8 FIG. 4: Graphs of the type that led to a violation of TMD-factorization in Ref. [17]. Gluons radiated from H 1 are illustrated in (a) and (b); gluons radiated from H 2 are illustrated in (c) and (d). The Hermitian conjugate graphs should also be included. These graphs vanish in our model because of their trivial color factor. deformation l can be made collinear to the minus direc- tion, and a vector n µ 2 = (1 , 0 , 0$_{t}$ ) is used instead of n µ $_{1}$. The eikonal factors are, The remaining type of single-gluon graphs are those in Fig. 4. For the lines collinear to H$_{1}$ the eikonal propaga- tors are for Fig. 3(c) and from Fig. 4(a) and for Fig. 3(d). In our calculation of the SSA, p$_{2}$ is unpolarized so there is no factor of i coming from the attachment to the up- per spectator. Therefore, it is only the real principal value contributions that are kept in Eqs. (12,13). Note that there can be no double Sivers effect at the level of one extra gluon because there must be at least one extra gluon from each of the hadrons for their PDFs to have spin dependence. The t $^{a}$s in Eqs. (8,10) combine with the t $^{a}$s in Eq. (9,11) to produce an overall color factor equal to Exactly the same color factor also arises when l is collinear to p$_{2}$ as in Figs. 3(c,d). If the color factor in Eq. (17) did not vanish, then it would contribute to an SSA when the factor of i combines with the factor of i in Eq. (9). The same result is obtained from the Hermitian conjugate graphs. In a totally un- polarized cross section, there would be no contribution from Fig. 4(b). The sum of these graphs gives a contri- bution equal to t $^{a}$gnν 2 − l − + iϵ = − gt $^{a}$nν $_{2}$P . V . 1 l − − igt $^{a}$nν $_{2}$π δ ( l $^{−}$) (12) Tr$_{C}$ [ t $^{a}$] gn µ 1 − l + + iϵ = − g Tr$_{C}$ [ t $^{a}$] n µ $_{1}$P . V . 1 l + − ig Tr$_{C}$ [ t $^{a}$] n µ $_{1}$π δ ( l $^{+}$) , (15) t $^{a}$gnν 2 − l − − iϵ = − gt $^{a}$nν $_{2}$P . V . 1 l − + igt $^{a}$nν $_{2}$π δ ( l $^{−}$) (13) Tr$_{C}$ [ t $^{a}$] gn µ 1 l + + iϵ = + g Tr$_{C}$ [ t $^{a}$] n µ $_{1}$P . V . 1 l + − ig Tr$_{C}$ [ t $^{a}$] n µ $_{1}$π δ ( l $^{+}$) , (16) − 2 ig Tr$_{C}$ [ t $^{a}$] πn µ $_{1}$δ ( l $^{+}$) . (17) T$_{F}$ ( N 2 c − 1) . (14) k 2 p 2 p 1 k 3 k 4 l k 1 − l k 4 − l p 2 p 1 k 3 k 4 k $_{2}$+ l l k 1 − l (a) (b) p 2 p 1 k 3 k 4 k 2 − l k 3 − l l k 1 p 2 p 1 k 3 k 4 k 2 − l l (c) (d)