  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.
