Patterns of $C$- and $CP$-violation in hadronic $\eta$ and ${\eta }^{\prime }$ three-body decays

Abstract

We construct hadronic amplitudes for the three-body decays ${\eta }^{(\prime )}\to {\pi }^{+}{\pi }^{-}{\pi }^0$ and ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$ 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 ${\eta }^{\prime }\to \eta {\pi }^{+}{\pi }^{-}$, while the strongly different sensitivities to these operators can be understood from chiral power counting arguments.

Comments: 43 pages, 9 figures, 5 tables; v2: discussion extended, version published in JHEP; v3: incorporates an erratum that fixes the phases of the C-violating norms by hermiticity and time reversal