Web worlds, web-colouring matrices, and web-mixing matrices

Abstract

We introduce a new combinatorial object called a web world that consists of a set of web diagrams. The diagrams of a web world are generalizations of graphs, and each is built on the same underlying graph. Instead of ordinary vertices the diagrams have pegs, and edges incident to a peg have different heights on the peg. The web world of a web diagram is the set of all web diagrams that result from permuting the order in which endpoints of edges appear on a peg. The motivation comes from particle physics, where web diagrams arise as particular types of Feynman diagrams describing scattering amplitudes in non-Abelian gauge (Yang-Mills) theories. To each web world we associate two matrices called the web-colouring matrix and web-mixing matrix. The entries of these matrices are indexed by ordered pairs of web diagrams (D1, D2), and are computed from those colourings of the edges of D1 that yield D2 under a transformation determined by each colouring.We show that colourings of a web diagram (whose constituent indecomposable diagrams are all unique) that lead to a reconstruction of the diagram are equivalent to order-preserving mappings of certain partially ordered sets (posets) that may be constructed from the web diagrams. For web worlds whose web graphs have all edge labels equal to 1, the diagonal entries of web-mixing and web-colouring matrices are obtained by summing certain polynomials determined by the descents in permutations in the Jordan-Hölder set of all linear extensions of the associated poset. We derive tri-variate generating generating functions for the number of web worlds according to three statistics and enumerate the number of different web diagrams in a web world. Three special web worlds are examined in great detail, and the traces of the web-mixing matrices calculated in each case.

From webs to polylogarithms

“…This has been further used in refs. [105,106] to establish a relation with partially ordered sets, and deduce all-order solutions for certain classes of webs.…”

“…Progress was achieved there owing to the replica trick of statistical physics which led to a general algorithm for determining web mixing matrices. The study of these [93,[103][104][105][106] proceeded using both combinatorial methods and the connection with the renormalizability of the Wilson-line vertex. The most striking feature of webs is that -despite the fact that they contain disconnected, often reducible diagrams -their colour factors always correspond to connected graphs [93].…”

Progress on soft gluon exponentiation and long-distance singularities

“…The work of Ref. [17][18][19][20][21][22][23] shows that it does in fact generalise but in a rather non-trivial way. A few hints can already be drawn from the simple example above: first, it is convenient to consider together sets of diagrams which are related by permutation of the order of gluon attachments to the Wilson lines; we refer to the entire set as a single web.…”

“…This was done in the context of a Wilson loop, or two Wilson lines, corresponding to a colour singlet form factor (for a review see [16]). The generalization to a product of more than two Wilson lines, as relevant for QCD hard scattering, was only made over the last three years [17][18][19][20][21][22][23].…”

New insights into soft gluons and gravitons

“…A further conjecture involves a weighted sum of column entries [14]. A pure combinatoric expression for web mixing matrix elements has been given in [15], and may be related to order-preserving maps on partially ordered sets (posets) [16]. The latter are used in computer science applications, and thus progress in understanding webs can be made with or without any field theory knowledge.…”