Cross section measurements of $e^{+}{e}^{-}\to \mathrm{\Phi }{K}^{+}{K}^{-}$ and $e^{+}{e}^{-}\to \mathrm{\Phi }{K}_S^0{K}_S^0$ 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 $e^{+}{e}^{-}\to \phi K^{+}{K}^{-}$ and $e^{+}{e}^{-}\to \phi K_S^0{K}_S^0$ are measured for the first time. No indication of resonant production through an intermediate vector state $V$ is observed, and the upper limits on the product of the electronic width ${\mathrm{\Gamma }}_{e^{+}{e}^{-}}$ and the branching fraction $Br(V\to \phi K\bar{K})$ of the processes $e^{+}{e}^{-}\to V\to \phi K^{+}{K}^{-}$ and $e^{+}{e}^{-}\to V\to \phi K_S^0{K}_S^0$ at the $90\mathrm{\%}$ confidence level are obtained for a large parameter space in resonance masses and widths. For the current world average mass and width of the $\psi (4230)$ of $m=4.2187$ GeV$\mathrm{/}{c}^2$ and $\mathrm{\Gamma }=44$ MeV, we set upper limits on the $\phi K^{+}{K}^{-}$ and $\phi K_S^0{K}_S^0$ final states of 1.75 eV and 0.47 eV at the $90\mathrm{\%}$ confidence level, respectively.