Evidence for $e^{+}{e}^{-}\to \gamma {\eta }_c(1S)$ at center-of-mass energies between 4.01 and 4.60 GeV

We present first evidence for the process $e^{+}{e}^{-}\to \gamma {\eta }_c(1S)$ at six center-of-mass energies between 4.01 and 4.60~GeV using data collected by the BESIII experiment operating at BEPCII. These data sets correspond to a total integrated luminosity of 4.6~fb${}^{-1}$. We measure the Born cross section at each energy using a combination of twelve ${\eta }_c(1S)$ decay channels. Because the significance of the signal is marginal at each energy ($\le 3.0\sigma$), we also combine all six energies under various assumptions for the energy-dependence of the cross section. If the process is assumed to proceed via the $Y(4260)$, we measure a peak Born cross section $\sigma_{\mathrm{ peak}}(e^+e^-\to\gamma\eta_c(1S)) = 2.11 \pm 0.49 ($\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.}$) \pm 0.36 ($\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}$)$pbwithastatisticalsignificanceof4.2$\sigma$.