Scattering of unstable particles in a finite volume: the case of pi rho scattering and the a1(1260) resonance

TL;DR

This paper develops a method to extract infinite-volume scattering observables for unstable particles from finite-volume lattice data, demonstrated through $ ho ext{-} ho$ scattering and the $a_1(1260)$ resonance.

Contribution

The paper introduces a novel approach to analyze unstable particle scattering in finite volumes, enabling the determination of physical scattering amplitudes from lattice simulations.

Findings

Finite-volume energy levels differ significantly when considering the $ ho$ as stable versus unstable.

The method allows extraction of phase shifts and scattering amplitudes from lattice data.

Differences in energy levels highlight the importance of accounting for resonance instability.

Abstract

We present a way to evaluate the scattering of unstable particles quantized in a finite volume with the aim of extracting physical observables for infinite volume from lattice data. We illustrate the method with the $\pi\rho$ scattering which generates dynamically the axial-vector $a_1(1260)$ resonance. Energy levels in a finite box are evaluated both considering the $\rho$ as a stable and unstable resonance and we find significant differences between both cases. We discuss how to solve the problem to get the physical scattering amplitudes in the infinite volume, and hence phase shifts, from possible lattice results on energy levels quantized inside a finite box.