Chiral anomaly in the $\eta^{(\prime)}\to\pi^+\pi^-\gamma$ and $\eta^{(\prime)}\to\pi^+\pi^-l^+l^-$ decays

Abstract

We report the presence of a flavor-violating correction to the $VAAA$-type anomaly, $\delta^{(\prime)}$, induced by the surface terms of the anomalous quark triangle diagram, previously found in the $\eta,\eta'\to\pi^+\pi^-\gamma$ decay amplitudes, and investigate its impact on the corresponding semileptonic decay modes of $\eta$ and $\eta'$. The magnitude of $\delta^{(\prime)}$ can be set from the experimental data on the $\eta^{(\prime)}\to \pi^+\pi^-\gamma$ decay width. We then estimate another low-energy constant, the slope parameter $\alpha^{(\prime)}$. The impact of different schemes for describing $\eta$-$\eta^\prime$ mixing on the value of $\delta^{(\prime)}$ and $\alpha^{(\prime)}$ is discussed. The predictions are shown to be in complete agreement with the available experimental data.