Mulitgraded DysonSchwinger systems
Abstract
We study systems of combinatorial DysonSchwinger 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 preLie 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 DysonSchwinger systems in the one coupling constant case, called fundamental.We also give a generalization of the second family, called cyclic.
 Publication:

arXiv eprints
 Pub Date:
 November 2015
 DOI:
 10.48550/arXiv.1511.06859
 arXiv:
 arXiv:1511.06859
 Bibcode:
 2015arXiv151106859F
 Keywords:

 Mathematics  Rings and Algebras
 EPrint:
 40 pages