Mass-Generation by Weyl-Symmetry Breaking

TL;DR

This paper develops a Weyl-invariant electroweak theory in a Weyl space, demonstrating how explicit symmetry breaking leads to mass generation for gauge and fermion fields, and explores the emergence of Einstein's equations post-symmetry breaking.

Contribution

It introduces a Weyl-invariant formulation of electroweak theory with explicit symmetry breaking, linking Weyl geometry to mass generation and gravity.

Findings

Massive gauge and fermion fields emerge after symmetry breaking.

Einstein's equations arise in the broken symmetry phase, connecting Weyl geometry to gravity.

The theory relates the gravitational constant to scalar field properties.

Abstract

A massless electroweak theory for leptons is formulated in a Weyl space, W_4, yielding a Weyl invariant gauge dynamics allowing for conformal rescalings of the metric and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields representing the D(1) or dilatation gauge fields. To study the appearance of nonzero masses this theory is explicitly broken by a term in the Lagrangean involving the curvature scalar R of the W_4 and a mass term for the scalar field. Thereby also the gauge fields as well as the charged fermion field acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D Phi^2=0, where Phi is the modulus of the scalar field and D denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory, is compared to the so-called spontanous symmetry breaking in the standard electroweak theory which is, actually, the choice of a particular (nonlinear) gauge obtained by adopting an origin in the coset space representing the scalar field which is invariant under the electromagnetic gauge group. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl-symmetry breaking yielding a pseudo-Riemannian space V_4 from a W_4 and a scalar field with a constant modulus which in turn affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory.