Chiral anomaly in the $η^{(\prime)}\toπ^+π^-γ$ and $η^{(\prime)}\toπ^+π^-l^+l^-$ decays

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