
positions of the other components relative to component A are given in
Table~\ref{tab_radpos} (see Figure \ref{fig_vla_x} for component
labels). The positional uncertainties for the different components
are estimated as the beam size at each frequency divided by the
signal-to-noise ratio of the component flux densities; the positional
errors in the 8.5~GHz map are given in Table~\ref{tab_radpos}. The
largest separation in the system is 1\farcs9. At all frequencies,
five point components are adequate to fit the data. No sign of
extended emission is seen, even in the deep 8.5~GHz map. The final
component flux densities and RMS noise levels for the maps are given
in Table~{\ref{tab_radflux}}. In all cases the noise levels are
within 5\% of the thermal noise expected in the maps.
\begin{figure}
\plotone{Fassnacht.fig1.ps}
\caption{\label{fig_vla_x} December 1996 8.5 GHz map. The contours are
($-$2.5, 2.5, 5, 10, 20, 40, 80, 160, 320, 640) times the RMS noise level
of 0.0145~mJy/beam. Map made by fitting point source components,
with flux densities listed in Table~\ref{tab_radflux}, to the $(u,v)$ data,
and restoring with a
0\farcs27$\times$0\farcs23 restoring beam.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig2.ps}
\caption{\label{fig_vla_c} December 1996 4.9 GHz map. The contours are
($-$2.5, 2.5, 5, 10, 20, 40, 80, 160, 320) times the RMS noise level
of 0.0621~mJy/beam. Map made by fitting point source components,
with flux densities listed in Table~\ref{tab_radflux}, to the $(u,v)$ data,
and restoring with a
0\farcs44$\times$0\farcs41 restoring beam.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig3.ps}
\caption{\label{fig_vla_u} December 1996 15 GHz map. The contours are
($-$2.5, 2.5, 5, 10, 20, 40) times the RMS noise level
of 0.139~mJy/beam. Map made by fitting point source components,
with flux densities listed in Table~\ref{tab_radflux}, to the $(u,v)$ data,
and restoring with a
0\farcs14$\times$0\farcs13 restoring beam.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig4.ps}
\caption{\label{fig_vla_l} December 1996 1.4 GHz map. The contours are
($-$2.5, 2.5, 5, 10, 20, 40, 80, 160, 320) times the RMS noise level
of 0.0784~mJy/beam. Map made by fitting point source components,
with flux densities listed in Table~\ref{tab_radflux}, to the $(u,v)$ data,
and restoring with a
1\farcs5$\times$1\farcs4 restoring beam.}
\end{figure}
The 8.5~GHz data were also analyzed to search for polarized
emission. The polarization calibration was carried out in AIPS, using
3C~286 as the calibrator. The phase calibrator 2115+295 was observed
over a range of parallactic angles and was used to determine the
instrumental polarization. The final maps are consistent with
no polarized emission above 56$\mu$Jy/beam (0.34\% of the peak unpolarized
intensity).
Radio spectra for the five components are shown in
Fig. \ref{fig_radio_spec}. Components A --
C have very similar spectra, with spectral indices of
$\alpha^{4.9}_{1.4} \sim -0.2$\, and $\alpha^{15}_{4.9} \sim -0.6$
($S_{\nu} \propto \nu^{\alpha}$). The spectra of components D and E differ
from those of the three brighter components. In order to determine
which, if either, of these two components is the counter image to
components A, B, and C, flux density ratios are computed for each
component with respect to component A. By computing flux density
ratios at each frequency, it is possible to avoid uncertainties due to
errors in the absolute flux calibration. The resulting curves (Fig.\
\ref{fig_flux_ratio}), normalized to their values at 8.5~GHz, clearly
indicate that the spectrum of component E differs from those of the
other four components.
\begin{figure}
\plotone{Fassnacht.fig5.ps}
\caption{\label{fig_radio_spec} Radio spectra of the five components in the
B2045+265 system. The error bars are 3 $\sigma$ errors, based on the RMS
noise in the maps.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig6.ps}
\caption{\label{fig_flux_ratio} Flux density ratios for components B
(open squares), C (filled triangles), D (open triangles) and E (open
circles) with respect to component A. All curves have been
normalized to their 8.5~GHz point. Error bars are calculated by
propagating the RMS noise in the maps at each frequency.}
\end{figure}
\begin{deluxetable}{crrr}
\tablewidth{0pt}
\scriptsize
\tablecaption{\label{tab_radpos}Component Positions}
\tablehead{
\colhead{Component}
& \colhead{$\Delta \alpha$\tablenotemark{a}}
& \colhead{$\Delta \delta$\tablenotemark{a}}
& \colhead{$\sigma_x$\tablenotemark{b}}
}
\startdata
A & 0.000 & 0.000 & \nodata \\
B & $-0.134$ & $-0.248$ & 0.001 \\
C & $-0.288$ & $-0.789$ & 0.001 \\
D & +1.628 & $-1.007$ & 0.006 \\
E & +1.121 & $-0.824$ & 0.005 \\
\enddata
\tablenotetext{a}{Positions relative to component A are taken from
the 1996 Dec 31 8.5~GHz map. Component A is at 20:47:20.29, +26:44:02.7
(J2000)}
\tablenotetext{b}{Uncertainties in relative positions calculated using
the 1996 Dec 31 8.5~GHz data and assuming that the uncertainties in the
component flux densities are three times the RMS noise level in the map.}
\end{deluxetable}
\begin{deluxetable}{llrllllll}
\tablewidth{0pt}
\scriptsize
\tablecaption{\label{tab_radflux}Component Flux Densities}
\tablehead{
\colhead{}
& \colhead{}
& \colhead{$\nu$}
& \colhead{$S_A$}
& \colhead{$S_B$}
& \colhead{$S_C$}
& \colhead{$S_D$}
& \colhead{$S_E$}
& \colhead{RMS} \\
\colhead{Date}
& \colhead{Array}
& \colhead{(GHz)}
& \colhead{(mJy)}
& \colhead{(mJy)}
& \colhead{(mJy)}
& \colhead{(mJy)}
& \colhead{(mJy)}
& \colhead{(mJy/beam)}
}
\startdata
1995 Sep 02
& VLA
& 8.5
& 18.4
& \phantom{0}9.42
& 14.8
& 2.41
& 1.83
& 0.28 \\
1995 Sep 11
& VLA
& 14.9
& 15.5
& \phantom{0}9.18
& 11.8
& 1.22
& 2.36
& 0.32 \\
1995 Nov 12
& VLBA
& 5.0
& 15.8
& \phantom{0}8.09
& \phantom{0}8.75
& \nodata
& \nodata
& 0.20 \\
1996 Dec 31
& VLA
& 1.4
& 29.02
& 15.73
& 19.92
& 2.81
& 2.18
& 0.08 \\
&
& 4.9
& 23.40
& 12.65
& 15.19
& 2.68
& 3.00
& 0.06 \\
&
& 8.5
& 16.55
& \phantom{0}8.90
& 10.99
& 1.77
& 2.34
& 0.01 \\
&
& 14.9
& 10.69
& \phantom{0}5.75
& \phantom{0}7.13
& 1.05
& 1.90
& 0.14 \\
1997 Nov 30
& MERLIN
& 5.0
& 16.8
& \phantom{0}9.82
& 14.8
& 2.02
& 1.53
& 0.15 \\
\enddata
\end{deluxetable}
\begin{deluxetable}{llllll}
\tablewidth{0pt}
\scriptsize
\tablecaption{\label{tab_fluxrats}Component Flux Density Ratios}
\tablehead{
\colhead{$\nu$ (GHz)}
& \colhead{Date}
& \colhead{B/A}
& \colhead{C/A}
& \colhead{D/A}
& \colhead{E/A}
}
\startdata
\phantom{0}1.4 GHz
& 1996 Dec 31
& 0.54 $\pm$ 0.009
& 0.69 $\pm$ 0.010
& 0.097 $\pm$ 0.008
& 0.075 $\pm$ 0.008 \\
\phantom{0}4.9 GHz
& 1996 Dec 31
& 0.54 $\pm$ 0.009
& 0.65 $\pm$ 0.009
& 0.11\phantom{0} $\pm$ 0.008
& 0.13\phantom{0} $\pm$ 0.008 \\
\phantom{0}8.5 GHz
& 1995 Sep 02
& 0.51 $\pm$ 0.051
& 0.80 $\pm$ 0.059
& 0.13\phantom{0} $\pm$ 0.046
& 0.099 $\pm$ 0.046 \\
\phantom{0}8.5 GHz
& 1996 Dec 31
& 0.54 $\pm$ 0.003
& 0.66 $\pm$ 0.003
& 0.11\phantom{0} $\pm$ 0.003
& 0.14\phantom{0} $\pm$ 0.003 \\
14.9 GHz
& 1995 Sep 11
& 0.59 $\pm$ 0.072
& 0.76 $\pm$ 0.078
& 0.079 $\pm$ 0.062
& 0.15\phantom{0} $\pm$ 0.062 \\
14.9 GHz
& 1996 Dec 31
& 0.54 $\pm$ 0.044
& 0.67 $\pm$ 0.047
& 0.098 $\pm$ 0.046
& 0.18\phantom{0} $\pm$ 0.063 \\
\enddata
\end{deluxetable}
\section{MERLIN and VLBA Observations}
A 5~GHz MERLIN observation of B2045+265 was made on 1997 November 30,
with approximately 12.5~hr total integration on source. The final map
(Fig.\ \ref{fig_merlin}) has an RMS noise level of 0.146~mJy/beam and
an angular resolution of 60~milliarcsec (mas). The data are well
fitted by 5 point sources, with flux densities given in
Table~\ref{tab_radflux}. However, there are some indications of
additional emission around components A and C in the residual map, at
the 2 -- 3\,$\sigma$ level. The excess emission may indicate that the
images are slightly resolved. If mas-scale structure can be detected
in the images, it can be used to put constraints on the lens model.
The transformation matrix between corresponding positions in the
resolved images provides crucial limits on the lensing potential by
fixing its second derivatives at those points (e.g., \cite{vlbi0957}).
Observations with high angular resolution and dynamic range are needed
to search for such structure. A 5~GHz VLBA snapshot (35~min total
integration on source) of this system was made on 1995 November 12.
In this observation, which has an angular resolution of $\sim$1~mas,
the three brightest components are detected. The flux density of each
component is only $\sim$60\% of its flux density in the 5~GHz VLA
observations. Once again, this may indicate the presence of extended
mas-scale structure in this source which could be detected in more
sensitive observations. Deep VLBA observations of B2045+265 have now
been scheduled which may produce maps with sufficient dynamic range to
detect extended structure in the images.
\begin{figure}
\plotone{Fassnacht.fig7.ps}
\caption{\label{fig_merlin} MERLIN map. Components A, B, and C may
be slightly resolved.}
\end{figure}
\section{Infrared Imaging}
\subsection{NIRC}
The system was imaged on 1996 July 31 using the Near Infrared Camera
(NIRC; \cite{nirc}) on the W.\ M.\ Keck I Telescope.
Images were taken in both $J$ and $K$ bands, with 45 one-minute
exposures in $K$\ and 27 one-minute exposures in $J$. In order to
estimate the point-spread function (PSF), 18 exposures of a star at a
distance of 1.47~arcmin were interleaved with the exposures on the
lens in each band.
The seeing was 0\farcs45--0\farcs90 in $K$ and 0\farcs75--1\farcs15 in
$J$\ during the observations.
The dark current level was subtracted from each image, and then
sky-subtraction and gain correction were performed. The sky and gain
frames used for each image were constructed from images observed
directly before and after it. The individual frames were aligned by
centroiding on a star which appeared in each frame. For the highest
sensitivity in the final images, all 45 frames in $K$\ and all 27
frames in $J$\ were combined into mosaics. The $K$ band mosaic is
shown in Fig.~\ref{fig_k_mos} and close-ups of the lens system in the
two bands are shown in Figs.~\ref{fig_nirc_k} and \ref{fig_nirc_j}.
The two main features in the images of the lens system are a short
arc, corresponding to the three brightest radio images, and the
lensing galaxy, which is located 1\farcs2 from the arc. Spectroscopy
has shown that the object seen 2\arcsec\ to the left of the lens
system in Fig.~\ref{fig_k_mos} is a star.
\begin{figure}
\plotone{Fassnacht.fig8.ps}
\caption{\label{fig_k_mos} NIRC $K$ band mosaic of the B2045+265
field. The lens system is marked with the crosshairs, with the
lensing galaxy to the left and an arc of emission from the background
source to the right. The object 2\arcsec\ to the left of the galaxy
is a star. The circled objects are extended and may be galaxies in
a group at the redshift of the lens (see \S~7.3).}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig9.ps}
\caption{\label{fig_nirc_k} NIRC $K$ band image of B2045+265. The lensing
galaxy is to the left, perhaps with some emission being
contributed from component D. An arc consisting of emission from components
A, B and C is to the right.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig10.ps}
\caption{\label{fig_nirc_j} NIRC $J$ band image of B2045+265.}
\end{figure}
\subsection{NICMOS\label{nicobs}}
B2045+265 was observed with the {\em Hubble Space Telescope} using
the Near Infrared Camera/Multi-Object Spectrometer (NICMOS) on 1997
July 14. The NIC-1 camera was used, which has a pixel scale of
43~mas. Two exposures were taken giving a total exposure time of
2624~sec. The images were subjected to the standard NICMOS calibration
pipeline involving bias and dark current subtraction, linearity and
flat-field correction, photometric calibration and cosmic ray
identification and removal. The final image (Fig.~\ref{fig_nicmos})
clearly shows the three brightest lensed images and the lensing
galaxy. The relative positions of components A, B, and C match those
seen at radio wavelengths to within 0\farcs01. In addition, the galaxy
location matches the position of radio component E to within the
errors. There appears to be low surface brightness emission between
images B and C, indicating the possibility that extended optical
emission from the background source is being lensed into an arc-like
structure. The expected position of component D is marked in the
figure.
\begin{figure}
\plotone{Fassnacht.fig11.ps}
\caption{\label{fig_nicmos} NICMOS F160W image of B2045+265. The extended
object near the center of the image is the lensing galaxy. The
crosshairs mark the expected position of component D.}
\end{figure}
\subsection{Photometry\label{photometry}}
Because the NICMOS image has high spatial resolution, it is possible
to compute magnitudes for the individual images of the background
source in the F160W bandpass (roughly corresponding to the
ground-based $H$ band). Magnitudes were calculated using both the
DAOPHOT package (\cite{daophot}) in IRAF\footnote{IRAF (Image
Reduction and Analysis Facility) is distributed by the National
Optical Astronomy Observatories, which are operated by the Association
of Universities for Research in Astronomy under cooperative agreement
with the National Science Foundation.} and the SExtractor package
(\cite{SExtractor}). First, the magnitudes of the three bright lensed
images were calculated using 0\farcs26 diameter apertures. Next, an
empirical PSF, constructed from radial profiles of stars in the field,
was fitted to the emission from the lensed images and the star near
the lensing galaxy. The scaled PSFs were then subtracted from the
data and the lens magnitude was calculated using a 1\farcs9 diameter
aperture. This aperture was chosen to match the largest image
separation in the system, which is approximately equal to twice the
Einstein radius of the lens. The aperture diameter corresponds to
7.9\,$h^{-1}$~kpc at the redshift of the lens. The PHOTFNU header
card was used to convert count rates in the image into flux densities
into Janskys and then a Vega zero-point of 1087~Jy was assumed to get
the F160W magnitudes. For comparison with the NIRC photometry (see
below), the magnitude of the ``arc'' defined by the three lensed
images of the background source was also computed.
The DAOPHOT and SExtractor packages produced magnitudes that were
consistent within the errors of the sky determination. The final
magnitudes are given in Table~\ref{tab_irphot}. It should be possible
to search for extinction caused by the lensing galaxy by comparing the
F160W and radio flux density ratios of the lensed images. However,
the uncertainties in the NICMOS component magnitudes are large ($\sim
0.2^m$) and thus the F160W flux density ratios, B/A = 0.69 and C/A =
0.76, are consistent with the radio flux density ratios at the
1~$\sigma$ level. The upper limit on the brightness of a
point-source at the location of component D is $m_{F160W} > 22^m$.
This is not surprising since component D should be $2.5^m$ fainter
than A.
It is considerably more difficult to do photometry on the NIRC images
because the seeing disk is large compared to the component
separations. Instead of finding magnitudes of the individual lensed
images, the magnitude of the ``arc'' was calculated by centering a
rectangular aperture of width 0\farcs9 and height 1\farcs8 on the
``arc.'' The sky was estimated from regions directly to the north,
west and south of the aperture. Once again, the DAOPHOT package was
used to subtract the emission from the lensed images and the star from
the data before the lens galaxy magnitude was calculated. The
photometric zero points were estimated by observing the infrared
standard star SJ9184 ($K = 11.82$, $J = 12.18$; \cite{irstds}), which
was observed in each band prior to the lens and PSF star exposures.
The zero points in the two observed bands were $K_0 = 22.44,$ and $J_0
= 22.92$. The $J$ and $K$ magnitudes of the lensing galaxy and
``arc'' are given in Table~\ref{tab_irphot}.
\begin{deluxetable}{crrccc}
\tablewidth{0pt}
\scriptsize
\tablecaption{\label{tab_irphot}Component Data (Infrared Observations)}
\tablehead{
\colhead{Component}
& \colhead{$\Delta \alpha$\tablenotemark{a}}
& \colhead{$\Delta \delta$\tablenotemark{a}}
& \colhead{$J$}
& \colhead{$m_{F160W}$\tablenotemark{b}}
& \colhead{$K$}
}
\startdata
A\tablenotemark{c} & $ 0.00$ & $ 0.00$ & \nodata & 20.6 & \nodata \\
B\tablenotemark{c} & $-0.13$ & $-0.23$ & \nodata & 21.0 & \nodata \\
C\tablenotemark{c} & $-0.28$ & $-0.78$ & \nodata & 20.9 & \nodata \\
Lens Galaxy\tablenotemark{d} & $+1.10$ & $-0.80$ & 19.2 & 18.6 & 17.6 \\
``Arc'' & \nodata & \nodata & 19.6 & 18.7 & 17.3 \\
\enddata
\tablenotetext{a}{Relative positions in arcseconds calculated
with respect to component A after a $-23^{\circ}$ rotation was performed to
match the radio image orientation.}
\tablenotetext{b}{F160W magnitudes computed assuming that the Vega zero point
is 1087~Jy.}
\tablenotetext{c}{Magnitudes computed in a 0\farcs26 diameter aperture.}
\tablenotetext{d}{Magnitudes computed in a 1\farcs9 diameter aperture.}
\end{deluxetable}
\section{Optical Spectroscopy\label{optspec}}
Spectra of B2045+265 were taken with the Low Resolution Imaging
Spectrograph (LRIS; \cite{lris}) on the Keck I Telescope on 1996 June
18--19. A 1\arcsec\ longslit and the 300~gr/mm grating were used,
giving a pixel scale of 2.44~\AA/pix. The slit was positioned at
P.A. = 112$^{\circ}$, in order to have the highest probability of
spatially separating the emission of the background source from that
of the lensing galaxy. Seven 1500~sec exposures were taken, four on
the first night and three on the second night.
The spectra were reduced using standard IRAF tasks.
The average seeing over the two nights was sufficiently good to enable
the extraction of separate spectra at the positions of the lensing
galaxy and the brightest source emission. The spatial separation
between lens and source spectra is 1\farcs3, matching the separation
between the lens and the three bright lensed images seen in the NICMOS
image.
The final spectra have a wavelength range $\lambda\lambda$4309--9361\AA.
Flux calibration was performed using the standard star Feige 110
(\cite{okestds}) which was observed at the end of each night. An
atmospheric absorption template was created by fitting a power law to
the flux-calibrated spectrum of the BL Lac object 2155--304. This
template was used to remove atmospheric features from the B2045+265
spectra. The individual exposures were weighted by the squares of their
signal-to-noise ratios and combined to create the final lens and
source spectra shown in Figs. \ref{fig_spec_lens} and
\ref{fig_spec_source}.
\begin{figure}
\plotone{Fassnacht.fig12.ps}
\caption{\label{fig_spec_lens} LRIS spectrum of the lensing galaxy
from data taken on 1996 June 18--19.}
\end{figure}
\begin{figure}
\plotone{Fassnacht.fig13.ps}
\caption{\label{fig_spec_source} Spectrum of background source taken on
1996 June 18--19. There is some contamination from light from the lensing
galaxy, which produces the [\ion{O}{2}] emission line.}
\end{figure}
The lens system was re-observed with LRIS on 1996 September 15 with a
setup similar to that used in the earlier observations, but with
wavelength coverage $\lambda\lambda$2344--7398\AA. Two 2000~sec
exposures were taken and the data were reduced using a method similar to
that described above. In this case, however,
the chip response was not removed from the final spectrum. The seeing
was worse than in the earlier observations; as a result, the source and
lens spectra could not be extracted separately and both source and lens
features are seen in the final spectrum (Fig. \ref{fig_spec_jlc}).
\begin{figure}
\plotone{Fassnacht.fig14.ps}
\caption{\label{fig_spec_jlc} Wavelength calibrated spectrum taken on
1996 Sept.\ 15. Both source and lens features are seen in the spectrum.
The position of the expected \ion{C}{3}] emission is marked. The signal
drops to zero blueward of 4000\AA.}
\end{figure}
The reduced spectrum of the lensing object is typical of a Sa galaxy
(e.g., \cite{galspecs}). It shows [\ion{O}{2}] $\lambda$ 3727\AA\
emission at 6962\AA, \ion{Ca}{2} K $\lambda$3934\AA\ absorption at
7345\AA, \ion{Ca}{2} H $\lambda$3968\AA\ absorption (possibly blended
with H$\epsilon$ $\lambda$3970\AA) at 7411\AA, and H$\eta$\ and
H$\theta$ $\lambda\lambda$ 3835, 3797\AA\ absorption at 7159\AA\ and
7092\AA, respectively. These spectral features establish the lens
redshift as $z_{\ell} = 0.8673 \pm 0.0005$, where the uncertainty in
the redshift is determined from the RMS scatter in the redshifts derived
from the [\ion{O}{2}], \ion{Ca}{2} K, H$\eta$\ and H$\theta$ lines.
The source spectrum shows a broad emission line at $\lambda =
6396$\AA. No other broad emission features are seen in the spectrum,
which ranges from 4300 to nearly 9400\AA. We identify the line as
\ion{Mg}{2} $\lambda$2800\AA, which implies $z_s = 1.28$, because it
is the only broad emission line that is so isolated in typical quasar
spectra (e.g., \cite{qspec1}; \cite{qspec2}). The second set of
spectra were taken in order to search for \ion{C}{3}] $\lambda$1909
emission, expected to fall at $\sim 4350$\AA. No line was seen at
this position, although the sensitivity in this part of the spectrum
is low and the presence of a weak line is not ruled out. Other
possible identifications for the observed emission line
(e.g., H$\beta$, \ion{C}{3}], \ion{C}{4}, Ly$\alpha$) would imply the
presence of strong emission lines in parts of the spectrum in which
the signal-to-noise ratio is high, none of which we see. We conclude that
the line is correctly identified and the source lies at redshift $z_s
= 1.28 \pm$0.01, where the uncertainty in the redshift is estimated from
uncertainties in finding the line centroid in both the emitted and the
observed frames.
\section{Source Variability}
In order for a lens system to be used to measure $H_0$, the background
source must be variable so that time delays can be measured. We use
two methods to search for evidence of variability in the background
source in B2045+265. The first is to examine the component flux
density ratios at different epochs. The ratios are not affected by
errors in the absolute flux calibration and so any detected changes in
the flux density ratios reflect actual changes in the flux density of
the background source. A comparison of the 8.5~GHz observations made
in September 1995 with those made in December 1996 shows small changes
in the ratios of the component flux densities, but not at a
significant level.
The second method is to monitor the total 22~GHz flux density of the
B2045+265 lens system. Daily observations of the system were made
with the 40~m telescope of the Owens Valley Radio Observatory during
the periods 1996 May 18 -- September 03 and 1996 November 04 -- 22.
The data were calibrated using the CMBPROG package (\cite{emlthesis}).
A 15\% $\pm$ 3\%
change in total flux density was observed in November 1996 over a
period of less than a week. Because several other sources being
monitored showed no change in flux density during the same period, the
observed change in flux density may indicate variability in the
background source. However, it is also possible that the detected
variability is due to changes in component E, which has a radio
spectrum that is flatter that that of the lensed images. To determine
which of the components is varying requires high angular resolution
monitoring of the system with the VLA.
\section{Discussion}
\subsection{The Nature of Component E\label{counterim}}
The detection of five unresolved radio components in the B2045+265
system raises the possibility that component E is the fifth lensed
image of the background source. We believe, however, that component E
is instead associated with the lensing galaxy. Although standard
lensing models can produce five images in configurations similar to
that seen in the B2045+265 system (e.g., \cite{bn}), the central fifth
image tends to be highly demagnified with respect to the other images.
This behavior is not seen in B2045+265; in fact, component E is
brighter than component D at high frequencies. In addition, the radio
spectrum of component E (Figs.\ \ref{fig_radio_spec} and
\ref{fig_flux_ratio}) differs sufficiently from the spectra of the
other four components to suggest that component E is not related to
the other components. Finally, the radio emission of component E is
spatially coincident with the infrared emission from the lensing
galaxy (see \S\ref{nicobs}). We therefore conclude that component E
is indeed associated with the lensing galaxy. It thus appears that
we have discovered a radio galaxy lensing a radio-loud quasar. Of
previously known lens systems, only 2016+112 (\cite{2016_discovery})
may have radio emission associated with a lensing galaxy. However,
recent observations of 2016+112 suggest that the radio emission of its
component C1 may be lensed emission from a (separate) background
source (\cite{2016_evn}).
The flat radio spectrum of component E in B2045+265 is typical of an
active galactic nucleus. If component E is, indeed, associated with
the lensing galaxy, then the radio position of E gives an accurate
location for the nucleus of the lensing galaxy. The location of the
center of the lensing galaxy relative to the lensed images is a key
component in models of the lensing potential (see \S\ref{model}). An
{\em a priori} knowledge of the relative position of the center of the
lensing galaxy provides strong constraints on lens models.
\subsection{Preliminary Lens Model\label{model}}
We model the lens potential as a singular isothermal sphere
potential with ``mixed'' shear (\cite{csk91}), in which the scaled
lensing potential can be expressed as:
$$
\psi = b r + \gamma b r \cos 2(\theta - \theta_{\gamma}).
$$
Locations in the image plane relative to the lens center are given by
$\vec{x} = (r,\theta) = (x,y)$. The critical radius of the isothermal
sphere is given by $b$, and $\gamma$\ and $\theta_{\gamma}$ give the
magnitude of the shear and its position angle. In the modeling
process we have used an {\em astronomical} rather than mathematical
coordinate system such that $x \equiv \Delta \alpha$ and $y \equiv
\Delta \delta$. The polar coordinates are defined in the usual way,
with $r = (x^2 + y^2)^{1/2}$ and $\theta = \arctan (y/x)$. This
results in a {\em left-handed} coordinate system.
Although the image positions are defined relative to the
position of component A, they are treated as absolute positions.
Thus, both the lens position $(x_{\ell},y_{\ell})$\ and source
position $(x_s,y_s)$ are included as parameters in the model. We
assume that component E marks the nucleus of the lensing galaxy and
fix the lens position at that location, leaving five varying
parameters and eight observational constraints ($x$\ and $y$\ for each
of the four images).
A downhill simplex routine (\cite{numrec}) is used to minimize the
difference between the observed and model positions, expressed as:
$$
\chi^2 = \sum_i \left[ \frac{|\vec{x}_m - \vec{x}_o|^2}{\sigma_x^2}\right]
$$
where the subscripts $m$\ and $o$ refer to the model and observed
values, respectively. The positional errors are defined to be the
8.5~GHz beam size divided by the signal-to-noise ratio of the
component in the 8.5~GHz map. To avoid biases introduced by the
choice of starting values, we repeat the process with a grid of
choices of initial conditions containing $3^n$ values, where $n$\ is
the number of varying parameters in the model fitting. The best-fit
parameters for the model are given in Table~\ref{tab_models}, and the
model is shown graphically in Fig.\,\ref{fig_model}. The observed
image positions are recovered by this simple model of the lensing
potential, with RMS image displacements of $<$20~mas.
\begin{deluxetable}{lc}
\tablewidth{0pt}
\scriptsize
\tablecaption{Lens Model\label{tab_models}}
\tablehead{
\colhead{Parameter}
& \colhead{Value}
}
\startdata
$b$ & 1\farcs076 \\
$\gamma$ & 0.104 \\
$\theta_{\gamma}$\tablenotemark{a} & 69\fdg4 \\
$x_{\ell}$\tablenotemark{a} & +1\farcs12\tablenotemark{b} \\
$y_{\ell}$\tablenotemark{a} & $-$0\farcs82\tablenotemark{b} \\
$x_s$\tablenotemark{a} & +0\farcs72 \\
$y_s$\tablenotemark{a} & $-$0\farcs67 \\
\enddata
\tablenotetext{a}{The positions relative to component A ($x,y$) and
$\theta_{\gamma}$ are defined in a {\em left-handed} coordinate system.
See \S\ref{model}.}
\tablenotetext{b}{Held fixed during model fitting.}
\end{deluxetable}
\begin{figure}
\plotone{Fassnacht.fig15.ps}
\caption{\label{fig_model} Lens model showing critical curve (broken
line), caustic curve (solid line), lens center ($\oplus$), source position
(filled diamond) and model image positions ($\star$). The four circles are
centered on the observed image positions.}
\end{figure}
Radio images are unaffected by microlensing, so it is possible to use
the flux density ratios of the images to constrain the model of the
lensing potential further. The observed flux density ratios, however,
are difficult to reproduce with a simple model such as the
one presented above. Although this simple model is adequate to
recover the four image positions, it does not reproduce the relative
magnifications of images A, B, C. The reason for this is easy to
see. If, as we have assumed, the potential is smooth, then we can
perform a Taylor expansion of it about the location of image B. If we
approximate the images as collinear, then the highest order essential
terms in this expansion are quadratic in the coordinate perpendicular
to the A -- C line and quartic parallel to this line. Higher order
terms are, by assumption, ignorable. It is then possible to show that
in the limit of small image separations there are several scaling laws
including $S_B \sim S_A + S_C$, i.e., the flux of the central image
approaches the sum of the fluxes of the other two images
(\cite{kandp}; \cite{rdb_qjras}). This is clearly violated in this
source, which indicates strongly that the potential is not smooth on
the scale of the A -- C image separation.
In order to improve upon our model, we have introduced higher order
terms in the potential such as might be produced by a local mass
perturbation. Specifically we expand the potential around images A and
C to third order along the A -- C line and to fourth order around image
B. We then impose continuity of the potential and its first two
derivatives midway between A and B and midway between B and C. There
are then three observable quantities, the ratio of the image
separations and the ratios of the image fluxes, that can be used to
solve for the coefficients in these Taylor expansions. Performing this
exercise leads to the conclusion that adding a small, positive mass
perturbation centered on a point located 0\farcs11 from B towards C
suffices to recover the flux ratios as well as the image locations.
(This perturbation might be caused by a spiral arm, for example (cf
\cite{lenssub})). The ratio of the (almost certainly unmeasurable) time
delays $\Delta t_{BA}/\Delta t_{BC}$ is now found to be 0.2; however,
the delay $\Delta t_{BD}$ should be quite robust to this perturbation
within a given global model. In conclusion, none of the existing
observations challenge the gravitational lens interpretation that we
present here.
\subsection{Properties of the Lensing Galaxy}
Properties of the lensing galaxy can be derived by using the fact that
it is acting as a gravitational lens. For example, the mass within
the Einstein ring of a lens is (e.g., \cite{bn}):
$$
M_E \approx 1.24 \times 10^{11}
\left( \frac{\theta_E}{{\rm 1\arcsec}} \right)^2
\left( \frac{D}{\rm 1 Gpc} \right ) M_{\sun},
\quad D \equiv \frac{D_{\ell} D_s}{D_{\ell s}}
$$
where $\theta_E$ is the angular radius of the Einstein ring and
$D_{\ell}$, $D_s$, and $D_{\ell s}$ are the angular diameter distances
between observer and lens, observer and source and lens and source,
respectively. The angular diameter distance between two objects at
$z_i$ and $z_j$ ($z_i < z_j$) is given by (\cite{lambda}):
$$
D_{ij} = \frac{c}{H_0 (1+z_j)|\Omega_k|^{1/2}}
{\rm sinn} \left ( |\Omega_k|^{1/2} {\cal I}(z_i,z_j) \right ),
$$
where
$$
{\cal I} (z_i,z_j) =
\int_{z_i}^{z_j}
\frac{dz}{[(1+z)^2(1+\Omega_M z) - z(2+z)\Omega_{\Lambda}]^{-1/2}}
$$
and ${\rm sinn}(x)$ is defined as $\sinh (x)$ for $\Omega_k<0$, $x$
for $\Omega_k=0$, or $\sin (x)$ for $\Omega_k>0$, respectively. The
$\Omega$ terms are defined in the standard fashion:
$$
\Omega_M = \frac{8 \pi G}{3 H_0^2} \rho_0, \quad
\Omega_{\Lambda} = \frac{\Lambda}{3 H_0^2}, \quad
\Omega_k = -\frac{k c^2}{R_0^2 H_0^2}.
$$
The integral $\cal I$ has an analytic solution for $\Omega_{\Lambda} =
0$, and if $\Omega_M = 1$ the angular diameter distance acquires the
simple form
$$
D_{ij} =
\frac{2 c}{H_0} \frac{1}{(1+z_j)}
\left( \frac{1}{(1+z_i)^{1/2}} - \frac{1}{(1+z_j)^{1/2}} \right ).
$$
For B2045+265, the lens redshift is secure. The source redshift,
based on one emission line, is less certain. For this reason, we
express the physical quantities derived in this section as functions
of the source redshift. We will also assume, for ease of discussion,
an Einstein-de~Sitter cosmology. The effects of varying the source
redshift and the cosmological model are presented in
Table~\ref{tab_lenspars}. The angular diameter distances for this system
are $D_{\ell} = 860\,h^{-1}$Mpc and
$$
\frac{D_{s}}{D_{\ell s}} =
\frac{1.8673[(1+z_s)^{1/2} - 1]}{(1+z_s)^{1/2} - 1.8673}.
$$
The mass enclosed within the Einstein ring radius is calculated by
assuming that the maximum image separation is a measure of
$2\theta_E$; i.e., that $\theta_E$=0\farcs95. This assumption yields
$M_E \approx 9.6 \times 10^{10} (D_s / D_{\ell s}) h^{-1} M_{\odot}$
for the B2045+265 system. Note that $M_E$ represents the {\em total}
mass contained in the cylinder with projected radius $R_E$. The speed
of a particle moving in a circular orbit ($v_{circ}$) at the Einstein
ring radius is derived from $M_E$ by assuming that the mass
distribution of the lensing galaxy is an isothermal sphere. A simple
calculation is used to convert $M_E$ to the mass inside the sphere of
radius $R_E$. The value for $v_{circ}$ follows directly from the mass
inside the sphere. The luminosity of the lens inside $R_E$ can be
estimated by converting its $K$ magnitude to rest-frame $M_B$ and
$M_V$ magnitudes. The $k$-correction for the lens redshift and
rest-frame ($V - K$) and ($B - V$) colors for typical Sa galaxies are
taken from Poggianti (1997). The resulting luminosities are $(L_V)_E
= 2.92 \times 10^{10} h^{-2} L_{\sun ,V}$ and $(L_B)_E = 2.36 \times
10^{10} h^{-2} L_{\sun ,B}$. The rest-frame $B$ band mass-to-light
ratio becomes $(M/L_B)_E = 4.1 (D_s / D_{\ell s}) h (M/L_B)_{\sun}$.
Inserting the source redshift derived from the optical spectra
(\S\ref{optspec}) yields a lens mass of $M_E \approx 4.7 \times
10^{11} h^{-1} M_{\odot}$ within the Einstein ring radius ($\sim4
h^{-1}$~kpc at the redshift of the lens). This mass implies a
circular velocity at $R_E$ of 570~km/sec, two to three times the
velocities seen in both nearby (e.g., \cite{sa_rotation}) and more
distant ($0.1 < z < 1$; \cite{vogt}) spiral galaxies. The Einstein
ring mass-to-light ratios are $(M/L_V)_E = 16 h (M/L_V)_{\sun}$ and
$(M/L_B)_E = 20 h (M/L_B)_{\sun}$. These mass-to-light ratios are not
unusual for lens systems (e.g., Fig.~7 of \cite{lensgal}), although
they are higher than expected given the redshift of the lens and
assumptions about the luminosity evolution of lens galaxies (Keeton et
al.\ 1998). The large lensing mass required to produce the observed
image separation is a result of the relative closeness of the
background source to the lens in redshift space. There are several
possible explanations for the unusually high projected mass.
Compact groups of galaxies associated with the primary lensing
galaxies have been discovered in recent observations of two lens
systems (1115+080 and B1422+231; Kundi\'{c} et al.\ 1997a,b;
\cite{t1115_1422}). We are conducting observations of the B2045+265
field to search for evidence of a group associated with the lensing
galaxy (see also Fig.~\ref{fig_k_mos}). The high contamination rate
by field galaxies ($\sim$85\% at $z \sim 1$; \cite{cluster_spec}),
however, makes high-redshift groups difficult to detect. If such a
group exists and is gravitationally bound, its gravitational potential
can enhance the image splitting by the lensing galaxy. In such a
scenario, we would overestimate the mass of the lensing galaxy.
Another explanation for the high projected mass inside the Einstein
ring is that the lens may be one of a close pair of possibly merging
galaxies. This situation has been observed in the case of the CLASS
lens 1608+656. The initial observations revealed an Einstein ring
projected mass of $M_E \approx 3.1 \times 10^{11} h^{-1} M_{\odot}$,
corresponding to $v_{circ} = 440\,{\rm km/sec}$ (\cite{m1608};
\cite{f1608}). Later HST imaging showed that the lens is composed of
a very close pair of galaxies, explaining the high mass and
ellipticity required for the 1608+656 system (\cite{hstlens}). We are
unable to determine unequivocally whether two galaxies are present in
the B2045+265 system because the sensitivity of our NICMOS observation
is too low. More sensitive NICMOS and WFPC2 observations of B2045+265
are needed to test the two-galaxy hypothesis.
A third possibility is that the source redshift or the assumed
cosmology (or both) are in error. We examine the results of varying
the source redshift by assigning different identifications to the
emission line seen in Fig.~\ref{fig_spec_source}. We then
re-calculate the physical parameters associated with the lens in three
different cosmological models: $(\Omega_M, \Omega_\Lambda) =$ (1, 0),
(0.2, 0), and (0.2, 0.8). We find that the effects of changing $z_s$
are larger than those produced by changing the cosmological model. Of
the alternative identifications of the emission line in the source
spectrum, the choice of Ly$\alpha$ produces lens galaxy properties
that are the closest to those associated with normal nearby spirals
(Table~\ref{tab_lenspars}). However, no sign of a Ly$\alpha$ forest
is seen on the short-wavelength side of the emission line, even after
subtracting the estimated contribution of the lens galaxy from the
source spectrum. The absence of other emission lines in the spectrum
(\S\ref{optspec}) and the lack of a Ly$\alpha$ forest argue against
the identification of the emission line as Ly$\alpha$. More sensitive
spectroscopy of this system must be conducted to determine the source
redshift unambiguously.
\begin{deluxetable}{cccccccccc}
\tablewidth{0pt}
\scriptsize
\tablecaption{Lens Parameters\label{tab_lenspars}}
\tablehead{
\colhead{}
& \colhead{}
& \colhead{}
& \colhead{}
& \colhead{$D_{\ell}$}
& \colhead{$D_s$}
& \colhead{$D_{\ell s}$}
& \colhead{$M_E$}
& \colhead{$(M/L_B)_E$}
& \colhead{$v_{circ}$} \\
\colhead{Line ID}
& \colhead{$z_s$}
& \colhead{$\Omega_M$}
& \colhead{$\Omega_\Lambda$}
& \colhead{($h^{-1}$ Mpc)}
& \colhead{($h^{-1}$ Mpc)}
& \colhead{($h^{-1}$ Mpc)}
& \colhead{($10^{11} h^{-1} M_\odot$)}
& \colhead{($h (M/L_B)_\odot$)}
& \colhead{(km/sec)}
}
\startdata
\ion{Mg}{2} & 1.28 & 1.0 & 0.0 & \phantom{1}860 & \phantom{1}890 & 180 & 4.7 & 20 & 570 \\
& & 0.2 & 0.0 & 1000 & 1100 & 240 & 5.3 & 16 & 560 \\
& & 0.2 & 0.8 & 1200 & 1300 & 340 & 5.1 & 11 & 500 \\
\ion{C}{3}] & 2.35 & 1.0 & 0.0 & \phantom{1}860 & \phantom{1}810 & 330 & 2.3 & 10 & 400 \\
& & 0.2 & 0.0 & 1000 & 1200 & 470 & 2.8 & \phantom{1}9 & 400 \\
& & 0.2 & 0.8 & 1200 & 1300 & 650 & 2.6 & \phantom{1}6 & 360 \\
\ion{C}{4} & 3.13 & 1.0 & 0.0 & \phantom{1}860 & \phantom{1}740 & 350 & 2.0 & \phantom{1}8 & 370 \\
& & 0.2 & 0.0 & 1000 & 1100 & 530 & 2.5 & \phantom{1}8 & 380 \\
& & 0.2 & 0.8 & 1200 & 1200 & 700 & 2.3 & \phantom{1}5 & 340 \\
Ly $\alpha$ & 4.26 & 1.0 & 0.0 & \phantom{1}860 & \phantom{1}640 & 337 & 1.8 & \phantom{1}7 & 360 \\
& & 0.2 & 0.0 & 1000 & 1100 & 540 & 2.3 & \phantom{1}7 & 370 \\
& & 0.2 & 0.8 & 1200 & 1100 & 690 & 2.1 & \phantom{1}5 & 330 \\
\enddata
\end{deluxetable}
\subsection{Prospects for Measuring $H_0$}
A lens system can be used to estimate $H_0$ by comparing the time
delays predicted by the lens model to the observed time delays. The
time delays are a function of the lensing potential, the
source and image positions, the redshifts of the source and lens,
and the world model (through the angular diameter distances):
$$
\Delta t_i = (1 + z_{\ell})\frac{D_{\ell} D_s}{c D_{\ell s}}
\left[\frac{1}{2} |\vec{x}_i - \vec{\beta}|^2 - \psi(\vec{x}_i) \right],
$$
where $\vec{x}_i = (x_i,y_i)$ is the angular position of the $i^{th}$
lensed image and $\vec{\beta} = (x_s,y_s)$ is the position of the source.
The observable quantity is the difference in the time delays, $\Delta
t_{ij} \equiv \Delta t_i - \Delta t_j$. The predicted delays are
proportional to $h^{-1}$ through the ratio of angular diameter
distances, so if it is possible to measure delays, the ratio of
the predicted and observed delays will give $h$. For the model given
in \S\,\ref{model}, the components are expected to vary in the order:
C, A, B, D. The predicted delays between components A, B and C are
small: $\Delta t_{AC} = 6.5 h^{-1}$\,hr and $\Delta t_{BC} = 7.7
h^{-1}$\,hr. The delays between each of the three bright components
and component D are all on the order of 142\,$h^{-1}$\,d. It will be
challenging to measure the short time delays since radio loud quasars
typically do not vary significantly on those time scales. However,
with high-sensitivity observations, it will be quite possible to
measure the long delays in this system, if the background source is
variable.
\section{Summary and Future Work}
We have discovered a new four-image gravitational lens in the second
phase of CLASS. The B2045+265 lens system is unusual in several
respects. First, it is, with the possible exception of the puzzling
2016+112 system, the first known gravitational lens system in which
both the background source and the lensing galaxy are radio sources.
The radio emission from the lens, presumably from an active nucleus,
can be used to locate the center of the lensing galaxy with high
precision, which provides a strong constraint for lens models.
Secondly, the flux density ratios of the three brightest lensed images
are inconsistent with the predictions of simple models of lensing
potentials. This result may indicate that there is significant
structure in the mass distribution. Finally, the projected mass (and
mass-to-light ratio) inside the Einstein ring of the lens is unusually
high for a Sa galaxy. This may imply that there is a significant
amount of dark matter associated with this galaxy, that the lens
actually consists of a close pair of galaxies, or that the overall
mass distribution of an associated group of galaxies is contributing
to the lensing potential. Alternatively, the source redshift may be
misidentified.
There are intriguing hints that the images of the background source
may be resolved by high angular resolution observations. We have
obtained high dynamic range VLBA observations which can be used to
search for this possible extended mas-scale structure. If such
structure is detected, the transformation matrices between the lensed
images of the background source provide strong constraints on the
lensing model. These data may explain the unusual flux density ratios
among the three bright images. The VLBA observations also will
provide a more accurate position for the center of the lensing galaxy.
In addition, more sensitive {\em Hubble Space Telescope} and
spectroscopic observations can provide further details about the
lensing galaxy and background source. The goal of these observations
is to develop a well-constrained model of the lensing potential. A
program of monitoring can be used to search for time delays in the
system. Any measured delays can be combined with the lens models to
yield a measurement of $H_0$.
\acknowledgments
We are indebted to the VLA analysts and operators, to Terry Stickel
and Wayne Wack for heroic operation of the Keck Telescopes, and to the
Keck Observatory and STScI staff. CDF thanks Erik Leitch for
generously spending many hours in instruction in the use of the OVRO
40~m telescope and for writing the CMBPROG software. The operation of
the 40~m telescope would be impossible without the dedication and
knowledge of Russ Keeney and Mark Hodges. We are grateful to Gerry
Neugebauer, Lee Armus and Aaron Evans for their expert assistance
during the reduction of the NIRC and NICMOS data. We thank Lori
Lubin, D. Wardell Hogg, Mike Pahre, Mark Metzger, and Chung-Pei Ma for
useful discussions and comments on the manuscript. We thank the
anonymous referee for helpful suggestions on how to improve the paper.
This work is supported by the NSF under grant \#AST 9420018 and by the
European Commission, TMR Program, Research Network Contract
ERBFMRXCT96-0034 ``CERES.''
| 2024-02-18T23:40:16.025Z | 1998-11-11T00:10:39.000Z | algebraic_stack_train_0000 | 1,867 | 8,421 |
|
proofpile-arXiv_065-9173 | \section{INTRODUCTION}
\label{s-intro}
This bi-annual series of workshops is an outstanding opportunity
to focus on the enormous breadth and depth of fundamental physics
accessible from the study of the production and decay
of the tau lepton and the tau neutrino.
At each meeting, we have seen tremendous progress in the precision
with which Standard Model physics is measured,
and increasing sensitivity to new physics; and
this meeting, fifth in the series, is no exception.
Tau physics continues to be wonderfully rich and deep!
The study of the tau contributes to the field of particle physics
in many ways. I think of the ``sub-fields'' as follows:
\begin{itemize}
\item Precision electroweak physics:
neutral current ($Z^0$) and charged current ($W^\pm$) couplings,
Michel parameters, leptonic branching fractions
and tests of universality. These measurements typically
have world-average errors better than 1\%.
They give indirect information on new physics at high mass scales
($Z^\prime$, $W_R$, $H^\pm$, {\it etc.}).
The higher the precision, the better chance of seeing
the effects of new physics, so continued improvement is necessary.
\item Very rare (``upper limit'') physics:
Direct searches for new physics, such as
processes forbidden in the Standard Model:
lepton-flavor-violating neutrinoless decays,
lepton-number-violating decays such as $\tau^-\to\mu^+ X$,
and CP-violating effects that can result from
anomalous weak and electromagnetic dipole moments.
\item Non-perturbative hadronic physics:
Our inability to reliably predict the properties
and dynamics of light mesons and baryons
at intermediate energies is the greatest failing
of particle physics. Tau decays provide a clean beam
of intermediate energy light mesons,
including vectors, axial-vectors, scalars and tensors.
One can measure mesonic couplings, and tune models
of resonances and Lorentz structure.
Topics of current interest are the presence
and properties of radial excitations such as
the $a_1^\prime$, and $K^{*\prime}$,
mixing, SU(3)$_f$ violation, and isospin decomposition
of multihadronic final states.
\item Inclusive QCD physics: the total and differential
inclusive rate for tau decays to hadrons (spectral functions),
can be used to measure $\alpha_S(s)$,
non-perturbative quark and gluon condensates,
and quark masses; and can be used to test QCD sum rules.
Here, in particular, there is a
rich interaction between theory and experiment.
\item Neutrino mass and mixing physics:
Aside from its fundamental significance, the presence
of tau neutrino mass and mixing has important implications
for cosmology and astrophysics $-$
it is our window on the universe.
\end{itemize}
The study of taus is thus an important tool in many fields,
and continual progress is being made in all of them,
as evidenced in the many contributions to this workshop
which I review in the following sections.
Some of the more exciting future goals in tau physics
were discussed in the first talk of the workshop,
by Martin Perl \cite{ref:perl}.
This was followed by a very comprehensive
overview of the theory of tau physics,
including future prospects for testing the theory,
by J.~Kuhn \cite{ref:kuhn}.
These talks are reviews in and of themselves;
I will focus, in this review, on the subsequent presentations.
\section{$\mathbf{Z^0}$ COUPLINGS}
\label{s-Z0couplings}
SLD and the four LEP experiments study the reaction
$e^+e^- \to Z^0 \to \tau^+\tau^-$
to extract a wealth of information
on rates and asymmetries,
and ultimately, on the neutral current
vector and axial couplings $v_f$ and $a_f$
for each fermion species $f$,
and from these, values for the effective weak mixing angle
$\sin^2\theta_W^f$.
Lepton universality is the statement that
these couplings (and the charged current couplings
to be discussed later) are the same for the electron,
muon and tau leptons.
The simplest such observable is
the partial width of the $Z^0$ into fermion pairs,
which to lowest order is:
$$ \Gamma_f \equiv \Gamma (Z^0 \to f\bar{f}) =
\frac{G_F M_Z^3}{6\sqrt{2}\pi}
\left( v_f^2 + a_f^2 \right). $$
Equivalently, the ratio of the total
hadronic to leptonic width $R_\ell \equiv \Gamma_{had}/\Gamma_\ell$,
with $\ell=e, \mu, \tau$, can be measured with high precion.
The angular distribution of the outgoing fermions
exhibit a parity-violating forward-backward asymmetry
$A_f^{FB} = \frac{3}{4}{\cal A}_e{\cal A}_f$
which permits the measurement of the asymmetry parameters
$${\cal A}_f \equiv
\frac{2 v_f a_f}{v_f^2 + a_f^2} .$$
From these one can extract
the neutral current couplings $v_f$ and $a_f$.
The Standard Model predicts that the
outgoing fermions from $Z^0$ decay are polarized,
with a polarization that depends on the scattering angle
$\cos\theta_f$ with respect to the incoming electron or positron:
$${\cal P}_f(\cos\theta_f) =
- \frac{{\cal A}_f(1+\cos^2\theta_f) + 2{\cal A}_e\cos\theta_f}
{(1+\cos^2\theta_f) + 2{\cal A}_f{\cal A}_e\cos\theta_f} .$$
If the incoming electron beam is also polarized,
as at the SLC, it further modifies the polarization
of the outgoing fermions.
In the case of the tau ($f = \tau$), this polarization can be
measured at the statistical level by analyzing the
decay products of the tau, providing an independent way
to determine ${\cal A}_\tau$ and thus $v_\tau$ and $a_\tau$.
LEP and SLD have used almost all of the major tau decay modes
($e$, $\mu$, $\pi$, $\rho$, $a_1$)
to analyze the tau spin polarization as a function of $\cos\theta_\tau$.
SLD has measured the dependence of these polarizations on beam ($e^-$)
polarization.
SLD and all four LEP experiments measure
all these quantities.
At TAU 98, the results for the ratios of partial widths $R_\ell$
and forward-backward asymmetries $A_{FB}^\ell$ for $\ell=e, \mu, \tau$
are reviewed in \cite{ref:sobie} and shown in Fig.~\ref{fig:rl_afb}.
The tau polarization measurements at LEP are presented
in \cite{ref:alemany}.
The beam polarization dependent asymmetries at SLD
are described in \cite{ref:reinertsen}, and shown in Fig.~\ref{fig:sld}.
The procedure for combining the LEP results on ${\cal P}_\tau$
is reviewed in \cite{ref:roney},
in which it is concluded that the results from the four
LEP experiments are consistent but not too consistent.
These are nearly the final results from LEP on this subject.
\begin{figure}[ht]
\psfig{figure=figs/rl_afb.ps,width=2.6in}
\caption[]{LEP-I averages for
$A_{FB}$ versus $R_\ell$ for the three
lepton species and for the combined result.
The Standard Model prediction is given by the lines \cite{ref:sobie}.}
\label{fig:rl_afb}
\end{figure}
\begin{figure}[ht]
\psfig{figure=figs/sld_costh.ps,width=2.6in}
\caption[]{Polar angle distributions for leptonic final states,
from SLD with polarized beams \cite{ref:reinertsen}.}
\label{fig:sld}
\end{figure}
Note that LEP and SLD have completely consistent results
on the leptonic neutral current couplings;
the addition of measurements of the
heavy quark couplings to their
Standard Model averages for the Weinberg angle
$\sin^2\theta_W$ pull the LEP average away from that of SLD's
(and that discrepancy is shrinking as LEP and SLD results are updated).
The partial widths for the three charged leptons
agree with one another (and therefore with lepton universality
in the neutral current) to 3 ppm.
These results, along with those from the $e^+e^-$ and $\mu^+\mu^-$
final states, fit well to the Standard Model predictions
with a rather light Higgs mass:
$m_H < 262$ GeV/c$^2$ at 95\%\ C.L.
The results for the vector and axial couplings
of the three leptons is shown in Fig.~\ref{fig:gvga}.
Note that the tau pair contour is smaller than that for
mu pairs, because of the added information
from tau polarization.
\begin{figure}[ht]
\psfig{figure=figs/gvga.ps,width=2.6in}
\caption[]{Results on the vector and axial couplings of the leptons
to the $Z^0$, from LEP-I \cite{ref:sobie}.}
\label{fig:gvga}
\end{figure}
\section{$\mathbf{W\to \tau\nu}$}
\label{s-Wtotau}
This TAU workshop is the first to see results from LEP-II,
including new results on
the production of taus from real $W$ decays.
All four LEP experiments identify the leptonic decays of
$W$ bosons from $e^+e^-\to W^+W^-$ at center-of-mass energies
from 160 to 189 GeV.
The results are consistent between experiments (see Fig.~\ref{fig:wbrlep})
and the averaged branching fractions they obtain
are summarized in \cite{ref:moulik}:
\begin{eqnarray*}
{\cal B}(W\to e\nu) &=& 10.92\pm0.49\% \\
{\cal B}(W\to \mu\nu) &=& 10.29\pm0.47\% \\
{\cal B}(W\to \tau\nu) &=& 9.95\pm0.60\% \\
{\cal B}(W\to \ell\nu) &=& 10.40\pm0.26\%
\end{eqnarray*}
where the last result assumes universality of the
charged current couplings.
These results, and results on the measured cross-sections
for $W^+W^-$ as a function of center-of-mass energy,
are in good agreement with Standard Model predictions.
Lepton universality from real $W$ decays
is tested at the 4\%\ level at LEP:
$g_\mu/g_e = 0.971\pm0.031$,
$g_\tau/g_e = 0.954\pm0.040$.
\begin{figure}[ht]
\psfig{figure=figs/wbrlep.ps,width=2.6in}
\caption[]{Branching fractions for $W\to \ell\nu$ from the
4 LEP experiments \cite{ref:moulik}.}
\label{fig:wbrlep}
\end{figure}
The CDF and D0 experiments at the Tevatron
can also identify $W\to \tau\nu$ decays,
and separate them from a very large background
of QCD jets.
The methods and results are summarized in \cite{ref:protop}.
The average of results from UA1, UA2, CDF and D0,
shown in Fig.~\ref{fig:gtauge},
confirm lepton universality in real $W$ decays
at the 2.5\%\ level:
$g_\tau/g_e = 1.003\pm0.025$.
The LEP and Tevatron
results are to be compared with the ratios of couplings
to virtual $W$ bosons in $\tau$ decays,
summarized in the next section.
\begin{figure}[ht]
\psfig{figure=figs/gtauge.ps,width=2.6in}
\caption[]{Measurements of $g_\tau^W/g_e^W$
at hadron colliders \cite{ref:protop}.}
\label{fig:gtauge}
\end{figure}
CDF has also looked for taus produced from
decays of top quarks,
charged Higgs bosons, leptoquarks, and techni-rhos.
Limits on these processes are reviewed in \cite{ref:Gallinaro}.
For the charged Higgs searches, they are shown
in Fig.~\ref{fig:chiggs}.
\begin{figure}[ht]
\psfig{figure=figs/chiggs.ps,width=2.6in}
\caption[]{Excluded values for the charged Higgs boson mass,
as a function of $\tan\beta$, are shaded \cite{ref:Gallinaro}.}
\label{fig:chiggs}
\end{figure}
\section{TAU LIFETIME AND
LEPTONIC BRANCHING FRACTIONS}
\label{s-leptonic}
The primary properties of the tau lepton are its mass, spin,
and lifetime. That its spin is 1/2 is well established,
and its mass is well measured \cite{ref:pdg98}:
$m_\tau = 1777.05^{+0.29}_{-0.26}$ GeV/c$^2$.
The world average tau lifetime has changed considerably
in the last 10 years,
but recent results have been in good agreement,
converging to a value that is stable and of high precision.
At TAU 98, new measurements were presented from L3 \cite{ref:Colijn}
and DELPHI.
The new world average tau lifetime represents the work of
6 experiments (shown in Fig.~\ref{fig:taulife}),
each utilizing multiple techniques,
and each with $\stackrel{<}{\scriptstyle\sim} 1\%$ precision.
The result is presented in \cite{ref:wasserbach}:
$\tau_\tau = (290.5\pm 1.0)$ fs.
\begin{figure}[ht]
\psfig{figure=figs/taulife.ps,width=2.6in}
\caption[]{Recent measurements of the $\tau$ lifetime \cite{ref:wasserbach}.}
\label{fig:taulife}
\end{figure}
We now turn to the decays of the tau.
The leptonic decays of the tau comprise 35\%\ of the total,
and can be both measured and predicted with high accuracy.
In the Standard Model,
$$\Gamma(\tau\to \ell\nu_\tau\antibar{\nu}_\ell) =
\frac{G_{\ell\tau}^2 m_\tau^5}{192\pi^3}
f\left(\frac{m_\ell}{m_\tau}\right) (1+\delta).$$
Here, $f$ is a known function of the masses,
$f = 1$ for $e\nu\nu$ and 0.9726 for $\mu\nu\nu$;
$\delta$ is a small correction of $-0.4\%$
due to electromagnetic and weak effects,
and $G_{\ell\tau}$ defines the couplings:
$$G_{\ell\tau} = \frac{g_\ell g_\tau}{4\sqrt{2} m_W^2} = G_F?$$
By comparing the measured leptonic branching fractions
to each other and to the decay rate of the muon to
$e\nu\nu$, we can compare the
couplings of the leptons
to the weak charged current, $g_e$, $g_\mu$, and $g_\tau$.
At TAU 98, new measurements on leptonic branching fractions
were presented by DELPHI \cite{ref:stugu}, L3 and OPAL \cite{ref:robertson}.
The results are summarized in \cite{ref:stugu}:
\begin{eqnarray*}
{\cal B}(\tau\to e\nu\antibar{\nu}) &=& {\cal B}_e = 17.81\pm0.06\% \\
{\cal B}(\tau\to \mu\nu\antibar{\nu}) &=& {\cal B}_\mu = 17.36\pm0.06\%.
\end{eqnarray*}
These new world average branching fractions
have an accuracy of 3 ppm,
and are thus beginning to probe the corrections
contained in $\delta$, above.
The resulting ratios of couplings are consistent
with unity (lepton universality in the charged current couplings)
to 2.5 ppm:
\begin{eqnarray*}
g_\mu/g_e &=& 1.0014\pm0.0024 \\
g_\tau/g_\mu &=& 1.0002\pm0.0025 \\
g_\tau/g_e &=& 1.0013\pm0.0025.
\end{eqnarray*}
Some of the consequences of these precision measurements
are reviewed in \cite{ref:swain}.
The Michel parameter $\eta$ (see next section)
is constrained to be near zero to within 2.2\%.
One can extract a limit on the tau neutrino mass
(which, if non-zero, would cause the function $f$
defined above to depart from its Standard Model value)
of 38 MeV, at 95\%\ C.L.
One can also extract limits on mixing of the tau neutrino
to a 4$^{th}$ generation (very massive) neutrino,
and on anomalous couplings of the tau.
One can even measure fundamental properties of the strong
interaction through precision measurement of these
purely leptonic decays. The total branching fraction
of the tau to hadrons is assumed to be $1-{\cal B}_e -{\cal B}_\mu$.
This inclusive rate for semi-hadronic decays
can be formulated within QCD:
\begin{eqnarray*}
R_\tau &\equiv& \frac{{\cal B}_h}{{\cal B}_e} =
\frac{1-{\cal B}_e-{\cal B}_\mu}{{\cal B}_e} = 3.642 \pm 0.019 \\
&=& 3 (V_{ud}^2 + V_{us}^2) S_{EW} (1+\delta_{pert}+\delta_{NP}),
\end{eqnarray*}
where $V_{ud}$ and $V_{us}$ are Cabibbo-Kobayashi-Maskawa (CKM)
matrix elements,
and $S_{EW}$ is a small and calculable electroweak correction.
The strong interaction between the final state quarks
are described by
$\delta_{pert}$, the prediction from perturbative QCD
due to radiation of hard gluons, expressible as a
perturbation expansion in the strong coupling constant
$\alpha_S(m_\tau^2)$,
and $\delta_{NP}$, which describes non-perturbative
effects in terms of incalculable expectation values
of quark and gluon operators (condensates).
The value of $\delta_{NP}$ is estimated to be small,
and values for these expectation values can be extracted
from experimental measurements of the spectral functions
in semi-hadronic decays (see section~\ref{s-spectral}).
The extraction of $\alpha_S(m_\tau^2)$ from $R_\tau$
depends on an accurate estimate of $\delta_{NP}$
and on a convergent series for $\delta_{pert}(\alpha_S)$.
Recent improvements in the techniques for calculating
this series (``Contour-improved Perturbation Theory'', CIPT)
and extrapolating to the $Z^0$ mass scale,
are reviewed in ~\cite{ref:Maxwell}.
The resulting values of $\alpha_S$ evaluated
at the tau mass scale, and then run up to the $Z^0$ mass scale, are:
\begin{eqnarray*}
\alpha_S(m_\tau^2) &=& 0.334\pm 0.010 \\
\alpha_S(m_Z^2 ) &=& 0.120\pm 0.001.
\end{eqnarray*}
We will return to this subject in section \ref{s-spectral}.
\section{LORENTZ STRUCTURE}
\label{s-lorentz}
The dynamics of the leptonic decays
$\tau^-\to \ell^-\nu_\ell\antibar{\nu}_\tau$
are fully determined in the Standard Model,
where the decay is mediated by the $V-A$ charged current
left-handed $W_L^-$ boson.
In many extensions to the Standard Model,
additional interactions can modify the
Lorentz structure of the couplings, and thus the dynamics.
In particular, there can be
weak couplings to scalar currents
(such as those mediated by the charged Higgs of the
Minimal Supersymmetric extensions
to the Standard Model, MSSM),
or small deviations from maximal parity violation
such as those mediated by a right-handed $W_R$ of
left-right symmetric extensions.
The effective lagrangian for the 4-fermion interaction
between $\tau-\nu_\tau-\ell-\nu_\ell$
can be generalized to include such interactions.
Michel and others in the 1950's
assumed the most general, Lorentz invariant,
local, derivative free, lepton number conserving,
4 fermion point interaction.
Integrating over the two unobserved neutrinos, they
described the differential distribution for the
daughter charged lepton ($\ell^-$) momentum
relative to the parent lepton ($\mu$ or $\tau$)
spin direction, in terms of the so-called Michel parameters:
\begin{eqnarray*}
\lefteqn{\frac{1}{\Gamma} \frac{d\Gamma}{dxd\cos\theta}
= \frac{x^2}{2} \times } \\
&& \left[ \left( 12(1-x) + \frac{4{ \rho}}{3}(8x-6)
+ 24{ \eta}{ \frac{m_\ell}{m_\tau}}
\frac{(1-x)}{x}\right) \right. \\
& & \left. \pm { P_\tau }
{ \xi} { \cos\theta} \left( 4(1-x)+
\frac{4}{3}{ \delta}(8x-6) \right)\right] \\
&& \propto x^2\left[ I(x\vert {\rho , \eta} )
\pm { P_\tau} A( x,{ \theta} \vert
{\xi ,\delta}) \right],
\end{eqnarray*}
where $\rho$ and $\eta$ are the spectral shape Michel
parameters and $\xi$ and $\delta$ are the spin-dependent
Michel parameters~\cite{ref:michel};
$x=E_{\ell}/E_{max}$ is the daughter charged lepton energy
scaled to the maximum energy $E_{max} = (m_{\tau}^2 +
m_{\ell}^2)/2m_{\tau}$ in the $\tau$ rest frame;
$\theta$ is the angle between the tau spin direction and the
daughter charged lepton momentum in the $\tau$ rest frame;
and $P_\tau$ is the polarization of the $\tau$.
In the Standard Model
(SM), the Michel Parameters have the values
$\rho=3/4$, $\eta=0$, $\xi = 1$ and $\delta = 3/4$.
There are non-trivial extensions to this approach.
In SUSY models, taus can decay into scalar neutralinos
instead of fermionic neutrinos.
These presumably are massive,
affecting the phase space for the decay
as well as the Lorentz structure of the dynamics.
In addition, there exists a non-trivial extension
to the Michel formalism that admits anomalous
interactions with a tensor leptonic current that includes
derivatives; see \cite{ref:seager} for details.
Such interactions will produce distortions
of the daughter charged lepton spectrum
which cannot be described with the Michel parameters.
DELPHI has used both leptonic and semihadronic decays
to measure the tensor coupling $\kappa_\tau^W$,
with the result
$\kappa_\tau^W = -0.029\pm 0.036\pm 0.018$
(consistent with zero).
There are new or updated results on Michel parameter measurements
for TAU 98 from ALEPH, OPAL, DELPHI, and CLEO
\cite{ref:michel}.
The world averages, summarized in \cite{ref:stahl},
also include results from ARGUS, L3, and SLD.
All results are consistent with the Standard Model,
revealing no evidence for departures from the $V-A$ theory.
These measurements have now reached rather high precision,
but they are still not competitive with the precision
on Michel parameters obtained from muon decay,
$\mu \to e \nu \antibar{\nu}$.
Results from the two decays $\tau\to e\nu\nu$ and $\tau\to \mu\nu\nu$
can be combined under the assumption of $e-\mu$ universality.
Such an assumption is clearly not called for
when one is searching for new physics that explicitly
violates lepton universality, such as charged Higgs
interactions, which couple to the fermions
according to their mass.
However, such couplings mainly affect the Michel parameter $\eta$,
and it is clear from the Michel formula above
that $\eta$ is very difficult to measure
in $\tau\to e\nu\nu$, since it involves a chirality flip
of the daughter lepton, which is suppressed for the light electron.
Thus, lepton universality is usually invoked to constrain
the $\rho$, $\xi$, and $\xi\delta$ parameters to be the same
for the two decays. Since the measurements of $\rho$ and $\eta$
in $\tau\to \mu\nu\nu$ decays are strongly correlated,
the constraint that $\rho_e = \rho_\mu \equiv \rho_{e\mu}$
significantly improves the errors on $\eta_\mu$.
Invoking universality in this sense,
the world averages for the four Michel parameters \cite{ref:stahl},
shown in Fig.~\ref{fig:michwa}, are:
\begin{eqnarray*}
\rho_{e\mu} &=& 0.7490\pm 0.0082 \q (SM = 3/4) \\
\eta_{e\mu} &=& 0.052 \pm 0.036 \q (SM = 0) \\
\xi_{e\mu} &=& 0.988 \pm 0.029 \q (SM = 1) \\
(\xi\delta)_{e\mu} &=& 0.734 \pm 0.020 \q (SM = 3/4).
\end{eqnarray*}
\begin{figure}[ht]
\psfig{figure=figs/michwa.ps,width=2.6in}
\caption[]{New world averages for the Michel parameters in leptonic $
\tau$ decays, assuming $e-\mu$ universality in the couplings \cite{ref:stahl}.}
\label{fig:michwa}
\end{figure}
A measurement of the spin-dependent Michel parameters
allows one to distinguish the Standard Model $V-A$ interaction
(left-handed $\nu_\tau$) from $V+A$ (right-handed $\nu_\tau$).
The probability that a right-handed (massless) tau neutrino
participates in the decay can be expressed as
$$P^\tau_R = 1/2 \left[1+ 1/9 \left(3 {\xi}
-16{\xi\delta} \right)\right], $$
and $P^\tau_R = 0$ for the SM $V-A$ interaction.
The Michel parameters measured in all experiments provide
strong constraints on right-handed $(\tau - W - \nu)_R$ couplings,
as shown in Fig.~\ref{fig:michcoup}.
However, they are unable to distinguish left-handed
$(\tau - W - \nu)_L$ couplings,
for example, between scalar, vector, and tensor currents,
without some additional information, such as
a measurement of the cross-section
$\sigma(\nu_\tau e^-\to \tau^- \nu_e)$.
\begin{figure}[ht]
\psfig{figure=figs/michcoup,width=2.6in}
\caption[]{Limits on the coupling constants $g^\kappa_{\epsilon\rho}$
in $\tau$ decays, assuming $e-\mu$ universality.
Here, $\kappa = S,V,T$ for scalar, vector, and tensor couplings,
and $\epsilon$ and $\rho$ are the helicities ($L$ or $R$)
of the $\nu_\tau$ and $\nu_\ell$, respectively.
The black circles are the corresponding
limits from $\mu$ decays \cite{ref:stahl}.}
\label{fig:michcoup}
\end{figure}
In the minimal supersymmetric extension to the
Standard Model (MSSM), a charged Higgs boson
will contribute to the decay of the tau
(especially for large mixing angle $\tan\beta$),
interfering with the left-handed $W^-$ diagram,
and producing a non-zero value for $\eta$.
Since the world average value of $\eta$
(from spectral shapes and the indirect limit from
${\cal B}(\tau\to \mu\nu\antibar{\nu})$)
is consistent
with zero, one can limit the mass of a charged Higgs
boson to be \cite{ref:stahl}:
$M(H^\pm) > 2.1 \tan\beta$ (in GeV/c$^2$)
at 95\%\ C.L., which is competitive with direct searches
only for $\tan\beta > 200$.
In left-right symmetric models,
there are two sets of weak charged bosons $W^\pm_1$ and $W^\pm_2$,
which mix to form the observed ``light''
left-handed $W^\pm_L$ and a heavier (hypothetical)
right-handed $W^\pm_R$.
The parameters in these models are
$\alpha = M(W_1)/M(W_2)$ (= 0 in the SM), and
$\zeta = $ mixing angle, $= 0$ in the SM.
The heavy right-handed $W^\pm_R$
will contribute to the decay of the tau,
interfering with the left-handed $W^-$ diagram,
and producing deviations from the Standard Model values
for the Michel parameters $\rho$ and $\xi$.
The limit on $M(W_R)$ is obtained from a
likelihood analysis \cite{ref:stahl}
which reveals a very weak minimum (less than 1$\sigma$)
at around 250 GeV, so that the 95\%\ C.L.~limit on the mass
of 214 GeV (for a wide range of mixing angles $\zeta$)
is actually slightly worse than it was at TAU 96.
The limit from muon decay Michel parameters is 549 GeV.
It is worth continuing to improve the precision
on the tau Michel parameters, to push the
limits on charged Higgs and right-handed $W$'s,
and perhaps open a window on new physics
at very high mass scales.
\section{SEARCHES FOR NEW PHYSICS}
\label{s-searches}
One can look directly for physics beyond the Standard Model
in tau decays by searching for decays which violate
lepton flavor (LF) conservation or lepton number (LN) conservation.
These two conservation laws are put into the Standard Model
by hand, and are not known to be the result of some symmetry.
The purported existence of neutrino mixing implies that
LF is violated at some level,
in the same sense as in the quark sector.
Four family theories,
SUSY, superstrings,
and many other classes of models also predict
LF violation (LFV).
If LF is violated, decays such as $\tau\to\mu\gamma$
become possible; and in general, decays containing
no neutrino daughters are possible (neutrinoless decays).
In theories such as GUTs, leptoquarks, {\it etc.},
a lepton can couple directly to a quark,
producing final states where LN is violated (LNV)
but $B-L$ (baryon number minus lepton number) is conserved.
In most cases, LFV and LNV are accompanied by
violation of lepton universality.
Examples of lepton flavor violating decays
which have been searched for at CLEO, ARGUS, SLC, and LEP include: \\
$\tau^- \to \ell^-\gamma$, $\ell^-\ell^+\ell^-$ \\
$Z^0 \to \tau^- e^+$, $\tau^-\mu^+$ \\
$\tau^- \to \ell^- M^0$, $\ell^- P_1^+ P_2^-$ \\
where the $P$'s are pseudoscalar mesons.
Decays which violate lepton number but conserve $B-L$ include
$\tau^- \to \bar{p} X^0$,
where $X^0$ is some neutral, bosonic hadronic system.
Decays which violate lepton number and $B-L$ include:
$\tau^- \to \ell^+ P_1^- P_2^-$.
A broad class of R-parity violating
SUSY models which predict LFV or LNV
through the exchange of lepton superpartners
were reviewed at this conference, in \cite{ref:kong}.
A different class of models containing heavy
singlet neutrinos, which produce LFV,
is discussed in \cite{ref:ilakovac}.
In many cases, branching fractions for
neutrinoless tau decay can be as high as $10^{-7}$
while maintaining consistency with existing data from
muon and tau decays.
CLEO has searched for
40 different neutrinoless decay modes \cite{ref:stroynowski},
and has set upper limits on the branching fractions
of $\stackrel{<}{\scriptstyle\sim}$ few $\times 10^{-6}$.
A handful of modes that have not yet been studied by CLEO,
including those containing anti-protons,
have been searched for by the Mark II and ARGUS experiments,
with branching fraction upper limits
in the $10^{-4} - 10^{-3}$ range.
Thus, the present limits are approaching levels
where some model parameter space can be excluded.
B-Factories will push below $10^{-7}$;
it may be that the most important results coming from
this new generation of ``rare $\tau$ decay experiments''
will be the observation of lepton flavor or lepton number violation.
\section{CP VIOLATION IN TAU DECAYS}
\label{s-cpv}
The minimal Standard Model contains no mechanism for
CP violation in the lepton sector.
Three-family neutrino mixing can produce
(presumably extremely small) violations of CP
in analogy with the CKM quark sector.
CP violation in tau production can occur
if the tau has a non-zero electric dipole moment or
weak electric dipole moment,
implying that the tau is not a fundamental
(point-like) object.
Although other studies of taus (such as production cross section,
Michel parameters, {\it etc.}) are sensitive to tau substructure,
``null'' experiments such as the search for CP violating
effects of such substructure can be exquisitely sensitive.
I discuss searches for dipole moments in the next section.
CP violation in tau decays can occur, for example,
if a charged Higgs with complex couplings
(which change sign under CP)
interferes with the dominant $W$-emission process:
$$|A(\tau^-\to W^-\nu_\tau) + g e^{i\theta} A (\tau^-\to H^- \nu_\tau)|^2 .$$
If the dominant process produces a phase shift
(for example, due to the $W\to \rho$, $a_1$, or $K^*$ resonance),
the interference will be of opposite sign
for the $\tau^+$ and $\tau^-$,
producing a measurable CP violation.
The effect is proportional to isospin-violation
for decays such as $\tau\to \pi\pi\nu_\tau$ $3\pi\nu_\tau$;
and SU(3)$_f$ violation for decays such as
$\tau\to K\pi\nu_\tau$, $K\pi\pi\nu_\tau$.
The various signals for CP violation in tau decays
are reviewed in \cite{ref:tsai}.
CLEO has performed the first direct search for CP violation
in tau decays \cite{ref:kass},
using the decay mode $\tau\to K\pi\nu_\tau$.
The decay is mediated by the usual p-wave vector exchange,
with a strong interaction phase shift provided by the $K^*$ resonance.
CP violation occurs if there is interference with an
s-wave scalar exchange, with a complex weak phase $\theta_{CP}$
and a different strong interaction phase.
The interference term is CP odd, so CLEO searches for
an asymmetry in a CP-odd angular observable
between $\tau^+$ and $\tau^-$.
They see no evidence for CP violation,
and set a limit on the imaginary part of the complex coupling
of the tau to the charged Higgs ($g$, in units of $G_F/2\sqrt{2}$):
$g\sin\theta_{CP} < 1.7$.
Tests of CP violation in tau decays can also be
made using $2\pi\nu_\tau$ and $3\pi\nu_\tau$,
and we can look forward to results from such analyses in the future.
\section{DIPOLE MOMENTS}
\label{s-dipole}
In the Standard Model, the tau couples to the photon and to the $Z^0$
via a minimal prescription, with a single coupling constant
and a purely vector coupling to the photon
or purely $v_f V + a_f A$ coupling to the $Z^0$.
More generally, the tau can couple to the neutral currents
with $q^2$ dependent vector or tensor couplings.
The most general Lorentz-invariant form of the coupling
of the tau to a photon of 4-momentum $q_\mu$
is obtained by replacing the usual $\gamma^\mu$ with
\begin{eqnarray*}
\Gamma^\mu &=& F_1(q^2)\gamma^\mu + \\
&& F_2(q^2)\sigma^{\mu\nu}q_\nu
- F_3(q^2)\sigma^{\mu\nu}\gamma_5 q_\nu .
\end{eqnarray*}
At $q^2 = 0$, we interpret $F_1(0) = q_\tau$ as the electric charge
of the tau, $F_2(0) = a_\tau = (g_\tau-2)/2$ as the
anomalous magnetic moment, and
$F_3(0) = d_\tau/q_\tau$, where $d_\tau$ is the
electric dipole moment of the tau.
In the Standard Model, $a_\tau$ is non-zero due to
radiative corrections, and has the value
$a_\tau \approx \alpha/2\pi \approx 0.001177$.
The electric dipole moment $d_\tau$ is zero
for pointlike fermions; a non-zero value
would violate $P$, $T$, and $CP$.
Analogous definitions can be made for the
{\it weak} coupling form factors
$F^w_1$, $F^w_2$, and $F^w_3$,
and for the weak static dipole moments $a^w_\tau$ and $d^w_\tau$.
In the Standard Model, the weak dipole moments
are expected to be small (the weak electric dipole moment
is tiny but non-zero due to CP violation in the CKM matrix):
$a_\tau^W = -(2.1 + 0.6i)\times 10^{-6}$, and
$d_\tau^W \approx 3\times 10^{-37}$ e$\cdot$cm.
Some extensions to the Standard Model predict
vastly enhanced values for these moments:
in the MSSM, $a_\tau^W$ can be as large as $10^{-5}$
and $d_\tau^W$ as large as a few $\times 10^{-20}$.
In composite models, these dipole moments
can be larger still.
The smallness of the Standard Model expectations
leaves a large window for discovery of
non-Standard Model couplings.
At the peak of the $Z^0$, the reactions
$Z^0\to \tau^+\tau^-$ are sensitive to the
weak dipole moments, while the electromagnetic dipole moments
can be measured at center of mass energies far below the $Z^0$
(as at CLEO or BES), and/or through the study of
final state photon radiation in
$e^+e^-\to(\gamma^*, Z^0)\to \tau^+\tau^-\gamma$.
Extensive new or updated
results on searches for weak and electromagnetic
dipole moments were presented at TAU 98.
In all cases, however, no evidence for non-zero
dipole moments or CP-violating couplings were seen,
and upper limits on the dipole moments were
several orders of magnitude larger than Standard Model expectations,
and is even far from putting meaningful limits
on extensions to the Standard Model.
There is much room for improvement in both technique
and statistical power, and, as in any search for
very rare or forbidden phenomena,
the potential payoffs justify the effort.
An anomalously large weak magnetic dipole moment
will produce a transverse spin polarization
of taus from $Z^0$ decay, leading to (CP-conserving)
azimuthal asymmetries in the subsequent tau decays.
The L3 experiment searched for these asymmetries
using the decay modes $\tau\to\pi\nu_\tau$ and
$\tau\to\rho\nu_\tau$, and observed none \cite{ref:vidal}.
They measure values for the real and imaginary parts of
the weak magnetic dipole moment
which are consistent with zero, and set upper limits
(at 95\%\ C.L.):
\begin{eqnarray*}
|Re(a_\tau^w)| &<& 4.5 \times 10^{-3} \\
|Im(a_\tau^w)| &<& 9.9 \times 10^{-3}.
\end{eqnarray*}
They measure, for the real part of the weak electric
dipole moment, a value which is consistent with zero
within a few $\times 10^{-17}$ e$\cdot$cm.
The ALEPH, DELPHI, and OPAL experiments search for
CP violating processes induced by a
non-zero weak electric dipole moment,
by forming CP-odd observables from the 4-vectors
of the incoming beam and outgoing taus, and the outgoing tau spin vectors.
These ``optimal observables'' \cite{ref:zalite}
pick out the CP-odd terms in the cross section for production
and decay of the $\tau^+\tau^-$ system;
schematically, the optimal observables are defined by:
$$d\sigma \propto |M_{SM} + d^w_\tau M_{CP}|^2,$$
$$O^{Re} = Re(M_{CP})/M_{SM}, \, O^{Im} = Im(M_{CP})/M_{SM}.$$
These observables are essentially CP-odd triple products,
and they require that the spin vectors of the taus be determined
(at least, on a statistical level).
Most or all of the decay modes of the tau
$(\ell,\pi,\rho,a_1)$ are used to spin-analyze the taus.
ALEPH, DELPHI, and OPAL measure the expectation values
$<O^{Re}>$ and $<O^{Im}>$ of these observables,
separately for each tau-pair decay topology.
From these, they extract measurements of the
real and imaginary parts of $d_\tau^w$.
The combined limits (at 95\%\ C.L.) \cite{ref:zalite} are:
\begin{eqnarray*}
|Re(d_\tau^w)| &<& 3.0 \times 10^{-18} \q\mbox{e$\cdot$cm} \\
|Im(d_\tau^w)| &<& 9.2 \times 10^{-18} \q\mbox{e$\cdot$cm} .
\end{eqnarray*}
They are consistent with the Standard Model,
and there is no evidence for CP violation.
SLD makes use of the electron beam polarization
to enhance its sensitivity to Im($d_\tau^W$).
Rather than measure angular asymmetries or expectations
values of CP-odd observables,
they do a full unbinned likelihood fit to the observed event
(integrating over unseen neutrinos)
using tau decays to leptons, $\pi$, and $\rho$,
in order to extract limits on the real and imaginary parts
of both $a_\tau^w$ and $d_\tau^w$.
They obtain preliminary results \cite{ref:barklow}
which are again consistent with zero, but which have
the best sensitivity to $Im(a_\tau^w)$ and $Im(d_\tau^w)$:
\begin{eqnarray*}
|Re(d_\tau^w)| &=& (18.3\pm7.8) \times 10^{-18} \q\mbox{e$\cdot$cm}; \\
|Im(d_\tau^w)| &=& (-6.6\pm4.0) \times 10^{-18} \q\mbox{e$\cdot$cm}; \\
|Re(a_\tau^w)| &=& (0.7\pm1.2) \times 10^{-3}; \\
|Im(a_\tau^w)| &=& (-0.5\pm0.6) \times 10^{-3}.
\end{eqnarray*}
\subsection{EM dipole moments}
\label{ss-emdipole}
The anomalous couplings to photons can be probed,
even on the peak of the $Z^0$, by searching for
anomalous final state photon radiation in
$e^+e^- \to \tau^+\tau^- \gamma$.
The L3 experiment
studies the distribution of photons
in such events, as a function of the photon energy,
its angle with respect to the nearest reconstructed tau,
and its angle with respect to the beam.
In this way, they can distinguish anomalous
final state radiation from initial state radiation,
photons from $\pi^0$ decays, and other backgrounds.
The effect on the photon distribution
due to anomalous electric couplings is very similar
to that of anomalous magnetic couplings,
so they make no attempt to extract values for
$a_\tau$ and $d_\tau$ simultaneously, but instead
measure one while assuming the other
takes on its Standard Model value.
They see no anomalous photon production \cite{ref:taylor},
and set the limits (at 95\%\ C.L.)
$-0.052 < a_\tau < 0.058$ and
$|d_\tau| < 3.1\times 10^{-16}$ e$\cdot$cm.
These results should be compared with those for the muon:
\begin{eqnarray*}
a_\mu^{theory} &=& 0.00116591596(67) \\
a_\mu^{expt} &=& 0.00116592350(780) \\
d_\mu^{expt} &=& (3.7\pm3.4) \times 10^{-19} \q\mbox{e$\cdot$cm}.
\end{eqnarray*}
Theoretical progress on the evaluation of $a_\mu$ in the
Standard Model is reviewed in \cite{ref:Czarnecki},
including the important contribution
from tau decays (see section \ref{ss-gminus2} below).
Progress in the experimental measurement at Brookhaven
is described in \cite{ref:Grosse}.
Clearly, there is much room for improvement of the
measurements of the anomalous moments of the tau.
\section{SPECTRAL FUNCTIONS}
\label{s-spectral}
The semi-hadronic decays of the tau are dominated by
low-mass, low-multiplicity hadronic systems:
$n\pi$, $n\le 6$; $Kn\pi$, $K\bar{K}$, $K\bar{K}\pi$, $\eta\pi\pi$.
These final states are dominated by resonances
($\rho$, $a_1$, $\rho^\prime$, $K^*$, $K_1$, {\it etc.}).
The rates for these decays, taken individually,
cannot be calculated from fundamental theory (QCD),
so one has to rely on models, and on
extrapolations from the chiral limit using
chiral perturbation theory.
However, appropriate sums of final states
with the same quantum numbers can be made,
and these semi-inclusive measures of the
semi-hadronic decay width of the tau
can be analyzed using perturbative QCD.
In particular, one can define spectral functions
$v_J$, $a_J$, $v_J^S$ and $a_J^S$, as follows:
\begin{eqnarray*}
\lefteqn{
\frac{d\Gamma}{dq^2} (\tau \to \hbox{hadrons} + {\nu_\tau}) =
{{G_F^2}\over{32 \pi^2 m_\tau^3}} (m_\tau^2-q^2)^2} \\
& \times & \left\{ |V_{ud}|^2 \left[
(m_\tau^2+2 q^2) \left(v_1(q^2)+a_1(q^2)\right) \right. \right. \\
& & \left. + m_\tau^2 \left(v_0(q^2)+a_0(q^2)\right) \right] \\
& & + |V_{us}|^2 \left[
(m_\tau^2+2 q^2) \left(v_1^S(q^2)+a_1^S(q^2)\right) \right. \\
& & \left.\left. + m_\tau^2 \left(v_0^S(q^2)+a_0^S(q^2)\right) \right]
\right\} .
\end{eqnarray*}
The spectral functions $v$ and $a$ represent the
contributions of the vector and axial-vector hadronic currents
coupling to the $W$.
The subscripts on these functions denote the spin $J$ of the hadronic system,
and the superscript $S$ denotes states with net strangeness.
The hadronization information contained in the spectral functions
falls in the low-energy domain of strong interaction
dynamics, and it cannot be calculated in QCD.
Nonetheless, many useful relations between the spectral functions
can be derived.
For example, in the limit of exact $SU(3)_L\times SU(3)_R$ symmetry,
we have $v_1(q^2) = a_1(q^2) = v_1^S(q^2) = a_1^S(q^2)$
and $v_0(q^2) = a_0(q^2) = v_0^S(q^2) = a_0^S(q^2) = 0$.
Relations amongst the spectral functions
depend on assumptions about how the $SU(3)$ symmetry is broken.
The Conserved Vector Current (CVC) hypothesis
requires that $v_0(q^2) = 0$, and that
$v_1(q^2)$ can be related to the total cross-section for
$e^+e^-$ annihilations into hadrons.
Several sum rules relate integrals of these spectral functions,
as described below.
From arguments of parity and isospin,
the final states containing an even number of pions
arise from the vector spectral function,
and those with an odd number of pions
arise from the axial-vector spectral functions $a_1$,
or in the case of a single pion, $a_0$.
Final states containing one or more kaons
can contribute to both types of spectral functions,
since $SU(3)_f$ is violated
(and because of the chiral anomaly).
Experimentally, one can determine
whether a final state contributes to $v$ or $a$
through a careful analysis of its dynamics.
The spectral functions can be measured experimentally
by adding up the differential distributions
from all the exclusive final states that
contribute to $v$ or $a$,
in a ``quasi''-inclusive analysis:
$$ v_1 = \frac{{\cal B}_{v}}{{\cal B}_e}
\frac{1}{N_{v}}
\frac{dN_{v}}{dq^2}
\frac{M_\tau^8}{\left(m_\tau^2-q^2\right)^2
\left(m_\tau^2+2q^2\right)} ,
$$
and similarly for $a$.
For $v(q^2)$, the result is dominated by the $2\pi$ and $4\pi$
final states, with small contributions from $6\pi$, $K\bar{K}$, and others.
For $a(q^2)$, the result is dominated by the $3\pi$ and $5\pi$
final states, with small contributions from $K\bar{K}\pi$ and others.
The $\pi\nu$ and $K\nu$ final states are delta functions
and must be handled separately.
OPAL and ALEPH have presented new or updated measurements
of these non-strange spectral functions \cite{ref:Menke,ref:Hoecker}.
In both cases,
the small contributions mentioned above were obtained
from Monte Carlo estimates, not the data.
the ALEPH results are shown in Fig.~\ref{fig:spec_aleph}.
\begin{figure}[!ht]
\psfig{figure=figs/specv_aleph.ps,width=2.6in}
\psfig{figure=figs/speca_aleph.ps,width=2.6in}
\caption[]{Total vector and axial-vector spectral functions from ALEPH.
The contributions from the exclusive channels, from data and MC,
are indicated \cite{ref:Hoecker}.}
\label{fig:spec_aleph}
\end{figure}
These spectral functions can be used to study many
aspects of QCD, as described in the following subsections.
\subsection{Moments of the Spectral functions}
\label{ss-moments}
Although the spectral functions themselves cannot be predicted
in QCD, the moments $R_{kl}$ of those functions:
$$ R_{kl}^{v/a} = \int^{m_\tau^2}_0 ds \left(1 - \frac{s}{m_\tau^2}\right)^k
\left(\frac{s}{m_\tau^2}\right)^l
\frac{1}{N_{v/a}}
\frac{dN_{v/a}}{ds},
$$
with $k = 1$, $l = 0 \cdots 3$ {\it are} calculable.
In direct analogy with $R_\tau$ (section \ref{s-leptonic}),
the moments (for non-strange final states) can be expressed as:
$$R_{kl}^{v/a} = \frac{3}{2} V_{ud}^2 S_{EW} (1+\delta_{pert}
+\delta_{mass}^{v/a}+\delta_{NP}^{v/a}),$$
where $V_{ud}$ is the CKM matrix element,
and $S_{EW}$ is a small and calculable electroweak correction.
$\delta_{pert}$ is a calculable polynomial
in the strong coupling constant $\alpha_S(m_\tau^2)$,
$\delta_{mass}^{v/a}$ is a quark mass correction:
$$\delta_{mass}^{v/a} \simeq - 16 \frac{\bar{m}_q^2}{m_\tau^2},$$
and $\delta_{NP}$ describes non-perturbative
effects in terms of incalculable expectation values
of quark and gluon operators in the operator product expansion (OPE):
$$\delta_{NP}^{v/a} \simeq
C_4^{v/a} \frac{\vev{\cal{O}}^4}{m_\tau^4}
+ C_6^{v/a} \frac{\vev{\cal{O}}^6}{m_\tau^6}
+ C_8^{v/a} \frac{\vev{\cal{O}}^8}{m_\tau^8}.$$
The $C_n$ coefficients discribe short-distance effects,
calculable in QCD; and the expectation values for the operators
are the non-perturbative condensates. For example,
$$\vev{\cal{O}}^4 \sim \vev{\frac{\alpha_S}{\pi} GG}
+ \vev{m \bar{\psi}_q \psi_q} .$$
The important point is that one can calculate
distinct forms for $\delta_{pert}$ and $\delta_{NP}$
for each of the moments (values of $k$ and $l$),
separately for $V$ and $A$.
One can measure several different moments,
and from these, extract values for $\alpha_S(m_\tau^2)$
and for each of the non-perturbative condensates.
The result depends only on the method used to obtain
the QCD perturbation expansion; several methods are available,
including the CIPT mentioned in section \ref{s-leptonic}.
Both OPAL and ALEPH measure the moments of their
quasi-inclusive spectral functions, and fit to extract
values for $\alpha_S(m_\tau^2)$
and for the non-perturbative condensates.
The results are presented in \cite{ref:Menke,ref:Hoecker}.
The value of $\alpha_S(m_\tau^2)$ is in good agreement
with the one determined solely from the electronic branching fraction
(section \ref{s-leptonic}), but without the assumption that
$\delta_{NP}$ is small. It extrapolates to a value
at the $Z^0$ pole, $\alpha_S(m_Z^2)$, which agrees well
with measurements made there from hadronic event shapes
and other methods.
More importantly, the non-perturbative condensates
indeed are measured to be small ($\sim 10^{-2}$).
\subsection{QCD Chiral Sum Rules}
One can use the structure of QCD, and/or chiral perturbation theory,
to predict the moments of the difference
$v(s)-a(s)$ of the spectral functions (with $s = q^2$).
The physics of these sum rules is reviewed in \cite{ref:Rafael}.
Four sum rules have been studied with tau decay data:
\begin{itemize}
\item First Weinberg sum rule:
$$\frac{1}{4\pi^2}
\int_0^\infty ds \left(v_1(s) - a_1(s)\right) = f_\pi^2 $$
\item Second Weinberg sum rule:
$$\frac{1}{4\pi^2}
\int_0^\infty ds \cdot s \left(v_1(s) - a_1(s)\right) = 0$$
\item Das-Mathur-Okubo sum rule:
$$\frac{1}{4\pi^2}
\int_0^\infty \frac{ds}{s}
\left(v_1(s) - a_1(s)\right) = f_\pi^2 \frac{\vev{r_\pi^2}}{3} - F_A$$
\item Isospin-violating sum rule:
\begin{eqnarray*}
\frac{1}{4\pi^2}
\int_0^\infty ds && s \ln\frac{s}{\Lambda^2}
\left(v_1(s) - a_1(s)\right) = \\
&& -\frac{16\pi^2 f_\pi^2}{3\alpha}
\left( m_{\pi^\pm}^2 - m_{\pi^0}^2 \right).
\end{eqnarray*}
\end{itemize}
The first, second, and fourth sum rule listed above
have definite predictions on their right-hand side,
and the data can be used to test those predictions.
However, the spectral functions measured in tau decay
extend up to $s = m_\tau^2$, not infinity.
So in practice, the tests only allow one to address the
question, is $m_\tau^2$ close enough to infinity;
is it ``asymptotia''?
So far, the data are {\it consistent} with the
sum rule predictions and with the assumption
that $m_\tau^2$ is sufficiently close to infinity
(see Fig.~\ref{fig:qcdsum} for the OPAL results);
however, the data are not yet
sufficiently precise to
provide a quantitative test of these predictions.
However, ALEPH has studied the evolution of
the integral of the spectral functions and their moments
as a function of the cutoff $s \le m_\tau^2$,
and compared them with the theoretical prediction
for the perturbative and non-perturbative terms
as a function of their renormalization scale $s$
(fixing them at $s=m_\tau^2$ to the values they
obtain from their fits).
In all cases, they find \cite{ref:Hoecker}
that the experimental
distributions and the theoretical predictions
overlap and track each other well
before $s=m_\tau^2$. It appears that $s=m_\tau^2$
{\it is} asymptotia.
One can use the third (DMO) sum rule to extract
a value for the pion electric polarizability
$$\alpha_E = \frac{\alpha F_A}{m_\pi f_\pi^2}.$$
This can be compared with predictions from
the measured value of the axial-vector form factor $F_A$,
which give $\alpha_E = (2.86\pm 0.33)\times 10^{-4}$ fm$^3$.
OPAL \cite{ref:Menke}
obtains $\alpha_E = (2.71\pm 0.88)\times 10^{-4}$ fm$^3$,
in good agreement with the prediction.
\begin{figure}[ht]
\centerline{
\psfig{figure=figs/qcdsum,width=1.2in}
\psfig{figure=figs/qcdsum2,width=1.2in}}
\centerline{
\psfig{figure=figs/qcdsum3,width=1.2in}
\psfig{figure=figs/qcdsum4,width=1.2in}}
\caption[]{QCD sum rule integrals versus the upper integration limit
from OPAL data, for the four sum rules given in the text.
The chiral prediction is given by the lines \cite{ref:Menke}.}
\label{fig:qcdsum}
\end{figure}
\subsection{$\mathbf{(g-2)_\mu}$ from $\mathbf{v(s)}$ and CVC}
\label{ss-gminus2}
As noted in section \ref{ss-emdipole},
the muon's anomalous magnetic moment $a_\mu$ is
measured with far higher precision than that of the tau,
and is in excellent agreement with the precise
theoretical prediction.
The experimental precision will soon improve considerably
\cite{ref:Grosse},
and threatens to exceed the precision with which
the theoretical prediction is determined.
Until recently, the contribution to $a_\mu$,
$$\left(\frac{g-2}{2}\right)_\mu \equiv a^\gamma_\mu
= a_\mu^{QED} + a_\mu^{W} + a_\mu^{had} ,$$
from virtual hadronic effects $a_\mu^{had}$ had large uncertainties.
The contribution from weak effects
($a_\mu^{W}$, from $W$-exchange vertex correction)
is small:
$a_\mu^W = (151\pm 40)\times 10^{-11}$.
Observing this contribution is one of the goals of the
current round of measurements.
(A more important goal is to look for effects of the same scale
due to physics beyond the Standard Model).
The contribution from hadronic effects
(quark loops in the photon propagator in the EM vertex correction,
and, to a lesser extent, ``light-by-light'' scattering \cite{ref:Czarnecki})
is much larger, and its uncertainty was larger than the
entire contribution from $a_\mu^W$:
$a_\mu^{had} = (7024\pm 153)\times 10^{-11}$.
This value for $a_\mu^{had}$ was obtained by relating
the quark loops in the photon propagator
to the total rate for $\gamma^*\to q\bar{q}$
as measured in $e^+e^-$ annihilation experiments at low
$s = q^2 <$ (2 GeV)$^2$.
Unfortunately, these experiments had significant
overall errors in their measured values for the
total hadronic cross section $\sigma(s)$.
These results are currently being improved,
as reported in \cite{ref:Eidelman}.
In the meantime, one can use the
vector part of the total decay rate
of the tau to non-strange final states, $v(s)$,
to determine $a_\mu^{had}$.
One must assume CVC, which relates the
vector part of the weak charged current
to the isovector part of the electromagnetic current;
and one must correct for the isoscalar part of the current
which cannot be measured in tau decay.
In addition, one must use other data
to estimate the contribution to $a_\mu^{had}$
from $s > m_\tau^2$; however, the contribution
from $s < m_\tau^2$ dominates the value and the error.
Using the vector spectral function $v(s)$
measured by ALEPH, one obtains \cite{ref:Davier}
a value for $a_\mu^{had}$ with improved errors:
$a_\mu^{had} = (6924\pm 62)\times 10^{-11}$.
Now the error is smaller than the contribution
from $a_\mu^W$, and the sensitivity of the
forthcoming precision experimental result
to new physics is greatly improved.
However, the use of tau data to determine
$a_\mu^{had}$ with high precision
relies on CVC to 1\%.
Is this a valid assumption?
\subsection{Testing CVC}
\label{ss-cvc}
To test the validity of CVC at the per cent level,
one can compare the new
VEPP-II data on $e^+e^-\to 2n\pi$ \cite{ref:Eidelman}
to data from tau decays (from ALEPH, DELPHI, and CLEO).
When this is done, small discrepancies appear,
both in individual channels ($2\pi$ and $4\pi$)
and in the total rate via the vector current.
Discrepancies are expected, at some level,
because of isospin violation.
These comparisons are made in \cite{ref:Eidelman},
where the data from VEPP-II are converted (using CVC)
into predictions for the branching fractions of the tau
into the analogous (isospin-rotated)
final states:
$$ \frac{{\cal B}(\tau\to\pi\pi\nu) - B_{CVC}}{{\cal B}(\tau\to\pi\pi\nu)}
= (3.2\pm 1.4)\%, $$
$$\frac{\Delta{\cal B}}{{\cal B}} (2\pi+4\pi+6\pi+\eta\pi\pi+K\bar{K})
= (3.6\pm 1.5)\%. $$
In addition, a comparison of the spectral function {\it shape}
extracted from $\tau\to 2\pi\nu_\tau$
and that extracted from $e^+e^-\to 2\pi$
shows discrepancies at the few per cent level.
It is not clear whether these comparisons mean that
CVC is only good to $\sim 3\%$,
or whether the precision in the data used for the comparison
needs improvement.
To be conservative, however, results that rely on
CVC should be quoted with an error
that reflects these discrepancies.
\section{EXCLUSIVE FINAL STATES}
\label{s-structure}
The semi-hadronic decays of the tau to
exclusive final states is the realm of low energy meson dynamics,
and as such cannot be described with perturbative QCD.
In the limit of small energy transfers,
chiral perturbation theory can be used to predict rates;
but models and symmetry considerations must be used to
extrapolate to the full phase space of the decay.
In general, the Lorentz structure of the decay
(in terms of the 4-vectors of the final state pions, kaons,
and $\eta$ mesons) can be specified; models are then required
to parameterize the a priori unknown form factors
in the problem.
Information from independent measurements in low energy
meson dynamics can be used to reduce the number of free
parameters in such models; an example is given in \cite{ref:bingan}.
Measurements of the total branching fractions
to different exclusive final states
({\it e.g.}, $h n\pi^0{\nu_\tau}$, $n = 0 ... 4$,
$3h n\pi^0{\nu_\tau}$, $n = 0 ... 3$,
$5h n\pi^0{\nu_\tau}$, $n = 0 ... 1$, with $h = \pi^\pm$ or $K^\pm$;
or $\eta n\pi^0{\nu_\tau}$, $n = 2 ... 3$)
have been refined for many years,
and remain important work;
recent results from DELPHI are presented in \cite{ref:Lopez}.
Branching fractions for final states containing kaons
($K^\pm$, $K^0_S$, and $K^0_L$)
are presented in \cite{ref:Kravchenko,ref:Andreazza,ref:Chen}
and are discussed in more detail in section \ref{s-kaons}.
The world average summaries of all the
semi-hadronic exclusive branching fractions
are reviewed in \cite{ref:Heltsley}.
Since the world average branching fractions for all
exclusive tau decays now sum to one with small errors,
emphasis has shifted to the detailed study of
the structure of exclusive final states.
At previous tau workshops, the focus was on the simplest
final states with structure: $\pi\pi^0{\nu_\tau}$ and $K\pi{\nu_\tau}$
\cite{ref:rhostructure}.
At this workshop, the attention has shifted to the
$3\pi$, $K\pi\pi$, and $K\bar{K}\pi$ final states,
which proceed dominantly through the axial-vector current.
The results for the $K\pi\pi$, and $K\bar{K}\pi$ final states
are discussed in section \ref{s-kaons};
here we focus on $3\pi$.
The $4\pi$ final state remains to be studied in detail.
Final states with 5 or 6 pions
contain so much resonant substructure,
and are so rare in tau decays, that detailed
fits to models have not yet been attempted.
However, isospin can be used to characterize
the pattern of decays; this is discussed,
using data from CLEO, in \cite{ref:Gan}.
Recent results on $\tau\to 3\pi{\nu_\tau}$ from
OPAL, DELPHI, and CLEO are discussed in \cite{ref:Schmidtler}.
There are two complementary approaches that can be taken:
describing the decay in a Lorentz-invariant way,
parameterized by form-factors which model
intermediate resonances; or via model-independent
structure functions, defined in a specified angular basis.
The model-dependent approach gives a simple picture
in terms of well-defined decay chains, such as
$a_1 \to \rho\pi\to 3\pi$ or $K_1\to (K^*\pi, K\rho) \to K\pi\pi$;
but the description is only as good as the model,
and any model is bound to be incomplete.
The structure function approach results in large tables of numbers
which are harder to interpret (without a model);
but it has the advantage that some of the functions,
if non-zero, provide model-independent evidence for sub-dominant processes
such as pseudoscalar currents ({\it e.g.}, $\pi^\prime(1300)\to 3\pi$)
or vector currents ({\it e.g.}, $K^{*\prime} \to K\pi\pi$).
OPAL and DELPHI present fits of their $3\pi$ to two simple models
for $a_1\to \rho\pi$,
neither of which describe the data in detail \cite{ref:Schmidtler}.
DELPHI finds that in order to fit the high $m_{3\pi}$ region
with either model, a radially-excited $a_1^\prime(1700)$ meson
is required, with a branching fraction
${\cal B}(\tau\to a_1^\prime{\nu_\tau}\to 3\pi{\nu_\tau})$
of a few $\times 10^{-3}$, depending upon model.
Even then, the fits to the Dalitz plot variables are poor.
The presence of an enhancement at high mass
(over the simple models) has important consequences
for the extraction of the tau neutrino mass using
$3\pi{\nu_\tau}$ events.
OPAL also analyzes their data in terms of structure functions,
and from these, they set limits on scalar currents:
$$\Gamma^{scalar}/\Gamma^{tot} (3\pi{\nu_\tau}) < 0.84\%,$$
and make a model-independent determination of the {\it signed}
tau neutrino helicity:
$$h_{\nu_\tau} = -1.29\pm 0.26\pm 0.11$$
(in the Standard Model, $h_{\nu_\tau} = -1$).
CLEO does a model-dependent fit to their
$\tau^-\to\pi^-\pi^0\pi^0{\nu_\tau}$ data.
They have roughly 5 times the statistics
of OPAL or DELPHI.
This allows them to consider contributions
from many sub-dominant processes, including:
$a_1\to \rho^\prime\pi$, both S-wave and D-wave;
$a_1\to f_2(1275)\pi$, $\sigma\pi$, and $f_0(1370)\pi$;
and $\pi^\prime(1300)\to 3\pi$.
Here, the $\sigma$ is a broad scalar resonance
which is intended to ``mock up'' the complex structure
in the S-wave $\pi\pi$ scattering amplitude above threshold,
according to the Unitarized Quark Model.
CLEO also considers the process $a_1\to K^*K$,
as a contribution to the total width and therefore
the Breit Wigner propagator for the $a_1$
(of course, the final state that is studied
does not receive contributions from $K^*K$).
CLEO finds significant contributions from all of these processes,
with the exception of $\pi^\prime(1300)\to 3\pi$.
All measures of goodness-of-fit are excellent,
throughout the phase space for the decay.
There is also excellent agreement with the
data in the $\pi^-\pi^+\pi^-{\nu_\tau}$ final state,
which, because of the presence of isoscalars
in the substructure, is non-trivial.
There is strong evidence for a $K^*K$ threshold.
There is only very weak evidence for an $a_1^\prime(1700)$.
They measure the radius of the $a_1$ meson to be $\approx 0.7$ fm.
They set the 90\%\ C.L. limit
$$\Gamma(\pi^\prime(1300)\to\rho\pi)/\Gamma(3\pi) < 1.0\times 10^{-4},$$
and make a model-dependent determination of the signed
tau neutrino helicity:
$$h_{\nu_\tau} = -1.02\pm 0.13\pm 0.03 \q(model).$$
{\it All} of these results are model-dependent;
but the model fits the data quite well.
\section{KAONS IN TAU DECAY}
\label{s-kaons}
Kaons are relatively rare in tau decay,
and modes beyond $K{\nu_\tau}$ and $K^*{\nu_\tau}$
are only being measured with some precision
in recent years.
At TAU 98, ALEPH presented \cite{ref:Chen}
branching fractions for 27 distinct modes with $K^\pm$, $K^0_S$,
and/or $K^0_L$ mesons, including $K3\pi$;
DELPHI presented \cite{ref:Andreazza}
12 new (preliminary) branching fractions,
and CLEO presented \cite{ref:Kravchenko}
an analysis of
four modes of the form $K^- h^+\pi^-(\pi^0)\nu_\tau$.
In the $K\pi$ system, ALEPH
sees a hint of $K^{*\prime}(1410)$,
with an amplitude (relative to $K^*(892)$)
which is in good agreement with
the analogous quantity from $\tau\to (\rho,\rho^\prime){\nu_\tau}$.
CLEO sees no evidence for anything beyond the $K^*(892)$.
ALEPH and CLEO both study the $K\pi\pi$ system.
Here, one expects contributions from:
the axial-vector $K_1(1270)$, which decays to $K^*\pi$, $K\rho$,
and other final states;
the axial-vector $K_1(1400)$, which decays predominately to $K^*\pi$;
and, to a much lesser extent, the vector $K^{*\prime}$,
via the Wess-Zumino parity-flip mechanism.
Both ALEPH and CLEO see more $K_1 (1270)$ than $K_1(1400)$,
with significant signals for $K\rho$ as well as $K^*\pi$
in the Dalitz plot projections.
The two $K_1$ resonances are quantum mechanical mixtures
of the $K_{1a}$ (from the $J^{PC} = 1^{++}$ nonet, the strange
analog of the $a_1$),
and the $K_{1b}$ (from the $J^{PC} = 1^{+-}$ nonet, the strange
analog of the $b_1$).
The coupling of the $b_1$ to the $W$ is a second-class current,
permitted in the Standard Model only via isospin violation.
The coupling of the $K_{1b}$ to the $W$ is permitted
only via $SU(3)_f$ violation.
CLEO extracts the $K_{1a} - K_{1b}$ mixing angle
(with a two-fold ambiguity)
and $SU(3)_f$-violation parameter $\delta$ in $\tau \to K_{1b}\nu_\tau$,
giving results consistent with
previous determinations from hadroproduction experiments
\cite{ref:Kravchenko}.
ALEPH studies the $K\bar{K}\pi$ structure, and finds \cite{ref:Chen}
that $K^*K$ is dominant, with little contribution
from $\rho\pi$, $\rho\to K\bar{K}$.
The $K\bar{K}\pi$ mass spectrum is consistent
with coming entirely from $a_1\to K\bar{K}\pi$,
although there may be a large vector component.
ALEPH also analyzes the isospin content of the
$K\pi$, $K\pi\pi$, and $K\bar{K}\pi$
systems. Finally, they classify the net-strange final states
as arising from the vector or axialvector current,
and construct the strange spectral function
$(v+a)^S_1(s)$ (using the data for the $K\pi$ and $K\pi\pi$
components, and the Monte Carlo for the small contributions
from $K3\pi$, $K4\pi$, {\it etc.}) \cite{ref:Chen}.
This function, shown in Fig.~\ref{fig:specs_aleph},
can then be used for QCD studies, as discussed
in the next section.
\begin{figure}[ht]
\psfig{figure=figs/specs_aleph.ps,width=2.6in}
\caption[]{Total $V+A$ spectral function from $\tau$ decays into
strange final states. from ALEPH.
The contributions from the exclusive channels, from data and MC,
are indicated \cite{ref:Chen}.}
\label{fig:specs_aleph}
\end{figure}
\subsection{$\mathbf{m_s}$ from $\mathbf{R_{\tau,s}}$}
\label{ss-ms}
The total strange spectral function can be used
to extract QCD parameters, in direct analogy
with the total and non-strange rates and moments
as described in sections \ref{s-leptonic} and \ref{ss-moments}.
In the strange case, we have
\begin{eqnarray*}
R_\tau^s &\equiv& \frac{{\cal B}_{Kn\pi}}{{\cal B}_e} \\
&=& 3 V_{us}^2 S_{EW} (1+\delta_{pert}+\delta_{mass}^s+\delta_{NP}),
\end{eqnarray*}
and we focus on the quark mass term:
$$\delta_{mass}^s \simeq
- 8 \frac{\bar{m}_s^2}{m_\tau^2} \left[
1 +\frac{16}{3}\frac{\alpha_S}{\pi} +
{\cal O}\left(\frac{\alpha_S}{\pi}\right)^2 \right],$$
where $\bar{m}_s = m_s(m_\tau^2)$ is the $\overline{\mbox{MS}}$
running strange quark mass, evaluated at the tau mass scale.
For $m_s(m_\tau^2) \approx 150$ MeV/c$^2$, we expect
$\delta_{mass}^s \approx -10\%$, and $R_\tau^s \approx 0.16$
(but with a large uncertainty due to poor convergence of
the QCD expansion).
The history of the calculation of the
${\cal O}\left(\frac{\alpha_S}{\pi}\right)^2$ term,
and the apparent convergence of the series, has been rocky.
But considerable progress has been made in the last year,
and the theoretical progress is reviewed in \cite{ref:Prades}.
The value of the strange quark mass appears in many
predictions of kaon properties;
most importantly, it appears in the theoretical expression
for the parameter governing direct CP violation
in the kaon system, $\epsilon^\prime/\epsilon$.
Thus we need to know its value in order to extract information
on the CKM matrix from measurements of direct CP violation.
ALEPH has constructed the strange spectral function
as described in the previous section
and shown in Fig.~\ref{fig:specs_aleph},
and has calculated its integral:
$$R_\tau^s = 0.1607\pm0.0066$$
and its moments $R_{\tau,s}^{kl}$.
By comparing with the non-strange moments,
they cancel, to lowest order,
the mass-independent non-perturbative terms.
They fit for $m_s(m_\tau^2)$
and the residual non-perturbative condensates.
They obtain \cite{ref:Hoecker}
the strange quark running mass
$m_s(m_\tau^2) = (163^{+34}_{-43})$ MeV/c$^2$.
At the (1 GeV)$^2$ scale,
$m_s(1\,\mbox{GeV}^2) = (217^{+45}_{-57})$ MeV/c$^2$,
which compares well with estimates from
sum rules and lattice calculations.
The uncertainty is rather large, especially
the component that comes from the uncertainty
in the convergence in the QCD series;
but improvements are expected.
\section{TAU NEUTRINO MASS}
\label{s-mnutau}
In the Standard Model, the neutrinos are assumed to be massless,
but nothing prevents them from having a mass.
If they do, they can mix (in analogy with the down-type quarks),
exhibit CP violation, and potentially decay.
In the so-called ``see-saw'' mechanism, the tau neutrino
is expected to be the most massive neutrino.
Indirect bounds from cosmology and big-bang nucleosynthesis
imply that
if the $\nu_\tau$ has a lifetime long enough to have not decayed
before the period of decoupling,
then a mass region between 65 eV/c$^2$ and 4.2 GeV/$c^2$
can be excluded.
For unstable neutrinos, these bounds can be evaded.
A massive Dirac neutrino will have a right-handed component,
which will interact very weakly with matter via the
standard charged weak current.
Produced in supernova explosions, these right-handed neutrinos
will efficiently cool the newly-forming neutron star,
distorting the time-dependent neutrino flux.
Analyses of the detected neutrino flux
from supernova 1987a,
results in allowed ranges $m_{\nu_\tau} < 15-30$ KeV/$c^2$
or $m_{\nu_\tau} > 10-30$ MeV/$c^2$,
depending on assumptions.
This leaves open a window for an MeV-range mass for $\nu_\tau$
of 10-30 MeV/$c^2$, with lifetimes on the order of
$10^5 - 10^9$ seconds.
The results from Super-K
(\cite{ref:Nakahata}, see section \ref{s-neuosc} below)
suggest neutrino mixing, and therefore, mass.
If they are observing $\nu_\mu \leftrightarrow \nu_\tau$
oscillation, then \cite{ref:McNulty} $m_{\nu_\tau} < 170$ KeV/c$^2$,
too low to be seen in collider experiments.
If instead they are observing oscillations
of $\nu_\mu$ to some sterile neutrino,
then there is no information from Super-K
on the tau neutrino mass.
If neutrino signals are observed from a galactic supernova,
it is estimated that neutrino masses as low as of 25 eV/$c^2$
could be probed by studying the dispersion in arrival time
of the neutrino events in a large underground detector
capable of recording neutral current interactions.
Very energetic neutrinos from a distant active galactic nucleus (AGN)
could be detected at large underground detectors (existing or planned).
If neutrinos have mass and therefore a magnetic moment,
their spin can flip in the strong magnetic field of the AGN,
leading to enhanced spin-flip and flavor oscillation effects
\cite{ref:Husain}.
The detectors must have the ability to measure the
direction of the source, the energy of the neutrino,
and its flavor.
At TAU 98, CLEO presented \cite{ref:Duboscq} two new limits,
one using $\tau\to 5\pi^\pm{\nu_\tau}$ and $3\pi^\pm2\pi^0{\nu_\tau}$,
and a preliminary result using $3\pi^\pm\pi^0{\nu_\tau}$.
Both results are based on the distribution of events in the
2-dimensional space of $m_{n\pi}$ {\it vs.}\ $E_{n\pi}^{lab}$,
looking for a kinematical suppression in the
high-mass, high energy corner
due to a 10 MeV-scale neutrino mass (see Fig.~\ref{fig:duboscq1}).
This technique has been used previously by ALEPH, DELPHI, and OPAL.
\begin{figure}[ht]
\psfig{figure=figs/duboscq1,width=2.6in}
\caption[]{The scaled hadronic energy $E_{n\pi}^{lab}/E_{beam}$
{\it vs.}\ hadronic mass for (a) the $5\pi$ and
(b) the $3\pi^\pm2\pi^0$ event candidates from CLEO.
Ellipses represent the resolution countours \cite{ref:Duboscq}.}
\label{fig:duboscq1}
\end{figure}
A summary of the best direct tau neutrino mass limits
at 95\%\ C.L.~is given \cite{ref:McNulty} in Table~\ref{tab:numass}.
\begin{table}[!ht]
\centering
\label{tab:numass}
\caption[]{Limits on the $\tau$ neutrino mass.}
\begin{tabular}{llr} \hline
Indirect & from ${\cal B}_e$ & 38 MeV \\
ALEPH & $5\pi(\pi^0)$ & 23 MeV \\
ALEPH & $3\pi$ & 30 MeV \\
ALEPH & both & 18.2 MeV \\
OPAL & $5\pi$ & 43 MeV \\
DELPHI & $3\pi$ & 28 MeV \\
OPAL & $3\pi - vs - 3\pi$ & 35 MeV \\
CLEO (98) & $5\pi$, $3\pi2\pi^0$ & 30 MeV \\
CLEO (98p) & $4\pi$ & 31 MeV \\
\hline
\end{tabular}
\end{table}
The limits are usually dominated by a few ``lucky'' events
near the endpoint, which necessarily have a low probability;
they are likely to be upward fluctuations
in the detector's mass {\it vs.}\ energy resolution response.
Therefore, it is essential to understand and model
that response, especially the tails.
Extracting meaningful limits on the neutrino mass
using such kinematical methods is made difficult by
many subtle issues regarding resolution,
event migration, modeling of the spectral functions,
and certainly also {\it luck}.
Pushing the discovery potential to or below 10 MeV
will require careful attention to these issues,
but most importantly, much higher statistics.
We look forward to the high-statistics event samples
obtainable at the B Factories soon to come on line.
\section{NEUTRINO OSCILLATIONS}
\label{s-neuosc}
If neutrinos have mass,
then they can mix with one another,
thereby violating lepton family number conservation.
In a two-flavor oscillation situation,
a beam of neutrinos that are initially of one pure flavor,
{\it e.g.}, $\nu_\mu$
will oscillate to another flavor, {\it e.g.}, $\nu_\tau$
with a probability given by
$$P(\nu_\mu\to\nu_\tau) =
\sin^2(2\theta_{\mu\tau})\sin^2(\pi L/L_0) ,$$
where the strength of the oscillation is governed by
the mixing parameter $\sin^2 2\theta_{\mu\tau}$.
$L$ is the distance from the source of initially pure $\nu_\mu$
in meters, and
the oscillation length $L_0$ is given by
$$ L_0 = \frac{2.48 E_\nu\,[\mbox{GeV}]}{
\Delta m^2\,[\mbox{eV}^2] }. $$
$\Delta m^2$ is the difference of squared masses
of the two flavors measured in eV$^2$, and
$E_\nu$ is the neutrino energy measured in GeV.
This formula is only correct in vacuum.
The oscillations are enhanced if the neutrinos are travelling
through dense matter, as is the case for neutrinos
from the core of the Sun.
This enhancement, known as the MSW effect, is invoked
as an explanation
for the deficit of $\nu_e$ from the Sun's
core as observed on Earth.
In such a scenario, all three neutrino flavors mix with one another.
Evidence for neutrino oscillations
has been seen in the solar neutrino deficit
($\nu_e$ disappearance),
atmospheric neutrinos (apparently, $\nu_\mu$ disappearance),
and neutrinos from $\mu$ decay ($\nu_\mu\to \nu_e$
and $\bar{\nu}_\mu\to \bar{\nu}_e$ appearance, at LSND).
At this workshop, upper limits were presented
for neutrino oscillations from the $\nu_\mu$ beam at CERN, from
NOMAD \cite{ref:Paul} and CHORUS \cite{ref:Cussans}.
It appears that, if the evidence for neutrino oscillations
from solar, atmospheric, and LSND experiments are {\it all} correct,
the pattern cannot be explained with only three
Standard Model Dirac neutrinos.
There are apparently three distinct $\Delta m^2$ regions:
\begin{eqnarray*}
\Delta m^2_{solar} &=& 10^{-5} \ \mbox{or}\ 10^{-10}\ \mbox{eV}^2 \\
\Delta m^2_{atmos} &=& 10^{-2} \ \mbox{to}\ 10^{-4} \mbox{eV}^2 \\
\Delta m^2_{LSND} &=& 0.2 \ \mbox{to}\ 2 \ \mbox{eV}^2 .
\end{eqnarray*}
This mass hierarchy is difficult (but not impossible)
to accomodate in a 3 generation model.
The addition of a
$4^{th}$ (sterile? very massive?) neutrino
can be used to describe either the solar neutrino
or atmospheric neutrino data \cite{ref:Gonzalez}
(the LSND result requires $\nu_\mu\to \nu_e$).
The introduction of such a $4^{th}$ neutrino makes it
relatively easy to describe all the data.
In addition, a light sterile neutrino is a candidate
for hot dark matter, and a heavy neutrino, for cold dark matter.
\subsection{Results from Super-K}
\label{ss-superk}
We turn now to the results on neutrino oscillations
from Super-Kamiokande, certainly
the highlight of {\it any} physics conference in 1998.
Neutrinos produced in atmospheric cosmic ray showers
should arrive at or below the surface of the earth
in the ratio $(\nu_\mu+\bar{\nu}_\mu)/(\nu_e+\bar{\nu}_e) \simeq 2$
for neutrino energies $E_\nu < 1$ GeV, and somewhat higher
at higher energies. There is some uncertainty in the
flux of neutrinos of each flavor from the atmosphere,
so Super-K measures \cite{ref:Nakahata}
the double ratio:
$$ R =
\left(\frac{\nu_\mu+\bar{\nu}_\mu}{\nu_e+\bar{\nu}_e}\right)_{observed}
\left/
\left(\frac{\nu_\mu+\bar{\nu}_\mu}{\nu_e+\bar{\nu}_e}\right)_{calculated}.
\right.
$$
They also measure the
zenith-angle dependence of the flavor ratio;
upward-going
neutrinos have traveled much longer since they were produced,
and therefore have more time to oscillate.
Super-K analyzes several classes of events
(fully contained and partially contained,
sub-GeV and multi-GeV),
classifying them as ``e-like'' and ``$\mu$-like''.
Based on the flavor ratio, the double-ratio,
the lepton energy,
and the zenith-angle dependence,
they conclude that they are observing effects
consistent with $\nu_\mu$ disappearance and thus neutrino oscillations
(see Fig.~\ref{fig:superkz}).
Assuming $\nu_\mu\to\nu_\tau$, their best fit gives
$\Delta m^2 = 2.2\times 10^{-3}\,\mbox{eV}^2$, $\sin^2 2\theta_{\mu\tau} = 1$
(see Fig.~\ref{fig:icarustau}).
They also measure the
upward through-going and stopping muon event rates
as a function of zenith angle, and see consistent results.
Whether these observations imply that muon neutrinos are oscillating
into tau neutrinos or into some other (presumably sterile) neutrino
is uncertain.
\begin{figure}[ht]
\psfig{figure=figs/superkz.ps,width=2.6in}
\caption[]{Zenith angle distribution of atmospheric
neutrino events:
(a) sub-GeV $e$-like;
(b) sub-GeV $\mu$-like;
(c) multi-GeV $e$-like;
(d) multi-GeV $\mu$-like and partially-contained.
Shaded histograms give the MC expectations
without oscillations, dotted histograms
show the best fit assuming neutrino oscillations \cite{ref:Nakahata}.}
\label{fig:superkz}
\end{figure}
Super-K also observes \cite{ref:Nakahata}
some 7000 low-energy $\nu_e$ events
which point back to the sun, presumably produced during
$^8B$ synthesis in the sun.
They measure a flux which is significantly smaller
than the predictions from standard solar models
with no neutrino mixing
($\approx 40\%$, depending on the model),
with a hint of energy dependence in the (data/model) ratio.
They see no significant difference
between solar neutrinos which pass through the earth
(detected at night) and those detected during the day.
\subsection{NOMAD and CHORUS}
\label{ss-nomad}
Two new accelerator-based
``short-baseline'' neutrino oscillation experiments
are reporting null results at this workshop.
The NOMAD and CHORUS detectors are situated in the
$\nu_\mu$ beam at CERN, searching for $\nu_\mu\to \nu_\tau$ oscillations.
They have an irreducible background from $\nu_\tau$ in the beam
(from $D_s$ production and decay),
but it is suppressed relative to the $\nu_\mu$ flux by $5\times 10^{-6}$.
NOMAD is an electronic detector which searches for
tau decays to $e\nu\nu$, $\mu\nu\nu$,
$h n\pi^0\nu$, and $3\pi n\pi^0\nu$.
They identify them as decay products of a tau
from kinematical measurements,
primarily the requirement of
missing $p_t$ due to the neutrino(s).
They see no evidence for oscillations \cite{ref:Paul},
and set the following limits at 90\%\ C.L.:
$$P(\nu_\mu \to \nu_\tau) < 0.6\times 10^{-3}$$
$$\sin^22\theta_{\mu\tau} < 1.2\times 10^{-3}$$
for $\Delta m^2 > 10^2$ eV$^2$.
The exclusion plot is shown in Fig.~\ref{fig:nomad}.
The CHORUS experiment triggers on
oscillation candidates with an electronic detector,
then aims for the direct observation of production and decay
of a tau in a massive nuclear emulsion stack target.
they look for tau decays to $\mu\nu\nu$ or $h n\pi^0\nu$,
then look for a characteristic pattern in their emulsion
corresponding to: nothing, then a
tau track, a kink, and a decay particle track.
They also see no evidence for oscillations \cite{ref:Cussans},
and set limits that are virtually identical to those of NOMAD;
the exclusion plot is shown in Fig.~\ref{fig:nomad}.
\begin{figure}[ht]
\psfig{figure=figs/nomad.ps,width=2.6in}
\caption[]{Exclusion plot at 90\%\ C.L.~for $\nu_\mu\leftrightarrow\nu_\tau$
oscillations, from CHORUS and NOMAD \cite{ref:Paul}.}
\label{fig:nomad}
\end{figure}
CHORUS has also reconstructed a beautiful event,
shown in Fig.~\ref{fig:chorusevt},
in which a $\tau^-$ lepton is tracked
in their emulsion \cite{ref:Cussans}.
They infer the decay chain:
$\nu_\mu N \to \mu^- D^{*+}_s N$,
$D^{*+}_s \to D^{+}_s \gamma$,
$D^{+}_s \to \tau^+\nu_\tau$,
$\tau^+\to \mu^+\nu_\mu\bar{\nu}_\tau$.
In their emulsion, they track the $D^{+}_s$, the $\mu^-$,
the $\tau^+$, and the decay $\mu^+$.
\begin{figure}[ht]
\psfig{figure=figs/chorusevt.ps,width=2.6in}
\caption[]{Tracks in the CHORUS emulsion
from a $D_s\to \tau\nu$ candidate \cite{ref:Cussans}.}
\label{fig:chorusevt}
\end{figure}
\subsection{OBSERVATION OF $\mathbf{\nu_\tau}$}
The appearance of a $\tau$ and its subsequent decay
in a detector exposed to a neutrino beam
would constitute direct observation
of the tau neutrino.
Fermilab experiment 872 ({\bf D}irect {\bf O}bservation
of {\bf NU\_T}au, DONUT) is designed to directly see
for the first time, such events.
The experiment relies on the production of $D^+_s$
mesons, which decay to $\tau^+\nu_\tau$ with
branching fraction $\simeq 4\%$.
A detector downstream from a beam dump
searches for $\tau^-$ production and subsequent
decay (with a kink) in an emulsion target.
They have collected, and are presently analyzing, their data;
they expect to find $40\pm12$ $\nu_\tau$ interactions \cite{ref:Thomas}.
At TAU 98, they showed \cite{ref:Thomas} an event
consistent with such an interaction; see Fig.~\ref{fig:donutevt}.
If/when a sufficient number of such events are
observed and studied, there will finally be
direct observational evidence for the tau neutrino.
\begin{figure}[ht]
\psfig{figure=figs/donutevt.ps,width=2.6in}
\caption[]{Candidate event for $\nu_\tau\to \tau X$, $\tau\to \mu\nu\nu$,
in the DONUT emulsion.
There is a 100 mrad kink 4.5 mm from the interaction vertex.
The scale units are microns \cite{ref:Thomas}.}
\label{fig:donutevt}
\end{figure}
\section{WHAT NEXT?}
\label{s-whatnext}
The frontiers of tau physics center on ever-higher precision,
farther reach for rare and forbidden phenomena,
and a deeper study of neutrino physics.
The next generation of accelerator-based
experiments will search for
neutrino oscillations with the small $\Delta m^2$
suggested by the Super-K results,
and do precision studies of $\tau$ decays
with samples exceeding $10^8$ events.
\subsection{Long-baseline neutrino oscillations}
\label{ss-longbase}
Several ``long-baseline'' experiments are planned,
in which $\nu_\mu$ beams are produced at accelerators,
allowed to drift (and hopefully, oscillate
into $\nu_e$, $\nu_\tau$, or sterile neutrinos)
for many km, and are then detected.
The motivation comes from the Super-K results,
which suggest $\nu_\mu \leftrightarrow \nu_\tau$ oscillations
with $\Delta m^2$ in the $10^{-2} - 10^{-3}$ eV$^2$ range.
For such small $\Delta m^2$ values,
accelerator-produced $\nu_\mu$ beams must travel many kilometers
in order to get appreciable $\nu_\mu\to \nu_\tau$ conversion.
At Fermilab, an intense, broad-band $\nu_\mu$ beam
is under construction (NuMI).
The MINOS experiment consists of
two nearly identical detectors which will search for
neutrino oscillations with this beam \cite{ref:Thomas}.
The near detector will be on the Fermilab site.
The neutrino beam will pass through Wisconsin,
and neutrino events will be detected
at a far detector
in the Soudan mine in Minnesota, 720 km away.
The experiment is approved, and is scheduled
to turn on in 2002.
From the disappearance of $\nu_\mu$'s between the
near and far detectors,
MINOS will be sensitive \cite{ref:Thomas}
to oscillations with mixing angle
$\sin^2 2\theta \stackrel{>}{\scriptstyle\sim} 0.1$
for $\Delta m^2 \stackrel{>}{\scriptstyle\sim} 2\times 10^{-3}$ eV$^2$.
From the appearance of an excess of $\nu_e$ events
in the far detector, they are sensitive to
$\nu_\mu\to\nu_e$ mixing down to
$\sin^2 2\theta \stackrel{>}{\scriptstyle\sim} 0.002$
for $\Delta m^2 \stackrel{>}{\scriptstyle\sim} 2\times 10^{-2}$ eV$^2$.
MINOS may be able to detect the appearance
of $\nu_\tau$'s from
$\nu_\mu\to\nu_\tau$ oscillations
using the decay mode
$\tau\to\pi\nu_\tau$.
They expect a sensitivity for such oscillations of
$\sin^2 2\theta \stackrel{>}{\scriptstyle\sim} 0.21$
for $\Delta m^2 \stackrel{>}{\scriptstyle\sim} 2\times 10^{-2}$ eV$^2$.
At CERN, the {\bf N}eutrino Beam to {\bf G}ran {\bf S}asso
(NGS) project plans on building several neutrino detectors
in the Gran Sasso Laboratory in central Italy,
732 km from the CERN wide-band $\nu_\mu$ beam.
The energy and flux of the CERN $\nu_\mu$ beam
are both somewhat higher than is planned for NuMi at Fermilab.
The ICARUS experiment \cite{ref:Bueno},
using liquid argon TPC as a target and detector,
is optimized for $\nu_\tau$ and $\nu_e$ appearance.
It is approved and expects to be taking data
as soon as the beam is available (2003?).
They can search for $\nu_\mu\to \nu_e$ appearance down to
$\sin^2 2\theta \stackrel{>}{\scriptstyle\sim} 1\times 10^{-3}$,
and $\nu_\mu\to \nu_\tau$ appearance down to
$\sin^2 2\theta \stackrel{>}{\scriptstyle\sim} 5\times 10^{-3}$,
for $\Delta m^2 \stackrel{>}{\scriptstyle\sim} 2\times 10^{-2}$ eV$^2$.
A projected exclusion plot is shown in Fig.~\ref{fig:icarustau}.
\begin{figure}[ht]
\psfig{figure=figs/icarustau.ps,width=2.6in}
\caption[]{ICARUS excluded region
for $\nu_\mu \to \nu_\tau$ oscillations
if no signal is observed \cite{ref:Bueno}.
The region favored by Super-K
is shown near $\sin^2 2\theta = 1$,
with $\Delta m^2$ between $10^{-1} - 10^{-3}$ eV$^2$.}
\label{fig:icarustau}
\end{figure}
Several other detectors are being proposed: OPERA \cite{ref:Bueno}
(a lead-emulsion stack),
NICE (iron and scintillator),
AQUA-RICH (water $\check{\mbox{C}}$erenkov), and NOE.
The NOE detector \cite{ref:Scapparone}
will be consist of TRD and calorimeter modules,
optimized for the detection of electrons from
$\nu_\mu\to \nu_e\to e X$,
and also and from $\nu_\mu\to\nu_\tau\to \tau X$,
$\tau\to e\nu\nu$ (with missing $p_t$).
They also hope to measure the rate for neutral current events
relative to charged current events.
If the interpretation of the Super-K data is correct,
we will soon have a wealth of data to pin down
the oscillation parameters for
$\nu_\mu\leftrightarrow\nu_e$, $\nu_\tau$, and $\nu_{sterile}$.
\subsection{High luminosity $e^+e^-$ Colliders}
\label{ss-collider}
The IHEP Laboratory in Beijing has been operating
the BEPC collider and the BES detector,
with $e^+e^-$ collisions at or above tau pair threshold,
for many years. They are pursuing the physics of taus
(their measurement of the tau mass totally dominates
the world average), $\psi$ spectroscopy and decay,
and charm.
They have proposed the construction
of a much higher luminosity tau-charm factory (BTCF).
However,
because of the limitation of funds, the BTCF will
not be started for at least 5 years \cite{ref:Qi}.
The plan for the near term is to continue to improve
the luminosity of the existing BEPC collider.
In tau physics, they
hope to produce results from the BES experiment at BEPC
on $m_{\nu_\tau}$, where they favor \cite{ref:Qi}
the use of the decay mode $\tau\to K\bar{K}\pi\nu_\tau$.
Within the next two years, three new ``B Factories'',
high luminosity $e^+e^-$ colliders with
center of mass energies around
10.58 GeV (the $\Upsilon(4S) \to B\bar{B}$ resonance)
will come on line:
the upgraded CLEO III detector at the CESR collider;
BaBar at SLAC's PEP-II collider \cite{ref:Seiden};
and BELLE at KEK's TRISTAN collider \cite{ref:Oshima}.
All these colliders and detectors expect to begin operation
in 1999. The BaBar and BELLE experiments operate with asymmetric beam
energies, to optimize the observation of CP violation in the $B$
mixing and decay; CLEO will operate with symmetric beams.
At design luminosities, these experiments expect to collect
between $10^7$ and $10^8$ tau pair events per year.
In a few years, one can expect more than $\sim 10^{8}$ events
from BaBar, BELLE, and CLEO-III.
The asymmetric beams at BaBar and BELLE should not present
too much of a problem (or advantage) for tau physics;
it may help for tau lifetime measurements.
The excellent $\pi$-$K$ separation that the detectors
require for their $B$ physics goals will also
be of tremendous benefit in the study of
tau decays to kaons.
BaBar and BELLE will also have improved ability
to identify low-momentum muons,
which is important for Michel parameter measurements.
The high luminosities will make it possible
to improve the precision of almost all the
measurements made to date in tau decays,
including the branching fractions, Michel parameters,
and resonant substructure in multi-hadronic decays.
In particular, they will be able to
study rare decays (such as $\eta X\nu_\tau$ or $7\pi\nu_\tau$)
with high statistics.
They will search with higher sensitivity for
forbidden processes, such as LFV neutrinoless decays
and CP violation due to an
electric dipole moment \cite{ref:Seiden,ref:Oshima}
in tau production or a charged Higgs in tau decay.
And they will be able to search for neutrino masses
down to masses below the current 18 MeV/c$^2$ best limit.
At the next Tau Workshop, we look forward to a raft of new results
from DONUT, BES, the B Factories, the Tevatron, and the long-baseline
neutrino oscillation experiments.
I expect that that meeting will be full of
beautiful results, and maybe some surprises.
\section{Acknowledgements}
\label{s-ack}
I thank the conference organizers for a thoroughly stimulating
and pleasant conference.
This work is supported by the U.S.~Department of Energy.
| 2024-02-18T23:40:16.031Z | 1998-11-25T23:04:48.000Z | algebraic_stack_train_0000 | 1,869 | 14,531 |
|
proofpile-arXiv_065-9281 | \section{\label{intro} Introduction}
The high-precision determination of the machine luminosity at
{\sc lep/slc} is an essential ingredient of the success of
precision tests of the electroweak interactions on top of the $Z$
resonance \cite{review}.
As well known, the Bhabha scattering process at small angle (of the
order of a few degrees) is the reference reaction used for luminosity
monitoring at {\sc lep/slc}, owing to its large cross section (dominated by
$t$-channel photon exchange) and its substantial independence of
purely electroweak effects. Experimental efforts in the development of
efficient, dedicated luminometry detectors, as well as precision
calculations of the small-angle Bhabha (hereafter {\sc sabh}) scattering cross
section both contribute to achieve a measurement of the ``$Z$ factories'' luminosity
with a total relative error at the $0.1\%$ level \cite{review,exp,common}.
On the experimental side, the present total uncertainty is smaller than
$0.1\%$ \cite{exp}, close to the $0.05$ level \cite{ward}. As far as the theory
contribution to the luminosity measurement is concerned, the
estimate of the theoretical errors, used by the {\sc lep} collaborations,
is summarized in table \ref{sabs} \cite{common} for centre of mass
energies around and above the $Z$ resonance.
\begin{table}[ht]
\caption[sabs]{\label{sabs}
Theoretical error in {\sc sabh} scattering according
to ref.~\cite{common} at typical {\sc lep1} and {\sc lep2}
energies.}
\medskip
\begin{center}
\begin{tabular}{|l||c|c|} \hline
Type of correction/error & {\sc lep1} ($\%$) & {\sc lep2} ($\%$)\\ \hline \hline
missing photonic $O(\alpha^2L)$ & $0.100 $ & $0.200$ \\
missing photonic $O(\alpha^3L^3)$ & $0.015 $ & $0.030$ \\
vacuum polarization & $0.040 $ & $0.100$ \\
light pairs & $0.030 $ & $0.050$ \\
$Z$-exchange & $0.015 $ & $0.000$ \\ \hline
total & $0.110 $ & $0.250$ \\ \hline
\end{tabular}
\end{center}
\end{table}
Some comments on table \ref{sabs} are in order. The components of the theoretical
error refer to the {\sc sabh} scattering cross section, for any typical event
selection of {\sc lep} experiments, as computed by the program {\tt
BHLUMI v4.03} \cite{bhl}.
The largely dominating source of theoretical error
is due to the missing part of $O(\alpha^2 L)$ subleading photonic corrections,
where $L = \ln(-t/m^2)$ is the collinear logarithm in $t$-channel scattering.
Also the contribution of the missing part of the leading $O(\alpha^3L^3)$
corrections is of photonic nature. The vacuum polarization entry
is the effect of the uncertainty in the hadronic contribution
to the running of $\alpha_{\rm QED}$, when considering the parameterization and
relative error estimate of ref.~\cite{oldvacuum}.
The next contribution is the uncertainty introduced by the corrections due
to the production of light pairs, chiefly $e^+ e^-$ ones. The last
entry refers to the uncertainty associated to the treatment of the
$\gamma$-$Z$ interference.
More details about the strategy adopted in order to estimate the various
sources of theoretical error can be found in ref.~\cite{common}.
After the analysis of ref. \cite{common}, important theoretical
developments took place. Additional work in the sector of two-loop
photonic corrections \cite{pv,kr} led to the conclusion that the
perturbative contribution due to the uncontrolled part of $O(\alpha^2L)$
corrections does not exceed the $0.03\%$ level. This conclusion has been
very recently reinforced by the detailed analysis of ref.~\cite{ward}. Furthermore,
new determinations \cite{vacuum} of $\alpha_{\rm QED}$ lower the error
on hadronic contribution to vacuum polarization in $s$-channel
processes at $\sqrt{s}=M_Z$. This might affect {\sc sabh}
scattering too, although no dedicated analysis for the low-angle
regime exists yet.
As a consequence of this progress, it is relevant
to reduce the uncertainty associated to the light pair
contribution. At present, the calculations available in the
literature and used to estimate the light pair uncertainty
as given in table \ref{sabs} concern a Monte Carlo (hereafter
{\sc mc}) computation based on an approximate $t$-channel matrix
element \cite{jadach} and to an analytical approach with fixed
event selections \cite{russi}. Previous leading logarithmic
evaluations of the dominant light pair contribution to {\sc sabh}
can be found in ref.~\cite{llog}.
In order to improve the existing situation and
contribute to the lowering of the light pair error, in
the present paper a {\sc mc} calculation is drawn with the
exact $e^+ e^- \to e^+ e^-e^+ e^-$ matrix element and taking
into account realistic event selections. The analysis is chiefly
presented at {\sc lep1/slc} energies ($\sqrt{s}=92{\rm~GeV}$),
but numerical results are shown at {\sc lep2} energies
($\sqrt{s}=176{\rm~GeV}$) too.
The impact of the present calculation in the reduction of
the theoretical error for {\sc lep/slc} luminosity measurement
is also discussed.
The outline of the paper is as follows. The details concerning the treatment of
phase space are described in section \ref{phsp}, while in section \ref{caloes}
the selection criteria considered in the present study are reviewed.
Section \ref{algdyn} is devoted to describe the calculation.
The last sections contain a discussion of numerical
results, including comparison with existing analytical calculations
(section \ref{ancomp}) and approximate {\sc mc} results (section \ref{tchn}),
as well as study of the effect of initial-state radiation (hereafter {\sc isr})
(section \ref{isrcorr}). The conclusions and possible developments are
given in section \ref{end}.
\section{\label{phsp} Phase Space Generation}
Pair corrections to Bhabha scattering lead to a four-body kinematics and to
an 8-dimensional phase space. Expansion in few body processes can greatly
simplify the phase space parameterization. The choice between different equivalent
expansions should be suggested by the relevant dynamics.
At high energies and small momentum transfer, which is of interest for
luminosity measurements at {\sc lep}, the leading contribution
to cross section is given by bremsstrahlung Feynman graphs sketched
in figure \ref{tch}, i.e. by $t$-channel photon exchange dynamics.
Indeed, since the relevant dynamics is dominated by pure {\sc qed} processes,
weak effects, such as $Z$-exchange or multiperipheral graphs mediated
by at least one $Z$ boson, will be neglected in the following.
Further the present study deals mainly with electron pair production,
because it is known \cite{common,russi} that heavier particles give a
much smaller contribution.\\
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=140 600 254 681,scale=1.]{fig1.ps}
\end{center}
\caption[bremsstrahlung]{\label{tch}
One of the sixteen bremsstrahlung graphs representing
the leading $t$-channel dynamics.}
\end{figure}
Let us define the photon momentum $k\equiv q_3+q_4$ and its energy
by $\omega\equiv k_0$.
The core of bremsstrahlung contribution is
given by the soft pair approximation, i.e. the limit
$|t|\gg \omega,|\vec k|$. In this regime the emitted pair is almost collinear
to the photon $k$. Thus the phase space configurations in which $q_3$ and $q_4$
are back-to-back are highly suppressed by $t$-channel dynamics.
However, the selection criteria for kinematic events, used by
the {\sc lep} collaborations and reviewed in section \ref{caloes}, scan also
the hard region. When bremsstrahlung processes get smaller, the next
to leading Feynman graph topology is represented by multiperipheral
dynamics shown in figure \ref{mul}. Notice that this contribution is relevant
also for $\gamma\gamma$ physics, being described in its bulk by
the Weizs\"acker-Williams approximation \cite{WW} for which the internal
photons become quasi-real.
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=140 600 254 681,scale=1.]{fig2.ps}
\end{center}
\caption[multiperipheral]{\label{mul}
One of the eight Feynman diagrams for
multiperipheral dynamics.}
\end{figure}
Bremsstrahlung and multiperipheral graphs do not complete
all the Feynman graph topologies. Other two classes of diagrams can
be drawn, namely the annihilation and conversion ones, which are shown in
figure \ref{ext}. Their contribution is less important at high
energies and small momentum transfer. Thus in this paper phase space
parameterization and importance sampling does not deal with these configurations.\\
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=140 600 394 681,scale=1.]{fig3.ps}
\end{center}
\caption[extra]{\label{ext}
Two of the twelve Feynman diagrams representing conversion and
annihilation dynamics, respectively.}
\end{figure}
The two following subsections show how the kinematics is treated according to
the previous considerations about the dynamics.
\subsection{Phase Space for Bremsstrahlung Events}
\subsubsection{Phase Space Parameterization}
Let us consider the four-body phase space for two incoming particles of momentum
$p_-$ and $p_+$, and four outgoing particles of momentum $q_1 \dots q_4$.
Let us define $P\equiv p_-+p_+$.
\begin{equation}
\label{phspvel}
d^8R_4(P;q_1,q_2,q_3,q_4) \equiv \prod_{i=1}^4{d^3 q_i \over 2E_i}
\delta^{(4)}(P-\sum_{i=1}^4 q_i)
\end{equation}
The four-body kinematics can be split into the product of simpler
processes. $t$-channel dynamics suggests to think of the event as
formed by a fermionic current interacting with a particle which
radiates a pair. Thus let us deal with the event as a three body
plus a two body kinematics. The following decomposition can be
introduced
\begin{equation}
d^8R_4(P;q_1,q_2,q_3,q_4) = d^5R_3(P;q_1,q_2,q_{34})d^2R_2(q_{34};q_3,q_4)ds_{34}
\end{equation}
\noindent where $s_{34}$ is the squared mass of $q_{34} \equiv q_3+q_4$.
The three body kinematics can be expressed in terms of invariants. Let us define
$s_{ij\dots} \equiv (q_i+q_j+\dots)^2$ and
$t_{\pm j\dots} \equiv (p_\pm-q_j-\dots)^2$.
A straightforward calculation leads to
\begin{equation}
d^5R_3(P;q_1,q_2,q_{34}) = {1\over 32\lambda^{1/
2}(s,m^2,m^2)\sqrt{-\Delta_4}} \;ds_{12}dt_{-34}ds_{134}dt_{+2}d\varphi
\end{equation}
\noindent where $d\varphi$ integration is over the beam direction, $\lambda$
is the K\"allen function and $\Delta_4$ is the fourth order symmetric Gram
determinant.
The two body kinematics gets the simpler form
\begin{equation}
d^2R_2(q_{34};q_3,q_4) = {\lambda^{1/2}(s_{34},m^2,m^2)\over
8s_{34}}d^2\Omega_3^*
\end{equation}
\noindent where $\Omega_3^*$ is the solid angle of $q_3$ in the frame in which
$\vec q_{34} = 0$.
\subsubsection{Physical Region}
The physical region depends on the choice of the order of integration for the phase
space variables. Let us consider the following choice (read up-down then left-right)
\begin{tabbing}
\hspace*{12em} \= $ds_{34}$\hspace*{1em} \= $dR_3$\hspace*{1em} \= $dR_2$ \\
\> \makebox[8.5em][c]{\hrulefill}\\
\> $s_{34}$ \> $s_{12}$ \> $\cos\theta_3^*$ \\
\> \> $t_{-34}$ \> $\varphi_3^*$ \\
\> \> $s_{134}$ \\
\> \> $t_{+2}$
\end{tabbing}
\noindent With this ordering the limits for the phase space variables are
\begin{eqnarray}
s_{34} & \in & [4m^2,(s^{1/2}-2m)^2] \nonumber \\
s_{12} & \in & [4m^2,(s^{1/2}-s_{34}^{1/2})^2] \nonumber \\
t_{-34} & \in & \left[m^2+s_{34}-\Big({s+s_{34}-s_{12}\over 2}+
\sqrt{{1\over 4}-{m^2\over s}}\lambda^{1/2}
(s,s_{34},s_{12})\Big),\right. \nonumber \\
& & \left.,m^2+s_{34}-\Big({s+s_{34}-s_{12}\over 2}-
\sqrt{{1\over 4}-{m^2\over s}}\lambda^{1/2}
(s,s_{34},s_{12})\Big)\right] \\
s_{134} & \in & \left[m^2+s_{34}-\Big({s_{12}+s_{34}-s\over 2}+
\sqrt{{1\over 4}-{m^2\over s_{12}}}\lambda^{1/2}
(s,s_{34},s_{12})\Big),\right. \nonumber \\
& & \left.,m^2+s_{34}-\Big({s_{12}+s_{34}-s\over 2}-
\sqrt{{1\over 4}-{m^2\over s_{12}}}\lambda^{1/2}
(s,s_{34},s_{12})\Big)\right] \nonumber \\
t_{+2} & \in & \{t_{+2}: \Delta_4 \leq 0 \} \nonumber
\end{eqnarray}
\noindent These limits are exact, thus the phase space generation has efficiency
equal to one if no selection criterion is set on.
\subsubsection{Importance Sampling}
\label{bremwg}
The soft pair limit provides a relatively simple analytical approximation
for the $t$-channel contribution to the cross section.
This expression can be reached either from direct
calculation or from analytic results available in the literature \cite{soft}.
The resulting integral gives a guideline to sample the full cross section formula.
According to the choice for the phase space variables ordering, the weights $w$ follow
\begin{eqnarray}
w(s_{34}) & = & {1\over s_{34}}\nonumber\\
w(s_{12}) & = & \left\{\begin{array}{ll}
1/(s+s_{34}-s_{12})\propto(1/E_{34})
& ,s_{12}\geq s'_{12}\\
1 & ,s_{12}\leq s'_{12}
\end{array}\right.\nonumber\\
w(t_{-34}) & = & {1\over m^2-t_{-34}}\\
w(s_{134}) & = & {1\over s_{134}-m^2}\nonumber\\
w(t_{+2}) & = & \left\{\begin{array}{ll}
1 & ,t_{+2}\leq t'_{+2}\\
1/(t_{+2}^2) & ,t'_{+2}\leq t_{+2}\leq t''_{+2}\\
1/(-t_{+2}) & ,t_{+2}\geq t''_{+2}
\end{array}\right.\nonumber
\end{eqnarray}
\noindent where $w(s_{34})$ and $w(s_{12})$ deal with the infrared pole,
$w(t_{-34})$ and $w(s_{134})$ with the collinear pole, and
$w(t_{+2})$ with the coulomb pole. The boundaries $s'_{12}$, $t'_{+2}$
and $t''_{+2}$ should be set according to the selection criteria.
\subsection{Phase Space for Multiperipheral Events}
Multiperipheral events require true four-body kinematics, since no natural
expansion of the phase space can fit the dynamics, because of the three
propagators between the ingoing particles. Thus a description based on
energies and angles can be preferred to an invariant picture.
\subsubsection{Phase Space Parameterization}
\noindent Let us integrate out $q_4$ and put $E_i\equiv q_{i,0}$
\begin{equation}
d^8R_4(P;q_1,q_2,q_3,q_4) = \delta(E-\sum_{i=1}^4 E_i){d^3 q_1 \over 2E_1}
{d^3 q_2 \over 2E_2}{d^3 q_3 \over 2E_3}{1\over 2E_4}
\end{equation}
\noindent where $E_1$, $E_2$ and $E_3$ are given by mass shell relations and
$E_4$ by energy-momentum conservation.
Let us consider the decay $k\rightarrow q_3,q_4$ where
$k\equiv q_3+q_4$, $\omega\equiv k_0$, and $s_{34}\equiv k^2$.
The three momentum $|\vec q_3|$ and the cosine of the polar angle
$\theta_3$, with respect to a fixed direction $\hat {\vec u_z}$,
are needed as phase space variables.
Let us write $\vec k$ as $\vec k = k_z \hat {\vec u_z} + k_x \hat {\vec u_x}$
the third axis being defined as $\hat {\vec u_y}=\hat {\vec u_z} \times \hat {\vec u_x}$.
Energy conservation implies that
\begin{eqnarray}
\omega^2 + E_3^2 - 2 \omega E_3 &=& m^2 + |\vec q_3|^2 + |\vec k|^2 -2 k_z |\vec q_3| \cos\theta_3 + \nonumber \\
& & - 2 k_x |\vec q_3| \sin\theta_3\sin\varphi_3
\end{eqnarray}
\noindent namely
\begin{equation}
\sin\varphi_3 = {1\over 2 k_x |\vec q_3| \sin\theta_3} ( 2 \omega E_3 - s_{34}
-2 k_z |\vec q_3| \cos\theta_3 )
\end{equation}
\noindent The Jacobian is easily obtained as
\begin{equation}
J^{-1} = { k_x |\vec q_3| \sin\theta_3\cos\varphi_3 \over E_4}
\end{equation}
\noindent This leads to the following phase space parameterization
\begin{eqnarray}
d^8R_4(P;q_1,q_2,q_3,q_4) &=& {1\over 16 k_x E_1 E_2 E_3 |\vec q_3|
\sin\theta_3\cos\varphi_3} \times \nonumber \\
& & d|\vec q_1| d^2\Omega_1 d|\vec q_2| d^2\Omega_2
d|\vec q_3| d\cos\theta_3
\end{eqnarray}
\noindent where $k_x=(\vec P - \vec q_1 - \vec q_2)\cdot \hat {\vec u_x}$,
$E_i = \sqrt{m^2+ |\vec q_i|}$, and $d^2\Omega_i\equiv d\cos\theta_i d\varphi_i$.
\subsubsection{Physical Region}
\noindent The physical region is given by
\begin{equation}
| 2 k_x |\vec q_3| \sin\theta_3 | \ge | 2 \omega E_3 - \omega^2 + s_{34}
-2 k_z |\vec q_3| \cos\theta_3 |
\end{equation}
\noindent which means ($\sin\theta_3\ge 0$)
\begin{equation}
\begin{array}{lll}
2 ( k_x |\vec q_3| \sin\theta_3 - \omega E_3 + k_z |\vec q_3| \cos\theta_3 ) & \ge &
- \omega^2 + s_{34} \\
2 ( k_x |\vec q_3| \sin\theta_3 + \omega E_3 - k_z |\vec q_3| \cos\theta_3 ) & \ge &
\omega^2 - s_{34}
\end{array}
\end{equation}
\noindent By taking into account that $|\vec q_3|\le\sqrt{|\vec q_3|^2+m^2}\le |\vec q_3|+m$
it follows that the range of interest is certainly contained into the domain
\begin{equation}
\begin{array}{lll}
2 ( k_x \sin\theta_3 - \omega + k_z \cos\theta_3 ) |\vec q_3| & \ge &
- \omega^2 + s_{34} \\
2 ( k_x \sin\theta_3 + \omega - k_z \cos\theta_3 ) |\vec q_3| & \ge &
\omega^2 - s_{34} -2 \omega m
\end{array}
\end{equation}
\noindent which leads to
\begin{equation}
{ \omega^2 - s_{34} \over 2 ( k_x \sin\theta_3 + \omega - k_z \cos\theta_3 ) }
\le |\vec q_3| \le
{ \omega^2 - s_{34} \over 2 ( \omega - k_x \sin\theta_3 - k_z \cos\theta_3 ) }
\end{equation}
\noindent if $\omega - k_x \sin\theta_3 - k_z \cos\theta_3 > 0$
\noindent The maximum and minimum of $|\vec q_3|$ are achieved for $\vec q_3$ collinear to
$\vec k$ in the forward and backward direction, respectively.
\subsubsection{Importance Sampling}
\label{multiwg}
The Weizs\"acker-Williams equivalent photon approximation \cite{WW} provides
a useful guideline to deal with the contribution of multiperipheral
diagrams. The resulting integral shows how to sample the full cross section
formula by introducing the weights
\begin{equation}
w(\cos\theta_1) = {1\over -t_{-1}(\cos\theta_1)} \quad,\quad
w(\cos\theta_2) = {1\over -t_{+2}(\cos\theta_2)}
\end{equation}
\noindent where $w(\cos\theta_1)$ and $w(\cos\theta_2)$ mimic the most singular
behaviour of the Weiz\-s\"ac\-ker-Williams spectrum. $t_{-1}$ and $t_{+2}$ are
equal to $(p_- - q_1)^2$ and $(p_+ - q_2)^2$, respectively, as previously
defined. Further a flat importance sampling can be made on $\cos\theta_3$
to match the selection criteria better.
\section{\label{caloes} Selection Criteria}
Two types of selection criteria are considered: {\tt BARE1} and {\tt CALO2}.
They were defined by the "Event Generators for Bhabha Scattering" working
group during the {\sc cern} Workshop "Physics at {\sc lep2}" (1994/1995),
see ref.~\cite{common} for more details.
These algorithms are tailored to Bhabha scattering with photon
emission. Thus they must be modified to fit with pair emission, chiefly
to deal with identical particles and with fermion clusters.
\subsection{A Simple Setup: {\tt BARE1}}
This non-calorimetric criterion selects events with one or more particles per
calo. If more than one particle hits a calo it chooses the most energetic
hit, the reference particle. Thus for each kinematic event two particles
are labelled as the emitted pair and the other two as the beam particles.
\begin{table}[ht]
\caption[acceptance]{\label{angle}
Angular acceptances in degrees for {\tt BARE1}
and {\tt CALO2} algorithms.}
\medskip
\begin{center}
\begin{tabular}{|c||c|c||c|c|} \hline
& \multicolumn{2}{c||}{\tt BARE1} & \multicolumn{2}{c|}{\tt CALO2} \\
\cline{2-5}
& Left Calo & Right Calo & Left Calo & Right Calo \\
\hline\hline
WW & $2.70\rightarrow7.00$ & $2.70\rightarrow7.00$ & $2.97\rightarrow6.73$ & $2.97\rightarrow6.73$ \\
\hline
NN & $3.30\rightarrow6.30$ & $3.30\rightarrow6.30$ & $3.49\rightarrow6.11$ & $3.49\rightarrow6.11$ \\
\hline
NW & $3.30\rightarrow6.30$ & $2.70\rightarrow7.00$ & $3.49\rightarrow6.11$ & $2.97\rightarrow6.73$ \\
\hline
\end{tabular}
\end{center}
\end{table}
Then the beam particles are selected by their energies and angles.
The angular cut can be made with the same acceptance for the two caloes
(wide-wide and narrow-narrow), or with different acceptances (narrow-wide).
\begin{equation}
\begin{array}{lll}
3.3^\circ \leq\theta\leq 6.3^\circ &-& {\rm narrow\;acceptance}\\
2.7^\circ \leq\theta\leq 7.0^\circ &-& {\rm wide\;acceptance}
\end{array}
\end{equation}
\noindent Angular acceptances are summarized in table \ref{angle}.
Further the energy cut is imposed by considering
the adimensional variable $z$ as usually done in the literature
\cite{jadach}
\begin{equation}
\label{zed}
z \equiv 1-{E_{\rm 1}E_{\rm 2}\over E^2_{\rm beam}} \leq z_{\rm max}
\end{equation}
It is worth noticing that small values of $z$ means soft pair
emission, while large values allows for hard pair radiation.
\subsection{A Calorimetric Setup: {\tt CALO2}}
This selection criterion looks for any particle in two caloes,
the former is along the ingoing
electron direction, the latter is along the ingoing positron direction.
Then it scans the detected particles in each calo to pick up the most
energetic one, the reference particle.
\begin{figure}[ht]
\begin{center}
\includegraphics[scale=.8]{fig4.ps}
\end{center}
\caption[calo2]{\label{calo2}
Geometry and acceptances of {\tt CALO2} calorimetric setup.
From ref.~\cite{common}.}
\end{figure}
It defines a cluster for each reference particle. The shape of the cluster
is a square in the $(\theta,\varphi)$ plane of the corresponding calo, its
size is of $3\Delta\theta/16 \times 3\pi/16$ radiants, where
$\Delta\theta$ is the calo polar width.
The cluster center is pinned to the reference particle. The cluster energy
is the sum of the energies of the particles in the cluster itself.
The last step of the algorithm is to reject the events according to
an angular and an energy cut, where angles are referred to the cluster center
and energies to the cluster energy.
Each polar width must be reduced to an effective acceptance,
as shown in figure \ref{calo2}, to prevent the cluster square
to exceed the calo size, i.e. should the reference particle be out
of the effective acceptance but within the calo total width, yet the
event is rejected.
The angular cut can be made with the same polar width for the two
caloes, defining a wide-wide and a narrow-narrow acceptance, or
with different widths for the two caloes, defining a narrow-wide
acceptance
\begin{equation}
\begin{array}{lll}
3.3^\circ \leq\theta\leq 6.3^\circ &-& {\rm width\; for\; narrow\;acceptance}\\
2.7^\circ \leq\theta\leq 7.0^\circ &-& {\rm width\; for\; wide\;acceptance}
\end{array}
\end{equation}
\noindent Angular acceptances are summarized in table \ref{angle}.
Further the energy cut is imposed by considering the adimensional
variable $z$ as given by eq.~(\ref{zed}).
\section{\label{algdyn} Dynamics Calculation}
Generators including the full set, or part, of the {\sc qed} diagrams
for $e^+e^-\rightarrow e^+e^-l^+l^-$ ($l=e,\mu,\tau$)
was already described in ref.~\cite{bdk} and they are used for analysis and
simulation of $\gamma\gamma$ collision processes at {\sc lep} and other
$e^+e^-$ colliders.
In the present paper the {\sc mc} program {\tt PAIRS}, written in {\tt FORTRAN},
was built to compute the $e^+e^-\rightarrow e^+e^-e^+e^-$ process.
It implements the importance samplings and the selection criteria sketched
previously, and computes the exact matrix element, including all
mass terms, by using the {\tt ALPHA} algorithm and the resulting code
\cite{alpha}, that is conceived for the automatic computation of tree-level
multi-particle production amplitudes without any need of
Feynman graphs expansion.
Without entering the details of the algorithm {\tt ALPHA} it is worth
noticing, for the aim of the present study, that the predictions
of this automatic algorithm have been already compared with
diagrammatic results for processes with four fermions in the final
state \cite{fferm} showing an excellent agreement, and also
successfully used to obtain original results for other reactions
\cite{photons}.
The main features of the program {\tt PAIRS} can be summarized
as follows. The code computes the phase space integral by means
of an importance sampling both for bremsstrahlung and
multiperipheral graphs. Since the contribution of the other
topologies is small, as discussed in section \ref{phsp}, there is
no need of a specific strategy to reduce the associated variance.
The integration is performed in two steps. In the former the matrix
element is sampled by using the bremsstrahlung weights, see
section \ref{bremwg}. Then a rejection algorithm selects the events
belonging to the phase space region $\Omega_B$ which is preferred by the
bremsstrahlung dynamics. In the latter the same calculation scheme
is performed in terms of the multiperipheral weights, see
section \ref{multiwg}, and of a rejection algorithm selecting
the remaining phase space region $\Omega_M$, which is preferred
by the multiperipheral dynamics. The treatment of the interference
between the two dynamics is addressed below.
After phase space generation the program deals with identical particles and
other symmetries of the amplitude. Feynman expansion in pure {\sc qed}
shows sixteen bremsstrahlung graphs, eight with initial-state emission
and eight with final-state emission of a fermion pair.
These two families can be sampled at the same time
leading only to eight different weights, which can
be obtained, fixed one of them, by repeated application of
identical fermion exchange symmetry ($ID$ symmetry)
or of $CP$ symmetry. Let us note that under these symmetries
the squared amplitude does not change. There are also eight multiperipheral
graphs, four with an amplitude value and four, obtained twisting
the kernel fermion lines ($TW$ symmetry), with a different value.
The two multiperipheral squared amplitudes are left invariant
by the $ID$ symmetry.
The symmetries of the integrand allow us to cast the cross section
formula in a simpler form, because the phase space jacobians and
the selection criterion characteristic functions share these symmetries too.
An example can clarify the importance of the integrand symmetries. Let us
consider a function $f(x,y)$, symmetric under the change $x \rightarrow y$,
which is integrated over the unit square in the $(x,y)$ plane.
In order to draw the calculation some weights are introduced, for definiteness
say $\omega_1\equiv 1/(x+a)$ ($a>0$) is a suited weight because
of some pathology in $x=-a$, but, since $f(x,y)$ is symmetric under coordinate
exchange, it must also exhibit the same pathology in $y=-a$. Thus the weight
$\omega_2\equiv 1/(y+a)$ must be introduced too.
The integration by using importance sampling follows.
\begin{eqnarray}
& & \int_0^1 \,d x \int_0^1 \,d y \; f(x,y) =
\int_0^1 \,d x \int_0^1 \,d y \; \left( \frac{\omega_1 f(x,y)}{\omega_1+\omega_2}+\frac{\omega_2 f(x,y)}{\omega_1+\omega_2} \right) = \nonumber \\
&=& \;\int_0^1 \, \frac{dx}{x+a} \int_0^1 \,d y \; \frac{(x+a)(y+a)}{x+y+2a}f(x,y) + \nonumber \\
& & + \int_0^1 \,d x \int_0^1 \, \frac{dy}{y+a} \; \frac{(x+a)(y+a)}{x+y+2a}f(x,y) = \nonumber \\
&=& \; 2 \int_0^1 \,d x \int_{\ln a}^{\ln (a+1)} \,d \mu \; \frac{(x+a)(y+a)}{x+y+2a}f(x,y)
\end{eqnarray}
\noindent In the last step coordinate exchange symmetry both of $f(x,y)$ and
integration region was used, and the integration measure changed in order to include
the weight by putting $d\mu\equiv dy/(y+a)$.
Hence the illustrated two channel sampling can be collapsed into a single integration,
because the integrand and the integration region share the same symmetries. The
same strategy can also be adopted for the more complex integral involved in the present
calculation.
Let us name by $x$ the phase space coordinates, by
$d^8R_4(x)$ the phase space volume element defined by eq.~(\ref{phspvel}),
by $N$ the cross section normalization,
by $\chi(x)$ the characteristic function of
the selection criterion, by $p^B_i(x)$ the eight sets of weights for
the bremsstrahlung amplitude and by $p^M_i(x)$ and $p^M_i(x_{\rm TW})$
the four weights for each of the multiperipheral amplitudes.
These quantities share the symmetry properties
\begin{equation}
\begin{array}{llll}
d^8R_4(x_i) &=& d^8R_4(x) & \forall i \in \{ID,CP,TW\}\\
\chi(x_i) &=& \chi(x) & \forall i \in \{ID,CP,TW\} \\
|M(x_i)|^2 &=& |M(x)|^2 & \forall i \in \{ID,CP\} \\
p^{B,M}_i(x_j) &=& p^{B,M}_j(x_i) & \forall i,j \in \{ID,CP,TW\}
\end{array}
\end{equation}
\noindent where $x_i$ is a phase space point obtained from $x$ applying
a suited symmetry among $ID$, $CP$, and $TW$.
\noindent In terms of these quantities the cross section integral becomes
\begin{eqnarray}
\label{master1}
\sigma &=& 8N\int_{\Omega^B}\,{d^8R_4(x)p^B(x)\chi(x)\over\sum_{i\in\{ID,CP\}}p^B_i(x)}
\sum_{\rm spin}|M(x)|^2 + \nonumber \\
& & 4N\int_{\Omega^M}\,{d^8R_4(x)p^M(x)\chi(x)\over\sum_{i\in\{ID,TW\}}p^M_i(x)}
\sum_{\rm spin}\left(|M(x)|^2+|M(x_{TW})|^2\right)
\end{eqnarray}
\noindent This technique is very useful because it permits to treat twenty-four
channels as they were three.
The above division of phase space leaves some problem in the
bremsstrahlung region because of the interferences between
$t$-channel and multiperipheral dynamics. Thus it is useful
to add an extra flat channel to the first integral of eq.~(\ref{master1}).
The flat channel deals also with a multiperipheral
contribution so it must be sampled further by the $p^M_i(x)$ weights.
Hence the cross section integral is
\begin{eqnarray}
\label{master2}
\sigma &=& 8N\int_{\Omega^B}\,{d^8R_4(x)p^B(x)\chi(x)\over\sum_{i\in\{ID,CP\}}p^B_i(x)+\eta}
\sum_{\rm spin}|M(x)|^2 + \nonumber \\
& & 8N\eta\int_{\Omega^B}\,{d^8R_4(x)p^M(x)\chi(x)\over p^M(x)\left(
\sum_{i\in\{ID,CP\}}p^B_i(x)+\eta\right)}
\sum_{\rm spin}|M(x)|^2 + \nonumber \\
& & 4N\int_{\Omega^M}\,{d^8R_4(x)p^M(x)\chi(x)\over\sum_{i\in\{ID,TW\}}p^M_i(x)}
\sum_{\rm spin}\left(|M(x)|^2+|M(x_{TW})|^2\right)
\end{eqnarray}
\noindent where $\eta$ is a suitable weight for the flat channel.
The matrix element is computed by functionally integrating the {\sc qed}
tree level effective lagrangian with the iterative algorithm {\tt ALPHA},
which gives directly the total amplitude $M(x)$.
After the cross section is computed for the real contribution, also the virtual
correction \cite{burgers} corresponding to pair emission is added. The resulting
pair contribution is then normalized against the tree level Bhabha
scattering cross section. The final result can be corrected for initial-state
radiation via collinear structure functions too \cite{alta}.
\section{\label{ancomp} Comparison between {\sc mc} Results and Analytical Calculations}
Before the numerical results being worked out, the {\sc mc}
program is now tested against analytical results already present
in the literature \cite{russi}. Such calculations are available for fixed
selection criteria, among which one resembling {\tt BARE1} algorithm.
A direct comparison, which could be affected by some bias because of
analytical approximations and possibly small differences in the selection
criterion, is shown in figure \ref{rus} and table \ref{rust} for the absolute
value of the pair correction. Solid line represents analytical
computation of ref.~\cite{russi}, while markers show the results of the
present {\sc mc} calculation. The error bars are within
the markers. Entry values are in $\mathrm{pb}$, sum up both real and
virtual part, and are computed for the {\tt BARE1}
setup with symmetric angular acceptance $1.375^\circ\leq\theta\leq 3.323^\circ$
at $\sqrt{s}=92.3{\rm~GeV}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=55 385 280 610,scale=.8]{fig5.ps}
\end{center}
\caption[analytical]{\label{rus}
Comparison between {\sc mc} integration of the exact
matrix element
(markers) and analytical computation of ref.~\cite{russi}
(solid line) as a function of the energy cut $z$, as given
by eq.(\ref{zed}). Entry values
sum up real plus virtual cross sections, and are computed
for {\tt BARE1} setup with symmetric acceptance
$1.375^\circ\leq\theta\leq 3.323^\circ$ at
$\sqrt{s}=92.3{\rm~GeV}$.}
\end{figure}
\begin{table}[ht]
\caption[analyticalt]{\label{rust}
Comparison between present numerical results and analytical
ones of ref.~\cite{russi} with the same settings of figure \ref{rus}.
Entry values are in $\mathrm{pb}$. In this setup Bhabha
tree level cross section is $\sigma_0=175426(14) \;\mathrm{pb}$.
The error on $\sigma_0$ is due to numerical integration.}
\medskip
\begin{center}
\begin{tabular}{|l||c|c|c|c|c|} \hline
& $z=0.1$ & $z=0.3$ & $z=0.5$ & $z=0.7$ & $z=0.9$ \\ \hline \hline
Present {\sc mc} & $-162\pm 4$ & $-95\pm 1$ & $-60\pm 2$ & $-40\pm 3$ & $-13\pm 3$ \\ \hline
Analytical & $-142$ & $-90$ & $-58$ & $-33$ & $ +7$ \\ \hline
\end{tabular}
\end{center}
\end{table}
\noindent Table \ref{rust} shows that the relative difference between
numerical and analytical results is within $15\%$ for the experimentally
relevant values of $z$ \break ($0.3\leq z\leq0.7$), thus providing a rather satisfactory
test of the {\sc mc} program. Further checks were performed with the results of
ref.~\cite{russi} for {\tt CALO2} event selection leading to a similar behaviour.
\section{\label{tchn} Comparison between Exact Matrix Element and $t$-Channel Approximation}
Numerical computations already present in the literature are limited
to bremsstrahlung graphs without fermion exchange and up-down
interference \cite{jadach}. Let us define such an approach as $t$-channel
approximation. This approximation, together with the analytical work of
ref.~\cite{russi}, is an essential ingredient to estimate
the theoretical error associated with light pair correction to {\sc sabh}
scattering.
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=49 230 550 555,scale=.75]{fig6.ps}
\end{center}
\caption[vary]{\label{vary}
In the upper row: exact matrix element (markers)
vs $t$-channel approximation (solid line) as a function
of the energy cut $z$, as given by eq.(\ref{zed}). Entry values
sum up real plus virtual cross sections, and are computed
by varying the angular acceptances of {\tt CALO2} setup
at {\sc lep1/slc} energies.
In the lower row: relative difference between the exact matrix
element and $t$-channel approximation still as a function
of the energy cut. Entry values refer only to the real
pair production cross section.}
\end{figure}
As far as the {\sc mc} calculation of ref.~\cite{jadach} is concerned,
the uncertainty is the sum of a physical error, due to the incomplete
matrix element calculation, and a technical error,
due to algorithm stability and finite {\sc cpu} time.
In ref.~\cite{jadach} the physical error associated to the matrix element
for real pair production is estimated to be $30\%$. Once the virtual
correction is added, the cancellation against the real part can increase
the magnitude of the error. In ref.~\cite{jadach} summing up all the sources of
physical and technical uncertainty leads to a conservative estimate of
the light pair correction of $-1.3\times10^{-4} \pm 2\times10^{-4}$
for narrow-wide {\sc calo2} acceptance at $z=0.5$ at {\sc lep1/slc} energies.
Thus the comparison of $t$-channel approximation with the full matrix
element computation can size the magnitude of the physical error, and
therefore lowers the theoretical error.
In order to remove any spurious effect in the comparison between
such calculations and the present one, the amplitude has been
directly calculated in $t$-channel approximation by using
the {\tt ALPHA} algorithm\footnotemark
\footnotetext{Since the ALPHA approach does not make use of Feynman
graph expansion, this is not entirely straightforward. The goal
is achieved as follows: the input lagrangian is modified by introducing
two distinct photons $\gamma_1$ and $\gamma_2$. The electron couples
to both photons, whereas the muon and the tau lepton couples
to $\gamma_1$ and $\gamma_2$, respectively. Within this modified
lagrangian $m_\tau=m_\mu=m_e$ is assigned and the process
$e^+ \mu^- \rightarrow e^+ \mu^- \tau^+ \tau^-$ is studied. It is immediately
seen that this reproduces the $t$-channel approximation.}
and comparing that amplitude with the full amplitude within the same
selection criteria. Further the same virtual correction
\cite{burgers} for both the cross sections is implemented.
The results are shown in figure \ref{vary} for {\tt CALO2}
setup and different angular acceptances at {\sc lep1/slc} energies.
In the first row the solid line represents the $t$-channel approximation,
while the markers the results of the present full calculation.
Entries are normalized to Bhabha
tree level cross section ($\sigma_0$) and sum up both real and
virtual part. In the second row the markers represent the relative difference
between the real part of the $t$-channel approximation
($\sigma_t^{\rm R}$) and the real part of the full calculation
($\sigma_{\rm tot}^{\rm R}$).
The comparison shows appreciable differences only for $z\geq 0.7$,
without much sensitivity to angular acceptance variation.
By increasing $z$ greater than $0.7$, the relative difference between
the real part of the cross sections can grow up to $15\%$.
\begin{table}[ht]
\caption[finalt]{\label{finalt}
Comparison between exact and $t$-channel approximation
numerical results for narrow-wide {\tt CALO2} setup, see figure \ref{final}.
Entry values are in $\mathrm{pb}$ except for the last column.
In this setup Bhabha tree level cross section
is $\sigma_0=21939(1) \;\mathrm{pb}$.
The error on $\sigma_0$ is due to numerical integration.}
\medskip
\begin{center}
\begin{tabular}{|c|r|r||r|r|} \hline
& Full Dynamics & $t$-Channel & Abs.Diff. & Rel.Diff. ($\%$) \\ \hline \hline
$z=0.1$ & $-25.36 \pm 0.01$ & $-25.48 \pm 0.02$ & $ 0.12 \pm 0.02$ & $ 0.48 \pm 0.08$ \\ \hline
$z=0.3$ & $-12.85 \pm 0.05$ & $-13.34 \pm 0.02$ & $ 0.49 \pm 0.05$ & $ 3.67 \pm 0.38$ \\ \hline
$z=0.5$ & $ -7.14 \pm 0.05$ & $ -8.43 \pm 0.02$ & $ 1.29 \pm 0.05$ & $15.30 \pm 0.63$ \\ \hline
$z=0.7$ & $ -4.98 \pm 0.12$ & $ -7.06 \pm 0.06$ & $ 2.08 \pm 0.14$ & $29.46 \pm 2.23$ \\ \hline
$z=0.9$ & $ -1.75 \pm 0.17$ & $ -6.90 \pm 0.07$ & $ 5.15 \pm 0.19$ & $74.64 \pm 3.51$ \\ \hline
\end{tabular}
\end{center}
\end{table}
A more refined computation is shown in figure \ref{final},
allowing a higher statistics, to get an improved accuracy in order to
establish a better evaluation of the physical error. The result is shown
in figure \ref{final} and table \ref{finalt} for {\tt CALO2} narrow-wide
setup. The solid line represents $t$-channel approximation, while the
markers the full calculation. The error bars are within the markers.
Entry values are in $\mathrm{pb}$, and sum up both real and virtual
part. Table \ref{finalt} shows the numerical values for the two
calculations and the absolute and relative difference. Entry
values are in $\mathrm{pb}$.
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=55 385 280 610,scale=.8]{fig7.ps}
\end{center}
\caption[final]{\label{final}
Results of a refined computation of the exact matrix element (markers)
vs $t$-channel approximation (solid line) as a function
of the energy cut $z$, as given by eq.(\ref{zed}). Entry values
sum up real plus virtual cross sections, and are computed
for narrow-wide {\tt CALO2} setup.}
\end{figure}
The relative difference between the $t$-channel approximation and
the full calculation listed in table \ref{finalt} shows that
the physical error, introduced by using the $t$-channel approximation,
is within $30\%$ for the relevant values of $z$.
In the soft region (small $z$ values) the relative difference gets
smaller since the bremsstrahlung diagrams are the largely dominating
contribution at this regime. On the other hand, in the hard region (large $z$ values)
the multiperipheral diagrams become more and more relevant, worsening
the agreement with the $t$-channel approximation.
Some tests performed at {\sc lep2} energies show the physical error behaviour
given in table \ref{lep2t}. The difference between the exact calculation and
the $t$-channel approximation is within $10\%$ for the experimentally relevant
$z$ values.
\begin{table}[ht]
\caption[lep2]{\label{lep2t}
Comparison between exact and $t$-channel approximation numerical results
for narrow-wide {\tt CALO2} setup at {\sc lep2} energies.
Entry values are in $\mathrm{pb}$ except for the last column.
In this setup Bhabha tree level cross section
is $\sigma_0=6087.0(3) \;\mathrm{pb}$.
The error on $\sigma_0$ is due to numerical integration.}
\medskip
\begin{center}
\begin{tabular}{|c|r|r||r|r|} \hline
& Full Dynamics & $t$-Channel & Abs.Diff. & Rel.Diff. ($\%$) \\ \hline \hline
$z=0.3$ & $-4.47 \pm 0.01$ & $-4.27 \pm 0.01$ & $-0.20 \pm 0.02$ & $-4.61 \pm 0.24$ \\ \hline
$z=0.5$ & $-2.91 \pm 0.01$ & $-2.68 \pm 0.01$ & $-0.23 \pm 0.02$ & $-8.70 \pm 0.62$ \\ \hline
$z=0.7$ & $-2.43 \pm 0.01$ & $-2.28 \pm 0.01$ & $-0.15 \pm 0.02$ & $-6.62 \pm 0.64$ \\ \hline
\end{tabular}
\end{center}
\end{table}
The previous results refer to the {\tt CALO2} calorimetric setup
explained in section~\ref{caloes}, where the calo acceptances
are set according to ref.~\cite{jadach}. Yet some tests were drawn also
with different angular acceptances to match the recent angular standard
of {\sc lep} luminometers, as previously considered in refs.~\cite{common,ward}.
The new ranges are $(1.50,3.20)$ degrees for the wide acceptance and $(1.62,2.84)$
degrees for the narrow acceptance at {\sc lep1/slc} energies. The results are
shown in table \ref{newcalo} confirming the same behaviour previously obtained.
\begin{table}[ht]
\caption[newcalo]{\label{newcalo}
Comparison between exact and $t$-channel approximation numerical results
for narrow-wide {\tt CALO2} setup at {\sc lep1/slc} energies with a
different choice for the angular acceptances: wide $(1.50^\circ,3.20^\circ)$,
narrow $(1.62^\circ,2.84^\circ)$. Entry values are in $\mathrm{pb}$
except for the last column.
In this setup Bhabha tree level cross section
is $\sigma_0=87203(4) \;\mathrm{pb}$.
The error on $\sigma_0$ is due to numerical integration.}
\medskip
\begin{center}
\begin{tabular}{|c|r|r||r|r|} \hline
& Full Dynamics & $t$-Channel & Abs.Diff. & Rel.Diff. ($\%$) \\ \hline \hline
$z=0.3$ & $-41.01 \pm 0.30$ & $-43.62 \pm 0.20$ & $ 2.61 \pm 0.36$ & $ 5.99 \pm 0.85$ \\ \hline
$z=0.5$ & $-23.91 \pm 0.42$ & $-28.91 \pm 0.25$ & $ 5.31 \pm 0.49$ & $18.36 \pm 1.84$ \\ \hline
$z=0.7$ & $-16.99 \pm 0.69$ & $-24.99 \pm 0.25$ & $ 8.00 \pm 0.74$ & $32.02 \pm 3.27$ \\ \hline
\end{tabular}
\end{center}
\end{table}
It must be emphasized that now a {\sc mc} program, that includes
all the {\sc qed} Feynman diagrams for $e^+e^-\rightarrow e^+e^-l^+l^-$
($l=e,\mu,\tau$), is available to calculate pair production
corrections. Hence the physical error for electron real pair
production can be reduced well below the $30\%$ quoted
in the literature. Further it is worth noticing that muon pair production
was checked to be one order of magnitude smaller than the electron one.
\section{\label{isrcorr} Initial-State Radiation}
Corrections by {\sc isr} can be evaluated by the aid of collinear {\sc qed}
structure functions in the non-singlet approximation
\cite{alta}. Numerical computations for the previously discussed
$t$-channel approximation are known in the literature \cite{jadach}.
Moreover, corrections of the order of $O(\alpha^3L^3)$, as due to
single bremsstrahlung to pair production, was analytically calculated
in ref.~\cite{russi}.
\begin{figure}[ht]
\begin{center}
\includegraphics[bb=35 305 535 475,scale=.75]{fig8.ps}
\end{center}
\caption[isr]{\label{isr}
First picture: exact matrix element (markers)
vs $t$-channel approximation (solid line) as a function
of the energy cut $z$, as given by eq.(\ref{zed}), in the presence
of {\sc isr}. Entry values sum up real plus virtual cross sections,
and are computed for narrow-wide angular acceptances of {\tt CALO2} setup
at {\sc lep1/slc} energies.
Second picture: relative difference between the exact matrix
element and $t$-channel approximation till as a function
of the energy cut in the presence of {\sc isr}. Entry values
refer only to the real pair production cross section.
Third picture: relative effect of {\sc isr} correction for exact matrix
element and $t$-channel approximation.}
\end{figure}
In figure \ref{isr} the comparison between the exact matrix element and
$t$-channel approximation is made in the presence of {\sc isr} with the {\tt CALO2}
setup in the narrow-wide acceptance at {\sc lep1/slc} energies and with the same notation
previously used. An apex $\gamma$ means that the relative correction is dressed
by {\sc isr}.
As can been seen, the presence of {\sc isr} becomes more and more important in the soft region
(small $z$ values) up to modifying pair correction by a $25\%$, but it does not
alter the analysis in tree level approximation discussed in the previous section.
\section{\label{end} Conclusions}
The progress in the reduction of the sources of theoretical error
to Bhabha cross section at small angle, driven by increased
experimental accuracy, showed the need of reducing the uncertainty
associated to the light pair contribution.
Till now the theoretical error to {\sc sabh} scattering due to pair
production is evaluated by using {\sc mc} results based on $t$-channel approximation
and with the aid of analytical means. In order to improve the present situation,
a new {\sc mc} program for $e^+e^- \rightarrow e^+e^-e^+e^-$ process
was presented. It includes the exact {\sc qed} four-fermion matrix element, {\sc isr} in
the collinear approximation, and realistic selection criteria.
The results obtained by means of that program were compared with
the best approximations present in the literature, either analytical or numerical.
Two kinds of contribution form the theoretical error: the physical error,
due to matrix element approximation, and the technical error, due to
numerical integration. The comparison between the exact matrix element
calculation performed in the present paper and the $t$-channel approximation
adopted in the literature showed that the physical error committed
by using such approximation at {\sc lep1/slc} energies is within $30\%$
with the exception of the hard region. Indeed in such region it grows up
to $75\%$ of the correction, while in the soft and intermediate region it
can be much lower. Thus the calculation drawn with the $t$-channel approximation
is within the physical error quoted in the literature \cite{jadach}
in the experimentally relevant kinematic region. At {\sc lep2} energies
the physical error due to the approximation is smaller and it is within $10\%$.
Concerning the present calculation, the physical error associated to electron pairs
amounts in neglecting $Z$-boson contributions to the tree level matrix element,
in approximating {\sc isr} by the collinear structure functions,
and in omitting pair correction exponentiation as for instance done in
ref.~\cite{jadach}. Relatively to the whole correction, the physical error
due to these sources can be estimated within $5\%$,
while the technical error amounts only to a few per cent, leading to
a conservative $10\%$ of the correction for the total theoretical error.
Further light pair production sums up the electron pair
production with other contributions, e.g. muon pair production. Such
effect amounts for muon pair production to a $10$-$20\%$ of
the electron contribution. However, since the present algorithm can
compute also the effect of muon pairs, the total theoretical error
can be estimated at $30\%$ level of the whole correction, as due to
approximations in electron and muon pair corrections and to neglecting
heavier pair contributions.
A review for the contribution to the theoretical error at {\sc lep1/slc}
is shown in table \ref{error}. In the first column are listed
the values of table \ref{sabs} \cite{common} and in the second column
the updated values to the latest results for photonic $O(\alpha^2L)$
corrections \cite{ward,pv,kr}, while in the third column the
values are further updated to the result of the present analysis, when
taking into account electron and muon pair corrections and a total
error estimate of $30\%$.
\begin{table}[ht]
\caption[error]{\label{error}
Theoretical error in {\sc sabh} scattering at {\sc lep1/slc}.
The first column (see also table \ref{sabs}) refers to ref.~\cite{common},
the second column takes into account the results of refs.~\cite{ward,pv,kr},
the third column updates to the present analysis.}
\medskip
\begin{center}
\begin{tabular}{|l||c|c||c|} \hline
Type of correction/error & ref.~\cite{common} ($\%$) & ref.~\cite{ward} ($\%$) & updated ($\%$) \\ \hline \hline
missing photonic $O(\alpha^2L)$ & $0.100 $ & $0.027 $ & $0.027$ \\
missing photonic $O(\alpha^3L^3)$ & $0.015 $ & $0.015 $ & $0.015$ \\
vacuum polarization & $0.040 $ & $0.040 $ & $0.040$ \\
light pairs & $0.030 $ & $0.030 $ & $0.010$ \\
$Z$-exchange & $0.015 $ & $0.015 $ & $0.015$ \\ \hline
total & $0.110 $ & $0.061 $ & $0.054$ \\ \hline
\end{tabular}
\end{center}
\end{table}
Table \ref{error} shows that the theoretical error for {\sc sabh}
scattering, because of the reduction of the pair production error,
is now close to $0.05\%$ and so comparable with the experimental accuracy.
The conclusions, drawn for {\sc lep1/slc}, can also be extended to {\sc lep2}
energies and lead to the reduction of the theoretical error shown in table
\ref{error2}.
\begin{table}[ht]
\caption[error2]{\label{error2}
Theoretical error in {\sc sabh} scattering at {\sc lep2}.
The first column (see also table \ref{sabs}) refers to ref.~\cite{common},
the second column takes into account the results of refs.~\cite{ward,pv,kr},
the third column updates to the present analysis.}
\medskip
\begin{center}
\begin{tabular}{|l||c|c||c|} \hline
Type of correction/error & ref.~\cite{common} ($\%$) & ref.~\cite{ward} ($\%$) & updated ($\%$) \\ \hline \hline
missing photonic $O(\alpha^2L)$ & $0.200 $ & $0.040 $ & $0.040$ \\
missing photonic $O(\alpha^3L^3)$ & $0.030 $ & $0.030 $ & $0.030$ \\
vacuum polarization & $0.100 $ & $0.100 $ & $0.100$ \\
light pairs & $0.050 $ & $0.050 $ & $0.015$ \\
$Z$-exchange & $0.000 $ & $0.000 $ & $0.000$ \\ \hline
total & $0.250 $ & $0.122 $ & $0.113$ \\ \hline
\end{tabular}
\end{center}
\end{table}
It is worth noticing that now the major contribution to the theoretical
error is due to the hadronic uncertainty in the vacuum polarization correction. Thus,
if the new determinations \cite{vacuum} of $\alpha_{\rm QED}(M_Z)$
could be drawn to the $1{\rm\;GeV}$ mass scale too, the total error could decrease
further.
The present work was conceived to lower the theoretical error on
{\sc sabh} scattering due to light pair production.
Yet, in order to manage the numerical and theoretical questions,
techniques of a wider range of applicability were implemented.
Chiefly, to deal with multiperipheral dynamics, Weizs\"acker-Williams
approximation played an important r\^ole as a guideline to develop a suited
importance sampling and it led to a numerical recipe that can be further
applied to perform phenomenological studies of processes involving
particles lost in the beam pipe.
Therefore, future developments could deal with the evaluation of pair correction
to two fermion production large angle processes at present and future $e^+e^-$
colliders. Experimentally relevant processes are $e^+e^-\rightarrow e^+e^-$, whose
interest is for luminosity measurements at {\sc da$\Phi$ne} and {\sc nlc}, and
$e^+e^-\rightarrow$~hadrons in the context of precision studies of the electroweak
interaction at {\sc lep} and beyond.
Other fields of interest concern background processes for $WW$ physics and searches
for new physics, as very recently addressed in ref.~\cite{giamp} for the single $W$
production case, and two photon collision processes.
All these studies will require the upgrade of the {\sc qed} matrix elements considered in the
present paper to the full electroweak ones. It is worth noticing that this generalization
could be worked out quite directly by virtue of the automatic algorithm employed in the
present approach.
\renewcommand{\thesection}{$\!\!\!\!\!\!\!$} \section{Acknowledgements}
M.Moretti was supported by a Marie Curie fellowship ({\sc tmr-erbfmbict} 971934).
A.Pallavicini wishes to thank {\sc infn} Sezione di Pavia for
the support and for computer facilities.
| 2024-02-18T23:40:16.419Z | 1998-11-23T12:27:04.000Z | algebraic_stack_train_0000 | 1,889 | 8,549 |
|
proofpile-arXiv_065-9347 | \section{Introduction}
The use of the Baldwin et al. (1981) or Veilleux \& Osterbrock (1987) diagnostic
diagrams generally yields an immediate classification of the nuclear
emission-line clouds; ``transition objects'' exist however, which cannot be
classified unambiguously from their line ratios (Heckman et al. 1983; Keel
1984; Veilleux \& Osterbrock 1987; Ho et al. 1993a).
When observed with sufficient spectral resolution, such objects show
diffe\-rent profiles for the permitted and forbidden lines (Heckman et al.
1981; Busko \& Steiner 1990; V\'eron et al. 1981a,b; V\'eron-Cetty \&
V\'eron 1985, 1986b).
In a previous paper (V\'eron et al. 1997, hereafter Paper I), we
presented high-dispersion
(66 \Am) spectroscopic observations of 15 ``transition objects''
selected for having an ambiguous location on the Veilleux \& Osterbrock (1987)
diagnostic diagrams, and showed that most of them are in fact ``composite".
This was done by modelling the $\rm H\alpha$, \lbrack\ion{N}{ii}\rbrack$\lambda\lambda$\,6548, 6583 and/or
$\rm H\beta$, \lbrack\ion{O}{iii}\rbrack$\lambda\lambda$\,4959, 5007 emission lines with
Gaussian profiles, allowing for the contribution of several components; best
fits showed these components to have different line stren\-gths and widths,
as the result of the lines being produced in regions that are kinematically
and spatially distinct, usually a Seyfert 2 or Liner cloud and a \ion{H}{ii}\ region.
We have found in the literature 88 emission-line galaxies located north of
$\delta \sim$ $-$20\degr, with $z <$ 0.100 and $B <$ 17.0, for which
the published line ratios gave indication of a ``transition'' spectrum,
constituting an unbiased sample of such objects. Here we report results for
53 of these galaxies, including seven already observed in Paper I,
bringing up to 61 the total number of observed objects, or 70\% of
our unbiased sample of galaxies with a ``transition'' spectrum.
\begin{table*}
\begin{center}
\caption{\label{x_names}
Cross-reference names for the galaxies studied in this paper.}
\begin{flushleft}
\begin{tabular}{crcrrllr}
\hline
NGC & UGC & Zw & MCG\verb+ + & Mark & \verb+ +KUG &
\verb+ +IRAS & Misc.\verb+ + \\
\hline
0034 & --\verb+ + & -- & $-$02.01.032 & 938 &
\verb+ +-- & \verb+ +00085$-$1223 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & 957 &
\verb+ +-- & \verb+ +00391$+$4004 & 5C 3.100 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +01346$-$0924 & --\verb+ + \\
-- & 2456$\;$ & 524.040 & 06.07.027 & 1066 &
\verb+ +-- & \verb+ +02568$+$3637 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +03355$+$0104 & HS 0335$+$0104 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +04210$+$0400 & --\verb+ + \\
-- & --\verb+ + & 420.015 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +04507$+$0358 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +06256$+$6342 & VII Zw 73 \\
-- & 4229$\;$ & 207.033 & --\verb+ + & 622 &
0804$+$391 & \verb+ +08043$+$3908 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +$\,-$ & 3C 198.0 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
0825$+$248 & \verb+ +$\,-$ & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +09111$-$1007 & --\verb+ + \\
-- & --\verb+ + & 238.066 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +09277$+$4917 & SBS 0927$+$493 \\
-- & 5101$\;$ & 289.011 & 10.14.025 & --\verb+ + &
\verb+ +-- & \verb+ +09320$+$6134 & --\verb+ + \\
2989 & --\verb+ + & -- & $-$03.25.020 & --\verb+ + &
\verb+ +-- & \verb+ +09430$-$1808 & ESO 566$-$G09 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +09581$+$3126 & CG 49 \\
3185 & 5554$\;$ & 123.034 & 04.24.024 & --\verb+ + &
\verb+ +-- & \verb+ +10148$+$2156 & --\verb+ + \\
\verb+ + -- & 5984$\;$ & 155.031 & 05.26.024 & --\verb+ + &
\verb+ +-- & \verb+ +$\,-$ & Arp 107A \\
3504 & 6118$\;$ & 155.049 & 05.26.039 & --\verb+ + &
1100$+$282 & \verb+ +11004$+$2814 & TEX 1100$+$282 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +11058$-$1131 & --\verb+ + \\
3642 & 6385$\;$ & 291.062 & 10.16.128 & --\verb+ + &
\verb+ +-- & \verb+ +11194$+$5920 & --\verb+ + \\
3660 & --\verb+ + & -- & $-$01.29.016 & 1291 &
\verb+ +-- & \verb+ +11210$-$0823 & --\verb+ + \\
-- & --\verb+ + & 291.074 & 10.17.004 & --\verb+ + &
\verb+ +-- & \verb+ +11258$+$5806 & SBS 1125$+$581 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- &\verb+ +11285$+$8240\,A & --\verb+ + \\
3758 & --\verb+ + & 126.110 & 04.27.073 & 739 &
\verb+ +-- & \verb+ +11338$+$2152 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +$\,-$ & SBS 1136$+$594 \\
3994 & 6936$\;$ & 186.074 & 06.26.059 & --\verb+ + &
1155$+$325\,A & \verb+ +$\,-$ & Arp 313 \\
4102 & 7096$\;$ & 269.036 & 09.20.094 & --\verb+ + &
\verb+ +-- & \verb+ +12038$+$5259 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- &\verb+ +12474$+$4345\,S & --\verb+ + \\
-- & 8621$\;$ & 218.030 & 07.28.041 & --\verb+ + &
\verb+ +-- & \verb+ +13354$+$3924 & --\verb+ + \\
5256 & 8632$\;$ & 246.021 & 08.25.031 & 266 &
\verb+ +-- & \verb+ +13362$+$4832 & I Zw 67 \\
-- & --\verb+ + & 073.074 & --\verb+ + & 1361 &
\verb+ +-- & \verb+ +13446$+$1121 & --\verb+ + \\
-- & 8718$\;$ & 190.055 & 06.30.085 & 461 &
1345$+$343 & \verb+ +$\,-$ & CG 1190 \\
-- & --\verb+ + & 162.010 & 05.33.005 & --\verb+ + &
\verb+ +-- & \verb+ +$\,-$ & 4C 26.42 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +14063$+$4905 & I Zw 81 \\
-- & --\verb+ + & 273.023 & --\verb+ + & 477 &
1439$+$537 & \verb+ +14390$+$5343 & I Zw 92 \\
-- & --\verb+ + & 221.050 & --\verb+ + & 848 &
\verb+ +-- & \verb+ +15163$+$4255 & I Zw 107 \\
-- & --\verb+ + & 077.080 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +15184$+$0834 & --\verb+ + \\
5953 & 9903$\;$ & 107.008 & 03.40.005 & 1512 &
\verb+ +-- & \verb+ +15322$+$1521 & Arp 91 \\
-- & --\verb+ + & 319.034 & 11.19.030 & --\verb+ + &
\verb+ +-- & \verb+ +15564$+$6359 & Kaz 49 \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +16129$-$0753 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +16382$-$0613 & --\verb+ + \\
-- & 10675$\;$ & 169.035 & 05.40.034 & 700 &
1701$+$315 & \verb+ +17013$+$3131 & --\verb+ + \\
-- & --\verb+ + & 112.010 & 03.45.003 & --\verb+ + &
\verb+ +-- & \verb+ +17334$+$2049 & --\verb+ + \\
-- & --\verb+ + & 142.019 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +18101$+$2152 & PGC 61548 \\
-- & --\verb+ + & 341.006 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +18462$+$7207 & Kaz 214 \\
6764 & 11407$\;$ & 256.007 & 08.35.003 & --\verb+ + &
\verb+ +-- & \verb+ +19070$+$5051 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +22114$-$1109 & --\verb+ + \\
-- & --\verb+ + & 452.043 & 03.57.031 & 308 &
2239$+$199 & \verb+ +22395$+$2000 & --\verb+ + \\
-- & --\verb+ + & -- & --\verb+ + & 522 &
2257$+$161 & \verb+ +$\,-$ & --\verb+ + \\
7465 & 12317$\;$ & 453.050 & 03.58.024 & 313 &
2259$+$156 & \verb+ +22595$+$1541 & PG 2259$+$156 \\
-- & --\verb+ + & 453.062 & --\verb+ + & --\verb+ + &
\verb+ +-- & \verb+ +23024$+$1916 & --\verb+ + \\
-- & --\verb+ + & 475.036 & 04.54.038 & --\verb+ + &
\verb+ +-- & \verb+ +23135$+$2516 & IC 5298 \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table*}
\begin{table*}
\begin{center}
\caption{\label{line_ratios}
Published line intensity ratios for the 53 emission-line galaxies
studied in this paper. References: (1) Aguero et al. (1995), (2) Anton
(1993), (3) Augarde et al. (1994), (4) Boller et al. (1994), (5) Cohen \&
Osterbrock (1981), (6) de Grijp et al. (1992), (7) Delgado \& Perez (1996),
(8) Duc et al. (1997), (9) Fruscione \& Griffiths (1991), (10) Goodrich \&
Osterbrock (1983), (11) Hill et al. (1988), (12) Ho et al. (1997a),
(13) Keel et al. (1985),
(14) Kim et al. (1995), (15) Klaas \& Elsasser (1991), (16)
Kollatschny et al. (1983), (17) Koratkar et al. (1995), (18) Koski (1978),
(19) Martel \& Osterbrock (1994), (20) Netzer et al. (1987),
(21) Osterbrock \& Pogge (1987), (22) Osterbrock \& Martel (1993),
(23) Phillips et al. (1983), (24) Rafanelli et al. (1990),
(25) Salzer et al. (1995),
(26) Shuder \& Osterbrock (1981),
(27) Ulvestad \& Wilson (1983),
(28) Veilleux \& Osterbrock (1987),
(29) V\'eron et al. (1997), (30) Vogel et al. (1993).}
\begin{flushleft}
\begin{tabular}{lllcc|lllcc}
\hline
Name & \underline{$\lambda$5007$\:$} \verb+ +
& \underline{$\lambda$6300$\:$} \verb+ +
& \underline{$\lambda$6583$\:$} & Ref. &
Name & \underline{$\lambda$5007$\:$} \verb+ +
& \underline{$\lambda$6300$\:$} \verb+ +
& \underline{$\lambda$6583$\:$} & Ref. \\
& \verb+ +$\rm H\beta$ & \verb+ +$\rm H\alpha$ & $\rm H\alpha$ & &
& \verb+ +$\rm H\beta$ & \verb+ +$\rm H\alpha$ & $\rm H\alpha$ & \\
\hline
Mark 938 & \verb+ +3.98 & 0.09 & 1.29 & (14) &
IRAS 12474$+$4345\,S & \verb+ +2.93 & \verb+ +-- & 0.42 & (6) \\
Mark 957 & \verb+ +0.63 & 0.04 & 0.46 & (19) &
UGC 8621 & \verb+ +5.25 & 0.081 & 0.95 & (22) \\
IRAS 01346$-$0924 & \verb+ +2.90 & \verb+ +-- & 0.41 & (6) &
Mark 266\,NE & \verb+ +1.41 & 0.15 & 0.66 & (22) \\
Mark 1066 & \verb+ +4.35 & 0.084 & 0.88 & (10) &
Mark 266\,SW & \verb+ +3.98 & 0.05 & 0.50 & (22) \\
IRAS 03355$+$0104 & 13.52 & 0.06 & 0.18 & (30) &
Mark 1361 & \verb+ +4.93 & 0.038 & 0.33 & (14) \\
\verb+ +"
& 12.41 & \verb+ +-- & 0.58 & (6) &
Mark 461 & \verb+ +-- & \verb+ +-- & -- & -- \\
IRAS 04210$+$0400 & 14.2 & 0.13 & 0.35 & (11) &
4C 26.42 & \verb+ +0.40 & 0.22 & 0.81 & (2) \\
IRAS 04507$+$0358 & 11.77 & \verb+ +-- & 0.28 & (6) &
I Zw 81 & \verb+ +3.14 & 0.066 & 0.67 & (18) \\
VII Zw 73 & \verb+ +3.96 & \verb+ +-- & 0.56 & (6) &
Mark 477 & 10.42 & 0.17 & 0.36 & (26) \\
Mark~622 & \verb+ +6.25 & 0.064 & 0.94 & (26) &
Mark 848\,S & \verb+ +1.39 & 0.070 & 0.78 & (14) \\
3C 198.0 & \verb+ +2.13 & 0.04 & 0.35 & (5) &
IRAS 15184$+$0834 & \verb+ +5.60 & \verb+ +-- & 0.42 & (6) \\
KUG 0825$+$248 & \verb+ +0.84 & 0.13 & 0.28 & (3) &
NGC 5953 & \verb+ +3.04 & 0.10 & 1.24 & (13) \\
IRAS 09111$-$1007\,E & \verb+ +3.94 & 0.07 & 0.74 & (8) &
\verb+ +"
& \verb+ +2.08 & 0.058 & 0.78 & (14) \\
Zw 238.066 & \verb+ +1.73 & 0.065 & 0.82 & (14) &
\verb+ +"
& \verb+ +4.3 & 0.11 & 1.38 & (7) \\
UGC 5101 & \verb+ +2.86 & 0.089 & 1.34 & (14) &
\verb+ +"
& \verb+ +4.98 & 0.10 & 1.12 & (24) \\
NGC 2989 & \verb+ +2.50 & 0.037 & 0.52 & (23) &
Kaz 49 & \verb+ +2.58 & 0.025 & 0.56 & (4) \\
CG 49 & 11.68 & \verb+ +-- & 0.30 & (25) &
IRAS 16129$-$0753 & \verb+ +2.03 & \verb+ +-- & 0.64 & (6) \\
NGC 3185 & \verb+ +3.42 & 0.045 & 0.70 & (12) &
IRAS 16382$-$0613 & \verb+ +6.67 & 0.09 & 1.09 & (1) \\
Arp 107A & 13.62 & 0.38 & 3.00 & (13) &
Mark 700 & \verb+ +0.55 & 0.11 & 1.75 & (18) \\
NGC 3504 & \verb+ +0.53 & 0.023 & 0.59 & (12) &
MCG 03.45.003 & \verb+ +9.74 & \verb+ +-- & 0.42 & (6) \\
IRAS 11058$-$1131 & \verb+ +9.10 & 0.05 & 0.38 & (29) &
PGC 61548 & \verb+ +1.44 & 0.11 & 0.69 & (9) \\
NGC 3642 & \verb+ +1.32 & 0.08 & 0.71 & (12) &
Kaz 214 & \verb+ +5.23 & \verb+ +-- & 0.39 & (6) \\
Mark 1291 & \verb+ +3.18 & \verb+ +-- & 0.48 & (16) &
NGC 6764 & \verb+ +0.53 & 0.045 & 0.68 & (18) \\
IRAS 11285$+$8240\,A & \verb+ +8.25 & 0.106 & 0.46 & (15) &
IRAS 22114$-$1109 & \verb+ +4.22 & 0.077 & 0.62 & (14) \\
Mark 739\,W & \verb+ +1.14 & \verb+ +-- & 0.43 & (20) &
Mark 308 & \verb+ +4.8 & 0.05 & 0.40 & (29) \\
\verb+ +"
& \verb+ +1.18 & \verb+ +-- & 0.69 & (27) &
Mark 522 & \verb+ +3.23 & 0.068 & 0.93 & (29) \\
SBS 1136$+$594 & 12.30 & 0.11 & 0.25 & (19) &
Mark 313 & \verb+ +3.52 & 0.10 & 0.52 & (21) \\
NGC 3994 & \verb+ +0.56 & 0.19 & 0.89 & (13) &
Zw 453.062 & \verb+ +1.23 & 0.12 & 1.23 & (14) \\
NGC 4102 & \verb+ +0.99 & 0.041 & 0.92 & (12) &
IC 5298 & \verb+ +4.68 & 0.05 & 0.95 & (14) \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table*}
\section{Observations and data analysis}
\subsection{Observations}
The 53 observed galaxies suspected to have
a ``transition'' spectrum are listed in
Table \ref{x_names} with the various names under which they are
known, and in Table \ref{line_ratios} with the published line intensity ratios
$\lambda$5007$/$H$\beta$, $\lambda$6300$/$H$\alpha$\ and $\lambda$6583$/$H$\alpha$. Table \ref{opt_pos} gives their optical positions
measured on the Digitized Sky Survey\footnotemark[1]
(V\'eron-Cetty \& V\'eron 1996).
\footnotetext[1]{The Digitized Sky Survey was produced at the Space
Telescope Science Institute (STScI) under U.S. Government grant NAG W-2166.}
\begin{table*}[t]
\begin{center}
\caption{\label{opt_pos}
B1950 optical positions of the observed objects measured on the
Digitized Sky Survey. The r.m.s. error is 0\farcs6 in each coordinate;
`` * '' indicates objects with larger errors due to their location near
one edge of the Schmidt plate (V\'eron-Cetty \& V\'eron 1996). References
for finding charts:
(1) Andreasian \& Alloin (1994), (2) Arp (1966), (3) Bowen et al. (1994),
(4) Carballo et al. (1992), (5) de Grijp et al. (1987),
(6) Delgado \& Perez (1996), (7) Duc et al. (1997), (8) Kazarian (1979),
(9) Keel (1996), (10) Markarian \& Lipovetski
(1971), (11) Markarian \& Lipovetski (1973), (12) Markarian \& Lipovetski
(1974), (13) Markarian et al. (1979a),
(14) Markarian et al. (1979b), (15) Markarian \& Stepanian (1983),
(16) Mazzarella \& Boroson (1993), (17) Olsen (1970),
(18) Pesch \& Sanduleak (1983), (19) Rubin et al. (1975),
(20) Sandage \& Bedke (1994), (21) Takase \& Miyauchi-Isobe (1986),
(22) Takase \& Miyauchi-Isobe (1990),
(23) Vogel et al. (1993), (24) Wyndham (1966).}
\begin{flushleft}
\begin{tabular}{lcrcc|lcrcc}
\hline
Name & $\alpha$ & $\delta\; \; \; \; \; \; \;$ & Ref. & mag. &
Name & $\alpha$ & $\delta\; \; \; \; \; \; \;$ & Ref. & mag. \\
\hline
Mark 938 & 00 08 33.41 & $\/-\,$12 23 06.6 & (16) & 13.5 &
IRAS 12474$+$4345\,S& 12 47 25.08 & $\; $ 43 45 16.6 & (5) & 15.4 \\
Mark 957 & 00 39 09.65 & $\; $ 40 04 51.6 & (3) & 15.1 &
UGC 8621 & 13 35 28.44 & $\; $ 39 24 30.8 & -- & 14.2 \\
IRAS 01346$-$0924& 01 34 37.53 & $\/-\,$09 24 12.9 & (5) & 15.8 &
Mark 266\,SW & 13 36 14.50 & $\; $ 48 31 47.4 & (16) & 13.4 \\
Mark 1066 & 02 56 49.91 & $\; $ 36 37 21.1 & (16) & 14.0 &
Mark 266\,NE & 13 36 14.99 & $\; $ 48 31 53.5 & (16) & 13.4 \\
IRAS 03355$+$0104& 03 35 35.77 & $\; $ 01 04 34.0 & (23) & 14.5 &
Mark 1361 & 13 44 36.53 & $\; $ 11 21 20.1 & (14) & 15.3 \\
IRAS 04210$+$0400& 04 21 02.69 & $\; $ 04 01 08.2 & (5) & 16.3 &
Mark 461 & 13 45 04.29 & $\; $ 34 23 51.9 & (9) & 14.6 \\
IRAS 04507$+$0358& 04 50 47.50 & $\; $ 03 58 48.9 & (5) & 15.0 &
4C 26.42 & 13 46 33.84 & $\; $ 26 50 26.3 & (17) & 15.2 \\
VII Zw 73 & 06 25 37.78 & $\; $ 63 42 42.9 & (5) & 14.8 &
I Zw 81 & 14 06 20.29 & $\; $ 49 05 56.1 & -- & 16.5 \\
Mark 622 & 08 04 21.03 & $\; $ 39 08 57.4 & (12) & 14.1 &
Mark 477 & 14 39 02.52 & $\; $ 53 43 03.3 & (9) & 15.0 \\
3C 198.0 & 08 19 52.43 & $\; $ 06 06 45.7 & (24) & 17.3 &
Mark 848\,S & 15 16 19.40 & $\; $ 42 55 35.9 & (16) & 15.0 \\
KUG 0825$+$248 & 08 25 29.98 & $\; $ 24 48 31.9 & (21) & 16.0 &
IRAS 15184$+$0834& 15 18 27.10 & $\; $ 08 34 33.9 & (5) & 13.9 \\
IRAS 09111$-$1007\,E& 09 11 13.06 & $\/-\,$10 06 54.6 & (7) & 16.1 &
NGC 5953 & 15 32 13.23 & $\; $ 15 21 35.9 & (6) & 13.1 \\
Zw 238.066 & 09 27 45.68 & $\; $ 49 18 00.4 & -- & 16.5 &
Kaz 49 & 15 56 26.70 & $\; $ 63 59 00.8 & (8) & 15.3 \\
UGC 5101 & 09 32 04.95 & $\; $ 61 34 36.5 & (9) & 15.5 &
IRAS 16129$-$0753& 16 12 58.38 & $\/-\,$07 53 07.2 & (4) & -- \\
NGC 2989 & 09 43 03.79 & $\/-\,$18 08 35.1 & (20) & 14.5 &
IRAS 16382$-$0613& 16 38 11.57 & $\/-\,$06 13 07.6 & (4) & 14.7 \\
CG 49 & 09 58 07.76 & $\; $ 31 26 44.7 & (18) & 16.0 &
Mark 700 & 17 01 21.49 & $\; $ 31 31 37.8 & (9) & 15.5 \\
NGC 3185 & 10 14 53.07 & $\; $ 21 56 18.8 & (20) & 12.3 &
MCG 03.45.003$^{*}$ & 17 33 25.27 & $\; $ 20 49 37.6 & (5) & 13.4 \\
Arp 107A & 10 49 29.66 & $\; $ 30 19 25.1 & (2) & 14.6 &
PGC 61548 & 18 10 07.06 & $\; $ 21 52 15.9 & -- & 14.2 \\
NGC 3504 & 11 00 28.55 & $\; $ 28 14 31.6 & (20) & 12.9 &
Kaz 214 & 18 46 15.77 & $\; $ 72 07 42.9 & (5) & 15.5 \\
IRAS 11058$-$1131& 11 05 49.65 & $\/-\,$11 31 56.8 & (5) & 14.9 &
NGC 6764 & 19 07 01.23 & $\; $ 50 51 08.5 & (19) & 13.2 \\
NGC 3642 & 11 19 25.03 & $\; $ 59 20 54.9 & (20) & 14.1 &
IRAS 22114$-$1109& 22 11 26.01 & $\/-\,$11 09 21.1 & -- & -- \\
Mark 1291 & 11 21 00.13 & $\/-\,$08 23 01.5 & (13) & 15.5 &
Mark 308 & 22 39 30.53 & $\; $ 20 00 00.1 & (10) & 14.6 \\
IRAS 11285$+$8240\,A & 11 28 41.22 & $\; $ 82 40 16.0 & -- & 15.6 &
KUG 2239$+$200\,A & 22 39 33.13 & $\; $ 20 00 38.4 & (22) & 15.5 \\
Mark 739\,W$^{*}$ & 11 33 52.49 & $\; $ 21 52 22.2 & (16) & 14.0 &
Mark 522 & 22 57 50.44 & $\; $ 16 06 50.7 & (11) & 17.0 \\
SBS 1136$+$594 & 11 36 24.27 & $\; $ 59 28 31.4 & (15) & 16.0 &
Mark 313 & 22 59 32.07 & $\; $ 15 41 44.3 & (10) & 14.0 \\
NGC 3994$^{*}$ & 11 55 02.44 & $\; $ 32 33 21.1 & (20) & 12.9 &
Zw 453.062 & 23 02 28.55 & $\; $ 19 16 55.2 & -- & 15.1 \\
NGC 4102 & 12 03 51.33 & $\; $ 52 59 21.2 & (20) & 12.6 &
IC 5298 & 23 13 33.13 & $\; $ 25 17 01.9 & (1) & 14.9 \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table*}
Spectroscopic observations were carried out during several observing runs
in May, June and July 1996 and January, March, October and November 1997
with the spectrograph CA\-RELEC (Lema\^{\i}tre et al. 1989) attached to
the Cas\-se\-grain focus of the Observatoire de Haute-Pro\-vence (OHP) 1.93 m
telescope. The detector was a 512 $\times$ 512 pixels, 27 $\times$ 27 $\mu$m
Tektronic CCD. We used a 600 l\,mm$^{-1}$ grating resulting in a dispersion
of 66 \Am; the spectral range was
$\lambda\lambda$\,6700--7600 \AA\ in the red (with a Schott GG 435 filter)
and $\lambda\lambda$\,4860--5760 \AA\ in the blue. In each case, the galaxy
nucleus was centered on the slit.
Usually five columns
of the CCD ($\sim$ 5\arcsec) were extracted.
The slit width was 2\farcs1, corres\-ponding to a projected slit width on
the detector of 52 $\mu$m or 1.9 pixel. The slit position angle was not always
the same for the blue and red spectra; as the aperture used
is rectangular (2\farcs1 $\times$ 5\arcsec), this may introduce some
inconsistencies when the line emitting regions are extended.
The resolution, as measured on the
night sky emission lines, was $\sim$ 3.4 \AA\ FWHM. The spectra were
flux calibrated using the standard stars given in Table \ref{std_stars},
taken from Oke (1974), Stone (1977), Oke \& Gunn (1983) and
Massey et al. (1988). The journal of observations is given in
Table \ref{journal_obs}.
\subsection{Line profile fitting}
Morgan (1958, 1959) has introduced a classification of galaxies based on
their nuclear region stellar population. Classes ``a'' and ``af''
are dominated by early-type stars. The main absorption features are the
Balmer lines, which are usually filled up by emission as these objects
invariably contain a \ion{H}{ii}\ region. Classes ``g'', ``gk'' and ``k'' are
dominated by a bulge of old population II stars. Intermediate classes
``f'' and ``fg'' have, in addition to a population of young stars, a faint
bulge of old stars. The old star population have similar spectra in all
classes (Bica 1988). AGN activity is exceptional in classes ``a'' and
``af'' but frequent in all other classes (V\'eron \& V\'eron-Cetty 1986);
in consequence, the nuclear region of most AGNs contains a population of old
stars with many strong absorption lines which can make the line fitting
analysis ra\-ther difficult, especially for the
blue spectra. Therefore, when the absorption blend \ion{Mg}{i}\,b $\lambda$5175
was prominent, we subtracted a suitable fraction of the spectrum of the
elliptical galaxy NGC 5982, used as a template, to remove the old stellar
population contribution. Another elliptical, NGC 4365, was used as a template
for the red spectra, while NGC 821 was used for both the red and blue
spectra obtained in October and November 1997. All templates were observed
with the same spectral resolution as the emission-line galaxies.
Whenever
ne\-cessary, we have added a $\rm H\alpha$\ or $\rm H\beta$\ absorption component; as,
usually, the $\rm H\alpha$\ absorption line is completely filled up by the
emission lines, we assumed its intensity to be 1.8 times the intensity
of the nearby absorption \ion{Ca}{i}\,$\lambda$6495 line (V\'eron-Cetty \&
V\'eron 1986b). Whenever a template and/or absorption component was used
in a fit, this is indicated in Table \ref{fits} which contains the line
fitting analysis results for the 53 observed galaxies.
\begin{table}[h]
\begin{center}
\caption{\label{std_stars}
Spectrograph settings and standard stars.}
\begin{flushleft}
\begin{tabular}{rcl}
\hline
Date~~~~~~~~~ & $\lambda$ Range (\AA) & \verb+ +Standard stars \\
\hline
21 -- 22.03.95 & 6500 -- 7400 & \verb+ +BD~26\degr2606 \\
09 -- 10.05.96 & 6700 -- 7600 & \verb+ +GD~140, BD~26\degr2606\\
11 -- 13.05.96 & 4860 -- 5760 & \verb+ +Feige~98, Kopff 27\\
08.06.96 & 4860 -- 5760 & \verb+ +Feige~66, Kopff 27\\
09.06.96 & 6700 -- 7600 & \verb+ +Feige~66, BD~28\degr4211\\
15 -- 16.07.96 & 4675 -- 5575 & \verb+ +BD~28\degr4211\\
23 -- 25.07.96 & 6335 -- 7235 & \verb+ +BD~28\degr4211\\
07.01.97 & 4720 -- 5620 & \verb+ +EG~247 \\
10.01.97 & 6175 -- 7075 & \verb+ +EG~247 \\
04 -- 07.03.97 & 4825 -- 5725 & \verb+ +Feige~66 \\
08 -- 12.03.97 & 6310 -- 7210 & \verb+ +Feige~66 \\
13.03.97 & 4825 -- 5725 & \verb+ +Feige~66 \\
31.10.97 & 6455 -- 7365 & \verb+ +Feige~24 \\
01 -- 02.11.97 & 4655 -- 5560 & \verb+ +Feige~24 \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table}
\begin{table*}
\begin{center}
\caption{\label{journal_obs}
Journal of observations. A: 66 \Am, red;
B: 66 \Am, blue. A ``*'' indicates objects published in
Paper I.}
\begin{flushleft}
\begin{tabular}{lcccr|lcccr}
\hline
Name & Disp. & Date & Exp. time & PA &
Name & Disp. & Date & Exp. time & PA \\
& & & (min) & (\degr) & & & & (min) & (\degr) \\
\hline
Mark 938 & A & 31.10.97 & 20 & 90 &
UGC 8621 & A & 10.05.96 & 20 & 180 \\
& B & 01.11.97 & 20 & 180 &
& B & 11.05.96 & 20 & 179 \\
Mark 957 & A & 10.01.97 & 20 & 90 &
Mark 266\,SW & A & 10.01.97 & 20 & 180 \\
& B & 01.11.97 & 20 & 270 &
& B & 07.03.97 & 20 & 215 \\
IRAS 01346$-$0924$^{*}$ & A & 10.01.97 & 20 & 90 &
Mark 266\,NE & B & 07.03.97 & 20 & 215 \\
Mark 1066 & A & 10.01.97 & 20 & 90 &
Mark 1361$^{*}$ & B & 06.03.97 & 20 & 180 \\
& B & 07.01 97 & 20 & 90 &
Mark 461 & A & 22.03.95 & 20 & 90 \\
IRAS 03355$+$0104 & A & 10.01.97 & 20 & 90 &
& B & 08.06.96 & 20 & 89 \\
& B & 04.03.97 & 20 & 90 &
4C 26.42 & A & 09.06.96 & 20 & 90 \\
IRAS 04210$+$0400 & A & 10.01.97 & 20 & 90 &
& B & 13.05.96 & 20 & 0 \\
& B & 07.01.97 & 20 & 90 &
I Zw 81 & A & 10.05.96 & 20 & 180 \\
IRAS 04507$+$0358 & A & 10.01.97 & 20 & 90 &
& B & 11.05.96 & 20 & 84 \\
& B & 06.03.97 & 20 & 180 &
Mark 477$^{*}$ & B & 11.05.96 & 20 & 120 \\
VII Zw 73 & A & 10.01.97 & 20 & 180 &
Mark 848\,S & A & 22.06.96 & 20 & 171 \\
& B & 07.01.97 & 20 & 180 &
& B & 08.06.96 & 20 & 90 \\
Mark 622 & A & 10.01.97 & 20 & 180 &
IRAS 15184$+$0834 & A & 08.03.97 & 20 & 270 \\
& B & 07.01.97 & 20 & 180 &
& B & 07.03.97 & 20 & 270 \\
3C 198.0 & A & 12.03.97 & 20 & 270 &
NGC 5953 & A & 10.05.96 & 20 & 180 \\
KUG 0825$+$248 & A & 08.03.97 & 20 & 220 &
& B & 11.05.96 & 20 & 180 \\
& B & 07.03.97 & 20 & 217 &
Kaz 49 & A & 22.06.96 & 20 & 180 \\
IRAS 09111$-$1007\,E & A & 08.03.97 & 20 & 258 &
& B & 23.06.96 & 20 & 180 \\
& B & 05.03.97 & 20 & 252 &
IRAS 16129$-$0753 & A & 22.06.96 & 20 & 180 \\
Zw 238.066 & A & 08.03.97 & 20 & 180 &
& B & 15.07.96 & 20 & 180 \\
& B & 07.01.97 & 20 & 90 &
IRAS 16382$-$0613 & A & 09.05.96 & 20 & 180 \\
UGC 5101 & A & 10.03.97 & 20 & 270 &
& B & 13.05.96 & 20 & 0 \\
& B & 06.03.97 & 20 & 180 &
Mark 700 & A & 09.05.96 & 20 & 180 \\
NGC 2989 & A & 09.03.97 & 20 & 270 &
& B & 07.06.96 & 20 & 90 \\
& B & 06.03.97 & 20 & 180 &
& B & 08.06.96 & 20 & 90 \\
CG 49 & A & 10.05.96 & 20 & 180 &
MCG 03.45.003$^{*}$ & B & 13.05.96 & 20 & 0 \\
& B & 01.11.97 & 20 & 270 &
PGC 61548$^{*}$ & A & 09.05.96 & 20 & 180 \\
NGC 3185 & A & 08.03.97 & 20 & 270 &
& B & 13.05.96 & 20 & 0 \\
& B & 06.03.97 & 20 & 180 &
Kaz 214 & A & 09.06.96 & 20 & 139 \\
Arp 107A & A & 09.03.97 & 20 & 270 &
& B & 13.05.96 & 20 & 0 \\
NGC 3504 & A & 08.03.97 & 20 & 270 &
NGC 6764 & A & 09.06.96 & 20 & 253 \\
& B & 07.03.97 & 20 & 180 &
& B & 08.06.96 & 20 & 72 \\
IRAS 11058$-$1131$^{*}$ & B & 06.03.97 & 20 & 180 &
IRAS 22114$-$1109 & A & 24.07.96 & 20 & 180 \\
NGC 3642 & A & 08.03.97 & 20 & 270 &
& A & 25.07.96 & 20 & 180 \\
& B & 07.03.97 & 20 & 180 &
& B & 15.07.96 & 20 & 180 \\
Mark 1291 & A & 10.01.97 & 20 & 180 &
& B & 16.07.96 & 20 & 180 \\
& B & 06.03.97 & 20 & 180 &
Mark 308$^{*}$ & A & 09.06.96 & 20 & 44 \\
IRAS 11285$+$8240\,A & A & 10.05.96 & 20 & 180 &
Mark 522 & A & 23.07.96 & 20 & 180 \\
Mark 739\,W & A & 09.01.97 & 20 & 90 &
& B & 15.07.96 & 20 & 180 \\
& B & 07.03.97 & 20 & 272 &
Mark 313 & A & 23.07.96 & 20 & 180 \\
SBS 1136$+$594 & A & 10.01.97 & 20 & 180 &
& B & 15.07.96 & 20 & 180 \\
& B & 05.03.97 & 20 & 180 &
Zw 453.062 & A & 25.07.96 & 20 & 225 \\
NGC 3994 & A & 08.03.97 & 20 & 270 &
& B & 01.11.97 & 20 & 270 \\
& B & 13.03.97 & 20 & 180 &
& B & 02.11.97 & 60 & 270 \\
NGC 4102 & A & 21.03.95 & 20 & 90 &
IC 5298 & A & 25.07.96 & 15 & 204 \\
& B & 04.03.97 & 15 & 270 &
& B & 02.11.97 & 60 & 270 \\
IRAS 12474$+$4345\,S & A & 11.03.97 & 20 & 345 &
& & & & \\
& B & 13.03.97 & 20 & 345 &
& & & & \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table*}
\onecolumn
\begin{sidewaystable}
\centering
\small{
\caption{\label{fits}
Fitting profile analysis results. Col. 1 gives the name of the
object, col. 2 the adopted redshift, cols. 4 and 9 the velocities for each
set of components measured on the blue and red spectra, respectively, and
de-redshifted using the redshift given in col.~2; cols. 5 and 10 the
corresponding FWHM, cols. 6, 11 and 12 the intensity ratios $\lambda$5007$/$H$\beta$,
$\lambda$6583$/$H$\alpha$\ and $\lambda$6300$/$H$\alpha$\ respectively, and cols. 7 and 13 the
fraction of the $\rm H\beta$\ emission
flux (respectively $\rm H\alpha$) in each component with respect to the
total flux of the line in each object. A ``\,T\,'' in col. 3
(or 8) indicates that the blue (or red) spectrum has been corrected for
starlight using a suitable fraction of a template (in the blue, we have used
the elliptical galaxy NGC 5982 and in the red, the elliptical galaxy NGC 4365;
NGC 821 was used as a template for the objects observed in October and November
1997); in the same columns, an ``A'' indicates that a $\rm H\beta$\ (or $\rm H\alpha$)
absorption component was added to the model. In col. 14 we give the
velocity difference between the blue and red systems, and in col. 15 the
spectroscopic classification of each component in
the model; Gaussian profiles were
used throughout, except when indicated by ``$lz$'' (Lorentzian profile).
Values in parenthesis have been fixed.}
\begin{flushleft}
\begin{tabular}{ll|crrrr|crrccr|rc}
\hline
Name & \verb+ +$z$ & Stellar & $V\; \; \; \; \; \;$ &
FWHM\/ & \underline{$\lambda$5007$\:$} & $\rm H\beta$ & Stellar &
$V\; \; \; \; \; \;$ & FWHM\/ & \underline{$\lambda$6583$\:$} &
\underline{$\lambda$6300$\:$} & $\rm H\alpha$ & $\Delta V \verb+ + $ &
Class. \\
& & corr. & (\kms) &
(\kms) & $\rm H\beta$$\;\;$ & (\%) & corr. &
(\kms) & (\kms) & $\rm H\alpha$$\;\;$ & $\rm H\alpha$$\;\;$ &
(\%) & (\kms) & \\
\hline
Mark 938 & 0.019 & A & 111 \verb+ + & 240 \verb+ + &
1.18\verb+ + & 91 & T & 173 \verb+ + & 270 \verb+ + &
0.71 & 0.07 & 89 & $-$62 \verb+ + &
\ion{H}{ii}\ \\
& & &($-$79)$\;$ \verb+ +& 890 \verb+ + &
(10)$\;\verb+ +$ & 9 & & $-$11 \verb+ + & 648 \verb+ + &
6.78 & $<$ 0.3\, \verb+ + & 11 & $-$68 \verb+ + &
Sey2 \\
Mark 957 & 0.0732 & & $-$402 \verb+ + & 710 \verb+ + &
9.00\verb+ + & 8 & & $-$201 \verb+ + & 726 \verb+ + &
1.35 & $<$ 0.3\, \verb+ + & 18 & $-$201 \verb+ + &
Sey2 \\
& & & $-$43 \verb+ + & 200 \verb+ + &
0.15\verb+ + & 92 & & 3 \verb+ + & 192 \verb+ + &
0.35 & $<$ 0.02\, \verb+ + & 82 & $-$46 \verb+ + &
\ion{H}{ii}\ \\
IRAS 01346$-$0924& 0.070 & & 114 \verb+ + & 178 \verb+ + &
2.13\verb+ + & 38 & & 165 \verb+ + & 148 \verb+ + &
0.70 & $<$ 0.04\, \verb+ + & 23 & $-$51 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$83 \verb+ + & $<$ 80 \verb+ + &
0.83\verb+ + & 39 & & $-$33 \verb+ + & 219 \verb+ + &
0.46 & 0.07 & 37 & $-$50 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$224 \verb+ + & 1069 \verb+ + &
(10)$\;\verb+ +$ & 23 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 219 \verb+ + & 2640 \verb+ + &
-- & -- & 40 & -- \verb+ + &
Sey1 \\
Mark 1066 & 0.0122 & & $-$116 \verb+ + & 348 \verb+ + &
12.38\verb+ + & 11 & & $-$51 \verb+ + & 270 \verb+ + &
0.77 & 0.07 & 35 & $-$65 \verb+ + &
Sey2 \\
& & & 30 \verb+ + & 220 \verb+ + &
1.54\verb+ + & 66 & & 54 \verb+ + & 205 \verb+ + &
0.89 & 0.08 & 39 & $-$24 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$262 \verb+ + & 875 \verb+ + &
5.76\verb+ + & 23 & & $-$165 \verb+ + & (875)\verb+ + &
1.49 & 0.15 & 26 & $-$97 \verb+ + &
Sey2 \\
IRAS 03355$+$0104& 0.0398 & & $-$1 \verb+ + & 364 \verb+ + &
13.69\verb+ + & 100 & & 12 \verb+ + & 354 \verb+ + &
0.49 & 0.12 & 66 & $-$13 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 264 \verb+ + & 1930 \verb+ + &
-- & -- & 34 & -- \verb+ + &
Sey1 \\
IRAS 04210$+$0400& 0.0462 & & 14 \verb+ + & 178 \verb+ + &
13.30\verb+ + & 41 & & 63 \verb+ + & 219 \verb+ + &
0.40 & 0.17 & 47 & $-$49 \verb+ + &
Sey2 \\
& & & 0 \verb+ + & 541 \verb+ + &
13.03\verb+ + & 59 & & 41 \verb+ + & 615 \verb+ + &
0.34 & 0.06 & 53 & $-$41 \verb+ + &
Sey2 \\
IRAS 04507$+$0358& 0.0296 & & $-$28 \verb+ + & 240 \verb+ + &
11.00\verb+ + & 82 & & 14 \verb+ + & 282 \verb+ + &
0.47 & 0.09 & 64 & $-$42 \verb+ + &
Sey2 \\
& & & $-$146 \verb+ + & 603 \verb+ + &
14.57\verb+ + & 18 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$88 \verb+ + & 1254$_{lz}$ &
-- & -- & 36 & -- \verb+ + &
Sey1 \\
VII Zw 73 & 0.0405 & & 0 \verb+ + & 307 \verb+ + &
(15)$\;\verb+ +$ & 28 & & 17 \verb+ + & 272 \verb+ + &
0.80 & 0.10 & 37 & $-$17 \verb+ + &
Sey2 \\
& & & $-$186 \verb+ + & 794 \verb+ + &
(15)$\;\verb+ +$ & 12 & & $-$171 \verb+ + & 685 \verb+ + &
0.52 & 0.22 & 16 & 15 \verb+ + &
Sey2 \\
& & & 3 \verb+ + & 151 \verb+ + &
1.28\verb+ + & 60 & & 14 \verb+ + & 111 \verb+ + &
0.48 & 0.02 & 47 & $-$11 \verb+ + &
\ion{H}{ii}\ \\
Mark 622 & 0.0233 &T, A& 6 \verb+ + & 178 \verb+ + &
(0.1)\,\verb+ + & 59 & T & 12 \verb+ + & 205 \verb+ + &
0.79 & 0.06 & 49 & $-$6 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$58 \verb+ + & 1172 \verb+ + &
(10)$\;\verb+ +$ & 41 & & $-$22 \verb+ + & 748 \verb+ + &
1.44 & $<$ 0.5\, \verb+ + & 51 & $-$36 \verb+ + &
Sey2 \\
3C 198.0 & 0.081 & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 125 \verb+ + & 294 \verb+ + &
0.28 & 0.05 & 100 & -- \verb+ + &
? \\
KUG 0825$+$248 & 0.083 & & 108 \verb+ + &$<$ 80 \verb+ + &
0.62\verb+ + & 100 & & 135 \verb+ + & 95 \verb+ + &
0.29 & 0.02 & 100 & $-$27 \verb+ + &
\ion{H}{ii}\ \\
IRAS 09111$-$1007\,E & 0.055 & & $-$113 \verb+ + &$<$ 80 \verb+ + &
(0.1)\,\verb+ + & 16 & & $-$230 \verb+ + & 95 \verb+ + &
0.55 & $<$ 0.09\, \verb+ + & 12 & 117 \verb+ + &
\ion{H}{ii}\ \\
& & &($-$76)$\;$ \verb+ +& 259 \verb+ + &
(0.1)\,\verb+ + & 53 & & $-$41 \verb+ + & 257 \verb+ + &
0.86 & $<$ 0.05\, \verb+ + & 40 & $-$35 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$113 \verb+ + & 430 \verb+ + &
(10)$\;\verb+ +$ & 31 & & $-$96 \verb+ + & 626 \verb+ + &
0.95 & 0.12 & 48 & $-$17 \verb+ + &
Sey2 \\
Zw 238.066 & 0.034 &T, A& 95 \verb+ + & 220 \verb+ + &
0.60\verb+ + & 54 & A & 149 \verb+ + &$<$ 80 \verb+ + &
0.69 & $<$ 0.05\, \verb+ + & 14 & $-$54 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$134 \verb+ + & 154 \verb+ + &
1.33\verb+ + & 37 & & $-$71 \verb+ + & 307 \verb+ + &
0.79 & 0.05 & 78 & $-$63 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$295 \verb+ + & 935 \verb+ + &
(10)$\;\verb+ +$ & 9 & & $-$529 \verb+ + & 1130 \verb+ + &
3.75 & $<$ 0.6\, \verb+ + & 8 & 234 \verb+ + &
Sey2 \\
\hline
\end{tabular}
\end{flushleft}}
\end{sidewaystable}
\addtocounter{table}{-1}
\newpage
\onecolumn
\begin{sidewaystable}
\centering
\small{
\caption{Fitting profile analysis results (continued).}
\begin{flushleft}
\begin{tabular}{ll|crrrr|crrccr|rc}
\hline
Name & \verb+ +$z$ & Stellar & $V\; \; \; \; \; \;$ &
FWHM\/ & \underline{$\lambda$5007$\:$} & $\rm H\beta$ & Stellar &
$V\; \; \; \; \; \;$ & FWHM\/ & \underline{$\lambda$6583$\:$} &
\underline{$\lambda$6300$\:$} & $\rm H\alpha$ & $\Delta V \verb+ + $ &
Class. \\
& & corr. & (\kms) &
(\kms) & $\rm H\beta$$\;\;$ & (\%) & corr. &
(\kms) & (\kms) & $\rm H\alpha$$\;\;$ & $\rm H\alpha$$\;\;$ &
(\%) & (\kms) & \\
\hline
UGC 5101 \#1 & 0.039 & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 254 \verb+ + & 95 \verb+ + &
0.51 & 0.05 & 100 & -- \verb+ + &
\ion{H}{ii}\ \\
UGC 5101 \#2 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 222 \verb+ + & 163 \verb+ + &
0.51 & 0.05 & 85 & -- \verb+ + &
\ion{H}{ii}\ \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 101 \verb+ + & 526 \verb+ + &
2.57 & $<$ 0.4\, \verb+ + & 15 & -- \verb+ + &
Sey2 \\
UGC 5101 \#3 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 146 \verb+ + & 132 \verb+ + &
0.51 & 0.07 & 28 & -- \verb+ + &
\ion{H}{ii}\ \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 92 \verb+ + & 481 \verb+ + &
4.50 & 0.14 & 14 & -- \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 197 \verb+ + & 1033$_{lz}$ &
-- & -- & 58 & -- \verb+ + &
Sey1 \\
UGC 5101 \#4 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$108 \verb+ + & 148 \verb+ + &
0.70 & $<$ 0.13\, \verb+ + & 15 & -- \verb+ + &
\ion{H}{ii}\ \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 87 \verb+ + & 114 \verb+ + &
0.38 & $<$ 0.09\, \verb+ + & 21 & -- \verb+ + &
\\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 36 \verb+ + & 492 \verb+ + &
5.33 & $<$ 0.4\, \verb+ + & 10 & -- \verb+ + &
\\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 324 \verb+ + & 1438$_{lz}$ &
-- & -- & 54 & -- \verb+ + &
Sey1 \\
UGC 5101 \#5 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$3 \verb+ + & 401 \verb+ + &
1.31 & $<$ 0.12\, \verb+ + & 34 & -- \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$119 \verb+ + & 148 \verb+ + &
0.52 & 0.06 & 66 & -- \verb+ + &
\ion{H}{ii}\ \\
UGC 5101 & 0.039 & A & 92 \verb+ + & 364 \verb+ + &
2.17\verb+ + & 100 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
\\
NGC 2989 & 0.014 & A & $-$105 \verb+ + & 128 \verb+ + &
1.70\verb+ + & 100 & & $-$108 \verb+ + & 148 \verb+ + &
0.53 & 0.04 & 100 & 3 \verb+ + &
\ion{H}{ii}\ \\
CG 49 & 0.044 & A & $-$67 \verb+ + & 259 \verb+ + &
7.20\verb+ + & 100 & & $-$40 \verb+ + & 257 \verb+ + &
0.80 & 0.09 & 100 & $-$27 \verb+ + &
Sey2 \\
NGC 3185 & 0.004 & A & 12 \verb+ + & 154 \verb+ + &
4.68\verb+ + & 74 & A & 68 \verb+ + & 148 \verb+ + &
0.81 & 0.04 & 66 & $-$56 \verb+ + &
? \\
& & & $-$164 \verb+ + & 154 \verb+ + &
2.03\verb+ + & 26 & & $-$78 \verb+ + & 114 \verb+ + &
0.64 & $<$ 0.03\, \verb+ + & 34 & $-$86 \verb+ + &
\ion{H}{ii}\ \\
Arp 107A & 0.034 & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 141 \verb+ + & 178 \verb+ + &
1.41 & 0.20 & 54 & -- \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 108 \verb+ + & 492 \verb+ + &
1.23 & $<$ 0.1\, \verb+ + & 46 & -- \verb+ + &
Sey2 \\
NGC 3504 & 0.005 & T, A& $-$47 \verb+ + & 200 \verb+ + &
(0.1)\,\verb+ + & 95 & A & 0 \verb+ + & 192 \verb+ + &
0.58 & 0.02 & 93 & $-$47 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$10 \verb+ + & 398 \verb+ + &
(10)$\;\verb+ +$ & 5 & & 12 \verb+ + & 582 \verb+ + &
1.87 & 0.08 & 7 & $-$22 \verb+ + &
Sey2 \\
IRAS 11058$-$1131& 0.0547 & & $-$9 \verb+ + & 259 \verb+ + &
7.57\verb+ + & 100 & & 65 \verb+ + & 205 \verb+ + &
0.34 & 0.05 & 76 & $-$74 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 399 \verb+ + & 2071 \verb+ + &
-- & -- & 24 & -- \verb+ + &
Sey1 \\
NGC 3642 & 0.005 & T & 44 \verb+ + & 98 \verb+ + &
0.14\verb+ + & 88 & & 71 \verb+ + & 95 \verb+ + &
0.48 & 0.07 & 39 & $-$27 \verb+ + &
\ion{H}{ii}\ \\
& & & 50 \verb+ + & (330)\verb+ + &
(10)$\;\verb+ +$ & 12 & & 36 \verb+ + & 330 \verb+ + &
1.00 & $<$ 0.3\, \verb+ + & 9 & 14 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$10 \verb+ + & 2168 \verb+ + &
-- & -- & 52 & -- \verb+ + &
Sey1 \\
Mark 1291 & 0.0122 & & $-$70 \verb+ + & 154 \verb+ + &
3.84\verb+ + & 100 & & $-$4 \verb+ + & 178 \verb+ + &
0.73 & 0.04 & 50 & $-$66 \verb+ + &
? \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$54 \verb+ + & 1941 \verb+ + &
-- & -- & 50 & -- \verb+ + &
Sey1 \\
IRAS 11285$+$8240\,A & 0.028 & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 141 \verb+ + & 205 \verb+ + &
0.45 & 0.11 & 100 & -- \verb+ + &
Sey2 \\
Mark 739\,W & 0.0297 & A & 9 \verb+ + &$<$ 80 \verb+ + &
0.27\verb+ + & 90 & A & 14 \verb+ + & 95 \verb+ + &
0.49 & 0.02 & 44 & $-$5 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$4 \verb+ + & 259 \verb+ + &
8.77\verb+ + & 10 & & $-$38 \verb+ + & 307 \verb+ + &
0.67 & 0.10 & 18 & 34 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$57 \verb+ + & 2103 \verb+ + &
-- & -- & 38 & -- \verb+ + &
Sey1 \\
SBS 1136$+$594 & 0.0613 & & $-$38 \verb+ + & 154 \verb+ + &
11.66\verb+ + & 10 & & 6 \verb+ + & 192 \verb+ + &
0.10 & 0.18 & 9 & $-$44 \verb+ + &
Sey2 \\
& & & $-$105 \verb+ + & 4470 \verb+ + &
-- \verb+ + & 90 & & $-$21 \verb+ + & 3871 \verb+ + &
-- & -- & 83 & $-$84 \verb+ + &
Sey1 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$23 \verb+ + & 1010 \verb+ + &
-- & -- & 8 & -- \verb+ + &
Sey1 \\
NGC 3994 & 0.010 & T & 98 \verb+ + & 200 \verb+ + &
0.82\verb+ + & 89 & T & 77 \verb+ + & 192 \verb+ + &
0.53 & 0.05 & 70 & 21 \verb+ + &
\ion{H}{ii}\ \\
& & & $-$64 \verb+ + & 905 \verb+ + &
6.00\verb+ + & 11 & & 33 \verb+ + & 537 \verb+ + &
1.41 & 0.53 & 30 & $-$99 \verb+ + &
Liner \\
NGC 4102 & 0.0025 & T & $-$25 \verb+ + & 154 \verb+ + &
0.43\verb+ + & 93 & A & 57 \verb+ + & 192 \verb+ + &
0.87 & -- & 79 & $-$82 \verb+ + &
H II \\
& & &$-$126 \verb+ + & 557 \verb+ + &
(10)$\;\verb+ +$ & 7 & & $-$19 \verb+ + & (557)\verb+ + &
1.57 & -- & 6 &$-$107 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 273 \verb+ + & 132 \verb+ + &
0.94 & -- & 15 & -- \verb+ + &
? \\
\hline
\end{tabular}
\end{flushleft}}
\end{sidewaystable}
\addtocounter{table}{-1}
\newpage
\onecolumn
\begin{sidewaystable}
\centering
\small{
\caption{Fitting profile analysis results (continued).}
\begin{flushleft}
\begin{tabular}{ll|crrrr|crrccr|rc}
\hline
Name & \verb+ +$z$ & Stellar & $V\; \; \; \; \; \;$ &
FWHM\/ & \underline{$\lambda$5007$\:$} & $\rm H\beta$ & Stellar &
$V\; \; \; \; \; \;$ & FWHM\/ & \underline{$\lambda$6583$\:$} &
\underline{$\lambda$6300$\:$} & $\rm H\alpha$ & $\Delta V \verb+ + $ &
Class. \\
& & corr. & (\kms) &
(\kms) & $\rm H\beta$$\;\;$ & (\%) & corr. &
(\kms) & (\kms) & $\rm H\alpha$$\;\;$ & $\rm H\alpha$$\;\;$ &
(\%) & (\kms) & \\
\hline
IRAS 12474$+$4345\,S & 0.062 & & 182 \verb+ + & 200 \verb+ + &
2.24\verb+ + & 53 & & 128 \verb+ + & 178 \verb+ + &
0.45 & 0.05 & 47 & 54 \verb+ + &
H II \\
& & & 125 \verb+ + & 478 \verb+ + &
4.43\verb+ + & 47 & & 77 \verb+ + & 458 \verb+ + &
0.40 & 0.03 & 53 & 48 \verb+ + &
? \\
UGC 8621 & 0.020 & T & 3 \verb+ + & $<$ 80 \verb+ + &
0.28\verb+ + & 83 & & 54 \verb+ + & $<$ 80 \verb+ + &
0.42 & $<$ 0.01\, \verb+ + & 55 & $-$51 \verb+ + &
H II \\
& & & $-$141 \verb+ + & 664 \verb+ + &
(10)$\;\verb+ +$ & 17 & & $-$21 \verb+ + & (664)\verb+ + &
1.78 & $<$ 0.3\, \verb+ + & 9 & $-$120 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$238 \verb+ + & 2987 \verb+ + &
-- & -- & 36 & -- \verb+ + &
Sey1 \\
Mark 266 NE & 0.0283 & & $-$88 \verb+ + & 313 \verb+ + &
0.96\verb+ + & 63 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
H II \\
& & & $-$278 \verb+ + & 1009 \verb+ + &
2.17\verb+ + & 37 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
Liner? \\
Mark 266\,SW & 0.0278 & & $-$125 \verb+ + & 200 \verb+ + &
3.65\verb+ + & 46 & & $-$61 \verb+ + & 270 \verb+ + &
0.58 & 0.06 & 58 & $-$64 \verb+ + &
? \\
& & & 114 \verb+ + & 295 \verb+ + &
0.36\verb+ + & 33 & & 186 \verb+ + & 270 \verb+ + &
0.58 & 0.04 & 31 & $-$72 \verb+ + &
H II \\
& & & $-$110 \verb+ + & 603 \verb+ + &
13.30\verb+ + & 21 & & $-$259 \verb+ + & (603)\verb+ + &
0.49 & 0.12 & 11 & 149 \verb+ + &
Sey2 \\
Mark 1361 & 0.0224 & & 108 \verb+ + & 178 \verb+ + &
(0.1)\,\verb+ + & 51 & & 260 \verb+ + & 95 \verb+ + &
0.54 & 0.05 & 15 & $-$152 \verb+ + &
H II \\
& & & 17 \verb+ + & 277 \verb+ + &
9.17\verb+ + & 31 & & 117 \verb+ + & 192 \verb+ + &
0.66 & 0.04 & 49 & $-$100 \verb+ + &
Sey2 \\
& & & $-$142 \verb+ + & 785 \verb+ + &
(10)$\;\verb+ +$ & 18 & & 54 \verb+ + & 548 \verb+ + &
1.04 & 0.09 & 26 & $-$196 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 68 \verb+ + & 2427 \verb+ + &
-- & -- & 10 & -- \verb+ + &
Sey1 \\
Mark 461 & 0.016 & T & 114 \verb+ + & 313 \verb+ + &
0.99\verb+ + & 66 & & 74 \verb+ + & 219 \verb+ + &
0.60 & -- & 37 & 40 \verb+ + &
H II \\
& & & $-$31 \verb+ + & 920 \verb+ + &
$>$ 2.6\verb+ +& 34 & & 41 \verb+ + & 692 \verb+ + &
0.64 & -- & 63 & $-$72 \verb+ + &
Sey2? \\
4C 26.42 & 0.063 & T & $-$267 \verb+ + & (342)\verb+ + &
0.42\verb+ + & 52 & & $-$173 \verb+ + & (342)\verb+ + &
0.83 & 0.20 & 62 & $-$94 \verb+ + &
Liner \\
& & & 47 \verb+ + & (220)\verb+ + &
0.62\verb+ + & 48 & & 159 \verb+ + & (220)\verb+ + &
0.86 & 0.27 & 38 & $-$112 \verb+ + &
Liner \\
I Zw 81 & 0.052 & A & 17 \verb+ + & (132)\verb+ + &
2.05\verb+ + & 53 & & 98 \verb+ + & 132 \verb+ + &
0.78 & $<$ 0.04\, \verb+ + & 43 & $-$81 \verb+ + &
H II? \\
& & & $-$226 \verb+ + & (132)\verb+ + &
1.41\verb+ + & 47 & & $-$122 \verb+ + & 132 \verb+ + &
0.56 & $<$ 0.03\, \verb+ + & 57 & $-$104 \verb+ + &
H II \\
Mark 477 & 0.038 & & $-$108 \verb+ + & 200 \verb+ + &
8.00\verb+ + & 30 & & $-$53 \verb+ + & 163 \verb+ + &
0.38 & 0.15 & 31 & $-$55 \verb+ + &
Sey2 \\
& & & $-$83 \verb+ + & 478 \verb+ + &
11.05\verb+ + & 47 & & $-$26 \verb+ + & 481 \verb+ + &
0.31 & 0.22 & 47 & $-$57 \verb+ + &
Sey2 \\
& & & $-$238 \verb+ + & 1600 \verb+ + &
13.01\verb+ + & 23 & & $-$379 \verb+ + & (1600)\verb+ + &
0.74 & 0.23 & 22 & 141 \verb+ + &
Sey2 \\
Mark 848\,S & 0.040 & & 81 \verb+ + & 154 \verb+ + &
0.83\verb+ + & 93 & & 128 \verb+ + & 132 \verb+ + &
0.43 & 0.03 & 81 & $-$47 \verb+ + &
H II \\
& & & $-$92 \verb+ + & 680 \verb+ + &
4.20\verb+ + & 7 & & 122 \verb+ + & 469 \verb+ + &
0.71 & 0.14 & 19 & $-$214 \verb+ + &
Sey2 \\
IRAS 15184$+$0834 & 0.031 & A & $-$286 \verb+ + & (178)\verb+ + &
1.35\verb+ + & 18 & & $-$203 \verb+ + & 178 \verb+ + &
0.54 & 0.01 & 21 & $-$83 \verb+ + &
H II \\
& & & $-$6 \verb+ + & 220 \verb+ + &
4.70\verb+ + & 82 & & 9 \verb+ + & 257 \verb+ + &
0.69 & 0.05 & 79 & $-$15 \verb+ + &
? \\
NGC 5953 & 0.007 & T, A& $-$92 \verb+ + & 200 \verb+ + &
0.55\verb+ + & 88 & & $-$36 \verb+ + & 205 \verb+ + &
0.60 & 0.03 & 54 & $-$56 \verb+ + &
H II \\
& & & $-$185 \verb+ + & 398 \verb+ + &
(10)$\;\verb+ +$ & 12 & & $-$128 \verb+ + & 412 \verb+ + &
1.96 & 0.18 & 22 & $-$57 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 311 \verb+ + & 1768 \verb+ + &
-- & -- & 24 & -- \verb+ + &
Sey1 \\
Kaz 49 & 0.030 & & 57 \verb+ + & 259 \verb+ + &
2.21\verb+ + & 88 & & 144 \verb+ + & 232 \verb+ + &
0.55 & 0.05 & 88 & $-$87 \verb+ + &
H II \\
& & & $-$252 \verb+ + & 1069 \verb+ + &
(10)$\;\verb+ +$ & 12 & & $-$218 \verb+ + & 879 \verb+ + &
2.05 & $<$ 0.5\, \verb+ + & 12 & $-$34 \verb+ + &
Sey2 \\
IRAS 16129$-$0753 & 0.033 & & $-$166 \verb+ + & 154 \verb+ + &
0.33\verb+ + & 84 & & $-$85 \verb+ + & 205 \verb+ + &
0.59 & $<$ 0.05\, \verb+ + & 100 & $-$81 \verb+ + &
H II \\
& & & $-$218 \verb+ + & 649 \verb+ + &
11.25\verb+ + & 16 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
Sey2 \\
IRAS 16382$-$0613 & 0.028 & & $-$76 \verb+ + & 348 \verb+ + &
3.94\verb+ + & 40 & & $-$72 \verb+ + & 257 \verb+ + &
0.91 & $<$ 0.03\, \verb+ + & 24 & $-$6 \verb+ + &
? \\
& & & $-$310 \verb+ + & 1157 \verb+ + &
4.06\verb+ + & 60 & & $-$188 \verb+ + & 1064 \verb+ + &
1.41 & $<$ 0.12\, \verb+ + & 24 & $-$122 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 144 \verb+ + & 4980 \verb+ + &
-- & -- & 52 & -- \verb+ + &
Sey1 \\
\hline
\end{tabular}
\end{flushleft}}
\end{sidewaystable}
\addtocounter{table}{-1}
\newpage
\onecolumn
\begin{sidewaystable}
\centering
\small{
\caption{Fitting profile analysis results (end).}
\begin{flushleft}
\begin{tabular}{ll|crrrr|crrccr|rc}
\hline
Name & \verb+ +$z$ & Stellar & $V\; \; \; \; \; \;$ &
FWHM\/ & \underline{$\lambda$5007$\:$} & $\rm H\beta$ & Stellar &
$V\; \; \; \; \; \;$ & FWHM\/ & \underline{$\lambda$6583$\:$} &
\underline{$\lambda$6300$\:$} & $\rm H\alpha$ & $\Delta V \verb+ + $ &
Class. \\
& & corr. & (\kms) &
(\kms) & $\rm H\beta$$\;\;$ & (\%) & corr. &
(\kms) & (\kms) & $\rm H\alpha$$\;\;$ & $\rm H\alpha$$\;\;$ &
(\%) & (\kms) & \\
\hline
Mark 700 & 0.034 & A & $-$74 \verb+ + & 618 \verb+ + &
1.86\verb+ + & 100 & & $-$28 \verb+ + & 435 \verb+ + &
2.19 & 0.19 & 23 & $-$46 \verb+ + &
Liner \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 369 \verb+ + & 1595 \verb+ + &
-- & -- & 61 & -- \verb+ + &
Sey1 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$1137 \verb+ + & 1097 \verb+ + &
-- & -- & 16 & -- \verb+ + &
Sey1 \\
MCG 03.45.003 & 0.024 & & 87 \verb+ + & 98 \verb+ + &
14.01\verb+ + & 47 & & 77 \verb+ + & 132 \verb+ + &
0.51 & 0.17 & 54 & 10 \verb+ + &
Sey2 \\
& & & 27 \verb+ + & 364 \verb+ + &
8.71\verb+ + & 53 & & 39 \verb+ + & 435 \verb+ + &
0.61 & $<$ 0.07\, \verb+ + & 46 & $-$12 \verb+ + &
Sey2 \\
PGC 61548 & 0.018 & T & 20 \verb+ + & 277 \verb+ + &
0.41\verb+ + & 92 & & 68 \verb+ + & 282 \verb+ + &
0.50 & 0.04 & 93 & $-$48 \verb+ + &
H II \\
& & & $-$5 \verb+ + & 541 \verb+ + &
(10)$\;\verb+ +$ & 8 & & 14 \verb+ + & 604 \verb+ + &
3.94 & 0.49 & 7 & $-$19 \verb+ + &
Sey2 \\
Kaz 214 & 0.046 & A & 0 \verb+ + &$<$ 80 \verb+ + &
5.32\verb+ + & 10 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
? \\
& & & 98 \verb+ + & 330 \verb+ + &
3.60\verb+ + & 73 & & -- \verb+ + & -- \verb+ + &
-- & -- & --$\;$ & -- \verb+ + &
? \\
& & & 195 \verb+ + & 740 \verb+ + &
(10)$\;\verb+ +$ & 17 & & -- \verb+ + & -- \verb+ + &
& -- & --$\;$ & -- \verb+ + &
Sey2 \\
Kaz 214 \#1 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$25 \verb+ + & 95 \verb+ + &
0.32 & $<$ 0.06\, \verb+ + & 100 & -- \verb+ + &
H II \\
Kaz 214 \#2 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & $-$28 \verb+ + & 148 \verb+ + &
0.38 & $<$ 0.05\, \verb+ + & 100 & -- \verb+ + &
H II \\
Kaz 214 \#3 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 12 \verb+ + & 245 \verb+ + &
0.52 & $<$ 0.05\, \verb+ + & 68 & -- \verb+ + &
H II \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & (120)\verb+ + & (525)\verb+ + &
(0.60) & 0.17 & 32 & -- \verb+ + &
Sey2? \\
Kaz 214 \#4 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 77 \verb+ + & 205 \verb+ + &
0.33 & $<$ 0.03\, \verb+ + & 60 & -- \verb+ + &
H II \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & (120)\verb+ + & (525)\verb+ + &
(0.60) & 0.16 & 40 & -- \verb+ + &
Sey2? \\
Kaz 214 \#5 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 114 \verb+ + & 132 \verb+ + &
0.28 & $<$ 0.02\, \verb+ + & 45 & -- \verb+ + &
H II \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & (120)\verb+ + & (525)\verb+ + &
(0.60) & 0.09 & 55 & -- \verb+ + &
Sey2? \\
Kaz 214 \#6 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 135 \verb+ + & 114 \verb+ + &
0.31 & 0.05 & 44 & -- \verb+ + &
H II \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & (120)\verb+ + & (525)\verb+ + &
(0.60) & 0.16 & 56 & -- \verb+ + &
Sey2? \\
Kaz 214 \#7 & & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 141 \verb+ + & 232 \verb+ + &
0.35 & $<$ 0.06\, \verb+ + & 100 & -- \verb+ + &
H II \\
NGC 6764 & 0.008 & & 47 \verb+ + & 330 \verb+ + &
0.62\verb+ + & 72 & & $-$7 \verb+ + & 319 \verb+ + &
0.65 & 0.04 & 66 & 54 \verb+ + &
H II \\
& & & $-$158 \verb+ + & 430 \verb+ + &
0.44\verb+ + & 28 & & $-$101 \verb+ + & 537 \verb+ + &
0.96 & 0.14 & 34 & $-$57 \verb+ + &
Liner \\
IRAS 22114$-$1109 & 0.054 & & 44 \verb+ + & 154 \verb+ + &
1.33\verb+ + & 46 & & $-$16 \verb+ + & 219 \verb+ + &
0.70 & $<$ 0.07\, \verb+ + & 57 & 60 \verb+ + &
H II \\
& & & 195 \verb+ + & 510 \verb+ + &
(10)$\;\verb+ +$ & 54 & & 203 \verb+ + & 319 \verb+ + &
0.60 & 0.12 & 43 & $-$8 \verb+ + &
Sey2 \\
Mark 308 & 0.023 & & 155 \verb+ + & 178 \verb+ + &
(0.1)\,\verb+ + & 56 & & 87 \verb+ + & 132 \verb+ + &
0.30 & 0.06 & 44 & 68 \verb+ + &
H II \\
& & & 243 \verb+ + & 240 \verb+ + &
(10)$\;\verb+ +$ & 24 & & 63 \verb+ + & 412 \verb+ + &
0.49 & 0.03 & 30 & 180 \verb+ + &
Sey2? \\
& & & 182 \verb+ + & 995 \verb+ + &
(10)$\;\verb+ +$ & 20 & & $-$34 \verb+ + & 1097 \verb+ + &
0.50 & 0.19 & 19 & 216 \verb+ + &
Sey2 \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & (23)\verb+ + & (1725)\verb+ + &
-- & -- & 7 & -- \verb+ + &
Sey1 \\
Mark 522 & 0.032 & A & 95 \verb+ + &$<$ 80 \verb+ + &
0.63\verb+ + & 86 & & 114 \verb+ + & 114 \verb+ + &
0.53 & $<$ 0.04\, \verb+ + & 75 & $-$19 \verb+ + &
H II \\
& & & $-$15 \verb+ + & 220 \verb+ + &
7.87\verb+ + & 14 & & 30 \verb+ + & 270 \verb+ + &
1.50 & $<$ 0.2\, \verb+ + & 25 & $-$45 \verb+ + &
Sey2 \\
Mark 313 & 0.006 & A & 128 \verb+ + & 154 \verb+ + &
2.29\verb+ + & 100 & & 117 \verb+ + & 114 \verb+ + &
0.44 & 0.10 & 74 & 11 \verb+ + &
H II \\
& & & -- \verb+ + & -- \verb+ + &
-- \verb+ + & --$\;$ & & 84 \verb+ + & 424 \verb+ + &
0.71 & 0.28 & 26 & -- \verb+ + &
Sey2? \\
Zw 453.062 & 0.025 & A & 20 \verb+ + & 200 \verb+ + &
0.31\verb+ + & 94 & & $-$42 \verb+ + & 163 \verb+ + &
0.46 & 0.05 & 91 & 62 \verb+ + &
H II \\
& & & $-$274 \verb+ + & 541 \verb+ + &
(10)$\;\verb+ +$ & 6 & & $-$160 \verb+ + & 514 \verb+ + &
3.33 & 0.4 $\;$ & 9 & 114 \verb+ + &
Sey2 \\
IC 5298 & 0.027 & T & 95 \verb+ + & 313 \verb+ + &
0.53\verb+ + & 76 & & 44 \verb+ + & 307 \verb+ + &
0.83 & 0.12 & 86 & 51 \verb+ + &
H II \\
& & & $-$68 \verb+ + & 557 \verb+ + &
(10)$\;\verb+ +$ & 24 & & 33 \verb+ + & 537 \verb+ + &
2.28 & $<$ 0.4\, \verb+ + & 14 & $-$101 \verb+ + &
Sey2 \\
\hline
\end{tabular}
\end{flushleft}}
\end{sidewaystable}
\newpage
\twocolumn
The line fitting analysis of the spectra was done in terms of Gaussian
components as described in V\'eron et al. (1980, 1981b,c). The three
emission lines, $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack$\lambda\lambda$\,6548, 6583 (or $\rm H\beta$\ and
\lbrack\ion{O}{iii}\rbrack$\lambda\lambda$\,4959, 5007)
were fitted by one or several sets of three Gaussian components;
whenever necessary, two components were added to fit the
\lbrack\ion{O}{i}\rbrack $\lambda\lambda$\,6300, 6363 lines. The width and redshift of each
component in a set were supposed to be the same. The intensity ratios
of the \lbrack\ion{N}{ii}\rbrack$\lambda\lambda$\,6548, 6583, \lbrack\ion{O}{iii}\rbrack$\lambda\lambda$\,4959, 5007 and
\lbrack\ion{O}{i}\rbrack$\lambda\lambda$\,6300, 6363 lines were taken to be
equal to 3.00, 2.96 and 3.11, respectively (Osterbrock 1974). Whenever a fit
resulted in a small negative intensity for a $\rm H\beta$\ component, we set the
corresponding $\lambda$5007$/$H$\beta$\ ratio to 10, the mean value for Seyfert 2s.
All line widths given in this paper have been
corrected for the instrumental broadening. The spectra and best
fits are plotted in Fig. \ref{spectra}, the parameters describing the
individual components required by the analysis being given in Table \ref{fits}.
\subsection{Notes on individual objects}
{\bf Mark 938}. This galaxy is apparently undergoing a merger as evidenced by
the presence of tidal tails (Mulchaey et al. 1996; Mazzarella \& Boroson 1993).
The nature of its emission-line spectrum has been rather controversial.
Afa\-nasjev et al. (1980) classified it as a Seyfert 2, Osterbrock \& Dahari
(1983) claimed that it is not a Seyfert; Dahari (1985), V\'eron-Cetty \&
V\'eron (1986a) and Veilleux et al. (1995) called it again a Seyfert 2, while
Mulchaey et al. (1996), observing a weak emission of \lbrack\ion{O}{iii}\rbrack$\lambda$5007 and
a strong
$\rm H\alpha$\ over the entire galaxy, suggested that there is no Seyfert activity
in this object, in agreement with Mazzarella \& Boroson (1993) who called it
a \ion{H}{ii}\ region. The line ratios published by Veilleux et al. (1995) and Vaceli
et al. (1997) indicate a ``transition'' spectrum, the \lbrack\ion{O}{iii}\rbrack\ lines
being weak for a Seyfert 2 galaxy ($\lambda$5007$/$H$\beta$\ = 4). The high resolution
spectroscopic observations of Busko \& Steiner (1990), showing complex emission
line profiles with great differences in width and shape between $\rm H\alpha$\ and
\lbrack\ion{N}{ii}\rbrack$\lambda$6583 (the measured line widths are 264 $\pm$7 and
384 $\pm$12 \kms\ for $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack$\lambda$6583, respectively), suggest
a ``composite'' spectrum. To fit our spectra, two components are needed: one
is a \ion{H}{ii}\ region with narrow lines ($\sim$ 255 \kms\ FWHM);
the other is a Seyfert 2 with much broader
lines ($\sim$ 760 \kms\ FWHM). For this component, we find a very high and
unlikely $\lambda$6583$/$H$\alpha$\ ratio ($\sim$ 6.8); however, there is a very strong and broad
$\rm H\beta$\ absorption line. It is probable that the broad $\rm H\alpha$\ emission component
intensity is greatly reduced by the presence of a $\rm H\alpha$\ absorption line
which has not been accounted for.
{\bf Mark 957}. This galaxy has been identified with the radiosource 5C 3.100
(Antonucci 1985) and a ROSAT X-ray sour\-ce (Boller et al. 1998).
Dahari \& de Robertis (1988) called it a Seyfert 2. However, Koski (1978)
and Halpern \& Oke (1987) have observed strong \ion{Fe}{ii}\ emission lines in
this object; furthermore, the continuum is very flat, extending far into the
blue (Koski 1978), accounting
for the classification of this object as a Narrow Line Seyfert 1 galaxy. This
classification is supported by Boller et al. (1996) who have found a steep
soft X-ray component (photon index $\Gamma$ = 2.9 $\pm$ 0.2) with a variable
flux (by a factor 1.9 over 18\,900 sec). The $\rm H\beta$\ line is very narrow
(FWHM $<$ 685 \kms) (Goodrich 1989); narrow $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack\ lines are observed
as far as 10\arcsec\ from the nucleus (with $\lambda$6583$/$H$\alpha$\ $\sim$ 0.4) (Halpern \&
Oke 1987), suggesting the presence of an extended \ion{H}{ii}\ region. In the
nucleus, the high ionization lines (\lbrack\ion{O}{iii}\rbrack\ and \lbrack\ion{Ne}{iii}\rbrack$\lambda$3869) are
found to be blueshifted by $\sim$ 280 \kms\ with respect to the low
ionization lines. Although having a relatively low signal-to-noise ratio,
our spectra are quite interesting. In the blue, there is a very narrow $\rm H\beta$\
emission line (FWHM $\sim$ 200 \kms) associated with very weak ($\lambda$5007$/$H$\beta$\
$\sim$ 0.15) and relatively broad (FWHM $\sim$ 710 \kms) \lbrack\ion{O}{iii}\rbrack\ lines; the
associated broad $\rm H\beta$\ component is
weak ($\lambda$5007$/$H$\beta$\ $\sim$ 9) and accounts for only 8\% of the total $\rm H\beta$\ flux.
The \lbrack\ion{O}{iii}\rbrack\ lines are blueshifted by $\sim$ 360 \kms\ with respect to $\rm H\beta$.
The red spectrum is also reasonably fitted with two sets of components; one
is narrow with weak \lbrack\ion{N}{ii}\rbrack\ lines, while the second is broader with
relatively strong \lbrack\ion{N}{ii}\rbrack\ lines. This is in satisfactory agreement with Halpern
\& Oke's results, and suggests that the nuclear spectrum is dominated by a
strong \ion{H}{ii}\ region superimposed onto a relatively weak Seyfert 2 nucleus.
{\bf IRAS 01346\,-\,0924} was identified by de Grijp
et al. (1987) with a galaxy they called MCG $-$02.05.022, which se\-ems to be
erroneous. It was classified a Seyfert 2 by de Grijp et al. (1992) on the
basis of its emission-line ratios. We discussed this object in Paper I,
giving it the wrong name (MCG $-$02.05.022); we suggested, on the basis of a
blue spectrum, that it was ``composite''. The best blue spectrum fit
is obtained with three sets of three Gaussians,
two being typical of a \ion{H}{ii}\ region and the third of a weak Seyfert 2
nebulosity. A weak broad (FWHM $\sim$ 2\,640 \kms) $\rm H\alpha$\ component may also
be present. The Seyfert 2 cloud is so weak that it is not detected on our
red spectrum.
\begin{figure*}
\resizebox{18cm}{!}{\includegraphics{SPECTRA_1BIT.EPS}}
\caption{\label{spectra}
Blue and red spectra for the 53 galaxies studied in this paper.
For 52 of the observed AGNs we present 66 \Am\ spectra; in the case of
VII Zw 73, we give 33 \Am\ spectra. For 4 of the observed objects, only
one spectral region is available. All the spectra were de-redshifted to
rest wavelengths. The spectral ranges displayed are
$\lambda\lambda$\,4750--5120 \AA\ and $\lambda\lambda$\,6480--6650 \AA. In
each frame the data points (small crosses), the best fit (solid line) and
the residuals (lower solid line) are shown. For the red spectrum of
SBS 1136$+$594, the individual components of the fit are also given as
an example (dotted lines).}
\end{figure*}
\addtocounter{figure}{-1}
\begin{figure*}
\resizebox{18cm}{!}{\includegraphics{SPECTRA_2BIT.EPS}}
\caption{Blue and red spectra for the 53 galaxies studied in this paper
(continued).}
\end{figure*}
\addtocounter{figure}{-1}
\begin{figure*}
\resizebox{18cm}{!}{\includegraphics{SPECTRA_3BIT.EPS}}
\caption{Blue and red spectra for the 53 galaxies studied in this paper
(continued).}
\end{figure*}
\addtocounter{figure}{-1}
\begin{figure*}
\resizebox{18cm}{!}{\includegraphics{SPECTRA_4BIT.EPS}}
\caption{Blue and red spectra for the 53 galaxies studied in this paper
(continued).}
\end{figure*}
\addtocounter{figure}{-1}
\begin{figure*}
\resizebox{18cm}{!}{\includegraphics{SPECTRA_5BIT.EPS}}
\caption{Blue and red spectra for the 53 galaxies studied in this paper (end).}
\end{figure*}
{\bf Mark 1066} is an early-type spiral galaxy (Afanasjev et al. 1981;
Mazzarella \& Boroson 1993). It was classified as a Seyfert 2 by Afanasjev et
al. (1980) and as a Seyfert 1.9 by Afanasjev et al. (1981) on the basis of
weak broad components in the Balmer lines; the existence of these broad
components has not been confirmed by Goodrich \& Osterbrock (1983) who
concluded, on the basis of the line ratios ($\lambda$5007$/$H$\beta$\ = 4.35, $\lambda$6583$/$H$\alpha$\ = 0.88,
$\lambda$6300$/$H$\alpha$\ = 0.08), that
this object is a Seyfert 2. However, Osterbrock \& Dahari (1983), on the
basis of the same data, called it a ``marginal'' Seyfert 2, the $\lambda$5007$/$H$\beta$\
ratio being relatively weak. The spectra
published by Wilson \& Nath (1990) and Veilleux (1991a) show that the
emission lines have a broad blue wing extending up to velocities of
1\,000 \kms\ with respect to the line peaks, the \lbrack\ion{O}{iii}\rbrack\ lines being
significantly broader (403 \kms\ FWHM) than the Balmer lines (280 \kms)
(Veilleux 1991b,c). De Robertis \& Osterbrock (1986) noted a good correlation
between the width of the line at half-peak intensity and the critical
density, suggesting that the narrow line region (NLR) is density stratified;
however, the density
stratification mostly affects the high velocity gas producing the wings of
the line profiles (Veilleux 1991c). Haniff et al. (1988) have published an
\lbrack\ion{O}{iii}\rbrack\ image suggestive of a double structure with a separation of
0\farcs8, and
the continuum nucleus in between. There are two emission peaks in the
core of the low-ionization lines; these peaks are separated by about 0\farcs5,
the velocity difference between them being $\Delta$V = 125 $\pm$ 20
\kms\ (Veilleux 1991c). High-resolution (0\farcs1) HST images (Bower et al.
1995) reveal that the $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack\ emission comes from a 3\arcsec -long
region centered on the nucleus, while the \lbrack\ion{O}{iii}\rbrack-emitting gas is concentrated
in a bright ``jet-like'' structure extending 1\farcs4 NW of the nucleus.
Long-slit spectroscopy suggests the existence of two kinematically distinct
regions: the first, of low-excitation, lies in the plane of the galaxy and
is normally rotating, while the second, of high-excitation, would be
inclined with respect to the disk. Bower et al. (1995) suggested that the
high-ionization cloud is a Seyfert 2 (with $\lambda$5007$/$H$\beta$\ $\sim$ 10--15) and the
low-ionization cloud is a Liner ($\lambda$5007$/$H$\beta$\ $\sim$ 2--3); it seems, however, that
the \lbrack\ion{O}{i}\rbrack\ lines are weak and that this region could be a \ion{H}{ii}\ region instead.
Three sets of lines are needed to fit our spectra; one set originates,
most probably, from a \ion{H}{ii}\ region, the two others having line ratios typical
of Seyfert 2 clouds. The velocity difference between the two Seyfert components
is $\Delta$V = 146 \kms\ in the blue and 105 \kms\ in the red. The complexity
of the line emission regions in this object, revealed by the HST observations,
makes the measured line ratios for each individual component rather
inaccurate. It seems likely that the density-stratified cloud emitting
relatively broad lines is compact and coincides with the nucleus. This is,
therefore, a ``composite-spectrum object''.
{\bf IRAS 03355\,+\,0104} has been identified by de Grijp et al.
(1987) with a galaxy shown to be a Seyfert 2 by de Grijp et al. (1992) who
have measured $\lambda$6583$/$H$\alpha$\ = 0.58, a normal value for such an
object; however, Vogel et al. (1993) have found much weaker \lbrack\ion{N}{ii}\rbrack\ lines,
with $\lambda$6583$/$H$\alpha$\ = 0.18. Our red spectrum gives $\lambda$6583$/$H$\alpha$\ = 0.49, in agreement with
de Grijp et al. (1992), and $\lambda$6300$/$H$\alpha$\ = 0.12, so this object
is a Seyfert 2 galaxy. In addition, a weak broad $\rm H\alpha$\ component
seems to be present, in which case it would be a Seyfert 1.9 galaxy.
{\bf IRAS 04210\,+\,0400} has been identified with a compact blue galaxy with a
faint blue, spiral companion (Moorwood et al. 1986). It is associated with a
double lobed radio source, 20--30 kpc in size (Beichman et al. 1985; Hill et
al. 1988) . The galaxy has an apparent spiral structure (Beichman et al.);
however, these features are dominated by emission lines, and the galaxy
is probably an elliptical (Hill et al. 1988; Steffen et al. 1996).
We have searched the {\it Hubble Space Telescope} archives and found images
obtained with the Wide Field Planetary Camera 2, on January 31, 1995
through medium and broad band filters isolating several emission lines and
a line-free continuum. We retrieved and analysed these unpublished images, the
{\it HST} observing log being given in Table \ref{hst_log}.
The galaxy was imaged on
the Planetary Camera, a 800 $\times$ 800 pixels CCD with a readout noise
of $\sim$ 5 e$^{-}$\,pixel$^{-1}$. The pixels size is 15 $\times$ 15 $\mu$m,
which corresponds to 0\farcs0455 on the sky; the field is 36\farcs4 $\times$
36\farcs4 (Trauger et al. 1994; Holzman et al. 1995).
Both the $\rm H\alpha$+\lbrack\ion{N}{ii}\rbrack\ and the \lbrack\ion{O}{iii}\rbrack\ images (after subtraction of the
continuum) show a
very complex structure with a bright unresolved nucleus, a relatively bright
elongated central region extending over $\sim$ 2\farcs4, made of several
distinct clouds, and a thin spiral feature with a total extent of about
15\arcsec\ (Fig. \ref{i04210_hst}).
\begin{figure}
\resizebox{8.8cm}{!}{\includegraphics{I04210_HSTBIT.EPS}}
\caption{\label{i04210_hst}
$\rm H\alpha$+\lbrack\ion{N}{ii}\rbrack\ {\it HST} image of IRAS 04210$+$0400, after removal of the
continuum. The nucleus is indicated by an arrow. 10\arcsec, given as a
reference on the image, correspond
to 13.5 kpc at the distance of the
galaxy (assuming $H\rm_{o}$ = 50 \kms\ Mpc$^{-1}$).}
\end{figure}
The Balmer decrement observed
over a 2\arcsec $\times$ 2\arcsec\ area centered on the nucleus is relatively
large ($\rm H\alpha$/$\rm H\beta$\ = 5.5) (Hill et al. 1988), implying a large extinction
($A\rm_{V}$ $\sim$ 1.4 mag). We cannot exclude the possibility that the
extinction varies over the emission nebulosity; therefore, the ratio
($\rm H\alpha$ $+$ \lbrack\ion{N}{ii}\rbrack)$/\lambda$5007, which is approximately equal
to $\rm H\alpha$$/ \lambda$5007 as \lbrack\ion{N}{ii}\rbrack$\lambda$6583 $\sim$ 1/3 $\rm H\alpha$, cannot
be taken as an estimate for the excitation
parameter $\lambda$5007$/$H$\beta$. Our entrance aperture (2\farcs1 $\times$ 5\farcs0), with
the slit oriented in the E-W direction, basically includes the central point
source and the bright central nebulosity.
Published nuclear line ratios (Hill et al.) led to the classification of
this object as a Seyfert 2 (Beichman et al.1985; Holloway et al. 1996)
although $\lambda$6583$/$H$\alpha$\ = 0.35, a low value for this class. The core region shows
asymmetric spatial structure with several separate
components in velocity and space; there are systematic shifts between peak
positions for the different lines (Holloway et al. 1996). Our spectra
basically confirm the line ratios obtained in the nuclear region. This is a
Seyfert 2 galaxy with abnormally weak $\lbrack \rm NII \rbrack$ emission lines.
\begin{table}[h]
\begin{center}
\caption{\label{hst_log}
{\it HST} observing log of IRAS 04210$+$0400.}
\begin{flushleft}
\begin{tabular}{llcrll}
\hline
ID label & Band & $\lambda\rm_{c}$ & $\Delta\lambda$ & Filter & Time \\
& & (\AA) & (\AA) & & \verb+ +(s)\\
\hline
u2mg0401t & \lbrack\ion{O}{iii}\rbrack & 5479 & 486 & F547\,M & ~300 \\
u2mg0402t & \lbrack\ion{O}{iii}\rbrack & 5479 & 486 & F547\,M & ~300 \\
u2mg0403t & continuum & 7940 & 1531 & F814\,W & ~600 \\
u2mg0404t & $\rm H\alpha$+\lbrack\ion{N}{ii}\rbrack & 6814 & 877 & F675\,W & ~600 \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table}
{\bf IRAS 04507\,+\,0358} has been identified with an elliptical galaxy
(de Grijp et al. 1987) shown to be a Seyfert 2 by de Grijp et al. (1992) who,
however, have measured $\lambda$6583$/$H$\alpha$\ = 0.28, a very low value
for an object of such a class. Our red spectrum gives $\lambda$6583$/$H$\alpha$\ = 0.47 with,
perhaps, a weak broad Lorentzian $\rm H\alpha$\ component. Therefore, this is
not a ``composite-spectrum object'', but rather a Seyfert 2 galaxy or perhaps a
Seyfert 1.9, if the broad component is confirmed.
{\bf VII Zw 73} has been classified as a
Seyfert 2 galaxy by de Grijp et al. (1992) on the basis of its emission line
ratios ($\lambda$5007$/$H$\beta$\ = 3.96, $\lambda$6583$/$H$\alpha$\ = 0.56);
however, as in the case of Mark 1066, the \lbrack\ion{O}{iii}\rbrack$\lambda$5007 line is rather
weak for a Seyfert 2. Our blue spectrum shows $\rm H\beta$\ to be clearly narrower
than the \lbrack\ion{O}{iii}\rbrack\ lines, an indication of the probable
``composite'' nature of this galaxy. To obtain a good fit
to our blue and red spectra, three components are needed. However, two of
these components turned out to have similar velocities and widths, making
the determination of the line ratios rather uncertain. We therefore re-observed
this galaxy with a higher resolution (33 \Am) in the red on
October 29, 1997 and in the blue on October 30. Again three components were
needed to fit the spectra, but this time they were clearly
identified as corresponding to two Seyfert 2 nebulosities and a
\ion{H}{ii}\ region.
{\bf Mark 622}. The line ratios published by Shuder \& Osterbrock (1981) show
that it is a Seyfert 2 galaxy, although the \lbrack\ion{O}{i}\rbrack$\lambda$6300 relative flux is
too low for such a class ($\lambda$6300$/$H$\alpha$\ = 0.06). Furthermore,
these authors found that the \lbrack\ion{O}{iii}\rbrack\ lines are much
broader (FWHM $\sim$ 1\,050 \kms) than the $\rm H\alpha$, \lbrack\ion{N}{ii}\rbrack\ and \lbrack\ion{O}{ii}\rbrack\
li\-nes (FWHM $\sim$ 350 \kms) suggesting the presence of two emission-line
regions of different ionization. Wilson \& Nath (1990) have shown that,
in the nucleus, the \lbrack\ion{N}{ii}\rbrack\ lines are broader than $\rm H\alpha$\ (FWHM = 340$\pm$20
and 240$\pm$20 \kms\ respectively); moreover, the \lbrack\ion{O}{iii}\rbrack\ image of this object
is only slightly resolved, while $\sim$ 60\% of the $\rm H\alpha$\ emission comes from
an extended component (Mulchaey et al. 1996). Our spectra confirm these
results; in fact, two components are needed in order to obtain a good fit:
one is representative of a \ion{H}{ii}\ region, the other is typical of a
Seyfert 2 cloud.
{\bf 3C 198.0}. The line ratios in this elliptical radio galaxy
are those of a \ion{H}{ii}\ region, as discussed in Paper I. Our measurements are
in agreement with the published values: $\lambda$6300$/$$\rm H\alpha$\ = 0.05
and $\lambda$6583$/$H$\alpha$\ = 0.28. This is therefore a really puzzling
object.
{\bf KUG 0825\,+\,248}. The published emission-line relative intensities
(Augarde et al. 1994) are typical of a \ion{H}{ii}\ region except for the
\lbrack\ion{O}{i}\rbrack\ lines, which are too strong ($\lambda$6300$/$$\rm H\alpha$\ = 0.13). On our
red spectrum, we measured $\lambda$6300$/$$\rm H\alpha$\ = 0.02. Our values
for $\lambda$5007$/$H$\beta$\ and $\lambda$6583$/$H$\alpha$\ are in agreement
with the published values. This object is, therefore, a typical \ion{H}{ii}\ region.
{\bf IRAS 09111\,-\,1007} has been identified with the
western component of a galaxy pair (Murphy et al. 1996). The eastern
component, separated by 40\arcsec, is called IRAS 09111$-$1007\,E, although
it is probably unrelated to the IR source; it has a ``transition'' spectrum
with $\lambda$5007$/$H$\beta$\ = 3.94, $\lambda$6583$/$H$\alpha$\ = 0.74 and $\lambda$6300$/$H$\alpha$\ = 0.07 (Duc et al. 1997). Our
observations suggest that the spectrum of this object is indeed
``composite'', the \lbrack\ion{O}{iii}\rbrack\ lines being broader than $\rm H\beta$. The
blue spectrum CCD image clearly shows the $\rm H\beta$\ line to be double and spatially
resolved. In fact, to obtain a good fit to the $\rm H\alpha$+\lbrack\ion{N}{ii}\rbrack\ and $\rm H\beta$+\lbrack\ion{O}{iii}\rbrack\
lines, three components are needed: one with line ratios typical of a
Seyfert 2 cloud, and the two others typical of \ion{H}{ii}\ regions.
{\bf Zw 238.066}. On the basis of the published line
intensity ratios, Veilleux et al. (1995) have called this object a Liner;
however, the \lbrack\ion{O}{i}\rbrack\ lines are weak ($\lambda$6300$/$H$\alpha$\ = 0.07).
Our observations suggest that it has a ``composite'' nature.
The red spectrum image clearly shows the $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack\ lines to be extended,
with a low relative intensity of the \lbrack\ion{N}{ii}\rbrack\ lines. The blue spectrum shows that
the \lbrack\ion{O}{iii}\rbrack\ lines have a broad blue wing not seen in $\rm H\beta$. We have therefore
fitted both the red and blue spectra with three sets of lines. The blue
spectrum has a rather poor signal-to-noise ratio which probably explains
why the parameters of the blue and red fits (especially the line widths) are
not in good agreement; however, the differences of profile of the
different lines leave no doubt about the ``composite'' nature of this
spectrum. Indeed, the fits show that the emission-line spectrum is dominated
by two \ion{H}{ii}\ clouds
separated by $\sim$ 220 \kms; but there is, in addition, a weak Seyfert 2
nebulosity with relatively broad lines ($\sim$ 1\,030 \kms\ FWHM).
{\bf UGC 5101}. This is an ultra-luminous infrared galaxy
($L\rm_{IR}$ $\geq$ 10$^{12}$ $L\rm_{\sun}$); it is a peculiar galaxy with
a large ring and a 15 kpc tidal tail extending to the west, which suggests a
merger, although no companions are known to this galaxy. It has a single
bright nucleus (Sanders et al. 1988). Optical spectra have been
published by Sanders et al. who classified it as a Seyfert 1.5 galaxy on the
basis of a relatively broad $\rm H\alpha$\ line, by Veilleux et al. (1995) and Wu et al.
(1998a,b) who called it
a Liner, and by Liu \& Kennicutt (1995). However, the published line ratios
($\lambda$5007$/$H$\beta$\ = 2.9--4.7, $\lambda$6583$/$H$\alpha$\ = 1.17--1.35, $\lambda$6300$/$H$\alpha$\ = 0.06--0.09) rather indicate a
``transition'' spectrum. Our red slit spectrum (PA = 270\degr)
shows spatially extended narrow emission lines with a steep velocity gradient
across the nucleus in addition to broader, spatially unresolved lines in
the nucleus itself. We have extracted five columns centered on
the nucleus and analysed separately the five spectra. In each case, we have
found a narrow component (FWHM $\sim$ 100--150 \kms)
with $\lambda$6583$/$H$\alpha$\ $\sim$ 0.50 and $\lambda$6300$/$H$\alpha$\ $\sim$ 0.05, the velocity decreasing
from $\sim$ 250 (West) to $-$120 \kms\ (East).
On three spectra, we detected relatively broad (FWHM $\sim$ 500 \kms)
lines, with $\lambda$6583$/$H$\alpha$\ $>$ 1 and $\lambda$6300$/$H$\alpha$\ $<$ 0.40. In addition, on the
two central spectra, there is a weak, broad (FWHM $\sim$ 1\,200 \kms)
$\rm H\alpha$\ component. It was not possible to perform such a detailed analysis on
the blue spectrum, which has a much lower signal-to-noise ratio due
to a large Balmer decrement ($\rm H\alpha$/$\rm H\beta$\ = 8.3; Sanders et al. 1988).
Nevertheless, we can draw some conclusions: the medium width
$\rm H\alpha$\ component flux, coadded on the three central columns, represents
24\% of the total $\rm H\alpha$\ flux on the same three columns
(excluding the broad $\rm H\alpha$\ component); assuming that the Balmer
decrement is the same for the narrow and medium width components, and that
the narrow component is a \ion{H}{ii}\ region (this component
having $\lambda$6583$/$H$\alpha$\ $\sim$ 0.50, must have $\lambda$5007$/$H$\beta$\ $<$ 2), we conclude that the
intermediate width lines set must have $\lambda$5007$/$H$\beta$\ $>$ 3.5 and is, therefore, a
Seyfert 2 cloud. So, UGC 5101 has a ``composite'' spectrum with a rather strong
starburst component and a Seyfert 1.8 or 1.9 nucleus
(we are not able to decide between 1.8 or 1.9 as, because of the rather
poor signal-to-noise ratio around $\rm H\beta$, it is not possible to put a significant
upper limit to the broad $\rm H\beta$\ component flux).
{\bf NGC 2989} is a Sc galaxy (Sandage \& Bedke 1994). Published data (Table
\ref{line_ratios}) indicate a ``transition'' spectrum; Phillips et al. (1983) called it a
intermediate object, but V\'eron-Cetty \& V\'eron (1984), on the basis of
the same line ratios, classified it as a \ion{H}{ii}\ region. Our measured flux
ratios ($\lambda$5007$/$H$\beta$\ = 1.70, $\lambda$6300$/$H$\alpha$\ = 0.04 and $\lambda$6583$/$H$\alpha$\ = 0.53) are in good agreement
with Phillips et al. and show that this is indeed a pure \ion{H}{ii}\
region with, perhaps a marginally large $\lambda$6583$/$H$\alpha$\ ratio.
{\bf CG 49}. This galaxy has been shown to have a Seyfert 2 spectrum
by Salzer et al (1995); however, they have measured a low relative
intensity for the \lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ = 0.30) as the redshifted wavelength
of \lbrack\ion{N}{ii}\rbrack$\lambda$6583 is 6873 \AA\ and falls on the atmospheric B band.
After correction for this absorption, our spectrum gives $\lambda$6583$/$H$\alpha$\ = 0.80,
a normal value for a Seyfert 2 galaxy.
{\bf NGC 3185} is a SBa galaxy (Sandage \& Bedke 1994). Its
emission-line spectrum is power-law photoionized
according to Stauffer (1982). Ho et al. (1997a) called it a Seyfert 2,
although the published line ratios indicate a ``transition'' spectrum.
We have fitted the lines with two sets of Gaussian profiles:
one system is a \ion{H}{ii}\ region; the other corresponds to a Seyfert nebulosity,
although the \lbrack\ion{O}{i}\rbrack\ lines are quite weak ($\lambda$6300$/$H$\alpha$\ = 0.04).
{\bf Arp 107\,A} is the SW component of an interacting galaxy pair (Arp 1966).
It exhibits Seyfert 2 activity and its spectrum shows very strong \lbrack\ion{N}{ii}\rbrack\
lines ($\lambda$6583$/$H$\alpha$\ = 3.0) according to Keel et al. (1985). Our spectrum shows a
more normal value.
In fact, the lines have a complex profile which can best be fitted by
two sets of Gaussians having different widths, but similar $\lambda$6583$/$H$\alpha$\ ratios,
namely 1.41 and 1.23.
{\bf NGC 3504}. This Sb galaxy (Sandage \& Bedke 1994) has a ``composite''
nucleus showing both non-thermal activity and recent star formation, the
optical spectrum being dominated by the regions of stellar activity (Keel
1984). Ho et al. (1993a), who published line ratios for this object,
suggested that it could be a ``transition'' between a \ion{H}{ii}\ region and a Liner,
but Ho et al. (1997a) called it a \ion{H}{ii}\ region. Fitting our red spectrum
with a single set of Gaussians does not give a satisfactory fit and indicates
the presence of weak broad wings in the \lbrack\ion{N}{ii}\rbrack\ lines; two sets of Gaussians
are needed, revealing the presence of a weak Seyfert-like nebulosity with
$\lambda$6583$/$H$\alpha$\ = 1.87. The blue spectrum was also fitted by two sets of Gaussians
satisfying, respectively, $\lambda$5007$/$H$\beta$\ = 0.1 and $\lambda$5007$/$H$\beta$\ = 10. The NGC 3504
spectrum is therefore ``composite'' and dominated by a starburst; a weak Seyfert
feature is also present. No component showing Liner characteristics was
detected in this object.
{\bf IRAS 11058\,-\,1131}. In Paper I we concluded, on the basis of a red
spectrum, that this object, classified as a Seyfert 2 by various authors
(de Grijp et al. 1992, Osterbrock \& de Robertis 1985), has
weak \lbrack\ion{N}{ii}\rbrack\ lines. Re-analyzing the red spectrum, we have found
the possible presence of a weak broad (FWHM $\sim$ 2\,100 \kms)
$\rm H\alpha$\ component with 24\% of the total $\rm H\alpha$\ flux. Our blue
spectrum confirms that this is indeed an AGN with $\lambda$5007$/$H$\beta$\ = 7.6 and
a relatively strong \ion{He}{ii}\ $\lambda$4686 line ($\lambda$4686$/$H$\beta$\ = 0.18).
Our conclusion is that IRAS 11058$-$1131 belongs to the relatively rare
class of Seyfert 2 galaxies having weak \lbrack\ion{N}{ii}\rbrack\ lines, discussed in
Sect. \ref{weak_NII_lines};
other galaxies belonging to this class are UM 85 and 3C 184.1 (Paper I).
{\bf NGC 3642} is an Sb galaxy (Sandage \& Bedke 1994). It was classified as a
Liner by Heckman (1980). Filippenko \& Sargent (1985) noted the presence
of very narrow emission lines (FWHM $\sim$ 110 \kms) superposed on weak,
significantly broader components; a weak broad $\rm H\alpha$\ component was also
detected. Koratkar et al. (1995) confirmed the presence of the broad $\rm H\alpha$\
component and called this object a Liner, although their published line ratios
rather point to a ``transition'' spectrum. Our spectra confirm the
presence of a broad $\rm H\alpha$\ component (FWHM $\sim$ 2\,160 \kms) and show, in
addition, that the narrow line spectrum is made of two clouds, one with
line ratios typical of a \ion{H}{ii}\ region and the other of a Seyfert 2 cloud, this
object being, therefore, a ``composite''.
{\bf Mark 1291}. Spectroscopic observations of this barred
spiral by Kollatschny et al. (1983) show it to be a ``transition object''
between Seyfert 2s and \ion{H}{ii}\ regions; however, high excitation lines
characteristic of Seyfert 2s such as \lbrack\ion{Fe}{xiv}\rbrack\,$\lambda$5303
and \lbrack\ion{Fe}{x}\rbrack\,$\lambda$6374 are detected. The nuclear emission
is compact (Gonzalez Delgado et al. 1997).
Our optical spectra show a weak broad (FWHM $\sim$ 1\,950 \kms) $\rm H\alpha$\ component
suspected by Kollatschny et al. The narrow lines are well fitted with a
single Gaussian profile ($\sim$ 160 \kms\ FWHM). The line ratios are
$\lambda$5007$/$H$\beta$\ = 3.84, $\lambda$6300$/$H$\alpha$\ = 0.04 and $\lambda$6583/ $\rm H\alpha$\ = 0.73
(significantly larger than the published value, 0.48); the classification
of this object is therefore ambiguous: it is a Seyfert 2 in the the $\lambda$5007$/$H$\beta$\
{\it vs.} $\lambda$6583$/$H$\alpha$\ diagram and a \ion{H}{ii}\ region in the $\lambda$5007$/$H$\beta$\ {\it vs.} $\lambda$6300$/$H$\alpha$\
diagram. It would be of interest to obtain high-resolution, high
signal-to-noise spectra of this object to confirm its ``transition'' nature.
{\bf IRAS 11285\,+\,8240\,A} has been classified as a Seyfert 2 gala\-xy
by Klaas \& Elsasser (1991), with $\lambda$5007$/$H$\beta$\ = 8.25 and $\lambda$6583$/$H$\alpha$\ = 0.46. Our
red spectrum is well fitted by a single set of components with $\lambda$6583$/$H$\alpha$\ = 0.45,
in excellent agreement with the published value, and $\lambda$6300$/$H$\alpha$\ = 0.11. This is
another example of Seyfert 2 galaxy with marginal\-ly weak \lbrack\ion{N}{ii}\rbrack\ lines.
{\bf Mark 739} has a double nucleus (Petrosian et al. 1978).
The eastern nucleus has a Seyfert 1 spectrum (Petrosian et al. 1979;
Netzer et al. 1987). The western component, Mark 739\,W, has a starburst
spectrum according to Netzer et al.; however, Rafanel\-li et al. (1993)
have remarked that $\rm H\beta$\ is unresolved with a resolution of 130 \kms,
while the \lbrack\ion{O}{iii}\rbrack\ lines are significantly broader
($\sim$ 300 \kms\ FWHM). Our observations show that this spectrum is
indeed ``composite'' with a Seyfert 2 nucleus and a \ion{H}{ii}\ region and, in
addition, a weak broad $\rm H\alpha$\ component.
{\bf SBS 1136\,+\,594} is a Seyfert 1.5 galaxy (Markarian et al.
1983; Martel \& Osterbrock 1994). The narrow-line spectrum, however, has
very weak \lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ = 0.25; Martel \& Osterbrock);
this is confirmed by our spectra, with even weaker
\lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ = 0.10). This object belongs to the class of AGNs
with very weak \lbrack\ion{N}{ii}\rbrack\ lines discussed in Sect. \ref{weak_NII_lines}.
The individual components used to fit the red spectrum of this
object are plotted in Fig. \ref{spectra}.
{\bf NGC 3994} is an Sbc galaxy (Sandage \& Bedke 1994) in a triple system,
interacting both with NGC 3991 and NGC 3995. Based on the observed line
ratios, Keel et al. (1985) classified it as a
Liner; the $\lambda$5007$/$H$\beta$\ and $\lambda$6583$/$H$\alpha$\ values rather suggest a \ion{H}{ii}\ region. We found
that its spectrum is ``composite'', the main contribution to the Balmer lines
coming from a starburst. The relative
strength of the \lbrack\ion{O}{i}\rbrack\ lines is large in the other component ($\lambda$6300$/$H$\alpha$\ = 0.53),
suggesting that it is a Liner.
{\bf NGC 4102} is a Sb galaxy (Sandage \& Bedke 1994). For Ho et. al.
(1997a) it is a \ion{H}{ii}\ region, although its UV spectrum does not resemble that
of a starburst galaxy (Kinney et al. 1993). The spectrum published by Ho et al.
(1995) rather indicates a ``transition object''. Our blue spectrum shows
a $\rm H\beta$~line obviously narrower than the \lbrack\ion{O}{iii}\rbrack\ lines, indicating that the
spectrum is ``composite''. The blue spectrum was fitted with two sets of three
Gaussians. The broadest $\rm H\beta$\ profile in the fit contains 7\% of
the total $\rm H\beta$\ flux. The red spectrum having a very high signal-to-noise
ratio, we needed three sets of three Gaussians to get a good fit;
we forced one set to have the same width as the broadest
set in the blue fit. The fitting analysis gives two strong narrow
components and a weak broad one, containing $\sim$ 5\% of the total
$\rm H\alpha$\ flux and having $\lambda$6583$/$H$\alpha$\ = 1.57.
There was no need to use two narrow components to fit
the $\rm H\beta$\ line, as the spectrum was taken under poor transparency conditions
and its signal-to-noise ratio is much lower. Our conclusion is that the nucleus
of NGC 4102 is dominated by a starburst, but that a weak Seyfert 2 component
is present and detected mainly by the broadening of the \lbrack\ion{O}{iii}\rbrack\ lines.
{\bf IRAS 12474\,+\,4345\,S}. For de Grijp et al. (1992), this object is a
\ion{H}{ii}\ region; however, the published $\lambda$5007$/$H$\beta$\ line ratio (2.93) is slightly
high for the corresponding $\lambda$6583$/$H$\alpha$\ ra\-tio (0.42). Fitting our spectra with
a single set of lines gives an unsatisfactory result, while the solution
with two sets of lines is acceptable, with small residuals. One set of
lines corresponds to a \ion{H}{ii}\ region, while the other has $\lambda$5007$/$H$\beta$\ = 4.43,
$\lambda$6300$/$H$\alpha$\ = 0.03 and $\lambda$6583$/$H$\alpha$\ = 0.40; these values are intermediate between
those corresponding to \ion{H}{ii}\ regions and Seyfert 2 nebulosities. Moreover,
the \ion{He}{ii}\,$\lambda$4686 line is detected with $\lambda$4686$/$H$\beta$\ = 0.17, if all the
\ion{He}{ii}\ flux is attributed to the ``transition'' component. We are unable
to conclude concerning the nature of this second component.
{\bf UGC 8621} is a Seyfert 1.8 galaxy according to Osterbrock \& Martel
(1993). Our blue spectrum shows the $\rm H\beta$\ line to be much narrower
($<$ 80 \kms) than the \lbrack\ion{O}{iii}\rbrack\ lines ($\sim$ 665 \kms),
but no evidence of a broad component. To account for the different
widths observed, we fitted the blue spectrum with two sets of lines; for
the narrow component, we found $\lambda$5007$/$H$\beta$\ = 0.28, while we have
forced the broader component to have $\lambda$5007$/$H$\beta$\ = 10. We fitted the red
spectrum with two sets of three Gaussians, imposing
to one of them to have the same width as the broader \lbrack\ion{O}{iii}\rbrack\ component; an
additional Gaussian was added to fit the broad $\rm H\alpha$\ wings.
We find no trace of \lbrack\ion{O}{i}\rbrack\ emission
($\lambda$6300$/$H$\alpha$\ $\leq$ 0.01) for the narrow component and $\lambda$6300$/$H$\alpha$\ $<$ 0.3 for the
Seyfert cloud. This is a ``composite object'', with a relatively strong \ion{H}{ii}\
region and a weak Seyfert 1.9 nebulosity.
{\bf Mark 266} is a merging system with
two nuclei separated by 10\arcsec\ (Hutchings \& Neff 1988; Wang et al. 1997).
It is a Luminous Infrared Galaxy (LIG), i.e.,
10$^{11.2}$ $< L\rm_{IR} <$ 10$^{12}$ $L_{\sun}$ (Goldader et al. 1997).
Line intensity ratios in the two nuclei have been measured by Osterbrock \&
Dahari (1983), Veilleux \& Osterbrock (1987), Mazzarella \& Boroson (1993),
Osterbrock \& Martel (1993) and Kim et al. (1995); all these measurements
are in good agreement, if we make the assumption that Kim et al. have
inverted the NE and SW components. On the basis of these line ratios,
Mark 266\,SW has been classified as a Seyfert 2 by Mazzarella \& Boroson,
Kim et al. and Wu et al. (1998b), and as a ``marginal'' Seyfert 2 by
Osterbrock \& Dahari, while Mark 266\,NE has been called a Liner by
Mazzarella \& Boroson, Kim et al. and Wu et al., and a
``narrow emission-line galaxy'' (NELG) by Osterbrock \& Dahari and
Veilleux \& Osterbrock. NELGs, for
these authors, are emission-line galaxies that may be either Liners or
\ion{H}{ii}\ regions. For Mark 266\,NE, we have only a blue spectrum which, by simple
visual inspection, shows quite different $\rm H\beta$\ and \lbrack\ion{O}{iii}\rbrack\ line profiles. Our
line profile analysis reveals two clouds, one with narrow lines
(FWHM $\sim$ 300 \kms) and
$\lambda$5007$/$H$\beta$\ = 0.96, the other with broader lines (FWHM $\sim$ 1\,000 \kms) and
$\lambda$5007$/$H$\beta$\ = 2.2 and containing 37\% of the total $\rm H\beta$\ flux. The published
value of the intensity ratio $\lambda$6300$/$H$\alpha$\ is 0.15. If the narrow component is
associated with a \ion{H}{ii}\ region, it contributes in a small amount to the
observed \lbrack\ion{O}{i}\rbrack$\lambda$6300 flux and therefore the $\lambda$6300$/$H$\alpha$\ ratio for the broader
component is likely to be significantly larger than 0.12, which means that this
component could be a Liner. We have fitted the blue spectrum of Mark 266\,SW
with three sets of Gaussians: one of them corresponds, most probably, to a
\ion{H}{ii}\ region with narrow lines (295 \kms\ FWHM) and $\lambda$5007$/$H$\beta$\ = 0.36;
the two other sets have widths of 200 and 600 \kms\ and $\lambda$5007$/$H$\beta$\ = 3.7
and 13.3, respectively. We also fitted the red spectrum with three sets
of Gaussians,
forcing, in addition, one of the sets to have a width of 600 \kms. The result
is a set of narrow lines with $\lambda$6583$/$H$\alpha$\ = 0.58 corresponding to the
narrow blue lines, confirming that this system is indeed coming from a
\ion{H}{ii}\ region. The set having the broadest lines has intensity ratios typical
of a Seyfert cloud. The third set, with $\lambda$5007$/$H$\beta$\ = 3.65 and $\lambda$6583$/$H$\alpha$\ = 0.58, has
still an intermediate spectrum.
{\bf Mark 1361} was called a Seyfert 2 galaxy by Kim et al. (1995). Our
analysis of a red spectrum (Paper I) lead to the conclusion that it is a
``composite object''. Our blue spectrum confirms this result. Three sets of
three components were needed to get a good fit.
In one set we had to impose
$\lambda$5007$/$H$\beta$\ = 10, in another $\lambda$5007$/$H$\beta$ = 0.1. The best fit resulted in a narrow set
of lines with very weak \lbrack\ion{O}{iii}\rbrack\ lines and two sets of lines with
strong \lbrack\ion{O}{iii}\rbrack\ contribution. We then re-analyzed the red spectrum using three
sets of three lines; we had to add a weak broad $\rm H\alpha$\ component
(FWHM $\sim$ 2\,400 \kms, with $\sim$ 10\% of the total $\rm H\alpha$\ flux) in
order to obtain a good fit. For the
narrowest set of three lines, we found $\lambda$6583$/$H$\alpha$\ = 0.54, for the other two,
0.66 and 1.04 respectively. The conclusion is that Mark 1361 has a
``composite'' spectrum with a \ion{H}{ii}\ cloud contributing half of the
$\rm H\beta$\ line and a Seyfert 2
nebulosity with complex line profiles (two Gaussians were needed for the fit).
If the presence of the broad $\rm H\alpha$\ component is confirmed, this object could be
a Seyfert 1.9 galaxy.
{\bf Mark 461} is a Seyfert 2 galaxy according to Huchra \& Burg (1992);
however, Cruz-Gonzalez et al. (1994) have measured $\lambda$5007$/$H$\beta$\ = 1.13. The emission
is concentrated in the nuclear region (Gonzalez Delgado et al. 1997).
No other line ratios have been published for this object. The $\rm H\beta$\ and
\lbrack\ion{O}{iii}\rbrack\ lines obviously do not have the same profile. To get
a good fit, two sets of lines were necessary. The object is ``composite'', one
component being a \ion{H}{ii}\ region and the other probably a Seyfert 2 nucleus.
{\bf 4C 26.42}. This object has been identified with a cD galaxy,
MCG 05.33.005 (Carswell \& Wills 1967; Olsen 1970; Merkelijn 1972), the
brightest member of Abell 1975 (Parma et al. 1986; Pilkington 1964); it is
a FR~I, Z-shaped radiosource (van Breugel et al. 1984; Ge \& Owen 1993).
Emission lines have been detected in the nuclear region, with $\lambda$5007$/$H$\beta$\ = 0.4,
$\lambda$6583$/$H$\alpha$\ = 0.8 and $\lambda$6300$/$H$\alpha$\ = 0.2 (Anton 1993). These
values are similar to the ones usually observed in Liners, but for
the low \lbrack\ion{O}{iii}\rbrack$\lambda$5007 line intensity. Examination of
the red spectrum shows that the lines are obviously double. Fitting the lines
with two sets of components revealed two clouds with a velocity difference of
$\sim$ 330 \kms. Their line ratios are very similar and typical of Liners
with exceptionally weak \lbrack\ion{O}{iii}\rbrack\ lines.
{\bf I Zw 81}. Koski (1978) observed narrow (FWHM = 225 $\pm$ 200 \kms)
emission lines in this galaxy, with $\lambda$5007/$\rm H\beta$\ = 3.14,
$\lambda$6583/$\rm H\alpha$\ = 0.67 and $\lambda$6300$/$H$\alpha$\ = 0.07; the
narrowness of the emission lines and
the relative weakness of the \lbrack\ion{O}{iii}\rbrack\ lines led him to
conclude that this is not a Seyfert galaxy, but rather a ``transition''
case between a \ion{H}{ii}\ galaxy and a Seyfert 2. Shuder \& Osterbrock (1981)
and Veilleux \& Osterbrock (1987) called it a narrow-line Seyfert.
Our red spectrum shows the lines to be double with a separation of 220
\kms. Fitting both the red and blue spectra with two sets of three Gaussians,
we found $\lambda$5007$/$H$\beta$\ = 2.05 (1.41) and $\lambda$6583$/$H$\alpha$\ = 0.78 (0.56) for the high (low)
velocity clouds. In both cases, the \lbrack\ion{O}{i}\rbrack\ lines are undetected with
$\lambda$6300$/$H$\alpha$\ $<$ 0.04. The two clouds are most probably \ion{H}{ii}\ regions.
{\bf Mark 477}. This object was discussed in Paper I, where we concluded from
the published data and the analy\-sis of a red spectrum that its nature
was unclear. Heckman et al. (1997) have argued that the observed UV through
near-IR continuum in the nucleus of Mark 477 is dominated by light from a
starburst. The narrow Balmer emission lines would then be excited by ionizing
radiation from both the hidden Seyfert 1 nucleus and from the hot stars in
the starburst. We have re-analyzed our old red spectrum together with our new
blue spectrum. Three sets of components were needed to fit these very high
signal-to-noise data. The three line-systems are typical of Seyfert 2s,
although two have relatively weak \lbrack\ion{N}{ii}\rbrack\ intensities, with
$\lambda$6583/ $\rm H\alpha$\ = 0.31 and 0.38, respectively. We found
no evidence for the presence of broad Balmer components.
{\bf Mark 848\,S} is a LIG (Goldader et al. 1997) belonging to a pair of
interacting galaxies (Armus et al. 1990). The northern galaxy is a \ion{H}{ii}\
region (Wu et al. 1998a,b); the southern component has been called a
Liner (Mazzarella \& Boroson 1993; Veilleux et al. 1995), though its
line ratios (Kim et al. 1995) are ambiguous, the \lbrack\ion{O}{i}\rbrack$\lambda$6300 line being
rather weak for a Liner. The line profile analy\-sis of our spectra shows it
to be a ``composite object'' with two distinct emission-line clouds: a narrow
one (FWHM $\sim$ 140 \kms, with $\lambda$5007$/$H$\beta$\ = 0.83, $\lambda$6583$/$H$\alpha$\ = 0.43 and
$\lambda$6300$/$H$\alpha$\ = 0.03), identified with a \ion{H}{ii}\ region, and a somewhat broader one
(FWHM $\sim$ 580 \kms), with line ratios ty\-pical of a Seyfert 2
($\lambda$5007$/$H$\beta$\ = 4.20, $\lambda$6583/ $\rm H\alpha$\ = 0.71 and $\lambda$6300$/$H$\alpha$\ = 0.14).
{\bf IRAS 15184\,+\,0834}. De Grijp et al. (1992) called this object a
Seyfert 2; however they found relatively weak \lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ = 0.42).
Our spectra show that the \lbrack\ion{N}{ii}\rbrack\ lines are significantly stronger than the
published values. A reasonable fit is obtained with two sets of components:
one is a \ion{H}{ii}\ region; the other could be a Seyfert 2, although the \lbrack\ion{O}{i}\rbrack\
lines are relatively weak.
{\bf NGC 5953} is a peculiar S0 (Rampazzo et al. 1995) or Sa
(Delgado \& Perez 1996) galaxy interacting with NGC 5954 (Arp 1966). It
has a Seyfert 2 nucleus (Rafanelli et al. 1990; Delgado \& Perez)
surrounded by a ring of star formation with a radius of $\sim$ 4\arcsec\
(Delgado \& Perez). Rafanelli et al. and Delgado \& Perez
studied this object using a slit width of 2\farcs0 and 1\farcs5 respectively.
The seeing was $\sim$ 1\arcsec\ during Delgado \& Perez' observations, while
it was not specified by Rafanelli et al. who, however, easily separated
the galaxy nucleus from a star located 3\arcsec\ away.
We may reasonably assume that, in both cases, the nuclear spectrum
corresponds to a 2\farcs0 $\times$ 2\farcs0, or smaller, aperture.
The line ratios given by these authors (see Table \ref{line_ratios}) are
typical of a
Seyfert 2 region, although \lbrack\ion{O}{i}\rbrack$\lambda$6300 may be somewhat weak for
this type of objects; but as stressed before, we cannot exclude some
contamination by the circumstellar emission region.
Both Keel et al. (1985) and Kim et al. (1995) have observed the NGC 5953
nuclear region with a relatively large aperture: Keel et al. used a
$\phi$ = 4\farcs7 circular aperture, while Kim et al. used a long
2\arcsec\ $\times$ 14\arcsec\ slit. It is clear
that these two large aperture spectra must contain a significant
contribution from the circumstellar emission region and, indeed, the
published line intensity ratios are those of ``transition'' spectra.
We used a 2\farcs1 slit and the seeing was 2\farcs6; we extracted
7 columns, i.e., 7\farcs6, so some
contamination from the circumstellar emission region was expected.
Effectively, by doing the line profile fitting analy\-sis, we were able to
identify two components of different line widths and strengths, one of
which can be associated with a \ion{H}{ii}\ region
($\lambda$5007$/$H$\beta$\ = 0.55, $\lambda$6583$/$H$\alpha$\ = 0.60, $\lambda$6300$/$H$\alpha$\ = 0.03 and FWHM $\sim$ 200 \kms); the
other component, broader (FWHM $\sim$ 400 \kms), reveals the
presence of a Seyfert 2 nebulosity, the measured line intensities being:
$\lambda$6583$/$H$\alpha$\ = 1.96 and $\lambda$6300$/$H$\alpha$\ = 0.18, with $\lambda$5007$/$H$\beta$\ fixed to 10.
Lines of \lbrack\ion{Fe}{iii}\rbrack$\lambda$4659 and \lbrack\ion{Fe}{vii}\rbrack$\lambda$5158 are clearly visible in
the blue spectrum. A very weak broad $\rm H\alpha$\ line is possibly detected,
which would qualify NGC 5953 as a Seyfert 1.9 galaxy.
{\bf Kaz 49} has been classified as a Seyfert 1 by Yegiazarian \& Khachikian
(1988), as a Seyfert 1.9 by Moran et al. (1994), who have detected a weak broad
$\rm H\alpha$\ component (FWHM = 1\,150 \kms), and as a \ion{H}{ii}\ region by Boller et al.
(1994). The latter classification was based on measured line ratios
($\lambda$5007$/$H$\beta$\ = 2.58, $\lambda$6583$/$H$\alpha$\ = 0.56, $\lambda$6300$/$H$\alpha$\ = 0.025) that rather point to a
``transition'' spectrum. The line profile analysis of our spectra shows a
strong \ion{H}{ii}\ region ($\lambda$5007$/$H$\beta$\ = 2.21, $\lambda$6300$/$H$\alpha$\ = 0.05, $\lambda$6583$/$H$\alpha$\ = 0.55) and a
weak Seyfert 2 component for which we have fixed $\lambda$5007$/$H$\beta$\ = 10. There is no
evidence for the presence of a broad $\rm H\alpha$\ component; however, the blended
weak $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack\ components, each having a FWHM $\sim$ 880 \kms\ may be
easily mistaken for a broad $\rm H\alpha$\ line.
{\bf IRAS 16129\,-\,0753} has been classified as a possible Liner by de
Grijp et al. (1992) on the basis of the measured line intensity ratios
($\lambda$5007$/$H$\beta$\ = 2.03, $\lambda$6583$/$H$\alpha$\ = 0.64), although \lbrack\ion{O}{i}\rbrack$\lambda$6300 was very weak.
The line fitting analysis of our blue spectrum shows this object to be
``composite''. The red spectrum, which has a relatively low signal-to-noise
ratio, is well fitted by a single set of lines corresponding to the \ion{H}{ii}\
region; the Seyfert component is undetected.
{\bf IRAS 16382\,-\,0613} has been called a Seyfert 2 by Aguero et al.
(1995) and a possible Seyfert 2 by de Grijp et al. (1992); however, the
\lbrack\ion{O}{i}\rbrack$\lambda$6300 line is marginally weak for a Seyfert 2, with $\lambda$6300$/$H$\alpha$\ =
0.09 (Aguero et al.). The line profiles on the blue spectrum are obviously
complex. Fitting these lines with two sets of Gaussians reveals a narrow
component (FWHM $\sim$ 350 \kms) with $\lambda$5007$/$H$\beta$\ = 3.94, and a broader
component (FWHM $\sim$ 1\,160 \kms) with $\lambda$5007$/$H$\beta$\ = 4.06. The red spectrum
fit gives a solution compatible with the blue solution plus a broad Balmer
line (FWHM $\sim$ 5\,000 \kms). The two components have strong \lbrack\ion{N}{ii}\rbrack\ lines,
but the \lbrack\ion{O}{i}\rbrack\ lines are weak. For the broadest set of lines, we find $\lambda$6300$/$H$\alpha$\
$<$ 0.12, compatible with a Seyfert 2 nebulosity; however, the narrow
component has $\lambda$6300$/$H$\alpha$\ $<$ 0.03 and seems therefore to have a genuine
``transition'' spectrum.
{\bf Mark 700} was called a Seyfert 1 galaxy by Denisyuk et al. (1976), who
found a broad $\rm H\alpha$\ component. For Koski (1978), it is a weak-lined Seyfert
galaxy with Balmer absorption lines, very similar to ``normal''
emission-line galaxies. Ferland \& Netzer (1983) included it in a
Liner list, on the basis of the intensity ratios published by Koski.
Our observations show that this object is, indeed, a Liner; the
broad $\rm H\alpha$\ component seen by Denisyuk et al. is confirmed.
{\bf MCG 03.45.003}. The \lbrack\ion{N}{ii}\rbrack\ lines measured by de Grijp et al. (1992) are
rather weak for a Seyfert 2 galaxy ($\lambda$6583$/$H$\alpha$\ = 0.42) and, on the basis of a red
spectrum, we concluded in Paper I that this object could have a ``composite''
spectrum. Our analysis of both the blue and red spectra show
that two kinematically distinct clouds are present in this object,
both of them having Seyfert 2 characteristics.
{\bf PGC 61548}. The red spectrum is ``composite'' and confirms the
result presented in Paper I. The line profile analysis reveals the presence
of both a \ion{H}{ii}\ region ($\lambda$5007/$\rm H\beta$\ = 0.41,
$\lambda$6583/$\rm H\alpha$\ = 0.50, $\lambda$6300/$\rm H\alpha$\ = 0.04, FWHM $\sim$ 250 \kms)
and a Seyfert 2 nebulosity $\lambda$5007$/$H$\beta$\ fixed to 10.0,
$\lambda$6583$/$H$\alpha$\ $\sim$ 3.9, $\lambda$6300$/$H$\alpha$\ $\sim$ 0.5 and FWHM $\sim$ 570 \kms).
{\bf Kaz 214} is a Seyfert 2 galaxy for de Grijp et al. (1992), with
$\lambda$5007$/$H$\beta$\ = 5.23 and $\lambda$6583$/$H$\alpha$\ = 0.39; however, the \lbrack\ion{N}{ii}\rbrack\ lines are weak for
a Seyfert 2. On our red exposure, the slit position angle was PA = 139\degr.
By simple visual inspection of the CCD image, we see that the lines are double:
in one of the line-systems the lines are spatially extended and narrow,
with relatively weak \lbrack\ion{N}{ii}\rbrack; in the other, the lines are spatially unresolved,
but relatively broad, and \lbrack\ion{N}{ii}\rbrack\ is stronger. The spectrum is obviously
``composite'' with a \ion{H}{ii}\ region and a Seyfert component. However, when analysing
the spectrum obtained by extracting three columns centered on the nucleus,
we were unable to get a satisfactory fit confirming the visual impression.
We then extracted individually seven columns (numbered 1 to 7, from SE to NW)
containing obvious emission lines; the continuum was brightest in columns
4 and 5. Columns 1, 2 and 7 were fitted with a single set of lines, while for
columns 3 to 6, two sets of lines were necessary. We have made the assumption
that the Seyfert component is really spatially unresolved and, consequently,
forced the redshift, width and the $\lambda$6583$/$H$\alpha$\ ratio of this component to be the
same on all columns (that is, 120 \kms, 525 \kms\ FWHM and 0.60,
respectively), the
only free parameter being the $\rm H\alpha$\ intensity. In addition to this Seyfert
component, we have found, on all columns, a narrow component with relatively
weak \lbrack\ion{N}{ii}\rbrack\ lines; the velocity of this narrow component increases from
$-$25 \kms\ to 140 \kms\ from column 1 to 7. The blue spectrum was taken with
the slit oriented N-S. As the seeing was rather poor, seven columns were
added together when extracting the spectrum. The best fit was obtained with
three sets of lines: for one of them, we forced $\lambda$5007$/$H$\beta$\ = 10 (this turns to be
the broadest component); the two other sets have narrow lines, with moderate
$\lambda$5007$/$H$\beta$\ ratios. We therefore conclude that Kaz 214 has a ``composite'' spectrum.
But this example shows that it may not be possible to show that
a ``transition'' spectrum is a ``composite'' spectrum when
the spatial resolution is insufficient,
this being due to the large velocity dispersion gradient sometimes present
in the nuclear region which broadens the lines.
{\bf NGC 6764} has been called a Seyfert 2 galaxy by Rubin et al. (1975),
in spite of $\rm H\beta$\ being stronger than \lbrack\ion{O}{iii}\rbrack$\lambda$5007; this classification
was based on the width of the $\rm H\alpha$\ and \lbrack\ion{N}{ii}\rbrack\ lines ($\sim$ 750 \kms)
but Wilson \& Nath (1990) found these lines to be much narrower
($\sim$ 380 \kms\ FWHM). Koski (1978) noticed the presence of weak
\ion{H}{i}\ absorption lines and, from the line intensity ratios, concluded
that it was very much like ``normal'' galaxies, while for Shuder \&
Osterbrock (1981), it is not a Seyfert 2.
Using the line ratios published by Koski (1978), Ferland \& Netzer (1983)
classified it as a Liner. Osterbrock \& Cohen (1982) have detected in the
spectra of this object the $\lambda$4650 Wolf-Rayet emission feature. For
Ashby et al. (1992), it is a starburst galaxy. Line profile fitting of our
spectra revealed the ``composite'' nature of this object.
Two systems were identified: a narrow one (FWHM = 325 \kms)
with line ratios compatible with those usually observed in \ion{H}{ii}\ regions
($\lambda$5007$/$H$\beta$\ = 0.62, $\lambda$6583$/$H$\alpha$\ = 0.65 and $\lambda$6300$/$H$\alpha$\ = 0.04) and a broader system
(FWHM = 480 \kms) with line ratios similar to those of Liners
($\lambda$5007$/$H$\beta$\ = 0.44, $\lambda$6583$/$H$\alpha$\ = 0.96 and $\lambda$6300$/$H$\alpha$\ = 0.14). This is, therefore,
a ``composite object''.
{\bf IRAS 22114\,-\,1109} was classified a Seyfert 2 by Veilleux et al.
(1995); however, the \lbrack\ion{O}{iii}\rbrack\ lines are relatively weak
for this type of objects ($\lambda$5007$/$H$\beta$\ = 4.22; Kim et al. 1995).
A line profile analysis was performed on the red and blue
spectra. The measured line intensities and widths are compatible with the
simultaneous presence on the slit of both a \ion{H}{ii}\ region
($\lambda$5007$/$H$\beta$\ = 1.33, $\lambda$6583$/$H$\alpha$\ = 0.70, $\lambda$6300$/$H$\alpha$\ $<$ 0.07 and FWHM = 185 \kms) and a
Seyfert 2 nebulosity ($\lambda$5007$/$H$\beta$\ fixed to 10.0, $\lambda$6583$/$H$\alpha$\ = 0.60,
$\lambda$6300$/$H$\alpha$\ = 0.12 and FWHM = 415 \kms), so this is another example of a
``composite-spectrum object''.
{\bf Mark 308} was called a Seyfert 2 galaxy by Popov \& Khach\-ikian
(1980) and Zamorano et al. (1994); V\'eron-Cetty \& V\'eron
(1986a) classified it as a \ion{H}{ii}\ region, although the published
line ratios (Table \ref{line_ratios}) are unlikely
for either classes. The analysis of our blue spectrum (Paper I) showed this
object to be ``composite'' with one narrow component with weak \lbrack\ion{O}{iii}\rbrack\ lines and
two broader components with strong \lbrack\ion{O}{iii}\rbrack\ lines; the analysis of our red
spectrum confirms this result, the narrow system (FWHM $\sim$ 155 \kms) being
typical of a \ion{H}{ii}\ region (with $\lambda$5007$/$H$\beta$\ fixed to 0.1,
$\lambda$6583$/$H$\alpha$\ = 0.30 and $\lambda$6300$/$H$\alpha$\ = 0.06)
and the two broader line-sets (FWHMs $\sim$ 325 and 1\,045 \kms, respectively)
of Seyfert-like clouds. Moreover, we have detected a weak broad
(1\,725 \kms\ FWHM)
$\rm H\alpha$\ component containing $\sim$ 7\% of the total $\rm H\alpha$\ flux. The companion
galaxy, KUG 2239$+$200\,A, at $z$ = 0.024 (Keel \& van Soest 1992) and located
53\arcsec\ NE of Mark 308, has a \ion{H}{ii}-like emission-line spectrum.
{\bf Mark 522} is a Seyfert 2 galaxy according to Veilleux \& Osterbrock
(1987); however, the \lbrack\ion{O}{iii}\rbrack\ and \lbrack\ion{O}{i}\rbrack\ lines are relatively weak ($\lambda$5007$/$H$\beta$\
= 3.23, $\lambda$6300$/$H$\alpha$\ =0.07). Our observations show this object to be ``composite'',
with two different line systems: one, ``narrow'' (FWHM $\sim$ 100 \kms),
typical of a \ion{H}{ii}\ region ($\lambda$5007$/$H$\beta$\ = 0.63, $\lambda$6583$/$H$\alpha$\ = 0.53 and $\lambda$6300$/$H$\alpha$\ $<$ 0.04),
the other, somewhat broader (FWHM $\sim$ 240 \kms), associated with a
Seyfert 2 nebulosity ($\lambda$5007/ $\rm H\beta$\ = 7.87, $\lambda$6583$/$H$\alpha$\ = 1.50 and
$\lambda$6300$/$H$\alpha$\ $<$ 0.2).
{\bf Mark 313} is a Seyfert 2 galaxy according to Osterbrock \& Pogge (1987)
(with $\lambda$5007$/$H$\beta$\ = 3.52, $\lambda$6583$/$H$\alpha$\ = 0.52 and $\lambda$6300$/$H$\alpha$\ = 0.10), and Moran et al.
(1996); the \lbrack\ion{O}{iii}\rbrack\ lines are relatively weak for a Seyfert 2.
Images in \lbrack\ion{O}{iii}\rbrack\ and $\rm H\alpha$+\lbrack\ion{N}{ii}\rbrack\ show a very complex structure, with high
excitation gas restricted to a symmetric, linear feature (Mulchaey et al.
1996). From a two-component Gaussian fitting of high-dispersion spectra
of the nucleus of this object, Maehara \& Noguchi (1988) concluded that
it is a ``composite object'' with a \ion{H}{ii}\ region and a Liner nebulosity.
Line profile analysis of our spectra reveals the contribution of two
different line-emitting regions: one, with $\lambda$5007$/$H$\beta$\ = 2.29, $\lambda$6583$/$H$\alpha$\ = 0.44,
$\lambda$6300/ $\rm H\alpha$\ = 0.10 and narrow width (135 \kms\ FWHM), is typical
of a \ion{H}{ii}\ region; the other, much weaker, is not detected in the blue
and its line ratios are $\lambda$6583$/$H$\alpha$\ = 0.71 and $\lambda$6300$/$H$\alpha$\ = 0.28; it could be either
a Seyfert 2 or a Liner, depending on the $\lambda$5007$/$H$\beta$\ ratio.
{\bf Zw 453.062} is a LIG (Goldader et al. 1997); it was called a Liner by
Veilleux et al. (1995) on the basis of the measured emission line ratios,
while Wu et al. (1998a,b) found its properties to be intermediate between
\ion{H}{ii}\ regions and Liners, although the \lbrack\ion{O}{i}\rbrack\ lines are very weak. Our spectra
suggest that it is a ``composite object'', one component being a Seyfert 2
nebulosity and the other a \ion{H}{ii}\ region.
{\bf IC 5298} is a LIG (Goldader et al. 1997). Wu et al. (1998a,b) found
that its properties are intermediate between \ion{H}{ii}\ regions and Liners;
it was classified as a Seyfert 2 by Veilleux et al. (1995) although the
\lbrack\ion{O}{i}\rbrack\ lines are rather weak ($\lambda$6300$/$H$\alpha$\ = 0.05). Our observations suggest
that the spectrum is ``composite'', being dominated by a \ion{H}{ii}\ region.
\section{Results}
In Fig. \ref{fwhms} we have plotted the FWHMs (corrected for the instrumental
broadening) of each individual com\-ponent, i.e. of each
set of lines used to fit the blue and red spectra, as listed in Table
\ref{fits} (cols. 5 and 10, respectively). The good correlation found
between the blue and red FWHMs gives confidence in the fitting analysis.
Figure \ref{dd} shows the log($\lambda$5007$/$H$\beta$) {\it vs.}
log($\lambda$6583$/$H$\alpha$) and log($\lambda$5007$/$H$\beta$) {\it vs.}
log($\lambda$6300$/$H$\alpha$) diagrams traditionally used to classify nuclear emission-line
regions into \ion{H}{ii}\ regions, Liners or Sey\-fert 2s. We have delimited in
the two diagrams three regions, each corresponding to one of these classes.
In Figs. \ref{dd}a and \ref{dd}b we have plotted all
objects for which line ratios are
available in the literature and which are unambiguously
classified as \ion{H}{ii}\ regions (crosses), Seyfert 2s (open circles) or
Liners (open squares); we have also plotted the 61 observed objects
suspected of having a ``transition'' spectrum (filled circles):
they fall, at least in one of the diagrams, in a
``zone of avoidance'', i.e. outside the regions arbitrary assigned to the
classical emission-line regions. In figures \ref{dd}c and \ref{dd}d, which are
the same as \ref{dd}a and \ref{dd}b respectively, we have plotted
the individual components used to fit the spectra, as given in Table \ref{fits}.
\begin{figure}
\resizebox{8.4cm}{!}{\includegraphics{FWHM_AREF.EPS}}
\caption{\label{fwhms}
FWHM of all the individual line-components measured on
the red spectra {\it vs.} the FWHM of the individual components measured on
the blue spectra.}
\end{figure}
\begin{figure*}
\resizebox{12cm}{!}{\includegraphics{DD_AREF.EPS}}
\hfill
\parbox[b]{55mm}{
\caption{\label{dd}
Diagnostic diagrams showing the log($\lambda$5007$/$H$\beta$) {\it vs.} log($\lambda$6583$/$H$\alpha$) --
boxes (a) and (c) -- and log($\lambda$5007$/$H$\beta$) {\it vs.} log($\lambda$6300$/$H$\alpha$) -- boxes
(b) and (d). \ion{H}{ii}\ regions are represented by crosses,
Seyfert 2 galaxies by open circles,
and Liners by open squares. In (a) and (b), filled circles represent
``transition objects'', i.e. objects which, in at least one of the
diagrams, fall outside the arbitrarily delimited regions assigned to
\ion{H}{ii}\ regions, Seyfert 2s and Liners. In (c) and (d) we plotted
the individual components. The symbols are the same as in the upper panels;
open triangles represent objects which could not be classified (``?'' in
Table \ref{fits}).}
}
\end{figure*}
It is apparent that most of the
``transition objects'' belong to one of the three following categories:
\begin{enumerate}
\item{A few objects fall into the ``zones of avoidance'' only because they
have inaccurate published line ratios, appearing to be ``normal'' when more
accurate measurements are obtained; this is the case, for instance, for
IRAS 04507$+$0358, KUG 0825$+$248, NGC 2989, CG 49 and Arp 107A.}
\item{A few objects have Seyfert 2 spectra with abnormal\-ly weak \lbrack\ion{N}{ii}\rbrack\ lines.
They constitute a rare but interesting class of objects which is further
discussed below.}
\item{Most ``transition'' spectra turn out to be ``composite'',
due to the simultaneous presence on the slit of a \ion{H}{ii}\ region and a Seyfert 2
nebulosity. We have observed 70\% of all the objects in an unbiased sample of
galaxies displaying a ``transition'' nuclear spectrum. Modeling of the data
revealed that most of them have in fact a ``composite'' spectrum,
suggesting that genuine ``transition objects'' do not exist at all. However,
in a few cases such as NGC 3185, Mark 1291,
IRAS 12474$+$4345\,S, Mark 266\,SW or IRAS 15184$+$0834, we
cannot prove that the spectra are ``composite''; the classification
is ambiguous. Further studies are needed to find out the true nature of these
``transition objects''.}
\end{enumerate}
Fig. \ref{histo_oi_oiii} is the histogram of the parameter log
($\lambda$6300/$\lambda$5007) for 159 Seyfert 2s and Liners after correction
of the line fluxes for reddening,
assuming that the intrinsic Balmer decrement is $\rm H\alpha$ /$\rm H\beta$ = 3.1 (Osterbrock
\& Dahari 1983) [Binette et al. (1990) suggested an even higher value for the
intrinsic Balmer decrement in AGNs: $\rm H\alpha$ /$\rm H\beta$\ = 3.4]. The histogram
has two main peaks showing a clear separation between strong \lbrack\ion{O}{iii}\rbrack$\lambda$5007
objects (Seyfert 2s) and weak \lbrack\ion{O}{iii}\rbrack$\lambda$5007 objects (Liners). Although
our sample is heterogeneous and incomplete, this suggests that there is
no continuity between the two classes of objects. Heckman (1980)
originally defined Liners as objects with $\lambda$6300$/\lambda$5007\ $>$ 0.33; it seems that
$\lambda$6300$/\lambda$5007\ $>$ 0.25 would be a more realistic definition, as the observed
distribution of this ratio really shows a minimum centered around this value.
According to Ho et al. (1997a), the separation between the two principal
ionization sources (young stars {\it vs.} AGNs) and between the two AGN
excitation classes (Seyfert 2 {\it vs.} Liners) does not have sharp,
rigorously defined boundaries. Fig. \ref{dd} shows that this is not the
case. In fact, the three areas containing the \ion{H}{ii}\ regions, the Seyfert 2s
and the Liners are clearly separated; almost every ``transition object'' turns
out to be ``composite'' when observed with sufficient resolution.
Several authors had already suspected this to be the case. Kennicutt et
al. (1989) and Ho et al. (1997c) have shown that the distribution of
\ion{H}{ii}\ nuclei in the $\lambda$5007$/$H$\beta$\ {\it vs.} $\lambda$6583/ $\rm H\alpha$\ plane parallels
the disk \ion{H}{ii}\ region sequence, the most striking feature being a clear
offset between the two classes of objects, the \ion{H}{ii}\ nuclei having larger
$\lambda$6583$/$H$\alpha$\ ratios for the same excitation; this effect could be due to the
presence of a weak active nucleus in many of these galaxies.
Binette (1985) also suggested that mixed cases of starburst and
Liner spectra might be relatively common, providing a possible interpretation
for objects which have an unusually strong $\lambda$6300$/$H$\alpha$\ ratio compared to \ion{H}{ii}\
regions (NGC 3994, for example). Filippenko \& Terlevich (1992)
suggested that Liners with weak \lbrack\ion{O}{i}\rbrack\ emission ($\lambda$6300$/$H$\alpha$\ $<$ 1/6) might be powered
by hot main-sequence stars; however, Ho et al. (1993a) showed that these
objects are most probably ``composite''.
Ho et al. (1993b) reported the discovery of a non random trend in the
dispersion of emission-line intensity ratios for Sey\-fert 2s. $\lambda$6300$/$H$\alpha$\ and
$\lambda$6583$/$H$\alpha$\ were found to be correlated with $\lambda$5007$/$H$\beta$, suggesting the influence
of a single underlying physical parameter -- the hardness of the ionizing
continuum. Our data do not show these correlations, which could be artifacts
due to the inclusion in the sample of ``composite'' spectra.
Examination of Fig. \ref{dd} shows that the points representative of Seyfert 2
galaxies are not distributed at random in the region assigned to them.
Figure \ref{histo_nii_ha} is the histogram of the
quantity log($\lambda$6583$/$H$\alpha$); it shows a
sharp maximum at $\sim$ $-$0.05, with broad wings. Our sample of
(131) Seyfert 2 galaxies is not complete in any sense and this could
therefore be due to observational biases although this seems unlikely, as
the $\lambda$6583$/$H$\alpha$\ ratio is not used for finding Seyfert 2 galaxies. We have no
explanation for this fact.
\section{Discussion}
\subsection{The blue continuum in Seyfert 2 galaxies}
Koski (1978) and Kay (1994) found that all Seyfert 2 galaxies show
an ultraviolet excess and
weak absorption lines when compared with galaxies with no emission lines,
indicating the presence of a blue featureless continuum. Boisson
\& Durret (1986) and Vaceli et al. (1997) suggested that this continuum is
a non-thermal power-law continuum. Kinney et
al. (1991) argued that most of the Seyfert 2s in which a blue continuum has
been observed are of type Sb or earlier, suggesting that it is truly
associated with the Seyfert nucleus. Shuder (1981)
showed that its strength and the $\rm H\alpha$\ luminosity are
strongly correlated suggesting that a direct physical connection exists between
the two; studying a sample of 28 Seyfert 2s, Yee (1980) found that the
$\rm H\beta$\ and continuum fluxes (rather than luminosities) are proportional over
two orders of magnitude, with, however, a relatively large dispersion; but
a number of those objects are now known to be Seyfert 1 galaxies.
Martin et
al. (1983) discovered that a small fraction of all Seyfert 2 galaxies have a
highly polarized continuum. Subsequently, Antonucci \& Miller (1985),
Miller \& Goodrich (1990) and Tran et al. (1992) showed that
these objects harbour a hidden Seyfert 1
nucleus, the observed polarized continuum arising from scattering of the
nuclear continuum by dust or warm electrons. But most Seyfert 2s have very
little polarization (Martin et al. 1983), much less than expected in the
reflection model (Miller \& Goodrich 1990).
On the other hand, Terlevich et
al. (1990) showed that in Seyfert 2 galaxies, the IR \ion{Ca}{ii}\ triplet is equal
or, in some cases, higher than in normal elliptical galaxies, which is most
naturally explained by the presence of young stars contributing heavily to
the nuclear light at near-IR wavelengths.
\begin{figure}
\resizebox{8.4cm}{!}{\includegraphics{HISTO_OI_OIII.EPS}}
\caption{\label{histo_oi_oiii}
Histogram of log($\lambda$6300$/\lambda$5007) for the 159 Seyfert 2 and Liners plotted
in Fig. \ref{dd}.}
\end{figure}
Heckman et al. (1995) used {\it International Ultraviolet Explorer (IUE)}
spectra of 20 of the brightest type 2 Seyfert nuclei to build an ultraviolet
template for this class; while the continuum was
well detected in the template, there was no detectable broad line region (BLR),
implying that no more than 20\%\ of the template continuum could be light
from a hidden Seyfert 1 nucleus scattered by dust; they suggested
that either most of the nuclei in their sample were ``pure'' type 2 Seyfert
galaxies for which we have a direct view of the central engine and which
simply lack of BLR, or that most of the observed ultraviolet
continuum is produced by starbursts. From the absence of polarization of the
continuum of most Seyfert 2 galaxies and of broad Balmer lines,
Cid Fernandez \& Terlevich (1995) concluded that, most probably, this
continuum was due to a population of young stars in the vicinity of the
nucleus. Colina et al. (1997) obtained
ultraviolet {\it HST} images of four nearby Seyfert 2 galaxies known to
have circumstellar star-forming rings, providing direct empirical evidence
that the UV flux emitted by these galaxies is dominated by radiation
coming from clusters of young hot stars distributed along the star-forming
ring. If similar rings are a common characteristic of Seyfert 2
galaxies, the large {\it IUE} aperture would include both the Seyfert 2 nucleus
and the rings for distances larger than 25 Mpc. Gonzalez Delgado
et al. (1998) presented {\it HST} images and ultraviolet spectra of three
Seyfert 2 nuclei (IC 3639, NGC 5135 and IC 5135); the data show the existence
of nuclear starbursts (with absorption features formed in the photosphere of
late O and early B stars) dominating the ultraviolet light. It is remarkable
that, of the three observed galaxies, two (NGC 5135 and IC 5135) have a
``composite'' nuclear emission spectrum, while the third (IC 3639), which has
the largest UV nuclear flux (associated with the Seyfert nucleus) relative to
the total UV flux, has a pure Seyfert 2 spectrum due to the relative weakness
of the starburst emission component.
We conclude that there is ample evidence for the presence of young, hot
stars in the nuclear region of many Seyfert 2 galaxies. When the
continuum is relatively bright,
the associated \ion{H}{ii}\ region could be strong enough to displace the
object into the
``transition'' zone in the diagnostic diagrams.
AGNs are more frequent in
early type galaxies while starbursts
are more often found in late-type galaxies (V\'eron \& V\'eron-Cetty 1986;
Ho et al. 1997b; Vaceli et al. 1997). It is therefore rather surprising to
find almost systematically a population of young stars in Seyfert 2 galaxies;
perhaps the nuclear activity triggers the star formation?
\begin{figure}
\resizebox{8.4cm}{!}{\includegraphics{HISTO_NII_HA.EPS}}
\caption{\label{histo_nii_ha}
Histogram of log($\lambda$6583/$\rm H\alpha$) for the 131 Seyfert 2 galaxies
plotted in Fig. \ref{dd}.}
\end{figure}
\subsection{Excitations and abundances in \ion{\it H}{\it II} nuclei and AGNs}
The $\lambda$5007$/$H$\beta$\ and $\lambda$6583$/$H$\alpha$\ ratios are strongly correlated in \ion{H}{ii}\ regions.
Theoretical studies show that the hea\-vy-metal abundances
change continuously along this
sequence, a low $\lambda$5007$/$H$\beta$\ ratio indicating a high metal abundance and a high
$\lambda$5007$/$H$\beta$\ ratio, a low metal abundance, with the heavy metal abundances changing
from 1.5 Z$_{\sun}$ at the lower right of Fig. \ref{dd}a to
0.25 Z$_{\sun}$ at the
upper left (see for instance Dopita \& Evans 1986; Ho et al. 1997b).
However, Stasi\'nska \& Leitherer (1996) have shown that most startbusts
and \ion{H}{ii}\ galaxies can be described as being produced by an evolving
starburst with an universal initial mass function embedded in a gas cloud
of the same metallicity. The emission line ratios depend mainly on two
independent parameters: the age of the starburst and the metallicity. In
this scenario, the $\lambda$5007$/$H$\beta$\ ratio effectively changes with these two parameters
and therefore is not a direct measurement of metallicity.
The metallicity is strongly correlated with luminosity, luminous
galaxies having higher metallicities; this correlation is also valid for
elliptical galaxies, for which the metallicity is determined from absorption
lines with [O/H] $\sim$ 1 at $M\rm_{B} = -$21
(Salzer et al. 1989; Zaritsky et al. 1994).
AGNs are known to occur preferentially in high luminosity (Ho et al. 1997b),
early-type (V\'eron \& V\'eron-Cetty 1986; Vacali et al. 1997) galaxies;
they are therefore expected to have high metallicities.
Indeed, the NLRs of active galactic nuclei have enhanced nitrogen abundances
(Storchi-Bergmann \& Pastoriza 1989, 1990; Storchi-Bergmann et al. 1992;
Schmitt et al. 1994). In these NLRs, [N/O] correlates with [O/H] in a
manner identical to \ion{H}{ii}\ regions in normal galaxies, with nuclear [O/H]
and [N/O] values ranging from 1 Z$_{\sun}$ to 2 Z$_{\sun}$ (Storchi-Bergmann et
al. 1996b). Storchi-Bergmann et al. (1996b,c) have determined the chemical
composition of the \ion{H}{ii}\ regions in the ring surrounding the nucleus of several
AGNs, as well as in the nuclei; high metallicities were found ([O/H]
$\sim$ 2 Z$_{\sun}$ and
[N/O] $\sim$ 3 Z$_{\sun}$) both in the \ion{H}{ii}\ regions and in the AGNs,
these abundances being
similar to those found in the nuclei of non-active galaxies with the same
morphological type and absolute magnitude. Further work by
Storchi-Bergmann et al. (1998) has shown that, in fact, oxygen abundances
derived for Seyfert 2 nebulosities and neighbouring \ion{H}{ii}\ regions (assuming
that the emission lines in the active nucleus are due to photoionization
by a typical active galactic nucleus continuum) are well correlated, while
this is not the case for Liners. This suggests that the gas in AGNs
and in the neighbouring \ion{H}{ii}\ regions has the same origin and that the
scatter observed in the Seyfert 2 region in the
diagnostic diagrams, involving the $\lambda$6583$/$H$\alpha$\ ratio, is due to variations in the
nitrogen abundance. In NGC 6300, in which $\lambda$6583$/$H$\alpha$\ = 3.4, the
nitrogen abundance is estimated to be $\sim$ 5 Z$_{\sun}$.
We have seen that nuclear \ion{H}{ii}\ regions and Seyfert 2 nebulosities, when
appearing in the same galaxy, have the same high metallicity; as a
result of their metallicity, the \ion{H}{ii}\ regions have a low excitation, while
the Seyfert 2 nebulosities have a high excitation. This explains why it is
relatively easy to separate the two components in ``transition'' spectra.
\subsection{Objects with weak [\ion{\it N}{\it II}] lines}
\label{weak_NII_lines}
Figure \ref{dd} shows a small number of objects which
have very weak \lbrack\ion{N}{ii}\rbrack\ lines for Seyfert 2 galaxies; their \lbrack\ion{O}{i}\rbrack\ lines
are however normal for this class of objects.
The first photoionization models invoked to explain the narrow emission lines
in AGNs assumed a single density cloud. However, new observations quickly
sug\-gested the presence of several emitting clouds, ruling out single
component models. Most of the multicloud models first studied
were such that the emitting
gas, as a whole, was ionization-bounded and thus the \ion{He}{ii}\,$\lambda$4686
line intensity relative to $\rm H\beta$\ was
determined by the hardness of the ionizing spectrum. In these models, the
extreme values reached by the $\lambda$4686$/$H$\beta$\ ratio are not
well reproduced. A number of
objects have $\lambda$4686$/$H$\beta$\ of the order of 0.2 or more; such high values cannot be
accounted for unless the line emitting clouds are matter-bounded
(Stasi\'nska 1984). On the
basis of a weak trend for the low excitation lines to become weaker as
$\lambda$4686$/$H$\beta$\ gets larger, Viegas-Aldrovandi (1988) and Viegas \& Prieto (1992) argued
in favor of a model in which matter-bounded clouds are present; indeed, if the
gas is not optically thick to all the ionizing continuum (i.e., is matter
bounded), the H$^{+}$ emitting volume is smaller, but the He$^{++}$ volume
is not, leading to a higher $\lambda$4686$/$H$\beta$\ line ratio. Moreover, Viegas-Aldrovandi
\& Gruenwald (1988) and Rodriguez-Ardila et al. (1998) showed that, for most
AGNs, the observed low-excitation lines are better explained by matter-bounded
mo\-dels with about 50\% of the $\rm H\beta$\ luminosity produced in
ioniza\-tion-bounded clouds.
Storchi-Bergmann et al. (1996a) have obtained long-slit spe\-ctra of five
active galaxies showing extended high excitation lines. At some positions,
two of the objects (PKS 0349$-$27 and PKS 0634$-$20) show quite peculiar
line ratios, with a strong \ion{He}{ii}\,$\lambda$4686 line ($\lambda$4686$/$H$\beta$\ $>$ 0.3) and
weak \lbrack\ion{N}{ii}\rbrack\ lines (that is, $\lambda$6583$/$H$\alpha$\ $<$ 0.3). In fact, there seems to be
a correlation between $\lambda$6583$/$H$\alpha$\ and $\lambda$4686$/$H$\beta$, weak \lbrack\ion{N}{ii}\rbrack\ lines being associated
with strong \ion{He}{ii}\ emission, suggesting that very small $\lambda$6583$/$H$\alpha$\ ratios
(as observed in the two above mentioned radiogalaxies) are not necessarily
a signature of star-forma\-tion, but a natural consequence of having a
region dominated by matter-bounded clouds (Binette et al. 1996, 1997).
However, in the extranuclear regions of PKS 0349$-$278 in which strong
\ion{He}{ii}\,$\lambda$4686 and weak \lbrack\ion{N}{ii}\rbrack$\lambda$6583 lines are observed, the
\lbrack\ion{O}{i}\rbrack\ $\lambda$6300 line is also reduced ($\lambda$6300$/$H$\alpha$\ $\sim$ 0.05), which is
a natural consequence of the model (Viegas-Aldrovandi 1988), while in
our sample of weak \lbrack\ion{N}{ii}\rbrack$\lambda$6583 galaxies, we verify that the
\lbrack\ion{O}{i}\rbrack\ $\lambda$6300 line is not weakened in most of the objects.
In Table \ref{nii_line_ratios} we give the list of known AGNs with relatively
weak \lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ $<$ 0.45) with published values of the $\lambda$4686$/$H$\beta$\ and
$\lambda$6300$/$H$\alpha$\ ratios. Three objects in this table (UM 85, MS 04124$-$0802 and Mark
699) have both weak \lbrack\ion{N}{ii}\rbrack\ lines ($\lambda$6583$/$H$\alpha$\ $<$ 0.20) and a strong \ion{He}{ii}\ line
($\lambda$4686/ $\rm H\beta$\ $>$ 0.30).
In the last two, the \lbrack\ion{O}{i}\rbrack\ lines are also relatively weak ($\lambda$6300$/$H$\alpha$\ $\le$ 0.05);
these two objects could be dominated by matter-bounded clouds. Alternatively,
in the other objects, the weakness of the \lbrack\ion{N}{ii}\rbrack\ lines could be due to a
selective under-abun\-dance of nitrogen. For a photoionized single cloud model
with $U \sim$ 10$^{-2.5}$, Ferland \& Netzer (1983) predicted $\lambda$6583$/$H$\alpha$\
$\sim$ 1.0 for solar nitrogen abundances and $\sim$ 0.3 for nitrogen
abundances $\sim$ 0.3 solar.
\begin{table}[h]
\begin{center}
\caption{\label{nii_line_ratios}
Known AGNs with weak \lbrack\ion{N}{ii}\rbrack\ lines.}
\begin{flushleft}
\begin{tabular}{lllll}
\hline
Name & \verb+ +Short & \underline{$\lambda$6583$\:$}
& \underline{$\lambda$4686$\:$} & \underline{$\lambda$6300$\:$} \\
& \verb+ +position & \verb+ + $\rm H\alpha$ & \verb+ + $\rm H\beta$ & \verb+ + $\rm H\alpha$ \\
\hline
UM 85 & 01\,04$+$06 & \verb+ +0.19 & \verb+ +0.37 & \verb+ +0.09 \\
MS 04124$-$0802 & 04\,12$-$08 & \verb+ +0.10 & \verb+ +0.35 & \verb+ +0.05 \\
IRAS 04210$+$0400& 04\,21$+$04 & \verb+ +0.35 & \verb+ +0.21 & \verb+ +0.13 \\
3C 184.1 & 07\,34$+$80 & \verb+ +0.22 & \verb+ +0.26 & \verb+ +0.07 \\
IRAS 11058$-$1131& 11\,05$-$11 & \verb+ +0.38 & \verb+ +0.17 & \verb+ +0.05 \\
SBS 1136$+$594 & 11\,36$+$59 & \verb+ +0.10 & \verb+ +0.18 & \verb+ +0.11 \\
Mark 477 & 14\,39$+$53 & \verb+ +0.36 & \verb+ +0.13 & \verb+ +0.17 \\
Mark 699 & 16\,22$+$41 & \verb+ +0.20 & \verb+ +0.34 & \verb+ +0.03 \\
\hline
\end{tabular}
\end{flushleft}
\end{center}
\end{table}
\subsection{Seyfert 2s and Liners}
It has been suggested by several authors (see for instance
Ferland \& Netzer 1983; Shields 1992; Ho et
al. 1993a) that in Seyfert 2s, as well as in Liners, the ionized gas is
excited by a non-thermal continuum, the only differences being the value of
the ionizing parameter which would be $\sim$ 10$^{-3.5}$ for Liners, and
$\sim$ 10$^{-2.5}$ for Seyfert 2s. If this is the case, the discontinuity
between Seyfert 2s and Liners is not easily understood. No reliable detection
of the \ion{He}{ii}\ line in {\it {bona fide}} Liners has been reported suggesting
that there could be a serious problem with the picture of simply reducing
$U$ in a standard power-law photoionization model predicting
$\lambda$4686$/$H$\beta$\ $>$ 0.15 (Viegas-Aldrovandi \& Gruenwald, 1990), as the weakness of
\ion{He}{ii}\ indicates that the continuum illuminating the NLR clouds must contain
few photons more energetic than
54.4 eV, the ionization potential of He$^{+}$ (P\'equignot 1984). Binette
et al. (1996) proposed that the emission spectrum
of Liners is due to ionization-bounded clouds illuminated by a ionization
spectrum filtered by matter-bounded clouds hidden from view by obscuring
material. In this case, the \ion{He}{ii}\ emission is reduced ($\lambda$4686$/$H$\beta$\ $<$ 0.01).
However, a nearly total obscuration of the matter-bounded component must then
be invoked in order to keep the emission from \ion{He}{ii}\ at an acceptable low
level, a scenario which seems to be rather unlikely to Barth et al. (1996).
\section{Conclusions}
We have shown that:
\begin{itemize}
\item{Nuclear \ion{H}{ii}\ regions, Seyfert 2s and Liners lie in distinct, well
separated regions in the log($\lambda$5007$/$H$\beta$) {\it {vs.}} log($\lambda$6583/ $\rm H\alpha$)
and log($\lambda$5007$/$H$\beta$) {\it {vs.}} log($\lambda$6300$/$H$\alpha$) diagrams. The\-re is no continuity
between Liners and Seyfert 2s, with an apparent deficit of objects with
$\lambda$6300$/\lambda$5007\ = 0.25.}
\item{A number of objects have ``transition'' spectra, falling outside the
regions assigned to the three types of emission nebulosities. They probably
all have a ``composite'' spectrum.}
\item{We have isolated a class of Seyfert 2 galaxies with weak \lbrack\ion{N}{ii}\rbrack\ lines.
This weakness could be due to an under-abun\-dance of nitrogen or to the
presence of matter-bounded clo\-uds in these objects.}
\end{itemize}
\begin{acknowledgements}
This research has made use of the NASA / IPAC
extragalactic database (NED) which is ope\-rated by the Jet Propulsion
Laboratory, Caltech, under contract with the National Aeronautics and Space
Administration. A.\,C. Gon\c{c}al\-ves acknowledges support from the
{\it Funda\c{c}\~ao para a Ci\^encia e a Tecnologia}, Portugal, during the
cou\-rse of this work (PhD. grant ref. PRAXIS XXI/BD/5117/95).
\end{acknowledgements}
| 2024-02-18T23:40:16.613Z | 1998-11-20T23:31:02.000Z | algebraic_stack_train_0000 | 1,908 | 22,141 |
|
proofpile-arXiv_065-9361 | \section{Introduction}
An important problem of nowadays high energy physics is searching
for deviation from the Standard Model (SM) of elementary particles
which may appear due to heavy virtual states entering the extended
models and having the masses much greater than the W-boson mass
$m_W$ \cite{1}. One of approaches for the description of such
phenomena is the construction of the effective Lagrangians (EL)
appearing owing to decoupling of heavy particles. In principle, it
is possible to write down a lot of different EL describing effects
of new physics beyond the SM. In Ref. \cite{2} the EL generated at
a tree level in a general renormalizable gauge theory have been
derived. These objects by construction contain a great number of
arbitrary parameters responsible for specific processes. But it is
well known that a renormalizable theory includes a small number of
independent constants due to relations between them. The
renormalizability of the theory is resulted in the renormalization
group (RG) equations for scattering amplitudes \cite{3}. In Ref.
\cite{4} it has been proven that RG equation can be used to obtain
a set of relations between the parameters of the EL. Two main
observations were used. First, it has been shown that a heavy
virtual state may be considered as an external field scattering SM
light particles. Second, the renormalization of the vertices,
describing scattering on the external field, can be determined by
the $\beta $- and $\gamma $- functions calculated with light
particles, only. Hence, the relations mentioned above follow. As
an example the SM with the heavy Higgs scalar has been
investigated. In the decoupling region the RG equations for
scattering amplitudes have been reduced to the ones for vertices
describing the scattering of light particles on the external field
substituting the corresponding virtual heavy field. In
Ref.\cite{4} the only scalar field of the theory was taken as the
heavy particle, and no mixing between the heavy and the light
fields at the one-loop level has been considered. Here, we are
going to investigate the Yukawa model with a heavy scalar field
$\chi$ and a light scalar field $\varphi$. The purposes of our
investigation are two fold: to derive the one-loop RG relation for
the four-fermion scattering amplitude in the decoupling region and
to find out the possibility of reducing this relation in the
equation for vertex describing the scattering of light particles
on the external field when the mixing between heavy and light
virtual states takes place. In Ref.\cite{4} the specific algebraic
identities originated from the RG equation for scattering
amplitude have been derived. When the explicit couplings in EL are
unknown and represented by the arbitrary parameters, one may treat
the identities as the equations dependent on the parameters and
appropriate $\beta -$ and $\gamma -$ functions. If due to a
symmetry the number of $\beta -$ and $\gamma -$ functions is less
than the number of RG relations, one can obtain non trivial system
of equations for the parameters mentioned. This was shown for the
gauge couplings \cite{4}. In present paper we derive RG relations
for the EL parameters in the model including one-loop mixing of
heavy and light fields.
\section{Renormalization group relation for amplitude}
The Lagrangian of the model reads
\begin{eqnarray}\label{1}
{\cal L}&=&\frac{1}{2}{\left( \partial_{\mu}\varphi \right) }^{2}-
\frac{m^{2}}{2}{\varphi}^{2}-\lambda{\varphi}^{4}+ \frac{1}{2}{\left(
\partial_{\mu}\chi \right) }^{2}- \frac{{\Lambda}^{2}}{2}{\chi}^{2}-
\xi{\chi}^{4}+\nonumber \\
&&\rho {\varphi}^{2}{\chi}^{2}+{\bar\psi}\left(
i\partial_{\mu}\gamma_{\mu}-M-G_{\varphi}\varphi- G_{\chi}\chi \right) \psi,
\end{eqnarray}
where $\psi$ is a Dirac spinor field.
The $S$-matrix element for the four-fermion scattering at the one-loop level
is given by
\begin{eqnarray}\label{2}
{\hat S}&=&-\frac{i}{2}\int\frac{dp_{1}}{{\left( 2\pi\right)
}^{4}}...\frac{dp_{4}}{{\left( 2\pi\right) }^{4}}{\left( 2\pi\right) }^{4}\delta\left(
p_{1}+...+p_{4}\right) {\cal N}\left[ S_{1PR}+ S_{box}\right],\nonumber \\
S_{1PR}&=&\sum\limits_{{\phi}_{1},{\phi}_{2}=\varphi,
\chi}G_{{\phi}_{1}} G_{{\phi}_{2}}\left(
\frac{{\delta}_{{\phi}_{1}{\phi}_{2}}}{s-m_{{\phi}_{1}}}+\frac{1}{s-
m_{{\phi}_{1}}}{\Pi}_{{\phi}_{1}{\phi}_{2}}\left( s \right) \frac{1}{s-
m_{{\phi}_{2}}}\right) \times\nonumber \\
&& {\bar\psi}\left( p_{1}\right) \left(1+2\Gamma\left( p_{2},
-p_{1}- p_{2}\right) \right) \psi\left( p_{2}\right)\times {\bar\psi}\left(
p_{4}\right) \psi\left( p_{3}\right),
\end{eqnarray}
where $s={\left( p_{1}+p_{2}\right) }^{2}$, $S_{1PR}$ is the
contribution from the one-particle reducible diagrams shown in the
Figs.\ref{fig:tree}-\ref{fig:loop} and $S_{box}$ is the
contribution from the box diagram. The one-loop polarization
operator of scalar fields ${\Pi}_{{\phi}_{1}{\phi}_{2}}$ and the
one-loop vertex function $\Gamma$ are usually defined trough the
Green functions:
\begin{eqnarray}\label{3}
D_{{\phi}_{1}{\phi}_{2}}\left( s \right)
&=&\frac{{\delta}_{{\phi}_{1}{\phi}_{2}}}{s-m_{{\phi}_{1}}}+\frac{1}{s-
m_{{\phi}_{1}}}{\Pi}_{{\phi}_{1}{\phi}_{2}}\left( s \right) \frac{1}{s-
m_{{\phi}_{2}}},\nonumber \\
G_{\phi\phi\psi}\left( p,q \right)&=&-\sum\limits_{{\phi}_{1}
}G_{{\phi}_{1}} D_{{\phi}_{1}\phi}\left( q^{2} \right) S_{\psi}\left( p \right)
\left(1+\Gamma\left( p, q\right) \right) S_{\psi} \left( -p-q\right),
\end{eqnarray}
where $ S_{\psi}$ is the spinor propagator in the momentum representation. The
renormalized fields, masses and charges are defined as follows
\begin{eqnarray}\label{4}
\psi &=&Z_{\psi}^{-1/2}{\psi}_{0},\quad\left( \begin{array}{c}\varphi \\ \chi
\end{array}\right)= Z_{\phi}^{-1/2}\left( \begin{array}{c}{\varphi}_{0} \\
{\chi}_{0} \end{array}\right),\quad \left( \begin{array}{c}G_{\varphi} \\ G_{\chi}
\end{array}\right)= Z_{G}^{-1}\left( \begin{array}{c}G_{{\varphi}_{0}} \\
G_{{\chi}_{0}} \end{array}\right),\nonumber \\
M^{2}&=& M_{0}^{2}-{\delta M}^{2},\qquad\quad m^{2}= m_{0}^{2}-{\delta
m}^{2},\qquad\quad {\Lambda}^{2}= {\Lambda}_{0}^{2}-{\delta \Lambda}^{2},
\end{eqnarray}
Using the dimensional regularization (the dimension of the momentum space is
$D=4-\varepsilon$) and the $\overline{MS}$ renormalization scheme \cite{5} one can
compute the renormalization constants
\begin{eqnarray}\label{5}
Z_{\psi}&=&1-\frac{1}{16{\pi}^{2}\varepsilon}\left( {G_{\varphi}}^{2}+
{G_{\chi}}^{2}\right) ,\quad {\delta M}^{2}=\frac{3}{8{\pi}^{2}\varepsilon}\left(
{G_{\varphi}}^{2}+ {G_{\chi}}^{2}\right) M^{2},\nonumber \\
Z_{\phi}^{1/2}&=&1-\frac{1}{8{\pi}^{2}\varepsilon}\left(
\begin{array}{cc}{G_{\varphi}}^{2}&
2G_{\varphi}G_{\chi}\frac{{\Lambda}^{2}-6 M^{2}}{{\Lambda}^{2}-
m^{2}}\\-2G_{\varphi}G_{\chi}\frac{m^{2}-6 M^{2}}{{\Lambda}^{2}-
m^{2}}& {G_{\chi}}^{2}\end{array}\right) , \nonumber \\
{\delta m}^{2}&=&\frac{1}{4{\pi}^{2}\varepsilon}\left( \left(
{G_{\varphi}}^{2}+6\lambda\right) m^{2}-6 {G_{\varphi}}^{2}M^{2}-
\rho{\Lambda}^{2}\right),\nonumber \\
{\delta \Lambda}^{2}&=&\frac{1}{4{\pi}^{2}\varepsilon}\left( \left(
{G_{\chi}}^{2}+6\xi\right) {\Lambda}^{2}-6 {G_{\chi}}^{2}M^{2}-\rho
m^{2}\right),\nonumber \\
Z_{G}^{-1}&=&\left[ 1-\frac{3}{16{\pi}^{2}\varepsilon}\left(
G_{\varphi}^{2}+ G_{\chi}^{2}\right) \right] {\left( Z_{\phi}^{1/2}\right) }^{T}.
\end{eqnarray}
From Eq.(\ref{5}) we obtain the appropriate $\beta -$ and $\gamma -$ functions
\cite{5} at the one-loop level:
\begin{eqnarray}\label{6}
&& {\beta}_{\varphi} =\frac{dG_{\varphi}}{d ln \kappa}=\frac{1}{16{\pi}^{2}}\left(
5{G_{\varphi}}^{3}+3G_{\varphi} {G_{\chi}}^{2}-4\frac{m^{2}-6 M^{2}}{{\Lambda}^{2}-
m^{2}}G_{\varphi} {G_{\chi}}^{2} \right) ,\nonumber \\
&& {\beta}_{\chi} =\frac{dG_{\chi}}{d ln \kappa}=\frac{1}{16{\pi}^{2}}\left(
5{G_{\chi}}^{3}+3G_{\chi} {G_{\varphi}}^{2}+4\frac{{\Lambda}^{2}-6
M^{2}}{{\Lambda}^{2}-m^{2}}G_{\chi} {G_{\varphi}}^{2} \right) ,\nonumber \\
&& {\gamma}_{m} =-\frac{d ln m^{2}}{d ln \kappa}=-\frac{1}{4{\pi}^{2}}\left(
{G_{\varphi}}^{2}\frac{m^{2}-6 M^{2}}{m^{2}}+6\lambda-\rho
\frac{{\Lambda}^{2}}{m^{2}} \right) ,\nonumber \\
&& {\gamma}_{\Lambda} =-\frac{d ln {\Lambda}^{2}}{d ln \kappa}=-
\frac{1}{4{\pi}^{2}}\left( {G_{\chi}}^{2}\left( 1-
6\frac{M^{2}}{{\Lambda}^{2}}\right) +6\xi-\rho
\frac{m^{2}}{{\Lambda}^{2}} \right) ,\nonumber \\
&& {\gamma}_{\psi} =-\frac{d ln \psi}{d ln
\kappa}=\frac{1}{32{\pi}^{2}}\left({G_{\varphi}}^{2}+{G_{\chi}}^{2} \right) .
\end{eqnarray}
Then, the $S$-matrix element can be expressed in terms of the renormalized
quantities (\ref{4}). The contribution from the one-particle reducible diagrams becomes
\begin{eqnarray}\label{7}
S_{1PR}&=&\sum\limits_{{\phi}_{1},{\phi}_{2}}G_{{\phi}_{1}}
G_{{\phi}_{2}}\left( \frac{{\delta}_{{\phi}_{1}{\phi}_{2}}}{s-
m_{{\phi}_{1}}}+\frac{1}{s-
m_{{\phi}_{1}}}{\Pi}_{{\phi}_{1}{\phi}_{2}}^{fin}\left( s \right) \frac{1}{s-
m_{{\phi}_{2}}}\right) \nonumber \\
&& {\bar\psi}\left( p_{1}\right)
\left(1+2{\Gamma}^{fin}\left( p_{2}, -p_{1}- p_{2}\right) \right) \psi\left(
p_{2}\right)\times {\bar\psi}\left( p_{4}\right) \psi\left( p_{3}\right),
\end{eqnarray}
where the functions ${\Pi}_{{\phi}_{1}{\phi}_{2}}^{fin}$ and
${\Gamma}^{fin}$ are the expressions ${\Pi}_{{\phi}_{1}{\phi}_{2}}$ and
$\Gamma$ without the terms proportional to $1/\varepsilon$. Since the quantity
$S_{box}$ is finite, the renormalization leaves it without changes.
Introducing the RG operator at the one-loop level \cite{6}
\begin{eqnarray}\label{8}
{\cal D}&=&\frac{d}{d ln \kappa}=\frac{\partial}{\partial ln \kappa}+{\cal
D}^{(1)}=\nonumber\\
&&\frac{\partial}{\partial ln \kappa}+\sum\limits_{\phi}
{\beta}_{\phi}\frac{\partial}{\partial G_{\phi}}-{\gamma}_{m} \frac{\partial}{\partial ln m^{2}}-{\gamma}_{\Lambda}
\frac{\partial}{\partial ln {\Lambda}^{2}}-{\gamma}_{\psi} \frac{\partial}{\partial ln \psi}
\end{eqnarray}
we determine that the following relation holds for the $S$-matrix element
\begin{equation}\label{9}
{\cal D}\left( S_{1PR}+S_{box} \right) =\frac{\partial
S_{1PR}^{(1)}}{\partial ln \kappa}+{\cal D}^{(1)} S_{1PR}^{(0)}=0,
\end{equation}
where the $S_{1PR}^{(0)}$ and the $S_{1PR}^{(1)}$ are the contributions to the
$S_{1PR}$ at the tree level and at the one-loop level, respectively:
\begin{equation}\label{10}
S_{1PR}^{(0)}=\left( \frac{{G_{\varphi}}^{2}}{s-m^{2}}+\frac{{G_{\chi}}^{2}}{s-
{\Lambda}^{2}} \right) {\bar\psi}\psi\times{\bar\psi}\psi ,
\end{equation}
\begin{eqnarray}\label{11}
\frac{\partial S_{1PR}^{(1)}}{ \partial ln
\kappa}&=&\frac{{\bar\psi}\psi\times{\bar\psi}\psi }{4{\pi}^{2}}\left( -\left(
{G_{\varphi}}^{2}+{G_{\chi}}^{2}\right) \left( \frac{{G_{\varphi}}^{2}}{s-
m^{2}}+\frac{{G_{\chi}}^{2}}{s-{\Lambda}^{2}} \right) + \right. \nonumber \\
&& \frac{{G_{\varphi}}^{2}\left( \rho{\Lambda}^{2}-6\lambda
m^{2}+{G_{\varphi}}^{2}\left( 6M^{2}-s\right) \right) }{{\left( s-m^{2}\right) }^{2}}+
\frac{2{G_{\varphi}}^{2}{G_{\chi}}^{2}\left( 6M^{2}-s\right) }{\left( s-m^{2}\right)
\left( s-{\Lambda}^{2}\right) }+\nonumber \\
&&\left. \frac{{G_{\chi}}^{2}\left( \rho m^{2}-6\xi{\Lambda}^{2}+{G_{\chi}}^{2}\left(
6M^{2}-s\right) \right) }{{\left( s-{\Lambda}^{2}\right) }^{2}} \right) .
\end{eqnarray}
The first term in Eq.(\ref{11}) is originated from the one-loop
correction to the fermion-scalar vertex. The rest terms are
connected with the polarization operator of scalars. The third
term describes the one-loop mixing between the scalar fields. It
is canceled in the RG relation (\ref{9}) by the mass-dependent
terms in the $\beta -$ functions produced by the non-diagonal
elements in $Z_{\phi}$. Eq.(\ref{9}) is the consequence of the
renormalizability of the model. It insures the leading logarithm
terms of the one-loop $S$-matrix element to reproduce the
appropriate tree-level structure. In contrast to the familiar
treatment we are not going to improve scattering amplitudes by
solving Eq.(\ref{9}). We will use it as an algebraic identity
implemented in the renormalizable theory. Naturally if one knows
the explicit couplings expressed in terms of the basic set of
parameters of the model, this RG relation is trivially fulfilled.
But the situation changes when the couplings are represented by
unknown arbitrary parameters as it occurs in the EL approach
\cite{1},\cite{2}. In this case the RG relations are the algebraic
equations dependent on these parameters and appropriate $\beta -$
and $\gamma -$ functions. In the presence of a symmetry the number
of $\beta -$ and $\gamma -$ functions is less than the number of
RG relations. So, one has non trivial system of equations relating
the parameters of EL. Such a scenario is realized for the gauge
coupling as it has been demonstrated in \cite{4}. Although the
considered simple model has no gauge couplings and no relation
between the EL parameters occurs, we are able to demonstrate the
general procedure of deriving the RG relations for EL parameters
in the theory with one-loop mixing. This is essential for dealing
with the EL describing deviations from the SM. At energies $s\ll
{\Lambda}^{2}$ the heavy scalar field $\chi$ is decoupled. So, the
four-fermion scattering amplitude consists of the contribution of
the model with no heavy field $\chi$ plus terms of the order $s/
{\Lambda}^{2}$. The expansion of the heavy scalar propagator
\begin{equation}\label{12}
\frac{1}{s-{\Lambda}^{2}}\to -
\frac{1}{{\Lambda}^{2}}\left(1+O\left(\frac{s}{{\Lambda}^{2}}\right) \right)
\end{equation}
in Eq.(\ref{10}) is resulted in the effective contact four-fermion interaction
\begin{equation}\label{13}
{\cal L}_{eff}=-\alpha{\bar\psi}\psi\times{\bar\psi}\psi ,\quad\alpha=
\frac{{G_{\chi}}^{2}}{{\Lambda}^{2}},
\end{equation}
and the tree level contribution to the amplitude becomes
\begin{equation}\label{14}
S_{1PR}^{(0)}=\left( \frac{{G_{\varphi}}^{2}}{s-m^{2}}-
\alpha+O\left(\frac{s}{{\Lambda}^{4}}\right) \right)
{\bar\psi}\psi\times{\bar\psi}\psi .
\end{equation}
In the decoupling region the lowest order effects of the heavy
scalar are described by the parameter $\alpha$, only. The method
of constructing the RG equation in terms of the low energy
quantities $G_{\varphi}, \lambda, m, M, \alpha$ was proposed in
\cite{6}. As it has been demonstrated in \cite{6}, the
redefinition of the parameters of the model allows to remove all
the heavy particle loop contributions to Eq.(\ref{11}). Let us
define a new set of fields, charges and masses ${\tilde\psi}$, ${\tilde G}_{\varphi}$,
${\tilde G}_{\chi}$, ${\tilde\Lambda }$, ${\tilde m }$, ${\tilde M }$
\begin{eqnarray}\label{15}
&& {G_{\varphi}}^{2}={{\tilde G}_{\varphi}}^{2}\left( 1+\frac{3{{\tilde G}_{\chi}}^{2}}{16{\pi}^{2}}
ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right) ,
\quad{G_{\chi}}^{2}={{\tilde G}_{\chi}}^{2}\left( 1+\frac{3{{\tilde G}_{\chi}}^{2}}{16{\pi}^{2}}
ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right) , \nonumber \\
&& m^{2} ={{\tilde m }}^{2}\left( 1-\frac{\tilde\rho}{8{\pi}^{2}}
\frac{{{\tilde\Lambda }}^{2}}{{{\tilde m }}^{2}} ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right) ,
\quad{\Lambda}^{2}={{\tilde\Lambda }}^{2}\left( 1+\frac{3\tilde\xi}{4{\pi}^{2}}
ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right) ,\nonumber \\
&& \psi={\tilde\psi} \left( 1-\frac{{{\tilde G}_{\chi}}^{2}}{64{\pi}^{2}}
ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right) .
\end{eqnarray}
One is able to rewrite the differential operator (\ref{8}) in
terms of these new low-energy parameters:
\begin{equation}\label{16}
{\cal D}=\frac{\partial}{\partial ln \kappa}+{\tilde{\cal D}}^{(1)}=
\frac{\partial}{\partial ln \kappa}+\sum\limits_{\phi} {\tilde\beta}_{\phi}\frac{\partial}{\partial
{\tilde G}_{\phi}}-{\tilde\gamma}_{m} \frac{\partial}{\partial ln {{\tilde m }}^{2}}-{\tilde\gamma}_{\Lambda} \frac{\partial}{\partial ln
{{\tilde\Lambda }}^{2}}-{\tilde\gamma}_{\psi} \frac{\partial}{\partial ln {\tilde\psi}}
\end{equation}
where ${\tilde\beta}-$ and ${\tilde\gamma}-$ functions are obtained from the one-
loop relations (\ref{6}) and (\ref{15})
\begin{eqnarray}\label{17}
&& {\tilde\beta}_{\varphi} =\frac{1}{16{\pi}^{2}}\left( 5{{\tilde G}_{\varphi}}^{3}-4\frac{{{\tilde m }}^{2}-6
{{\tilde M }}^{2}}{{{\tilde\Lambda }}^{2}-{{\tilde m }}^{2}}{\tilde G}_{\varphi} {{\tilde G}_{\chi}}^{2} \right) ,\nonumber \\
&& {\tilde\beta}_{\chi} =\frac{1}{16{\pi}^{2}}\left( 2{{\tilde G}_{\chi}}^{3}+\left( 3+
4\frac{{{\tilde\Lambda }}^{2}-6 {{\tilde M }}^{2}}{{{\tilde\Lambda }}^{2}-{{\tilde m }}^{2}}\right) {\tilde G}_{\chi} {{\tilde G}_{\varphi}}^{2}
\right) ,\nonumber \\
&& {\tilde\gamma}_{m} =-\frac{1}{4{\pi}^{2}}\left( {{\tilde G}_{\varphi}}^{2}\frac{{{\tilde m }}^{2}-6
{{\tilde M }}^{2}}{{{\tilde m }}^{2}}+6{\tilde\lambda} \right) ,\nonumber \\
&& {\tilde\gamma}_{\Lambda} =-\frac{1}{4{\pi}^{2}}\left( {{\tilde G}_{\chi}}^{2}\left( 1-
6\frac{{{\tilde M }}^{2}}{{{\tilde\Lambda }}^{2}}\right) -{\tilde\rho}\frac{{{\tilde m }}^{2}}{{{\tilde\Lambda }}^{2}}
\right) ,\nonumber \\
&& {\tilde\gamma}_{\psi} =\frac{1}{32{\pi}^{2}}{{\tilde G}_{\varphi}}^{2}.
\end{eqnarray}
Hence, one immediately notices that ${\tilde\beta}-$ and
${\tilde\gamma}-$ functions contain only the light particle loop
contributions, and all the heavy particle loop terms are
completely removed from them. The $S$-matrix element expressed in
terms of new parameters satisfies the following RG relation
\begin{equation}\label{18}
{\cal D}\left( S_{1PR}+S_{box} \right) =\frac{\partial {\tilde
S}_{1PR}^{(1)}}{\partial ln \kappa}+{\tilde{\cal D}}^{(1)} {\tilde
S}_{1PR}^{(0)}=0,
\end{equation}
\begin{equation}\label{19}
{\tilde S}_{1PR}^{(0)}=\left( \frac{{{\tilde G}_{\varphi}}^{2}}{s-{{\tilde m }}^{2}}-
{\tilde\alpha}+O\left( \frac{s^{2}}{{{\tilde\Lambda }}^{4}} \right) \right)
{\bar{\tilde\psi}}{\tilde\psi}\times{\bar{\tilde\psi}}{\tilde\psi} ,
\end{equation}
\begin{eqnarray}\label{20}
\frac{\partial {\tilde S}_{1PR}^{(1)}}{ \partial ln
\kappa}&=&\frac{{\bar{\tilde\psi}}{\tilde\psi}\times{\bar{\tilde\psi}}{\tilde\psi} }{4{\pi}^{2}}\left( -
\frac{{{\tilde G}_{\varphi}}^{4}}{s-{{\tilde m }}^{2}}+\frac{{{\tilde G}_{\varphi}}^{2}\left( -
6{\tilde\lambda}{{\tilde m }}^{2}+{{\tilde G}_{\varphi}}^{2}\left( 6{{\tilde M }}^{2}-s\right) \right) }{{\left(
s-{{\tilde m }}^{2}\right) }^{2}}+\right. \nonumber \\
&&\left. {\tilde\alpha}{{\tilde G}_{\varphi}}^{2}-
\frac{2{{\tilde G}_{\varphi}}^{2}{\tilde\alpha}\left( 6{{\tilde M }}^{2}-s\right) }{s-m^{2}}+
O\left( \frac{s^{2}}{{{\tilde\Lambda }}^{4}} \right) \right) ,
\end{eqnarray}
where ${\tilde\alpha}={{\tilde G}_{\chi}}^{2}/{{\tilde\Lambda }}^{2}$ is the redefined effective four-
fermion coupling. As one can see, Eq.(\ref{20}) includes all the terms of Eq.(\ref{11}) except
for the heavy particle loop contributions. It depends on the low energy quantities
${\tilde\psi}$, ${\tilde G}_{\varphi}$, $\tilde\alpha$, $\tilde\lambda$, ${\tilde m }$, ${\tilde M }$. The first and the
second terms in Eq.(\ref{20}) are just the one-loop amplitude calculated within the
model with no heavy particles. The third and the fourth terms describe the light
particle loop correction to the effective four-fermion coupling and the mixing of
heavy and light virtual fields.
\section{Elimination of one-loop scalar field mixing}
Due to the mixing term it is impossible to split the RG relation
(\ref{18}) for the S- matrix element into the one for vertices.
Hence, we are not able to consider Eq.(\ref{18}) in the framework
of the scattering of light particles on an external field induced
by the heavy virtual scalar as it has been done in \cite{4}. But
this is an important step in deriving the RG relation for EL
parameters. Fortunately, there is a simple procedure allowing to
avoid the mixing in Eq.(\ref{20}). The way is to diagonalize the
leading logarithm terms of the scalar polarization operator in the
redefinition of the $\tilde\varphi$, $\tilde\chi$, ${\tilde G}_{\varphi}$,
${\tilde G}_{\chi}$
\begin{eqnarray}\label{21}
&&\left( \begin{array}{c}\varphi \\ \chi \end{array}\right)=
{\zeta}^{1/2}\left( \begin{array}{c}{\tilde\varphi} \\ {\tilde\chi}
\end{array}\right),\quad \left( \begin{array}{c}G_{\varphi} \\ G_{\chi}
\end{array}\right)=\left[ 1+\frac{3{{\tilde G}_{\chi}}^{2}}{32{\pi}^{2}}
ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}} \right]
{\left( {\zeta}^{-1/2}\right) }^{T}\left(
\begin{array}{c}{\tilde G}_{\varphi} \\ {\tilde G}_{\chi} \end{array}\right)
,\nonumber \\
&& {\zeta}^{1/2}=1- \frac{{\tilde G}_{\varphi}{\tilde G}_{\chi}}{8{\pi}^{2} \left({{\tilde\Lambda }}^{2}-
{{\tilde m }}^{2}\right) }ln\frac{{\kappa}^{2}}{{{\tilde\Lambda }}^{2}}\left( \begin{array}{cc}0 &
{{\tilde\Lambda }}^{2}-6 {{\tilde M }}^{2}\\-{{\tilde m }}^{2}-6 {{\tilde M }}^{2}& 0\end{array}\right) .
\end{eqnarray}
The appropriate ${\tilde\beta}-$ functions
\begin{equation}\label{22}
{\tilde\beta}_{\varphi} =\frac{5{{\tilde G}_{\varphi}}^{3}}{16{\pi}^{2}},\quad
{\tilde\beta}_{\chi} =\frac{1}{16{\pi}^{2}}\left( 2{{\tilde G}_{\chi}}^{3}+3{\tilde G}_{\chi} {{\tilde G}_{\varphi}}^{2}
\right)
\end{equation}
contain no terms connected with mixing between light and heavy scalars. So, the
fourth term in Eq.(\ref{20}) is removed, and the RG relation for the $S$-matrix element
becomes
\begin{equation}\label{23}
{\cal D}\left( S_{1PR}+S_{box} \right) =\frac{\partial {\tilde
S}_{1PR}^{(1)}}{\partial ln \kappa}+{\tilde{\cal D}}^{(1)} {\tilde
S}_{1PR}^{(0)}=0,
\end{equation}
\begin{equation}\label{24}
{\tilde S}_{1PR}^{(0)}=\left( \frac{{{\tilde G}_{\varphi}}^{2}}{s-{{\tilde m }}^{2}}-
{\tilde\alpha}+O\left( \frac{s^{2}}{{{\tilde\Lambda }}^{4}} \right) \right)
{\bar{\tilde\psi}}{\tilde\psi}\times{\bar{\tilde\psi}}{\tilde\psi} ,
\end{equation}
\begin{eqnarray}\label{25}
\frac{\partial {\tilde S}_{1PR}^{(1)}}{ \partial ln
\kappa}&=&\frac{{\bar{\tilde\psi}}{\tilde\psi}\times{\bar{\tilde\psi}}{\tilde\psi} }{4{\pi}^{2}}\left( -
\frac{{{\tilde G}_{\varphi}}^{4}}{s-{{\tilde m }}^{2}}+\frac{{{\tilde G}_{\varphi}}^{2}\left( -
6{\tilde\lambda}{{\tilde m }}^{2}+{{\tilde G}_{\varphi}}^{2}\left( 6{{\tilde M }}^{2}-s\right) \right) }{{\left(
s-{{\tilde m }}^{2}\right) }^{2}}+\right. \nonumber \\
&&\left. {\tilde\alpha}{{\tilde G}_{\varphi}}^{2}+
O\left( \frac{s^{2}}{{{\tilde\Lambda }}^{4}} \right) \right) .
\end{eqnarray}
At ${\tilde\alpha}=0$ Eq.(\ref{23}) is just the RG identity for the scattering amplitude
calculated in the absence of the heavy particles. The terms of order $\tilde\alpha$
describe the RG relation for the effective low-energy four-fermion interaction in the
decoupling region. The last one can be reduced in the RG relation for the vertex
describing the scattering of the light particle (fermion) on the external field
$\sqrt{\tilde\alpha}$ substituting the virtual heavy scalar:
\begin{equation}\label{26}
{\cal D}\left( \sqrt{\tilde\alpha}{\bar{\tilde\psi}}{\tilde\psi} \right)
=\frac{{{\tilde G}_{\varphi}}^{2}}{8{\pi}^{2}}\sqrt{\tilde\alpha}{\bar{\tilde\psi}}{\tilde\psi}+{\tilde{\cal
D}}^{(1)} \left( \sqrt{\tilde\alpha}{\bar{\tilde\psi}}{\tilde\psi} \right) =0,
\end{equation}
where
\begin{eqnarray}\label{27}
&&{\tilde{\cal D}}^{(1)}= {\tilde\beta}_{\varphi}\frac{\partial}{\partial {\tilde G}_{\varphi}}-
{\tilde\gamma}_{\alpha}\frac{\partial}{\partial ln {\tilde\alpha}}-{\tilde\gamma}_{m}
\frac{\partial}{\partial ln {{\tilde m }}^{2}}-{\tilde\gamma}_{\psi} \frac{\partial}{\partial ln
{\tilde\psi}},\nonumber \\
&& {\tilde\gamma}_{\alpha}=-{\cal D}{\tilde\alpha}=-
\frac{1}{8{\pi}^{2}}\left( 3{{\tilde G}_{\varphi}}^{2}+O\left({\tilde\alpha}\right) \right) .
\end{eqnarray}
Eqs.(\ref{23})-(\ref{27}) is the main result of our investigation.
One can derive them with only the knowledge about the EL
(\ref{13}) and the Lagrangian of the model with no heavy
particles. One also has to ignore all the heavy particle loop
contributions to the RG relation and the one-loop mixing between
the heavy and the light fields. Eqs.(\ref{23})-(\ref{27}) depend
on the effective low-energy parameters, only. But as the
difference between the original set of parameters and the
low-energy one is of one-loop order, one may freely substitute
them in Eqs.(\ref{23})-(\ref{26}).
\section{Discussion}
Let us discuss the results obtained. The RG relation for the
four-fermion scattering amplitude is derived in the decoupling
region $s\ll {\Lambda}^{2}$. It was shown that one can redefine
the parameters and the fields of the model in order to remove all
the heavy particle loop contributions to the RG relation. Then the
RG relation becomes dependent on the low-energy physics
parameters, only. As the RG operator coefficients and the
difference between the original parameters and the redefined ones
are of the one-loop order one can substitute one set of parameters
by another at the lowest level. Thus, we extend the result of
Ref.\cite{4} to the case when mixing terms are present. The
additional transformation of fields and charges allows one to
diagonalize the leading logarithm terms of the scalar polarization
operator and to avoid the contributions to the RG relation
originated from the one-loop mixing between heavy and light field.
Since the difference between the diagonalized fields and charges
and the original ones is of one-loop order, one may simply omit
one-loop mixing terms in the RG relation at the lower level. Then
it is possible to reduce the RG relation for $S$-matrix element to
the one for vertex describing the scattering of light particles on
the external field induced by the heavy virtual particle. In fact,
this result is independent on the specific features of the
considered model, as it was shown in \cite{4}.
The RG relations of the considered type may be used in searching
for the dependences between the parameters of EL describing
physics beyond the SM. For example, let a symmetry requires the
same charge structure for some effective Lagrangians. Then the
number of unknown ${\tilde\beta}-$ and ${\tilde\gamma}-$ functions
is less than the number of RG relations, and it is possible to
derive non-trivial solutions for the parameters. The present
results allow to omit the one-loop mixing diagrams in construction
of the RG relations for the tree-level EL.
\newpage
| 2024-02-18T23:40:16.672Z | 1998-11-10T13:02:15.000Z | algebraic_stack_train_0000 | 1,911 | 4,492 |
|
proofpile-arXiv_065-9488 | \section{Introduction}\label{intro}
The inner few hundred parsecs of the Milky Way, known as the ``Central Molecular Zone" (CMZ), contains $\sim$\,80\,per cent of the Galaxy's dense molecular gas (2\,-\,6\,$\times10^{7}$\,{\rm M$_\odot$}; \citealp{morris_1996}). The conditions within this region are extreme compared to those within the Galactic disc: the average density, temperature, pressure, velocity dispersion, interstellar radiation field and cosmic ray ionisation rate are factors of a few to several orders of magnitude larger. However, it has been noted for several decades that despite harbouring this vast reservoir of dense gas, the CMZ appears to be underproducing stars with respect to nearby star-forming regions in the disc (e.g. \citealp{guesten_1983, caswell_1983, taylor_1993, longmore_2013}). Understanding this dearth of star formation has wider implications, as the extreme properties of the CMZ are similar to those observed in the centres of nearby galaxies, starburst galaxies, and high redshift galaxies at the epoch of peak star formation density at z\,$\sim\,1-3$ \citep{kruijssen_2013}.
There could be several possible explanations for the apparent lack of star formation within the Galactic Centre, which can be split into two categories. Either the low star formation rate is a result of observational bias or uncertainty, or is the product of a physical mechanism.
The first observational explanation, could be that the gas is less dense than commonly assumed, and therefore should not form stars at such a high rate. The gas density is a difficult quantity to determine, as inferences of the three-dimensional structure have to be made in order to convert two-dimensional line-of-sight column density measurements. The measured column density of hydrogen in the CMZ appears to be at least an order of magnitude higher than clouds within the disc ($>\,10^{22}$\,cm$^{-2}$; \citealp{rathborne_2014}; Battersby et al. in prep), implying an average gas volume density above $\sim$\,$10^{4}$\,cm$^{-3}$. However, if the gas is more extended along the line of sight than in the plane of the sky, the density would be overestimated. Surveys using ATCA, APEX, and the SMA have shown that high critical density molecular gas tracers are widespread and spatially trace the peaks in column density (e.g. \citealp{jones_2012}). Additionally, these tracers have line of sight velocities which are consistent with being at the distance of the Galactic Centre (\citealp{ginsburg_2016, henshaw_2016}; Keto et al. in prep; Battersby et al. in prep). Recent radiative transfer modelling of the emission from dense molecular gas have shown that the gas has a density of the order $\sim\,10^{4}$\,cm$^{-3}$ \citep{Armijos-Abendano_2015, ginsburg_2016}. We conclude it is reasonable to assume that a significant fraction of the gas has a density $>\,10^{4}$\,cm$^{-3}$, and therefore remove this as a potential explanation for the apparent dearth of star formation.
The second possible observational explanation for the apparent dearth in star formation is that methods to determine the star formation rate have systematic differences when applied to the CMZ compared to other environments. Star formation rates within local clouds are primarily determined by counting the embedded young-stellar population (YSO counting; refer to section\,\ref{YSO counting}). However, it is not possible to use this technique in external galaxies, as the individual sites of star formation cannot be resolved. Instead, the star formation rate is determined from integrated light measurements (e.g. infrared and free-free emission; refer to section\,\ref{free-free emission} and \ref{Infrared luminosities}). Our proximity to the centre of the Galaxy means that it is the only {\it extreme} environment in which comparison between YSO counting and integrated light measurement methods can be made. However, compared to the solar neighbourhood star-forming regions, the visual extinction is orders of magnitude higher (some positions have A$_\mathrm{V}>1000$\,mag), and contamination from non-associated (e.g. bulge) stars are more of an issue. This could result in systematic uncertainties in the YSO counting method. Furthermore, as we observe the Galactic Centre through the disc of the Milky Way, contamination of sources along the line-of-sight may also be an issue for the integrated light methods. A combination of these systematic uncertainties could lead to unreliable star formation estimates from any given method.
The first physical explanation for the apparent lack of star formation within the Galactic Centre may be that star formation is episodic \citep{kruijssen_2014a}. \citet{krumholz_2015} and \citet{krumholz_2016} have modelled the dynamics of gas flows funnelled into the CMZ from large radii as acoustic instabilities within the bar's inner Lindblad resonance \citep{montenegro_1999}. In this model, when the gas reaches a radius of $\sim$\,100\,pc, and the rotation curve turns from flat to near-solid body, there is a decrease in shear which stops the inward flow and gas begins to accumulate. This accumulation of mass proceeds until the density is high enough for the gas to become gravitationally unstable, at which point there is an episode of intense star formation. The feedback from the recently formed high-mass stars then begins to drive turbulence and thereby increase the virial ratio of the gas, which quenches the star formation. Then as feedback from these stars' fades, gas can again accumulate and the cycle repeats. The estimated cycle timescale for a Milky Way-like galaxy is $\sim\,10-20$\,Myr. \citet{emsellem_2015} have conducted high-resolution, numerical simulations of the large scale gas motions within a galaxy similar to the Milky Way. These authors also find that gas is funnelled along the bar into the central $\sim$\,100\,pc, where transient star-forming complexes are observed, with timescales of a few Myr. \citet{torrey_2016} have tested the stability of feedback-regulated star formation for different environmental properties (e.g. ambient density, pressure) in the centres of galaxies. These authors find that a steady equilibrium state of star formation, where the energy input from feedback (which stops gravitational collapse) is balanced by the energy dissipation (which allows gravitational collapse), cannot be reached within the Galactic Centre, again requiring some degree of episodicity. \citet{suzuki_2015} also predict time-dependent flows, but these are instead driven via magnetic instabilities generated by differential rotation of the galaxy. Although these models and simulations differ in many aspects, the predicted trends in star formation activity are broadly similar in that they follow the scenario proposed by \citet{kruijssen_2014a}: gas steadily accumulates until a critical point is reached, when it becomes gravitationally unstable, collapses, and rapidly forms stars. Star formation continues until it is quenched by feedback, and the cycle restarts.
\citet{leroy_2013} selected a sample of 30 nearby galaxies from the HERACLES survey to study the distribution of gas and stars on scales of $\sim$\,1\,kpc. They find a $\sim$\,1\,dex scatter on the gas depletion time (i.e. the time taken for all the gas to be converted to stars at the current star formation rate) towards the central $<0.5$\,kpc of the galaxies within their sample. This is a $\sim$\,0.3\,dex increase when compared to similar measurements in the disc of the same galaxies. This could be suggestive that episodic star formation is not limited to the centre of the Milky Way, but is also present within centres of other galaxies.
The second physical explanation for the apparent lack of observed star formation may be that the comparison to the predictions from star formation models may need revision (see section\,\ref{models} for discussion). These models have been benchmarked against regions in the solar neighbourhood (see \citealp{federrath_2012} and references within), so the predictions may not be directly applicable to extreme environments (e.g. as is found in the Galactic Centre).
In this paper we investigate the three outstanding (observational and physical) explanations for the low star formation rate observed within the Galactic Centre: 1) inconsistent star formation rate measurements, 2) episodic star formation, 3) inappropriate comparison to the predictions of theoretical star formation models. To do this, we use infrared luminosities to determine the star formation rate over global and local (cloud) scales, and compare these to existing measurements and predictions from star formation models. In section\,\ref{method} we discuss how both the luminosities and column densities are determined. In section\,\ref{global} we determine the global star formation rate, compare this to previous measurements and predictions from star formation models, and discuss the implications. In section\,\ref{embedded} we determine the gas mass, embedded stellar mass and associated uncertainties of several sources within the Galactic Centre. In section\,\ref{sec:SFE} we determine the star formation efficiencies and star formation rates, and compare to the predictions of star formation models. In section\,\ref{conclusions} we discuss and summarise the implications of these results.
\section{Bolometric luminosity maps of the Galactic Centre}\label{method}
\begin{figure}
\centering
\includegraphics[trim = 0mm 6mm 0mm 0mm, clip,angle=0,width=1\columnwidth]{\dir 20percenterr-absfalse_sed_average}
\caption{The average background subtracted spectral energy distribution for the $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ region (see Figure\,\ref{sfr_maps}). The points show the flux density for each wavelength (70{\rm \,$\mu$m}\ point is shown but is not used in the fitting), and the lines represent the warm component fit in dashed red (TIR$_{warm}$), the cool component in dashed blue (TIR$_{cool}$) and the total fit in solid black (TIR$_{tot}$\,=\,TIR$_{warm}$\,+\,TIR$_{cool}$). Error bars show the estimated $\sim$\,20\,per cent uncertainty on each point, and the shaded region represents the uncertainty on each fit.}
\label{sed}
\end{figure}
\begin{figure*}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip,angle=0,width=1\textwidth]{\dir lum_maps_with_warm_and_cold-raster.pdf}
\caption{Maps of the extinction-corrected 24{\rm \,$\mu$m} \ luminosity (upper panel, purple), 70{\rm \,$\mu$m} \ luminosity (upper middle panel, green), and warm (lower middle panel, blue) and cool (lower panel, red) components of the bolometric luminosity. Overlaid are the warm component column density contours in grey of {$N_\mathrm{H_2}^{warm}$}\,=\,$\{1, 1.9, 2.6\}\times10^{17}$\,cm$^{-2}$, and cool component contours in white of {$N_\mathrm{H_2}^{cool}$}\,=\,$\{5, 10, 24, 75\}\times10^{22}$\,cm$^{-2}$. These contours levels are used to define the various sources (see Table\,\ref{masses_table}). The thick white contour shown in the lower two panels is of {$N_\mathrm{H_2}^{cool}$}\,=\,$1\times10^{22}$\,cm$^{-2}$. This contour highlights the widespread distribution of the cool column density component, which dominates the total column density {$N_\mathrm{H_2}$}\,=\,{$N_\mathrm{H_2}^{warm}$}\,+\,{$N_\mathrm{H_2}^{cool}$}. The 24{\rm \,$\mu$m}\ luminosity map (upper panel) has several sources labeled, and the 70{\rm \,$\mu$m}\ map has the ``Galactic Centre Bubble'' shown (upper middle panel; e.g. \citealp{bally_2010}). Each panel has a scale bar located in the top right which represents 40\,pc at a distance of $\sim$\,8.5\,kpc \citep{reid_2014} and a circle in the bottom left which represents the beam size of the observations.}
\label{sfr_maps}
\end{figure*}
\begin{table*}
\caption{Summary of survey data. Columns show the wavelength of the band, the beam full width at half maximum (FWHM), the image pixel size, the extinction with respect to the K-band, and the survey from which the observations were taken.}
\centering
\begin{tabular}{ c c c c c }
\hline
Band ({\rm \,$\mu$m}) & FWHM ({\rm $^{\prime\prime}$}) & R$_{pix}$ ({\rm $^{\prime\prime}$}) & A$_\lambda$/A$_k$ & Survey \\ \hline
5.8 & 2 & 1.2 & 0.44$^a$ & GLIMPSE$^c$ \\
8 & 2 & 1.2 & 0.43$^a$ & GLIMPSE$^c$ \\
24 & 6 & 2.4 & 0.61$^a$ & MIPSGAL$^d$ \\
70 & 5 & 3.2 & 0.06$^b$ & Hi-GAL$^e$ \\
160 & 11 & 4.5 & \dots & Hi-GAL$^e$ \\
250 & 18 & 6.0 & \dots & Hi-GAL$^e$ \\
350 & 25 & 8.0 & \dots & Hi-GAL$^e$ \\
500 & 36 & 11.5 & \dots & Hi-GAL$^e$ \\ \hline
\end{tabular}
\label{surveys}
{ \vspace{0.2cm}}
\begin{minipage}{\textwidth}
\vspace{1mm}
$^a$ Relations taken directly from \citet{chapman_2009}.\\
$^b$ Calculated using conversion from \citet{suutarinen_2013}.\\
$^c$ The {\it Spitzer} GLIMPSE is presented by \citet{churchwell_2009}.\\
$^d$ The {\it Spitzer} MIPSGAL survey is presented by \citet{carey_2009}.\\
$^e$ The {\it Herschel} Hi-GAL survey is presented by \citet{molinari_2010}.\\
\vspace{0mm}
\end{minipage}
\end{table*}
To create the infrared luminosity maps of the Galactic Centre needed to derive the star formation rates, we use {\it Spitzer} and {\it Herschel} telescope observations. The wavelengths and resolutions of these observations are presented in Table\,\ref{surveys}. To account for interstellar extinction, we adopt an average K-band extinction of A$_\mathrm{K}\sim2$\,mag from \citet{finger_1999}, \citet{dutra_2003} and \citet{schodel_2010}, who determined the extinction for several objects within the CMZ region. This is applied to the $5.8-70$\,{\rm \,$\mu$m}\ wavelength maps using the extinction relations presented by \citet[][see Table\,\ref{surveys}]{chapman_2009}. To apply the extinction to the 70\,-\,500\,{\rm \,$\mu$m} \ emission, we use the extinction relation A$_\mathrm{\lambda}$/A$_\mathrm{K}\,\sim\,0.0042\,(250/\lambda$\,[{\rm \,$\mu$m}])$^2$ \citep{suutarinen_2013}. From this, the 70{\rm \,$\mu$m}\ luminosity is corrected by A$_\mathrm{70\,\mu m}$\,=\,0.06\,mag, whilst for wavelengths larger than 70{\rm \,$\mu$m}\ the correction is negligible. To remove the background emission from the $70-500$\,{\rm \,$\mu$m}\ data, we follow the method outlined by \citet[][background removal of these data will be presented by Battersby et al. in prep]{battersby_2011}.
Figure\,\ref{sed} displays the extinction corrected average spectral energy distribution for an example region ($|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$). This shows that the Galactic Centre shows two distinct temperature components (see Figure\,\ref{sed}). The first peaks at wavelengths $\sim$\,$100-200$\,{\rm \,$\mu$m}, and is thought to originate from the cool dust (T\,$\sim$\,30\,K). The second peaks at $\sim$\,10\,{\rm \,$\mu$m}\ (T\,$\sim$\,300\,K), and becomes prominent towards known star-forming regions, and originates from warm dust. To measure the total column density of hydrogen ({$N_\mathrm{H_2}$}), the dust temperatures and the total bolometric luminosity, we fit a two component modified Planck function to the spectral energy distribution at each pixel (after smoothing all wavelengths to the largest resolution of $\sim$\,36{\rm $^{\prime\prime}$}). The modified Planck function with respect to frequency, ${\rm S}_\nu$, takes the form,
\begin{equation}
{\rm S}_\nu = \frac{2 h \nu^3}{c^2 \left( e^{h\nu/kT} - 1 \right)} \left( 1 - e^{-\tau_\nu} \right),
\end{equation}
where the opacity is given by,
\begin{equation}
\tau_\nu = \mu_\mathrm{H_2} m_\mathrm{H} \kappa_\nu N_\mathrm{H}.
\end{equation}
We assume a mean molecular weight of $\mu_\mathrm{H_2}$=2.8\,a.m.u, a dust opacity of $\kappa_{\nu}\,=\,\kappa \left( \nu / \nu_{0} \right)^\beta$ at $\nu$\,=\,505\,GHz with a $\kappa_0$\,=\,4.0 and $\beta$\,=\,1.75 \citep{battersby_2011}, and a constant gas-to-dust ratio of 100.
To separate the two temperature components, we consider wavelengths between 5.8\,-\,24\,{\rm \,$\mu$m}\ for the warm component, and 160\,-\,500\,{\rm \,$\mu$m}\ for the cool component (cool component is provided by Battersby et al. in prep). We do not consider {\it Spitzer} data with wavelengths smaller than 5.8\,{\rm \,$\mu$m}\ when fitting the spectral energy distribution, as it is not clear how much flux at these wavelengths is from recently formed stars or the older stellar population in the bulge of the Galaxy. Overlaid on Figure\,\ref{sed} are the average warm (TIR$_{warm}$) and cool component (TIR$_{cool}$) fits, and the total fit (TIR$_{tot}$\,=\,TIR$_{warm}$\,+\,TIR$_{cool}$).
The shaded region for the average fits represents the instrumental uncertainty, which has been estimated as a conservative $\sim$\,20\,per cent on the flux density measurements. However, we expect the absolute uncertainty to be higher than this, due to variations in the dust properties across the region. For example, several authors have shown that there is a gradient of decreasing gas-to-dust ratio with decreasing galactocentric radius \citep{schlegal_1998, watson_2011}, a trend which has also been observed in several other star-forming galaxies \citep{sandstrom_2013}. Assuming that the gas-to-dust ratio is inversely proportional to the metallicity, the gas-to-dust ratio within the central kpc of the Galaxy would be $\sim$\,50 (e.g. \citealp{sodroski_1995}), which would cause the column densities, and gas mass measurements later in this work (see section\,\ref{embedded}), to be a factor of two lower. Given this, we estimate the absolute column density measurements should be reliable to within a factor of two.
The flux-densities, ${\rm S}_\nu$, are converted into fluxes, S (in units of MJy\,sr$^{-1}$), by integrating the two component modified black body at each position. To convert these into luminosities (units of erg\,s$^{-1}$ or {\rm L$_\odot$}) requires an accurate measurement of the source distance. This analysis, therefore, has been restricted to the $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ region, as parallax measurements \citep{reid_2009, reid_2014} and modelling \citep{molinari_2011, kruijssen_2015} have shown that the majority of the dense molecular gas and star-forming regions are close to the Galactic Centre. Additionally, the extreme environment within this region has many identifiable features (e.g. large velocity dispersion), which have been used to show that there is little contamination from non-associated material along the line-of-sight (e.g. \citealp{henshaw_2016}). We are, therefore, confident with the distance measurement to the $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ region of $\sim$\,8.5\,kpc \citep{reid_2014}. The integrated flux, S (MJy\,sr$^{-1}$), is converted to luminosity, L (erg\,s$^{-1}$), with the units shown in parenthesis using,
\begin{equation}
{\rm L} ({\rm erg\,s}^{-1}) = 2.8\,\times\,10^{10}\,{\rm S}_\nu({\rm MJy\,sr}^{-1})\,{\rm R}_{pix}^{2}(^{\prime\prime})\,{\rm D}^{2}({\rm pc}),
\end{equation}
where R$_{pix}$ is the pixel size and D is the distance to the region. Figure\,\ref{sfr_maps} presents the 24{\rm \,$\mu$m}, 70{\rm \,$\mu$m}, and the warm and cool component bolometric luminosity maps (TIR$_{warm}$ and TIR$_{cool}$, respectively) for the $|l|<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ region. Over-plotted are grey and white contours of the warm and cool gas column densities, respectively. Labels shows the positions of the main objects of interest. The total luminosities within $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ are L(24{\rm \,$\mu$m})\,=\,9.4\,$\pm$\,1.9\,$\times\,10^{7}$\,{\rm L$_\odot$}, L(70{\rm \,$\mu$m})\,=\,3.4\,$\pm$\,0.7\,$\times\,10^{8}$\,{\rm L$_\odot$}, L(TIR$_{tot}$)\,=\,5.7\,$\pm$\,1.7\,$\times\,10^{8}$\,{\rm L$_\odot$}.
Throughout this work, we make the standard assumption that all the emission from the embedded stellar population is reprocessed by the surrounding dust to infrared wavelengths, which is emitted at much shorter wavelengths at the interface where the dust becomes optically thin. In this scenario, the total infrared luminosity directly corresponds to the bolometric luminosity produced by the embedded population. We estimate the measurement uncertainty on the total bolometric luminosity as the maximum variation after changing the flux densities at each wavelength by $\pm$\,20\,per cent (an upper limit estimate of the flux uncertainties) and re-fitting the spectral energy distribution. These uncertainties are, however, small when compared to the systematic uncertainties, for example: i) leakage of high-energy photons (in which case the infrared luminosity is not equal to the total bolometric luminosity), ii) heating of dust via other sources, and iii) emission produced from the older embedded population within the CMZ (e.g. \citealp{calzetti_2010}). It is difficult to estimate the amount of energy leakage, as this requires an accurate description of the three-dimensional density structure of the individual star-forming regions. Similarly, for the dust heating from other sources, such as from the central super massive black hole (Sgr A$^*$), it is difficult to estimate given our limited knowledge of the radiation field within this region. However, an estimate of the contribution from the older embedded population can be made using the Besan\c{c}on model \citep{robin_2003}. To estimate the expected bolometric luminosity from the field stellar population we use the online\footnote{\url{http://model.obs-besancon.fr}} catalogue simulation, which includes all the stellar luminosity classes and ages. Within the region $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$, taking all stars with a distance between 8.4\,-\,8.6\,kpc (i.e. forming a $\sim$\,200$\times$100$\times$200\,pc box containing the CMZ), we find that the total bolometric luminosity from the population older than 0.15\,Gyr is $\sim\,2.5\times\,10^{8}$\,L$_{\odot}.$\footnote{We note that \citet{robin_2003} do not fit the model to the observed stellar density within $|l|$$<$1$^{\circ}$. Rather the stellar densities are predicted by extrapolating a power-law, from the DENIS survey within $-8^{\circ}<l<12^{\circ}$, $|b|<$4$^{\circ}$ (see \citealp{epchtein_1997}).}
The \citet{robin_2003} bulge population luminosity of $\sim\,2.5\times\,10^{8}$\,L$_{\odot}$ is similar to that determined by \citet[][see their Table 6, 1.9\,$\times\,10^{8}$\,{\rm L$_\odot$}]{launhardt_2002}, from $2.2-240${\rm \,$\mu$m}\ IRAS and COBE data. The \citet{robin_2003} and \citet{launhardt_2002} estimates are around half of the total measured luminosity we find within $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ (5.7\,$\pm$\,1.7\,$\times\,10^{8}$\,{\rm L$_\odot$}). However, the spectral energy distribution of old stars will peak at wavelengths $<5$\,{\rm \,$\mu$m}. To estimate the direct contribution of the old stellar population to the infrared luminosity in the wavelength range between $5.8-500$\,{\rm \,$\mu$m}\ we integrate the black-body spectral energy distribution with the average effective temperature of all stars from the \citet{robin_2003} stellar population model ($\sim$\,3000\,K). We find that the fraction of the luminosity produced by these stars emitted between $5.8-500$\,{\rm \,$\mu$m}\ is $\sim$\,3\,per cent. This suggests that approximately $\sim$\,$1-2$\,per cent of the total infrared luminosity within $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ is directly produced by the old bulge star population. We, therefore, do not remove this contribution from the luminosity and conclude the bolometric luminosity between $5.8-500$\,{\rm \,$\mu$m}\ is dominated by the emission from young stars.
\section{Global star formation}\label{global}
\subsection{Determining the global star formation rate}\label{totalsfr}
In order to estimate the total star formation rate across the CMZ, we apply several infrared luminosity-to-star formation rate (luminosity-SFR) relations to the bolometric and monochromatic infrared luminosities (see \citealp{kennicutt_2012} and references therein). These relations are based on the assumption that high-mass ``young stellar objects'' (YSOs) classified as having ages in the range $\sim$\,$0.1 - 5$\,Myr (see section\,\ref{litsfr}), are still heavily embedded within their parent molecular clouds when they first reach the zero age main sequence (ZAMS). Therefore, the majority of their prodigious short wavelength (e.g. ultra-violet) emission is absorbed by dust within their surrounding medium, and re-emitted at longer infrared wavelengths ($\sim$\,$1-1000${\rm \,$\mu$m}). Hence, the infrared luminosity can be used, similar to a calorimeter, to estimate the underlying embedded population. Given that high-mass stars have a characteristic age of a few Myr, the star formation rate can then be estimated. One advantage of this method is that it does not require the individual sites of star formation to be resolved ($<$\,0.05\,pc). By using the integrated luminosity of an entire stellar population, over scales of $\goa\,$100\,pc, the luminosity-SFR relations can be used to determine the star formation rates within extragalactic sources for which it is impossible to resolve individual forming stars. A sample of the most widely used monochromatic and bolometric luminosity-SFR relations are summarised in Table\,\ref{SFR_conv}, with the luminosity limits over which they are considered to be reliable. Table\,\ref{SFR_conv} also shows the global star formation rates within the region $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ derived using these relations. We find that the average star formation rates derived from the 24{\rm \,$\mu$m}, 70{\rm \,$\mu$m}, and TIR luminosities are 0.09$\pm$\,0.02, 0.10$\pm$\,0.02, and 0.09$\pm$\,0.03\,{\rm M$_\odot$\,yr$^{-1}$}, respectively. The uncertainties shown here are from the measurement uncertainties on the luminosity (see section\,\ref{method}). We note, however, that the systematic uncertainty on the luminosities are significantly larger, and the luminosity-SFR relations have an uncertainty of around a factor two. Taking these uncertainties into account, we estimate that the measured star formation rates are reliable to within a factor of two. Hence the average {\it global} star formation rate within the $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ region derived from the luminosity-SFR relations has a mean of $0.09\pm0.02\,${\rm M$_\odot$\,yr$^{-1}$}.
\begin{table*}
\caption{Summary of various SFR conversion factors. Columns show the wavelengths to which the conversions apply, the conversion reference, the conversion factors (where L($\lambda$) is the luminosity at wavelength $\lambda$), the reliability limits, and the calculated star formation rate within within $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$.}
\centering
\begin{tabular}{ c c c c c}
\hline
Wavelengths for & Conversion & Luminosity range$^{a}$ & Reference for & SFR within $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ \\
conversion ({\rm \,$\mu$m}) & (erg s$^{-1}$ {\rm M$_\odot$}$^{-1}$ yr) & (erg\,s$^{-1}$) & conversion & ({\rm M$_\odot$} yr$^{-1}$) \\ \hline
24 & 2.75\,$\times\,10^{-43}$\,L(24{\rm \,$\mu$m}) & (0.01\,-\,1)\,$\times\,10^{44}$ & \citet{wu_2005} & 0.10\\
& 2.46\,$\times\,10^{-43}$\,L(24{\rm \,$\mu$m}) & (0.004\,-\,2)\,$\times\,10^{44}$ & \citet{zhu_2008} & 0.09\\
& 2.04\,$\times\,10^{-43}$\,L(24{\rm \,$\mu$m}) & (0.4\,-\,5)\,$\times\,10^{43}$ & \citet{rieke_2009} & 0.07$^{b}$\\
& 9.01\,$\times\,10^{-34}$\,L(24{\rm \,$\mu$m})$^{0.768}$ & (0.001\,-\,3)\,$\times\,10^{41}$ & \citet{perez-gonzalez_2006} & 0.07\\
& 5.83\,$\times\,10^{-38}$\,L(24{\rm \,$\mu$m})$^{0.871}$ & (0.0001\,-\,3)\,$\times\,10^{44}$ & \citet{alonso-herrero_2006} & 0.09\\
& 1.31\,$\times\,10^{-38}$\,L(24{\rm \,$\mu$m})$^{0.885}$ & (0.0001\,-\,3)\,$\times\,10^{44}$ & \citet{calzetti_2007} & 0.08\\
& 5.66\,$\times\,10^{-36}$\,L(24{\rm \,$\mu$m})$^{0.826}$ & (0.000001\,-\,3)\,$\times\,10^{44}$ & \citet{relano_2007} & 0.12 \\ \hline
70 & 5.88\,$\times\,10^{-44}$\,L(70{\rm \,$\mu$m}) & $>$\,1.4\,$\times\,10^{42}$ & \citet{calzetti_2010} & 0.07\\
& 9.37\,$\times\,10^{-44}$\,L(70{\rm \,$\mu$m}) & (0.005\,-\,5)\,$\times\,10^{43}$ & \citet{li_2010} & 0.12\\
& 9.70\,$\times\,10^{-44}$\,L(70{\rm \,$\mu$m}) & \dots & \citet{lawton_2010} & 0.12\\ \hline
TIR & 4.5\,$\times\,10^{-44}$\,L(TIR) & \dots & \citet{kennicutt_1998} & 0.10$^{c}$\\
& 3.88\,$\times\,10^{-44}$\,L(TIR) & (0.02\,-\,2)\,$\times\,10^{43}$ & \citet{murphy_2011} & 0.09\\ \hline
\end{tabular}
\label{SFR_conv}
{ \vspace{0.4cm}}
\begin{minipage}{\textwidth}
\vspace{2mm}
$^a$ Note, most of the authors do not specify a luminosity range of the validity of the SFR conversion. Therefore, following \citet{calzetti_2010}, we define the luminosity range as the limits of the sample in each work. \\
$^b$ \citet{yusef_2009} estimate a star formation rate of 0.07{\rm M$_\odot$\,yr$^{-1}$}, using this conversion with a 24\,{\rm \,$\mu$m}\ luminosity over an area of $|l|$$<$1.3$^{\circ}$, $|b|<$0.17$^{\circ}$.\\
$^c$ \citet{crocker_2011} estimate a star formation rate of 0.08{\rm M$_\odot$\,yr$^{-1}$}, using this conversion with a bolometric luminosity found from 2.2\,-\,240{\rm \,$\mu$m} \ IRAS data.
\end{minipage}
\end{table*}
\subsection{Comparison to star formation rates within the literature}\label{litsfr}
In this section, we discuss the methods which have been previously used to determine the total star formation rate within the Galactic Centre, and compare to the values determined in section\,\ref{totalsfr}.
\subsubsection{YSO counting}\label{YSO counting}
The first method uses infrared emission to measure the masses of individual high-mass YSOs (ages $\sim$\,$0.1-1$\,Myr; M\,$\ga$\,10\,{\rm M$_\odot$}). As previously mentioned, high-mass stars reach the ZAMS whilst they are still heavily embedded within their parent molecular cloud (and are most likely still accreting material). They inject a significant amount of energy into their surrounding environment, and can therefore be identified from their strong, compact near/mid infrared emission. Once identified, their masses can be estimated from their bolometric luminosity. The total embedded stellar population mass of a region can then be inferred by extrapolating the stellar initial mass function down to lower masses assuming an appropriate initial mass function (IMF). Using this method, \citet{yusef_2009} identified potential YSOs as sources which show excess 24{\rm \,$\mu$m}\ emission with respect to 8{\rm \,$\mu$m}\ emission within the region $|l|<1.3^{\circ}$, $|b|<0.17^{\circ}$. By modelling the spectral energy distributions, they distinguish which are the young sources, and measure their luminosities to estimate the masses. They find the total embedded stellar population after IMF extrapolation is $\sim$\,1.4\,$\times\,10^{4}$\,{\rm M$_\odot$}. These authors assume a YSO lifetime of 0.1\,Myr, which they use to estimate a global star formation rate of $\sim\,0.14$\,{\rm M$_\odot$\,yr$^{-1}$}\ ($|l|<1.3^{\circ}$, $|b|<0.17^{\circ}$).
There are several inherent difficulties with the YSO counting method. For example, it is not trivial to determine the YSO ages from either the infrared excess, or the spectral energy distributions. Furthermore, dusty, bright, asymptotic giant branch stars are also known to emit at infrared wavelengths with similar colours to YSOs \citep{habing_1996}. Therefore, YSO identification can be plagued with source contamination. In light of this, \citet{koepferl_2015} re-examined the \citet{yusef_2009} YSO sample, comparing in-depth radiative transfer modelling of both embedded YSOs and embedded older main sequence stars (ages of $>$\,1\,Myr). These authors conclude the \citet{yusef_2009} sample suffers from significant contamination from embedded main sequence stars (which produce very similar emission profiles within the wavelengths used by \citealp{yusef_2009}). They propose the star formation rate is factors of several lower: $\sim\,0.06$\,{\rm M$_\odot$\,yr$^{-1}$}. \citet{immer_2012} used infrared spectral features to attempt to disentangle the young and more evolved stars. These authors initially identified candidate YSOs as compact sources with excess 7{\rm \,$\mu$m}/15{\rm \,$\mu$m}\, emission, which are verified by examining the infrared spectral features of a test sample. Within $|l|$$<$1.5$^{\circ}$, $|b|<$0.5$^{\circ}$, \citet{immer_2012} estimate a total embedded stellar population mass of $\sim$\,7.7\,$\times\,10^{4}$\,{\rm M$_\odot$}. Given the slightly different selection criteria used by \citet{immer_2012}, these authors assume a longer YSO lifetime of $\sim$\,1\,Myr compared to \citet{yusef_2009} and \citet{koepferl_2015}, yet calculate a comparable average SFR of $\sim\,0.08$\,{\rm M$_\odot$\,yr$^{-1}$}.
\subsubsection{Free-free emission}\label{free-free emission}
The second method to determine star formation rates involves using cm-(mm-)-continuum emission to measure the mass of the YSO population. Along with heating the surrounding environment, YSOs with ages of $\sim$\,3\,Myr and masses $>$\,$8-10$\,{\rm M$_\odot$}\ emit a significant amount of high-energy ionising radiation (i.e. photons with $h \nu$$>$13.6\,eV), which produces {\ion{H}{II}}\ regions. The free-free emission from the ionised gas (i.e. bremsstrahlung radiation) can be observed at cm-wavelengths, when the medium is optically thin. Cm-continuum emission observations, therefore, provide a reliable way to determine the rate at which ionising photons are produced from massive stars within a region, which in turn can be used to estimate their mass. The total embedded stellar population can then be extrapolated using an IMF, from which the star formation rate can be estimated (e.g. \citealp{murray_2010}). \citet{lee_2012} used {\it Wilkinson Microwave Anisotropy Probe (WMAP)} continuum observations at wavelengths of $\sim$\,$3-150$\,mm, to identify and measure sources of free-free emission within the Milky Way. \citet{longmore_2013} used this catalogue and the free-free-to-SFR conversion presented by \citet{murray_2010}, to estimate a global star formation rate of $\sim\,0.06$\,{\rm M$_\odot$\,yr$^{-1}$}\ within the region $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ (the same region adopted by this work).
\subsubsection{Infrared luminosities}\label{Infrared luminosities}
The third method to estimate star formation rates involves using the bulk infrared emission and the luminosity-SFR relations. This method follows a similar basis to the YSO counting method, where the near/mid infrared dust emission is modelled to determine the embedded population. The key difference is that the luminosity-SFR relations use the integrated emission from entire stellar populations, hence sample star formation rates over larger times, which usually translates to larger spatial scales, than to YSOs counting (see Table\,\ref{areas}).
\citet{launhardt_2002} fit the spectral energy distribution of 2.2\,-\,240{\rm \,$\mu$m}\ IRAS and COBE data, and estimate the total infrared bolometric luminosity within $|l|<0.8^{\circ}$, $|b|<0.3^{\circ}$ is L(TIR$_{tot}$)\,=\,4.2$\times\,10^{8}$\,{\rm L$_\odot$}. \citet{crocker_2011} use this, with the relation of \citet[][ see Table\,\ref{SFR_conv}]{kennicutt_1998} to calculate a SFR of $\sim$\,0.08{\rm M$_\odot$\,yr$^{-1}$}. \citet{yusef_2009} have used the 24{\rm \,$\mu$m}\ monochromatic luminosity from {\it Spitzer}, of L(24{\rm \,$\mu$m})\,=\,9\,$\times\,10^{7}$\,{\rm L$_\odot$}\ within $|l|$$<$1.3$^{\circ}$, $|b|<$0.17$^{\circ}$, with the luminosity-SFR relation presented by \citet[][ also see Table\,\ref{SFR_conv}]{rieke_2009}, to estimate a star formation rate of $\sim$\,0.07\,{\rm M$_\odot$\,yr$^{-1}$}.
\subsubsection{Comparison of star formation rates}
In Table\,\ref{areas} we tabulate the measurements of the star formation rate within the Galactic Centre. Apart from the YSO counting measurement of \citet{yusef_2009}\footnote{This likely suffers from contamination, see section\,\ref{YSO counting}.}, we find agreement to within $\sim$\,35\,per cent between the various measurements. To attempt a fairer comparison between these star formation rates and those found in this work, we re-determine the star formation rates within the areas adopted by the works listed in Table\,\ref{areas}, when using the total bolometric luminosity and using a single luminosity-SFR relation \citep{kennicutt_1998}. These measurements are shown in Table\,\ref{areas}. As the total bolometric luminosity measurements are self-consistent, they exclusively reflect the effect of changing the considered area. Given this, we can conclude that the there is no systematic uncertainty in any one measurement method which is causing an underestimation of the star formation rate.
\begin{table*}
\caption{Summary of star formation rate measurements within the literature. Shown is the measurement method, the characteristic age range traced by the method, the CMZ area over which the star formation rate has been measured, the measured star formation rate, the star formation rate determined using the infrared bolometric luminosity within the different areas of the CMZ and the luminosity-star formation-rate relation from \citep{kennicutt_1998}.}
\centering
\begin{tabular}{c c c c c c}
\hline
Method used to & Characteristic & Area of CMZ over & SFR determined from & SFR determined using the infrared \\
determine the SFR & age probed by & which the SFR is & the corresponding method & bolometric luminosity with the \\
& method (Myr) & calculated & and area ({\rm M$_\odot$\,yr$^{-1}$}) & \citet{kennicutt_1998} relation from the \\
& & & & corresponding area ({\rm M$_\odot$\,yr$^{-1}$}) \\ \hline
YSO counting & $\sim$\,0.1 & $|l|<1.3^{\circ}$, $|b|<0.17^{\circ}$ & 0.14$^{b}$ & 0.07 \\
YSO counting & $\sim$\,0.1 & $|l|<1.3^{\circ}$, $|b|<0.17^{\circ}$ & 0.06$^{c}$ & 0.07\\
Infrared luminosities & 0-5-100$^{a}$ & $|l|$$<$1.3$^{\circ}$, $|b|<$0.17$^{\circ}$ & 0.07$^{b}$ & 0.07 \\
Infrared luminosities & 0-5-100$^{a}$ & $|l|$$<$0.8$^{\circ}$, $|b|<$0.3$^{\circ}$ & 0.08$^{d}$ & 0.07\\
YSO counting & $\sim$\,1 & $|l|$$<$1.5$^{\circ}$, $|b|<$0.5$^{\circ}$ & 0.08$^{e}$ & 0.12 & \\
free-free emission & $\sim$\,0-3-10$^{a}$ & $|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$ & 0.06$^{f}$ & 0.10 & \\
\hline
\end{tabular}
\label{areas}
{ \vspace{0.4cm}}
\begin{minipage}{\textwidth}
\vspace{2mm}
$^{a}$ The second number indicates the mean age of the stellar population contributing to the emission, the third number shows the age below which 90\,per cent of emission is contributed \citep{kennicutt_2012}. \\
$^{b}$ \citet{yusef_2009}. \\
$^{c}$ \cite{koepferl_2015}. \\
$^{d}$ \citet{crocker_2011}. \\
$^{e}$ \citet{immer_2012}. \\
$^{g}$ \citet{longmore_2013}. \vspace{-3mm}
\end{minipage}
\end{table*}
\subsection{Comparison to star formation rates predicted from theoretical models}\label{models}
Given that we now have a set of consistent measurements for the global star formation rate within the Galactic Centre, comparison can be made to different star formation theories within the literature. Specifically, we focus on a ``column density'' threshold relation, a relation between the gas mass and the star formation rate, and three theoretical models based on ``volumetric'' gas density scalings.
\vspace{3mm}
We firstly consider the ``column density'' threshold relation presented by \citet{lada_2010, lada_2012}. \citet{lada_2010} found that, for local clouds, a correlation exists between the gas mass at high extinctions (A$_K$\,$>$\,0.8\,mag) and the number of embedded YSOs identified in the infrared. These authors estimate that this gas has a hydrogen column density of the order 6.7\,$\times10^{21}$\,cm$^{-2}$ (or $\sim\,10^4$\,cm$^{-3}$; assuming no line of sight contamination, and a typical spherical core of radius $\sim$\,0.1\,pc). They measure the star formation rates of clouds from the number of embedded YSOs, assuming an initial mass function median mass ($\sim$\,0.5\,{\rm M$_\odot$}), and a median age spread of the clouds ($t_{\rm age}\sim$\,2\,Myr), such that SFR({\rm M$_\odot$\,yr$^{-1}$})~=~N$_{\rm YSO}$\,M$_{{\rm IMF},median}$({\rm M$_\odot$})\,/\,$t_{\rm age}$(yr). \citet{lada_2010} (and \citealp{lada_2012} who incorporate extragalactic sources) show that the amount of dense gas and the level of star formation are correlated. The measured depletion time of the dense gas is 20\,Myr, which implies,
\begin{equation}
\mathrm{SFR} =
\begin{cases}
0 & \text{if } N_\mathrm{H_2} < 6.7\times10^{21}\,\mathrm{cm}^{-2} \\
4.6\times10^{-8}\,\mathrm{M_{gas}} & \text{if } N_\mathrm{H_2} \geq 6.7\times10^{21}\,\mathrm{cm}^{-2}
\end{cases}
\label{lada}
\end{equation}
where the star formation rate is in units of {\rm M$_\odot$\,yr$^{-1}$}, and mass is in units of {\rm M$_\odot$}.
We now consider the ``volumetric'' models for star formation (see \citealp{padoan_2014} for a comprehensive review). These are based on the assumption that the density distribution of the gas in star-forming regions follows a log-normal probability distribution when turbulence is dominant. These models use the dimensionless star formation efficiency per free-fall time, $\epsilon_\mathrm{ff}$, to describe the level of star formation within a region. This can be expressed as the integral of the probability distribution function, $p(x = \rho/\bar{\rho})$, above the critical overdensity of collapse, $x_{crit}=\rho_{crit}/\bar{\rho}$, where $\bar{\rho}$ is the mean density and $\rho_{crit}$ is the density when gravitational collapse begins to dominate. This integral is given as,
\begin{equation}
\epsilon_{\rm ff} = \frac{\epsilon_{core}}{\phi_{t}}\int_{x_{crit}}^{\infty} \frac{t_{\rm ff}(\bar{\rho})}{t_{\rm ff}(\rho)}\,p(x)\,dx,
\label{pdf}
\end{equation}
where $\epsilon_{core}$ is the fraction of core mass which forms the protostar, and $\phi_{t}$ is the gas replenishment factor, such that $\phi_{t}\,t_\mathrm{ff}$ is the replenishment time \citep{krumholz_2005}. There are two physical interpretations of this integral, which differ in the treatment of the density dependence of the free-fall time ($t_{\rm ff}(\bar{\rho})/t_{\rm ff}(\rho)$). Assuming a constant free-fall time based on the mean properties of the cloud simplifies the integral, and gives a ``single-free-fall'' solution where all the gas collapses over the same timescale ($t_{\rm ff}(\bar{\rho})/t_{\rm ff}(\rho)$ = constant). On the other hand, assuming that the free-fall time varies as a function of density, as suggested initially by \citet{hennebelle_2011}, requires the integral to be solved over the log-normal probability distribution. In this ``multi-free-fall'' time solution, smaller, denser structures can decouple from their lower density environment, and collapse on shorter timescales than the global free-fall time ($t_{\rm ff}(\rho>\bar{\rho})\,<\,t_{\rm ff}(\bar{\rho})$). This may be more representative of the hierarchical structure observed in star-forming regions.
The solutions to the integral shown in equation\,\ref{pdf}, require the calculation of the dispersion of the density probability distribution function with respect to the mean density, $\sigma_{x}$ (see \citealp{federrath_2012}). This can be estimated as,
\begin{equation}
\sigma_{x}\,\approx\,b\,\mathcal{M}\,(1\,+\,\beta^{-1})^{-1/2},
\label{sigma}
\end{equation}
where $b$ is the turbulence driving parameter. The turbulence driving parameter was introduced by \citet{federrath_2008} to distinguish solenoidal (divergence-free, $b\,\sim\,0.33$) driving of the turbulence from compressive (curl-free, $b\,\sim\,1$) driving. The turbulence driving varies from as low as $b\,=\,0.22\,\pm\,0.12$ in the ``Brick'' \citep{federrath_2016} to typical values of $b\,\approx\,0.5$ in several clouds in the Galactic disc (Taurus, IC5146, GRSMC43.30-0.33; see \citealp{federrath_2016}). In equation\,\ref{sigma}, $\mathcal{M}$ is the three-dimensional turbulent sonic Mach number,\footnote{The three-dimensional Mach number, denoted by $\mathcal{M}$, is used throughout this work.} and $\beta=2\mathcal{M}^2_\mathrm{A}/\mathcal{M}^2$ is used to quantify the importance of the magnetic field within the plasma, where $\mathcal{M}_\mathrm{A}$ is the Alfv{\'e}nic Mach number. Strong magnetic fields are, therefore, represented by low values of $\beta$, and a hydrodynamical (rather than magnetohydrodynamic) expression of equation\,\ref{sigma} can be obtained by setting $\beta\rightarrow\infty$. Shown in Table\,1 of \citet{federrath_2012} are the several definitions for the critical density for collapse, $x_{crit}$, and resultant solution to the integral of the density probability distribution function (equation\,\ref{pdf}).
\citet{longmore_2013} use two models to determine the predicted {\it global} star formation rate within the CMZ ($|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$): the column density threshold relation of \citet{lada_2010}, as described above, and the model of \citet{krumholz_2012}, which is an evolution of the \citet{krumholz_2005} model. For the column density limit model, they estimate that $\sim$\,95\,per cent of the gas within the CMZ lies above the threshold for collapse, which gives a predicted star formation rate of 0.78\,{\rm M$_\odot$\,yr$^{-1}$}. For the volumetric model, they then calculate the volume density by assuming that the gas at $|l|$$<$1$^{\circ}$ deg is distributed in a ring-like stream with a radius of $\sim$\,100\,pc \citep{molinari_2011}. This structure has a mass of 1.8$\,\times\,10^7${\rm M$_\odot$}, which gives a predicted star formation rate of 0.4\,{\rm M$_\odot$\,yr$^{-1}$}. These predictions are significantly higher than the average observed star formation rate of $\sim$\,$0.08$\,{\rm M$_\odot$\,yr$^{-1}$}. Table\,\ref{areas} shows the conclusion of \citet{longmore_2013} holds no matter which method is used to determine the star formation rate.
\subsection{Implications for the global star formation rate}\label{impsfr}
In summary, we find that the star formation rates for the CMZ measured from the infrared luminosity, YSO counting, and free-free emission have a mean value across all measurements of 0.09\,$\pm$\,0.02\,{\rm M$_\odot$\,yr$^{-1}$}, and given their uncertainties are in agreement to within a factor two. Furthermore, in agreement with the conclusion of \citet{longmore_2013}, we find that this is factors of a few to more than an order of magnitude smaller than is predicted from star formation models. In section\,\ref{intro} we speculated three possible causes for this apparent dearth in star formation within the Galactic Centre: 1) inaccurate star formation rate measurements, 2) episodic star formation, or 3) inappropriate comparison to the predictions from star formation relations/models. The results found in this section have shown:
\begin{itemize}
\item[i)] The star formation rates determined from the infrared luminosity-SFR relations are within a factor two of previous measurements. This allows us to rule out that systematic uncertainties in the measurements are causing the apparent low star formation rate, unless this uncertainty affects all the methods in the same way, which seems unlikely.
\item[ii)] The luminosity-SFR relations, which use the integrated light from the whole stellar population, and YSO counting methods, which require the sites of star formation to be resolved, are consistent in the Galactic Centre. As the Galactic Centre is the most extreme environment for which is it possible to resolve individual forming stars and make this measurement, the results here provide confidence that the luminosity-SFR relations reliably trace the star formation rate over kpc scales within similar environments present in starburst galaxies, and high-redshift galaxies.
\item[iii)] The various methods to determine the star formation rates are in agreement, despite being sensitive to star formation over different time scales over the past few Myr. Therefore, the global star formation rate has not changed over this time by more than a factor of two to three of its current rate. This is consistent with recent theoretical models predicting that the star formation rate in the CMZ is episodic on a timescale of $\sim$\,$10 - 20$\,Myr, much longer than the mean timescales covered by the adopted star formation rate tracers of $\sim$\,$0.1 - 5$\,Myr.
\end{itemize}
Returning to the discussion in the introduction, we have ruled out the first of the possibilities for the apparent low star formation rate within the Galactic Centre, that it results from inaccurate star formation rate measurements. Furthermore, in agreement with recent theoretical work, we find that the Galactic Centre could be in a low point in a star formation cycle. \citet{kruijssen_2014a} have suggested that the majority of gas within the CMZ is not bound by self-gravity, rather it is bound by the potential produced from the embedded bulge stars. Therefore, despite the gas being very dense, it will not gravitationally collapse to form stars as it would in the Galactic disc, and the CMZ is therefore at a star formation minimum. This idea has been quantified further by \citet{krumholz_2015} and \citet{krumholz_2016}, who predict that significant star formation should take place once the gas becomes self-gravitating. To investigate this, we examine the star formation rates on parsec scales (rather than global scales) within molecular clouds and star formation regions which are believed to be bound by self-gravity.
\section{The gas mass and embedded stellar populations within individual clouds}\label{embedded}
\subsection{Determining the gas and embedded stellar masses from infrared observations}\label{determing_embedded}
It is clear from Figure\,\ref{sfr_maps} that the luminosity in the Galactic Centre varies significantly over scales as little as a few parsecs, implying that the instantaneously measured local star formation rate and efficiency also vary over similar scales. However, as the individual clouds only harbour specific stages of star formation, we can not apply the SFR-luminosity relations, as these require continuous star formation over $\gg$\,5\,Myr \citep{kruijssen_2014}. In this section we therefore propose an alternative method to measure the embedded stellar masses, which will be used in the following sections to estimate the star formation rate.
Firstly, the individual clouds which are believed to be bound by self-gravity are identified, and their boundaries are determined. Next, we measure the enclosed infrared bolometric luminosity. This is used to estimated the mass of the most massive star, from which the total embedded population can be extrapolated using a stellar initial mass function. We choose to limit the sources to those within the region $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$, as this region has both significant cool and warm gas emission, and is known to contain both quiescent (the so called ``dust-ridge'') and actively star-forming regions (e.g. Sgr B2), whilst not suffering significant line-of-sight confusion from prominent sources within the Galactic Centre (e.g. Sgr A$^{*}$). Despite being limited to this ``simple'' region of the Galactic Centre, a certain level of ambiguity is present when identifying the extent for the sources. The interstellar medium is intrinsically hierarchical and the three-dimensional structure of the gas within the Galactic Centre is complex (e.g. \citealp{rathborne_2015, walker_2015, henshaw_2016}; see section\,\ref{boundaries}). It is therefore difficult to impose physically meaningful cloud boundaries in the same way one can separate individual stars.
We define the boundaries using various warm and cool column density contours (see Figure\,\ref{sfr_maps}), which have be chosen by-eye to best separate different sources. The column density limits, radii and enclosed gas masses for each source is displayed in Table\,\ref{masses_table}. These masses are within a factor of two to those presented by \citet{immer_2012a}, \citet{longmore_2012}, \citet{walker_2015} and \citet{federrath_2016}. The moderate difference is a result of our higher column density boundaries. We investigate the effect of changing the source boundaries in section\,\ref{Uncertainties}, and show that this does not affect the results of this work.
To estimate the embedded population within each cloud, we first assume that the total infrared luminosity represents the bolometric luminosity from a single massive embedded star. This is a reasonable assumption since the most massive star should dominate the luminosity of a simple stellar population (M\,$\propto$\,L$^{x}$, where ${x}\,\sim\,1-3.5$; e.g. \citealp{mould_1982, salaris_2005}). To estimate the mass of this embedded object (M$_\mathrm{*, max}$), we adopt the bolometric luminosity-mass conversions presented by \citet{davies_2011}. The total embedded population mass (M$_\mathrm{*, tot}$), is extrapolated by solving the following two equations:
\begin{equation}
1 = \int_{M_\mathrm{*, max}}^{\infty} m ^{-\alpha}\,dm,
\end{equation}
where $\alpha$ = 2.3, and,
\begin{equation}
M_\mathrm{*, tot} = \int_{0.001\,M_\odot}^{\infty} m ^{1 -\alpha}\,dm,
\end{equation}
where $\alpha$\,=\,0.3 for 0.001\,$<$\,$m/$\,{\rm M$_\odot$}\,$<$\,0.08, $\alpha$\,=\,1.3 for 0.08\,$<$\,$m/$\,{\rm M$_\odot$}\,$<$\,0.5, and $\alpha$\,=\,2.3 for $m$\,$>$\,0.5\,{\rm M$_\odot$}\ \citep{kroupa_2001}. Figure\,\ref{rgb} presents a red-green-blue map of the region containing the sources, where the quiescent clouds are in red and (proto-)clusters in blue, over which the embedded and gas masses are labeled. The calculated masses are summarised in Table\,\ref{masses_table}, which are used to determine star formation rates and efficiencies in Section\,\ref{sec:SFE}.
\subsection{Determining the embedded stellar masses from additional observations}\label{lit_masses}
In addition to measurements from infrared observations, the embedded stellar mass can be inferred from cm/mm wavelengths observations. \citet{walker_2015} have determined the embedded stellar mass within the Sgr B2 region which encompasses the Sgr B2 ``main'', ``north'' and ``south'' {\ion{H}{II}}\ region complexes. These authors estimate the mass of high-mass stars embedded within the ultra compact {\ion{H}{II}}\ regions within these complexes (UC{\ion{H}{II}}) from their 1.3\,cm continuum emission (\citealp{gaume_1995} identified $\sim$\,40 regions). Given that the cm-observations are only sensitive to high-mass stars, the full population is extrapolated using a Kroupa IMF. \citet{walker_2015} estimate a total embedded stellar population mass of $\sim$\,3500\,{\rm M$_\odot$}. \citet{belloche_2013} determined a similar mass of $\sim$\,3900\,{\rm M$_\odot$} when using this same method, with the data of \citet{gaume_1995}. More recently, \citet{schmiedeke_2016} have complied all the available cm-continuum data from the literature (\citealp{mehringer_1993, gaume_1995, depree_1998}; $\sim$~70 regions identified) and follow the above method to determine the total embedded stellar population mass. These authors estimate a mass of $\sim\,3.3\,\times\,10^4\,${\rm M$_\odot$}\ (shown in parentheses in Table\,\ref{masses_table}), which is an order of magnitude larger than the previously derived values.
To determine the embedded stellar masses within G0.6-0.005 (henceforth G0.6) and Sgr B1, we follow the method used by \citet{walker_2015}, \citet{belloche_2013}, and \citet{schmiedeke_2016}, and the spectral classifications of the UC{\ion{H}{II}}\ regions as determined by \citet{mehringer_1992}. To convert these into masses, we use the spectroscopic masses of zero age main sequence stars given by \citet{vacca_1996}. We obtain the total mass using a Kroupa IMF. We find that the total embedded masses within G0.6 and Sgr B1 are 3300\,{\rm M$_\odot$}\ and 7200\,{\rm M$_\odot$}, respectively.
Along with the above VLA observations, we use {\it WMAP} observations to calculate the stellar mass within Sgr B1.\footnote{{\it WMAP} observations can only be used for Sgr B1, as this is the only source identified by \citet{lee_2012}.} {\it WMAP} data has better absolute flux precision, suffers less from spatial filtering and covers a larger frequency range than the VLA observations. From this data an accurate spectral index of the emission can be calculated allowing accurate determination of the relative contributions from free-free, non-thermal and spinning dust emission. From the WMAP source catalogue presented by \citet{lee_2012}, we find that Sgr B1 has an inferred ionising flux of Q\,=\,$0.5\pm\,0.19\,\times\,10^{53}$\,s$^{-1}$. \citet{murray_2010} showed that the ionising flux per stellar mass averaged over the initial mass function is $<{\rm q}>/<{\rm m_{*}}>\,=\,6.3\,\times\,10^{46}$\,{\rm M$_\odot$}$^{-1}$s$^{-1}$. Using this, we find that the total embedded stellar mass within Sgr B1 is $\sim$\,8000\,{\rm M$_\odot$}.
\subsection{Uncertainties}\label{Uncertainties}
This section includes a discussion of the uncertainties present when estimating the embedded young stellar mass and gas mass.
\subsubsection{Source boundaries}\label{boundaries}
\begin{figure}
\centering
\includegraphics[trim = 4mm 0mm 0mm 0mm, clip,angle=0,width=0.95\columnwidth]{\dir source_vs_radius.pdf}
\caption{Plots of radius against the total mass of the gas (upper panel) and embedded young stellar population (lower panel). The radial profiles of the CMZ sources are shown by the coloured solid and dashed lines (see the legend in the upper panel). }
\label{radius_vs}
\end{figure}
In section\,\ref{determing_embedded} we attempted to determine the boundaries of sources within the region $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$, using several column density contours. This is not trivial as the interstellar medium is intrinsically fractal and hierarchically structured. A similar difficulty in defining the sources was noted by \citet{walker_2015, walker_2016}, who showed that the properties of sources vary depending on the choice of boundary. To test the effect of changing the boundaries, Figure\,\ref{radius_vs} shows how the enclosed gas and embedded stellar masses vary as a function of radius (assuming a spherical geometry). Varying the radius of each cloud by approximately $\pm$\,30\,per cent gives on average a difference of M$_\mathrm{gas}$$^{+90\%}_{-40\%}$, and M$_\mathrm{*, tot}$$^{+50\%}_{-20\%}$.
We find the Sgr B2 region is particularly difficult to define with a single column density contour, as it is thought to contain both dense gas and actively star forming regions \citep[also see][]{schmiedeke_2016}. We choose to define the Sgr B2 region by a cool component column density of 7.5\,$\times$10$^{23}$\,\,cm$^{-2}$, as this contour separates it from neighbouring sources (see section\,\ref{determing_embedded}). However, this contour does not contain the warm luminosity component to the south-west ($l\approx0.64^{\circ}$, $b\approx\,$-$0.8^{\circ}$) or the extended cool envelope to the north ($l\approx0.6^{\circ}$, $b\approx0.0^{\circ}$), which have both been previously attributed to Sgr B2 (e.g. \citealp{yusef_2009}). Increasing the column density limit to include these would increase the gas and embedded stellar masses by factors of two to three.
Changing the source boundaries varies both the gas mass and embedded young stellar mass on average by a factor of two, however the gradient of both parameters as a function of radius are similar. As the star formation efficiencies per free-fall time which are calculated in the later sections of this work (section\,\ref{sec:SFE}), are essentially ratios of the gas mass and embedded young stellar mass, these are are not sensitive to the choice boundary over a few parsecs.
\subsubsection{Field Star contribution}
The sample of clouds within $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ are thought to reside $\sim$\,100\,pc from the centre of the galaxy. There is high number density of old population bulge stars within this region, which may contribute to the measured infrared luminosity and cause an overestimation of mass of the embedded stellar population. To investigate this, we use the Besan\c{c}on model \citep{robin_2003}. We use the same catalogue simulation from section\,\ref{totalsfr}, with a distance range of 8.3\,$-$\,8.5\,kpc and a step of 50\,pc, towards the coordinates of the clouds within the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region. This gives an average bolometric luminosity density of 28\,$\pm$\,5\,L$_\odot$\,pc$^{-3}$ ($\sim$15\,stars\,pc$^{-3}$). Therefore, we find that the old stellar population stars will contribute to $<$\,1\,per cent of the total bolometric luminosity measured for each cloud.
\subsubsection{Background contribution}
There is a smoothly varying infrared component to the diffuse Galactic emission along the line-of-sight to the CMZ, which contributes to the luminosity. This has been subtracted for the emission at wavelengths $>$70{\rm \,$\mu$m}\ using the method presented in \citet{battersby_2011}, which causes an average decrease of $\sim$20\,per cent in the measured bolometric luminosity at each pixel. There may be some residual diffuse emission that varies on small scales associated with the CMZ itself, which has not been removed when using the background subtraction routine and could contaminate the bolometric luminosity measurements for each cloud. We, however, do not expect this to be a significant effect.
We have not taken into account background subtraction for the line-of-sight emission at wavelengths $<$70{\rm \,$\mu$m}, which could lead to an overestimation of the luminosity. This is expected to be more of a problem for the sources which have less emission at these wavelength, such as those in the dust-ridge. We estimate the magnitude of this to be on the order of the $>$70{\rm \,$\mu$m}\ Galactic diffuse emission: $\sim$20\,per cent. On the other hand, we do not believe this will be a significant effect to the brighter star-forming clouds (e.g. Sgr B2), as their luminosities are clearly dominated by the embedded young stellar population.
\subsubsection{Luminosity from external heating}
The dust ridge clouds are thought to be externally heated \citep{longmore_2013, longmore_2013a, ott_2014, rathborne_2014a}, therefore we consider that some fraction of their luminosity may be produced by the heating from bright nearby sources. As an example, we estimate how much luminosity would be produced by the heating from the two brightest clusters within the Galactic Centre, the Arches and Quintuplet clusters. Using the three-dimensional structure of the Galactic Centre determined by \citet[][see section\,\ref{sec:SFE}]{kruijssen_2015}, we estimate that the ``Brick'' is the closest cloud, residing at a distance of $\sim$\,25\,pc from these clusters, hence use this as the example subject. \citet{figer_1999, figer_2002} estimate that the luminosity within the Arches and Quintuplet clusters is $10^{7.8}$ and $10^{7.5}$\,{\rm L$_\odot$}, respectively. Assuming that the luminosity is isotropically radiated from each cluster, and is completely absorbed and re-emitted by the ``Brick'' (assuming a circular geometry with a radius of $\sim$\,3.1\,pc), the luminosity contribution from the Arches and Quintuplet clusters is $\sim\,2\,\times\,10^{5}$\,{\rm L$_\odot$}\ and $10^{5}$\,{\rm L$_\odot$}, respectively. This is $\sim$\,30 per cent of the total luminosity of the ``Brick''. We suggest that this is an upper limit to the affect of external radiation on the measured luminosity of the sources considered here.
\subsubsection{Accretion luminosity}
We consider that some fraction of the bolometric luminosity from these clouds may be caused by the accretion of material onto the embedded stars. It is thought that the accretion luminosity for low-mass young stars can be around an order of magnitude higher than the intrinsic stellar luminosity, whereas for high-mass stars the stellar luminosity dominates over the accretion luminosity for all reasonable accretion rates ($>$\,10\,{\rm M$_\odot$}; e.g. \citealp{hosokawa_2009}). In this work we assume that the infrared luminosity from each of the embedded stellar populations is dominated by high-mass stars, so assume the contribution of the accretion luminosity should be insignificant.
\subsubsection{Variation in the embedded stellar population}\label{IMF}
We calculate the total embedded stellar population mass within each source in the Galactic Centre by extrapolating from a high mass sample of the population. The three caveats which may affect our measurement of the embedded stellar population mass are the choice of the initial mass function, the sampling of the initial mass function and the upper mass limit of the initial mass function (i.e. the maximum stellar mass of the population), and are discussed below.
Throughout this work we estimate the embedded stellar population assuming a Kroupa IMF \citep{kroupa_2001}. However, many IMFs are used in the literature, which despite years of major scrutiny, share broadly similar properties: a power-law at the high-mass end with a slope of roughly -2.3, and a turnover (Kroupa) or Gaussian \citep{chabrier_2003} low-mass end. This shape turns out to be universal, from the solar neighbourhood \citep{bastian_2010} to the CMZ (in the Arches cluster; \citealp{habibi_2013}). However, we note that it has been suggested that the nuclear star cluster (i.e. central parsec around Sgr A$^*$) has a top-heavy initial mass function \citep{bartko_2010}, and there is some indirect evidence for a bottom-heavy IMF in early-type galaxies, which harbour a similar environment to the CMZ \citep{conroy_2012, cappellari_2012}.
When using IMFs it is important to consider their stochastic nature, particularly for stellar populations with low number statistics - i.e. in regions where star formation has recently begun. When stars form they stochastically populate the IMF, such that each star has finite probability of having any mass between a given mass range. More massive stellar populations will in general have more stars, and hence will have a higher chance of being fully populated \citep[e.g. ][]{gilmore_2001, fumagalli_2011}. Populations with masses of $\sim$\,10$^{4}$\,{\rm M$_\odot$}\ should fully sample the initial mass function \citep{bruzual_2003}. In this work we estimate embedded stellar masses significantly less than this, for example dust-ridge clouds have M$_\mathrm{*, tot}$\,$\sim$\,10$^3$\,{\rm M$_\odot$} ($\sim$\,400\,-\,1000\,stars). The initial mass function for these sources will, therefore, most likely not be fully sampled, and stochastic effects may be significant. \citet{elmegreen_1999} have shown that low number sampling (1000\,stars) of an IMF could cause the power-law slope to vary by $\pm$\,0.1. This effect is inherently ``random'' and is therefore difficult to quantify for each source. Nevertheless, changing a single power-law slope of $\alpha\,=\,2.3$ by $\pm$\,0.1, would vary the total embedded mass by a factor of two to three.
The final caveat in calculating the total embedded stellar population mass for each source is that the bolometric luminosity is produced solely by the most massive star in the population. There may be, however, a non-negligible contribution by the second, and progressively lower massive stars to the measured bolometric luminosity, which could cause an overestimation of the most massive star in the population. To investigate how this affects the total mass of the embedded population, we plot the bolometric luminosity as a function of total embedded stellar population mass, with the assumption that all the luminosity is produced by the most massive star. On the same axis we plot the bolometric luminosity as a function of total embedded stellar population mass produced from the synthetic stellar population model {\sc starburst99} (see Figure\,\ref{mass_lum}).\footnote{\url{http://www.stsci.edu/science/starburst99/docs/default.htm}} The model input parameters are an instantaneous star formation burst populating a Kroupa IMF with total cluster masses ranging from 1000 to 100,000\,{\rm M$_\odot$}. We note that, the {\sc starburst99} is unable to produce stellar populations with masses below 1000\,{\rm M$_\odot$}. We find that the {\sc starburst99} mass-luminosity relation has a power-law slope which is shallower than if we assume all the luminosity is produced from the most massive star, with the intersection between the two relations at $\sim$\,5000\,{\rm M$_\odot$}. As the star-forming sources G0.6, Sgr B1 and G0.3-0.056 (henceforth G0.3) have masses close to this value, these should not be significantly affected by this uncertainty. However, the dust ridge clouds have embedded masses much lower than this. The {\sc starburst99} modelling shows that stochastic sampling may lead to an overestimation of the total embedded stellar mass by up to factor of 3 (see Appendix\,\ref{Appendix B}).
In summary, we estimate that the main sources of uncertainty on the embedded population within the Galactic Centre clouds are the stochastic nature of the IMF and the form of the adopted IMF. This leads to an uncertainty in the embedded stellar mass estimate of at least a factor of two.
\subsubsection{Saturation}
Several of the {\it Spitzer} and {\it Herschel} maps used in this work contain saturated pixels, which we treat these as having the maximum observed value within each map. This was not considered a problem when determining the global star formation rates as these pixels did not significantly contribute to the total luminosities. However, this is not the case for the individual clouds, where the majority of saturated pixels are located. We find that 8, 5, 40,\footnote{We highlight that the high fraction of saturated pixels in the bolometric luminosity maps towards the Sgr B1 region is a result of saturation in the {\it Spitzer} 24\,{\rm \,$\mu$m}\ map.} and 3\,per cent of the pixels within Sgr B2, G0.6, Sgr B1, and G0.3, respectively, are saturated. The luminosity towards these saturated pixels is considered a lower limit, and therefore contributes to an underestimation of the measured embedded stellar mass of each source. We expect that this will cause a more severe underestimation of the embedded stellar mass when a high concentration of the embedded sources are within the saturated pixels. Despite Sgr B1 having the majority of saturated pixels, we find only eight {\ion{H}{II}}\ regions towards these pixels \citep{mehringer_1992}, whereas within the saturated pixels towards Sgr B2 contain more than sixty {\ion{H}{II}}\ regions \citep{schmiedeke_2016}. We therefore expect pixel saturation to cause a significant underestimation of the embedded stellar mass towards the Sgr B2 region.
\subsubsection{Summary of embedded stellar population mass uncertainties}
To summarise, several of the uncertainties discussed above can significantly affect the estimate of the embedded stellar population masses for all the sources. Some are only applicable to either the quiescent clouds (high gas mass/low embedded stellar mass), or the star-forming clouds (low gas mass/high embedded stellar mass). The uncertainties which affect all the sources are the choice of arbitrary boundary ($\pm$50\,per cent) and the contribution from accretion luminosity (negligible). The uncertainties which primarily affect the quiescent clouds are the contribution of CMZ diffuse background luminosity and the stochasticity in the IMF, as their low masses do not enable full/significant sampling of the IMF. As the diffuse background luminosity will cause the stellar mass to be overestimated, we conclude that the embedded stellar mass estimates for the quiescent clouds (``Brick'', ``b'', ``c'', ``d'', ``e'' and ``f'') should be considered as upper limits. On the other hand, the main uncertainty on the bolometric luminosity, hence the embedded stellar mass, which affects the star-forming clouds is pixel saturation, yet it is difficult to quantify the magnitude of the uncertainty this induces. To approximate severity of this, we compare the embedded stellar masses determined from the infrared bolometric luminosity and cm-continuum observations (section\,\ref{lit_masses} and Table\,\ref{masses_table}). We find that the discrepancy in embedded stellar mass determined from these observations for G0.6 and Sgr B1 is small ($20-30$\,per cent). Therefore, given the other uncertainties on the embedded stellar mass discussed in this section, we conservatively estimate that the embedded stellar masses determined for these two sources are reliable to within a factor of $2-3$. As no previous measurements of embedded stellar mass toward G0.3 are available, we suggest that this is also reliable to within a factor of $2-3$. We find that the embedded stellar mass determined from infrared measurements of Sgr B2 is an order of magnitude below the value determined by \citet{schmiedeke_2016}, which we expect is due to the high concentration of UC\ion{H}{II} regions towards the saturated pixels within this source. We suggest that this has caused a significant underestimation of the embedded stellar mass within the Sgr B2, hence this infrared measurement should be considered a lower limit.
\begin{figure*}
\centering
\includegraphics[trim = 0mm 0mm 0mm 0mm, clip,angle=0,width=1\textwidth]{\dir rgb_fig_new.pdf}
\caption{Three colour zoom-in of the ``dust-ridge'', shown in blue and red are the warm and cool component luminosities, and in green is the 70{\rm \,$\mu$m} \ emission. Over-plotted are contours identical to Figure\,\ref{sfr_maps}. Labeled are the sources with their total gas and embedded masses determined from infrared measurements (see Table\,\ref{masses_table} from embedded masses determined via additional methods). The transparent curved arrow represents the path of the orbital model of \citet{kruijssen_2015}, with labels of time since pericentre passage according to this model.}
\label{rgb}
\end{figure*}
\section{Deriving the star formation efficiency and star formation rates}\label{sec:SFE}
Having measured the embedded stellar populations and gas masses for each of the regions we now seek to determine the efficiency with which gas is converted into stars in the Galactic Centre. We are interested in deriving two numbers in particular: (i) the ratio of a cloud's gas mass, M$_{\rm gas}$, to the mass in stars, M$_\mathrm{*, tot}$, which is defined as the star formation efficiency, SFE\,$\equiv$\,M$_\mathrm{*, tot}$/(M$_\mathrm{gas}$ + M$_\mathrm{*, tot}$); (ii) the fraction of the cloud's gas mass which is converted into stars per free-fall time, $\epsilon_\mathrm{ff}\,=\,\mathrm{SFE}\times\,(t_\mathrm{ff}$\,/\,$t_0$), where $t_0$ is the time since the onset of star formation.
Table\,\ref{masses_table} summarises the properties of gas and young stars in several of the most massive and dense clouds and (proto-)clusters within the Galaxy. Figure\,\ref{rgb} shows the cool molecular gas (red) and hot (ionised) gas (blue and green) towards the region containing these sources of interest; $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$. Over-plotted are contours of cool and warm gas column densities in white and grey, respectively (see section\,\ref{method}). Labeled on this plot are the sources and their respective gas and embedded stellar population masses within the radius defined in Table\,\ref{masses_table}.
From the observed M$_{\rm gas}$ and M$_\mathrm{*, tot}$ it is straightforward to derive and compare the SFE for each region. We know that all the regions lie in the same environment, hence we can remove a major source of uncertainty that has hampered previous comparisons of SFEs for sources that may have formed in (potentially very) different environments. As shown in Table\,\ref{masses_table}, we find that the ``dust-ridge" clouds have SFE\,$<4$\,per cent (upper limits), and the star-forming sources have SFE\,$\sim$\,7\,per cent. Taking these numbers at face value it appears that, despite these regions containing some of the most vigorous star formation activity in the Galaxy, only a small fraction of their total gas mass has so far been converted to stars.
\subsection{Deriving the star formation efficiency per free-fall time assuming a model of tidally triggered star formation}
\label{sub:SFE_triggered}
Other than assuming all the sources lie at the same distance and in the same general environment, all of the analysis until this point has implicitly treated all regions as independent. We now attempt to infer the possible relationship between gas clouds and (proto-)clusters by interpreting pertinent observational facts.
Firstly, observations of dense gas molecular line tracers (e.g. HNCO, N$_2$H$^+$) towards the region from Figure\,\ref{rgb}, clearly show the quiescent clouds (red) and (proto-)clusters (blue) are all linked along a coherent velocity structure, or ``stream", in position-position-velocity (PPV) space (e.g. \citealp{kruijssen_2015, henshaw_2016}). The quiescent clouds found towards the north of this region lie at one end of the PPV stream. These clouds show very little sign of star formation, despite their similarly large gas masses and small radii \citep[e.g.][]{lis_1999, immer_2012, longmore_2013a, walker_2015, ginsburg_2015}. Following the stream to higher galactic longitudes and velocities, and lower latitudes lies the mini-starburst complex Sgr B2 (e.g. \citealp{bally_1988, hasegawa_1994, sato_2000}). Continuing on the stream from Sgr B2 to lower longitudes, latitudes and velocities, the {\ion{H}{II}}\ region complexes G0.6, Sgr B1 and G0.3 are found (e.g. \citealp{mehringer_1992, lang_2010}). The first assumption we make is that as the quiescent clouds and (proto-)clusters are all part of the same stream.
Secondly, the quiescent gas clouds all have very similar masses and radii, and are close to virial equilibrium and therefore likely to collapse \citep{walker_2015}. Thirdly, the gas column density probability distribution functions of the quiescent gas clouds are indicative of imminent star formation (\citealp{rathborne_2014}; Battersby in prep.), and the radial distributions of the mass surface densities suggest that collapse of these clouds could produce Arches or Quintuplet-like clusters in the future \citep{walker_2016}. Lastly, there is a general progression of star formation along the stream from quiescent gas clouds to (proto-)clusters.
If the clouds and (proto-)clusters within the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region do represent an evolutionary sequence, one has an estimation of the initial conditions for star formation within the Galactic Centre. With a measure of SFE from section\,\ref{sec:SFE} and $t_{\rm ff}$, the only thing needed to derive $\epsilon_\mathrm{ff}$ is an absolute timescale linking the clouds and (proto-)clusters.
\citet{kruijssen_2015} have developed a dynamical orbital solution to interpret the PPV structure of the molecular emission throughout the central $\sim\,$200\,pc of the Galaxy, from which ``Streams 2 and 3'' have been over-plotted in Figure\,\ref{rgb}. The focal point of this open, elliptical orbit coincides with the position of the supermassive black hole at the centre of the Galaxy, Sgr A$^*$, and all the sources within the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region are downstream past pericentre passage on this orbit.
The observed star formation activity increases with time past pericentre (shown as the increasing spatial extent of the hot gas component in Figure\,\ref{rgb}). In the scenario presented by \citet{longmore_2013a}, \citet{kruijssen_2015}, and \citet{longmore_2016}, gas clouds will experience strong tidal forces close to pericentre passage, which will compress the gas along the vertical direction. This adds turbulent energy into the gas which can be quickly radiated away by shocks due to the high density. In the model, this implies the clouds become self-gravitating and allows them to initiate gravitational collapse, eventually triggering star formation. Further downstream, as star formation continues, feedback from massive embedded stars begins to blow cavities in the surrounding environment, and eventually cause the dissipation of the host cloud (Barnes et al. in prep).
Assuming that star formation within each cloud was triggered at pericentre passage, we can calculate the time-averaged star formation rate as SFR\,({\rm M$_\odot$}\,yr$^{-1}$)\,=\,M$_\mathrm{*, tot}$({\rm M$_\odot$})\,$/t _\mathrm{p,last}$\,(yr), where $t _\mathrm{p,last}$ is the time since the last pericentre passage. In Figure\,\ref{rgb} we label the time since pericentre passage for the orbital model presented by \citet{kruijssen_2015}, where the pericentre of the orbit is located just upstream from the ``Brick''. We find the star formation rates of these clouds are in the range $0.001-0.045$\,{\rm M$_\odot$\,yr$^{-1}$}. The total star formation rate within these clouds sums to $0.03-0.071$\,{\rm M$_\odot$\,yr$^{-1}$}. These clouds, therefore, contribute a quarter to three-quarters of the total star formation within the CMZ ($|l|$$<$1$^{\circ}$, $|b|<$0.5$^{\circ}$; $\sim$\,0.09\,{\rm M$_\odot$\,yr$^{-1}$}). The values for the individual clouds are presented in Table\,\ref{SFR_table}, and are in agreement with those determined through independent methods by \citet[][also see Table\,\ref{SFR_table}]{kauffmann_2016a} and Lu et al. (in prep).
The calculated star formation rates are used to determine the fraction of the cloud's mass which is converted into stars per free-fall time. This can be described as $\epsilon_\mathrm{ff}\,=\,\mathrm{SFE}\times\,(t_\mathrm{ff}$\,/\,$t_\mathrm{p, last}$). We find star formation efficiencies per free-fall time in the range of $1 - 5$\,per cent,\footnote{Note that these values change by less than a factor of two when using typical timescales for star formation in young massive clusters (e.g. the free-fall time; see the review by \citealp{longmore_2014}), rather than the timescales from the \citet{kruijssen_2015} orbital model.} which are again listed in Table 5.
\subsection{Comparison to theoretical models}\label{Comparison to theoretic models}
We now compare these measurements of the star formation rate to the predictions of star formation theories within the literature. As discussed in section\,\ref{models}, we limit our comparison to the column density threshold relation and volumetric models for star formation (see reviews by \citealp{federrath_2012} and \citealp{padoan_2014}). The column density threshold relation predicts the star formation rate solely from the dense gas mass \citep{lada_2010, lada_2012}. On the other hand, the volumetric models predict the dimensionless star formation efficiency per free-fall time, $\epsilon_\mathrm{ff}$, from the physical global properties of the cloud: the sonic Mach number, Alfv{\'e}nic Mach number, virial parameter, and the turbulence driving parameter. \citet{padoan_2014} show that a comparison between the predictions from these relations/models and observations is not trivial. Scatter of more than an order of magnitude exists in the observed star formation rate for a given gas mass, due to the range of environments, evolutionary stages, and spatial scales probed by observations of different clouds.
In section\,\ref{sub:SFE_triggered} we discussed several means by which we hope to overcome several of these limitations: i) directly determining the gas mass and embedded stellar population mass within several sources in the same environment and at the same spatial resolution, ii) using the \citet{kruijssen_2015} orbital model to estimate the absolute time which has passed since star formation within each source was plausibly triggered, and iii) limiting our study to sources within the extreme environment of the Galactic Centre, which may be representative of the environment in which the majority of stars have formed \citep{kruijssen_2013}.
\vspace{0.5cm}
\citet{lada_2010, lada_2012} propose that the gas above a column density of $6.7\,\times\,10^{21}\,$cm$^{-2}$ has a universal depletion timescale of $\sim$\,20\,Myr ($\sim$\,$50$\,$t_\mathrm{ff}$). They propose that the star formation rate can be predicted using Equation\,\ref{lada}. We find that the constraints on the star formation rate within the dust-ridge clouds ``d'', ``e'', and ``f'' are too poor to test the \citet{lada_2010, lada_2012} predictions with any significance. \citet{kauffmann_2016a} have inferred an upper limit of the star formation rate within the ``Brick'' of $<$\,0.0008\,{\rm M$_\odot$\,yr$^{-1}$}, from the non-detection of the star formation tracers (e.g. {\ion{H}{II}}\ regions and masers). This is an order of magnitude below the upper limit found here, and is most likely more representative of the true star formation rate within this source. The star formation rate predicted by the column density limit relation for the ``Brick'' is $\sim$\,0.006{\rm M$_\odot$\,yr$^{-1}$}, which is significantly higher that measured by \citet{kauffmann_2016a}. The predictions for the star-forming sources Sgr B2, G0.6, Sgr B1 and G0.3 show better agreement with the observed star formation rates.
Predictions from volumetric star formation models within the Galactic Centre have been previously determined, however their comparison to observations have been limited. \citet{rathborne_2014} and \citet{rathborne_2015} used The Atacama Large Millimeter/submillimeter Array to measure 3\,mm continuum dust emission at high-spatial resolution towards the ``Brick''. These authors measure a column density probability function dispersion of $\sigma_N$\,=\,0.34, and place lower limits on the critical over-density of collapse of $x_{crit}>100$ (see equation\,\ref{pdf}). Although the star formation rate was not calculated, these results are consistent with the an environmentally dependent {\it absolute} density threshold for star formation, which is orders of magnitude higher than that derived for clouds within the disc of the Milky Way. Recently, \citet{federrath_2016} have conducted a further analysis of these continuum data, with the addition of molecular line observations from \citet{rathborne_2015} and dust polarisation observations from \citet{pillai_2015}. These observations were used to determine the dimensionless parameters required for the volumetric star formation models: the three-dimensional turbulent Mach number, the virial parameter including both turbulence and shear, the turbulent magnetic field parameter, and the turbulence driving parameter (see section\,\ref{models}). These authors used the multi-free-fall model of \citet{krumholz_2005}, with the fiducial parameters determined by \citet{federrath_2012}, to predict a star formation efficiency per free-fall time within the brick of $\epsilon_\mathrm{ff}$\,=\,4\,$\pm$\,3\,per cent. This is consistent with our measurement of $\epsilon_\mathrm{ff}\,\sim$\,2\,per cent towards the ``Brick'' (see Table\,\ref{masses_table}). We note, however, that the observed $\epsilon_\mathrm{ff}$ within the ``Brick'' should be considered an upper limit. Taking the embedded stellar population inferred by \citet{kauffmann_2016a} would give $\epsilon_\mathrm{ff}\,\sim$\,1\,per cent, which is only just within the lower uncertainty range of the value predicted by \citet{federrath_2016}. Here we attempt to expand on this comparison by testing all the volumetric star formation models presented by \citet{federrath_2012} against the observed star formation within a number of Galactic centre clouds.
To compare the measured star formation rates to the volumetric models, we first need to determine the initial conditions for star formation for each source. As discussed in section\,\ref{sub:SFE_triggered} we assume that the ``dust-ridge'' clouds should represent the early evolutionary stages of the star-forming sources. The ``Brick'' is the most recent of the ``dust-ridge'' clouds to pass pericentre, hence its properties should best represent the initial conditions for star formation within this region. This cloud has the benefit of being the most well studied of the dust-ridge clouds (i.e. its properties are the most well constrained), and is known to contain little-to-no active star formation (e.g. \citealp{kauffmann_2013, johnston_2014, rathborne_2014, kauffmann_2016a}).
However, we note that the volumetric models are limited to predicting the $\epsilon_\mathrm{ff}$ over the next free-fall time. In the orbital model of \citet{kruijssen_2015}, the ``Brick'' and Sgr B2 are separated by a time along the orbit that is similar to the free-fall time of the ``Brick'' (see Tables\,\ref{masses_table} and \ref{SFR_table}), and thus the comparison for these regions should be robust. Given that there are several free-fall times between the ``Brick'' and G0.3, using the initial conditions present within the ``Brick'' to predict $\epsilon_\mathrm{ff}$ in the more evolved sources (G0.6, Sgr B1 and G0.3) may not be ideal. Nevertheless, given the relatively small range in gas properties of the ``dust-ridge'' clouds and of the progenitor condensations upstream from pericentre \citep{henshaw_2016e}, the conditions within the ``Brick'' should provide at least approximate predictions for these more evolved sources, which can be compared to the observations.
The star formation rates determined for the quiescent clouds are not considered for the comparison to the volumetric models, as their embedded masses measured from infrared observations (and hence $\epsilon_\mathrm{ff}$) are considered as upper limits (see section\,\ref{Uncertainties}). We, therefore, limit our analysis to the star-forming sources (Sgr B2, G0.6, Sgr B1, and G0.3), for which we are confident in the measurement of the embedded stellar mass.
\defcitealias{krumholz_2005}{KM05}
\defcitealias{padoan_2011}{PN11}
\defcitealias{hennebelle_2013}{HC13}
We use the volumetric models given in the form presented by the \citet{federrath_2012} review (see their Table\,1; also see \citealp{padoan_2014}). This includes the derivation of the log-normal volume density probability distribution function (equation\,\ref{sigma}), and the critical density for collapse and solution to the $\epsilon_\mathrm{ff}$ for each model (equation\,\ref{pdf}). We refer to each model based on its original reference -- \citet[][KM05]{krumholz_2005}, \citet[][PN11]{padoan_2011}, and \citet[][HC13]{hennebelle_2013} -- and whether the derivation is from the single-free-fall or multi-free-fall form (see section\,\ref{models}). Figure\,\ref{sfr_ff_vs_beta} shows how the predicted star formation efficiency per free-fall time varies as a function of the three-dimensional turbulent sonic Mach number, $\mathcal{M}$, the (turbulence+shear) virial parameter, $\alpha$, and the parameter which describes the strength of the turbulent magnetic field, $\beta$ (the purely hydrodynamical scenario is retrieved by setting $\beta\rightarrow\infty$). Here we use a value of $b$\,=\,0.33 which represents a purely solenoidal turbulent driving mode, as \citet{krumholz_2015} suggest that shear is the typical driving mode of turbulence within the dust-ridge clouds, and within clouds at the centres of other galaxies. The vertical dash lines shows the result of varying either $\mathcal{M} = 11 \pm 3 $, $\alpha = 4.3 \pm 2.3$, and $\beta = 0.11-0.61$,\footnote{The probability distribution function of $\beta$ within the ``Brick'' is asymmetric, with a mean value and standard deviation of $\beta = 0.34\,(0.35)$ \citep{federrath_2016}. Here we use the 16$^{\rm th}$ and 84$^{\rm th}$ percentiles of this distribution ($\beta = 0.11-0.61$; C.~Federrath, private communication).} which are representative of the conditions derived for the ``Brick'' \citep{federrath_2016}. Each model line (shown in colour) is plotted as a function of one of the above three parameters, where the shaded region around each line indicates the uncertainty on the predicted $\epsilon_{\rm ff}$ when varying the two remaining parameters over the ranges specified above. The horizontal dotted lines show the calculated ranges of $\epsilon_\mathrm{ff}$, within star-forming regions Sgr B2, G0.6, Sgr B1, and G0.3.\footnote{As $\epsilon_\mathrm{ff}$ measured from infrared observations for Sgr B2 is a lower limit, henceforth we use only the more accurate $\epsilon_\mathrm{ff}$ determined using the VLA embedded stellar mass \citep{schmiedeke_2016}. \label{footnote1} } Models passing through the shaded region enclosed by the dotted and dashed lines correctly predict the star formation rate per free-fall time in the star-forming clouds, when assuming that their initial conditions were similar to the current properties of the ``Brick''. We find that both single-free-fall and multi-free-fall models of \citetalias{padoan_2011} and \citetalias{hennebelle_2013} enter the shaded region on Figure\,\ref{sfr_ff_vs_beta}.
When testing the volumetric star formation theories, we need to choose the value for several of their free parameters ($\epsilon_{core}$, $\phi_t$, $\phi_x$, $\theta$, and $y_\mathrm{cut}$). In Figure\,\ref{sfr_ff_vs_beta} we adopted the fiducial parameters from the original papers, which have been summarised in Table\,\ref{fiducial values}. \citet{federrath_2012} have constrained these parameters in a different way, by fitting them to magneto-hydrodynamical turbulent box simulations, resulting in the substantially different values, which are also shown in Table\,\ref{fiducial values}.\footnote{We note that the \citet{federrath_2012} determine values of $y_\mathrm{cut}\,>\,1$, however a physical interpretation of $y_\mathrm{cut}$ can only be made for $0\,>\,y_\mathrm{cut}\,>\,1$ (\citetalias{hennebelle_2013}).}
The result of using the \citet{federrath_2012} parameters is shown in Figure\,\ref{sfe_ff_models-vol-FK12values-b033}, where we find {\it all} of the volumetric models appear to overpredict the star formation rate by factors of several. Recently, \citet{federrath_2016} used the \citet{federrath_2012} fiducial values and a turbulent driving parameter of $b\,=\,0.22$\footnote{The idealised case presented by \citet{federrath_2008}, $b\,=\,0.33$ represents purely isotropic solenoidal turbulence driving. However, lower values of $b$ may be possible when the driving is anisotropic, whereby a particular vortex direction is continuously driven (e.g. by anisotropic shear).}
in the \citetalias{krumholz_2005} multi-free-fall model to predict the $\epsilon_\mathrm{ff}$ within the ``Brick''. Figure\,\ref{sfe_ff_models-vol-FK12values-b022} shows the result of using these values for all the volumetric star formation theories. Here we find that both the single-free-fall and multi-free-fall models of \citetalias{krumholz_2005} and \citetalias{padoan_2011} enter the shaded region, and are therefore consistent with the observed star formation rate.
The analysis in Appendix\,\ref{Appendix C}, shows that varying the turbulent driving parameter in the range determined for the ``Brick'' ($b = 0.10 - 0.34$; \citealp{federrath_2016}), and adopting either the fiducial values from the original papers or those determined by \citet{federrath_2012} gives a range of predicted star formation rate which spans several orders of magnitude (see Figure\,\ref{sfe_ff_models-vol-b}). In view of the above discussion, we point out that the verification or falsification of these star formation theories is fundamentally obstructed by the lack of consensus on the values of these free parameters.
\begin{figure*}
\centering
\includegraphics[trim = 0mm 7mm 0mm 3mm, clip,angle=0,width=0.86\textwidth]{\dir sfe_ff_models-vol-b033.pdf}
\vspace{-2.5mm}
\caption{Plots of the star formation efficiency per free-fall time, $\epsilon_\mathrm{ff}$, as predicted from the single-free-fall (left panels) and multi-free-fall (right panels) models, as a function of the Mach number (upper panels), the virial parameter (middle panels), and the magnetic field strength (lower panels). The purely hydrodynamical scenario is retrieved by setting $\beta\rightarrow\infty$. These are calculated using the fiducial values of $\phi_x\approx1.12$ \citep[KM05]{krumholz_2005}, $\theta\approx0.35$ \citep[PN11]{padoan_2011}, and $y_\mathrm{cut}\approx0.1$ \citep[HC13]{hennebelle_2013}. The coloured lines represent the model predictions. The shaded coloured regions represent the upper and lower limits within the range of our adopted initial conditions (as shown in the legend of each plot). The vertical dashed lines show the result of varying the variable on the x-axis by the range assumed to represent the properties present within the ``Brick'', i.e. the initial conditions for star formation within this region: $\beta = 0.11-0.61$, $\mathcal{M} = 11 \pm 3 $, $\alpha = 4.3 \pm 2.3$ \citep{federrath_2016} and $b = 0.33$ (see text). Here we adopt values of $\epsilon_{core}\approx0.5$ and $\phi_{t}\approx1.91$ for each model \citep{federrath_2012}. The horizontal dotted regions represent the range of $\epsilon_\mathrm{ff}$ for the star-forming sources within the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region (determined from both infrared and VLA embedded stellar mass estimates, see Table\,\ref{SFR_table})\textsuperscript{\ref{footnote1}}, accounting for the approximate factor of two uncertainty in $\epsilon_\mathrm{ff}$ (i.e. $\epsilon_\mathrm{ff}$ = 0.5 -- 8 per cent; see section\,\ref{Uncertainties}).}
\label{sfr_ff_vs_beta}
\end{figure*}
\section{DISCUSSION AND CONCLUSIONS}\label{conclusions}
In this work, we have utilised infrared {\it Spitzer} and {\it Herschel} observations, with the aim of investigating the lack of star formation within the extreme environment of the Galactic Centre (see \citealp{longmore_2013}). To do this we have determined the star formation rate for the Galactic Centre as a whole (defined as $|l|$$<$1$^{\circ}$ and $|b|<$0.5$^{\circ}$), by using a variety of extra-galactic luminosity-star formation-rate conversions (which trace star formation within the last $\sim$\,5\,Myr). From the 24{\rm \,$\mu$m}, 70{\rm \,$\mu$m}, and the total infrared bolometric luminosity (determined from two component modified Planck function fits between $5.8 - 500$\,{\rm \,$\mu$m}), we find average global star formation rates of $\sim$\,$0.09\pm0.02\,${\rm M$_\odot$\,yr$^{-1}$}. These are comparable to previous measurements made from YSO counting and the free-free emission, which are sensitive to star formation over the last $\sim\,0.1 - 3$\,Myr. The conclusions that can be drawn from this analysis, are discussed below.
\begin{itemize}
\item[i)] We can rule out that systematic uncertainties in the star formation rate measurements are causing the apparent low star formation rate, unless this uncertainty affects all the methods in the same way, which seems unlikely.
\item[ii)] The luminosity-SFR relations and YSO counting methods are consistent in the Galactic Centre. As the Galactic Centre is the most extreme environment for which is it possible to resolve individual forming stars and make this measurement, the results here provide confidence that the luminosity-SFR relations reliably trace the star formation rate over kpc scales within similar environments present in starburst galaxies, and high-redshift galaxies.
\item[iii)] The global star formation rate has not changed by more than a factor of two to three from its current rate over the past few Myr. This is quantitatively consistent with models predicting that the star formation rate is episodic \citep{kruijssen_2014a}, with a timescale of $\sim$\,$10-20$\,Myr \citep{krumholz_2015, krumholz_2016}.
\end{itemize}
In an attempt to better understand the origin of the presently low star formation within the Galactic Centre, we have investigated the properties of several individual clouds and {\ion{H}{II}}\ region complexes. These are thought to be at differing evolutionary stages and connected along a coherent gas stream within Galactic Centre \citep[e.g.][]{henshaw_2016}. In order to interpret the observations two assumptions are made about how the gas clouds are related, and how star formation proceeds in this environment.
We make the assumption that the sources are orbiting along a coherent gas stream with known orbital parameters \citep{kruijssen_2015}, and that star formation within these sources is triggered at the pericentre of the orbit (i.e. when compressive tidal forces are strongest; see \citealp{longmore_2013a, kruijssen_2015}). The direct consequence of this scenario is that the regions reside on a common evolutionary timeline, which allows us to derive their star formation timescales and efficiencies. We estimate that the dense molecular clouds remain relatively quiescent for $0.3-0.5$\,Myr after star formation is triggered, as these contain a stringent upper limit of few hundred solar masses of embedded stars (see Table\,\ref{SFR_table}, ``Brick'' to cloud ``f'', i.e. the ``dust-ridge'' clouds). These then transition to very actively star-forming clouds, which contain a few thousand solar masses of stars, within only $0.2-0.4$\,Myr (as seen towards Sgr B2). The feedback from these stars provides sufficient pressure to remove the remaining dense gas over a timescale of $\sim$\,0.9\,Myr (Barnes et al. in prep), which reveals the later stages of star formation (e.g. diffuse {\ion{H}{II}}\ regions). We note that this division in phases represents the broad brush strokes according to which star formation proceeds in this region, and that the detailed physical picture will be considerably more complex. Nevertheless, this serves as a general model which can be refined in the future.
We take this simple star formation model for the CMZ and determine star formation rates for the sources in the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region. We find that on average the quiescent ``dust-ridge'' clouds have stringent upper limits of $<$\,0.007\,{\rm M$_\odot$\,yr$^{-1}$}, whereas the ``star-forming clouds'', Sgr B2, G0.6, Sgr B1 and G0.3, have star formation rates in the range $\sim$\,$0.002-0.045$\,{\rm M$_\odot$\,yr$^{-1}$}. We find that $\sim$\,$1 - 5$\,per cent of a clouds gas mass is converted to stellar mass per free-fall time.
We use this Galactic Centre gas cloud data to quantitatively test the predictions of different star formation models/relations. We find that the \citet{lada_2010, lada_2012} column density limit relations significantly under predict the observed star formation rate in the quiescent clouds. The predictions for the star-forming sources (Sgr B2 etc), are in better agreement with the observed values.
As a first comparison to the volumetric models, we take the predictions presented by \citet{federrath_2016}. These authors use the multi-free-fall model of \citetalias{krumholz_2005} with the fiducial values of \citet{federrath_2012} to predict $\epsilon_\mathrm{ff}$\,=\,4\,$\pm$\,3\,per cent within the ``Brick'', which is consistent with the observed value of $\epsilon_\mathrm{ff}\,\sim$\,2\,per cent.
Expanding on this, we compare our observed star formation rates to all the volumetric relations. Figure\,\ref{sfr_ff_vs_beta} shows that the \citetalias{krumholz_2005} model does not accurately predict the star formation rate for any set of initial conditions, when using the fiducial values from the original models.
The middle row of Figure\,\ref{sfr_ff_vs_beta} shows that the \citetalias{hennebelle_2013} model is much more sensitive to variations in ${\mathcal M}$ and $\beta$ than the other models, which is a key signature of the \citetalias{hennebelle_2013} theory (and highlights the role of (shock)-turbulence in star-formation). The observational uncertainty in these properties for the ``Brick'', therefore produce a large range of predicted $\epsilon_\mathrm{ff}$ values for a fixed $\alpha$. This makes verifying/falsifying the \citetalias{hennebelle_2013} model predictions more difficult than for the other models.
The figures in Appendix\,\ref{Appendix C} show the effect of using the fiducial values of \citet{federrath_2012}, and choosing a value for the turbulent driving parameter of 0.33 and 0.22. The former value represents total solenoidal turbulent driving, whilst the latter value is that determined for the ``Brick'' by \citet{federrath_2016}. The $\epsilon_\mathrm{ff}$ typically changes by an order of magnitude when adopting this range of the turbulent driving parameter. This again highlights the importance for self-consistently determining the values of the model free parameters.
In general, we find better agreement with the multi-free-fall models over the single-free-fall models, which could ultimately reflect their more accurate description of the hierarchical collapse of star-forming regions. In the future, we aim to use the unique laboratory of the Galactic Centre environment to test additional star formation models.
We find that the most promising range of parameter space for verification or falsification of the models is at the high ${\mathcal M}$, high $\beta$, and low $\epsilon_\mathrm{ff}$ end. We are currently working to identify clouds as far into this regime as possible. Combining newly available high resolution and sensitivity sub-mm interferometry data (Longmore et al in prep; Battersby et al submitted) of such clouds with detailed hydrodynamic simulations of gas clouds on the known orbit in the Galactic Centre environment (Kruijssen et al. in prep.), as well as accounting for a more detailed propagation of the observational uncertainties on the dimensionless ratios (taking into account the covariance of uncertainties), we hope to unambiguously distinguish between the competing theories. We point out that the falsification of these star formation theories is currently obstructed by the lack of consensus on the values of their free parameters.
\vspace{5mm}
To summarise, we suggest that the total {\it global (hundred parsec) scale} star formation rate for the Galactic Centre appears to be overpredicted by the star formation models \citep{longmore_2013}, as the majority of the gas is unbound (super-virial), despite it being very dense \citep{kruijssen_2014a}. When investigating {\it local (parsec) scales} within gravitationally bound clouds, we find that several of the models accurately predict the star formation rate. However, a consensus on the free parameters of these models is required before reliable comparison to observations are possible.
\begin{table*}
\centering
\caption{The properties of the sources within the $0.18<l<$0.76$^{\circ}$, $-0.12<b<$0.13$^{\circ}$ region. These do not depend on any relationship between these regions, other than that they all reside at the same distance. The columns show the column density limits used to define the sources, the masses and the radii, bolometric luminosities, maximum embedded object mass, total embedded stellar mass, the free-fall time and the star formation efficiency$^{1}$. Shown in parentheses are the embedded stellar masses, and the resulting cloud properties, determined from VLA, and WMAP observations (see corresponding footnotes for references).}
\begin{tabular}{c c c c c c c c c }
\hline
Source & $N_{H_2}$ boundary & M$_\mathrm{gas}$$^{1}$ & R & L$_\mathrm{bol}$$^{1}$ & M$_\mathrm{*, max}$ & M$_\mathrm{*, tot}$$^{1}$ & $t_\mathrm{ff}$$^{1}$ & SFE$^{1}$ \\
& (cm$^{-2}$)& ($10^4$\,{\rm M$_\odot$})& (pc) & ($10^5$\,L$_\odot$) & ({\rm M$_\odot$}) & ({\rm M$_\odot$}) & (Myr) & (per cent)\\
\hline
``Brick" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,8\,\times10^{22}$ & 11 & 3.1 & 9.1 & 80 & $<$2.1\,$\times10^{3}$ (8.8\,$\times10^{2}$)$^a$ & 0.27 & 2 \\
``b" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,5\,\times10^{22}$ & 1.5 & 1.8 & 2.2 & 40 & $<$9.3\,$\times10^{2}$ & 0.31 & 6 \\
``c" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,10\,\times10^{22}$ & 0.51 & 0.8 & 0.59 & 20 & $<$4.2\,$\times10^{2}$ & 0.16 & 8 \\
``d" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,10\,\times10^{22}$ & 4 & 2.0 & 2.4 & 40 & $<$9.8\,$\times10^{2}$ & 0.23 & 2 \\
``e" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,24\,\times10^{22}$ & 4.8 & 1.5 & 1.8 & 40 & $<$8.1\,$\times10^{2}$ & 0.13 & 2 \\
``f" & {$N_\mathrm{H_2}^{cool}$}\,=\,$\,24\,\times10^{22}$ & 0.9 & 0.7 & 0.36 & 20 & $<$3.2\,$\times10^{2}$ & 0.10 & 3 \\
Sgr B2 &{$N_\mathrm{H_2}^{cool}$}\,=\,$\,75\,\times10^{22}$ & 65 & 2.7 & 23 & 120 & $>$3.6\,$\times10^{3}$ (3.3\,$\times10^{4}$)$^b$ & 0.09 (0.09) & 1 (5) \\
G0.6 & {$N_\mathrm{H_2}^{warm}$}\,=\,$\,2.6\,\times10^{17}$ & 4.6 & 2.8 & 15 & 100 & 2.8\,$\times10^{3}$ (3.3\,$\times10^{3}$)$^c$ & 0.35 (0.35) & 6 (7) \\
Sgr B1 & {$N_\mathrm{H_2}^{warm}$}\,=\,$\,2.6\,\times10^{17}$ & 8.7 & 5.8 & 66 & 180 & 6.0\,$\times10^{3}$ (7.2\,$\times10^{3}$, 8.0\,$\times10^{3}$)$^{c, d}$ & 0.77 (0.77) & 6 (8) \\
G0.3 & {$N_\mathrm{H_2}^{warm}$}\,=\,$\,1.9\,\times10^{17}$ & 9.3 & 6.5 & 69 & 180 & 6.2\,$\times10^{3}$ & 0.86 & 6 \\
\hline
\end{tabular}
{\vspace{0.3cm}}
\begin{minipage}{\textwidth}
\vspace{1mm}
$^a$ \citet{kauffmann_2016a} \\
$^b$ Mass determined from high-resolution VLA observations \citet{schmiedeke_2016}. \\
$^c$ Mass determined from medium-resolution VLA observations \citet{mehringer_1992}. \\
$^d$ Mass determined from WMAP observations \citet{lee_2012}. \\
$^{1}$ These values represent the instantaneous source properties: M$_\mathrm{gas}$, M$_\mathrm{*, tot}$, $t_\mathrm{ff}$, SFE (see section\,\ref{sec:SFE}).
\end{minipage}
\label{masses_table}
\end{table*}
\begin{table*}
\centering
\caption{The source properties based on the assumptions adopted in section\,\ref{sub:SFE_triggered}. The first assumption is that the``dust-ridge'' clouds are representative of the early evolutionary stages of the star-forming clouds, Sgr B2, G0.6, Sgr B1 and G0.3. The second assumption is that the clouds are on an elliptical orbit around the CMZ, which tidally triggers star formation at the point of pericentre passage. Tabulated in the first column are the times since pericentre passage (i.e. triggering of star formation) as defined from the orbital model of \citep{kruijssen_2015}, which have been used to determine the star formation rates and star formation efficiencies per free-fall time. Shown in parentheses are the properties determined when using the embedded stellar masses calculated from VLA and WMAP observations (see Table\,\ref{masses_table}). }
\begin{tabular}{c c c c c c c c c}
\hline
Source & $t _\mathrm{p,last}$ & SFR & $\epsilon_\mathrm{ff}$ \\
& (Myr) & ({\rm M$_\odot$\,yr$^{-1}$}) & (per cent) \\
\hline
``Brick" & 0.3 & 0.007 & 2 & \\
``b" & 0.4 & 0.002 & 5 & \\
``c" & 0.4 & 0.001 & 3 & \\
``d" & 0.5 & 0.002 & 1 & \\
``e" & 0.5 & 0.002 & 0.5 &\\
``f'' & 0.5 & 0.001 & 1 &\\
Sgr B2 & 0.7 & 0.005 (0.045) & 0.1 (1) \\
G0.6 & 1.4 & 0.002 (0.002) & 1 (2) \\
Sgr B1 & 1.6 & 0.004 (0.005) & 3 (4) \\
G0.3 & 1.8 & 0.004 & 3 \\
\hline
\end{tabular}
\label{SFR_table}
\end{table*}
\section*{ACKNOWLEDGEMENTS}
We would like to thank the anonymous referee for their constructive comments. We greatly appreciate discussions with Gilles Chabrier, Paolo Padoan, Patrick Hennebelle and Christoph Federrath, who provided detailed feedback on the comparison to the volumetric models. Furthermore, we thank Christoph Federrath for providing the probability distribution function of turbulence driving parameter from \citet{federrath_2016}. ATB would like to acknowledge the funding provided by Liverpool John Moores University and the Max Planck Institute for Extraterrestrial Physics. JMDK gratefully acknowledges funding in the form of an Emmy Noether Research Group from the Deutsche Forschungsgemeinschaft (DFG), grant number KR4801/1-1. This research has made use of NASA's Astrophysics Data System.
\bibliographystyle{mnras}
| 2024-02-18T23:40:17.185Z | 2017-04-13T02:02:47.000Z | algebraic_stack_train_0000 | 1,935 | 17,816 |
|
proofpile-arXiv_065-9662 | \section{Introduction}
Let k be an algebraically closed field in characteristic $p > 0$, and
let $\varphi : S \longrightarrow B$ be an elliptic surface
with Kodaira dimension $\kappa (S) = 1$ over $k$.
Let $K_{S}$ be a canonical
divisor of $S$. We consider the multicanonical system $\vert mK_S \vert$.
In Katsura-Ueno \cite{KU} and Katsura \cite{K}, we considered
the following question:
\begin{question}
Is there a positive integer $M$ such that if $m\geq M$,
then the multicanonical system $\vert mK_S \vert$ gives a structure
of elliptic surface for any elliptic surface $S$ over $k$ with $\kappa (S) = 1$?
\end{question}
For the complex analytic case, Iitaka showed that $M = 86$
and 86 is best possible (cf. Iitaka \cite{I}). Namely, if
$m$ is smaller than 86, then there exists an elliptic surface
$\varphi : S \longrightarrow B$
with $\kappa (S) = 1$ such that $\vert mK_S \vert$
does not give the structure of elliptic surface.
In the case of algebraic
elliptic surfaces, if the characteristic $p = 0$ or $p \geq 3$,
we showed that $M = 14$ and 14 is best possible
(Katsura-Ueno \cite{KU} and Katsura \cite{K}). If $p =2$, then we showed
that $M = 12$ and 12 is best possible (cf. Katsura \cite{K}).
In this paper, we treat quasi-elliptic surfaces $\varphi : S \longrightarrow B$
and consider the following similar question.
\begin{question}
Is there a positive integer $M$ such that if $m\geq M$,
then the multicanonical system $\vert mK_S \vert$ gives the structure
of quasi-elliptic surface for any quasi-elliptic surface $S$ over $k$ with $\kappa (S) = 1$?
\end{question}
Note that
quasi-elliptic surfaces exist only in characterisitics $p = 2$ and $3$
(cf. Bombieri-Mumford \cite{BM}). We show that
if $p = 3$, then we have $M = 5$ and the number $5$ is best possible.
In characteristic $ p = 2$, we have still some difficulties
to determine the best possible number.
In Section 2, we summarize basic facts on quasi-elliptic surfaces.
In Section 3, we examine the multicanonical system of
quasi-elliptic surfaces in characteristic 3 and show our main theorem.
\section{Some lemmas for quasi-elliptic surfaces}
Let k be an algebraically closed field in characteristic $p > 0$ and let
$\varphi : S \longrightarrow B$ be a quasi-elliptic surface defined over $k$.
Throughout this paper, we assume that any exceptional curve of the first kind
is not contained in fibers.
Such a surface exists if and only if $p$ is equal to 2 or 3, and
the multiplicities of multiple fibers are all equal to $p$ (cf. Bombieri-Mumford \cite{BM}).
We denote by $g$ the genus of the curve $B$.
In this section,
we recall some facts on quasi-elliptic surfaces.
We denote by $pF_{i}$ $(i = 1, \ldots,\lambda)$ the multiple fibers.
Let ${\mathcal T}$ be the torsion part of $R^{1}\varphi_{*}{\mathcal O}_{S}$.
Then, there exists a Cartier divisor ${\bf f}$ on $B$ such that
$R^{1}\varphi_{*}{\mathcal O}_{S}/{\mathcal T} \cong {\mathcal O}_{B}({\bf f})$.
The canonical divisor formula of $S$ is given by
$$
K_{S} \sim \varphi^{*}(K_{B} - {\bf f}) + \sum_{i= 1}^{\lambda}a_{i}F_{i},
$$
where $- {\text{deg}} {\bf f} = \chi({{\mathcal O}_{S}}) + t$ with
$t =$ the length of
the torsion part of $R^{1}\varphi_{*}{\mathcal O}_{S}$ and $0 \leq a_{i} \leq p - 1$.
Here, $\sim$ means linear equivalence.
If $pF_i$ is a tame multiple fiber, then we have $a_i = p -1$.
For details, see Bombieri-Mumford~\cite{BM}.
\begin{lemma}\label{lm:alb}
The Albanese variety ${\rm Alb}(S)$ of $S$ is isomorphic to the Jacobian variety ${\rm J}(B)$
of $B$.
\end{lemma}
\proof{Let $\psi : S \longrightarrow {\rm Alb}(S)$ be the Albanese mapping.
If ${\rm Alb}(S)$ is a point, then by the universality of Albanese variety
we see that the Jacobian variety ${\rm J}(B)$ of $B$ is also a point.
Now, assume ${\rm Alb}(S)$ is not a point.
Since the general fiber of $\varphi$ is a rational curve with one cusp,
the fibers are contracted by $\psi$. Therefore, $\psi (S)$ is a curve.
We have, by the universality of Albanese variety, a commutative diagram:
$$
\begin{array}{rccc}
\varphi : & \quad S & \longrightarrow & B\\
& \psi \downarrow & & \downarrow \\
& {\rm Alb}(S) & \longrightarrow & {\rm J}(B) \\
& \quad \cup & & \cup \\
& \quad \psi (S) & & B.
\end{array}
$$
By this diagram, we have a morphism $\psi (S) \longrightarrow B$.
Therefore, by the Stein factorization theorem, we see that
$\psi (S)$ is isomorphic to $B$. Therefore, by the universality of Jacobian variety,
we conclude ${\rm Alb}(S) \cong {\rm J}(B)$ (see also Katsura-Ueno \cite{KU}, Lemma 3.4).}
We find the following lemma and corollary
in Lang \cite{L} and Raynaud \cite{R}.
We give here an easy proof for the lemma.
\begin{lemma}\label{lm:chi}
Let $\varphi : S \longrightarrow B$ be a quasi-elliptic surface over a non-singular complete curve
$B$ with genus $g$. Then, we have the inequality
$\chi ({\mathcal O}_{S}) \geq (1 -g)/3$.
\end{lemma}
\proof{By Noether's formula and the self-intersection number $c_1(S)^2 = 0$ of the first Chern class of $S$, we have
$$
12 \chi ({\mathcal O}_{S}) = c_{1}(S)^{2} + c_{2}(S) = 2 - 4q(S) + b_{2}(S).
$$
By Lemma~\ref{lm:alb}, we have $q(S) = g$. Denoting by $\rho (S)$
the Picard number of $S$, we have also $b_{2}(S) \geq \rho (S) \geq 2$.
Hence, we have $\chi ({\mathcal O}_{S}) \geq (1 -g)/3$.}
\begin{corollary}\label{cor:chi}
{\rm (i)} If $g = 1$, then $\chi ({\mathcal O}_{S}) \geq 0$.
{\rm (ii)} If $g = 0$, then $\chi ({\mathcal O}_{S}) \geq 1$.
\end{corollary}
\section{Multicanonical systems}
In this section, let $k$ be an algebraically closed field of
characteristic 3.
Let $\varphi: S \longrightarrow B$ be a quasi-elliptic surface defined over $k$.
\begin{example}\label{ex:3}
In characteristic 3, we consider the quasi-elliptic surface $\varphi : S \longrightarrow {\bf P}^1$
which is given by
a non-singular complete model of the surface defined by
$$
t^{2}(t-1)^{2}z + t^{2}(t -1)^{2}+x^{3} +tz^{3} = 0
$$
Here, $t$ is a parameter of the base curve ${\bf P}^{1}$.
By Lang \cite{L} p.485, this surface has
two tame multiple fibers at $t = 0, 1$,
and we have $\chi({\mathcal O}_{S}) = 1$. We denote the two tame multiple
fibers by $3F_0$ and $3F_1$. The canonical divisor $K_S$ is given by
$$
\begin{array}{rl}
K_S & \sim \varphi^* (K_{{\bf P}^{1}} - {\bf f}) + 2F_0 + 2F_1\\
& \sim - F + 2F_0 + 2F_1.
\end{array}
$$
Here, ${\bf f}$ is a Cartier divisor on ${\bf P}^1$ with $- {\text{deg}} {\bf f} = \chi({\mathcal O}_S) = 1$
such that
${\mathcal O}_{{\bf P}^1}({\bf f}) \cong R^{1}\varphi_{*}{\mathcal O}_{S}$, and $F$ is a general fiber of
$\varphi : S \longrightarrow {\bf P}^1$. Since we have $F \sim 3F_0 \sim 3F_1$,
we see $4K_S \sim 2F_0 + 2F_1$. Therefore, we have $\dim {\rm H}^0(S, {\mathcal O}_{S}(4K_S)) = 1$, and $\vert 4 K_{S} \vert$ does not give
the structure of quasi-elliptic surface. If $m \geq 5$, then we have
$\dim {\rm H}^0(S, {\mathcal O}_{S}(mK_S)) \geq 2$, and $\vert m K_{S} \vert$ gives
the structure of quasi-elliptic surface.
\end{example}
We have the following theorem.
\begin{theorem}
Assume that the characteristic $p = 3$. Then, for any quasi-elliptic surface
$f : S \to B$ with $\kappa (S) = 1$ over $k$ and
for any $ m \geq 5$, the multicanonical system $\vert mK_S \vert$
gives the unique structure of quasi-elliptic surface, and the number 5 is best possible.
\end{theorem}
\proof{The method of the proof is similar to the one in Iitaka~\cite{I},
Katsura-Ueno \cite{KU} and Katsura~\cite{K}. Since the Kodaira dimension
is equal to 1, the structure of quasi-elliptic surface is unique.
The Kodaira dimension of $S$ is equal to 1 if and only if
$$
(*) \quad 2g -2 + \chi({\mathcal O}_{S}) + t + \sum_{i=1}^{\lambda}(a_{i}/3) > 0.
$$
Therefore, we need to find the least integer $m$ such that
$$
(**)\quad m(2g -2 + \chi({\mathcal O}_{S}) + t) + \sum_{i = 1}^{\lambda}[ma_{i}/3]
\geq 2g + 1
$$
holds under the condition $(*)$. Here, $[r]$ means the integral part of
a real number $r$.
We have the following 6 cases:
Case (I) $g \geq 2$
Case (II-1) $g = 1, \chi({\mathcal O}_{S}) + t \geq 1$
Case (II-2) $g = 1, \chi({\mathcal O}_{S}) = 0, t = 0$
Case (III-1) $g = 0, \chi({\mathcal O}_{S}) + t \geq 3$
Case (III-2) $g = 0, \chi({\mathcal O}_{S}) + t = 2$
Case (III-3) $g = 0, \chi({\mathcal O}_{S})= 1, t = 0$
\noindent
We check $(**)$ under the condition $(*)$ for each case.
In Case (I), by Lemma~\ref{lm:chi}, we have $2g - 2 + \chi({\mathcal O}_{S})\geq 5(g - 1)/3$.
Hence, if $m \geq 3$, $(**)$ holds.
In Case (II-1), if $m \geq 3$, $(**)$ holds.
In Case (II-2),
all multiple fibers are tame in this case, and we have at least one multiple fiber by $(*)$.
Since $a_i = 2$, $(**)$ holds for $m \geq 5$.
In Case (III-1), $(**)$ holds for $m \geq 1$.
In Case (III-2),
since $\chi ({\mathcal O}_{S}) \geq 1$ by Corollary \ref{cor:chi}, we have $t \leq 1$.
Therefore, the number of wild fibers is less than or equal to $1$.
If there exists at least one tame multiple fiber, then $(**)$ holds for $m \geq 2$.
If there exist no tame fibers and only one wild fiber, then by $(*)$
we have $a_1 \geq 1$. Therefore,
$(**)$ holds for
$m \geq 3$.
In Case (III-3), all multiple fibers are tame, and we have $\lambda \geq 2$ by $(*)$.
Therefore, $(**)$ holds for $m \geq 5$.
The result on the best possible number in characteristic 3 follows
from Example \ref{ex:3}.
}
In characteristic 2, we can also consider a similar question to the one in characteristic 3.
We have still difficulties to decide the best possible number. For example we need to solve
the following question.
\begin{question}
Does there exist a quasi-elliptic surface over an elliptic curve
with only one tame multiple fiber and with $\chi({\mathcal O}_{S}) = 0$ in characteristic 2?
\end{question}
If there don't exist such quasi-elliptic surfaces, then we can show that
in characteristic 2, $(**)$ holds for $m \geq 4$
and that the best possible number is equal to $4$. Namely, we have $M = 4$ in characteristic 2.
\frenchspacing
| 2024-02-18T23:40:17.781Z | 2017-04-13T02:02:22.000Z | algebraic_stack_train_0000 | 1,968 | 1,801 |
|
proofpile-arXiv_065-9707 | \section{Introduction and Preliminaries}
\label{intro}
To understand the current accelerating expansion history of the universe, mainly two different approaches are
considered. The first one is to introduce some dark energy fluid in the framework of general relativity, while the later
introduces some modifications into the gravitational sector. Now, considering the dark energy concept, observations
favor the existence of $\Lambda$CDM cosmology where $\Lambda$ is the cosmological constant and it is considered to be responsible for
this accelerated expansion. However, this cosmology has one serious issue known as cosmological constant problem
which inspired several authors to consider some time varying dark energy fluid in the name of quintessence, phantom,
Chaplygin gas and so on, although it should me mentioned that they introduced some other issues, for instance the
cosmic coincidence problem and so on. Probably, both the cosmological constant as well as the time varying dark
energy models inspired cosmologists to introduce the modifications into the gravitational sector which finally appeared
with a large number of gravitational modifications, such as $f (R)$, $f (T )$, etc. However, the basic problem in all such
modified gravity theories is to first fix a functional form of $f (R)$ and $f (T )$ and then we study their viability with
the theoretical bounds and finally with the observational data\footnote{Exactly what we do while dealing with several dark energy fluids.}. That means essentially in these two formalisms, we
start with some phenomenological grounds. Thus, no one can exclude the possibility of a new dark energy fluid or some
new modified gravity theory, or even some new direction of research which may positively account of the current
astrophysical and cosmological issues.
In this article we have discussed the evolution of the universe in the context of Finslerian geometry which is a
generalization of the traditional Riemannian geometry \cite{chern, bao}\footnote{We remark that this geometry was in fact suggested by Riemann himself in his book [20]} . The basic concept in such theories is to consider the
violation of Lorentz symmetry which is one of the common features in the quantum gravitational regime and it may
introduce some new directions in the modern cosmological research. The metric in this space is defined by a norm
$F (x, y)$ (where $y \in T_x M$ is the tangent vector on a spacetime point $x$) on a tangent bundle of the spacetime instead
of an usual inner product structure on the corresponding spacetime, for a detailed description we refer \cite{PCS, Vacaru}. It
has been found that during last couple of years a considerable attention has been paid on this extended geometry
to address some issues related to astrophysics and cosmology \cite{fn1, fn2, fn3, fn4, fn5, fn6, Li2014, XWC, FR1, FR2}, specifically, it has been argued that the dark
matter and the dark energy problem can be addressed in such a context \cite{fn2,fn3,fn6}. So, essentially, the cosmology in
this space-time will be worth exploring for new physical results.
A Finslerian structure on a smooth $4-$dimensional manifold $M$ is defined on the tangent bundle $TM$ of $M$ by Finsler metric $F=\tilde {T}M = TM - \{0\}$ and $F$ is smooth on $\tilde TM$. In addition $F$ is positively homogeneous of degree one with respect to $(y^a)$, i.e. $F(x,ky) = kF(x,y)$, $k > 0$. The Finslerian metric tensor $g_{ij}$ is given by
\begin{equation}
g_{ij} = \dfrac{1}{2}\dfrac{\partial^2F^2(x,y)}{\partial y^i \partial y^j}
\end{equation}
defined on $\tilde TM$.
The causality in Finsler spacetime can be given by the metric function $F$. A positive, zero or negative value of $F$ corresponds to timelike, null or space-like curves. In many cases it is useful that a Lorentz signature to be taken under consideration.
A non-linear connection $N$ on $TM$ is a distribution on $TM$, supplementary to the vertical distribution $V$ on $TM$ as
\begin{equation}
T_{(x,y)}(TM) = N_{(x,y)}\oplus V_{(x,y)}.
\end{equation}
A non-linear connection can be defined as
\begin{equation}
N^a_j = \dfrac{\partial G^a}{\partial y^j},
\end{equation}
where $G^a$ is given by
\begin{equation}
G^a = \frac{1}{4} g^{aj}\left(\frac{\partial^2F^2}{\partial y^j \partial y^k}y^k - \partial_jF^2 \right).
\end{equation}
The geodesic equation for the Finsler space follows from the Euler-Lagrange equations
\begin{equation}
\frac{d}{ds}\left( \frac{\partial F}{\partial y^a}\right) - \frac{\partial F}{\partial x^a} = 0,
\end{equation}
and it has the form
\begin{equation}
\frac{dy^a}{ds} + 2G^a(x,y) = 0.
\end{equation}
A complete description of Finslerian structure of spacetime manifold $M$ is usually given with a metric function $F$, a nonlinear connection $N$ and compatible (metrical) connections in the framework of a tangent bundle $TM$ of spacetime \cite{Vacaru}. However in some cases we can restrict our consideration in order to describe the local anisotropic ansantz of gravitational field equations on a base $4-$dimensional manifold \cite{fn1}. In such a case we are able to compare different scenarios of Riemannian-type cosmological models as FLRW with Finslerian ones, e.g \cite{fn7}.
Thus, being motivated by the new generalized spacetime governed by a Finslerian metric, in the present work we
have tried to address the current evolution of the universe. Considering the background matter distribution to be
anisotropic we wrote the gravitational equations in such spacetime. Then we introduce an interesting equation of state which for the isotropic limit simply assumes the barotropic equation of state. We show that under such equation of state, the field equations can be analytically solved which depending on the barotropic index of the fluid exhibits
several cosmological issues. We found that depending on the barotropic index of the fluid the cosmological solution
can predict a finite time past singularity and currently it can exhibit an accelerating scenario of our universe. On the
other hand, the model can offer a singularity free universe in the past while it can not escape from a finite time future singularity. Moreover, we also show that the violation of the energy conditions can lead to the cosmic bounce in such geometric structure. Finally, we establish that the first law of thermodynamics can hold under a simple energy conservation relation. Overall, the current picture offers some interesting possibilities.
We have organized the paper as follows: In section \ref{field-equations-Finsler}, we have presented the modified field equations in Finsler space-time. Section \ref{the_model} deals with a toy model of an equation of state connecting the anisotropic matter distribution and evaluates the cosmological solutions. In \ref{energy}, we briefly describe the energy conservation relation and the first law of thermodynamics. Finally, we close our work in section \ref{discussion} with a brief summary.
\section{Gravitational equations in Finsler space-time}
\label{field-equations-Finsler}
To explain the dynamics of the universe, one desires to introduce a metric specifying
the geometry of the space-time which connects with its matter distribution by the Einstein's gravitational equation.
In order to realize the dynamics, we consider that our universe is described
by the Finsler metric which is of the form \cite{Li2014}
\begin{equation}\label{metric}
F^2=y^ty^t-R^2(t)y^ry^r-r^2R^2(t)\bar{F}^2(\theta,\phi,y^\theta, y^\phi),
\end{equation}
which has been taken inspired by the known fact that at large scales our universe is
well described by the flat Friedmann-Lema\^{i}tre-Robertson-Walker (FLRW) line element.
We construct a cosmological model by inserting $\bar{F}^2$ as a quadric in $ y^\theta$ and $y^\phi $.
Note that the present Finsler space (\,for the case $\bar{F}^2$ as quadric in $ y^\theta$ and $y^\phi $\,)
can be obtained from a Riemannian manifold $( M, g_{\mu \nu}(x))$ as we have
\[F(x,y) =\sqrt{g_{\mu \nu}(x)y^\mu y^\nu }.\]
One can notice that, this is a semi-definite Finsler space-time and consequently, one can use the covariant
derivative of the Riemannian space. It is to be noted that the Bianchi identities
overlap with those of the Riemannian space (being the
covariant conservation of Einstein tensor). Since the present Finsler space
reduces to the Riemannian space, therefore, the gravitational
field equations can be obtained readily.
The base manifold of the Finsler space regulates the gravitational field
equation in Finsler space and the fiber coordinates $y^i$ play the role of
the velocities of the cosmic components i.e. velocities in the energy
momentum tensor. Hence, one can derive
the gravitational field equations in Finsler space from the Einstein gravitational field equation in
the Riemannian space-time with the metric (\ref{metric}) in which the metric $\bar{g}_{ij}$ is given by (see Appendix A)
\begin{align}
\bar{g}_{ij} = \mbox{diag}(1,\sin^2 \sqrt{\lambda}\,\, \theta),\nonumber
\end{align}
i.e.
\begin{align}
g_{\mu\nu }= \mbox{diag}(1,-R^2(t),-r^2R^2(t),-r^2R^2(t) \sin^2 \sqrt{\lambda}\,\, \theta),\nonumber
\end{align}
where $\lambda > 0$ and it plays a very crucial role in the derived field equations in Finsler spacetime and hence in the background cosmology.
The two dimensional Finsler space $\bar{F}$ is specified as a
constant flag curvature space, that is it is assume that
$\bar{R}\mbox{ic}=\lambda$. This Ricci scalar ``$\bar{R}$ic'' is that of the
Finsler structure $\bar{F}$ and two dimensional Finslerian
structure is specified by the constancy of the flag curvature.
This flag curvature of Finsler space is in fact, the generalization of
the sectional curvature of Riemannian space. It will be apparent latter
that the solution of vacuum field equation must lead to the constancy
of the flag curvature with its value $\lambda = 1$. But for more general
case of Finsler structure $\bar{F}$, it can be specified by the constant
flag curvature having any real value for $\lambda $. In fact, for
$\lambda = 1$, we can get the usual Friedmann equations of the FLRW universe.
Let us now assume the general energy-momentum tensor for the matter sector as
\begin{equation}\label{em-tensor}
T^\mu_\nu =(\rho +p_t)u^\mu u_\nu -p_tg^\mu_\nu+(p_r-p_t)\eta^\mu \eta_\nu,
\end{equation}
where $u^\mu u_\mu = -\eta^\mu \eta_\mu = 1$, $p_r$, $p_t$ are respectively
denote the pressures of the anisotropic fluid in the radial and transversal directions.
The modified gravitational field equations in Finsler space-time are obtained as (see Appendix B)
\begin{eqnarray}
8\pi_F G\rho &=& \frac{3\dot{R}^2}{R^2}+\frac{\lambda-1}{r^2R^2},\label{F1}\\
8\pi_F G p_r &=& -\frac{2\ddot{R}}{R}-\frac{\dot{R}^2}{R^2}-\frac{\lambda -1}{r^2R^2},\label{F2}\\
8\pi_F G p_t &=& -\frac{2\ddot{R}}{R}-\frac{\dot{R}^2}{R^2}.\label{F3}
\end{eqnarray}
Note that, $\lambda = 1$ implies $p_r=p_t$, that means, it helps to recover the gravitational
field equations for flat FLRW universe. On the
other hand, if we put $p_r= p_t$ in the above field equations, we readily find $\lambda= 1$. Hence,
we find that, $\lambda= 1 \Longleftrightarrow$ flat FLRW universe. Moreover, one can easily see
that when $t\longrightarrow \infty$, we again find that both the anisotropic pressure
components become equal and the usual Friedmann equations in the spatially flat FLRW
universe in presence of a perfect fluid with energy-momentum
tensor $T_{\mu \nu}= (p+\rho) u_{\mu} u_{\nu}+ p g_{\mu \nu}$ are recovered.
Now, introducing the Hubble parameter $H= \dot{R}/ R$,
the field equation (\ref{F1}) can be written as
(for simplicity, from now we work in the units where $8 \pi_F G= 1$)
\begin{eqnarray}\label{F1a}
\rho &=& 3 \left(H^2+ \frac{\lambda-1}{3\, r^2\, R^2}\right)\Longleftrightarrow~ \Omega+ \Omega_k = 1,
\end{eqnarray}
where $\Omega= \rho/3H^2$ is the density parameter representing the matter sector, and in
compared to the Friedmann universe, the quantity $\Omega_k= -\frac{(\lambda-1)}{3 (r\,R\,H)^2}$
can be looked as the density parameter for the scalar curvature in the Finslerian geometry.
Also, for both the directions, the Raychaudhuri equation can be written as
\begin{eqnarray}
\dot{H} &=& -\frac{1}{2} (\rho+ p_r),\label{F2a}\\
\dot{H} &=& -\frac{1}{2} (\rho+ p_t)+ \frac{\lambda-1}{2\, r^2\,R^2},\label{F3a}
\end{eqnarray}
where again we note that, for $\lambda= 1$,
it reduces to only one equation.
\section{Evolution and Dynamics: A toy model}
\label{the_model}
In general, the equation of state of this anisotropic fluid takes a general form $f (p_r,~p_t,~\rho)= 0$. The exact form of the equation of state is not known and hence, it is a challenge for modern cosmology to derive the cosmological evolution correctly. Still, we adopt mainly two possible ways. One is to assume a very simple formulation of the equation of state in order to derive the evolutionary parameters so that we can match them with the observational data, and on the other hand, the reconstruction of any quantity from observed data is of worth exploring. However, in the present work, we adopt the first possibility, and thus, we start with the following equation of state
\begin{equation}\label{eos}
\gamma (p_t- p_r)+p_r =(\gamma-1) \rho
\end{equation}
where $\gamma \in \mathbb{R}$ is simply a constant. The essence of this equation of state is that
for $p_t= p_r= p$ (say), the equation of state in (\ref{eos})
is simply reduced to $p= (\gamma - 1) \rho$, representing the barotropic equation of state. Further, we notice that, for $\gamma= 0$, equation (\ref{eos}) implies $p_r= -\rho$, and hence $p_t= - 3 \dot{R}^2/R^2$. We are interested in the cosmological solutions for the above choice of the equation of state. Now, using the field equations (\ref{F1}), (\ref{F2}), (\ref{F3}), we can exactly solve the scale factor $R$ as
\begin{align}
R &= R_0 \left[1+ \frac{3\,\gamma}{2} H_0 (t- t_0)\right]^{\frac{2}{3\, \gamma}},~(\gamma \neq 0)\label{scale-factor}\\
R &= R_0 \exp\left(H_0 (t-t_0)\right),~~~~(\gamma = 0)\label{sf2}
\end{align}
where $t_0$, $H_0$ are respectively the present cosmic time and present day value of the Hubble parameter, and it is worth noting that the solutions obtained in Eqns (\ref{scale-factor}), (\ref{sf2}) exactly match with the solutions obtained for the isotropic matter distribution in the FLRW geometry with the equation of state $p= (\gamma -1 ) \rho$. Consequently, the Hubble parameter can be derived as
\begin{eqnarray}
H &=& \frac{H_0}{1+ \frac{3\, \gamma}{2} H_0 (t- t_0)},~~~~(\gamma \neq 0)\label{Hubble}\\
H &=& H_0= \mbox{Constant},~~~~~(\gamma = 0)\label{Hubble2}
\end{eqnarray}
Therefore, it is clear that for $\gamma= 0$ we realize an exponential expansion of
the universe. On the other hand, we can divide $\gamma \neq 0$ into the following two conditions
when $\gamma > 0$ and $\gamma < 0$.
\subsection{The case for $\gamma > 0$}
In this case, the solutions for the scale factor and the Hubble parameter take the forms as
in equations (\ref{scale-factor}) and (\ref{Hubble}). The solutions offer the following
scenario of our universe.
\begin{eqnarray}
\label{BB-singularity}
\mbox{At}~t_f = t_0- \frac{2}{3\,H_0\, \gamma},~~~~~~~~~~~\nonumber\\
~~\lim_{t \rightarrow t_f} R (t)= 0,~~\mbox{and}~~\lim_{t \rightarrow t_f} H (t)= \infty\, ,
\end{eqnarray}
which clearly shows that the universe attains a big bang singularity in the past ($t_f < t_0$).
On the other hand, at late time we find that
\begin{eqnarray}
\label{future-evolution}
\lim_{t \rightarrow \infty} R (t)&=& \infty,~~~~\mbox{and}~~~~\lim_{t \rightarrow \infty} H (t)= 0.
\end{eqnarray}
Now, introducing the deceleration parameter $q= -1-\dot{H}/H^2$,
we find that, for this cosmological solution one has $q= -1 + 3\gamma/2$, which
represents an accelerating universe (i.e. $q< 0$) for $\gamma < 2/3$. Thus,
the model with the equation of state in (\ref{eos}) presents a model of
our universe which predicts the big-bang singularity (a finite time
singularity, but independent of $\gamma$), and describes an accelerating
universe for $\gamma < 2/3$.
\subsection{The case for $\gamma < 0$}
Now, we consider the cosmological solutions for $\gamma< 0$. For a clear image let us consider
$\gamma= -\alpha$ (where $\alpha> 0$). We rewrite the scale factor and the Hubble parameter as
\begin{eqnarray}
R&=& \frac{R_0}{\Bigl[1- \frac{3\,\alpha}{2} H_0 (t-t_0)\Bigr]^{\frac{2}{3\alpha}}},\label{SF-emergent}\\
H &=& \frac{H_0}{1- \frac{3\, \alpha}{2} H_0 (t- t_0)}.
\end{eqnarray}
We find that the scale factor can not be zero at any finite cosmic time in the past evolution of the universe, in other words it gives a solution to the nonsingular universe that has been proposed in several contexts in modern cosmology with great interests, see Refs. \cite{BV1988, EM2004, Ellis2004, Mulryne2005, Nunes2005,Lidsey2006,Banerjee2007,Banerjee2008}, but on the other hand, from the solution of the scale factor in eqn. (\ref{SF-emergent}), it is seen that it diverges
at some finite cosmic time in future, that means for $t_s= t_0+ \frac{2}{3\alpha H_0}$, $R(t) \longrightarrow \infty$.
Thus, we find that this cosmological solution gives a realization of a nonsingular universe in the past which consequently predicts a ``big rip'' singularity.
Now, we find that we arrive at two different cosmological scenarios,
one which starts with big bang but at late time it does not have any singularity, the other
has a nonsingular nature in past but at future it has a big rip singularity. Therefore, in the
next section we constrain the model parameters for the first cosmological solution (i.e. for $\gamma > 0$).
Let us look at a particular case when
both the pressure components satisfy barotropic equations of state, that means, $p_r= \omega_r \rho$, $p_t= \omega_t \rho$, and where we assume that $\omega_r$ and $\omega_t$ are the constants. Now, it is easy to see that under such conditions, one can exactly solve the energy density using the above two equations, where the explicit form for $\rho$ is
\begin{equation}
\rho= \rho_0 \left( \frac{r^{2(\omega_t-\omega_r)}}{R^{(3+\omega_r+ 2\omega_t)}}\right)
\end{equation}
where $\rho_0 > 0$ is an integration constant and it recovers the standard evolution equations in Friedmann cosmology for $\omega_r = \omega_t$.
Additionally, the energy conditions in this spacetime can be written in the following way:\\
\textbf{WEC:} $\rho \geq 0$, and $\rho+ p_r \geq 0$, $\rho+ p_t \geq 0$\\
In the Finsler space-time, using the gravitational equations (\ref{F1}), (\ref{F2}), (\ref{F3}), the conditions respectively reduced to the following inequalities as \\
$$3\left(\frac{\dot{R}^2}{R^2}\right)+ \frac{\lambda-1}{r^2\,R^2} \geq 0,\,\,~~\frac{\ddot{R}}{R} \leq \frac{\dot{R}^2}{R^2},~~\mbox{and}~~\frac{\ddot{R}}{R} \leq \frac{\dot{R}^2}{R^2}+ \frac{\lambda-1}{2\,r^2\,R^2}$$
\textbf{SEC:} $\rho+ p_r \geq 0$, $\rho+ p_r+ 2 p_t \geq 0$\\
In this case, the conditions are reduced to $$\frac{\ddot{R}}{R} \leq \frac{\dot{R}^2}{R^2},~~~~\mbox{and}~~~~\frac{\ddot{R}}{R} \leq 0$$
\textbf{NEC:} $\rho+ p_r \geq 0$, and $\rho+ p_t \geq 0$\\
Here, using the gravitational equations, these conditions respectively reduced to
$$\frac{\ddot{R}}{R} \leq \frac{\dot{R}^2}{R^2},\,\,~~\mbox{and}~~\frac{\ddot{R}}{R} \leq \frac{\dot{R}^2}{R^2}+ \frac{\lambda-1}{2\,r^2\,R^2}$$
\textbf{DEC:} $\rho \geq 0$, and $-\rho \leq p_r \leq \rho$, $-\rho \leq p_t \leq \rho$\\
In a similar way, the inequalities respectively reduced to
$$3\left(\frac{\dot{R}^2}{R^2}\right)+ \frac{\lambda-1}{r^2\,R^2} \geq 0,\,\,$$
\begin{align}
-2\left(\frac{\dot{R}^2}{R^2}\right)- \frac{\lambda-1}{r^2\,R^2}\,\leq \, \frac{\ddot{R}}{R}\, \leq \, \frac{\dot{R}^2}{R^2},\,\,~~~~~~~~~\nonumber\\~~\mbox{and}~~-2\left(\frac{\dot{R}^2}{R^2}\right)- \frac{\lambda-1}{r^2\,R^2} \, \leq \, \frac{\ddot{R}}{R} \, \leq \, \frac{\dot{R}^2}{R^2}+ \frac{\lambda-1}{2\,r^2\,R^2}\nonumber
\end{align}
Clearly the term $\left(\frac{\lambda-1}{r^2 R^2}\right)$ makes a significant contribution to the energy conditions, where we note that for $\lambda= 1$, the above conditions are simply reduced to the energy conditions as we find for a flat FLRW universe with a perfect fluid distribution.
\subsection{Bouncing conditions}
In this section we will seek for bouncing conditions in this spacetime. Since the expansion scalar in this spacetime is defined to be $H = \dot{R}/R$, therefore one may recall the bounce condition \cite{CDS}
\begin{eqnarray}
H (t_b) = 0,\,\,\,\,\mbox{and}\,\,\,\,\dot{H}(t_b) > 0 \label{bounce_conditions}
\end{eqnarray}
where $t_b$ is the time where universe has bounced. Actually, the above conditions can also be written as $\dot{R} (t_b) = 0$ and $\ddot{R} (t_b)> 0$. From the above energy conditions, following the bounce conditions, an immediate solution for the bouncing universe is that $\lambda > 1$.
By using bounce conditions \eqref{bounce_conditions} in (9), (10), (11) we get for the energy conditions
\begin{itemize}
\item WEC : $\rho+p(r)<0, \rho+p(t)<0, \lambda>1 $, so WEC is violated.
\item SEC : $\rho+p(r)<0, \rho+p(r)+2p(t)<0$, so SEC is violated.
\item NEC and DEC are also violated.
\end{itemize}
These cases are necessary conditions in order to have a cosmic bounce.
\section{Energy Conservation relations}
\label{energy}
In this section we shall devote our discussions on the energy conservation relation. Let us
propose the energy conservation equation as follows
\begin{equation}\label{conservation}
d(\rho V) = -P dV -V F_a dr.
\end{equation}
The additional term in R.H.S. is due to the anisotropic force $F_a$ given by
\begin{equation}
F_a = \frac{2(p_t-p_r)}{r} =\frac{2(\lambda -1)}{8 \pi_FG}\frac{1}{r^3R^2},
\end{equation}
and the pressure $P$ is the average pressure which is given by
\begin{equation}\label{pressure}
P=\frac{p_r+p_t+p_t}{3}=\frac{p_r}{3}+\frac{2p_t}{3}.
\end{equation}
The proposed conservation equation (\ref{conservation}) can be written as
\[Vd\rho +\rho dV+PdV+VF_adr=0,
\]
or, alternatively as
\[d\rho+\rho \frac{dV}{V}+P\frac{dV}{V}+F_adr=0,
\]
which then turns into
\[\frac{\partial \rho}{\partial t}dt+\frac{\partial \rho}{\partial
r}dr+(\rho+ P)\frac{dV}{V}+F_a dr = 0,\]
which again can be recast as
\begin{align}
dt\left(\frac{\partial \rho}{\partial t}+(\rho
+P)\frac{\dot{V}}{V}\right)+dr\left(\frac{\partial \rho}{\partial
r}+F_a\right)=0,\nonumber
\end{align}
which clearly reports
the following two separate equations:
\begin{eqnarray}
\frac{\partial \rho}{\partial t} + 3H(\rho+P) &= & 0,\label{cons1}\\
\frac{\partial \rho}{\partial r}& = & -F_a.\label{cons2}
\end{eqnarray}
Note that, the first equation is the usual energy conservation equation for the homogeneous and isotropic universe with the effective pressure $P$. The second equation can be derived from the above field equations (\ref{F1}), (\ref{F2}), (\ref{F3}). That means, the proposed conservation relation (\ref{conservation}) is consistent and well motivated with the energy conservation relation in the context of general relativity.
Also, if we define a pressure $P$ as the weighted average of $p_t$ and $p_r$, i.e. if $P= \gamma p_t-(\gamma- 1) p_r$ (note that the sum of the weight is $\gamma- (\gamma-1)=1$) then we see that the present equation of state is the barotropic equation of state $P=(\gamma- 1)\rho$. In equation (\ref{pressure}) we have taken $P$ as the simple arithmetic average of the radial and transverse pressure, i.e. the pressure in three orthogonal directions.
This case corresponds $\gamma =\frac{2}{3} $.
With this, the conservation equation becomes
\begin{equation}\label{cons-sp}
d(\rho V)+PdV+\left(\gamma-\frac{2}{3}\right)(p_r-p_t)dV+\tilde{F}dr=0
\end{equation}
where, $\tilde{F}=VF_a $ is an anisotropic force. In fact, the equation (\ref{cons-sp}) can be written as \begin{equation}
dU+dW=dQ=0,
\end{equation}
where $U=\rho V$ is the internal energy, $W$ is the work done and $Q$ is the heat transfer and using these terminology, eqn. (\ref{cons-sp}) is nothing but the following equation
\begin{equation}
dW= PdV +\left(\gamma- \frac{2}{3}\right)(p_r-p_t)dV+\tilde{F}dr.
\end{equation}
The first two terms in the right hand side being the change of work from pressures and the third being that due to the anisotropic force. This equation represents the first law of thermodynamics for the case of adiabatic heat transfer.
\section{Concluding remarks}
\label{discussion}
In this work we considered the spacetime of our universe described by the Finsler geometry, a generalization of the Riemannian geometry \cite{chern, bao}. A number of studies \cite{fn1, fn2, fn3, fn4, fn5, fn6, Li2014, XWC, FR1, FR2} in this framework have been performed in order to offer some explanations on some recent astrophysical and cosmological evidences. Now, considering a general matter distribution which by construction stands for an anisotropic matter distribution, we rewrote the modified gravitational field equations. We solved the dynamics of the universe for a simple equation of state of the matter sector (see eq. (\ref{eos})) characterized by a sole parameter $\gamma$ and which is motivated from the fact that in case of a perfect fluid matter distribution the barotropic equation of state, is recovered. Since $\gamma$ has been kept free for our analysis, so depending on it, we found three distinct cosmological scenarios with current interests:
\begin{itemize}
\item For $\gamma = 0$, we realize an exponential expansion of the universe. But, we do not have any such other information for such specific value of $\gamma$.
\item $\gamma > 0$ is of special importance in this study since the model perfectly indicates big bang singularity in the past and finally there is an accelerating phase for $\gamma < 2/3$. So, this model can trace back the early phase of the universe as well as stands for the late time accelerated phase.
\item The case $\gamma < 0$ is also interesting because the model in this case does not encounter any finite time singularity in the past. That means, an existence of nonsingular universe is supported as observed in other cosmological theories \cite{BV1988, EM2004, Ellis2004, Mulryne2005, Nunes2005,Lidsey2006,Banerjee2007,Banerjee2008}. Additionally, the late accelerating universe is also realized, but finally at some finite future time the scale factor diverges (Big rip singularity). Therefore, the model starts from a nonsigular phase, exhibits an accelerating universe and finally hints for the divergence in the scale factor at finite time in future.
\end{itemize}
Further, we show that the violation of the energy conditions can lead to the cosmic bounce. We also further established the energy conservation relation which show that the model is in well agreement with the first law of thermodynamics.
Thus, in summary considering the background spacetime of our universe as Finsler geometry instead of Riemannian geometry it is observed that a simple barotropic like fluid can offer some interesting cosmological solutions which accommodate the past and present scenarios of the universe evolution. Thus, as a beginning, the current study may be considered as a viable one for the next complicated models in Finsler cosmology.
\section*{Acknowledgments}
FR would like to thank the authorities of the Inter-University Centre
for Astronomy and Astrophysics, Pune, India for providing research facilities. FR is also grateful to DST-SERB and DST-PURSE, Govt. of India for financial support.
We wish to thank Panayiotis C. Stavrinos and Supriya Pan for helpful discussion.
| 2024-02-18T23:40:17.975Z | 2018-02-15T02:04:05.000Z | algebraic_stack_train_0000 | 1,977 | 4,632 |
|
proofpile-arXiv_065-9796 | \section{Introduction}
The field of radio astronomy is growing rapidly. Since the year 2000, the number of known radio sources has grown from a few hundred thousand to over two million. This is expected to increase further to 70 million with the development of the ASKAP-EMU~\cite{johnston2008science} telescope, and beyond with the development of the SKA~\cite{bourke2015advancing}. Furthermore, global 21cm experiments such as EDGES ~\cite{bowman2018absorption}, REACH~\cite{de2022reach} and SARAS~\cite{singh2017first} aim to probe into yet unseen redshift ranges to gain knowledge on the Cosmic Dawn, Dark Ages and Re-ionisation Era.
A major constraint on the design of all such telescopes is radio frequency interference (RFI), which is typically orders of magnitude brighter in amplitude than the sky signal and cannot be modelled as Gaussian noise. RFI is typically anthropogenic, narrow band and comes in various species that can be either constant in time or transient~\cite{ellingson2006rfi}. Sources of RFI include FM Radio and Digital TV signals.
Building antennae in remote locations on earth can alleviate but never completely avoid such nuisance signals. Furthermore, RFI always increases the amplitude of the detected signal and is often transient in nature. Therefore, simply averaging the sky over long time spans (for example in global experiments) is not sufficient to prevent contaminated data leading to systematic errors.
Currently, RFI is flagged and then excised prior to and separately from the data analysis stage of of the radio experiment. In older telescopes this may have been done manually by the astronomer. However with modern telescopes gathering terabytes of data per second, an automated approach is required. Various approaches have been taken in the past including wavelet based methods~\cite{oslick1998general}, \textsc{cumsum}~\cite{baan2004radio}, and the \textsc{aoflagger}~\cite{offringa2010aoflagger} package used for LOFAR, which implements Singular Value Decomposition~\cite{offringa2010post}. Another approach, used by the HERA project, is to flag the data using a watershed segmentation algorithm~\cite{kerrigan2019optimizing}. More recently, Deep Learning models using convolutional neural networks trained on manually flagged and/or and simulated data~\cite{akeret2017radio};~\cite{vafaei2020deep};~\cite{sun2022robust} have been shown to be able to effectively flag varying species of RFI. ~\cite{2022arXiv221105088K} takes a partially Bayesian approach where contaminated frequency channels are flagged and then excised using other methods, after which they are mitigated for using Gaussian constrained realisations and Gibbs sampling.
In this paper we propose a first-of-its-kind RFI mitigation algorithm that takes a Bayesian Approach, where RFI is both flagged and managed at the likelihood level. The paper is structured as follows: we introduce the basic theory behind Bayesian Inference in~\cref{sec:bayesianinftheory}; then we introduce our Bayesian RFI Correction method in~\cref{sec:rficorrtheory}; followed by an analysis of the model when tested on a simple toy scenario in~\cref{sec:toyex}; our methodology is then tested in a real use case in the REACH pipeline in~\cref{sec:reachex}; finally we discuss our conclusions in \cref{sec:conclusions}. The techniques described in this paper can be incorporated into a Bayesian data analysis pipeline with just a few lines of code.
\section{Theory}
\subsection{Bayesian Inference}\label{sec:bayesianinftheory}
Bayesian methods can be used to perform parameter estimates and model comparison. A model $\mathcal{M}$ uses data $\mathcal{D}$ to infer its free parameters $\theta$. Using Bayes Theorem,
\begin{align}
P(\mathcal{D}|\theta) \times P(\theta) &= P(\theta|\mathcal{D}) \times P(\mathcal{D}), \\
\mathcal{L} \times \pi &= \mathcal{P} \times \mathcal{Z},
\end{align}
the prior $\pi$ is updated onto the posterior $\mathcal{P}$ in light of the likelihood $\mathcal{L}$ and furthermore the Bayesian Evidence $\mathcal{Z}$ can be inferred by computing the integral
\begin{equation}
\mathcal{Z} = \int \mathcal{L}(\theta) \times \pi(\theta) \; d\theta.
\end{equation}
In practice, $\mathcal{P}$ and $\mathcal{Z}$ can be determined simultaneously using a Bayesian numerical solver. We use the Nested Sampling algorithm \textsc{polychord}~\cite{handley2015polychord}; where a series of points generated within $\pi$ are updated such that they sequentially contract around the peak(s) of the likelihood, forming the posterior which can be used to generate parameter estimates. The artefacts of this process can then be used to compute $\mathcal{Z}$, which is used for model comparison. For a more detailed description of Bayesian Inference and Nested Sampling see \cite{sivia2006data}.
\subsection{RFI correction likelihood}\label{sec:rficorrtheory}
A data point contaminated by RFI can be considered corrupted. Any information relevant to the model is lost and furthermore it cannot be modelled as Gaussian noise. Assuming $\mathcal{D}$ is uncorrelated, the likelihood
\begin{equation}
\begin{aligned}
\mathcal{L} &= P(\mathcal{D}|\theta) = \prod_{i} \mathcal{L}_{i}(\theta),\label{l1} = \prod_{i} P(\mathcal{D}_i|\theta)
\end{aligned}
\end{equation}
where $i$ represents the $i$'th data point, is insufficient to model such contaminated data.
It is therefore necessary to model the likelihood that each data point is corrupted.
Thus, we introduce a piecewise likelihood including the possibility of corruption of data
\begin{equation}
P(\mathcal{D}_i|\theta) = \begin{cases}
\mathcal{L}_i(\theta) &: \text{uncorrupted}\\
\Delta^{-1}[ 0<\mathcal{D}_i<\Delta] &: \text{corrupted}.\\
\end{cases}
\end{equation}
Corruption is modelled as the data becoming completely unreliable and therefore being distributed uniformly within some range $\Delta$ (which, as a scale of corruption, has the same dimensions as the data).
An efficient way to write this likelihood is
\begin{equation}
P(\mathcal{D}|\theta, \varepsilon) = \prod_{i} \mathcal{L}_{i}^{\varepsilon_{i}} \Delta^{\varepsilon_i-1}
\label{eq:li2}
\end{equation}
where the Boolean mask vector $\varepsilon$ has a $i$th component which takes the value $1$ if the datum $i$ is uncorrupted and value $0$ if corrupted.
We do not know before the data arrive whether or not they are corrupted by RFI. We may infer this in a Bayesian fashion, by ascribing a Bernoulli prior probability $p_i$ of corruption (which has dimensions of probability) i.e:
\begin{equation}
P(\varepsilon_i) = p_i^{(1-\varepsilon_i)}(1-p_i)^{\varepsilon_i}.\label{eq:pei}
\end{equation}
Both $\Delta$ and $p_i$ are required for a dimensionally consistent analysis. It should be noted that above we assume the a-priori probability that each bin contains RFI is uncorrelated, i.e $P(\varepsilon)=\prod _i P(\varepsilon_i)$, which in practice will almost certainly not be true. We will discuss later the extent to which this assumption can be considered valid.
Multiplying \cref{eq:pei,eq:li2} yields
\begin{equation}
P(\mathcal{D},\varepsilon|\theta) = \prod_{i} \left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon_{i}} \left[p_i/\Delta\right]^{(1-\varepsilon_i)}
\label{eqn:likelihood_eps}
\end{equation}
and to recover a likelihood independent of $\varepsilon$ we formally can marginalise out:
\begin{align}
P(\mathcal{D}|\theta) &=\sum_{\varepsilon \in \{ 0, 1 \} ^N}P(\mathcal{D},\varepsilon|\theta) \\
&= \sum_{\varepsilon \in \{ 0, 1 \} ^N} \prod_{i} \left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon_{i}} \left[p_i/\Delta_i\right]^{(1-\varepsilon_i)}.
\label{eq:pdtheta2}
\end{align}
This would require the computation of the all $2^N$ terms in \cref{eq:pdtheta2}. For realistic values of $N$, this computation becomes impractical. However, if it is assumed that the most likely model (i.e the maximum term in \cref{eq:pdtheta2}) dominates over the next to leading order terms, we can make the approximation
\begin{equation}
P(\mathcal{D},\varepsilon|\theta) \approx \delta_{\varepsilon \varepsilon^\mathrm{max}} \times P(\mathcal{D},\varepsilon^{\mathrm{max}}|\theta)\label{eqn:postierioreps}
\end{equation}
where $\delta_{ij}$ is the usual Kroneker delta function, and
$\varepsilon^\mathrm{max}$ is the mask vector which maximises the likelihood $P(D,\varepsilon|\theta)$, namely:
\begin{equation}
\varepsilon^{\mathrm{max}}_{i}=
\begin{cases}
1, & \mathcal{L}_i(1-p_i) > p_i/\Delta_i \\
0, & \text{otherwise}.
\end{cases}
\label{eqn:emax}
\end{equation}
Under this approximation we find that the sum in \cref{eq:pdtheta2} becomes
\begin{equation}
P(\mathcal{D}|\theta) \approx P(\mathcal{D},\varepsilon^{\mathrm{max}}|\theta).\label{eq:approx}
\end{equation}
In practice the approximation in \cref{eq:approx} is only valid if the next to leading order term is much smaller, such that
\begin{equation}
P(\mathcal{D}|\theta, \varepsilon_{\mathrm{max}}) \gg \mathrm{max}_j P(\mathcal{D}|\theta,\varepsilon^{(j)})\label{eq:nlo},
\end{equation}
where $\varepsilon^{(j)}$ is $\varepsilon^\mathrm{max}$ with its $j$th bit flipped:
\begin{equation}
\varepsilon^{(j)}_k =
\begin{cases}
1-\varepsilon^{\mathrm{max}}_k & k=j \\
\varepsilon^{\mathrm{max}}_k & k\ne j \\
\end{cases}
\end{equation}
and we may use \cref{eq:nlo} as a consistency check.
To summarise, we can correct for RFI under these approximations by replacing the original likelihood $\mathcal{L} = \prod_i\mathcal{L}_i$ in \cref{l1} with
\begin{equation}
P(\mathcal{D}|\theta) = \prod_i\left[\mathcal{L}_{i}(1-p_i)\right]^{\varepsilon^{\mathrm{max}}_{i}} \left[p_i/\Delta\right]^{(1-\varepsilon^\mathrm{max}_i)}
\label{eqn:likelihood}
\end{equation}
where $\varepsilon^{\mathrm{max}}$ is defined by \cref{eqn:emax}.
\subsection{Computing the posterior}
The posterior and evidence are computed from \cref{eqn:likelihood} via Nested Sampling (although any numerical Bayesian sampling method could be used). Taking logs for convenience gives
\begin{equation}
\begin{aligned}
\log{P(\mathcal{D}|\theta)} = \sum_{i} &[\log{\mathcal{L}_i}+\log({1-p_i})]\varepsilon^{\mathrm{max}}\\
&+ [\log{p}_i - \log{\Delta}](1 - \varepsilon^\mathrm{max}_i), \label{eq:loglikelihood}
\end{aligned}
\end{equation}
yielding a masked chi squared like term which can be used to distinguish whether there is a statistically significant difference between the classes of data, i.e corrupted or non corrupted. Furthermore, the second term in \cref{eq:loglikelihood} introduces an Occam penalty. Each time a data point is predicted to be RFI its likelihood is replaced with the penalty rather than being completely removed. Without this term, the likelihood where all data points are flagged would be larger and thus `more likely' than all other possibilities. Therefore, flagging all datum would always be preferable.
We compute this by imposing the condition in \cref{eqn:emax} on \cref{eq:loglikelihood} as follows, \begin{equation}
\log{P(\mathcal{D}|\theta)} =
\begin{cases}
\log \mathcal{L}_i + \log (1-p_i), &
\begin{aligned}
&[\log{\mathcal{L}_i} + \log({1-p_i}) \\
&> \log p_i - \log \Delta]
\end{aligned}\\
\log p_i - \log \Delta, & \text{otherwise}.
\end{cases}
\label{eqn:loglcompute}
\end{equation}
The effect that this correction has on a log-likelihood with a corrupted datapoint is shown in \cref{fig:theorysketch}.
The corrected likelihood is then updated iteratively via the selected Bayesian sampling method, compressing the prior onto the posterior while simultaneously correcting for RFI. One may also notice that the right hand side of \cref{eq:loglikelihood} is a Logit function. Logit functions are used routinely as an activation function in binary classification tasks, which hints at the potential of a future extension of this work using machine learning.
\begin{figure}
\includegraphics[width=\columnwidth]{f_theory_sketch.pdf}
\caption{Visualising the effect of a correction on the likelihood for a simple 5 point data set where the 3rd point is corrupted.}
\label{fig:theorysketch}
\end{figure}
\section{Toy Example}\label{sec:toyex}
We will initially test this approach on a simple toy model, consisting of a straight line with Gaussian noise and RFI injected. Then, we move onto a more realistic and complex case in \cref{sec:reachex}.
\subsection{Dataset}\label{sec:toyexdataset}
Two datasets of 25 data points are generated for comparison. The first is a line with Gaussian noise $\sigma=5$, gradient $m=1$ and intercept $c=1$. It does not contain RFI thus can be used as a ground truth. The second dataset is the same, with the random noise identically seeded so the data is reproducible, but with two RFI spikes injected. This is shown in the top pane of \cref{fig:posterior_frac_toy}.
\subsection{Initial Testing}\label{sec:initialsetup}
We fit the data described in \cref{sec:toyexdataset} in a Bayesian sense and attempt to recover the two free parameters $m$ and $c$ using the correcting likelihood in \cref{eqn:loglcompute} with
\begin{equation}
\mathcal{L}_i = -\frac{\log(2\pi \sigma^{2})}{2} - \frac{[y_{i} - y_{\mathcal{S}}(x_i;m, c)]^2}{2\sigma^2},
\end{equation}
where $\theta = m, c, \sigma$, $y_i$ is the simulated data and $y_\mathcal{S}(x_i; m, c)$ is the output when the estimated parameters from the current sampling iteration are used to compute the model $y_i=mx_i + c$. We set $\Delta = \mathcal{D}_\textsc{max}$, to encapsulate the full range of possible data values. $\Delta$ could likely be fit as a free parameter, as will be investigated further in future works.
Sampling the posterior $P(\theta|\mathcal{D})$ via a Bayesian sampling procedure, the prior distribution is compressed and the belief that each point fits (or is corrupted and does not fit) the model is updated. Thus by evaluating the subsequent posterior on $\varepsilon$, as shown in \cref{fig:posterior_frac_toy}, we can assess how many times across the entire sampling run each data point was believed to fit (non corrupted) or not fit (corrupted) the model. As seen in \cref{fig:posterior_frac_toy}, it is evident that the points containing RFI make up a near zero fraction of the posterior, because as the prior was iteratively updated onto the posterior they were frequently believed to be corrupted. Conversely, points that do not contain RFI often fit the model and as such contribute significantly to the final posterior distribution. There are also some points that lie somewhere in between, which the model is less confident are uncontaminated. When comparing the data in the top pane to the bottom pane in the probability domain, it appears as though the points the model has less confidence in are the ones that deviate the most from the true signal due to Gaussian noise.
\begin{figure}
\includegraphics[width=\columnwidth]{f_test.pdf}
\caption{The top pane shows the dataset described in~\cref{sec:toyexdataset}, with the true signal marked in green and points corrupted by RFI marked in red. The bottom pane shows the mean posterior on $\varepsilon$, $P(\varepsilon_i|\mathcal{D})$, produced during the sampling run.}
\label{fig:posterior_frac_toy}
\end{figure}
It should be emphasised that although $\varepsilon_i$ is constrained to binary values, the subsequent mask on $\varepsilon$ is not. Unlike traditional RFI flagging algorithms, points are not simply flagged as `RFI' and `Not RFI'. The mask takes the weighted mean across the posterior. Thus, points more likely to contain RFI will have less `impact' on the final posterior distribution than points believed to be uncontaminated. In practice, as seen in \cref{fig:posterior_frac_toy}, points predicted to contain RFI will have a near zero contribution on the subsequent posterior. However, it is possible that points which the model is less confident in, such as at $x=4$ and $x=22$ will have a lesser but non-zero contribution to the final distribution. The mask could be thought of as being slightly opaque to these data points, accounting for the models uncertainty. The ability to incorporate the models confidence in its correction directly into the subsequent parameter estimates and model comparison makes this mitigation approach unique in comparison with its counterparts.
\subsection{Model Evaluation}
To further examine the effectiveness of the our methodology it is necessary to develop a simple toy model, similar to the above but also simulating various other scenarios. Both datasets described in \cref{sec:toyexdataset} will be evaluated when fit using the likelihood capable of correcting for RFI. They will also be fit using a traditional likelihood, which cannot account for RFI. This will generate a total of four posterior distributions for comparison.
We initially fit a baseline case to be used as a ground truth, using the clean dataset and a traditional likelihood that does not facilitate the modeling of RFI. The new likelihood which is capable of correcting for RFI is then used to fit the clean dataset, to confirm that it performs correctly even in the absence of RFI. The dataset containing RFI is then fit using the new likelihood and the traditional likelihood, to examine how much better the model performs when the correction is applied. All but the contaminated, uncorrected case would be expected to perform similarly if RFI has been effectively mitigated. It should be noted however that from a Bayesian standpoint, the simplest model will always be preferable. Thus, for the clean dataset it would be expected that the standard likelihood would be preferred slightly over the correcting likelihood.
Fig.~\ref{fig:anesthetic} shows the parameter distributions inferred from the data in \cref{sec:toyexdataset} in the four cases described above and their associated $1\sigma$ and $2\sigma$ confidences. The bottom left pane shows that both $m$ and $c$ are inferred to within $1\sigma$ of their true values. The `No RFI No Correction' case is similar to the `RFI Corrected' case, indicating that the model has effectively corrected the corrupted regions of data. This is particularly evident when comparing these two cases to the uncorrected, contaminated case.
According to Bayes theorem, the simplest model will always be favoured. This is clear when comparing the Bayes Factor computed from the log evidences in~\cref{tab:tab1}. For the uncontaminated data, the model that does not correct for RFI is slightly preferred. Conversely the correcting model is strongly preferred on the contaminated data indicating that the correction is working effectively.
\begin{table}
\begin{center}
\begin{tabularx}{0.4\textwidth} {
| >{\raggedright\arraybackslash}X
| >{\centering\arraybackslash}X
| >{\raggedleft\arraybackslash}X | }
\hline
& No RFI & RFI \\
\hline
No Correction ($\log \mathcal{Z}$) & -83.9 $\pm$ 0.2 & -120 $\pm$ 0.4 \\
\hline
Correction ($\log \mathcal{Z})$ & -85.3 $\pm$ 0.2 & -95 $\pm$ 0.2 \\
\hline
$\log$ Bayes Factor & -1.4 $\pm$ 0.3 & 25.0 $\pm$ 0.4 \\
\hline
\end{tabularx} \label{tab:tabletoymodel}
\caption{Evidences when running the four cases described in~\cref{sec:toyex} and the subsequent Bayes factor.}
\end{center}
\label{tab:tab1}
\end{table}
\begin{figure*}
\includegraphics[width=\textwidth]{f_4pane_samples2.pdf}
\caption{Showing the parameter distributions inferred from the dataset described in Section~\ref{sec:toyexdataset}. The top left to bottom right panes show probability distribution functions for $m$, $c$ and $\sigma$, respectively. Plots generated using posterior plotting tool \textsc{anesthetic}~\protect\cite{anesthetic}.}
\label{fig:anesthetic}
\end{figure*}
\begin{figure*}
\includegraphics[width=\textwidth]{f_4pane_toy_sidebar.pdf}
\caption{Showing the inferred parameter estimates in a contour plot, where darker tones indicate higher $\sigma$ confidence in the parameter estimates. Generated from the dataset described in Section~\ref{sec:toyexdataset}. The plots are generated using the functional posterior plotter \textsc{fgivenx}~\protect\cite{fgivenx}.}
\label{fig:fgx}
\end{figure*}
It is also useful to view the posterior plots of $P(y|x, \mathcal{D})$ plotted from the inferred parameter estimates and the associated confidences. As seen in \cref{fig:fgx}, when RFI is not corrected, the true value is well outside the $1\sigma$ and sometimes $2\sigma$ confidence bounds. Conversely the other three cases fit almost entirely within the $1\sigma$ bounds, indicating that the RFI has been mitigated.
\subsection{Evaluating the $\log p$ dependence}\label{sec:logpdependence}
Proper selection of the probability thresholding term $\log p$ is essential to the efficacy of the mitigation process. Set too high, the model will require such a high confidence of a data point fitting the model that deviations due to Gaussian noise will be predicted as RFI. Set too low, nothing will be corrected. From a Bayesian standpoint $\log p$ would be set to represent our prior degree of belief in there being RFI in each datum.
We firstly assess the question in a qualitative manor, viewing the posterior on $\varepsilon_{\mathrm{max}}$ for a selection of $p $ values. This is computed by evaluating $P(\mathcal{D}|\theta, \varepsilon_{\mathrm{max}})$, avoiding the sum to preserve each data point in $x$ for each posterior sample. We then weighted average across the samples, giving the probability that each data point fits the model. More specifically, the probability that each data point is uncontaminated by RFI.
When $\log p$ is set appropriately as seen in the left pane of \cref{fig:tri_plot}, the model has a high confidence in its predictions. Across the sampling run the probability threshold frequently laid between the likelihood for contaminated and uncontaminated datapoints, whilst not being so high (and therefore sensitive) as to flag any deviations caused by higher order Gaussian noise. At $\log p = -3.0$ the RFI still appears to have been accurately modelled, however the confidence in some of the predictions are lower. As $\log p$ decreases further, the model breaks down. Points are predicted incorrectly in many of the sampling iterations and as such the model is less able to confidently predict their class. Viewing \cref{fig:tri_plot} from left to right, we observe the effects of this thresholding term moving to the peak of the likelihood, sweeping up points the model is less confident in due to Gaussian noise.
\begin{figure}
\includegraphics[width=\columnwidth]{f_tri_plot_3_horiz.pdf}
\caption{Showing the mean mask on $\varepsilon$, ie $P(\varepsilon_i|\mathcal{D})$, where $\log p$ is varied to the point at which the model breaks down.}
\label{fig:tri_plot}
\end{figure}
We assess the $\log p$ dependence while varying $\log p$ as a function of the RMSE on the fit generated from the parameter estimates, the $\log$ Bayesian Evidence and the mean number of points flagged across all samples.
\begin{figure}
\includegraphics[width=\columnwidth]{f_approx_current_sig5_2.pdf}
\caption{Assessing how various methods of model evaluation vary as function of $\log p$. From top to bottom: the RMSE, the $\log$ of the Bayesian Evidence, then the weighted average number of points flagged and finally the radio between $P_{\textsc{max}}$ and $P_{\textsc{NLO}}$. All dependant variables (excluding $\log \mathcal{Z}$) are averaged over the weighted posterior samples. The noise observable in these plots is sampling noise; the noise in the simulated data is seeded.}
\label{fig:4pane}
\end{figure}
For high $\log p$, the RMSE is high and we observe in \cref{fig:4pane} that the model generates less accurate parameter estimates. Here the threshold is so high that the model is more confident that any of the points are RFI than non RFI. This matches the corresponding low evidence. The RMSE drops as $\log p$ decreases to near its minimum. The model incorrectly flags $\approx 5$ data points while the RMSE is low, showing that the model is able to generate accurate parameter estimates whilst over flagging, indicating it is insensitive to false positives. As $\log p$ decreases further, the model is better able to distinguish between higher order Gaussian noise and as such the average number of points predicted to be RFI approaches the true value. As this happens the evidence also reaches its maximum, which indicates that the Bayesian Evidence is appropriately showing how well each of the many models created by different $\log p$ values fit the data.
\subsection{To what extent is $P(\mathcal{D}|\theta) \approx P(\mathcal{D}|\theta, \varepsilon_{\mathrm{max}})$ valid?}
A key assumption is made in \cref{eq:approx} is that the leading order term, \cref{eq:loglikelihood}, is considerably larger than all the other possible terms for $\varepsilon \in (0, 1)^N$. If this approximation is not valid, the mathematical framework behind this approach breaks down.
Therefore it is necessary to test the validity of this approximation by computing \cref{eqn:loglcompute} and comparing the result ($P_{\textsc{max}}$) with the next leading order term ($P_{\textsc{NLO}}$) as calculated by \cref{eq:nlo}. For $-5 < \log p < -0.1$. These results are displayed in the bottom pane of \cref{fig:4pane}. $P_{\textsc{max}}$ is 18 times larger than $P_{\textsc{NLO}}$ at peak $\log \mathcal{Z}$ and increases linearly for $\log p$ below this. Depending on the $\log p$ selection strategy, $P_{\textsc{max}}$ is at least 11 times more likely than the next leading order term. Assuming an appropriate $\log p$ selection strategy, the ratio would be higher, indicating that $P(\mathcal{D}|\theta) \approx P(\mathcal{D}|\theta \varepsilon_{\mathrm{max}})$ is valid.
\subsection{Selection Strategy for $\log p$}
Various selection strategies could be taken to select the optimal $\log p$ value. For each $\log p$, the model changes. As such, selecting the $\log p$ that maximises the evidence seems to be the most obvious selection strategy. In the case of the toy model, the peak $\log \mathcal{Z}$ occurs where $\log p = -2.7$ as shown in \cref{fig:4pane}. Here, $P_{\textsc{max}}$ is 18 times larger than $P_{\textsc{NLO}}$. Another possible strategy could be to select $\log p$ where the number of points flagged is at its minimum.
It is also possible to ascribe a prior to $\log p$, fitting it as a free parameter thus fully automating the approach. This will be examined further in future works.
\vfill
\section{REACH Example}\label{sec:reachex}
\begin{figure*}
\includegraphics[width=\textwidth]{f_4pane_reach_sidebar.pdf}
\caption{Showing the results when the RFI correction is applied on simulated data in the REACH data analysis pipeline.}\label{fig:reach_dual_plot}
\end{figure*}
Finally, we examine a real use case for this method. The REACH~\cite{de2022reach} radio telescope is designed to detect the faint 21cm signal from the Cosmic Dawn. This signal is estimated to be 5 orders of magnitude dimmer than the foreground, therefore a highly precise measurement is required. The REACH data analysis pipeline takes a Bayesian approach to antenna calibration, foreground modelling and chromaticity correction~\cite{anstey2021general}. As such it is a useful environment to test our RFI mitigation framework. The 21cm signal is expected to take the shape of an inverted Gaussian, so the model takes the form
\begin{equation}
f(x) = A \exp{\bigg( -\frac{(x-\mu)^{2}}{2 \sigma^{2}}\bigg)}
\end{equation}
with center frequency $\mu$, standard deviation $\sigma$ and magnitude $A$ all free parameters. The four cases discussed in \cref{sec:toyex} are then examined, but this time on a simulated sky data set containing a 21cm signal with two RFI spikes injected. The evidences and subsequent log Bayes Factors are shown in~\cref{tab:tab2} below.
\begin{table}
\begin{center}
\begin{tabularx}{0.4\textwidth} {
| >{\raggedright\arraybackslash}X
| >{\centering\arraybackslash}X
| >{\raggedleft\arraybackslash}X | }
\hline
& No RFI & RFI \\
\hline
No Correction ($\log \mathcal{Z}$) & 296.0 $\pm$ 0.4 & -99525038.0 $\pm$ 0.5\\
\hline
Correction ($\log \mathcal{Z}$) & 295.4 $\pm$ 0.4 & 251.0 $\pm$ 0.4 \\
\hline
log Bayes Factor & =1.4 $\pm$ 0.6 & 99525289.0 $\pm$ 0.6 \\
\hline
\end{tabularx}
\end{center}
\caption{Evidences when running the four cases described in~\cref{sec:reachex} and the subsequent Bayes factor.}
\label{tab:tab2}
\end{table}
The No RFI Correction and ground truth cases are very similar with the simpler (ground truth) case marginally preferred as expected. The RFI Corrected case is again similar with a slightly lower evidence due to the penalties incurred during the corrections. The above is also evident when viewing \cref{fig:reach_dual_plot}. The reconstructed signal is within $2\sigma$ of the true signal for all but the RFI No Correction case.
\section{Conclusions}\label{sec:conclusions}
In this paper, which serves as a proof-of-concept, we show that RFI can be both flagged and corrected in a fully Bayesian sense at the likelihood level. This approach to RFI mitigation may be a highly effective tool for future radio telescopes, given the growing prevalence of modern data analysis pipelines taking a Bayesian approach and the need for a system that can be incorporated into these increasingly complex data analysis systems. Our methods can effectively perform inference on data hampered by RFI, as shown when tested on a simple toy model and also in a real use case in the pipeline of a global 21cm experiment, all as part of a single step Bayesian fitting process.
In future works, it will be necessary to further examine to approximation made in \cref{eq:approx} to confirm that it is valid in other more complex scenarios. Further analysis is also required to assess the efficacy of this technique when compared to other modern RFI mitigation approaches. As mentioned in \cref{sec:rficorrtheory} we assume that each data point is separable and uncorrelated. It is unlikely that this is true for real data~\cite{wilensky2021improving}. Therefore, the case where data points are dependent on each other should be investigated in future works. Development is required to incorporate this method when applied to time integrated data and furthermore on different species of RFI, such as transient and broadband.
Our methodologies yield a means by which one can leverage their belief that certain parameters fit the data, therefore they will be useful in other fields. For example in the field of transient (Pulsar, Fast Radio Burst, etc) detection, RFI looks very similar in real space to the object of interest but very different in parameter space. One could thus leverage their belief in a Pulsars periodicity (for example) against similarly shaped (in real space) but non periodic RFI, generating a means by which the two phenomena can be distinguished from each other.
\section{Data Availability}
The methods described in this paper can be implemented into a Bayesian data analysis pipeline with just a few lines of code. An example can be found at \href{https://github.com/samleeney/Publications}{github.com/samleeney/publications}. All other data and codes related to these works are available on request.
\bibliographystyle{mnras}
| 2024-02-18T23:40:18.294Z | 2022-11-29T02:26:38.000Z | algebraic_stack_train_0000 | 2,005 | 4,911 |
|
proofpile-arXiv_065-9829 | \section{Introduction}
Spectral imaging technologies have rapidly evolved during the past decades. The light captured by the camera sensor contains information across a wide range of the electromagnetic spectrum, which can potentially be used to identify the texture and the material / molecular composition of any given object of interest. Hyperspectral sensors with up to 150 or even 256 spectral channels within and outside the visible spectral range have been developed in the last few years for various applications within health care \cite{calin2014hyperspectral,Lu2014}, industrial imaging \cite{shafri2012hyperspectral} or agriculture \cite{jung2006hyperspectral,moghadam2017plant}. However, current acquisition methods for such devices (as filter-wheels \cite{Wisotzky2018}, line-scanning \cite{WisotzkyComparision2021}) have decisive disadvantages, trading the acquisition of hyperspectral images (HSI) with high costs or long acquisition times. Alternative approaches are Multi Spectral Filter Arrays (MSFA) based on spectral masking on pixel-level using a single sensor plane similar to the Bayer pattern for single chip RGB cameras \cite{bayer1976color}. This single-camera-one-shot (mosaic snapshot) technique allows multiple spectral bands to be captured simultaneously in a simple and compact system. The captured image data is defined by its moxel (mosaic element) corresponding to the occurring filter pattern and stored in a hypercube representation with three dimensions, two spatial (width and height) and one spectral (wavelength $\lambda$). The simplest example is similar to the Bayer pattern, with one green filter element being replaced by another filter resulting in a $2\times2$ pattern \cite{hershey2008multispectral}. These systems can be extended to $3\times3$, $4\times4$, $5\times5$ or even non-quadratic $2\times3$ mosaic patterns for recording wavelengths in near-ultraviolet, visible and near-infrared spectral range. However, a mosaic pattern is always a compromise between spatial and spectral resolution, as with increasing mosaic size (for higher spectral resolution) the spatial resolution decreases. This spectral-spatial trade-off can be resolved through interpolation or prediction of the missing spectral values, such that the final HSI exhibits higher resolution (spatial and spectral). Techniques for spectral reconstruction include bilinear and nonlinear filtering methods and are referred to as demosaicing.
Typically, demosaicing is achieved by interpolation based on the information from neighboring pixels. Traditional algorithms such as bilinear, bicubic interpolation are popular choices in the field of image processing, where missing pixels are calculated from their neighborhood. Further, interpolation methods based on image fusion have been proposed, where spatial-resolution is enhanced through fusion with Pseudo Panchromatic Images (PPI), statistics based fusion techniques, e.g.~Maximum a posteriori (MAP) estimation, stochastic mixing model \cite{eismann2004application,eismann2005hyperspectral,hardie2004map}, and dictionary-based fusion techniques, e.g.~spectral and spatial dictionary based methods \cite{bendoumi2014hyperspectral,yokoya2011coupled,zhang2014spatial}.
Fusion based methods usually require the availability of a guiding image with higher spatial resolution, which is difficult to obtain in many scenarios.
Demosaicing by interpolation based techniques, both traditional as well as fusion-based, is easy to achieve, however, these methods suffer from color artifacts and lead to lower spatial resolution.
Especially at edges, they do not take into account the spectral correlations between adjacent bands as well as due to crosstalk. This results in spectral distortions in the demosaiced image, especially for increasing mosaic filter size.
Alternatively, deep neural networks can be trained to account for scene information as well as correlations between individual spectral bands.
Demosaicing using convolutional neural networks (CNN) for images with $2\times2$ Bayer pattern was first proposed in \cite{wang2014multilayer} and \cite{gharbi2016deep}. In recent years, CNN based color image super resolution (SR) has gained popularity. Examples of such networks include SRCNN \cite{dong2014learning}, DCSCN \cite{yamanaka2017fast} and EDSR \cite{lim2017enhanced}. Due to their success, these networks have been extended to HSI super resolution \cite{li2018single}. The underlining aspect of all CNN based HSI demosaicing networks is the utilization of spatial and spectral context from the data during training.
However, the need of high quality ground truth data leads to challenges. In such a dataset, each pixel should contain the entire spectral information, which is difficult to acquire in a natural environment.
The contribution of this work is as follows. We present a new ground-truth dataset acquired and generated using a shifting unit to achieve a 1-pixel movement on the camera side in order to obtain a full resolution image for all color channels. Further, we propose a new demosaicing network and compare it to three relevant network architectures, performing demosaicing on a dataset combining captured and publicly available data.
The remainder of this paper is as follows. The next chapter gives an overview on related publications relevant for this work. Chapter 3 describes the proposed network architecture, before chapter 4 explains the acquisition of the ground truth data. Chapter 5 introduces training and evaluation parameters. Chapter \ref{sec:res} describes experiments and results, followed by a thorough discussion and conclusion.
\section{Related Work}
In the last years, several HSI demosaicing algorithms have been presented. Some methods require presence of a dominant-band (as in the Bayer pattern) \cite{monno2012multispectral}, but of more interest are methods without such explicit assumptions designed for mosaic pattern having no redundant band \cite{ogawa2016demosaicking}.
Dijkstra et al.~\cite{dijkstra2019hyperspectral} proposed a similarity maximization network for HSI demosaicing, inspired by single image SR. This network learns to reconstruct a downsampled HSI by upscaling via deconvolutional layers. The network results are presented for a $4\times4$ mosaic pattern and the demosaiced HSI showed high spatial and spectral resolution. Habtegebrial et al.~\cite{habtegebrial2019deep} use residual blocks to learn the mapping between low and high resolution HSI, inspired by the HCNN+ architecture \cite{shi2018hscnn+}.
These two networks use 2D convolutions in order to learn the spectral-spatial correlations. An important characteristic of HSI is the correlation between adjacent spectral bands which are not taken into account when using 2D convolutional based networks. These correlations can be incorporated by using 3D convolutional networks. Mei et al.~\cite{mei2017hyperspectral} proposed one of the first 3D CNNs for hyperspectral image SR addressing both the spatial context between neighboring pixels as well as the spectral correlation between adjacent bands of the image. A more effective way to learn the spatial and spectral correlations is through a mixed 2D/3D convolutional network, as proposed in \cite{li2020mixed}.
One major challenge for the task of snapshot mosaic HSI demosaicing using neural networks is the lack of real world ground truth data. Publicly available real world datasets, such as CAVE \cite{yasuma2010generalized} or HyTexiLa \cite{khan2018hytexila}, were recorded either using a push-broom technique or by spatio-spectral-line scans.
Hence, the data has different characteristics than snapshot mosaic data (e.g., missing cross talk) and can therefore not be used to adequately train a robust network for demosaicing. One alternative is a downsampling strategy from captured snapshot mosaic data as presented in \cite{dijkstra2019hyperspectral}. However, simple downsampling leads to differences in distances of adjacent pixels, which affects the network results
\section{Network Architecture}
We propose a new neural network architecture to generate a full-spectrum hypercube (dimension $[L \times W \times H]$) with $L$ wavelength bands from a captured mosaic image (dimension $[1\times W \times H]$), represented as a low resolution or sparse hypercube. Here, $W$ and $H$ correspond to the spatial width and height of the image and $L$ represents the spectral resolution. In this work, a snapshot mosaic camera with a $4\times4$ pattern, i.e.~$L = 16$ bands, is used as shown in Fig.\ref{fig:filter_pattern}.
\begin{figure}[htp]
\centering
\includegraphics[width=0.2\columnwidth]{filter_pattern.png}
\caption{ $4\times 4$ filter pattern with 16 wavelengths}
\label{fig:filter_pattern}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[Full network\label{fig:network_general}]{\includegraphics[width=0.4\columnwidth]{Network_general.pdf}}\
\subfloat[Resblock\label{fig:network_resnet}]{\includegraphics[width=0.25\columnwidth]{Network_resnet.pdf}}
\caption{(a) Architecture of the hyperspectral demosaicing network. (b) Architecture of the resnet block. The full network includes four consecutive resnet blocks. The resnet block is build up of two convolutional layers holding $16$ convolutional filters with size $3\times 3$ each. These two layers are separated by a ReLU activation function.}
\end{figure}
The building blocks of our demosaicing network (Fig.~\ref{fig:network_general}) are a feature extraction layer along with a feature addition (\textit{FeatureAdd}) block and two deconvolution (\textit{deconv}) layers. The feature extraction layer of the network is split into two parallel parts: (1) a mosaic to cube converter (M2C) implemented by a convolutional layer (\textit{conv}) and (2) feature extraction using four residual blocks (\textit{Resblocks}). These two paths learn the spatial spectral correlations from different representations of the input mosaic image. The input to the M2C is a 2D mosaic image $MI_{2D}$ of dimension $[1, W, H]$, while the input to the \textit{Resblocks} is a 3D cube representation $C_{M2C}$ of dimension $[L, W/\sqrt{L}, H/\sqrt{L}]$ created by resampling the one channel input image as follows:
\begin{equation}
C_{M2C}(x,y,z) = MI_{2D}(u, v),
\label{eq:M2Chandcraft}
\end{equation}
with
\begin{align}
u &= x \cdot \sqrt{L} + z\ \text{mod}\ \sqrt{L},\\
v &= y \cdot \sqrt{L} + z\ \text{div}\ \sqrt{L},
\end{align}
where $u, v$ correspond to the 2D mosaic pixel coordinate, $x, y, z$ correspond to the 3D multispectral voxel coordinate and $\sqrt{L}$ is the dimension of the mosaic pattern; in our case $\sqrt{L} = 4$. The operation $\text{div}$ describes integer division. The M2C composed of a convolutional layer is defined as
\begin{equation}
C_{nn}(x,y,z) = MI_{2D} \otimes_4 \mathcal{G}^{4\times 4}_{16},
\label{eq:M2Cconv}
\end{equation}
where $\otimes$ and $\mathcal{G}$ represent the convolutional operator with stride of $4$, equal to mosaic size, and the set of $16$ filters with a size of $4\times 4$ respectively.
The features of $C_{M2C}$ are extracted by four consecutive residual blocks, where a single block consists of two convolutional layers separated by a rectified linear unit (ReLU) activation function and interconnected through skip connection avoiding the problem of vanishing gradients \cite{hochreiter1998vanishing}. In addition, the ReLU function clips values below zero as negative spectral responses cannot exist. Each convolutional layer uses a filter of dimension $3\times 3$ with stride and padding of $1$, see Fig.~\ref{fig:network_resnet}. Finally, a ReLU activation function is applied to the obtained feature map and the resulting feature map is passed onto the next residual block as input.
The extracted features are of the same size and concatenated in the FeatureAdd block. The combined feature map is passed to two upsampling layers, which upsample by a factor of $4\times 4 = 16$ to produce a fully defined hyperspectral cube of dimension $[L, W, H]$ in a non-linear fashion. The first deconv layer determines the capacity of the network with $128$ filters, while the second layer has the same amount of filters as the number of required spectral bands
\begin{equation}
O(C) = \Phi(\Phi(C_{add} \oslash_2 \mathcal{F}^{8\times 8}_{128}) \oslash_2 \mathcal{F}^{8\times 8}_{16}),
\label{eq:Upsampling}
\end{equation}
where $\oslash$ and $\mathcal{F}$ represent the deconvolutional operators with stride of $2$ and the filter sets with a size of $8\times 8$ respectively. The ReLU activation function $\Phi$ after each \textit{deconv} layer accounts for non-linearity in the interpolation process.
\section{Data Acquisition}
In order to train the network with real ground truth images, we created a customized dataset from publicly available as well as new self-recorded ground truth data.
A HSI captured with a $4\times 4$ mosaic pattern contains information for $1$ out of $16$ wavelength bands in each pixel only. For training the demosaicing algorithm, ground-truth data is necessary such that for each pixel, full spectral information exists across all $16$ wavelength bands. Hence, we captured an unprecedented new HSI dataset, providing accurate ground truth upsampling information. We capture in an controlled environment with a Ximea snapshot 4x4-VIS camera using an IMEC CMOS sensor. The sensor resolution is $2048\times 1088$ px, with an active area of $2048\times 1024$ px and the $4\times 4$ mosaic filter pattern captures $16$ wavelength bands in the spectral range from $463$ nm to $638$ nm.
For the generation of the ground-truth information, we captured $16$ images with precise pixel-wise shifting, meandering along the $4\times 4$ mosaic pattern for each scene, using a computer-controlled shifting unit. To achieve accurate 1-pixel shifts, the camera was calibrated and the scene-camera distance was measured. After acquiring the $4\times 4 = 16$ images, a full resolution image with $L = 16$ wavelengths was created by resampling the image stack.
At each position, we captured and averaged several images to decrease the influence of sensor noise. In order to account for illumination differences and work with reflectance intensity, all captured images were corrected using a white reference image according to \cite{Wisotzky2019SPIEMI}. Further, snapshot mosaic images contain spectral crosstalk, which influences the reflectance behavior \cite{Wisotzky2020JMI}. To learn this behavior during demosaicing, the ground-truth data was crosstalk corrected in addition to white reference correction, while the training and test input data were white reference corrected only.
\begin{figure}[!ht]
\centering
\subfloat[Experimental setup with a color chart as captured object]{\includegraphics[width=0.7\columnwidth]{Setup.png}}
\\
\subfloat[Scene 2]{\includegraphics[width=0.394\columnwidth]{Scene3.png}}\
\subfloat[Scene 3]{\includegraphics[width=0.405\columnwidth]{Scene1.png}}
\caption{(a) Experimental setup for recording hyperspectral images using a computer-controlled shifting unit, allowing exact 1-px shifts of the camera. Examples for two captured scenes containing (b) a combination of vegetables and meat and (c) different types of vegetables.}
\label{fig:experimental_setup}
\end{figure}
We captured a total of twelve scenes, six with a color chart and two of each showing different vegetables, different meats, as well as a combination of vegetables and meats, see Fig.~\ref{fig:experimental_setup}.
It is important to note, that a correct 1-pixel shift can only be achieved for the focus plane and points in front or behind that plane will always show slightly divergent pixel shifts of ($1\text{-px}\pm \epsilon$). For this reason, we added a flat color chart to half of the entire dataset and filtered the remaining images with a Gaussian filter with $\sigma = 1.5$ to smooth the present shifting error.
In addition, to increase the dataset, our captured data was combined with synthetic images generated from the publicly available CAVE dataset \cite{yasuma2010generalized}. The CAVE dataset has 32 reflectance images of real-world objects. In total $18$ images of the CAVE data sections 'skin and hair', 'food and drinks', 'real and fake' as well as 'stuff' were added to our dataset by interpolating the needed spectral bands and building up a simulated mosaic pattern representation of each scene.
The entire dataset will be available upon publication.
\section{Training and Network Evaluation Metrics}
During training, we used image patches of size $[1\times 100\times 100] = [16\times 25\times 25]$ as input for the network. We split the dataset into a training set with $1000$ patches of eleven captured images and 17 images of the CAVE data as well as a test set including $75$ patches of each of the two datasets.
We used the ADAM optimizer with an adaptive learning rate strategy and an initial learning rate of $0.001$ \cite{kingma2014adam}. The learning rate was reduced after each epoch until a value of $0.0001$ was reached, using at least $30$k epochs. A batch size of $20$ was used for updating the weights during the network training. The M2C block was initialized with uniform random weight. The loss function for calculating the difference between the ground truth and the predicted full-spectrum hyperspectral cube is defined by the mean squared error (MSE)
\begin{equation}
MSE(o,p) = \frac{1}{N} \sum_{i=0}^{N}{|o_i-p_i|^2},
\end{equation}
where $o$ is the ground truth and $p$ is the predicted value.
\section{Experiments and Results} \label{sec:res}
For quantitative analysis, we use the structural similarity index (SSIM), measuring the similarity between spectral cubes \cite{wang2004image} from predicted and ground-truth data as well as the peak signal to noise ratio (PSNR) \cite{hore2010image}.
Both, SSIM and PSNR are calculated individually for each spectral channel and averaged over all channels for the test images.
We compare the results of our proposed network to two state-of-the-art demosaicing approaches DCCNN \cite{dijkstra2019hyperspectral} and DeepCND \cite{habtegebrial2019deep}. Further, we analyze the resulting images visually and, to show the usability of our work, we visually analyze intraoperative snapshot images acquired during a parotidectomy.
\subsection{Quantitative Results}
\begin{table}[b]
\caption{SSIM and PSNR results of our proposed network and the DCCNN \cite{dijkstra2019hyperspectral} using $32$ and $128$ filters.}
\label{tab:res_filter}
\centering
\begin{tabular}{l|cc|cc|r}
\hline
\# of filters & \multicolumn{2}{c}{32} & \multicolumn{2}{c}{128} & \\
Networks & SSIM & PSNR & SSIM & PSNR & epochs \\
\hline
DCCNN \cite{dijkstra2019hyperspectral} & 0.755 & 40.05 & 0.825 & 41.54 & $30$k \\
Ours & 0.776 & 40.50 & 0.836 & 42.11 & $30$k \\
Ours & 0.784 & 40.95 & 0.841 & 42.63 & $40$k \\
\hline
\end{tabular}
\end{table}
All networks, our proposed network as well as the two reference networks, were trained on the created dataset and the networks learned to predict a full spectral cube of dimension $[16\times 100\times 100]$ from the input mosaic image of dimension $[1\times 100\times 100]$. As the quality of the results depends on the number of filters in the first deconvolutional layer, with best results reported between $32$ and $256$ filters \cite{dijkstra2019hyperspectral}, we trained two versions of our as well as the DCCNN \cite{dijkstra2019hyperspectral} network using $32$ and $128$ filters in the first deconvolutional layer. Table \ref{tab:res_filter} reports the SSIM and PSNR results of our network and the DCCNN using $32$ and $128$ filters, respectively. Using $32$ filters and $1000$ images as well as $30$k epochs for training, the DCCNN network shows results comparable to the initial results reported in Dijkstra et al.~\cite{dijkstra2019hyperspectral}. Our proposed network outperforms the reference network approximately by $2\%$ using the same parameter and training set, see Tab.~\ref{tab:res_filter}. Further, the results show that using $128$ filters yields better results as with $32$ filters as recommended by Dijkstra et al.~\cite{dijkstra2019hyperspectral}. Therefore, for further analysis, we have used $128$ filters in the first \textit{deconv} layer for the other experiments.
\begin{table}[t]
\caption{PSNR results of all analyzed demosaicing methods. }
\label{tab:res_final}
\centering
\begin{tabular}{l|c}
\hline
Networks & PSNR [dB] \\
\hline
Ours & 43.06 \\
DeepCND \cite{habtegebrial2019deep} & 42.38 \\
DCCNN \cite{dijkstra2019hyperspectral} & 41.54 \\
2D-3D-Net \cite{Charul21} & 41.33 \\
Intensity Difference \cite{Mihoubi2015} & 40.43 \\
Bilinear Interpolation & 40.23 \\
Bicubic Interpolation & 39.20 \\
\hline
\end{tabular}
\end{table}
In addition, we noticed that with $30$k epochs, the training error was still decreasing for our network, while it converged for the reference networks. Therefore, the quality of our results increases further when using up to $40$k epochs during training. The final results, shown in Tab.~\ref{tab:res_final}, present the PSNR at lowest training loss showing that our model outperforms the two reference networks as well as traditional interpolation methods by approximately $1$ dB (compared to DeepCND \cite{habtegebrial2019deep}) to $4$ dB (compared to bicubic interpolation).
\subsection{Qualitative Results}
\begin{figure}[b]
\centering
\subfloat[Ground Truth Image]{\includegraphics[width=0.49\columnwidth]{rgb2wp_2veg_GT.png}}\
\subfloat[Our Network]{\includegraphics[width=0.49\columnwidth]{rgb2wp_2veg_ownNet.png}}
\caption{RGB representation of one captured HSI scene as (a) high resolution ground truth imag
and (b) demosaiced images using our network.}
\label{fig:resVeg2_entire}
\end{figure}
In the following, we visually analyze of differences between individual result images. This qualitative analysis includes a spectral signature analysis, analyzing the spectral behavior of the predicted hyperspectral cube across all 16 wavelengths. Additionally, we provide a visual comparison for the single spectral bands as well as calculated RGB image of the predicted hyperspectral cube. Fig.~\ref{fig:resVeg2_entire} shows full-resolution RGB images
ground truth image and demosaiced result with our approach) of one example image in our captured dataset (dimension of $1024\times 2048$ pixels). The RGB images have been calculated using the CIE color matching functions \cite{stockman2019cone} with standard illuminant D65. Note, that the calculated RGB images show missing red components, since the snapshot camera is only sensitive up to $638$ nm and thus information is missing in the red channel of the RGB images.
\begin{figure}[tb]
\centering
\subfloat[Original Input]{\includegraphics[width=0.248\columnwidth]{rgb2wp_2veg_orig_zoom2_sampled.png}}\
\subfloat[Ground Truth]{\includegraphics[width=0.25\columnwidth]{rgb2wp_2veg_GT_zoom2.png}}\
\subfloat[Our Network]{\includegraphics[width=0.25\columnwidth]{rgb2wp_2veg_tukNet_zoom2.png}}
\\
\subfloat[DeepCND \cite{habtegebrial2019deep}]{\includegraphics[width=0.244\columnwidth]{rgb2wp_2veg_ownNet_zoom2.png}}\
\subfloat[DCCNN \cite{dijkstra2019hyperspectral}]{\includegraphics[width=0.244\columnwidth]{rgb2wp_2veg_openHS_zoom2.png}}\
\subfloat[Lin. Interpolation]{\includegraphics[width=0.244\columnwidth]{rgb2wp_2veg_linint_zoom2.png}}\
\subfloat[Cubic Interpol.]{\includegraphics[width=0.244\columnwidth]{rgb2wp_2veg_cubint_zoom2.png}}
\caption{RGB representation of a crop of Fig.~\ref{fig:resVeg2_entire} (center of the right image half). Our result is most similar to the ground truth, while traditional interpolation shows color artifacts at the edges.}
\label{fig:resVeg2_zoom}
\end{figure}
Fig.~\ref{fig:resVeg2_zoom} shows enlarged cutouts of the scene depicted in Fig.~\ref{fig:resVeg2_entire} for the ground truth images as well as the input image and demosaiced versions of our network as well as linear and bilinear interpolation and the reference demosaicing networks.
Compared to the original input image, our demosaiced result shows more structure while maintaining edges and color. In the reference networks the mosaic pattern is visible over the entire demosaiced image, c.f.~Fig.~\ref{fig:resVeg2_zoom}(d) and (e). This effect is drastically reduced by our approach, where this artifact only slightly appears especially around specular reflections as well as some homogeneous but noisy regions, which are not perfectly in focus, c.f. Fig.~\ref{fig:resVeg2_zoom}(c). Classical linear and cubic interpolation increase noise present in the input images, while single-pixel information is lost resulting in not correctly reconstructed edges. For the interpolation-based results, color artifacts appear around the edges (Fig.~\ref{fig:resVeg2_zoom}(f) and (g)) due to wrong spectral demosaicing. On the other hand, all neural network approaches learned to denoise the image as well as to correct the crosstalk.
Fig.~\ref{plot:resVeg2_spectral} plots the spectral signatures of two example regions in the image for all compared demosaicing approaches. For homogeneous regions, the spectral signatures of all analyzed methods are close to that of the ground truth image region, c.f.~Fig~\ref{plot:resVeg2_spectral}(a). For edges, the spectral signature produced by our network almost completely follows that of the ground truth, followed closely by DeepCND \cite{habtegebrial2019deep} and then DCCNN \cite{dijkstra2019hyperspectral}, while the classical interpolation methods are not able to reconstruct similar characteristics of the spectral curves, resulting in wrong color appearance in the RGB images.
\begin{figure}[t]
\centering
\subfloat[Homogeneous Area]{\includegraphics[width=0.45\columnwidth]{SpectralPlot373_1082.pdf}}
\subfloat[Edge]{\includegraphics[width=0.45\columnwidth]{SpectralPlot385_1428.pdf}}
\caption{Spectral plots of (a) a homogeneous area of the centered backmost red pepper under the total reflection and (b) an edge of the green and yellow parts visible in Fig.~\ref{fig:resVeg2_zoom}.}
\label{plot:resVeg2_spectral}
\end{figure}
\subsection{Intraoperative Image Demosaicing}
We also applied our approach to intraoperative image data, as HSI can be used in intraoperative settings in order to differentiate between different tissue types \cite{Wisotzky2020JMI} or to extract vital information \cite{WisotzkyComparision2021,Kossack_2022_CVPR}. Fig.~\ref{fig:resMed_entire} shows an example of a demosaiced image acquired intraoperatively. Our network is able to demosaic the original input and perform denoising as well as crosstalk correction at the same time, while the overall texture and color appearance is preserved. In the overall impression, all visual analyses from the test data are confirmed, c.f.~Fig.~\ref{fig:resMed_zoom}.
\begin{figure}[ht]
\centering
\includegraphics[width=0.7\columnwidth]{rgb2wp_ownNet.png}
\caption{RGB representation of a captured surgical scene using the $4\times 4$ snapshot camera and demosaiced with our network.}
\label{fig:resMed_entire}
\end{figure}
\begin{figure}[t]
\centering
\subfloat[Original Input]{\includegraphics[width=0.3\columnwidth]{rgb2wp_orig_zoom_scaled.png}}\
\subfloat[Our Network]{\includegraphics[width=0.3\columnwidth]{rgb2wp_tukNet_zoom.png}}\
\subfloat[DeepCND \cite{habtegebrial2019deep}]{\includegraphics[width=0.3\columnwidth]{rgb2wp_ownNet_zoom.png}}\\
\subfloat[DCCNN \cite{dijkstra2019hyperspectral}]{\includegraphics[width=0.3\columnwidth]{rgb2wp_openHS128_zoom.png}}\
\subfloat[Linear Interpolation]{\includegraphics[width=0.3\columnwidth]{rgb2wp_linint_zoom.png}}\
\subfloat[Cubic Interpolation]{\includegraphics[width=0.3\columnwidth]{rgb2wp_cubint_zoom.png}}
\caption{RGB represenation of a crop of Fig.~\ref{fig:resMed_entire} (image center).}
\label{fig:resMed_zoom}
\end{figure}
\section{Discussion and Conclusion}
In this work, we propose a neural network architecture for hyperspectral demosaicing for snapshot $4\times 4$ mosaic cameras.
Additionally, we present a new unprecendened ground-truth dataset of real scenes for training, acquired with a sensor shifting unit and pixel reordering strategy. This dataset is combined with synthetic data generated from the CAVE \cite{yasuma2010generalized} dataset. The performance of the network is evaluated using SSIM and PSNR scores and compared to traditional interpolation methods as well as to two demosaicing networks \cite{dijkstra2019hyperspectral,habtegebrial2019deep}. The results show that our proposed network with parallel feature extraction outperforms the reference networks with nearly $2\%$ and $0.4\%$ increase in SSIM score and $1.05$ dB and $0.6$ dB increase in PSNR. The scores of the classical interpolation methods are lowest as no spectral-spatial correlations are taken into consideration during demosaicing, through which they fail to preserve the spectral signature. This behavior visually emerges at edges.
All network-based approaches are able to preserve the spectral signatures.
In detail, the results show that increasing the number of filters allows the network to learn more features (e.g., spatial-spectral correlations and edges) resulting in improved demosaicing. The improved performance for our network can be attributed to the feature addition layer, which combines the features from the M2C layer and the deep features extracted through the four residual blocks from the hand-crafted M2C input. This indicates that the upsampling performed on extracted feature maps (from four residual blocks) might yield better spatial-spectral resolution in the full spectrum mosaic cube. Currently, our network is only designed for $4\times 4$ mosaic pattern but similar architectures are possible for $3\times 3$ or $5\times 5$ patterns through adaptions at the deconvolutional layers.
Visual inspection of the results show a slight presence of the $4\times 4$ mosaic pattern in the demosaiced images, due to the challenge of creating a real world dataset using the 1-pixel shifting of a captured 3D scene. This 1-pixel shifting is dependent on the distance between camera and object, while the 3D nature of the scene did not allow an exact 1-pixel shift for the entire scene leaving a shadowing mosaic pattern as artifact. One approach to decrease or overcome this problem would be to use synthetic data, which would be completely aligned to (specific) mosaic snapshot cameras. We assume that this will further improve the results, since demosaicing results are heavily dependent on the quality of the full spectrum hyperspectral data available for training.
\section*{Acknowledgment}
This work was funded by the German Federal Ministry of Education and Research (BMBF) under Grant No.~16SV8061 (MultiARC) and the German Federal Ministry for Economic Affairs and Climate Action (BMWi) under Grant No.~01MK21003 (NaLamKI). Only tissue that has been exposed during normal surgical treatment has been scanned additionally with our described camera. This procedure has been approved by Charit\'e--Universit\"atsmedizin Berlin, Germany.
\bibliographystyle{splncs04}
| 2024-02-18T23:40:18.416Z | 2022-11-29T02:26:24.000Z | algebraic_stack_train_0000 | 2,012 | 4,617 |
|
proofpile-arXiv_065-9936 | \section{Introduction}
\label{sec:intro}
Coupled nonlinear oscillators constitute an excellent framework to unravel and understand a plethora of intriguing collective dynamics/patterns
observed in a wide variety of natural systems~\cite{Kuramoto:1984,winfree,Strogatz:2000,Pikovsky:2001,Acebron:2005}. In particular, the phenomenon of synchronization has been widely studied in the past two decades
due to its manifestation in several natural and man-made systems~\cite{Pikovsky:2001,Acebron:2005,Gupta:2014,Gupta:2018}. For instance, collective synchrony includes synchronized firing
of cardiac pacemaker cells~\cite{Peskin:1975}, synchronous emission of light pulses by groups of fireflies~\cite{Buck:1988}, chirping of crickets~\cite{Walker:1969},
synchronization in ensembles of electrochemical oscillators~\cite{Kiss:2002}, synchronization in human cerebral connectome~\cite{Schmidt:2015}, and
synchronous clapping of audience~\cite{Neda:2000}.
Incredibly, the Kuramoto model has been employed as a paradigmatic model to
understand diverse emerging nonlinear phenomena across various disciplines, including physics, biology, chemistry, ecology, electrical engineering,
neuroscience, and sociology~\cite{Kuramoto:1984,winfree,Strogatz:2000,Pikovsky:2001,Acebron:2005,Pikovsky:2001}, as it allows
for an exact analytical treatment in most cases in explaining macroscopic dynamics.
The Kuramoto model comprises of $N$ globally-coupled phase oscillators with distributed natural frequencies interacting symmetrically with one another
through the sine of their phase differences. Considering symmetric interaction in the dynamics is only an approximation that may
simplify the theoretical analysis, which indeed may fail to capture important phenomena occurring in real systems. In contrast to the standard Kuramoto model, interactions between
oscillators may be asymmetric, in general. For example, asymmetric interaction leads to novel features such as families of traveling wave
states~\cite{Iatsenko:2013,Petkoski:2013}, glassy states and super-relaxation~\cite{Iatsenko:2014}, and so forth, and has been invoked to
discuss coupled circadian neurons~\cite{Gu:2016}, dynamic interactions~\cite{Yang:2020,Sakaguchi:1988}, etc. A
generalization of the Kuramoto model that accounts for asymmetric interaction is the so-called Sakaguchi-Kuramoto model, whose
dynamics can be described by the equation of motion~\cite{Sakaguchi:1986,Oleh-SK-1,Oleh-SK-2}
\begin{equation}
\frac{{\rm d}\theta_j}{{\rm d}t}=\omega_j+\frac{K}{N}\sum_{k=1}^N
\sin(\theta_k-\theta_j +\alpha),
\label{eq:eom-SK}
\end{equation}
where $0\le\alpha < \pi/2$ is the asymmetry parameter. The model~(\ref{eq:eom-SK}) and its variants have been successfully employed to study a variety of dynamical
scenarios such as disordered Josephson series array~\cite{Wiesenfeld:1996}, multiplex network~\cite{sj:1,sj:2,sj:3,sj:4}, time-delayed interactions~\cite{Yeung:1999}, hierarchical populations of coupled
oscillators~\cite{Pikovsky:2008}, chaotic transients~\cite{Wolfrum:2011}, dynamics of pulse-coupled oscillators~\cite{Pazo:2014}, etc.
Majority of the investigations in either Kuramoto or Sakaguchi-Kuramoto models were carried out with pairwise interactions.
Nevertheless, in many realistic systems, such as Huygens pendulum,
neuronal oscillators, genetic networks, globally coupled photochemical oscillators, etc., ~\cite{o:2,o:3,o:4,o:5}
higher order Fourier harmonics in the coupling function~\cite{ho:1,Bick2011} or higher order couplings~\cite{Bick2016,ho:2}
play a predominant role in shaping the collective dynamics.
Recently, it has been shown that higher order couplings lead to added nonlinearity in the macroscopic system
dynamics that induce abrupt synchronization transitions via hysteresis and bistability~{\cite{o:5}}.
Further, higher order interactions are shown to stabilize strongly synchronized states even when the pairwise coupling is repulsive, which is otherwise unstable~{\cite{ho:3}}.
Abrupt or explosive synchronization was shown to manifest in networks in which the degrees of the nodes are positively correlated with the
frequency of the node dynamics. In contrast, higher oder interactions are shown to be responsible
for the rapid switching to synchronization, leading to explosive synchronization, in many biological and other systems without
the need for particular correlation mechanism between the oscillators and the topological
structure~\cite{ho:4}.
In this work, we unravel the influence of the asymmetry parameters in the phase diagram of the Sakaguchi-Kuramoto model with
pairwise and higher order couplings under the influence of both the unimodal and bimodal distributions of the natural frequencies. We employ
two different asymmetry parameters, namely $\alpha_1$ in the pairwise coupling and $\alpha_2$ in the higher order coupling.
The effect of interplay of the asymmetry parameters and the higher order coupling on the collective dynamical behavior of the Sakaguchi-Kuramoto model will be captured in
the two parameter phase diagrams. We consider five different cases, namely
(i) $\alpha_1$~=~$\alpha_2$~=~0 (ii) $\alpha_1$~=~$\alpha_2~\ne~0$, (iii) $\alpha_1~\ne~0$; $\alpha_2$~=~0, (iv) $\alpha_2~\ne~0$; $\alpha_1$~=~0, and
(v) $\alpha_1~>0~\&~\alpha_2>0$.
to unravel the emerging collective dynamics and their respective phase diagrams. We observe incoherent state (IC), partially synchronized state-I (PS-I),
partially synchronized state-II (PS-II), and standing wave (SW) in the phase diagrams along with various bistable and multistable regions.
We also deduce the evolution equations for the macroscopic order
parameters by employing the Ott-Antonsen ansatz~\cite{Ott:2008,Ott:2009}. We derive analytical stability conditions for the incoherent state, which results in
the pitchfork and Hopf bifurcation curves, from the governing equations of motion of the macroscopic order parameters. Furthermore, we obtain
the saddle-node and homoclinic bifurcation curves using the software package XPPAUT~\cite{xpp}, which leads to several bifurcation transitions across the various dynamical states.
We find that the higher order coupling essentially facilitates enlargement of bistable states.
Higher order coupling also facilitates the onset of the bistability between the IC and PS-I states
even for the unimodal frequency distribution, a phenomenon which cannot be seen in the Sakaguchi-Kuramoto model with pairwise coupling and unimodal distribution.
Furthermore, a low value of $\alpha_1$ for $\alpha_2=0$ and a large value of $\alpha_2$ for $\alpha_1=0$ facilitate
the onset of PS-II and bistable region R3 (bistability between PS-I and PS-II)
in the phase diagram. Very large values of $\alpha_1$ and $\alpha_2$ allow the phase diagrams to admit only the monostable dynamical states despite the fact that appropriate
values of the asymmetry parameters induce bistable and multistable states. It is to be noted that bistable (multistable) regions are characterized by abrupt transitions among
the dynamical states.
The paper is organized as follows. We introduce the Sakaguchi-Kuramoto model in Sec.~\ref{model}. We deduce the evolution equations corresponding to
the macroscopic order parameters using the Ott-Antonsen ansatz in Sec.~\ref{sec:analysis}. In Sec.~\ref{sec:numerics}, we illustrate the phase diagrams
of the model with both unimodal and bimodal frequency distribution for various possible combinations of the asymmetry parameters $\alpha_1$ and $\alpha_2$ and discuss the dynamical transitions across various bifurcation scenarios
demarcating the dynamical states in the phase diagrams. Finally, we will provide a summary and conclusions in Sec.~\ref{sec:conclusions}.
\section{Model}
\label{model}
The $N$-coupled Sakaguchi-Kuramoto model with a specific higher order interaction is governed by the set of $N$ coupled first order nonlinear ordinary differential equations (ODEs),
\begin{align}
\label{eq:km2}
\dot{\theta}_i&\,=\omega_i+{k}\big[\frac{1}{N}\sum_{j=1}^{N}\sin(\theta_j-\theta_i-\alpha_1)\\ \nonumber
&\,+\frac{1}{N^3}\sum_{j=1}^{N}\sum_{k=1}^{N}\sum_{l=1}^{N}\sin(\theta_j+\theta_k-\theta_l-\theta_i-\alpha_2)\big],~i=1,2,\ldots, N,
\end{align}
where $\theta_i$ is the phase of the $i$th oscillator, $\omega_i$ is its natural frequency, which is typically assumed to be drawn from a well behaved distribution $g(\omega)$. $\alpha_1$ and $\alpha_2$ are the asymmetry parameters of pairwise and higher order interactions, respectively. $k$ is the coupling strength of both pairwise and higher order interactions~\cite{ho:1,ho:2,ho:3,ho:4}. The Kuramoto model with higher-order interactions is known to describe topological structures such as higher-order simplexes or a simplicial complex~\cite{ex:1,ex:2}, which are relevant to brain dynamics, neuronal networks, and biological transport networks~\cite{ex:3,ex:4}. In recent times, neuroscience studies have confirmed the existence of higher-order interactions between neurons. For example, astrocytes and other glial cells are thought to be a biological source of high-order interactions since they interact with hundreds of synapses and actively regulate their activity~\cite{ex:5,ex:6}.\\
We consider a bimodal frequency distribution for $g(\omega)$ in our system. Specifically, we consider the Lorentzian distribution of unimodal and bimodal frequency distribution,
\begin{align}
g(\omega)&=\frac{\gamma}{\pi((\omega-\omega_0)^2+\gamma^2)};~~\gamma >0.
\label{eq:lor}\\
g(\omega)&=\frac{\gamma}{\pi}\left[\frac{1}{((\omega-\omega_0)^2+\gamma^2)}+\frac{1}{((\omega+\omega_0)^2+\gamma^2)}\right],~\gamma >0.
\label{eq:bil}
\end{align}
Here $\gamma$ is the width parameter (half width at half maximum)
of each peak and $\pm\omega_0$ are the location of their peaks. A more physically relevant interpretation
of $\omega_0$ is that it defines the detuning in the system (which is proportional to the
separation between the two central frequencies).
Note that the form of the distribution $g(\omega)$ given in (\ref{eq:bil})
is symmetric about zero. Another
point to observe is that $g(\omega)$ is bimodal if and only if the
peaks are sufficiently far apart compared to their widths.
Specifically, one needs $\omega_0 > \gamma/\sqrt{3}$. Otherwise, the distribution is unimodal and the classical results still apply.\\
\section{Evolution equation of the macroscopic order parameters}
\label{sec:analysis}
In the thermodynamic limit ($N \to \infty$ ), the system of equations (\ref{eq:km2}) can be reduced to a finite set of macroscopic variables
in terms of the macroscopic order parameters governing the dynamics of the original system of equations. In this limit, the discrete set of equations
can be extended to a continuous formulation using the probability density function $f(\theta,\omega,t)$,
where $f(\theta,\omega,t){\rm d}\theta$ characterizes the fraction of the oscillators with phases between $[\theta,\theta+{\rm d}\theta]$ along with
the natural frequency $\omega$ at a time $t$.
The distribution is
$2\pi$-periodic in $\theta$ and obeys the normalization condition
\begin{equation}
\int_0^{2\pi} {\rm d}\theta~f(\theta,\omega,t)=g(\omega)~\forall~\omega.
\label{eq:norm}
\end{equation}
The evolution of $f(\theta,\omega,t)$ follows the continuity equation
\begin{equation}
\frac{\partial f}{\partial t}+\frac{\partial(fv) }{\partial
\theta}=0,
\label{eq:continuity-equ}
\end{equation}
where $v(\theta,\omega, t)=\frac{d\theta}{dt}$ is the angular velocity at position $\theta$ at time $t$. From Eq. (\ref{eq:km2}), one can get
\begin{align}
v(\theta,\omega,t)=\omega+&\frac{k}{2i}\big[(Ze^{-i(\theta+\alpha_1)}-Z^\star e^{i(\theta+\alpha_1)})\nonumber\\+&({Z^2} {Z^\star} e^{-i(\theta+\alpha_2)}-{Z^\star}^2 Z e^{i(\theta+\alpha_2)})\big],
\end{align}
where $Z(t)$ is the macroscopic order parameter defined as
\begin{equation}
Z=\int_{-\infty}^{\infty} g(\omega) \int_0^{2\pi} f(\theta, \omega, t)e^{i\theta}d\theta d\omega,
\label{eq:mo}
\end{equation}
and $Z^\star$ is its complex conjugate.
Expanding $f(\theta, \omega, t)$ in Fourier series, we have
\begin{equation}
f(\theta,\omega,t)=\frac{g(\omega)}{2\pi}\left[1+\sum_{n=1}^\infty
\left(a_n(\omega,t) e^{\mathrm{i} n\theta}\right)+{\rm c.c.}\right],
\label{eq:f-Fourier}
\end{equation}
where the prefactor of $g(\omega)$ ensures that the normalization
(\ref{eq:norm}) is satisfied, $a_n(\omega,t)$ is the $n$-th Fourier
coefficient, while c.c. denotes the complex conjugation
of the preceding sum within the brackets. Using the Ott-Antonsen ansatz ~\cite{Ott:2008,Ott:2009}
\begin{equation}
a_n(\omega,t)=\left[a(\omega,t)\right]^n,
\label{eq:OA}
\end{equation}
one can obtain,
\begin{equation}
\frac{\partial a}{\partial t}+i\omega a+\frac{k}{2}\big[(Z {a^2}e^{-i\alpha_1}-Z^\star e^{i\alpha_1})+{|Z|^2}( {Z} {a^2} e^{-i\alpha_2}-{Z^\star} e^{i\alpha_2})\big],
\label{eq:12}
\end{equation}
where
\begin{equation}
Z=\int_{-\infty}^{\infty}a^\star(t,\omega)g(\omega)d\omega.
\label{eq:13}
\end{equation}
\begin{figure*}[ht]
\hspace*{-1cm}
\includegraphics[width=18.5cm]{ho-u.eps}
\caption{Phase diagrams in the ($k/\gamma-\omega_0/\gamma$) plane. (a) $\alpha_{1,2}$=0, (b) $\alpha_1=\alpha_2$, (c) $\alpha_1=1.0, \alpha_2=0.0$, and
(d) $\alpha_2=1.0, \alpha_1=0.0$. Incoherent state and standing wave are denoted by
IC and SW, respectively. Phase space with bistability (grey shaded region) between IC and SW is denoted as R1.
The Hopf bifurcation (line connected by filled squares) curves are the analytical stability curves.
Homoclinic (dotted-dashed line) bifurcation curve is obtained from XPPAUT.
}
\label{fig:1a}
\end{figure*}
\begin{figure}[ht]
\hspace*{-1.2cm}
\includegraphics[width=10cm]{ho-ut.eps}
\caption{Order parameter (R), obtained from numerical analysis of Eq.~(2) for unimodal distribution, illustrating the nature of the
dynamical transitions for (a) $\alpha_{1,2}=0$; (b) $\alpha_1=1, \alpha_2=0.0$.
}
\label{fig:1t}
\end{figure}
\begin{figure}[!ht]
\hspace*{-1cm}
\includegraphics[width=10cm]{ho-1.eps}
\caption{Phase diagram in ($k/\gamma-\omega_0/\gamma$) plane for $\alpha_1=\alpha_2=0$. Incoherent state, partially synchronized state-I and standing wave are denoted by
IC, PS-I and SW, respectively. Phase space with bistability between IC and PS-I states is denoted as R1 and that between SW and PS-I is denoted as R2.
The pitchfork (solid black), Hopf bifurcation (line connected by filled squares) and saddle-node (dashed line) curves are the analytical stability curves.
Homoclinic (dotted-dashed line) bifurcation curve is obtained from XPPAUT.}
\label{fig:1}
\end{figure}
\begin{figure*}[ht]
\hspace*{-1.2cm}
\includegraphics[width=18.5cm]{ho-bt.eps}
\caption{Order parameter (R), obtained from numerical analysis of (2) with bimodal distribution, illustrating the nature of dynamical transitions for $\alpha_{1,2}=0$ for various ratios of $\omega_0/\gamma$: (a) 0.75; (b) 1.5, and (c) 2.
}
\label{fig:2t}
\end{figure*}
\subsection{Unimodel Frequency Distribution}
The arbitrary function $a(\omega,t)$ is assumed to satisfy
$|a(\omega,t)| < 1$, together with the requirements that
$a(\omega,t)$ may be analytically
continued in the whole of the complex-$\omega$ plane and it has no singularities in the lower-half complex-$\omega$ plane. Further, $|a(\omega,t)| \to 0$ as
${\rm Im}(\omega) \to -\infty$. If these conditions are satisfied for $a(\omega,0)$, then as shown in (\ref{eq:OA}), they continue to be satisfied by $a(\omega,t)$
as it evolves under Eqs. (\ref{eq:12}) and (\ref{eq:13}). Expanding the unimodal frequency distribution $g(\omega)$, Eq. (\ref{eq:lor}), in partial fractions as
\begin{align}
g(\omega)=\frac{1}{4\pi i}\bigg[&\frac{1}{((\omega-\omega_0)-i\gamma)}-\frac{1}{((\omega-\omega_0)+i\gamma)}\bigg],
\end{align}
and evaluating Eq. (\ref{eq:13}) using the appropriate contour integral, the order parameter becomes,
\begin{equation}
Z(t)=a^\star(\omega_0-i\gamma,t).
\end{equation}
Substituting the above in Eq. (\ref{eq:12}), one obtains a
complex ODE, describing the evolution of the suborder parameter,
\begin{align}
\frac{\partial Z}{\partial t}+(\gamma+i&\omega_0)Z+\frac{k}{2} Z\bigg[(|Z|^2e^{-i\alpha_{1}}-e^{i\alpha_{1}})\nonumber\\&+|Z|^2 (|Z|^2 e^(-i\alpha_2)-e^{i\alpha_2})\bigg]
\label{eq:z-dynamics}
\end{align}
Rewriting the above equation in terms of $r$ and $\psi$ as $Z=re^{i\psi}$, one obtains the evolution equations for $r$ and $\psi$ as
\begin{align}
\dot{r}=&-\gamma r-\frac{k}{2}r((r^2-1)\cos(\alpha_1) +r^2\cos(\alpha_2)(r^2-1) )\nonumber\\
\dot{\psi}=&-\omega_0-\frac{k}{2}((r^2+1)sin(\alpha_1)+r^2sin(\alpha_2)(r^2+1))
\label{uni}
\end{align}
The above reduced low-dimensional equations describe the dynamics of the model (\ref{eq:km2}) with unimodal frequency distribution. Then, $r(t)=\vert \frac{1}{N}\sum_{j=1}^Ne^{\mathrm{i}\theta_j(t)} \vert$ takes either a null value, when the dynamics corresponds to the incoherent state, or oscillating values corresponding to the standing wave behavior
of the Sakaguchi-Kuramoto oscillators.
\subsection{Bimodal Frequency Distribution}
Now, we will deduce the governing equations for the macroscopic variables for the
model (\ref{eq:km2}) corresponding to the bimodal frequency distribution.
Expanding the bimodal frequency distribution $g(\omega)$, Eq.~(\ref{eq:bil}), in partial fractions as
\begin{align}
g(\omega)=\frac{1}{4\pi i}\bigg[&\frac{1}{((\omega-\omega_0)-i\gamma)}-\frac{1}{((\omega-\omega_0)+i\gamma)}\nonumber\\&+\frac{1}{((\omega+\omega_0)-i\gamma)}-\frac{1}{((\omega+\omega_0)+i\gamma)}\bigg],
\end{align}
and evaluating Eq. (\ref{eq:13}) using the appropriate contour integral, the order parameter becomes,
\begin{equation}
Z(t)=\frac{1}{2}[z_{_1}(t)+z_{_2}(t)],
\end{equation}
where
\begin{equation}
z_{_{1,2}}(t)=a^\star(\pm\omega_0-i\gamma,t).
\end{equation}
Substituting the above in Eq. (\ref{eq:12}), one obtains two
coupled complex ODEs, describing the evolution of two suborder parameters,
\begin{align}
\dot{z}_1=&-(\gamma+i\omega_0)z_1+\frac{k}{4}\bigg((z_1+z_2)e^{-i\alpha_1}-{z_1^2}(z_1^\star+z_2^\star) e^{i\alpha_1}\nonumber\\&+\frac{|z_1+z_2|^2}{4}\big( (z_1+z_2)e^{-i\alpha_2}-{z_1^2}(z_1^\star+z_2^\star) e^{i\alpha_2}\big)\bigg),\label{eq:z1}\\
\dot{z}_2=&-(\gamma-i\omega_0)z_2+\frac{k}{4}\bigg((z_1+z_2)e^{-i\alpha_1}-{z_2^2}(z_1^\star+z_2^\star) e^{i\alpha_1}\nonumber\\&+\frac{|z_1+z_2|^2}{4}\big( (z_1+z_2)e^{-i\alpha_2}-{z_2^2}(z_1^\star+z_2^\star) e^{i\alpha_2}\big)\bigg),\label{eq:z2}
\end{align}
where overdot represents the time derivative.
Rewriting Eq.~(\ref{eq:z1})~and ~(\ref{eq:z2}) in terms of $r_{_{1,2}}$ and $\psi_{_{1,2}}$, as $z_{_1,2}=r_{_{1,2}}e^{-\mathrm{i}{\psi_{1,2}}}$ and defining the phase difference as $\psi$ = $\psi_{{_1}}-\psi_{{_2}}$ , the dimensionality can be further reduced to
three as follows:
\begin{widetext}
\begin{subequations}
\begin{align}
\dot{r}_1&=-\gamma r_{_1} - \frac{k}{16}(r_{_1}^2-1)((r_1^2+r_2^2+2 r_1 r_2 \cos[\psi]) (\cos(\alpha_2) r_1+ \cos(\psi+\alpha_2) r_2)+4(\cos(\alpha_1) r_1+\cos(\psi+\alpha_1)r_2)), \\
\dot{r}_2&=-\gamma r_{_2} - \frac{k}{16}(r_{_2}^2-1)((r_1^2+r_2^2+2 r_1 r_2 \cos[\psi]) (\cos(\psi-\alpha_2) r_1+ \cos(\alpha_2) r_2)+4(\cos(\psi-\alpha_1) r_1+\cos(\alpha_1)r_2)), \\
\dot{\psi}&=-2\omega-\frac{k}{16 r_2}(1+r_2^2)(4 r_2\sin(\alpha_1)-4 r_1 \sin(\psi-\alpha_{1})+(r_2 \sin(\alpha_2)-r_1\sin(\psi-\alpha_2))(r_1^2+r_2^2+2 r_1 r_2 \cos[\psi]))\nonumber\\&~~~~~~~-\frac{k}{16 r_1}(1+r_1^2)(4 r_2\sin(\psi+\alpha_1)+4 r_1 \sin(\alpha_{1})+(r_2 \sin(\psi+\alpha_2)+r_1\sin(\alpha_2))(r_1^2+r_2^2+2 r_1 r_2 \cos[\psi])).
\end{align}
\label{eq:ps}
\end{subequations}
\end{widetext}
The above system of three coupled nonlinear ordinary differential equations are the evolution equations for the macroscopic variables of the
model (\ref{eq:km2}) and describes its dynamics faithfully. Note that the partially synchronized states and standing wave patterns of the
Sakaguchi-Kuramoto model (\ref{eq:km2}) correspond to the periodic and quasi-periodic
orbits, respectively, in the above reduced model (that is the system of three coupled ordinary
differential equations governing the evolution of the macroscopic order parameters) for nonzero $\alpha_{1,2}$.
However, for the null value of the asymmetry parameters, the partially synchronized states and standing wave patterns
correspond to the steady states and periodic orbits, respectively.
\begin{figure*}[ht]
\hspace*{-1.5cm}
\includegraphics[width=20cm]{ho-2.eps}
\caption{Phase diagrams in the ($k/\gamma-\omega_0/\gamma$) plane for $\alpha_1~=~\alpha_2=\alpha$. (a) $\alpha=0.1$, (b) $\alpha=0.5$, and (c) $\alpha=1.0$.
Bifurcation curves and dynamical states are represented similar to those in Fig.~\ref{fig:1}. Partially synchronized state II (PS-II) is observed in the region enclosed by pitchfork,
Hopf and saddle-node bifurcation curves (see Fig.~\ref{fig:3}(a)). PS-II and R3 in Fig.~\ref{fig:3}(b) are enclosed by saddle-node bifurcation curves.
Here, R3 corresponds to the region of bistability between IC, and PS-II states. }
\label{fig:3}
\end{figure*}
\section{Phase diagrams of the Sakaguchi-Kuramoto model with higher order coupling}
\label{sec:numerics}
In this section, we will proceed to understand the dynamics of the generalized Sagakuchi-Kuramoto model by constructing appropriate two parameter phase
diagrams and classifying the underlying states from a numerical analysis of the evolution equations of the macroscopic order parameters
Eqs.~(\ref{uni}) and (\ref{eq:ps}) corresponding to unimodal and bimodal frequency distributions, respectively.
We also solve the associated Sakaguchi-Kuramoto model by numerically integrating Eq.~(\ref{eq:km2}) to verify the dynamical transitions in the phase
diagrams. Specifically,
we will unravel the phase diagrams of the Sakaguchi-Kuramoto model with higher order coupling and unimodal frequency distribution
and as well as that with bimodal frequency distribution for various possible
combinations of the asymmetry parameters. The number of oscillators is fixed as $N=10^4$ and we use the standard fourth-order Runge-Kutta integration
scheme with integration step size $h=0.01$ to solve the Sakaguchi-Kuramoto model (\ref{eq:km2}).
\subsection{Unimodal Frequency Distribution}
The reduced low-dimensional equations (\ref{uni}), describing the dynamics of the Sakaguchi-Kuramoto model with higher order coupling and unimodal frequency distribution,
is characterized by a trivial steady state $(r = 0)$, corresponding to the incoherent state (IC) and an oscillatory state corresponding to the standing wave (SW) nature of the
Sakaguchi-Kuramoto oscillators. The stability determining eigenvalues of the trivial steady state can be obtained as
\begin{equation}
\lambda_{1,2}=\frac{-2\gamma+k \cos(\alpha_1)\pm \sqrt{\Delta}}{2},
\label{eq:M-eigenvalues}
\end{equation}
where
$\Delta=k\sin(\alpha_1)(4\omega_0 +k \sin(\alpha_1))+4\omega_0^2$. The stability condition/curve for the onset of IC is
obtained as
\begin{align}
&k_{HB}=2 \gamma \sec (\alpha_1).
\label{eq:stability-ISS}
\end{align}
Phase diagrams of the Sakaguchi-Kuramoto model with higher order coupling and unimodal frequency distribution for different combinations of the asymmetry parameters
$\alpha_1$ and $\alpha_2$ are depicted in Fig.~\ref{fig:1a}. The line connected by filled squares corresponds to the Hopf bifurcation condition (\ref{eq:stability-ISS}).
In the absence of both the asymmetry parameters, that is for $\alpha_1=\alpha_2=0$, there is a transition
from the incoherent state to the standing wave pattern as a function of $k$ (see Fig.~\ref{fig:1a}(a)) via the Hopf bifurcation curve. Similar dynamical transition
is also observed for the other choices of the asymmetry parameters, namely for $\alpha_1=\alpha_2=1.0$ (see Fig.~\ref{fig:1a}(b)) and
for $\alpha_1=0$ and $\alpha_2=1.0$ (see Fig.~\ref{fig:1a}(d)) except for the region shift. For $\alpha_1=1.0$ and $\alpha_2=0.0$, one can observe
bistability between the IC and SW (indicated by grey shaded region, marked as R1) in Fig.~\ref{fig:1a}(c). The bistable region is bounded by the
homoclinic (indicated by dotted-dashed line) and Hopf bifurcation curves. Note that the homoclinic bifurcation curve is obtained from XPPAUT.
It is also to be noted that the dynamical transition is independent of $\omega_0$ in this case of unimodal frequency distribution, in general.
Now, the time-averaged order paramter $R=\lim_{T\rightarrow\infty}\frac{1}{T}\int_{0}^{T}dt'r(t')$ estimated from the simulation of
the Sakaguchi-Kuramoto model, by numerically integrating Eq.~(\ref{eq:km2}), for the unimodal frequency distribution is
depicted in Fig.~\ref{fig:1t} for two different values of the asymmetry parameters. The line connected by open circles corresponds to the forward trace,
while the line connected by filled circles corresponds to the backward trace.
For $\alpha_1=\alpha_2=0$, there is a
transition from the incoherent state (characterized by the null value of $R$) to the standing wave pattern, corroborated by a finite value of $R$ (seeFig.~ \ref{fig:1t}(a)),
which is in accordance with the phase diagram in Fig.~\ref{fig:1a}(a) that is obtained from the reduced low-dimensional systems (\ref{uni}).
The dotted line is the analytical Hopf bifurcation curve $k_{HB}$ across which there is a transition. Similar dynamical transition will be
observed for the other combinations of the asymmetry parameters except for the region shift as in the phase diagrams (see Fig.~\ref{fig:1a})
and hence they are not shown here to avoid repetitions.
Nevertheless, there is a bistability between IC and SW as in the phase diagram for $\alpha_1=1.0$ and $\alpha_2=0.0$ (see \ref{fig:1t}(b))
bounded by the homoclinic and Hopf bifurcation curves. Thus, direct numerical simulation of the model equation agrees well with the dynamical
transitions observed from their reduced low-dimensional equations corresponding to the macroscopic order parameters.
\subsection{Bimodal Frequency Distribution}
\subparagraph{Case I ($\alpha_1=\alpha_2$~=~0):}
In order to appreciate and understand the effect of the asymmetry parameters $\alpha_1$ and $\alpha_2$ on the dynamics as represented by
the phase diagram, one should first familiarize with the
phase diagram of the Sakaguchi-Kuramoto model with higher order coupling and bimodal frequency distribution in the absence of the asymmetry parameters.
The phase diagram in the ($\omega_0/\gamma$-$k/\gamma$) plane for the case $\alpha_1=\alpha_2=0$ is depicted in Fig.~\ref{fig:1}. The dynamical states
in the phase diagram are distinguished by features which are essentially based on the asymptotic behavior of $r(t)$. Incoherent state (IC), partially synchronized state (PS-I) and standing wave (SW)
along with the bistable regimes (R1 and R2) among the observed dynamical states are depicted in the phase diagram. The parameter space marked as R1 corresponds
to the bistable regime between IC and PS-I states, while that indicated as R2 corresponds to the bistable regime between SW and PS-I states. The null value of $r(t)$
characterizes the incoherent state, while a finite value of $r(t)$ indicates partially synchronized states. Oscillating nature of $r(t)$
confirms the standing wave.
The stable regions of the incoherent state in the phase diagram can be inferred from the dynamical equations of the reduced macroscopic variables
given in Eqs.~(\ref{eq:z1})~and~(\ref{eq:z2}). The phases of the oscillators are uniformly distributed between $0$ to $2\pi$ for
the incoherent state and hence it is characterized by $z_1 = z_2 = 0$. Performing a linear stability analysis of the fixed point $(z_1,~z_2)=(0,~0)$,
one obtains the condition for stability as
\begin{align}
k_{PF}=\frac{2(\gamma^2+\omega_0^2)}{\gamma}, ~~~~\text{for} ~\omega_0/\gamma~~<~1,\\
k_{HB}=4\gamma ~~~~~~~~~ \text{for} ~\omega_0/\gamma~~\ge~1.
\label{eq:pf}
\end{align}
Here, $K_{PF}$ corresponds to the pitchfork bifurcation curve across which the fixed point $(z_1,~z_2)=(0,~0)$ (incoherent state) loses
its stability leading to the inhomogeneous steady state (PS-I state), while $K_{HB}$ corresponds to the Hopf bifurcation curve across
which the incoherent state loses its stability resulting in the standing wave pattern. The pitchfork bifurcation curve,
indicated by the solid line in Fig.~\ref{fig:1}, serves as the
boundary between the incoherent and partially synchronized state
for $\omega_0/\gamma~~<~1$. The Hopf bifurcation curve, denoted by the line connected by filled squares, demarcates the incoherent state and standing wave region
of the phase diagram. The dashed line in Fig.~\ref{fig:1} corresponds to the saddle-node bifurcation curve, while the homoclinic bifurcation curve is denoted as the dotted-dashed line.
The latter is obtained from the software XPPAUT~\cite{xpp}, while the former is determined as follows.
The inhomogeneous steady state of the PS-I region in the phase diagram is characterized by $r_1=r_2=r=Const.$ and $\psi_1=-\psi_2=\phi$,
and hence from Eqs.~(\ref{eq:ps}) one can obtain
\begin{subequations}
\begin{align}
\sin(2\phi)&=\frac{8\omega_0}{k(1+r^2)(2+r^2+r^2\cos(2\phi))},\\
\cos(2\phi)&=\frac{k-kr^2-a}{(k r^2(r^2-1))},
\end{align}
\end{subequations}
where $a$~=$\sqrt{k^2-2k^2r^2+8k\gamma r^2+k^2r^4-8k\gamma r^4}$. The above equations give the following solutions for the stationary $r$ and $\phi$:
\begin{subequations}
\begin{align}
1&=\frac{64 \omega_0^2(r^2-1)^2}{(r^2+1)^2(k-k r^2+ a)^2}+\frac{(k r^4-k+a)^2}{(k^2 r^4(r^2-1)^2)}\\
\tan(2\phi)&=\frac{8 k \omega_0 r^2 (r^2-1)^2}{(r^2+1)(k-k r^2+a)(k r^4-k+a)}.
\end{align}
\label{sn}
\end{subequations}
Now, one can numerically solve the above equations for fixed values
of the parameters to obtain $r$ and $\phi$, which can be substituted back in the original equation of motion of the order parameters, Eqs.~(\ref{eq:ps}),
to deduce the characteristic eigenvalue equation. The resulting eigenvalues determine the saddle-node bifurcation curves
in the ($\omega_0/\gamma$-$k/\gamma$) parameter space.
The standing wave pattern loses its stability across the homoclinic bifurcation curve resulting in the PS-I state.
Upon decreasing the value of $k/\gamma$ in the phase diagram, the PS-I state (inhomogeneous steady states of $z_1$ and $z_2$) loses its stability via the saddle-node
bifurcation curve resulting in the incoherent state ($z_1 = z_2 = 0$) up to $\omega_0/\gamma=1.6$ and in the standing wave patterns
for $\omega_0/\gamma>1.6$. Hence, the bistability between the IC and PS-I states is enclosed by the saddle-node and pitchfork bifurcation curves in the phase diagram
in the region denoted as R1. Saddle-node and homoclinic bifurcation curves enclose the bistable region between the standing wave and PS-I state, which is denoted as R2 in the
phase diagram. It is to be noted that the phase diagram of the Sakaguchi-Kuramoto model with higher order coupling and bimodal frequency distribution
in the absence of asymmetry parameters resembles closely that of the Sakaguchi-Kuramoto model with pairwise interactions and bimodal frequency distribution~\cite{bim}. The
higher order coupling has essentially enlarged the bistable regions of the phase diagram. Further, the Sakaguchi-Kuramoto model with higher order coupling and bimodal frequency distribution is characterized by PS-I, R1 and R2 when compared to the Sakaguchi-Kuramoto model with higher order coupling and unimodal frequency distribution
(compare Figs.~\ref{fig:1} and ~ \ref{fig:1t}(a)). Similar rich dynamical states are also observed for the other choices of the asymmetry parameters in the
presence of bimodal frequency distribution as will be elucidated in the following cases.
Now, the order parameter $R$ estimated from
the Sakaguchi-Kuramoto model by numerically integrating Eq.~(\ref{eq:km2}) for the bimodal frequency distribution is
depicted in Fig.~\ref{fig:2t} for the asymmetry parameters $\alpha_1=\alpha_2=0$ and for three different values of $\omega_0/\gamma$.
Here, the line connected by open circles corresponds to the forward trace, while the line connected by filled circles corresponds to the backward trace
as in Fig.~\ref{fig:1t}. The dotted vertical line in Fig.~\ref{fig:2t}(a) corresponds to the analytical pitch-fork bifurcation curve,
the dotted-dashed line corresponds to the analytical saddle-node bifurcation curve, and the dashed line in Fig.~\ref{fig:2t}(b) corresponds to the analytical
Hopf bifurcation curve, while the solid line corresponds to the homoclinic bifurcation curve obtained using XPPAUT.
There is a transition from the incoherent state to the standing wave via the pitch-fork bifurcation during the forward trace, whereas there is
a transition from the SW to IC via the saddle-node bifurcation during the reverse trace (see Fig.~\ref{fig:2t}(a)) for $\omega_0/\gamma=0.75$.
Similarly, there is a transition from
IC(SW) to SW(IC) via the homoclinic(saddle-node) bifurcation curve during the forward(backward) trace for $\omega_0/\gamma=1.5$
as depicted in Fig.~\ref{fig:2t}(b). For $\omega_0/\gamma=2.0$, there is a similar transitions via the homoclinic and saddle-node bifurcation curves
during the forward and backward traces, respectively. These transitions, obtained by numerically solving the model equation (\ref{eq:km2}),
perfectly correlate with the dynamical transitions observed in the phase diagram (see Fig.~\ref{fig:1}), which are obtained by solving the
reduced low-dimensional evolution equations for the macroscopic order parameters (\ref{eq:ps}).
\begin{figure*}[ht!]
\hspace*{-1cm}
\includegraphics[width=19.2cm]{ho-3.eps}
\caption{ Phase diagrams in the ($k/\gamma-\omega_0/\gamma$) plane for $\alpha_2~=~0$ and for various values of the asymmetry parameter in the
pairwise coupling. (a) $\alpha_1=0.1$, (b) $\alpha_1=0.5$, (c) $\alpha_1=1.0$ and (d) $\alpha_1=1.5$. Bifurcation curves and dynamical states are similar to those in
Fig.~\ref{fig:3}(a). Here, R4 corresponds to the region of multistability between IC, PS-I and PS-II states.}
\label{fig:4}
\end{figure*}
\subparagraph{Case II ($\alpha_1=\alpha_2~\ne$~0):}
In order to analyze the effect of the asymmetry parameters on the phase diagram (see Fig.~\ref{fig:1}), we have next considered the case where the asymmetry parameters
$\alpha_1~=~\alpha_2~=~\alpha$ for simplicity.
We have depicted the corresponding phase diagrams in the ($k/\gamma-\omega_0/\gamma$) plane in Figs.~\ref{fig:3}(a)-\ref{fig:3}(c) for $\alpha=0.1, 0.5$, and $1$, respectively.
The dynamical sates
and the bifurcation curves are similar to those in Fig.~\ref{fig:1} without any asymmetry parameter. However for $\alpha=0.1$, partially synchronized state-II (PS-II)
is characterized by a different set of inhomogeneous steady states corresponding to nonzero values of ($z_1,~z_2$) in addition to the dynamical states observed in Fig.~\ref{fig:1}.
A linear stability analysis of the fixed point $(z_1,~z_2)=(0,~0)$ results in the stability condition
\begin{align}
\hspace{-0.5cm}
\omega_0^2 = \frac{{(32 \gamma^3 k +
\gamma k^3) \cos(\alpha_1) -32 \gamma^4 - 6 \gamma^2 k^2 - 4 \gamma^2 k^2 \cos(2 \alpha_1)}}{{
32 \gamma^2 + k^2 - 16 \gamma k \cos(\alpha_1) + k^2 \cos(2 \alpha_1])}}.
\end{align}
The above algebraic expression can be further simplified as
\begin{align}
\gamma &k^3 \cos(\alpha_1)-32 \gamma^4 - 6 \gamma^2 k^2 + 32 \gamma^3 k \cos(\alpha_1)\nonumber\\&-
2 \omega_0^2 ( k \cos(\alpha_1)-4 \gamma )^2 - 4 \gamma^2 k^2 \cos(2 \alpha_1)=0,
\label{eq:pf1}
\end{align}
which actually corresponds to the pitchfork bifurcation curve across which the fixed point $(z_1,~z_2)=(0,~0)$ (incoherent state) loses
its stability leading to the partially synchronized states PS-I and PS-II. Note that the incoherent state loses it stability only through the pitchfork bifurcation curve
in the entire explored range of $\omega_0/\gamma$ (see Fig.~\ref{fig:3}(a)).
All other bifurcation curves are obtained from XPPAUT. One may observe that the PS-II state
is enclosed by pitchfork, Hopf and homoclinic bifurcation curves, whereas the region corresponding to the bistability between PS-I and PS-II (denoted by R3)
is enclosed by pitchfork, Hopf and saddle-node bifurcation curves. The other dynamical transition and bistable regions are similar to that discussed in
Fig.~\ref{fig:1} in the absence of the asymmetry parameters.
Thus, a rather low value of the asymmetry parameters results in an additional partially synchronized state (PS-II state) with a region
of multistability between PS-I and PS-II.
However, a slight increase in the values of the asymmetry parameters results in drastic changes in the phase diagram (see Fig.~\ref{fig:3}(b) for $\alpha=0.5$). It is evident
from the figure that the bistable regions (R2 and R3) and the parameter space with standing wave are reduced drastically with increase in the PS-I state.
The PS-II state coexists with the PS-I state in the region enclosed by the two saddle-node bifurcation curves, while the bistable region R1 is completely wiped off from the phase diagram.
A large asymmetry parameter results in the loss of bistable regions and standing wave regions completely from the phase diagram, while retaining only
the incoherent state and partially synchronized state-I as illustrated in Fig.~\ref{fig:3}(c) for $\alpha=1$. Further increase in the asymmetry parameter results in
similar phase diagrams as in Fig.~\ref{fig:3}(c).
\begin{figure*}[ht!]
\hspace*{-1cm}
\centering
\includegraphics[width=19.5cm]{ho-4.eps}
\caption{Phase diagrams in the ($k/\gamma-\omega_0/\gamma$) plane for $\alpha_1~=~0$ and for various values of the asymmetry parameter in the
higher order coupling. (a) $\alpha_2=0.1$, (b) $\alpha_2=0.5$, (c) $\alpha_2=1.0$ and (d) $\alpha_2=1.5$. Bifurcation curves and dynamical states are similar to those in
Fig.~\ref{fig:3}(a). }
\label{fig:2}
\end{figure*}
\subparagraph{Case III ($\alpha_1~\ne$~~0; $\alpha_2$~=~~0): }
Now, we analyse the nature of the phase diagram with asymmetry parameter only in the pairwise coupling by switching off the asymmetry parameter in the
higher order coupling, so that $\alpha_1~\ne~~0$ and $\alpha_2~=~~0$. The phase diagrams for $\alpha_1=0.1, 0.5, 1$ and $1.5$
are shown in Figs.~\ref{fig:4}(a)-\ref{fig:4}(d), respectively.
For $\alpha_1=0.1$, the dynamics and the dynamical transitions in the phase diagram (see Fig.~\ref{fig:4}(a)) are similar to those observed in Fig.~\ref{fig:3}(a) for
$\alpha_1~=~\alpha_2=0.1$, which elucidates that the onset of PS-II state is facilitated by the asymmetry parameter in the pairwise coupling and is independent of the
asymmetry parameter in the higher order coupling. Increasing $\alpha_1$ to $\alpha_1=0.5$ results in an enhancement of the PS-II state
and the bistability between both the partially synchronized states in the phase diagram (see Fig.~\ref{fig:4}(b)). It is to be noted that R3 is enclosed by the
saddle-node and Hopf bifurcation curves. The spread of SW and R2 in the phase diagram is decreased appreciably for increasing values of $\alpha_1$, whereas
that of PS-I remains almost unaffected. The bistability between the IC and PS-I (region R1) states is completely destroyed.
Next, the phase diagram for $\alpha=1$ is depicted in Fig.~\ref{fig:4}(c), where the spread of SW and R2 is completely eliminated.
Further, the spread of R3 enclosed by the saddle-node bifurcation curves in the phase diagram is considerably reduced.
It is to be noted that there is a reemergence of the bistable region R1 even for $\omega_0/\gamma~<~1/\sqrt{3}$, where bimodal frequency distribution
becomes unimodal, which elucidates that the bistable region R1 has its manifestation in the phase diagram essentially due to the higher order coupling.
Otherwise, the phase diagram is almost equally shared by IC and PS-I states. Further increase in the asymmetry parameter in the pairwise coupling
results in the increase in the R1 region to a large extent, where IC and PS-I states coexist and are bounded by the saddle-node and pitchfork bifurcation curves. It is to
be noted that a new multistable region enclosed by the saddle-node bifurcation curves appears (denoted as R4 in Fig.~\ref{fig:4}(d) for $\alpha=1.5$), where IC, PS-I and PS-II states
coexist. Thus, it is evident that the asymmetry parameter in the pairwise coupling facilitates several interesting multistable states in the phase diagram mediated by
various types of bifurcations.
\begin{figure*}[ht!]
\hspace*{-1cm}
\centering
\includegraphics[width=19.5cm]{ho-5.eps}
\caption{Phase diagrams in the ($\alpha_1-\alpha_2$) plane for $\omega_0/\gamma~=~2$.
(a) $k/\gamma=4.5$, and (b) $k/\gamma=5.0$. Bifurcation curves and dynamical states are similar to those in
Fig.~\ref{fig:3}(a). }
\label{fig:5}
\end{figure*}
\subparagraph{Case IV ($\alpha_2~\ne$~~0; $\alpha_1$~=~~0): }
In order to analyze the effect of the asymmetry parameter in higher order interactions alone, we have fixed $\alpha_1=0$ and depicted the phase diagrams
in Figs.~\ref{fig:2}(a)-\ref{fig:2}(d) for $\alpha_2=0.1, 0.5, 1.0$ and $1.5$, respectively. The phase diagram (see Fig.~\ref{fig:2}(a)) for $\alpha_2=0.1$ is similar to
the phase diagram in Fig.~\ref{fig:1}, which is depicted for the choice $\alpha_1=\alpha_2=0$, but now with an enlarged bistable region R2 enclosed by saddle-node and homoclinic
bifurcation curves. Thus, it is again evident that
the asymmetry parameters largely contribute to the onset of multistability and facilitate the latter to a large extent. Note that the PS-II state and consequently the region R3 are
absent in the phase diagram for $\alpha_1=0$, which is actually facilitated by intermediate values of $\alpha_1$ (see Figs.~\ref{fig:4}(a) and ~\ref{fig:4}(b)).
Increasing $\alpha_2$ to $0.5$ (see Fig.~\ref{fig:2}(b)), the spread of the bistability region shrinks compared to that in Fig.~\ref{fig:2}(a). Further increase
in the value of the asymmetry parameter in the higher order coupling results in a decrease in the spread of R2 with the onset of R3, where PS-I and PS-II coexist,
via the saddle-node bifurcation as depicted in Fig.~\ref{fig:2}(c) for $\alpha=1.0$. For further larger values of $\alpha_2$, the spread of R1 and R3 in the phase
diagram decreases to a large extent resulting in the monostable regions of IC, PS-I, PS-II and SW states as depicted in Fig.~\ref{fig:2}(d) for $\alpha=1.5$.
The spread of R2 is completely wiped off from the phase diagram for $\alpha_2=1.5$. Thus, it is evident that large values of $\alpha_2$ facilitate the onset of
PS-II and eventually R3, while smaller values of $\alpha_2$ favor the spread of bistable regions to a large extent.
\subparagraph{Case V ($\alpha_1~>0~\&~\alpha_2>0$): }
Now, we consider $\alpha_1>0$ and $\alpha_2>0$ in order to analyze the dynamical states and their transitions due to the trade-off between the asymmetry parameters
in both pair-wise and higher order couplings. Phase diagrams in the asymmetry parameter ($\alpha_1, \alpha_2$) space for $\omega_0/\gamma=2$ and
for two different values of $k/\gamma$ are depicted in Figs.~\ref{fig:5}. The dynamical states and their bifurcation transitions are found to be similar to those
in the previous figures. For low values of $\alpha_2$, there is a transition from R2 to R3 via the homoclinic bifurcation and then to PS-II state via the saddle-node bifurcation and
finally to IC state through the pitchfork bifurcation as a function of $\alpha_1$.
For larger values of $\alpha_2$, there is a transition from SW to PS-II via the homoclinic bifurcation and then to IC via the pitchfork bifurcation as
a function of $\alpha_1$. Small to intermediate values of $\alpha_1$ and $\alpha_2$ favor bistable states R2 and R3, while larger values of the asymmetry parameters
$\alpha_1$ and/or $\alpha_2$ result in monostable states (see Figs.~\ref{fig:5} and ~\ref{fig:3}). Increasing $k/\gamma$ from $4.5$ to $5$ results in increase in the spread of bistable regions R2 and R3 (compare
Figs.~\ref{fig:5}(a) and ~\ref{fig:5}(b) ).
\section{Summary and Conclusion}
\label{sec:conclusions}
Higher order interactions have physical relevance in physics and neuroscience and they have gained recent interest in network theory. In this work, we have investigated the
phase diagrams of the Sakaguchi-Kuramoto model along with a higher order interaction, and unimodal and bimodal distributions of the natural frequencies of the individual phase oscillators.
We have also introduced asymmetry parameters both in the pairwise and higher order couplings to elucidate their role in the dynamical transitions in the phase diagram.
We have investigated the effects of five possible combinations of the asymmetry parameters $\alpha_1$ and $\alpha_2$ on the phase diagram along with the higher order interaction.
Using the Ott-Antonsen ansatz, we have obtained the coupled evolution equations corresponding to the macroscopic order parameters.
We have deduced the analytical stability condition for the linear stability of the incoherent state, resulting in the pitchfork bifurcation curve, using the governing equations of the
macroscopic order parameters. Further, we have also analytically deduced the Hopf bifurcation curve for $\alpha_1=0$, while the saddle-node and homoclinic bifurcation
curves are obtained using the software package XPPAUT.
The Sakaguchi-Kuramoto model along with a higher order interaction and unimodal frequency distribution displays only IC and SW states,
and bistability among them for $\alpha_1=1.0$ and $\alpha_2=0.0$. In contrast,
we have observed rich phase diagrams with dynamical states such as IC, PS-I, PS-II and SW states along with the
bistable (R1, R2 and R3) and multistable (R4) states with the bimodal frequency distribution.
In the absence of asymmetry parameters, higher order couplings favor the spread of the bistable states R1 and R2 to a large extent when compared to the
Sakaguchi-Kuramoto model with pairwise coupling alone and bimodal frequency distribution. Further, the asymmetry parameters favor the onset of the
bistable regions R3 and R4 which are generally absent in the Sakaguchi-Kuramoto model with pairwise coupling and bimodal frequency distribution. It is to be
noted that rather low values of the asymmetry parameter in the pairwise coupling for $\alpha_2=0$ and
relatively larger values of the asymmetry parameter in the higher order coupling for $\alpha_1=0$ favors the onset of PS-II state and eventually the region R3 in the phase diagrams.
However, very large values of both the asymmetry parameters render the phase diagram only with monostable dynamical states.
It is to be noted that there exists bistable region R1 even for $\omega_0/\gamma~<~1/\sqrt{3}$ in the phase diagrams, where the bimodal frequency distribution
breaks down to unimodal one, which is purely a manifestation of the higher order coupling as the bistable region R1, which has not yet been observed in the
Sakaguchi-Kuramoto model with pairwise coupling only along with unimodal frequency distribution. We sincerely believe that the above results, with rich phase diagrams
comprising of bistable and monostable regions of
the Sakaguchi-Kuramoto model due to the tradeoff between the asymmetry parameters and the higher order coupling, provide valuable new insights on
the dynamical nature of the model. Note that the presence of bistable (multistable) regions denote the regions across which abrupt dynamical transition occurs,
a typical nature of biological systems and, in particular, in neuroscience where bistability and fast switching between states are very relevant.
\section*{Acknowledgements}
M.M. thanks the Department of
Science and Technology, Government of India, for providing financial support through an INSPIRE Fellowship No.
DST/INSPIRE Fellowship/2019/IF190871.
DVS is supported by the DST-SERB-CRG Project under Grant No. CRG/2021/000816.
The work of V.K.C. is supported
by the SERB-DST-MATRICS Grant No. MTR/2018/000676 and DST-SERB-CRG Project under Grant No. CRG/2020/004353 and VKC wish to thank DST, New Delhi for computational
facilities under the DST-FIST programme (SR/FST/PS- 1/2020/135) to the Department of Physics. ML is supported by the DST-SERB National Science Chair program.
| 2024-02-18T23:40:18.977Z | 2022-03-08T02:38:43.000Z | algebraic_stack_train_0000 | 2,030 | 7,898 |
|
proofpile-arXiv_065-10035 | \section{Introduction}
In this paper,
we are interested in a kind of semilinear wave equations with the inverse-square potential
and small, spherically symmetric initial data, which has the form
\begin{Eq}\label{Eq:U_o}
\begin{cases}
\partial_t^2 U -\Delta U+Vr^{-2}U=|U|^p, \quad r=|x|,~(t,x)\in{\mathbb{R}}_+\times{\mathbb{R}}^n;\\
U(0,x)=\varepsilon U_0(r),\quad U_t(0,x)=\varepsilon U_1(r);
\end{cases}
\end{Eq}
where $p>1$,
$n\geq 2$,
$0<\varepsilon\ll 1$
and $V\geq -(n-2)^2/4$ is a constant.
We will study the long-time existence and global solvability of \eqref{Eq:U_o}.
Specifically,
setting $T_\varepsilon$ to be the lifespan of the solution to \eqref{Eq:U_o},
we want to know its relation with $n$, $V$, $p$ and $\varepsilon$.
When $V=0$,
this problem reduces to the well known \emph{Strauss} conjecture,
which has been extensively studied in a long history.
See, e.g., \cite{MR1481816}, \cite{MR1804518}, \cite{MR3247303}, \cite{MR3169791} and the references therein for more information.
Let $p_S(n)$ be the positive root of $h_S(p;n)=0$,
where
\begin{Eq*}
h_S(p;n):=(n-1)p^2-(n+1)p-2.
\end{Eq*}
From the early researches,
under some natural requirements of $(U_0,U_1)$,
it is known that
\begin{Eq*}
\kl\{\begin{aligned}
T_\varepsilon\approx &\varepsilon^{\frac{2p(p-1)}{h_S(p)}},&&\max(1,\frac{2}{n-1})<p<p_S;\\
\ln T_\varepsilon\approx& \varepsilon^{-p(p-1)},&&p=p_S;\\
T_\varepsilon=&\infty, && p>p_S.
\end{aligned}\kr.
\end{Eq*}
Here and in what follows,
we denote $x\lesssim y$ and $y\gtrsim x$ if $x\leq Cy$ for some $C>0$,
independent of $\varepsilon$,
which may change from line to line.
We also denote $x\approx y$ if $x\lesssim y\lesssim x$.
When there exists a potential, i.e.,
$V\neq 0$,
the problem becomes much more complicated.
This is partly because that the inverse-square potential is in the same scaling as the wave operator,
which means that it provides a comparable effect to the evolution of the solution.
Meanwhile, the extra singularity at the origin also needs to be taken care of.
The elliptic operator $-\Delta+V|x|^{-2}$ has been studied in several different equations related to physics and geometry,
such as in heat equations (see, e.g., \cite{MR1760280}), in quantum mechanics (see, e.g., \cite{MR0397192}), in Schr\"odinger equations and wave equations.
Among others, the \emph{Strichartz} estimates for wave equations with the inverse square potential have been well-developed in many works.
Such result was firstly developed in \cite{MR1952384} for the wave equations with radial data.
Shortly afterwards, the radial requirement was removed by \cite{MR2003358}.
A decade later, the \emph{Strichartz} estimates with angular regularity were developed in \cite{MR3139408}.
Despite these results, we expect that these kind of estimates still have room to improve and generalize.
Turn back to the equation \eqref{Eq:U_o}.
Note that the initial data of \eqref{Eq:U_o} are spherically symmetric,
which suggest that the solution $U$ is also spherically symmetric.
Let $$A:=2+\sqrt{(n-2)^2+4V},~u(t,r):=r^{\frac{n-A}{2}}U(t,x).$$
A formal calculation shows that $u$ satisfies the equation
\begin{Eq}\label{Eq:u_o}
\begin{cases}
\partial_t^2 u -\Delta_Au=r^{\frac{(A-n)p+n-A}{2}}|u|^p,\quad (t,r)\in {\mathbb{R}}_+^2,\\
u(0,x)=\varepsilon r^{\frac{n-A}{2}}U_0(r),\quad u_t(0,x)=\varepsilon r^{\frac{n-A}{2}}U_1(r),
\end{cases}
\end{Eq}
where $\Delta_A:=\partial_r^2+(A-1)r^{-1}\partial_r$.
When $A\in {\mathbb{Z}}_+$, the operator $\Delta_A$ agrees with the $A$-dimensional Laplace operator (for radial functions),
from which we consider the parameter $A$ as the spatial ``dimension" for the equation after the transformation.
The blow-up result of \eqref{Eq:U_o} has been systematically considered in the previous paper \cite{MR4130094} by the first author and his collaborators.
Here we define
\begin{Eq*}
h_F(p;n):=np-(n+2)
\end{Eq*}
with $p_F(n)$ be the root of $h_F(p;n)=0$, and use abbreviations
\begin{Eq*}
&p_d=p_d(A):=\frac{2}{A-1},\quad p_F=p_F((n+A-2)/2),\quad p_S=p_S(n),\\
&h_S=h_S(p;n),\quad h_F=h_F(p;(n+A-2)/2),
\end{Eq*}
if these do not lead to ambiguity.
Then, under some requirements of initial data, there exists a constant $C=C(p;n,A)$
such that when $(3-A)(A+n-2)<8$, where $p_d<p_F<p_S$, we have
\begin{Eq*}
T_\varepsilon\leq
\begin{cases}
C \varepsilon^{\frac{p-1}{h_F}},&p\leq p_d;\\
C\varepsilon^{\frac{2p(p-1)}{h_S}},&p_d<p<p_S;\\
\exp\kl(C \varepsilon^{-p(p-1)}\kr),&p=p_S.
\end{cases}
\end{Eq*}
When $(3-A)(A+n+2)=8$, where $p_d=p_F=p_S$, we have
\begin{Eq*}
T_\varepsilon\leq
\begin{cases}
C \varepsilon^{\frac{p-1}{h_F}},&p<p_F;\\
\exp\kl(C \varepsilon^{-(p-1)}\kr),&p=p_F.
\end{cases}
\end{Eq*}
When $(3-A)(A+n+2)>8$, where $p_d>p_F>p_S$, we have
\begin{Eq*}
T_\varepsilon\leq
\begin{cases}
C \varepsilon^{\frac{p-1}{h_F}},&p<p_F;\\
\exp\kl(C \varepsilon^{-(p-1)}\kr),&p=p_F.
\end{cases}
\end{Eq*}
This result suggests that two effects will impact the lifespan.
For simplicity we call one \emph{Strauss} effect
and the other \emph{Fujita} effect, since $p_S$ is the \emph{Strauss} exponent and $p_F$ is the \emph{Fujita} exponent. On the other hand, we remark that $p_F((n+A-2)/2)=p_G((n+A)/2)$, where $p_G(n)=\frac{n+1}{n-1}$ is the \emph{Glassey} exponent.
The \emph{Glassey} exponent appears in the wave equations with derivative nonlinearity $|\partial_t u|^p$, which suggests that there may exist some relation between the \emph{Glassey} conjecture (see, e.g., \cite{W15}) and our problem.
For the existence part, there are also a few studies of \eqref{Eq:U_o}.
Using \emph{Strichartz} estimates, the global existence result was shown in \cite{MR1952384, MR2003358} if
\begin{Eq*}
p\geq \frac{n+3}{n-1},\quad \frac{A-2}{2}>\frac{n-2}{2}-\frac{2}{p-1}+\max\kl\{\frac{1}{2p},\frac{1}{(n+1)(p-1)}\kr\}.
\end{Eq*}
Later, the result was further extended in \cite{MR3139408}, where the global result in the radial case was obtained for $1+\frac{4n}{(n+1)(n-1)}<p<\frac{n+3}{n-1}$,
\begin{Eq*}
&V>\max\kl\{\frac{1}{(n-1)^2}-\frac{(n-2)^2}{4},\frac{n}{q_0}\kl(\frac{n}{q_0}-n+2\kr),\kl(\frac{n}{r_0}-n\kr)\kl(\frac{n}{r_0}-2\kr)\kr\},\\
&q_0=\frac{(p-1)(n+1)}{2},\qquad r_0=\frac{(n+1)(p-1)}{2p}.
\end{Eq*}
However,
compared with the result of the problem without potential, in general,
it seems that the sharp result for \eqref{Eq:U_o}
could not be obtained by the \emph{Strichartz} estimates without weight.
On the other hand, there is also a gap between these results and the blow-up result we mentioned before.
Now,
we are in a juncture to state our main results in this paper.
Firstly, we give the definition of the solution,
and see \Se{Se:2} for further discussions.
\begin{definition}\label{De:U_w}
We call $U$ is a weak solution of \eqref{Eq:U_o} in $[0,T]\times {\mathbb{R}}^n$
if $U$ satisfies
\begin{Eq}\label{Eq:U_i}
\int_0^T\int_{{\mathbb{R}}^n} |U|^p\Phi \d x\d t
=&\int_0^T\int_{{\mathbb{R}}^n} U \kl(\partial_t^2-\Delta+\frac{V}{r^2}\kr)\Phi \d x\d t\\
&-\varepsilon\int_{{\mathbb{R}}^n} (U_1\Phi(0,x)-U_0\partial_t\Phi(0,x))\d x,
\end{Eq}
for any $\Phi(t,x)\in \kl\{r^{\frac{A-n}{2}}\phi(t,x):\phi\in C_0^\infty((-\infty,T)\times {\mathbb{R}}^n)\kr\}$.
\end{definition}
For convenience we introduce the notations
\begin{Eq*}
p_m:=\frac{n+1}{n-1},\qquad
p_M:=\begin{cases}\frac{n+1}{n-A}&n>A\\\infty&n\leq A\end{cases},\qquad
p_t:=\frac{n+A}{n-1},\qquad
p_{conf}:=\frac{n+3}{n-1}.
\end{Eq*}
Then, we give the existence results for $A\in[2,3]$.
\begin{theorem}\label{Th:M_1}
Set $2\leq n$,
$2\leq A\leq 3$ and $p_m<p<p_M$.
Assume that the initial data satisfy
\begin{Eq}\label{Eq:ir_1}
\|r^\frac{n-A+2}{2}U_0'(r)\|_{L_r^\infty}+\|r^\frac{n-A}{2}U_0(r)\|_{L_r^\infty}+\|r^\frac{n-A+2}{2}U_1(r)\|_{L_r^\infty}<\infty,
\end{Eq}
and supported in $[0,1)$, where $L_r^p$ stands for $L^p((0,\infty),\d r)$.
Then, there exists an $\varepsilon_0>0$ and a constant $c=c(p;n,A)$,
such that for any $0<\varepsilon<\varepsilon_0$, there is a weak solution $U$ of \eqref{Eq:U_o} in $[0,T_*)\times {\mathbb{R}}^n$ which satisfies
\begin{Eq*}
r^{\frac{n-A}{2}}U\in L_{loc;t,x}^\infty ([0,T_*)\times {\mathbb{R}}^n).
\end{Eq*}
Where, when $(3-A)(A+n+2)<8$, we have $p_d<p_F<p_S$, then
\begin{Eq}\label{Eq:Main_7}
T_*=\begin{cases}
c\varepsilon^\frac{p-1}{h_F},&p<p_d;\\
c\varepsilon^\frac{p-1}{h_F} |\ln\varepsilon|^{\frac{1}{h_F}},&p=p_d;\\
c\varepsilon^\frac{2p(p-1)}{h_S},&p_d<p<p_S;\\
\exp\kl(c\varepsilon^{p(1-p)}\kr),&p=p_S;\\
\infty,&p>p_S.
\end{cases}
\end{Eq}
When $(3-A)(A+n+2)=8$, we have $p_d=p_F=p_S$, then
\begin{Eq}\label{Eq:Main_8}
T_*=\begin{cases}
c\varepsilon^\frac{p-1}{h_F},&p<p_d;\\
\exp\kl(c\varepsilon^{\frac{1-p}{2}}\kr),&p=p_d;\\
\infty,&p>p_d.
\end{cases}
\end{Eq}
When $(3-A)(A+n+2)>8$, we have $p_d>p_F>p_S$, then
\begin{Eq}\label{Eq:Main_9}
T_*=\begin{cases}
c\varepsilon^\frac{p-1}{h_F},&p<p_F;\\
\exp\kl(c\varepsilon^{1-p}\kr),&p=p_F;\\
\infty,&p>p_F.
\end{cases}
\end{Eq}
\end{theorem}
Next, we give the existence results for $A\in[3,\infty)$.
\begin{theorem}\label{Th:M_2}
Set $2\leq n$, $A\geq 3$ and $1<p<p_{conf}$
and define $T_*$ by
\begin{Eq}\label{Eq:Main_6}
T_*=\begin{cases}
c\varepsilon^\frac{2p(p-1)}{h_S},&1<p<p_S;\\
\exp\kl(c\varepsilon^{p(1-p)}\kr),&p=p_S;\\
\infty,&p>p_S,
\end{cases}
\end{Eq}
which is the same as \eqref{Eq:Main_7} since that $p_d\leq 1$ when $A\geq3$.
Assume that $1<p\leq p_m$ and
the initial data $(U_0,U_1)$ satisfy
\begin{Eq}\label{Eq:ir_21}
\|r^\frac{n-1}{2}U_0(r)\|_{L_r^p}+\|r^\frac{n+1}{2}U_1(r)\|_{L_r^p}<\infty,
\end{Eq}
and supported in $[0,1)$.
Then there exists an $\varepsilon_0>0$ and a constant $c$,
such that for any $\varepsilon<\varepsilon_0$,
\eqref{Eq:U_o} has a weak solution in $[0,T_*]\times {\mathbb{R}}^n$ verifying
\begin{Eq*}
\|(1+t)^{\frac{(n-1)p-n-1}{2p}}r^{\frac{n+1}{2p}}U\|_{L_t^\infty L_r^p ([0,T_*]\times {\mathbb{R}}_+)}<\infty,
\end{Eq*}
with $T_*$ defined in \eqref{Eq:Main_6}.
Assume that $p_m\leq p< p_S$, the initial data satisfy \eqref{Eq:ir_21} and
\begin{Eq}\label{Eq:ir_22}
\|r^{\frac{n-1}{2}+\frac{1}{p}}U_0(r)\|_{L_r^\infty}+\|r^{\frac{n+1}{2}+\frac{1}{p}}U_1(r)\|_{L_r^\infty}<\infty,
\end{Eq}
with no compact support requirement.
Then there exists an $\varepsilon_0>0$ and a constant $c$,
such that for any $\varepsilon<\varepsilon_0$,
\eqref{Eq:U_o} has a weak solution in $[0,T_*]\times {\mathbb{R}}^n$ verifying
\begin{Eq*}
\|t^{\frac{(n-1)p-n-1}{2p}}r^{\frac{n+1}{2p}}U\|_{L_t^\infty L_r^p ([0,T_*]\times {\mathbb{R}}_+)}<\infty,
\end{Eq*}
with $T_*$ defined in \eqref{Eq:Main_6}.
Assume that $p=p_S$,
and the initial data satisfy \eqref{Eq:ir_21} and \eqref{Eq:ir_22} for $p=p_S$ as well as some $p>p_S$.
Then there exists an $\varepsilon_0>0$ and a constant $c$,
such that for any $\varepsilon<\varepsilon_0$,
\eqref{Eq:U_o} has a weak solution in $[0,T_*]\times {\mathbb{R}}^n$ verifying
\begin{Eq*}
\|r^{\frac{n+1}{2p_S}}U\|_{L_t^{p_S^2}L_r^{p_S}([0,1]\times {\mathbb{R}}_+)}+
\|t^{\frac{1}{p_S^2}}r^{\frac{n+1}{2p_S}}U\|_{L_t^\infty L_r^{p_S} ([1,T_*]\times {\mathbb{R}}_+)}<\infty,
\end{Eq*}
with $T_*$ defined in \eqref{Eq:Main_6}.
Assume that $p>p_S$ and the initial data satisfy
\begin{Eq*}
\|r^\frac{n-1}{2}U_0(r)\|_{L_r^q}+\|r^\frac{n+1}{2}U_1(r)\|_{L_r^q}<\infty,\qquad q:=\frac{2(p-1)}{(n+3)-(n-1)p}.
\end{Eq*}
Then there exists an $\varepsilon_0>0$,
such that for any $\varepsilon<\varepsilon_0$,
\eqref{Eq:U_o} has a weak solution in ${\mathbb{R}}_+\times {\mathbb{R}}^n$ verifying
\begin{Eq*}
\|r^{\frac{n+1}{2p}}U\|_{L_t^{pq} L_r^p({\mathbb{R}}_+\times {\mathbb{R}}_+)}+\|r^{\frac{n-1}{2}}U\|_{L_t^{\infty} L_r^q({\mathbb{R}}_+\times {\mathbb{R}}_+)}<\infty.
\end{Eq*}
\end{theorem}
\begin{remark}
Here we use a graph with $n\in[4,8]$ as an example
to describe the results we got.
\begin{figure}[H]
\centering
\includegraphics[width=0.99\textwidth]{n=7.png}
\end{figure}
The white area stands for the region that the solution is global,
the light gray area ($p>p_d$) stands for the region that the \emph{Strauss} effect plays role,
the dark gray area ($p<p_d$) stands for the region that \emph{Fujita} effect plays role,
and the chessboard area stands for the region that we can not deal with due to the technical difficulty.
When $n\in[2,3]$,
we find $\frac{3n-1}{n+1}\leq 2$,
which means that the dark gray area does not exist for $p\geq p_m=\frac{n+1}{n-1}$.
When $n=2$ we have $p_M=\infty$ for all $A\geq 2$.
This means that the lower right chessboard area does not exist.
When $n\in[9,\infty)$, we find $1+\frac{4}{n}>\frac{n+1}{n-2}$,
which means that the dark gray area will be slightly blocked by the lower right chessboard area.
Here we list these situations as the figures below.
\begin{figure}[H]
\centering
\includegraphics[width=0.99\textwidth]{n=2,3,10.png}
\end{figure}
\end{remark}
\begin{remark}
The nonlinear term $|U|^p$ in \eqref{Eq:U_o} can be replaced by any $F_p(U)$ which satisfies
\begin{Eq*}
|F_p(U)|\lesssim |U|^p,\qquad |F_p(U_1)-F_p(U_2)|\lesssim |U_1-U_2|\max(|U_1|,|U_2|)^{p-1},
\end{Eq*}
and typical examples include $F_p(U)=\pm|U|^p$ and $F_p(U)=\pm|U|^{p-1}U$.
The only difference is that the constants in the result and proof need to be changed.
\end{remark}
In lower dimension, the weighted $L^\infty$ norm estimate,
which firstly appeared in \cite{MR535704},
is very useful to prove the long-time existence result.
In \cite{MR2025737}, the authors showed long-time existence results for a two-dimensional wave system,
where they use a trick that they take different weights in different zones.
In this paper, we further develop such method, adapted for the wave equations with potential,
and finally show the long-time existence result for $A\in[2,3]$.
On the one hand,
our result is sharp in general, in the sense that
our lower bound of the lifespan has the same order as the upper bound estimates as appeared in the blow-up results, except for the borderline case $p=p_d$.
On the other hand, we notice that $A=2$, which means $V=-(n-2)^2/4$, is an extreme case to the operator $-\Delta+Vr^{-2}$.
In this case, the operator is still non-negative but not positive any more, which
makes the implementation of the classical energy method more difficult.
However, our approach could handle this extreme case as well as the usual case that $V>-(n-2)^2/4$, without any additional difficulties.
In higher dimension ($n\geq 3$),
it is well known that the weighted \emph{Strichartz} estimates
is a helpful tool for the Strauss conjecture, particularly for the high dimensional case (see, e.g., \cite{MR1408499},
\cite{MR1481816} and \cite{MR1804518}).
In this paper, we adapt the approach of \cite{MR1408499} to the `fractional dimension' $A\geq 3$,
and give the long-time existence result for $A\in[3,\infty)$, which gives the sharp lower bound of the lifespan.
After comparing all the results we knew, we find that
the determination of the exact lifespan can be considered by a competition between
the \emph{Strauss} effect and the \emph{Fujita} effect.
When $p>p_d$, the \emph{Strauss} effect is stronger,
where the final result is only determined by the \emph{Strauss} exponent.
When $p<p_d$, the \emph{Fujita} effect is stronger,
the final result is only determined by the \emph{Fujita} exponent.
Compared with the results for the problem without potential,
it seems that the requirement $p>p_m$ in \Th{Th:M_1} is only a technical restriction.
Also, compared with the result of $2$-dimensional \emph{Strauss} conjecture with $p=2$ (though $2\leq p_m(2)=3$),
we expect that the both the lower bound and the upper bound of lifespan for $p=p_d$ can be further improved.
On the other hand, it will be interesting to investigate
the problem with non-radial data, as well as the problem with more general potential functions.
It is known that when $n=3, 4$ and the potential function is of short range, the similar long-time existence results (including non-radial case) as in
\Th{Th:M_2} are available in \cite{MW17}.
When the potential function has asymptotic behavior $Vr^{-2}$ with $V>0$, the subcritical blow-up result ($p<\max(p_S, p_F)$) was recently obtained in \cite{lai2021lifespan}.
The existence theory for the corresponding problem remains largely open.
Another interesting problem is whether or not the Cauchy problem \eqref{Eq:U_o} admits global solutions for initial data with lowest possible regularity.
For example, when $V=0$ and $2\leq n\leq 4$, such a result is available for $p\in(p_S,p_{conf}]$
with small, spherically symmetric data in the scale-invariant $(\dot H^{s_c}\times \dot H^{s_c-1})$ space ($s_c=n/2-2/(p-1)$).
See \cite{MR2455195,MR2333654,MR2769870} for more discussion.
Here we should remark that the global result in \Th{Th:M_2} reaches the lowest regularity requirement (in the sense of scale invariance, though not in $\dot H^s$ space), but \Th{Th:M_1} still has room for improvement.
The rest of the paper is organized as follows. In \Se{Se:2} we give a detailed discussion of \eqref{Eq:u_o} and its solution.
In \Se{Se:3}, we restrict $A\in[2,3]$ and show the long-time existence of the solution by weighted $L^\infty$ norm estimate.
In \Se{Se:4}, we move to situation $A\in[3,\infty)$ and establish the long-time existence result through weighted \emph{Strichartz} type estimate.
\section{Some preparations}\label{Se:2}
In this section,
we transfer \eqref{Eq:U_o} into the equivalent equation \eqref{Eq:u_o},
and explain the rationality of \De{De:U_w}.
After that, we show the formula of the solution and analyze the properties of this solution.
\subsection{The definition of weak solution}
As we said before, after introducing $u(t,r):=r^{\frac{n-A}{2}}U(t,x)$,
a formal calculation shows that $u$ satisfied the equation \eqref{Eq:u_o}.
We pause here and consider its linear form equation
\begin{Eq}\label{Eq:u_l}
\begin{cases}
\partial_t^2 u -\Delta_Au=F(t,r),\quad r\in {\mathbb{R}}_+\\
u(0,r)=f(r),\quad u_t(0,r)=g(r),
\end{cases}
\end{Eq}
with $f$, $g$, $F$ good enough.
When $A\in {\mathbb{Z}}_+$,
equation \eqref{Eq:u_l} can be considered as an $A$-dimensional spherically symmetric wave equation,
where $u(t,r)$ is a classical solution in $[0,T]$
if $u(t,r)$ satisfies \eqref{Eq:u_l} and $u(t,|x|)\in C^2([0,T]\times {\mathbb{R}}^A)$.
Thus,
for general situation,
we should say $u$ is a classical solution of \eqref{Eq:u_l} in $[0,T]$
if $u\in C^2([0,T]\times {\mathbb{R}}_+)$, $\partial_ru(t,0)=0$ and $u$ satisfies \eqref{Eq:u_l} point wise.
Here we give a quick proof to show that such classical solution is unique.
When $f,g,F=0$, multiplying $r^{A-1}u_t$ to both sides of \eqref{Eq:u_l} and integrating them in $\Omega:=\{(t,r):0<t<T, 0<r<R+T-t\}$, we see
\begin{Eq*}
0=\frac{1}{2}\kl.\int_0^Rr^{A-1}\kl(u_t^2+u_r^2\kr)\d r\kr|_{t=T}+\frac{1}{2\sqrt{2}}\int\limits_{t+r=T+R\atop 0<t<T}r^{A-1}(u_t-u_r)^2\d \sigma_{t,r}.
\end{Eq*}
This gives the uniqueness.
After the discussion of the classical solution,
we naturally say $u$ is the weak solution of \eqref{Eq:u_l} if $u$ satisfies
\begin{Eq}\label{Eq:u_i}
\int_0^T\int_0^\infty F\phi r^{A-1}\d r\d t
=&\int_0^T\int_0^\infty u \kl(\partial_t^2-\Delta_A\kr)\phi r^{A-1}\d r\d t\\
&-\int_0^\infty (g\phi(0,r)-f\partial_t\phi(0,r))r^{A-1}\d r,
\end{Eq}
for any $\phi(t,r)\in C_0^\infty\kl((-\infty,T)\times {\mathbb{R}}\kr)$
with $\partial_r^{1+2k} \phi(t,0)=0$ for any $k\in {\mathbb{N}}_0$.
Also, set $u=r^{\frac{n-A}{2}}U$, $\phi=r^\frac{n-A}{2}\Phi$,
$f=\varepsilon r^{\frac{n-A}{2}} U_0$, $g=\varepsilon r^{\frac{n-A}{2}} U_1$
and $F=r^{\frac{(A-n)p+n-A}{2}}|u|^p$,
it is obvious that $u$ satisfies \eqref{Eq:u_i} is equivalent to that
$U$ satisfies \eqref{Eq:U_i} with
$\Phi(t,x)\in \kl\{r^{\frac{A-n}{2}}\phi(t,x):\phi\in C_0^\infty((-\infty,T)\times {\mathbb{R}}^n)\kr\}$.
That's the reason we use \De{De:U_w} as the definition of weak solution of \eqref{Eq:U_o}.
\subsection{The formula of classical solution}
In this section we are going to give the formula of solution to \eqref{Eq:u_l}.
We denote by $u_g$, $u_f$ and $u_F$ the solution of \eqref{Eq:u_l} with only $g\neq 0$, $f\neq 0$ and $F\neq 0$, respectively.
\begin{lemma}\label{Le:u_e}
Assume that $f,g, F$ are smooth enough. Then, the classical solution of \eqref{Eq:u_l} is $u=u_f+u_g+u_F$ with
\begin{Eq*}
u_g=&r^{\frac{1-A}{2}} \int_{0}^{t+r}\rho ^{\frac{A-1}{2}}g(\rho) I_A(\mu) \d \rho,\quad
\mu=\frac{r^2+\rho^2-t^2}{2r\rho},\\
u_f=&r^{\frac{1-A}{2}} \partial_t\int_{0}^{t+r}\rho ^{\frac{A-1}{2}}g(\rho) I_A(\mu) \d \rho,\quad
\mu=\frac{r^2+\rho^2-t^2}{2r\rho},\\
u_F=&r^{\frac{1-A}{2}} \int_0^t\int_{0}^{t-s+r}\rho ^{\frac{A-1}{2}}F(s,\rho) I_A(\mu) \d \rho\d s,\quad
\mu=\frac{r^2+\rho^2-(t-s)^2}{2r\rho},\\
I_A(\mu):=&\frac{2^{\frac{1-A}{2}}}{\Gamma\kl(\frac{A-1}{2}\kr)} \int_{-1}^{1} \mathcal{X}_+^{\frac{1-A}{2}}\kl(\lambda-\mu\kr)\sqrt{1-\lambda^2}^{A-3}\d \lambda.
\end{Eq*}
\end{lemma}
\begin{remark}
Here $\mathcal{X}_+^\alpha$ is a distribution, which has the expression
\begin{Eq*}
\mathcal{X}_+^\alpha(x)=\begin{cases}
0,&x<0,~\alpha>-1,\\
\frac{x^\alpha}{\Gamma(\alpha+1)},& x>0,~\alpha>-1,\\
\frac{\d~}{\d x}\mathcal{X}_+^{\alpha+1}(x), &\alpha\leq -1,\\
\end{cases}
\end{Eq*}
with $\Gamma$ the Gamma function and $\frac{\d~}{\d x}$ the weak derivative.
\end{remark}
\begin{remark}
Consider $\mu=\frac{r^2+\rho^2-t^2}{2r\rho}$. When $r> t$, we have
\begin{Eq*}
\mu|_{0<\rho<r-t}~>~\mu|_{\rho=r-t}=1~>~\mu|_{r-t<\rho<t+r}(>0)~<~\mu|_{\rho=t+r}=1~<~\mu|_{t+r<\rho},
\end{Eq*}
and when $r< t$, we have
\begin{Eq*}
\mu|_{0<\rho<t-r}~<~\mu|_{\rho=t-r}=-1~<~\mu|_{t-r<\rho<t+r}~<~\mu|_{\rho=t+r}=1~<~\mu|_{t+r<\rho}.
\end{Eq*}
By \Le{Le:I_p} below, the $\int_0^{t+r}$ in the formula of $u_g$ and $u_f$ in \Le{Le:u_e} can be replaced by $\int_{\max\{0,r-t\}}^{t+r}$, and $\int_0^{t-s+r}$ in $u_F$ can be replaced by $\int_{\max\{0,r+s-t\}}^{t-s+r}$ .
\end{remark}
To show \Le{Le:u_e}, we need to explore some properties of $I_A(\mu)$.
\begin{lemma}\label{Le:I_p}
For $A>1$ and $I_A(\mu)$ defined in \Le{Le:u_e}, we have
\begin{Eq}
I_A(1-)=\frac{1}{2},\qquad I_A(\mu)=0~for~\mu>1.\label{Eq:I_p_1}
\end{Eq}
Moreover,
for $A\not\in 1+2{\mathbb{Z}}_+$ with some constants $C_0$ and $C_{1}$ depending on $A$,
we have
\begin{alignat}{2}
|\partial_{\mu}^mI_A(\mu)|\lesssim&(1-\mu)^{\frac{1-A}{2}-m}, &\qquad& \mu\leq -2,m=0,1;\label{Eq:I_p_2}\\
I_A(\mu)=&C_0\ln|1+\mu|+O(1),&\qquad& -2<\mu<1;\label{Eq:I_p_3}\\
\partial_\mu I_A(\mu)= &C_1(1+\mu)^{-1}+O\kl(|\ln|1+\mu||+1\kr),&\qquad& -2<\mu<1.\label{Eq:I_p_4}
\end{alignat}
On the other hand, for $A\in 1+2{\mathbb{Z}}_+$,
we have
\begin{alignat}{2}
I_A(\mu)=&0,&\qquad &\mu<-1;\label{Eq:I_p_5}\\
|\partial_\mu^{m}I_A(\mu)|\lesssim& 1,&\qquad &-1<\mu<1,~m=0,1.\label{Eq:I_p_6}
\end{alignat}
\end{lemma}
\subsection{Proof of \Le{Le:u_e}}\label{Pf_u_e}
Here we only show the proof of $u=u_g$ with $f=F=0$, the other formulas can be demonstrated by a direct calculation and \emph{Duhamel}'s principle. Without loss of generality we only deal with the case $A\not\in {\mathbb{Z}}_+$.
\setcounter{part0}{0}
\value{parta}[Alternative expressions of $u_g$]
Before the proof, we give an alternative expressions of $u_g$ constructed in \Le{Le:u_e}.
We first introduce a change of the variables
\begin{Eq*}
&(\rho,\lambda)=\kl(\sqrt{r^2+\tilde \rho^2-2r\tilde \rho\tilde\lambda},~\frac{r-\tilde \rho\tilde\lambda}{\sqrt{r^2+\tilde \rho^2-2r\tilde \rho\tilde\lambda}}\kr)\\
\Leftrightarrow&(\tilde\rho,\tilde\lambda)=\kl(\sqrt{r^2+ \rho^2-2r \rho\lambda},~\frac{r- \rho\lambda}{\sqrt{r^2+ \rho^2-2r \rho\lambda}}\kr).
\end{Eq*}
A direct calculation shows that the map $(\rho,\lambda)\mapsto(\tilde\rho,\tilde\lambda)$ satisfies the relation
\begin{Eq*}
\kl|\frac{\d(\rho,\lambda)}{\d(\tilde\rho,\tilde\lambda)}\kr|=\frac{\tilde\rho^2}{r^2+\tilde \rho^2-2r\tilde \rho\tilde\lambda}=\frac{\tilde\rho^2}{\rho^2},\qquad \rho^2(1-\lambda^2)=\tilde\rho^2(1-\tilde\lambda^2),
\end{Eq*}
and is a bijection from $(0,\infty)\times(-1,1)$ to itself.
For $A=1+2k+2\theta$ with $k\in {\mathbb{N}}_0$ and $\theta\in(0,1)$, we substitute $I_A(\mu)$ into $u_g$.
Noticing $\mathcal{X}_+^{\alpha}$ is a homogeneous distribution of degree $\alpha$, we find
\begin{Eq*}
u_g=&\frac{1}{\Gamma\kl(\frac{A-1}{2}\kr)} \int_{0}^{\infty}\int_{-1}^{1} g(\rho)\mathcal{X}_+^{\frac{1-A}{2}}\kl(t^2-r^2-\rho^2+2r\rho\lambda\kr)\rho^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\rho.\\
=&\frac{1}{\Gamma\kl(k+\theta\kr)} \int_{0}^{\infty}\int_{-1}^{1} g\kl(\sqrt{r^2+\tilde\rho^2-2r\tilde\rho\tilde\lambda}\kr)\\
&\phantom{\frac{1}{\Gamma\kl(k+\theta\kr)} \int_{0}^{\infty}\int_{-1}^{1}}
\times\mathcal{X}_+^{-k-\theta}\kl(t^2-\tilde\rho^2\kr)\tilde\rho^{A-1}\sqrt{1-\tilde\lambda^2}^{A-3}\d \tilde\lambda\d\tilde\rho\\
=&\frac{1}{\Gamma\kl(k+\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^k\int_{0}^{\infty}\int_{-1}^{1} g\kl(\sqrt{r^2+\tilde\rho^2-2r\tilde\rho\tilde\lambda}\kr)\\
&\phantom{\frac{1}{\Gamma\kl(k+\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^k\int_{0}^{\infty}\int_{-1}^{1}}
\times\mathcal{X}_+^{-\theta}\kl(t^2-\tilde\rho^2\kr)\tilde\rho^{A-1}\sqrt{1-\tilde\lambda^2}^{A-3}\d \tilde\lambda\d\tilde\rho.
\end{Eq*}
Set $\tilde\rho=t\sigma$ and $\tilde\lambda=\lambda$. Considering the definition of $\mathcal{X}_+^{-\theta}$ we finally reach
\begin{Eq}\label{Eq:u_g_c}
u_g=&\frac{1}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^k\Bigg(t^{1+2k}\int_{0}^{1}\int_{-1}^{1} \frac{g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^\theta}\\
&\phantom{\frac{1}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^k\Bigg(t^{1+2k}\int_{0}^{1}\int_{-1}^{1}}\times\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma\Bigg).
\end{Eq}
\value{parta}[Differentiability, boundary requirement and initial requirement]
Now we begin the proof.
Firstly, using the expression we just obtained,
we can easily check that $u\in C^2({\mathbb{R}}_+^2)$ while $g\in C_0^\infty((0,\infty))$.
We can also calculate that
\begin{Eq*}
\partial_ru=&\frac{1}{\Gamma\kl(k+\theta\kr)\Gamma(1-\theta)} \kl(\frac{\partial_t}{2t}\kr)^kt^{1+2k}\int_{0}^{1}\int_{-1}^{1}\frac{r-t\sigma\lambda}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}} \frac{g'\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^\theta}\\
&\phantom{\frac{1}{\Gamma\kl(k+\theta\kr)\Gamma(1-\theta)} \kl(\frac{\partial_t}{2t}\kr)^kt^{1+2k}\int_{0}^{1}\int_{-1}^{1}}
\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Let $r=0$.
Since the integrand is an odd function of $\lambda$,
such $u$ satisfies the boundary requirement.
To check the initial conditions we temporarily use the original expression in \Le{Le:u_e}.
Using \Le{Le:I_p} we know $I_A(\mu)=0$ when $\mu>1$, which happens when $\rho<r-t$ with $r>t$.
Then for any $r>t>0$, we have
\begin{Eq*}
u(t,r)=&r^{\frac{1-A}{2}} \int_{r-t}^{t+r}\rho^{\frac{A-1}{2}}g(\rho) I_A\kl(\mu\kr) \d \rho\\
u_t(t,r)=&r^{\frac{1-A}{2}}\kl((t+r)^{\frac{A-1}{2}}g(t+r)+(r-t)^{\frac{A-1}{2}}g(r-t)\kr)I_A(1-)\\
&+r^{\frac{1-A}{2}} \int_{r-t}^{t+r}\rho^{\frac{A-1}{2}}g(\rho) \frac{-t}{r\rho}I_A'\kl(\mu\kr) \d \rho.
\end{Eq*}
Let $t\rightarrow 0$.
Using \Le{Le:I_p} again
we find $u(0,r)=0$ and $u_t(0,r)=g(r)$.
\value{parta}[Differential equation requirement]
Finally,
we need to check that $u$ satisfies \eqref{Eq:u_l}.
By a calculation trick
\begin{Eq*}
\partial_t^2 \kl(\frac{\partial_t}{t}\kr)^{k}t^{1+2k}=\kl(\frac{\partial_t}{t}\kr)^{k+1}t^{2k+2}\partial_t,
\end{Eq*}
(see e.g. \cite[Lemma 2 in Section 2.4]{MR2597943}) we calculate that
\begin{Eq*}
\partial_t^2 u=&\frac{2}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^{k+1}\kl(t^{2k+2}w_1\kr),\\
w_1:=&\int_{0}^{1}\int_{-1}^{1}\frac{t\sigma^2-r\sigma\lambda}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}} \frac{g'\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^\theta}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
On the other hand, a similar process as that deduced \eqref{Eq:u_g_c} also shows
\begin{Eq*}
u=&\frac{1}{\Gamma\kl(k+\theta\kr)\Gamma\kl(2-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^{k+1}\kl(t^{2k+2} \tilde w_2\kr),\\
\tilde w_2:=&t\int_{0}^{1}\int_{-1}^{1} \frac{g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta-1}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Then, we see
\begin{Eq*}
\partial_r\tilde w_2=&t\int_{0}^{1}\int_{-1}^{1}\frac{r-t\sigma\lambda}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}}\frac{g'\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta-1}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma\\
=&-\int_{0}^{1}\int_{-1}^{1}\frac{\partial_\lambda g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta-1}}\sigma^{A-2}\sqrt{1-\lambda^2}^{A-1}\d \lambda\d\sigma\\
&-\int_{0}^{1}\int_{-1}^{1}\lambda \frac{\partial_\sigma g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta-1}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Using integration by parts, we get
\begin{Eq*}
\partial_r\tilde w_2=&-2(1-\theta)\int_{0}^{1}\int_{-1}^{1}\lambda \frac{g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta}}\sigma^{A}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Thus we have
\begin{Eq*}
\partial_r u=&\frac{2}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^{k+1}\kl(t^{2k+2}w_2\kr)\\
w_2:=&-\int_{0}^{1}\int_{-1}^{1}\lambda \frac{g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta}}\sigma^{A}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Taking the derivative again, we also have
\begin{Eq*}
\partial_r^2 u=&\frac{2}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^{k+1}\kl(t^{2k+2}w_3\kr)\\
w_3:=&\int_{0}^{1}\int_{-1}^{1}\frac{t\sigma^2\lambda^2-r\sigma\lambda}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}} \frac{g'\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\theta}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma.
\end{Eq*}
Gluing $\partial_t^2u$, $\partial_r^2 u$ and $\partial_r u$ together, we finally calculate
\begin{Eq*}
&(\partial_t^2-\partial_r^2-(A-1)r^{-1}\partial_r)u\\
=&\frac{2r^{-1}}{\Gamma\kl(k+\theta\kr)\Gamma\kl(1-\theta\kr)} \kl(\frac{\partial_t}{2t}\kr)^{k+1}\kl(t^{2k+2}\kl(rw_1-rw_3-(A-1)w_2\kr)\kr)
\end{Eq*}
where
\begin{Eq*}
rw_1-rw_3=&-\int_{0}^{1}\int_{-1}^{1}\frac{\partial_\lambda g\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^\theta}\sigma^{A}\sqrt{1-\lambda^2}^{A-1}\d \lambda\d\sigma\\
=&(A-1)w_2.
\end{Eq*}
This finishes the proof.
\subsection{Proof of \Le{Le:I_p}}
We begin with the second half of \eqref{Eq:I_p_1},
it is trivial since $\mathcal{X}_+^{\frac{1-A}{2}}\kl(\lambda-\mu\kr)=0$ for $\lambda\in[-1,1]$ and $\mu>1$.
As for other results, we need to divide $A$ into two cases.
\setcounter{part0}{0}
\value{parta}[$A$ is not odd]
We begin with the case that $A$ is not odd, i.e. $A=1+2k+2\theta$ with $k\in {\mathbb{N}}_0$ and $0<\theta<1$.
By definition we see that when $\mu\leq -2$ and $\lambda\in[-1,1]$, we have
\begin{Eq*}
\partial_\mu^m \mathcal{X}_+^{\frac{1-A}{2}}\kl(\lambda-\mu\kr)\approx (1-\mu)^{\frac{1-A}{2}-m},
\end{Eq*}
which gives \eqref{Eq:I_p_2}. When $-1<\mu<1$, $I_A$ has the formula
\begin{Eq*}
I_A(\mu)=&\frac{2^{-k-\theta}}{\Gamma\kl(k+\theta\kr)} \int_{-1}^{1} \partial_\lambda^k\mathcal{X}_+^{-\theta}\kl(\lambda-\mu\kr)(1-\lambda^2)^{k+\theta-1}\d \lambda\\
=&\frac{2^{-k-\theta}}{\Gamma\kl(k+\theta\kr)}\int_{-1}^{1} \mathcal{X}_+^{-\theta}\kl(\lambda-\mu\kr)(-\partial_\lambda)^k(1-\lambda^2)^{k+\theta-1}\d \lambda\\
=&\frac{2^{-k-\theta}}{\Gamma\kl(k+\theta\kr)\Gamma(1-\theta)}\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\sum_{j=0}^{\lfloor k/2\rfloor }C_{j,k,\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda
\end{Eq*}
with some constants $C_{j,k,\theta}$. Here $\lfloor a\rfloor$ stands for the integer part of $a$.
\subpart[$\mu$ close to $1-$]
Firstly we let $\mu$ close to $1-$. Introducing $\lambda=(1-\mu)\sigma+\mu$, we have
\begin{Eq*}
&\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
=&(1-\mu)^j\int_0^1 \sigma^{-\theta}(1-\sigma)^{j+\theta-1}\kl(\sigma+\mu-\mu\sigma\kr)^{k-2j}\kl(\sigma+\mu-\mu\sigma+1\kr)^{j+\theta-1}\d\sigma.
\end{Eq*}
Let $\mu\rightarrow 1-$. Using dominated convergence theorem, we find the limit is nonzero only if $j=0$, where
\begin{Eq*}
&\lim_{\mu\rightarrow 1-}\int_0^1 \sigma^{-\theta}(1-\sigma)^{\theta-1}\kl(\sigma+\mu-\mu\sigma\kr)^{k}\kl(\sigma+\mu-\mu\sigma+1\kr)^{\theta-1}\d\sigma\\
=&2^{\theta-1}\int_0^1 \sigma^{-\theta}(1-\sigma)^{\theta-1}\d\sigma.
\end{Eq*}
Now, we calculate
\begin{Eq*}
C_{0,k,\theta}=\begin{cases}
1,&k=0;\\
2^k(k+\theta-1)(k+\theta-2)\cdots \theta,&k>0,
\end{cases}
\end{Eq*}
then
\begin{Eq*}
\lim_{\mu\rightarrow 1-}I_A(\mu)=&\frac{1}{2\Gamma\kl(\theta\kr)\Gamma(1-\theta)}\int_{0}^{1} \sigma^{-\theta}(1-\sigma)^{\theta-1}\d \sigma=\frac{1}{2}.
\end{Eq*}
This finishes the first half of \eqref{Eq:I_p_1} for non odd $A$.
For derivative, we calculate
\begin{Eq*}
&\partial_\mu\kl((1-\mu)^j\int_0^1 \sigma^{-\theta}(1-\sigma)^{j+\theta-1}\kl(\sigma+\mu-\mu\sigma\kr)^{k-2j}\kl(\sigma+\mu-\mu\sigma+1\kr)^{j+\theta-1}\d\sigma\kr)\\
=&j(1-\mu)^{j-1}\int_0^1 \sigma^{-\theta}(1-\sigma)^{j+\theta-1}\kl(\sigma+\mu-\mu\sigma\kr)^{k-2j}\kl(\sigma+\mu-\mu\sigma+1\kr)^{j+\theta-1}\d\sigma\\
&+(k-2j)(1-\mu)^j\int_0^1 \sigma^{-\theta}(1-\sigma)^{j+\theta}\kl(\sigma+\mu-\mu\sigma\kr)^{k-2j-1}\kl(\sigma+\mu-\mu\sigma+1\kr)^{j+\theta-1}\d\sigma\\
&+(j+\theta-1)\int_0^1 \sigma^{-\theta}(1-\sigma)^{j+\theta}\kl(\sigma+\mu-\mu\sigma\kr)^{k-2j}\kl(\sigma+\mu-\mu\sigma+1\kr)^{j+\theta-2}\d\sigma,
\end{Eq*}
with no singularity in all these integrals. This means $\partial_\mu I_A(\mu)=O(1)$ for $\mu$ close to $1-$, which corroborates with \eqref{Eq:I_p_4}.
\subpart[$\mu$ close to $-1+$]
Then we let $\mu$ close to $-1+$, without loss of generality we assume $\mu<-1/2$, then
\begin{Eq*}
&\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
=&\int_{\mu}^{0} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda+\int_{0}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
=&\int_{\mu}^{0} (\lambda-\mu)^{-\theta}(1+\lambda)^{j+\theta-1}h(\lambda)\d \lambda+O(1),
\end{Eq*}
where $h(\lambda):=\lambda^{k-2j}(1-\lambda)^{j+\theta-1}$ satisfying $h(\lambda)\in C^\infty([-1,0])$. For the first integral, we split it to
\begin{Eq*}
&\int_{\mu}^{0} (\lambda-\mu)^{-\theta}(1+\lambda)^{j+\theta-1}(h(\lambda)-h(-1))\d \lambda
+h(-1)\int_{\mu}^{0} (1+\lambda)^{j-1}\d \lambda\\
&+h(-1)\kl(\int_{\mu}^{1+2\mu}+\int_{1+2\mu}^0\kr) \kl((\lambda-\mu)^{-\theta}-(1+\lambda)^{-\theta}\kr)(1+\lambda)^{j+\theta-1}\d \lambda\\
\equiv& J_1+J_2+J_3+J_4.
\end{Eq*}
Using the mean value theorem, it is easy to find that
\begin{Eq*}
|J_1|\lesssim& \int_{\mu}^{0} (\lambda-\mu)^{-\theta}(1+\lambda)^{j+\theta}\d \lambda \lesssim 1,\\
J_2=&C_j\ln(1+\mu)+O(1),\\
|J_3|\lesssim& (1+\mu)^{j+\theta-1}\int_{\mu}^{1+2\mu} (\lambda-\mu)^{-\theta}-(1+\lambda)^{-\theta}\d \lambda\lesssim (1+\mu)^j\lesssim 1,\\
|J_4|\lesssim&(1+\mu)\int_{1+2\mu}^{0}(1+\lambda)^{j-2}\d \lambda\lesssim 1.
\end{Eq*}
Adding together, we find
\begin{Eq*}
\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda
=& C_j\ln(1+\mu)+O(1),
\end{Eq*}
which gives \eqref{Eq:I_p_3} for $-1<\mu$.
As for the derivative, we introduce the change of variable $\lambda=\sigma(1+\mu)-1$, then
\begin{Eq*}
&\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
=&(1+\mu)^{j}\int_{1}^{(1+\mu)^{-1}} (\sigma-1)^{-\theta}\sigma^{j+\theta-1}h(\sigma(1+\mu)-1)\d \sigma\\
&+\int_{0}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda.
\end{Eq*}
Taking derivative and splitting it similarly as above, we also find
\begin{Eq*}
\partial_\mu\int_{\mu}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda=C_j'(1+\mu)^{-1}+O\kl(|\ln(1+\mu)|+1\kr),
\end{Eq*}
which gives \eqref{Eq:I_p_4} for $-1<\mu$.
\subpart[$\mu$ close to $-1-$]
To get another part of \eqref{Eq:I_p_3}, we only need to control $I_A(\mu)-I_A(-2-\mu)$ for $-3/2< \mu<-1$. Here, for $-2\leq \mu<-1$, $I_A$ has the formula
\begin{Eq*}
I_A(\mu)=&\frac{2^{-k-\theta}}{\Gamma\kl(k+\theta\kr)\Gamma(1-\theta)}\int_{-1}^{1} (\lambda-\mu)^{-\theta}\sum_{j=0}^{\lfloor k/2 \rfloor}C_{j,k,\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda.
\end{Eq*}
Thus, to show \eqref{Eq:I_p_3}, we only need to estimate
\begin{Eq*}
&\int_{-1}^{1} (\lambda-\mu)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda-\int_{-\mu-2}^{1} (\lambda+\mu+2)^{-\theta}\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
=&\int_{-1}^{-\mu-2} (\lambda-\mu)^{-\theta}(1+\lambda)^{j+\theta-1}h(\lambda)\d \lambda\\
&+\kl(\int_{-\mu-2}^{-2\mu-3}+\int_{-2\mu-3}^{0}\kr) \kl((\lambda-\mu)^{-\theta}-(\lambda+\mu+2)^{-\theta}\kr)(1+\lambda)^{j+\theta-1}h(\lambda)\d \lambda\\
&+\int_{0}^{1} \kl((\lambda-\mu)^{-\theta}-(\lambda+\mu+2)^{-\theta}\kr)\lambda^{k-2j}(1-\lambda^2)^{j+\theta-1}\d \lambda\\
\equiv& J_1+J_2+J_3+J_4.
\end{Eq*}
Here we have
\begin{Eq*}
|J_1|\lesssim &(-1-\mu)^{-\theta}\int_{-1}^{-\mu-2} (1+\lambda)^{j+\theta-1}|h(\lambda)|\d \lambda\lesssim 1,\\
|J_2|\lesssim &(-1-\mu)^{j+\theta-1}\int_{-\mu-2}^{-2\mu-3}(\lambda-\mu)^{-\theta}-(\lambda+\mu+2)^{-\theta}\d\lambda\lesssim(-1-\mu)^j\lesssim 1,\\
|J_3|\lesssim &(-1-\mu)\int_{-2\mu-3}^{0}(1+\lambda)^{j-2}\d\lambda\lesssim 1,\\
|J_4|\lesssim &\int_{0}^{1}(1-\lambda)^{j+\theta-1}\d\lambda\lesssim 1.\\
\end{Eq*}
In summary, we finish the proof of \eqref{Eq:I_p_3}.
As for the derivative, we introduce the change of variable $\lambda=\mu-\sigma(1-\mu)$ for $J_2$.
A similar approach as above we find $|\partial_\mu(J_1+J_2+J_3+J_4)|\lesssim |\ln(1+\mu)|+1$.
This finishes the proof of \eqref{Eq:I_p_4}.
\value{parta}[$A$ is odd]
Next, we consider the case $A=1+2k$ with $k\in {\mathbb{Z}}_+$. In this case we have
\begin{Eq*}
\supp \mathcal{X}_+^{\frac{1-A}{2}}\kl(x\kr)=\supp \delta^{(k-1)}\kl(x\kr)=\{0\},
\end{Eq*}
which gives \eqref{Eq:I_p_5}. On the other hand, when $-1<\mu<1$, we have
\begin{Eq*}
I_A(\mu)=&\frac{2^{-k}}{\Gamma\kl(k\kr)} \int_{-1}^{1} \partial_\lambda^{k-1}\delta\kl(\lambda-\mu\kr)(1-\lambda^2)^{k-1}\d \lambda\\
=&\frac{2^{-k}}{\Gamma\kl(k\kr)}\int_{-1}^{1} \delta\kl(\lambda-\mu\kr)(-\partial_\lambda)^{k-1}(1-\lambda^2)^{k-1}\d \lambda\\
=&\frac{2^{-k}}{\Gamma\kl(k\kr)}\sum_{j=0}^{\lfloor (k-1)/2 \rfloor}C_{j,k}'\mu^{k-1-2j}(1-\mu^2)^{j}.
\end{Eq*}
This means there is no singularity both for $I_A(\mu)$ and its derivate, which lead to \eqref{Eq:I_p_6}.
Here we also find $C_{0,k}'=2^{k-1}(k-1)!$,
which implies the first half of \eqref{Eq:I_p_1} for odd $A$. Now we finish the proof of \Le{Le:I_p}.
\subsection{Additional discussion of weak solutions}
In light of the fact that the framework we take is slightly different from the usual one,
we will discuss a bit more of the weak solution.
We will show that when
\begin{Eq}\label{Eq:u_i_r}
r^{A-1}f\in L_{loc;r}^1,~r^{A-1}g\in L_{loc;r}^1,~ r^{A-1}F\in L_{loc;t,r}^1,~r^{A-1}u\in L_{loc;t,r}^1
\end{Eq}
with $u$ calculated by \Le{Le:u_e}, then \eqref{Eq:u_i} holds.
To show this result,
we divide $u$ to $u_g$, $u_f$ and $u_F$.
We begin with $u_g$ part. Noticing $I_A(\mu)=0$ while $\mu>1$, in this case we have
\begin{Eq*}
&\int_0^T\int_0^\infty u\cdot \kl(\partial_t^2-\Delta_A\kr)\phi r^{A-1}\d r\d t\\
=&\int_0^T\int_0^\infty \int_{0}^\infty r^{\frac{A-1}{2}}\rho ^{\frac{A-1}{2}}g(\rho) I_A\kl(\frac{r^2+\rho^2-t^2}{2r\rho}\kr) \kl(\partial_t^2-\Delta_A\kr)\phi(t,r) \d \rho\d r\d t.
\end{Eq*}
Set $t=T-s$, swap $r$ and $\rho$ then exchange the order of integration. It goes to
\begin{Eq*}
&\int_0^\infty \int_0^T \int_0^\infty r^{\frac{A-1}{2}}\rho ^{\frac{A-1}{2}}g(r) I_A\kl(\frac{r^2+\rho^2-(T-s)^2}{2r\rho}\kr) \kl(\partial_s^2-\Delta_A\kr)\phi(T-s,\rho) \d \rho\d s\d r.
\end{Eq*}
Here $\phi(T-s,\rho)$ has zero initial data at $s=0$ and regular enough, by expression of $u_F$ deduced in \Le{Le:u_e}, we know
\begin{Eq*}
&r^{\frac{1-A}{2}}\int_0^T \int_0^\infty \rho ^{\frac{A-1}{2}} I_A\kl(\frac{r^2+\rho^2-(T-s)^2}{2r\rho}\kr) \kl(\partial_s^2-\Delta_A\kr)\phi(T-s,\rho) \d \rho\d s\\
=&\kl.\phi(T-s,r)\kr|_{s=T}=\phi(0,r).
\end{Eq*}
This gives \eqref{Eq:u_i} with $f=F=0$. The proof of $u_f$ and $u_F$ parts is similar, we leave them to the interested reader.
\section{Long-time existence for $A\in[2,3]$}\label{Se:3}
In this section, we will consider the case $A\in[2,3]$, and show the proof of \Th{Th:M_1}. Without loss of generality we assume $n\neq A$, otherwise $V=0$ then \eqref{Eq:U_o} reduced to the equation of \emph{Strauss} conjecture.
By the discussion in the last section, we begin to study the equation \eqref{Eq:u_o} and \eqref{Eq:u_l}.
\subsection{Estimate for homogeneous solution}
In this subsection, we will give an estimate of the homogeneous solution to \eqref{Eq:u_l}.
\begin{lemma}\label{Le:u0_e}
Let $A\in[2,3]$, $n\geq 2$ and assume $\supp(f,g)\subset [0,1)$. We have
\begin{Eq*}
|u_f+u_g|\lesssim& \kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl(\|\rho g(\rho)\|_{L_\rho^\infty}+\|f(\rho)\|_{L_\rho^\infty}+\|\rho f'(\rho)\|_{L_\rho^\infty}\kr).
\end{Eq*}
Here and throughout the paper, $\kl<a\kr>$ stands for $\sqrt{|a|^2+4}$.
\end{lemma}
\begin{proof}[Proof of \Le{Le:u0_e}]
Here we define
\begin{Eq*}
\Omega_0:=\{\rho:0<\rho<t-r\},\qquad \Omega_1:=\{\rho:|t-r|<\rho<\min(1,t+r)\},
\end{Eq*}
with $\Omega_0=\emptyset$ when $t<r$.
\setcounter{part0}{0}
\value{parta}[Estimate of $u_g$ with $A\in[2,3)$]
Firstly we consider $u_g$ with $A\in[2,3)$. For $u_g$, by \Le{Le:u_e} and \eqref{Eq:I_p_1} we have
\begin{Eq*}
u_g(t,r)=&r^{\frac{1-A}{2}}\kl(\int_{\Omega_0}+\int_{\Omega_1}\kr)I_A(\mu)\rho^{\frac{A-1}{2}}g(\rho)\d\rho
\equiv J_{0}+J_{1}.
\end{Eq*}
Also, for $A\in[2,3)$ we have
\begin{Eq*}
|\ln|1+\mu||\lesssim |1+\mu|^{\frac{A-3}{2}}\lesssim |1+\mu|^{\frac{1-A}{2}},\qquad -2<\mu<1.
\end{Eq*}
\subpart[$t+r\leq 4$]
In this part, we have $\kl<t+r\kr>\approx \kl<t-r\kr>\approx 1$ and $r\lesssim 1$.
In the region of $\Omega_0$ where $\mu<-1$,
by \eqref{Eq:I_p_2} and \eqref{Eq:I_p_3} we see
\begin{Eq*}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r+\rho)(t-r-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim r^{\frac{A-1}{2}}(t-r-\rho)^{\frac{1-A}{2}}.
\end{Eq*}
Then we have
\begin{Eq*}
|J_{0}|\lesssim& \int_{0}^{t-r}(t-r-\rho)^{\frac{1-A}{2}} \rho^{\frac{A-3}{2}}\rho|g(\rho)|\d\rho\\
\lesssim&\kl\|\rho g(\rho)\kr\|_{L^\infty}
\lesssim\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}.
\end{Eq*}
In the region of $\Omega_1$ where $\mu>-1$, by \eqref{Eq:I_p_3} we see
\begin{Eq*}
|I_A(\mu)|\lesssim (1+\mu)^{\frac{A-3}{2}}
=\kl(\frac{2r\rho}{(t+r+\rho)(r+\rho-t)}\kr)^{\frac{3-A}{2}}
\lesssim \rho^{\frac{3-A}{2}}(r+\rho-t)^{\frac{A-3}{2}}.
\end{Eq*}
Thus we get
\begin{Eq*}
|J_{1}|\lesssim& r^{\frac{1-A}{2}}\int_{|t-r|}^{t+r}(r+\rho-t)^{\frac{A-3}{2}} \rho|g(\rho)|\d\rho\\
\lesssim&\kl\|\rho g(\rho)\kr\|_{L^\infty}
\lesssim\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho^{\frac{A-1}{2}}g(\rho)\kr\|_{L^\infty}.
\end{Eq*}
\subpart[$t+r\geq 4$, $-1\leq t-r\leq 2$]
In this part, we have $\kl<t-r\kr>\approx 1$ and $r\approx \kl<t+r\kr>$.
In the region of $\Omega_0$ where $\mu<-1$, we have
\begin{Eq*}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r+\rho)(t-r-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim \rho^{\frac{A-1}{2}}(t-r-\rho)^{\frac{1-A}{2}}.
\end{Eq*}
Then we get
\begin{Eq*}
|J_{0}|\lesssim& \kl<t+r\kr>^{\frac{1-A}{2}}\int_{0}^{t-r}(t-r-\rho)^{\frac{1-A}{2}} \rho^{A-2}\rho|g(\rho)|\d\rho\\
\lesssim&\kl<t+r\kr>^{\frac{1-A}{2}}(t-r)^{\frac{A-1}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}
\lesssim\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}.
\end{Eq*}
In the region of $\Omega_1$ where $\mu>-1$, we also have
\begin{Eq*}
|I_A(\mu)|\lesssim (1+\mu)^{\frac{A-3}{2}}
=\kl(\frac{2r\rho}{(t+r+\rho)(r+\rho-t)}\kr)^{\frac{3-A}{2}}
\lesssim \rho^{\frac{3-A}{2}}(r+\rho-t)^{\frac{A-3}{2}}.
\end{Eq*}
Then we see
\begin{Eq*}
|J_{1}|\lesssim& \kl<t+r\kr>^{\frac{1-A}{2}}\int_{|t-r|}^{2}(r+\rho-t)^{\frac{A-3}{2}} \rho|g(\rho)|\d\rho\\
\lesssim&\kl<t+r\kr>^{\frac{1-A}{2}}(r+2-t)^{\frac{A-1}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}
\lesssim\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}.
\end{Eq*}
\subpart[$t+r\geq 4$, $t-r\geq 2$]
In this part, we have $t+r\gtrsim\kl<t+r\kr>$ and $t-r-1\gtrsim\kl<t-r\kr>$. Here $\Omega_1=\emptyset$ so we only need to consider $\Omega_0$ where $\mu<-1$. Again we see
\begin{Eq*}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r+\rho)(t-r-\rho)}\kr)^{\frac{A-1}{2}}\\
\lesssim& r^{\frac{A-1}{2}}\rho^{\frac{A-1}{2}}\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}.
\end{Eq*}
Then we have
\begin{Eq*}
|u_g|\lesssim& \kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\int_{0}^{1} \rho^{A-1}|g(\rho)|\d\rho\\
\lesssim&\kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho g(\rho)\kr\|_{L^\infty}.
\end{Eq*}
In summary, we finish the estimate of $u_g$ when $A<3$.
\value{parta}[Estimate of $u_f$ with $A\in[2,3)$]
Next we consider $u_f$. For simplicity let $u_{g=\phi}$ stand for $u_g$ with $g=\phi$.
By \Le{Le:u_e} and the expression of $u_g$ \eqref{Eq:u_g_c}, we know
\begin{Eq*}
u_f=&\partial_t \kl(u_{g=f}\kr)\\
=&\frac{1}{\Gamma\kl(\frac{A-1}{2}\kr)\Gamma\kl(\frac{3-A}{2}\kr)} \partial_t\kl(t\int_{0}^{1}\int_{-1}^{1} \frac{f\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)}{(1-\sigma^2)^{\frac{A-1}{2}}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma\kr)\\
\lesssim &t\int_{0}^{1}\int_{-1}^{1} \frac{\kl|t\sigma^2-r\sigma\lambda\kr|}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}}\frac{\kl|f'\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)\kr|}{(1-\sigma^2)^{\frac{A-1}{2}}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma\\
&+\int_{0}^{1}\int_{-1}^{1} \frac{\kl|f\kl(\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}\kr)\kr|}{(1-\sigma^2)^{\frac{A-1}{2}}}\sigma^{A-1}\sqrt{1-\lambda^2}^{A-3}\d \lambda\d\sigma\\
\equiv&H_1+H_2.
\end{Eq*}
Since $A\in[2,3)$, we can easily find that $H_2\lesssim \|f\|_{L^\infty}$. Meanwhile, we find that
\begin{Eq*}
H_2\approx&t^{-1}u_{g=|f|}\lesssim t^{-1} \kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\|f(\rho)\|_{L_\rho^\infty}.
\end{Eq*}
Adding together, we finish the estimate of $H_2$. Next we consider $H_1$. Noticing that when $\lambda\in[-1,1]$, we have
\begin{Eq*}
\kl(\frac{t\sigma^2-r\sigma\lambda}{\sqrt{r^2+t^2\sigma^2-2rt\sigma\lambda}}\kr)^2=&\frac{r^2\sigma^2\lambda^2+t^2\sigma^4-2rt\sigma^3\lambda}{r^2+t^2\sigma^2-2rt\sigma\lambda}\leq\sigma^2\leq 1.
\end{Eq*}
Then we calculate
\begin{Eq*}
H_1\lesssim u_{g=|f'|}\lesssim \kl<t+r\kr>^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{1-A}{2}}\kl\|\rho f'(\rho)\kr\|_{L_\rho^\infty}.
\end{Eq*}
Adding together, we finish the estimate of $u_f$ when $A<3$.
\value{parta}[Estimate of $u_g$ and $u_f$ for $A=3$]
When $A=3$, the estimate for $u_g$ is similar, where we use \eqref{Eq:I_p_5} and \eqref{Eq:I_p_6} instead of \eqref{Eq:I_p_2} and \eqref{Eq:I_p_3}. As for $u_f$, we easily calculate
\begin{Eq*}
u_f=\frac{(t+r)f(t+r)-(t-r)f(|t-r|)}{2r}
\end{Eq*}
with $\supp u_f\subset \{(t,r):|t-r|\leq 1\}$. When $r< t$, by mean value theorem we see
\begin{Eq*}
|u_f|=\kl|\frac{(t+r)f(t+r)-(t-r)f(t-r)}{2r}\kr|\leq \|\partial_\rho(\rho f(\rho))\|_{L_\rho^\infty}.
\end{Eq*}
When $t<r$, we see
\begin{Eq*}
|u_f|\leq|f(t+r)|+|f(r-t)|\lesssim \|f(\rho)\|_{L_\rho^\infty}.
\end{Eq*}
In summary, we obtain the desired estimate of $u_g$ and $u_f$ for $A=3$, and finish the proof of \Le{Le:u0_e}.
\end{proof}
\subsection{Estimate for non-homogeneous solution}
In this subsection, we will omit the initial data and give the estimate of solution to the nonlinear equation \eqref{Eq:u_o}. For simplicity, we shift the time variable and consider the equation
\begin{Eq}\label{Eq:v_o}
\begin{cases}
\kl(\partial_t^2-\partial_r^2-(A-1)r^{-1}\partial_r\kr)v(t,r)=r^{\frac{(A-n)p+n-A}{2}}G(t,r)\\
v(4,r)=0,\qquad v_t(4,r)=0.
\end{cases}
\end{Eq}
\begin{lemma}\label{Le:v_e}
Define $\Omega:=\{(t,r)\in [4,\infty)\times {\mathbb{R}}_+: t>r+2\}$ and $\Lambda(t,r):=\{(s,\rho)\in\Omega:s+\rho<t+r,s-\rho<t-r\}$. We will show that, if $v$ solves the equation \eqref{Eq:v_o} with $\supp v\subset \Omega$, then for any $(t,r)\in\Omega$ and $k=1,2,3$ we have
\begin{Eq}\label{Eq:v_e}
\kl<t+r\kr>^{\frac{A-1}{2}} |v|\lesssim N_k(t-r)\|\omega_k^p G\|_{L_{s,\rho}^\infty(\Lambda)}.
\end{Eq}
Here
\begin{Eq*}
\omega_k(t,r):=&\kl<t+r\kr>^{\frac{A-1}{2}}\beta_k(t-r),\qquad k=1,2,3;\\
\beta_k(t-r):=&\begin{cases}
\kl<t-r\kr>^{\frac{A-1}{2}},&k=1,~for~ p_m<p<p_M,\\
\kl<t-r\kr>^{\frac{(n-1)p-n-1}{2}},& k=2,~for~ p_m<p<p_t,\\
\kl<t-r\kr>^{\frac{A-1}{2}}(\ln\kl<t-r\kr>)^{-1},&k=3,~for~p=p_t>p_d;
\end{cases}
\end{Eq*}
\begin{Eq*}
N_1(t-r):=&\begin{cases}
\kl<t-r\kr>^{\frac{1-A}{2}},&p>\max(p_d,p_t)~or~p=p_d>p_t~or~ p_F<p<p_d,\\
\kl<t-r\kr>^{\frac{1-A}{2}}\ln\kl<t-r\kr>,&p=p_t>p_d~or~p=p_F<p_d,\\
\kl<t-r\kr>^\frac{(1-n)p+n+1}{2},&p_t>p>p_d,\\
\kl<t-r\kr>^\frac{(-n-A+2)p+n+3}{2},&p<\min(p_d,p_F),\\
\kl<t-r\kr>^{\frac{(-n-A+2)p+n+3}{2}}\ln\kl<t-r\kr>,& p=p_d< p_t,\\
\kl<t-r\kr>^{\frac{1-A}{2}} (\ln \kl<t-r\kr>)^2,& p=p_d= p_t;
\end{cases}\\
N_2(t-r):=&\begin{cases}
\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}}&p_S<p<p_t,\\
\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}}\ln \kl<t-r\kr>&p=p_S<p_t,\\
\kl<t-r\kr>^{\frac{(1-n)p^2+2p+n+3}{2}}&p_m<p<\min(p_S,p_t);
\end{cases}\\
N_3(t-r):=&
\kl<t-r\kr>^{\frac{1-A}{2}}\ln\kl<t-r\kr>,\qquad p=p_t>p_d.
\end{Eq*}
\end{lemma}
\begin{remark}
By the definition of $\beta_k$, we can easily find that for any $\xi, \eta>2$
with $\xi/\eta\in (1/2, 2)$, we have
\begin{Eq*}
\beta_k(\xi)\approx \beta_k(\eta).
\end{Eq*}
Also if $2<\eta_1<\eta_2$, we have
\begin{Eq*}
\beta_k(\eta_1)\lesssim\beta_k(\eta_2).
\end{Eq*}
\end{remark}
\begin{proof}[Proof of \Le{Le:v_e}]
Using \Le{Le:u_e}, we find
\begin{Eq*}
v=r^{\frac{1-A}{2}} \int_{\Lambda} I_A(\mu) \rho^{\frac{(A-n)p+n-1}{2}}G(s,\rho)\d\rho\d s,
\end{Eq*}
with $\mu:=\frac{r^2+\rho^2-(t-s)^2}{2 r \rho}$. To reach \eqref{Eq:v_e}, we calculate
\begin{Eq*}
|\kl<t+r\kr>^\frac{A-1}{2} v|\leq& r^{\frac{1-A}{2}}\kl<t+r\kr>^\frac{A-1}{2} \|\omega_k^p G\|_{L_{s,\rho}^\infty (\Lambda)}\int_{\Lambda} |I_A(\mu)| \rho^{\frac{(A-n)p+n-1}{2}}\omega_k^{-p}\d\rho\d s.
\end{Eq*}
Thus, we only need to show that
\begin{Eq}\label{Eq:Jijk}
J_{ij;k}:=r^{\frac{1-A}{2}}\kl<t+r\kr>^\frac{A-1}{2} \int_{\Lambda_{ij}} |I_A(\mu)|\rho^{\frac{(A-n)p+n-1}{2}}\omega_k^{-p}\d\rho\d s\lesssim N_k(t-r)
\end{Eq}
for $i=1,2,3$, $j=1,2$ with
\begin{Eq*}
\Lambda_{11}:=&\{(s,\rho)\in\Lambda: s+\rho\in(t-r,t+r), \rho\leq s/2\};\\
\Lambda_{12}:=&\{(s,\rho)\in\Lambda: s+\rho\in(t-r,t+r), \rho\geq s/2\};\\
\Lambda_{21}:=&\{(s,\rho)\in\Lambda: s+\rho\in\kl(\frac{t-r}{2},t-r\kr), \rho\leq s/2\};\\
\Lambda_{22}:=&\{(s,\rho)\in\Lambda: s+\rho\in\kl(\frac{t-r}{2},t-r\kr), \rho\geq s/2\};\\
\Lambda_{31}:=&\{(s,\rho)\in\Lambda: s+\rho\in\kl(3,\frac{t-r}{2}\kr), \rho\leq s/2\};\\
\Lambda_{32}:=&\{(s,\rho)\in\Lambda: s+\rho\in\kl(3,\frac{t-r}{2}\kr), \rho\geq s/2\}.
\end{Eq*}
\begin{figure}[H]
\centering
\includegraphics[width=0.99\textwidth]{Omega.png}
\end{figure}
It's easy to find that
\begin{Eq}\label{Eq:srho_r}
\begin{cases}
s+\rho\leq 3(s-\rho)\leq 3(s+\rho),&(s,\rho)\in \Lambda_{11}\cup\Lambda_{21}\cup\Lambda_{31},\\
s+\rho\leq 3\rho\leq 3(s+\rho),&(s,\rho)\in \Lambda_{12}\cup\Lambda_{22}\cup\Lambda_{32}.
\end{cases}
\end{Eq}
Then, a quick calculation shows
\begin{Eq}\label{Eq:trsrho_r}
\begin{cases}
t-r\leq s+\rho \leq \min\{t+r,3(t-r)\}, \quad (t-r)/3\leq s-\rho\leq t-r, &(s,\rho)\in \Lambda_{11},\\
t-r\leq s+\rho\leq t+r,\quad 2\leq s-\rho\leq \min\{(t+r)/3,t-r\}, &(s,\rho)\in \Lambda_{12},\\
(t-r)/2\leq s+\rho \leq t-r, \quad (s+\rho)/3\leq s-\rho\leq s+\rho, &(s,\rho)\in \Lambda_{21},\\
(t-r)/2\leq s+\rho\leq t-r,\quad 2\leq s-\rho\leq (t-r)/3, &(s,\rho)\in \Lambda_{22},\\
4\leq s+\rho \leq (t-r)/2, \quad (s+\rho)/3\leq s-\rho\leq s+\rho, &(s,\rho)\in \Lambda_{31},\\
6\leq s+\rho\leq (t-r)/2,\quad 2\leq s-\rho\leq (s+\rho)/3, &(s,\rho)\in \Lambda_{32}.
\end{cases}
\end{Eq}
From now on, we introduce $\xi:=s+\rho$ and $\eta:=s-\rho$.
We will always adopt \eqref{Eq:srho_r} and \eqref{Eq:trsrho_r} in each region.
\setcounter{part0}{0}
\value{parta}[Preparation for $A\in[2,3)$ with $r\leq t/2$]
We will firstly consider $A\in[2,3)$,
notice that $\mu=\frac{r^2+\rho^2-(t-s)^2}{2r\rho}< -1$ when $s+\rho< t-r$, and $\mu> -1$ when $s+\rho> t-r$.
In this part we have
\begin{Eq*}
t+r\leq 3(t-r)\leq 3(t+r).
\end{Eq*}
In the region of $\Lambda_{11}$,
using \eqref{Eq:I_p_3} we have
\begin{Eq}\label{Eq:I_e1}
|I_A(\mu)|\lesssim& (1+\mu)^{\frac{A-3}{2}}
=\kl(\frac{2r\rho}{(r+\rho+t-s)(r+\rho-t+s)}\kr)^{\frac{3-A}{2}}
\lesssim \kl(\frac{\rho}{r+\rho-t+s}\kr)^{\frac{3-A}{2}}.
\end{Eq}
Then we find
\begin{Eq*}
J_{11}\lesssim & r^{\frac{1-A}{2}} \kl<t-r\kr>^{\frac{A-1}{2}} \int_{\Lambda_{11}} \rho^{\frac{(A-n)p+n-A+2}{2}}\kl<s+\rho\kr>^{-\frac{A-1}{2}p}\beta(s+\rho)^{-p}\kl({r+\rho-t+s}\kr)^{\frac{A-3}{2}}\d\rho\d s\\
\lesssim &r^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{(1-A)p+A-1}{2}}\beta(t-r)^{-p}\int_{t-r}^{t+r}\int_{(t-r)/3}^{t-r} (\xi-\eta)^{\frac{(A-n)p+n-A+2}{2}}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d \eta\d\xi\\
\lesssim& r^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{(1-n)p+n+3}{2}}\beta(t-r)^{-p}\int_{t-r}^{t+r}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d\xi\\
\lesssim&\kl<t-r\kr>^{\frac{(1-n)p+n+3}{2}}\beta(t-r)^{-p},
\end{Eq*}
where we noticed $\frac{(A-n)p+n-A+2}{2}> -1$ while $p<p_M$.
In the region of $\Lambda_{12}$, we also have \eqref{Eq:I_e1}. Then we find
\begin{Eq*}
J_{12}\lesssim & r^{\frac{1-A}{2}} \kl<t-r\kr>^{\frac{A-1}{2}}\int_{\Lambda_{12}}\kl<s+\rho\kr>^{\frac{(1-n)p+n-A+2}{2}}\beta(s-\rho)^{-p}\kl({r+\rho-t+s}\kr)^{\frac{A-3}{2}}\d\rho\d s\\
\lesssim & r^{\frac{1-A}{2}}\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}}\int_{t-r}^{t+r}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d\xi \int_2^{(t+r)/3} \beta(\eta)^{-p}\d\eta\\
\lesssim &\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}} \int_2^{t-r} \beta(\eta)^{-p}\d\eta.
\end{Eq*}
In the region of $\Lambda_{21}$,
using \eqref{Eq:I_p_2} and \eqref{Eq:I_p_3}
we have
\begin{Eq}\label{Eq:I_e2}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r-s+\rho)(t-r-s-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim \kl(\frac{r}{t-r-s-\rho}\kr)^{\frac{A-1}{2}}.
\end{Eq}
Then we find
\begin{Eq*}
J_{21}\lesssim & \kl<t-r\kr>^{\frac{A-1}{2}} \int_{\Lambda_{21}} \rho^{\frac{(A-n)p+n-1}{2}}\kl<s+\rho\kr>^{\frac{(1-A)p}{2}}\beta(s+\rho)^{-p}\kl({t-r-s-\rho}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{(1-A)p+A-1}{2}}\beta(t-r)^{-p} \int_{(t-r)/2}^{t-r}\int_{\xi/3}^{\xi} (\xi-\eta)^{\frac{(A-n)p+n-1}{2}}\kl({t-r-\xi}\kr)^{\frac{1-A}{2}}\d \eta\d\xi\\
\lesssim &\kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\beta(t-r)^{-p} \int_{(t-r)/2}^{t-r} \kl(t-r-\xi\kr)^{\frac{1-A}{2}}\d\xi\\
\lesssim&\kl<t-r\kr>^{\frac{(1-n)p+n+3}{2}}\beta(t-r)^{-p},
\end{Eq*}
where we require $p<p_M$ to ensure that $\frac{(A-n)+n-1}{2}> -1$.
In the region of $\Lambda_{22}$, we still have
\eqref{Eq:I_e2}. Then we find
\begin{Eq*}
J_{22}\lesssim & \kl<t-r\kr>^{\frac{A-1}{2}} \int_{\Lambda_{22}}\kl<s+\rho\kr>^{\frac{(1-n)p+n-1}{2}}\beta(s-\rho)^{-p}\kl({t-r-s-\rho}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\lesssim & \kl<t-r\kr>^{\frac{(1-n)p+n+A-2}{2}}\int_{(t-r)/2}^{t-r}\kl(t-r-\xi\kr)^{\frac{1-A}{2}}\d\xi \int_2^{(t-r)/3}\beta(\eta)^{-p}\d\eta\\
\lesssim &\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}} \int_2^{(t-r)/3} \beta(\eta)^{-p}\d\eta.
\end{Eq*}
In the region of $\Lambda_{31}$
we have
\begin{Eq}\label{Eq:I_e3}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r-s+\rho)(t-r-s-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim\kl(\frac{r\rho}{(t-r)^2}\kr)^{\frac{A-1}{2}}.
\end{Eq}
Then we find
\begin{Eq*}
J_{31}\lesssim &\kl<t-r\kr>^{\frac{A-1}{2}} \int_{\Lambda_{31}} \rho^{\frac{(A-n)p+n+A-2}{2}}\kl<s+\rho\kr>^{\frac{(1-A)p}{2}}\beta(s+\rho)^{-p}\kl({t-r}\kr)^{-A+1}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}}\int_4^{(t-r)/2}\int_{\xi/3}^{\xi} (\xi-\eta)^{\frac{(A-n)p+n+A-2}{2}}\kl<\xi\kr>^{\frac{(1-A)p}{2}}\beta(\xi)^{-p}\d \eta\d\xi\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}}\int_4^{(t-r)/2}\kl<\xi\kr>^{\frac{(1-n)p+n+A}{2}}\beta(\xi)^{-p} \d\xi,
\end{Eq*}
where we noticed $\frac{(A-n)p+n+A-2}{2}>-1$ while $p<p_M$.
In the region of $\Lambda_{32}$ we still have \eqref{Eq:I_e3}.
Then we find
\begin{Eq*}
J_{32}\lesssim &\kl<t-r\kr>^{\frac{A-1}{2}} \int_{\Lambda_{32}}\kl<s+\rho\kr>^{\frac{(1-n)p+n+A-2}{2}}\beta(s-\rho)^{-p}\kl({t-r}\kr)^{-A+1}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\int_2^{\xi/3}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\beta(\eta)^{-p}\d \eta\d\xi.
\end{Eq*}
\value{parta}[Preparation for $A\in[2,3)$ with $r\geq t/2$]
In this part we have
\begin{Eq*}
t+r\leq 3r\leq 3(t+r).
\end{Eq*}
In the region of $\Lambda_{11}$, we have \eqref{Eq:I_e1}.
Then we find
\begin{Eq*}
J_{11}'\lesssim & \int_{\Lambda_{11}} \rho^{\frac{(A-n)p+n-A+2}{2}}\kl<s+\rho\kr>^{\frac{(1-A)p}{2}}\beta(s+\rho)^{-p}\kl({r+\rho-t+s}\kr)^{\frac{A-3}{2}}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{(1-A)p}{2}}\beta(t-r)^{-p} \int_{t-r}^{3(t-r)}\int_{(t-r)/3}^{t-r} (\xi-\eta)^{\frac{(A-n)p+n-A+2}{2}}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d \eta\d\xi\\
\lesssim &\kl<t-r\kr>^{\frac{(1-n)p+n-A+4}{2}}\beta(t-r)^{-p} \int_{t-r}^{3(t-r)}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d\xi\\
\lesssim&\kl<t-r\kr>^{\frac{(1-n)p+n+3}{2}}\beta(t-r)^{-p},
\end{Eq*}
again $\frac{(A-n)p+n-A+2}{2}> -1$ since $p<p_M$.
In the region of $\Lambda_{12}$, we have \eqref{Eq:I_e1}.
Then we find
\begin{Eq*}
J_{12}'\lesssim & \int_{\Lambda_{12}}\kl<s+\rho\kr>^{\frac{(1-n)p+n-A+2}{2}}\beta(s-\rho)^{-p} \kl({r+\rho-t+s}\kr)^{\frac{A-3}{2}}\d\rho\d s\\
\leq & \int_{t-r}^{t+r}\kl<\xi\kr>^{\frac{(1-n)p+n-A+2}{2}}\kl({\xi-(t-r)}\kr)^{\frac{A-3}{2}}\d\xi \int_2^{t-r} \beta(\eta)^{-p}\d\eta\\
\lesssim &
\kl(\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}} +\int_{3(t-r)}^{t+r}\kl<\xi\kr>^{\frac{(1-n)p+n-1}{2}}\d\xi\kr)\int_2^{t-r} \beta(\eta)^{-p}\d\eta\\
\lesssim & \kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}} \int_2^{t-r} \beta(\eta)^{-p}\d\eta,
\end{Eq*}
where we require $p>p_m$ so that $\frac{(1-n)p+n-1}{2}<-1$.
In the region of $\Lambda_{21}$ we have
\begin{Eq}\label{Eq:I_e4}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r-s+\rho)(t-r-s-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim \kl(\frac{\rho}{t-r-s-\rho}\kr)^{\frac{A-1}{2}}.
\end{Eq}
Then we find
\begin{Eq*}
J_{21}'\lesssim & \int_{\Lambda_{21}} \rho^{\frac{(A-n)p+n+A-2}{2}}\kl<s+\rho\kr>^{\frac{(1-A)p}{2}}\beta(s+\rho)^{-p}\kl({t-r-s-\rho}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\leq &\kl<t-r\kr>^{\frac{(1-A)p}{2}}\beta(t-r)^{-p}\int_{(t-r)/2}^{t-r}\int_{\xi/3}^{\xi} (\xi-\eta)^{\frac{(A-n)p+n+A-2}{2}}\kl({t-r-\xi}\kr)^{\frac{1-A}{2}}\d \eta\d\xi\\
\lesssim& \kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\beta(t-r)^{-p}\int_{(t-r)/2}^{t-r} \kl(t-r-\xi\kr)^{\frac{1-A}{2}}\d\xi\\
\lesssim& \kl<t-r\kr>^{\frac{(1-n)p+n+3}{2}}\beta(t-r)^{-p},
\end{Eq*}
where $\frac{(A-n)p+n+A-2}{2}>-1$ since $p<p_M$.
In the region of $\Lambda_{22}$, we have \eqref{Eq:I_e4}.
Then we find
\begin{Eq*}
J_{22}'\lesssim & \int_{\Lambda_{22}}\kl<s+\rho\kr>^{\frac{(1-n)p+n+A-2}{2}}\beta(s-\rho)^{-p}\kl({t-r-s-\rho}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\leq &\kl<t-r\kr>^{\frac{(1-n)p+n+A-2}{2}}\int_{(t-r)/2}^{t-r}\kl(t-r-\xi\kr)^{\frac{1-A}{2}}\d\xi \int_2^{(t-r)/3} \beta(\eta)^{-p}\d\eta\\
\lesssim &\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}} \int_2^{(t-r)/3} \beta(\eta)^{-p}\d\eta.
\end{Eq*}
In the region of $\Lambda_{31}$, we have
\begin{Eq}\label{Eq:I_e5}
|I_A(\mu)|\lesssim& (-1-\mu)^{\frac{1-A}{2}}
=\kl(\frac{2r\rho}{(t+r-s+\rho)(t-r-s-\rho)}\kr)^{\frac{A-1}{2}}
\lesssim \kl(\frac{\rho}{t-r}\kr)^{\frac{A-1}{2}}.
\end{Eq}
Then we find
\begin{Eq*}
J_{31}'\lesssim & \int_{\Lambda_{31}} \rho^{\frac{(A-n)p+n+A-2}{2}}\kl<s+\rho\kr>^{\frac{(1-A)p}{2}}\beta(s+\rho)^{-p}\kl({t-r}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}} \int_4^{(t-r)/2}\int_{\xi/3}^{\xi} (\xi-\eta)^{\frac{(A-n)p+n+A-2}{2}}\kl<\xi\kr>^{\frac{(1-A)p}{2}}\beta(\xi)^{-p}\d \eta\d\xi\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}} \int_4^{(t-r)/2}\kl<\xi\kr>^{\frac{(1-n)p+n+A}{2}}\beta(\xi)^{-p}\d\xi,
\end{Eq*}
where $\frac{(A-n)p+n+A-2}{2}>-1$ since $p<p_M$.
In the region of $\Lambda_{32}$, we have \eqref{Eq:I_e5}.
Then we find
\begin{Eq*}
J_{32}'\lesssim& \int_{\Lambda_{32}}\kl<s+\rho\kr>^{\frac{(1-n)p+n+A-2}{2}}\beta(s-\rho)^{-p}\kl({t-r}\kr)^{\frac{1-A}{2}}\d\rho\d s\\
\lesssim &\kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\int_2^{\xi/3}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\beta(\eta)^{-p} \d \eta\d\xi.
\end{Eq*}
\value{parta}[Estimate for $A\in[2,3)$]
Turning to the proof of \eqref{Eq:Jijk},
we will only present the estimate of $J_{32}$,
and the other terms can be estimated in a similar manner.
At first,
for $J_{32;1}$ with $p_m<p<p_M$,
we have
\begin{Eq*}
J_{32;1}\lesssim \kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\int_2^{\xi/3}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\kl<\eta\kr>^{-\frac{A-1}{2}p}\d\eta\d\xi.
\end{Eq*}
When $p>p_d=2/(A-1)$,
it is easy to see that
\begin{Eq*}
J_{32;1}\lesssim \kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\d\xi \lesssim N_1(t-r).
\end{Eq*}
Similarly, when $p\le p_d$,
we have
\begin{Eq*}
J_{32;1}\lesssim
\begin{cases}
\kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\ln\xi\d\xi, & p=p_d,\\
\kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\kl<\xi\kr>^{\frac{(-n-A+2)p+n+A}{2}}\d\xi,& p<p_d,
\end{cases}
\end{Eq*} which are controlled by $N_1(t-r)$ and
this finishes the proof of \eqref{Eq:v_e} with $k=1$.
For $J_{32;2}$ with $p_m<p<p_t$, we have
\begin{Eq*}
J_{32;2}\lesssim& \kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\kl<\xi\kr>^{\frac{(1-n)p+n+A-2}{2}}\d\xi\int_2^{(t-r)/6}\kl<\eta\kr>^{\frac{(1-n)p^2+(n+1)p}{2}} \d \eta\\
\lesssim&\kl<t-r\kr>^{\frac{(1-n)p+n+1}{2}}\int_2^{(t-r)/6}\kl<\eta\kr>^{\frac{(1-n)p^2+(n+1)p}{2}} \d \eta\lesssim N_2(t-r).
\end{Eq*}
Finally, for $J_{32;3}$ with $p=p_t>p_d$, we have
\begin{Eq*}
J_{32;3}\lesssim& \kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\int_2^{\xi/3}\kl<\xi\kr>^{-1}\kl<\eta\kr>^{\frac{(1-A)p}{2}} (\ln\eta)^{p}\d \eta\d\xi\\
\lesssim& \kl<t-r\kr>^{\frac{1-A}{2}}\int_6^{(t-r)/2}\kl<\xi\kr>^{-1} \d\xi\lesssim \kl<t-r\kr>^{\frac{1-A}{2}}\ln\kl<t-r\kr>= N_3(t-r).
\end{Eq*}
In conclusion, this completes the proof for $A\in[2,3)$.
\value{parta}[Estimate for $A=3$]
The case $A=3$ is much simpler, thanks to \eqref{Eq:I_p_5},
we only need to consider $\Lambda_{11}$ and $\Lambda_{12}$.
By \eqref{Eq:I_p_6} and a similar approach as above we get the desired estimate.
\end{proof}
\subsection{Long-time existence}
In this subsection,
we will construct a Cauchy sequence to approximate the desired solution.
We set $u_{-1}=0$ and let $u_{j+1}$ be the solution of the equation
\begin{Eq}\label{Eq:uj_o}
\begin{cases}
\partial_t^2 u_{j+1} -\Delta_Au_{j+1}=r^{\frac{(A-n)p+n-A}{2}}|u_{j}|^p,\quad r\in {\mathbb{R}}_+,\\
u_{j+1}(0,x)=\varepsilon r^{\frac{n-A}{2}}U_0(r),\quad \partial_t u_{j+1}(0,x)=\varepsilon r^{\frac{n-A}{2}}U_1(r).
\end{cases}
\end{Eq}
By \Le{Le:u0_e} and \Le{Le:v_e},
noticing that for any $p>1$, we have
\begin{Eq*}
\kl||a|^p-|b|^p\kr|\lesssim |a-b|\max(|a|,|b|)^{p-1},
\end{Eq*}
then we see
\begin{Eq*
\kl<t+r\kr>^\frac{A-1}{2}|u_{j+1}|\leq& \varepsilon C_0\kl<t-r\kr>^\frac{1-A}{2}\Psi+C_0N_k(t-r)\|\omega_k u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\kl<t+r\kr>^\frac{A-1}{2}|u_{j+1}-u_{j}|\leq& C_0N_k(t-r)\|\omega_k (u_j-u_{j-1})\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_k u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1},
\end{Eq*}
with $k=1,2,3$ and $C_0$ large enough.
Here
\begin{Eq*}
\Psi:=&\|r^\frac{n-A+2}{2}U_0'(r)\|_{L_r^\infty}+\|r^\frac{n-A}{2}U_0(r)\|_{L_r^\infty}+\|r^\frac{n-A+2}{2}U_1(r)\|_{L_r^\infty},\\
\Lambda(t,r):=&\kl\{(s,\rho)\in\Omega: s+\rho<t+r,s-\rho<t-r\kr\},\\
\Omega:=&\kl\{(t,r)\in {\mathbb{R}}_+^2:t>r-1\kr\}.
\end{Eq*}
To prove \Th{Th:M_1}, we need to separate $p\in(p_m,p_M)$ into more parts rather than that in \eqref{Eq:Main_7}, \eqref{Eq:Main_8} or \eqref{Eq:Main_9}. For the reader's convenience, we list them as below. When $(3-A)(A+n+2)<8$, we have $p_d<p_F<p_S<p_t$. Then the proof for $p<p_d$ will be found in \Pt{Pt:4}, $p=p_d$ in \Pt{Pt:6}, $p_d<p<p_S$ in \Pt{Pt:8}, $p=p_S$ in \Pt{Pt:9}, $p_S<p<p_t$ in \Pt{Pt:3}, $p=p_t$ in \Pt{Pt:2} and $p>p_t$ in \Pt{Pt:1}. When $(3-A)(A+n+2)=8$, we have $p_d=p_F=p_S=p_t$. The proof for $p<p_d$ will be found in \Pt{Pt:4}, $p=p_d$ in \Pt{Pt:7} and $p>p_d$ in \Pt{Pt:1}. Finally when $(3-A)(A+n+2)>8$, we have $p_d>p_F>p_S>p_t$. The proof for $p<p_F$ will be found in \Pt{Pt:4}, $p=p_F$ in \Pt{Pt:5} and $p>p_F$ in \Pt{Pt:1}.
Now, we are prepared to give the proofs for each part.
\setcounter{part0}{0}
\value{parta}[$\max(p_t,p_F)<p$]\label{Pt:1}
In this part, we choose $k=1$. For $(t,r)\in\Omega$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Taking the $L_{t,r}^\infty(\Omega)$ norm on both sides and we get
\begin{Eq*}
\kl\|\omega_1u_{j+1}\kr\|_{L_{t,r}^\infty(\Omega)}\leq& \varepsilon C_0\Psi+C_0\|\omega_1 u_j\|_{L_{t,r}^\infty(\Omega)}^p,\\
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega)}\leq& C_0\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{t,r}^\infty(\Omega)}^{p-1}.
\end{Eq*}
For any $\varepsilon>0$ satisfying $(2\varepsilon C_0\Psi)^p<\varepsilon \Psi$, we find
\begin{Eq*
\kl\|\omega_1u_{j}\kr\|_{L_{t,r}^\infty(\Omega)}\leq 2\varepsilon C_0\Psi
\end{Eq*}
holds for any $j$ since $u_{-1}=0$. Meanwhile, it also gives us
\begin{Eq*}
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega)}\leq& C_0\kl(2\varepsilon C_0\Psi\kr)^{p-1}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega)}\\
\leq &\frac{1}{2}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega)}.
\end{Eq*}
This means $\{u_j\}$ is a Cauchy sequence in weighted $L^\infty$ norm.
Set the limit as $u$. It is easy to check $u$ and $F=r^{\frac{(A-n)p+n-A}{2}}|u|^p$ satisfy \eqref{Eq:u_i_r} while $p<p_F$. Thus we get the desired global weak solution.
\value{parta}[$p_S<p_t=p$]\label{Pt:2}
In this part we take $k=3$. For $(t,r)\in\Omega$ we find
\begin{Eq*}
\omega_3|u_{j+1}|\leq& \varepsilon C_0(\ln\kl<t-r\kr>)^{-1}\Psi+C_0\|\omega_3 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_3|u_{j+1}-u_j|\leq& C_0\|\omega_3 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_3 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Noticing $\kl<t-r\kr>\geq 2$, by a similar process as above we get the desired global solution.
\value{parta}[$p_S<p<p_t$]\label{Pt:3}
In this part we take $k=2$. For $(t,r)\in\Omega$ we find
\begin{Eq*}
\omega_2|u_{j+1}|\leq& \varepsilon C_0\kl<t-r\kr>^{\frac{(n-1)p-n-A}{2}}\Psi+C_0\|\omega_2 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_2|u_{j+1}-u_j|\leq& C_0\|\omega_2 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_2 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Here $\frac{(n-1)p-n-A}{2}\leq 0$ since $p<p_t$, by a similar process again we get the desired global solution.
\value{parta}[$p<\min(p_d,p_F)$]\label{Pt:4}
In this case, we choose $T_*=T_*(\varepsilon)$ which satisfies $\varepsilon^{p-1}T_*^{\frac{(-n-A+2)p+n+A+2}{2}}=a$ with $a$ to be fixed later. Here we define $\Omega_*:=\Omega\cap \{t:t<T_*\}$ and choose $k=1$. Then for $(t,r)\in\Omega_*$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\kl<t-r\kr>^{\frac{(-n-A+2)p+n+A+2}{2}}\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\kl<t-r\kr>^{\frac{(-n-A+2)p+n+A+2}{2}}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\\
&\times\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Taking the $L_{t,r}^\infty(\Omega_*)$ norm on both sides,
when $T_*\geq 3$ we get
\begin{Eq*}
\kl\|\omega_1u_{j+1}\kr\|_{L_{t,r}^\infty(\Omega_*)}\leq& \varepsilon C_1\Psi+C_1T_*^{\frac{(-n-A+2)p+n+A+2}{2}}\|\omega_1 u_j\|_{L_{t,r}^\infty(\Omega_*)}^p,\\
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_*)}\leq& C_1T_*^{\frac{(-n-A+2)p+n+A+2}{2}}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_*)}\\
&\times\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{t,r}^\infty(\Omega_*)}^{p-1},
\end{Eq*}
with some $C_1$ large enough.
Considering $\varepsilon$ such that $T_*(\varepsilon)\geq 3$
and defining $a=(2C_1)^{-p}\Psi^{1-p}$,
we conclude that
\begin{Eq*}
\kl\|\omega_1u_{j}\kr\|_{L_{t,r}^\infty(\Omega_*)}\leq& 2\varepsilon C_1\Psi,\\
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_*)}\leq& \frac{1}{2}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_*)}
\end{Eq*}
for any $j$, which are sufficient to get the desired solution.
\value{parta}[$p=p_F<p_d$]\label{Pt:5}
In this case,
we choose $T_*$ which satisfies $\varepsilon^{p-1}\ln T_*=a$
with $a$ to be fixed later,
$\Omega_*$ as above and $k=1$.
For $(t,r)\in \Omega_*$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\ln\kl<t-r\kr>\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\ln\kl<t-r\kr>\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Similar as above,
taking $\varepsilon$ such that $T_*(\varepsilon)\geq 3$,
defining $a:=(2C_1)^{-p}\Psi^{1-p}$ with $C_1$ large enough,
we get the Cauchy sequence $\{u_j\}$ and the desired solution.
\value{parta}[$p=p_d<p_F$]\label{Pt:6}
In this case,
we choose $T_*$ which satisfies
\begin{Eq}\label{Eq:T*_pd}
\varepsilon^{p-1} T_*^\frac{(-n-A+2)p+n+A+2}{2}\ln T_*=a
\end{Eq}
and $\Omega_*$ same as above.
Taking $k=1$,
for $(t,r)\in\Omega_*$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\kl<t-r\kr>^{\frac{(-n-A+2)p+n+A+2}{2}}\ln\kl<t-r\kr>\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\kl<t-r\kr>^{\frac{(-n-A+2)p+n+A+2}{2}}\ln\kl<t-r\kr>\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\\
&\times\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Choosing $\varepsilon$ such that $T_*(\varepsilon)\geq3$,
$a=(2C_1)^{-p}\Psi^{1-p}$ with $C_1$ large enough,
we get the Cauchy sequence $\{u_j\}$ and the desired solution.
To finish the proof of \Th{Th:M_1} for this part,
we introduce the following claim and postpone its proof to the end of this section.
\begin{claim}\label{Cl:T*_pd}
Assume that $T_*$ satisfies $\varepsilon^{p-1} T_*^\frac{-h_F(p)}{2}\ln T_*=a$ with some constant $a$, then there exists two constant $c_1$, $c_2$ such that $c_1\varepsilon^{\frac{2(p-1)}{h_F(p)}}|\ln \varepsilon |^{\frac{2}{h_F(p)}}\leq T_*\leq c_2\varepsilon^{\frac{2(p-1)}{h_F(p)}}|\ln \varepsilon |^{\frac{2}{h_F(p)}}$ for $\varepsilon$ small enough.
\end{claim}
\value{parta}[$p=p_d=p_F$]\label{Pt:7}
In this case,
we choose $T_*$ which satisfies $\varepsilon^{p-1} (\ln T_*)^2=a$
and $\Omega_*$ same as above.
Taking $k=1$,
for $(t,r)\in\Omega_*$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0(\ln \kl<t-r\kr>)^2\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0(\ln \kl<t-r\kr>)^2\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Choosing $\varepsilon$ such that $T_*(\varepsilon)\geq 3$,
$a=(2C_1)^{-p}\Psi^{1-p}$ and $C_1$ large enough,
we get the Cauchy sequence $\{u_j\}$ and finish the proof.
\value{parta}[$p_d<p<p_S$]\label{Pt:8}
In this case,
we choose $T_*$ which satisfies $\varepsilon^{p(p-1)}T_*^{\frac{(1-n)p^2+(n+1)p+2}{2}}=a$
with $a$ to be fixed later and $\Omega_*$ as above.
Moreover,
we separate the region $\Omega_*$ to
\begin{Eq*}
\Omega_{*1}:=&\Omega_*\cap\{(t,r):\kl<t-r\kr>\leq (b \varepsilon)^{\frac{2(p-1)}{(n-1)p-n-A}}\},\\
\Omega_{*2}:=&\Omega_*\cap\{(t,r):\kl<t-r\kr>\geq (b \varepsilon)^{\frac{2(p-1)}{(n-1)p-n-A}}\},
\end{Eq*}
with $b$ to be fixed later. Firstly we take $k=1$,
for $(t,r)\in\Omega_{*1}$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
Taking the $L_{t,r}^\infty(\Omega_{*1})$ norm on both sides we get
\begin{Eq*}
\kl\|\omega_1u_{j+1}\kr\|_{L_{t,r}^\infty(\Omega_{*1})}\leq& \varepsilon C_0\Psi+C_0(b\varepsilon)^{1-p}\|\omega_1 u_j\|_{L_{t,r}^\infty(\Omega_{*1})}^p,\\
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_{*1})}\leq& C_0(b\varepsilon)^{1-p}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_{*1})}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{t,r}^\infty(\Omega_{*1})}^{p-1}.
\end{Eq*}
Choosing $b$ such that $(2C_0)^pb^{1-p}\Psi^{p-1}=1$, we find
\begin{Eq}\label{Eq:u_i_1}
\kl\|\omega_1u_{j}\kr\|_{L_{t,r}^\infty(\Omega_{*1})}\leq& 2\varepsilon C_0\Psi,\\
\kl\|\omega_1(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_{*1})}\leq& \frac{1}{2}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_{*1})}
\end{Eq}
holds for any $j$.
This means $\{u_j\}$ is a Cauchy sequence in weighted $L^\infty$ norm on $\Omega_{*1}$ region.
On the other hand, for $(t,r)\in \Omega_{*2}$, we separate $\Lambda$ into $\Lambda_1:=\Lambda\cap \Omega_{*1}$ and $\Lambda_2:=\Lambda\cap \Omega_{*2}$.
Because of the linearity of function, we can separate $u_{j+1}(t,r)$ to two terms which determined by the nonlinear terms supported in $\Lambda_1$ and $\Lambda_2$ respectively.
Taking $k=1$ in the former region and $k=2$ in the latter region, for $(t,r)\in \Omega_{*2}$ we find
\begin{Eq*}
\omega_2|u_{j+1}|\leq& \varepsilon C_0\kl<t-r\kr>^\frac{(n-1)p-n-A}{2}\Psi+C_0\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda_1)}^p\\
&+C_0\kl<t-r\kr>^{\frac{(1-n)p^2+(n+1)p+2}{2}}\|\omega_2 u_j\|_{L_{s,\rho}^\infty(\Lambda_2)}^p\\
\omega_2|u_{j+1}-u_j|\leq& C_0\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda_1)} \max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda_1)}^{p-1},\\
+C_0\kl<t-r\kr>&{}^{\frac{(1-n)p^2+(n+1)p+2}{2}}\|\omega_2 (u_j-u_{j-1})\|_{L_{s,\rho}^\infty(\Lambda_2)}\max_{l\in\{j,j-1\}}\|\omega_2 u_l\|_{L_{s,\rho}^\infty(\Lambda_2)}^{p-1}.
\end{Eq*}
Taking the $L_{t,r}^\infty(\Omega_{*2})$ norm on both sides,
noticing $\kl<t-r\kr>\geq (b \varepsilon)^{\frac{2(p-1)}{(n-1)p-n-A}}$ for $(t,r)\in \Omega_{*2}$
and $(2C_0)^pb^{1-p}\Psi^{p-1}=1$,
and using the estimate in $\Omega_{*1}$,
when $T_*\geq 3$ we get
\begin{Eq*}
\kl\|\omega_2u_{j+1}\kr\|_{L_{t,r}^\infty(\Omega_{*2})}\leq&C_1(\varepsilon \Psi)^p+C_1T_*^{\frac{(1-n)p^2+(n+1)p+2}{2}}\|\omega_2 u_j\|_{L_{t,r}^\infty(\Omega_{*2})}^p,\\
\kl\|\omega_2(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_{*2})}\leq& C_1(\varepsilon \Psi)^{p-1}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_{*1})}\\
+C_1T_*^{\frac{(1-n)p^2+(n+1)p+2}{2}}&\|\omega_2 (u_j-u_{j-1})\|_{L_{t,r}^\infty(\Omega_{*2})}\max_{l\in\{j,j-1\}}\|\omega_2 u_l\|_{L_{t,r}^\infty(\Omega_{*2})}^{p-1},
\end{Eq*}
with some $C_1$ large enough.
Choosing $\varepsilon$ such that $T_*(\varepsilon)\geq 3$ and $a=(2C_1)^{-p}\Psi^{p(1-p)}$, we find
\begin{Eq*}
\kl\|\omega_2u_{j+1}\kr\|_{L_{t,r}^\infty(\Omega_{*2})}\leq&2C_1(\varepsilon \Psi)^p,\\
\kl\|\omega_2(u_{j+1}-u_j)\kr\|_{L_{t,r}^\infty(\Omega_{*2})}\leq& C_1(\varepsilon \Psi)^{p-1}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{t,r}^\infty(\Omega_{*1})}\\
&+\frac{1}{2}\|\omega_2 (u_j-u_{j-1})\|_{L_{t,r}^\infty(\Omega_{*2})},
\end{Eq*}
holds for any $j$.
Now, since we already know $\{u_j\}$ is a Cauchy sequence on $\Omega_{*1}$ region,
we can also know $\{u_j\}$ is a Cauchy sequence in weighted $L^\infty$ norm on $\Omega_{*2}$ region.
In summary, we get the desired solution.
\value{parta}[$p_d<p=p_S$]\label{Pt:9}
In this part,
we choose $T_*$ which satisfies $\varepsilon^{p(p-1)}\ln T_*=a$
and $\Omega_{*1}$, $\Omega_{*2}$ as above with $a,b$ to be fixed latter.
Similarly,
for $(t,r)\in \Omega_{*1}$ we find
\begin{Eq*}
\omega_1|u_{j+1}|\leq& \varepsilon C_0\Psi+C_0\kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda)}^p,\\
\omega_1|u_{j+1}-u_j|\leq& C_0\kl<t-r\kr>^{\frac{(1-n)p+n+A}{2}}\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda)}\max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda)}^{p-1}.
\end{Eq*}
For $(t,r)\in \Omega_{*2}$ we find
\begin{Eq*}
\omega_2|u_{j+1}|\leq& \varepsilon C_0\kl<t-r\kr>^\frac{(n-1)p-n-A}{2}\Psi+C_0\|\omega_1 u_j\|_{L_{s,\rho}^\infty(\Lambda_1)}^p\\
&+C_0\ln\kl<t-r\kr>\|\omega_2 u_j\|_{L_{s,\rho}^\infty(\Lambda_2)}^p\\
\omega_2|u_{j+1}-u_j|\leq& C_0\|\omega_1 \kl(u_j-u_{j-1}\kr)\|_{L_{s,\rho}^\infty(\Lambda_1)} \max_{l\in\{j,j-1\}}\|\omega_1 u_l\|_{L_{s,\rho}^\infty(\Lambda_1)}^{p-1}\\
&+C_0\ln\kl<t-r\kr>\|\omega_2 (u_j-u_{j-1})\|_{L_{s,\rho}^\infty(\Lambda_2)}\max_{l\in\{j,j-1\}}\|\omega_2 u_l\|_{L_{s,\rho}^\infty(\Lambda_2)}^{p-1}.
\end{Eq*}
Taking $b$ satisfying $(2C_0)^pb^{1-p}\Psi^{p-1}=1$,
choosing $\varepsilon$ such that $T_*(\varepsilon)\geq 3$,
$a=(2C_1)^{-p}\Psi^{p(1-p)}$ and $C_1$ large enough, we get the Cauchy sequence $\{u_j\}$ and the desired solution.
Before the end of this section, we show the proof of \Cl{Cl:T*_pd}.
Following \eqref{Eq:T*_pd}, we can easily find
\begin{Eq*}
\varepsilon^{\frac{2(p-1)}{h_F(p)}+\delta}\lesssim T_*\lesssim \varepsilon^{\frac{2(p-1)}{h_F(p)}}
\end{Eq*}
for any $0<\delta\ll1$ and $\varepsilon$ small enough. This suggests us to define
\begin{Eq*}
S(\varepsilon):=\varepsilon^{-\frac{2(p-1)}{h_F(p)}}T_*,\qquad \varepsilon^{\delta}\lesssim S\lesssim 1.
\end{Eq*}
Then, \eqref{Eq:T*_pd} goes to
\begin{Eq*}
a=S^\frac{-h_F(p)}{2}\ln\kl(\varepsilon^{\frac{2(p-1)}{h_F(p)}}S\kr)\approx S^\frac{-h_F(p)}{2}|\ln\varepsilon|,
\end{Eq*}
and then
\begin{Eq*}
S\approx |\ln \varepsilon|^{\frac{2}{h_F(p)}},\qquad T_*\approx \varepsilon^{\frac{2(p-1)}{h_F(p)}}|\ln \varepsilon|^{\frac{2}{h_F(p)}},
\end{Eq*}
which finishes the proof.
\section{Long-time existence for $A\in[3,\infty)$}\label{Se:4}
In this section, we will consider the case $A\in[3,\infty)$, and show the proof of \Th{Th:M_2}. Again, we only need to consider the equation \eqref{Eq:u_o}.
\subsection{Estimate for linear solution}
In this subsection, we will construct some prior estimates of the solution to the linear equation \eqref{Eq:u_l}. Firstly we give the following argument.
\begin{lemma}\label{Le:ul_e1}
Let $u$ be the solution of \eqref{Eq:u_l}. We have
\begin{Eq}\label{Eq:ul_e11}
\kl\|r^{\frac{A-1}{2}}u\kr\|_{L_t^\infty L_r^q}\lesssim& \|r^{\frac{A+1}{2}}g\|_{L_r^q}+ \|r^{\frac{A-1}{2}}f\|_{L_r^q}+\|r^{\frac{A+1}{2}}F\|_{L_t^qL_r^1},
\end{Eq}
for any $1<q<\infty$, and
\begin{Eq}\label{Eq:ul_e12}
\kl\|r^{\frac{A-1}{2}-\alpha}u\kr\|_{L_t^\sigma L_r^p}\lesssim& \|r^{\frac{A+1}{2}}g\|_{L_r^q}+ \|r^{\frac{A-1}{2}}f\|_{L_r^q}+\|r^{\frac{A+1}{2}}F\|_{L_t^qL_r^1},
\end{Eq}
provided that
\begin{Eq}\label{Eq:sigpq_r}
1<\frac{q}{p}<\frac{\sigma}{p}<\infty,\qquad~\alpha p=1-\frac{p}{q}+\frac{p}{\sigma}.
\end{Eq}
\end{lemma}
We also have a modification result.
\begin{lemma}\label{Le:ul_e2}
Let $u$ be the solution of \eqref{Eq:u_l}. If $1<p\leq (n+1)/(n-1)$, then
\begin{Eq}\label{Eq:ul_e21}
&(T+1)^{\frac{(n-1)p-n-1}{2p}}\kl\|r^{\frac{(A-n)p+n+1}{2p}}u(T,r)\kr\|_{L_r^p(r<T+1)}\\
\lesssim& \|r^{\frac{A+1}{2}}g\|_{L_r^p}+ \|r^{\frac{A-1}{2}}f\|_{L_r^p}+\|r^{\frac{A+1}{2}}F\|_{L_t^pL_r^1(t<T)}.
\end{Eq}
Also,
if $(n+1)/(n-1)\leq p\leq p_S$,
then
\begin{Eq}\label{Eq:ul_e22}
&T^{\frac{(n-1)p-n-1}{2p}}\kl\|r^{\frac{(A-n)p+n+1}{2p}}u(T,r)\kr\|_{L_r^p}\\
\lesssim& \|r^{\frac{A+1}{2}+\frac{1}{p}}g\|_{L_r^\infty}+ \|r^{\frac{A-1}{2}+\frac{1}{p}}f\|_{L_r^\infty}+T^{\frac{1}{p}}\|r^{\frac{A+1}{2}}F\|_{L_t^\infty L_r^1(T/4<t<T)}\\
&+\|r^{\frac{A+1}{2}}g\|_{L_r^p}+ \|r^{\frac{A-1}{2}}f\|_{L_r^p}+\|r^{\frac{A+1}{2}}F\|_{L_t^pL_r^1(t<T/4)}.
\end{Eq}
\end{lemma}
\begin{proof}[Proof of \Le{Le:ul_e1}]
When $A\in {\mathbb{Z}}_+$,
this result is exactly the same with the Theorem 4.7 of \cite{MR1408499}
since we can take $\kappa$ in Theorem 4.7 arbitrary close to $2$.
So we only deal with the non-integer case.
Here we introduce $\delta< (A-3)/2$ which will be fixed later.
Firstly we consider $u=u_g$.
Using \Le{Le:u_e} and \Le{Le:I_p} with $\mu=\frac{r^2+\rho^2-t^2}{2r\rho}$,
we have
\begin{Eq*}
r^{\frac{A-1}{2}}|u_g|\lesssim&
\int_{|t-r|}^{t+r}(1+\mu)^{-\delta}\rho^{\frac{A-1}{2}}|g(\rho)|\d\rho+\int\limits_{0<\rho<t-r\atop -2<\mu<-1}|1+\mu|^{-\delta}\rho^{\frac{A-1}{2}}|g(\rho)|\d\rho\\
&+\int\limits_{0<\rho<t-r\atop \mu<-2}(1-\mu)^{\frac{1-A}{2}}\rho^{\frac{A-1}{2}}|g(\rho)|\d\rho\\
\lesssim&\int_{|t-r|}^{t+r}(1+\mu)^{-\delta}\rho^{-1}|\rho^{\frac{A+1}{2}}g(\rho)|\d\rho\\
&+\int\limits_{0<\rho<t-r}(1-\mu)^{-1}|1+\mu|^{-\delta}\rho^{-1}|\rho^{\frac{A+1}{2}}g(\rho)|\d\rho,
\end{Eq*}
where the second integral does not appear for $t<r$.
Using Proposition 2.7 and Proposition 4.4 in \cite{MR1408499}, we obtain the estimate as we desired.
Next we consider $u=u_f$. Here for simplicity we denote $h(\rho):=\rho^{\frac{A-1}{2}}f(\rho)$.
Using \Le{Le:u_e} and \Le{Le:I_p} again with $\mu=\frac{r^2+\rho^2-t^2}{2r\rho}$,
for $r<t$ we find
\begin{Eq*}
r^{\frac{A-1}{2}}u_f=&\frac{1}{2}h(t+r)-P.V.\int_0^{t+r}\frac{t}{r\rho}I_A'(\mu)h(\rho)\d\rho,\\
r^{\frac{A-1}{2}}|u_f|\lesssim& \frac{1}{2}|h(t+r)|+\kl|P.V.\int\limits_{0\leq\rho\leq t+r\atop -2<\mu<1}\frac{t}{r\rho}(1+\mu)^{-1}h(\rho)\d\rho\kr|\\
&+\int\limits_{0\leq\rho\leq t+r\atop -2<\mu<1}\frac{t}{r\rho}|1+\mu|^{-\delta}|h(\rho)|\d\rho+\int\limits_{0\leq\rho\leq t-r\atop \mu<-2}\frac{t}{r\rho}|1-\mu|^{\frac{-1-A}{2}}|h(\rho)|\d\rho\\
\lesssim&\frac{1}{2}|h(t+r)|+\kl|P.V.\int_{0}^{t+r}\frac{t}{r\rho}(1+\mu)^{-1}h(\rho)\d\rho\kr|\\
&+\int_{0}^{t-r}\frac{t}{r\rho}|1-\mu|^{-1}|h(\rho)|\d\rho+\int_{t-r}^{t+r}\frac{t}{r\rho}|1+\mu|^{-\delta}|h(\rho)|\d\rho\\
&+\int_0^{t-r}\frac{t}{r\rho}(1-\mu)^{-1}|1+\mu|^{-\delta}|h(\rho)|\d\rho\\
\equiv& K_1+K_2+K_3+K_4+K_5.
\end{Eq*}
It's easy to find that
\begin{Eq*}
\|K_1\|_{L_t^\infty L_r^q}\lesssim \|h\|_{L_r^q}.
\end{Eq*}
On the other, we introduce the well known \emph{Hardy-Littlewood} inequality
\begin{Eq*}
\kl\||y|^{-\alpha}f(x\pm y)\kr\|_{L_x^\sigma L_y^p({\mathbb{R}}^2)} \lesssim \|f\|_{L^q({\mathbb{R}})},
\end{Eq*}
with \eqref{Eq:sigpq_r}. Now, taking $f(x)=|h(x)|\chi_{[0,\infty)}(x)$, we also find
\begin{Eq*}
\|r^{-\alpha}K_1\|_{L_t^\sigma L_r^p}\lesssim \|h\|_{L_r^q},
\end{Eq*}
provided that \eqref{Eq:sigpq_r}.
Meanwhile, adopting Proposition 2.5 in \cite{MR1408499} to deal with $K_4$, and adopting Proposition 4.4 in \cite{MR1408499} for $K_3$ and $K_5$, we find both of them have the same control as $K_1$.
As for $K_2$, noticing
\begin{Eq*}
\frac{t}{r\rho}(1+\mu)^{-1}=\frac{1}{r+\rho-t}-\frac{1}{t+r+\rho},
\end{Eq*}
we can control $K_2$ by
\begin{Eq*}
K_2\lesssim &\kl|P.V.\int_0^\infty \frac{1}{\rho+r-t}h(\rho)\d\rho\kr|+\int_{t+r}^\infty \frac{1}{\rho+r-t}|h(\rho)|\d\rho\\
&+\frac{1}{t+r}\int_0^{t+r}|h(\rho)|\d\rho\\
\equiv&K_{2,1}(t-r)+K_{2,2}(t,r)+K_{2,3}(t+r).
\end{Eq*}
For $K_{2,1}$, we introduce the estimate of \emph{Hilbert}-transform
\begin{Eq*}
\kl\|P.V.\int \frac{1}{x-y}f(y)\d y\kr\|_{L_x^q}\lesssim \|f\|_{L_x^q}
\end{Eq*}
with $1<q<\infty$. Taking $f(x)=h(x)\chi_{[0,\infty)}(x)$, we find
\begin{Eq*}
\kl\|K_{2,1}(t-r)\kr\|_{L_t^\infty L_r^q}=\kl\|K_{2,1}(r)\kr\|_{L_r^q}\lesssim \|h\|_{L^q}.
\end{Eq*}
Also, using \emph{Hardy-Littlewood} inequality again, we get the dominate of $K_{2,1}(t-r)$ same as that of $K_1$ provided that \eqref{Eq:sigpq_r}. As for $K_{2,3}$, we introduce the \emph{Hardy-Littlewood} maximal inequality
\begin{Eq*}
\kl\|\sup_{y>0}\frac{1}{2y}\int_{x-y}^{x+y} f(z)\d z\kr\|_{L_x^q}\lesssim \|f\|_{L^q}
\end{Eq*}
with $1<q<\infty$. Taking $f(x)=h(x)\chi_{[0,\infty)}(x)$, we find
\begin{Eq*}
\kl\|K_{2,3}(t+r)\kr\|_{L_t^\infty L_r^q}=\kl\|K_{2,3}(r)\kr\|_{L_r^q}\lesssim \|h\|_{L^q}.
\end{Eq*}
Using \emph{Hardy-Littlewood} inequality again, we get the dominate of $K_{2,3}(t+r)$ same as that of $K_1$ provided that \eqref{Eq:sigpq_r}.
Finally, for $K_{2,2}(t,r)$, we have
\begin{Eq*}
K_{2,2}(t,r)=\int_r^\infty \frac{1}{\rho+r}|h(\rho+t)|\d\rho\leq \int_r^\infty \rho^{-1}|h(\rho+t)|\d\rho.
\end{Eq*}
Then, using \emph{Hardy's} inequality we find
\begin{Eq*}
\|K_{2,2}\|_{L_t^\infty L_r^q}\lesssim \|h\|_{L_r^q}.
\end{Eq*}
On the other hand, for any $G(t,r)$ with $\|G\|_{L_t^{\sigma'}L_r^{p'}}\leq 1$ we see
\begin{Eq*}
&\int_0^\infty\int_0^\infty r^{-\alpha}K_{2,2}(t,r)G(t,r)\d r\d t\\
=&\int_0^\infty\int_t^\infty \int_{0}^{\rho-t} \frac{r^{-\alpha}}{\rho+r-t}|h(\rho)|G(t,r)\d r\d\rho\d t\\
\lesssim &\int_0^\infty\int_t^\infty |h(\rho)|\kl\|\frac{r^{-\alpha}}{\rho+r-t} \kr\|_{L_r^p(0,\rho-t)}\|G\|_{L_r^{p'}}\d\rho\d t\\
\approx &\int_0^\infty\int_t^\infty |h(\rho)||\rho-t|^{-\alpha-1+\frac{1}{p}}\|G\|_{L_r^{p'}}\d\rho\d t\\
\lesssim &\kl\|\int_t^\infty |h(\rho)||\rho-t|^{-\alpha-1+\frac{1}{p}}\d\rho\kr\|_{L_t^{\sigma}}\\
\lesssim &\|h\|_{L_r^q},
\end{Eq*}
where in the last step we use the \emph{Hardy-Littlewood} inequality. Now, we find
\begin{Eq*}
\|r^{-\alpha}K_{2,2}\|_{L_t^\sigma L_r^p}\leq\sup\limits_{\|G\|_{L_t^{\sigma'}L_r^{p'}}\leq 1}\kl<r^{-\alpha}K_{2,2},G\kr>\lesssim \|h\|_{L_r^q}.
\end{Eq*}
Mixing these results, we obtain the estimate for $r<t$ part. For $r>t$, we have
\begin{Eq*}
r^{\frac{A-1}{2}}u_f=&\frac{1}{2}h(t+r)+\frac{1}{2}h(r-t)-\int_{r-t}^{t+r}\frac{t}{r\rho}I_A'(\mu)h(\rho)\d\rho\\
\equiv &K_1'+K_2'+K_3'.
\end{Eq*}
The estimate of $K_1'$ and $K_2'$ is the same as that of $K_1$ in $r<t$ part, and the estimate of $K_3'$ is the same as that of $K_4$. Adding all together, we finish the proof of $u_f$ part.
Finally, we consider $u=u_F$.
Using \Le{Le:u_e} and \Le{Le:I_p} again with $\mu=\frac{r^2+\rho^2-(t-s)^2}{2r\rho}$,
similarly we have
\begin{Eq*}
r^{\frac{A-1}{2}}u_F=&\int_0^t\int_{0}^{r+t-s}I_A(\mu)\rho^{\frac{A-1}{2}}F(s,\rho)\d\rho\d s\\
r^{\frac{A-1}{2}}|u_F|\lesssim&\int_0^t\int_{|r-t+s|}^{r+t-s}(1+\mu)^{-\delta}\rho^{-1}|G(s,\rho)|\d\rho\d s
\\
&+\iint\limits_{0\leq s\leq t\atop 0\leq \rho\leq t-s-r}(1-\mu)^{-1}|1+\mu|^{-\delta}\rho^{-1}|G(s,\rho)|\d\rho\d s
\end{Eq*}
where $G(s,\rho):=\rho^{\frac{A+1}{2}}F(s,\rho)$ and the second integral does not appear for $t-s<r$. Here we choose $\delta$ small enough. Using Proposition 4.5 in \cite{MR1408499}, we obtain the desired estimate for $u_F$.
Now, we finish the proof of \Le{Le:ul_e1}.
\end{proof}
\begin{proof}[Proof of \Le{Le:ul_e2}]
The proof of \Le{Le:ul_e2} is almost the same with that of Theorem 6.4 in \cite{MR1408499}. Thus, we only give a sketch of the proof.
The estimate \eqref{Eq:ul_e21} and part of estimate \eqref{Eq:ul_e22} are direct consequence of \eqref{Eq:ul_e11} with $q=p$.
We only need to show the estimate of
$T^{\frac{(n-1)p-n-1}{2p}}\kl\|r^{\frac{(A-n)p+n+1}{2p}}u(T,r)\kr\|_{L_r^p(r<T/4)}$.
To dominate $u_f$, we separate $f=f_0+f_1$ with $f_0=\chi_{[0,T/4]}f$.
Then, $u_{f=f_1}$ depends on $f_1(\rho)$ with $T/4<\rho<5T/4$.
So, we obtain the weight of $T$ by extracting the weight of $\rho$.
For $u_{f=f_0}$, we find that $\rho,r<T/4$ in the expression of $u_{f=f_0}$
where $|1\pm\mu|^{-1}\approx r\rho/T^2$.
Then we get the desired estimate by a direct calculation.
The estimate of $u_g$ is similar to that of $u_f$. Finally for $u_F$, we separate the integral in $u_F$ into three parts: $\{r\geq(T-s)/4\}$, $\{r,\rho\leq(T-s)/4\}$ and $\{r\leq (T-s)/4\leq \rho\}$. Then we get the estimate by a similar discussion.
\end{proof}
\subsection{Long-time existence for $1<p<p_{conf}$}
In this subsection, we will give the proof of \Th{Th:M_2}.
The main process of proof is almost the same as that of \cite[Theorem 5.1, Theorem 6.1 and Theorem 6.3]{MR1408499}.
So we only prove global existence in $p_S<p<p_{conf}$ and long-time existence in $p_m\leq p<p_S$ to show such processes fit our frame.
\setcounter{part0}{0}
\value{parta}[Proof of $p_S< p<p_{conf}$]
Similar to the last section,
we will construct a Cauchy sequence to approach the weak solution.
We set $u_{-1}=0$ and let $u_{j+1}$ be the solution of the equation \eqref{Eq:uj_o}.
We are going to use the estimate \eqref{Eq:ul_e12},
where we set
\begin{Eq*}
q=\frac{2(p-1)}{(n+3)-(n-1)p},\qquad \sigma=pq,\qquad \alpha=\frac{(n-1)p-n-1}{2p}.
\end{Eq*}
Here $p<p_{conf}$ so $0<q<\infty$ and $p>p_S$ so $q>p$. Then, we conclude
\begin{Eq*}
\kl\|r^{\frac{(A-n)p+n+1}{2p}}u_{j+1}\kr\|_{L_t^{pq} L_r^p}\leq& \varepsilon C_0\Psi+C_0\|r^{\frac{(A-n)p+n+1}{2}}|u_{j}|^p\|_{L_t^qL_r^1}\\
\leq& \varepsilon C_0\Psi+C_0\|r^{\frac{(A-n)p+n+1}{2p}}u_{j}\|_{L_t^{pq}L_r^p}^p\\
\Psi:=&\kl\|r^\frac{n+1}{2}U_1\kr\|_{L_r^q}+\|r^{\frac{n-1}{2}}U_0\|_{L_r^q}\\
\end{Eq*}
for some $C_0$ large enough.
Then, for any $\varepsilon$ satisfies $(2\varepsilon C_0\Psi)^p<\varepsilon \Psi$ we find
\begin{Eq*}
\kl\|r^{\frac{(A-n)p+n+1}{2p}}u_{j}\kr\|_{L_t^{pq} L_r^p}\leq 2\varepsilon C_0\Psi
\end{Eq*}
holds for any $j\geq 0$. By this result and \eqref{Eq:ul_e12} we also find
\begin{Eq*}
&\kl\|r^{\frac{(A-n)p+n+1}{2p}}\kl(u_{j+1}-u_j\kr)\kr\|_{L_t^{pq} L_r^p}\\
\leq& C_0\|r^{\frac{(A-n)p+n+1}{2}}\kl(|u_{j}|^p-|u_{j-1}|^p\kr)\|_{L_t^qL_r^1}\\
\leq& C_1\|r^{\frac{(A-n)p+n+1}{2p}}(u_{j}-u_{j-1})\|_{L_t^{pq}L_r^p}\max_{k\in\{j,j-1\}}\|r^{\frac{(A-n)p+n+1}{2p}}u_k\|_{L_t^{pq}L_r^p}^{p-1}\\
\leq& C_1(2\varepsilon C_0\Psi)^{p-1}\|r^{\frac{(A-n)p+n+1}{2p}}(u_{j}-u_{j-1})\|_{L_t^{pq}L_r^p}
\end{Eq*}
with some $C_1$ large enough.
Now, for any $0<\varepsilon$ with $C_1(2\varepsilon C_0\Psi)^{p-1}<1/2$,
we find $\{u_j\}$ is a Cauchy sequence in its space.
Set the limit as $u$.
Using \eqref{Eq:ul_e11} we can also find
$\|r^{\frac{A-1}{2}}u\|_{L_t^\infty L_r^q}\leq 2\varepsilon C_2\Psi$ with some $C_2$.
To check it is the weak solution of \eqref{Eq:u_o} indeed,
we need to show \eqref{Eq:u_i_r}.
For any compact set $K_t\times K_r\subset {\mathbb{R}}_+^2$, we find
\begin{Eq*}
\kl\|r^{\frac{(A-n)p+n+A}{2}-1}|u|^p\kr\|_{L_{t,r}^1(K_t\times K_r)}\leq& C(K_t)\kl\|r^{\frac{(A-n)p+n+1}{2p}}u\kr\|_{L_t^{pq} L_r^p}^p\kl\|r^{\frac{A-3}{2}}\kr\|_{L_r^{\infty}(K_r)}\\
\leq& (2\varepsilon C_0\Psi)^pC(K_t,K_r)<\infty\\
\|r^{A-1}u\|_{L_{t,r}^1(K_t\times K_r)}\leq &C(K_t)\kl\|r^{\frac{A-1}{2}}u\kr\|_{L_t^\infty L_r^q}\kl\|r^{\frac{A-1}{2}}\kr\|_{L_r^{q'}(K_r)}\\
\leq& (2\varepsilon C_1\Psi)C(K_t,K_r)<\infty.
\end{Eq*}
This finishes the proof.
\value{parta}[Proof of $p_m\leq p<p_S$]
Next we consider $(n+1)/(n-1)\leq p< p_S$. Set
\begin{Eq*}
A_j(T):=T^{\frac{(n-1)p-n-1}{2p}}\kl\|r^{\frac{(A-n)p+n+1}{2p}}u_{j}(T,r)\kr\|_{L_r^p},
\end{Eq*}
using \eqref{Eq:ul_e22} we see
\begin{Eq*}
A_{j+1}(T)\leq& \varepsilon C_0\Psi+ C_0T^{\frac{1}{p}}\|r^{\frac{(A-n)p+n+1}{2}}|u_j|^p\|_{L_t^\infty L_r^1(T/4<t<T)}
\\
&+C_0\|r^{\frac{(A-n)p+n+1}{2}}|u_j|^p\|_{L_t^pL_r^1(t<T/4)}\\
\leq& \varepsilon C_0\Psi+ C_0T^{\frac{1}{p}}\|t^{\frac{(1-n)p+n+1}{2p}}A_j(t)\|_{L_t^\infty(T/4<t<T)}^p
\\
&+C_0\|t^{\frac{(1-n)p+n+1}{2p}}A_j(t)\|_{L_t^{p^2}(t<T/4)}^p
\end{Eq*}
for some $C_0$ large enough and
\begin{Eq*}
\Psi:=\|r^{\frac{n+1}{2}+\frac{1}{p}}U_1\|_{L_r^\infty}+ \|r^{\frac{n-1}{2}+\frac{1}{p}}U_0\|_{L_r^\infty}
&+\|r^{\frac{u+1}{2}}U_1\|_{L_r^p}+ \|r^{\frac{n-1}{2}}U_0\|_{L_r^p}.
\end{Eq*}
When $\sup_{0\leq T\leq T_*}A_j(T)\leq 2\varepsilon C_0\Psi$ holds for some $j$ and $T_*$ defined in \eqref{Eq:Main_6} with $c$ to be fixed later, we find
\begin{Eq*}
\sup_{0\leq T\leq T_*} A_{j+1}(T)\leq&\varepsilon C_0\Psi+ C_1(2\varepsilon C_0 \Psi)^pT_*^{\frac{(1-n)p^2+(n+1)p+2}{2p}}\\
\leq&\varepsilon C_0\Psi+ \varepsilon C_1(2 C_0 \Psi)^p c^{\frac{-h_S}{2p}}
\end{Eq*}
with some $C_1$. Choosing $c$ small enough such that $C_1(2 C_0 \Psi)^p c^{\frac{-h_S}{2p}}\leq C_0\Psi$ and noticing $A_{-1}(T)\equiv 0$, we find such estimate holds for any $j$. A similar manner also shows
\begin{Eq*}
\sup_{0\leq T\leq T_*} B_{j+1}(T)\leq& \frac{1}{2}\sup_{0\leq T\leq T_*} B_{j}(T),\\
B_{j}(T):=&T^{\frac{(n-1)p-n-1}{2p}}\kl\|r^{\frac{(A-n)p+n+1}{2p}}\kl(u_{j}-u_{j-1}\kr)(T,r)\kr\|_{L_r^p}.
\end{Eq*}
Then, we get the desired solution $u$ as the limit of $\{u_j\}$.
Now we check \eqref{Eq:u_i_r}. For any compact set $K_t\times K_r\subset {\mathbb{R}}_+^2$, we find
\begin{Eq*}
\kl\|r^{\frac{(A-n)p+n+A}{2}-1}|u|^p\kr\|_{L_{t,r}^1(K_t\times K_r)}\leq& \kl\|t^{\frac{(n-1)p-n-1}{2p}}r^{\frac{(A-n)p+n+1}{2p}}u\kr\|_{L_t^\infty L_r^p}^p\\
&\times \kl\|t^{\frac{(1-n)p+n+1}{2}}r^{\frac{A-3}{2}}\kr\|_{L_t^1 L_r^\infty(K_t\times K_r)},\\
\|r^{A-1}u\|_{L_{t,r}^1(K_t\times K_r)}\leq &\kl\|t^{\frac{(n-1)p-n-1}{2p}}r^{\frac{(A-n)p+n+1}{2p}}u\kr\|_{L_t^\infty L_r^p}\\
&\times\kl\|t^{\frac{(1-n)p+n+1}{2p}}r^{\frac{(n+A-2)p-n-1}{2p}}\kr\|_{L_t^1 L_r^{p'}(K_t\times K_r)}.
\end{Eq*}
Noticing $\frac{(n+A-2)p-n-1}{2p}>\frac{A-3}{2p}\geq 0$ and $\frac{(1-n)p+n+1}{2}>-1$ due to $p<p_S<p_{conf}$, we find both of the above two terms are finite.
This finishes the proof.
\subsection*{Acknowledgments}
The authors would like to thank the anonymous referee for the careful reading and valuable comments.
The authors were supported by NSFC 11671353 and NSFC 11971428.
| 2024-02-18T23:40:19.403Z | 2021-11-23T02:15:44.000Z | algebraic_stack_train_0000 | 2,051 | 19,894 |
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