Search for ${\eta }_c(2S)\to 2({\pi }^{+}{\pi }^{-})$ and improved measurement of ${\chi }_{cJ}\to 2({\pi }^{+}{\pi }^{-})$

We search for the hadronic decay ${\eta }_c(2S)\to 2({\pi }^{+}{\pi }^{-})$ in the $\psi (3686)\to \gamma {\eta }_c(2S)$ radiative decay using $(27.12\pm 0.14)\times {10}^8$ $\psi (3686)$ events collected by the BESIII detector at the BEPCII collider. No significant signal is found, and the upper limit of $\mathcal{B}[\psi (3686)\to \gamma {\eta }_c(2S)]\mathcal{B}[{\eta }_c(2S)\to 2({\pi }^{+}{\pi }^{-})]$ is determined to be $0.78\times {10}^{-6}$ at the 90% confidence level. Using $\psi (3686)\to \gamma {\chi }_{cJ}$ transitions, we also measure the branching fractions of $\mathcal{B}[{\chi }_{cJ(J=0,1,2)}\to 2({\pi }^{+}{\pi }^{-})]$, which are $\mathcal{B}[{\chi }_{c0}\to 2({\pi }^{+}{\pi }^{-})]=(2.127\pm 0.002(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.101(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))$%, $\mathcal{B}[{\chi }_{c1}\to 2({\pi }^{+}{\pi }^{-})]=(0.685\pm 0.001(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.031\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))$%, and $\mathcal{B}[{\chi }_{c2}\to 2({\pi }^{+}{\pi }^{-})]=(1.153\pm 0.001(\mathrm{s}\mathrm{t}\mathrm{a}\mathrm{t}\mathrm{.})\pm 0.063(\mathrm{s}\mathrm{y}\mathrm{s}\mathrm{t}\mathrm{.}))$%.