The Non-Abelian Exponentiation theorem for multiple Wilson lines

TL;DR

The paper proves a generalized non-Abelian exponentiation theorem for correlators of any number of Wilson lines, showing that all colour factors in the exponent are fully connected. It introduces an effective-vertex formalism combined with the replica trick to reorganize soft-gluon webs, providing a natural, basis-fixed colour structure and simplifying the extraction of kinematic factors. By classifying three-loop webs (connecting four and three lines) and presenting explicit colour-kinematic decompositions, the authors lay groundwork for computing the three-loop multiparton soft anomalous dimension. The results deepen our understanding of infrared singularities in non-Abelian gauge theories and offer practical avenues for high-order resummations and factorization analyses.

Abstract

We study the structure of soft gluon corrections to multi-leg scattering amplitudes in a non-Abelian gauge theory by analysing the corresponding product of semi-infinite Wilson lines. We prove that diagrams exponentiate such that the colour factors in the exponent are fully connected. This completes the generalisation of the non-Abelian exponentiation theorem, previously proven in the case of a Wilson loop, to the case of multiple Wilson lines in arbitrary representations of the colour group. Our proof is based on the replica trick in conjunction with a new formalism where multiple emissions from a Wilson line are described by effective vertices, each having a connected colour factor. The exponent consists of connected graphs made out of these vertices. We show that this readily provides a general colour basis for webs. We further discuss the kinematic combinations that accompany each connected colour factor, and explicitly catalogue all three-loop examples, as necessary for a direct computation of the soft anomalous dimension at this order.

Figures (21)

• Figure 1: 2-loop web connecting 3 Wilson lines and its exponentiated colour factor.
• Figure 2: Examples of the tree colour structures associated with nested commutators: each corresponds to a connected colour factor. Here (a) illustrates the structures obtained from a single hierarchy of nested commutators as in eq. (\ref{['BT']}) and (b) illustrates additional structures occurring in (\ref{['CKj']}) owing to the double hierarchy of nested commutators.
• Figure 3: Examples of diagrams formed out of the effective Feynman rules appearing in eq. (\ref{['Zrepdef3']}). For the multiple-emission effective vertices we indicated the type of vertex by $V_K$. The colour part of each such vertex is internally fully connected. We also encircled all vertices on a given Wilson line by an ellipse and a ${\rm +}$ sign, to recall that one should symmetrise over all orderings of these, reflecting the ordinary (i.e. non-ordered) exponential in eq. (\ref{['Zrepdef3']}).
• Figure 4: An effective-vertex connected graph representing the 1-2-1 web of fig. \ref{['fig:121']}.
• Figure 5: The (1,1,1,3) web, together with a representation for the corresponding connected colour factors. Note that in each of the web diagrams $s$, $t$ and $u$ indicate respectively the positions along line 4 of the attachments of the gluons connecting to lines $1$, $2$ and $3$.

Theorems & Definitions (1)

• Theorem 2.1