Dispersion relation for hadronic light-by-light scattering: pion pole

Understand

The pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon $(g-2)_\mu$ is fully determined by the doubly-virtual pion transition form factor.

• Although this crucial input quantity is, in principle, directly accessible in experiment, a complete measurement covering all kinematic regions relevant for $(g-2)_\mu$ is not realistic in the foreseeable future.
• Here, we report in detail on a reconstruction from available data, both space- and time-like, using a dispersive representation that accounts for all the low-lying singularities, reproduces the correct high- and low-energy limits, and proves convenient for the evaluation of the $(g-2)_\mu$ loop integral.
• We concentrate on the systematics of the fit to $e^+e^-\to 3\pi$ data, which are key in constraining the isoscalar dependence, as well as the matching to the asymptotic limits.