Next-to-leading order prediction for the decay $\mu \to e(e^{+}{e}^{-})\nu \bar{\nu }$

We present the differential decay rates and the branching ratios of the muon decay with internal conversion, $\mu \to e(e^{+}{e}^{-})\nu \bar{\nu }$, in the Standard Model at next-to-leading order (NLO) in the on-shell scheme. This rare decay mode of the muon is among the main sources of background to the search for $\mu \to eee$ decay. We found that in the phase space region where the neutrino energies are small, and the three-electron momenta have a similar signature as in the $\mu \to eee$ decay, the NLO corrections decrease the leading-order prediction by about $10-20\mathrm{\%}$