Local conformal symmetry in non-Riemannian geometry and the origin of physical scales

TL;DR

This paper develops a conformally invariant extension of the Standard Model and General Relativity using non-Riemannian Weyl geometry, proposing a mechanism for the emergence of physical scales through spontaneous symmetry breaking.

Contribution

It introduces a novel geometric framework with a scalar and vector field, demonstrating how physical scales can arise without dimensionful parameters.

Findings

Test particles follow Levi-Civita geodesics, addressing Einstein's criticism.

Physical scales emerge from symmetry breaking in a conformally invariant theory.

The approach unifies gravity and matter fields within a Weyl geometric setting.

Abstract

We introduce an extension of the Standard Model and General Relativity built upon the principle of local conformal invariance, which represents a generalization of a previous work by Bars, Steinhardt and Turok. This is naturally realized by adopting as a geometric framework a particular class of non-Riemannian geometries, first studied by Weyl. The gravitational sector is enriched by a scalar and a vector field. The latter has a geometric origin and represents the novel feature of our approach. We argue that physical scales could emerge from a theory with no dimensionful parameters, as a result of the spontaneous breakdown of conformal and electroweak symmetries. We study the dynamics of matter fields in this modified gravity theory and show that test particles follow geodesics of the Levi-Civita connection, thus resolving an old criticism raised by Einstein against Weyl's original proposal.