Broken Weyl-Invariance and the Origin of Mass

TL;DR

This paper explores how breaking Weyl-invariance in a geometric framework leads to the emergence of mass for scalar and spinor fields, integrating a Higgs-like mechanism within Weyl geometry.

Contribution

It introduces a Weyl-invariant model with explicit symmetry breaking that results in mass generation through a geometric Higgs mechanism in Weyl space.

Findings

Massless Weyl-invariant dynamics formulated in Weyl space.

Explicit symmetry breaking induces mass for scalar and spinor fields.

The resulting field equations are generally covariant and coupled to Einstein's gravity.

Abstract

A massless Weyl-invariant dynamics of a scalar, a Dirac spinor, and electromagnetic fields is formulated in a Weyl space, $W_4$, allowing for conformal rescalings of the metric and of all fields with nontrivial Weyl weight together with the associated transformations of the Weyl vector fields $\ka_\mu$ representing the D(1) gauge fields with D(1) denoting the dilatation group. To study the appearance of nonzero masses in the theory the Weyl-symmetry is broken explicitly and the corresponding reduction of the Weyl space $W_4$ to a pseudo-Riemannian space $V_4$ is investigated assuming the breaking to be determined by an expression involving the curvature scalar $R$ of the $W_4$ and the mass of the scalar, selfinteracting field. Thereby also the spinor field acquires a mass proportional to the modulus $\Phi$ of the scalar field in a Higgs-type mechanism formulated here in a Weyl-geometric setting with $\Phi$ providing a potential for the Weyl vector fields $\ka_\mu$. After the Weyl-symmetry breaking one obtains generally covariant and U(1) gauge covariant field equations coupled to the metric of the underlying $V_4$. This metric is determined by Einstein's equations, with a gravitational coupling constant depending on $\Phi$, coupled to the energy-momentum tensors of the now massive fields involved together with the (massless) radiation fields.