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

## TL;DR

This paper develops a Lorentz-covariant formalism for relating finite-volume matrix elements to two-hadron transition amplitudes, with detailed implementation guidance and an example involving pion scattering and form factors.

## Contribution

It introduces a covariant finite-volume formalism for two-hadron states, including a new finite-volume function and detailed numerical evaluation methods.

## Key findings

- Derived a Lorentz-covariant formalism for finite-volume matrix elements.
- Introduced a new finite-volume function, G, for numerical evaluation.
- Applied the formalism to pion scattering, incorporating form factors and phase shifts.

## Abstract

In this work we develop a Lorentz-covariant version of the previously derived formalism for relating finite-volume matrix elements to $\textbf 2 + \mathcal J \to \textbf 2$ 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 $\pi \pi + \mathcal J \to \pi \pi$ amplitude in the $\rho$ channel, for which the single-pion form factor, $F_\pi(Q^2)$, as well as the scattering phase, $\delta_{\pi\pi}$, 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.