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

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 Γ_eff,∓^X (X=LNC, LNV) of these rare decays, where we account for N₁-N₂ 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 ∆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.