Rare tau decays via exchange of on-shell almost degenerate Majorana neutrinos, τ∓ → π∓Nj → π∓μ∓π± and τ∓ → π∓Nj → π∓μ±π∓

A bstract 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), τ ∓ → π ∓ N j → π ∓ μ ∓ π ± , or lepton number violating (LNV), τ ∓ → π ∓ N j → π ∓ μ ± π ∓ . 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 Γeff,∓(X) Γ eff , ∓ 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 Δ ΓCP(X)=(Γeff,-(X)-Γeff,+(X)) ∆ Γ CP X = Γ eff , − X − Γ eff , + 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.