Hadronic vacuum polarization and vector-meson resonance parameters from $e^{+}{e}^{-}\to {\pi }^0\gamma$

We study the reaction $e^{+}{e}^{-}\to {\pi }^0\gamma$ based on a dispersive representation of the underlying ${\pi }^0\to \gamma {\gamma }^{\ast }$ 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 $\varphi$. We observe good agreement with the $e^{+}{e}^{-}\to 3\pi$ 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 ${\bar{M}}_{\varphi }=1019.457(20)\text{MeV}$ for the masses including vacuum-polarization corrections. The $\varphi$ 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.