Resummation of Large Logarithms in $\gamma^* \pi^0 \to \gamma $

Abstract

In the collinear factorization of the form factor for the transition $\gamma^* \pi^0 \to \gamma$ the hard part contains double log terms as $\ln^2 x$ with $x$ as the momentum fraction of partons from 0 to 1. A simple exponentiation for resummation leads to divergent results. We study the resummation of these $\ln^2 x$ terms. We show that the $\ln^2 x$ terms come partly from the light-cone wave function(LCWF) and partly from the form factor. We introduce a jet factor to factorize the $\ln^2 x$ term in the form factor. To handel the $\ln^2 x$ terms from the LCWF we introduce a nonstandard light-cone wave function(NLCWF) with the gauge links off the light-cone direction. An interesting relation between two wave function is found. With the introduced NLCWF and the jet factor we can re-factorize the form factor and obtain a new hard part which does not contain terms with $\ln^2 x$. Beside the renormalization scale $\mu$ the introduce NLCWF and jet factor have extra scales to characterize their $x$-behaviors. Using the evolutions of the extra scales and the relation we can do the resummation perturbatively in sense that the LCWF is the only nonpertubative object in the resumed formula. Our results with some models of LCWF show that there is a significant difference between numerical predictions with the resummation and that without the resummation, and the resummed predictions can describe the experimental data.