Multiple-channel generalization of Lellouch-Lüscher formula

Abstract

We generalize the Lellouch-L\"uscher formula, relating weak matrix elements in finite and infinite volumes, to the case of multiple strongly coupled decay channels into two scalar particles. This is a necessary first step on the way to a lattice quantum chromodynamics calculation of weak decay rates for processes such as $D\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ and $D\ensuremath{\rightarrow}K\overline{K}$. We also present a field theoretic derivation of the generalization of L\"uscher's finite-volume quantization condition to multiple two-particle channels. We give fully explicit results for the case of two channels, including a form of the generalized Lellouch-L\"uscher formula expressed in terms of derivatives of the energies of finite-volume states with respect to the box size. Our results hold for arbitrary total momentum and for degenerate or nondegenerate particles.