kTfactorization of exclusive processes

We prove the $k_T$ factorization theorem in perturbative QCD (PQCD) for exclusive processes by considering $\pi {\gamma }^{*}\to \gamma \left(\pi \right)$ and $\overrightarrow{B}\gamma \left(\pi \right)l\overline{\nu }.$ The relevant form factors are expressed as the convolution of hard amplitudes with two-parton meson wave functions in the impact parameter b space, b being conjugate to the parton transverse momenta $k_T.$ The point is that on-shell valence partons carry longitudinal momenta initially, and acquire $k_T$ through collinear gluon exchanges. The b-dependent two-parton wave functions with an appropriate path for the Wilson links are gauge-invariant. The hard amplitudes, defined as the difference between the parton-level diagrams of on-shell external particles and their collinear approximation, are also gauge-invariant. We compare the predictions for two-body nonleptonic B meson decays derived from $k_T$ factorization (the PQCD approach) and from collinear factorization (the QCD factorization approach).