Quadratic Gravity Overview

• Quadratic gravity is a higher-derivative extension of general relativity that incorporates curvature-squared invariants like R² and Weyl², ensuring perturbative renormalizability.
• Its improved ultraviolet behavior, marked by a graviton propagator scaling as 1/q⁴, introduces extra modes including a massive spin-2 ghost, raising challenges about unitarity.
• Multiple formulations—from metric and BRST approaches to affine and bimetric methods—offer insights into its nonlinear dynamics, cosmological applications, and exact solution structures.

Searching arXiv for recent and foundational papers on quadratic gravity to support the article. arxiv.search(query="quadratic gravity review renormalizable asymptotically free ghost Weyl squared R^2", max_results=10) Quadratic gravity is a higher-derivative extension of general relativity in which the gravitational action contains local terms quadratic in curvature in addition to, or in some formulations instead of, the Einstein–Hilbert term. In four dimensions, modulo the Euler–Poincaré or Gauss–Bonnet density, the independent curvature-squared structures can be taken as $R^2$ and either $R_{\mu\nu}R^{\mu\nu}$ or $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$. The subject is defined by an unusual combination of properties: perturbative renormalizability, dimensionless ultraviolet couplings, and, in several commonly used formulations, asymptotic freedom, together with a longstanding controversy over the interpretation of the extra massive spin-2 mode that appears in perturbation theory (Donoghue et al., 2021, Arık et al., 2024, Holdom et al., 2016).

1. Definition and invariant content

A standard four-dimensional metric action is

$$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$

while an equivalent parameterization used elsewhere is

$$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$

In both cases the defining feature is the presence of curvature-squared operators with dimensionless couplings (Donoghue et al., 2021, Holdom et al., 2016).

In four dimensions, the irreducible curvature decomposition implies that only two independent quadratic curvature structures contribute dynamically once total derivatives are removed. In the exterior-calculus formulation, the most general four-dimensional quadratic-curvature Lagrangian considered is

$$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$

which can be rewritten, up to the Euler–Poincaré term, as

$$L_{QC}=\frac12\Big(\alpha+\frac{\beta}{3}\Big)R^2*1+\frac{\beta}{2}C_{ab}*C^{ab}.$$

This makes explicit the equivalence between $R^2+R_{\mu\nu}R^{\mu\nu}$ and $R^2+C^2$ bases in four dimensions (Arık et al., 2024).

The topic also includes several systematic extensions. One class couples all independent quadratic invariants to a scalar field,

$$f_1(\phi)R^2+f_2(\phi)R_{\mu\nu}R^{\mu\nu}+f_3(\phi)R_{\mu\nu\rho\sigma}R^{\mu\nu\rho\sigma}+f_4(\phi)R_{\mu\nu\rho\sigma}{}^{\!*}R^{\mu\nu\rho\sigma},$$

and treats quadratic gravity as an effective field theory in which scalar–Gauss–Bonnet and dynamical Chern–Simons sectors arise as special cases (Finch et al., 2016). Another class studies $R_{\mu\nu}R^{\mu\nu}$0-dimensional actions of the form

$R_{\mu\nu}R^{\mu\nu}$1

where the Gauss–Bonnet combination is dynamical for $R_{\mu\nu}R^{\mu\nu}$2 and topological for $R_{\mu\nu}R^{\mu\nu}$3 (Ilkhchi et al., 16 Jul 2025).

2. Perturbative ultraviolet structure and linearized spectrum

The central perturbative fact is that curvature-squared terms generate fourth-order equations and a graviton propagator that behaves as

$R_{\mu\nu}R^{\mu\nu}$4

at large momentum. This improves ultraviolet convergence relative to Einstein gravity, whose propagator behaves as $R_{\mu\nu}R^{\mu\nu}$5, and is the power-counting reason quadratic gravity is perturbatively renormalizable (Donoghue et al., 2021). In the formulation emphasized by Holdom and Ren, quadratic gravity was already known more than three decades ago as a perturbative, renormalizable, asymptotically free theory of quantum gravity (Holdom et al., 2016).

The linearized propagator around flat space exhibits extra poles. In the spin-2 sector one finds

$R_{\mu\nu}R^{\mu\nu}$6

so the massive pole appears with opposite-sign residue (Donoghue et al., 2021). The standard linearized particle content is therefore the healthy massless spin-2 graviton, a massive scalar associated with the $R_{\mu\nu}R^{\mu\nu}$7 sector, and a massive spin-2 mode associated with the Weyl-squared sector. In the differential-form review, this is summarized as eight degrees of freedom: healthy massless spin-2, a massive scalar degree of freedom, and a massive spin-2 ghost (Arık et al., 2024).

The ultraviolet running depends on conventions and sign choices. In one description the quadratic couplings $R_{\mu\nu}R^{\mu\nu}$8 are asymptotically free, so $R_{\mu\nu}R^{\mu\nu}$9 as $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$0 and the theory becomes weakly coupled in the ultraviolet (Holdom et al., 2016). In another review, with the sign choice adopted to avoid a high-mass scalar tachyon, the Weyl-squared coupling $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$1 is asymptotically free whereas the $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$2 coupling $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$3 is not (Donoghue et al., 2021). This difference is explicitly tied to sign conventions and the treatment of the scalar sector, not to a disagreement over the basic renormalizability result.

The same perturbative structure also implies that the usual Källén–Lehmann assumptions do not straightforwardly apply. Since the spin-2 propagator falls faster than $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$4, some combination of ordinary positivity, causality, and standard analyticity must fail if the theory is to remain consistent as a quantum field theory (Donoghue et al., 2021).

3. Geometric, canonical, and affine formulations

Quadratic gravity admits several inequivalent formulations. In the purely metric approach, the equations are fourth order and can be organized geometrically. Using differential forms on a pseudo-Riemannian manifold, the metric field equations for the general four-dimensional model can be written compactly as

$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$5

where $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$6 are the Bach 1-forms associated with $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$7, and $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$8 are the analogous divergence-free 1-forms associated with $C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}$9. In explicit form,

$$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$0

This formulation makes the split between Weyl, traceless-Ricci, and scalar-curvature sectors transparent and extends naturally to $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$1 models with $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$2 (Arık et al., 2024).

A very different route is the manifestly covariant BRST–canonical quantization developed in harmonic gauge. There the classical action is written as Einstein–Hilbert plus $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$3 plus $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$4, and auxiliary fields are introduced so that the higher-derivative system can be quantized in a second-order form. In that framework, the physical content is identified explicitly as a massless graviton, a massive scalar, and a massive spin-2 ghost, while the equal-time commutators between the metric and all of its time derivatives are found to vanish identically (Oda, 14 May 2025). This result shifts the canonical dynamical content away from the metric alone and toward a larger BRST field complex.

First-order or metric–affine formulations are not generically equivalent to second-order metric formulations once curvature-squared terms are present. In the most general torsionless, Weyl-invariant, quadratic-in-curvature action in four dimensions, the independent connection remains dynamical, there are 12 independent quadratic invariants, and there are no propagators falling faster than $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$5. In that approach, the interaction between external sources is conveyed mainly by the three-index connection field, and the theory is in a conformal invariant phase until Weyl invariance is broken by coupling to matter, at which point an Einstein–Hilbert term can be generated by quantum corrections (Alvarez et al., 2017).

A related recent development is bimetric-affine quadratic gravity, where two dynamical metrics are supplemented by independent connections and an additional $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$6 term. For a wide range of interaction parameters, the resulting theory is free of ghosts and retains the standard bimetric spectrum of one massless and one massive spin-2 field (Gialamas et al., 2023).

4. Ghost problem and competing interpretations

The massive spin-2 mode is the central controversy of quadratic gravity. In the standard perturbative reading, the minus sign in

$$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$7

signals either a negative-norm state or a negative-energy excitation, and therefore nonunitarity or catastrophic instability (Donoghue et al., 2021). The canonical BRST analysis makes this diagnosis explicit: it identifies a massive scalar and a massive spin-2 ghost as physical modes and concludes that the theory, in that operator construction, is not unitary, while ghost confinement is presented only as a possibility for recovering the unitarity of the physical S-matrix (Oda, 14 May 2025).

A different interpretation, developed by Donoghue and Menezes, treats the heavy spin-2 excitation as an unstable Lee–Wick-type resonance rather than an asymptotic state. In that picture the dressed propagator near the resonance takes the form

$$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$8

with the sign structure characteristic of what they call a “Merlin mode.” The resonance propagates with a reversed causal arrow at microscopic scales, but the imaginary part of the full propagator has the sign required for unitarity, and the $$S=\int d^4x\,\sqrt{-g}\,\Bigg[ -\frac{1}{2K^2}R+\frac{1}{4\xi^2}C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}+\frac{1}{6f_0^2}R^2\Bigg],$$9 partial-wave amplitude satisfies the unitarity relation explicitly (Donoghue et al., 2021, Donoghue et al., 2018).

A third line of argument is nonperturbative rather than contour-theoretic. Holdom and Ren consider the regime $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$0, in which the perturbative poles fall into a strongly interacting domain. By analogy with QCD, they conjecture that the full graviton propagator

$$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$1

is reshaped nonperturbatively so that the perturbative ghost pole disappears and a healthy $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$2 graviton propagator emerges in the infrared. In that scenario the effective Planck mass is generated dynamically,

$$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$3

and general relativity appears as the infrared effective theory. The proposal remains conjectural: the exact form of $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$4 is not derived, and the argument is supported mainly by the analogy with gluon propagators and by evidence for Gribov copies in gravity (Holdom et al., 2016).

These positions are not mutually reducible. The literature therefore treats the ghost problem as unresolved: perturbative canonical analyses, Lee–Wick-type interpretations, and nonperturbative infrared-emergence scenarios each preserve different parts of the standard quantum-field-theoretic package.

5. Exact solutions and nonlinear dynamics

Quadratic gravity has a broad exact-solution sector. For the action

$$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$5

all vacuum general-relativistic solutions remain exact vacuum solutions of quadratic gravity, because if $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$6 then the quadratic tensors $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$7 and $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$8 vanish (Medeiros et al., 2024). This simple observation is central to both black-hole and cosmological analyses.

An important exact family is the AdS-wave sector. For the most general $$S_{\mathrm{QG}}=\int d^4x\,\sqrt{-g}\left(\frac{1}{2}M^2R-\frac{1}{2f_2^2}C_{\mu\nu\alpha\beta}C^{\mu\nu\alpha\beta}+\frac{1}{3f_0^2}R^2\right).$$9-dimensional quadratic action with cosmological constant, the Kerr–Schild-type AdS-wave metric solves the full nonlinear field equations, and the same profile function also solves the linearized equations, including the linearized equations of critical gravity. A subset of these solutions has logarithmic behavior and changes the asymptotic structure of anti-de Sitter space (Gullu et al., 2011).

Nonlinear evolution has also become accessible numerically. In spherical symmetry, a stable nonlinear initial-value formulation has been constructed by combining harmonic gauge, the Cartoon method, and an order reduction to first order in time via auxiliary variables. Randomly perturbed flat-space and Schwarzschild initial data exhibit convergent, numerically stable evolution, which provides proof-of-principle evidence for a well-posed nonlinear formulation of leading-order gravitational effective-field-theory quadratic gravity in that sector (Held et al., 2021).

The $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$0-dimensional extension goes further. A numerically stable system of evolution equations for nonlinear quadratic gravity has been implemented, and it recovers a known linear instability while also providing evidence for a physically stable Ricci-flat subsector. In particular, Teukolsky-wave perturbations of Schwarzschild and a full binary inspiral evolved up to merger remain Ricci flat throughout evolution, suggesting that, at least in vacuum, classical quadratic gravity can mimic general relativity even in the fully nonlinear strong-gravity regime (Held et al., 2023).

6. Cosmology, matter couplings, phenomenology, and current directions

Cosmological applications are diverse and often sensitive to which quadratic sector dominates. In the anisotropic Bianchi V model with a tilted fluid, numerical backward evolution shows that, in quadratic gravity, universes with higher and smaller matter densities fall into Kasner or isotropic singularity attractors to the past, respectively. The Kasner attractor has zero vorticity in both GR and quadratic gravity, whereas the isotropic singularity attractor in quadratic gravity may have divergent vorticity. The same study reports an initial kinematic singularity substance that approaches a perfect fluid source under time reversal, a behavior not obtained there in GR (Medeiros et al., 2024).

Matter-coupled and phenomenological sectors are equally active. In scalar-coupled quadratic gravity, slowly rotating axisymmetric solutions modify circular orbits, geodetic precession, and Lense–Thirring precession. The resulting Kepler law and gravitomagnetic effects reduce to their general-relativistic forms when the modified couplings are set to zero, while the corrections are largest in the strong-field regime and fall with radius (Finch et al., 2016). A different matter-driven scenario, studied in an asymptotic-safety FRG truncation with the Higgs field, proposes a UV fixed point with $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$1, $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$2, $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$3, and $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$4, together with an IR fixed point with $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$5, $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$6, and $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$7. In that picture, gravity is purely quadratic in the ultraviolet, Einstein gravity emerges at low energy, and the Fermi scale is generated through tachyonic condensation and Higgs symmetry breaking (Mehrafarin, 14 Apr 2025).

Thermodynamic reformulations have also been developed. In a $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$8-dimensional FLRW universe, quadratic gravity admits a generalized Misner–Sharp energy, an apparent-horizon equation of state $$L_{QC}=\frac{\alpha}{2}R^2*1+\frac{\beta}{2}R_a *R^a,$$9, and a Wald entropy modified by the quadratic parameter $$L_{QC}=\frac12\Big(\alpha+\frac{\beta}{3}\Big)R^2*1+\frac{\beta}{2}C_{ab}*C^{ab}.$$0. The same framework yields critical temperature, critical radius, specific heat, enthalpy, and Gibbs free energy, and the analysis finds that quadratic terms change the stability conditions and can lead to new thermodynamic behaviors compared to general relativity (Ilkhchi et al., 16 Jul 2025).

Taken together, these developments portray quadratic gravity not as a single settled theory but as a family of renormalizable higher-curvature frameworks with several internally consistent realizations. The ultraviolet behavior, the role of the Einstein–Hilbert term, the status of the massive spin-2 excitation, and the infrared interpretation remain formulation-dependent. This suggests that the phrase “quadratic gravity” now denotes a research program unified by curvature-squared dynamics and ultraviolet control, but divided by competing answers to the questions of unitarity, causality, and emergence of general relativity.