Evidence for the ${\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta $ decay

The decay ${\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta $ is searched for through the radiative transition $\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)$ using 448 million $\psi \left(3686\right)$ events accumulated at the BESIII detector. The first evidence of ${\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta $ is found with a statistical significance of $3.5\sigma $. The product of the branching fractions of $\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)$ and ${\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta $ is measured to be $\mathrm{Br}\left(\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)\right)×\mathrm{Br}\left({\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta \right)=\phantom{\rule{0ex}{0ex}}\left(2.97±0.81±0.26\right)×{10}^{-6}$, where the first uncertainty is statistical and the second one is systematic. The branching fraction of the decay ${\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta $ is determined to be $\mathrm{Br}\left({\eta }_{c}\left(2S\right)\to {\pi }^{+}{\pi }^{-}\eta \right)=\phantom{\rule{0ex}{0ex}}\left(42.4±11.6±3.8±30.3\right)×{10}^{-4}$, where the third uncertainty is transferred from the uncertainty of the branching fraction of $\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)$.