Local RG and Weyl Consistency Conditions

The local renormalization group promotes couplings to spacetime-dependent sources and asks how the generating functional responds to a local change of scale. Because Weyl rescalings commute, their anomaly must satisfy integrability conditions. These Weyl consistency conditions constrain beta functions and anomaly coefficients, but they become monotonicity statements only after an additional positive metric or spectral input is established.

Required background. The Trace Ward Identity and Weyl Anomaly fixes the local source convention. Local RG and Trace Identities supplies coupling renormalization. Helpful background. Beta Functions and Anomalous Dimensions reviews ordinary RG flow.

Couplings as local sources

Let $g^I(x)$ couple to scalar operators $\mathcal O_I$, and let $A_\mu^A(x)$ source flavor currents. A general local RG operator contains

$$\Delta_\sigma =\int d^dx\sqrt g\left[ 2\sigma g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}} -\sigma\beta^I\frac{\delta}{\delta g^I} -\big(\sigma\rho_I^A\nabla_\mu g^I-\partial_\mu\sigma\,S^A\big) \frac{\delta}{\delta A_\mu^A} +\cdots\right].$$

The coefficients $\rho_I^A$ and $S^A$ encode current mixing and flavor rotations. The trace identity is most naturally expressed through the flavor-covariant flow vector

$$B^I=\beta^I-(S^AT_Ag)^I,$$

rather than through $\beta^I$ alone. A beta function that is a pure flavor rotation can describe a redundant direction instead of a physically distinct scale dependence.

For constant sources in flat space, the operator identity has schematic form

$$T^\mu{}_{\mu} =B^I\mathcal O_I +P_I^A\partial_\mu g^I J_A^\mu +\nabla_\mu V^\mu +\text{contacts}.$$

Setting $\partial_\mu g^I=0$ before deriving this equation erases precisely the terms needed to distinguish physical flow from source reparametrization.

Wess–Zumino consistency

Local Weyl rescalings form an abelian group, so

$$[\Delta_\sigma,\Delta_{\sigma'}]W=0.$$

Expand the anomaly in a basis of local curvature and source-derivative terms, apply the two transformations, and match independent structures such as

$$(\sigma\partial_\mu\sigma'-\sigma'\partial_\mu\sigma) \,\partial^\mu g^I.$$

The coefficients must then obey differential relations on coupling space. A representative gradient-type equation has the form

$$\partial_I\widetilde A =\big(\chi_{IJ}+\partial_Iw_J-\partial_Jw_I\big)B^J +\text{flavor-covariant terms}.$$

Contracting with $B^I$ removes the antisymmetric piece:

$$B^I\partial_I\widetilde A =\chi_{(IJ)}B^IB^J.$$

This is an identity in the chosen local-RG scheme. It gives a monotone quantity only where $\chi_{(IJ)}$ is positive in the relevant directions and the flow parameter orientation has been fixed. Wess–Zumino consistency by itself does not prove that positivity; the four-dimensional gradient-type relation and its perturbative domain are analyzed in Jack and Osborn 1990.

Scheme covariance

Adding a finite local functional $C[g_{\mu\nu},g^I,A_\mu]$ changes the anomaly by $\Delta_\sigma C$. Consequently $\widetilde A$, $w_I$, and $\chi_{IJ}$ can shift while the consistency equation retains its form. A coupling redefinition $g^I\mapsto g'^I(g)$ also changes their components as tensors or connections on theory space.

Quantities at a fixed point are often simpler: $B^I=0$, and the nontrivial Euler or Weyl coefficients become invariant after their density normalization is fixed. Away from a fixed point, the robust object is the complete covariant equation, not an isolated coefficient in one coordinate system. Shore 2017 reviews this distinction between consistency identities, scheme dependence, and positivity input.

The chapter’s scheme matrix is:

Object	Category	Allowed change	Robust statement
Type-A fixed-point coefficient	Universal	density normalization only	comparable between fixed points in one convention
Type-B fixed-point coefficient	Universal	tensor/density basis change	matched to normalized separated-point data where known
Total-derivative anomaly	Scheme dependent	finite local curvature counterterm	meaningful only in a declared scheme
$B^I$	Scheme covariant	coupling coordinates and flavor redundancy	zeros modulo redundant directions are physical
$w_I$, $\chi_{IJ}$	Scheme covariant local-RG data	finite source counterterms	enter consistency relations as a combination
Contact term in an integrated correlator	Local convention	operator/source counterterm	cannot be inferred from separated points alone
Endpoint inequality	Theorem when hypotheses hold	no arbitrary local shift of endpoint invariant	compares specified UV and IR fixed points

The table is deliberately categorical: universal, scheme-dependent, and contact data should never occupy the same numerical column without their transformation rules.

A derivation workflow

To derive a representative consistency condition:

1. choose a complete basis of local anomaly terms at the derivative order of interest;
2. retain spacetime-dependent scalar and vector sources;
3. calculate $\Delta_\sigma(\Delta_{\sigma'}W)-(\sigma\leftrightarrow\sigma')$;
4. reduce by integrations by parts and algebraic identities;
5. set every independent coefficient to zero;
6. test covariance under finite counterterms and coupling redefinitions;
7. only then ask whether a positive two-point function or spectral representation makes a contracted relation monotone.

Osborn’s four-dimensional construction supplies the canonical detailed example Osborn 1991, §§2–4. The procedure, not a single preferred coefficient basis, is the reusable result.

Common pitfalls

Equating $\beta^I=0$ with a coordinate-invariant fixed point. Flavor rotations and redundant operators can move couplings without changing observables. Use the covariant $B^I$ and quotient redundant directions.

Reading a gradient formula as a theorem of positivity. Consistency supplies the differential identity. Positivity of its quadratic form is an additional, dimension- and regime-dependent input.

Comparing $\widetilde A(g)$ between schemes away from fixed points. Finite local counterterms can change the interpolating function. Fixed-point differences are safer when their anomaly normalization is held fixed.

Exercises

Show why the antisymmetric part of $\partial_Iw_J$ drops out after contraction with $B^IB^J$.

Solution

$B^IB^J$ is symmetric under $I\leftrightarrow J$, while $\partial_Iw_J-\partial_Jw_I$ is antisymmetric. Their contraction therefore vanishes identically.

References

• Jack, I., and Osborn, H. “Analogs for the $c$ Theorem for Four-Dimensional Renormalisable Field Theories.” Nuclear Physics B 343 (1990): 647–688. DOI.
• Osborn, H. “Weyl Consistency Conditions and a Local Renormalization Group Equation for General Renormalisable Field Theories.” Nuclear Physics B 363 (1991): 486–526. DOI.
• Shore, G. M. “The $c$ and $a$-Theorems and the Local Renormalisation Group.” SpringerBriefs in Physics (2017). arXiv. DOI.