Search for the rare decay of $B^+ \to \ell^{\,+} \nu_{\ell} \gamma$ with improved hadronic tagging

We present the result of the search for the rare $B$ meson decay of $B^+ \to \ell^{\,+} \nu_{\ell} \gamma$ with $\ell =e,\mu$. For the search the full data set recorded by the Belle experiment of $711 \, \mathrm{fb}^{-1}$ integrated luminosity near the $\Upsilon (4S)$ resonance is used. Signal candidates are reconstructed for photon energies $E_{\gamma}$ larger than $1 \, \mathrm{GeV}$ using a novel multivariate tagging algorithm. The novel algorithm fully reconstructs the second $B$ meson produced in the collision using hadronic modes and was specifically trained to recognize the signal signature in combination with hadronic tag-side $B$ meson decays. This approach greatly enhances the performance. Background processes that can mimic this signature, mainly charmless semileptonic decays and continuum processes, are suppressed using multivariate methods. The number of signal candidates is determined by analyzing the missing mass squared distribution as inferred from the signal side particles and the kinematic properties of the tag-side $B$ meson. No significant excess over the background-only hypothesis is observed and upper limits on the partial branching fraction $ \Delta \mathcal{B} $ with $E_{\gamma}> 1 \, \mathrm{GeV}$ individually for electron and muon final states as well as for the average branching fraction of both lepton final states are reported. We find a Bayesian upper limit of $\Delta \mathcal{B}( B^{+} \to \ell^{\, +} \nu_{\ell} \gamma) < 3.0 \times 10^{-6}$ at 90% CL and also report an upper limit on the first inverse moment of the light-cone distribution amplitude of the $B$ meson of $\lambda_B$ at 90% CL.