We study the reaction $e^+e^-\to\pi^0\gamma$ based on a dispersive representation of the underlying $\pi^0\to\gamma\gamma^*$ transition form factor. As a first application, we evaluate the contribution of the $\pi^0\gamma$ channel to the hadronic-vacuum-polarization correction to the anomalous magnetic moment of the muon. We find $a_\mu^{ \pi^0\gamma}\big|_{ \leq 1.35\,\text{ GeV}}=43.8(6)\times 10^{ -11}$, in line with evaluations from the direct integration of the data. Second, our fit determines the resonance parameters of $\omega$ and $\phi$. We observe good agreement with the $e^+e^-\to3\pi$ channel, explaining a previous tension in the $\omega$ mass between $\pi^0\gamma$ and $3\pi$ by an unphysical phase in the fit function. Combining both channels we find $\bar M_\omega=782.736(24)\,\text{ MeV}$ and $\bar M_\phi=1019.457(20)\,\text{ MeV}$ for the masses including vacuum-polarization corrections. The $\phi$ mass agrees perfectly with the PDG average, which is dominated by determinations from the $\bar K K$ channel, demonstrating consistency with $3\pi$ and $\pi^0\gamma$. For the $\omega$ mass, our result is consistent but more precise, exacerbating tensions with the $\omega$ mass extracted via isospin-breaking effects from the $2\pi$ channel.
Categories: hep-ph hep-ex hep-lat nucl-th