On Geometric Scaling of Light-Like Wilson Polygons: Higher Orders in $\alpha_s$

TL;DR

This paper investigates the higher-order scaling behavior of light-like Wilson polygons' UV singularities in gauge theories, proposing a geometric approach that extends beyond leading order calculations.

Contribution

It introduces a novel geometric framework linking shape variations of Wilson polygons to their renormalization-group evolution at higher perturbative orders.

Findings

Identifies a special class of shape variations related to Fréchet differentials.

Derives a combined geometric and RG evolution equation applicable beyond leading order.

Supports the approach with simple arguments and theoretical reasoning.

Abstract

We address the scaling behaviour of contour-shape-dependent ultra-violet singularities of the light-like cusped Wilson loops in Yang-Mills and ${\cal N} = 4$ super-Yang-Mills theories in the higher orders of the perturbative expansion. We give the simple arguments to support the idea that identifying of a special type of non-local infinitesimal shape variations of the light-like Wilson polygons with the Fr\'echet differentials results in the combined geometric and renormalization-group evolution equation, which is applicable beyond the leading order exponentiated Wilson loops.