Estimating Form Factors of $B_s\to D_s^{(\ast )}$ and their Applications to Semi-leptonic and Non-leptonic Decays

$B_s^0\to D_s^{-}$ and $B_s^0\to D_s^{\ast -}$ weak transition form factors are estimated for the whole physical region with a method based on an instantaneous approximated Mandelstam formulation of transition matrix elements and the instantaneous Bethe-Salpeter equation. We apply the estimated form factors to branching ratios, CP asymmetries and polarization fractions of non-leptonic decays within the factorization approximation. And we study the non-factorizable effects and annihilation contributions with the perturbative QCD approach. The branching ratios of semi-leptonic $B_s^0\rightarrow D_s^{(*)-}l^+\nu_l$ decays are also evaluated. We show that the calculated decay rates agree well with the available experimental data. The longitudinal polarization fraction of $B_s\to D_s^{\ast }V(A)$ decays are $\sim 0.8$ when $V(A)$ denotes a light meson, and are $\sim 0.5$ when $V(A)$ denotes a $D_q$ ($q=d,s$) meson.