Generating function for web diagrams

TL;DR

This paper establishes a generating-function framework for the exponentiation of Wilson-line amplitudes in both Abelian and non-Abelian gauge theories. By identifying a broad class of operators of the form $\mathcal{O}[\phi]=\exp \mathcal{Y}$, it proves that their correlators exponentiate with $Y$ composed of source-connected diagrams, and presents a matrix-generalized contour of Wilson lines that yields a matrix exponent $W[\int t^a V_a]$. For multiple Wilson lines, the formalism shows that the logarithm of their correlator is a sum of source-connected contributions with fully connected color factors, thereby recovering and clarifying the non-Abelian exponentiation theorem and the structure of web diagrams. The work provides a transparent, generating-function approach to infrared singularities, renormalization, and the factorization properties of multi-Wilson-line amplitudes, applicable to half-infinite straight-line configurations and beyond.

Abstract

We present the description of the exponentiated diagrams in terms of generating function within the universal diagrammatic technique. In particular, we show the exponentiation of the gauge theory amplitudes involving products of an arbitrary number of Wilson lines of arbitrary shapes, which generalizes the concept of web diagrams. The presented method gives a new viewpoint on the web diagrams and proves the non-Abelian exponentiation theorem.

Figures (3)

• Figure 1: Exponentiation in Abelian gauge theory. The blobs denote all connected diagrams, loops denote the (same) Wilson line. The photon line ending on Wilson loop is integrated over the path of the Wilson line, while several photons ending on the same Wilson loop are integrated over the Wilson loop preserving path-ordering.
• Figure 2: Web diagrams contributing to the expectation value of three Wilson lines at $\mathcal{O}(g^6)$ order and connecting all three Wilson lines (diagrams with permutations of Wilson lines should be added, as well as, web diagrams of $\mathcal{O}(g^4)$ order with loop corrections). Blobs with $V_n$ denote the vertices (\ref{['nonabel:op_V']}), while the empty blob denotes the all possible four-gluon tree interaction. Ovals with "plus" sign denote the symmetrization of the vertices.
• Figure 3: Color factors that appear in the web diagrams shown in fig.2. The color factors shown in the first (second) row appear in the diagrams shown in the first (second) row of fig.2.