Novel Master Formula for Twist-3 Soft-Gluon-Pole Mechanism to Single Transverse-Spin Asymmetry

We prove that twist-3 soft-gluon-pole (SGP) cross section for single spin asymmetries is determined by a certain ``primordial’’ twist-2 cross section up to kinematic and color factors in the leading order perturbative QCD. This unveils universal structure behind the SGP cross sections in a variety of hard processes, and also the special role of the scale invariance in the corresponding primordial cross section, which leads to remarkable simplification of the SGP cross sections for the production of massless particle, such as those for pion production p^\uparrow p \to \pi X and direct-photon production p^\uparrow p \to \gamma X.