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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 $e^+e^-\to 3π$ cross section, generalizing previous studies on $ω,ϕ\to3π$ decays and $γπ\toππ$ scattering, and verify our result by comparing to $e^+e^-\toπ^0γ$ data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below 1 GeV, and extract the slope of the form factor at vanishing momentum transfer $a_π=(30.7\pm0.6)\times 10^{-3}$. 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.