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

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.

32 pages, 21 figures, JLAB-THY-18-2657, CERN-TH-2018-046; version 2: corrected typos, updated references, minor stylistic changes---consistent with published version

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