Diagrammatic Exponentiation for Products of Wilson Lines

TL;DR

This work delivers a general, coordinate-space framework for exponentiating gauge-theory amplitudes built from products of Wilson lines and loops. By deriving a coordinate-identity-based recursion, it extends the web formalism to arbitrary multi-eikonal vertices and line configurations, including non-commuting color structures that mix lower-order subdiagrams. The authors develop an all-order renormalization scheme using BCH theory, defining renormalized webs and relating them to anomalous dimensions, with explicit two- and three-loop examples. The approach clarifies how to organize soft radiation effects in complex processes and points to potential simplifications in special limits such as commuting color factors or massless cases, with implications for cross sections and resummation.

Abstract

We provide a recursive diagrammatic prescription for the exponentiation of gauge theory amplitudes involving products of Wilson lines and loops. This construction generalizes the concept of webs, originally developed for eikonal form factors and cross sections with two eikonal lines, to general soft functions in QCD and related gauge theories. Our coordinate space arguments apply to arbitrary paths for the lines.

Figures (4)

• Figure 1: Representation of the eikonal amplitude $A$ discussed in the text. Each gluon line represents an arbitrary number of connections to each Wilson line, indicated by $e_a$ for line $a$, with $a=1\dots L$ for $L$ Wilson lines.
• Figure 2: Illustration of the coordinate identity, Eq. (\ref{['eq:coordidentity']}), where each line attaching $W_1$ and $W_2$ to a Wilson line stands for an arbitrary number of gluons ($e^{(a)}_i$ in the text.) The sum on the right represents the sum over all mutual orderings of the external gluons of $W_1$ and $W_2$, preserving the orderings internal to each $W$ along the Wilson line. The color-dependent product of the web internal factors, $W_1$ and $W_2$ are the same on both sides of the figure.
• Figure 3: Double exchange diagrams discussed in connection with Eq. (\ref{['eq:w2w3']}). The shaded circle represents a multi-eikonal vertex.
• Figure 4: Examples of two-web diagrams for the form factor.