NLO prediction for the decays $\tau \to \ell \ell'\ell' \nu \bar \nu$ and $\mu \to e e e \nu \bar\nu$

These proceedings review the differential decay rates and the branching ratios of the tau and muon decays \tau \to \ell \ell' \ell' \nu \bar\nuτ→ℓℓ′ℓ′νν‾ (with \ell,\ell'=\mu,eℓ,ℓ′=μ,e) and \mu \to e e e \nu \bar \nuμ→eeeνν‾ 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 e e eμ→eee and \tau \to \ell \ell' \ell'τ→ℓℓ′ℓ′.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 eeeμ→eee and \tau \to \ell \ell'\ell'τ→ℓℓ′ℓ′, the NLO corrections decrease the leading-order prediction by about 10 - 20%.