Multichannel one-to-two transition amplitudes in a finite volume

TL;DR

This paper extends finite-volume matrix element formulas to include arbitrary momentum transfer and coupled two-particle states, aiding calculations in heavy meson decays and meson photo production.

Contribution

It provides a model-independent, relativistic, non-perturbative generalization of the Lellouch-Luscher formula for complex two-particle states and arbitrary external momenta.

Findings

Generalized Lellouch-Luscher formula for multiple partial waves

Applicable to heavy meson decay and meson photo production

Discussed complications with unstable resonance states

Abstract

We derive a model-independent expression for finite-volume matrix elements. Specifically, we present a relativistic, non-perturbative analysis of the matrix element of an external current between a one-scalar in-state and a two-scalar out-state. Our result, which is valid for energies below higher-particle inelastic thresholds, generalizes the Lellouch-Luscher formula in two ways: we allow the external current to inject arbitrary momentum into the system and we allow for the final state to be composed an arbitrary number of strongly coupled two-particle states with arbitrary partial waves (including partial-wave mixing induced by the volume). We also illustrate how our general result can be applied to some key examples, such as heavy meson decays and meson photo production. Finally, we point out complications that arise involving unstable resonance states, such as $B\rightarrow K^*\ell^+\ell^-$ when staggered or mixed-action/partially-quenched calculations are performed.