Three particles in a finite volume

TL;DR

This paper extends the Lüscher formula to three-particle systems in finite volumes, providing a theoretical framework for analyzing three-particle interactions and spectra in lattice simulations.

Contribution

It derives a generalized Lüscher formula including inelastic channels and discusses Faddeev equations in finite volume, linking finite and infinite volume scattering data.

Findings

Generalized Lüscher formula for three particles with inelastic channels

Finite-volume spectrum determined by infinite-volume S-matrix elements

Corrections exponentially suppressed at large volumes

Abstract

Within the non-relativistic potential scattering theory, we derive a generalized version of the L\"uscher formula, which includes three-particle inelastic channels. Faddeev equations in a finite volume are discussed in detail. It is proved that, even in the presence of the three-particle intermediate states, the discrete spectrum in a finite box is determined by the infinite-volume elements of the scattering S-matrix up to corrections, exponentially suppressed at large volumes.