Observation of ${\chi }_{c0}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta $ and evidence for ${\chi }_{c1,2}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta $

Using $\left(27.12±0.14\right)×{10}^{8}$ $\psi \left(3686\right)$ events collected with the BESIII detector, the decay ${\chi }_{c0}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta $ is observed for the first time with a signal significance of $7.0\sigma $, and evidence for ${\chi }_{c1}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta $ and ${\chi }_{c2}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta $ is found with signal significances of $4.3\sigma $ and $4.6\sigma $, respectively. The branching fractions are determined to be $B\left({\chi }_{c0}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta \right)=\left(1.26±0.20±0.13\right)×{10}^{-4}$, $B\left({\chi }_{c1}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta \right)=\left(5.10±1.21±\phantom{\rule{0ex}{0ex}}0.67\right)×{10}^{-5}$, and $B\left({\chi }_{c2}\to {\Sigma }^{+}{\overline{\Sigma }}^{-}\eta \right)=\left(5.46±1.18±0.50\right)×{10}^{-5}$, where the first uncertainties are statistical, and the second ones are systematic.