Pion-Pole Contribution to Hadronic Light-By-Light Scattering in the Anomalous Magnetic Moment of the Muon

Abstract

The ${\ensuremath{\pi}}^{0}$ pole constitutes the lowest-lying singularity of the hadronic light-by-light (HLBL) tensor, and thus, it provides the leading contribution in a dispersive approach to HLBL scattering in the anomalous magnetic moment of the muon $(g\ensuremath{-}2{)}_{\ensuremath{\mu}}$. It is unambiguously defined in terms of the doubly virtual pion transition form factor, which in principle, can be accessed in its entirety by experiment. We demonstrate that, in the absence of a direct measurement, the full spacelike doubly virtual form factor can be reconstructed very accurately based on existing data for ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}3\ensuremath{\pi}$, ${e}^{+}{e}^{\ensuremath{-}}\ensuremath{\rightarrow}{e}^{+}{e}^{\ensuremath{-}}{\ensuremath{\pi}}^{0}$, and the ${\ensuremath{\pi}}^{0}\ensuremath{\rightarrow}\ensuremath{\gamma}\ensuremath{\gamma}$ decay width. We derive a representation that incorporates all the low-lying singularities of the form factor, matches correctly onto the asymptotic behavior expected from perturbative QCD, and is suitable for the evaluation of the $(g\ensuremath{-}2{)}_{\ensuremath{\mu}}$ loop integral. The resulting value, ${a}_{\ensuremath{\mu}}^{{\ensuremath{\pi}}^{0}\text{-pole}}=62.{6}_{\ensuremath{-}2.5}^{+3.0}\ifmmode\times\else\texttimes\fi{}{10}^{\ensuremath{-}11}$, for the first time, represents a complete data-driven determination of the pion-pole contribution with fully controlled uncertainty estimates. In particular, we show that already improved singly virtual measurements alone would allow one to further reduce the uncertainty in ${a}_{\ensuremath{\mu}}^{{\ensuremath{\pi}}^{0}\text{-pole}}$.