Three-gluon contribution to the single spin asymmetry in Drell-Yan and direct-photon processes

We derive the single-spin-dependent cross section for the Drell-Yan lepton-pair production and the direct-photon production in the $pp$-collision induced by the twist-3 three-gluon correlation functions in the transversely polarized nucleon in the leading order with respect to the QCD coupling constant. Combined with the contribution from the twist-3 quark-gluon correlation functions in the literature, this completes the twist-3 cross section for these processes. We also present a model calculation of the asymmetry for the direct photon production at the RHIC energy, demonstrating the sensitivity of the asymmetry to the form of the three-gluon correlation functions. In particular, we show that the asymmetry in the backward direction of the polarized nucleon is determined by the small-$x$ behavior of the correlation functions.