Form factors of two-hadron states from a covariant finite-volume formalism

Abstract

In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $\mathbf{2}+\mathcal{J}\ensuremath{\rightarrow}\mathbf{2}$ transition amplitudes. We also give various details relevant for the implementation of this formalism in a realistic numerical lattice QCD calculation. Particular focus is given to the role of single-particle form factors in disentangling finite-volume effects from the triangle diagram that arise when $\mathcal{J}$ couples to one of the two hadrons. This also leads to a new finite-volume function, denoted $G$, the numerical evaluation of which is described in detail. As an example we discuss the determination of the $\ensuremath{\pi}\ensuremath{\pi}+\mathcal{J}\ensuremath{\rightarrow}\ensuremath{\pi}\ensuremath{\pi}$ amplitude in the $\ensuremath{\rho}$ channel, for which the single-pion form factor, ${F}_{\ensuremath{\pi}}({Q}^{2})$, as well as the scattering phase, ${\ensuremath{\delta}}_{\ensuremath{\pi}\ensuremath{\pi}}$, are required to remove all power-law finite-volume effects. The formalism presented here holds for local currents with arbitrary Lorentz structure, and we give specific examples of insertions with up to two Lorentz indices.