On unique predictions for single spin azimuthal asymmetry

Theoretically there are two approaches to predict single spin azimuthal asymmetries. One is to take transverse momenta of the partons into account by using transverse momentum dependent parton distributions, while another is to take asymmetries as twist-3 effects. The non-perturbative effects in these approaches are parameterized with different matrix elements, and predictions can be different. Recently, gauge invariant definitions of transverse momentum dependent parton distributions were derived. With these definitions it can be shown that there are relations between non-perturbative matrix elements in these two approaches. These relations may enable us to unify two approaches and to have unique predictions for single spin azimuthal asymmetries. In this letter we derive these relations by using time-reversal symmetry and show that even with these relations the single spin azimuthal asymmetry in a Drell-Yan process is predicted differently in different approaches.