NLO prediction for the decays $\tau \to \mathrm{\ell }{\mathrm{\ell }}^{\mathrm{\prime }}{\mathrm{\ell }}^{\mathrm{\prime }}\nu \bar{\nu }$ and $\mu \to eee\nu \bar{\nu }$

These proceedings review the differential decay rates and the branching ratios of the tau and muon decays $\tau \to \mathrm{\ell }{\mathrm{\ell }}^{\mathrm{\prime }}{\mathrm{\ell }}^{\mathrm{\prime }}\nu \bar{\nu }$ (with $\mathrm{\ell },{\mathrm{\ell }}^{\mathrm{\prime }}=\mu ,e$) and $\mu \to eee\nu \bar{\nu }$ in the Standard Model at NLO. These five-body leptonic decays are a tool to study the Lorentz structure of weak interactions and to test lepton flavour universality. They are also a source of SM background to searches for the lepton-flavour-violating decays $\mu \to eee$ and $\tau \to \mathrm{\ell }{\mathrm{\ell }}^{\mathrm{\prime }}{\mathrm{\ell }}^{\mathrm{\prime }}$. Even if the shift in the branching ratios induced by radiative corrections turns out to be small and of order 1% — mainly due to a running effect of the fine structure constant — locally in the phase space these corrections can reach the 5 - 10% level, depending on the applied cuts. We found for instance that in the phase space region where the neutrino energies are small, and the momenta of the three charged leptons have a similar signature as in $\mu \to eee$ and $\tau \to \ell \ell’\ell’$, the NLO corrections decrease the leading-order prediction by about 10 - 20%.