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Gauge links for transverse momentum dependent correlators at tree-level

In this paper we discuss the incorporation of gauge links in hadronic matrix elements that describe the soft hadronic physics in high energy scattering processes. In this description the matrix elements appear in soft correlators and they contain non-local combinations of quark and gluon fields. In our description we go beyond the collinear approach in which case also the dependence on transverse momenta of partons is taken into consideration. The non-locality in the transverse direction leads to a complex gauge link structure for the full process, in which color is entangled, even at tree-level. We show that at tree-level in a 1-parton unintegrated (1PU) situation, in which only the transverse momentum of one of the initial state hadrons is relevant, one can get a factorized expression involving transverse momentum dependent (TMD) distribution functions. We point out problems at the level of two initial state hadrons, even for relatively simple processes such as Drell-Yan scattering.