Abstract

A massless electroweak theory for leptons is formulated in a Weyl space, W4, yielding a Weyl invariant dynamics of a scalar field φ, chiral Dirac fermion fields ψL and ψR, and the gauge fields κμ, Aμ, Zμ, Wμ, and Wμ †, allowing for conformal rescalings of the metric gμν and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, κμ, representing the D(1) or dilatation gauge fields. The local group structure of this Weyl electroweak (WEW) theory is given by $G = SO(3,1) \otimes D(1) \otimes \tilde G$ —or its universal coverging group $\bar G$ for the fermions—with $\tilde G$ denoting the electroweak gauge group SU(2)W × U(1)Y. In order to investigate the appearance of nonzero masses in the theory the Weyl symmetry is explicitly broken by a term in the Lagrangean constructed with the curvature scalar R of the W4 and a mass term for the scalar field. Thereby also the Zμ and Wμ gauge fields as well as the charged fermion field (electron) acquire a mass as in the standard electroweak theory. The symmetry breaking is governed by the relation D μ Φ 2 = 0, where Φ is the modulus of the scalar field and Dμ denotes the Weyl-covariant derivative. This true symmetry reduction, establishing a scale of length in the theory by breaking the D(1) gauge symmetry, is compared to the so-called spontaneous symmetry breaking in the standard electroweak theory, which is, actually, the choice of a particular (nonlinear ) gauge obtained by adopting an origin, ${\hat \phi }$ , in the coset space representing ϕ, with ${\hat \phi }$ being invariant under the electromagnetic, gauge group U(1)e.m.. Particular attention is devoted to the appearance of Einstein's equations for the metric after the Weyl symmetry breaking, yielding a pseudo-Riemannian space, V4, from a W4 and a scalar field with a constant modulus $\hat \phi _0$ . The quantity $\hat \phi _0^2$ affects Einstein's gravitational constant in a manner comparable to the Brans-Dicke theory. The consequences of the broken WEW theory are worked out and the determination of the parameters of the theory is discussed