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

TL;DR

This paper investigates $C$ and $CP$ violation in $ ext{eta} o ext{pi}^+ ext{pi}^- ext{pi}^0$ decay using effective field theory and experimental data, providing new operator formulations and estimating new physics scales.

Contribution

It introduces the first hadron-level operators for $I=0$ contributions at order $p^6$ and fits experimental data to constrain new physics parameters.

Findings

Determined coefficients of $I=0$ and $I=2$ amplitudes from data.

Estimated the energy scale of potential new physics.

Analyzed the impact of future experiments on $C$ and $CP$ violation studies.

Abstract

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.