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

TL;DR

This paper generalizes the Lellouch-Luscher formula to multi-channel systems with external currents in finite volume, enabling precise extraction of transition amplitudes relevant for meson decay and production studies.

Contribution

It provides a model-independent, exact relation between finite-volume matrix elements and infinite-volume transition amplitudes for multi-channel systems with arbitrary external currents.

Findings

Derived a generalized Lellouch-Luscher formula for multiple channels.

Applied the formula to meson decay and production processes.

Validated the approach with example scenarios.

Abstract

We perform a model-independent, non-perturbative investigation of two-point and three-point finite-volume correlation functions in the energy regime where two-particle states can go on-shell. We study three-point functions involving a single incoming particle and an outgoing two-particle state, relevant, for example, for studies of meson decays (e.g., B-to-pi Kll) or meson photo production (e.g., pi gamma-to-pi pi). We observe that, while the spectrum solely depends on the on-shell scattering amplitude, the correlation functions also depend on off-shell amplitudes. The main result of this work is a generalization of the Lellouch-Luscher formula relating matrix elements of currents in finite and infinite spatial volumes. We extend that work by considering a theory with multiple, strongly-coupled channels and by accommodating external currents which inject arbitrary four-momentum as well as arbitrary angular momentum. The result is exact up to exponentially suppressed corrections governed by the pion mass times the box size. We also apply our master equation to various examples, including the two processes mentioned above as well as examples where the final state is an admixture of two open channels.