# Lattice QCD and Three-particle Decays of Resonances

## TL;DR

This paper reviews the development of a three-particle quantization condition in lattice QCD to analyze three-particle decay channels of resonances, addressing finite-volume effects and comparing different theoretical approaches.

## Contribution

It provides a comprehensive review and comparison of three different methods for relating finite-volume energies to infinite-volume scattering amplitudes for three-particle systems in lattice QCD.

## Key findings

- Comparison of three approaches to three-particle quantization.
- Numerical implementation examples for all three methods.
- Identification of key unresolved issues in the field.

## Abstract

Most strong-interaction resonances have decay channels involving three or more particles, including many of the recently discovered $X$, $Y$ and $Z$ resonances. In order to study such resonances from first principles using lattice QCD, one must understand finite-volume effects for three particles in the cubic box used in calculations. Here we review efforts to develop a three-particle quantization condition that relates finite-volume energies to infinite-volume scattering amplitudes. We describe in detail the three approaches that have been followed, and present new results on the relationship between the corresponding results. We show examples of the numerical implementation of all three approaches and point out the important issues that remain to be resolved.