Factorization and Gauge Invariance of Twist-3 Cross Section for Single Spin Asymmetry

We prove factorization and gauge invariance of the twist-3 single-spin-dependent cross section in the leading order perturbative QCD, which was missing in the previous literature. We emphasize that the consistency relation from the Ward identities for color gauge invariance is crucial to guarantee the cancelation among the various gauge-noninvariant contributions in the cross section. This relation also proves the absence of the "derivative" terms in the cross section corresponding to the hard-pole and soft-fermion-pole contributions. Applying the formalism to SIDIS, $ep^\uparrow\to e\pi X$, we have derived the complete cross section associated with the twist-3 distribution for the transversely polarized nucleon.