Radiative return meets GVMD

We improve the description of pion-photon interactions in the radiative return process $e^+e^-\to \pi^+\pi^-\gamma$ at the next-to-leading order by including the pion form factor in the Feynman rules. We present a general calculation of the new amplitudes, and provide an implementation easy to interface with any Monte Carlo generator. We incorporate this framework into the event generator $\texttt{Phokhara}$ and study several experimental configurations. Overall, we find percent-level effects appearing in angular differential cross section distributions at colliders whose centre-of-mass energies lie near the peak of the pion form factor. By contrast, total cross sections and distributions in charge-even variables show effects only at the permille level, or no visible differences at all. Finally, we compare the new predictions with KLOE measurements of the forward-backward asymmetry in order to assess the predictive power of the modifications.