Extracting the Sivers function from polarized SIDIS data and making predictions

The most recent data on the weighted transverse single spin asymmetry $A_{UT}^{\sin(\phi_h-\phi_S)}$ from HERMES and COMPASS collaborations are analysed within LO parton model with unintegrated parton distribution and fragmentation functions; all transverse motions are taken into account, with exact kinematics, in the elementary interactions. The overall quality of the data is such that, for the first time, a rather well constrained extraction of the Sivers function for $u$ and $d$ quarks is possible and is performed. Comparisons with models are made. Based on the extracted Sivers functions, predictions for $A_{UT}^{\sin(\phi_h-\phi_S)}$ asymmetries at JLab are given; suggestions for further measurements at COMPASS, with a transversely polarized hydrogen target and selecting favourable kinematical ranges, are discussed. Predictions are also presented for Single Spin Asymmetries (SSA) in Drell-Yan processes at RHIC and GSI.