Radiative return at NLOPS accuracy

The radiative return, together with the energy scan, is the method used at flavour factories to measure the pion form factor, which is a crucial input for the data-driven dispersive computation of the leading-order hadronic contribution to the muon anomalous magnetic moment. We consider the radiative hadronic and leptonic channels of main experimental interest, namely the processes $e^+e^-\to X^+X^-\gamma$, with $X = \{\pi \, , \mu \}$. For such processes, we compute the exact next-to-leading order (NLO) corrections matched to a Parton Shower (PS) to describe exclusive multiple photon emission. All sources of radiative corrections from initial-state and final-state radiation, as well as their interference, are considered according to QED for $e^+e^-\to\mu^+\mu^-\gamma$ and QED$\oplus$F$\times$sQED (Factorised scalar QED) for $e^+e^-\to\pi^+\pi^-\gamma$. We describe in detail the novel features of our PS approach to compute the fixed-order corrections in association with higher-order contributions to $2\to3$ processes, with a hard photon in the final state. We present validation tests and comparisons with NLO predictions available in the literature to cross-check various ingredients of our formulation. We also show numerical results at NLOPS accuracy according to realistic event selection criteria for precision measurements at flavour factories. Our calculation is implemented in an updated version of the Monte Carlo event generator BabaYaga@NLO, which can be used for fully exclusive simulations and data analysis in radiative return experiments.