[Paper Review] Mass-Generation by Weyl-Symmetry Breaking

Wolfgang Drechsler|CERN Bulletin|Jan 11, 1999

Cosmology and Gravitation Theories3 references5 citations

TL;DR

This paper proposes a mechanism for mass generation in electroweak theory via explicit breaking of Weyl symmetry in a Weyl space, where a curvature scalar term and a scalar field mass term induce nonzero masses for gauge and fermion fields. The resulting theory yields Einstein's equations in a pseudo-Riemannian space with a scalar field of constant modulus, linking the gravitational constant to the scalar field, analogous to Brans-Dicke theory, while avoiding spontaneous symmetry breaking in the standard model sense.

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.

Motivation & Objective

• To develop a mass-generating mechanism in electroweak theory without relying on spontaneous symmetry breaking.
• To explore the consequences of explicit Weyl-symmetry breaking in a Weyl space (W₄) for gauge and fermion fields.
• To derive Einstein’s equations in a pseudo-Riemannian space (V₄) after symmetry breaking, linking the gravitational constant to a scalar field.
• To compare this explicit breaking mechanism to the standard model’s nonlinear gauge choice, clarifying its physical distinction from spontaneous symmetry breaking.

Proposed method

• Formulate a Weyl-invariant gauge theory for leptons in a Weyl space W₄, allowing conformal rescalings of the metric and fields with nonvanishing Weyl weights.
• Introduce an explicit symmetry-breaking term in the Lagrangian involving the Weyl space curvature scalar R and a scalar field mass term.
• Implement the condition DΦ² = 0, where D is the Weyl-covariant derivative and Φ is the scalar field modulus, to enforce true symmetry reduction and establish a physical length scale.
• Derive the effective metric and gravitational equations in the resulting pseudo-Riemannian space V₄ after Weyl symmetry breaking.
• Show that the scalar field acquires a constant modulus, leading to a running gravitational constant analogous to Brans-Dicke theory.
• Demonstrate that gauge and charged fermion fields acquire nonzero masses through the explicit breaking mechanism, not through spontaneous symmetry breaking.

Experimental results

Research questions

• RQ1 How can mass generation in the electroweak sector be achieved through explicit Weyl-symmetry breaking rather than spontaneous symmetry breaking?
• RQ2 What are the geometric and dynamical consequences of breaking Weyl symmetry in a Weyl space W₄ on the metric and gravitational sector?
• RQ3 How does the condition DΦ² = 0 lead to a physical scale and mass generation for gauge and fermion fields?
• RQ4 What is the resulting form of Einstein’s equations after Weyl symmetry breaking, and how does the gravitational constant depend on the scalar field?
• RQ5 How does this mechanism differ fundamentally from the standard model’s treatment of electroweak symmetry breaking via a nonlinear gauge choice?

Key findings

• Explicit Weyl-symmetry breaking via a term involving the curvature scalar R and a scalar field mass term successfully generates nonzero masses for gauge bosons and charged fermions.
• The condition DΦ² = 0 enforces a true reduction of Weyl symmetry, establishing a physical length scale and distinguishing it from the standard model’s gauge-fixing procedure.
• After symmetry breaking, the theory reduces to a pseudo-Riemannian space V₄ where Einstein’s equations are recovered with a gravitational constant dependent on the scalar field modulus.
• The scalar field acquires a constant modulus, leading to a running gravitational constant analogous to the Brans-Dicke theory, with the scalar field acting as a dynamical gravitational coupling.
• The resulting theory avoids the standard model’s use of spontaneous symmetry breaking, offering an alternative geometric mechanism for mass generation.
• The gauge and fermion fields acquire masses through explicit breaking, not through vacuum expectation values, providing a new perspective on the origin of mass in gauge theories.