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-six Standard Model effective field theory. The corresponding hadron-level operators that generate a nonvanishing $I=0$ amplitude at order $p^6$ in the chiral effective theory are presented for the first time, to the best of our knowledge, in addition to the leading-order operators ascribed 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.