[Paper Review] Weyl-invariant Higher Curvature Gravity Theories in n Dimensions and Mass Generation by Symmetry Breaking

Suat Dengiz|arXiv (Cornell University)|Sep 18, 2014

Black Holes and Theoretical Physics91 references4 citations

TL;DR

This paper constructs Weyl-invariant higher curvature gravity theories in $n$ dimensions, including New Massive Gravity and Einstein-Gauss-Bonnet gravity, by eliminating dimensionful parameters. Spontaneous symmetry breaking via the Coleman-Weinberg mechanism generates mass scales radiatively, leading to unitary, ghost-free spectra: massless gravitons, massive or massless vectors, and scalars, with the $n$-dimensional Einstein-Gauss-Bonnet theory being the only unitary Weyl-invariant quadratic curvature theory.

ABSTRACT

Weyl-invariant extensions of three-dimensional New Massive Gravity, generic n-dimensional Quadratic Curvature Gravity theories and three-dimensional Born-Infeld gravity theory are analyzed in details. As required by Weyl-invariance, the actions of these gauge theories do not contain any dimensionful parameter; therefore the local symmetry is spontaneously broken in (Anti) de Sitter vacua in analogy with the Standard Model Higgs mechanism. About the flat vacuum, symmetry breaking mechanism is more complicated: The conformal symmetry is radiatively broken (at two loop level in 3-dimensions and at one-loop level in 4-dimensions) a la Coleman-Weinberg mechanism and hence the dimensionful parameters come from dimensional transmutation in the quantum field theory. In the broken phases, save for New Massive Gravity, the theories generically propagate with a unitary (tachyon and ghost-free) massless tensor, massive (or massless) vector and massless scalar particles for the particular intervals of the dimensionless parameters. For New Massive Gravity, there is a massive Fierz-Pauli-type graviton. Finally, it is shown that n-dimensional Weyl-invariant Einstein-Gauss-Bonnet theory is the only unitary higher dimensional Weyl-invariant Quadratic Curvature Gravity theory.

Motivation & Objective

• To construct Weyl-invariant extensions of higher curvature gravity in $n$ dimensions without dimensionful parameters.
• To analyze spontaneous symmetry breaking in (A)dS vacua and flat spacetime via Higgs and Coleman-Weinberg mechanisms.
• To determine the unitarity and particle content of the broken phase, especially the graviton spectrum.
• To identify the unique Weyl-invariant quadratic curvature gravity theory that is unitary in $n$ dimensions.

Proposed method

• Imposes Weyl invariance on $n$-dimensional quadratic curvature gravity actions, ensuring no explicit mass scales.
• Performs perturbative expansion around (A)dS and flat vacua using scale-invariant gauge-fixing and field redefinitions.
• Applies the Coleman-Weinberg mechanism to generate dimensionful parameters radiatively at one- or two-loop level.
• Analyzes the quadratic action to extract physical degrees of freedom and check for tachyons or ghosts.
• Uses the Gauss-Bonnet combination and curvature invariants to classify unitary theories in $n$ dimensions.
• Derives effective Lagrangians for metric, vector, and scalar fluctuations up to quadratic order in perturbations.

Experimental results

Research questions

• RQ1 Which $n$-dimensional Weyl-invariant higher curvature gravity theories are unitary and free of ghosts and tachyons?
• RQ2 How does spontaneous symmetry breaking generate mass scales in Weyl-invariant gravity, particularly in flat spacetime?
• RQ3 What is the particle spectrum (graviton, vector, scalar) in the broken phase of New Massive Gravity and Born-Infeld gravity?
• RQ4 Why is the $n$-dimensional Einstein-Gauss-Bonnet theory the only unitary Weyl-invariant quadratic curvature gravity theory?
• RQ5 How do the Coleman-Weinberg and Higgs mechanisms differ in their role in generating mass in $n=3$ and $n=4$ dimensions?

Key findings

• In (A)dS vacua, Weyl symmetry is spontaneously broken via the Higgs mechanism, generating mass scales without explicit parameters.
• In flat spacetime, the Coleman-Weinberg mechanism generates a mass scale radiatively: at one-loop in 4D and two-loop in 3D.
• For generic Weyl-invariant quadratic curvature gravity, the broken phase propagates a unitary, ghost-free massless tensor, massive or massless vector, and massless scalar.
• New Massive Gravity in the broken phase features a massive Fierz-Pauli-type graviton, consistent with unitarity.
• The $n$-dimensional Einstein-Gauss-Bonnet theory is the only Weyl-invariant quadratic curvature gravity theory that is unitary.
• The effective Lagrangian at quadratic order shows explicit mixing between metric, vector, and scalar fluctuations, with mass terms generated via symmetry breaking.