Measurement of the Decays and in Fully Reconstructed Events at Belle

We report branching fraction measurements of the decays $B^{+}\to \eta {\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ and $B^{+}\to {\eta }^{\mathrm{\prime }}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ based on 711~fb${}^{-1}$ of data collected near the $\mathrm{{\rm Y}}(4S)$ resonance with the Belle experiment at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider. This data sample contains 772 million $B\bar{B}$~events. One of the two $B$~mesons is fully reconstructed in a hadronic decay mode. Among the remaining (“signal-$B$”) daughters, we search for the $\eta$~meson in two decay channels, $\eta \to \gamma \gamma$ and $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$, and reconstruct the ${\eta }^{\mathrm{\prime }}$~meson in ${\eta }^{\mathrm{\prime }}\to \eta {\pi }^{+}{\pi }^{-}$ with subsequent decay of the $\eta$ into $\gamma \gamma$. Combining the two $\eta$ modes and using an extended maximum likelihood, the $B^{+}\to \eta {\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$ branching fraction is measured to be $(4.2\pm 1.1(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.3(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))\times {10}^{-5}$. For $B^{+}\to {\eta }^{\mathrm{\prime }}{\mathrm{\ell }}^{+}{\nu }_{\mathrm{\ell }}$, we observe no significant signal and set an upper limit of $0.72\times {10}^{-4}$ at 90% confidence level.