Dispersive analysis of the pion transition form factor

TL;DR

This paper employs dispersion theory to analyze the pion transition form factor, providing predictions in the space-like region and establishing a formalism for the doubly-virtual case relevant to muon g-2 calculations.

Contribution

It introduces a dispersive formalism for the pion transition form factor and extends previous analyses to the doubly-virtual case for muon g-2.

Findings

Predicted the space-like form factor below 1 GeV.

Extracted the slope at zero momentum transfer: a_π=(30.7±0.6)×10⁻³.

Verified results with e⁺e⁻→π⁰γ data.

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 $\omega,\phi\to3\pi$ 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 $a_\pi=(30.7\pm0.6)\times 10^{-3}$. 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.