Two-parton twist-3 factorization in perturbative QCD

We prove factorization theorem for the process $\pi\gamma^*\to\pi$ at the twist-3 level in the covariant gauge by means of the Ward identity and gauge invariance, 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 $B$ meson decay $B\to\pi l\bar\nu$, assuming the hard scale to be of $O(\sqrt{\bar\Lambda M_B})$, where $\bar\Lambda\equiv M_B-m_b$ is the $B$ meson and $b$ quark mass difference. We explain the distinction between the above collinear factorization and the soft factorization for the $B$ meson distribution amplitudes. The gauge invariance and universality of the two-parton twist-3 pion distribution amplitudes are confirmed. The proof presented here can also accommodate the leading twist-2 case.