QED corrections to neutrino-electron scattering

We evaluate the $O\left(\alpha \right)$ QED corrections to the recoil electron energy spectrum in the process ${\nu }_l+\overrightarrow{e}{\nu }_l+e\hspace{0.5em}(+\gamma ),$ where $(+\gamma )$ indicates the possible emission of a photon and $l=e,$ $\mu$ or $\tau .$ The soft and hard bremsstrahlung differential cross sections are computed for an arbitrary value of the photon energy threshold. We also study the $O\left(\alpha \right)$ QED corrections to the differential cross section with respect to the total combined energy of the recoil electron and a possible accompanying photon. Their difference from the corrections to the electron spectrum is investigated. We discuss the relevance and applicability of both radiative corrections, emphasizing their role in the analysis of precise solar neutrino-electron scattering experiments.