Decay amplitudes to three hadrons from finite-volume matrix elements

TL;DR

This paper extends finite-volume lattice QCD methods to relate three-hadron decay amplitudes to finite-volume matrix elements, enabling the study of complex three-particle decays like $K o 3 ext{pi}$ and $ ext{eta} o 3 ext{pi}$.

Contribution

It generalizes the Lellouch-L"uscher relation to three-particle decays and introduces integral equations for extracting physical amplitudes from finite-volume data.

Findings

Derived relations for three-hadron decay amplitudes from finite-volume matrix elements.

Presented a strategy for lattice QCD calculations of three-particle decays.

Applied the formalism to processes like $K o 3 ext{pi}$, $ ext{eta} o 3 ext{pi}$, and $ ext{gamma}^* o 3 ext{pi}$.

Abstract

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-L\"uscher 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 $K \to 3\pi$ weak decay, the isospin-breaking $\eta \to 3\pi$ QCD transition, and the electromagnetic $\gamma^*\to 3\pi$ amplitudes that enter the calculation of the hadronic vacuum polarization contribution to muonic $g-2$.