15 factor analogous to Eq. (11) (but for a gluon attaching at the p$_{2}$ -spectator) on the right side of the cut, giving an overall factor of i ( − i ) = +1. So, combined with Eq. (36) the relevant factor from extra gluon attachments is again 4 π $^{2}$δ ( l + 1 ) δ ( l − 2 ). itly, one can use Eqs. (36) (extracting the overall fac- tor of i ( − i ) = 1 that comes with the two spectator at- tachments) to explicitly calculate the contribution from graphs with gluons on opposite sides of the cut: Summing all graphs of the type shown in Fig. 8, there- fore, results in just a single integral. To check it explic- Here we have dropped the irrelevant factors of (2 E$_{3(4)}$ ), λ$_{1(2)}$ , and (2 π ) 6 which appeared in Eq. (4). We have also dropped the l$_{1}$ and l$_{2}$ dependence inside the factor in braces. This is permitted because the δ -functions will set l + 1 and l − 2 to zero, and because the other components yield power suppressed corrections. The δ -functions can be used to evaluate the k + 1 , k − 1 , k + 2 , k − 2 , l + 1 , l − 2 and k$_{2}$$_{T}$ integrals, and the l − 1 and l + 2 integrals can be evaluated by contour integration. The result is The factor H ( k$_{1}$, k$_{3}$, k$_{4}$ ) is now the hard part in Eq. (6). The symbol “ · · · ” in the denominators refers to terms that involve only x$_{1}$ , x$_{2}$ and masses. For graphs with both gluons on the same side of the cut, exactly the same result (with the same overall sign) is obtained by using Eq. (35) and taking into account the overall factor of i 2 = − 1 that comes from the two spectator attachments. There are, again, identical results coming from the Her- mitian conjugate graphs. The integral in Eq. (38) is not generally zero as can be checked by considering a fixed q ≡ k$_{3}$$_{T}$ + k$_{4}$$_{T}$ . How- ever, there is no corresponding factorization anomaly term contained in an expansion of Eq. (32) up to or- der g 2 in the Wilson lines. The modified Wilson lines in the TMD PDFs of the generalized TMD-factorization formula are specified by the factorization anomaly terms that arise when gluons are radiated from just one hadron at a time, as in Sect. IV. However, the resulting general- ized TMD-factorization formula is in contradiction with graphs like Fig. 8, where there are simultaneously glu- ons radiated from H$_{1}$ and H$_{2}$ . Thus, generalized TMD- factorization breaks down at the two-gluon level. Indeed, at up to two gluons, the generalized TMD- factorization formula Eq. (32) gives no contribution to a DSA beyond what is predicted by the standard TMD- factorization formula Eq. (24). The only factorization anomaly contribution from Eq. (32) is to the unpolar- ized cross section, and arises from graphs with both extra gluons radiated from the same hadron. There must be at least one gluon coming from each hadron at the same time to have spin dependence in both hadrons simulta- neously. So, the only contribution to a double Sivers effect that violates standard TMD-factorization is from the sum of graphs in Fig. 8, and these are not accounted for in Eq. (32). To recover the factorization anomaly in the DSA, one would have to somehow allow each TMD PDF to have a non-vanishing factorization anomaly con- tribution at the level of one extra gluon. This, however, C. Together 2 T 2 $_{F}$( N 2 c − 1) g 4 2 s ∫ d $^{4}$k$_{1}$ (2 π ) 4 ∫ d $^{4}$k$_{2}$ (2 π ) 4 ∫ d $^{4}$l$_{1}$ (2 π ) 4 ∫ d $^{4}$l$_{2}$ (2 π ) $_{4}$(2 π ) $^{4}$δ$^{4}$( k$_{1}$ + k$_{2}$ − k$_{3}$ − k$_{4}$ )(2 π ) $^{2}$δ ( l + 1 ) δ ( l − 2 ) × × { λ ′ 1 $^{2}$λ ′ 2 $_{2}$( k$_{1}$ + k$_{3}$ ) · ( k$_{2}$ + k$_{4}$ ) ( k$_{1}$ − k$_{3}$ ) 2 − M 2 X } 2 2 ϵ$_{jk}$s j $_{1}$l k $_{1}$p + 1 ( m$_{H}$$_{1}$ (1 − x$_{1}$ ) + m$_{ψ}$$_{1}$ ) [ l 2 1 + iϵ ][( k$_{1}$ − l$_{1}$ ) 2 − m 2 q$_{1}$ + iϵ ][( p$_{1}$ − k$_{1}$ + l$_{1}$ ) 2 − m 2 ψ$_{1}$ + iϵ ][ k 2 − m 2 q$_{2}$ + iϵ ] × × 2 ϵ$_{j}$ ′ k ′ s j ′ 2 l k ′ 2 p − 2 ( m$_{H}$$_{2}$ (1 − x$_{2}$ ) + m$_{ψ}$$_{2}$ ) [ l 2 − iϵ ][( k$_{2}$ − l$_{2}$ ) 2 − m 2 q$_{2}$ − iϵ ][( p$_{2}$ − k$_{2}$ + l$_{2}$ ) 2 − m 2 ψ$_{2}$ − iϵ ][ k 2 1 − m 2 q$_{1}$ − iϵ ] (2 π ) $^{2}$δ (( p$_{1}$ − k$_{1}$ ) $^{2}$− m 2 $_{ψ$_{1}$}$) δ (( p$_{2}$ − k$_{2}$ ) $^{2}$− m 2 $_{ψ$_{2}$}$) . (37) T 2 $_{F}$( N 2 c − 1) g 4 s ∫ d $^{2}$k$_{1}$$_{T}$ (2 π ) 2 ∫ d $^{2}$l$_{1}$$_{T}$ (2 π ) 2 ∫ d $^{2}$l$_{2}$$_{T}$ (2 π ) 2 H ( k$_{1}$, k$_{3}$, k$_{4}$ ) × × [ ϵ$_{jk}$s j $_{1}$l k $_{1}$( m$_{H}$$_{1}$ (1 − x$_{1}$ ) + m$_{ψ}$$_{1}$ )][ ϵ$_{j}$ ′ k ′ s j ′ 2 l k ′ 2 ( m$_{H}$$_{2}$ (1 − x$_{2}$ ) + m$_{ψ}$$_{2}$ )] l 2 1 $_{T}$l 2 $_{T}$[ k 2 1 T + · · · ][( l$_{1}$$_{T}$ − k$_{1}$$_{T}$ ) 2 + · · · ][( k$_{3}$$_{T}$ + k$_{4}$$_{T}$ − k$_{1}$$_{T}$ ) 2 + · · · ][( l$_{2}$$_{T}$ − k$_{3}$$_{T}$ − k$_{4}$$_{T}$ + k$_{1}$$_{T}$ ) 2 + · · · ] . (38)