Abstract
We consider aspects of the noncommutative approach to the standard model based on the spectral action principle. We show that as a consequence of the incorporation of the Clifford structures in the formalism, the spectral action contains an extended scalar sector, with respect to the minimal standard model. This may have interesting phenomenological consequences. Some of these new scalar fields carry both weak isospin and color indices. We calculate the new terms in spectral action due to the presence of these fields. Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions. Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action. Their contributions to the Dirac operator correspond to couplings with the spurious fermions, which are projected out.
Highlights
The standard model of particle interactions can be efficiently described by a particular noncommutative geometry: an “almost commutative geometry.” Over the years the model has been developing both in its mathematical and physical aspects
Our analysis demonstrates that the fermionic doubling in the noncommutative geometry is not just a presence of spurious degrees of freedom, but it is an interesting and peculiar property of the formalism, which leads to physically valuable conclusions
Some of the new fields do not contribute to the physical fermionic action, but they appear in the bosonic spectral action
Summary
The standard model of particle interactions can be efficiently described by a particular noncommutative geometry: an “almost commutative geometry.” Over the years the model has been developing both in its mathematical and physical aspects. The numbers produced in [5], encouraging, are not in agreement with present data; in particular, the model requires the unification of all couplings at a single energy, and one calculates the Higgs boson mass around 170 GeV. Model is by nature Euclidean and exhibits spurious degrees of freedom, known as “fermion doubling” [24]; for physical applications a Wick (anti)rotation accompanied by an elimination of these spurious degrees of freedom is necessary We have described this procedure in detail in [25]. We find that not all of these extra bosons behave upon this procedure in a standard way: some of the new scalar fields present in the Euclidean Dirac operator are absent in the corresponding (Lorentzian) physical action for fermions.
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