Search for ${\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{\eta }_c$ and ${\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{K}_S^0{K}^{\pm }{\pi }^{\mp }$ decays

Based on $(27.12\pm 0.14)\times {10}^8$ $\psi (2S)$ events collected with the BESIII detector, we search for the decay ${\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{\eta }_c$ with ${\eta }_c\to K_S^0{K}^{\pm }{\pi }^{\mp }$ and ${\eta }_c\to K^{+}{K}^{-}{\pi }^0$. No significant signal is observed, and the upper limit on the product branching fraction $\mathcal{B}(\psi (2S)\to \gamma {\eta }_c(2S))\times \mathcal{B}$(${\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{\eta }_c$) is determined to be $2.21\times {10}^{-5}$ at the 90% confidence level. In addition, the analysis of the process $\psi (2S)\to \gamma {\eta }_c(2S),{\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{K}_S^0{K}^{\pm }{\pi }^{\mp }$ gives a clear ${\eta }_c(2S)$ signal with a statistical significance of $10\sigma$ for the first time, %The product branching fraction $\mathcal{B}(\psi (2S)\to \gamma {\eta }_c(2S))\times \mathcal{B}({\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{K}_S^{0}K\pi )$ is measured to be $(9.31\pm 0.72\pm 2.77)\times {10}^{-6}$, and and the branching fraction $\mathcal{B}({\eta }_c(2S)\to {\pi }^{+}{\pi }^{-}{K}_S^0{K}^{\pm }{\pi }^{\mp })$ is determined to be ($1.33\pm 0.11\pm 0.4\pm 0.95$)$\times {10}^{-2}$, where the first uncertainty is statistical, the second is systematic, and the third uncertainty is due to the quoted $\mathcal{B}(\psi (2S)\to \gamma {\eta }_c(2S))$.