Updated Measurement of $\mathcal{B}(B_s\to D_s^{(\ast )+}{D}_s^{(\ast )-})$ and Determination of $\mathrm{\Delta }{\mathrm{\Gamma }}_s$

Using fully reconstructed $B_s$ mesons, we measure exclusive branching fractions for the decays $B_s\to D_s^{(\ast )+}{D}_s^{(\ast )-}$. The results are ${\cal B}(B^0_s\ra D^+_s D^-s)=(0.58,^{+0.11}{-0.09},\pm 0.13)%$,${\cal B}(B^0_s\ra D^{\pm}_s D^{\mp}_s)=(1.8, \pm 0.2,\pm 0.4)%$,and${\cal B}(B^0_s\ra D^{+}_s D^{-}_s)=(2.0,\pm 0.3,\pm 0.5)%$;thesumis${\cal B}(B^0_s\ra D^{()+}_s D^{(*)-}_s)=(4.3,\pm 0.4,\pm 1.0)%$. Assuming these decay modes saturate decays to CP-even final states, the branching fraction determines the relative width difference between the $B_s$ CP-odd and CP-even eigenstates. Taking \cp\ violation to be negligibly small, we obtain $\dgs/\gs = 0.090,\pm 0.009,{\rm(stat.)},\pm 0.022 ,{\rm (syst.)}$,where$\gs$ is the mean decay width. The results are based on a data sample collected with the Belle detector at the KEKB $e^{+}{e}^{-}$ collider running at the $\mathrm{{\rm Y}}(5S)$ resonance with an integrated luminosity of 121.4 fb${}^{-1}$.