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

Work in four Euclidean dimensions with dimensionless scalar sources $g^I(x)$ in a renormalizable theory. Relevant sources require additional terms and are suppressed here. Use $W=-\log Z$, positive covariant-metric stress variation, and the real action source $+\int\sqrt g\,g^I\mathcal O_I$. The ordinary mass-scale beta function is $\beta_\mu^I=\mu\,dg^I/d\mu$ at fixed bare theory. Functional derivatives are coordinate densities: for example, $\delta W/\delta g^I(x)=\sqrt{g(x)}\langle\mathcal O_I(x)\rangle$. The local generator therefore has no second $\sqrt g$ in its integration measure.

When background flavor sources $A_\mu^A$ are retained, write

$$\begin{aligned} \Delta_\sigma=\int d^4x\,\bigg[ &2\sigma g_{\mu\nu}\frac{\delta}{\delta g_{\mu\nu}} +\sigma\beta_\mu^I\frac{\delta}{\delta g^I} \\ &+\big(\sigma\rho_I^A D_\mu g^I-\partial_\mu\sigma\,S^A\big) \frac{\delta}{\delta A_\mu^A} +\cdots\bigg]. \end{aligned}$$

The coefficients $\rho_I^A$ and $S^A$ encode current mixing and flavor rotations. Define $D_\mu g^I=\partial_\mu g^I+A_\mu^A(T_Ag)^I$, with background transformations $\delta_\omega g=-\omega^AT_Ag$ and $\delta_\omega A_\mu=D_\mu\omega$. In a flavor-anomaly-free prescription, their Ward identity is $D_\mu J_A^\mu=-(T_Ag)^I\mathcal O_I$, modulo equations of motion. Adding this vanishing background gauge generator with parameter $\omega^A=\sigma S^A$ replaces the scalar and vector coefficients by

$$B^I=\beta_\mu^I-(S^AT_Ag)^I, \qquad P_I^A=\rho_I^A+\partial_I S^A.$$

Here $S$ is flavor equivariant in the chosen local coupling basis. The derivative of $\sigma$ cancels, leaving the vector-source term $+\sigma P_I^A D_\mu g^I\,\delta/\delta A_\mu^A$. A beta component tangent to a flavor orbit is thus redundant rather than an independent scalar breaking. The spurion construction and reduction are given in Osborn 1991, printed preprint pp. 15–16, Eqs. (3.38)–(3.41), PDF, with the generating-functional translation below.

For local sources the reduced trace identity has the form

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

The anomaly $\mathcal A$ contains curvature and source-derivative terms; $V^\mu$ represents retained improvements or a virial contribution. This is a renormalized identity with the appropriate equation-of-motion and insertion-contact prescription. In flat space with constant scalar couplings and vanishing background flavor connection, the displayed source-gradient and smooth curvature terms vanish, but setting those sources constant before deriving the identity would erase the information needed for the consistency calculation. The local-RG prerequisite derives the source and contact signs.

Wess–Zumino consistency

Local Weyl rescalings form an abelian group, so on a closed manifold, or for variations of compact support away from a boundary,

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

Writing $\Delta_\sigma W=\mathcal A_\sigma$, this requires $\Delta_\sigma\mathcal A_{\sigma'}-\Delta_{\sigma'}\mathcal A_\sigma=0$. Expand the anomaly in a complete basis at the chosen derivative order and match independent structures. The following representative four-dimensional calculation uses only scalar sources: flavor sources and rotations are absent, so $B^I=\beta_\mu^I$.

Keep the coefficient basis of Osborn’s anomaly functional. He uses $W_O=+\log Z$ and $\Delta_{\sigma,O}=\Delta^W_\sigma-\Delta^\beta_\sigma$, whose metric part varies the inverse metric positively. Thus $W=-W_O$ and $\Delta_\sigma=-\Delta_{\sigma,O}$, leaving the raw anomaly functional unchanged. Its coefficients $\beta_b,w_I,\chi^g_{IJ}$ are not individually negated Osborn 1991, printed preprint pp. 5–6, Eqs. (2.1)–(2.5), PDF.

The terms that define the required coefficients are

$$\begin{aligned} \mathcal A_\sigma\supset\int d^4x\sqrt g\,\bigg[ &\sigma\beta_b E_4 +\frac12\sigma\chi^g_{IJ}G^{\mu\nu} \partial_\mu g^I\partial_\nu g^J \\ &+(\partial_\mu\sigma)w_I G^{\mu\nu}\partial_\nu g^I \bigg], \end{aligned}$$

where $G^{\mu\nu}=R^{\mu\nu}-\tfrac12g^{\mu\nu}R$ is the Einstein tensor and $E_4$ is the Euler density. The full anomaly also contains other four-derivative structures Osborn 1991, printed preprint p. 9, Eqs. (3.1)–(3.4), PDF. Matching the coefficient of

$$(\sigma\partial_\mu\sigma'-\sigma'\partial_\mu\sigma) \,G^{\mu\nu}\partial_\nu g^I$$

gives the one-form relation

$$8\partial_I\beta_b-\chi^g_{IJ}B^J =-\mathcal L_B w_I, \qquad (\mathcal L_Bw)_I=B^J\partial_Jw_I+(\partial_I B^J)w_J.$$

The component matching is carried out in Osborn 1991, printed preprint p. 10, Eqs. (3.9)–(3.10a), PDF. To obtain the gradient form, define

$$\widetilde A=8\widetilde\beta_b=8\beta_b+w_I B^I, \qquad \chi_{IJ}=\chi^g_{IJ}.$$

Differentiating this definition and using the one-form equation cancels the terms $w_J\partial_I B^J$, giving

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

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

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

This is Osborn’s scalar-source relation with its factor of eight made explicit Osborn 1991, printed preprint p. 11, Eq. (3.13), PDF. His Euler density $G$ is $E_4$, and the coefficient in the raw anomaly is $\beta_b$. At a fixed point in this chapter’s trace convention, $\beta_b=a/(4\pi)^2$ and hence $\widetilde A=8a/(4\pi)^2$; $\widetilde A$ is not identically the conventionally normalized $a$.

With flow parameter $t=\log\mu$, a positive $\chi_{(IJ)}$ on the relevant directions makes $\widetilde A$ nondecreasing toward the ultraviolet and nonincreasing toward the infrared. Wess–Zumino consistency alone does not establish that positivity. The gradient relation and its perturbative domain are analyzed by Jack and Osborn 1990; the equation here is not a global nonperturbative monotonicity theorem.

When flavor sources are active, the vector-source commutator also requires $B^I P_I^A=0$ in the stated source basis. Its coefficient multiplies $\sigma\partial_\mu\sigma'-\sigma'\partial_\mu\sigma$. The general consistency equations use a modified Lie derivative and additional flavor-covariant terms Osborn 1991, printed preprint p. 16, Eqs. (3.43)–(3.45), PDF. Those structures must be retained before deriving a general gradient or contracted equation; the scalar result above does not justify dropping an unspecified flavor remainder.

Scheme covariance

Adding a finite local functional $C[g_{\mu\nu},g^I,A_\mu]$ to $W$ changes the anomaly by $\Delta_\sigma C$. In Osborn’s convention the corresponding addition to $W_O$ is $-C$, so component counterterm formulas must be translated with that sign. 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 all spacetime-dependent sources in the chosen sector, including vector sources when flavor rotations are present;
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.

The scalar example above uses the consistency component in Osborn 1991, printed preprint pp. 9–11, Eqs. (3.1)–(3.13), PDF; his Eqs. (3.11)–(3.12) also give its finite-counterterm transformations. A complete source basis is needed for other components. The reusable procedure is the commutator calculation and its convention translation, rather than one preferred set of coefficient names.

Common pitfalls

Equating $\beta_\mu^I=0$ 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. Open PDF. The locators above use the printed pages of author preprint DAMTP/91-1.
• Shore, G. M. “The $c$ and $a$-Theorems and the Local Renormalisation Group.” SpringerBriefs in Physics (2017). arXiv. DOI.