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BkiUd47xK7FjYAb_7pWN | 5 | 1 | \section{Introduction} \label{seintro}
Let $\mathcal{N}$ be a non-Archimedean ordered field extension of $\ensuremath{\mathbb{R}}$ that is real closed and
complete in the order topology and whose Hahn group $S_\mathcal{N}$ is Archimedean, i.e. (isomorphic to) a subgroup of $\ensuremath{\mathbb{R}}$. Recall that $S_{\mathcal{N}}$ is the set of equivalence classes under the relation $\sim$ defined on $\mathcal{N}^*:=\mathcal{N}\setminus\{0\}$ as follows: For $x,y\in \mathcal{N}^*$, we say that $x$ is of the same order as $y$ and write $x\sim y$ if there exist
$n,m\in\ensuremath{\mathbb{N}}$
such that $n|x|>|y|$ and $m|y|>|x|$, where $|\cdot|$ denotes the ordinary absolute value on $\mathcal{N}$:
$ |x|=\max\left\{x,
-x\right \}$.
$S_{\mathcal{N}}$ is naturally endowed with an addition via $[x]+[y]=[x\cdot y]$ and an order via
$[x]<[y]$ if $|y|\ll|x|$ (which means $n|y|< |x|$ for all $n\in\mathbb{N}$), both of which are readily checked to be well-defined.
It follows that $(S_{\mathcal{N}},+,<)$ is an ordered group,
often referred to as the Hahn group or skeleton group, whose neutral element is $[1]$, the class of $1$.
The theorem of Hahn \cite{hahn} provides a complete classification
of non-Archimedean ordered field extensions of $\ensuremath{\mathbb{R}}$ in terms
of their skeleton groups. In fact, invoking the axiom of choice it
is shown that the elements of our field $\mathcal{N}$ can be written as
(generalized) formal power series (also called Hahn series) over its skeleton group $S_{\mathcal{N}}$ with real
coefficients, and the set of appearing exponents forms a
well-ordered subset of $S_{\mathcal{N}}$. That is, for all $x\in \mathcal{N}$, we have that
$x=\sum_{q\in S_{\mathcal{N}}}a_qd^{q}$;
with $a_q\in\setR$ for all $q$, $d$ a positive infinitely small element of $\mathcal{N}$, and the support of $x$, given by
$\mbox{supp}(x):=\{q\in S_\mathcal{N}: a_q\ne 0\}$,
forming a well-ordered subset of $S_{\mathcal{N}}$.
We define for $x\ne 0$ in $\mathcal{N}$,
$ \lambda(x)=\min\left(\mbox{supp}(x)\right)$,
which exists since $\mbox{supp}(x)$ is well-ordered. Moreover, we set $\lambda(0)=\infty$. Given a nonzero $x=\sum_{q\in \mbox{supp}(x)}a_qd^q$, then $x>0$ if and only if $a_{\lambda(x)}>0$.
The smallest such
field $\mathcal{N}$ is the Levi-Civita field $\mathcal{R}$, first introduced in \cite{levicivita1,levicivita2}. In this case
$S_\mathcal{R}=\ensuremath{\mathbb{Q}}$, and for any element $x\in\mathcal{R}$, supp$(x)$ is a left-finite
subset of $\ensuremath{\mathbb{Q}}$, i.e. below any rational bound $r$ there are only finitely many exponents in the Hahn representation of $x$. The Levi-Civita field
$\mathcal{R}$ is of particular interest because of its practical usefulness. Since the supports of the elements of $\mathcal{R}$ are left-finite, it is possible to represent these
numbers on a computer. Having infinitely
small numbers allows for many computational
applications; one
such application is the computation of derivatives of real functions representable on a computer \cite{rsdiffsf,dabul00}, where both the accuracy of
formula manipulators and the speed of classical numerical methods are achieved. For a review of the Levi-Civita field $\mathcal{R}$, see
\cite{rsrevitaly13} and references therein.
In the wider context of valuation theory, it is
interesting to note that the topology induced by the order on $\mathcal{N}$ is the same as the valuation topology $\tau_v$ introduced via the ultrametric $\Lambda:\mathcal{N}\times\mathcal{N}\rightarrow\mathbb{R}$, given
by $\Lambda(x,y)=\exp{(-\lambda(x-y))}$. It follows therefore that the field $\mathcal{N}$ is just a special
case of the class of fields discussed in \cite{schikhofbook}. For a general
overview of the algebraic properties of formal power series fields, we refer to the comprehensive overview by Ribenboim \cite{ribenboim92}, and
for an overview of the related valuation theory the book by Krull \cite{krull32}. A thorough and complete treatment of ordered structures can
also be found in \cite{priessbook}. A more comprehensive survey of all non-Archimedean fields can be found in \cite{barria-sham-18}.
\section{Weak Local Uniform Differentiability and Review of Recent Results}
Because of the total disconnectedness of the field $\mathcal{N}$ in the order topology, the standard theorems of real calculus like the
intermediate value theorem, the inverse function theorem, the mean value theorem, the implicit function theorem and Taylor's theorem require stronger smoothness criteria of the functions involved in order for the theorems to hold.
In this section we will present one such criterion: the so-called \lq weak local uniform differentiability\rq,
we will review recent work based on that smoothness criterion and then present new results.
In \cite{boo-sham-18}, we focus our attention on $\mathcal{N}$-valued functions of one variable. We study the properties of weakly locally uniformly differentiable (WLUD)
functions at a point $x_0\in\mathcal{N}$ or on an open subset $A$ of $\mathcal{N}$. In particular, we show that WLUD functions are $C^1$, they include all polynomial functions,
and they are closed under addition, multiplication and composition. Then we generalize the definition of weak local uniform differentiability to any order. In particular,
we study the properties of WLUD$^2$ functions at a point $x_0\in\mathcal{N}$ or on an open subset $A$ of $\mathcal{N}$; and we show that WLUD$^2$ functions are $C^2$,
they include all polynomial functions, and they are closed under addition, multiplication and composition. Finally, we
formulate and prove an inverse function theorem as well as a local intermediate value theorem and a local mean value theorem for these functions.
Here we only recall the main definitions and results (without proofs) in \cite{boo-sham-18} and refer the reader to that paper for the details.
\begin{defn}
Let $A\subseteq \mathcal{N}$ be open, let $f:A\rightarrow \mathcal{N}$, and let $x_0\in A$ be given. We say that $f$ is weakly locally uniformly differentiable (abbreviated as WLUD)
at $x_0$ if $f$ is differentiable in a neighbourhood $\Omega$ of $x_0$ in $A$ and if for every $\epsilon > 0$ in $\mathcal{N}$ there exists $\delta > 0$ in $\mathcal{N}$ such that $(x_0 - \delta, x_0 + \delta) \subset \Omega$, and
for
every $x,y \in (x_0 - \delta, x_0 + \delta)$ we have that $\abs{f(y) - f(x) - f^\prime (x)(y-x)} \le \epsilon \abs{y-x}$. Moreover, we say that $f$ is WLUD on $A$ if $f$
is WLUD at every point in $A$.
\end{defn}
We extend the WLUD concept to higher orders of differentiability and we define WLUD$^k$ as follows.
\begin{defn}\label{def:wludn}
Let $A\subseteq \mathcal{N}$ be open, let $f:A\rightarrow \mathcal{N}$, let $x_0\in A$, and let $k\in\mathbb{N}$ be given. We say that $f$ is WLUD$^k$ at $x_0$ if $f$ is $k$ times
differentiable in a neighbourhood $\Omega$ of $x_0$ in $A$ and if for every $\epsilon > 0$ in $\mathcal{N}$ there exists $\delta > 0$ in $\mathcal{N}$ such that $(x_0-\delta,x_0+\delta) \subset \Omega$, and for every
$x,y \in (x_0-\delta,x_0+\delta)$ we have that
\[
\left|f(y) - \sum\limits_{j=0}^k \frac{f^{(j)}(x)}{j!}(y-x)^j\right|\le \epsilon \left|y-x\right|^k.
\]
Moreover, we say that $f$ is WLUD$^k$ on $A$ if $f$ is WLUD$^k$ at every point in $A$. Finally, we say that $f$ is WLUD$^\infty$ at $x_0$ (respectively, on $A$) if $f$ is WLUD$^k$ at $x_0$ (respectively, on $A$) for every
$k\in\mathbb{N}$.
\end{defn}
\begin{theorem}[Inverse Function Theorem]
Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow \field $ be WLUD on $A$, and let $x_0 \in A$ be such that $f^\prime(x_0) \neq 0$. Then there exists a neighborhood $\Omega$
of $x_0$
in $A$ such that
\begin{enumerate}
\item $\left.f\right|_\Omega$ is one-to-one;
\item $f(\Omega)$ is open; and
\item $f^{-1}$ exists and is WLUD on $f(\Omega)$ with $(f^{-1})^\prime = 1/\left(f^\prime \circ f^{-1}\right)$.
\end{enumerate}
\end{theorem}
\begin{theorem}[Local Intermediate Value Theorem]\label{tlivt}
Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow \field $ be WLUD on $A$, and let $x_0 \in A$ be such that $f^\prime(x_0) \neq 0$. Then there exists a neighborhood $\Omega$
of $x_0$ in $A$ such that for
any $a<b$ in $f(\Omega)$ and for any $c\in (a,b)$, there is an $x\in\left(\min\left\{f^{(-1)}(a),f^{(-1)}(b)\right\},\max\left\{f^{(-1)}(a),f^{(-1)}(b)\right\}\right)$
such that $f(x)=c$.
\end{theorem}
\begin{theorem}[Local Mean Value Theorem]\label{thmvt}
Let $A\subseteq\mathcal{N}$ be open, let $f:A\rightarrow \field $ be WLUD$^2$ on $A$, and let $x_0 \in A$ be such that $f^{\prime\prime}(x_0) \neq 0$. Then there exists a neighborhood $\Omega$ of
$x_0$ in $A$ such that $f$ has the mean value property on $\Omega$. That is, for every $a,b\in \Omega$ with $a<b$, there exists $c\in (a,b)$ such that
\[
f^\prime(c) = \frac{f(b) - f(a)}{b-a}.
\]
\end{theorem}
As in the real case, the mean value property can be used to prove other important results. In particular, while L'H\^opital's rule does not hold for differentiable functions on
$\mathcal{N}$, we prove the result under similar conditions to those of the local mean value theorem.
\begin{theorem}[L'H\^opital's Rule]
Let $A\subset \mathcal{N}$ be open, let $f,g:A\rightarrow \mathcal{N}$ be WLUD$^2$ on $A$, and let $a\in A$ be such that $f^{\prime\prime}(a) \neq 0$ and $g^{\prime\prime}(a) \neq 0$.
Furthermore, suppose that $f(a) = g(a) = 0$, that there exists a neighborhood
$\Omega$ of $a$ in $A$ such that $g^{\prime}(x) \neq 0$ for every $x\in \Omega\setminus\{a\}$, and that $\lim\limits_{x\rightarrow a} f^{\prime}(x)/g^{\prime}(x)$ exists. Then
\[
\lim_{x\rightarrow a} \frac{f(x)}{g(x)} = \lim_{x\rightarrow a} \frac{f^{\prime}(x)}{g^{\prime}(x)}.
\]
\end{theorem}
In \cite{MultivWLUD}, we formulate and prove a Taylor theorem with remainder for WLUD$^k$ functions from $\mathcal{N}$ to $\mathcal{N}$. Then we extend the concept of WLUD to functions from $\mathcal{N}^n$ to
$\mathcal{N}^m$ with $m,n\in\mathbb{N}$ and study the properties of those functions as we did for functions from $\mathcal{N}$ to $\mathcal{N}$. Then we formulate and prove
the inverse function theorem for WLUD functions from $\mathcal{N}^n$ to
$\mathcal{N}^n$ and the implicit function theorem for WLUD functions from $\mathcal{N}^n$ to $\mathcal{N}^m$ with $m<n$ in $\mathbb{N}$.
As in the real case, the proof of Taylor's theorem with remainder uses the mean value theorem.
However, in the non-Archimedean setting, stronger conditions on the function are needed than in the real case for the formulation of the theorem.
\begin{theorem}\label{thmtaylor}
(Taylor's Theorem with Remainder) Let $A \subseteq \mathcal{N}$ be open, let $k\in\mathbb{N}$ be given, and let $f : A \rightarrow \mathcal{N}$ be WLUD$^{k+2}$ on $A$.
Assume further that $f^{(j)}$ is WLUD$^{2}$ on $A$ for $0\leq j \leq k$. Then, for every $x\in A$, there exists a neighborhood $U$ of $x$ in $A$ such that, for any $y \in U$,
there exists
$c \in \left[ \min(y, x),\max(y, x) \right]$ such that
\begin{equation}\label{eqtaylor}
f(y)=\sum_{j=0}^{k} \frac{f^{(j)}\left(x\right)}{j !}\left(y-x\right)^{j}+\frac{f^{(k+1)}(c)}{(k+1) !}\left(y-x\right)^{k+1}.
\end{equation}
\end{theorem}
Before we define weak local uniform differentiability for functions from $\mathcal{N}^n$ to $\mathcal{N}^m$ and then state the inverse function theorem and the implicit function theorem, we introduce the following notations.
\begin{notation} Let $A\subset\mathcal{N}^n$ be open, let $\boldsymbol{x_0}\in A$ be given, and let $\boldsymbol{f}:A\rightarrow\mathcal{N}^m$ be such that all the first order partial derivatives of $\boldsymbol{f}$ at $\boldsymbol{x_0}$ exist. Then $\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x_0})$ denotes the linear map from $\mathcal{N}^n$ to $\mathcal{N}^m$ defined by the $m\times n$
Jacobian matrix of $\boldsymbol{f}$ at $\boldsymbol{x_0}$:
\[
\begin{pmatrix} \boldsymbol{f}^1_1(\boldsymbol{x_0}) & \boldsymbol{f}^1_2(\boldsymbol{x_0})& \ldots & \boldsymbol{f}^1_n(\boldsymbol{x_0})
\\ \boldsymbol{f}^2_1(\boldsymbol{x_0}) &
\boldsymbol{f}^2_2(\boldsymbol{x_0}) & \ldots & \boldsymbol{f}^2_n(\boldsymbol{x_0}) \\ \vdots & \vdots &\ddots & \vdots \\
\boldsymbol{f}^m_1(\boldsymbol{x_0}) & \boldsymbol{f}^m_2(\boldsymbol{x_0}) & \ldots & \boldsymbol{f}^m_n(\boldsymbol{x_0}) \end{pmatrix}
\]
with $\boldsymbol{f}^i_j(\boldsymbol{x_0})=\frac{\partial f_i}{\partial x_j}(\boldsymbol{x_0})$ for $1\le i\le m$ and $1\le j\le n$. Moreover, if $m=n$ then the determinant of the $n\times n$ matrix $\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x_0})$ is denoted by $J\boldsymbol{f}(\boldsymbol{x_0})$.
\end{notation}
\begin{defn}[WLUD]\label{defWLUDnm}
Let $A\subset\mathcal{N}^n$ be open, let $\boldsymbol{f}:A \to \mathcal{N}^m$, and let $\boldsymbol{x_0}\in A$ be given. Then we say that $\boldsymbol{f}$ is weakly
locally uniformly differentiable (WLUD) at $\boldsymbol{x_0}$ if $\boldsymbol{f}$ is differentiable in a neighborhood
$\Omega$ of $\boldsymbol{x_0}$ in $A$ and if for every $\epsilon>0$ in $\mathcal{N}$ there exists $\delta>0$ in $\mathcal{N}$ such that $B_{\delta}(\boldsymbol{x_0}):=\left\{\boldsymbol{t}\in\mathcal{N}:\left|\boldsymbol{t}-\boldsymbol{x_0}\right|<\delta\right\}\subset\Omega$, and
for all $\boldsymbol{x},\boldsymbol{y}\in B_{\delta}(\boldsymbol{x_0})$ we have that
\[
\left|\boldsymbol{f}(\boldsymbol{y}) - \boldsymbol{f}(\boldsymbol{x}) -
\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})(\boldsymbol{y} - \boldsymbol{x})\right| \le \epsilon \vert \boldsymbol{y} - \boldsymbol{x} \vert.
\]
Moreover, we say that $\boldsymbol{f}$ is WLUD on $A$ if $\boldsymbol{f}$ is WLUD at every point in $A$.
\end{defn}
We show in \cite{MultivWLUD} that if $\boldsymbol{f}$ is WLUD at $\boldsymbol{x_0}$ (respectively on $A$) then $\boldsymbol{f}$ is C$^1$ at $\boldsymbol{x_0}$ (respectively on $A$). Thus, the class of WLUD functions at a point $\boldsymbol{x_0}$ (respectively on an open set $A$) is a subset of
the class of
$C^1$ functions at $\boldsymbol{x_0}$ (respectively on $A$). However, this is still large enough to include all polynomial functions. We also show in \cite{MultivWLUD} that if $\boldsymbol{f},\boldsymbol{g}$ are WLUD at $\boldsymbol{x_0}$ (respectively on $A$) and if $\alpha\in\mathcal{N}$ then $\boldsymbol{f}+\alpha\boldsymbol{g}$ and $\boldsymbol{f}\cdot\boldsymbol{g}$ are WLUD at $\boldsymbol{x_0}$ (respectively on $A$). Moreover, we show that if $\boldsymbol{f}:A \to \mathcal{N}^m$ is WLUD at $\boldsymbol{x_0}\in A$ (respectively on $A$) and if $\boldsymbol{g}:C \to \mathcal{N}^p$ is WLUD at $\boldsymbol{f}(\boldsymbol{x_0})\in C$ (respectively on $C$),
where $A$ is an open subset of $\mathcal{N}^n$, $C$ an open subset of $\mathcal{N}^m$ and $\boldsymbol{f}(A) \subseteq C$, then
$\boldsymbol{g} \circ \boldsymbol{f}$ is WLUD at $\boldsymbol{x_0}$ (respectively on $A$).
\begin{theorem}[Inverse Function Theorem]\label{IFT}
Let $A\subset\mathcal{N}$ be open, let $\boldsymbol{g}:A\rightarrow\mathcal{N}^n$ be WLUD on $A$ and let $\boldsymbol{t_0}\in A$ be such that
$J\boldsymbol{g}(\boldsymbol{t_0})\neq0$. Then there is a neighborhood $\Omega$ of $\boldsymbol{t_0}$ such that:
\begin{enumerate}
\item $\boldsymbol{g}|_\Omega$ is one-to-one;
\item $\boldsymbol{g}(\Omega)$ is open;
\item the inverse $\boldsymbol{f}$ of $\boldsymbol{g}|_\Omega$ is WLUD on $\boldsymbol{g}(\Omega)$; and
$\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})=\left[\boldsymbol{D}\boldsymbol{g}(\boldsymbol{t})\right]^{-1}$ for $\boldsymbol{t}\in\Omega$ and
$\boldsymbol{x}=\boldsymbol{g}(\boldsymbol{t})$.
\end{enumerate}
\end{theorem}
As in the real case, the inverse function theorem is used to prove the implicit function theorem. But before we state the implicit function theorem, we introduce the following notations.
\begin{notation}Let $A\subseteq\mathcal{N}^n$ be open and let $\boldsymbol{\Phi}: A\rightarrow\mathcal{N}^m$ be WLUD on $A$. For $\boldsymbol{t}=(t_1,...,t_{n-m},t_{n-m+1},...,t_{n} )\in A$, let
\begin{equation*}
\hat{\boldsymbol{t}}=(t_1,...,t_{n-m})\text{ and }\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t})=\det\left(\dfrac{\partial(\Phi_1,...,\Phi_m)}{\partial(t_{n-m+1},...,t_{n})}\right).
\end{equation*}
\end{notation}
\begin{theorem}[Implicit Function Theorem]
Let $\boldsymbol{\Phi}:A\rightarrow\mathcal{N}^m$ be WLUD on $A$, where $A\subseteq\mathcal{N}^n$ is open and $1\leq m<n.$
Let $\boldsymbol{t_0}\in A$ be such that $\boldsymbol{\Phi}(\boldsymbol{t_0})=\boldsymbol{0}$ and
$\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t_0})\neq0$. Then there exist a neighborhood $U$ of $\boldsymbol{t_0}$, a neighborhood $R$ of
$\hat{\boldsymbol{t_0}}$ and $\boldsymbol{\phi}:R\rightarrow\mathcal{N}^m$ that is WLUD on $R$ such that
\[
\tilde{J}\boldsymbol{\Phi}(\boldsymbol{t})\neq0
\text{ for all } \boldsymbol{t}\in U,
\]
and
\begin{equation*}
\{\boldsymbol{t}\in
U:\boldsymbol{\Phi}(\boldsymbol{t})=\boldsymbol{0}\}=\{(\hat{\boldsymbol{t}},\boldsymbol{\phi}(\hat{\boldsymbol{t}})):\hat{\boldsymbol{t}}\in
R\}.
\end{equation*}
\end{theorem}
\section{New Results}
This paper is a continuation of the work done in
\cite{boo-sham-18,MultivWLUD}. In the following section, we will generalize in Definition \ref{defWLUDn1k} and Definition \ref{defWLUDn1infty} the concepts of WLUD$^k$ and WLUD$^\infty$ to functions from $\mathcal{N}^n$ to $\mathcal{N}$; and we will formulate (in Theorem \ref{thmtaylorseries1} and Theorem \ref{thmtaylorseriesn} and their proofs) conditions under which a WLUD$^\infty$ $\mathcal{N}$-valued function at a point $x_0\in \mathcal{N}$ or a WLUD$^\infty$ $\mathcal{N}$-valued function at a point $\boldsymbol{x_0} \in \mathcal{N}^n$ will be analytic at that point.
\begin{theorem}\label{thmtaylorseries1}
Let $A \subseteq \mathcal{N}$ be open, let $x_0\in A$, and let $f : A \rightarrow \mathcal{N}$ be WLUD$^{\infty}$ at $x_0$. For each $k\in\mathbb{N}$, let $\delta_k>0$ in $\mathcal{N}$ correspond to $\epsilon=1$ in Definition \ref{def:wludn}. Assume that
\[
\limsup_{j\rightarrow\infty}\left(\frac{-\lambda\left(f^{(j)}(x_0)\right)}{j}\right)<\infty \text{ and }
\limsup _{k\rightarrow\infty}\lambda\left(\delta_k\right)<\infty.
\]
Then there exists a neighborhood $U$ of $x_0$ in $A$ such that, for any $x,y \in U$, we have that
\[
f(y)=\sum_{j=0}^{\infty} \frac{f^{(j)}\left(x\right)}{j!}\left(y-x\right)^{j}.
\]
That is, the Taylor series $\sum\limits_{j=0}^{\infty} \frac{f^{(j)}\left(x\right)}{j!}\left(y-x\right)^{j}$ converges in $\mathcal{N}$ to $f(y)$; and hence $f$ is analytic in $U$.
\end{theorem}
\begin{proof}
Let
\[
\lambda_0=\limsup_{j\rightarrow\infty}\left(\frac{-\lambda\left(f^{(j)}(x_0)\right)}{j}\right).
\]
Then $\lambda_0\in\mathbb{R}$ and $\lambda_0<\infty$; and, by \cite[Page 59]{schikhofbook}, we have that $\sum\limits_{j=0}^{\infty} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}$ converges in $\mathcal{N}$ for all $x\in \mathcal{N}$ satisfying $\lambda(x-x_0)>\lambda_0$.
For all $k\in \mathbb{N}$, we have that $(x_0-\delta_k, x_0+\delta_k)\subset A$, $f$ is $k$ times differentiable on $(x_0-\delta_k, x_0+\delta_k)$, and
\[
\left|f(x)-\sum\limits_{j=0}^{k} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\right|\le \left|x-x_0\right|^k \text{ for all }x\in (x_0-\delta_k, x_0+\delta_k).
\]
Since $\limsup\limits _{k\rightarrow\infty}\lambda\left(\delta_k\right)<\infty$, there exists $t>0$ in $\mathbb{Q}$ such that $\limsup\limits _{k\rightarrow\infty}\lambda\left(\delta_k\right)<t<\infty$. Thus, there exists $N\in\mathbb{N}$ such that
\begin{equation}\label{eqtaylor1:1}
\lambda(\delta_k)<t\text{ for all }k>N.
\end{equation}
Let
$\delta>0$ in $\mathcal{N}$ be such that $\lambda(\delta)>\max\{\lambda_0, t,0\}$; this is possible since $\max\{\lambda_0, t,0\}<\infty$. It follows from (\ref{eqtaylor1:1})
that $\lambda(\delta)>\lambda(\delta_k)$ and hence $0<\delta\ll\delta_k$ for all $k>N$. Thus,
$(x_0-\delta, x_0+\delta)\subset A$, $f$ is infinitely often differentiable on $(x_0-\delta, x_0+\delta)$, and
\begin{equation}\label{eqtaylor1:2}
\left|f(x)-\sum\limits_{j=0}^{k} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\right|\le \left|x-x_0\right|^k \forall\ x\in (x_0-\delta, x_0+\delta)\text{ and }\forall\ k>N.
\end{equation}
Moreover, for all $x\in (x_0-\delta, x_0+\delta)$, we have that $\lambda(x-x_0)\ge \lambda(\delta)>\lambda_0$ and hence $\sum\limits_{j=0}^{\infty} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}$ converges in $\mathcal{N}$. Let $U=(x_0-\delta, x_0+\delta)$.
First we show that
\[
f(x)=\sum_{j=0}^{\infty} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\text{ for all }x\in U.
\]
Let $x\in U$ be given. Taking the limit in (\ref{eqtaylor1:2}) as $k\rightarrow\infty$, we get:
\[
0\le\lim_{k\rightarrow\infty}\left|f(x)-\sum\limits_{j=0}^{k} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\right|\le \lim_{k\rightarrow\infty} \left|x-x_0\right|^k,
\]
from which we obtain
\[
0\le\left|f(x)-\lim_{k\rightarrow\infty}\sum\limits_{j=0}^{k} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\right|\le \lim_{k\rightarrow\infty} \left|x-x_0\right|^k.
\]
Since $\lambda(x-x_0)\ge \lambda(\delta)>0$, we obtain that
$\lim\limits_{k\rightarrow\infty} \left|x-x_0\right|^k=0$.
It follows that
\[
0\le\left|f(x)-\sum\limits_{j=0}^{\infty} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}\right|\le 0
\]
from which we infer that
$f(x)=\sum\limits_{j=0}^{\infty} \frac{f^{(j)}\left(x_0\right)}{j!}\left(x-x_0\right)^{j}$ or, equivalently,
\begin{equation}\label{eqtaylor1:3}
f(x)=\sum\limits_{l=0}^{\infty} \frac{f^{(l)}\left(x_0\right)}{l!}\left(x-x_0\right)^{l}.
\end{equation}
Since the convergence of the Taylor series above is in the order (valuation) topology, we will show that the derivatives of $f$ at $x$ to any order are obtained by differentiating the power series in Equation (\ref{eqtaylor1:3}) term by term. That is, for all $j\in\mathbb{N}$,
\begin{equation}\label{eqtaylor1:4}
f^{(j)}(x)=\sum_{l=j}^{\infty} l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_0\right)}{l!}\left(x-x_0\right)^{l-j}.
\end{equation}
First note that, since $\lambda\left(l(l-1)\ldots(l-j+1)\right)=0$, it follows that $\sum_{l=j}^{\infty} l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_0\right)}{l!}\left(x-x_0\right)^{l-j}$ converges in $\mathcal{N}$ for all $j\in\mathbb{N}$. Using induction on $j$, it suffices to show that
\[
f^\prime(x)= \sum_{l=1}^{\infty} l\frac{f^{(l)}\left(x_0\right)}{l!}\left(x-x_0\right)^{l-1}=\sum_{l=1}^{\infty} \frac{f^{(l)}\left(x_0\right)}{(l-1)!}\left(x-x_0\right)^{l-1}.
\]
Let $h\in \mathcal{N}$ be such that $x+h\in U$. We will show that
\[
\lim_{h\rightarrow0}\left\{\frac{f(x+h)-f(x)}{h}\right\}=\sum_{l=1}^{\infty} \frac{f^{(l)}\left(x_0\right)}{(l-1)!}\left(x-x_0\right)^{l-1}.
\]
Thus,
\begin{eqnarray*}
&&\lim_{h\rightarrow0}\left\{\frac{f(x+h)-f(x)}{h}\right\}=\lim_{h\rightarrow0}\left\{\sum\limits_{l=0}^{\infty}\frac{f^{(l)}\left(x_0\right)}{l!}\frac{ \left(x+h-x_0\right)^{l}-\left(x-x_0\right)^{l}}{h}\right\}\\
&=&\lim_{h\rightarrow0}\left\{\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_0\right)}{l!}\frac{ \left(x+h-x_0\right)^{l}-\left(x-x_0\right)^{l}}{h}\right\}\\
&=&\lim_{h\rightarrow0}\left\{\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_0\right)}{l!} \left[(x+h-x_0)^{l-1}+(x+h-x_0)^{l-2}(x-x_0)+\cdots+(x-x_0)^{l-1}\right]\right\}\\
&=&\sum\limits_{l=1}^{\infty}\frac{f^{(l)}\left(x_0\right)}{l!} \left[l(x-x_0)^{l-1}\right]\\
&=&\sum_{l=1}^{\infty} \frac{f^{(l)}\left(x_0\right)}{(l-1)!}\left(x-x_0\right)^{l-1}.
\end{eqnarray*}
Now let $y\in U$ be given. Then
\begin{eqnarray*}
f(y)&=&\sum_{l=0}^{\infty} \frac{f^{(l)}\left(x_0\right)}{l!}\left(y-x_0\right)^{l}\\
&=&\sum_{l=0}^{\infty} \frac{f^{(l)}\left(x_0\right)}{l!}\left[(y-x)+(x-x_0)\right]^{l}\\
&=&\sum_{l=0}^{\infty} \sum_{j=0}^{l} \frac{f^{(l)}\left(x_0\right)}{l!}\left(\begin{array}{c}l\\j\end{array}\right) (y-x)^j(x-x_0)^{l-j}\\
&=&\sum_{l=0}^{\infty} \sum_{j=0}^{l} \frac{l(l-1)\ldots(l-j+1)}{j!}\frac{f^{(l)}\left(x_0\right)}{l!}(x-x_0)^{l-j}(y-x)^j.
\end{eqnarray*}
Since convergence in the order topology (valuation topology) entails absolute convergence, we can interchange the order of the summations in the last equality \cite{shamseddinephd,rspsio00}. We get:
\begin{eqnarray*}
f(y)&=&\sum_{j=0}^{\infty}\frac{1}{j!}\left[ \sum_{l=j}^{\infty}l(l-1)\ldots(l-j+1)\frac{f^{(l)}\left(x_0\right)}{l!}(x-x_0)^{l-j}\right](y-x)^j\\
&=&\sum_{j=0}^{\infty}\frac{f^{(j)}(x)}{j!}(y-x)^j
\end{eqnarray*}
where we made use of Equation (\ref{eqtaylor1:4}) in the last step.
\end{proof}
Replacing $m$ by $1$ in Definition \ref{defWLUDnm}, then the $m\times n$ matrix $\boldsymbol{D}\boldsymbol{f}(\boldsymbol{x})$ is replaced by the gradient of $\boldsymbol{f}$ at $\boldsymbol{x}$: $\boldsymbol{\nabla}f(\boldsymbol{x})$, and we readily obtain the definition of a WLUD $\mathcal{N}$-valued function at a point $\boldsymbol{x_0}$ or on an open subset $A$ of $\mathcal{N}^n$.
\begin{defn}\label{defWLUDn1}
Let $A\subset\mathcal{N}^n$ be open, let $f:A \to \mathcal{N}$, and let $\boldsymbol{x_0}\in A$ be given. Then we say that $f$ is WLUD at $\boldsymbol{x_0}$ if $f$ is differentiable in a neighborhood
$\Omega$ of $\boldsymbol{x_0}$ in $A$ and if for every $\epsilon>0$ in $\mathcal{N}$ there exists $\delta>0$ in $\mathcal{N}$ such that $B_{\delta}(\boldsymbol{x_0})\subset\Omega$, and
for all $\boldsymbol{x},\boldsymbol{y}\in B_{\delta}(\boldsymbol{x_0})$ we have that
\[
\left|f(\boldsymbol{y}) - f(\boldsymbol{x}) -
\boldsymbol{\nabla}f(\boldsymbol{x})\cdot(\boldsymbol{y} - \boldsymbol{x})\right| \le \epsilon \vert \boldsymbol{y} - \boldsymbol{x} \vert.
\]
Moreover, we say that $f$ is WLUD on $A$ if $f$ is WLUD at every point in $A$.
\end{defn}
Using Defintion \ref{def:wludn} and Definition \ref{defWLUDn1}, the natural way to define $k$ times weak local uniform differentiability (WLUD$^k$) at a point $\boldsymbol{x_0}$ or on an open subset $A$ of $\mathcal{N}^n$ is as follows.
\begin{defn}\label{defWLUDn1k}
Let $A\subset\mathcal{N}^n$ be open, let $f:A \to \mathcal{N}$, and let $\boldsymbol{x_0}\in A$ be given. Then we say that $f$ is WLUD$^k$ at $\boldsymbol{x_0}$ if $f$ is $k$-times differentiable in a neighborhood
$\Omega$ of $\boldsymbol{x_0}$ in $A$ and if for every $\epsilon>0$ in $\mathcal{N}$ there exists $\delta>0$ in $\mathcal{N}$ such that $B_{\delta}(\boldsymbol{x_0})\subset\Omega$, and
for all $\boldsymbol{\xi},\boldsymbol{\eta}\in B_{\delta}(\boldsymbol{x_0})$ we have that
\[
\left|f(\boldsymbol{\eta})-f(\boldsymbol{\xi})-\sum_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{\xi})\cdot\nabla\right]^{j}f(\boldsymbol{\xi})\right| \le \epsilon \vert \boldsymbol{\eta} - \boldsymbol{\xi} \vert^k,
\]
where
\begin{eqnarray*}
\left[(\boldsymbol{\eta}-\boldsymbol{\xi})\cdot\nabla\right]^{j}f(\boldsymbol{\xi})&=&
\left.\left[(\eta_1-\xi_1)\frac{\partial}{\partial x_1}+\cdots+(\eta_n-\xi_n)\frac{\partial}{\partial x_n}\right]^{j}f(\boldsymbol{x})
\right|_{\boldsymbol{x}=\boldsymbol{\xi}}\\
&=&\sum_{l_{1},\ldots,l_{j}=1}^{n} \left(
\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{\xi}}
\prod_{m=1}^{j}\left( \eta_{l_{m}}-\xi_{l_{m}}\right) \right).
\end{eqnarray*}
Moreover, we say that $f$ is WLUD$^k$ on $A$ if $f$ is WLUD$^k$ at every point in $A$.
\end{defn}
\begin{defn}\label{defWLUDn1infty}
Let $A\subset\mathcal{N}^n$ be open, let $f:A \to \mathcal{N}$, and let $\boldsymbol{x_0}\in A$ be given. Then we say that $f$ is WLUD$^\infty$ at $\boldsymbol{x_0}$ if $f$ is WLUD$^k$ at $\boldsymbol{x_0}$ for every
$k\in\mathbb{N}$. Moreover, we say that $f$ is WLUD$^\infty$ on $A$ if $f$ is WLUD$^\infty$ at every point in $A$.
\end{defn}
Now we are ready to state and prove the analog of Theorem \ref{thmtaylorseries1} for functions of $n$ variables.
\begin{theorem}\label{thmtaylorseriesn}
Let $A \subseteq \mathcal{N}^n$ be open, let $\boldsymbol{x_0}\in A$, and let $f : A \rightarrow \mathcal{N}$ be WLUD$^{\infty}$ at $\boldsymbol{x_0}$. For each $k\in\mathbb{N}$, let $\delta_k>0$ in $\mathcal{N}$ correspond to $\epsilon=1$ in Definition \ref{defWLUDn1k}. Assume that
\begin{eqnarray*}
\limsup_{{\tiny\begin{array}{l}j\rightarrow\infty\\l_1=1,\ldots,n\\
\vdots\\
l_j=1,\ldots,n
\end{array}}}\left(\frac{-\lambda\left(\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}} \right)}{j}\right)&<&\infty\\
&\mbox{ }&\\
\text{ and }
\limsup _{k\rightarrow\infty}\lambda\left(\delta_k\right)&<&\infty.
\end{eqnarray*}
Then there exists a neighborhood $U$ of $\boldsymbol{x_0}$ in $A$ such that, for any $\boldsymbol{\eta}\in U$, we have that
\[
f(\boldsymbol{\eta})=f(\boldsymbol{x_0})+\sum_{j=1}^{\infty} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0}).
\]
\end{theorem}
\begin{proof}
Let
\[
\lambda_0=\limsup_{{\tiny\begin{array}{l}j\rightarrow\infty\\l_1=1,\ldots,n\\
\vdots\\
l_j=1,\ldots,n
\end{array}}}\left(\frac{-\lambda\left(\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}} \right)}{j}\right).
\]
Then $\lambda_0\in\mathbb{R}$ and $\lambda_0<\infty$.
For all $k\in \mathbb{N}$, we have that $B_{\delta_k}(\boldsymbol{x_0})\subset A$, $f$ is $k$ times differentiable on $B_{\delta_k}(\boldsymbol{x_0})$, and
\[
\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_0})-\sum_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right| \le \vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert^k \text{ for all }\boldsymbol{\eta}\in B_{\delta_k}(\boldsymbol{x_0}).
\]
Since $\limsup\limits _{k\rightarrow\infty}\lambda\left(\delta_k\right)<\infty$, there exists $t>0$ in $\mathbb{Q}$ such that $\limsup\limits _{k\rightarrow\infty}\lambda\left(\delta_k\right)<t<\infty$. Thus, there exists $N\in\mathbb{N}$ such that
\begin{equation}\label{eqtaylorn:1}
\lambda(\delta_k)<t\text{ for all }k>N.
\end{equation}
Let
$\delta>0$ in $\mathcal{N}$ be such that $\lambda(\delta)>\max\{\lambda_0, t,0\}$. It follows from (\ref{eqtaylorn:1})
that $\lambda(\delta)>\lambda(\delta_k)$ and hence $0<\delta\ll\delta_k$ for all $k>N$. Thus,
$B_{\delta}(\boldsymbol{x_0})\subset A$, $f$ is infinitely often differentiable on $B_{\delta}(\boldsymbol{x_0})$, and
\begin{equation}\label{eqtaylorn:2}
\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_0})-\sum_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right| \le \vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert^k \ \forall \boldsymbol{\eta}\in B_{\delta}(\boldsymbol{x_0})\text{ and }\forall k>N.
\end{equation}
Let $U=B_{\delta}(\boldsymbol{x_0})$; and let
$\boldsymbol{\eta}\in U$ be given.
Then we have that $\lambda(\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert)\ge \lambda(\delta)>\lambda_0$. We will show first that $\sum_{j=1}^{\infty} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})$ converges in $\mathcal{N}$. Since
$\lambda(\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert)>\lambda_0$, there exists $q>0$ in $\mathbb{Q}$ such that $\lambda(\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert)-q>\lambda_0$. Hence there exists $M\in\mathbb{N}$ such that
\[
\lambda(\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert)-q> \frac{-\lambda\left(\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}} \right)}{j}
\]
for all $j>M$ and for $l_1=1, \ldots, n$, $l_2=1, \ldots, n$, \ldots, $l_j=1, \ldots, n$. It follows that
\begin{eqnarray*}
\lambda\left(
\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}}
\prod_{m=1}^{j}\left( \eta_{l_{m}}-x_{0,l_{m}}\right) \right)
&\ge&\lambda \left(
\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}}
\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert^j\right)\\
&=&\lambda \left(
\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}}\right)+j\lambda\left(
\vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert\right)\\
&>&jq
\end{eqnarray*}
for all $j>M$ and for $l_1=1, \ldots, n$, $l_2=1, \ldots, n$, \ldots, $l_j=1, \ldots, n$. Thus,
\begin{eqnarray*}
\lambda\left(\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right)&=&
\lambda\left(\sum_{l_{1},\ldots,l_{j}=1}^{n} \left(
\left.\frac{\partial^j f(\boldsymbol{x})}{\partial_{x_{l_{1}}}\cdots\partial_{x_{l_{j}}}}\right|_{\boldsymbol{x}=\boldsymbol{x_0}}
\prod_{m=1}^{j}\left( \eta_{l_{m}}-x_{0,l_{m}}\right) \right)\right)\\
&>&jq
\end{eqnarray*}
for all $j>M$; and hence
\begin{eqnarray*}
\lim_{j\rightarrow\infty}\lambda\left(\frac1{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right)&=&
\lim_{j\rightarrow\infty}\lambda\left(\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right)\\
&\ge&q\lim_{j\rightarrow\infty}j=\infty.
\end{eqnarray*}
Thus,
\[
\lim_{j\rightarrow\infty}\left(\frac1{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right)=0
\]
and hence $\sum_{j=1}^{\infty} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})$ converges in $\mathcal{N}$; that is,
\[
\lim\limits_{k\rightarrow\infty}\sum\limits_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\text{ exists in }\mathcal{N}.
\]
Taking the limit in (\ref{eqtaylorn:2}) as $k\rightarrow\infty$, we get:
\[
0\le\lim_{k\rightarrow\infty} \left|f(\boldsymbol{\eta})-f(\boldsymbol{x_0})-\sum_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right|\le \lim_{k\rightarrow\infty} \vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert^k,
\]
from which we obtain
\[
0\le \left|f(\boldsymbol{\eta})-f(\boldsymbol{x_0})-\lim_{k\rightarrow\infty}\sum_{j=1}^{k} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right|\le \lim_{k\rightarrow\infty} \vert \boldsymbol{\eta} - \boldsymbol{x_0} \vert^k.
\]
Since $\lambda(\vert\boldsymbol{\eta} - \boldsymbol{x_0}\vert)\ge \lambda(\delta)>0$, we obtain that
$\lim\limits_{k\rightarrow\infty} \left|\boldsymbol{\eta} - \boldsymbol{x_0}\right|^k=0$.
It follows that
\[
0\le\left|f(\boldsymbol{\eta})-f(\boldsymbol{x_0})-\sum_{j=1}^{\infty} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0})\right|\le 0
\]
from which we infer that
\[
f(\boldsymbol{\eta})=f(\boldsymbol{x_0})+\sum_{j=1}^{\infty} \frac{1}{j!}\left[(\boldsymbol{\eta}-\boldsymbol{x_0})\cdot\nabla\right]^{j}f(\boldsymbol{x_0}).
\]
\end{proof}
| train/arxiv |
BkiUfuXxK0wg05VB91M5 | 5 | 1 | \section{INTRODUCTION}
Motivated by the Callan-Rubakov effect in the context of magnetic
monopoles \cite{callan}, studies have been carried out recently
on the possibility that cosmic strings can also catalyze
baryon-number violation with strongly enhanced cross sections.
It has been shown that the wave function of a fermion scattering
off a cosmic string can acquire a large amplification factor near
the core of the string, leading to enhancement of the processes
that violate baryon number inside the string \cite{alford,perkin}.
The catalysis processes that have been studied include those
mediated by scalar fields and by the grand-unified X and Y gauge
bosons in the string core. Although strings, in contrast to
monopoles, have no magnetic fields outside, fermions can interact
quantum-mechanically with the long-range gauge fields via the
Aharonov-Bohm effect. Depending on the flux of the string and
the core model used, the enhanced catalysis cross sections (per
length) can be of the scale of strong interactions in comparison to
the much smaller geometrical cross section $\sim \Lambda_{GUT}^{-1}$,
where $\Lambda_{GUT} \sim 10^{16}$ GeV. In the early universe when
the density of cosmic strings is high, such processes can play
important roles, washing out any primordially-generated baryon
asymmetry \cite{RB1}, or conceivably even generating the baryon to
entropy ratio observed today.
Cosmic strings can be produced during certain phase transitions
when a gauge group G is broken down to a subgroup H by the vacuum
expectation value of some scalar field $\phi$. The topological
criterion for the existence of a string is a nontrivial fundamental
homotopy group of the vacuum manifold G/H, denoted by
$\pi_1(\hbox{G}/\hbox{H})$. For a connected and simply-connected
G, the general construction of the scalar field at large distances
from the string is given by
\begin{equation}
\phi(\theta) = g(\theta) \phi_0\,,
\quad g(\theta) = e^{i\tau\theta}\,.
\end{equation}
Here $\tau$ is some generator of G, $\theta$ is the azimuthal
angle measured around the string, and $g(0)$ and $g(2\pi)$
belong to two disconnected pieces of H. In the papers
referenced in the previous paragraph, the scalar field
responsible for the formation of the string is taken to
have the simple form $\phi(\theta) = e^{i\tau\theta} \phi_0
= e^{i\theta} \phi_0$. As a result, a non-Abelian string can be
modeled by a U(1) vortex, and the scattering of fermions in the
background fields of the string is governed by the Abelian Dirac
equation. In general however, for a given $\phi_0$, the generator
$\tau$ can be chosen such that $e^{i\tau\theta} \phi_0$ ``twists''
around the string in more complicated fashion than a phase
$e^{i\theta}$ times $\phi_0$. This gives rise to dynamically
different strings which are intrinsically non-Abelian
\cite{leandros}. One expects the complexity and rich
structure of such strings to lead to interesting effects
on fermions traveling around them. In particular, we will
demonstrate in this paper that for certain $\tau$'s, the
twisting of $\phi(\theta)$ can result in mixing of lepton
and quark fields, providing a mechanism for baryon number
violations distinct from the processes in Abelian strings
studied previously.
Since no strings are formed in the minimal SU(5) model, we choose
the gauge group SO(10) \cite{so10} in this paper as an example of
grand unified theories in investigating the B-violating process.
We will construct string configurations, solve numerically for the
undetermined functions, and study the baryon catalysis in the SO(10)
theory, although we expect such processes to occur in other
non-Abelian theories as well. In SO(10), stable strings can
be formed when Spin(10) --- the simply-connected covering group
of SO(10) --- is broken down to SU(5)$\times {\cal Z}_2$ by the
vacuum expectation value of a Higgs field $\phi$ in the {\bf 126}
representation \cite{kibble}. The generators of SO(10) transform
as the adjoint {\bf 45}, which transforms as {\bf 24} + {\bf 1}
+ {\bf 10} + $\bf{\bar{10}}$ under SU(5). The {\bf 24} and {\bf 1}
generate the subgroup SU(5)$\times$U(1), where the U(1) includes
simultaneous rotations in the 1-2, 3-4, 5-6, 7-8, and 9-10 planes.
We are interested in the generators outside SU(5) because to have
noncontractible loops at all, $g(\theta)$ in Eq.~(1) has to be
outside the unbroken H for some $\theta$. We will refer to the
U(1) generator as $\tau_{\rm all}$ and to any of the other 20
basis generators outside SU(5) as $\tau_1$; we name the
associated strings as string-$\tau_{\rm all}$ and string-$\tau_1$,
respectively. As we shall see, the scalar field of string-$\tau_1$
causes mixing of leptons and quarks while string-$\tau_{\rm all}$
is effectively Abelian and no such mixing occurs. Properties of
string-$\tau_{\rm all}$ such as the string mass per unit length
\cite{everett} and its superconducting capability in terms of
fermion zero modes \cite{witten} have been studied. We will
compare it with string-$\tau_1$, which will be the main subject
of study of this paper.
In Sec.~II, we give more detailed discussion of the Higgs {\bf 126}
and the breaking of Spin(10) to SU(5)$\times {\cal Z}_2$, and
elaborate on the B-violating mechanism due to the nontrivial
winding of the Higgs field. In Sec.~III, we write down an
{\it ansatz\ } for the field configuration of each string and
derive the corresponding equations of motion. The numerical
solutions and the energy of the strings are presented in Sec.~IV,
where we find that $\tau_1$-strings have lower energy than
$\tau_{\rm all}$-strings, probably for the entire range of the
parameters in the theory. Having shown that such strings are
energetically favorable, we turn to the scattering problem in
Sec.~V, where the Dirac equation in the background fields of
the strings is solved, and the differential cross section for
the B-violating processes in string-$\tau_1$ is calculated.
We also comment on the role of the self-adjoint parameters and
compute their values using our string solutions. To establish
a common notation and to facilitate reading of this paper,
we include in the Appendix a discussion about the relevant
aspects of the spinor representation {\bf 16} of SO(10), which
accommodates a single generation of left-handed fermions.
\section{SO(10) strings}
There is considerable freedom in the breakings of SO(10) down
to the low energy gauge group SU(3)$\times$U(1). Two commonly
studied examples include the breaking via an intermediate SU(5),
SO(10)$\rightarrow$SU(5), and the one via an intermediate
Pati-Salam SU(4)$\times$SU(2)$_L\times$SU(2)$_R$ \cite{pati}.
Details of the symmetry breaking patterns and the Higgs fields
inducing the breakings can be found in Ref.~6 and the papers
by Slansky and Rajpoot \cite{slansky}. Kibble, Lazarides and
Shafi argued that the strings formed during the phase transition
SO(10) $\rightarrow$SU(4)$\times$SU(2)$_L\times$SU(2)$_R$ become
boundaries of domain walls \cite{kibble}. Thus in this paper we
choose the SU(5) breaking pattern instead for its simplicity.
More precisely, we study strings formed when
Spin(10)$\rightarrow$SU(5)$\times{\cal Z}_2$ by the vacuum
expectation value of a Higgs {\bf 126} $\phi$. The nontrivial
element of ${\cal Z}_2$ corresponds to rotation by 2$\pi$ in SO(10).
The homotopy group $\pi_1(\hbox{Spin(10)}/\hbox{SU(5)}\times
{\cal Z}_2)$ is ${\cal Z}_2\,$; therefore a ${\cal Z}_2$ string
is formed during this phase transition. The subsequent symmetry
breakings can be implemented by the adjoint {\bf 45} of SO(10) and
the fundamental {\bf 10} in the usual fashion:
\begin{eqnarray}
\hbox{Spin(10)} &\stackrel{\bf 126}{\longrightarrow}&
\hbox{SU(5)}\times{\cal Z}_2 \nonumber\\
& \stackrel{\bf 45}{\longrightarrow} &
\hbox{SU(3)}\times\hbox{SU(2)}\times\hbox{U(1)}
\times{\cal Z}_2 \nonumber\\
& \stackrel{\bf 10}{\longrightarrow} &
\hbox{SU(3)}\times\hbox{U(1)}_{\hbox{em}}\times{\cal Z}_2\,.
\end{eqnarray}
This ${\cal Z}_2$ string survives all the symmetry breakings
since ${\cal Z}_2$ is preserved at low energies.
The {\bf 126} representation consists of fifth\--rank
anti-symmetric tensors satisfy\-ing the self\--duality condition
\begin{equation}
\phi_{i_1...i_5} = \frac{i}{5!} \epsilon_{i_1....i_{10}}
\phi_{i_6...i_{10}}.
\end{equation}
The component which acquires an expectation value $\langle\phi
\rangle$ transforms as an SU(5) singlet, and to write it down
explicitly, we first specify how the SU(5) subgroup is embedded
in SO(10). The fundamental representation of SO(10) consists of
10$\times$10 matrices, which can be labeled by indices $i, i = 1,
\ldots ,10\,.$ The generators of SO(10) in this representation
can be written as antisymmetric, purely imaginary matrices. The
generators of SU(5) in the fundamental representation are hermitian,
traceless 5$\times$5 matrices which can be written as
\begin{equation}
\tau_{\alpha \beta} = S_{\alpha \beta} + iA_{\alpha \beta}\,,
\end{equation}
where $\alpha,\beta =1,..,5$ label the matrix elements, and $S, A$
are real 5$\times$5 matrices, representing the real and imaginary
parts of $\tau$. Hermiticity and tracelessness of $\tau$ require
$S_{\alpha \beta} = S_{\beta \alpha}, A_{\alpha \beta} =
-A_{\beta\alpha}$, and $TrS=0$. A natural way to embed SU(5)
in SO(10) is to treat five-dimensional complex vectors as
ten-dimensional real vectors, {\it i.e.} replace the paired
indices ($\alpha, a$), where $\alpha = 1, \ldots ,5$ label a
five-dimensional vector and $a=1,2$ label its real and imaginary
parts, by the index $i,\,i=1, \ldots ,10$. Then, the generators
of the subgroup SU(5) of SO(10) can be expressed as
\begin{equation}
\tau_{\alpha a,\,\beta b} = i( A_{\alpha \beta}I_{ab} +
S_{\alpha \beta}M_{ab})\,,
\end{equation}
where $I$ is the 2$\times$2 identify matrix and $M = i \sigma_2\,,
\sigma_2$ being the second 2$\times$2 Pauli matrix. One can
convince oneself that in this $(\alpha, a)$ notation, the
rank-five antisymmetric Levi-Civita tensor
$\epsilon_{\alpha_1 \alpha_2 \alpha_3 \alpha_4 \alpha_5 }$ which
transforms as an SU(5) singlet in the SU(5) notation becomes
\begin{equation}
i^{f(a_1...a_5)} \epsilon_{\alpha_1\alpha_2\alpha_3
\alpha_4\alpha_5}\,,
\end{equation}
where $f(a_1 \ldots a_5)$ is defined to equal the number of $a_i$
that takes the value 2. It is also straightforward to check that
this expression satisfies the self-duality condition (Eq.~(3)).
Thus $\langle\phi\rangle$ is written as
\begin{equation}
\langle \phi_{\alpha_1 a_1...\alpha_5 a_5} \rangle
= \mu\ i^{f(a_1...a_5)}
\epsilon_{\alpha_1 \alpha_2 \alpha_3 \alpha_4 \alpha_5 }\,,
\end{equation}
where $\mu$ is a parameter.
Some words about our notation. The tensor indices $i_1,\ldots ,i_5$
of $\phi_{i_1 \ldots i_5}$ will be suppressed for convenience and
legibility whenever no ambiguity should arise. In the expressions
like $\tau\phi$ and $e^{i\tau\theta} \phi$ where $\tau$ operates
on $\phi$, $\tau$ is understood to be in the same representation
of $\phi$, {\it i.e.} $\tau$ is the short-hand for
\FL
\begin{equation}
\tau_{i_1 \ldots i_5j_1 \ldots j_5} =
\tau_{i_1j_1} \delta_{i_2j_2} \ldots \delta_{i_5j_5}
+ \delta_{i_1j_1} \tau_{i_2j_2} \ldots \delta_{i_5j_5}
+ \ldots
\end{equation}
With the symmetry breaking Spin(10)$\rightarrow$SU(5)$
\times{\cal Z}_2$, strings are formed. At spatial infinity,
the general form of $\phi$ is given by Eq.~(1). For the energy
to be finite, the co\-variant derivative of $\phi$,
$D_\mu \phi \equiv \partial_\mu \phi + eA_\mu \phi\ $, has to
vanish at spatial infinity; therefore the gauge field $A_\mu$
takes the form $A^\theta = i\frac{1}{er} \tau$, $A^r = 0\,,$ as
$r \rightarrow \infty$. In the core of the string, there is a
magnetic flux $\oint \vec{A} \cdot d\vec{l} = \frac{2\pi}{e}\tau$
pointing in the direction of $\tau$ in group space. Strings
carrying flux pointing in different directions in group space
are topologically equivalent since the only nontrivial winding
number here is one, but dynamically they can differ. Because
the scalar field $\phi(\theta)$ varies with $\theta$, the
embedding of the unbroken subgroup SU(5) in SO(10) outside
the string also varies with $\theta$. More precisely,
the generators $\tau^a_\theta, a=1, \ldots ,24$ of the unbroken
SU(5) at $\theta$ are related to the generators $\tau^a_0$ of
the unbroken SU(5) at $\theta=0$ by the similarity transformation
\begin{equation}
\tau^a_{\theta}= g(\theta)\tau_0^a g^{-1}(\theta)\,,\ \
g(\theta)=e^{i\tau\theta}\,.
\end{equation}
Consequently, the fermion fields which transform as {\bf 1},
$\bf{\bar 5}$ and {\bf 10} under SU(5) are also rotated as
one goes around the string. How the fields mix depends on
which direction in group space $\phi(\theta)$ winds.
The SO(10) generators can be written as 10$\times$10 matrices
of the form $(\tau^{ab})_{ij} = -i(\delta^a_i \delta^b_j
- \delta^b_i\delta^a_j)\,,$ where $a,b$ label the group
indices, $i,j$ label the matrix elements, and $a,b,i,j$ all
run from 1 to 10. In this notation $\tau_{\rm all}$ is given by
\begin{equation}
\tau_{\rm all}\equiv \frac{1}{5} (\tau^{12} +
\tau^{34} + \ldots +\tau^{9\,10})\,,
\end{equation}
where the factor of 1/5 is included for $\phi(\theta)$ to have a
$2\pi$ rotational period. It takes a little more effort to write
down the $\tau_1$'s. Let us first write the SU(5) generators
specified by Eq.~(5) in terms of $\tau^{ab}$ given above.
The four diagonal generators are trivial. For the other twenty
generators, one can group the 10$\times$10 space into 2$\times$2
blocks, and write the 45 $\tau^{ab}$'s as $\tau^{2\alpha-1,\,
2\beta-1}, \tau^{2\alpha-1,\,2\beta}, \tau^{2\alpha,\, 2\beta-1}$
and $\tau^{2\alpha, 2\beta}$, where $\alpha, \beta$ both run from
1 to 5. Then it is not hard to see that the twenty linear
combinations
\begin{eqnarray}
&& \frac{1}{2} (\tau^{2\alpha-1,\,2\beta}
-\tau^{2\alpha,\,2\beta-1})\,,\nonumber\\
&& \frac{1}{2} (\tau^{2\alpha-1,\,2\beta-1}
+\tau^{2\alpha,\,2\beta})\,,\quad \alpha < \beta
\end{eqnarray}
are all of the form of Eq.~(5), and therefore can be chosen
to be the twenty off-diagonal generators of SU(5). Note that
the superscripts $\alpha, \beta$ above label the group indices
while the subscripts $\alpha, \beta$ in Eq.~(5) label the
matrix elements. The twenty $\tau_1$'s outside SU(5) then can
be expressed by the other twenty linear combinations as
\begin{eqnarray}
\tau_1 &\equiv& \frac{1}{2}(\tau^{2\alpha-1,\,2\beta}
+\tau^{2\alpha,\,2\beta-1})\,,\nonumber\\
&& \frac{1}{2}(\tau^{2\alpha-1,\,2\beta-1} -
\tau^{2\alpha,\,2\beta})\,,\quad \alpha < \beta\,.
\end{eqnarray}
Other than the SU(5) group properties, the linear combinations
above can also be classified under the group SO(4), which is
locally isomorphic to SU(2)$\times$SU(2). For a given $\alpha$
and $\beta$ where $\alpha < \beta$, the two generators of
Eq.~(11) plus the diagonal
\begin{equation}
\frac{1}{2} (\tau^{2\alpha-1,\,2\beta-1}-\tau^{2\alpha,\,2\beta})
\end{equation}
can be easily shown to obey the SU(2) algebra. Similarly,
the two generators of Eq.~(12) plus
\begin{equation}
\frac{1}{2} (\tau^{2\alpha-1,\,2\beta-1}+\tau^{2\alpha,\,2\beta})
\end{equation}
generate another SU(2). Thus, for a given $\alpha$ and $\beta$
$(\alpha < \beta)$, the six generators of Eqs.~(11-14) generate
rotations in the 4-dimensional space spanned by vectors in the
$2\alpha-1, 2\alpha, 2\beta-1, 2\beta$ directions.
\section{Field Configurations}
The relevant part of the Lagrangian for the SO(10) theory is
given by
\begin{equation}
{\cal L} = \frac{1}{4} trF_{\mu \nu} F^{\mu \nu} +
(D_\mu \phi )^\ast (D^\mu \phi) - V(\phi)
\end{equation}
where $F_{\mu \nu} = -iF_{\mu \nu}^a \tau_a\,, A_{\mu} = -iA_{\mu}^a
\tau_a\,, F_{\mu \nu} = \partial_\mu A_\nu - \partial_\nu A_\mu +
e[A_\mu ,A_\nu ]\,, D_\mu = \partial_\mu + e A_\mu\ $;
$A^a_\mu, a=1, \ldots ,45$, are the SO(10) gauge fields and
$\phi$ is the Higgs {\bf 126}. The most general gauge-invariant
and renormalizable potential $V(\phi)$ contains all the distinct
contractions of two and four $\phi$'s:
\FL
\begin{eqnarray}
V(\phi) & = & v_1 \phi_{i_1 \ldots i_5} \phi^{\ast}_{i_1 \ldots i_5}
+ v_2 (\phi_{i_1 \ldots i_5} \phi^{\ast}_{i_1 \ldots i_5})^2
\nonumber\\
& + & v_3 \phi_{i_1 n_2 n_3 n_4 n_5}
\phi^\ast_{j_1 n_2 n_3 n_4 n_5}
\phi_{i_1 \ell_2 \ell_3 \ell_4 \ell_5}
\phi^\ast_{j_1 \ell_2 \ell_3 \ell_4 \ell_5} \nonumber\\
& + & v_4 \phi_{i_1 i_2 n_3 n_4 n_5}
\phi^\ast_{j_1 j_2 n_3 n_4 n_5}
\phi_{i_1 i_2 \ell_3 \ell_4 \ell_5}
\phi^\ast_{j_1 j_2 \ell_3 \ell_4 \ell_5} \nonumber\\
& + & v_5 \phi_{i_1 j_2 n_3 n_4 n_5}
\phi^\ast_{j_1 i_2 n_3 n_4 n_5}
\phi_{i_1 i_2 \ell_3 \ell_4 \ell_5}
\phi^\ast_{j_1 j_2 \ell_3 \ell_4 \ell_5} \nonumber\\
& + & v_6 \phi_{i_1 i_2 j_3 n_4 n_5}
\phi^\ast_{j_1 j_2 i_3 n_4 n_5}
\phi_{i_1 i_2 i_3 \ell_4 \ell_5}
\phi^\ast_{j_1 j_2 j_3 \ell_4 \ell_5}\,.\ \
\end{eqnarray}
In writing down the $v_3$ through $v_6$ terms above, one has to
consider two things: (1) the possible ways to contract the
indices, and (2) which $\phi$'s are to be complex conjugated.
One can deal with (1) without the complication of (2) by adopting
an equivalent real 252 representation for $\phi$ because a complex,
self-dual 126-dimensional tensor can be thought of as a real,
252-dimensional tensor by dropping the self-duality condition
and taking the real parts of the resulting complex, 252-dimensional
tensor. One can see there are only four distinct terms and they
are terms $v_3$ through $v_6$ in Eq.~(16) above. Then when $\phi$
is taken to be complex, two out of the four $\phi$'s have to be
complex conjugated to make the potential real. There are three
possibilities: $\phi\phi^{\ast} \phi\phi^{\ast},
\ \phi^{\ast}\phi\phi\phi^{\ast},
\ \phi\phi \phi^{\ast}\phi^{\ast}\ $, for each of the four
contractions $\phi\phi\phi\phi$ when $\phi$ is real. But
after the self-duality condition is applied, one can show
that only one of the three terms is actually independent.
The Euler-Lagrange equations of motion for $\phi$ and $A_\mu$
are given by
\begin{eqnarray}
&& D_\mu D^\mu \phi = -\frac{\partial V}{\partial \phi^\ast}\,,
\label{eq:EOMI}\\
&& Tr(\tau^{a\,2})(\partial_\mu F^{a\,\mu \nu} +
ef^{abc} A_\mu ^b F^{c\,\mu\nu}) \nonumber\\
&& \qquad = ie\{(D^\nu \phi)^\ast (\tau^a \phi) -
(\tau^a \phi)^\ast (D^\nu \phi)\} \,,
\end{eqnarray}
where $a$ is not summed over, and where a basis has been chosen
so that $Tr(\tau^a \tau^b)=\delta^{ab} Tr(\tau^{a\,2})$.
We construct for string-$\tau_{\rm all}$ a solution of
the following form:
\newline {\em Ansatz I\ }:
\begin{eqnarray}
\phi & = & f(r) e^{i\tau_{\rm all}\theta} \phi_0
= f(r) e^{i\theta} \phi_0\,, \nonumber\\
A^\theta & = & i\frac{g(r)}{er} \tau_{\rm all}\,,
\label{eq:ansI}\\
A^r & = & 0\,, \nonumber \
\end{eqnarray}
where $\phi_0 \equiv \langle\phi\rangle$ as defined in
Eq.~(7). The boundary conditions on the functions are
\begin{eqnarray}
f(0) = 0\,,\qquad &
f(r) \stackrel{r\rightarrow \infty}{\longrightarrow} \mu\,,
\nonumber\\
g(0) = 0\,,\qquad &
g(r) \stackrel{r\rightarrow\infty}{\longrightarrow} 1\,;
\end{eqnarray}
$V(\phi)$ is minimized at $f=\mu$. Inserting this {\it ansatz\ }
into the equations of motion and using the relations $\tau_{\rm all}
\tau_{\rm all} \phi_0 = \phi_0\ $ and $(\tau_{\rm all}\phi_0)^\ast
(\tau_{\rm all}\phi_0) = \phi_0^\ast \phi_0 = 3840 \equiv N$,
we obtain two coupled differential equations for $f(r)$ and $g(r)$:
\begin{eqnarray}
f^{\prime\prime} + \frac{1}{r} f^\prime
- \frac{(1-g)^2}{r^2} f & = & f(v_1+2Nv_2 f^2)
\,,\nonumber\\
Tr(\tau_{\rm all}^2) \left( g^{\prime\prime} - \frac{1}{r}
g^\prime \right) & = & -2N e^2 (1-g) f^2 \,,
\end{eqnarray}
where the prime denotes differentiation with respect to $r$, and
$Tr(\tau_{\rm all}^2) = \frac{2}{5}$ from Eq.~(10). An expansion
of $f(r)$ and $g(r)$ in powers of $r$ around the origin reveals
that $f(r)$ is odd in $r$ with a linear leading term, whereas
$g(r)$ is even in $r$ with a quadratic leading term.
Inserting {\it Ansatz I\ } for string-$\tau_{\rm all}$ into the
Lagrangian gives
\begin{eqnarray}
-{\cal L}^{\rm all} &=& \frac{Tr(\tau_{\rm all}^2)}{2e^2 r^2}
g^{\prime\,2}
+ N f^{\prime\,2} + N \frac{(1-g)^2}{r^2} f^2 \nonumber\\
&& + N(v_1 f^2 + Nv_2 f^4)\,.
\end{eqnarray}
As a consistency check, note that the equations of motion
obtained by varying ${\cal L}^{\rm all}$ with respect to the
functions $g$ and $f$ are identical to those in Eq.~(21).
Note that the parameters $v_3$ through $v_6$ in the potential
$V$ are absent from Eq.~(21) and ${\cal L}^{\rm all}$ above.
This is because whenever one index of a given $\phi$ is
contracted with one index of another $\phi$, this index is
summed over from 1 through 10, or in the $(\alpha, a)$ notation
discussed earlier, from $\alpha = 1$ through 5 and $a=1,2$. For
a given $\alpha$, the term with $a=2$ by definition has an extra
factor of $i^2=-1$ compared to the term with $a=1$. These two terms
cancel each other when they are added. Because this is true for
every $\alpha$, the third through the sixth terms in $V$ vanish
identically for the string-$\tau_{\rm all}$ {\it ansatz}.
To construct an {\it ansatz\ } for string-$\tau_1$, we need to
consider separately the two sets of generators in Eq.~(12),
which will be referred to as
\begin{eqnarray}
\tau_{1+} &=& \frac{1}{2}(\tau^{2\alpha-1,\,2\beta}
+\tau^{2\alpha,\,2\beta-1})\,, \nonumber\\
\tau_{1-} &=& \frac{1}{2}(\tau^{2\alpha-1,\,2\beta-1}
-\tau^{2\alpha,\,2\beta})\,,
\ \ \alpha < \beta\,.
\end{eqnarray}
As we shall see, it is sufficient to derive the equations of motion
for an {\it ansatz\ } based on a generator of the form $\tau_{1+}$.
By a simple redefinition, it will then be possible to construct
an {\it ansatz\ } based on a generator of the form $\tau_{1-}$.
For now, we consider the case when $\tau_1$ has the form $\tau_{1+}$.
The simple extension of {\it Ansatz I} with $\tau_{\rm all}$
replaced by $\tau_1$ does not work for string-$\tau_1$.
The problem arises from the term $\tau_1\tau_1 \phi$ on the
left-hand side of Eq.~(17) in which a new tensor $\phi_0^A$,
\begin{equation}
\tau_1\tau_1 \phi_0 = \phi^A_0 \,,
\end{equation}
is generated, where
\FL
\begin{equation}
\phi^A_{0\,i_1 \ldots i_5} \equiv
\left\{ \begin{array}{ll}
\phi_{0\,i_1 \ldots i_5}\,,\ &
\mbox{if two indices take the values} \\
& \mbox{$(2\alpha-1, 2\beta-1)$
or $(2\alpha, 2\beta)$}\,,\\
0 \,, & \mbox{otherwise}.
\end{array} \right.
\end{equation}
As a result, the differential equations for $g(r)$ and $f(r)$
are satisfied only if $g(r)=1$ or $f(r)=0$ everywhere, which
is not consistent with the boundary conditions given by Eq.~(20).
(Note that the solution $g=1$ and $f=\mu$ is the vacuum field
configuration expressed in a singular gauge.)
We construct a nontrivial solution for string-$\tau_1$ by
replacing $f(r)\phi_0$ and $\tau_{\rm all}$ in {\it Ansatz I}
with $(f_1(r)\phi_0 + f_2(r)\phi^A_0)$ and $\tau_1$ respectively.
Note that $\phi_0$ is not orthogonal to $\phi^A_0$ because
$\phi^A_{0\,i_1 \ldots i_5} \phi^\ast_{0\,i_1 \ldots i_5} \neq 0$.
Therefore instead of expanding $\phi$ in $\phi_0$ and $\phi^A_0$,
we will use the more convenient basis $\phi^A_0$ and $\phi^B_0$
where
\begin{equation}
\phi^B_0 \equiv \phi_0 - \phi^A_0\
\end{equation}
and $\phi^B_0$ is orthogonal to $\phi^A_0$:
\begin{equation}
\phi^A_{0\,i_1 \ldots i_5} \phi^{B\,\ast}_{0\,i_1 \ldots i_5}
= 0\,.
\end{equation}
{}From the definition of $\phi^A_0$ (Eq.~(25)) and the properties of
$\phi_0$, one can see that
\FL
\begin{equation}
\phi^B_{0\, i_1 \ldots i_5} =
\left\{ \begin{array}{ll}
\phi_{0\,i_1 \ldots i_5}\,,\ &
\mbox{if two indices take the values} \\
& \mbox{$(2\alpha-1, 2\beta)$ or $(2\alpha, 2\beta-1)$} \,,\\
0 \,, & \mbox{otherwise}
\end{array} \right.
\end{equation}
and $\phi^B_0$ is annihilated by $\tau_1$:
\begin{equation}
\tau_1 \phi^B_0 = 0\,.
\end{equation}
The solution constructed for string-$\tau_1$ is
\newline {\em Ansatz II\ }:
\begin{eqnarray}
\phi & = & e^{i\tau_1 \theta} \left\{ f_o(r) \phi^A_0 +
f_e(r) \phi^B_0 \right\} \,, \nonumber\\
A^\theta & = & i\frac{g(r)}{er} \tau_1 \,, \\
A^r & = & 0\,, \nonumber
\end{eqnarray}
where as will become clear in the next two paragraphs, the
functions $f_o(r)$ and $f_e(r)$ are named after their odd
and even parities in $r$.
At the origin, we require the fields to be regular. Since
$\phi^B_0$ is left invariant by $e^{i\tau_1\theta}$ (Eq.~(29))
but $\phi^A_0$ is not, at the origin $f_e(0)$ can be any
constant but $f_o(0)$ has to vanish. At large $r$,
the scalar field $\phi$ has to take the form
\begin{equation}
\phi \stackrel{r\rightarrow \infty}{\longrightarrow} \mu
\ e^{i\tau_1 \theta} \phi_0 = \mu\ e^{i\tau_1 \theta}
(\phi^A_0 + \phi^B_0)
\end{equation}
for the unbroken gauge group to be SU(5), so both $f_o(r)$
and $f_e(r)$ approach $\mu$ at large $r$. The boundary
conditions on the functions are
\begin{eqnarray}
& f_o(0)=0\,,\qquad &
f_o(r) \stackrel{r\rightarrow \infty}{\longrightarrow} \mu\,,
\nonumber\\
& f_e(0)= a_0\,,\qquad &
f_e(r) \stackrel{r\rightarrow \infty}{\longrightarrow} \mu\,,
\nonumber\\
& g(0)=0\,,\qquad &
g(r)\stackrel{r\rightarrow \infty}{\longrightarrow} 1\,,
\end{eqnarray}
where $a_0$ is a constant.
The equations of motion for $\phi$ and $A_\mu$ are closed when
the fields take the form in {\it Ansatz II\ }. We obtain three
coupled differential equations for $f_o(r),f_e(r)$ and $g(r)$.
The algebra involved in extracting these three equations, however,
is considerably more tedious than in the $\tau_{\rm all}$ case
mainly because the forms of $\phi^A_0, \phi^B_0$ and $\tau_1$ are
less symmetric. We will not present the algebra involved and simply
quote the results:
\FL
\begin{eqnarray}
f_e^{\prime\prime} + \frac{1}{r} f_e^\prime
& = & f_e \left\{ v_1 + N v_2 (f_o^2 + f_e^2)
-\frac{N}{25} e^2 \lambda_3 (f_o^2 - f_e^2) \right\}
\nonumber\\
f_o^{\prime\prime} + \frac{1}{r} f_o^\prime
& - & \frac{(1-g)^2}{r^2} f_o \nonumber\\
& = & f_o \left\{ v_1 + N v_2 (f_o^2 + f_e^2)
+ \frac{N}{25} e^2 \lambda_3 (f_o^2 - f_e^2) \right\}
\nonumber
\end{eqnarray}
\FL
\begin{equation}
Tr(\tau_1^2) \left( g^{\prime\prime} -
\frac{1}{r} g^\prime \right) = -N e^2 (1-g) f_o^2 \,,
\end{equation}
where $e^2 \lambda_3 \equiv v_3 + \frac{v_4}{4} + \frac{v_5}{4}
+ \frac{v_6}{12}$, and $Tr(\tau_1^2)=1$ from Eq.~(12). An expansion
of $g, f_o$ and $f_e$ in powers of $r$ around the origin gives
\begin{eqnarray}
f_o(r) & = & a_1 r + a_3 r^3 + \ldots \,,\nonumber\\
f_e(r) & = & a_0 + a_2 r^2 + \ldots \,,\nonumber\\
g(r) & = & b_2 r^2 + b_4 r^4 + \ldots \,,
\end{eqnarray}
where the coefficients of all the higher terms are related to
$a_0, a_1$ and $b_2$ recursively. The function $f_o$ is indeed
odd and $f_e$ even in $r$ as claimed earlier.
Inserting {\it Ansatz II\ } for string-$\tau_1$ into the
Lagrangian gives
\FL
\begin{equation}
-{\cal L}^1 = \frac{Tr(\tau_1^2)}{2e^2 r^2} g^{\prime\,2}
+ \frac{N}{2} \left( f_e^{\prime\,2} + f_o^{\prime\,2} \right)
+ \frac{N}{2} \frac{(1-g)^2}{r^2} f_o^2 + V_{ans}
\end{equation}
where
\begin{eqnarray}
V_{ans} &=& \frac{N}{2} \left\{ v_1 (f_o^2 + f_e^2)
+ \frac{N}{2} v_2 (f_o^2 + f_e^2)^2 \right. \nonumber\\
&& \left. +\frac{N}{50} e^2 \lambda_3 (f_o^2 - f_e^2)^2 \right\}\,.
\end{eqnarray}
Here again, note that the equations of motion obtained
by varying ${\cal L}^1$ with respect to the functions
$g, f_o$ and $f_e$ are identical to those in Eq.~(33).
Now let us consider the other case when $\tau_1$ has the form of
$\tau_{1-}$. One can show that Eq.~(24) now is $\tau_1\tau_1\phi_0
=\phi^B_0$, and instead of $\tau_1 \phi^B_0=0$, one has
$\tau_1 \phi^A_0=0$. Therefore by switching the definitions of
$\phi^A_0$ and $\phi^B_0$ in Eqs.~(25) and (28), all the equations
between (24) and (32) are preserved, and one can show that the
equations of motion are unchanged. We conclude that {\it Ansatz II}
applies to all twenty $\tau_1$'s, where for $\tau_{1+}$, $\phi^A_0$
and $\phi^B_0$ are defined by Eqs.~(25) and (28) respectively, but
for $\tau_{1-}$, the definitions of the two are reversed.
The equations of motion are given by Eq.~(33) for all cases.
\section{Numerical Calculations}
In this section we present the numerical solutions to the two
sets of differential equations (21) and (33) with the appropriate
boundary conditions at the origin and some large value of $r$.
We implemented two methods: the ``shooting'' and the relaxation
methods to handle this two-point boundary value problem. In the
``shooting'' method \cite{num rec}, an initial guess for the free
parameters at $r=0$ was made and then the equations were integrated
out to large $r$ where the boundary conditions were specified. As
the name of the method suggests, the true solutions were found by
adjusting the parameters at $r=0$ in the beginning of each iteration
to reduce the discrepancies from the desired boundary conditions at
large $r$ computed in the previous iteration. For string-$\tau_1$,
the small-$r$ expansion of the functions in Eq.~(34) gives
$g(0) = g^\prime(0) = 0\,, f_o(0) = f_o^{\prime\prime}(0) =
f_e^\prime(0) = 0\,$, and $f_e^{\prime\prime}(0)=2a_2\,,$ where
$a_2$ is related to $a_0$, $a_1$ and $b_2$, but the values of
\begin{eqnarray}
f_e(0) &=& a_0\,, \nonumber\\
f_o^\prime(0) &=& a_1\,, \nonumber\\
g^{\prime\prime}(0) &=& 2b_2\,,
\end{eqnarray}
were adjusted to match the boundary conditions at large $r$.
For string-$\tau_{\rm all}$, we have shown that $f(r)$ is odd and
$g(r)$ is even in $r$, with $f(r)=ar+\ldots$ and $g(r)=br^2+\ldots$.
Thus only the two values $f^\prime(0), g^{\prime\prime}(0)$ were free
parameters. At large $r$, discrepancies from the boundary condition
were corrected by the multi-dimensional Newton-Raphson method which
computed the corrections to the initial parameters. With an initial
guess for the parameters at $r=0$, this ``shooting'' process was
iterated until the ``targets'' were met. The fourth-order
Runge-Kutta method was used to integrate the equations.
We have also implemented a relaxation scheme for comparison.
In this method the first step is to express the string
energy as a function of the values of the functions $f$ and
$g$ (or $f_e$, $f_o$, and $g$) defined on an evenly spaced
mesh of points. While a Simpson's rule approximation worked
well for the middle range of parameters, a more sophisticated
approximation was used to extend the range of parameters that
could be treated. For each interval of two lattice spacings,
smooth functions $\tilde f$ and $\tilde g$ were defined by 2nd
order polynomial interpolation from the three mesh points
(midpoint and two end points); with the help of a symbolic
integration program, the integral defining the energy was
carried out exactly for the interpolated functions. (By this
method the energy obtained is a rigorous upper limit on
the true ground state string energy.) To avoid divergences
caused by the explicit factors of $1/r^2$ in the energy density,
the first interval had to be treated more carefully--- instead
of fitting the functions with a 2nd order polynomial, we fitted
the coefficients of the analytically determined power series,
such as Eq.~(34). Trial functions $f$ and $g$ were chosen,
and then the energy was minimized by varying each mesh point
one at a time, successively going through the lattice many
times. We found it efficient to begin with a coarse mesh
which was made successively finer by factors of 2,
interpolating the solution at each stage to obtain the first
trial solution for the next stage. For the final run in
each case we used 2048 points.
We found the results by the two methods to agree to
approximately one part in a million or better. In general we
were able to explore a wider parameter range with the relaxation
method than with the ``shooting'' method, but the qualitative
features given by the ``shooting'' method remained the same.
(The author wishes to thank Alan Guth for implementing the
relaxation part of the calculations.)
The dependence of the equations on the parameters in the theory can
be simplified if $r, f, f_o$ and $f_e$ are rescaled as ($v_1 < 0$)
\begin{eqnarray}
r & \rightarrow & \sqrt{-v_1} r\,, \nonumber\\
\{f\,, f_o\,, f_e\} & \rightarrow & \sqrt{\frac{2Nv_2}{-v_1}}
\{f\,, f_o\,, f_e\}\,.
\end{eqnarray}
Then only the following combinations of parameters appear
in the differential equations:
\begin{eqnarray}
\lambda_2 & \equiv & \frac{v_2}{e^2} \,,\nonumber\\
\lambda_3 & \equiv & \frac{1}{e^2} \left(
v_3 + \frac{v_4}{4} + \frac{v_5}{4} + \frac{v_6}{12}\right)\,.
\end{eqnarray}
The Hamiltonian densities ${\cal H}^{\rm all}$ and ${\cal H}^1$ for
the two strings are simply $-{\cal L}^{\rm all}$ and $-{\cal L}^1$
given by Eqs.~(22) and (35) because all fields are assumed to be
time-independent. With the same rescaling, one obtains
\begin{eqnarray}
\frac{v_2}{(-v_1)^2} {\cal H}^{\rm all} &=& \frac{1}{2} \left\{
\frac{2\lambda_2}{5r^2} g^{\prime\,2}
+ f^{\prime\,2} + \frac{(1-g)^2}{r^2} f^2 \right.\nonumber\\
&& \left. + \frac{1}{2}(1-f^2)^2 \right\}
\end{eqnarray}
and
\begin{eqnarray}
&&\frac{v_2}{(-v_1)^2} {\cal H}^1 = \frac{1}{2} \left\{
\frac{\lambda_2}{r^2} g^{\prime\,2}
+ \frac{ f_o^{\prime\,2} + f_e^{\prime\,2}}{2}
+ \frac{(1-g)^2}{2r^2} f_o^2 \right. \nonumber\\
&&\ + \left.
\frac{1}{2} \left( 1 - \frac{f_o^2 + f_e^2}{2} \right)^2
+ \frac{\lambda_3}{200\lambda_2} (f_o^2 - f_e^2)^2 \right\}
\end{eqnarray}
where the $\tau_{\rm all}$ equation depends on $\lambda_2$ only but
the $\tau_1$ equation depends on both $\lambda_2$ and $\lambda_3$.
Typical solutions for the two strings calculated from the
``shooting'' method are shown in Figs.~1 and 2, where
$\lambda_2 = 0.132$ and $\lambda_3 = 10.25$. For the same
$\lambda_2$ and $\lambda_3$, the solutions given by the
relaxation method appear indistinguishable visually from
those in Figs.~1 and 2. For string-$\tau_{\rm all}$, we
were able to find solutions in the approximate range $10^{-2}
< \lambda_2 < 10$ using the ``shooting'' method and $10^{-4}
< \lambda_2 <10^3$ using the relaxation method. For
string-$\tau_1$, we explored the range $5\times 10^{-2} <
\lambda_2 < 1$ and $0.5 < \lambda_3 < 10^2$. In general,
the functions converged more slowly near the two ends of each
range above, and we did not attempt to find solutions beyond
these limits. We numerically integrated ${\cal H}^{\rm all}$
and ${\cal H}^1$ for the solutions we computed, and found
string-$\tau_1$ to have the lower energy for all the parameters
we explored. In Fig.~3, the energy density $2\pi r {\cal H}$
of the two solutions shown in Figs.~1 and 2 is plotted, and
the energy of string-$\tau_1$ is clearly lower. For comparison,
we point out that the energy per unit length of
string-$\tau_{\rm all}$ in the range $0.9 < \lambda_2 < 4.0$ has
been calculated by Aryal and Everett \cite{everett}. Our values
in this range of parameters agree with theirs to within 1\%.
One of the most important properties of the two strings we
investigate in this paper is whether string-$\tau_1$ has
lower energy than string-$\tau_{\rm all}$. We just showed
that this is true for some range of the parameters. To
systematically explore a wider parameter range, however, it
is very laborious and time-consuming to calculate the $\tau_1$
solutions for different $\lambda_2$ and $\lambda_3$ first and
then compute the corresponding energy. Instead, we employ an
upper-bound argument to reduce the two-dimensional parameter space
$(\lambda_2, \lambda_3)$ to one. We set $f_o = f_e \equiv f_1$ in
the Lagrangian and take $g(r), f_1(r)$ as trial functions for
string-$\tau_1$. The advantage in using $f_o = f_e$ is that the
last term in Eq.~(41) vanishes, and the equations no longer depend
on $\lambda_3$. Moreover, Eqs.~(40) and (41) then have the same
functional form, differing only in the coefficients of the first
and the third terms, and one can solve the equations for
string-$\tau_1$ the same way as for string-$\tau_{\rm all}$ using
different values of $\lambda_2$. The corresponding energy,
denoted by $E_1(f_o=f_e)$, gives an upper bound on the true
energy of string-$\tau_1$ by the variational principle.
If $E_1(f_o=f_e) < E_{\rm all}$ for a given $\lambda_2$, then one
can conclude that string-$\tau_1$ has the lower energy for that
value of $\lambda_2$ and all values of $\lambda_3$.
(Note that in the limit of $\lambda_3 \rightarrow \infty$, the
trial functions approach the true string solution because for the
energy to be finite, the last term in Eq.~(41) requires $f_o
\rightarrow f_e$.) Our result is presented in Fig.~4, where the
ratio $E_1(f_o=f_e)/E_{\rm all}$ is plotted as a function of
log$\,\lambda_2$ for $10^{-4} < \lambda_2 < 2.5\times 10^3$.
Note that $E_1(f_o=f_e)/E_{\rm all} < 1$ for all 7 decades of
$\lambda_2$, and is approaching an asymptote of 1 (or possibly
less than 1) as $\lambda_2 \rightarrow 0$. For large $\lambda_2$,
we find the individual curves of $E_{\rm all}$ vs. log$\,\lambda_2$
and $E_1$ vs. log$\,\lambda_2$ approach straight lines,
suggesting that the ratio $E_1(f_o=f_e)/E_{\rm all}$ levels off at a
constant for large $\lambda_2$. We conclude that string-$\tau_1$ has
lower energy than string-$\tau_{\rm all}$ for $10^{-4} < \lambda_2
< 2.5\times 10^3$ and all $\lambda_3$, and probably is the ground
state for the entire range of the parameters in the theory.
\section{Scattering Solutions}
To study the scattering of fermions by an SO(10) cosmic string,
one first needs to understand the 16-dimensional spinor
representation of SO(10) to which the left-handed fermions
are assigned. Spinor representations certainly have been discussed
in the literature \cite{spin}, but to establish a common notation,
we discuss in the Appendix the construction of the generators,
the sixteen states and the identification of states with fermions
that are relevant to this paper.
Now we proceed to study the Dirac equation
\begin{equation}
(i\not\!\partial - e\not\! A^a \tau^a - m)\psi = 0
\end{equation}
in the background fields of string-$\tau_{\rm all}$ and $\tau_1$:
$A^a_\mu \tau^a = A^{\rm all}_\mu \tau_{\rm all}$ and $A^1_\mu
\tau_1$. As shown in the Appendix, the fermion fields can be
written as a 16-dimensional column vector where each component
is identified with a fermion given by Eq.~(A.16). The generators
$\tau_{\rm all}$ and $\tau_1$ can be written as 16$\times$16
hermitian matrices, where $\tau_{\rm all}$ is diagonal with one
diagonal entry equal to $\frac{1}{2}$, ten entries equal to
$\frac{1}{10}$ and five entries equal to $-\frac{3}{10}$. For
$\tau_1$, we choose $\tau_1 = \frac{1}{2} (\tau^{58}+\tau^{67})$ for
illustration. We find that $\tau_1$ takes the block-diagonal form
\begin{equation}
-\tau_1 = \frac{1}{2} \left( \begin{array}{cc}
B\ & 0\ \\
0\ & B\
\end{array} \right) \,,
\end{equation}
where
\begin{equation}
B = \left( \begin{array}{cccc}
0 \ & 0 \ & 0 \ & I \\
0 \ & 0 \ & 0 \ & 0 \\
0 \ & 0 \ & 0 \ & 0 \\
I \ & 0 \ & 0 \ & 0
\end{array} \right) \,,
\end{equation}
and $I$ is the 2$\times$2 identity matrix.
For string-$\tau_{\rm all}$, since $\tau_{\rm all}$ is diagonal,
Eq.~(42) decouples into sixteen equations, one for each component
of the wave function, and there is no mixing of leptons and quarks
due to twisting of the Higgs. However, since the sixteen
eigenvalues of $\tau_{\rm all}$ are all fractional, all sixteen
fermions scatter nontrivially off the string via the Aharonov-Bohm
effect. As pointed out by previous studies, the wave functions of
these fermions can be strongly enhanced near the core of the string,
leading to strong B-violating processes inside the string.
In the case of string-$\tau_1$, upon diagonalizing $\tau_1$ by a
unitary matrix $U$ and simultaneously rotating the fermion basis
$\psi$ in Eq.~(A.16) to $\tilde{\psi} \equiv U\psi$, we can
write $\tilde{\psi}$ as
\FL
\begin{eqnarray}
\tilde{\psi}
& = &( e^- + u_1^c\,, e^- - u_1^c\,, \nu^c+d_1\,, \nu^c-d_1\,,
u^c_2\,, u^c_3\,, d_3\,, d_2\,, \nonumber\\
& & u_3 + d_2^c\,, u_3 - d_2^c\,, u_2+d_3^c\,, u_2-d_3^c\,,
u_1\,, \nu\,, e^+\,, d_1^c)_L \nonumber
\end{eqnarray}
\begin{equation}
\quad \quad
\end{equation}
and Eq.~(42) again decouples into sixteen equations of
the form
\begin{equation}
(i\not\!\partial + e\lambda_i\not\! A^1 - m)
\tilde{\psi}_i = 0\,,
\label{eq:dede}
\end{equation}
where each $\tilde{\psi}_i$ interacts with the gauge field with
coupling strength $e\lambda_i\,; \lambda_i$ are the eigenvalues
of $-\tau_1$. The eigenvalues are $\lambda_i = \frac{1}{2}$ for
$e + u^c_1\,,\nu^c + d_1\,,u_3 + d^c_2\,,u_2 + d^c_3\,, \lambda_i =
-\frac{1}{2}$ for $e - u^c_1\,,\nu^c -d_1\,,u_3 - d^c_2\,,u_2 -
d^c_3\,,$ and $\lambda_i = 0$ for all others. Since the $e + u^c$
and $e - u^c$ components have opposite eigenvalues, we expect a pure
$e$ or $u^c$ to turn into a mixture of $e$ and $u^c$ as it
propagates around the string, producing baryon-number violation.
Before calculating the scattering amplitude, we first comment on
the choice of gauge in this problem. The fields in {\it Ansatz II}
(See Eq.~(30)) for string-$\tau_1$ were constructed in a gauge where
the scalar field $\phi$ winds with $\theta$ and the gauge field
falls off as $r^{-1}$ at large $r$. The particle content, however,
is probably most transparent in a different gauge where $\phi$ does
not wind with $\theta$ and $A_\mu \rightarrow 0$ at large $r$
everywhere except on a sheet of singularities at $\theta=0$.
We will refer to the former as the $1/r$-gauge and the latter
as the ``sheet'' gauge, in analogy with the ``string'' gauge of
a magnetic monopole. Continuing to work in the diagonalized basis,
the fermion fields in the ``sheet'' gauge, $\tilde{\psi}_0$,
are related to those in the $1/r$-gauge, $\tilde{\psi}$,
by the gauge transformation
\begin{equation}
\tilde{\psi}_0 = e^{-i\tau_1 (\pi-\theta)} \tilde{\psi}\,.
\label{eq:gauge}
\end{equation}
We will solve the Dirac equation and calculate the scattering
amplitude in the $1/r$-gauge, and then write down the baryon-number
violating cross section in the ``sheet'' gauge.
In the presence of an infinitely-thin $\tau_1$-string along the
$z$-axis, the gauge field $A^1_\mu$ takes the form
$A^{1\,r} = A^{1\,z}=0, A^{1\,\theta}= \frac{1}{er}\,,$
where $(r, \theta)$ denote the usual polar coordinates with
$\theta$ running counter-clockwise from the positive $x$-axis.
Owing to the symmetry along the $z$-axis, the matrix $\gamma_3$ in
Eq.~(46) drops out, and with the choice for the $\gamma$-matrices
\begin{eqnarray}
\gamma_0 &= \left( \begin{array}{cc}
\sigma_3 & 0 \\
0 & -\sigma_3
\end{array} \right) \,,
&\quad \gamma_1 = \left( \begin{array}{cc}
i\sigma_2 & 0 \\
0 & -i\sigma_2
\end{array} \right) \,, \nonumber\\
\gamma_2 &= \left( \begin{array}{cc}
-i\sigma_1 & 0 \\
0 & i\sigma_1
\end{array} \right) \,,
&\quad \gamma_3 = \left( \begin{array}{cc}
0\ \ &\ 1 \\
-1\ \ &\ 0
\end{array} \right) \,,
\end{eqnarray}
Eq.~(46) decouples into two independent equations
for the upper and lower 2-component spinors of $\tilde{\psi}_i$,
where the two equations differ by the sign of the mass term.
Writing the upper spinor of $\tilde{\psi}_i$ as
\begin{equation}
\left( \begin{array}{c}
\chi_1 (r) \\
\chi_2 (r) e^{i\theta}
\end{array} \right)
e^{in\theta - iEt} \,,
\end{equation}
one can show
\begin{equation}
\left( \begin{array}{cc}
m-E & -i\left( \partial_r + \frac{n+\lambda_i+1}{r} \right) \\
-i\left( \partial_r - \frac{n+\lambda_i}{r} \right) & -m-E
\end{array} \right)
\left( \begin{array}{c}
\chi_1 \\
\chi_2
\end{array} \right) = 0 \,,
\end{equation}
and the solutions are Bessel functions of order $(n+\lambda_i)$
and $-(n+\lambda_i)$:
\FL
\begin{equation}
\left( \begin{array}{c}
\chi_1 \\
\chi_2
\end{array} \right) =
\left( \begin{array}{c}
J_{\pm(n+\lambda_i)} (kr) \\
\pm \frac{ik}{E+m} J_{\pm(n+\lambda_i+1)} (kr)
\end{array} \right) \,,\ k=\sqrt{E^2-m^2}\,.
\end{equation}
The appropriate boundary conditions to impose, as pointed out
in Ref.~14, are the square-integrability of the wave functions
near the origin and a self-adjoint Hamiltonian. The usual
requirement that wave functions be regular at the origin is
sometimes too strong and has to be relaxed. Since $J_\nu (r)
\sim r^\nu / (2^\nu \nu!)$ for small $r$, one can see that
the solutions above are square-integrable if the $+$ sign
is chosen for the modes $n+\lambda_i > 0$, and the $-$ sign for
$n+\lambda_i < -1$. For the mode $ -1 < n+\lambda_i < 0$,
however, both choices are square-integrable albeit neither is
regular at the origin, and the solution takes the form
\FL
\begin{equation}
\left( \begin{array}{c}
\chi_1 \\
\chi_2
\end{array} \right) =
\left( \begin{array}{c}
\sin\mu\,J_{n+\lambda_i} + \cos\mu\,J_{-(n+\lambda_i)} \\
\frac{ik}{E+m}
(\sin\mu\,J_{n+\lambda_i+1} - \cos\mu\,J_{-(n+\lambda_i+1)})\\
\end{array} \right) \,,
\end{equation}
where $\mu$ is the self-adjoint parameter.
The scattering amplitude $f^{\lambda_i}(\theta)$ for the $i$th
fermion in $\tilde{\psi}$ appears in the asymptotic wave function
written as the sum of the incident plane wave and the scattered
part:
\begin{eqnarray}
\tilde{\psi}_i &\sim &
u_E e^{-i\lambda_i(\pi-\theta)} e^{i(kx - Et)} \nonumber\\
&& + \sqrt{\frac{i}{r}} v_E e^{-i\lambda_i(\pi-\theta)}
f^{\lambda_i}(\theta) e^{i(kr - Et)} \,,
\end{eqnarray}
where $u_E$ and $v_E$ are given by
\begin{equation}
u_E = \left( \begin{array}{c}
1 \\
\frac{k}{E+m}
\end{array} \right) \,, \quad
v_E = \left( \begin{array}{c}
1 \\
\frac{k}{E+m} e^{i\theta}
\end{array} \right) \,.
\end{equation}
Expanding $e^{ikx}=e^{ikr\cos\theta}$ and $e^{ikr}$ in Bessel
functions using
\begin{equation}
e^{ikr\cos\theta}
= \sum_{n=-\infty}^{\infty} i^n J_n(kr) e^{in\theta}\,,
\end{equation}
and with
\begin{equation}
f^{\lambda_i}(\theta) = \sum_{n=-\infty}^{\infty}
f_n^{\lambda_i} e^{in\theta}\,,
\end{equation}
Eq.~(53) can be matched to the solutions in Eq.~(51) mode by mode
at large $r$. Then the scattering amplitude can be calculated:
\begin{equation}
f^{\lambda_i} (\theta)
= \frac{i}{\sqrt{2\pi k}} e^{-i([\lambda_i]+1)\theta}
\left( \frac{ \sin\left( \frac{\theta}{2} - \pi\lambda_i
\right)} { \sin \frac{\theta}{2} }
- e^{2i\delta} \right)\,,
\end{equation}
where $[\lambda_i]$ denotes the largest integer less than
or equal to $\lambda_i$, and $\delta$ is related to $\lambda_i$
and the self-adjoint parameter $\tan\mu$ by \cite{gerbert}
\begin{equation}
\tan \delta
= \frac{1-\tan\mu}{1+\tan\mu}\,\tan\frac{\lambda_i \pi}{2}\,.
\end{equation}
With the gauge transformation Eq.~(47), one can easily see that
$(\tilde{\psi}_0)_i$ in the ``sheet'' gauge is given by Eq.~(53)
without the phase $e^{-i\lambda_i(\pi-\theta)}$.
To illustrate the processes that violate the baryon number, we
consider an incident beam of electrons propagating in the fields
of the string. We will study the $(e, u^c)$-subspace and ignore
other fermions since $e$ in $\psi$ is mixed with $u^c$ only. In
the ``sheet'' gauge, the eigenstates of $\tau_1$ can be written as
\begin{equation}
e + u^c = \left( \begin{array}{c}
1 \\
0
\end{array} \right) \,,\quad
e - u^c = \left( \begin{array}{c}
0 \\
1
\end{array} \right) \,,
\end{equation}
and the electron is simply given by
\begin{equation}
e = \left( \begin{array}{c}
\frac{1}{2} \\
\frac{1}{2}
\end{array} \right) \,.
\end{equation}
An incident wave of electrons can be written as
\begin{equation}
\tilde{\psi}^e_{0\,inc} = u_E \left( \begin{array}{c}
\frac{1}{2} \\
\frac{1}{2}
\end{array} \right) e^{i(kx-Et)} \,,
\end{equation}
which scatters into
\FL
\begin{equation}
\tilde{\psi}_{0\,sca} = \sqrt{\frac{i}{r}} v_E
\left\{ f^{\frac{1}{2}}(\theta)
\left( \begin{array}{c}
\frac{1}{2} \\
0
\end{array} \right)
+ f^{-\frac{1}{2}}(\theta)
\left( \begin{array}{c}
0 \\
\frac{1}{2}
\end{array} \right) \right\} e^{i(kr-Et)}\,.
\end{equation}
Note that the suppressed index on the 2-component spinors
$u_E$ and $v_E$ should not be confused with the index associated
with the 2-component eigenvectors used here to label the
$e + u^c$ and $e-u^c$ components of the Dirac field.
Rewriting $\tilde{\psi}_{0\,sca}$ above as
\FL
\begin{eqnarray}
\tilde{\psi}_{0\,sca} &=& \sqrt{\frac{i}{r}} v_E
\left\{ \left(
\frac{f^{\frac{1}{2}}(\theta) + f^{-\frac{1}{2}}(\theta)}{2}
\right)
\left( \begin{array}{c}
\frac{1}{2} \\
\frac{1}{2}
\end{array} \right)
\right. \nonumber\\
&& \left. + \left(
\frac{f^{\frac{1}{2}}(\theta) - f^{-\frac{1}{2}}(\theta)}
{2} \right)
\left( \begin{array}{c}
\frac{1}{2} \\
-\frac{1}{2}
\end{array} \right) \right\} e^{i(kr-Et)}\,,
\end{eqnarray}
one finds that the scattered wave consists of a mixture of electrons
and $u^c$-quarks.
The differential cross section per unit length for the production of
$u$-quark is defined by
\begin{equation}
\frac{d\sigma}{d\theta} = \lim_{r\rightarrow \infty}
\frac{\vec{J}_{sca}^u\cdot \vec{r}}{J_{inc}}
\end{equation}
where $J^i = \bar{\psi}\gamma^i \psi\ $. Substituting
$\tilde{\psi}_{0\,inc}$ and $\tilde{\psi}_{0\,sca}$
into the currents, one obtains
\begin{equation}
\frac{d\sigma}{d\theta} =
\frac{1}{4} \left| f^{\frac{1}{2}}(\theta)
-f^{-\frac{1}{2}}(\theta) \right|^2\,,
\end{equation}
which can be written out using Eq.~(57) as
\begin{equation}
\frac{d\sigma}{d\theta} = \frac{1}{2\pi k}
\left\{ \frac{\cos^4 \frac{\theta}{2}}{\sin^2 \frac{\theta}{2}}
+ \sin^2 \left( \frac{\theta}{2} - 2\delta \right)\right\}\,.
\end{equation}
The calculation above was done in the limit of zero string width.
Now let us examine the string core. The structure of the string
core is ``encoded'' in the self-adjoint parameter $\delta$ (or
$\mu$, related to $\delta$ by Eq.~(58)), which appears in the
differential cross section above. In general the self-adjoint
parameter is determined either from physical properties at the
origin or sometimes by symmetry arguments. Since the string
solutions have already been obtained in the previous section,
we can find $\mu$ by solving Eq.~(50) numerically for the mode
$-1 < n+\lambda_i < 0$, using the realistic form $g(r)/r$ for the
gauge field computed earlier in place of the $1/r$ in Eq.~(50).
As we have shown, $\lambda_i=\pm\frac{1}{2}$ for the fermions that
scatter nontrivially off the $\tau_1$-string. Thus the special
mode satisfying $-1 < n+\lambda_i < 0$ takes the value $n+\lambda_i
=-\frac{1}{2}$, where $n=-1$ for $\lambda_i=\frac{1}{2}$ and
$n=0$ for $\lambda_i=-\frac{1}{2}$. Recall that in the calculation
of $g(r)$, the radial distance $r$ was rescaled to the dimensionless
$\sqrt{-v_1} r\ (v_1 < 0)$, where $v_1$ is the quadratic coupling
in the Higgs potential in Eq.~(16). Rescaling $\chi_2$ and $r$ by
\begin{eqnarray}
\chi_2 & \rightarrow & i\frac{E+m}{k}\chi_2\,,\nonumber\\
r & \rightarrow & \sqrt{-v_1} r\,,
\end{eqnarray}
and replacing $\lambda_i$ in Eq.~(50) by $\lambda_i g(r)$,
Eq.~(50) can be rewritten as
\begin{eqnarray}
\partial_r \chi_1 & = & \frac{g(r)-2}{2r}\chi_1+\beta\chi_2
\nonumber\\
\partial_r \chi_2 & = & - \frac{g(r)}{2r}\chi_2
-\beta\chi_1
\end{eqnarray}
for $\lambda_i=\frac{1}{2}, n=-1$, and
\begin{eqnarray}
\partial_r \bar{\chi}_1 & = & -\frac{g(r)}{2r}\bar{\chi}_1
+ \beta\bar{\chi}_2 \nonumber\\
\partial_r \bar{\chi}_2 & = & \frac{g(r)-2}{2r}\bar{\chi}_2
-\beta\bar{\chi}_1
\end{eqnarray}
for $\lambda_i=-\frac{1}{2}, n=0$. The parameter $\beta$
is defined by
\begin{equation}
\beta \equiv k/\sqrt{-v_1}\,,
\end{equation}
and the bars over $\chi_1, \chi_2$ are used to distinguish the
solutions of $\lambda_i=-\frac{1}{2}$ from those of $\lambda_i
=\frac{1}{2}$. Upon closer inspection of the two sets of equations
above, one finds that Eq.~(69) is in fact identical to Eq.~(68) if
$\bar{\chi}_1$ is identified with $\chi_2$ and $\bar{\chi}_2$
with $-\chi_1$. What about the boundary conditions at the origin?
In Eq.~(49), for $n=-1$, the upper component depends on $\theta$
but the lower component does not, and vice versa for $n=0$.
Therefore $\chi_1$ and $\bar{\chi}_2$ must vanish at the origin for
the solution to be continuous, but $\chi_2$ and $\bar{\chi}_1$ can
be nonzero at $r=0$. One thus has $\bar{\chi}_1 = \chi_2$
and $\bar{\chi}_2 = -\chi_1$. Since Eq.~(68) is linear, the
value of $\chi_2(0)$ can be chosen arbitrarily when integrating
the differential equations.
The self-adjoint parameters $\mu$ for $\lambda_i=\frac{1}{2}$ and
$\bar\mu$ for $\lambda_i=-\frac{1}{2}$ are determined by matching
the solutions of Eq.~(68) to the asymptotic expression in Eq.~(52)
at some radius $r$. For $n+\lambda_i = -\frac{1}{2}$, the Bessel
functions in Eq.~(52) are simply $J_{\pm\frac{1}{2}}$, which have
the analytic forms
\begin{equation}
J_{\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}} \sin x\,,\quad
J_{-\frac{1}{2}}(x)=\sqrt{\frac{2}{\pi x}} \cos x\,.
\end{equation}
Then Eq.~(52) leads to the simple expression for $\mu$ and
$\bar\mu$:
\begin{eqnarray}
\frac{\chi_1}{\chi_2} &=& \tan(\mu + \beta r) \,,\nonumber\\
\frac{\bar{\chi}_1}{\bar{\chi}_2} &=&
\tan(\bar\mu + \beta r) \,,
\end{eqnarray}
which can be inverted to give $\mu$ and $\bar\mu$ at a given $r$,
using $\chi_1$ and $\chi_2$ computed from Eq.~(68).
Using Eq.~(72) and trigonometric identities, one finds
\begin{equation}
\bar\mu = \mu + \frac{\pi}{2} \,.
\end{equation}
Note that the solutions depend on $\beta$ which appears
in Eq.~(68), and the quartic couplings $\lambda_2, \lambda_3$ in
the Higgs potential. The parameter $\beta$ defined in Eq.~(70)
measures the ratio of the incident fermion momentum $k$ to
the Higgs mass parameter $\sqrt{-v_1}$, which is of the order of
GUT energy scale. To put it another way, $\beta$ measures
the string width relative to the wavelength of the incident fermion.
In Fig.~5, we set $\beta = 1$ and plot $\mu$ computed from Eq.~(72)
at a given $r$ for three sets of $\lambda_2$ and $\lambda_3$.
The true value of $\mu$ is given by the limit $r \rightarrow
\infty$. In Fig.~6, we choose the same set of parameter as in
Figs. 1-3: $\lambda_2 = 0.132$ and $\lambda_3 = 10.25$; $\mu$
is shown for five values of $\beta$ ranging from 0.1 to 2.0.
One can see that as $\beta$ decreases, {\it i.e.} when the
wavelength of the fermion becomes large compared to the string
width, $\mu$ decreases.
\section{CONCLUSIONS}
We constructed two types of strings, string-$\tau_{\rm all}$
and string-$\tau_1$, in the SO(10) grand unified theory.
They are topologically equivalent but dynamically different
strings, produced during the phase transition $\hbox{Spin(10)}
\rightarrow\hbox{SU(5)}\times{\cal Z}_2$ in the early universe.
String-$\tau_{\rm all}$ is effectively Abelian, and can catalyze
baryon number violation with a strong cross section via grand-unified
processes inside the string. It has been the subject of study in
several recent papers. The richer Higgs structure of
string-$\tau_1$, on the other hand, has been shown in this paper
to induce baryon catalysis by mixing components in the
fermion multiplet, turning leptons into quarks as they travel
around the string. The underlying B-violating mechanism is
the ``twisting'' of the scalar field, which leads to different
unbroken SU(5) subgroups around the string. This mechanism is
distinct from the grand-unified processes which can only occur
inside the string core where the GUT symmetry is restored.
The corresponding string solutions have been calculated numerically
with both the ``shooting'' and the relaxation methods. The energy
of both strings was computed. With an additional upper bound
argument, we found string-$\tau_1$ to have lower energy than
string-$\tau_{\rm all}$ in a wide range of parameters:
$10^{-4} < \lambda_2 < 2.5\times 10^3$ and all $\lambda_3$. The
ratio of the upper bound on $\tau_1$ energy to the $\tau_{\rm all}$
energy increases as $\lambda_2$ decreases, and possibly approaches
one from below as $\lambda_2 \rightarrow 0$. Scattering of fermions
in the fields of string-$\tau_1$ has also been analyzed, and the
B-violating cross section is given by Eq.~(66). We conclude that
string-$\tau_1$ is more stable than string-$\tau_{\rm all}$, and
can catalyze baryon decay with strong cross sections via the
interesting mechanism of Higgs field twisting.
\nonum
\section{ACKNOWLEDGMENTS}
I wish to thank Alan Guth for many valuable suggestions on
this work and a critical reading of the manuscript. I am also
grateful for advice from Ed Bertschinger, Robert Brandenberger,
Jeffrey Goldstone, Roman Jackiw and Leandros Perivolaropoulos,
and assistance from Roger Gilson.
\nonum
\section{APPENDIX}
The generators of SO(2n) in the spinor representation can be
constructed from a set of $2^n \times 2^n$ hermitian matrices
$\Gamma_a^{(n)}, a=1, \ldots ,2n\,$, which satisfy the Clifford
algebra
\begin{equation}
\{\Gamma_a^{(n)},\Gamma_b^{(n)}\} = 2\delta_{ab}\,.
\eqnum{A.1}
\end{equation}
Starting with the two Pauli matrices for $n=1$
\begin{equation}
\Gamma_1^{(1)} = \left( \begin{array}{cc}
0\ & 1\ \\
1\ & 0\
\end{array} \right) \,, \quad
\Gamma_2^{(1)} = \left( \begin{array}{cc}
0 & -i \\
i & 0
\end{array} \right) \,, \eqnum{A.2}
\end{equation}
one can iteratively build the higher-dimensional
$\Gamma^{(n+1)}_a\ $ from the $\Gamma^{(n)}_a\ $ by
\begin{eqnarray}
\Gamma_a^{(n+1)} &=& \left( \begin{array}{cc}
\Gamma_a^{(n)} & 0 \\
0 & -\Gamma_a^{(n)}
\end{array} \right)\,,\ a=1,\ldots ,2n
\nonumber\\
\Gamma_{2n+1}^{(n+1)} &=& \left( \begin{array}{cc}
0\ \ & 1 \\
1\ \ & 0
\end{array} \right)\,, \nonumber\\
\Gamma_{2n+2}^{(n+1)} &=& \left( \begin{array}{cc}
0\ & -i \\
i\ & 0
\end{array} \right)\,. \eqnum{A.3}
\end{eqnarray}
One can check that these $\Gamma$ matrices satisfy the Clifford
algebra. The $\frac{2n(2n-1)}{2}$ generators of SO(2n) are
constructed by
\begin{equation}
M_{ab} = \frac{1}{4i} [\Gamma_a,\Gamma_b]\,,\ \ a,b=1,
\ldots ,2n \eqnum{A.4}
\label{eq:clifford}
\end{equation}
where $M_{ab}$ satisfy the SO(2n) commutation relations
\FL
\begin{equation}
[M_{ab},M_{cd}]=-i(\delta_{bc} M_{ad}+\delta_{ad} M_{bc}
-\delta_{ac} M_{bd}-\delta_{bd} M_{ac})\,. \eqnum{A.5}
\end{equation}
Thus far, we have used the explicit matrix notation to
construct $\Gamma$ and $M$. For convenience, however, we
will use an alternative notation in which each of the
$2^n \times 2^n$ matrices is written as a tensor product
of $n$ independent Pauli matrices, each acting on a different
two-dimensional space. We choose the convention that the first
matrix from the right in the tensor product acts on the largest
2$\times$2 block in the matrix notation, while the second from
the right acts on the next, and so on, with the matrix on the left
acting on the smallest 2$\times$2 block. In this notation,
the 10 $\Gamma$'s of SO(10) given by Eq.~(A.3) become
\begin{eqnarray}
\Gamma_1 &= \sigma_1 \sigma_3 \sigma_3 \sigma_3 \sigma_3\,,\ &
\Gamma_2 = \sigma_2 \sigma_3 \sigma_3 \sigma_3 \sigma_3\,,
\nonumber\\
\Gamma_3 &= I\ \sigma_1 \sigma_3 \sigma_3 \sigma_3\,,&
\Gamma_4 = I\ \sigma_2 \sigma_3 \sigma_3 \sigma_3\,,\nonumber\\
\Gamma_5 &= I\ I\ \sigma_1 \sigma_3 \sigma_3\,,&
\Gamma_6 = I\ I\ \sigma_2 \sigma_3 \sigma_3\,,\nonumber\\
\Gamma_7 &= I\ I\ I\ \sigma_1 \sigma_3\,,&
\Gamma_8 = I\ I\ I\ \sigma_2 \sigma_3\,,\nonumber\\
\Gamma_9 &= I\ I\ I\ I\ \sigma_1\,,&
\Gamma_{10} = I\ I\ I\ I\ \sigma_2 \,,
\eqnum{A.6}
\end{eqnarray}
and the 45 generators $M$ can be found accordingly.
Furthermore, one can write down the five diagonal $M$'s
that generate the Cartan sub-algebra:
\begin{eqnarray}
M_{12} &=& \frac{1}{2}\ \sigma_3 I I I I\,, \nonumber\\
M_{34} &=& \frac{1}{2}\ I \sigma_3 I I I\,, \nonumber\\
M_{56} &=& \frac{1}{2}\ I I\sigma_3 I I\,, \nonumber\\
M_{78} &=& \frac{1}{2}\ I I I\sigma_3 I\,, \nonumber\\
M_{9\,10} &=& \frac{1}{2}\ I I I I\sigma_3 \,.
\eqnum{A.7}
\end{eqnarray}
The eigenvalues of the five generators above can be used
to label the states in the spinor representation. Let
$\frac{1}{2}\epsilon_1, \ldots , \frac{1}{2}\epsilon_5$
be the eigenvalues of $M_{12}, \ldots ,M_{9\,10}$ respectively
with $\epsilon_i = +1$ or $-1$, and denote the states by
\begin{equation}
|\,\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4 \epsilon_5
\,\rangle\,. \eqnum{A.8}
\end{equation}
This 32-dimensional representation is reducible to two 16-dimensional
irreducible representations because there exists a chirality operator
\begin{eqnarray}
\chi &\equiv & (-i)^5 \Gamma_1 \Gamma_2 \ldots \Gamma_{10}
\nonumber\\
& = & \sigma_3\sigma_3\sigma_3\sigma_3\sigma_3\,,
\eqnum{A.9}
\end{eqnarray}
which satisfies the commutation relations
\begin{equation}
\{\,\chi, \Gamma_i\,\} = 0\,,\quad [\,\chi, M_{ab}\,]=0\,.
\eqnum{A.10}
\end{equation}
Moreover,
\begin{equation}
\chi |\,\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4 \epsilon_5
\,\rangle \ = \prod_{i} \epsilon_i
|\,\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4
\epsilon_5\,\rangle\,, \eqnum{A.11}
\end{equation}
where the eigenvalue $\prod_{i} \epsilon_i$ is $+1$ or $-1$
depending on whether the number of spins that are down
$(\epsilon_i=-1)$ is even or odd.
We assign the sixteen left-handed fermions to the states
of positive chirality, {\it i.e. } states with even number of
$\epsilon_i = -1$. The explicit identification of states
to fermions can be achieved by first breaking the SO(10)
10$\times$10 representation into an upper 6$\times$6 and
a lower 4$\times$4 blocks for the subgroups SO(6) and SO(4),
and then embedding SU(3) in SO(6) and SU(2) in SO(4).
The generators for SO(4) are $M_{ab}, a, b = 7,8,9,10$,
and with the choice \cite{spin}
\begin{equation}
\tau_i=\frac{1}{2} \epsilon_{ijk} M_{jk} - M_{i\,10}\,,
\ \ i,j,k = 7,8,9 \eqnum{A.12}
\end{equation}
for the generators of SU(2), one can easily verify that
the last two spins in $|\,\epsilon_1 \epsilon_2 \epsilon_3
\epsilon_4 \epsilon_5\,\rangle$ label the SU(2) states with
$|+ -\,\rangle, |- +\,\rangle$ labeling the doublets and
$|+ +\,\rangle, |- -\,\rangle$ the singlets.
Similarly, the first three spins in
$|\epsilon_1 \epsilon_2 \epsilon_3 \epsilon_4 \epsilon_5 \rangle$
label the SU(3) states with $|+ + +\,\rangle, |- - - \,\rangle$
labeling the singlets, and $|+ + -\,\rangle, |- + + \,\rangle$
with their permutations labeling the SU(3) triplets. One also
needs the charge operator $Q$ to make the assignment unique.
In SU(5), $Q = diag(1/3,1/3,1/3,0,-1)$, which takes the form
\begin{equation}
Q = \frac{1}{3} ( M_{12} + M_{34} + M_{56} ) - M_{9\,10}\,.
\eqnum{A.13}
\end{equation}
In the SO(10) spinor representation,
\begin{equation}
Q |\,\epsilon_1...\epsilon_5\,\rangle
= \left\{ \frac{1}{6} (\epsilon_1 + \epsilon_2 + \epsilon_3)
- \frac{\epsilon_5}{2} \right\} |\,\epsilon_1 \ldots \epsilon_5
\,\rangle\,. \eqnum{A.14}
\end{equation}
Putting all the above together one obtains
\begin{eqnarray}
|+ + + + +\,\rangle &= \nu^c\,,\ & |+ + + - -\,\rangle = e^+
\nonumber\\
|- - + + +\,\rangle &= u^c_1\,,\ & |- - + - -\,\rangle = d^c_1
\nonumber\\
|- + - + +\,\rangle &= u^c_2\,,\ & |- + - - -\,\rangle = d^c_2
\nonumber\\
|+ - - + +\,\rangle &= u^c_3\,,\ & |+ - - - -\,\rangle = d^c_3
\nonumber\\
|- - - + -\,\rangle &= \nu\,,\ & |- - - - +\,\rangle = e^-
\nonumber\\
|+ + - + -\,\rangle &= u_1\,,\ & |+ + - - +\,\rangle = d_1
\nonumber\\
|+ - + + -\,\rangle &= u_2\,,\ & |+ - + - +\,\rangle = d_2
\nonumber\\
|- + + + -\,\rangle &= u_3\,,\ & |- + + - +\,\rangle = d_3\,.
\eqnum{A.15}
\end{eqnarray}
Since we already know how to express the generators $M_{ab}$
as matrices, we can write the states as a single 32-dimensional
column vector which is projected into two 16-dimensional vectors
of positive and negative chirality by the operator
$P_\pm \equiv \frac{1}{2} (1\pm\chi)$. We find
\FL
\begin{equation}
\psi = (\nu^c\ u^c_1\ u^c_2\ u^c_3\ d_3\ d_2\ d_1\ e^-
\ u_3\ u_2\ u_1\ \nu\ e^+\ d^c_1\ d^c_2\
d^c_3)_L\,. \eqnum{A.16}
\end{equation}
In this paper, we studied two types of strings:
string-$\tau_{\rm all}$, where $\tau_{\rm all}$ is given by
Eq.~(10), and string-$\tau_1$, where $\tau_1$ can be any of
the generators in Eq.~(12). It is easy to see that in terms
of $M_{ab}, \tau_{\rm all}$ is written as
\begin{equation}
\tau_{\rm all} = \frac{1}{5}
(M_{12}+M_{34}+M_{56}+M_{78}+M_{9\,10})\,, \eqnum{A.17}
\end{equation}
and $|\,\epsilon_1 \ldots \epsilon_5\,\rangle$ is an eigenstate of
$\tau_{\rm all}$ with eigenvalue $\frac{1}{10} \sum_i \epsilon_i\,.$
For the left-handed fermions above, $\frac{1}{10} \sum_i \epsilon_i
= \frac{1}{2}$ for $\nu^c$, $\frac{1}{10}$ for $e^+, u, d, u^c$,
and $-\frac{3}{10}$ for $\nu, e^-, d^c$.
To study how $\tau_1$ act on the fermions, we write $\tau_{1+}$
and $\tau_{1-}$ defined in Eq.~(23) as a product of five Pauli
matrices using Eqs.~(A.4) and (A.6), and then replace the
matrices $\sigma_1$ and $\sigma_2$ by the usual raising and
lowering operators $\sigma_\pm=\frac{1}{2} (\sigma_1 \pm i\sigma_2)$.
One obtains
\begin{eqnarray}
\tau_{1+} & = & \frac{1}{2} (\tau^{2\alpha-1, 2\beta}
+ \tau^{2\alpha, 2\beta-1}) \nonumber\\
& = & I \ldots I \sigma_+ \sigma_3 \ldots \sigma_3
\sigma_+ I \ldots I \nonumber\\
&& + I \ldots I \sigma_- \sigma_3 \ldots \sigma_3
\sigma_- I \ldots I \eqnum{A.18}
\end{eqnarray}
and
\begin{eqnarray}
\tau_{1-} & = & \frac{1}{2} (\tau^{2\alpha-1, 2\beta-1}
- \tau^{2\alpha, 2\beta}) \nonumber\\
& = & I \ldots I \sigma_+ \sigma_3 \ldots \sigma_3
\sigma_- I \ldots I \nonumber\\
&& - I \ldots I \sigma_- \sigma_3 \ldots \sigma_3
\sigma_+ I \ldots I \eqnum{A.19}
\end{eqnarray}
where $\alpha,\beta=1, \ldots 5, \alpha < \beta$, and
the two $\sigma_\pm$ matrices in each term occupy the
$\alpha$th and $\beta$th positions from the left. Now one
can read off from the list of fermions above which particles
are mixed by a given $\tau_1$. For generators of the form
$\tau_{1+}$, one immediately finds that except for the case
$\alpha=4, \beta=5$, all mix leptons with quarks; when
$\alpha=4, \beta=5$, the generator mixes $(e^+, \nu^c),
(u_1^c, d_1^c), (u_2^c, d_2^c),$ and $(u_3^c, d_3^c)$.
For generators of the form $\tau_{1-}$, leptons are mixed
with quarks when $\alpha$ = 1, 2, or 3 and $\beta$ = 4 or 5.
\newpage
| train/arxiv |
BkiUdB7xK7IDF1Ddtq1d | 5 | 1 | \section{\protect\large \bf Introduction}
\hspace{2em}Since the discovery of high-T$_c$ superconductivity,$^1$
intensive theoretical work has been carried out to understand its properties.
Much of this effort was devoted to the analysis of two dimensional
electronic models,$^2$ in particular, the Hubbard$^3$ and
$t - J$ models.$^4$
In spite of their apparent simplicity, these models are very
difficult to study with analytical techniques. Actually,
there are no exact solutions of these models except in one dimension
(and even in this case, for the $t - J$ model only $J = 0$ and
$J = 2t$, i.e. the supersymmetric point, can be solved exactly).
In the parameter regime of interest for high-T$_c$
superconductivity,
these models can be regarded as strongly correlated electronic systems.
It is well known that most analytical methods, like Hartree-Fock $^5$
or RPA approximations, which are reliable for
weak coupling systems, have difficulties in dealing with
strongly correlated electrons. The same problem arises in
approximations like slave boson mean-field techniques.$^6$
In particular, for the
$t - J$ model it is not easy to decouple the charge and spin degrees
of freedom.
One should also note that in mean field calculations it is
necessary to make assumptions about ground state properties.
Numerical methods, on the other hand, are not biased by
any ``a priori" assumptions, and they have provided much of the reliable
information available for these models, as well as a useful check
of predictions formulated by analytical approximations.
Among the most widely used numerical techniques are the Monte Carlo
algorithms.$^7$ In particular, the version that uses the Hubbard-
Stratonovich transformation has been applied to the Hubbard
model$^8$ and several important results have been obtained.
An alternative to Monte Carlo techniques is the Lanczos method $^9$
which essentially gives the ground state of a given model for a
finite lattice.
{}From the ground state, we can compute all static and dynamical properties,
and in this sense, we obtain a complete characterization of a model at
zero temperature except for finite size effects.$^{10}$
This technique has provided important information about models of
correlated electrons.
For example, let us consider
a very recent work$^{11}$ where the $t - J$ model at quarter filling
has been studied.
In this work, strong signals of $d_{x^2-y^2}$
superconductivity close to the phase separation
border were found. These indications come from the study of
pairing correlations, Meissner effect and flux quantization
in the $4 \times 4$ lattice.
At quarter filling there are an equal number of holes and electrons
and we expect that at this point the finite size effects are
small.
However, if we consider the region physically relevant for
high-T$_c$ superconductivity which is close to half filling (doping
fraction $x \cong 0.10$), the number of
holes is very small (2 for the $4 \times 4$ lattice ) and then we
would expect
a weak signal for hole superconductivity.
Actually, most of the exact diagonalization studies of the $t - J$ model
on this lattice, using realistic couplings, have not found any
indications of superconductivity.
Then, in order to study the phase diagram of the $t-J$ model,
its properties, and the relation superconductivity-phase separation
in the physically
relevant region, it appears to be necessary to analyze larger clusters.
However, the 32 sites lattice with 4 holes requires the
diagonalization of a matrix of $\sim 2.25
\times 10^{10}$ states, which is unreachable with present-day computers.
Similar Hamiltonian matrix dimensions appear in many other situations.
In this paper we want to stress the need for developing new methods in
the context of diagonalization in a reduced basis set
in order to answer quantitatively the important
questions posed by models of high-T$_c$ superconductivity.
There are strong reasons why we should attempt to improve
diagonalization
schemes, rather than other approaches like
Monte Carlo methods. It is well known that
Monte Carlo simulations of fermionic models
present ``the minus sign problem",$^{12}$
which makes very difficult the study of these systems at the physically
interesting densities.
It is also well known that there are difficulties in the
analytical continuation procedure that is necessary to perform in
Monte Carlo calculations of dynamical
properties, and thus these techniques are not well developed.
The diagonalization procedures are free from the minus
sign problem, and as we mentioned at the beginning, all quantities
static and dynamical, can be computed from the ground state. Thus, it
is very important to extend these techniques to large clusters, and the
attempt discussed in this paper corresponds to a systematic expansion
of the Hilbert space.
\section{\protect\large \bf Systematic expansion of the basis set}
\hspace{2em}As it was described in the Introduction, the sizes of
the Hilbert
space necessary to study quantitatively problems relevant to high-$T_c$
superconductivity
are considerably larger than the dimensions that can
be reached with present computers (although the currently available
results for small clusters seem to be qualitatively reliable).
In this context, here we want to show that significant results can be
obtained by diagonalization of the Hamiltonian in a truncated
or reduced Hilbert space.$^{13}$ Some variations of this procedure have
been used for
many years in other fields such as chemical physics (see, for example,
Ref.$\:$14) where similar work has been recently discussed
by Wenzel and Wilson.$^{15}$
The method of diagonalization in a truncated basis is of course
justified only if a few coefficients $x_i$ of the ground state:
\begin{eqnarray}
\Psi_0 = \sum_{\scriptstyle i } x_i \phi_i,
\end{eqnarray}
\noindent
have significant weight. In some cases, fairly accurate properties of
the ground state
can be reached even with a small fraction of the total Hilbert space.
There are two questions that must be addressed to implement the proposed
technique:
\begin{enumerate}
\item It is necessary to choose an appropriate basis $\{\phi_i\}$ according to
the
physics of the problem. For example,
\begin{itemize}
\item real space $S^z$ representation for the $t-J$ model,
\item momentum space representation for the one-band Hubbard model in
weak coupling.
\end{itemize}
\item The algorithm must be able to find the most significant states
that contribute to the ground state wave function.
\end{enumerate}
The outline of the method we have developed and present in this paper,
which we call ``systematic
expansion of the Hilbert space'' (SEHS), is the following:
\begin{enumerate}
\item start from as few as possible states chosen according to the
expected behavior of the system (knowing quantum numbers of the ground
state greatly simplifies the work);
\item at each step $i$ expand the Hilbert space by applying the
Hamiltonian, or at least part of it, to the current set of states;
\item diagonalize in the new enlarged Hilbert space using the Lanczos
method;
\item retain the states with the largest weight, such that the
dimension of the Hilbert space is $N_i = \lambda N_{i-1}$, $1 < \lambda
\leq 2$ (``slow growth'' approach);
\item go back to step 2 until convergence in the physical quantities
is achieved,
or until the largest available dimension in the computer is reached.
\end{enumerate}
In an ideal situation, the states chosen at the starting point should
correspond to those
that carry most of the weight in the exact ground state. For some sets of
parameters (couplings, densities)
it is possible to guess these states.
However, different sets of parameters may have different behaviors,
and usually it is not possible to predict at which point the crossover
between them will occur. For example, in the $t-J$ model, for $J \gg t$ the
holes are bound together in pairs, so we can take as starting point
states where the holes are in nearest neighbor sites. On the
other hand, for $J \ll t$ the holes are not bound and move
around independently of each other and then it is not correct to
take the same states as before as the initial state for the iterations.
In this situation, the ``pruning" of the Hilbert space retaining
the most weighted states as indicated in point 4 is essential to
improve or correct the initial starting set of states. As we
discuss below, this procedure effectively works as a systematic method to
obtain and improve variational states. Moreover, it allows the
dimension of the Hilbert space to grow at a slow rate and the
behavior of the energy results smoother than in the case of the
straight application of the Hamiltonian. Below we will apply the
proposed
method to several cases relevant to theories of high-T$_c$ superconductors.
\section{\protect\large \bf Study of the $t-J$ model}
\hspace{2em}Let us apply the SEHS method
to the $t-J$ model$^4$ which is
defined by the Hamiltonian:
\begin{eqnarray}
H = - t \sum_{\scriptstyle <i j>, \sigma}
(\tilde{c}^{\dagger}_{i,\sigma}\tilde{c}_{j,\sigma} +
\tilde{c}^{\dagger}_{j,\sigma}\tilde{c}_{i,\sigma})
+ J \sum_{ <i j>}
({\bf S}_{i}\cdot {\bf S}_{j} - \frac{1}{4} n_{i} n_{j}),
\end{eqnarray}
\noindent
where the notation is standard. The first term describes the
hopping of holes or kinetic energy, while the second one corresponds to the
antiferromagnetic Heisenberg interaction. In this model the size of
the Hilbert space grows roughly as $3^{N_s}$, where $N_s$ is the number of
sites of the lattice, after taking into account the constraint of no
double occupancy.
In two dimensions (2D), this model has been studied at all fillings on
the $4 \times 4$ cluster.$^{10}$
Up to 2 holes, clusters of up to 26 sites have also been considered.
$^{16}$
First, let us briefly discuss the application of this
method
to the two dimensional $t-J_z$ model$^{17}$ which is obtained from the $t-J$
model by eliminating the spin exchange term in the Heisenberg
interaction.
Consider the case of one hole. In the limit of $J_z/t \gg
1$, the ground state of this model consists of a state in
which the hole is located at an arbitrary site surrounded by
an otherwise perfect N\'{e}el state.
In this limit the dimension of the Hilbert space
needed to get the physics of the problem is just equal to one
(plus all states translationally equivalent).
Now, as $J_z/t$ is reduced to the most interesting region, i.e. $J_z/t
\leq 1$, the hole gains kinetic energy at the expense of magnetic
energy and starts to move away from its initial position. As the hole
hops, it leaves behind a trail of overturned spins called a
``string".$^{18}$
As $J_z/t$ is
lowered, one must take into account longer and longer strings.
However the important string excitations are still of finite length,
and then in this case it is enough to keep a fraction of the total Hilbert
space
to describe it.
For this model, the application of the Hamiltonian, i.e. the hopping
term, to expand the Hilbert
space at each step has a direct physical meaning.
As we have shown in a previous paper, $^{17}$ it is possible to converge
to the ground state energy with several digits of accuracy
by retaining a small fraction of the full Hilbert space. As an example,
in Table I, the energy of the system for two holes is shown
for a cluster of 50 sites and $J_z/t = 0.3$ as a function of the
dimension of the Hilbert space. It is clear that the new technique works
very
well in this case. For more details see Ref. 17.
Let us now consider the $t - J$ model with the full Heisenberg
interaction (Eq. (2)). In this case, even in the absence of
holes, the ground state is characterized by the presence of
spin wave excitations that reduce the antiferromagnetic order from
its N\'{e}el (classical) value. Thus, in principle, we not only need to
physically
describe the modification of the spin background in the vicinity
of the holes, but also the spin exchanges that take place at arbitrary
distances from the holes which contribute significantly to the spin
background. This qualitative difference between the $t-J$ and $t-J_z$
models
can be detected by measuring
the distribution of weights $S(x)$ defined as the sum of the
weights $\mid x_i \mid ^2$ belonging to the interval $\left[ x, x +
\Delta \right]$. In Fig.1, we show $S(x)$ in the exact ground
state of the $4 \times 4$ lattice with two holes at $J_z = 0.6$ and $J=0.6$
(in general we take t=1), for the $t - J_z$ (Fig.1a) and $t - J$
(Fig.1b) models, respectively. It can be seen that in the latter, there is more
weight for very small absolute values of the coefficients $x_i$
of the ground state $\Psi_0$ (Eq. (1)).
Let us start the expansion of the Hilbert space from the same sets of
states considered for the $t-J_z$ model. At each step, the Hilbert
space is expanded by the application of both the hopping term and
the spin exchange term of the Heisenberg interaction.$^{19}$
In the language of perturbation theory, this is like a double expansion
around the Ising limit ($t-J_z$) with static holes, namely one or two
holes in an otherwise perfect N\'{e}el state. The expansion
with the spin exchange term of the Heisenberg interaction could be
regarded as a perturbation in the spin anisotropic parameter.
In Figs. 2-5, we show results for the $4 \times 4$ lattice .
These can be compared with results for the
exact ground state which can be easily computed.
In Fig. 2, the energy is shown as a function of the
dimension of the basis set, for two holes at $J = 0.2$. The
energies obtained with the ``truncation'' procedure (dot-dashed line)
are much better than the energies obtained without it (dashed line)
namely diagonalizing at step 3 of the method, but without truncating in
step 4. As explained before, this improvement helps in discarding states
with very small weight.
Finally, both
are much better than the energies obtained at each iteration
of the conventional Lanczos algorithm (full line).
In Fig. 3, the overlap between the variational wave functions
in the truncated Hilbert space with the exact ground state are
shown for both procedures with (dot-dashed line) and without
(dashed line) the elimination of the less weighted states or
``truncation''.
In Fig. 4, the evolution of the
hole-hole correlations at the {\em maximum} distance in this lattice
is shown as a function of the dimension of the Hilbert space. It can be
seen that the convergence with the ``truncation'' procedure is much faster
than without it, even for correlation functions. The notation in these
figures is the
same as for Fig. 2. A similar behavior was also obtained for the
spin-spin correlation at the maximum distance.
Finally, to complete the preliminary study on the
$4 \times 4$ lattice, we show in Fig. 5 the energies obtained
with the full basis set expansion procedure starting from the
N\'{e}el state (curve labeled 0); from the N\'{e}el state and all the
states obtained from it by one spin exchange (curve labeled 1);
from the N\'{e}el state and all the states obtained from it by two
spin exchanges (curve labeled 2); and so on. The energies at the
beginning of each set correspond to the variational states
discussed in Ref. 20. We see that the energies
obtained with the new method starting from the N\'{e}el state are
considerably better,
even for a very small number of iterations, than those corresponding
to Dagotto-Schrieffer's variational states. As a conclusion, even though we
cannot
reach the ground state as accurately as we did for the $t-J_z$ model, we still
can obtain a very good variational state compared with other
states discussed in the literature for finite lattices.
Now let us discuss clusters that cannot be studied with the conventional
Lanczos approach for lack of enough memory in present-day computers.
We will show results obtained for the
$t - J$ model on the $6 \times 6$ lattice with two holes, and $J = 0.4$.
The dimension of the Hilbert space is, in this case, $2.55 \times 10^9$ states
using translational and spin reversal symmetries.
In Fig. 6, the energy is plotted as a function of the dimension of
the Hilbert space (in a logarithmic scale).
With a full line we show the energies obtained at each step of the conventional
Lanczos algorithm, while with a dashed line we plot the energies obtained
expanding the Hilbert space by applying the Hamiltonian, and at each step
diagonalizing in the enlarged space using the Lanczos method, i.e.
steps 2 and 3 of
the method described
above. Finally, with circles and diamonds, we show the points
obtained by retaining the most weighted states, i.e. step 4 of our
method. The long-dashed line in zig-zag shows the order in which
every point is obtained starting with the circle at the top.
It is clear that a better convergence is achieved with the full
procedure of the SEHS method. After reaching the maximum
dimension that can be handled with the available computer, it is also
possible to use an extrapolation procedure to extract results at the dimension
of the total Hilbert space, but we have not attempted such an analysis
in the present paper.
(The energy for this particular system has been estimated
with a Green's Function Monte Carlo technique$^{21}$ to be
near $\sim -20.0$.) In principle, one should also compute other
physical quantities of interest at each coupling,
and then also extrapolate them to the full dimension.
Presumably, we can attribute the slow convergence of the ground state
energy with the size of the Hilbert space to the highly nontrivial (and
fluctuating) spin-1/2
background. Then, the convergence is not going to deteriorate if we
put more holes on
the lattice. On the other hand, Monte Carlo algorithms typically encounter
increasingly severe problems as the number of holes is increased, at
least if one remains close to half-filling.
The number of off-diagonal transitions for
both the hopping (dashed line) and the exchange (full line)
parts of the Hamiltonian as a function of the number of states
included in the basis set can be computed
at each step.
The result is that successive sets generated
during the process of enlargement of the Hilbert space are
increasingly more interacting, i.e. the Hamiltonian matrix becomes
more dense (See, for example, Fig. 11 in Ref. 13.)
\section{\protect\large \bf Application to the one-band Hubbard model}
\hspace{2em}The one-band Hubbard model is defined
by the Hamiltonian:
\begin{eqnarray}
H = - t \sum_{\scriptstyle <i j>, \sigma}
(c^{\dagger}_{i,\sigma}c_{j,\sigma} +
c^{\dagger}_{j,\sigma}c_{i,\sigma})
+ U \sum_{i} n_{i,\uparrow}n_{i,\downarrow},
\end{eqnarray}
\noindent where the notation is standard.
The size of the Hilbert space grows
as $4^{N_s}$, and thus it is even more difficult to study than the $t-J$
model from a numerical point of view.
In this case, the largest lattice considered in the literature
is the $4 \times 4$ lattice
for all dopings.$^{10,22}$
In momentum space, the Hamiltonian of the Hubbard model takes the
form:
\begin{eqnarray}
H = \sum_{\scriptstyle {\bf k }, \sigma} \epsilon ({\bf k })
c^{\dagger}_{ {\bf k } ,\sigma} c_{ {\bf k } ,\sigma} +
+ U \sum_{ {\bf k_1,k_2,k_3}} c^{\dagger}_{ {\bf k_1 },\uparrow}
c_{ {\bf k_2},\uparrow} c^{\dagger}_{ {\bf k_3},\downarrow}
c_{ {\bf k_1-k_2+k_3},\downarrow},
\end{eqnarray}
\noindent
where each {\bf k } runs over the Brillouin zone. The single
particle energies are given by $\epsilon ( {\bf k }) = - 2 t
(cos( k_x ) + cos(k_y ))$.
In the absence
of Coulomb repulsion, the model reduces to a tight binding
model which is easily solved. The total energy is the sum of
the single particle energies for all the momentum ${\bf k }$
up to the Fermi surface. Here, we have to distinguish between
two cases: the closed shell, in which the last shell is completely
occupied; and the open shell in which the last shell is partially
occupied. In the former case the ground state is not degenerate
while in the latter the degeneracy can be very large.
In the following,
we concentrate on the $6 \times 6$ cluster with 18 (9$\uparrow$ and
9$\downarrow$) and 26 (13$\uparrow$ and 13$\downarrow$) electrons
which correspond to $closed$ shell situations.
The dimensions of the Hilbert space for some closed shell cases
in this cluster are: for 10 electrons, $3.95 \times 10^{9}$; for 18 electrons,
$2.46 \times 10^{14}$; and for
26 electrons, $1.48 \times 10^{17}$ well beyond the reach of
techniques that fully diagonalize the full Hilbert space of the problem.
For the closed shell situations, our initial Hilbert space consists
of only one state, which is the ground state of the $U = 0$ case
(remember that we are working in momentum space).
The Hilbert space is expanded by applications of the second term
of Eq. (4), which contains the off diagonal transitions.
These terms create and annihilate pairs of
electrons in such a way that the total momentum is conserved.
In some other approaches the Hamiltonian is expanded through the
creation of single pair electron-hole excitations$^{23}$ but then
the total momentum is not conserved.
In the spirit of the general
procedure outlined in Section 2, we expand the Hilbert space by applying
the whole second term of Eq. (4). (Another possibility, which we have not
yet fully explored, is to expand the Hilbert space by taking only transitions
between the shells at both sides of the Fermi level, and then increase
successively the number of shells involved.)
The expansion of the Hilbert space by application of the Coulomb
term could also be considered as a weak-coupling perturbation
expansion in
a parameter which is proportional to $U$, but unlike other
perturbation schemes,$^{24}$ our procedure remains variational
in the sense that the energy is always an upper bound to the
exact ground state energy.$^{25}$
In Figs. 7 and 8, we show the convergence of the energy as a
function of the dimension of the Hilbert space for 18 and 26
electrons respectively, and for several values of $U$.
The energies are measured in units of $t$ as usual, and they
have been shifted in order to fit them into the same
plot and in order to compare their convergence.
It can be observed that the convergence is faster the fewer the electrons, and
as expected, the convergence is faster for smaller values of $U$.
For example, for the case of 26 electrons, for $U = 2$ we obtain
a value of -47.907, in good agreement with the Monte Carlo estimate
$^{26}$ of -47.87$\pm$0.05, i.e. the new technique reaches the same
accuracy as Monte Carlo methods.
The most important features in these plots are the presence of
discontinuities in the derivative of the energy,
and a ``wrong" concavity of the curves
(compared for example with the curvature in Figs. 5 and 6 of the
$t-J$ model). We do not have an explanation for this behavior,
although perhaps the long-range nature of the Coulomb interaction in
momentum space may matter.
The wrong curvature of the plots makes it difficult to assess
the convergence of the energy and to perform an extrapolation
procedure.
The points at which there are discontinuities in the derivative
are the points obtained by successive application of the
Hamiltonian starting from the initial state. All the other points
are obtained by pruning these
Hilbert spaces, and by applying the Hamiltonian to the reduced
spaces.
The somewhat strange behavior of the energy vs. the dimension of
the Hilbert space is an artifact of the momentum representation
chosen, and perhaps a manifestation of the shell structure
of the tight binding limit.
In the interval considered, i.e. $U \leq 4$, we found that the convergence
of the energies
obtained by working in the momentum representation is much faster
than the one obtained by working in real space. Presumably, the
opposite is true for larger values of $U$.
Finally, in Table II we provide comparisons of our estimates
with the results obtained using Quantum
Monte Carlo techniques,$^{26,27}$ as well as the results
obtained with a stochastic implementation of the modified Jacobi
method$^{28}$ also referred to as ``stochastic diagonalization'' (SD).
To obtain the results quoted in this Table, $N_R \sim 2 \times 10^4 $
important states were included in the SD calculation and a CPU
time of $\sim 10^4 $ seconds (for the $4 \times 4$ lattice) was required.
This CPU time is also what is required by our method for
$N_h \sim 10^6$. However, as reported in Ref. 28, and as it can be
seen in Table II, the energy is not yet converged and presumably $N_R$
has to be increased by a factor of $\sim 10$ in order to obtain the
same accuracy as our results. This translates to a factor of $\sim 100$
in the total CPU time, since in the SD algorithm the CPU time grows
quadratically with $N_R$. Besides, from the results reported in Ref.
28, it is also evident that for the SD method the convergence is
more difficult for larger values of the Coulomb repulsion.
In summary, it seems that at least in its current implementation, the
SD method is more expensive than the SEHS method reported in this
paper for a given accuracy.
\section{\protect\large \bf Application to the three-band Hubbard model}
\hspace{2em}Finally, and for completeness,
we briefly consider the three-band Hubbard model which contains
the Coulomb on-site repulsion for both the copper and
oxygen sites ($U_d$ and $U_p$ respectively), the energies of each ion
($e_d$ and $e_p$ for copper and oxygens ions), and a Coulomb
repulsion between copper and oxygens ions, $V$.$^{29}$
We study the $\sqrt{8} \times \sqrt{8}$ lattice (24 sites between
oxygens and
coppers) with
two doped holes (10 fermions), and the following set of parameters:
$U_d = 7$, $U_p = 0$, $e_p - e_d = 1.5$ and $V = 3$. As the initial
basis set, we took all the states with all the Cu sites having single
occupancy, and the remaining two holes located in O sites (also single
occupied). This is a good starting Hilbert space for the case
$V = 0$, but as the algorithm itself has shown it is not appropriate
for all values of the parameters.
In Fig. 9, we show the results obtained using the
Hilbert space expansion procedure. The dashed lines show the order
in which these points were obtained starting from the circle at the
top right in the same way as was explained in Fig. 6. In Fig. 10,
the best points
in the set of results shown in Fig. 9 are plotted with circles. In a
second stage, once we have reached $\sim 10^6$ states, we go
all the way back (points indicated with full
diamonds), finding that the initial guess was not appropriate (i.e. the
states with the highest weights were not those used in the starting Ansatz),
and then we increase the dimension of the basis set again
(empty squares). It can be seen that this last set of points
behaves very smoothly and the final part of the curve is fairly
flat indicating a reasonable convergence. From this set of
states, in principle, we could compute all quantities of interest and
eventually
extrapolate them to the full Hilbert space.
However, one should also notice that in this case the largest dimension that
we have considered ($\sim 2.5 \times 10^6$) is ``only" two orders of magnitude
smaller than the dimension of the full Hilbert space, and probably that
is the reason for the good convergence of the results.
In Fig. 11 we compare the energies for $V = 0$ and $V = 3$ as a
function of the dimension of the Hilbert space. The energies have been
shifted for the sake of comparison. It can be seen that the
convergence is better for the $V = 3$ case. For $V = 0$, following the
Zhang-Rice construction,$^4$ one can map this model to the one-band
$ t - J $ model. It is then reasonable to assume that, as in this
model, the spin background is responsible for the slow convergence.
The same pattern of convergence was also found for the other set of
parameters we have studied: $U_p = 3$, $e_p - e_d = 4$,
and $V = 0$, $V = 3$, and the same value of $U_d = 7$. In this
case, for $V = 3$ the convergence
is faster than for $V = 0$, reflecting the fact that it is easier for
the algorithm to find the most relevant states which contains double
occupied Cu sites.
Finally, we show in Fig. 12 the spin-spin correlation
at the maximum distance on the lattice, and the density of holes in
Cu sites as a function of the dimension of the Hilbert space for
the set of parameters $U_d = 7$, $U_p = 0$, $e_p - e_d = 1.5$ and
$V = 3$. These curves
indicate also a reasonable convergence. For this set of parameters
we obtain $n_{Cu} = 0.555$, while for $U_p = 3$, $e_p - e_d = 4$,
$n_{Cu} = 1.088$, indicating the presence of two different regimes
for large $V$. This result might be relevant to some speculation
regarding the nature of pairing and phase separation in $Cu-O$
planes.$^{30}$
In any case, it is quite encouraging to observe that the new technique
may work well in the realistic (and complicated) case of the three-band
Hubbard model.
\section{\protect\large \bf Discussion and conclusions}
\hspace{2em}
The procedure described in this paper can be regarded as a method to generate
and/or
improve variational wave functions. In the first place, it should be noted
that since no approximations are done on the Hamiltonian, and since
we work in a reduced Hilbert space, the energies obtained with
this procedure are rigorous upper bounds to the exact ground
state energies. The application to the $t-J$ model is one example
in which the initial set of states is ``corrected" by this algorithm.
In this case, a direct comparison with a variational
state was also given (see also Ref. 19).
Another application in which the elimination at each step of the
least weighted states leads to an improvement or to a correction
of the initial guess is the case of the three-band Hubbard model.
In this case, the initial state depends on the parameters
that determine the $Cu$ or $O$ occupancy when the nearest
neighbor Coulomb repulsion is large enough. In general, we believe that
the technique is promising and may compete against more standard Lanczos
and
Quantum Monte Carlo methods, at least for some particular Hamiltonians
and parameters. A clear example is the $t-J_z$ model in which the new
method has provided the more accurate results reported in the literature
thus far.$^{17}$
For the systems where we cannot arrive at a good approximation for the
ground state due to the slow rate of convergence of the results (for example
the $t-J$ model seems to converge only logarithmically), one should
resort to some extrapolation procedure to the full Hilbert dimension.
In this sense, we are in the same situation as the zero temperature
(Green's function or random walk) Monte Carlo algorithms that
cannot reach convergence before the noise becomes very high.$^{21}$
Besides the possible applications of this reduced Hilbert space
approach as indicated above, there are other situations that can also
be studied with the SEHS method. One of them is the quarter-filled
$t-J$ model on the 26 sites lattice, which is interesting to
study in order to analyze the finite size dependence of the results obtained in
Ref. 11 in the context of superconductivity in the $t-J$ model.
The method can also be applied to
coupled planes $t-t_\perp-J$
model.$^{31}$ For this system, one could start from the best states
of the ground state of each plane separately and then expand the
basis set by application of the interplane hopping term of the
Hamiltonian. This is equivalent to an expansion around
$t_\perp / t = 0$.
Finally, we want to comment that there are other algorithms that also
deal with truncated Hilbert spaces besides the stochastic
diagonalization approach and the presently described technique.
In an already mentioned paper,$^{23}$ the Hubbard model was studied
in momentum space with a truncation technique using concepts
of renormalization group theory.
Another stochastic truncation method has recently
been developed for the $Z_2$ gauge model.$^{32}$
The computational effort of the SEHS method of systematic
expansion of the Hilbert space grows roughly linearly with $N_h$, and
currently $N_h \sim 10^6$ for present-day computers.
These other methods use a smaller
size of the basis set, but the CPU time grows
as ${N_h}^3$ for the methods of references [23] and [32], and
quadratically in $N_h$ for the stochastic diagonalization algorithm.
Summarizing, a new algorithm has been discussed that has several
of the advantages of the Lanczos approach (specially the possibility of
studying dynamical responses), but that can be applied to large
clusters. The method works remarkably well in some special cases,
while in general it is competitive with other more standard algorithms.
\section{\protect\large \bf Acknowledgements}
\hspace{2em}
We thank Adriana Moreo for providing the Monte Carlo results
used in this paper, and for useful conversations. E. D. thanks
the Office of Naval Research for its partial support under
grant ONR-N00014-93-1-0495. J. R. wishes to acknowledge the support
from High Performance Computations grant from Vanderbilt
University.
Most of the calculations were done using the Cray YMP at the
Supercomputer Computations Research Institute in Tallahassee,
Florida. The research was sponsored in part by the U. S. Department
of Energy under contract No. DE-AC05-84OR21400 managed by Martin
Marietta Energy Systems, Inc.
\newpage
\section{\protect\large \bf References}
\begin{enumerate}
\item J. G. Bednorz and K. M\"uller, {\em Z. Phys.} {\bf B 64}, 189 (1986).
\item P. W. Anderson, {\em Science} {\bf 235}, 1196 (1987).
\item J. Hubbard, {\em Proc. R. Soc. London, Ser.} {\bf A 276}, 238 (1963).
\item F. Zhang and T. M. Rice, {\em Phys. Rev.} {\bf B 37}, 3759 (1988).
\item See for example, J. A. Verg\'{e}s, E. Louis, P. S. Lomdahl, F.
Guinea and A. R. Bishop, {\em Phys. Rev. } {\bf B 43}, 4462 (1989), and
references therein.
\item G. Kotliar and A. E. Ruckenstein, Phys. Rev. Lett. {\bf 57}, 1362 (1986).
\item W. von der Linden, {\em Phys. Rep. } {\bf 220 }, 53 (1992).
\item A. Moreo, D. J. Scalapino, R. L. Sugar, S. R. White and N. E.
Bickers, {\em Phys. Rev.} {\bf B 41}, 2313 (1990); and references therein.
\item B. N. Parlett ,{\em ``The symmetric eigenvalue problem"},
(Prentice Hall, 1980).
\item E. Dagotto, A. Moreo, F. Ortolani, D. Poilblanc
and J. Riera, {\em Phys. Rev. } {\bf B 45}, 10741 (1992).
\item E. Dagotto and J. Riera, {\em Phys. Rev. Lett.} {\bf 70},
682 (1993).
\item E. Y. Loh, et al., Phys. Rev. {\bf 41}, 9301 (1990).
\item An earlier discussion of this method was given in J. Riera, in
``Proceedings of the Mardi Gras '93 Conference on Concurrent Computing
in the Physical Sciences", World Scientific, 1993.
\item P. J. Knowles, { \em Chem. Phys. Letters} { \bf 155}, 513
(1989); P. J. Knowles and N. C. Hardy, { \em J. Chem. Phys.} { \bf 91},
2396 (1989).
\item W. Wenzel and K. G. Wilson, Phys. Rev. Lett. {\bf 69}, 800 (1992).
\item D. Poilblanc, J. Riera, and E. Dagotto, preprint, (1993).
\item J. Riera and E. Dagotto, {\em Phys. Rev. } {\bf B 47}, xxxxx (1993).
\item W. F. Brinkman and T. M. Rice, {\em Phys. Rev.} {\bf B 2}, 1324 (1970);
B. I. Schraiman and E. D. Siggia, {\em Phys. Rev. Lett.} {\bf 60},
740 (1988).
\item In a semi-analytical approach (S. Trugman, {\em Phys. Rev.} {\bf B 37},
1597 (1988); {\em Phys. Rev.} {\bf B 41}, 892 (1990)), the basis set was
expanded by
the application of the hopping term only (and the second neighbor double
hopping term present in the model considered by Trugman). See also J.
Inoue and
S. Maekawa, Prog. Theor. Phys., Suppl. {\bf 108}, 313 (1992).
Typically,
the Hilbert space was expanded to include a few hundred states. This is a very
small quantity compared with the $\sim 10^6$ one can reach with our
method, but Trugman's results are valid for the bulk limit. So, we obtain a
much better variational state but at the cost of limiting ourselves to
finite lattices.
\item E. Dagotto and J. R. Schrieffer, {\em Phys. Rev. }{\bf B 43},
8705 (1991).
\item M. Boninsegni and E. Manousakis, preprint (1992).
\item G. Fano, F. Ortolani and A. Parola, {\em Phys. Rev.} {\bf B 42}, 6878
(1990).
\item S. R. White, {\em Phys. Rev.} {\bf B 45}, 5752 (1992).
\item J. Gal\'{a}n and J. A. Verg\'{e}s, {\em Phys. Rev. } {\bf B 44},
10093 (1991).
\item A numerical, but more conventional, weak-coupling perturbative study
on the $6 \times 6$ lattice was reported by B. Friedman,
{ \em Europhysics Letters} { \bf 14}, 495 (1991).
\item A. Moreo, private communication.
\item N. Furukawa and M. Imada,
{\em J. Phys. Soc. Jpn.} {\bf 61}, 3331 (1992).
\item H. De Raedt and W. von der Linden, {\em Phys. Rev.} {\bf B 45},
8787 (1992); H. de Raedt and M. Frick, {\em Phys. Rep.}, to appear. See
also
P. Prelovsek, and X. Zotos, preprint.
\item V. Emery, {\em Phys. Rev. Lett.} {\bf 58}, 2794 (1987)
\item C. Varma,
S. Schmitt-Rink and E. Abrahams, {\em Solid State Commun.} {\bf 62}, 681
(1987).
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121 (1988).
\item C. J. Hamer and J. Court, preprint (1992).
\end{enumerate}
\newpage
\noindent
{\bf Table I}
\vskip 0.8cm
\begin{tabular}{rr} \hline\hline
${\rm H_D}$ & $E_{2h}$ \\ \hline
234 & -18.707940 \\
696 & -18.882805 \\
6204 & -19.026339 \\
18416 & -19.052528 \\
52672 & -19.066660 \\
106435 & -19.074957 \\
212486 & -19.079975 \\
673640 & -19.083531 \\
980681 & -19.084816 \\
1502829 & -19.085503 \\
2249454 & -19.085857 \\ \hline\hline
\end{tabular}
\noindent
\vskip 2cm
{\bf Table II}
\vskip 0.8cm
\begin{tabular}{llr} \hline\hline
method & 18 electrons & 26 electrons \\ \hline
QMC & -41.87$\pm$0.10 & -41.98$\pm$0.15 \\
SEHS & -41.69 & -41.49 \\
SD & -41.45 & -40.77 \\ \hline\hline
\end{tabular}
\newpage
\centerline {\bf TABLE CAPTIONS}
\vskip 2truecm
\noindent
{\bf Table I}
\noindent
Energy $E_{2h}$ of two holes in the ${\rm t-J_z}$ model, as a function
of the size of the Hilbert space, ${\rm H_D}$, for a cluster of
50 sites, and coupling $J_z/t = 0.3$.
\vskip 2truecm
\noindent
{\bf Table II}
\noindent
Comparison between ground state energies (in units of $t$) obtained
with the present method (SEHS), Quantum Monte Carlo (QMC), and
Stochastic Diagonalization (SD), for the $6 \times 6$ lattice and
$U = 4$.
\newpage
\centerline {\bf FIGURE CAPTIONS}
\vskip 2truecm
\noindent
{\bf Figure 1}
\noindent
Distribution of weights S(x) a) for the $t - J_z$ model, b) for
the $t - J$ model on the $4 \times 4$ lattice with 2 holes and $J/t = 0.6$.
\vskip 1truecm
\noindent
{\bf Figure 2}
\noindent
Energy vs dimension of the Hilbert space for the $4 \times 4$
lattice with two holes, J = 0.4. The full curve corresponds to
the energies obtained at each step of the conventional Lanczos
iteration. The dot-dashed (dashed) corresponds to the procedure
indicated in Sec. 3 with (without) including step 4.
\vskip 1truecm
\noindent
{\bf Figure 3}
\noindent
Overlap between the exact ground state and the states
generated during the procedure of expansion of the Hilbert space.
The meaning of the curves are as in Fig. 2.
\vskip 1truecm
\noindent
{\bf Figure 4}
\noindent
Hole-hole correlations at the maximum distance on the $4 \times 4$
lattice. The meaning of the curves are the same as for Fig. 2.
\vskip 1truecm
\noindent
{\bf Figure 5}
\noindent
Expansion of the Hilbert space starting from different
initial basis sets for the $4 \times 4$ lattice with 2 holes and J=0.2.
\vskip 1truecm
\noindent
{\bf Figure 6}
Energy vs dimension of the Hilbert space for the $6 \times 6$
lattice, 2 holes, J=0.4.
\noindent
\vskip 1truecm
\noindent
{\bf Figure 7}
\noindent
Energy of the Hubbard model on the $6 \times 6$ lattice with
18 electrons vs dimension of the Hilbert space. The asterisk
indicate the Monte Carlo estimates.
\vskip 1truecm
\noindent
{\bf Figure 8}
\noindent
Energy of the Hubbard model on the $6 \times 6$ lattice with
26 electrons vs dimension of the Hilbert space. The asterisk
indicate the Monte Carlo estimates.
\vskip 1truecm
\noindent
{\bf Figure 9}
Energy of the three-band Hubbard model on the
8 cells square lattice as obtained by application of the SEHS
procedure.
\vskip 1truecm
\noindent
{\bf Figure 10}
Energy of the three-band Hubbard model on the
8 cells square lattice vs the dimension of the Hilbert space.
The open circle points correspond to the filled square points
of Fig. 12. After reaching $\sim 10^6$ states, we truncate
the Hilbert space in successive steps (diamonds), and then
we start a new expansion of the basis set (squares).
\vskip 1truecm
\noindent
{\bf Figure 11}
Energy of the three-band Hubbard model on the
8 cells square lattice vs the dimension of the Hilbert space
for different values of the intersite Coulomb repulsion $V$.
\vskip 1truecm
\noindent
{\bf Figure 12}
Spin-spin correlation at the maximum distance and density of
holes at Cu sites for the three-band Hubbard model on the
8 cells square lattice vs the dimension of the Hilbert space.
\vskip 1truecm
\end{document}
| train/arxiv |
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BkiUdizxaKgQGdQApUFb | 5 | 1 | "\\section{Introduction}\\label{intro}\n\nPolar Ring/Disk Galaxies (PRGs) are multi-spin systems. Th(...TRUNCATED) | train/arxiv |
BkiUc_Y5qhLA5_F8juOy | 5 | 1 | "\\section{Introduction}\n\n\nLet $\\Gamma$ be \na finitely generated group \nand let $S$ be \na fin(...TRUNCATED) | train/arxiv |
BkiUbMU5qWTD6essZRcR | 5 | 1 | "\\section{Introduction}\nThis demo file is intended to serve as a ``starter file''\nfor IEEE confer(...TRUNCATED) | train/arxiv |
BkiUddQ4uBhi4In9Ya_Y | 5 | 1 | "\\section{Heavy hadron lifetimes}\n\nLifetimes are fundamental properties of particles, which conne(...TRUNCATED) | train/arxiv |
BkiUdwHxK0fkXQzmMBqp | 5 | 1 | "\\section{Introduction}\n\nIn 1975, T. Koornwinder (\\cite{Koor75}) introduced a non--trivial metho(...TRUNCATED) | train/arxiv |
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Top 30B token SlimPajama Subset selected by the Cleanliness rater
This repository contains the dataset described in the paper Meta-rater: A Multi-dimensional Data Selection Method for Pre-training Language Models.
Code: https://github.com/opendatalab/Meta-rater
Dataset Description
This dataset contains the top 30B tokens from the SlimPajama-627B corpus, selected using the Cleanliness dimension of the PRRC (Professionalism, Readability, Reasoning, Cleanliness) framework. Each document in this subset is scored and filtered by a ModernBERT-based rater fine-tuned to assess the formatting, completeness, and absence of noise or irrelevant content in the text.
- Source: SlimPajama-627B Annotated Dataset
- Selection: Top 30B tokens by PRRC-Cleanliness score
- Quality metric: Cleanliness (0–5 scale, see below)
- Annotation coverage: 100% of selected subset
Dataset Statistics
- Total tokens: 30B (subset of SlimPajama-627B)
- Selection method: Top-ranked by PRRC-Cleanliness ModernBERT rater
- Domains: Same as SlimPajama (CommonCrawl, C4, GitHub, Books, ArXiv, Wikipedia, StackExchange)
- Annotation: Each document has a cleanliness score (0–5)
Cleanliness Quality Metric
Cleanliness evaluates the formatting, completeness, and absence of noise or irrelevant content in the text. Higher scores indicate well-formatted, complete, and clean data, while lower scores reflect noisy, incomplete, or poorly formatted content.
- 0–1: Serious or obvious issues affecting fluency or completeness
- 2–3: Some problems, but not seriously affecting reading
- 4–5: Minor or no problems; text is clean and well-formatted
Scores are assigned by a ModernBERT model fine-tuned on Llama-3.3-70B-Instruct annotations, as described in the Meta-rater paper.
Annotation Process
- Initial annotation: Llama-3.3-70B-Instruct rated 500k+ SlimPajama samples for cleanliness
- Model training: ModernBERT fine-tuned on these annotations
- Scoring: All SlimPajama documents scored by ModernBERT; top 30B tokens selected
Citation
If you use this dataset, please cite:
@article{zhuang2025meta,
title={Meta-rater: A Multi-dimensional Data Selection Method for Pre-training Language Models},
author={Zhuang, Xinlin and Peng, Jiahui and Ma, Ren and Wang, Yinfan and Bai, Tianyi and Wei, Xingjian and Qiu, Jiantao and Zhang, Chi and Qian, Ying and He, Conghui},
journal={arXiv preprint arXiv:2504.14194},
year={2025}
}
License
This dataset is released under the same license as the original SlimPajama dataset. See the original SlimPajama repository for details.
Contact
- Project Lead: Ren Ma ([email protected])
- Corresponding Author: Conghui He ([email protected])
- Issues: GitHub Issues
Made with ❤️ by the OpenDataLab team
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