Studies of the resonance components in the $B_s$ decays into charmonia plus kaon pair

In this work, the decays of $B_s$ mesons to a charmonium state and a $K^+K^-$ pair are carefully investigated in the perturbative QCD approach. Following the latest fit from the LHCb experiment, we restrict ourselves to the case where the produced $K^+K^-$ pair interact in isospin zero $S$, $P$, and $D$ wave resonances in the kinematically allowed mass window. Besides the dominant contributions of the $\phi(1020)$ resonance in the $P$-wave and $f_2'(1525)$ in the $D$-wave, other resonant structures in the high mass region as well as the $S$-wave components are also included. The invariant mass spectra for most of the resonances in the $B_s\rightarrow J/\psi K^+K^-$ decay are well reproduced. The obtained three-body decay branching ratios can reach the order of $10^{-4}$, which seem to be accessible in the near future experiments. The associated polarization fractions of those vector-vector and vector-tensor modes are also predicted, which are compared with the existing data from LHCb Collaboration.