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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 thelatest fit from the LHCb experiment, we restrict ourselves to the case wherethe produced $K^+K^-$ pair interact in isospin zero $S$, $P$, and $D$ waveresonances in the kinematically allowed mass window. Besides the dominantcontributions of the $\phi(1020)$ resonance in the $P$-wave and $f_2'(1525)$ inthe $D$-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 ofthe 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. Theassociated polarization fractions of those vector-vector and vector-tensormodes are also predicted, which are compared with the existing data from LHCbCollaboration.

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