Abstract
We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal with no new fields beyond the SM spectrum and Weyl geometry. The action inherits a gauged scale symmetry D(1) (known as Weyl gauge symmetry) from the underlying geometry. The associated Weyl quadratic gravity undergoes spontaneous breaking of D(1) by a geometric Stueckelberg mechanism in which the Weyl gauge field (omega _mu ) acquires mass by “absorbing” the spin-zero mode of the {tilde{R}}^2 term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action emerges in the broken phase. In the presence of the SM, this mechanism receives corrections (from the Higgs) and it can induce electroweak (EW) symmetry breaking. The EW scale is proportional to the vev of the Stueckelberg field. The Higgs field (sigma ) has direct couplings to the Weyl gauge field (sigma ^2omega _mu omega ^mu ). The SM fermions only acquire such couplings for non-vanishing kinetic mixing of the gauge fields of D(1)times U(1)_Y. If this mixing is present, part of the mass of Z boson is not due to the usual Higgs mechanism, but to its mixing with massive omega _mu . Precision measurements of Z mass then set lower bounds on the mass of omega _mu which can be light (few TeV). In the early Universe the Higgs field can have a geometric origin, by Weyl vector fusion, and the Higgs potential can drive inflation. The dependence of the tensor-to-scalar ratio r on the spectral index n_s is similar to that in Starobinsky inflation but mildly shifted to lower r by the Higgs non-minimal coupling to Weyl geometry.
Highlights
We study the Standard Model (SM) in Weyl conformal geometry
In this work we consider the SM with a gauged scale symmetry [2,3,4] which we prefer to the more popular global scale symmetry, since the latter is broken by black-hole physics [5]
The SMW differs from the SM with conformal symmetry of [84] or [77,78,79] and from conformal gravity models [85,86,87] formulated in theRiemannian space and based on Cμ2νρσ term; these models are metric and do not have a gauged scale symmetry; in our case the Cμ2νρσ term is largely spectator and may even be absent in a first instance; it was included because its Weyl geometry counterpart contributed a threshold correction to α and it is needed at the quantum level
Summary
The Weyl geometry is defined by classes of equivalence (gαβ , ωμ) of the metric (gαβ ) and the Weyl gauge field (ωμ), related by the Weyl gauge transformation, see (a) below. D is the Weyl charge of gμν, α is the Weyl gauge coupling, g = | det gμν| and > 0. This is a noncompact gauged dilatation symmetry, denoted D(1). Since it is Abelian, the normalization of the charge d is not fixed.. A discussion on symmetry (1) and a brief introduction to Weyl geometry are found in Appendix A. To study the SM in Weyl geometry, all one needs to know for the purpose of this work is the expression of the connection ( ̃ ) of this geometry, which differs from the Levi–
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