We report the result for a search for the leptonic decay of $B^+ to mu^+ , u_{mu}$ using the full Belle data set of 711 fb${}^{-1}$ of integrated luminosity at the $Upsilon(4S)$ resonance. In the Standard Model leptonic $B$-meson decays are helicity and CKM suppressed. To maximize sensitivity an inclusive tagging approach is used to reconstruct the second $B$ meson produced in the collision. The directional information from this second $B$ meson is used to boost the observed $mu$ into the signal $B$ meson rest-frame, in which the $mu$ has a monochromatic momentum spectrum. Though its momentum is smeared by the experimental resolution, this technique improves the analysis sensitivity considerably. Analyzing the $mu$ momentum spectrum in this frame we find $mathcal{B}(B^+ to mu^+ , u_mu) = left( 5.3 pm 2.0 pm 0.9 right) times 10^{-7}$ with a one-sided significance of 2.8 standard deviations over the background-only hypothesis. This translates to a frequentist upper limit of $mathcal{B}(B^+ to mu^+ , u_{mu}) < 8.6 times 10^{-7}$ at 90% CL. The experimental spectrum is then used to search for a massive sterile neutrino, $B^+ to mu^+ , N$, but no evidence is observed for a sterile neutrino with a mass in a range of 0 - 1.5 GeV. The determined $B^+ to mu^+ , u_{mu}$ branching fraction limit is further used to constrain the mass and coupling space of the type II and type III two-Higgs-doublet models.
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