We study the transverse single-spin asymmetries for $p^uparrow pto pi, X$ and $p^uparrow pto gamma, X$ within the so-called color gauge invariant generalized parton model (CGI-GPM) which, in addition to spin and transverse momentum effects, includes initial and final state interactions with the polarized proton remnants. We compute all relevant contributions, focusing in particular on the process dependence of the gluon Sivers function, which, for these processes, can always be expressed as a linear combination of two independent, universal terms. This study extends and completes a previous one, where only quark initiated partonic processes were considered. We then perform a combined phenomenological analysis of RHIC data on transverse single-spin asymmetries in $p^uparrow pto pi, X$ and $p^uparrow pto D, X$, putting the first preliminary constraints on these two gluon Sivers functions. We show how their size can be estimated by means of these data, and use our results to provide predictions for the process $p^uparrow pto J/psi,X$, comparing them with data, and $p^uparrow pto gamma, X$, for which experimental information will soon become available. Corresponding estimates within the simpler GPM approach, without initial and final state interactions and with a single universal gluon Sivers function, are also given, showing that a clear discrimination between these two models is, for the moment, not possible.
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