Combinatorial Hopf algebraic description of the multiscale renormalization in quantum field theory

• DOI: 10.48550/arxiv.1211.4429
• Publication Date: 2012-01-01
• Authors: Krajewski, Thomas; Rivasseau, Vincent; Tanasa, Adrian
• Keywords: High Energy Physics - Theory; High Energy Physics - Theory (hep-th); [MATH.MATH-CO]Mathematics [math]/Combinatorics [math.CO]; [PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]; 0103 physical sciences; FOS: Mathematics; Mathematics - Combinatorics; FOS: Physical sciences; Combinatorics (math.CO); 01 natural sciences

We define in this paper several Hopf algebras describing the combinatorics of the so-called multi-scale renormalization in quantum field theory. After a brief recall of the main mathematical features of multi-scale renormalization, we define assigned graphs, that are graphs with appropriate decorations for the multi-scale framework. We then define Hopf algebras on these assigned graphs and on the Gallavotti-Nicol�� trees, particular class of trees encoding the supplementary informations of the assigned graphs. Several morphisms between these combinatorial Hopf algebras and the Connes-Kreimer algebra are given. Finally, scale dependent couplings are analyzed via this combinatorial algebraic setting.<br/>26 pages, 3 figures; the presentation of the results has been reorganized. Several details of various proofs are given and some references have been added<br/>