Expressing the three-particle finite-volume spectrum in terms of the three-to-three scattering amplitude

TL;DR

This paper completes the theoretical framework connecting finite-volume energy spectra to the three-to-three scattering amplitude in scalar quantum field theories, enabling more accurate extraction of scattering information from lattice simulations.

Contribution

It establishes the relation between the previously introduced non-standard quantity ${ m K}_{{ m df},3}$ and the physical three-to-three scattering amplitude ${ m M}_3$, completing the formalism.

Findings

Derived the relation between ${ m K}_{{ m df},3}$ and ${ m M}_3$

Provided a pathway for practical extraction of ${ m M}_3$ from finite-volume spectra

Enhanced the theoretical tools for analyzing three-particle interactions in lattice QFT

Abstract

In this article we complete our formalism relating the finite-volume energy spectrum of a scalar quantum field theory to the three-to-three scattering amplitude, ${\cal M}_3$. In previous work we found a quantization condition relating the spectrum to a non-standard infinite-volume quantity, denoted ${\cal K}_{{\rm df},3}$. Here we present the relation between ${\cal K}_{{\rm df},3}$ and ${\cal M}_3$. We then discuss briefly how our now completed formalism can be practically implemented to extract ${\cal M}_3$ from the finite-volume energy spectrum.