We discuss possible applications of the equations of motion in the generalized Wilson loop space to the phenomenology of the three-dimensional parton distribution functions in the large-$x_B$ approximation. This regime is relevant for future experime
ntal programs to be launched at the (approved) Jefferson Lab 12 GeV upgrade and the (planned) Electron-Ion Collider. We show that the geometrical evolution of the Wilson loops corresponds to the combined rapidity and renormalization-group equation of the transverse-distance dependent parton densities in the large-$x_B$ factorization scheme.
We address the geometrical structure of the skewed correlator of two space-like separated (almost) oppositely directed Wilson lines. Similar objects occur in the analysis of the transverse-momentum broadening probability function, the first moment of
which is associated with the jet quenching parameter. We start from the Euclidean space formulation and then transform the result to the Minkowski light-cone geometry, arguing that this procedure is consistent in the leading order of the perturbative expansion. We discuss as well the issues of the UV, rapidity and IR singularities, and possible use of the proposed approach in lattice simulations.
