  We discuss gauge theories of scale invariance beyond the Standard Model (SM)
and Einstein gravity. A consequence of gauging this symmetry is that their
underlying 4D geometry is non-metric ($\nabla_\mu g_{\alpha\beta}\not=0$).
Examples of such theories are Weyl's {\it original} quadratic gravity theory
and its Palatini version. These theories have spontaneous breaking of the
gauged scale symmetry to Einstein gravity. All mass scales have a geometric
origin: the Planck scale ($M_p$), cosmological constant ($\Lambda$) and the
mass of the Weyl gauge boson ($\omega_\mu$) of scale symmetry are proportional
to a scalar field vev that has an origin in the (geometric) $\tilde R^2$ term
in the action. With $\omega_\mu$ of non-metric geometry origin, the SM Higgs
field also has a similar origin, 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 local scale invariant theories (no $\omega_\mu$ present) with an
underlying pseudo-Riemannian geometry, in our case: 1) there are no ghosts and
no additional fields beyond the SM and underlying Weyl or Palatini geometry, 2)
the cosmological constant is positive and is small because gravity is weak, 3)
the Weyl or Palatini connection shares the Weyl (gauge) symmetry of the action,
and: 4) there exists a non-trivial, conserved Weyl current of this symmetry. An
intuitive picture of non-metricity and its relation to mass generation is also
provided from a solid state physics perspective where it is common and is
associated with point defects (metric anomalies) of the crystalline structure.
