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Journal Articles Journal of Mathematical Physics Year : 2020

Mulitgraded Dyson-Schwinger systems


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.
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hal-01231362 , version 1 (20-11-2015)



Loïc Foissy. Mulitgraded Dyson-Schwinger systems. Journal of Mathematical Physics, 2020, 61 (5), pp.051703. ⟨10.1063/5.0005480⟩. ⟨hal-01231362⟩
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