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gauge field couples with equal strength to the quarks in each hadron, g$_{1}$ = g$_{2}$ = g . 

We also will introduce one more field: The hard sub- process will be described by an exchange of a hypotheti- cal new massive color-neutral U(1) gauge boson which we will call X . It couples with strengths λ ′ 1(2) to quarks of flavor 1(2). Hence, the hard subgraph has a trivial color structure. This will allow us to study the role of extra gluons in a non-Abelian gauge theory, while limiting the number of Feynman graphs that need to be explicitly considered, and eliminating color flow directly through the hard part. It is important to emphasize that the general conclusions of this paper do not depend on this specific model for the hard subprocess. What is impor- tant is that color is carried by both the initial and final state partons. A proof of generalized TMD-factorization for such processes should be general and apply to any hard hadro-production process with observed hadrons in both the initial and final states. If it fails for a specific ex- ample, then no general proof exists. In cases where color is exchanged in the hard process, the Wilson line struc- tures for the TMD PDFs in Eq. (3) are determined after first expanding the color matrices of the hard subprocess into different routings following the procedure described in Ref. [12]. The steps in this paper then apply to each term in a series of color routings, and similar contradic- 

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tions with generalized TMD-factorization can be found. Our choice of hard subprocess is to make the illustra- tion of why generalized TMD-factorization fails as clear as possible. 

The basic graph contributing to hadro-production of high- p$_{t}$ hadrons, with no extra gluon attachments, is shown in Fig. 1. Because of the trivial color structure of the hard part, and the simplicity of our model of quarks and spectators, there is only one subprocess and one color flow. So a generalized TMD-factorization formula corre- sponding to Eq. (3) can only involve one term. 

We work in the center-of-mass frame where the incom- ing hadron H$_{1}$ is initially moving with large rapidity in the forward plus direction with no transverse momen- tum, while H$_{2}$ is initially moving with large rapidity in the minus direction with no transverse momentum. Each hadron splits into an active “quark” which enters the hard subgraph, and a spectator “diquark” which enters the final state. The quarks interact in the hard part by exchanging a hard colorless vector boson X , with momen- tum q where | q $^{2}$| is large, | q $^{2}$|  ≫ M 2 $_{X}$. Within the model, the final state high transverse momentum hadrons or jets are represented simply by on-shell final state quarks. 

The polarization dependent differential cross section at zeroth order in g is, 

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The factor of N 2 c comes from tracing over the unit color triplet matrices in the upper and lower color loops in Fig. 1. We will use the approximation that the struck partons are collinear to the directions of their parent hadrons. So, k + 1 ∼ p + 1 and k − 2 ∼ p − 2 . The minus component of k$_{1}$ and the plus component of k$_{2}$ are small (order Λ $^{2}$/p + 1 ) and can be neglected inside the hard part. We also define x$_{1}$ = k + 1 /p + 1 and x$_{2}$ = k − 2 /p − 2 , and s is the usual Mandelstam variable 

