Three-particle finite-volume formalism for $\pi^+\pi^+ K^+$ and related systems

TL;DR

This paper develops a relativistic finite-volume formalism to extract two- and three-particle scattering amplitudes for systems with two identical and one different spinless particles, aiding lattice QCD studies.

Contribution

It extends the three-particle finite-volume formalism to systems with two identical and one different particle, including derivations and threshold expansions.

Findings

Derived the formalism for two- and three-particle scattering amplitudes.

Provided threshold expansions for the three-particle K matrix.

Validated the formalism through a limit and projection check.

Abstract

We consider three-particle systems consisting of two identical particles and a third that is different, with all being spinless. Examples include $\pi^+\pi^+ K^+$ and $K^+K^+\pi^+$. We derive the formalism necessary to extract two- and three-particle infinite-volume scattering amplitudes from the spectrum of such systems in finite volume. We use a relativistic formalism based on an all-orders diagrammatic analysis in generic effective field theory, adopting the methodology used recently to study the case of three nondegenerate particles. We present both a direct derivation, and also a cross-check based on an appropriate limit and projection of the fully nondegenerate formalism. We also work out the threshold expansions for the three-particle K matrix that will be needed in practical applications, both for systems with two identical particles plus a third, and also for the fully nondegenerate theory.