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

Abstract

We present a simple scheme for solving relativistic integral equations for the partial-wave projected three-body amplitudes. Our techniques are used to solve a problem of three scalar particles with a formation of a -wave two-body bound state. We rewrite the problem in a form suitable for numerical solution and then explore three solving strategies. In particular, we discuss different ways of incorporating the bound-state pole contribution in the integral equations. All of them lead to agreement with previous results obtained using finite-volume spectra of the same theory, providing further evidence of the validity of the existing finite- and infinite-volume formalism for studying three-particle systems. We discuss an analytic and numerical estimate of the systematic errors and provide numerical evidence that the methods presented allow for determination of amplitude above the three-body threshold as well. In conjunction with the previously derived finite-volume formalism, this work furthers the objective for extracting three-hadron scattering amplitudes directly from lattice QCD.