Search for the radiative transitions $\psi(3770)\to\gamma\eta_c$ and $\gamma\eta_c(2S)$

By using a 2.92 fb$^{-1}$ data sample taken at $\sqrt{s} = 3.773$ GeV with the BESIII detector operating at the BEPCII collider, we search for the radiative transitions $\psi(3770)\to\gamma\eta_c$ and $\gamma\eta_c(2S)$ through the hadronic decays $\eta_c(\eta_c(2S))\to K^0_SK^\pm\pi^\mp$. No significant excess of signal events above background is observed. We set upper limits at a 90% confidence level for the product branching fractions to be $\mathcal{B}(\psi(3770)\to\gamma\eta_c)\times \mathcal{B}(\eta_c\to K^0_SK^\pm\pi^\mp)<1.6\times10^{-5}$ and $\mathcal{B}(\psi(3770)\to\gamma\eta_c(2S))\times \mathcal{B}(\eta_c(2S)\to K^0_SK^\pm\pi^\mp)<5.6\times10^{-6}$. Combining our result with world-average values of $\mathcal{B}(\eta_c(\eta_c(2S))\to K^0_SK^\pm\pi^\mp)$, we find the branching fractions $\mathcal{B}(\psi(3770)\to\gamma\eta_c)<6.8\times10^{-4}$ and $\mathcal{B}(\psi(3770)\to\gamma\eta_c(2S))<2.0\times10^{-3}$ at a 90% confidence level.