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Dirac Neutrino Masses in NCG

“…We will begin to discuss several models beyond SM: Model 1 considers the possibility of introducing right-hand neutrino in NCG in terms of changing the fermion representation H F ; model 2 is originally studied by others [10] to find whether there is another U(1) gauge boson, in which another C algebra was put in; model 3 comes from the mathematics consideration on quantum group, where quaternion algebra was changed to M 2 (C) ; model 4 is a combination of model 2 and 3. They all give mild changes to the Connes-Lott's version of SM, and we will introduce right-neutrinos in them.…”

“…Another extension of standard model is discussed in paper [10], in which the algebra A F has two C algebras C and C ′ . Now we put N generations right-handed neutrinos in this model, leaving the weak and strong sectors unchanged as it has been done in [10].…”

“…T r p is the trace on H F restricted to particle sector H + F and x is a unitary element of A F . Here we still use standard model as an example: we refer [10] to find details.…”

“…Another extension of standard model is discussed in paper [10], in which the algebra A F has two C algebras C and C ′ . Now we put N generations right-handed neutrinos in this model, leaving the weak and strong sectors unchanged as it has been done in [10]. The properties of NCG [4]: π(a), Jπ(b)J † = 0 , [D, π(a)] , Jπ(b)J † = 0 f or a, b ∈ A, make the representation of algebra A F on antiparticle part of leptons is vectorial [4] (all belong to algebra C or C ′ ).…”

“…In this paper, we only use the Connes-Lott's new scheme(a real spectral triplet in it) [6][7][8], which has achieved many successes in SM [4,9]. Because three generations of right-handed neutrinos can not exist in it [4], we must consider other models beyond SM, such as [10][11][12][13], if it is needed to discuss the neutrino mass problem. To our opinion, any new models in NCG must obey some physical and geometrical principles.…”