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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 $M^{'}=π, K, D, D_s$, and the charged leptons are $\ell_1, \ell_2 = e, μ$. It turns out that the CP-violating difference $S_{-}(M) \equiv [Γ(M^- \to \ell_1^- \ell_2^- M^{' +})-Γ(M^+ \to \ell_1^+ \ell_2^+ M^{' -})]$ 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 $ΔM_N$ becomes comparable with the (small) decay widths $Γ_N$ of these neutrinos: $ΔM_N \not\gg Γ_N$. We show that in such a case, the CP ratio ${\cal A}_{CP}(M) \equiv [Γ(M^- \to \ell_1^- \ell_2^- M^{' +})-Γ(M^+ \to \ell_1^+ \ell_2^+ M^{' -})]/[Γ(M^- \to \ell_1^- \ell_2^- M^{' +})+Γ(M^+ \to \ell_1^+ \ell_2^+ M^{' -})]$ becomes a quantity $\sim 1$. The observation of CP violation in these decays would be consistent with the existence of the well-motivated $ν$MSM model with two almost degenerate heavy neutrinos in the mass range between $M_N \sim 0.1$-$10^1$ GeV.