Abstract
We compute the leading terms of the spectral action for a noncommutative geometry model that has no fermion doubling. The spectral triple describing it, which is chiral and allows for CP-symmetry breaking, has the Dirac operator that is not of the product type. Using Wick rotation we derive explicitly the Lagrangian of the model from the spectral action for a flat metric, demonstrating the appearance of the topological θ-terms for the electroweak gauge fields.
Highlights
The geometry of the Standard Model [2,3,4] and its extensions [7,8,9,10,11]
The geometry of the entire enhanced space-time is determined by a Dirac operator that depends on the metric and the gauge connections, and includes the Higgs field, which plays a role of a connection over the finite noncommutative component
The story of the noncommutative model-building is, not yet complete as the most accepted model is in the Euclidean signature and requires additional assumptions to remove the possibility of the SU(3) symmetry breaking [13, 14] as well as an additional projection onto the physical space of fermions [15,16,17]
Summary
We begin by briefly reviewing the model as described in details in [18, 22]. The particle content in the one-generation Standard Model can be conveniently parametrized in the following form: νR u1R u2R u3R. It is convenient to encode local linear operator acting on the particle content of the model, at every point of M1,3, as an element of M4(C) ⊗ M2(C) ⊗ M4(C), where the first and the last matrices act by multiplication from the left and from the right, respectively, while the middle M2(C) matrix acts on the components of the Weyl spinor. Without referring to additional symmetries or assumptions the model preserves the SU(3) strong symmetry and allows for the natural breaking of the CP-symmetry, which is linked to the non-reality of the mixing matrices This is, on the level of the algebra of the model, equivalent to the failure of the finite part of the Krein-shifted Dirac operator to be J -real (see [18] for details)
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