High-energy electrons from the muon decay in orbit: radiative corrections

We determine the $\mathcal{O}(\alpha)$ correction to the energy spectrum of electrons produced in the decay of muons bound in atoms. We focus on the high-energy end of the spectrum that constitutes a background for the muon-electron conversion and will be precisely measured by the upcoming experiments Mu2e and COMET. The correction suppresses the background by about 15\%.