Evolution of light-like Wilson loops with a self-intersection in loop space

Abstract

Recently, we proposed a general evolution equation for single quadrilateral Wilson loops on the light-cone. In the present work, we study the energy evolution of a combination of two such loops that partially overlap or have a self-intersection. We show that, for a class of geometric variations, then evolution is consistent with our previous conjecture, and we are able to handle the intricacies associated with the self-intersections and overlaps. This way, a step forward is made towards the understanding of loop space, with the hope of studying more complicated structures appearing in phenomenological relevant objects, such as parton distributions.

Transverse Momentum Dependent (TMD) Parton Distribution Functions: Status and Prospects

“…The evolution of the gauge-invariant path-dependent TMDs with the light-like cusped Wilson lines can also be associated with the geometric evolution in the generalized space [189][190][191]. The differential shape variations of the underlying contours to the Wilson loops are formulated in terms of the Fréchet derivative [192,193] and the equations of motion in the loop space are dual to the energy and rapidity evolution of the TMDs having the same structure of the Wilson lines [194,195].…”

Generalized loop space and TMDs

“…Furthermore in [29] we demonstrated the validity, at leading order, of our conjecture for two simple extensions of our originally considered quadrilateral Wilson loops, one with overlapping segments and one with a self-intersection. Recently [30] we also demonstrated that the new derivative we introduced in [22] is a special case of the Fréchet derivative, itself having a perturbative expansion when applied to the lightlike quadrilateral supports our belief that (9) is valid for higher-orders.…”

Gauge-Invariant Gluon TMD and Evolution in the Coordinate Space

“…They can also be related to the energy/rapidity evolution of the TMDs with pure light-like Wilson lines [21,22]. In other words, starting from a loop having a given shape, one can come to a loop with another shape by solving the above evolution equations.…”

“…These differential operators in the loop space determine the evolution of the Wilson loops in the coordinate representation. They can also be related to the energy/rapidity evolution of the TMDs with pure light-like Wilson lines [22,23]. In other words, starting from a loop having a given shape, one can come to a loop with another shape by solving the above evolution equations.…”