General properties of multiparton webs: proofs from combinatorics

TL;DR

This paper proves fundamental combinatorial properties of web mixing matrices in multiparton soft-gluon exponentiation, revealing their projection nature and zero-sum row structure, which are crucial for understanding non-Abelian gauge theory exponentiation.

Contribution

It provides a rigorous proof that web mixing matrices are idempotent and have zero sum rows using combinatorics and replica trick methods, advancing the theoretical understanding of soft-gluon exponentiation.

Findings

Web mixing matrices are idempotent, acting as projection operators.

Rows of web mixing matrices sum to zero, removing symmetric components.

Zero sum property applies separately to planar and non-planar diagrams.

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.