Spectral action and the electroweak θ-terms for the Standard Model without fermion doubling

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.

Modave lectures on Noncommutative Geometry and its applications to physics

“…Furthermore, the lack of real structure can be interpreted as the source of CP violation in the physical Standard Model. One can also try to compute the spectral action [39] for such a model. But since it is purely Lorentzian, at the moment this can be done only for some simplified situations.…”

Octonion Internal Space Algebra for the Standard Model

“…Theories whose field algebra is a tensor product of a Dirac spinor bundle on a spacetime manifold with a finite dimensional internal space usually encounter the problem of fermion doubling [61] (still discussed over 20 years later, [62]). It was proposed in [34] as a remedy to consider the algebra P C 10 P, where P is the projector (42) on the 16 dimensional particle subspace (including the hypothetical right-handed sterile neutrino).…”