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

AbstractWe 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.