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

The cross sections of the $e^{+}{e}^{-}\to \phi {\eta }^{\mathrm{\prime }}$ process at center-of-mass energies from 3.508 to 4.951 GeV are measured with high precision using 26.1 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 $s^{-n\mathrm{/}2}$ with $n=4.35\pm 0.14$. This result is in agreement with the Nambu-Jona-Lasinio model prediction of $n=3.5\pm 0.9$. In addition, the charmless decay $\psi (3770)\to \phi {\eta }^{\mathrm{\prime }}$ is searched for by fitting the measured cross sections, yet no significant signal is observed. The upper limit of $\mathcal{B}(\psi (3770)\to \phi {\eta }^{\mathrm{\prime }})$ at the 90% confidence level is determined to be $2.3\times {10}^{-5}$.