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Based on 22.7 fb$^{-1}$ of $e^+e^-$ annihilation data collected at 33different center-of-mass energies between 3.7730 GeV and 4.7008 GeV with theBESIII detector at the BEPCII collider, Born cross sections of the twoprocesses $e^+e^-\to \phi K^+ K^-$ and $e^+ e^- \to \phi K_{S}^{0} K_{S}^{0}$are measured for the first time. No indication of resonant production throughan intermediate vector state $V$ is observed, and the upper limits on theproduct of the electronic width $\Gamma_{e^+e^-}$ and the branching fraction$Br(V\rightarrow \phi K \bar{K})$ of the processes $e^+e^- \to V \to \phi K^+K^-$ and $e^+e^- \to V \to \phi K_S^0K_S^0$ at the $90\%$ confidence level areobtained for a large parameter space in resonance masses and widths. For thecurrent world average mass and width of the $\psi(4230)$ of $m=4.2187$GeV$/c^2$ and $\Gamma=44$ MeV, we set upper limits on the $\phi K^+ K^-$ and$\phi K_S^0K_S^0$ final states of 1.75 eV and 0.47 eV at the $90\%$ confidencelevel, respectively.

physical review. d/physical review. d.

Cross section measurements of $e^+e^- \to \Phi K^+ K^-$ and $e^+ e^- \to  \Phi K_S^0 K_S^0$ at center-of-mass energies between 3.7730 GeV and 4.7008  GeV