Renormalization: A Quasi-shuffle Approach

  • Frédéric MenousEmail author
  • Frédéric Patras
Conference paper
Part of the Abel Symposia book series (ABEL, volume 13)


In recent years, the usual BPHZ algorithm for renormalization in perturbative quantum field theory has been interpreted, after dimensional regularization, as a Birkhoff decomposition of characters on the Hopf algebra of Feynman graphs, with values in a Rota-Baxter algebra of amplitudes. We associate in this paper to any such algebra a universal semigroup (different in nature from the Connes-Marcolli “cosmical Galois group”). Its action on the physical amplitudes associated to Feynman graphs produces the expected operations: Bogoliubov’s preparation map, extraction of divergences, renormalization. In this process a key role is played by commutative and noncommutative quasi-shuffle bialgebras whose universal properties are instrumental in encoding the renormalization process.



We acknowledge support from the CARMA grant ANR-12-BS01-0017, “Combinatoire Algébrique, Résurgence, Moules et Applications”and the CNRS GDR “Renormalisation”. We thank warmly K. Ebrahimi-Fard, from whom we learned some years ago already the meaningfulness of Rota–Baxter algebras and their links with quasi–shuffle algebras.


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Authors and Affiliations

  1. 1.Laboratoire de Mathmatiques d’Orsay, University of Paris-Sud, CNRSUniversit Paris-SaclayOrsayFrance
  2. 2.Laboratoire J.A. DieudonnéUniversité de la Côte d’Azur, CNRS, UMR 7531Nice Cedex 2France

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