Mass Generation by Weyl Symmetry Breaking

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 ψ_L and ψ_R, and the gauge fields κ_μ, A_μ, Z_μ, W_μ, and W_μ ^†, allowing for conformal rescalings of the metric g_μν and all fields with nonvanishing Weyl weight together with the corresponding transformations of the Weyl vector fields, κ_μ, 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 Z_μ and W_μ 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 D_μ 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.