The resonant $\pi^+\gamma\to\pi^+\pi^0$ amplitude from Quantum Chromodynamics

We present the first ab initio calculation of a radiative transition of a hadronic resonance within quantum chromodynamics (QCD). We compute the amplitude for $\pi\pi \to \pi\gamma^\star$, as a function of the energy of the $\pi\pi$ pair and the virtuality of the photon, in the kinematic regime where $\pi\pi$ couples strongly to the unstable $\rho$ resonance. This exploratory calculation is performed using a lattice discretization of QCD with quark masses corresponding to $m_\pi \approx 400$~MeV. We obtain a description of the energy dependence of the transition amplitude, constrained at 48 kinematic points, that we can analytically continue to the $\rho$ pole and identify from its residue the $\rho \to \pi\gamma^\star$ form factor.