Measurements of the branching fractions of \boldmath ${\chi }_{cJ}\to {\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0\eta$

Using $(2712.4\pm 14.3)\times {10}^6$ $\psi (3686)$ events collected with the BESIII detector at the BEPCII collider, we report the first observations of the decay ${\chi }_{cJ}(J=0,1,2)\to {\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0\eta$. The signal significances, after considering systematic uncertainties, are determined to be 7.2$\sigma$, 5.7$\sigma$, and 6.7$\sigma$ for ${\chi }_0$, ${\chi }_{c1}$, and ${\chi }_{c2}$, respectively. The branching fractions are measured to be $\mathcal{B}({\chi }_{c0}\to {\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0\eta )=(9.7\pm 1.7\pm 0.8)\times {10}^{-5}$, $\mathcal{B}({\chi }_{c1}\to {\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0\eta )=(4.9\pm 1.1\pm 0.4)\times {10}^{-5}$, and $\mathcal{B}({\chi }_{c2}\to {\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0\eta )=(6.1\pm 1.2\pm 0.3)\times {10}^{-5}$, where the first uncertainties are statistical and the second systematic. No significant structure in the ${\mathrm{\Sigma }}^0{\bar{\mathrm{\Sigma }}}^0,{\mathrm{\Sigma }}^0\eta ,{\bar{\mathrm{\Sigma }}}^0\eta$ mass distributions is observed under the current statistics.