16 would contradict the single-extra-gluon treatment of the SSA in Sect. III. Therefore, a TMD-factorization formula cannot be recovered by simply modifying the Wilson lines in each TMD PDF separately. We have thus illustrated a specific example of a violation of generalized TMD- factorization. One can interpret the breakdown of generalized TMD- factorization visually by imagining the flow of color in the factorized expression, Eq. (32), as compared to the original unfactorized graph. The graph in Fig. 8b, for example, must correspond to the product of two single- gluon contributions to the Wilson lines in Eq. (32), if that formula is correct. The resulting color structure can be visualized as in Fig. 9. The narrow double lines represent the Wilson lines in each TMD PDF. The box shaped Wil- son lines represent the extra color-traced Wilson loops in Eq. (32). The color trace over each of the Wilson loops means that only a color singlet gluon can be exchanged between a Wilson loop and a spectator at the single-gluon level, as illustrated by the factors representing the TMD PDFs in Fig. 9. Therefore, these contributions to the TMD PDFs must vanish. By contrast, in the original unfactorized graph, the interlaced color flow of the two gluons means there is no problem with exchanging non- singlet color, as illustrated by the color flow diagram in Fig. 10. This corresponds to the non-zero color factor in Eq. (34). We have shown that a generalized TMD-factorization formula like Eq. (3)/Eq. (32) does not in general exist for back-to-back hadro-production of high- p$_{t}$ hadrons due to the non-trivial interplay of gluons that are collinear to both hadron directions. The failure of generalized TMD-factorization calls into question results that use it as a starting assumption. This includes calculations of weighted cross sections which retain a memory of the non-universality of the TMD PDFs. It is interesting to note that the same complications do not seem to arise if the gauge field is Abelian. In that case, the Abelian “color factors” can be identi- fied with coupling constants, such as in the model from Ref. [17]. Then the contribution to a DSA from graphs like Fig. 8 is indeed just a product of single gluon contri- butions to Wilson loops from separate correlation func- tions (which are non-zero in the Abelian case). There- fore, there is no obvious contradiction with generalized TMD-factorization for the Abelian case, at least at the level of graphs discussed here. While a full proof of gen- eralized TMD-factorization for the Abelian gauge theory does not yet exist, the breakdown of generalized TMD- factorization that we have found is a specific consequence of the non-Abelian nature of QCD. It is worth stressing that the failure of generalized TMD-factorization occurs in a regime where factoriza- tion would ordinarily be expected to apply; namely for It is possible to understand the origin of the general- ized TMD-factorization breakdown intuitively as arising from non-linear effects in the phases acquired by partons as they pass through the A + and A − fields of the collid- While the observation of generalized TMD factoriza- tion breaking leads to frustrating practical difficulties in cross section calculations, it should not necessarily be regarded as a purely negative result. The question of whether transverse momentum effects can be mean- ingfully associated with parton transverse momentum in separate parton correlation functions for each hadron is intrinsically important in the search for an improved fun- damental understanding of QCD dynamics in hard col- lisions. A counter-proof implies the existence of effects which challenge normal partonic intuition, and suggests new avenues of research. While naive factorization fails, the fact that the extra gluons eikonalize suggests that Wilson lines still may play a role. Insight might be gained, for example, from methods currently being ap- plied to small- x physics (e.g. [28, 29]). Another possi- bility is to model factorization breaking effects by di- rectly calculating factorization breaking phase contribu- tions in perturbation theory, but with explicit infrared and collinear cutoffs. Furthermore, the contribution from extra gluon at- tachments (which lead to factorization breaking) should not be thought of as negligible higher order corrections. They correspond to soft and collinear divergences in higher order hard scattering calculations. In real QCD, they are non-perturbative gluons in the strong coupling regime. For the perturbation series to have sensible con- vergence properties, it must be possible to rearrange terms such that the extra soft and collinear gluons are resummed to all orders into TMD PDFs. We also remark that, at the level of many extra gluons, there are more graphs with gluons collinear to both hadrons simultane- ously (which led to a breakdown of generalized TMD- factorization) than graphs with gluons radiated from one hadron only. hard processes with large- p$_{t}$ . Moreover, it should not be thought that the breakdown of generalized TMD- factorization is specific to a DSA or SSA. We have calcu- lated for a DSA with color singlet boson exchange for the hard part in order to maximize the simplicity of the ar- gument, but the reasons for the breakdown are quite gen- eral. The counter-example we have provided in this paper is sufficient to show that a general proof does not exist. That is, it is not possible to consistently define TMD PDFs for each hadron separately, even if we allow for process dependence in the Wilson line structures. To en- counter the same complications with generalized TMD- factorization in an unpolarized cross section, one needs the type of non-cancellation that was found in Sect. IV B (which comes from having two gluons collinear to one hadron), but with at least one more gluon collinear to the opposite hadron. An example of the type of graph which can produce a violation of generalized TMD-factorization in the unpolarized cross section is shown in Fig. 11. VII. DISCUSSION AND CONCLUSION