$Z_3$ orbifold construction of $SU(3)^3$ GUT with $\sin^2\theta_W^0=3/8$

It is argued that a phenomenologically viable grand unification model from superstring is $SU(3)^3$, the simplest gauge group among the grand unifications of the electroweak hypercharge embedded in semi-simple groups. We construct a realistic 4D $SU(3)^3$ model with the GUT scale $\sin^2\theta_W^0= \frac38$ in a $Z_3$ orbifold with Wilson line(s). By two GUT scale vacuum expectation values, we obtain a rank 4 supersymmetric standard model below the GUT scale, and predict three more strange families.


A. Introduction and motivation
Supersymmetric standard models(SSM), if proven experimentally, need a theoretical explanation of why they become the effective theory below the Planck scale M P ≃ 2.44 × 10 18 GeV. A most probable scenario is that they result from compactifications of superstring models preserving one supersymmetry N = 1. The effective 4D N = 1 field theory models were extensively considered in this regard in the Calabi-Yau compactifications [1] and orbifold compactifications [2,3]. Furthermore, the standard-like models initiated more than 15 years ago opened up the search for SSM directly from superstring [4].
The initial standard-like models SU(3) × SU(2) × U(1) n were very attractive, in realizing the standard model(SM) gauge group and reasonable matter spectrum [4,5,6], with possible desirable physics on the strong CP problem [7] and cosmology with a hidden world [8].
Furthermore, the doublet-triplet splitting has been realized in some standard-like models [4].
However, these standard-like models failed because they generally do not predict correct weak mixing angle sin 2 θ 0 W at the string scale [9]. To predict the observed coupling constants at the electroweak scale successfully at least in ∼ 2.2σ level, the sin 2 θ 0 W at the unification scale ∼ (2 − 3) × 10 16 GeV is required to be ≃ 3 8 . The reason is very simple. In these standard-like models, the electroweak hypercharge group U(1) Y is one combination out of n U(1)'s. Thus, the singlet representations of the standard-like gauge group, not belonging to the family structure of the fifteen(or sixteen if we include a heavy Majorana neutrino), can have nonvanishing U(1) Y charges, which lowers the string scale weak mixing angle from the needed value of 3 8 , because the string scale weak mixing angle sin 2 θ 0 W is expressed if we assume α 0 2 = α 0 1 at the string scale, This sin 2 θ W problem can be resolved if the standard model gauge group is unified in a simple group GUT, for example SU(5), where U(1) Y is a subgroup of the GUT group. Then, the electroweak hypercharge generater is an SU(5) generator. Namely, SU(5) singlets do not carry nonvanishing electroweak hypercharges and we conclude that the string scale sin 2 θ 0 W is 3 8 . To obtain a supersymmetric standard model in 4D, SU(5) must be broken by a VEV of an adjoint Higgs field(24 H ). However, it is impossible to obtain an adjoint matter field at the level 1, i.e. k = 1. [23] If simplicity is any guidance to the truth of nature, one must break the GUT group without an adjoint matter representation. This leads us to GUT groups with a U(1) factor, notably SU(5) × U(1) which is now called flipped SU (5). The flipped SU(5) is an interesting rearrangement of a singlet field and fifteen chiral fields of SU(5) [11].
The symmetry breaking of the flipped SU(5) is particularly interesting in supersymmetric flipped SU(5) [12]. In this regards, the string compactifications toward flipped SU(5) is very interesting, since breaking of SU(5) × U(1) down to the standard model gauge group can be achieved without an adjoint Higgs representation [12]. Indeed, the fermionic construction of 4D flipped SU(5) was obtained already fifteen years ago [13]. As shown in many subsequent papers, the flipped SU(5) has many phenomenologically interesting features [13].
However, the flipped SU(5) generally fails in the aforementioned sin 2 θ W problem. The reason is the following. The flipped SU(5) needs three SU(5) singlet representations which carry +1 unit of the electric charge for the three singlet charged leptons of SSM. This implies, SU(5) singlets can carry electromagnetic charges, or the electroweak hypercharge Y . Since there appear numerous SU(5) singlets from string compactification, the charged singlets generally reduce dramatically sin 2 θ 0 W from the needed value 3 8 , viz. (1). In the orbifold construction, this sin 2 θ W problem has been really serious. In the literature, one can find many models with SU(5) × U(1) groups [14], and even it was claimed that there are flipped SU(5)'s [15], but as shown above these models ignored the sin 2 θ W problem.
However, one may argue that even if the flipped SU(5) contains a U(1) factor, the sin 2 θ W problem goes away if the representations are embeddable in SO (10). In this case, the U(1) Y generator belongs to SO(10) and hence SO(10) singlets do not carry the U(1) Y charge.
Then, the singlets of the flipped SU(5) carry only the needed electroweak hypercharges of the flipped SU(5), and hence the string scale sin 2 θ 0 W is 3 8 . However, this scenario is not realized generally in orbifold compactifications, which can be easily understood by remembering that orbifolds generally choose only part of the original complete representation. In fact, this property is the root for the solutions of the doublet-triplet splitting problem in the 4D orbifold compactifications [4].
However, if it happens that the extra fields beyond the complete multiplets conspire to contribute to T rT 2 3 and T rQ 2 em in the ratio 3/8, then we can obtain 3/8 as the string scale value of sin 2 θ 0 W . Therefore, the above argument is not a no-go theorem. It may be extremely difficult however, if not impossible, to find such a model with the electroweak hypercharge leaking to U(1) at the GUT scale.
Before considering our 4D string model, let us comment on the recent field theoretic orbifold breaking of grand unification group with extra dimensions [16]. One interesting feature here has been family unification groups with SO(2n) with n ≥ 7 [17]. In these extra-dimensional field theories, it is possible to allow fixed point fields as far as there are no anomalies, and hence it is not much achieved in the prediction of the matter representations at the orbifold fixed points. In this context, 6D string theoretic models were considered as an intermediate step toward a final 4D string theory construction [18]. In this paper, however, we attempt to obtain a more ambitious 4D model.

B. Z 3 orbifold with Wilson line
In 4D, if a GUT group containing a U(1), as in the SU(5)×U (1), is difficult to obtain, the next simple GUT groups to try are semi-simple groups. Therefore, we propose grand unified theories with the hypercharge embedded in a semi-simple group with no adjoint representation needed(HESSNA) as possible 4D string models toward a realistic SSM. For a realistic 4D superstring model, we must require that the factor groups of the HESSNA can be broken to SSM without an adjoint representation. In this regard, note that the Pati-Salam GUT group SU(4) × SU(2) L × SU(2) R is not a HESSNA because it has the same problem as that in the SU(5) model: one needs an adjoint representation. Therefore, the simplest HESSNA is SU(3) 3 . The next simple HESSNA is SU(3) × SU(3) × SU(4). If we find a realistic HESSNA, then it is a simple matter to find a SSM from this HESSNA, as the SU(5) model leads to the SM.
In the HESSNA also, the orbifold compactification is very much chiral, and may be too much chiral. But here at least it is easy to study the electroweak hypercharge concretely in a few steps.
At the phenomenological level, the group SU(3) 3 has been extensively considered [19].
Our objective in this paper is to realize a string theory SU(3) 3 . If we obtain such a model, it can be considered as a realistic superstring GUT.
We expect that one family in the SU(3) 3 HESSNA is composed of 27 chiral fields, Only two possible SU(3) 3 groups can be found in the extensive tables of Z N orbifold models [14]. They appear in Z 12 orbifold models. However, the fermionic spectrums of these Z 12 compactifications are not the one required in (2). This leads us to consider orbifold models with Wilson lines. [24] In a separate publication, we tabulate Z 3 orbifold models with one Wilson line [20].
In the remainder of this paper, we present a SU (3) The modular invariance condition requires in addition, 3 v · a i = (integer) for (i = 1, 3, 5), The notation is the same as those discussed in [21]. For an SU (3)

C. Untwisted sector
Gauge group : From the mass shell condition m 2 4 = p 2 2 − 1, we find the massless spectrum in the untwisted sector. For the gauge bosons, the p 2 = 2 root vectors, satisfying p · v = 0 and p · a 1 = 0 mod integer, are the nonvanishing roots. These are presented for the first E 8 subgroup in Table I. The second E ′ 8 gauge group is not broken.
In Table I gauge group with the above orbifold. Thus, it is possible to realize the model-independent axion as a quintessential axion [22].  Matter from the untwisted sector : The matter fields from the untwisted sector satisfy the condition In Table II, we present the root vectors satisfying these.

D. Matter from the twisted sectors
In In our model, the massless matter fields from the twisted sectors satisfy (p +ṽ) 2 = 2 3 , 4 3 , whereṽ = v, v + a 1 , v − a 1 , for T0, T1, and T2, respectively. Of course, the weights we present survive the GSO-like projection. For the vectors corresponding to 4 3 the multiplicity is 9 as described above, and for the vectors corresponding to 2 3 the multiplicity is 27 because of the three oscillator modes in this case.
In general, the matter fields from the twisted sectors make the theory extremely chiral which was the reason that we have not obtained yet any realistic SSM or flipped SU(5) model from orbifold compactification of the heterotic string. Since it is very chiral, there is a chance that the spectrum (2) can appear through orbifolding.
In Tables III, IV, and V, we list the massless spectrum from the twisted sectors. But note that the chirality of the twisted sector in the Z 3 orbifold is the opposite of the chirality of the untwisted sector matter fields.
where · · · represents 27 multiplets of the vectorlike combination       We also checked that the vectorlike representation contributes in the same ratio. There unfamiliar particles such as lepton doublets with Y = ± 1 6 appear, but they form a vectorlike representation, are removed at the GUT scale and do not alter sin 2 θ 0 W . This miraculous prediction of sin 2 θ 0 W is based on the fact that everything appears in the multiples of 3. The model given in Eq. (7) gives 9 families. But note that there appear additional 9 families with the opposite colors. By adding more Wilson line(s) in the hidden sector E ′ 8 part, the family number can be easily reduced to 3, not spoiling our precious spectrum obtained in (2). Below, we comment on six family models obtained by adding more Wilson line(s) at The spectrum (7) has two villages, each having three families. The family mixing is allowed inside the village but is forbidden between different villages, predicting two CP phases, one in each village. To explain the three light families, the members of the strange village are required to be heavy at the electroweak scale. With 6 families, the QCD coupling is not asymptotically free, but still perturbatively unifiable at the GUT scale. The rank-