Patterns of $C$- and $CP$-violation in hadronic $\eta$ and $\eta'$ three-body decays

TL;DR

This paper develops a non-perturbative framework for analyzing $C$- and $CP$-violation in three-body decays of $ ext{eta}$ and $ ext{eta'}$ mesons, constraining asymmetries with experimental data and extracting coupling strengths.

Contribution

It introduces a dispersive, analyticity- and unitarity-consistent approach to study $C$- and $CP$-violating effects in hadronic $ ext{eta}$ and $ ext{eta'}$ decays, linking theory with experimental Dalitz plot data.

Findings

Asymmetries are confined to a per mille level by current data.

Dispersive representation enables extraction of coupling strengths of symmetry-violating operators.

Sensitivity to different operators aligns with chiral power counting expectations.

Abstract

We construct hadronic amplitudes for the three-body decays $\eta^{(\prime)}\to\pi^+\pi^-\pi^0$ and $\eta'\to\eta\pi^+\pi^-$ in a non-perturbative fashion, allowing for $C$- and $CP$-violating asymmetries in the $\pi^+\pi^-$ distributions. These amplitudes are consistent with the constraints of analyticity and unitarity. We find that the currently most accurate Dalitz-plot distributions taken by the KLOE-2 and BESIII collaborations confine the patterns of these asymmetries to a relative per mille level. Our dispersive representation allows us to extract the individual coupling strengths of the $C$- and $CP$-violating contributions arising from effective isoscalar and isotensor operators in $\eta^{(\prime)}\to\pi^+\pi^-\pi^0$ and an effective isovector operator in $\eta'\to\eta\pi^+\pi^-$, while the strongly different sensitivities to these operators can be understood from chiral power counting arguments.