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We present a determination of the $P$-wave $\pi\pi\to\pi\gamma^\star$transition amplitude from lattice quantum chromodynamics. Matrix elements ofthe vector current in a finite-volume are extracted from three-pointcorrelation functions, and from these we determine the infinite-volumeamplitude using a generalization of the Lellouch-L\"uscher formalism. Wedetermine the amplitude for a range of discrete values of the $\pi\pi$ energyand virtuality of the photon, and observe the expected dynamical enhancementdue to the $\rho$ resonance. Describing the energy dependence of the amplitude,we are able to analytically continue into the complex energy plane and from theresidue at the $\rho$ pole extract the $\rho\to \pi \gamma^\star$ transitionform factor. This calculation, at $m_\pi\approx 400$ MeV, is the first todetermine the form factor of an unstable hadron within a first principlesapproach to QCD.

The $\pi\pi\to\pi\gamma^\star$ amplitude and the resonant $\rho\to\pi\gamma^\star$ transition from lattice QCD