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

Patrick Blackstone,Matteo Fael,Emilie Passemar

doi:10.1140/epjc/s10052-020-8059-7

Abstract

We present in a full analytic form the partial widths for the lepton flavour violating decays mu ^{pm } rightarrow e^{pm } e^+ e^- and tau ^{pm } rightarrow 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 (Sov J Nucl Phys 25:340, 1977), 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 Pham (Eur Phys J C8:513, 1999) that the tau rightarrow 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.

Highlights

It is reported by several experimental collaborations, e.g., by CMS [3], ATLAS [4], LHCb [5], BABAR [6,7,8,9] andBelle [10], that the branching ratios for the charged lepton flavour violating (CLFV) decays τ ± → ± + −, with, = e, μ, can be as large as 10−14 in the Standard Model extended with either a Dirac or a Majorana mass term for neutrinos
Reference [1] employed for the evaluation of the one-loop diagrams the zero-momentum-limit (ZML) approximation, which assumes vanishing momenta and masses of the external particles while it retains the dependence on the internal masses of neutrinos and the W boson
Ref. [2], on the contrary, argued that once the external momentum dependence is taken into account, the GIM cancellation in L →, with L = τ or μ, becomes much milder, with a suppression only of the form | i U i UL∗i log xi |2, which leads to branching ratio values of the order of 10−14

Summary

Introduction

Reference [1] employed for the evaluation of the one-loop diagrams the zero-momentum-limit (ZML) approximation, which assumes vanishing momenta and masses of the external particles while it retains the dependence on the internal masses of neutrinos and the W boson. [2], on the contrary, argued that once the external momentum dependence is taken into account, the GIM cancellation in L → , with L = τ or μ, becomes much milder, with a suppression only of the form | i U i UL∗i log xi |2, which leads to branching ratio values of the order of 10−14.

Results

Conclusion