On the $\gamma^*\gamma \to\pi(\eta ,\eta')$ transition form factors

The surprising results by the BarBar collaboration on the $\pi\gamma$ transition form factor require new thoughts about the high-$Q^2$ dependence of the form factors with virtual photons. We make use of the anomaly sum rule [J. Horejsi and O. Teryaev, Z. Phys. C65, 691 (1995).] which relates the hadron spectral density to the axial anomaly [S. Adler, Phys. Rev. 177, 2426 (1969); J. S. Bell and R. Jackiw, Nuovo Cimento A 60, 47 (1969).]. We study the quark-hadron duality relation for this sum rule and find out that the increase of the rescaled form factor $Q^2F_{\pi\gamma}(Q^2)\sim\log(Q^2)$ suggested by the BaBar data requires the presence of a $1/s$-correction term in the relation between the one-loop spectral density and the hadron-continuum spectral density.