Renormalization: a quasi-shuffle approach

Frédéric Menous (1) , Frédéric Patras (2)

1 Département de Mathématiques-Université de Paris XI
2 JAD - Laboratoire Jean Alexandre Dieudonné

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 semi-group (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.

Mots clés: Renormalization; Birkhoff decomposition; quasi-shuffle algebra; noncommutative quasi-shuffle algebra

Domaines: Anneaux et algèbres [math.RA]; Combinatoire [math.CO]

Frédéric Menous, Frédéric Patras. Renormalization: a quasi-shuffle approach. 2017. ⟨hal-01492967v1⟩