# Three-body scattering: Ladders and Resonances

## TL;DR

This paper analyzes the unitarity constraints in three-body scattering, separating short-range resonances from long-range forces, and derives a factorized algebraic form for the scattering amplitude.

## Contribution

It introduces a factorization approach to three-body unitarity equations, connecting short-range resonances with long-range exchange processes.

## Key findings

- Resonance amplitudes are dressed by initial- and final-state interactions.
- Factorization leads to algebraic unitarity equations similar to two-body cases.
- Approximates the narrow-resonance limit for simplified analysis.

## Abstract

We discuss unitarity constraints on the dynamics of a system of three interacting particles. We show how the short-range interaction that describes three-body resonances can be separated from the long-range exchange processes, in particular the one-pion-exchange process. It is demonstrated that unitarity demands a specific functional form of the amplitude with a clear interpretation: the bare three-particle resonances are dressed by the initial- and final-state interaction, in a way that is consistent with the considered long-range forces. We postulate that the resonance kernel admits a factorization in the energy variables of the initial- and the final-state particles. The factorization assumption leads to an algebraic form for the unitarity equations, which is reminiscent of the well-known two-body-unitarity condition and approaches it in the limit of the narrow-resonance approximation.