Generalized Covariant Derivative with Gauge and Higgs Fields in the Standard Model

Abstract

The concept of covariant derivatives for quark and lepton fields is generalized on algebras of internal symmetries and the Dirac matrices. Commutators of the covariant derivatives define field strengths for both the gauge fields and the Higgs field. The bosonic part of the Lagrangian is determined using a symmetric combination of quadratic invariants of the gauge and Higgs field strengths. The new version of the standard model thus formulated predicts the Higgs boson mass in terms of the top quark mass as mu'=f2m, at low energies and the Weinberg angle sin 2 In this article, we formulate a new unified theory of the gauge and Higgs fields without changing the notion of the ordinary Minkowski spacetime by generalizing the covariant derivative Dp. (rather than the Dirac operator l/J) on algebras of internal symmetries and the Dirac matrices in a Lorentz and gauge covariant way. Taking commutators of the covariant derivatives, field strengths are defined for both the gauge fields and the Higgs field. The bosonic part of the Lagrangian is determined by a symmetric combination of quadratic invariants of the gauge and Higgs field strengths. Essential results of the Connes theory are reproducible in this version of the standard model, where a strong parallelism is imposed between the gauge and Higgs field strengths. Three generations of leptons are represented by the chiral fields of the electr­ oweak doublets and singlets as t/Jlj=rh j (j=1, 2, 3). Similarly, those of quarks are described by the colored chiral fields of electroweak doublets t/Jqj= tPL (qJj, and the singlets t/Jw=t/JR j and t/Jdi=t/JR i. With these chiral fields, the fermionic part of the Lagrangian density of the standard model is given by ..( ~ ' ;r. ~'(·a + A ca>a 1 1 + A c2>a 1 + A co 1 y)-'· .