Weyl Invariant Standard Model and its Symmetry Breaking

TL;DR

This paper formulates a Weyl invariant Standard Model in a Weyl space, incorporating gravity and symmetry breaking, and explores its implications for particle masses and interactions within a unified geometric framework.

Contribution

It introduces a Weyl covariant formulation of the Standard Model including gravity and symmetry breaking, with novel dynamics for quark fields and a new approach to mass generation.

Findings

Weyl invariance leads to a current-current self-interaction of quarks.

Symmetry breaking establishes a universal mass scale via the scalar field.

Masses of weak bosons and electrons emerge from the energy-momentum tensor after symmetry breaking.

Abstract

A standard model is formulated in a Weyl space, $W_4$, yielding a Weyl covariant dynamics of massless chiral Dirac fermion fields for leptons and quarks as well as the gauge fields involved for the groups D(1)\,(Weyl), $U(1)_Y{\times} SU(2)_W$\,(electroweak), $SU(3)_c$\,(colour), SO(3,1)\,(gravity) and SO(4,1)\,(strong interaction, symmetry breaking). The dynamics is based on a gauge and Weyl invariant Lagrangean density ${\cal L}$. Gravitation is included from the beginning as the gauge aspect of the Lorentz group which is here extended in the hadronic sector of the model to the ten parameter SO(4,1) de Sitter group. A part of the dynamics is, as usual, a scalar isospinor field $\phi$ being a section on a bundle related to the electroweak gauge group and to symmetry breaking. In parallel to $\phi$ on the leptonic side a section $\tilde\xi^a$ on the hadronic side is considered as part of the dynamics, governing the symmetry breaking $SO(4,1)\longrightarrow SO(3,1)$ and recovering gravitation in the symmetry breaking limit outside the regions in space-time where strong interactions persist. Besides spin, isospin and helicity the Weyl weights determine the form of the contributions of fields in ${\cal L}$. Of particular interest is the appearance of a current-current self-interaction of quark fields allowed by the Weyl weight changing the debate about quark masses. In a second step the D(1)-Weyl symmetry is explicitly broken and a universal mass scale is established through the mass of the $\phi$-field appearing in the symmetry breaking Lagrangean ${\cal L}_B$. The Weyl symmetry breaking is governed by the relation $D_\mu \Phi^2{=}0$, where $\Phi$ is the norm of $\phi$. After D(1) symmetry breaking the masses of the weak bosons and of the electron appear on the scene through the energy-momentum tensor of the $\phi$-field.