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

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, M₃. In previous work [Phys. Rev. D 90, 116003 (2014)], we found a quantization condition relating the spectrum to a nonstandard infinite-volume quantity, denoted Kdf,3. Here we present the relation between Kdf,3 and M₃. We then discuss briefly how our now completed formalism can be practically implemented to extract M₃ from the finite-volume energy spectrum.