Extracting the chiral anomaly from $e^+e^-\to 3\pi$

The strength of the interaction of three pions and a photon, $F_{3\pi}$ is predicted by the axial anomaly in terms of the pion decay constant, a relation that is frequently used to constrain low-energy radiative processes involving pions, but only tested experimentally at the $10\%$ level. Here, we present a new avenue to test this prediction, via a fit of a dispersive description of the $\gamma^*\to3\pi$ amplitude to data for $e^+e^-\to 3\pi$. From the global fit to SND, CMD-2, and BaBar data we obtain $F_{3\pi}=33.1(1.7)\,\text{GeV}^{-3}$, in agreement with the chiral prediction at the level of $5\%$. We also consider dispersive fits to the recent data by Belle II, in which case we observe tensions with the dispersive constraints, the width parameters of $\omega$ and $\phi$, and the chiral anomaly.