  We consider the recent RHIC data on the transverse single spin asymmetry
(SSA) A_N, measured in p(transv. polarized) p --> pion X processes at
mid-rapidity by the PHENIX Collaboration. We analyze this experimental
information within a hard scattering approach based on a generalized QCD
factorization scheme, with unintegrated, transverse momentum dependent (TMD),
parton distribution and fragmentation functions. In this kinematical region,
only the gluon Sivers effect could give a large contribution to A_N; its
vanishing value is thus used to give approximate upper limits on the gluon
Sivers function (GSF). Additional constraints from the Burkardt sum rule for
the Sivers distributions are also discussed.
