Study of B s ⟶ ϕ ℓ + ℓ − Decays in the PQCD Factorization Approach with Lattice QCD Input

In this paper, we studied systematically the semileptonic decays with by using the perturbative QCD (PQCD) and the “PQCD+Lattice” factorization approach, respectively. We first evaluated all relevant form factors in the low- region using the PQCD approach, and we also took the available lattice QCD results at the high- region as additional input to improve the extrapolation of from the low- region to the endpoint . We then calculated the branching ratios and many other physical observables , , , and and the clean angular observables and . From our studies, we find the following points: (a) the PQCD and “PQCD+Lattice” predictions of are about , 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 and ; (c) the PQCD and “PQCD+Lattice” predictions of the longitudinal polarization , the CP-averaged angular coefficients , and the CP asymmetry angular coefficients agree with the LHCb measurements in all considered bins within the still large experimental errors; and (d) for those currently still unknown observables , and , we suggest LHCb and Belle-II Collaboration to measure them in their experiments.