2 ble to cases where observables are not sensitive to in- trinsic transverse parton momentum. In the collinear factorization theorems, transverse momentum is inte- grated over inside the definitions of the standard PDFs and FFs. These are the standard “integrated” PDFs and FFs which are expressible as well-defined, process- independent operator matrix elements [4]. The process- independence means that the integrated PDFs and FFs can be parameterized by experimental data and then later reused in calculations to make first-principle pre- dictions for future experiments. Thus, this universal- ity property of the standard integrated PDFs and FFs lends great predictive power to the standard pQCD ap- proaches, and is a basic component of the standard collinear factorization theorems. It is natural to hope that an analogous universality property applies to the TMD PDFs and TMD FFs in a TMD-factorization formula. In that case, we would say that standard TMD-factorization is valid, where we will explicitly use the word “standard” to refer to the universality condition. If a standard TMD-factorization formula were valid for the process in (1), then it would be possible to calculate cross sections using the same TMD PDFs and FFs that are parameterized in other processes like deep inelastic scattering (DIS) or the Drell- Yan (DY) process. However, as we will discuss in more detail, TMD-factorization is not generally valid for the process in Eq. (1). The essential complication arises from longitudinally polarized gluons that couple soft and collinear subgraphs to the hard part and which, at first sight, appear to break topological factorization graph-by-graph at lead- ing power. Dealing with these “extra” gluons is one of the main issues that must be dealt with in all factoriza- tion proofs in pQCD. In DIS, for example, it is found that after summing over all graphs (and applying appropriate approximations), a Ward identity argument allows the extra gluon contributions to be factored into contribu- tions which correspond to Wilson lines (also called gauge links) in the definitions of the PDFs or FFs [3]. How- ever, in order to justify the approximations that allow an application of the Ward identity, it is necessary to first perform certain contour deformations on the gluon mo- mentum [5]. For DIS, the contour deformations must be consistent with having extra gluon attachments between hadron spectators and a final state struck quark. Similar steps apply to DY, but there the contour deformations should correspond to extra attachments between hadron spectators and an initial state quark. (See also Ref. [6].) The result is that the Wilson line for the quark TMD PDF in DIS is future pointing while the Wilson line for a quark TMD PDF in DY is past pointing [7]. The dif- ference in Wilson line direction has been shown to result in a sign flip for the Sivers function (a particular type of TMD PDF) in DY as compared to DIS [7]. Strictly speaking, this could be regarded as a breakdown of stan- dard TMD-factorization because the Sivers function is not truly universal in DIS and DY. However, since a sign Here, H$_{c,j}$ is the hard part for subprocess j and color routing c . For each subprocess and routing of color through the hard part, there is in general a different set of TMD PDFs and TMD FFs corresponding to the dif- ferent Wilson line structures that they contain. In the above notation Φ [ WL c,j 1 ] H$_{1}$,j is a gauge invariant TMD PDF for hadron H$_{1}$ with a Wilson line [ W L c,j 1 ] correspond- ing to subprocess j and color routing c . Analogous no- tation is used for the other process-dependent correla- tion functions. (There are also possible soft factors not shown explicitly in Eq. (3)). Each term in Eq. (3) has To address the role of non-universal Wilson lines, the concept of a “generalized” TMD-factorization formula was later developed [11, 12]. In this approach, it is assumed that the only deviation from standard TMD- factorization is that the TMD parton correlation func- tions (PDFs and FFs) must contain process-dependent Wilson lines. Schematically, instead of Eq. (2) one must assume a more complicated expression of the form Similar problems are encountered in proofs of collinear factorization for hadron-hadron scattering, but there one is saved by cancellations between graphs that occur after integration over transverse momentum (see, e.g., Ref. [5] and also more recent work in Ref. [10]). Since these can- cellations are not point-by-point in transverse momen- tum, they do not generally apply to TMD-factorization. The problems that occur with TMD-factorization for Eq. (1) are much more complicated because there one must deal with extra gluons that connect spectators to both initial and final state partons. The result is that the contour deformations necessary for a factor- ization proof prevent a direct application of the usual Ward identity arguments. This problem was observed by Bomhof, Mulders and Pijlman [9] who found that the Wilson lines needed for gauge-invariant TMD PDFs and FFs in hadro-production of hadrons are not generally process-independent. Although the extra gluon attach- ments eikonalize, the resulting sums of eikonal factors do not correspond to the simple future or past pointing Wilson lines that are found in DIS and DY. In [9] it was shown at the level of a single extra gluon that the TMD PDFs and FFs (assuming consistent definitions exist) are non-universal because they require, at a minimum, pro- cess dependent Wilson lines. Hence, there is a violation of the universality property necessary for standard TMD- factorization. flip is easily accounted for, it is more appropriate to say that the Sivers function in DIS and DY possesses a type of “modified” universality. The relationship between cor- relation functions in DIS and DY is further discussed in Ref. [8]. dσ = ∑ j,c H$_{j,c}$ ⊗ Φ [ WL c,j 1 ] H$_{1}$,j ⊗ Φ [ WL c,j 2 ] H$_{2}$,j ⊗ Δ [ WL c,j 3 ] H$_{3}$,j ⊗ Δ [ WL c,j 4 ] H$_{4}$,j + p . s . c . (3)