Abstract
We reconsider the Adler-Bardeen theorem for the cancellation of gauge anomalies to all orders, when they vanish at one loop. Using the Batalin-Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards. Our approach is based on an order-by-order analysis of renormalization, and, differently from most derivations existing in the literature, does not make use of arguments based on the properties of the renormalization group. As a consequence, the proof we give also applies to conformal field theories and finite theories.
Highlights
The Adler–Bardeen theorem [1,2] is a crucial property of quantum field theory, and one of the few tools to derive exact results
Using the Batalin–Vilkovisky formalism and combining the dimensional-regularization technique with the higher-derivative gauge invariant regularization, we prove the theorem in the most general perturbatively unitary renormalizable gauge theories coupled to matter in four dimensions, and we identify the subtraction scheme where anomaly cancellation to all orders is manifest, namely no subtractions of finite local counterterms are required from two loops onwards
Summarizing, the statement we prove in this paper is Theorem In renormalizable perturbatively unitary gauge theories coupled to matter, there exists a subtraction scheme where gauge anomalies manifestly cancel to all orders, if they are trivial at one loop
Summary
The Adler–Bardeen theorem [1,2] is a crucial property of quantum field theory, and one of the few tools to derive exact results. Using the dimensional regularization and the minimal subtraction scheme the cancellation of two-loop and higher-order corrections to gauge anomalies in the standard model is not manifest, and finite local counterterms must be subtracted every time. Summarizing, the statement we prove in this paper is Theorem In renormalizable perturbatively unitary gauge theories coupled to matter, there exists a subtraction scheme where gauge anomalies manifestly cancel to all orders, if they are trivial at one loop. Once we have this result, we know that no matter what scheme we use, it is always possible to find ad hoc finite local counterterms that ensure the cancellation of gauge anomalies to all orders.
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