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We report the result for a search for the leptonic decay of $B^+ \to \mu^+ \,\nu_{\mu}$ using the full Belle data set of 711 fb${}^{-1}$ of integratedluminosity at the $\Upsilon(4S)$ resonance. In the Standard Model leptonic$B$-meson decays are helicity and CKM suppressed. To maximize sensitivity aninclusive tagging approach is used to reconstruct the second $B$ meson producedin the collision. The directional information from this second $B$ meson isused to boost the observed $\mu$ into the signal $B$ meson rest-frame, in whichthe $\mu$ has a monochromatic momentum spectrum. Though its momentum is smearedby the experimental resolution, this technique improves the analysissensitivity considerably. Analyzing the $\mu$ momentum spectrum in this framewe find $\mathcal{B}(B^+ \to \mu^+ \, \nu_\mu) = \left( 5.3 \pm 2.0 \pm 0.9\right) \times 10^{-7}$ with a one-sided significance of 2.8 standarddeviations over the background-only hypothesis. This translates to afrequentist upper limit of $\mathcal{B}(B^+ \to \mu^+ \, \nu_{\mu}) < 8.6\times 10^{-7}$ at 90% CL. The experimental spectrum is then used to search fora massive sterile neutrino, $B^+ \to \mu^+ \, N$, but no evidence is observedfor a sterile neutrino with a mass in a range of 0 - 1.5 GeV. The determined$B^+ \to \mu^+ \, \nu_{\mu}$ branching fraction limit is further used toconstrain the mass and coupling space of the type II and type IIItwo-Higgs-doublet models.

Search for $B^+ \to \mu^+\, \nu_\mu$ and $B^+ \to \mu^+\, N$ with inclusive tagging