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 B s ⟶ ϕ l + l − with l − = e − , μ − , τ − 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 FB l , F L ϕ , 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 ′ . From our studies, we find the following points: (a) the PQCD and “PQCD+Lattice” predictions of B B s ⟶ ϕ μ + μ − are about 7 × 1 0 − 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 ϕ e μ and R ϕ μ τ ; (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 ϕ e μ , R ϕ μ τ , A FB l , P 1 , 2 , 3 , and P 4 , 5 , 6 , 8 ′ , we suggest LHCb and Belle-II Collaboration to measure them in their experiments.