Measurement of the Decays $\boldsymbol{B\to\eta\ell\nu_\ell}$ and $\boldsymbol{B\to\eta^\prime\ell\nu_\ell}$ in Fully Reconstructed Events at Belle

We report branching fraction measurements of the decays $B^+\to\eta\ell^+\nu_\ell$ and $B^+\to\eta^\prime\ell^+\nu_\ell$ based on 711~fb$^{-1}$ of data collected near the $\Upsilon(4S)$ 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 $\eta^{\prime}$~meson in $\eta^\prime\to\eta\pi^+\pi^-$ with subsequent decay of the $\eta$ into $\gamma\gamma$. Combining the two $\eta$ modes and using an extended maximum likelihood, the $B^+\to\eta\ell^+\nu_\ell$ branching fraction is measured to be $(4.2\pm 1.1 (\rm stat.)\pm 0.3 (\rm syst.))\times 10^{-5}$. For $B^+\to\eta^\prime\ell^+\nu_\ell$, we observe no significant signal and set an upper limit of $0.72\times 10^{-4}$ at 90\% confidence level.