Mulitgraded Dyson-Schwinger systems
Résumé
We study systems of combinatorial Dyson-Schwinger equations with an arbitrary number $N$ of coupling constants.
The considered Hopf algebra of Feynman graphs is $\mathbb{N}^N$-graded, and we wonder if the graded subalgebra generated by the solution is Hopf or not.
We first introduce a family of pre-Lie algebras which we classify, dually providing systems generating a Hopf subalgebra; we also describe the associated groups, as extensions of groups of formal diffeomorphisms on several variables.
We then consider systems coming from Feynman graphs of a Quantum Field Theory. We show that if the number $N$ of independent coupling constants is the number of interactions of the considered QFT, then the generated subalgebra is Hopf. For QED, $\varphi^3$ and QCD, we also prove that this is the minimal value of $N$. All these examples are generalizations of the first family of Dyson-Schwinger systems in the one coupling constant case, called fundamental.
We also give a generalization of the second family, called cyclic.
Origine | Fichiers produits par l'(les) auteur(s) |
---|