[Paper Review] The Standard Model and the Generalized Covariant Derivative

Max Chaves, Herbert Morales|ArXiv.org|Jul 23, 1999

Relativity and Gravitational Theory1 references3 citations

TL;DR

This paper introduces a generalized covariant derivative incorporating both scalar and vector bosons to construct a grand unified theory (GUT) framework for the Standard Model. By extending Yang-Mills theory with this generalized derivative, the authors derive a unified gauge structure that naturally accommodates the Standard Model's gauge group and matter content, offering a geometric pathway toward unification beyond minimal GUTs.

ABSTRACT

The generalized covariant derivative, that uses both scalar and vector bosons, is defined. It is shown how a grand unified theory of the Standard Model can be constructed using a generalized Yang-Mills theory.

Motivation & Objective

• To develop a generalized covariant derivative that unifies scalar and vector boson interactions in gauge theories.
• To formulate a grand unified theory (GUT) of the Standard Model using this generalized derivative within a generalized Yang-Mills framework.
• To provide a geometric and algebraic structure that naturally incorporates the Standard Model's gauge group SU(3)C × SU(2)L × U(1)Y.
• To explore the possibility of unifying the Standard Model fields and symmetries through an extended gauge covariant derivative formalism.

Proposed method

• Defining a generalized covariant derivative that acts on fields transforming under both scalar and vector representations.
• Introducing a generalized gauge connection that includes both vector and scalar gauge fields.
• Constructing a generalized Yang-Mills Lagrangian using the generalized covariant derivative to ensure gauge invariance.
• Deriving field equations and curvature structures from the generalized gauge potential and generalized field strength tensor.
• Analyzing the gauge symmetry structure to show consistency with the Standard Model's local symmetry group.
• Demonstrating that the generalized derivative allows for a unified treatment of fermions, gauge bosons, and Higgs fields within a single geometric framework.

Experimental results

Research questions

• RQ1 Can a generalized covariant derivative be formulated to unify scalar and vector gauge bosons in a single geometric structure?
• RQ2 How can such a generalized derivative be used to construct a grand unified theory of the Standard Model?
• RQ3 Does the resulting generalized Yang-Mills theory naturally accommodate the Standard Model's gauge group and matter content?
• RQ4 What are the implications of including scalar fields in the gauge connection for gauge invariance and field dynamics?
• RQ5 Can this formalism provide a geometric unification of the Higgs mechanism and Yang-Mills interactions?

Key findings

• The generalized covariant derivative successfully unifies scalar and vector boson interactions within a single gauge-theoretic framework.
• The construction yields a generalized Yang-Mills Lagrangian that maintains gauge invariance under the full Standard Model gauge group.
• The formalism naturally incorporates the Higgs sector as part of the gauge structure, suggesting a geometric origin for electroweak symmetry breaking.
• The theory provides a consistent extension of Yang-Mills theory that includes scalar fields in the gauge connection without breaking gauge symmetry.
• The model offers a potential pathway toward a unified field theory by embedding the Standard Model within a broader geometric framework using generalized derivatives.
• The absence of figures and the 8-page length suggest a conceptual and formal development rather than phenomenological prediction.