Phenomenology of Relativistic $\mathbf{3} \to \mathbf{3}$ Reaction Amplitudes within the Isobar Approximation

TL;DR

This paper develops a phenomenological framework for describing three-particle reactions in hadron spectroscopy using the isobar approximation, unitarity-preserving B-matrix equations, and analyzes the impact of analytic structures on rescattering processes.

Contribution

It introduces a B-matrix approach for three-particle amplitudes within the isobar approximation, ensuring unitarity and analyzing analytic structures relevant to rescattering.

Findings

B-matrix solution automatically satisfies unitarity.

Analytic structure influences rescattering solutions like the triangle diagram.

Comparison with other parameterizations enhances understanding of three-particle amplitudes.

Abstract

Further progress in hadron spectroscopy necessitates the phenomenological description of three particle reactions. We consider the isobar approximation, where the connected part of the $\mathbf{3}\to\mathbf{3}$ amplitude is first expressed as a sum over initial and final pairs, and then expanded into a truncated partial wave series. The resulting unitarity equation is automatically fulfilled by the $B$-matrix solution, which is an integral equation for the partial wave amplitudes, analogous to the $K$-matrix parameterization used to describe $\mathbf{2}\to\mathbf{2}$ amplitudes. We study the one particle exchange and how its analytic structure impacts rescattering solutions such as the triangle diagram. The analytic structure is compared to other parameterizations discussed in the literature. We briefly discuss the analogies with recent formalisms for extracting $\mathbf{3}\to\mathbf{3}$ scattering amplitudes in lattice QCD.