Generating function for web diagrams

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.

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 (16), 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.