Favoured $B_c$ Decay modes to search for a Majorana neutrino

Recently, the LHCb collaboration reported the observation of the decay mode $B_{c}^{-}\rightarrow\overline{B}_{s}^{0}\pi^{-}$ with the largest exclusive branching fraction amongst the known decay modes of all the $B$ mesons. Here we propose a search for a few lepton-number violating decay modes of $B_c$ which can only be induced by Majorana neutrinos. Distinguishing between Dirac and Majorana nature of neutrinos is an outstanding problem and hence, all possible searches for Majorana neutrinos need to be carried out. Since the lepton number violating modes are expected to be rare, when using meson decay modes for these searches one expects CKM favoured modes to be the preferred ones; $B_{c}\rightarrow B_{s}$ is one such transition. With a resonance enhancement of the Majorana neutrino mediating the $B_{c}^{-}\rightarrow\overline{B}_{s}^{0}\ell_{1}{-}\ell_{2}^{-}\pi^{+}$ modes one can hope to observe these rare modes, or, even their non-observation can be used to obtain constraints on the mixing angles of the heavy Majorana singlet with the light flavour neutrinos from upper limits of the branching fractions which are tighter or compatible with results from earlier studies. Also, we find that the relatively suppressed mode $B_{c}^{-}\rightarrow J/\psi\ell_{1}^{-}\ell_{2}^{-}\pi^{+}$ can provide even tighter constraints on $\mid V_{e N}\mid^2$, $\mid V_{\mu N}\mid^2$, $\mid V_{e N} V_{\mu N}\mid$, and in a larger range of the heavy neutrino mass. Further, exclusion regions for $\mid V_{e N} V_{\tau N}\mid$, $\mid V_{\mu N} V_{\tau N}\mid$ can also be obtained for masses larger than those accessible in tau decays. Upper limits on $\mathscr{B}\left(B_{c}^{-}\rightarrow\pi^{+}\ell_{1}^{-}\ell_{2}^{-}\right)$ can also result in stringent exclusion curves for all the mixing elements, including that for $\mid V_{\tau N}\mid^2$ in a mass range where it is unconstrained thus far.