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We prove the Adler-Bardeen theorem in a large class of general gaugetheories, including nonrenormalizable ones. We assume that the gauge symmetriesare general covariance, local Lorentz symmetry and Abelian and non-AbelianYang-Mills symmetries, and that the local functionals of vanishing ghost numbersatisfy a variant of the Kluberg-Stern--Zuber conjecture. We show that if thegauge anomalies are trivial at one loop, for every truncation of the theorythere exists a subtraction scheme where they manifestly vanish to all orders,within the truncation. Outside the truncation the cancellation of gaugeanomalies can be enforced by fine-tuning local counterterms. The framework ofthe proof is worked out by combining a recently formulated chiral dimensionalregularization with a gauge invariant higher-derivative regularization. If thehigher-derivative regularizing terms are placed well beyond the truncation, andthe energy scale $\Lambda$ associated with them is kept fixed, the theory issuper-renormalizable and has the property that, once the gauge anomalies arecancelled at one loop, they manifestly vanish from two loops onwards by simplepower counting. When the $\Lambda$ divergences are subtracted away and$\Lambda$ is sent to infinity, the anomaly cancellation survives in a manifestform within the truncation and in a nonmanifest form outside. The standardmodel coupled to quantum gravity satisfies all the assumptions, so it is freeof gauge anomalies to all orders.

Adler-Bardeen theorem and cancellation of gauge anomalies to all orders in nonrenormalizable theories