Lepton number violation in $B_s$ meson decays induced by an on-shell Majorana neutrino

Lepton-number violation can be induced by the exchange of an on-shell Majorana neutrino $N$ in semileptonic $\mathrm{\mid }\mathrm{\Delta }L\mathrm{\mid }=2$ decays of the $B_s$ meson, $B_s^0\to P^{-}{\pi }^{-}{\mu }^{+}{\mu }^{+}$ with $P=K,D_s$. We investigate the production of such a heavy sterile neutrino through these four-body ${\mu }^{+}{\mu }^{+}$ channels and explore the sensitivity that can be reached at the LHCb and CMS experiments. For heavy neutrino lifetimes of ${\tau }_N$ = [1, 100, 1000] ps and integrated luminosities collected of 10 and 50 fb${}^{-1}$ at the LHCb and 30, 300, and 3000 fb${}^{-1}$ at the CMS, we find a significant sensitivity on branching fractions of the orders ${\rm BR}(B_s^0 \to K^- \pi^-\mu^+ \mu^+) \lesssim $\mathcal{O}$(10^{-9} - 10^{-8})$and${\rm BR}(B_s^0 \to D_s^- \pi^-\mu^+ \mu^+) \lesssim $\mathcal{O}$(10^{-8} - 10^{-7})$. In the kinematically allowed mass ranges of $m_N\in [0.25,4.77]$ GeV and $m_N \in [0.25,3.29]$ GeV, respectively, we exclude regions on the parameter space ($m_N,\mathrm{\mid }{V}_{\mu N}{\mathrm{\mid }}^2$) associated with the heavy neutrino, which could slightly improve the limits from $B^{-}\to {\pi }^{+}{\mu }^{-}{\mu }^{-}$ (LHCb).