Three-body spectrum in a finite volume: the role of cubic symmetry

TL;DR

This paper develops a method to analyze three-particle spectra in finite cubic volumes by leveraging lattice symmetry, aiding the interpretation of energy levels in bound and scattering states.

Contribution

It introduces a cubic symmetry-based projection of the three-body quantization condition, improving the understanding of three-particle states in finite-volume lattice calculations.

Findings

The projection simplifies the three-body problem in finite volume.

Numerical solutions clarify the nature of energy eigenvalues.

Method aids interpretation of bound and scattering states.

Abstract

The three-particle quantization condition is partially diagonalized in the center-of-mass frame by using cubic symmetry on the lattice. To this end, instead of spherical harmonics, the kernel of the Bethe-Salpeter equation for particle-dimer scattering is expanded in the basis functions of different irreducible representations of the octahedral group. Such a projection is of particular importance for the three-body problem in the finite volume due to the occurrence of three-body singularities above breakup. Additionally, we study the numerical solution and properties of such a projected quantization condition in a simple model. It is shown that, for large volumes, these solutions allow for an instructive interpretation of the energy eigenvalues in terms of bound and scattering states.