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Pion elastic and $(π^0,η,η')\toγγ^*$ transition form factors in a broad range of momentum transfers

We analyze $F_π(Q^2)$ and $F_{Pγ}(Q^2)$, $P=π,η,η'$, within the local-duality (LD) version of QCD sum rules, which allows one to obtain predictions for hadron form factors in a broad range of momentum transfers. To probe the accuracy of this approximate method, we consider, in parallel to QCD, a potential model: in this case, the exact form factors may be calculated from the solutions of the Schrödinger equation and confronted with the results from the quantum-mechanical LD sum rule. On the basis of our quantum-mechanical analysis we conclude that the LD sum rule is expected to give reliable predictions for $F_π(Q^2)$ and $F_{πγ}(Q^2)$ in the region $Q^2 \ge 5-6$ GeV$^2$. Moreover, the accuracy of the method improves rather fast with growing $Q^2$ in this region. For the pion elastic form factor, the data at small $Q^2$ indicate that the LD limit may be reached already at relatively low values of momentum transfers, $Q^2\approx 4-8$ GeV$^2$; we therefore conclude that large deviations from LD in the region $Q^2=20-50$ GeV$^2$ reported in some recent theoretical analyses seem unlikely. The data on the ($η,η')\toγγ^*$ form factors meet very well the expectations from the LD model. Surprisingly, the {\sc BaBar} results for the $π^0\toγγ^*$ form factor imply a violation of LD growing with $Q^2$ even at $Q^2\approx 40$ GeV$^2$, at odds with the $η,η'$ case and the experience from quantum mechanics.