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≪mν≪MW, we retain full dependence on P, the momenta and masses of external particles, and we determine the branching ratios in the physical limit mν≪P≪MW. We show that the claim presented in Pham (Eur Phys J C8:513, 1999) that the τ→ℓℓ′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.