# On the Equivalence of Three-Particle Scattering Formalisms

## TL;DR

This paper proves the equivalence of two major three-particle scattering formalisms in infinite volume, showing they are physically identical and connecting them to Faddeev equations in the non-relativistic limit.

## Contribution

The paper demonstrates the equivalence of Hansen-Sharpe and Mai et al. formalisms, unifying different approaches to three-particle scattering theory.

## Key findings

- Both formalisms are mathematically equivalent in infinite volume.
- The Faddeev equations are recovered in the non-relativistic limit.
- The physical content of the formalisms is shown to be identical.

## Abstract

In recent years, different on-shell $\mathbf{3}\to\mathbf{3}$ scattering formalisms have been proposed to be applied to both lattice QCD and infinite volume scattering processes. We prove that the formulation in the infinite volume presented by Hansen and Sharpe in Phys.~Rev.~D92, 114509 (2015) and subsequently Brice\~no, Hansen, and Sharpe in Phys.~Rev.~D95, 074510 (2017) can be recovered from the $B$-matrix representation, derived on the basis of $S$-matrix unitarity, presented by Mai {\em et al.} in Eur.~Phys.~J.~A53, 177 (2017) and Jackura {\em et al.} in Eur.~Phys.~J.~C79, 56 (2019). Therefore, both formalisms in the infinite volume are equivalent and the physical content is identical. Additionally, the Faddeev equations are recovered in the non-relativistic limit of both representations.