τ to 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 $\mu^\pm \to e^\pm e^+ e^-$ and $\tau^\pm \to \ell^\pm \ell'^{+} \ell'^{-}$, with $\ell,\ell'=\mu,e$, mediated by neutrino oscillations in the one-loop diagrams. Compared to the first result by Petcov in [1], obtained in the zero momentum limit $\mathcal{P}\ll m_{\nu} \ll M_W$, we retain full dependence on $\mathcal{P}$, the momenta and masses of external particles, and we determine the branching ratios in the physical limit $m_\nu \ll \mathcal{P} \ll M_W$. We show that the claim presented in [2] that the $\tau \to \ell \ell' \ell'$ branching ratios could be as large as $10^{-14}$, as a consequence of keeping the $\mathcal{P}$ 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.