Rummukainen-Gottlieb's formula on two-particle system with different mass

TL;DR

This paper generalizes Rummukainen-Gottlieb's finite size formula to two-particle systems with different masses, enabling more accurate extraction of scattering phases in lattice simulations for distinguishable particles.

Contribution

The work extends existing finite size formulas to systems with unequal masses and different symmetries, broadening the applicability of lattice scattering analysis.

Findings

Derived finite size formulas for $C_{4v}$ and $C_{2v}$ symmetries.

Facilitates study of resonances like kappa and vector kaon.

Provides analytical tools for lattice QCD simulations.

Abstract

L\"uscher established a non-perturbative formula to extract the elastic scattering phases from two-particle energy spectrum in a torus using lattice simulations. Rummukainen and Gottlieb further extend it to the moving frame, which is devoted to the system of two identical particles. In this work, we generalize Rummukainen-Gottlieb's formula to the generic two-particle system where two particles are explicitly distinguishable, namely, the masses of the two particles are different. The finite size formula are achieved for both $C_{4v}$ and $C_{2v}$ symmetries. Our analytical results will be very helpful for the study of some resonances, such as kappa, vector kaon, and so on.