7 FIG. 3: Extra gluon attachments that are consistent with the standard Wilson line structure. The difference in shading on the arrow at the eikonal attachments for (a,b) and (c,d) is to emphasize that the gluons are from different hadrons. They correspond to p 1 in (a,b) and to p 2 in (c,d). extra gluon. As in Ref. [17], we focus on the calculation of an SSA. We start with graphs of the type shown in Fig. 3, where the extra gluon attaches on the side of the hard part nearest to its parent hadron. Any spin asym- metry disappears in the zeroth order cross section, Fig. 1 because there are too few Dirac matrices to produce a non-zero result in the traces with γ$_{5}$ . Consider, for example, Fig. 3(a). The arrow on the gluon line indicates that it is collinear to H$_{1}$ . By first deforming the l integral out of the Glauber region to the H$_{1}$ -collinear region, one may replace the intermediate struck quark line of momentum k$_{3}$ − l by the eikonal factor where n µ 1 ≡ (0 , 1 , 0$_{t}$ ). The sign on the iϵ is determined by the direction of the contour deformation. For the spin- dependent part, the attachment of the extra gluon at the spectator produces a factor at leading power equal to Analogous steps apply to Figs. 3(c,d). The different shadings on the arrows is to emphasize that they are ra- diated from H$_{2}$ rather than H$_{1}$ . Now, after a contour When this is combined with the imaginary part of Eq. (10), the factors of i and − i combine and a contribu- tion to an SSA is again obtained. It is exactly equal to the one found in Fig. 3(a) as it must be since the graphs are related by Hermitian conjugation. The factor from the attachment at the spectator is, If the extra gluon is on the other side of the cut as in Fig. 3(b), the eikonal factor is When this expression is combined with the imaginary part of Eq. (8), the factors of − i and i combine and a contribution to an SSA is obtained. The ϵ$_{jk}$ is the two- dimensional Levi-Civita symbol with ϵ$_{12}$ = 1. t $^{a}$gn µ 1 − l + − iϵ = − gt $^{a}$n µ $_{1}$P . V . 1 l + + igt $^{a}$n µ $_{1}$π δ ( l $^{+}$) . (10) t a 2 Tr$_{D}$ [ ( / p$_{1}$ + m$_{H}$$_{1}$ ) γ$_{5}$/ s$_{1}$ ( / p$_{1}$ − / k$_{1}$ + m$_{ψ}$$_{1}$ ) × × γ $^{+}$( / p$_{1}$ − / k$_{1}$ + / l + m$_{ψ}$$_{1}$ ) ] ≈ − 2 it $^{a}$ϵ$_{jk}$ s j $_{1}$l $^{k}$p $^{+}$( m$_{H}$$_{1}$ (1 − x$_{1}$ ) + m$_{ψ}$$_{1}$ ) . (11) t $^{a}$gn µ 1 − l + + iϵ = − gt $^{a}$n µ $_{1}$P . V . 1 l + − igt $^{a}$n µ $_{1}$π δ ( l $^{+}$) (8) t a 2 Tr$_{D}$ [ ( / p$_{1}$ + m$_{H}$$_{1}$ ) γ$_{5}$/ s$_{1}$ ( / p$_{1}$ − / k$_{1}$ + / l + m$_{ψ}$$_{1}$ ) × × γ $^{+}$( / p$_{1}$ − / k$_{1}$ + m$_{ψ}$$_{1}$ ) ] ≈ 2 it $^{a}$ϵ$_{jk}$ s j $_{1}$l $^{k}$p $^{+}$( m$_{H}$$_{1}$ (1 − x$_{1}$ ) + m$_{ψ}$$_{1}$ ) . (9) k 2 p 2 p 1 k 3 k 4 l k 1 − l l k 2 p 2 p 1 k 3 k 4 k 1 − l (a) (b) p 2 p 1 k 3 k 4 k 1 l k 2 − l p 2 p 1 k 3 k 1 l k 2 − l k 4 (c) (d)