QCD Sum Rule Calculation of $\gamma\gamma^*\to\pi^0$ Transition Form Factor

Abstract

We develop a QCD sum rule analysis of the form factor $F_{\gamma^*\gamma^*\pi^0}(q^2,Q^2)$ in the region where virtuality of one of the spacelike photons is small $q^2 \ll 1 GeV^2$ while another is large: $Q^2 \gapprox 1 GeV^2$. We construct the operator product expansion suitable for this kinematic situation and obtain a QCD sum rule for $F_{\gamma^*\gamma^*\pi^0}(0,Q^2)$. Our results confirm expectation that the momentum transfer dependence of $F_{\gamma^*\gamma^*\pi^0}(0,Q^2)$ 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 $\varphi_{\pi}(x)$. The absolute value of the $1/Q^2$ term obtained in this paper favours $\varphi_{\pi}(x)$ close to the asymptotic form $\varphi_{\pi}^{as} (x)=6 f_{\pi} x (1-x)$.