Tom Mertens

Cusped light-like Wilson loops in gauge theories

We propose and discuss a new approach to the analysis of the correlation functions which contain light-like Wilson lines or loops, the latter being cusped in addition. The objects of interest are therefore the light-like Wilson null-polygons, the soft factors of the parton distribution and fragmentation functions, high-energy scattering amplitudes in the eikonal approximation, gravitational Wilson lines, etc. Our method is based on a generalization of the universal quantum dynamical principle by J. Schwinger and allows one to take care of extra singularities emerging due to lightlike or semi-light-like cusps. We show that such Wilson loops obey a differential equation which connects the area variations and renormalization group behavior of those objects and discuss the possible relation between geometrical structure of the loop space and area evolution of the light-like cusped Wilson loops.

Evolution of cusped light-like Wilson loops and geometry of the loop space

We discuss the possible relation between certain geometrical properties of the loop space and energy evolution of the cusped Wilson exponentials defined on the light-cone. Analysis of the area differential equations for this special class of the Wilson loops calls for careful treatment of the ultraviolet and rapidity divergences which make those loops non-multiplicatively-renormalizable. We propose to consider the renormalization properties of the light-cone cusped Wilson loops from the point of view of the universal quantum dynamical approach introduced by Schwinger. We discuss the relevance of the Makeenko-Migdal loop equations supplied with the modified Schwinger principle to the energy evolution of some phenomenologically significant objects, such as transversemomentum dependent distribution functions, collinear parton densities at large-x, etc.

Wilson lines in quantum field theory

The objective of this book is to get the reader acquainted with theoretical and mathematical foundations of the concept of Wilson loops in the context of modern quantum field theory. It teaches how to perform independently some elementary calculations on Wilson lines, and shows the recent development of the subject in different important areas of research.This revised second edition also includes historical notes on Wilson loops and lines.

On geometric scaling of light-like Wilson polygons: Higher orders in αs

Abstract We address the scaling behaviour of contour-shape-dependent ultraviolet singularities of the light-like cusped Wilson loops in Yang–Mills and 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 Frechet differentials results in the combined geometric and renormalization-group evolution equation, which is applicable beyond the leading order exponentiated Wilson loops.

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, the 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.

EVOLUTION OF TRANSVERSE-DISTANCE DEPENDENT PARTON DENSITIES AT LARGE-XB AND GEOMETRY OF THE LOOP SPACE

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-xB approximation. This regime is relevant for future experimental 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-xB factorization scheme.

Evolution and Dynamics of Cusped Light-Like Wilson Loops in Loop Space

We address a connection between the energy evolution of the polygonal light-like Wilson exponentials and the geometry of the loop space with the gauge invariant Wilson loops of a variety of shapes being the fundamental degrees of freedom. The renormalization properties and the differential area evolution of these Wilson polygons are studied by making use of the universal Schwinger quantum dynamical approach. We discuss the appropriateness of the dynamical differential equations in the loop space to the study of the energy evolution of the collinear and transverse-momentum dependent parton distribution functions.

Fréchet 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 Frechet 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 Frechet derivative coincides (at least in the leading perturbative order) 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.