Two-parton twist-3 factorization in perturbative QCD

We prove collinear factorization theorem for the process $\pi\gamma^*\to\pi$ at the twist-3 level in the covariant gauge by means of the Ward identity, concentrating on the two-parton case. It is shown that soft divergences cancel and collinear divergences are grouped into the pseudo-scalar and pseudo-tensor two-parton twist-3 pion distribution amplitudes. The delicate summation of a complete set of diagrams for achieving factorization in momentum, spin, and color spaces is emphasized. The proof is then extended to the exclusive semileptonic decay $B\to\pi l\bar\nu$, assuming the hard scale to be of $O(\sqrt{\bar\Lambda M_B})$, where $\bar\Lambda$ is a hadronic scale and $M_B$ the $B$ meson mass. We explain the distinction between the factorization of collinear divergences for a pion distribution amplitude and of soft divergences for a $B$ meson distribution amplitude. The gauge invariance and universality of the two-parton twist-3 pion distribution amplitudes are confirmed. The proof presented here can accommodate the leading twist-2 case. We then compare our proof with that performed in the framework of soft-collinear effective theory.