Solving relativistic three-body integral equations in the presence of bound states

TL;DR

This paper introduces a numerically systematic method for solving relativistic three-body integral equations, employing discretization and extrapolation techniques, validated through scalar particle models and applicable beyond the three-particle threshold.

Contribution

The paper develops a discretization and extrapolation approach for relativistic three-body equations, including pole contributions, with validation against previous finite-volume spectrum results.

Findings

Method accurately reproduces known results for scalar particles.

Systematic errors are quantitatively estimated.

Approach extends to energies above the three-particle threshold.

Abstract

We present a systematically improvable method for numerically solving relativistic three-body integral equations for the partial-wave projected amplitudes. The method consists of a discretization procedure in momentum space, which approximates the continuum problem with a matrix equation. It is solved for different matrix sizes, and in the end, an extrapolation is employed to restore the continuum limit. Our technique is tested by solving a three-body problem of scalar particles with an $S$ wave two-body bound state. We discuss two methods of incorporating the pole contribution in the integral equations, both of them leading to agreement with previous results obtained using finite-volume spectra of the same theory. We provide an analytic and numerical estimate of the systematic errors. Although we focus on kinematics below the three-particle threshold, we provide numerical evidence that the methods presented allow for determination of amplitude above this threshold as well.