The $\pi\gamma \to \pi\pi$ transition and the $\rho$ radiative decay width from lattice QCD

We report a lattice QCD determination of the $\pi\gamma \to \pi\pi$ transition amplitude for the $P$-wave, $I=1$ two-pion final state, as a function of the photon virtuality and $\pi\pi$ invariant mass. The calculation was performed with $2+1$ flavors of clover fermions at a pion mass of approximately $320$ MeV, on a $32^3 \times 96$ lattice with $L\approx 3.6$ fm. We construct the necessary correlation functions using a combination of smeared forward, sequential and stochastic propagators, and determine the finite-volume matrix elements for all $\pi\pi$ momenta up to $|\vec{P}|= \sqrt{3} \frac{2\pi}{L}$ and all associated irreducible representations. In the mapping of the finite-volume to infinite-volume matrix elements using the Lellouch-L\"uscher factor, we consider two different parametrizations of the $\pi\pi$ scattering phase shift. We fit the $q^2$ and $s$ dependence of the infinite-volume transition amplitude in a model-independent way using series expansions, and compare multiple different truncations of this series. Through analytic continuation to the $\rho$ resonance pole, we also determine the $\pi\gamma \to \rho$ resonant transition form factor and the $\rho$ meson photocoupling, and obtain $|G_{\rho\pi\gamma}| = 0.0802(32)(20)$.