Abstract
We study systems of combinatorial Dyson–Schwinger equations with an arbitrary number N of coupling constants. The considered Hopf algebra of Feynman graphs is NN-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 that 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 quantum field theory, then the generated subalgebra is Hopf. For QED, φ3, 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.
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