Drell-Yan hadron tensor: Contour gauge and gluon propagator

We consider the gauge invariant Drell-Yan hadron tensor which includes the standard and nonstandard diagram contributions. The nonstandard diagram contribution appeared owing to the complexity of the twist three $B^V(x_1,x_2)$-function where the gluon pole manifests. We use the contour gauge conception which allows us to fix easily the spurious uncertainties in the gluon propagator. The contour gauge condition is generated by the corresponding Wilson lines in both the standard and nonstandard diagrams. We demonstrate the substantial role of the nonstandard diagram for forming of the relevant contour in the Wilson path-ordered exponential that leads to the spurious singularity fixing.

Figure 1

The Feynman diagrams which contribute to the polarized Drell-Yan hadron tensor: the standard (left panel) and nonstandard diagrams (right panel).

Figure 2

The longitudinal Wilson lines related to the standard (red lines, the exponentials with $A^{+}$) and nonstandard (blue lines, the exponentials with $A^{-}$) Drell-Yan hadron tensor. The circles single out the interception points which the continuity conditions are defined for.