Study of $\bar{B}_s\to K^{(*)}\ell^+ \ell^-$ decays in the PQCD factorization approach with the lattice QCD input

In this paper, we studied systematically the semileptonic decays $\bar{B}_s \to K^{(*)} \ell^+ \ell^-$ with $l^-=(e^-,\mu^-,\tau^-)$ by using the perturbative QCD (PQCD) and the "PQCD+Lattice" factorization approach, respectively. We first evaluated all relevant form factors $F_i(q^2)$ in the low $q^2$ region using the PQCD approach, and we also take the available Lattice QCD results at the end point $q^2_{max}$ as additional inputs to improve the extrapolation of the form factors to the high $q^2$ region. We calculated the branching ratios and other twelve kinds of physical observables. From our studies, we find the following points: (a) for $\bar{B}_s \to K l^+ l^-$ decays, the PQCD and "PQCD+Lattice" predictions for branching ratios (BRs) ${\cal B}(\bar{B}_s \to K l^+ l^-)$, the ratios of the BRs $R_K^{e\mu}$ and $R_K^{\mu\tau }$, and the longitudinal polarization asymmetry of the leptons $P_L$ agree well within errors; (b) the PQCD and "PQCD+Lattice" predictions for the CP averaged branching ratio ${\cal B}(\bar{B}_s \to K^* \mu^+ \mu^-)$ are $(3.17^{+0.95}_{-0.78})\times 10^{-8}$ and $(2.48^{+0.56}_{-0.50})\times 10^{-8}$ respectively, which agree well with the LHCb measured value $(2.9\pm 1.1) \times 10^{-8}$ and the light-cone sum rule (LCSR) prediction; (c) for the ratios $R_{K^\ast}^{e\mu }$ and $R_{K^\ast}^{\mu \tau}$, the PQCD and "PQCD+Lattice" predictions agree well with each other and have a small error less than $10\%$; (d) for the direct CP asymmetries ${\cal A}_{CP}$ of all considered decay modes, they are always very small as expected: less than $5\%$ in magnitude; and (e) for the angular observables $P_{1,2,3}$ and $P^\prime_{4,5,6,8}$ , our theoretical predictions for each kind of lepton are consistent within errors. We believe that above predictions could be tested by future LHCb and Belle-II experiments.