Sensitivity to Majorana neutrinos in $\Delta L=2$ decays of $B_c$ meson at LHCb

The possible existence of Majorana neutrinos can be tested through the study of processes where the total lepton number $L$ is violated by two units ($\Delta L=2$). In this work, the production of an on-shell Majorana neutrino with a mass around $\sim$ 0.2 GeV to a few GeV is studied in $\Delta L= 2$ decays of the $B_c$ meson. We focus on the same-sign di-muon channels: three-body $B_c^- \to \pi^+\mu^-\mu^-$ and four-body $B_c^- \to J/\psi \pi^+\mu^-\mu^-$ and their experimental sensitivity at the LHCb. In both channels, we find that sensitivities on the branching fraction of the order $\lesssim 10^{-7} \ (10^{-8})$ might be accessible at the LHC run 2 (future LHC run 3), allowing us to set additional and complementary constraints on the parameter space associated with the mass and mixings of the Majorana neutrino. In particular, bounds can be obtained on the mixing $|V_{\mu N}|^2 \sim \mathcal{O}(10^{-5}-10^{-4})$ that are similar or better than the ones obtained from heavy meson $\Delta L=2$ decays: $D_{(s)}^- \to \pi^+\mu^-\mu^-$ and $B^- \to \pi^+\mu^-\mu^- \ (D^0\pi^+\mu^-\mu^-)$.