Abelian anomaly and neutral pion production

The process $\gamma^\ast \gamma \to \pi^0$ is fascinating because in order to explain the associated transition form factor within the Standard Model on the full domain of momentum transfer, one must combine, using a single internally-consistent framework, an explanation of the essentially nonperturbative Abelian anomaly with the features of perturbative QCD. The case for attempting this has received a significant boost with the publication of data from the BaBar Collaboration [Phys.\ Rev.\ D {\bf 80}, 052002 (2009)] because, while they agree with earlier experiments on their common domain of squared-momentum-transfer [CELLO - Z.\ Phys.\ C {\bf 49}, 401 (1991); CLEO - Phys.\ Rev.\ D {\bf 57}, 33 (1998)], the BaBar data are unexpectedly far \emph{above} the prediction of perturbative QCD at larger values of $Q^2$. I will elucidate the sensitivity of the $\gamma^\ast \gamma \to \pi^0$ transition form factor, $G_{\gamma^\ast \gamma \pi}(Q^2)$, to the pointwise behaviour of the interaction between quarks; and use existing Dyson-Schwinger equation calculations of this and the kindred $\gamma^\ast \gamma^\ast \to \pi^0$ form factor to characterize the $Q^2$-dependence of $G_{\gamma^\ast \gamma \pi} (Q^2)$. It will become apparent that in fully-self-consistent treatments of pion: static properties; and elastic and transition form factors, the asymptotic limit of the product $Q^2 G_{\gamma^\ast\gamma \pi^0}(Q^2)$, which is determined \emph{a priori} by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer: the product is a monotonically-increasing concave function. Studies exist which interpret the BaBar data as an indication that the pion's distribution amplitude, $\phi_\pi(x)$, deviates dramatically from its QCD asymptotic form, indeed, that $\phi_\pi(x)=\,$constant, or is at least flat and nonvanishing at $x=0,1$. I will explain that such a distribution amplitude characterises an essentially-pointlike pion; and show that, when used in a fully-consistent treatment, it produces results for pion elastic and transition form factors that are in striking disagreement with experiment. A bound-state pion with a pointlike component will produce the hardest possible form factors; i.e., form factors which become constant at large-$Q^2$. On the other hand, QCD-based studies produce soft pions, a valence-quark distribution amplitude for the pion that vanishes as $\sim (1-x)^2$ for $x\sim 1$, and results that agree well with the bulk of existing data. It can thus be argued that the large-$Q^2$ BaBar data is inconsistent with QCD and also inconsistent with a vector current-current contact interaction; and hence that the large- $Q^2$ data reported by BaBar is not a true representation of the $\gamma^\ast\gamma \to \pi^0$ transition form factor.