Conformality from Field-String Duality on Abelian Orbifolds

If the standard model is embedded in a conformal theory, what is the simplest possibility? We analyse all abelian orbifolds for discrete symmetry Zp with p ≤ 7, and find that the simplest such theory is indeed SU(3)7. Such a theory predicts the correct electroweak unification (sin2θ ≃ 0.231). A color coupling αC(M) ≃ 0.07 suggests a conformal scale M near to 10 TeV. Permanent address. e-mail: [email protected] 1 Since, in the context of field-string duality, there has been a shift regarding the relationship of gravity to the standard model of strong and electroweak interactions we shall begin by characterising how gravity fits in, then to suggest more specifically how the standard model fits in to the string framework. The descriptions of gravity and of the standard model are contained in the string theory. In the string picture in ten spacetime dimensions, or upon compactification to four dimensions, there is a massless spin-two graviton but the standard model is not manifest in the way we shall consider it. In the conformal field theory extension of the standard model, gravity is strikingly absent. The field-string duality does not imply that the standard model already contains gravity and, in fact, it does not. The situation is not analogous to the Regge-pole/resonance duality (despite a misleading earlier version of this introduction!). That quite different duality led to the origin [1] of string theory and originated from the realization [2–5] phenomenologically that adding Regge pole and resonance descriptions is double counting and that the two descriptions are dual in that stronger sense. The duality [6] between the field and string descriptions is not analogous because the CFT description does not contain gravity. A first step to combining gravity with the standard model would be adding the corresponding lagrangians. In the field theory description [7–10] used in this article, one will simply ignore the massless spin-two graviton. Indeed since we are using the field theory description only below the conformal scale of ∼ 1TeV ( or, as suggested later in this paper, 10TeV) and forgoing any requirement of grand unification, the hierarchy between the weak scale and theory-generated scales like MGUT or MPLANCK is resolved. Moreover, seeking the graviton in the field theory description is possibly resolvable by going to a higher dimension and restricting the range of the higher dimension. Here we are looking only at the strong and weak interactions at accessible energies below, say, 10TeV. Of course, if we ask questions in a different regime, for example about scattering of particles with center-of-mass energy of the order MPLANCK then the graviton will become crucial [11,12] and a string, rather than a field, description will be the viable one. 2 It is important to distinguish between the holographic description of the five-dimensional gravity in (AdS)5 made by the four-dimensional CFT and the origin of the four-dimensional graviton. The latter could be described holographically only by a lower three-dimensional field theory which is not relevant to the real world. Therefore the graviton of our world can only arise by compactification of a higher dimensional graviton. Introduction of gravity must break conformal invariance and it is an interesting question (which I will not answer!) whether this breaking is related to the mass and symmetry-breaking scales in the low-energy theory. That is all I will say about gravity in the present paper; the remainder is on the standard model and its embedding in a CFT. An alternative to conformality, grand unification with supersymmetry, leads to an impressively accurate gauge coupling unification [13]. In particular it predicts an electroweak mixing angle at the Z-pole, sinθ = 0.231. This result may, however, be fortuitous, but rather than abandon gauge coupling unification, we can rederive sinθ = 0.231 in a different way by embedding the electroweak SU(2)× U(1) in SU(N)× SU(N)× SU(N) to find sinθ = 3/13 ≃ 0.231 [9,10]. This will be a common feature of the models in this paper. The conformal theories will be finite without quadratic or logarithmic divergences. This requires appropriate equal number of fermions and bosons which can cancel in loops and which occur without the necessity of space-time supersymmetry. As we shall see in one example, it is possible to combine spacetime supersymmetry with conformality but the latter is the driving principle and the former is merely an option: additional fermions and scalars are predicted by conformality in the TeV range [9,10], but in general these particles are different and distinguishable from supersymmetric partners. The boson-fermion cancellation is essential for the cancellation of infinities, and will play a central role in the calculation of the cosmological constant (not discussed here). In the field picture, the cosmological constant measures the vacuum energy density. What is needed first for the conformal approach is a simple model and that is the subject of this paper. Here we shall focus on abelian orbifolds characterised by the discrete group Zp. Non3 abelian orbifolds will be systematically analysed elsewhere. The steps in building a model for the abelian case (parallel steps hold for non-abelian orbifolds) are: • (1) Choose the discrete group Γ. Here we are considering only Γ = Zp. We define α = exp(2πi/p). • (2) Choose the embedding of Γ ⊂ SU(4) by assigning 4 = (α1 , α2, α3, α4) such that ∑q=4 q=1 Aq = 0(modp). To break N = 4 supersymmetry to N = 0 ( or N = 1) requires that none (or one) of the Aq is equal to zero (mod p). • (3) For chiral fermions one requires that 4 6≡ 4 for the embedding of Γ in SU(4). The chiral fermions are in the bifundamental representations of SU(N) i=p ∑