Standard Model in conformal geometry: local vs gauged scale invariance

We discuss comparatively local versus gauged Weyl symmetry beyond StandardModel (SM) and Einstein gravity and their geometric interpretation. The SM andEinstein gravity admit a natural embedding in Weyl integrable geometry which isa special limit of Weyl conformal (non-metric) geometry. The theory has a {\itlocal} Weyl scale symmetry but no associated gauge boson. Unlike previousmodels with such symmetry, this embedding is truly minimal i.e. with noadditional fields beyond SM and underlying geometry. This theory is compared toa similar minimal embedding of SM and Einstein gravity in Weyl conformalgeometry (SMW) which has a full {\it gauged} scale invariance, with anassociated Weyl gauge boson. At large field values, both theories giverealistic, Starobinsky-Higgs like inflation. The broken phase of the currentmodel is the decoupling limit of the massive Weyl gauge boson of the brokenphase of SMW, while the local scale symmetry of the current model is part ofthe larger gauged scale symmetry of SMW. Hence, the current theory has a gaugeembedding in SMW. Unlike in the SMW, we note that in models with local scalesymmetry the associated current is trivial, which is a concern for the physicalmeaning of this symmetry. Therefore, the SMW is a more fundamental UVcompletion of SM in a full gauge theory of scale invariance that generatesEinstein gravity in the (spontaneously) broken phase, as an effective theory.