We study the anomalous electromagnetic pion form factor $F_{{\pi }^0{\gamma }^{*}{\gamma }^{*}}$ with a set of holographic models. By comparing with the measured value of the linear slope, some of these models can be ruled out. From the remaining models, we obtain predictions for the low-energy quadratic slope parameters of $F_{{\pi }^0{\gamma }^{*}{\gamma }^{*}}$, currently out of experimental reach but testable in the near future. We find it particularly useful to encode this low-energy information in a form factor able to satisfy also QCD short-distance constraints. We choose the form factor introduced by D’Ambrosio, Isidori, and Portoles in kaon decays, which has the right short distance for a particular value of the quadratic slope, which is later shown to be compatible with our holographic predictions. We then turn to a determination of the (dominant) pion exchange diagram in the hadronic light by light scattering contribution to the muon anomalous magnetic moment. We quantify the theoretical uncertainty in $(g-2{)}_{\mu }$ coming from the different input we use: QCD short distances, experimental input, and low-energy holographic predictions. We also test the pion-pole approximation. Our final result is $a_{\mu }^{{\pi }^0}=6.54(25)\times {10}^{-10}$, where the error is driven by the linear slope of $F_{{\pi }^0{\gamma }^{*}{\gamma }^{*}}$, soon to be measured with precision at KLOE-2. Our numerical analysis also indicates that large values of the magnetic susceptibility ${\chi }_0$ are disfavored, therefore pointing at a mild effect from the pion off-shellness. However, in the absence of stronger bounds on ${\chi }_0$, an additional $(10–15)\%$ systematic uncertainty on the previous value for $a_{\mu }^{{\pi }^0}$ cannot be excluded.

• Received 16 November 2010

DOI:https://doi.org/10.1103/PhysRevD.83.093006