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 $(10087\pm 44)\times {10}^6$ $J\mathrm{/}\psi$ decays collected with the BESIII detector. The decay $J\mathrm{/}\psi \to \gamma \eta$ is used to select clean samples of 631,686 $\eta \to {\pi }^{+}{\pi }^{-}{\pi }^0$ decays and 272,322 $\eta \to {\pi }^0{\pi }^0{\pi }^0$ decays. The matrix elements for both channels are in reasonable agreement with previous measurements. The non-zero $g{X}^{2}Y$ 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.