Perturbative QCD factorization of $\pi \gamma^*\to \gamma(\pi)$ and $B\to \gamma(\pi)l\bar \nu$

Abstract

We prove factorization theorem for the processes $\pi\gamma^*\to\gamma$ and $\pi\gamma^*\to\pi$ to leading twist in the covariant gauge by means of the Ward identity. Soft divergences cancel and collinear divergences are grouped into a pion wave function defined by a nonlocal matrix element. The gauge invariance and universality of the pion wave function are confirmed. The proof is then extended to the exclusive $B$ meson decays $B\to\gamma l\bar\nu$ and $B\to\pi l\bar\nu$ in the heavy quark limit. It is shown that a light-cone $B$ meson wave function, though absorbing soft dynamics, can be defined in an appropriate frame. Factorization of the $B\to\pi l\bar\nu$ decay in $k_T$ space, $k_T$ being parton transverse momenta, is briefly discussed. We comment on the extraction of the leading-twist pion wave function from experimental data.