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

Abstract

We construct hadronic amplitudes for the three-body decays $η^{(\prime)}\toπ^+π^-π^0$ and $η'\toηπ^+π^-$ in a non-perturbative fashion, allowing for $C$- and $CP$-violating asymmetries in the $π^+π^-$ 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 $η^{(\prime)}\toπ^+π^-π^0$ and an effective isovector operator in $η'\toηπ^+π^-$, while the strongly different sensitivities to these operators can be understood from chiral power counting arguments.