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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- 

[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]]. 

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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]. 

[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 

## Acknowledgments 

