diff --git "a/6tE2T4oBgHgl3EQfkwcv/content/tmp_files/load_file.txt" "b/6tE2T4oBgHgl3EQfkwcv/content/tmp_files/load_file.txt" new file mode 100644--- /dev/null +++ "b/6tE2T4oBgHgl3EQfkwcv/content/tmp_files/load_file.txt" @@ -0,0 +1,523 @@ +filepath=/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf,len=522 +page_content='The LÃŒscher scattering formalism on the 𝒕-channel cut André Baião Raposo𝑎,∗ and Maxwell T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Hansen𝑎 𝑎Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, UK E-mail: a.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='baiao-raposo@sms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='uk, maxwell.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='hansen@ed.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ac.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='uk The LÃŒscher scattering formalism, the standard approach for relating the discrete finite-volume energy spectrum to two-to-two scattering amplitudes, fails when analytically continued so far below the infinite-volume two-particle threshold that one encounters the 𝑡-channel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This is relevant, especially in baryon-baryon scattering applications, as finite-volume energies can be observed in this below-threshold regime, and it is not clear how to make use of them.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In this talk, we present a generalization of the scattering formalism that resolves this issue, allowing one to also constrain scattering amplitudes on the 𝑡-channel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 39th International Symposium on Lattice Field Theory - Lattice2022 8-13 August 2022 Bonn, Germany ∗Speaker © Copyright owned by the author(s) under the terms of the Creative Commons Attribution-NonCommercial-NoDerivatives 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='0 International License (CC BY-NC-ND 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='0).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' https://pos.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='sissa.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='it/ arXiv:2301.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='03981v1 [hep-lat] 10 Jan 2023 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Introduction & motivation In recent years, there has been considerable progress in the determination of two-nucleon and other two-baryon scattering amplitudes using numerical lattice QCD [1–9].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' One of the leading methods in these calculations is to first extract the finite-volume energy spectrum and subsequently the scattering amplitudes via the LÃŒscher formalism and its extensions [10–20].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In such calculations, each finite-volume energy level constrains or predicts the scattering matrix for all multi-hadron channels that can physically propagate at that energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' One limitation in all finite-volume formalisms published to date is the neglect of volume effects associated with the 𝑡-channel (also called left-hand) cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='1 This is most obviously a problem when the lattice calculation predicts energies that are on top of the cut, as has recently been seen in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The finite-volume formalism is manifestly not applicable here, as it leads to predictions of a real-valued K-matrix (equivalently, a real-valued scattering phase shift) in a region where the latter is known to be complex.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In this proceedings, we present an extension of the original formalism that can be applied on the left-hand cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We begin with a brief review of infinite-volume scattering theory as well as the standard LÃŒscher formalism, in sections 2 and 3, respectively.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In section 4, we illustrate how the 𝑡-channel cut becomes an issue and, in section 5, we briefly describe our approach to a solution, the full details of which will be presented in a publication (to appear).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Conclusions and an outlook are given in section 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Two-to-two scattering in infinite-volume We start by reviewing a few properties of scattering amplitudes in the infinite-volume context, making no reference yet to the finite-volume formalism.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Considering two-to-two elastic scattering of non-identical mass-degenerate spin-zero particles with physical mass 𝑀, we write the total four-momentum in a given frame as 𝑃 = (𝐞, 𝑷) and introduce the standard Mandelstam invariants 𝑠 and 𝑡.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Mandelstam 𝑠 satisfies 𝑠 = 𝑃2 = 𝐞2 − 𝑷2 = (𝐞★)2, where 𝐞★ denotes the centre-of-mass frame energy.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Note we will use ★ to denote quantities boosted to the centre-of-mass frame.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The scattering amplitude, which we write M(𝑠, 𝑡), can be formally expressed as the sum of all connected and amputated two-to-two Feynman diagrams, with legs amputated and set on the mass shell (i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' with external momenta 𝑝 having 𝑝2 = (𝑝0)2 − 𝒑2 = 𝑀2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This all-orders sum can be organized by introducing the Bethe-Salpeter kernel, defined as the sum of all connected and amputated two-to-two diagrams that are two-particle irreducible in the 𝑠-channel2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The amplitude is then expressible in terms of the Bethe-Salpeter kernels and pairs of dressed propagators of the scattering scalars, as shown in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Note that all propagators considered are taken with the standard 𝑖𝜖 prescription, and all loop momenta are integrated over all components.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 1For ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [21], the issue of the cut may be circumvented by working in the plane-wave basis, but this is not specifically discussed in those proceedings.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 2In other words, the Bethe-Salpeter kernel is built from diagrams that cannot be separated into two pieces by cutting through two propagators whose momenta sum to the total four-momentum 𝑃 = (𝐞, 𝑷).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 2 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Figure 1: (a) Diagrammatic representation of the two-to-two scattering amplitude using Bethe-Salpeter kernels and dressed propagators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (b) Definition of the Bethe-Salpeter kernel as the sum of all connected and amputated two-to-two diagrams which are two-particle irreducible in the 𝑠-channel;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' we assume that self-interactions of the scattering particles are Z2-invariant, such that we consider vertices with even number of legs only, while dashed lines denote other particles that might couple to the scattering channels of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (c) Definition of the dressed propagator in terms of bare propagators and self-energy kernels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (d) Definition of the self-energy as the sum of all one-particle irreducible diagrams, with dashed lines again denoting other particle types that can couple to the scattering particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' It is also instructive to define partial-wave amplitudes according to M(𝑠, 𝑡) = ∞ ∑ ℓ=0 𝑃ℓ(cos 𝜃★)Mℓ(𝑠) , (1) where 𝑃ℓ is a Legendre polynomial and 𝜃★ is the scattering angle in the centre-of-mass frame, satisfying sin2(𝜃★/2) = −𝑡/(𝑠 − 4𝑀2).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Using the all-orders Bethe-Salpeter representation (as discussed below) or constraints based on the unitarity of the scattering matrix, one can show that the imaginary part of Mℓ(𝑠)−1 is independent of the details of particle interactions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The real part is then typically parameterized using the scattering phase shift 𝛿ℓ(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We may write Im Mℓ(𝑠)−1 = −𝜌(𝑠) Θ(𝐞★ − 2𝑀) , (2) Re Mℓ(𝑠)−1 = 𝜌(𝑠) cot 𝛿ℓ(𝑠) ≡ Kℓ(𝑠)−1 , (3) where 𝜌(𝑠) ≡ 𝑝★ 8𝜋𝐞★ is the phase-space factor for non-identical particles, with 𝑝★ ≡ 1 2 √ 𝑠 − 4𝑀2 denoting each particle’s centre-of-mass spatial momentum magnitude, and we have defined the K-matrix Kℓ(𝑠).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This leads to the standard form of the partial-wave amplitude Mℓ(𝑠) = 1 Kℓ(𝑠)−1 − 𝑖𝜌(𝑠) = 8𝜋𝐞★ 𝑝★ cot 𝛿ℓ − 𝑖𝑝★ .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (4) One can also reach these results via the Bethe-Salpeter series of figure 1 if one defines the K-matrix K by the same series as the amplitude M, but in which all two-particle loops are evaluated with a principal-value prescription instead of the 𝑖𝜖 prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The definitions of M and K can 3 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo then be related by writing each 𝑖𝜖-loop in terms of its real and imaginary parts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The former coincides with the principal-value loop integral, and the latter leads to a Dirac delta function, allowing the loop integral to be exactly evaluated.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We should emphasize that these relations hold only for (2𝑀)2 < 𝑠 < (𝐞★ inel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' )2 where 𝐞inel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' is the energy of the lowest-lying inelastic threshold coupling to the channel of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This fact has received significant attention for energies above 𝐞inel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (in the form of three-particle finite-volume formalisms [22–27]), but in this work we are concerned with the range 𝑠 < (2𝑀)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' For these sub-threshold energies, one can analytically continue the amplitude by taking −𝑖𝜌(𝑠) → |𝜌(𝑠)| in order to remain on the physical Riemann sheet.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Such analytic continuation leads to a real-valued scattering amplitude, provided that the K-matrix is real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' When, however, we have a lighter particle that couples to the scattering channel of interest, the K-matrix partial waves become complex-valued due to a sub-threshold branch cut, the so-called 𝑡-channel or left-hand cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Before turning to the consequences of the cut, we review the standard finite-volume formalism of LÃŒscher and show that a real-valued K-matrix is implicitly assumed in the sub-threshold analytic continuation, and thus that the formalism is not applicable on the left-hand cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Review of the LÃŒscher formalism In this section, we review the derivation of the LÃŒscher quantization condition [10], which has been subsequently extended in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [11–20] to include all types of coupled two-particle channels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We focus here on the case of a single channel with two mass-degenerate but non-identical spin-zero particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Consider a quantum field theory defined in a finite cubic spatial volume of side-length 𝐿, with periodic boundary conditions.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This system then has a discrete 𝐿-dependent energy spectrum, and the energies lying below the lowest-lying three- or four-particle threshold can be used to extract the elastic two-to-two scattering amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We follow closely the derivation of Kim, Sachrajda and Sharpe [13], also used in refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [17, 28–30].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We begin by defining a two-point correlation function 𝐶𝐿(𝐞, 𝑷) = ∫ 𝑑𝑥0 ∫ 𝐿 𝑑3𝒙 𝑒−𝑖𝐞𝑥0𝑒𝑖𝑷·𝒙⟚0|TA(𝑥)A†(0)|0⟩𝐿 , (5) where the subscript 𝐿 in ∫ 𝐿 𝑑3𝒙 indicates that the integral runs over the finite volume, 𝐞 denotes the total energy, 𝑷 is the total spatial momentum, and A(𝑥) and A†(𝑥) are annihilation and creation operators carrying the quantum numbers of the scattering channel of interest.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' One can construct a diagrammatic representation for this correlator using the ingredients already introduced in the previous section, the Bethe-Salpeter kernel and the dressed propagator pairs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This is known as the skeleton expansion for the correlator and is shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In finite volume, spatial loop momenta are discretized as 𝒌 = 2𝜋 𝐿 𝒏, with 𝒏 ∈ Z3, and we have spatial loop momentum sums instead of integrals, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' we replace ∫ 𝑑3𝒌 (2𝜋)3 → 1 𝐿3 ï¿œ 𝒌 ∈(2𝜋/𝐿)Z3 in all loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The key observation for deriving the LÃŒscher formalism is that not all loops have the same volume dependence: loops with intermediate states that cannot go on shell in the energy range considered have exponentially suppressed volume effects O(𝑒−𝑚𝐿),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' with 𝑚 being the mass of the lightest particle coupled to the system,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' while loops with intermediate states that can go on shell 4 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Figure 2: Skeleton-expansion representation of the finite-volume correlator 𝐶𝐿(𝐞,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝑷),' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' in terms of Bethe- Salpeter kernels and dressed propagators,' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' as defined in figure 1.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The end-cap “blobs” stand for functions in momentum-space originating from the Fourier transforms of the creation and annihilation operators.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' As discussed, one can take the kernels and dressed propagators in the expansion to be the infinite-volume objects, as the difference between these and their finite-volume counterparts is exponentially suppressed, and thus only the loops explicitly shown need to be treated as finite-volume loops.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' receive power-like effects O(𝐿−𝑛), for some non-negative integer 𝑛.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the elastic regime, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' for centre-of-mass frame energies above the two-particle threshold but below the thresholds of all other channels, on-shell states come precisely from the loops left explicit in the skeleton expansion shown in figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Every other loop, implicitly included in either the Bethe-Salpeter kernels or the dressed propagators, may be replaced by its infinite-volume counterpart, as the difference, which we neglect, is exponentially suppressed in 𝐿.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Thus, we effectively replace the finite-volume Bethe-Salpeter kernels and dressed propagators with their infinite-volume counterparts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Consider one of the two-particle loops shown in the expansion of figure 2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Its contribution can be written as 𝐶loop 𝐿 (𝑃) = ∫ 𝑑𝑘0 2𝜋 1 𝐿3 ∑ 𝒌 L(𝑃, 𝑘) Δ(𝑘) Δ(𝑃 − 𝑘) R∗(𝑃, 𝑘) , (6) with 𝑃 ≡ (𝐞, 𝑷) and loop momentum 𝑘 ≡ (ￜￜￜ0, 𝒌).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The functions L and R∗ stand for the objects before and after the given loop, Δ is a fully dressed scalar propagator (defined to have unit residue at each pole).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Performing the 𝑘0-integral and decomposing L and R in spherical harmonics with 𝑘 individually put on shell, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' setting 𝑘 = (𝜔(𝒌), 𝒌) with 𝜔(𝒌) ≡ √ 𝒌2 + 𝑀2, we can obtain 𝐶loop 𝐿 (𝑃) = 1 𝐿3 ∑ 𝒌 Lℓ𝑚(𝑃, |𝒌★|) 𝑖Sℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃, 𝒌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) R∗ ℓ′𝑚′(𝑃, |𝒌★|) + 𝑟(𝑃) , (7) where we have introduced Sℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃, 𝒌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ≡ 4𝜋𝑌ℓ𝑚( ˆ𝒌★) 𝑌∗ ℓ′𝑚′( ˆ𝒌★) 𝐻(𝒌★) 2𝜔(𝒌) 2𝜔(𝑷 − 𝒌) (𝐞 − 𝜔(𝒌) − 𝜔(𝑷 − 𝒌)) ï¿œ |𝒌★| 𝑘★ os ￜℓ+ℓ′ , (8) for later convenience.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Note also that sums over the repeated indices ℓ, 𝑚 and ℓ′, 𝑚′ are implied.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The term 𝑟(𝑃) in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (7) is a sum over a smooth summand, leading to exponentially suppressed finite-volume corrections, which we may neglect.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The summand in the first term and, more specifically, the quantities Sℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃, 𝒌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿), contain the pole corresponding to the two-particle intermediate state going on-shell, as can be seen explicitly in eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (8), and thus this term contains all the power-like volume dependence arising from the loop we are considering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (8), we use the notation 𝑘★ os for the value of |𝒌★| which satisfies the intermediate two-particle state on-shell condition 𝐞★ = 2𝜔(𝒌★), equivalent to 𝐞 = 𝜔(𝒌) + 𝜔(𝑷 − 𝒌), given by the simple relation 𝑘★ os ≡ 1 2 √ 𝑠 − 4𝑀2 = 1 2 √ 𝐞2 − 𝑷2 − 4𝑀2 .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (9) 5 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Note that this relation fixes the magnitude of 𝒌★ at the pole, but not its direction.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The barrier factor ï¿œ|𝒌★|/𝑘★ os ￜℓ+ℓ′ is introduced to ensure that no singularities arise from the spherical harmonics 𝑌ℓ𝑚( ˆ𝒌★) and 𝑌∗ ℓ′𝑚′( ˆ𝒌★).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The function 𝐻(𝒌★) is a regulator function that takes a value of 1 for 4𝜔(𝒌★)2 = 4(|𝒌★|2+𝑀2) < (𝐞★ inel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' )2 and of 0 for 4𝜔(𝒌★)2 = 4(|𝒌★|2 + 𝑀2) > (𝐞★ uv)2, where again 𝐞inel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' is the lowest lying three- or four-particle threshold, and 𝐞★ uv is some chosen high ultraviolet cut-off.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the region between, 𝐻(𝒌★) interpolates smoothly between the two values.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This regulator function is similar to the one found in the three-body scattering formalism of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [22, 23], and corresponds to a separation of low-energy and high-energy parts of the sum, such that both parts are regulated and kept finite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='3 We next reduce eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (7) by expanding the functions Lℓ𝑚(𝑃, |𝒌★|) and R∗ ℓ′𝑚′(𝑃, |𝒌★|) about |𝒌★| = 𝑘★ os and subtracting and adding an integral to reach 𝐶loop 𝐿 (𝑃) = Lℓ𝑚(𝑃, 𝑘★ os) 𝑖𝐹ℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) R∗ ℓ′𝑚′(𝑃, 𝑘★ os) + 𝑟′(𝑃) , = Los(𝑃) 𝑖𝐹(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) R† os(𝑃) + 𝑟′(𝑃) , (10) where we introduce the sum-integral difference 𝐹ℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) = ï¿œ 1 𝐿3 ∑ 𝒌 − p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' ∫ 𝑑3𝒌 (2𝜋)3 ï¿œ Sℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃, 𝒌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (11) Here, p.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='v.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' means the integral is evaluated using a principal-value prescription.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The remainder term 𝑟′(𝑃) differs from 𝑟(𝑃), but still has the property that it contains the sum of a smooth summand together with integrals.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the second line of eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (10), we have defined a compact vector-matrix notation in the angular-momentum index space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Applying this procedure iteratively to all loops in the skeleton expansion diagrams, it can be shown that we may write the finite-volume correlator in the form: 𝐶𝐿(𝑃) = ∞ ∑ 𝑛=0 𝐎(𝑃) 𝑖𝐹(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) [𝑖K(𝑃) 𝑖𝐹(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿)]𝑛 𝐎†(𝑃) + 𝐶∞(𝑃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (12) The quantities in the first term are vectors or matrices in angular momentum index space: 𝐎(𝑃) and 𝐎†(𝑃) are smooth vectors loosely corresponding to the source and sink operators and K(𝑃) is the K-matrix introduced in the previous section (note that K is an infinite-volume scalar, and thus only depends on 𝑠 = 𝑃2, but we keep 𝑃 as an argument for compactness of notation).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Noting that the above is a geometric series, we can sum it to obtain 𝐶𝐿(𝑃) = 𝐎(𝑃) 𝑖 𝐹−1(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) + K(𝑃) 𝐎†(𝑃) + 𝐶∞(𝑃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (13) Given that we neglect exponentially suppressed volume effects, the volume dependence is entirely contained in the 𝐹(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) matrix.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 3It should be emphasized that we have renormalization and regularization schemes keeping the overall result finite, the regulator function here simply ensures we have a separation of high-energy and low-energy contributions to the sum such that both parts are finite and that the low-energy part, which contains the relevant singular behaviour, is tractable when implemented numerically.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐞uv will simply be a scheme-dependence of the formalism, but should be kept high, as setting it too low will lead to enhanced finite-volume effects.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 6 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Using a spectral representation of the correlator, it is straightforward to show that it must have poles at the energy levels of the finite-volume spectrum 𝐞𝑛(𝑷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' These poles in 𝐶𝐿(𝐞, 𝑷) can only arise from the 𝐿-dependent part of the first term, meaning that we must have det ï¿œ 𝐹−1(𝐞𝑛(𝑷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿), 𝑷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) + K(𝐞𝑛(𝑷;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿), 𝑷) ï¿œ = 0 , (14) at all finite-volume energy levels.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This is called the LÃŒscher quantization condition, and it can be used to determine K, and hence the scattering amplitude M, from the knowledge of the finite-volume spectrum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The matrices involved in the condition (14) are formally infinite-dimensional, since the set of possible angular momentum indices ℓ𝑚 is infinite.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' For practical use, we must truncate them to the lowest harmonics, making the approximation that K vanishes for ℓ > ℓmax.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We should note that this relies on a fast convergence of the partial-wave expansion of the amplitude, such that keeping the lowest harmonics still leads to a reasonable reconstruction of the amplitude.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 4.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The 𝑡-channel problem The finite-volume spectrum can sometimes include energy levels that drop below the infinite- volume elastic threshold at 𝑠 = (2𝑀)2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This can occur due to the appearance of a bound state (such that the 𝐿 → ∞ limit gives the bound-state mass) as well as to an attractive scattering state (such that the energy approaches 2𝑀 for 𝐿 → ∞).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In many cases, the LÃŒscher formalism can be analytically continued from 𝑠 > (2𝑀)2 and the sub-threshold finite-volume energy then provides an important constraint on the K-matrix below threshold.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' A subtlety arises, however, when the sub-threshold two-to-two scattering amplitude M(𝑠, 𝑡), and therefore also the K-matrix, has a nearby 𝑡-channel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This is generically the case in baryon-baryon systems, for example, where a light meson can be exchanged in the 𝑡-channel as shown in figure 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Taking 𝑚 and 𝑀 to be the meson and baryon masses, respectively, and assuming 𝑚 ≪ 𝑀, one finds that a pole arises in M(𝑠, 𝑡) at 𝑡 = 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Then, for a given fixed choice of the centre-of-mass frame scattering angle 𝜃★, this leads to a pole in 𝑠 at 𝑠 = 4𝑀2 − 𝑡 sin2(𝜃★/2) ￜￜￜￜ 𝑡=𝑚2 = 4𝑀2 − 𝑚2 sin2(𝜃★/2) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (15) The analytic structure of the scattering amplitude for such systems is shown in figure 4(a).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' From the expression, one sees that the pole position in 𝑠 varies from 𝑠 = 4𝑀2 − 𝑚2 to 𝑠 = −∞ as 𝜃★ is varied from 0 to 𝜋.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' As a result, the angular-momentum projection of the scattering amplitude leads to a branch cut running over this interval as shown in figure 4(b).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Multiple meson exchanges can also occur, leading to additional cuts in both the fixed-𝜃★ and the angular-momentum projected amplitudes.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the latter case, these run along 𝑠 ≀ (2𝑀)2 − (𝑛𝑚)2 for 𝑛 exchanged mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' As stressed above, finite-volume energies can arise in the region of the branch cuts (as has recently been identified in ref.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [7]) and a naive application of the analytically continued LÃŒscher formalism fails.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In this work we restrict attention to the region (2𝑀)2 − (2𝑚)2 < 𝑠 < (2𝑀)2 − 𝑚2 in which only the single-meson cut arises and derive a modified version of the scattering formalism that resolves this limitation.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 7 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Figure 3: Meson exchanges contributing to the baryon-baryon scattering amplitude, expressed diagrammati- cally as a dressed 𝑡-channel meson propagator (with time flowing horizontally).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The exchanged meson is a pion in the case of 𝑁𝑁 scattering or an 𝜂 meson in the case of ΛΛ scattering.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the latter case this is not the nearest singularity for physical quark masses but can become so for heavier-than-physical masses.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Figure 4: (a) Analytic structure of the two-to-two scattering amplitude M(𝑠, 𝑡) in the complex-𝑠 plane, for fixed centre-of-mass scattering angle 𝜃★, in the case where a lighter particle of mass 𝑚 couples to the scattering particles of mass 𝑀.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We show the infinite-volume elastic threshold (at 𝑠 = (2𝑀)2) and inelastic threshold (at 𝑠 = (2𝑀 + 𝑚)2) and corresponding branch cuts.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Below threshold, we see the 𝑡-channel exchange pole, corresponding to 𝑡 = 𝑚2, and a lower branch cut, corresponding to two mesons being exchanged in the 𝑡 channel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (b) Analytic structure of the amplitude when projected to definite angular momentum.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The so-called 𝑡-channel or left-hand cut, which runs down from the branch point at 𝑠 = (2𝑀)2 − 𝑚2, arises from the 𝑡-channel pole above.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Proposed solution The origin of the breakdown in the original formalism can be traced back to the steps between eq.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (7) and (10) in the review of section 3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the step of replacing Lℓ𝑚(𝑃, |𝒌★|) and R∗ ℓ′𝑚′(𝑃, |𝒌★|) with the on-shell quantities Lℓ𝑚(𝑃, 𝑘★ os) and R∗ ℓ′𝑚′(𝑃, 𝑘★ os), the derivation assumes that the product of the two-particle pole and a given difference, e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Lℓ𝑚(𝑃, |𝒌★|)−Lℓ𝑚(𝑃, 𝑘★ os), is a smooth function of 𝒌★.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This step fails in the sub-threshold region due to the 𝑡-channel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' To handle this issue, we separate out the problematic 𝑡-channel exchanges from the Bethe-Salpeter kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We define 𝑖𝑔2𝑇(𝒌★, 𝒌′★) ≡ −𝑖𝑔2 1 −(𝒌★ − 𝒌′★)2 − 𝑚2 + 𝑖𝜖 , (16) and define a modified kernel by subtracting this from the full Bethe-Salpeter kernel as shown in figure 5.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Here, 𝑔 denotes the effective baryon-meson-baryon coupling.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We emphasize that 𝑚 is the physical mass of the meson, and thus that −𝑖𝑇 corresponds to the singular part of the fully-dressed meson propagator.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The difference between the bare and fully dressed propagators is smooth, and is simply absorbed into the modified kernel.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 8 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo Figure 5: Separation of the standard Bethe-Salpeter kernel into a modified kernel (square box), which is safe to put on shell, and the problematic 𝑡-channel meson exchange, represented by a meson propagator at physical mass.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Examining the modified kernel using, for instance, the cutting rules of time-ordered perturbation theory, shows that it can be safely evaluated at |𝒌★| = 𝑘★ os and does not possess a singularity or cut in the region (2𝑀)2 − (2𝑚)2 < 𝑠 < (2𝑀)2 − 𝑚2.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Crucially, we also note that the exchange 𝑖𝑔2𝑇 is safe if kept partially off shell, namely if we keep at least one of the magnitudes |𝒌★| and |𝒌′★| to be real.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Recalling the expression (10) for the contribution of a skeleton expansion loop, we again emphasize that the finite-volume frame momentum 𝒌, and hence its centre-of-mass frame counterpart 𝒌★, are discretized and can be indexed by 𝒏 ∈ Z3.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Thus, we can then treat 𝒌★ as an extra index, writing L𝒌★ℓ𝑚 ≡ Lℓ𝑚(𝑃, |𝒌★|), R𝒌★ℓ𝑚 ≡ Rℓ𝑚(𝑃, |𝒌★|) and defining 𝑆𝒌★ℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='𝒌′★ℓ′𝑚′(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ≡ 1 𝐿3 𝛿𝒌★𝒌′★Sℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='ℓ′𝑚′(𝑃, 𝒌;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) , (17) such that we may rewrite (10) as: 𝐶loop 𝐿 (𝑃) = L𝒌★ℓ𝑚(𝑃) 𝑖𝑆𝒌★ℓ𝑚;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='𝒌′★ℓ′𝑚′(𝑃) R𝒌′★ℓ′𝑚′(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) + 𝑟(𝑃) , (18) = L(𝑃) 𝑖𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) R(𝑃) + 𝑟(𝑃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (19) Note that, in the first line, we are now also implicitly summing over the momentum indices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' In the second line, we again employ a compact vector-matrix notation, but now in the angular momentum plus spatial loop momentum index space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Applying this to all loops in the skeleton expansion diagrams and rearranging by factors of 𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿), we obtain the finite-volume correlator in the form 𝐶𝐿(𝑃) = ∞ ∑ 𝑛=0 𝐎(𝑃) 𝑖𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ï¿œ (𝑖 ¯𝐟 + 𝑖𝑔2𝑇(𝑃)) 𝑖𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ￜ𝑛 𝐎†(𝑃) + 𝐶 (𝑖) ∞ (𝑃) , (20) where all quantities in the first term are vectors or matrices in the angular momentum plus loop momentum index space.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Note that 𝐎(𝑃) and 𝐎†(𝑃) are different from those in (10).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The matrix 𝑖𝑔2𝑇 is the matrix of angular momentum projections of the 𝑡-channel exchange defined in (16), and ¯𝐟(𝑃) is the sum of all possible smooth contributions one can obtain between 𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) matrices.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The second term 𝐶 (𝑖) ∞ (𝑃) is a collection of 𝐿-independent terms.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' From the discussion above, we know it is safe to set |𝒌★| = 𝑘★ os for ¯𝐟(𝑃) and, therefore, we can expand ¯𝐟(𝑃) about the on-shell point.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We implement this by making use of a trivial vector 𝑢 in the momentum index space, whose elements are 𝑢𝒌★ = 1, and making the substitution ¯𝐟(𝑃) = 𝑢 ¯K(𝑃)𝑢† + ï¿œ ¯𝐟(𝑃) − 𝑢 ¯K(𝑃)𝑢†ￜ.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The matrix ¯K(𝑃) is a matrix in the angular momentum index space only and corresponds to ¯𝐟(𝑃) with the dependence on the magnitude of spatial momentum (through the momentum index) set to the on-shell momentum, i.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' with |𝒌★| = 𝑘★ os.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' The 9 The LÃŒscher scattering formalism on the 𝑡-channel cut André Baião Raposo different terms in brackets lead to terms that are sums of smooth summands and can be shuffled into the remainder term.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This yields the following expressions for the correlator: 𝐶𝐿(𝑃) = ∞ ∑ 𝑛=0 𝐎(𝑃) 𝑖𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ï¿œ (𝑖𝑢 ¯K(𝑃)𝑢† + 𝑖𝑔2𝑇(𝑃)) 𝑖𝑆(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) ￜ𝑛 𝐎†(𝑃) + 𝐶 (𝑖𝑖) ∞ (𝑃) , (21) = 𝐎(𝑃) 𝑖 𝑆−1(𝑃;' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 𝐿) + 𝑢 ¯K(𝑃)𝑢† + 𝑔2𝑇(𝑃) 𝐎†(𝑃) + 𝐶 (𝑖𝑖) ∞ (𝑃) .' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' (22) Using the same arguments as in the standard derivation, we can derive a modified quantization condition: det ï¿œ 𝑆−1(𝐞𝑛(𝑷, 𝐿), 𝑷) + 𝑢 ¯K(𝐞𝑛(𝑷, 𝐿), 𝑷)𝑢† + 𝑔2𝑇(𝐞𝑛(𝑷, 𝐿), 𝑷) ï¿œ = 0 , (23) at all finite-volume energy levels 𝐞𝑛(𝑷, 𝐿).' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Given that 𝑆−1(𝐞𝑛(𝑷, 𝐿), 𝑷) and 𝑇(𝐞𝑛(𝑷, 𝐿), 𝑷) can be calculated numerically, one can use the knowledge of the finite-volume spectrum to obtain ¯K(𝐞𝑛(𝑷, 𝐿), 𝑷) as well as the coupling 𝑔.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This object can then be linked back to the two-to- two scattering amplitude via integral equations, in a similar vein to the procedure used for the three-particle scattering formalism refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [22, 23, 31].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We leave further discussion to the upcoming paper.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' 6.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Summary & Outlook In this proceedings, we have described our progress in addressing issues arising in the LÃŒscher finite-volume scattering formalism [10] and extensions [11–20] in the case of sub-threshold finite- volume energies appearing on the 𝑡-channel cut.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' This work is motivated by recent lattice calculations in baryon-baryon systems that have observed such energy levels [7].' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' To present the extension we first reviewed the standard derivation, following the method of Kim, Sachrajda, and Sharpe [13] for the case on non-identical spin-zero particles.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' We then identified the step in the derivation that fails to correctly account for the sub-threshold cut and provided a modification to address the issue.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Our main result is an adapted quantization condition that applies above and below elastic threshold, including on the cut associated with single-meson exchange, though not on lower cuts arising from the exchange of multiple mesons.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' As we have in mind applications to baryon-baryon scattering, the next step, currently ongoing, is generalizing the derivation to particles with arbitrary intrinsic spin.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Once the theoretical work is concluded, future directions include numerical tests on mock data (e.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='g.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' in the spirit of refs.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' [19, 32]) and eventually applications to lattice QCD baryon-baryon data.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' Acknowledgements The authors thank Raúl Briceño, John Bulava, Evgeny Epelbaum, Drew Hanlon, Arkaitz Rodas, Fernando Romero-López, Maxim Mai, Steve Sharpe, and Hartmut Wittig for useful discussions including those in the context of the Bethe Forum on Multihadron Dynamics in a Box that took place at the Bethe Center for Theoretical Physics in Bonn, Germany.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='T.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content='H.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} +page_content=' is supported by UKRI Future Leader Fellowship MR/T019956/1, and both M.' metadata={'source': '/home/zjlab/wf/langchain-ChatGLM/knowledge_base/6tE2T4oBgHgl3EQfkwcv/content/2301.03981v1.pdf'} 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