Abstract
We construct an evolution equation for the pion wave function in the $k_T$ factorization theorem, whose solution sums the mixed logarithm $\ln x\ln k_T$ to all orders, with $x$ ($k_T$) being a parton momentum fraction (transverse momentum). This joint resummation induces strong suppression of the pion wave function in the small $x$ and large $b$ regions, $b$ being the impact parameter conjugate to $k_T$, and improves the applicability of perturbative QCD to hard exclusive processes. The above effect is similar to those from the conventional threshold resummation for the double logarithm $\ln^2 x$ and the conventional $k_T$ resummation for $\ln^2 k_T$. Combining the evolution equation for the hard kernel, we are able to organize all large logarithms in the $\gamma^{\ast} \pi^{0} \to \gamma$ scattering, and to establish a scheme-independent $k_T$ factorization formula. It will be shown that the significance of next-to-leading-order contributions and saturation behaviors of this process at high energy differ from those under the conventional resummations. It implies that QCD logarithmic corrections to a process must be handled appropriately, before its data are used to extract a hadron wave function. Our predictions for the involved pion transition form factor, derived under the joint resummation and the input of a non-asymptotic pion wave function with the second Gegenbauer moment $a_2=0.05$, match reasonably well the CLEO, BaBar, and Belle data.
Highlights
Namely, of ζP2 introduces a factorization-scheme dependence into the hadron wave function
The moderate x and small b regions are more highlighted compared to the case with the conventional threshold and kT resummations
We stress that the joint resummation, organizing all the important logarithms for an arbitrary rapidity parameter in the pion wave function and in the hard kernel, is a treatment more general than the conventional resummations
Summary
The TMD pion wave function Φ(x, kT ) is defined by the non-local hadron-to-vacuum matrix element. A TMD hadron wave function describes the distributions of a light parton in both light-ray and transverse directions. The non-light-like vector u, different from the usual Wilson line direction n+ = (1, 0, 0T ), plays a role of the regulator for the light-cone divergences [15]. The transverse gauge link Iu;y,0, unraveling the cusp obstruction in the contour of the Wilson lines at infinity, does not contribute in the covariant gauge [42]. The QCD correction to the pion wave function gives rise to the mixed logarithm ln x ln(ζ2P −2/kT2 ) [9,10,11], with the dimensionless rapidity parameter ζ2. Where ⊗ represents convolutions in the momentum fraction x and the transverse momentum kT , and the evolution kernel Γ involves the diagrams with the special vertex
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