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 $\tau^{\mp} \to \pi^{\mp} N_j \to \pi^{\mp} \mu^{\pm} \pi^{\mp}$

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 $\Gamma_{{\rm eff},\mp}^{\rm (X)}$ (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 $\Delta \Gamma_{\rm CP}^{\rm (X)} = (\Gamma_{{\rm eff},-}^{\rm (X)} - \Gamma_{{\rm eff},+}^{\rm (X)})$. 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.
Comment: 14 pages, 7 figures; v2: references added