Gauge unification in noncommutative geometry

Gauge unification is widely considered to be a desirable feature for extensions of the standard model. Unfortunately the standard model itself does not exhibit a unification of its running gauge couplings but it is required by grand unified theories as well as the noncommutative version of the standard model [2]. We will consider here the extension of the noncommutative standard model by vector doublets as proposed in [6]. Two consequences of this modification are: 1. the relations of the coupling constants at unification energy are altered with respect to the well known relation from grand unified theories. 2. The extended model allows for unification of the gauge couplings at Λ ∼ 10 GeV. PACS-92: 11.15 Gauge field theories MSC-91: 81T13 Yang-Mills and other gauge theories 1 Unité Mixte de Recherche (UMR 6207) du CNRS et des Universités Aix–Marseille 1 et 2 et Sud Toulon–Var, Laboratoire affilié à la FRUMAM (FR 2291) 2 also at Université Aix–Marseille 1, [email protected] It is generally believed that the standard model and the big dessert are not the final theory describing the particle content of our universe. A hint for an underlying, more profound structure is the observation that the running gauge couplings almost converge, missing each other by roughly five orders of magnitude between ∼ 10 GeV and ∼ 10 GeV. Grand unified theories require an exact convergence, but since the standard model cannot provide for this, extensions have to be considered. One of the most popular extensions is certainly supersymmetry which enlarges the particle contend of the standard model roughly by a factor of two, introducing supersymmetric partners. Due to cancellations in renormalisation this extension leads to an exact convergence of the gauge couplings. The price which has to be paid is a multitude of hitherto unobserved particles which should although be detectable at the LHC. A different approach to the standard model is noncommutative geometry [1] which, through the spectral action, also requires gauge unification [2, 3]. Here again the pure standard model cannot meet the conditions on the gauge couplings. The conditions on the gauge couplings coming from noncommutative geometry coincide for the standard model with the classical ones from grand unified theories. In noncommutative geometry this unification is not thought of as having its origin in the breaking of a simple unifying group like SU(5) or SO(10) but as a modification of space-time itself. Recently extensions of the standard model within the framework of noncommutative geometry have been discovered [19, 5, 6]. At least one of these extensions, the ACmodel, even has a viable dark matter candidate [7] and is compatible with high precision measurements in particle physics [8]. In this publication we will examine the extension presented in [6], investigating its ability to cure the unification problem. Here the particle content of the standard model is enlarged by particles coupling vectorially to the electro-weak U(1)Y × SU(2)w subgroup. A most interesting fact of these extensions is that the conditions of the gauge unification get modified. If the mass of these vector doublets is taken to be of unification scale, ∼ 10 GeV, the altered unification conditions are almost exactly fulfilled. And even if one prefers the classical conditions from grand unified theories, these vector doublets alter the running of the gauge couplings sufficiently to obtain a perfect convergence. 1 Vector doublets In noncommutative geometry the gauge group G is extracted from the spectral triple either via the unimodularity condition [9, 3] or via centrally extending the lift of the automorphism group of the associated algebra [10]. The two approaches coincide for a minimal central extension [10]. There are other constraints, on the fermionic representations, coming from the axioms of the spectral triple. They are conveniently captured in Krajewski diagrams which classify all possible finite dimensional spectral triples [11]. They do for spectral triples what the Dynkin and weight diagrams do for groups and representations. The model considered here is an extension of the standard model by a set of fermions which couple vectorially to the U(1)Y × SU(2)w subgroup of the standard model. They 2 are colour singlets and have gauge invariant masses mψ. For convenience we will call them vector doublets. A thorough presentation of this model containing the details of the construction of the spectral triple, the lift of the automorphisms, the Lagrangian and possible mass assignments, which could give viable dark matter candidates, can be found in [6]. Extensions of the standard model within the noncommutative framework are rare and only a few viable ones are known [19, 5, 6]. Therefore the vector doublet model is far from ad-hoc and its properties are quite remarkable. We will concentrate here on the ability of the model to achieve unification of the U(1)Y -, SU(2)wand SU(3)c-gauge couplings. Figure 1 shows the Krajewski diagram of the standard model in Lorentzian signature with one generation of fermions and one vector doublet represented by the dashed arrow.