Three-particle systems with resonant subprocesses in a finite volume

TL;DR

This paper extends the relativistic three-particle quantization condition to include cases where two-particle subprocesses involve resonant poles, enabling more accurate analysis of resonant three-particle systems in finite volume.

Contribution

It derives a new quantization condition for identical scalar particles with resonant two-particle subprocesses, broadening the applicability of previous models.

Findings

Derived the quantization condition with resonant poles in two-particle K matrices.

Connected finite-volume quantities to physical three-to-three scattering amplitudes.

Enabled rigorous treatment of resonances like the rho meson in three-particle systems.

Abstract

In previous work, we have developed a relativistic, model-independent three-particle quantization condition, but only under the assumption that no poles are present in the two-particle K matrices that appear as scattering subprocesses. Here we lift this restriction, by deriving the quantization condition for identical scalar particles with a G-parity symmetry, in the case that the two-particle K matrix has a pole in the kinematic regime of interest. As in earlier work, our result involves intermediate infinite-volume quantities with no direct physical interpretation, and we show how these are related to the physical three-to-three scattering amplitude by integral equations. This work opens the door to study processes such as $a_2 \to \rho \pi \to \pi \pi \pi$, in which the $\rho$ is rigorously treated as a resonance state.