We consider within a generalized QCD factorization approach, the high energy inclusive polarized process p p --> pion + X, including all intrinsic partonic motions. Several new spin and k_T-dependent soft functions appear and contribute to cross sections and spin asymmetries. We present here formal expressions for transverse single spin asymmetries and double longitudinal ones. The transverse single spin asymmetry, A_N, is considered in detail, and all contributions are evaluated numerically. It is shown that the azimuthal phase integrations strongly suppress most contributions, leaving at work mainly the Sivers effect.
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