τ→μμμ at a rate of one out of 1014 tau decays?

We present in a full analytic form the partial widths for the lepton flavour violating decays μ ± → e ± e + e - and τ ± → ℓ ± ℓ ′ + ℓ ′ - , with ℓ , ℓ ′ = μ , e , mediated by neutrino oscillations in the one-loop diagrams. Compared to the first result by Petcov (Sov J Nucl Phys 25:340, 1977), obtained in the zero momentum limit P ≪ m ν ≪ M W , we retain full dependence on P , the momenta and masses of external particles, and we determine the branching ratios in the physical limit m ν ≪ P ≪ M W . We show that the claim presented in Pham (Eur Phys J C8:513, 1999) that the τ → ℓ ℓ ′ ℓ ′ branching ratios could be as large as 10 - 14 , as a consequence of keeping the P dependence, is flawed. We find rates of order 10 - 55 , even smaller than those obtained in the zero momentum limit, as the latter prediction contains an unphysical logarithmic enhancement.