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 $\eta ,{\eta }^{\prime }\to {\pi }^{+}{\pi }^{-}\gamma$ decay amplitudes, and investigate its impact on the corresponding semileptonic decay modes of $\eta$ and ${\eta }^{\prime }$. 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.

Comments: 25 pages, 5 figures