17 FIG. 9: Color flow resulting from the single gluon contributions for each of the Wilson loops in the TMD-factorization formula Eq. (32). H is the standard zeroth order hard part and the second two factors are the TMD PDFs. The narrow double lines represent Wilson lines. The boxes associated with each of the TMD PDFs correspond to the Wilson loops. The thick solid red and dotted blue lines (color online) illustrate the flow of color in each TMD PDF. Each of the contributions to a TMD PDF shown here is exactly zero because each includes a factor Tr C [ t $^{a}$] = 0. FIG. 10: Color flow in the unfactorized graph with a single gluon collinear to each of the incoming hadrons as in Fig. 8. The thick solid red and dotted blue lines (color online) again illustrate the flow of color. Non-singlet color can easily be exchanged and results in a non-zero contribution. Compare with Fig. 9. ing hadrons. If the overall phase were simply the product of the phases induced by the A + fields from hadron H$_{1}$ and the A − fields from hadron H$_{2}$ , then one could asso- ciate any process-dependent phases induced by the A + field in hadron H$_{1}$ with a modified Wilson line for the TMD PDF of H$_{1}$ and, likewise, any process-dependent phases induced by the A − field from H$_{2}$ could be asso- ciated with a modified Wilson line for the TMD PDF of H$_{2}$ . However, in the non-Abelian theory the role of the A − gluons in H$_{2}$ is affected by the presence of the A + gluons from H$_{1}$ and visa-versa. A direct example of this is Fig. 8/Eq. (38), where a single A − gluon exchanged between H$_{2}$ and the opposite-side struck quark gives a non-zero contribution, but only because there is simul- taneously an A + gluon exchanged between H$_{1}$ and the other struck quark. This means that one cannot address the role of phases induced by the A + and A − fields inde- pendently, but instead must deal with them simultane- ously. The result is a kind of nonperturbative correlation which cannot be identified as arising strictly from gluons coming from either hadron independently, but only from the combination. We would especially like to thank D. Boer and C. Mantz for useful discussions. One of us (T. Rogers) would like to thank J. Collins for helpful discussions. Support was provided by the research program of the “Stichting voor Fundamenteel Onderzoek der Materie (FOM)”, which is financially supported by the “Ned- erlandse Organisatie voor Wetenschappelijk Onderzoek (NWO)”. All figures were made using Jaxodraw [30]. Acknowledgments [1] D. Boer, P. J. Mulders and C. Pisano, arXiv:0909.4652 [hep-ph]. [2] Z. Lu and I. Schmidt, Phys. Rev. D 78 , 034041 (2008) [arXiv:0805.4006 [hep-ph]]. [3] J. C. Collins, D. E. Soper and G. Sterman, Adv. Ser. Direct. High Energy Phys. 5 , 1 (1988) [arXiv:hep-ph/0409313]. [4] J. C. Collins and D. E. Soper, Nucl. Phys. B 194 , 445 p$_{1}$ ⊗ ⊗ H dσ ∼ p$_{2}$ p$_{2}$ p$_{1}$