Revisiting $C$ and $CP$ Violation in $\eta \to \pi^+\pi^-\pi^0$ Decay

The decay $\eta\to \pi^+\pi^-\pi^0$ is an ideal process in which to study flavor-conserving $C$ and $CP$ violation beyond the Standard Model. We deduce the $C$- and $CP$-odd quark operators that contribute to $\eta\to \pi^+\pi^-\pi^0$ originating from the mass-dimension 6 Standard Model effective field theory. The corresponding hadron-level operators that generate a non-vanishing $I=0$ amplitude at order $p^6$ in the chiral effective theory are presented for the first time, in addition to the leading order operatorsascribed to the $I=2$ final state. By fitting the KLOE-2 and the most recent BESIII experimental data, we determine the coefficients of the lowest order $I=0$ and $I=2$ amplitudes and estimate the potential new physics energy scale. We also perform an impact study of the future $\eta\to \pi^+\pi^-\pi^0$ experiments.