# Three-body Unitarity with Isobars Revisited

## TL;DR

This paper develops a relativistic three-dimensional scattering equation for three-body hadron interactions that explicitly satisfies unitarity, improving the theoretical description of such processes especially near resonances.

## Contribution

It introduces a Bethe-Salpeter based approach to derive a unitarity-preserving three-body scattering equation for isobar models.

## Key findings

- Equation maintains three- and two-body unitarity above breakup energies.
- Applicable even with resonances in sub-amplitudes.
- Provides a consistent relativistic framework for three-body scattering.

## Abstract

The particle exchange model of hadron interactions can be used to describe three-body scattering under the isobar assumption. In this study we start from the 3->3 scattering amplitude for spinless particles, which contains an isobar-spectator scattering amplitude. Using a Bethe-Salpeter Ansatz for the latter, we derive a relativistic three-dimensional scattering equation that manifestly fulfills three-body unitarity and two-body unitarity for the sub-amplitudes. This property holds for energies above breakup and also in the presence of resonances in the sub-amplitudes.