Mulitgraded Dyson-Schwinger systems
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 $\mathbb{N}^N$-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.
- Publication:
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arXiv e-prints
- Pub Date:
- November 2015
- DOI:
- 10.48550/arXiv.1511.06859
- arXiv:
- arXiv:1511.06859
- Bibcode:
- 2015arXiv151106859F
- Keywords:
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- Mathematics - Rings and Algebras
- E-Print:
- 40 pages