Three particles in a finite volume: The breakdown of spherical symmetry

TL;DR

This paper analyzes how finite cubic volumes affect three-boson systems in lattice simulations, focusing on symmetry breakdown and methods to extract infinite-volume properties from finite-volume spectra.

Contribution

It provides a model-independent framework for understanding finite volume effects on three-particle systems and addresses symmetry breakdown and partial wave mixing.

Findings

Finite volume shifts can be used to determine infinite-volume scattering parameters.

Spherical symmetry breakdown effects are sub-percent compared to S-wave volume dependence.

Transition to boson-diboson scattering occurs in shallow bound states as volume decreases.

Abstract

Lattice simulations of light nuclei necessarily take place in finite volumes, thus affecting their infrared properties. These effects can be addressed in a model-independent manner using Effective Field Theories. We study the model case of three identical bosons (mass m) with resonant two-body interactions in a cubic box with periodic boundary conditions, which can also be generalized to the three-nucleon system in a straightforward manner. Our results allow for the removal of finite volume effects from lattice results as well as the determination of infinite volume scattering parameters from the volume dependence of the spectrum. We study the volume dependence of several states below the break-up threshold, spanning one order of magnitude in the binding energy in the infinite volume, for box side lengths L between the two-body scattering length a and L = 0.25a. For example, a state with a three-body energy of -3/(ma^2) in the infinite volume has been shifted to -10/(ma^2) at L = a. Special emphasis is put on the consequences of the breakdown of spherical symmetry and several ways to perturbatively treat the ensuing partial wave admixtures. We find their contributions to be on the sub-percent level compared to the strong volume dependence of the S-wave component. For shallow bound states, we find a transition to boson-diboson scattering behavior when decreasing the size of the finite volume.