Precision measurement of the matrix elements for $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ and $\eta \to {\pi }^{0}{\pi }^{0}{\pi }^{0}$ decays

A precision measurement of the matrix elements for $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ and $\eta \to {\pi }^{0}{\pi }^{0}{\pi }^{0}$ decays is performed using a sample of $\left(10087±44\right)×{10}^{6}\text{}\text{}J/\psi $ decays collected with the BESIII detector. The decay $J/\psi \to \gamma \eta $ is used to select clean samples of $631,686\text{}\text{}\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ decays and $272,322\text{}\text{}\eta \to {\pi }^{0}{\pi }^{0}{\pi }^{0}$ decays. The matrix elements for both channels are in reasonable agreement with previous measurements. The nonzero $g{X}^{2}Y$ term for the decay mode $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ is confirmed, as reported by the KLOE Collaboration, while the other higher-order terms are found to be insignificant. Dalitz plot asymmetries in the $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^{0}$ decay are also explored and are found to be consistent with charge conjugation invariance. In addition, a cusp effect is investigated in the $\eta \to {\pi }^{0}{\pi }^{0}{\pi }^{0}$ decay, and no obvious structure around the ${\pi }^{+}{\pi }^{-}$ mass threshold is observed.