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

Abstract

In this paper, we studied systematically the semileptonic decays ${\bar{B}}_s\to K^{(\ast )}{\ell }^{+}{\ell }^{-}$ 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 $q_{max}^2$ as additional inputs to improve the extrapolation of the form factors to the high $q^2$ region. We calculated the branching ratios and 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) $\mathcal{B}({\bar{B}}_s\to K{l}^{+}{l}^{-})$, 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 $\mathcal{B}({\bar{B}}_s\to K^{\ast }{\mu }^{+}{\mu }^{-})$ 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 (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 ${\mathcal{A}}_{CP}$ 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 $P_{4,5,6,8}^{\prime }$ , 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.

arXiv comment40 pages, 13 tables and 8 figures, some corrections to the text and the figures

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