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Riegert's non-local effective action is a nonlocal functional of the spacetime metric that compactly captures the quantum trace (conformal) anomaly in four-dimensional classically Weyl-invariant field theories on curved backgrounds. Introduced in the early 1980s, the Riegert action and its variants are foundational in the analysis of anomaly-induced phenomena in gravitational effective theories, bridging renormalization group flow, cosmological dynamics, and quantum black hole physics. Its defining property is that its metric variation yields the full nontrivial structure of the one-loop trace anomaly as required by renormalized quantum field theory in curved spacetime.

1. Structure of the Trace Anomaly and Riegert Action

The trace anomaly for a general classically Weyl-invariant 4D QFT in curved space takes the form

$$\langle T^\mu_{\;\mu}\rangle = \frac{1}{16\pi^2} \left( \alpha\,C^2 + \beta\,E + \gamma\,\Box R \right)$$

where $C^2 = C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$ is the square of the Weyl tensor, $E = R_{\mu\nu\alpha\beta}^2 - 4R_{\mu\nu}^2 + R^2$ is the Euler density, and $\Box R$ is a total derivative representing the scheme-dependent part of the anomaly.

Riegert's non-local effective action $\Gamma_A[g]$ is constructed such that its metric variation yields the trace anomaly. Its canonical form is

$$\Gamma_A[g] = \frac{1}{64\pi^2} \int d^4x\,\sqrt{g} \left( \alpha\,C^2 + \frac{\beta}{2}\,\mathcal{E}_4 \right) \frac{1}{\Delta_4}\mathcal{E}_4 -\frac{1}{32\pi^2}\left(\frac{\gamma}{6} + \frac{\beta}{9}\right) \int d^4x\,\sqrt{g}\,R^2$$

where $\mathcal{E}_4 = E - \frac{2}{3}\Box R$ and $\Delta_4$ is the Paneitz operator: $$\Delta_4 = \Box^2 + 2R^{\mu\nu}\nabla_\mu\nabla_\nu - \frac{2}{3}R\,\Box + \frac{1}{3}(\nabla^\mu R)\nabla_\mu$$ The Green's function of $\Delta_4$ defines the inverse operator $1/\Delta_4$, encoding the action’s inherent nonlocality. This structure ensures the nonlocal action generates the full covariance and functional form of the 4D trace anomaly under local Weyl rescalings.

2. Conformal Gauge Fixing and the Family of Anomaly-Induced Actions

Any renormalized effective action may be decomposed as

$\Gamma_{\rm ren} = \Gamma_A + \Gamma^{\rm conf}$

where $\Gamma^{\rm conf}$ is Weyl-invariant. The dependence of the anomaly action on the choice of conformal gauge can be formalized by selecting a representative metric $\bar{g}_{\mu\nu}$ on a conformal orbit via

$$g_{\mu\nu} = e^{2\sigma} \bar{g}_{\mu\nu}, \qquad \chi[\bar{g}] = 0$$

The family of anomaly (Wess–Zumino) actions is then given by

$$\Gamma_\chi[g] = \frac{1}{16\pi^2} \int d^4x\,\sqrt{g} \left\{ (\alpha\,C^2 + \beta\,\mathcal{E}_4)\Sigma_\chi -2\beta\,\Sigma_\chi\,\Delta_4\Sigma_\chi \right\} - \frac{1}{32\pi^2}\left(\frac{\gamma}{6} + \frac{\beta}{9}\right) \int R^2$$

with $\Sigma_\chi$ the solution to the gauge-fixing condition. Two notable gauges arise:

• Riegert/Fradkin–Tseytlin (RFT) gauge ($\chi=\bar{\mathcal{E}}_4=0$): $$\Sigma_{\rm RFT}=\tfrac{1}{4}\Delta_4^{-1}\mathcal{E}_4$$, reproducing the classic nonlocal action.
• Fradkin–Vilkovisky (FV) gauge ($\chi=\bar{R}=0$): $$\Sigma_{\rm FV} = -\ln[1+\frac{1}{6} (\Box-\frac{1}{6}R)^{-1}R]$$, which eliminates $1/\Box^2$-type double poles.

The ambiguity in conformal gauge translates to the freedom to add Weyl-invariant nonlocal functionals to $\Gamma_A$. The physical effect of such additions is context-dependent, varying between, for instance, horizon-local and cosmological settings.

3. Covariant Curvature Expansion and Embedding in One-Loop Effective Action

The nonlocal anomaly-induced action is embedded in the general one-loop effective action using the covariant curvature expansion, as implemented in covariant perturbation theory: $$\Gamma_{\rm ren}^{(2)} = \frac{1}{32\pi^2} \int d^4x \sqrt{g} \left[ \frac{\alpha}{120} C\,\ln(-\Box/\mu^2)\,C - \frac{\gamma}{6}R^2 \right] + O(\Re^3)$$ where $O(\Re^3)$ subsumes 29 independent nonlocal cubic invariants and corresponding form factors. Through suitable choice of finite counterterms (e.g., $\gamma = \frac{2}{3}\alpha$), the curvature expansion reproduces the nonlocal structure of the Riegert or FV forms up to cubic order, with the remainder manifestly Weyl-invariant after gauge-fixing.

A key result is that RG running of local cosmological and Einstein-Hilbert couplings is "metamorphosed"—the RG scale-dependence is carried by nonlocal form factors in curvature-squared terms: $$\Gamma_{\rm ren}[g] \supset \int d^4x \sqrt{g} \left[ R_{\mu\nu} F_1(\Box) R^{\mu\nu} + R F_2(\Box) R \right]$$ where $F_{1,2}(\Box)$ interpolate between ultraviolet logarithms ($\ln(-\Box)$) and infrared masslike poles ($\Box^{-1}, \Box^{-2}$).

4. Implementation, Localization, and Physical Interpretation

The nonlocal Riegert action is usually localized using one or more auxiliary scalar fields. For the gravitational sector, this involves solving for a scalar $\phi$ (and, when extending to all anomaly terms, a second scalar for $C^2$): $$S[g, \phi] = \int d^4x \sqrt{-g}\left\{ \frac{1}{16\pi}R - \frac{b}{2}(\Box\phi)^2 - \frac{b}{3}R(\nabla\phi)^2 + b R^{\mu\nu} \nabla_\mu\phi \nabla_\nu\phi + \frac{\phi}{8\pi} \big[ (a+b)C^2 + \cdots \big] \right\}$$ Variation with respect to $\phi$ yields the Paneitz equation, and integrating $\phi$ out reconstructs the nonlocal action. In a Schwarzschild background, the relevant inhomogeneities are sourced by the Kretschmann scalar $R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}$, leading to explicit, finite-dimensional families of solutions for the auxiliary fields. Physically meaningful boundary conditions (regularity on the future horizon, absence of ingoing flux at past null infinity, and absence of unphysical $r^{-n}$ growth) select a unique "Unruh-like" quantum state.

Nonlocal terms such as $1/\Delta_4$ and $1/(\Box-\frac{1}{6}R)$ have important physical roles. In the long-wavelength or near-horizon regime, they can induce large infrared effects; in the ultraviolet, they reproduce standard RG scaling.

5. Stress Tensor Relation in Conformally Related Backgrounds

For metrics related by a Weyl rescaling $g_{\mu\nu} = e^{2\sigma}\bar{g}_{\mu\nu}$, the quantum stress tensors are related by a generalized Brown–Cassidy equation: $$\sqrt{g}\,\langle T^\alpha_{~\beta} \rangle - \sqrt{\bar{g}}\,\langle \bar{T}^\alpha_{~\beta} \rangle = 2\,\bar{g}_{\beta\gamma} \frac{\delta }{\delta \bar{g}_{\alpha\gamma}}\, \Delta\Gamma[\bar{g},\sigma]$$ where $\Delta\Gamma[\bar{g},\sigma]$ is the appropriate Wess–Zumino functional derived from the anomaly action. In the Riegert formulation, explicit calculations yield additional tensor structures that generalize Brown–Cassidy beyond the conformally flat case, relevant e.g. for spacetimes with nonvanishing Weyl tensor.

6. Applications: Black Hole Backreaction and Cosmology

In black hole physics, the anomaly-induced action, together with physically motivated boundary conditions on the auxiliary scalars, yields a unique, time-independent semiclassical stress tensor that is regular at the horizon and asymptotes to a purely outgoing Hawking flux at $\mathcal{I}^+$, precisely matching the Unruh state and precluding additional "quantum hair" or Hartle–Hawking–type solutions with divergent ADM energy. In cosmology, for large-$N$ conformal matter, the anomaly action supplies corrections to the Euclidean Friedmann action, generating "garland" instantons and a finite inflationary era, thereby avoiding infrared pathologies of the no-boundary (Hartle–Hawking) proposal.

The nonlocal kernels' freezing of long-wavelength conformal factor modes carries significant implications for IR modifications of gravitational dynamics, both in cosmological inflation and black hole backreaction.

7. Extensions, Weyl-Invariant Nonlocal Terms, and Contemporary Perspectives

While the Riegert action is uniquely fixed by the requirement of reproducing the trace anomaly, one may freely add purely conformal (Weyl-invariant) nonlocal functionals to the effective action without affecting the trace. A prototypical example is a nonlocal Weyl-squared term: $$S_{\rm Weyl}[g] = \alpha \int d^4x \sqrt{-g} C_{\mu\nu\rho\sigma} F(\Box) C^{\mu\nu\rho\sigma}$$ with $F(\Box)$ an entire function, e.g., $F(\Box)=1/\Box^2$ in flat space limit. The addition of such terms can remove unphysical pathologies of the pure Riegert model (e.g., negative energy flux), and the joint analysis of the coupled auxiliary field system yields a unique, regular solution with physically acceptable boundary conditions.

The interplay between explicit anomaly-induced functionals and the tower of Weyl-invariant nonlocal terms constitutes both a technical tool for holographic and gravitational EFT analyses and a source of theoretical ambiguity whose physical resolution depends on global spacetime structure and boundary conditions (Barvinsky et al., 2023, Liu et al., 7 Nov 2025).