Cross-section measurements of ${e}^{+}{e}^{-}\to \varphi {K}^{+}{K}^{-}$ and ${e}^{+}{e}^{-}\to \varphi {K}_{S}^{0}{K}_{S}^{0}$ at center-of-mass energies between 3.7730 GeV and 4.7008 GeV

Based on $22.7\text{}{\mathrm{fb}}^{-1}$ of ${e}^{+}{e}^{-}$ annihilation data collected at 33 different center-of-mass energies between 3.7730 and 4.7008 GeV with the BESIII detector at the BEPCII collider, Born cross sections of the two processes ${e}^{+}{e}^{-}\to \varphi {K}^{+}{K}^{-}$ and ${e}^{+}{e}^{-}\to \varphi {K}_{S}^{0}{K}_{S}^{0}$ 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 ${\Gamma }_{{e}^{+}{e}^{-}}$ and the branching fraction $\mathrm{Br}\left(V\to \varphi K\overline{K}\right)$ of the processes ${e}^{+}{e}^{-}\to V\to \varphi {K}^{+}{K}^{-}$ and ${e}^{+}{e}^{-}\to V\to \varphi {K}_{S}^{0}{K}_{S}^{0}$ 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 $\psi \left(4230\right)$ of $m=4.2187\text{}\text{}\mathrm{GeV}/{c}^{2}$ and $\Gamma =44\text{}\text{}\mathrm{MeV}$, we set upper limits on the $\varphi {K}^{+}{K}^{-}$ and $\varphi {K}_{S}^{0}{K}_{S}^{0}$ final states of 1.75 and 0.47 eV at the 90% confidence level, respectively.