# 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-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 $g-2$.

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