11 FIG. 7: Graphs that contribute to a factorization anomaly. Other graphs that should be included are all other graphs with both gluons attaching the lower spectator to the k 2 and k 4 lines at the upper part of the graph. In addition, the anomalous contribution to the TMD PDF of H 2 is obtained by including the graphs with both gluons radiated from p 2 and attaching the upper spectator to the k 1 and k 3 lines. There is a total of 32 graphs of this type, including graphs with both gluons radiated from p 2 . A. Graphs Contributing to Standard Factorization k 2 p 2 p 1 l 1 k 3 k 4 k 1 − l 1 k 1 − l 2 l 2 k 2 p 2 p 1 k 3 k 4 l 1 k 1 − l 1 − l 2 l 2 (a) (b) B. Violation of Standard TMD-Factorization Tr$_{C}$ [ t $^{a}$t$^{b}$] g $^{2}$n µ $_{1}$n ν 1 ( 1 − l + 1 + iϵ + 1 l + 1 + $_{iϵ}$) × × ( 1 − l + 2 − iϵ + 1 l + 2 − $_{iϵ}$) = 4 π $^{2}$g$^{2}$n µ $_{1}$n ν $_{1}$Tr$_{C}$ [ t $^{a}$t$^{b}$] δ ( l + 1 ) δ ( l + 2 ) . (25) By repeating the steps of Sect. III, graphs of the type shown in Figs. 6 lead to the Wilson lines in Eqs. (19, 23) for the TMD PDFs of the standard TMD-factorization formula. The eikonal factors that arise from Figs. 6(a,b) correspond to keeping the order g Wilson line contri- bution from each of the TMD PDFs individually. Fig- ures 6(c,d) (and related graphs) correspond to keeping the order g 2 Wilson line contributions in one of the TMD PDFs, and the zeroth order from the other. The graphs considered up to now exhaust the contri- butions obtained by expanding each of the Wilson lines in the standard TMD-factorization formula Eq. (24) up to order g 2 (in the Wilson line). Therefore, any re- maining uncanceled contributions from the graphs of the type shown in Figs. 7, 8 violate the standard TMD- factorization formula. The graphs of the type shown in Fig. 7, where two gluons attach the spectator of one hadron to the op- posite side of the hard subgraph, have been shown in Refs. [18, 20] to be non-vanishing in calculations of both the unpolarized cross section and an SSA for an Abelian gauge theory. These graphs, therefore, yield a violation of the standard TMD-factorization formula Eq. (24). The contribution to an anomalous SSA requires an imaginary part from the extra eikonal factors. In Ref. [18] the imaginary part came from graphs with one gluon at- taching near to the parent hadron and the other attach- ing near to the opposite hadron (e.g., Fig. 2(b)). These For the graphs with both extra gluons coupling on the same side of the cut (Fig. 7(b) and related graphs)), the steps are similar. Again including all ways of attaching l$_{1}$ and l$_{2}$ at the upper part of the graph, one finds For the graphs with the extra gluons on opposite sides of the cut (Fig. 7(a) and related graphs), the factorization anomaly for the unpolarized cross section comes from replacing the extra off-shell quark propagators by their eikonal propagators. Including all ways of attaching l$_{1}$ and l$_{2}$ to the k$_{2}$ and k$_{4}$ lines, one finds the same result as in Ref. [18], but with the Abelian g$_{1}$ and g$_{2}$ charges replaced by the appropriate color factor: The non-cancellation of the unpolarized factorization anomaly, however, remains. It arises from graphs with both extra gluons radiating from the same hadron and attaching at the side of the hard subgraph near to the opposite hadron, as in Fig. 7. To make the violation of standard TMD-factorization explicit, we will briefly re- view the steps of Ref. [18]. We are for the moment only concerned with the unpolarized cross section so we will temporarily simplify the model of Sect. II by making all the hadron and spectator fields scalars. We further sim- plify the calculation by assuming m$_{q}$$_{1}$ = m$_{q}$$_{2}$ = m$_{ψ}$$_{1}$ = m$_{ψ}$$_{2}$ = m$_{q}$ . graphs vanish in our model because they again include a color factor Tr$_{C}$ [ t $^{a}$] = 0.