Rare tau decays via exchange of on-shell almost degenerate Majorana neutrinos, ${\tau }^{\mp }\to {\pi }^{\mp }{N}_j\to {\pi }^{\mp }{\mu }^{\mp }{\pi }^{\pm }$ and ${\tau }^{\mp }\to {\pi }^{\mp }{N}_j\to {\pi }^{\mp }{\mu }^{\pm }{\pi }^{\mp }$

Abstract

We investigate rare decays of tau leptons that occur via exchange of heavy on-shell neutrinos $N_j$ ($j=1,2$). These neutrinos can be either Dirac or Majorana, and are considered to be almost degenerate in mass. The decays can thus be either lepton number conserving (LNC), ${\tau }^{\mp }\to {\pi }^{\mp }{N}_j\to {\pi }^{\mp }{\mu }^{\mp }{\pi }^{\pm }$, or lepton number violating (LNV), ${\tau }^{\mp }\to {\pi }^{\mp }{N}_j\to {\pi }^{\mp }{\mu }^{\pm }{\pi }^{\mp }$. If neutrinos are Dirac, only LNC decays are possible. If they are Majorana, both LNC and LNV are possible. We derive the corresponding expressions for the effective decay widths ${\Gamma }_{\mathrm{eff},\mp }^{(\mathrm{X})}$ (X=LNC, LNV) of these rare decays, where we account for $N_1$-$N_2$ overlap and oscillation effects and for the finite detector length effects. We then numerically evaluate these decay widths as well as the related CP violation asymmetry width $\Delta {\Gamma }_{\mathrm{CP}}^{(\mathrm{X})}=({\Gamma }_{\mathrm{eff},-}^{(\mathrm{X})}-{\Gamma }_{\mathrm{eff},+}^{(\mathrm{X})})$. We conclude that for certain, presently allowed, ranges of the heavy-light neutrino mixing parameters, such decays and asymmetries could be observed in Belle II experiment.

Comments: 14 pages, 7 figures; v2: references added