14 FIG. 8: Graphs that contribute to a violation of generalized TMD-factorization. Other graphs that should be included are those with all possible attachments of l 1 to the k 4 and k 2 lines, and all possible attachments of l 2 to the k 3 and k 1 lines, and all Hermitian conjugate graphs. In total there are 16 graphs of this type. A. Same Side of the Cut result also follows from the same steps for find- ing general Wilson line structures from low order graphs as discussed in Ref. [12]. In the sum of graphs like Fig. 8(a), where both gluons are on the same side of the cut, the eikonal factors give a total contribution equal to Since spin dependence is needed in both H$_{1}$ and H$_{2}$ for a DSA, then there are also two factors corresponding to Eq. (9) (but with one corresponding to a p$_{2}$ spectator attachment). Taking into account both of the resulting factors of i gives an overall factor of i 2 = − 1. Combined with Eq. (35), the relevant factor from extra collinear gluons is then 4 π $^{2}$δ ( l + 1 ) δ ( l − 2 ). The same result is obtained from the Hermitian conjugate graphs. The sum of graphs with one extra gluon on each side of the cut, as in Fig. 8(b), works in much the same way. The eikonal factors give For a DSA, there is a factor of i from a factor analogous to Eq. (9) for the p$_{1}$ -spectator attachment of the gluon on the left side of the cut and a factor of − i from a Next, we must ensure that there is no cancellation be- tween graphs. We will therefore prove that generalized TMD- factorization, Eq. (32), is violated by showing that the sum of graphs of the type illustrated in Fig. 8 give a non-vanishing contribution to a DSA. First, we note that all graphs of the type shown in Fig. 8 include the non-zero color factor B. Opposite Side of Cut • Because of Eq. (33), the generalized TMD- factorizaton formula Eq. (32) cannot include any other two-gluon contributions to Wilson lines. In particular, there can be no contribution that cor- responds to graphs like Fig. 8, where one gluon is radiated collinear to each hadron simultaneously. These graphs would correspond to separate order- g contributions from the Wilson lines in separate TMD PDFs. If they are non-zero, then they contribute to a violation of both standard TMD- factorization in Eq. (24) and the generalized TMD factorization formula in Eq. (32). To incorporate such a contribution, one would have to modify the Wilson line in each TMD PDF such that it in- cludes a single-gluon contribution to a factorization anomaly. But this would contradict Sect. III where it was shown that there is no violation of standard factorization with just one gluon. Hence, contribu- tions from graphs like Fig. 8 cannot be consistently incorporated into a generalization of factorization simply by modifying Wilson lines in separate corre- lation functions. If they give a non-vanishing con- tribution, then there is a clear violation of general- ized TMD-factorization. Tr$_{C}$ [ t $^{a}$t$^{b}$] Tr$_{C}$ [ t $^{b}$t$^{a}$] = T 2 $_{F}$( N 2 c − 1) . (34) ( 1 − l + 1 − $_{iϵ}$)( 1 − l − 2 + $_{iϵ}$) + ( 1 l + 1 − $_{iϵ}$)( 1 − l − 2 + $_{iϵ}$) + + ( 1 − l + 1 − $_{iϵ}$)( 1 l − 2 + $_{iϵ}$) + ( 1 l + 1 − $_{iϵ}$)( 1 l − 2 + $_{iϵ}$) = 4 π $^{2}$δ ( l + 1 ) δ ( l − 2 ) . (36) ( 1 − l + 1 + $_{iϵ}$)( 1 − l − 2 + $_{iϵ}$) + ( 1 l + 1 + $_{iϵ}$)( 1 − l − 2 + $_{iϵ}$) + + ( 1 − l + 1 + $_{iϵ}$)( 1 l − 2 + $_{iϵ}$) + ( 1 l + 1 + $_{iϵ}$)( 1 l − 2 + $_{iϵ}$) = − 4 π $^{2}$δ ( l + 1 ) δ ( l − 2 ) . (35) p 2 p 1 k 1 − l 1 k 3 k 4 k 1 k 2 l 1 l 2 k 2 − l 2 k 2 p 2 p 1 k 1 − l 1 k 3 k 4 k 1 k 2 − l 2 l 1 l 2 (a) (b)