Study of \(B_s \longrightarrow \phi \ell^+ \ell^-\) decays in the PQCD factorization approach with lattice QCD input (Q2073359)

Summary: In this paper, we studied systematically the semileptonic decays \(B_s\longrightarrow\phi l^+ l^-\) 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 took the available lattice QCD results at the high-\(q^2\) region as additional input to improve the extrapolation of \(F_i(q^2)\) from the low-\(q^2\) region to the endpoint \(q_{\max}^2\). We then calculated the branching ratios and many other physical observables \(A_{\mathrm{FB}}^l\), \(F_L^\phi\), \(S_{3,4,7}\), and \(A_{5,6,8,9}\) and the clean angular observables \(P_{1,2,3}\) and \(P_{4,5,6,8}^\prime\). From our studies, we find the following points: (a) the PQCD and ``PQCD+Lattice'' predictions of \(\mathcal{B}(B_s \longrightarrow \phi \mu^+ \mu^-)\) are about \(7\times 10^{-7}\), which agree well with the LHCb measured values and the QCD sum rule prediction within still large errors; (b) we defined and calculated the ratios of the branching ratios \(R_\phi^{e \mu}\) and \(R_\phi^{\mu \tau}\); (c) the PQCD and ``PQCD+Lattice'' predictions of the longitudinal polarization \(F_L\), the CP-averaged angular coefficients \(S_{3,4,7}\), and the CP asymmetry angular coefficients \(A_{5,6,8,9}\) agree with the LHCb measurements in all considered bins within the still large experimental errors; and (d) for those currently still unknown observables \(R_\phi^{e \mu}\), \(R_\phi^{\mu \tau}\), \(A_{\mathrm{FB}}^l\), \(P_{1,2,3}\), and \(P_{4,5,6,8}^\prime\), we suggest LHCb and Belle-II Collaboration to measure them in their experiments.