Relativistic, model-independent, multichannel $2\to2$ transition amplitudes in a finite volume

TL;DR

This paper develops a relativistic, model-independent formalism to extract infinite-volume two-particle transition amplitudes from finite-volume lattice QCD calculations, applicable to multiple coupled channels and relevant for studying unstable particles like the rho meson.

Contribution

It provides a new formalism relating finite-volume matrix elements to infinite-volume transition amplitudes for two-scalar states, accommodating multiple channels and particle degeneracies.

Findings

Formalism for 2→2 transition amplitudes derived

Applicable to multiple coupled scalar channels

Enables lattice QCD studies of unstable particles like the rho meson

Abstract

We derive formalism for determining $\textbf{2} + \mathcal J \to \textbf{2}$ infinite-volume transition amplitudes from finite-volume matrix elements. Specifically, we present a relativistic, model-independent relation between finite-volume matrix elements of external currents and the physically observable infinite-volume matrix elements involving two-particle asymptotic states. The result presented holds for states composed of two scalar bosons. These can be identical or non-identical and, in the latter case, can be either degenerate or non-degenerate. We further accommodate any number of strongly-coupled two-scalar channels. This formalism will, for example, allow future lattice QCD calculations of the $\rho$-meson form factor, in which the unstable nature of the $\rho$ is rigorously accommodated.