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 ${\tau }^{\pm }\to M_1{}^{\pm }{N}_j\to M_1^{\pm }{M}_2^{\pm }{\mathrm{\ell }}^{\mp }$. We calculate the Branching ratios $Br({\tau }^{\pm })$ and the CP asymmetry $(\mathrm{\Gamma }({\tau }^{+})-\mathrm{\Gamma }({\tau }^{-}))\mathrm{/}(\mathrm{\Gamma }({\tau }^{+})+\mathrm{\Gamma }({\tau }^{-}))$ 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 $\mathrm{B}\mathrm{r}({\tau }^{\pm })$ when their mass difference is comparable with their decay width, $\mathrm{\Delta }{M}_N\sim {\mathrm{\Gamma }}_N$. 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.