Decay amplitudes to three hadrons from finite-volume matrix elements

We derive relations between finite-volume matrix elements and infinite-volume decay amplitudes, for processes with three spinless, degenerate and either identical or non-identical particles in the final state. This generalizes the Lellouch-Luscher relation for two-particle decays and provides a strategy for extracting three-hadron decay amplitudes using lattice QCD. Unlike for two particles, even in the simplest approximation, one must solve integral equations to obtain the physical decay amplitude, a consequence of the nontrivial finite-state interactions. We first derive the result in a simplified theory with three identical particles, and then present the generalizations needed to study phenomenologically relevant three-pion decays. The specific processes we discuss are the CP-violating K -> 3 pi weak decay, the isospin-breaking eta -> 3 pi QCD transition, and the electromagnetic gamma (*) -> 3 pi amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic g - 2.

Automated summary

This paper discusses the relations between finite-volume matrix elements and infinite-volume decay amplitudes for processes with three spinless, degenerate particles, providing a strategy for extracting three-hadron decay amplitudes using lattice QCD. It is pointed out that solving integral equations is necessary to obtain the physical decay amplitude due to nontrivial finite-state interactions. The article derives results in a simplified theory with three identical particles and presents generalizations needed to study phenomenologically relevant three-pion decays.