Observation of $ {\eta }_{c}(1\mathrm{S},2\mathrm{S})$ and $ {\chi }_{cJ}$ decays to $ 2({\pi }^{+}{\pi }^{-})\eta $ via $ \psi (3686)$ radiative transitions

Based on $ (2712.4\pm 14.1)\times {10}^{6}$ $ \psi (3686)$ decays collected with the BESIII detector, we have observed, for the first time, the hadronic decays of S- and P-wave charmonium states into $ 2({\pi }^{+}{\pi }^{-})\eta $ via radiative transitions from $ \psi (3686)$. The branching fraction of the decay $ {\eta }_{c}(1S)\to 2({\pi }^{+}{\pi }^{-})\eta $ has a significant dependence on the interference pattern between $ {\eta }_{c}(1S)$ and non-$ {\eta }_{c}(1S)$ processes. We measure it in both the destructive and constructive interference scenarios for the first time. The mass and width of the $ {\eta }_{c}(1S)$ are measured to be $ M=(2984.14\pm 0.13\pm 0.38)\text{}\text{}\mathrm{MeV}/{c}^{2}$ and $ \mathrm{\Gamma }=(28.82\pm 0.11\pm 0.82)\text{}\text{}\mathrm{MeV}$, respectively. Clear signals for the decays of the $ {\chi }_{cJ}(J=0,1,2)$ and the $ {\eta }_{c}(2S)$ to $ 2({\pi }^{+}{\pi }^{-})\eta $ are also observed for the first time, and the corresponding branching fractions are measured. The ratio of the branching fractions between the $ {\eta }_{c}(2S)$ and $ {\eta }_{c}(1S)$ decays is significantly lower than the theoretical prediction, which might suggest different dynamics in their decays.