Riegert’s Non-Local Effective Action

• Riegert’s non-local action is a covariant functional that reproduces the trace anomaly in Weyl-invariant quantum field theories.
• It employs the inverse Paneitz operator and curvature expansion to bridge ultraviolet renormalization and infrared gravitational effects.
• Gauge fixing and auxiliary field localization enable practical semiclassical analysis of phenomena in black hole evaporation and early-universe cosmology.

Riegert's non-local effective action is a covariant, nonlocal functional of the metric in four-dimensional curved spacetime, constructed to reproduce the trace (conformal) anomaly in classically Weyl-invariant quantum field theories. It provides a unique gravitational effective action whose variation under local Weyl transformations yields the universal anomaly terms in the quantum stress-tensor trace. The Riegert action serves as a foundational component in the perturbative expansion of quantum effective actions in gravity and is essential for the consistent semiclassical analysis of phenomena such as Hawking radiation and the cosmological backreaction of conformal fields.

1. The Conformal Anomaly and Riegert’s Non-Local Action

In classically Weyl-invariant four-dimensional theories, the renormalized quantum stress tensor acquires a nonvanishing trace at one-loop: $$\langle T^\mu{}_\mu \rangle = \frac{1}{16\pi^2} \left( \alpha\, C^2 + \beta\, E + \gamma\, \Box R \right)$$ where $$C^2 \equiv C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$$ is the Weyl tensor squared term, $E = R_{\mu\nu\alpha\beta}^2 - 4R_{\mu\nu}^2 + R^2$ is the Euler density, and $\Box R$ is the total derivative term. The coefficients $(\alpha, \beta, \gamma)$ are determined by the spin content and matter spectrum.

The Riegert–Fradkin–Tseytlin (RFT) nonlocal action $\Gamma_A[g]$ is uniquely defined (up to addition of a Weyl-invariant functional) by the requirement that its metric variation yields this anomaly. In integral form, for Lorentzian or Euclidean signature,

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

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 inverse $\Delta_4^{-1}$ is defined via a suitable Green’s function, selected by physical boundary conditions.

2. Gauge-Fixing, Weyl Invariants, and the Family of Anomaly Actions

The anomaly action $\Gamma_A$ is not unique, as one may add any Weyl-invariant functional without affecting the trace anomaly. The full renormalized one-loop effective action splits as

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

where $\Gamma^\mathrm{conf}$ is Weyl-invariant.

A systematic resolution employs conformal gauge fixing: $$g_{\mu\nu} = e^{2\sigma}\,\bar{g}_{\mu\nu},\quad \chi[\bar{g}]=0$$ for a chosen gauge condition $\chi$, leading to a "Wess-Zumino" family of anomaly actions: $$\Gamma_\chi[g] = \frac{1}{16\pi^2}\int d^4x\sqrt{g}\left\{(\alpha C^2+\beta\mathcal{E}_4)\Sigma_\chi -2\beta\Sigma_\chi\Delta_4\Sigma_\chi \right\} - \frac{1}{32\pi^2}\left(\frac{\gamma}{6}+\frac{\beta}{9}\right)\int R^2$$ Different gauge choices yield:

• RFT gauge ($\chi_{\rm RFT}[\bar{g}]=\bar{\mathcal{E}}_4=0$): $$\Sigma_{\rm RFT} = \tfrac14 \Delta_4^{-1}\mathcal{E}_4$$ returning the nonlocal Riegert action.
• Fradkin–Vilkovisky (FV) gauge ($\chi_{\rm FV}[\bar{g}]=\bar{R}=0$): $\Sigma_{\rm FV}$ nonlocal in terms of $(\Box - \tfrac16 R)^{-1}R$, eliminating $1/\Box^2$ double poles.

The choice of gauge reflects the freedom to add Weyl-invariant nonlocal terms, which can have physical consequences depending on context, e.g., black hole horizon physics versus cosmological initial conditions (Barvinsky et al., 2023).

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

Within the covariant perturbation theory (CPT) developed by Barvinsky, Vilkovisky, and others, the one-loop effective action is expanded in powers of general curvature invariants ("curvature expansion"): $$\Gamma_\mathrm{ren} = \Gamma^{(2)} + O(\mathcal{R}^3)$$ At quadratic order,

$$\Gamma^{(2)}_\mathrm{ren} = \frac{1}{32\pi^2}\int d^4x\sqrt{g} \left[\frac{\alpha}{120} C\,\ln\left(-\frac{\Box}{\mu^2}\right)\,C - \frac{\gamma}{6} R^2 \right]$$

At cubic order ($O(\mathcal{R}^3)$), 29 independent nonlocal invariants appear. The Riegert action is embedded within this expansion, with the nonlocal $1/\Box^2$ and log form factors arising naturally; the anomaly arises from the $R^2$ sector when $\gamma = \frac{2}{3}\alpha$, as enforced by a finite $R^2$ counterterm.

The Riegert and FV-type anomaly actions are both present in the first three orders of the generic CPT expansion, with re-expansion in the FV gauge cleanly separating anomalous and Weyl-invariant parts (Barvinsky et al., 2023, Donoghue et al., 2015).

4. Nonlocal Form Factors, Renormalization Group Running, and Infrared/Ultraviolet Structure

The nonlocality in Riegert’s action is controlled by the inverse Paneitz operator $1/\Delta_4$, resulting in $1/\Box^2$-type kernels, while FV-like forms involve $(\Box - \frac16 R)^{-1}$. These nonlocal operators interpolate between ultraviolet (UV) and infrared (IR) regimes:

• In the UV, $1/\Delta_4$ reduces to $\ln(-\Box)$, encoding standard renormalization group (RG) running via dimensionless form factors $F(\Box)$ multiplying curvature quadratic invariants.
• In the IR, the form factors become effective "mass" modifications scaling as $\Box^{-1}$ or $\Box^{-2}$, with dominant contributions from near-horizon or long-wavelength modes. For example, Donoghue “partners” of the cosmological and Einstein terms appear as $M^4 R\,\Box^{-2} R$ and $M^2 R\,\Box^{-1} R$ (Barvinsky et al., 2023, Bautista et al., 2017).

The RG "metamorphosis" refers to the flow of the running of the cosmological constant and gravitational couplings away from the lowest-dimensional operators into nonlocal higher-derivative terms, ensuring consistency with covariance and anomaly structure.

5. Functional Variation, Stress Tensor, and the Generalized Brown–Cassidy Relation

The Riegert action is constructed so that its metric variation under an infinitesimal Weyl transformation $\delta_\sigma g_{\mu\nu}=2\sigma g_{\mu\nu}$ directly yields the trace anomaly. Explicitly,

$$\delta_\sigma S_R = \int d^4x\sqrt{-g} \,A(x)\, \delta\sigma(x) \implies g^{\mu\nu}\left\langle T_{\mu\nu} \right\rangle = \frac{1}{16\pi^2} \left[ a\,C^2 + b(E-\frac{2}{3}\nabla^2 R) \right]$$

The full variation also enables the calculation of the induced stress tensor in arbitrary geometries.

For two conformally related metrics $g_{\mu\nu} = e^{2\sigma} \bar{g}_{\mu\nu}$, the quantum stress-tensor relation generalizes the Brown–Cassidy equation to arbitrary backgrounds: $$\sqrt{g}\, \langle T^\alpha{}_\beta \rangle - \sqrt{\bar{g}}\, \langle \bar{T}^\alpha{}_\beta \rangle = 2\, \bar{g}_{\beta\gamma} \frac{\delta}{\delta \bar{g}_{\alpha\gamma}} \Delta\Gamma[\bar{g}, \sigma]$$ with $\Delta\Gamma[\bar{g},\sigma]$ the Wess–Zumino functional. This allows for explicit tracking of the anomaly-induced stress tensor under Weyl rescalings, crucial for both black hole and cosmological applications (Barvinsky et al., 2023).

6. Localization via Auxiliary Fields and Application to Black Hole Physics

While the Riegert action is innately nonlocal, practical computations are facilitated by localization with one or two auxiliary scalar fields $\varphi$ (and $\chi$ for Weyl-square extensions). The localized action—upon integrating out the auxiliary fields—recovers the original nonlocal form.

In Schwarzschild backgrounds, where $R_{\mu\nu}=0=R$, but $R_{\mu\nu\rho\sigma}^2 \neq 0$, the fourth-order equation for $\varphi$ can be solved in closed form. Physically motivated boundary conditions—regularity at the future horizon, vanishing incoming flux at past null infinity—fix all but one (irrelevant) integration constant and yield a unique, time-independent induced stress tensor. This stress tensor matches the Unruh vacuum state, realizing a steady outgoing Hawking flux at future null infinity: $$n_+^\mu T_{\mu\nu} n_+^\nu \Big|_{\mathscr{I}^+} = \frac{(a+b)^2}{8\pi^2 M^2 b}\frac{1}{r^2} + \cdots$$ There are no solutions with additional quantum hair; all physically sensible semiclassical states reduce to the Unruh state (Lowe et al., 12 May 2025, Liu et al., 7 Nov 2025). Additional Weyl-invariant terms (e.g., nonlocal $C^2$ kernels) can be included without affecting the trace anomaly but may influence other metric components (Liu et al., 7 Nov 2025).

7. Cosmological and Theoretical Implications

In cosmology, the anomaly-induced action is vital for understanding quantum corrections in early-universe models, particularly in scenarios with a large-$N$ conformal sector. In closed FRW backgrounds conformally related to $S^1\times S^3$, the Riegert action yields a modified Euclidean Friedmann action with additional curvature-dependent terms, producing self-consistent instanton solutions. These "garland" instantons realize an inflationary phase with finite duration, resolving issues with standard no-boundary proposals such as the infrared catastrophe of the Hartle–Hawking state (Barvinsky et al., 2023).

The mathematical structure of Riegert’s action establishes a precise correspondence between perturbative expansions and anomaly-induced nonlocal structures. It is, however, not uniquely fixed beyond the anomaly sector; physically meaningful predictions require careful accounting of additional Weyl-invariant terms, choice of Green’s function (boundary conditions), and auxiliary field dynamics (Barvinsky et al., 2023, Bautista et al., 2017).

Table: Key Elements of Riegert’s Non-Local Action

Term / Operator	Mathematical Definition	Role in Action
$C^2$	$C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}$	Type-B anomaly (Weyl squared)
$E$	$R_{\mu\nu\alpha\beta}^2 - 4R_{\mu\nu}^2 + R^2$	Type-A anomaly (Euler density)
$\Delta_4$	$$\Box^2 + 2R^{\mu\nu}\nabla_\mu\nabla_\nu - \frac{2}{3}R \Box + \frac{1}{3}(\nabla^\mu R)\nabla_\mu$$	Paneitz operator, defines nonlocality
$1/\Delta_4$	Green’s function inverse of $\Delta_4$	Nonlocal kernel in action
Weyl-invariant part	Arbitrary functional $\Gamma^\mathrm{conf}[g]$	Physical ambiguity in gauge-fixed formulation

This table summarizes the primary mathematical objects appearing in the Riegert anomaly action and its related gauge-fixed forms.

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The Riegert nonlocal effective action is thus an indispensable tool in quantum field theory in curved space, encoding the universal manifestations of the conformal anomaly, organizing covariant curvature expansions, and underpinning physically meaningful semiclassical solutions in gravitational backgrounds, both in black hole evaporation and in early universe cosmology (Barvinsky et al., 2023, Lowe et al., 12 May 2025, Liu et al., 7 Nov 2025, Bautista et al., 2017, Donoghue et al., 2015).