Weyl Consistency Conditions from a local Wilsonian Cutoff

TL;DR

This paper develops Weyl invariance conditions using a local Wilsonian cutoff, deriving exact gradient flow equations for anomalous dimensions and beta functions in scalar field theories in curved spacetime.

Contribution

It introduces a novel approach to Weyl invariance by incorporating a local UV cutoff, leading to new consistency conditions and gradient flow equations for anomalous dimensions.

Findings

Weyl invariance can be achieved with a local UV cutoff in scalar theories.

Weyl consistency conditions form an exact gradient flow for anomalous dimensions.

Conditions are derived under which beta functions follow an exact gradient flow.

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.