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Within the so-called color gauge invariant generalized parton model, a TMDscheme including initial- (ISI) and final-state (FSI) interactions, we presenta phenomenological analysis of available SSA data for pion and $D$-mesonproduction in $pp$ collisions. This allows us, for the first time, to put apreliminary constraint on the two universal types of gluon Sivers functionentering the model. Predictions for SSAs in $J/\psi$ and direct photonproduction, as well as a comparison with the simpler generalized parton model(without ISIs and FSIs), are also presented.

Process dependence of the gluon Sivers function in inclusive $pp$ collisions: phenomenology