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

Abstract

We present in a full analytic form the partial widths for the lepton flavour violating decays ${\mu }^{\pm }\to e^{\pm }{e}^{+}{e}^{-}$ and ${\tau }^{\pm }\to {\ell }^{\pm }{\ell }^{\prime +}{\ell }^{\prime -}$, with $\ell ,{\ell }^{\prime }=\mu ,e$, 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 $\tau \to \ell {\ell }^{\prime }{\ell }^{\prime }$ 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.

Comments: 6 pages, 2 figures, 1 table. v2 version published in EPJC, minor changes