Twist-3 Formalism for Single Transverse Spin Asymmetry Reexamined: Semi-Inclusive Deep Inelastic Scattering

We study the single spin asymmetry (SSA) for the pion production in semi-inclusive deep inelastic scattering, $ep^\uparrow\to e\pi X$, in the framework of the collinear factorization. We derive the complete cross section formula associated with the twist-3 quark-gluon correlation functions for the transversely polarized nucleon, including all types of pole (hard-pole, soft-fermion-pole and soft-gluon-pole) contributions which produce the strong interaction phase necessary for SSA. We prove that the partonic hard part from each pole contribution satisfies certain constraints from Ward identities for color gauge invariance. We demonstrate that the use of these new constraints is crucial to reorganize the collinear expansion of the Feynman diagrams into manifestly gauge-invariant form so as to obtain the factorization formula for the cross section in terms of a complete set of the twist-3 distributions without any double counting. It also provides a simpler method for the actual calculation. We also present a simple estimate of SSA based on our cross section formula, using a model for the ``soft-gluon-pole function'' that represents the relevant twist-3 quark-gluon correlation, and compare the magnitude of the terms involving the derivative of the soft-gluon-pole function with that of the ``non-derivative'' terms.