In this paper, we studied systematically the semileptonic decays ${\overline{B}}_s\to K^{(*)}{\ell }^{+}{\ell }^{-}$ with $l^{-}=(e^{-},{\mu }^{-},{\tau }^{-})$ by using the perturbative QCD (PQCD) and the “$\mathrm{PQCD}+\mathrm{Lattice}$” 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 took the available lattice QCD results at the end point $q_{\max }^2$ as additional inputs to improve the extrapolation of the form factors to the high-$q^2$ region. We calculated the branching ratios and twelve other kinds of physical observables. From our studies, we find the following points: (a) for ${\overline{B}}_s\to K{l}^{+}{l}^{-}$ decays, the PQCD and $\mathrm{PQCD}+\mathrm{Lattice}$ predictions for branching ratios $\mathcal{B}({\overline{B}}_s\to K{l}^{+}{l}^{-})$, the ratios of the branching ratios $R_K^{e\mu }$ and $R_K^{\mu \tau }$, and the longitudinal polarization asymmetry of the leptons $P_L$ agree well within errors; (b) the PQCD and $\mathrm{PQCD}+\mathrm{Lattice}$ predictions for the $CP$-averaged branching ratio $\mathcal{B}({\overline{B}}_s\to K^{*}{\mu }^{+}{\mu }^{-})$ are $(3.17_{-0.78}^{+0.95})\times {10}^{-8}$ and $(2.48_{-0.50}^{+0.56})\times {10}^{-8}$ respectively, which agree well with the LHCb measured value $(2.9\pm 1.1)\times {10}^{-8}$ and the light-cone sum rule prediction; (c) for the ratios $R_{K^{*}}^{e\mu }$ and $R_{K^{*}}^{\mu \tau }$, the PQCD and $\mathrm{PQCD}+\mathrm{Lattice}$ predictions agree well with each other and have a small error less than 10%; (d) for the direct $CP$ asymmetries ${\mathcal{A}}_{CP}$ of all considered decay modes, they are always very small as expected: less than 5% in magnitude; (e) for the angular observables $P_{1,2,3}$ and $P_{4,5,6,8}^{\prime }$, our theoretical predictions for each kind of lepton are consistent within errors; (f) the theoretical predictions of the angular observables $P_3$ and $P_6^{\prime }$ are less than ${10}^{-2}$ in size, but the magnitude of $P_{1,2}$ and $P_{4,5}^{\prime }$ are larger than 0.2; and (g) the PQCD and $\mathrm{PQCD}+\mathrm{Lattice}$ predictions of the binned values of all considered observables in the two $q^2$ bins $[0.1–0.98]{\mathrm{GeV}}^2$ and $[1.1–6]{\mathrm{GeV}}^2$ generally agree with each other and are also consistent with the light-cone sum rule results within errors. We believe that above predictions could be tested by future LHCb and Belle-II experiments.

• Received 31 March 2020
• Accepted 22 June 2020

DOI:https://doi.org/10.1103/PhysRevD.102.013001

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Published by the American Physical Society