Higher order quantization conditions for two-body scattering with spin

TL;DR

This paper extends the Lüscher quantization condition to high order for spin-involved two-body scattering in finite volume, providing a detailed framework for precise meson-baryon interaction studies.

Contribution

It derives high-order quantization conditions for spin-1/2 scattering in various geometries and frames, including spin-orbit coupling, and validates 19 conditions for improved finite-volume analysis.

Findings

Validated 19 quantization conditions across geometries and frames

Clarified incorporation of spin-orbit coupling into the formalism

Demonstrated convergence of quantization conditions order by order

Abstract

We examine the L\"uscher quantization condition to high order for the scattering of a spinless particle and a spin-1/2 particle in a periodic box. First, we derive the quantization conditions in a non-relativistic framework up to total angular momentum $J=11/2$ in both cubic and elongated geometries, and for both rest and moving frames. Then, we introduce a method to transparently cross-check their convergence, using both quantized energy levels in the box and infinite-volume phase shifts for the same potential. We clarify how to incorporate spin-orbit coupling into the formalism and show in detail how the quantization conditions converge order by order in the various irreducible representations. In all, we validated 19 quantization conditions (12 in cubic box, 7 in elongated box). This is a necessary step in applying the method in precision studies of systems in finite volume with half-integer spin, such as meson-baryon scattering.