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.
