Dispersive analysis of the pion transition form factor

Abstract

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 $e^{+}{e}^{-}\to 3\pi$ cross section, generalizing previous studies on $\omega ,\varphi \to 3\pi$ decays and $\gamma \pi \to \pi \pi$ scattering, and verify our result by comparing to $e^{+}{e}^{-}\to {\pi }^0\gamma$ data. We perform the analytic continuation to the space-like region, predicting the poorly-constrained space-like transition form factor below 1 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.

Comments: 13 pages, 5 figures, journal version