Measurement of ${e}^{+}{e}^{-}\to \varphi {\eta }^{\prime }$ cross sections at center-of-mass energies from 3.508 to 4.951 GeV and search for the decay $\psi \left(3770\right)\to \varphi {\eta }^{\prime }$

The cross sections of the ${e}^{+}{e}^{-}\to \varphi {\eta }^{\prime }$ process at center-of-mass energies from 3.508 to 4.951 GeV are measured with high precision using $26.1\text{}\text{}{\mathrm{fb}}^{-1}$ data collected with the BESIII detector operating at the BEPCII storage ring. The cross sections are of the order of a few picobarn and decrease as the center-of-mass energy increases as ${s}^{-n/2}$ with $n=4.35±0.14$. This result is in agreement with the Nambu-Jona-Lasinio model prediction of $n=3.5±0.9$. In addition, the charmless decay $\psi \left(3770\right)\to \varphi {\eta }^{\prime }$ is searched for by fitting the measured cross sections, yet no significant signal is observed. The upper limit of $B\left(\psi \left(3770\right)\to \varphi {\eta }^{\prime }\right)$ at the 90% confidence level is determined to be $2.3×{10}^{-5}$.