Khuri-Treiman equations for $3\pi$ decays of particles with spin

TL;DR

This paper generalizes Khuri-Treiman equations to analyze three-particle decays involving particles with arbitrary spin, parity, and charge conjugation, using helicity amplitudes for better handling of unitarity relations.

Contribution

It introduces a comprehensive formalism for applying Khuri-Treiman equations to particles with any spin and parity, emphasizing the use of helicity amplitudes and crossing matrices.

Findings

Helicity amplitudes effectively handle unitarity relations.

Crossing matrices relate isobar expansions across channels.

Kinematical singularities are systematically derived.

Abstract

Khuri-Treiman equations have proven to be a useful theoretical tool in the analysis of 3-body decays, specially into the $3\pi$ final state. In this work we present in full detail the necessary generalization of the formalism to study the decays of particles with arbitrary spin, parity, and charge conjugation. To this extent, we find it most convenient to work with helicity amplitudes instead of the so-called invariant amplitudes, specially when dealing with the unitarity relations. The isobar expansions in the three possible ($s$-, $t$-, and $u$-) final channels are related with the appropriate crossing matrices. We pay special attention to the kinematical singularities and constraints of the helicity amplitudes, showing that these can be derived by means of the crossing matrix.