Next-to-leading order FsQED corrections to radiative pion pair production

We compute the next-to-leading order corrections to the radiative return process $e^+ e^- \to \pi^+ \pi^- \gamma$ within the FsQED approach to embed the pion form factor in the calculation of loop integrals. We compare our results with those of the factorised scalar QED approach, as well as with previous predictions obtained by us in the generalised vector meson dominance model. We show the numerical impact of the structure-dependent corrections on various observables of interest for radiative return experiments at flavour factories. Following recent input from the literature, we include in our calculation also leading corrections beyond FsQED and we provide a first estimate of such contributions. We also investigate additional mechanisms contributing to final-state radiation at center-of-mass energies around the $\phi$-meson resonance, such as radiative $\phi$ decays, presenting numerical results that are relevant for the KLOE experiment. These novel features are implemented in the Monte Carlo event generator BabaYaga@NLO, which can now be used to evaluate the impact of the modelling of pion-photon interaction in radiative return measurements.