Webs and Posets

TL;DR

The paper advances the understanding of infrared structure in non-Abelian gauge theories by generalising diagrammatic exponentiation to correlators of multiple Wilson lines and focusing on the combinatorics of web-mixing matrices. It builds a bridge between these matrices and partially ordered sets (posets), enabling explicit all-order solutions for two special web families and deriving a general rank formula for a broad class of webs. The authors also reinterpret the weighted column sum rule for web-mixing matrices in the poset framework, finding complete results for one family and nontrivial but verified results for another. Together with ongoing work on the accompanying kinematic integrals, these results provide deeper insights into the all-order structure of infrared singularities in non-Abelian gauge theories and set the stage for broader generalisations to more intricate web topologies.

Abstract

The non-Abelian exponentiation theorem has recently been generalised to correlators of multiple Wilson line operators. The perturbative expansions of these correlators exponentiate in terms of sets of diagrams called webs, which together give rise to colour factors corresponding to connected graphs. The colour and kinematic degrees of freedom of individual diagrams in a web are entangled by mixing matrices of purely combinatorial origin. In this paper we relate the combinatorial study of these matrices to properties of partially ordered sets (posets), and hence obtain explicit solutions for certain families of web-mixing matrix, at arbitrary order in perturbation theory. We also provide a general expression for the rank of a general class of mixing matrices, which governs the number of independent colour factors arising from such webs. Finally, we use the poset language to examine a previously conjectured sum rule for the columns of web-mixing matrices which governs the cancellation of the leading subdivergences between diagrams in the web. Our results, when combined with parallel developments in the evaluation of kinematic integrals, offer new insights into the all-order structure of infrared singularities in non-Abelian gauge theories.

Figures (21)

• Figure 1: A (1,2,1) web, connecting three parton lines at two-loop order.
• Figure 2: The partitions of the diagram on the far left-hand side, with $m=1,2,2,2,3$ respectively. Note that we have used different symbols to distinguish gluons belonging to different partition elements (double line, thick line), in addition to colours.
• Figure 3: Alternative notation for the 3-partition of figure \ref{['partex']}, in which each element of the partition is represented by its own diagram.
• Figure 4: Examples of the multiplication rule $D_1\circ D_2$ for diagrams.
• Figure 5: Three-loop web containing the diagram of figure \ref{['partex']}.