Dispersive analysis of the pion transition form factor

We analyze the pion transition form factor using dispersion theory. We calculate the singly-virtual form factor in the time-like region based on data for the \(e^+e^-\rightarrow 3\pi \) cross section, generalizing previous studies on \(\omega ,\phi \rightarrow 3\pi \) decays and \(\gamma \pi \rightarrow \pi \pi \) scattering, and verify our result by comparing to \(e^+e^-\rightarrow \pi ^0\gamma \) data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below \(1\,\text {GeV}\), and extract the slope of the form factor at vanishing momentum transfer \(a_\pi =(30.7\pm 0.6)\times 10^{-3}\). We derive the dispersive formalism necessary for the extension of these results to the doubly-virtual case, as required for the pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon.