We study the $CP$ violation in lepton-number-violating meson decays $M^{\pm }\to {\ell }_1^{\pm }{\ell }_2^{\pm }{M}^{\prime \mp }$, where $M$ and $M^{\prime }$ are pseudoscalar mesons, $M=K,D,D_s,B,B_c$ and $M^{\prime }=\pi ,K,D,D_s$, and the charged leptons are ${\ell }_1,{\ell }_2=e,\mu$. It turns out that the $CP$-violating difference $S_{-}(M)\equiv [\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M}^{\prime +})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M}^{\prime -})]$ can become appreciable when two intermediate on-shell Majorana neutrinos $N_j$ ($j=1,2$) participate in these decays. Our calculations show that the asymmetry becomes largest when the masses of $N_1$ and $N_2$ are almost degenerate, i.e., when the mass difference $\mathrm{\Delta }{M}_N$ becomes comparable with the (small) decay widths ${\mathrm{\Gamma }}_N$ of these neutrinos: $\mathrm{\Delta }{M}_N\cancel{\gg }{\mathrm{\Gamma }}_N$. We show that in such a case, the $CP$ ratio ${\mathcal{A}}_{CP}(M)\equiv [\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M}^{\prime +})-\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M}^{\prime -})]/[\mathrm{\Gamma }(M^{-}\to {\ell }_1^{-}{\ell }_2^{-}{M}^{\prime +})+\mathrm{\Gamma }(M^{+}\to {\ell }_1^{+}{\ell }_2^{+}{M}^{\prime -})]$ becomes a quantity $\sim 1$. The observation of $CP$ violation in these decays would be consistent with the existence of the well-motivated $\nu \mathrm{MSM}$ model with two almost degenerate heavy neutrinos in the mass range $M_N\sim 0.1–{10}^1\mathrm{GeV}$.

• Received 13 March 2014

DOI:https://doi.org/10.1103/PhysRevD.89.093012