Role of heavy neutral lepton in lepton number violating $B$ meson decays

We study the phenomenology of heavy neutral leptons (HNLs) in $B$-meson decays as probes of physics beyond the Standard Model. Focusing on the leptonic channels $B \to \mu N$ and $B \to \tau N$, we constrain the allowed regions in the $M_N$--$|U_{\ell N}|^2$ plane using current experimental data. Using these constraints, we investigate lepton-number violating ($\Delta L=2$) processes mediated by on-shell HNLs, including $B_{(c)}^- \to \pi^+ \mu^- \mu^-$ and $B_c^- \to J/\psi\, \pi^+ \mu^- \mu^-$. For benchmark values $|U_{\mu N}|^2 = 10^{-6}$ and $M_N = 2$-- $3\,\mathrm{GeV}$, the predicted branching ratios lie in the range $\mathcal{O}(10^{-13})$--$\mathcal{O}(10^{-8})$. Among the channels, $B_c^- \to \pi^+ \mu^- \mu^-$ shows the largest enhancement, while $B_c^- \to J/\psi\, \pi^+ \mu^- \mu^-$ is strongly suppressed. These results indicate a clear channel dependence, with $B_c$ modes providing enhanced sensitivity to HNL effects and offering promising avenues for future searches of lepton number violation.