CP violation in lepton number violating semihadronic decays of $K,D,D_s,B,B_c$

We study the CP violation in lepton number violating meson decays $M^{\pm} \to \ell_1^{\pm} \ell_2^{\pm} M^{’ \mp}$,where$M$and$M^{‘}$ are pseudoscalar mesons, $M=K,D,D_s,B,B_c$ and $M^{{}^{\mathrm{\prime }}}=\pi ,K,D,D_s$, and the charged leptons are ${\mathrm{\ell }}_1,{\mathrm{\ell }}_2=e,\mu$. It turns out that the CP-violating difference $S_{-}(M) \equiv [\Gamma(M^- \to \ell_1^- \ell_2^- M^{’ +})-\Gamma(M^+ \to \ell_1^+ \ell_2^+ M^{’ -})]$ 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: $\mathrm{\Delta }{M}_N\gg ̸{\mathrm{\Gamma }}_N$. We show that in such a case, the CP ratio ${\cal A}_{CP}(M) \equiv [\Gamma(M^- \to \ell_1^- \ell_2^- M^{’ +})-\Gamma(M^+ \to \ell_1^+ \ell_2^+ M^{’ -})]/[\Gamma(M^- \to \ell_1^- \ell_2^- M^{’ +})+\Gamma(M^+ \to \ell_1^+ \ell_2^+ M^{’ -})]$ becomes a quantity $\sim 1$. The observation of CP violation in these decays would be consistent with the existence of the well-motivated $\nu$MSM model with two almost degenerate heavy neutrinos in the mass range between $M_N \sim 0.1$-$10^1$ GeV.