Universal Structure of Twist-3 Soft-Gluon-Pole Cross Sections for Single Transverse-Spin Asymmetry

We prove that twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetries (SSA) is determined by a certain ``primordial'' twist-2 cross section up to kinematic and color factors in the leading order perturbative QCD. In particular, for the processes in which the partonic hard scattering occurs among massless partons, the invariance of the ``primordial'' partonic cross section under scale transformation leads to remarkable simplification of the SGP cross section, reproducing compact form that was recently observed for pion production $p^\uparrow p\to \pi X$ and direct-photon production $p^\uparrow p\to \gamma X$.