Weak decays of $B_s$ to $D_s$ based on the helicity analysis

Employing dipole and exponential hadronic transition form factors and helicity analysis combined with the Lattice QCD input, we present a detailed study of the decays $B_s\to D_s^{(\ast )}\mathrm{\ell }{\nu }_{\mathrm{\ell }}$. $B_s\to D_s+P(V)$, where $P={\pi }^{+},K^{+},D_s^{+}$ and $V={\rho }^{+},K^{\ast +},D_s^{\ast +}$, are also investigated and their branching fractions are examined. At the large recoil point, we calculate $f_{+,0}^{B_s\to D_s}(0)=0.67\pm 0.01$ for both parameterizations. Then we evaluate the branching fractions $BR(B_s\to D_s\mathrm{\ell }\nu )$ and $BR(B_s\to D_s\mathrm{\ell }\nu )$, which leads to $R(D_s)=0.298\pm 0.123$. The ratios $\frac{\mathrm{\Gamma }(B_s^0\to D_s^{-}{\mu }^{+}{\nu }_{\mu })}{\mathrm{\Gamma }(B_s^0\to D_s^{\ast -}{\mu }^{+}{\nu }_{\mu })}$ are found to be $0.4415\pm 0.1860$ and $0.4406\pm 0.1854$, which are in good agreement of recent LHCb collaboration measurement. We also calculate the physical observables, $A_{FB}^{\mathrm{\ell }},P_L^{\mathrm{\ell }},P_T^{\mathrm{\ell }},C_F^{\mathrm{\ell }},F_L^{\mathrm{\ell }}$.