On Overlapping Divergences

Using set-theoretic considerations, we show that the forest formula for overlapping divergences comes from the Hopf algebra of rooted trees. Motivation and Introduction The process of renormalization is governed by the forest formula, as derived for example in [1]. The underlying combinatorics is directly related to the Hopf algebra structure of rooted trees. This is evident in the case of Feynman diagrams which only provide nested or disjoint subdivergences. It is the purpose of this paper to show that the same Hopf algebra appears in the study of overlapping divergences. This was already shown using Schwinger Dyson equations [2], or by explicit considerations of divergent sectors [3], or differential equations on bare Green functions [4]. At such a level, one obtains a resolution of overlapping divergent graphs into a sum of rooted trees, to which then the combinatorics of the Hopf algebra of rooted trees applies [2, 4]. It was suggested to construct a Hopf algebra which directly considers overlapping divergent graphs, without using external input as Schwinger Dyson equations [5]. However, as already mentioned in [4], this leads to the same Hopf algebra as for the case of non-overlapping divergences, as we will prove by set-theoretic considerations. Heisenberg Fellow, email: [email protected]