Standard Model in Weyl conformal geometry

Abstract

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is natural and truly minimal {\it with no new fields} required 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 ($ω_μ$) acquires mass by "absorbing" the spin-zero mode ($ϕ_0$) of the $\tilde R^2$ term in the action. This mode also generates the Planck scale and the cosmological constant. The Einstein-Proca action of $ω_μ$ 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 ($ϕ_0$). The Higgs field ($σ$) has direct couplings to the Weyl gauge field, and its mass may be protected at quantum level by the D(1) symmetry. The SM fermions can acquire couplings to $ω_μ$ only in the special case of a non-vanishing kinetic mixing of the gauge fields of $D(1)\times U(1)_Y$. If this mixing is indeed present, part of $Z$ boson mass is not due to the Higgs mechanism, but to its mixing with massive $ω_μ$. Precision measurements of $Z$ mass then set lower bounds on the mass of $ω_μ$ which can be light (few TeV). In the early Universe the Higgs field can have a {\it geometric} origin, by Weyl vector fusion, and the Stueckelberg-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 shifted to lower $r$ by the Higgs non-minimal coupling to Weyl geometry.