QCD Sum Rule Calculation of $\gamma {\gamma }^{\ast }\to {\pi }^0$ Transition Form Factor

Abstract

We develop a QCD sum rule analysis of the form factor $F_{{\gamma }^{\ast }{\gamma }^{\ast }{\pi }^0}(q^2,Q^2)$ in the region where virtuality of one of the spacelike photons is small $q^2\ll 1Ge{V}^2$ while another is large: $Q^2\text{gapprox}1Ge{V}^2$. We construct the operator product expansion suitable for this kinematic situation and obtain a QCD sum rule for $F_{{\gamma }^{\ast }{\gamma }^{\ast }{\pi }^0}(0,Q^2)$. Our results confirm expectation that the momentum transfer dependence of $F_{{\gamma }^{\ast }{\gamma }^{\ast }{\pi }^0}(0,Q^2)$ is close to interpolation between its $Q^2=0$ value fixed by the axial anomaly and $Q^{-2}$ pQCD behaviour for large $Q^2$. Our approach, in contrast to pQCD, does not require additional assumptions about the shape of the pion distribution amplitude ${\phi }_{\pi }(x)$. The absolute value of the $1/Q^2$ term obtained in this paper favours ${\phi }_{\pi }(x)$ close to the asymptotic form ${\phi }_{\pi }^{as}(x)=6f_{\pi }x(1-x)$.

Comments: 13 pages LaTex, uses epsf.tex to insert figures; with 3 figures in encapsulated postscript form