  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.
