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

TL;DR

This paper analyzes rare tau decays mediated by nearly degenerate heavy neutrinos, deriving decay widths and CP asymmetries, and assesses their observability in the Belle II experiment, considering both Dirac and Majorana neutrino scenarios.

Contribution

It provides new formulas for effective decay widths and CP asymmetries in tau decays involving heavy neutrinos, including oscillation and finite detector effects.

Findings

Decay widths and CP asymmetries can be sizable within current experimental bounds.

Both LNC and LNV processes are possible if neutrinos are Majorana.

Potential observability of these rare decays at Belle II.

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 $\Gamma_{{\rm eff},\mp}^{\rm (X)}$ (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.