# Relating the finite-volume spectrum and the two-and-three-particle $S$ matrix for relativistic systems of identical scalar particles

## TL;DR

This paper derives an exact relation between finite-volume energy spectra and infinite-volume scattering amplitudes for coupled two- and three-particle systems of identical scalar particles in relativistic quantum field theory, valid below the four-particle threshold.

## Contribution

It provides a new quantization condition linking finite-volume spectra to multi-particle scattering amplitudes for relativistic scalar particles, extending previous two-particle results to include three-particle interactions.

## Key findings

- Derives an exact quantization condition for coupled two- and three-particle systems.
- Relates finite-volume energy levels to infinite-volume scattering amplitudes.
- Applicable below the four-particle threshold with exponential accuracy.

## Abstract

Working in relativistic quantum field theory, we derive the quantization condition satisfied by coupled two- and three-particle systems of identical scalar particles confined to a cubic spatial volume with periodicity $L$. This gives the relation between the finite-volume spectrum and the infinite-volume $\textbf 2 \to \textbf 2$, $\textbf 2 \to \textbf 3$ and $\textbf 3 \to \textbf 3$ scattering amplitudes for such theories. The result holds for relativistic systems composed of scalar particles with nonzero mass $m$, whose center of mass energy lies below the four-particle threshold, and for which the two-particle $K$ matrix has no singularities below the three-particle threshold. The quantization condition is exact up to corrections of the order $\mathcal{O}(e^{-mL})$ and holds for any choice of total momenta satisfying the boundary conditions.