Mirror Fermions in Noncommutative Geometry

In a recent paper we pointed out the presence of extra fermionic degrees of freedom in a chiral gauge theory based on Connes Noncommutative Geometry. Here we propose a mechanism which provides a high mass to these mirror states, so that they decouple from low energy physics. ∗E-mail: [email protected], [email protected], [email protected], [email protected] Recently in Ref.[1] we have pointed out a doubling of the fermionic degrees of freedom in the Connes [2, 3] approach to chiral gauge theories based on Noncommutative Geometry [4]. In this brief report we suggest a mechanism which solves the problems posed by these extra degrees of freedom, by giving them a high mass. The mechanism on which the solution is based is loosely modelled in analogy with lattice gauge theories where a similar phenomenon occurs. The origin of the two phenomena are however quite distinct, at least with the present level of understanding. As is known the essential ingredients of Noncommutative Geometry are an algebra A, which encodes the topology of space–time (or its noncommutative generalization), an Hilbert space H which represents the fermionic mass degrees of freedom, and an operator D which generalizes the Dirac operator and which encodes the metric structure of the space. In addition to these three components of the so called spectral triple, there are other two essential elements: the real structure J , which represents charge conjugation, and a grading γ which generalizes the usual γ5. At present the noncommutative geometric structure of chiral gauge theories is understood as the product of continuous geometry representing the usual (commutative) space–time times an internal geometry. Thus the algebra is chosen as A = C(R,C) ⊗ AF , where C (R,C) is the algebra of smooth complex valued functions on R, and AF a matrix algebra whose unimodular group is the gauge group. Analogously H = L2(SR4)⊗HF , with L 2(SR4) the space of spinors and HF an internal Hilbert space which comprises all fermionic degrees of freedom. The other ingredients are obtained in a similar way, for details we refer to the literature on the subject in its various versions ([3, 5]–[9] and references therein). Other versions of gauge theory based on Noncommutative Geometry [10] have some of the basic ingredients of the construction which differ in essential ways from the ones treated here, and in general the considerations about mirror fermions will not apply. It is evident that the full power of noncommutative geometry is still used in a very limited way. The theory is some sort of Kaluza–Klein in which there is a continuous commutative space time, still made of usual points with the usual Hausdorff topology. At each point then there is a noncommutative space of the simplest kind possible, the one represented by finite dimensional matrix algebras. Despite the promising phenomenological features of the model [7, 11, 12], this simple choice of the space as a product creates some problems. The main one arises in H. For the consistency of the model it must be the tensor product of spinors times all fermionic degrees of freedom, and therefore some degrees of freedom will appear more than once. Moreover the chirality assignments of the extra degrees of freedom are incorrect. We will be more detailed in the following. The problem could be solved by projecting out the unwanted degrees of freedom, but as we showed in Ref.[1] this procedure is ambiguous and can only be made in a highly ad hoc fashion. In this paper we would like to explore another possibility, namely that the mirror fermionic degrees of freedom are actually real ones, but that the mass they have is