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

Abstract

The pion-pole contribution to hadronic light-by-light scattering in the anomalous magnetic moment of the muon (g − 2) μ 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) μ 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) μ loop integral. We concentrate on the systematics of the fit to e + e − → 3 π data, which are key in constraining the isoscalar dependence, as well as the matching to the asymptotic limits. In particular, we provide a detailed account of the pion transition form factor at low energies in the time- and space-like region, including the error estimates underlying our final result for the pion-pole contribution, $$ {a}_{\mu}^{\uppi^0-\mathrm{pole}}={62.6}_{-2.5}^{+3.0}\times {10}^{-11} $$ a μ π 0 − pole = 62.6 − 2.5 + 3.0 × 10 − 11 , and demonstrate how forthcoming singly-virtual measurements will further reduce its uncertainty.