Modified Pati-Salam Model from $Z_7$ orbifolded AdS/CFT

We consider models built on $AdS_5\otimes S^5/\Gamma$ orbifold compactifications of the type $IIB$ superstring, where $\Gamma$ is the abelian group $Z_n$. An attractive three family $\mathcal{N}=0$ SUSY model is found for $n=7$ that is a modified Pati--Salam Model which reduced to the Standard Model after symmetry breaking.


Introduction
The simplest compactification of a ten-dimensional superstring on a product of an AdS space with a five-dimensional spherical manifold leads to an N = 4 SU(N) supersymmetric gauge theory, well known to be conformally invariant [1]. By replacing the manifold S 5 by an orbifold S 5 /Γ one arrives at a theory with less supersymmetries corresponding to N = 2, 1 or 0 depending [2] on whether: (i) Γ ⊂ SU(2), (ii) Γ ⊂ SU(3) but not in SU(2), or (iii) Γ ⊂ SU(4) but not in SU(3) respectively, where Γ is in all cases a subgroup of SU(4) ∼ SO (6), the isometry of the S 5 manifold (for a review see [3]). It was conjectured in [4] that such SU(N) gauge theories are conformal in the N → ∞ limit. In [5] it was conjectured that at least a subset of the resultant nonsupersymmetric N = 0 theories are conformal even for finite N and that one of this subset may provide the correct extension of the Standard Model.
Recently, all N = 0 and N = 1 SUSY models have been classified [6,7] that come from orbifolding AdS 5 ⊗ S 5 with an abelian group Γ of order less than 12, where Γ embeds irreducibly in the SU(4) isometry or in an SU(3) subgroup of the SU(4) isometry, respectively. This means that, to achieve N = 0, rep(Γ) → 4 of SU(4) must be embedded as 4 = (r), where r is a nontrivial four dimensional representation of Γ; for N = 1, rep(Γ) → 4 of SU(4) must be embedded as 4 = (1, r), where 1 is the trivial singlet of Γ and r is nontrivial.
We want to focus on non-supersymmetric Pati-Salam (PS) type models (for a SUSY version see [8]). One motivation for studying the nonSUSY case is that the need for supersymmetry is less clear in CFT as: (1) the hierarchy problem is absent or ameliorated, (2) the difficulties involved in breaking the remaining N = 1 SUSY can be avoided if the orbifolding already results in N = 0 SUSY, and (3) many of the positive effects of SUSY are still present in the theory, although just hidden.
For N = 0 the fermions are given by i 4 ⊗ R i and the scalars by i 6 ⊗ R i where the set R i runs over all the irreps of Γ. For Γ abelian, for example Γ = Z n , the irreps are all one dimensional and as a consequence of the choice of N in the 1/N expansion, the gauge group is SU n (N) [9].
In this paper, starting from the classification of Kephart and Pas (2004), we have searched for a minimal (respect to the order of Γ = Z n ) nonSUSY model that have SM particles as a subset of its particle content. To do this we have used symmetry breaking paths that contain the Pati-Salam (PS) group as a subgroup before reaching the SM. The minimal model of this type has symmetry group SU 7 (4), hence orbifolding group is Z 7 , as we will discuss.
The running of the coupling constants predicted by the model depends strongly on the scalar content. In fact, since there are scalars in addition to the usual SM Higgs sector, they can contribute to the running of the beta functions. After a presentation of the model and of the SSB chain that leads to the SM particle content, we show that, with the use of a judicious choice of the scalar sector, unification can be achieved at the scale M GU T ∼ 10 3 GeV. We then conclude with a few comments on the phenomenology of the model including proton decay constraints and dark matter.

Description of the model
We have systematically gone through all chiral models with Γ = Z n . All fail to have a PS type intermediate stage until n = 7. Hence after considerable exploration, we are lead to choose Γ = Z 7 and N = 4 with orbifold group embedding 4 = (α, α, α 2 , α 3 ). This yield an N = 0 SUSY model based on the gauge group SU(4) 7 . The particle spectrum of the unbroken theory at the string scale is given by the fermion states and scalars 2[(4141111) S + · · · ] + 4 [(4114111) S + · · · ] + 4 [(4111411) S + · · · ] + 2[(4111141) S + · · · ] of SU(4) 7 , where the dots mean cyclic permutations. SU(4) 7 is broken down to SU(4) 3 via diagonal subgroups by sequentially assigning vacuum expectation values (VEVs) to (1414111) S , (114411) S , (11441) S and (1144) S , which leaves chiral fermions in the following bifundamental representations and scalars We continue the chain of spontaneous symmetry breaking toward the Pati-Salam model with a VEV for the (441) of the form This breaks the symmetry to SU(3) ⊗ SU(3) ⊗ SU(4) ⊗ U(1) A (see [10,11] for a detailed study of the phenomenology of this model without U(1) A charge) and gives three U(1) A neutral (331) 0 scalars. Finally, giving a VEV of the form A this stage the scalar content is given by Table 1.
In order to arrive at the Standard Model we will break SU(4) C → SU(3) C × U(1) X and SU(2) R → U(1) Z . This can be accomplished by giving a VEV to a scalar in the (124) −1/3,−1/2 representation, which leads to the group SU(2) L × SU(3) C and three U(1) factors. More precisely this would result in four U(1) factors, but one linear combination is broken due to the non-zero U(1) charges of (124) −1/3,−1/2 . Nevertheless we write these four charge as superscripts in order to fix the normalization later.     Table 2.

Phenomenology
In the previous section, the symmetry breaking of the initial SU(4) 7 towards to SU(4) R ⊗ SU(4) L ⊗ SU(4) C gauge group was performed by allowing the states (1414111) S , (114411) S , (11441) S , (1144) S to obtain VEVs. This makes clear that SU(4) R , SU(4) L and SU(4) C are embedded in diagonal subgroups SU(4) q , SU(4) p and SU(4) r of SU(4) 7 , respectively. We then embed all of SU(2) L in SU(4) L , but for U(1) Y the embedding is slightly more complicated. We need to go back to Eq.   of the coupling constants turns out to be 5 3 and sin 2 θ W satisfies (see [15] and references therein) . To find this energy scale we consider the renormalization-group evolution of the gauge couplings in leading order as given by where b i are the one-loop contributions to the beta function coefficients that are given in general by [12] b i = 11 3 Here n F is the number of chiral families, C 2 (G) is the quadratic Casimir invariant for the gauge group G and S 2 (F ) and S 2 (S) are the Dynkin indices for the fermion and scalar representations F and S respectively, and κ is 1 2 for Weyl fermions and 1 for Dirac fermions, see also [13,14,15]. For the case at hand We can choose the number of light scalar representations, i.e, use the S 2 's in the equations (3.7) as parameters to match ratio between the coupling constant at the GUT scale. As an example, this procedure leads to an unification scale M GU T = 5.0 × 10 3 GeV, for the choice of a single Higgs doublet plus 24 complex color triplet scalars of hypercharge 1/3. The evolution of the couplings from the weak to the unification scale is shown in Fig. 1.
Changing the choice of light scalars adjusts the unification scale, but given the experimental input at low energy and the requirement of unification at a higher scale, we necessarily need many scalars to be light below the unification scale. Increasing the triplet scalar masses (they would probably already have been detected, at least indirectly, if they were at the weak scale) to a few hundred GeV would likewise increase the unification scale to the 6 TeV range. Using extra vectorlike fermions instead of scalars can achieve similar results and with fewer particles, since fermions contribute more strongly to the β functions.

Conclusions
We have shown that it is possible to find a non-supersymmetric, "minimal", Pati-Salam type model based on the AdS/CF T orbifold compactifications of type IIB string theory on AdS 5 ⊗ S 5 /Z 7 . At the unification scale, this model contains bifundamental fermion and scalar representations of the gauge group SU(4) 7 , where the one loop, and perhaps higher loop β functions vanish, and conformality is partially, or fully restored.  Figure 1: Gauge coupling unification in the Modified Pati-Salam model. The curves has been rescaled as 69α 1 (Q), 40α 2 (Q) and 10α 3 (Q) in such a way that their ratio match to one at the unification scale. The plot is for values of Q from M Z to M GU T . Note that SU(3) C is no longer asymptotically free above the scalar triplet threshold, but it is asymptotically free at low energy as required. Generically, the unification is lowered by keeping more scalars light (similar results would hold if we replaced them with vectorlike fermions). Since our model is not supersymmetric, there is no natural LSP dark matter candidate, but one can still expect other options to be available, e.g., axionic dark matter, although we will not explore these possibility here.