# Unitarity of the infinite-volume three-particle scattering amplitude arising from a finite-volume formalism

## TL;DR

This paper proves that the three-particle scattering amplitude in infinite volume always satisfies unitarity when parametrized by the divergence-free K matrix, which is free of certain divergences, aiding phenomenological and lattice QCD analyses.

## Contribution

It establishes that the divergence-free K matrix parametrization ensures three-particle unitarity in the infinite-volume limit, independent of finite-volume considerations.

## Key findings

- $ ext{Re} \, ext{and} \, ext{unitarity}$ of $ ext{scattering amplitude}$ confirmed
- $ ext{Divergence-free K matrix}$ provides a practical parametrization tool
- Applicability to phenomenology and lattice QCD calculations

## Abstract

In a previous publication, two of us derived a relation between the scattering amplitude of three identical bosons, $\mathcal M_3$, and a real function referred to as the {divergence-free} K matrix and denoted $\mathcal K_{\text{df},3}$. The result arose in the context of a relation between finite-volume energies and $\mathcal K_{\text{df},3}$, derived to all orders in the perturbative expansion of a generic low-energy effective field theory. In this work we set aside the role of the finite volume and focus on the infinite-volume relation between $\mathcal K_{\text{df},3}$ and $\mathcal M_3$. We show that, for any real choice of $\mathcal K_{\text{df},3}$, $\mathcal M_3$ satisfies the three-particle unitarity constraint to all orders. Given that $\mathcal K_{\text{df},3}$ is also free of a class of kinematic divergences, the function may provide a useful tool for parametrizing three-body scattering data. Applications include the phenomenological analysis of experimental data (where the connection to the finite volume is irrelevant) as well as calculations in lattice quantum chromodynamics (where the volume plays a key role).