Fr\'echet derivative for light-like Wilson Loops

Abstract

We address the equations of motion for the light-like QCD Wilson exponentials defined in the generalized loop space. We attribute an important class of the infinitesimal shape variations of the rectangular light-like Wilson loops to the Fr\'echet derivative associated to a diffeomorphism in loop space what enables the derivation of the law of the classically conformal-invariant shape variations. We show explicitly that the Fr\'echet derivative coincides (at least in the leading perturbative or- der) with the area differential operator introduced in the previous works. We discuss interesting implications of this result which will allow one to relate the rapidity evolution and ultra-violet evolution of phenomenologically important quantum correlation functions (such as 3-dimensional parton distribution functions) and geometrical properties of the light-like cusped Wilson loops.