Standard Model in Weyl conformal geometry

We study the Standard Model (SM) in Weyl conformal geometry. This embedding is truly minimal, {\it 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. The Einstein-Hilbert 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 Higgs field has direct couplings to the Weyl gauge field while the SM fermions only acquire such couplings following the kinetic mixing of the gauge fields of $D(1)\times U(1)_Y$. One consequence is that 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 set lower bounds on the mass of $\omega_\mu$ which can be light (few TeV), depending on the mixing angle and Weyl gauge coupling. The Higgs mass and the EW scale are proportional to the vev of the Stueckelberg field. 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.