Stueckelberg breaking of Weyl conformal geometry and applications to gravity

摘要

Weyl conformal geometry may play a role in early cosmology where effective theory at short distances becomes conformal. Weyl conformal geometry also has a built-in geometric Stueckelberg mechanism: it is broken spontaneously to Riemannian geometry after a particular Weyl gauge transformation (of gauge fixing) while the Stueckelberg mechanism rearranges the degrees of freedom, conserving their number ( n d f ). The Weyl gauge field ( ω μ ) of local scale transformations acquires a mass after absorbing a compensator (dilaton), decouples, and Weyl connection becomes Riemannian. Mass generation has thus a dynamic origin, corresponding to a transition from Weyl to Riemannian geometry. In applications, we show that a gauge fixing symmetry transformation of the original Weyl’s quadratic gravity action immediately gives the Einstein-Proca action for the Weyl gauge field and a positive cosmological constant, plus matter action (if present). As a result, the Planck scale is an emergent scale, where Weyl gauge symmetry is spontaneously broken and Einstein action is a broken phase of Weyl action. This is in contrast to local scale invariant models (no gauging) where a negative kinetic term (ghost dilaton) remains present and n d f is not conserved when this symmetry is broken. The mass of ω μ , setting the nonmetricity scale, can be much smaller than M Planck , for ultraweak values of the coupling ( q ), with implications for phenomenology. If matter is present, a positive contribution to the Planck scale from a scalar field ( ? 1 ) VEV(vacuum expectation value) induces a negative ( mass ) 2 term for ? 1 and spontaneous breaking of the symmetry under which it is charged. These results are immediate when using Weyl-covariant (invariant) scalar (tensor) curvatures, respectively, instead of their Riemannian form. Briefly, Weyl gauge symmetry is physically relevant and its role in high scale physics should be reconsidered.