Radiative corrections to inverse muon decay for accelerator neutrinos

Inverse muon decay (${\nu }_{\mu }{e}^{-}\to {\nu }_e{\mu }^{-}$) is a promising tool to constrain neutrino fluxes with energies $E_{\nu }\ge 10.9\mathrm{G}\mathrm{e}\mathrm{V}$. Radiative corrections introduce percent-level distortions to energy spectra of outgoing muons and depend on experimental details. In this paper, we calculate radiative corrections to the scattering processes ${\nu }_{\mu }{e}^{-}\to {\nu }_e{\mu }^{-}$ and ${\bar{\nu }}_e{e}^{-}\to {\bar{\nu }}_{\mu }{\mu }^{-}$. We present the muon energy spectrum for both channels, double-differential distributions in muon energy and muon scattering angle and in photon energy and photon scattering angle, and the photon energy spectrum for the dominant ${\nu }_{\mu }{e}^{-}\to {\nu }_e{\mu }^{-}$ process. Our results clarify and extend the region of applicability of previous results in the literature for the double differential distribution in muon energy and photon energy, and in the muon energy spectrum with a radiated photon above a threshold energy. We provide analytic expressions for single, double and triple differential cross sections, and discuss how radiative corrections modify experimentally interesting observable distributions.