  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.
