QED radiative corrections in inverse beta decay from virtual pions

Inverse beta decay (IBD), $\overline{\nu}_e p \to e^+ n \left( \gamma \right)$, is the main detection channel for reactor and supernova antineutrinos. To provide precise IBD cross sections at antineutrino energies $E_{\overline{\nu}_e} \gtrsim 10~\mathrm{MeV}$, we evaluate radiative corrections from virtual pions within the framework of heavy baryon chiral perturbation theory. At leading order, only the pion isospin-splitting contributions are not suppressed by the electron mass. At next-to-leading order, besides recoil effects, only the Wilson coefficient $c_4$ contributes to the kinematic dependence. However, its precise value is not relevant for IBD at relatively low energies since all next-to-leading order radiative corrections are relatively small. We find the kinematic dependence of the pion-induced QED radiative corrections at the level and below the uncertainty from the momentum dependence of the nucleon form factors. Our results enable sub-permille theoretical precision of charged-current elastic (anti)neutrino-nucleon scattering at antineutrino energies $E_{\overline{\nu}_e} \gtrsim 10~\mathrm{MeV}$.