The puzzle of the π-> γγ* transition form factor

We study the $P\toγ\,γ^*$ ($P=π^0,η,η'$) transition form factors by means of the local-duality (LD) version of QCD sum rules. For the case of $η$ and $η'$, the conventional LD model provides a good description of the existing data. However, for the $π$ form factor we find disagreement with recent BaBar results for high $Q^2$ even though the accuracy of the LD approximation is expected to increase with $Q^2$. It remains mysterious why the $η$ and $η'$ form factors to virtual photons, on the one hand, and the $π$ form factor, on the other hand, show a qualitatively different behaviour corresponding to a rising with $Q^2$ violation of local duality in the pion case. In a quantum mechanical example we show that, for a bound-state size of about 1 fm, the LD sum rule provides an accurate prediction for the form factor for $Q^2\ge$ a few GeV$^2$.

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