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

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