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

The ${\pi }^0$ 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 $(g-2{)}_{\mu }$. 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 $e^{+}{e}^{-}\to 3\pi$, $e^{+}{e}^{-}\to e^{+}{e}^{-}{\pi }^0$, and the ${\pi }^0\to \gamma \gamma$ 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 $(g-2{)}_{\mu }$ loop integral. The resulting value, $a_{\mu }^{{\pi }^0\text{-pole}}=62.6_{-2.5}^{+3.0}\times {10}^{-11}$, 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 $a_{\mu }^{{\pi }^0\text{-pole}}$.

Figure 1

Diagrammatic representation of ($a$) HVP, ($b$) HLBL, and ($c$) the ${\pi }^0$ pole in HLBL. Solid, dashed, and wiggly lines denote muons, pions, and photons, respectively, and the short-dashed line indicates that the pion is to be taken on shell.

Figure 2

Fit to the $e^{+}{e}^{-}\to 3\pi$ cross section from SND [86, 87] and BABAR [88], in comparison to [55] (HKLNS14).

Figure 3

Resulting uncertainty band for the asymptotic spacelike singly virtual transition form factor (upper) and the corresponding prediction for the low-energy region (lower). The dashed line refers to the BL limit $2F_{\pi }$.

Figure 4

Two-dimensional representation of $(Q_1^2+Q_2^2)F_{{\pi }^0{\gamma }^{*}{\gamma }^{*}}(-Q_1^2,-Q_2^2)$, indicating how the respective asymptotic limits are approached.