ππ→πγ$^{∗}$ amplitude and the resonant ρ→πγ$^{∗}$ transition from lattice QCD

Abstract

We present a determination of the P-wave ππ→πγ$^{⋆}$ transition amplitude from lattice quantum chromodynamics. Matrix elements of the vector current in a finite volume are extracted from three-point correlation functions, and from these we determine the infinite-volume amplitude using a generalization of the Lellouch-Lüscher formalism. We determine the amplitude for a range of discrete values of the ππ energy and virtuality of the photon and observe the expected dynamical enhancement due to the ρ resonance. Describing the energy dependence of the amplitude, we are able to analytically continue into the complex energy plane and from the residue at the ρ pole extract the ρ→πγ$^{⋆}$ transition form factor. This calculation, at m$_{π}$ ≈ 400 MeV, is the first to determine the form factor of an unstable hadron within a first principles approach to QCD.