Abstract
A local UV cutoff Lambda (x) transforming under Weyl rescalings allows to construct Weyl invariant kinetic terms for scalar fields including Wilsonian cutoff functions. First we consider scalar fields in curved space-time with local bare couplings of any canonical dimension, and anomalous dimensions which describe their dependence on the UV cutoff. The local component of the UV cutoff plays the role of an additional coupling, albeit with a trivial constant beta function. This approach allows to derive Weyl consistency conditions for the corresponding anomalous dimensions which assume the form of an exact gradient flow. For renormalizable theories the Weyl consistency conditions are initially of the form of an approximate gradient flow for the beta functions, and we derive conditions under which it becomes the form of an exact gradient flow.
Highlights
Weyl consistency conditions have lead to remarkable insights into quantum field theories
In the form derived by Osborn and Jack and Osborn in [1,2,3] [JO] the Weyl consistency conditions imply a gradient flow for the renormalization group flow described by β functions in low orders in perturbation theory, a phenomenon observed earlier in [4,5]
We present a way to implement a Wilsonian cutoff respecting Weyl invariance, based on a local cutoff (x) which transforms under Weyl rescalings
Summary
Weyl consistency conditions have lead to remarkable insights into quantum field theories. The implementation of a Wilsonian cutoff in an otherwise Weyl invariant theory has been considered in [23,24,25], with the aim to study exact (functional) renormalization group equations (RGEs) in 2 and 4 dimensions. Local couplings have not been introduced, and Weyl consistency conditions have not been considered Another approach to implement a Wilsonian cutoff in a Weyl invariant way has been proposed in [26], again in the context of exact RGEs. A local renormalization group equation was considered as early as 1987 in [27] in two-dimensional curved spacetime sigma models in order to derive consis-. 2 we introduce Weyl invariant Wilsonian cutoff functions which allow to derive local RGEs for case 1 and case 2 in Sect.
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