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

TL;DR

This paper analytically calculates the decay rates of lepton flavor violating processes mediated by neutrino oscillations, showing that the branching ratios are extremely suppressed, around 10^{-55}, contradicting previous claims of higher rates.

Contribution

It provides a full analytic derivation of the decay widths including external momenta dependence, correcting prior approximations and clarifying the expected suppression of these rare decays.

Findings

Branching ratios are about 10^{-55}, much smaller than previous estimates.

Including full momentum dependence removes unphysical logarithmic enhancements.

The results challenge earlier claims of observable rates at current experimental sensitivities.

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 $\tau^\pm \to \ell^\pm \ell'^{+} \ell'^{-}$, 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.