# Wilsonian ward identities

## 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.

## 1 Introduction

In classical field theory, Noether’s theorem states that continuous symmetries of the action correspond to conserved currents. If such a symmetry is preserved at the quantum level then it implies relationships satisfied by the correlation functions: the Ward identities.

^{1}That it is conserved follows from recalling the equations of motion: \(\delta S/ \delta \varphi = 0\), whereupon it is apparent that, on the equations of motion,

This discussion may give the impression that the only diagnostic for quantum scale (or conformal) invariance comes from the correlation functions. However, the Exact Renormalization Group (ERG) provides a framework in which dilatation invariance is realized as a non-linear constraint on the Wilsonian effective action. The key point is that this non-linearity enables us to formulate a condition for dilatation invariance as a property of a regularized action and hence directly in the presence of a cutoff function. This point is central to [1] and has been recently emphasised in [2].

The purpose of this paper is to understand how the Ward identities arise within the ERG representation of the conformal algebra. As a bonus, a further point that is rather opaque in a more standard treatment will be clarified. Notably, in [3], it is carefully demonstrated how the dilatation Ward identity follows for correlation functions of the fundamental field. However, in general \(d\), no concrete statement is made about the other fields for which the Ward identity also holds; moreover, in \(d=2\) it is simply asserted that it has already been shown that the Ward identity holds for all Virasoro^{2} primary fields! (Subsequently, however, an independent derivation of the Ward Identity is supplied for \(d=2\).) In this paper, a precise statement will be demonstrated for scalar conformal primary fields in scalar field theory: the Ward identity, in any \(d\), holds for the subset of such fields which, in the limit that the regularization is removed (i.e. the classical limit), have no derivative terms. An analogous statement holds for fermionic fields.

Since, in this paper, the ERG is our tool of choice it will be introduced in Sect. 2. However, rather than providing a Wilsonian motivation for this framework [4], we instead follow [1] and introduce it as furnishing a particular representation of the conformal algebra. This viewpoint is more sympathetic to the thrust of the paper. After assembling some basic properties of the ERG, the sought-for Ward identity follows rather directly in Sect. 3.1, by constructing appropriate solutions to the ERG equation. Following an example in Sect. 3.2, our conclusions are presented.

## 2 The exact RG

### 2.1 Motivation and equations

Building on the examples presented in the introduction, in this section we will make more precise the notion of the ERG realizing dilatation invariance as a non-linear constraint on the Wilsonian effective action.

One approach to determine \(\delta \) is to return to a local formulation of the theory. However, if we do so, then we must accept a more complicated representation of the conformal algebra than furnished by (2.2). Indeed, this is precisely what is achieved by fixed-point version of the Exact Renormalization Group (ERG). In this approach, the functional generators for translations and rotations are untouched. However, the dilatation and special conformal generators are modified as follows.

It will prove profitable to spend a few moments understanding the limit in which these generators reduce to their classical forms. This is most readily seen by transforming to dimensionful variables and working in momentum space; this procedure is discussed at length in [6] and is revisited in appendix A. Denoting the energy scale by \(\Lambda \), the transformed field, \(\phi (p)\), is accompanied by a factor of \(\Lambda ^{d-{\delta }}\) from which it follows that \(\delta / \delta \phi (p)\) picks up a factor of \(\Lambda ^{\delta }\). Bearing in mind that a momentum integral acquires a factor of \(\Lambda ^{-d}\) after transferring to dimensionful units, it is clear that the classical contributions to (2.7a) and (2.7b) survive the classical limit \(\Lambda \rightarrow \infty \). What of the remaining terms?

Consider the second contribution to \(\mathcal {D}\). Compared to a raw \(\phi \cdot \delta / \delta \phi \) – which we have just seen survives the limit – this term has an extra contribution which, at leading order in momentum, goes like \( K(0) K'(0) p^2/\Lambda ^2 \). Manifestly, this is sub-leading in the \(\Lambda \rightarrow \infty \) limit. As for the final contribution to \(\mathcal {D}\), this goes like \(\Lambda ^{2\delta - d} = \Lambda ^{\eta - 2}\). This is sub-leading so long as \(\eta < 2\), which we now assume to be the case. The special conformal generator, (2.7b), can be similarly treated.

### 2.2 Properties of solutions

To provide some intuition, note that \(\mathcal {O}^{(\delta )}\) is morally the fundamental field: it is easy to show that, in the limit that the regularization is removed, it simply reduces to \(\varphi \) [6]. On the other hand, \(\mathcal {O}^{(d-\delta )}\) is the redundant field which, in the classical limit, defines the equations of motion.

### 2.3 Coupling sources

## 3 Wilsonian ward identity

### 3.1 Derivation

*a*this spoils the boundary condition.

### 3.2 Example

## 4 Conclusion

For conformal field theories, the Ward identity corresponding to dilatation invariance has been shown to be encoded in solutions of the ERG equation. A particular benefit of deriving the Ward identity in this way is that it avoids questions as to whether the introduction of a regulator potentially spoils conformal invariance: by deforming the classical representation of the conformal generators, conformal invariance can be stated directly as a property of a regularized Wilsonian effective action, without any reference to the (non-local) correlation functions. Furthermore, this derivation straightforwardly identifies the subset of conformal primary fields for which the Ward identity actually holds: those fields which, in the limit that the regularization is removed, do not contain any derivatives.

It is also noteworthy that, contrary to more standard derivations, no explicit use is made of the dilatation current. Indeed, only the trace the energy-momentum tensor appears in our analysis, cementing the increasingly central role that this object plays in understanding the mathematical structure of the ERG.

## Footnotes

- 1.
At least up to terms which are exactly conserved.

- 2.
In this paper, we shall refer to fields for which the correlation functions exhibit global conformal covariance as conformal primary fields; in \(d=2\) these are typically called quasi-primary, with the Virasoro primary fields being those whose correlation functions posses local conformal covariance.

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