CPviolation in lepton number violating semihadronic decays ofK,D,Ds,B,Bc

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Abstract

We study the $CP$ violation in lepton-number-violating meson decays ${M}^{\ifmmode\pm\else\textpm\fi{}}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{\ifmmode\pm\else\textpm\fi{}}{\ensuremath{\ell}}_{2}^{\ifmmode\pm\else\textpm\fi{}}{M}^{\ensuremath{'}\ensuremath{\mp}}$, where $M$ and ${M}^{\ensuremath{'}}$ are pseudoscalar mesons, $M=K,D,{D}_{s},B,{B}_{c}$ and ${M}^{\ensuremath{'}}=\ensuremath{\pi},K,D,{D}_{s}$, and the charged leptons are ${\ensuremath{\ell}}_{1},{\ensuremath{\ell}}_{2}=e,\ensuremath{\mu}$. It turns out that the $CP$-violating difference ${S}_{\ensuremath{-}}(M)\ensuremath{\equiv}[\mathrm{\ensuremath{\Gamma}}({M}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{\ensuremath{-}}{\ensuremath{\ell}}_{2}^{\ensuremath{-}}{M}^{\ensuremath{'}+})\ensuremath{-}\mathrm{\ensuremath{\Gamma}}({M}^{+}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{+}{\ensuremath{\ell}}_{2}^{+}{M}^{\ensuremath{'}\ensuremath{-}})]$ 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 $\mathrm{\ensuremath{\Delta}}{M}_{N}$ becomes comparable with the (small) decay widths ${\mathrm{\ensuremath{\Gamma}}}_{N}$ of these neutrinos: $\mathrm{\ensuremath{\Delta}}{M}_{N}\overline{)\ensuremath{\gg}}{\mathrm{\ensuremath{\Gamma}}}_{N}$. We show that in such a case, the $CP$ ratio ${\mathcal{A}}_{CP}(M)\ensuremath{\equiv}[\mathrm{\ensuremath{\Gamma}}({M}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{\ensuremath{-}}{\ensuremath{\ell}}_{2}^{\ensuremath{-}}{M}^{\ensuremath{'}+})\ensuremath{-}\mathrm{\ensuremath{\Gamma}}({M}^{+}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{+}{\ensuremath{\ell}}_{2}^{+}{M}^{\ensuremath{'}\ensuremath{-}})]/[\mathrm{\ensuremath{\Gamma}}({M}^{\ensuremath{-}}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{\ensuremath{-}}{\ensuremath{\ell}}_{2}^{\ensuremath{-}}{M}^{\ensuremath{'}+})+\mathrm{\ensuremath{\Gamma}}({M}^{+}\ensuremath{\rightarrow}{\ensuremath{\ell}}_{1}^{+}{\ensuremath{\ell}}_{2}^{+}{M}^{\ensuremath{'}\ensuremath{-}})]$ becomes a quantity $\ensuremath{\sim}1$. The observation of $CP$ violation in these decays would be consistent with the existence of the well-motivated $\ensuremath{\nu}\mathrm{MSM}$ model with two almost degenerate heavy neutrinos in the mass range ${M}_{N}\ensuremath{\sim}0.1--{10}^{1}\text{ }\text{ }\mathrm{GeV}$.

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