  Recently, the BABAR collaboration published (arXiv:0905.4778) data for the
photon-pion transition form factor $F_{\pi\gamma\gamma^{\ast}}(Q^{2})$, which
are in strong contradiction to the predictions of the standard factorization
approach to perturbative QCD. Immediately afterwards, two mechanisms were
suggested (A.E. Dorokhov, arXiv:0905.4577; A.V. Radyushkin, arXiv:0906.0323),
that logarithmically enhance the form factor asymptotics and therefore provide
a qualitatively satisfactory description of the BABAR data. However, the
physics of the BABAR effect was not fully clarified. In the present work, based
on a nonperturbative approach to the QCD vacuum and on rather universal
assumptions, we show that there exists two asymptotic regimes for the pion
transition form factor. One regime with asymptotics
$F_{\pi\gamma^{\ast}\gamma}(Q^{2})\sim1/Q^{2}$ corresponds to the result of the
standard QCD factorization approach, while other violates the standard
factorization and leads to asymptotic behavior as
$F_{\pi\gamma^{\ast}\gamma}(Q^{2})\sim\ln(Q^{2})Q^{2}$. Furthermore,
considering specific nonlocal chiral quark models, we find the region of
parameters, where the existing CELLO, CLEO and BABAR data for the pion
transition form factor are successfully described.
