Resonant CP violation in rare tau decay

In this work, we study the lepton number violating tau decays via intermediate on-shell Majorana neutrinos $N_j$ into two scalar mesons and a lepton $\tau^{\pm} \to M_1{}^{\pm} N_j \to M_1^{\pm} M_2^{\pm} \ell^{\mp}$. We calculate the Branching ratios $Br(\tau^{\pm})$ and the CP asymmetry $(\Gamma(\tau^+) - \Gamma(\tau^-))/(\Gamma(\tau^+) + \Gamma(\tau^-))$ for such decays, in a scenario that contain at least two heavy Majorana neutrinos. The results show that the CP asymmetry is small, but becomes comparable with the branching ratio ${\rm Br}(\tau^{\pm})$ when their mass difference is comparable with their decay width, $\Delta M_N \sim \Gamma_N$. On the other hand, we present regions of the heavy-light neutrino mixing elements, in which the $CP$ asymmetry could be explore in future tau factories.