Gauge Symmetries and Renormalization

DOI10.1007/S11040-022-09423-8arXiv2001.00104WikidataQ114224691 ScholiaQ114224691MaRDI QIDQ6332122

Publication date: 31 December 2019

Abstract: We study the perturbative renormalization of quantum gauge theories in the Hopf algebra setup of Connes and Kreimer. It was shown by van Suijlekom (2007) that the quantum counterparts of gauge symmetries -- the so-called Ward--Takahashi and Slavnov--Taylor identities -- correspond to Hopf ideals in the respective renormalization Hopf algebra. We generalize this correspondence to super- and non-renormalizable Quantum Field Theories, extend it to theories with multiple coupling constants and add a discussion on transversality. In particular, this allows us to apply our results to (effective) Quantum General Relativity, possibly coupled to matter from the Standard Model, as was suggested by Kreimer (2008). To this end, we introduce different gradings on the renormalization Hopf algebra and study combinatorial properties of the superficial degree of divergence. Then we generalize known coproduct and antipode identities to the super- and non-renormalizable cases and to theories with multiple vertex residues. Building upon our main result, we provide criteria for the compatibility of these Hopf ideals with the corresponding renormalized Feynman rules. A direct consequence of our findings is the well-definedness of the Corolla polynomial for Quantum Yang--Mills theory without reference to a particular renormalization scheme.

Mathematics Subject Classification ID

Unified quantum theories (81V22) Vertex operators; vertex operator algebras and related structures (17B69) Yang-Mills and other gauge theories in quantum field theory (81T13) Perturbative methods of renormalization applied to problems in quantum field theory (81T15) Feynman diagrams (81T18) Quantization of the gravitational field (83C45) Applications of Lie groups to the sciences; explicit representations (22E70) Renormalization group methods applied to problems in quantum field theory (81T17) Hopf algebras and their applications (16T05)

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