Gauge Symmetries and Renormalization

The preservation of gauge symmetries to the quantum level induces symmetries between renormalized Green's functions. These symmetries are known by the names of Ward-Takahashi and Slavnov-Taylor identities. On a perturbative level, these symmetries can be implemented as Hopf ideals in the Connes-Kreimer renormalization Hopf algebra. In this article, we generalize the existing literature to the most general case by first motivating these symmetries on a generic level and then proving that they indeed generate Hopf ideals, where we also include the more involved cases of super- and non-renormalizable local QFTs. Finally, we provide a criterion for their validity on the level of renormalized Feynman rules