Flavor violating leptonic decays of $\tau$ and $\mu$ leptons in the Standard Model with massive neutrinos

TL;DR

This paper recalculates flavor-violating leptonic decays of tau and mu leptons within the Standard Model with neutrino masses, finding much smaller branching ratios than previously claimed, thus any future detection would indicate new physics.

Contribution

It provides a refined calculation of leptonic decay branching ratios in the Standard Model, correcting earlier overestimations by including external momenta and masses.

Findings

Branching ratios are strongly suppressed compared to previous claims.

Results align with earlier approximations neglecting external momenta.

Future observations would signal physics beyond the Standard Model.

Abstract

We have revisited the computations of the flavor violating leptonic decays of the $\tau$ and $\mu$ leptons into three lighter charged leptons in the Standard Model with non-vanishing neutrino masses. We were driven by a claimed unnaturally large branching ratio predicted for the $\tau^-\to \mu^- \ell^+ \ell^-$ ($\ell=\mu, e$) decays \cite{Pham:1998fq}, which was at odds with the corresponding predictions for the $\mu^-\to e^- e^-e^+$ processes \cite{Petcov:1976ff}. In contrast with the prediction in \cite{Pham:1998fq}, our results are strongly suppressed and in good agreement with the approximation done in ref.~\cite{Petcov:1976ff}, where masses and momenta of the external particles were neglected in order to deal with the loop integrals. However, as a result of keeping external momenta and masses in the computation of the dominant penguin and box diagrams- we even find slightly smaller branching fractions. Therefore, we confirm that any future observation of such processes would be an unambiguous manifestation of new physics beyond the Standard Model