Higher order quantization conditions for two spinless particles

TL;DR

This paper derives higher-order quantization conditions for two spinless particles in lattice QCD, enabling more precise extraction of scattering phaseshifts and resonance parameters beyond the lowest order.

Contribution

It introduces a comprehensive set of higher-order quantization conditions up to =5 for various geometries and momenta, with a transparent cross-check method for validation.

Findings

Derived 45 quantization conditions for different geometries and partial waves.

Successfully validated all quantization conditions with cross-checks.

Provides tools for more accurate lattice QCD scattering analyses.

Abstract

Lattice QCD calculations of scattering phaseshifts and resonance parameters in the two-body sector are becoming precision studies. Early calculations employed L\"uscher's formula for extracting these quantities at lowest order. As the calculations become more ambitious, higher-order relations are required. In this study we derive higher-order quantization conditions and introduce a method to transparently cross-check our results. This is an important step given the involved derivations of these formulae. We derive quantization conditions up to $\ell=5$ partial waves in both cubic and elongated geometries, and for states with zero and non-zero total momentum. All 45 quantization conditions we include here (22 in cubic box, 23 in elongated box) pass our cross-check test.