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∓

Abstract

We investigate rare decays of tau leptons that occur via exchange of heavy on-shell neutrinos Nj (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∓Nj → π∓μ∓π±, or lepton number violating (LNV), τ∓ → π∓Nj → 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\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ {\\Gamma}_{\ extrm{eff},\\mp}^{\\left(\ extrm{X}\\right)} $$\\end{document} (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 ∆ΓCPX=Γeff,−X−Γeff,+X\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$ \\Delta {\\Gamma}_{\ extrm{CP}}^{\\left(\ extrm{X}\\right)}=\\left({\\Gamma}_{\ extrm{eff},-}^{\\left(\ extrm{X}\\right)}-{\\Gamma}_{\ extrm{eff},+}^{\\left(\ extrm{X}\\right)}\\right) $$\\end{document}. 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.