Gauge theories of Dirac type

Abstract

A specific class of gauge theories is geometrically described in terms of fermions. In particular, it is shown how the geometrical frame presented naturally includes spontaneous symmetry breaking of Yang-Mills gauge theories without making use of a Higgs potential. In more physical terms, it is shown that the Yukawa coupling of fermions, together with gravity, necessarily yields a symmetry reduction provided the fermionic mass is considered as a globally well-defined concept. The structure of this symmetry breaking is shown to be compatible with the symmetry breaking that is induced by the Higgs potential of the minimal Standard Model. As a consequence, it is shown that the fermionic mass has a simple geometrical interpretation in terms of curvature and that the (semi-classical) "fermionic vacuum" determines the intrinsic geometry of space-time. We also discuss the issue of "fermion doubling" in some detail and introduce a specific projection onto the "physical sub-space" that is motivated from the Standard Model.

Dirac-type gauge theories and the mass of the Higgs boson

“…These two representations are related by the Yukawa-coupling matrix G Y which can be considered as a linear mapping from the representation space of the Higgs field to the representation space of left-handed fermions. For a general discussion we again refer to [TT05]. In what follows, we will restrict ourselves to the specific case of the minimal Standard Model (see, for instance, [Nac90]).…”

“…Indeed, the Dirac potential is but a trace of an endomorphism which is uniquely determined by D (c.f. section 2.2 in [TT05] and Prop. 3.1 in [Tol07]).…”

“…We stress, however, that such a parametrization is not favored by the geometrical setup of GTDT (c.f. [TT05]). Disregarding geometry, however, such a non-geometrical parametrization may be still of interest for it yields the principal model bounds for the predicted value of the Higgs mass within the mathematical frame presented.…”

“…In this paper we discuss the Higgs mass within the STM using the geometrical frame of (parameterized) Dirac type gauge theories (GTDT). The mathematical background of GTDT is discussed in some detail in [TT05] and [Tol07]. However, in order to be self-contained we present a purely local description of GTDT which is also needed to derive the relations for the Higgs mass.…”

“…In the third section we summarize the STM as it is described as a special GTDT. There we also present a natural parametrization of the general mathematical scheme that is presented in [TT05]. In the fourth section we discuss the parameter relations between the appropriately parameterized GTDT of the STM with its usual mathematical description.…”