General properties of multiparton webs: proofs from combinatorics

TL;DR

This work addresses the all-order exponentiation of Wilson-line correlators in multiparton scattering by studying the web mixing matrices that couple kinematic and colour factors. It employs the replica trick to derive the exponent structure and proves two key properties: idempotence $R^2=R$ and zero-sum rows $\sum_{D'} R_{DD'}=0$, establishing that $R$ acts as a projection operator onto the exponent-contributing subspace. The authors further derive an explicit combinatorial formula for the mixing matrix in terms of overlap functions between diagram decompositions, enabling a constructive link between diagrams within a web. They also show stronger planar-limit constraints, where planar and non-planar subsets satisfy the zero-sum property separately, highlighting how large-$N_c$ simplifications constrain the web structure and subdivergence cancellations. Together, these results deepen the understanding of the all-order structure of non-Abelian exponentiation and set the stage for further mathematical and physical investigations into the substructure of webs.

Abstract

Recently, the diagrammatic description of soft-gluon exponentiation in scattering amplitudes has been generalized to the multiparton case. It was shown that the exponent of Wilson-line correlators is a sum of webs, where each web is formed through mixing between the kinematic factors and colour factors of a closed set of diagrams which are mutually related by permuting the gluon attachments to the Wilson lines. In this paper we use replica trick methods, as well as results from enumerative combinatorics, to prove that web mixing matrices are always: (a) idempotent, thus acting as projection operators; and (b) have zero sum rows: the elements in each row in these matrices sum up to zero, thus removing components that are symmetric under permutation of gluon attachments. Furthermore, in webs containing both planar and non-planar diagrams we show that the zero sum property holds separately for these two sets. The properties we establish here are completely general and form an important step in elucidating the structure of exponentiation in non-Abelian gauge theories.

Figures (10)

• Figure 1: Example diagrams in the replicated theory, where the indices label the replica number of each gluon.
• Figure 2: Decompositions of an example three loop diagram, with (a) $n(P)=1$ (equivalent to the original diagram); (b) $n(P)=2$; (c) $n(P)=3$.
• Figure 3: Example one loop graphs.
• Figure 4: Example of a product of colour factors of three diagrams.
• Figure 5: Three loop web to which the diagram of figure \ref{['partex2']}(a) belongs.