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

Abstract

We consider the (gauged) Weyl gravity action, quadratic in the scalar curvature ($\tilde R$) and in the Weyl tensor ($\tilde C_{μνρσ}$) 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 $m_ω\sim$ 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 $ϕ_1$ (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 $ξ_1ϕ_1^2 \tilde R$ to Weyl geometry, with Higgs mass $\proptoξ_1/ξ_0$ ($ξ_0$ is the coefficient of the $\tilde R^2$ term). In realistic models $ξ_1$ must be classically tuned $ξ_1\ll ξ_0$. We comment on the quantum stability of this value.