Standard Model in conformal geometry: local vs gauged scale invariance

Abstract

We discuss comparatively local versus gauged Weyl symmetry beyond Standard Model (SM) and Einstein gravity and their geometric interpretation. The SM and Einstein gravity admit a natural embedding in Weyl integrable geometry which is a special limit of Weyl conformal (non-metric) geometry. The theory has a local Weyl scale symmetry but no associated gauge boson. Unlike previous models with such symmetry, this embedding is truly minimal i.e. with no additional fields beyond SM and underlying geometry. This theory is compared to a similar minimal embedding of SM and Einstein gravity in Weyl conformal geometry (SMW) which has a full gauged scale invariance, with an associated Weyl gauge boson. At large field values, both theories give realistic, Starobinsky-Higgs like inflation. The broken phase of the current model is the decoupling limit of the massive Weyl gauge boson of the broken phase of SMW, while the local scale symmetry of the current model is part of the larger gauged scale symmetry of SMW. Hence, the current theory has a gauge embedding in SMW. Unlike in the SMW, we note that in models with local scale symmetry the associated current is trivial, which is a concern for the physical meaning of this symmetry. Therefore, the SMW is a more fundamental UV completion of SM in a full gauge theory of scale invariance that generates Einstein gravity in the (spontaneously) broken phase, as an effective theory.

Weyl conformal geometry vs Weyl anomaly

“…the Weyl gauge boson (ω µ ) of scale invariance becomes a massive Proca field, 3 after "eating" the dilaton propagated by the R2 term of Weyl quadratic gravity [1,2], and then decouples. Further, one can show that the SM (with higgs mass set to zero) has a natural, truly minimal embedding in Weyl geometry [3,8] with no new states beyond SM and Weyl geometry. This gives an interesting UV completion and "unification" of SM and Einstein gravity in a gauge theory of scale invariance.…”

“…23 After the massive Weyl gauge field decouples (together with the dilaton), the usual Weyl anomaly emerges in the broken phase of the quantum theory, as discussed (section 3.3). Note that the equation of motion of ϕ, ϕ 2 = − R, gives after the symmetry breaking R = −4Λ ̸ = 0 [3,8]. Then, ln ϕ 2 in (3.30) or (3.26) generates ln R terms which prevent one from taking an exactly flat metric.…”