Three-generation Asymmetric Orbifold Models from Heterotic String Theory

Using Z3 asymmetric orbifolds in heterotic string theory, we construct N=1 SUSY three-generation models with the standard model gauge group SU(3)_C \times SU(2)_L \times U(1)_Y and the left-right symmetric group SU(3)_C \times SU(2)_L \times SU(2)_R \times U(1)_{B-L}. One of the models possesses a gauge flavor symmetry for the Z3 twisted matter.


Introduction
String theory is a candidate for a unified theory of the four fundamental forces including quantum gravity. One of the main characteristic features of the standard model of particle physics is the three-generation chiral structure of quarks and leptons, and there have been many attempts to construct four-dimensional models with three generations by string compactification.
Heterotic string model building is one of the successful methods for a stringy realization of particle physics models. Especially, in the framework of heterotic Z N or Z N × Z M orbifold compactifications [1], embedding a supersymmetric standard model or a higher dimensional grand unified theory into heterotic string theory is considered [2,3,4,5,6,7,8,9] (also see a review [10]). Since strings on an orbifold can be described by a solvable world-sheet conformal field theory [11], it is possible to calculate Yukawa couplings and selection rules [12,13,14,15]. Furthermore, the geometrical structure of orbifold fixed points can be an origin of a discrete symmetry [16,17], which may lead to a hierarchical structure of masses/mixings of quarks and leptons.
Asymmetric orbifold compactifications [18] can be considered as an extension of symmetric orbifolds, in which orbifold actions for left-and right-movers are generalized to be independent of each other, without destroying modular invariance of closed string theory. We may expect that the generalization of the orbifold action will give us the possibility to construct a large number of four-dimensional string models, among which we can try to find phenomenologically viable models. However, in asymmetric orbifold constructions, model building for SUSY standard models or other GUT extended models with three matter generations has not been investigated thoroughly 1 .
In [21], a systematic approach for the construction of four-dimensional string models using Z 3 asymmetric orbifolds, which are the simplest orbifolds capable of realizing N = 1 SUSY in four dimensions, is shown 2 . First, one specifies a (22,6)-dimensional Narain lattice [23] and then applies the asymmetric orbifold action. In [21], Narain lattices are constructed from 24-dimensional Niemeier lattices [24] by the lattice engineering technique 3 [26]. Using this method, 106 types of (22,6)-dimensional Narain lattices with a right-moving non-Abalian factor are constructed. Group breaking patterns due to a Z N shift action are also analyzed by extending the argument of breaking patterns of the E 8 group [27]. Furthermore, possible gauge group patterns of Z 3 asymmetric orbifold models are analyzed.
The aim of this paper is to apply the asymmetric orbifolding procedure to the construction of three-generation models with the standard model group SU(3) C ×SU(2) L ×U(1) Y and other groups. In the next section, we review asymmetric orbifold model building. In section 3 and 4 we construct Z 3 asymmetric orbifold models with three generations. Section 5 is devoted to conclusions. In appendix A we show another three-generation model example. 1 In free-fermionic string constructions, which are related to Z 2 × Z 2 (a)symmetric orbifolds by fermionization, the first SUSY standard-like models and left-right symmetric models with three generations were found in [19,20]. 2 In [22], Z 3 asymmetric orbifold models with one Wilson line were studied. 3 In [25], the lattice engineering technique is applied to a GUT model building.

Heterotic asymmetric orbifold construction
A starting point for the asymmetric orbifold construction is a heterotic string theory compactified on some (22,6)-dimensional Narain lattice. The corresponding world-sheet theory splits into left-moving and right-moving degrees of freedom. Besides the ghost fields, there are 26 left-moving bosons X L as well as ten right-moving boson-fermion pairs (X R , Ψ R ). The momentum modes p = (p L , p R ) associated with the internal dimensions X 4...25 L and X 4...9 R lie on the Narain lattice Γ. Modular invariance of closed string theory implies that Γ is even (p 2 = p 2 L − p 2 R ∈ 2Z) and self-dual (Γ =Γ). Here, Γ ≡ i n i γ i andΓ ≡ i m iγi , where γ i and γ j are bases for the lattice Γ and its dual latticeΓ, respectively. The bases for the lattice and its dual satisfy γ i ·γ j = δ ij .
In Z N orbifold constructions, one specifies an orbifold action for the internal dimensions as follows: This action contains a twist θ = (θ L , θ R ) and a shift V = (V L , V R ). The twist has to be a lattice automorphism of order N, i.e. θ N = 1. The shift has to satisfy NV ∈ Γ. In asymmetric orbifolds, the left-mover twist θ L and the right-mover twist θ R can be chosen independently.
For the sake of simplicity we consider only Z 3 models with N = 1 SUSY and without a leftmoving twist, i.e. θ L = 1 (also, one can set V R = 0). By writing θ R = diag(e 2πit 1 R , e 2πit 2 R , e 2πit 3 R ) in some complex basis one can define a right-mover twist vector t R = (0, t 1 R , t 2 R , t 3 R ). When acting on fermions one has to embed the twist into the double cover of SO(6), so t i R is only defined modulo 2. If i t i R = 0 and t 1,2,3 R = 0 one realizes N = 1 SUSY. For our models we use the right-mover twist vector Now, we can fully specify a model by the following: • a (22,6)-dimensional Narain lattice Γ which contains a right-moving E 6 or A 3 2 lattice. These are the only lattices which allow for a N = 1 compatible Z 3 automorphism [3,28]. In [21], such lattices were constructed from the well known 24-dimensional even self-dual lattices by the lattice engineering method.
In heterotic string theory, spacetime gauge symmetry is realized by left-moving massless modes. Generally, a four-dimensional Narain model has a gauge symmetry of rank 22. An orbifold shift V L breaks the original group into a subgroup of same rank. The breaking patterns can be calculated by analyzing shift vectors and extended Dynkin diagrams [21].
In the case of Z 3 asymmetric orbifold models with N = 1 SUSY whose twist action for the left-mover is trivial, we can check that modular invariance of the closed string theory requires the shift vector V L to satisfy the condition The massless spectrum in the untwisted sector can be read off as in the case of symmetric orbifolds. In the light-cone formalism, the right-moving modes can be described in terms of H-momentum. In the untwisted sector, the orbifold phases for H-momentum modes are given by t R · q. Here, "+even" means that we should take only combinations with the even number of plus signs and the underline represents all cyclic permutations. The massless modes with non-trivial phases under the orbifold action t R are given by Note that the H-momentum states |q ′ and |q ′′ are CPT conjugate to each other. We define the four-dimensional chirality as "left-handed" if the first component of fermionic modes is 1/2, i.e. | 1 2 , 1 2 , − 1 2 , − 1 2 in |q ′ . For the lattice part, the orbifold action (θ, V ) acts on momentum modes p = (p L , p R ) ∈ Γ and oscillator modes. The right-moving modes with p R = 0 or oscillator excitations are massive, so we do not need to consider orbifold phases for them. In the left-mover part, orbifold phases V L · p L arise for massless states with p 2 L = 2. Left-mover oscillators are not affected by the orbifold action because θ L = 1. This is the reason for the observed rank preservation. Now, when combining left-mover and right-mover phases one concludes that only massless states which satisfy the condition remain in the untwisted spectrum after the orbifold projection.
To read off massless states in the Z 3 twisted sector, we define I θ (Γ) as the sublattice of the original (22,6)-dimensional Narain lattice Γ that is invariant under the Z 3 twist action θ. In our case where we have no twist action for the left-movers, I θ (Γ) is a 22-dimensional left-mover lattice which is spanned by 22 basis vectors α i=1... 22 . We also define the dual lattice of the invariant sublatticeĨ θ (Γ) ≡ 22 i=1 n iαi , where theα i satisfy α i ·α j = δ ij and n i are integers. In the α-twisted sector (α ∈ {1, 2}), the momenta p = (p L , 0) lie on the shifted latticeĨ θ (Γ) + αV . The massless left-mover modes in the α-twisted sector can be obtained by solving the equation Since in our case we do not consider any twist actions for the left-mover, we have ∆c L = 0. For the right-moving part, massless states are described solely in terms of H-momentum: These are combined with massless states from the lattice part |p α=1,2 0 that satisfy (10), giving The states from α = 1 and α = 2 are CPT conjugate to each other and together form a four-dimensional chiral supermultiplet. The degeneracy factor for the twisted sector is given by where η i are defined as 0 ≤ t i +n i = η i < 1 for suitable integers n i . The volume of the invariant lattice is determined as Vol(I θ (Γ)) = det(g ij ) = det(α i · α j ). In this paper we consider only orbifold models with the right-mover twist (4) and a right-mover E 6 or a A 3 2 lattice. For these lattices one retrieves from (15) a degeneracy of D = 3 and D = 1, respectively.
In [29], couplings of asymmetric orbifold models are calculated by rewriting a right-moving complex chiral boson X(z) in terms of exponentials of boson fields φ, where α corresponds to suitable root vectors. In order to reproduce the Z N orbifold condition for i∂X, for some integer k, the orbifold action on φ has to be a shift action: In this representation, the momenta associated to φ (the Q-charges) enter the right mover mass equation. In particular, the massless states satisfy where Q ∈ Γ 22,6 + αs Q for α = 1, 2. The number of solutions to this equation corresponds to the degeneracy factor D. Now, let us denote the simple roots and fundamental weights of a simple Lie group G as α G i and ω G i . Then, for models from Narain lattices with E 6 , the right-moving twist (4) is replaced by the following shift for the internal six dimensions: Here we use the decomposition A 3 2 ⊂ E 6 . For models from Narain lattices with A 1 3 sublattice invariant under the twist. In our case, the conjugacy classes of I θ (Γ) are given by the generators (0, 1, 1, 2, 0, 0, 1, 1/4), (1, 0, 1, 1, 2, 0, 0, 1/4), (0, 1, 0, 1, 1, 2, 0, 1/4), (0, 0, 1, 0, 1, 1, 2, 1/4) (24) of A 7 3 × U(1). In general, the shift vector V is composed of shift vectors for the seven A 3 parts and a shift vector for U(1) part. The shift vector for each A 3 belongs to one of the conjugacy classes 0, 1, 2 and 3. Now, before looking at the full broken gauge group, it is useful to consider A 3 group breaking patterns by Z 3 shift actions in general. The breaking patterns are listed in Table 1 (For the definition of n ′(k) i , see [21]). In the table, breaking patterns due to shift vectors which belong to conjugacy class 3 A 3 are not listed as these can be reproduced from shift vectors in 1 A 3 by suitable reflections. Then, with (5) in mind we can specify a modular invariant shift vector V by appropriately combining seven A 3 breaking patterns with a shift in the U(1) direction.
From this, one verifies that I θ (Γ) is spanned by the basis Also, we can evaluate a dual basisα i and find thatĨ θ (Γ) is given by the following conjugacy class generators with a U(1) normalization of 2/ √ 3. Now, we can read off the massless spectrum in the twisted sector by solving (10). Taking all linear combinations of (47) results in 256 conjugacy classes for the dual invariant sublattice. Among them, here we shall show only massless states that arise from the conjugacy class (1, 2, 1, 3, 0, 0, 0, −1/4). Namely, for this conjugacy class, momentum modes in the α twisted sector belong to , For α = 1, we can find a solution for (10) as lying in the unbroken group where Combined with the H-momentum states in (11) this leads to three chiral supermultiplets which transform under the non-Abelian group as (1, 1, 3, 1, 1, 1, 1). Here, the degeneracy "three" comes from (15), with in our case. We can read off all massless states from the other conjugacy classes (47) in the same way. The resulting massless spectrum of this model is listed in Table 2.
Among the ten U(1) groups, a non-anomalous U(1) Y gauge symmetry is taken from a combination of four U(1)s, By choosing this combination, we can see that this model contains three standard model generations of chiral matter multiplets, and the additional fields have vector-like structure.
Also this model has one anomalous U(1) A gauge symmetry that can be given by the following combination In a four-dimensional model with an anomalous U(1) A gauge symmetry, a string loop effect will generate a Fayet-Iliopoulos D-term [30,31,32]. For the anomalous U(1) A we can check that mixed anomalies satisfy the Green-Schwarz universality relation 4 where G a and k a are non-Abelian groups and their Kac-Moody levels, and 2T (R) is the Dynkin index of the representation R. Q B represents a non-anomalous U(1) group of level one that is orthogonal to U(1) A .
By the U(1) Y charge assignment (57) we can see that this model has net three standard model generations, and the other extra fields are vector-like to each other. In this model, three-generations of right-handed down-type quarks d come from the untwisted sector (field f 12 ), and further three generations of quark doublets q, right-handed down-type quarks u and up-type Higgs fields h u come from the Z 3 twisted sector 5 (fields f 47 , f 48 and f 49 ). We find two pairs of SU(3) C color exotics and some extra SU(2) L doublets. There are also 3 × 12 singlets of the non-Abelian group SU (3) (4), and all other SU(3) C × SU(2) L singlets are non-trivially charged under the additional non-Abelian group SU(2) 2 × SU(3) 2 × SU(4).
In this paper we do not consider explicitly all of the terms in the superpotential of this model and we do not perform detailed analysis of the VEV structure. That is our future task. In the following, we will comment on decoupling of some exotic fields and Yukawa couplings of this model. Regarding the color exotic fields, c 1 , c 2 , c 1 and c 2 have a three point coupling with singlets as so they are expected to decouple from the low energy theory if the singlets s 0 1 and s 0 2 get VEVs. Similar thing can happen for l u and l u since there is a three point coupling Fields f 44 , f 45 and f 46 contain four (1, 2) −1/2 fields and two (1, 2) 1/2 fields after breaking additional A 3 and A 2 groups. Then, we can expect that net two (1, 2) −1/2 remain massless fields, and these fields can be identified as the lepton doublet and down-type Higgs fields. We can see that the other exotic fields have vector-like structure.
By analyzing three point couplings allowed by Q-charge invariance, relevant couplings for quarks and Higgses are given by because the Q-charges are for X ∈ {q, h u , u}. Since in asymmetric orbifolds the starting point is a torus compactification at self-dual radius, string world-sheet instanton effects can be neglected. So, each coefficient y 123 etc. for the three point couplings is expected to be of O(1). From above couplings it turns out that the mass of the second generation quark will be of the same order as the top quark mass.

from asymmetric orbifolds
In this section we construct a Z 3 asymmetric orbifold model with the left-right symmetric group We use a Narain lattice with A 3 2 as our starting point, so the resulting model has a degeneracy factor D = 1.
We start from four-dimensional heterotic string theory compactified on a (20,4)-dimensional Using the lattice engineering technique, the (20,4)-dimensional lattice is made from the 24-dimensional A 6 4 Niemeier lattice as Here, the left-moving A 2 2 factor in A 2 4 is replaced by the right-moving A  Gauge symmetry breaking patterns for this model can be analyzed as in the previous section. Breaking patterns for A 1 , A 2 and A 4 groups are listed in Table 3, 4 and 5.
Regarding the top quark mass, a 3-point coupling for the top Yukawa interaction HQ L1 Q R3 is not allowed by the Q-charge invariance though this operator is gauge invariant. Then, to reproduce a suitable value for the top quark mass, we need to consider higher-dimensional operators and larger VEVs for some singlets.

Conclusion
From Z 3 heterotic asymmetric orbifolds, we construct several three-generation models with the standard model gauge group or the left-right symmetric group. The starting points for model building are Narain lattices with E 6 or A 3 2 which are obtained from 24-dimensional Niemeier lattices by the lattice engineering technique. By taking a modular invariant combination of Z 3 shift actions for the left-mover we obtain four-dimensional three-generation models with vector-like exotics.
For models from Narain lattices with E 6 , the number of "three" standard model generations originates from the degeneracy factor D = 3. Also, the up-type quark resides in the Z 3 twisted sector and its Yukawa coupling is expected to be realized through a coupling of three fields from the twisted sector. By Q-charge analysis it turns out that, for one of the models, there is a three point coupling for top Yukawa interaction. However, the other three-point couplings lead to a too heavy mass for the second generation quark. Regarding the model with degeneracy factor D = 1, even though this is a left-right symmetric model with three generations, charge conservation forbids a three-point coupling for top Yukawa interaction. So, in order to reproduce appropriate Yukawa couplings we will need to search models from other Narain lattices with A In asymmetric orbifold constructions, it turns out that it is possible to construct models with a gauge flavor symmetry. It will be important to consider Yukawa interaction properties (masses and mixings) arising from such kind of flavor gauge symmetries as well as those arising from discrete flavor symmetries. Analyzing moduli stabilization in this formalism will also be important.

A Another example of a three-generation model
Here, we show another example of a simple three-generation model with the standard model group. To specify the model, we take the (22,6)-dimensional Narain lattice A 11 2 × E 6 as a starting point, and also take a Z 3 shift vector as This shift vector belongs to the conjugacy class (2, 2, 1, 0, 2, 2, 2, 2, 1, 1, 0, 0) of A 11 2 × E 6 . By the orbifold action, the original gauge group SU(3) 11 breaks to and chiral supermultiplets of this model are summarized in Table 8. This model is a threegeneration model with SU ( It turns out that this model has no color exotic fields, and the number of extra lepton doublet fields is very small (3 × 2). However, by gauge invariance there is no three point interaction for an up-type Yukawa coupling in this model. Then, to reproduce a suitable value for the top quark mass, we have to take into account higher-dimensional operators and larger VEVs for some singlets. 1; 1, 1, 1, 1, 1) 1; 1, 1   No. C.C. Group breaking n No. C.C. Group breaking n     Table 6: Massless spectrum of three-generation SU(3) C × SU(2) L × SU(2) R × U(1) B−L model. Representations under the non-Abelian group SU(3) C ×SU(2) L ×SU(2) R ×SU(2) F ×SU(3) 2 × SU(4) 2 and U(1) charges are listed. U and T mean the untwisted and twisted sector respectively. Note that the degeneracy of untwisted fields is 3, while it is 1 for twisted fields. The gravity and gauge supermultiplets are omitted.