Renormalization loop

18.01.2019, 9:30 Uhr  –  Haus 9, Raum: 2.22
Arbeitsgruppenseminar Analysis

Alessandra Frabetti (Université Lyon 1)

In perturbative quantum field theory, the renormalization group is a  group of formal diffeomorphisms in the powers of the coupling  constant, with coefficients built on the counterterms of divergent  Feynman graphs. For scalar theories, such groups are proalgebraic  (functorial on the coefficients algebra) and are represented by Faà di  Bruno types of Hopf algebras. For non-scalar theories, even if the  counterterms are scalar, they cannot be functorially represented by a  Hopf algebra, because on some intermediate series with non-commutative  coefficients the associativity of the composition fails.

In the paper arXiv:1807.10477, with Ivan P. Shestakov, we extend the  group of formal diffeomorphisms to a functor on non-commutative  algebras by regarding it as a loop (a non-associative group) and  exhibit its representative Faà di Bruno coloop bialgebra. In this talk  I explain how, even losing associativity, this loop has enough good  properties to allow performing renormalization in Dyson's sense, and  how it could then give rise to a "renormalization loop" suitable for  non-scalar theories.