Feynman diagrams and polylogarithms: shuffles and pentagons

There is a universal combinatorial Hopf algebra structure hidden in the process of renormalization [ 1, 2, 3, 4, 5, 6]. The universality of this structure can be conveniently understood if one considers the UV singularities of Feynman graphs from the viewpoint of configuration spaces. Then, the combinatorial structure of renormalization can be summarized as follows. Each Feynman diagram has various sectors which suffer from shortdistance singularities. These sectors are stratified by rooted trees, from which the Hopf algebra structures of [ 1, 2, 3, 4, 5, 6] are obtained. Figure 1 gives an example. The corresponding Hopf algebra can be formulated on rooted trees or, equivalently, directly on Feynman graphs. Details of the relation to configuration spaces will be described elsewhere. Here, we want to use this representation to motivate further investigation in algebraic relations between Feynman graphs, and report some encouraging first results which will be discussed in much greater detail in future work. Figure 1 essentially shows how the short distance singularities are located in sectors stratified by rooted trees. This is no surprise as the singularities are constrained to (sub-)diagonals. Along such subdiagonals, one essentially confronts the product of distributions with coinciding support. This localization of singularities along diagonals conveniently allows to rely on suitable local subtractions, whose combinatorics can be de-