Let us consider an action $S[\phi,\partial\phi]$ which is classically invariant under a transformation group $G$. The associated Noether current $\mathcal{J}^\mu$ is classically conserved, namely $\partial_\mu \mathcal{J}^\mu=0$ holds as an operatorial identity.

The generating functional of this theory is

$$ Z[J] = \int [d\phi] \,e^{iS + i \int d^dx \,J(x)\phi(x)} $$

where we added an external source for $\phi(x)$.

If the theory is anomaly-free, we can derive the Ward-Identity for $\mathcal{J}^\mu$, which is

$$ \partial_\mu \langle \mathcal{J}^\mu(x)\rangle_J = J(x) \langle \delta \phi(x)\rangle_J\,,\qquad \qquad (1) $$

where $\langle ... \rangle_J$ means that the correlations must be computed with $J\neq 0$ and $\delta \phi$ is the variation of the field $\phi$ under $G$.

By differentiating Eq. (1) wrt $J(x_i)$ and then setting $J=0$, we get the usual Ward identities.

Usually Eq. (1) is the departing point to state that conserved currents do not acquire anomalous dimension. Indeed, the Ward-identity can be written roughly as

$$ \partial_\mu \langle \mathcal{J}^\mu(x)X(y)\rangle_{J=0} \propto \delta^d(x-y)\langle\delta X(y)\rangle $$ for some local product of fields $X(y)$. When the RHS is renormalized, the same is true for the LHS. Then, no renormalization of $\mathcal{J}^\mu$ is needed. This implies the dimension of $\mathcal{J}^\mu$ is fixed.

First question How does Eq.(1) change if the theory is anomalous? For example, put a conformally invariant theory on a curved background. We have a trace anomaly for the trace of the stress-energy momentum tensor, namely $\langle T^\mu_\mu\rangle \neq 0$. Consider as classically conserved current the dilatation current $D^\mu = x_\rho T^{\rho\mu}$. More precisely, If I know the matter content of the theory, I can see how the path-integral measure changes and then I can derive the anomalous Ward identity. What if instead I don't specify the matter content, but just the trace anomaly?

Second question If the theory is anomalous, is still true that classically conserved currents do not acquire anomalous dimension?


1 Answer 1


Simplest example of anomaly is chiral anomaly. Due to transformation of mesuare (see (3.34)) under chiral transformations, equation (1) will be modified (for background gauge field):

$$ \partial_\mu \langle \mathcal{J}^\mu(x)\rangle_J = J(x) \langle \delta \phi(x)\rangle_J\, + a \varepsilon^{\mu\nu\rho\sigma} F_{\mu\nu}F_{\rho\sigma}\langle 1\rangle_J $$

In case of conformal anomaly measure transforms under Weyl rescaling. It also will lead to additional term in Ward identity. You need also use specific regulator. See Path Integrals and Anomalies in Curved Space.

About dimension:

Among the local operators of the theory, special role will be played by a stress tensor $T_{\mu\nu}$ and conserved currents $J_{\mu}$ associated to global symmetries of a theory. QFT axioms (Wightman axioms) don’t require existence of the stress tensor as the energy and momentum density, but only of full energy and momentum charges, and analogously for the conserved currents. Thus the existence of the stress tensor and currents is an extra assumption. It means that the theory preserves some locality. In perturbation theory these two operators don’t renormalize, i.e, have zero anomalous dimensions: $$\gamma_T = \gamma_J = 0$$

Simple argument:


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