Anomaly-Induced Effective Action in Quantum Gravity

• Anomaly-induced effective action is a framework that localizes nonlocal trace anomalies via auxiliary fields to capture the quantum backreaction in curved spacetimes.
• It integrates the trace anomaly by reproducing the anomalous stress–energy tensor through boundary-sensitive auxiliary field formulations in two and four dimensions.
• The approach underpins semiclassical analyses of black hole backreaction and vacuum state selection, highlighting implications for horizon stability and mass inflation.

Anomaly-induced effective actions are nonlocal quantum effective actions constructed to encode the effects of conformal (trace) anomalies in quantum field theories in curved spacetimes. These anomalies, manifest as nonvanishing traces of the quantum stress–energy tensor, capture the quantum violation of classical scale (Weyl) invariance arising under renormalization. The anomaly-induced action provides a tool to represent the quantum gravitational backreaction and state-dependent features (such as vacuum polarization and Hawking radiation) through local auxiliary field dynamics, subject to appropriate boundary and regularity conditions.

1. Construction and Formal Structure

The anomaly-induced effective action is derived by integrating the trace anomaly, ensuring that the resulting functional reproduces the anomalous trace of the quantum stress–energy tensor under metric variations. In two dimensions, for a conformal matter content with anomaly coefficient $Q^2=-N/6$ (where $N$ is the number of scalar fields), this action is given in its nonlocal Polyakov form,

$$S_{\rm anom}^{(2)}[g] = \frac{Q^2}{16\pi} \int d^2x \sqrt{-g} \int d^2x' \sqrt{-g'}\, R(x)\,\Box^{-1}(x,x')\, R(x'),$$

where $R$ is the scalar curvature and $\Box^{-1}$ denotes the Green’s function of the d’Alembertian. The key step is the localization of this nonlocal structure using an auxiliary scalar field $\phi$ obeying $\Box \phi = -R$, allowing the action to be expressed as

$$S^{(2)}_{\rm anom}[g; \phi] = \frac{Q^2}{16\pi} \int d^2x \sqrt{-g}\, \left( g^{ab}\nabla_a \phi \nabla_b \phi - 2R\phi \right).$$

In four dimensions, the structure is richer due to the two principal curvature invariants entering the anomaly: the Euler (Gauss–Bonnet) density $E$ and the Weyl tensor squared $F$. The local auxiliary field formulation employs two scalar fields $\phi$ and $\psi$, schematically: $$S^{(4)}_{\rm anom}[g; \phi, \psi] = b' S^{(E)}[g; \phi, \psi] + b S^{(F)}[g; \phi, \psi],$$ where the equations of motion governing these fields are

$$\Delta_4 \phi = \tfrac{1}{2} (E - \tfrac{2}{3}\Box R), \quad \Delta_4 \psi = F,$$

with $\Delta_4$ the fourth-order Paneitz operator. The resulting stress–energy tensor, constructed by varying this action with respect to the metric, captures the trace anomaly and encodes the quantum backreaction in a semiclassical gravity setting (Numajiri et al., 19 Nov 2024).

2. Boundary Conditions, Vacuum States, and Regularity

The auxiliary field approach encodes not only the quantum anomaly but also allows boundary data and global spacetime structure to determine physical vacuum states. The classical equations for the auxiliary fields admit homogeneous solution parts with free parameters (integration constants) that are not fixed by the anomaly alone. Their specification reflects the choice of quantum vacuum, as regularity requirements for the renormalized stress–energy tensor (RSET) at boundaries, horizons, or the origin determine these constants:

• In two dimensions, with proper boundary terms included (e.g., Gibbons–Hawking-like surface terms), the auxiliary field $\phi$ must satisfy $n^\nu \nabla_\nu \phi = 2K$ on the boundary $\Sigma$, where $K$ is the extrinsic curvature. This boundary constraint naturally selects the quantum vacuum state appropriate to the spacetime—Minkowski, Unruh, or Boulware—ensuring the RSET is regular where physically desired (Shen et al., 2015).
• In four dimensions, for static horizonless regular spacetimes, the regularity at the center imposes homogenous field profiles for $\psi$ (and $\phi$) that differ from the standard Schwarzschild (Boulware) behavior, thereby forcing a departure from the Boulware vacuum in favor of a vacuum with a distinct RSET profile (Numajiri et al., 19 Nov 2024). In the presence of black hole horizons, the free parameters can be adjusted to approximate the Hartle–Hawking or Unruh states, but one cannot, in general, achieve regularity of the RSET simultaneously at both the event and Cauchy horizons for spacetimes like Reissner–Nordström (Arrechea et al., 22 Nov 2024).

3. State Dependence and Anomaly-Induced Stress–Energy Tensors

The anomaly-induced stress–energy tensor (ASET), derived from the local auxiliary field action, is manifestly state-independent—a property determined solely by local geometry and topology via the anomaly structure. However, by tuning the integration constants in the homogeneous solutions for the auxiliary fields, one can approximate the physically realized quantum states. Key features of ASETs in various scenarios include:

• Regularity at boundaries or horizons occurs only for certain vacuum choices (e.g., Minkowski, Rindler, Hartle–Hawking), enforced by appropriate auxiliary field boundary values.
• In black hole geometries, imposing the Boulware vacuum as boundary data leads to divergent RSET at the horizon, and demanding regularity at both the event and Cauchy horizons is impossible—reflecting the expected inner horizon instability and supporting strong cosmic censorship (Arrechea et al., 22 Nov 2024).
• For horizonless static spacetimes, the regular RSET’s leading asymptotics differ from those of the standard Boulware vacuum due to the necessity of nonzero homogeneous auxiliary field contributions. In four dimensions, this induces a slower spatial decay of the RSET, revealing how the anomaly-induced action predicts features inaccessible in minimally coupled field theory with Boulware boundary conditions (Numajiri et al., 19 Nov 2024).

4. Impact on Black Hole Interiors and Inner Horizon Stability

Applying the anomaly-induced effective action to black hole interiors, especially charged black holes with inner (Cauchy) horizons, yields direct insight into the semiclassical stability of these regions. Numerical and analytic studies exploiting the auxiliary field formulation indicate:

• The RSET generically diverges at the inner horizon, independently of state choices accessible to the anomaly-induced method, consistent with theoretical expectations for mass inflation and instability of Cauchy horizons (Arrechea et al., 22 Nov 2024).
• For the Hartle–Hawking vacuum, the method precisely reproduces the exterior and interior RSET, agreeing with exact calculations for minimally and conformally coupled fields. It fails, however, to simultaneously encode both thermal flux (the essence of the Unruh state) and regularity at all relevant boundaries, as limited by the two-field structure and available free integration constants.

The limitation to reproduce all physical vacuum states (especially time-asymmetric ones like the Unruh state) reflects the degrees of freedom present in the localized anomaly-induced effective action, suggesting the need for generalizations involving additional auxiliary fields or modifications to the action for complete semiclassical description (Arrechea et al., 22 Nov 2024).

5. Technical Implementation and Key Mathematical Formulations

The anomaly-induced approach in practice proceeds as follows:

1. Identify the anomaly: The quantum trace anomaly is specified for the matter content and spacetime. For four-dimensional theories,

$$\langle T^\mu{}_\mu \rangle = \frac{1}{(4\pi)^2}[c F - a E + b \Box R],$$

where $F$ and $E$ are the Weyl squared and Gauss–Bonnet invariants.

1. Localize the nonlocal action: Auxiliary fields $\phi$ and $\psi$ are introduced to rewrite the nonlocal effective action in local terms. Their equations of motion are governed by higher-derivative operators (e.g., the Paneitz operator $\Delta_4$).
2. Set physical boundary conditions: The integration constants in the solutions for $\phi$ and $\psi$ are fixed by requiring regularity of the RSET (or its physically motivated properties) at horizons, asymptotic infinities, or the origin.
3. Compute the ASET: The renormalized stress–energy tensor is obtained as a functional derivative of the local action with respect to the metric, and its trace matches the specified quantum anomaly.

Typical equations include: $$\Delta_4 \phi = \frac{1}{2}(E - \tfrac{2}{3}\Box R)$$

$$T_{\mu\nu}^{\rm anom}[g; \phi, \psi] = b' E_{\mu\nu}[\phi, \psi] + b F_{\mu\nu}[\phi, \psi]$$

with explicit forms for $E_{\mu\nu}$ and $F_{\mu\nu}$ expressing dependence on the auxiliary fields and curvature.

6. Broader Implications in Semiclassical and Quantum Gravity

• Black hole backreaction and evaporation: By encoding the anomaly, the anomaly-induced effective action offers a systematic tool for analyzing semiclassical backreaction, especially in evaluating vacuum polarization and quantum energy fluxes inside and outside black holes. The breakdown of regularity and covariance at Cauchy horizons supports expectations of mass inflation and spacetime breakdown, aligning with the strong cosmic censorship conjecture (Arrechea et al., 22 Nov 2024).
• State sensitivity and quantum vacuum selection: The framework clarifies how quantum state information, absent from the pure geometric (anomaly) sector, enters via global boundary conditions on auxiliary fields. Thus, quantum gravity modifications to the vacuum structure must be encoded through such data, beyond mere local curvature invariants.
• Applicability to horizonless and exotic compact objects: The ability of anomaly-induced actions to encode the nonperturbative impact of quantum stress at the origin or in regular spacetimes has implications for the paper of hypothetical horizonless ultra-compact objects and their stability.
• Foundations for further generalizations: Extensions to this approach may involve additional scalar fields (allowing more flexible state encoding), inclusion of matter couplings, or considerations of higher-dimensional analogues and more general gravitational backgrounds.

7. Summary Table: Anomaly-Induced Effective Action Features in Semiclassical Gravity

Aspect	Two Dimensions	Four Dimensions	Phenomenological Impact
Auxiliary fields in local action	$\phi$	$\phi$, $\psi$	Encodes anomaly and vacuum state via boundary data
Boundary condition role	Selects vacua, e.g. Minkowski, Rindler	Determines regularity at horizon/origin, fixes state	Selective regularity/singularity, modifies asymptotics
Reproduces trace anomaly	Yes	Yes	Ensures stress tensor matches quantum correction
Coverage of physical vacua	Full (with boundaries)	Limited (two fields)	Full for static/horizonless, incomplete for Unruh state
Regularity at all horizons	Achievable	Not always	Supports expected instability at Cauchy horizons

The anomaly-induced effective action remains central to the understanding of quantum field theory effects in curved space, providing both technical machinery for semiclassical calculations and conceptual clarity regarding the interplay of geometry, quantum anomalies, and vacuum state selection.