$$\tau \rightarrow \mu \mu \mu $$ τ→μμμ at a rate of one out of $$10^{14}$$ 1014 tau decays?

Abstract We present in a full analytic form the partial widths for the lepton flavour violating decays $$\mu ^{\pm } \rightarrow e^{\pm } e^+ e^-$$ μ±→e±e+e- and $$\tau ^{\pm } \rightarrow \ell ^{\pm } \ell '^{+} \ell '^{-}$$ τ±→ℓ±ℓ′+ℓ′- , with $$\ell ,\ell '=\mu ,e$$ 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 $${\mathcal {P}}\ll m_{\nu } \ll M_W$$ P≪mν≪MW , we retain full dependence on $${\mathcal {P}}$$ P , the momenta and masses of external particles, and we determine the branching ratios in the physical limit $$m_\nu \ll {\mathcal {P}} \ll M_W$$ mν≪P≪MW . We show that the claim presented in Pham (Eur Phys J C8:513, 1999) that the $$\tau \rightarrow \ell \ell ' \ell '$$ τ→ℓℓ′ℓ′ branching ratios could be as large as $$10^{-14}$$ 10-14 , as a consequence of keeping the $${\mathcal {P}}$$ P dependence, is flawed. We find rates of order $$10^{-55}$$ 10-55 , even smaller than those obtained in the zero momentum limit, as the latter prediction contains an unphysical logarithmic enhancement.