Finite-volume scattering on the left-hand cut

Published for SISSA by Springer Received: April 3, Revised: June 14, Accepted: July 11, Published: August 8, 2024 2024 2024 2024 Finite-volume scattering on the left-hand cut and M.T. Hansen Higgs Centre for Theoretical Physics, School of Physics and Astronomy, The University of Edinburgh, Edinburgh EH9 3FD, U.K. E-mail: , Abstract: The two-particle finite-volume scattering formalism derived by Lüscher and generalized in many subsequent works does not hold for energies far enough below the two-particle threshold to reach the nearest left-hand cut. The breakdown of the formalism is signaled by the fact that a real scattering amplitude is predicted in a regime where it should be complex. In this work, we address this limitation by deriving an extended formalism that includes the nearest branch cut, arising from single particle exchange. We focus on two-nucleon (N N → N N ) scattering, for which the cut arises from pion exchange, but give expressions for any system with a single channel of identical particles. The new result takes the form of a modified quantization condition that can be used to constrain an intermediate K-matrix in which the cut is removed. In a second step, integral equations, also derived in this work, must be used to convert the K-matrix to the physical scattering amplitude. We also show how the new formalism reduces to the standard approach when the N → N π coupling is set to zero. Keywords: Algorithms and Theoretical Developments, Hadronic Spectroscopy, Structure and Interactions, Effective Field Theories of QCD, Lattice QCD ArXiv ePrint: 2311.18793 Open Access, © The Authors. Article funded by SCOAP3 . https://doi.org/10.1007/JHEP08(2024)075 JHEP08(2024)075 A. Baião Raposo Contents 1 Introduction 2 5 5 9 12 3 Finite-volume formalism 3.1 Skeleton expansion for the finite-volume correlator 3.2 Classifying finite-volume effects 3.3 Reduction of the finite-volume two-particle loop 3.4 On-shell intermediate states 3.5 Subthreshold regime 3.6 Analytic structure of the Bethe-Salpeter kernel 3.7 Full decomposition of the finite-volume correlator 3.8 Result 3.9 Incorporating spin 15 15 17 19 22 23 24 28 30 31 os 4 Relating K to the scattering amplitude 4.1 Finite-volume auxiliary amplitude 4.2 Integral equations 4.3 Divergence-free amplitude 4.4 Analytic continuation 34 34 36 37 39 5 Exploring the new formalism 5.1 Alternative form 5.2 Recovering the standard formalism 5.3 S-wave dominance 5.4 Comparison to previous work 39 39 41 43 44 6 Conclusions 45 A Analyticity of the Bethe-Salpeter kernel A.1 The u-channel loop A.2 Shuffling t- and u-type subdiagrams 46 47 50 B Manipulating the finite-volume S function 52 C Details of the derivation 54 –1– JHEP08(2024)075 2 Infinite-volume scattering 2.1 Scattering amplitude, K-matrix, and phase space 2.2 Analytic continuation and the left-hand cut 2.3 Incorporating spin 1 Introduction 0 < En (L, P )2 − P 2 < (4Mπ )2 , (1.1) where Mπ is the physical, infinite-volume pion mass. Because G-parity prevents the coupling between even- and odd-number multi-pion states, the lowest inelastic threshold is 4Mπ , as indicated. The lower bound of eq. (1.1) is completely irrelevant in practical calculations, because the lowest finite-volume energy is generically either near or above 2Mπ or else near some bound state mass that, while perhaps well below 2Mπ , is still well above zero. For this work it is nevertheless instructive to recall the origin of the lower bound. Two reasons can be given (see also ref. [5]): One is that the boost matrices to the center-of-mass (CM) frame become ill-defined, or at the very least require a subtle analytic continuation, if the four-vector P µ = (E, P ) becomes either light-like or space-like. The other reason is that the two-to-two amplitude has a left-hand branch cut, with a branch point at Mandelstam s = P 2 = E 2 − P 2 = 0 and this is not taken into account in the derivation. In the extension to N N systems [12], the analogous restriction is (2MN )2 − Mπ2 < En (L, P )2 − P 2 < (2MN + Mπ )2 , (1.2) where MN is the nucleon mass. The upper bound here is simply the lowest-lying inelastic threshold, that of N N π production. The lower bound is due to a left-hand cut from single-pion exchange and is the focus of this work. Over the last decade, extending the range of validity for finite-volume scattering formulae has received much attention. One aspect of this is the generalization to three-particle amplitudes [13–47]. This has progressed rapidly in recent years and a theoretical framework is now in place, along with first numerical lattice QCD determinations of three-to-three scattering amplitudes [48–65]. With respect to (1.2), this can be understood as extending beyond the upper cutoff of (2MN + Mπ )2 .1 In this article, we are concerned with extending 1 For this particular system, the generalization is still outstanding. Given the recent work treating nonidentical and non-degenerate particles [35] and particles with intrinsic spin [46], no fundamental issues are expected in deriving the relevant formalism. –2– JHEP08(2024)075 A powerful method for reliably predicting properties of quantum chromodynamics (QCD), is the application of Monte Carlo importance sampling to numerically evaluate the imaginarytime, discretized, finite-volume QCD path integral. This approach, called lattice QCD, delivers estimates of imaginary-time, discretized, finite-volume correlation functions, and various theoretical frameworks are then applied to relate this data to physical observables. One example application of this general approach is the extraction of finite-volume energies in a given spatial volume, with periodicity L, defined with a particular set of internal quantum numbers and a specified total momentum P in the finite-volume frame. Generally speaking, the numerical values of such finite-volume energies depend on the interaction strength of the hadrons in the channel. Following the seminal work of Lüscher [1] and subsequent generalizations [2–12], this can be used to constrain the infinite-volume hadronic scattering amplitudes. Such relations between energies and amplitudes necessarily come with kinematic restrictions. Denoting the nth finite-volume energy for a given periodicity and momentum by En (L, P ), the original work of Lüscher and the extension to non-zero P of refs. [2, 4, 5] apply only for two-pion elastic scattering, i.e. for energies satisfying D⇤ <latexit sha1_base64="WqTOYIekN2iEwhVgK9WuBjMfOD0=">AAAB6nicbVDJSgNBEK1xjXGLevTSGATxEGbE7RhQwWNEs0Ayhp5OTdKkp2fo7hHCkE/w4kERr36RN//GznLQxAcFj/eqqKoXJIJr47rfzsLi0vLKam4tv76xubVd2Nmt6ThVDKssFrFqBFSj4BKrhhuBjUQhjQKB9aB/NfLrT6g0j+WDGSToR7QrecgZNVa6v348bheKbskdg8wTb0qKMEWlXfhqdWKWRigNE1Trpucmxs+oMpwJHOZbqcaEsj7tYtNSSSPUfjY+dUgOrdIhYaxsSUPG6u+JjEZaD6LAdkbU9PSsNxL/85qpCS/9jMskNSjZZFGYCmJiMvqbdLhCZsTAEsoUt7cS1qOKMmPTydsQvNmX50ntpOSdl87uTovlm2kcOdiHAzgCDy6gDLdQgSow6MIzvMKbI5wX5935mLQuONOZPfgD5/MHtG+NcQ==</latexit> (...truncated)