Sensitivity limits on heavy-light mixing $|U_{\mu N}|^2$ from lepton number violating $B$ meson decays

We consider the lepton number violating decays $B \to \mu^{\pm} \mu^{\pm} \pi^{\mp}$ and $B \to D^{(*)} \mu^{\pm} \mu^{\pm} \pi^{\mp}$ which may be detected at LHCb and Belle-II experiments; and $B \to \mu^{\pm} \mu^{\pm} e^{\mp} \nu$ and $B \to D^{(*)} \mu^{\pm} \mu^{\pm} e^{\mp} \nu$ decays which may be detected at Belle-II experiment. The projected total number of produced $B$ mesons is $4.8 \times 10^{12}$ at LHCb upgrade and $5 \times 10^{10}$ at Belle-II. For the case that the above decays are not detected, we deduce the new upper bounds (sensitivity limits) for the mixing parameter $|U_{\mu N}|^2$ of heavy sterile neutrino with sub-eV light neutrino, as a function of the sterile neutrino mass in the interval $1.75 \ {\rm GeV} < M_N < 5.0 \ {\rm GeV}$. We take into account the probability of decay of the sterile neutrino $N$ within the detector, taking as the effective detector length $L=2.3 \ m$ at LCHb upgrade and $L=1 \ m$ at Belle-II. In the interval $1.75 \ {\rm GeV} < M_N < 3 \ {\rm GeV}$, the most stringent bounds can be obtained with the decays $B \to \mu^{\pm} \mu^{\pm} \pi^{\mp}$ at LHCb upgrade. The sensitivity limits are expected to be in general more stringent at LHCb upgrade than at Belle-II, principally because the number of produced $B$ mesons in LHCb upgrade is expected to be by about two orders of magnitude larger than at Belle-II. We conclude that the LHCb upgrade and Belle-II experiments have the potential to either find a new heavy Majorana neutrino $N$, or to improve significantly the sensitivity limits (upper bounds) on the heavy-light mixing parameter $|U_{\mu N}|^2$, particularly in the mass range $1.75 \ {\rm GeV} < M_N < 3 \ {\rm GeV}$. This work is a continuation and refinement of our previous work [1] on the subject.