Spontaneous breaking of Weyl quadratic gravity to Einstein action and Higgs potential

We consider the (gauged) Weyl gravity action, quadratic in the scalar curvature ($\tilde{R}$) and in the Weyl tensor (${\tilde{C}}_{\mu \nu \rho \sigma }$) of the Weyl conformal geometry. In the absence of matter fields, this action has spontaneous breaking in which the Weyl gauge field ω_μ becomes massive (mass m_ω ∼ Planck scale) after "eating" the dilaton in the $\tilde{R}$² term, in a Stueckelberg mechanism. As a result, one recovers the Einstein-Hilbert action with a positive cosmological constant and the Proca action for the massive Weyl gauge field ω_μ. Below m_ω this field decouples and Weyl geometry becomes Riemannian. The Einstein-Hilbert action is then just a "low-energy" limit of Weyl quadratic gravity which thus avoids its previous, long-held criticisms. In the presence of matter scalar field ϕ₁ (Higgs-like), with couplings allowed by Weyl gauge symmetry, after its spontaneous breaking one obtains in addition, at low scales, a Higgs potential with spontaneous electroweak symmetry breaking. This is induced by the non-minimal coupling ${\xi }_1{\varphi }_1^2\tilde{R}$ to Weyl geometry, with Higgs mass ∝ ξ₁/ξ₀ (ξ₀ is the coefficient of the $\tilde{R}$² term). In realistic models ξ₁ must be classically tuned ξ₁ ≪ ξ₀. We comment on the quantum stability of this value.