Numerical study of the relativistic three-body quantization condition in the isotropic approximation

TL;DR

This paper demonstrates a numerical approach to solving the relativistic three-particle quantization condition in the isotropic approximation, enabling practical calculations of three-particle systems in finite volume.

Contribution

It introduces a numerical method for solving the relativistic three-particle quantization condition using the isotropic approximation and leading effective range terms, with validation against known analytic results.

Findings

Successfully reproduces known volume dependence of three-particle states

Provides a numerical solution framework for relating scattering quantities to finite-volume spectra

Identifies unphysical finite-volume energies arising from certain parameter choices

Abstract

We present numerical results showing how our recently proposed relativistic three-particle quantization condition can be used in practice. Using the isotropic (generalized $s$-wave) approximation, and keeping only the leading terms in the effective range expansion, we show how the quantization condition can be solved numerically in a straightforward manner. In addition, we show how the integral equations that relate the intermediate three-particle infinite-volume scattering quantity, $\mathcal K_{\text{df},3}$, to the physical scattering amplitude can be solved at and below threshold. We test our methods by reproducing known analytic results for the $1/L$ expansion of the threshold state, the volume dependence of three-particle bound-state energies, and the Bethe-Salpeter wavefunctions for these bound states. We also find that certain values of $\mathcal K_{\text{df},3}$ lead to unphysical finite-volume energies, and give a preliminary analysis of these artifacts.