Observation of ${\eta }_{c}\left(2S\right)\to 3\left({\pi }^{+}{\pi }^{-}\right)$ and measurements of ${\chi }_{cJ}\to 3\left({\pi }^{+}{\pi }^{-}\right)$ in $\psi \left(3686\right)$ radiative transitions

The hadronic decay ${\eta }_{c}\left(2S\right)\to 3\left({\pi }^{+}{\pi }^{-}\right)$ is observed with a statistical significance of 9.3 standard deviations using $\left(448.1±2.9\right)×{10}^{6}\text{}\text{}\psi \left(3686\right)$ events collected by the BESIII detector at the BEPCII collider. The measured mass and width of ${\eta }_{c}\left(2S\right)$ are $\left(3643.4±2.3\text{}\left(\mathrm{stat}\right)±4.4\text{}\left(\mathrm{syst}\right)\right)\text{}\text{}\mathrm{MeV}/{c}^{2}$ and $\left(19.8±3.9\text{}\left(\mathrm{stat}\right)±3.1\text{}\left(\mathrm{syst}\right)\right)\text{}\text{}\mathrm{MeV}$, respectively, which are consistent with the world average values within two standard deviations. The product branching fraction $B\left[\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)\right]×B\left[{\eta }_{c}\left(2S\right)\to \phantom{\rule{0ex}{0ex}}3\left({\pi }^{+}{\pi }^{-}\right)\right]$ is measured to be $\left(9.2±1.0\text{}\left(\mathrm{stat}\right)±1.2\text{}\left(\mathrm{syst}\right)\right)×{10}^{-6}$. Using $B\left[\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)\right]=\phantom{\rule{0ex}{0ex}}\left(7.{0}_{-2.5}^{+3.4}\right)×{10}^{-4}$, we obtain $B\left[{\eta }_{c}\left(2S\right)\to 3\left({\pi }^{+}{\pi }^{-}\right)\right]=\left(1.31±0.15\text{}\left(\mathrm{stat}\right)±0.17\text{}\left(\mathrm{syst}\right)\text{}\left(\frac{+0.64}{-0.47}\right)\text{}\left(\mathrm{extr}\right)\right)×\phantom{\rule{0ex}{0ex}}{10}^{-2}$, where the third uncertainty is from $B\left[\psi \left(3686\right)\to \gamma {\eta }_{c}\left(2S\right)\right]$. We also measure the ${\chi }_{cJ}\to 3\left({\pi }^{+}{\pi }^{-}\right)$ ($J=0$, 1, 2) decays via ${\psi }^{\prime }\to \gamma {\chi }_{cJ}$ transitions. The branching fractions are $B\left[{\chi }_{c0}\to 3\left({\pi }^{+}{\pi }^{-}\right)\right]=\phantom{\rule{0ex}{0ex}}\left(2.080±0.006\text{}\left(\mathrm{stat}\right)±0.068\text{}\left(\mathrm{syst}\right)\right)×{10}^{-2}$, $B\left[{\chi }_{c1}\to 3\left({\pi }^{+}{\pi }^{-}\right)\right]=\left(1.092±0.004\text{}\left(\mathrm{stat}\right)±0.035\text{}\left(\mathrm{syst}\right)\right)×{10}^{-2}$, and $B\left[{\chi }_{c2}\to 3\left({\pi }^{+}{\pi }^{-}\right)\right]=\left(1.565±0.005\text{}\left(\mathrm{stat}\right)±0.048\text{}\left(\mathrm{syst}\right)\right)×{10}^{-2}$.