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We study the $P\to\gamma\,\gamma^*$ ($P=\pi^0,\eta,\eta'$) transition formfactors by means of the local-duality (LD) version of QCD sum rules. For thecase of $\eta$ and $\eta'$, the conventional LD model provides a gooddescription of the existing data. However, for the $\pi$ form factor we finddisagreement with recent BaBar results for high $Q^2$ even though the accuracyof the LD approximation is expected to increase with $Q^2$. It remainsmysterious why the $\eta$ and $\eta'$ form factors to virtual photons, on theone hand, and the $\pi$ form factor, on the other hand, show a qualitativelydifferent behaviour corresponding to a rising with $Q^2$ violation of localduality in the pion case. In a quantum mechanical example we show that, for abound-state size of about 1 fm, the LD sum rule provides an accurate predictionfor the form factor for $Q^2\ge$ a few GeV$^2$.

The puzzle of the \pi -> \gamma \gamma* transition form factor