# Wilsonian Ward Identities

## TL;DR

This paper demonstrates how dilatation Ward identities in conformal field theories can be derived within a Wilsonian framework, clarifying conceptual points and identifying primary fields where these identities hold.

## Contribution

It shows how the conformal Ward identity arises directly from the Wilsonian action, avoiding formal regulator removal arguments, and identifies the primary fields satisfying the Ward identity across dimensions.

## Key findings

- Dilatation Ward identity derived in Wilsonian setting
- Clarification of conformal invariance in the presence of a regulator
- Identification of primary fields satisfying the Ward identity

## Abstract

For conformal field theories, it is shown how the Ward identity corresponding to dilatation invariance arises in a Wilsonian setting. In so doing, several points which are opaque in textbook treatments are clarified. Exploiting the fact that the Exact Renormalization Group furnishes a representation of the conformal algebra allows dilatation invariance to be stated directly as a property of the action, despite the presence of a regulator. This obviates the need for formal statements that conformal invariance is recovered once the regulator is removed. Furthermore, the proper subset of conformal primary fields for which the Ward identity holds is identified for all dimensionalities.