cause they are not directly related to the main reasons that generalized TMD-factorization fails. The most naive definitions of TMD correlation functions include extra divergences that need to be removed by appropriate re- definitions (see Ref. [22] and references therein for an overview of issues related to the precise definition of a TMD PDF). One particular complication is that the Wil- son lines in these definitions cannot be exactly light-like without containing extra “light-cone” divergences which correspond to partons moving with large rapidity oppo- site to the direction of their parent hadrons. Proposed solutions involve either tilting the Wilson line away from the exactly light-like direction [4], or dividing out by ex- tra gauge invariant factors [6, 23–25]. Furthermore, an exactly correct factorization formula requires a soft fac- tor to account for gluons with all components small. The other parton correlation functions tend to overlap in the soft region, and a fully correct definition of a TMD PDF requires extra factors to remove the overlap. 

There has been much significant recent work devoted to finding a fully consistent definition of the TMD PDF. However, our main concern in this paper is only with the general color structure of the Wilson lines inside the matrix elements for the external hadrons. Therefore, for our purposes it will be sufficient to continue to treat the direction of the main Wilson lines in the TMD PDFs as being light-like, and we will not address the role of a soft factor or the overlap of regions. Finally, we will restrict consideration to the limit of very large relative transverse momentum (large- p$_{t}$ ) where one expects standard pQCD methods to be most appropriate, and we do not consider the possibility of recovering a type of TMD-factorization appropriate in the small- x limit [26]. 

Having discussed the basic graph in Fig. 1, the next step is to consider graphs dressed with extra gluons. The graphs which can contribute to the Wilson line insertions are those in which extra, nearly on-shell gluons connect different subgraphs — for example, graphs with gluons connecting the p$_{1}$ -collinear lines to the outgoing struck quark lines or to the p$_{2}$ -collinear lines. Normally, one expects the sum of such graphs to contribute to the Wil- son line in the TMD PDF for H$_{1}$ , after application of a Ward identity. The primary issue is that the approx- imations that normally allow Ward identities to be ap- plied are only valid after certain contour deformations on the extra momentum integrals. Namely, to apply a Ward identity, it must be possible to approximate an ex- tra gluon by a longitudinally polarized one with a large component of longitudinal momentum. Then the extra gluon momentum can be contracted with the hard scat- tering matrix element and a Ward identity argument can be applied directly. However, when virtual gluons at- tach to a spectator line, they give contributions from the “Glauber” region, meaning that if l is the momen- tum of an extra gluon, then | l $^{+}$| and | l $^{−}$| are both much smaller than | l$_{T}$ | . In the Glauber region, the approxi- mations needed for the Ward identity are not valid. For factorization to work, it must be possible to first deform 

the contour out of the Glauber region (see, for example, Refs. [6]). Alternatively, factorization could be recovered if there is a cancellation between graphs, as in the stan- dard proofs of integrated (collinear) factorization. The basic problem with TMD-factorization found in previous work [11, 12, 15, 17, 18] is that the necessary contour deformations needed to treat the Glauber region are in- consistent with a direct application of the standard Ward identity arguments. Namely, they are in different direc- tions for different graphs depending on whether the in- teraction is in the initial or final state. 

So, we will only consider graphs that can yield con- tributions from the Glauber region. Since real gluons can never be in the Glauber region, we will only con- sider graphs with virtual gluons. Also, as long as no restrictions are placed on the target remnant momenta, graphs with spectator-spectator interactions cancel [5] in the integration over final states [32]. Similar cancella- tions occur between different cuts of the same graph for active-spectator interactions after parton transverse mo- mentum is integrated over, and are needed in the stan- dard proofs of collinear factorization [5]. A counter-proof of TMD-factorizaton therefore needs to show that such cancellations generally fail when transverse parton mo- mentum is explicitly taken into account. A specific ex- ample of such a non-cancellation was given in Ref. [18] and will be reviewed in Sect. IV. In graphs with attach- ments between active quarks, there are not enough Dirac γ -matrices to give spin dependence to the TMD PDFs. Such graphs will therefore not affect our discussion of sin- gle and double spin asymmetries at lowest non-vanishing order. Furthermore, graphs with a scalar-scalar-gluon- gluon vertex do not give leading power contributions to eikonal factors. 

We remark that, because the TMD factorization break- ing effects are due to the Glauber region where all com- ponents of gluon momentum are small, the interactions responsible for breaking TMD factorization are associ- ated with large distance scales. 

In our specific model, a large number of graphs van- ish simply because of the highly simple color structure involved. Examples are shown in Fig. 2. They vanish because their color factors include a trace around a color loop of a single SU( N$_{c}$ ) generator, Tr$_{C}$ [ t $^{a}$] = 0. (The C on the Tr$_{C}$ [ · · · ] denotes a trace over triplet color indices.) 

Hence, the relevant types of graphs are represented by Figs. 3 through 8. If a generalized TMD-factorization formula is possible, then the sum over all such graphs must produce a factorized form like Eq. (3) with a Wilson line structure in the TMD PDF or FF for each hadron separately. We will consider each type of graph in the following sections. 

We begin the investigation of diagrams by reviewing the steps for determining the contribution from a single 

## III. ONE EXTRA GLUON 

