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 the 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 the Belle-II experiment. The projected total number of produced $B$ mesons is $4.8\times 10^{12}$ at the LHCb upgrade and $5\times 10^{10}$ at Belle-II. For the case in which 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 $1.75\mathrm{GeV}<M_N<5.0\mathrm{GeV}$. We take into account the probability of decay of the sterile neutrino $N$ within the detector, taking as the effective detector length $L=2.3m$ at the LCHb upgrade and $L=1m$ at Belle-II. In the interval $1.75\mathrm{GeV}<M_N<3\mathrm{GeV}$, the most stringent bounds can be obtained with the decays $B\to {\mu }^{\pm }{\mu }^{\pm }{\pi }^{\mp }$ at the LHCb upgrade. The sensitivity limits are expected to be, in general, more stringent at the LHCb upgrade than at Belle-II, principally because the number of produced $B$ mesons in the LHCb upgrade is expected to be about 2 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 $1.75\mathrm{GeV}<M_N<3\mathrm{GeV}$. This work is a continuation and refinement of our previous work [Phys. Rev. D 94, 053001 (2016); Phys. Rev. D 95, 039901(E) (2017)] on the subject.

• Received 23 June 2017

DOI:https://doi.org/10.1103/PhysRevD.96.035025