Doubling bialgebras of graphs and Feynman rules
Confluentes Mathematici, Volume 8 (2016) no. 1, pp. 3-30.

In this article, we define a doubling procedure for the bialgebra of specified Feynman graphs introduced in a previous paper [1]. This is the vector space generated by the pairs (Γ ¯,γ ¯) where Γ ¯ is a locally 1PI specified graph of a perturbation theory 𝒯 with γ ¯Γ ¯ locally 1PI and where Γ ¯/γ ¯ is a specified graph of 𝒯. We also define a convolution product on the characters of this new bialgebra with values in an endomorphism algebra, equipped with a commutative product compatible with the composition. We then express in this framework the renormalization as formulated by A. Smirnov [13, §8.5, 8.6], adapting the approach of A. Connes and D. Kreimer for two renormalization schemes: the minimal renormalization scheme and the Taylor expansion scheme. Finally, we determine the finite parts of Feynman integrals using the BPHZ algorithm after dimensional regularization procedure, by following the approach by P. Etingof [9] (see also [11]).

Published online:
DOI: 10.5802/cml.26
Classification: 05C90, 81T15, 16T05, 16T10
Keywords: Bialgebra, Hopf algebra, Feynman Graphs, Birkhoff decomposition, renormalization, dimensional regularization
Mohamed Belhaj Mohamed 1, 2

1 Laboratoire de mathématiques, physique, fonctions spéciales et applications, Université de Sousse, rue Lamine Abassi 4011 H. Sousse, Tunisie
2 Université Blaise Pascal, Laboratoire de mathématiques UMR 6620, 63177 Aubière, France, and
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Mohamed Belhaj Mohamed. Doubling bialgebras of graphs and Feynman rules. Confluentes Mathematici, Volume 8 (2016) no. 1, pp. 3-30. doi : 10.5802/cml.26.

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