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By using a 2.92 fb$^{-1}$ data sample taken at $\sqrt{s} = 3.773$ GeV withthe BESIII detector operating at the BEPCII collider, we search for theradiative transitions $\psi(3770)\to\gamma\eta_c$ and $\gamma\eta_c(2S)$through the hadronic decays $\eta_c(\eta_c(2S))\to K^0_SK^\pm\pi^\mp$. Nosignificant excess of signal events above background is observed. We set upperlimits at a 90% confidence level for the product branching fractions to be$\mathcal{B}(\psi(3770)\to\gamma\eta_c)\times \mathcal{B}(\eta_c\toK^0_SK^\pm\pi^\mp) < 1.6\times10^{-5}$ and$\mathcal{B}(\psi(3770)\to\gamma\eta_c(2S))\times \mathcal{B}(\eta_c(2S)\toK^0_SK^\pm\pi^\mp) < 5.6\times10^{-6}$. Combining our result with world-averagevalues of $\mathcal{B}(\eta_c(\eta_c(2S))\to K^0_SK^\pm\pi^\mp)$, we find thebranching 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.

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