Abelian anomaly and neutral pion production

Abstract

We show that in fully self-consistent treatments of the pion, namely, its static properties and elastic and transition form factors, the asymptotic limit of the product ${Q}^{2}{G}_{{\ensuremath{\gamma}}^{*}\ensuremath{\gamma}{\ensuremath{\pi}}^{0}}({Q}^{2})$, determined a priori by the interaction employed, is not exceeded at any finite value of spacelike momentum transfer. Furthermore, in such a treatment of a vector-vector contact-interaction one obtains a ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ transition form factor that disagrees markedly with all available data. We explain that the contact interaction produces a pion distribution amplitude that is flat and nonvanishing at the endpoints. This amplitude characterizes a pointlike pion bound state. Such a state has the hardest possible form factors (i.e., form factors that become constant at large momentum transfers and hence are in striking disagreement with completed experiments). However, interactions with QCD-like behavior produce soft pions, a valence-quark distribution amplitude that vanishes as $~(1\ensuremath{-}x){}^{2}$ for $x~1$, and results that agree with the bulk of existing data. Our analysis supports a view that the large-${Q}^{2}$ data obtained by the BaBar Collaboration is not an accurate measure of the ${\ensuremath{\gamma}}^{*}\ensuremath{\gamma}\ensuremath{\rightarrow}{\ensuremath{\pi}}^{0}$ form factor.