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 ($nabla_mu g_{alphabeta}not=0$).
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.
Gauge Theories of Scale Invariance Beyond the Standard Model
In this article, we explore gauge theories of scale invariance beyond the Standard Model (SM) and Einstein gravity. These theories have a non-metric 4D geometry, meaning that the connection between the metric tensor and covariant derivative is non-zero.
Examples of Non-Metric Gauge Theories
Two examples of these theories are Weyl’s original quadratic gravity theory and its Palatini version. In these theories, the gauged scale symmetry is spontaneously broken, resulting in Einstein gravity. The mass scales in these theories, such as the Planck scale (Mp), cosmological constant (Λ), and the mass of the Weyl gauge boson (ωμ), are proportional to a scalar field vev originating from the geometric R^2 term in the action.
Origin of SM Higgs Field
In addition to the non-metricity originating from ωμ, the Standard Model Higgs field also has a similar origin. It is generated through Weyl boson fusion in the early Universe, providing a microscopic realization of “matter creation from geometry” discussed in thermodynamics of open systems applied to cosmology.
Key Features of Non-Metric Gauge Theories
Unlike local scale invariant theories without ωμ, the theories discussed here have several key features:
- No ghosts or additional fields beyond the SM and underlying Weyl or Palatini geometry.
- The cosmological constant is positive and small due to the weak gravity.
- The Weyl or Palatini connection shares the Weyl (gauge) symmetry of the action.
- There exists a non-trivial, conserved Weyl current associated with this symmetry.
Understanding Non-Metricity from a Solid State Physics Perspective
An intuitive understanding of non-metricity and its relation to mass generation can be gained from a solid state physics perspective. Non-metricity is common in solid state physics and is associated with point defects (metric anomalies) in the crystalline structure.
Future Roadmap for Readers
As our understanding of gauge theories of scale invariance beyond the Standard Model and Einstein gravity continues to evolve, several challenges and opportunities lie ahead:
- Further Theoretical Developments: Researchers can explore and develop new theoretical frameworks for understanding non-metric gauge theories and their implications for fundamental physics.
- Experimental Validation: Experimental tests and observations are needed to validate the predictions and implications of these gauge theories. This could involve particle physics experiments, cosmological observations, and precision measurements.
- Interdisciplinary Collaboration: Collaboration between physicists and solid-state researchers can lead to a deeper understanding of the connection between non-metricity and mass generation, potentially uncovering new phenomena and applications in both fields.
- Practical Applications: Insights from non-metric gauge theories could have practical applications beyond fundamental physics, such as in materials science, condensed matter physics, and quantum computing.
In summary, the study of gauge theories of scale invariance beyond the Standard Model and Einstein gravity opens up exciting possibilities for advancing our understanding of fundamental physics. While there are challenges to overcome, the potential for theoretical breakthroughs, experimental discoveries, interdisciplinary collaboration, and practical applications makes this field ripe with opportunities.