Measurement of the $e^{+}{e}^{-}\to B_s^0{\bar{B}}_s^{0}X$ cross section in the energy range from $10.63$ to $11.02$ GeV using inclusive $D_s^{\pm }$ and $D^0$ production

We report the first measurement of the inclusive $e^{+}{e}^{-}\to b\bar{b}\to D_s^{\pm }X$ and $e^{+}{e}^{-}\to b\bar{b}\to D^0\mathrm{/}{\bar{D}}^{0}X$ cross sections in the energy range from $10.63$ to $11.02$. Based on these results, we determine $\sigma (e^{+}{e}^{-}\to B_s^0{\bar{B}}_s^{0}X)$ and $\sigma (e^{+}{e}^{-}\to B\bar{B}X)$ in the same energy range. We measure the fraction of $B_s^0$ events at $\mathrm{{\rm Y}}(10860)$ to be $f_{\mathrm{s}}=(22.0_{-2.1}^{+2.0})\mathrm{\%}$. We determine also the ratio of the $B_s^0$ inclusive branching fractions $\mathcal{B}(B_s^0\to D^0\mathrm{/}{\bar{D}}^{0}X)\mathrm{/}\mathcal{B}(B_s^0\to D_s^{\pm }X)=0.416\pm 0.018\pm 0.092$. The results are obtained using the data collected with the Belle detector at the KEKB asymmetric-energy $e^{+}{e}^{-}$ collider.