Non-metric geometry as the origin of mass in gauge theories of scale invariance

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 (∇μgαβ≠0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\nabla _\mu g_{\alpha \beta }\!\not =\!0$$\end{document}). Examples of such theories are Weyl’s 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 (Mp\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$M_p$$\end{document}), cosmological constant (Λ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Lambda $$\end{document}) and the mass of the Weyl gauge boson (ωμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _\mu $$\end{document}) of scale symmetry are proportional to a scalar field vev that has an origin in the (geometric) R~2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\tilde{R}}^2$$\end{document} term in the action. With ωμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _\mu $$\end{document} 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 (with no ωμ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\omega _\mu $$\end{document} 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.