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

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, ${\mathcal{M}}_3$. In previous work [Phys. Rev. D 90, 116003 (2014)], we found a quantization condition relating the spectrum to a nonstandard infinite-volume quantity, denoted ${\mathcal{K}}_{\mathrm{df},3}$. Here we present the relation between ${\mathcal{K}}_{\mathrm{df},3}$ and ${\mathcal{M}}_3$. We then discuss briefly how our now completed formalism can be practically implemented to extract ${\mathcal{M}}_3$ from the finite-volume energy spectrum.