Precision measurement of the matrix elements for $η\toπ^+π^-π^0$ and $η\toπ^0π^0π^0$ decays

A precision measurement of the matrix elements for $η\toπ^+π^-π^0$ and $η\toπ^0π^0π^0$ decays is performed using a sample of $(10087\pm44)\times10^6$ $J/ψ$ decays collected with the BESIII detector. The decay $J/ψ\to γη$ is used to select clean samples of 631,686 $η\toπ^+π^-π^0$ decays and 272,322 $η\toπ^0π^0π^0$ decays. The matrix elements for both channels are in reasonable agreement with previous measurements. The non-zero $gX^2Y$ term for the decay mode $η\toπ^+π^-π^0$ is confirmed, as reported by the KLOE Collaboration, while the other higher-order terms are found to be insignificant. Dalitz plot asymmetries in the $η\toπ^+π^-π^0$ decay are also explored and are found to be consistent with charge conjugation invariance. In addition, a cusp effect is investigated in the $η\toπ^0π^0π^0$ decay, and no obvious structure around the $π^+π^-$ mass threshold is observed.