Weyl Geometric Gravity Theory

• Weyl geometric gravity theory is defined by local scale invariance that unifies the metric tensor and a dynamic Weyl gauge field into a gauge-invariant framework.
• The quadratic curvature action ensures quantum consistency and drives spontaneous symmetry breaking, reducing the theory to Einstein gravity with a massive Weyl boson.
• Phenomenological implications include natural generation of mass scales, a candidate for dark matter, and inflationary predictions consistent with observational data.

Weyl Geometric Gravity Theory generalizes Riemannian geometry by gauging local scale (dilatation) symmetry, leading to a geometric framework where both the metric tensor and a Weyl gauge field (a vector field, typically denoted ωμ) define the spacetime structure. This local conformal symmetry organizes the metric and ωμ into equivalence classes under Weyl gauge transformations, fundamentally extending the conceptual and dynamical landscape of classical and quantum gravity.

1. Weyl Conformal Geometry and Gauge Structure

Weyl geometry starts from a covariantization of Riemannian geometry with respect to local dilatations. The metric and Weyl vector field together define the structure, transforming as

$$g_{\mu\nu} \rightarrow \Omega^2(x)\,g_{\mu\nu}, \quad \omega_\mu \rightarrow \omega_\mu - q\,\partial_\mu\ln\Omega(x),$$

where q sets the normalization, and Ω(x) is a smooth, positive function on spacetime. The characteristic connection satisfies

$$\tilde{\nabla}_\lambda g_{\mu\nu} = -q\,\omega_\lambda\,g_{\mu\nu},$$

ensuring Weyl gauge covariance of geometric operations. Through suitable definition of the Weyl-covariant derivative, the geometry can be made “metric” in a generalized sense, obviating century-old objections to the physical relevance of non-metricity by making the theory manifestly gauge invariant (Ghilencea, 13 Aug 2024).

2. Quadratic Action, Physical Gauge Boson, and Quantum Consistency

The fundamental action of Weyl geometric gravity is quadratic in the relevant curvature invariants: $$S = \int d^4x\,\sqrt{g}\, [ a_0 R^2 + b_0 F_{\mu\nu}F^{\mu\nu} + c_0\,C_{\mu\nu\rho\sigma}C^{\mu\nu\rho\sigma} + d_0 G ].$$ Here, $R$ is the scalar curvature of the Weyl connection, $F_{\mu\nu}$ is the field strength of ωμ, $C_{\mu\nu\rho\sigma}$ is the Weyl tensor, and $G$ is the Euler-Gauss-Bonnet topological invariant. Each term is manifestly Weyl gauge invariant. This is the only “true” gauge theory of a spacetime symmetry with a dynamical, physical gauge boson (ωμ) (Ghilencea, 13 Aug 2024).

Weyl quadratic gravity (WQG) actions are fundamental: they preclude explicit dimensionful parameters and act as a leading order of the more general Dirac-Born-Infeld/Weyl (WDBI) action. The theory incorporates a Weyl gauge invariant geometric regularization, being free from Weyl anomalies in d dimensions unless spontaneous symmetry breaking reduces it to standard Riemannian geometry (Ghilencea, 13 Aug 2024).

3. Spontaneous Breaking and Emergence of Einstein Gravity

Weyl gauge symmetry is realized at high energies but spontaneously broken at lower scales—analogous to the Higgs mechanism. The “dilaton” mode (arising from the R² sector) is absorbed by the Weyl gauge field as a Stückelberg field, rendering ωμ massive. In the Einstein (unitary) gauge, the theory reduces to the Einstein-Hilbert action plus a positive cosmological constant: $$S_{\text{eff}} = \int d^4x\,\sqrt{g}\,[ M_\text{Pl}^2\,R - 2\Lambda + \cdots],$$ with R now computed from the Levi-Civita (Riemannian) connection. The mass of the Weyl boson is set by the symmetry breaking scale and can be as high as M_\text{Pl} or substantially lower, according to the gauge coupling. The spontaneous breaking also generates the entire hierarchy of physical masses: the Planck mass, cosmological constant, and ωμ mass emerge coherently from the same geometric sector (Ghilencea, 13 Aug 2024, Oda, 2020, Yang et al., 2022).

After symmetry breaking, ωμ is a propagating Proca field. As it decouples (or in vacua where the Weyl current vanishes), ordinary Einstein gravity is recovered, with Weyl geometry flattening to Riemannian geometry (Ghilencea, 13 Aug 2024).

4. Conformal Anomaly Structure and Quantum Properties

WQG provides a unique, gauge-invariant regularization. With all fields and the renormalization process respecting Weyl gauge symmetry, no conformal anomaly arises at the quantum level as long as the dilaton/stückelberg mode remains in the spectrum. This anomaly re-emerges in the broken phase (after ωμ decouples) reinstating the known trace anomalies of standard Riemannian field theory (Ghilencea, 13 Aug 2024).

This gauge theory of dilatations—unlike generic “conformal gravity” which lacks a propagating Weyl gauge boson—offers a consistent mechanism for quantum regularization by replacing the usual subtraction scale with the dynamical Weyl geometric invariant (e.g., by using the operator $\hat R$ as a geometric scale) (Ghilencea, 13 Aug 2024).

5. Connections to Conformal Gravity and Limiting Cases

In the vanishing Weyl current (Fμν=0) limit, the theory reduces to conformal gravity: the Weyl gauge field is pure gauge and geometry collapses to integrable Weyl geometry (conformal to Riemannian). The quadratic Weyl geometric action then contains only the C² and possibly R² terms, but no dynamical dilatational gauge boson is present (Ghilencea, 13 Aug 2024). The theory thus smoothly interpolates between quantum-consistent gauge theory of gravity and “pure” conformal gravity.

6. Matter Embedding, Standard Model, and Inflation

The Standard Model (SM), classically scale invariant for vanishing Higgs mass, embeds naturally in Weyl geometry. Kinetic terms and field strengths are promoted to Weyl-covariant forms. The Higgs sector is non-minimally coupled to the Weyl curvature, and covariant derivatives incorporate ωμ. The Higgs field then generically mixes with the Weyl gauge boson (including possible kinetic mixing with hypercharge), linking electroweak and gravitational symmetry breaking scales (Ghilencea, 13 Aug 2024).

Starobinsky-Higgs inflation is a direct prediction: upon linearization of R², the scalaron inherits a geometric interpretation, and mixed Higgs-dilaton inflation yields predictions consistent with observation (e.g., tensor-to-scalar ratio r in the range 0.0023–0.0030 for 60 N_e) (Ghilencea, 13 Aug 2024).

7. Physical and Phenomenological Implications

The geometric approach fixes all mass scales (including the cosmological constant) and offers candidate dark matter (via ωμ as a heavy Proca field, depending on mixing and coupling). The theory’s predictions are compatible with Einstein gravity in the broken phase, while allowing novel phenomena—such as modifications of dark sector dynamics or unique inflationary signatures—when residual Weyl geometric effects are present (Ghilencea, 13 Aug 2024, Yang et al., 2022). Physical tests of the possible observability of Weyl gauge bosons or small departures from Einsteinian behavior at cosmological or high-energy scales are a subject of ongoing research.

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Summary Table: Core Features of Weyl Geometric Gravity Theory

Aspect	Gauge Symmetry	Physical Implication
Geometry	Local dilatations (Weyl)	Metric + ωμ, scale-covariant connection
Gravity Action	Quadratic in curvature	Fundamental gauge theory; massless at high energies
Symmetry Breaking	Stückelberg/dilaton	Einstein gravity + massive ωμ + Λ for Λ > 0
Quantum Consistency	Gauge-inv. regularization	No conformal anomaly before symmetry breaking
SM Coupling	Minimal, Higgs mixing	Predicts Starobinsky–like inflation and unique couplings
Limiting Case	Fμν = 0 (pure gauge)	Reduces to conformal gravity without gauge boson
Mass Scales	Geometric origin	Planck scale, dark matter (ωμ), cosmological constant

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Weyl geometric gravity proposes a unified, quantum-consistent description of gravity and its couplings. Its geometric gauge symmetry, anomaly structure, and natural embedding of the SM and inflation make it a prime candidate for bridging quantum field theory and gravitation, while providing precise predictions for cosmology and potentially accessible phenomenology (Ghilencea, 13 Aug 2024).