Non-metricity as the origin of mass generation in gauge theories of
  scale invariance
  We discuss gauge theories of scale invariance beyond the Standard Model (SM)
and Einstein gravity. We show that the non-metricity of their underlying 4D
geometry is at the origin of mass generation and discuss phenomenological
implications. Examples of such theories are Weyl's original quadratic gravity
theory and its Palatini formulation. Non-metricity leads to spontaneous
breaking of this gauged scale symmetry to Einstein gravity. All mass scales:
the Planck scale, the cosmological constant and the mass ($m_\omega$) of the
Weyl gauge boson of scale symmetry ($\omega_\mu$) are proportional to a scalar
field vev that has a (non-metric) geometric origin, in the $\tilde R^2$ term.
With $\omega_\mu$ of geometric origin, the SM Higgs field has a (non-metric)
geometric origin too, being generated by Weyl boson fusion in the early
Universe. This appears as a microscopic realisation of "matter creation from
geometry" discussed in the thermodynamics of open systems applied to cosmology.
Unlike in a locally scale invariant theory (no $\omega_\mu$ present) with an
underlying (metric) pseudo-Riemannian geometry, here there are no ghost degrees
of freedom or additional fields beyond the SM and their underlying
Weyl/Palatini geometry, the cosmological constant is predicted positive and
their connection shares the symmetry of the action. An intuitive picture of
non-metricity in solid state physics is also provided, where it is associated
with point defects (metric anomalies) of the crystalline structure.
