13 The sequence of eikonal factors in Eqs. (25,26) corre- spond exactly to gluons attaching to the “ V ” Wilson lines in Eq. (30). As before, the transverse links at infinity do not contribute to leading power in Feynman gauge. The color trace corresponds to the trace over the color loop in the upper part of the graphs in Fig. 7 and is what is needed to get the factor of Tr$_{C}$ [ t $^{a}$t$^{b}$] in Eqs. (25,26). Hence, the terms that violate standard factorization arise from expanding the U n$_{1}$ ( □ ) operator in Eq. (29) up to order g 2 (see also Ref. [20]). An exactly analogous TMD PDF is obtained for hadron H$_{2}$ when both extra gluons are radiated from H$_{2}$ in the collinear minus direction and attach to the lower half of the graph. The main legs of the Wilson loop in the TMD PDF for H$_{2}$ point in the direction of a vector n$_{2}$ : Thus, the factorization anomaly terms from Sect. IV B specify the type of new Wilson line insertions that are needed if generalized TMD-factorization is to hold. The generalized TMD-factorization formula is: Again, H is the same hard factor that appeared at zeroth order in Sect. II. The only difference from Eq. (24) comes from the U n ( □ ) insertions in the TMD PDFs, indicated by the ( □ ) superscripts on the individual TMD PDFs in Eq. (32). We have again included a “!” over the equal sign to indicate that even this formula will ultimately fail when the remaining graphs are considered. Equation (32), with the TMD PDFs defined as in Eq. (29), follows the method proposed in [9, 12, 15], and is the most natural generalization of TMD-factorization. For the purpose of our counter-argument to generalized TMD-factorization, the crucial point is that the new Wil- son line insertion Eq. (30) involves a trace over color. If we instead tried contracting the color indices of the new Wilson lines with the quark fields at points 0 and ξ , then we would obtain the wrong color factor for the anomalous contribution in Eqs. (25,26). An even more serious problem would be that the generalized TMD- factorization formula would produce spurious terms at the one -gluon level, already treated in Sect. III. Namely, single-gluon attachments to the extra legs of the Wilson line would introduce factorization anomaly contributions analogous to Eq. (17), but with different non-vanishing color factors, contradicting the analysis from Sect. III which found no violation of standard TMD-factorization at order g in the expansion of the Wilson lines (polarized or unpolarized). Thus, consistency between the order- g and order- g 2 contributions to the Wilson lines requires that the new Wilson line structures include the trace over color as in Eq. (30). Other possible redefinitions of the TMD PDFs are un- related to the color structure of the Wilson lines in the main correlation functions. These include the modifica- tions mentioned in Sect. II, such as tilting the direction of the Wilson line slightly away from the light-like direc- tion. Thus, accounting for all contributions up to order g $^{2}$, in the Wilson line, and requiring consistency between different orders in g , strongly constrains the possibilities for a generalization of TMD-factorization. The order- g contribution from a Wilson loop in Eq. (32) is zero simply because, with one gluon attach- ing to the Wilson loop, the color trace in the definition of the Wilson line operator Eq. (30) involves only one color generator Tr$_{C}$ [ t $^{a}$] = 0. So, Therefore, there is no disagreement between Eq. (32) and standard factorization, Eq. (24), from graphs with only one extra gluon. To find a disagreement between the standard and generalized TMD-factorization formu- las (Eqs. (24) and (32)), at least two gluons need to be collinear to one of the hadrons as in Sect. IV B. We will now directly illustrate a breakdown of the gen- eralized TMD-factorization formula Eq. (32) by calculat- ing the anomalous two-gluon contribution to a double Sivers effect from the graphs in Fig. 8. First, let us sum- marize the situation so far: • Considering graphs with up to two extra glu- ons, there is a one-to-one correspondence between graphs of the type shown in Fig. 3/Fig. 6 and the contributions to the Wilson lines in the standard TMD-factorization formula Eq. (24). • The sum of graphs with two extra gluons radi- ated from one spectator and attaching on the oppo- site side of the hard subgraph (as in the graphs of Fig. 7) results in terms that violate standard TMD- factorization in unpolarized scattering. These are the contributions already discussed in Ref. [18]. • The two-gluon factorization anomaly terms found in Sect. IV B specify which modifications of the Wilson lines in the TMD PDFs are needed if a form of factorization is to be recovered. It is found that the modified TMD PDFs must each contain an ex- tra color-traced Wilson loop, as in Eq. (29). This VI. BREAKDOWN OF GENERALIZED TMD-FACTORIZATION Φ [ n$_{2}$, ( □ )] H$_{2}$ ( x$_{2}$, k$_{2}$$_{T}$ ) = x$_{2}$p − 2 ∫ dw $^{+}$d$^{2}$w$_{t}$ (2 π ) 3 e − ix$_{2}$p − 1 w $^{+}$+ i k$_{t}$ · $^{w$_{t}$}$× × 〈 H$_{2}$, s$_{2}$ | φ † 2 $_{,r}$(0 , w $^{+}$, w$_{t}$ ) U n$_{2}$ rs [0 , w ] U n$_{2}$ ( □ $_{)}$φ$_{2}$$_{,s}$ (0) | H$_{2}$, s$_{2}$ 〉 . (31) U n$_{1}$ ( □ ) ∣ O ( g ) = U n$_{2}$ ( □ ) ∣ O ( g ) = 0 . (33) dσ ! = H ⊗ Φ [ n$_{1}$, ( □ )] H$_{1}$ ( x$_{1}$, k$_{1}$$_{T}$ ) ⊗ Φ [ n$_{2}$, ( □ )] H$_{2}$ ( x$_{2}$, k$_{2}$$_{T}$ ) ⊗ ⊗ δ $^{(2)}$( k$_{1}$$_{T}$ + k$_{2}$$_{T}$ − k$_{3}$$_{T}$ − k$_{4}$$_{T}$ ) . (32)