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

Abstract

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-Lüscher 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 $|G_{\rho \pi \gamma }|=0.0802(32)(20)$.

28 pages, 19 figures. Corrected $\rho$ radiative decay width by factor of 2 and re-evaluated $\pi \gamma \to \pi \pi$ cross section for unsymmetrized ${\pi }^{+}{\pi }^0$ final state

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