Spectral geometry for the standard model without fermion doubling

Abstract

We propose a simple model of noncommutative geometry to describe the structure of the Standard Model, which satisfies spin c condition, has no fermion doubling, does not lead to the possibility of color symmetry breaking and explains the CP-violation as the failure of the reality condition for the Dirac operator. I. INTRODUCTIONThe Standard Model of particle interactions is certainly one of the most successful and best tested theory about the fundamental constituents of matter and the forces between them.Even though we still have no satisfactory description of the strong interactions in low-energy regime and there are some puzzles concerning masses and character of neutrinos as well as there are some experimental signs that could point out to new physics, the Standard Model appears to be robust and verified. Yet neither the content of fermion sector, the mixing between the families and the fundamentally different character of the Higgs boson from other gauge bosons appear to have a satisfactory geometrical explanation.One of the few theories that aimed to provide a sound geometrical basis for the structure of the Standard Model, explaining the appearance of the Higgs and symmetry-breaking potential, was noncommutative geometry (see [1][2][3]). Constructed with the core idea that spaces with points can be replaced with algebras provided a plausible explanation of the gauge group of the Standard Model and the particles in its representation as linked to the unitary group of a finite-dimensional algebra. Merged with the Kaluza-Klein idea that the physical spacetime has extra dimensions, the geometry of the finite-dimensional algebra (in the noncommutative sense) gave rise to the Higgs field understood as a connection, and the Higgs symmetry-breaking potential appeared as the usual Yang-Mills term in the action.The original model, which is based on the construction of a product geometry, with the resulting geometry being the tensor product of a usual "commutative" space with the finitedimensional noncommutative geometry suffers from two problems. Firstly, in the original

Superselection of the weak hypercharge and the algebra of the Standard Model

“…3 Particle subspace, Higgs field and associated Lie superalgebra A general problem in theories, whose configuration space is a product of a commutative algebra of (continuous) functions on space-time with a finite dimensional quantum algebra, is the problem of fermion doubling [18] (still discussed over twenty years later, [7]). It was proposed in [13], as a remedy, to consider the algebra P 0 C 10 P 0 where P 0 is the projector on the 16 dimensional particle subspace including the right handed sterile neutrino ν R (see (2.17)).…”

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

“…The story of the noncommutative model-building is, however, 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 (due to the fermion quadrupling in the model) [15][16][17]. In the analysis of the Lorentzian case with slight modifications of the spectral triple rules we proved that there exists a model without the fermion doubling and with exact colour SU(3) symmetry [18]. Moreover, the non-product Dirac operator satisfied a slightly modified first-order condition which is tantamount to the spin c one under certain requirements for mass spectra of fermions.…”

“…As it was demonstrated in [18] the model has interesting algebraic properties, like the Morita duality (which means that the generalized Clifford algebra is a full commutant of the algebra A) provided that both pairs of matrices (Υ ν , Υ e ) and (Υ u , Υ d ) have pairwise different eigenvalues. Furthermore, 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.…”

“…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:…”

“…In the paper, we compute the spectral action for the model we presented in [18]. It needs to be stressed that this model is not of the product-type geometry and therefore the computations and results are not automatically identical to those performed in the series of papers computing the spectral action [12,19].…”