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

## TL;DR

This paper demonstrates how Weyl quadratic gravity naturally breaks to Einstein gravity with a cosmological constant and generates a Higgs potential, linking Weyl geometry, spontaneous symmetry breaking, and low-energy effective theories.

## Contribution

It shows that Weyl quadratic gravity undergoes spontaneous breaking, producing Einstein gravity and a Higgs potential, providing a novel geometric origin for these features.

## Key findings

- Weyl gauge field becomes massive via a Stueckelberg mechanism.
- Einstein-Hilbert action emerges as a low-energy limit of Weyl quadratic gravity.
- A Higgs potential with spontaneous electroweak symmetry breaking is generated.

## Abstract

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 $\omega_\mu$ becomes massive (mass $m_\omega\sim$ 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 $\omega_\mu$. Below $m_\omega$ 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 $\phi_1$ (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 $\xi_1\phi_1^2 \tilde R$ to Weyl geometry, with Higgs mass $\propto\xi_1/\xi_0$ ($\xi_0$ is the coefficient of the $\tilde R^2$ term). In realistic models $\xi_1$ must be classically tuned $\xi_1\ll \xi_0$. We comment on the quantum stability of this value.