Pion elastic and $({\pi }^0,\eta ,{\eta }^{\prime })\to \gamma {\gamma }^{\ast }$ transition form factors in a broad range of momentum transfers

Abstract

We analyze $F_{\pi }(Q^2)$ and $F_{P\gamma }(Q^2)$, $P=\pi ,\eta ,{\eta }^{\prime }$, within the local-duality (LD) version of QCD sum rules, which allows one to obtain predictions for hadron form factors in a broad range of momentum transfers. To probe the accuracy of this approximate method, we consider, in parallel to QCD, a potential model: in this case, the exact form factors may be calculated from the solutions of the Schrödinger equation and confronted with the results from the quantum-mechanical LD sum rule. On the basis of our quantum-mechanical analysis we conclude that the LD sum rule is expected to give reliable predictions for $F_{\pi }(Q^2)$ and $F_{\pi \gamma }(Q^2)$ in the region $Q^2\ge 5-6$ GeV${}^2$. Moreover, the accuracy of the method improves rather fast with growing $Q^2$ in this region. For the pion elastic form factor, the data at small $Q^2$ indicate that the LD limit may be reached already at relatively low values of momentum transfers, $Q^2\approx 4-8$ GeV${}^2$; we therefore conclude that large deviations from LD in the region $Q^2=20-50$ GeV${}^2$ reported in some recent theoretical analyses seem unlikely. The data on the ($\eta ,{\eta }^{\prime })\to \gamma {\gamma }^{\ast }$ form factors meet very well the expectations from the LD model. Surprisingly, the {\sc BaBar} results for the ${\pi }^0\to \gamma {\gamma }^{\ast }$ form factor imply a violation of LD growing with $Q^2$ even at $Q^2\approx 40$ GeV${}^2$, at odds with the $\eta ,{\eta }^{\prime }$ case and the experience from quantum mechanics.

8 pages

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