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

We present first evidence for the process $e^+e^-to gammaeta_c(1S)$ 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 ($le3.0sigma$), 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 $sigma_{mathrm{ peak}}(e^+e^-togammaeta_c(1S)) = 2.11 pm 0.49 (mathrm{stat.}) pm 0.36 (mathrm{syst.})$~pb with a statistical significance of 4.2$sigma$.