Weyl Quadratic Gravity

• Weyl Quadratic Gravity is a curvature-squared gravitational theory that employs the Weyl tensor to introduce extra spin-2, vector, and scalar degrees of freedom beyond General Relativity.
• It encompasses both the Einstein–Weyl truncation and fully Weyl-gauge invariant formulations, emphasizing dimensionless couplings and dual descriptions via metric and gauge-theoretic approaches.
• The theory provides insights into black-hole solutions, cosmological inflation, and potential quantum resolutions to ghost instabilities, making it relevant for high-energy gravitational research.

Weyl quadratic gravity designates a family of curvature-squared gravitational theories in which the Weyl tensor, or the Weyl-geometric curvature built from a dilatational gauge connection, provides the defining quadratic sector. In recent usage, the label covers both the Einstein–Weyl truncation

$${\cal S}=\frac{M_{\rm pl}^2}{2}\int d^4x\,\sqrt{-g}\left(R-\alpha\, C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma}\right),$$

and fully Weyl-gauge-invariant actions in Weyl conformal geometry such as

$$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$

Across these formulations, the recurrent themes are dimensionless couplings in four dimensions, extra spin-2/vector/scalar sectors beyond General Relativity, and the claim that Einstein gravity can emerge as a broken or infrared phase of a fundamentally quadratic theory (Felice et al., 2023, Ghilencea, 2018, Condeescu et al., 2023).

1. Action principles and invariant structures

A basic four-dimensional identity underlying the subject is

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$

with ${\cal G}$ the Gauss–Bonnet term, topological in four dimensions. In the Einstein–Weyl sector this identity isolates the effect of the Weyl-squared correction relative to the Einstein–Hilbert term, and on Minkowski background it introduces the mass scale

$m_W^2=\frac{1}{2\alpha}.$

In the Weyl-gauge-invariant formulation, the quadratic basis is enlarged to $\hat R^2$, $\hat C^2$, $\hat F^2$, and the Weyl-geometric Euler density $\hat G$, all built from Weyl-covariant curvature tensors and the Weyl gauge field strength (Felice et al., 2023, Ghilencea, 14 Aug 2025).

In first-order quadratic gravity, the operator basis is larger than in purely metric formulations. The most general parity-even quadratic action built from the curvature of an independent torsionless connection contains twelve independent Weyl scalar operators: five from two full Riemann tensors, six from the two inequivalent Ricci-type traces, and one scalar-curvature-squared term,

$$S = \int d^n x\, \sqrt{g}\, \sum_{I=1}^{12} g_I\, \mathcal O_I.$$

The couplings satisfy $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$0, so in $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$1 they are dimensionless. The same work emphasizes that the purely quadratic sector is the conformal invariant phase: it contains no Einstein–Hilbert term and no cosmological constant term, and counterterms remain within the same quadratic operator basis (Alvarez et al., 2017).

Dimensional specializations sharpen the notion further. In three-dimensional Riemann–Cartan–Weyl spacetime, the locally Weyl-covariant quadratic curvature theory reduces consistently to the original quadratic model only on the coupling locus

$$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$2

which includes New Massive Gravity, $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$3, $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$4, $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$5 (Dereli et al., 2019).

2. Geometric formulations and gauge-theory interpretation

A central bifurcation in the literature is between metric quadratic gravity and metric-affine or first-order quadratic gravity. For the Einstein–Hilbert action, varying an independent connection forces it to become Levi-Civita on shell, but for quadratic curvature gravity this equivalence fails: the connection is not forced to be Levi-Civita, even on shell, and becomes a genuinely dynamical field. In that setting, the interaction with external sources is conveyed mainly by the three-index connection field, the $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$6 interaction is typically contact-like, the $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$7 mixing is ultralocal, and the $$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$8 interaction can mediate a long-range potential (Alvarez et al., 2017).

The Weyl-geometric version gauges local dilatations. Its basic transformation law is

$$S_{\bf w}=\int d^4x\,\sqrt{g}\left\{ \frac{1}{4!\,\xi^2}\hat R^2 -\frac{1}{\eta^2}\hat C_{\mu\nu\rho\sigma}^2 -\frac{1}{4\alpha^2}\hat F_{\mu\nu}^2 +\hat G \right\}.$$9

with Weyl-covariant derivative

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$0

In the Weyl gauge-covariant metric formulation one has

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$1

while the corresponding Weyl connection is

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$2

Its scalar curvature satisfies

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$3

and the Weyl tensor obeys

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$4

This identity explains why the $$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$5 sector can be written in the familiar Riemannian form even though the underlying geometry is Weyl-geometric (Ghilencea, 14 Aug 2025, Condeescu et al., 2023).

Recent gauge-theoretic work presents three equivalent realizations of the same Weyl quadratic action: a torsion-free non-metric formulation, a Weyl gauge-covariant metric formulation, and a metric formulation with vectorial torsion. The non-metric and torsionful connections are related by a projective transformation, and the equivalence is interpreted as a duality between vectorial non-metricity and vectorial torsion. The same literature argues that gauging the full conformal group does not generate a true gauge theory with a physical propagating gauge boson, whereas the Weyl group does, with the Weyl gauge field as the gauge boson of local dilatations (Condeescu et al., 2023, Condeescu et al., 2024).

3. Degrees of freedom, residual symmetries, and breaking to Einstein gravity

On black-hole backgrounds, the dynamical content of the Einstein–Weyl theory is more intricate than on homogeneous backgrounds, but it does not collapse. For static and spherically symmetric Schwarzschild background, the odd-parity sector has three propagating dynamical degrees of freedom and the even-parity sector has four, for a total of seven. Because this matches the count on Minkowski and isotropic cosmological backgrounds, the theory does not suffer from a strong-coupling problem associated with missing kinetic terms. The odd sector always contains at least one ghost for $$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$6, but all three odd-parity propagation speeds are luminal: $$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$7 The even sector also contains at least one ghost mode, irrespective of the sign of $$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$8 (Felice et al., 2023).

Quadratic gravity also exhibits a weaker symmetry, restricted Weyl symmetry, defined by a Weyl factor satisfying

$$C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} =2R_{\mu\nu}R^{\mu\nu}-\frac{2}{3}R^2+{\cal G},$$9

Around Minkowski space this residual symmetry removes one component of the scalar dipole field ${\cal G}$0, leaving a single massless scalar state. On static asymptotically flat backgrounds, however, the zero-energy theorem and the falloff conditions force ${\cal G}$1 to be constant, so the restricted Weyl symmetry reduces to a global scale transformation and does not remove local dynamical degrees of freedom. The same conclusion holds for Euclidean backgrounds under the corresponding asymptotic assumptions (Kamimura et al., 2021).

A different branch of the subject studies spontaneous or gauge-fixed breaking of Weyl symmetry. In gauged Weyl quadratic gravity built from ${\cal G}$2 and ${\cal G}$3, the ${\cal G}$4 term contains an extra scalar mode that can be linearized with an auxiliary scalar ${\cal G}$5. After the Einstein-frame transformation

${\cal G}$6

and the shifted Weyl field

${\cal G}$7

the action contains the Einstein–Hilbert term, a positive cosmological constant, and a Proca mass term. The Stueckelberg mechanism is explicit: the scalar mode hidden in ${\cal G}$8 is absorbed by the Weyl gauge field, whose mass is

${\cal G}$9

Below $m_W^2=\frac{1}{2\alpha}.$0, the field decouples and Weyl geometry becomes Riemannian (Ghilencea, 2018).

A BRST formulation reaches a closely related result by imposing the Weyl gauge condition $m_W^2=\frac{1}{2\alpha}.$1. With a specific gauge choice, the gauge-fixed quadratic action becomes

$m_W^2=\frac{1}{2\alpha}.$2

with

$m_W^2=\frac{1}{2\alpha}.$3

This formulation interprets Weyl conformal gravity as quantum mechanically equivalent to Einstein gravity plus a massive Weyl gauge field (Oda, 2020).

4. Exact solutions and the black-hole sector

In four-dimensional quadratic gravity containing both $m_W^2=\frac{1}{2\alpha}.$4 and $m_W^2=\frac{1}{2\alpha}.$5, all Einstein spacetimes are vacuum solutions. Under the constant-Ricci-scalar assumption, the field equations reduce to a relation between the traceless Ricci tensor and the Bach tensor. Within that framework, vacuum solutions with traceless Ricci tensor of type N and aligned Weyl tensor of any Petrov type are necessarily Kundt spacetimes, and traceless Ricci type III with aligned Weyl tensor of Petrov type II or more special are again necessarily Kundt. A second construction starts from any Bach-flat background and generates new exact solutions by solving one nonlinear partial differential equation for a conformal factor; geometries conformal to Kundt are either Kundt or Robinson–Trautman (Pravda et al., 2016).

The static, spherically symmetric sector of Einstein–Weyl gravity has become a focal point because the Schwarzschild metric remains an exact solution for any $m_W^2=\frac{1}{2\alpha}.$6: $m_W^2=\frac{1}{2\alpha}.$7 In addition, a non-Schwarzschild branch exists only in a restricted coupling range. Writing

$m_W^2=\frac{1}{2\alpha}.$8

the non-Schwarzschild black hole appears for

$m_W^2=\frac{1}{2\alpha}.$9

with the Schwarzschild and non-Schwarzschild branches intersecting at $\hat R^2$0; for $\hat R^2$1 its mass becomes negative. A complementary phase-diagram description organizes static, spherically symmetric solutions by ADM mass $\hat R^2$2 and Yukawa charge $\hat R^2$3, with asymptotic metric

$\hat R^2$4

The diagram contains the Schwarzschild line $\hat R^2$5, a non-Schwarzschild black-hole curve, and regions of naked singularities and wormholes (Felice et al., 2023, Silveravalle et al., 2022).

Linear stability distinguishes sharply between coupling regimes. In the WKB-accessible regime $\hat R^2$6, even-parity perturbations of Schwarzschild develop both radial and angular Laplacian instabilities. For large radial and angular momentum, the instability timescales are much shorter than the horizon light-crossing time: $\hat R^2$7 and, in the angular sector,

$\hat R^2$8

By contrast, for $\hat R^2$9 the WKB approximation fails, and the stability of both Schwarzschild and non-Schwarzschild branches is left open (Felice et al., 2023).

Weyl conformal geometry supports a separate black-hole program in the scalar–vector–tensor system obtained from $\hat C^2$0 plus the Weyl field strength. Numerical black-hole solutions have been obtained for Weyl vectors with purely radial, purely temporal, and mixed temporal-radial components, and an exact solution exists for a purely radial Weyl vector. The thermodynamic quantities—horizon temperature, specific heat, entropy, and evaporation time—were analyzed within that framework (Yang et al., 2022).

5. Cosmology, inflation, and phenomenological bounds

For the cosmological theory

$\hat C^2$1

the homogeneous FLRW background is unchanged because $\hat C^2$2 vanishes on FLRW. The background equations therefore reduce to the standard $\hat C^2$3CDM form. The perturbation sector is not GR-like: it contains four propagating tensorial degrees of freedom, two propagating vector degrees of freedom, and two propagating scalar degrees of freedom, $\hat C^2$4 and $\hat C^2$5. During matter domination, classical stability in the vector and tensor sectors requires $\hat C^2$6, while recovery of structure growth in the scalar sector leads to the bound

$\hat C^2$7

The same analysis notes that the extra vector mode is a ghost for $\hat C^2$8, classically stable but requiring a prescription such as the fakeon approach if one demands unitarity (Sutton et al., 21 Apr 2025).

A different cosmological realization arises from quadratic gravity in generalized Weyl geometry with linear vector distortion. Ghost freedom selects the one-parameter family

$\hat C^2$9

which generalizes ordinary Weyl geometry by allowing a torsion trace. The resulting four-dimensional vector-tensor theory propagates five degrees of freedom—two tensors, two transverse vectors, and one scalar longitudinal mode—and admits de Sitter attractors, vector-driven dark energy, bounded Hubble evolution, bounces, and self-tuning only in special non-flat cases. The perturbative stability conditions constrain the tensor, vector, and scalar sectors separately (Jimenez et al., 2016).

When Weyl gauge symmetry is coupled to matter through a non-minimal scalar coupling, the post-breaking scalar potential is Higgs-like at small field values and inflationary at large field values. In the Weyl versus Palatini comparison, both theories have a small tensor-to-scalar ratio because of their $\hat F^2$0 term,

$\hat F^2$1

with larger $\hat F^2$2 in the Palatini case. For fixed spectral index $\hat F^2$3, reducing $\hat F^2$4 increases $\hat F^2$5, while in the Weyl theory $\hat F^2$6 is bounded above by the Starobinsky value; for $\hat F^2$7, the Weyl case yields an $\hat F^2$8 dependence similar to Starobinsky inflation (Ghilencea, 2020). In the matter-coupled Stueckelberg construction, the same symmetry-breaking mechanism generates a Higgs mass

$\hat F^2$9

so a light Higgs requires $\hat G$0; the quantum stability of this hierarchy is explicitly left open (Ghilencea, 2018).

6. Renormalizability, unitarity, and current controversies

Quadratic gravity has long been motivated by power-counting renormalizability, but the status of its extra spin-2 sector remains contested. In the Hamiltonian analysis of curvature-squared theories, Weyl gravity is singled out because the number of local constraints equals the number of unstable directions in phase space, suggesting that its conformal symmetry could, in principle, eliminate all unstable directions in the full nonlinear theory. The same work finds that other quadratic theories are unstable and that adding a cosmological constant directly to Weyl gravity is pathological, because consistency of the conformal constraint leads to

$\hat G$1

hence $\hat G$2 (Klusoň et al., 2013).

The first-order program offers a different route. In the most general first-order quadratic theory there are no propagators falling faster than $\hat G$3, and the independent connection carries the main propagating content, which is one reason the framework has been proposed as a candidate for a unitary and renormalizable theory. Yet a specific study of first-order Weyl gravity concludes that using only the Weyl connection does not remove the quartic graviton behavior: the ultraviolet propagator remains effectively quartic, so the unitarity problem of the restricted Weyl-only model persists (Alvarez et al., 2017, Alvarez et al., 2016).

A recent proposal reinterprets the extra spin-2 sector of quadratic gravity with positive Weyl-squared coefficient as a dual inverted harmonic oscillator rather than a ghost. In that construction, the Wightman spectrum condition excludes the spacelike spin-2 pole from the physical Källén–Lehmann spectrum, the spectral density vanishes,

$\hat G$4

and the propagator is fixed to principal value form,

$\hat G$5

The claim is that the mode is not an asymptotic state, contributes only virtually, satisfies the optical theorem, and preserves renormalizability (Kumar et al., 21 Apr 2026). This suggests a nonstandard resolution of the traditional ghost problem, but it should be read alongside the older Hamiltonian and perturbative diagnoses rather than as a settled consensus.

Weyl-gauge-covariant extensions push the subject further. The Weyl–Dirac–Born–Infeld construction is an exact determinant action in arbitrary dimension whose leading-order expansion reproduces the Weyl quadratic gravity action, and whose broken phase recovers Einstein–Hilbert gravity and the Standard Model in a Riemannian geometry. Because the action is built to be Weyl invariant in arbitrary $\hat G$6, it is presented as free of Weyl anomaly, with $\hat G$7 acting as a geometric regulator; related review work states that the anomaly is recovered only in the broken phase after the massive Weyl boson decouples (Ghilencea, 14 Aug 2025, Ghilencea, 2024).

Taken together, these developments leave several technical questions open. The stability of black-hole branches in the regime $\hat G$8 remains unresolved because the WKB approximation fails there; the radiative stability of hierarchies such as $\hat G$9 is not established; and the relation between renormalizability, spectral positivity, and the physical interpretation of the Weyl-induced spin-2 sector remains an active point of dispute across metric, metric-affine, and Weyl-gauge formulations (Felice et al., 2023, Ghilencea, 2018, Kumar et al., 21 Apr 2026).