Semi-inclusive studies of semileptonic $B_s$ decays at Belle

We present an analysis of the semi-inclusive decays $B_s \to D_s^- X \ell^+ \nu$ and $B_s \to D_s^{*-} X \ell^+ \nu$, where $X$ denotes a final state that may consist of additional hadrons or photons and $\ell$ is an electron or muon. The studied $B_s$ decays are contained in the $121.4~{\rm fb}^{-1}$ $\Upsilon(5S)$ data sample collected by the Belle detector at the KEKB asymmetric-energy $e^+e^-$ collider. The branching fractions of the decays are measured to be $\mathcal{B}(B_s \to D_s^- X \ell^+ \nu) = [8.2 \pm 0.2 (\text{stat}) \pm 0.6 (\text{syst}) \pm 1.4 (\text{ext})]\%$ and $\mathcal{B}(B_s \to D_s^{*-} X \ell^+ \nu) = [5.4 \pm 0.4 (\text{stat}) \pm 0.4 (\text{syst}) \pm 0.9 (\text{ext})]\%$, where the first two uncertainties are statistical and systematic and the last is due to external parameters. The measurement also provides an estimate of the $B_s^{(*)}\bar{B}_s^{(*)}$ production cross-section, $\sigma(e^+e^- \to B_s^{(*)}\bar{B}_s^{(*)}) = (53.8 \pm 1.4 (\text{stat}) \pm 4.0 (\text{syst}) \pm 3.4 (\text{ext}))\,{\rm pb}$, at the center-of-mass energy $\sqrt{s} = 10.86\,{\rm GeV}$.