Partial-wave projection of relativistic three-body amplitudes

TL;DR

This paper develops integral equations for relativistic three-body scattering amplitudes using partial-wave projection, applicable to spinless particles, and provides parameterizations for amplitude analysis, with applications to toy models of 3π systems.

Contribution

It introduces a new framework for partial-wave projection of relativistic three-body amplitudes, including simplified expressions for short-distance interactions and parameterizations for amplitude analysis.

Findings

Derived integral equations for three-body scattering amplitudes.

Provided parameterizations for factorizable short-distance interactions.

Analyzed toy models of 3π systems at heavy pion masses.

Abstract

We derive the integral equations for partial-wave projected three-body scattering amplitudes, starting from the integral equations for three-body amplitudes developed for lattice QCD analyses. The results, which hold for generic three-body systems of spinless particles, build upon the recently derived partial-wave projected one-particle exchange, a primary component of the relativistic framework proven to satisfy S matrix unitarity. We derive simplified expressions for factorizable short-distance interactions, $\mathcal{K}_{3}$, in two equivalent formalisms - one symmetric under particle interchange and one asymmetric. For the asymmetric case, we offer parameterizations useful for amplitude analysis. Finally, we examine toy models for $3\pi$ systems at unphysically heavy pion masses with total isospins 0,1, and 2.