Three-generation Asymmetric Orbifold Models from Heterotic String Theory
aa r X i v : . [ h e p - t h ] N ov CYCU-HEP-13-11KUNS-2468
Three-generation Asymmetric OrbifoldModels from Heterotic String Theory
Florian Beye ∗ , Tatsuo Kobayashi † and Shogo Kuwakino ‡ Department of Physics, Nagoya University, Furo-cho, Chikusa-ku, Nagoya 464-8602, Japan Department of Physics, Kyoto University, Kitashirakawa, Kyoto 606-8502, Japan Department of Physics, Chung-Yuan Christian University, 200, Chung-Pei Rd.Chung-Li,320, Taiwan
Abstract
Using Z asymmetric orbifolds in heterotic string theory, we construct N = 1 SUSYthree-generation models with the standard model gauge group SU (3) C × SU (2) L × U (1) Y and the left-right symmetric group SU (3) C × SU (2) L × SU (2) R × U (1) B − L . One of themodels possesses a gauge flavor symmetry for the Z twisted matter. ∗ Electronic address: [email protected] † Electronic address: [email protected] ‡ Electronic address: [email protected] Introduction
String theory is a candidate for a unified theory of the four fundamental forces includingquantum gravity. One of the main characteristic features of the standard model of particlephysics is the three-generation chiral structure of quarks and leptons, and there have beenmany attempts to construct four-dimensional models with three generations by string com-pactification.Heterotic string model building is one of the successful methods for a stringy realizationof particle physics models. Especially, in the framework of heterotic Z N or Z N × Z M orbifoldcompactifications [1], embedding a supersymmetric standard model or a higher dimensionalgrand unified theory into heterotic string theory is considered [2, 3, 4, 5, 6, 7, 8, 9] (also see areview [10]). Since strings on an orbifold can be described by a solvable world-sheet conformalfield 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 discretesymmetry [16, 17], which may lead to a hierarchical structure of masses/mixings of quarksand leptons.Asymmetric orbifold compactifications [18] can be considered as an extension of symmetricorbifolds, in which orbifold actions for left- and right-movers are generalized to be independentof each other, without destroying modular invariance of closed string theory. We may expectthat the generalization of the orbifold action will give us the possibility to construct a largenumber of four-dimensional string models, among which we can try to find phenomenologicallyviable models. However, in asymmetric orbifold constructions, model building for SUSYstandard models or other GUT extended models with three matter generations has not beeninvestigated thoroughly .In [21], a systematic approach for the construction of four-dimensional string models using Z asymmetric orbifolds, which are the simplest orbifolds capable of realizing N = 1 SUSYin four dimensions, is shown . First, one specifies a (22 , [26]. Usingthis method, 106 types of (22,6)-dimensional Narain lattices with a right-moving non-Abalianfactor are constructed. Group breaking patterns due to a Z N shift action are also analyzedby extending the argument of breaking patterns of the E group [27]. Furthermore, possiblegauge group patterns of Z asymmetric orbifold models are analyzed.The aim of this paper is to apply the asymmetric orbifolding procedure to the constructionof three-generation models with the standard model group SU (3) C × SU (2) L × U (1) Y and othergroups. In the next section, we review asymmetric orbifold model building. In section 3 and4 we construct Z asymmetric orbifold models with three generations. Section 5 is devoted toconclusions. In appendix A we show another three-generation model example. In free-fermionic string constructions, which are related to Z × Z (a)symmetric orbifolds by fermioniza-tion, the first SUSY standard-like models and left-right symmetric models with three generations were foundin [19, 20]. In [22], Z asymmetric orbifold models with one Wilson line were studied. 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 com-pactified on some (22,6)-dimensional Narain lattice. The corresponding world-sheet theorysplits into left-moving and right-moving degrees of freedom. Besides the ghost fields, thereare 26 left-moving bosons X L as well as ten right-moving boson-fermion pairs ( X R , Ψ R ). Themomentum modes p = ( p L , p R ) associated with the internal dimensions X ... and X ... lieon the Narain lattice Γ. Modular invariance of closed string theory implies that Γ is even( p = p − p ∈ Z ) and self-dual (Γ = ˜Γ). Here, Γ ≡ P i n i γ i and ˜Γ ≡ P i m i ˜ γ i , where γ i and˜ γ j are bases for the lattice Γ and its dual lattice ˜Γ, respectively. The bases for the lattice andits dual satisfy γ i · ˜ γ j = δ ij .In Z N orbifold constructions, one specifies an orbifold action for the internal dimensionsas follows: X L → θ L X L + V L , (1) X R → θ R X R + V R , (2)Ψ R → θ R Ψ R . (3)This action contains a twist θ = ( θ L , θ R ) and a shift V = ( V L , V R ). The twist has to be alattice automorphism of order N , i.e. θ N = 1. The shift has to satisfy N V ∈ Γ. In asymmetricorbifolds, 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 models with N = 1 SUSY and without a left-moving twist, i.e. θ L = 1 (also, one can set V R = 0). By writing θ R = diag( e πit , e πit , e πit )in some complex basis one can define a right-mover twist vector t R = (0 , t , t , t ). Whenacting on fermions one has to embed the twist into the double cover of SO(6), so t i R is onlydefined modulo 2. If P i t i R = 0 and t , , = 0 one realizes N = 1 SUSY. For our models weuse the right-mover twist vector t R = (0 , , , −
23 ) . (4)Now, we can fully specify a model by the following: • a (22,6)-dimensional Narain lattice Γ which contains a right-moving E or A lattice.These are the only lattices which allow for a N = 1 compatible Z automorphism [3, 28].In [21], such lattices were constructed from the well known 24-dimensional even self-duallattices by the lattice engineering method. • a Z shift vector V = ( V L ,
0) that satisfies 3 V ∈ Γ.In heterotic string theory, spacetime gauge symmetry is realized by left-moving masslessmodes. Generally, a four-dimensional Narain model has a gauge symmetry of rank 22. Anorbifold shift V L breaks the original group into a subgroup of same rank. The breaking patternscan be calculated by analyzing shift vectors and extended Dynkin diagrams [21].3n the case of Z asymmetric orbifold models with N = 1 SUSY whose twist action for theleft-mover is trivial, we can check that modular invariance of the closed string theory requiresthe shift vector V L to satisfy the condition3 V ∈ Z . (5)The massless spectrum in the untwisted sector can be read off as in the case of symmetricorbifolds. 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 | q i ∈ {|± , , , i , | ± , ± , ± , ± i +even } (6)are given by t R · q . Here, ”+even” means that we should take only combinations with theeven number of plus signs and the underline represents all cyclic permutations. The masslessmodes with non-trivial phases under the orbifold action t R are given by | q ′ i ∈ {| , , , i , | , , − , − i} : t R · q ′ ∼ / , (7) | q ′′ i ∈ {| , − , , i , | − , − , , i} : t R · q ′′ ∼ − / . (8)Note that the H -momentum states | q ′ i and | q ′′ i are CPT conjugate to each other. We definethe four-dimensional chirality as ”left-handed” if the first component of fermionic modes is 1 / | , , − , − i in | q ′ i . For the lattice part, the orbifold action ( θ, V ) acts on momentummodes p = ( p L , p R ) ∈ Γ and oscillator modes. The right-moving modes with p R = 0 oroscillator excitations are massive, so we do not need to consider orbifold phases for them. Inthe left-mover part, orbifold phases V L · p L arise for massless states with p = 2. Left-moveroscillators are not affected by the orbifold action because θ L = 1. This is the reason for theobserved rank preservation. Now, when combining left-mover and right-mover phases oneconcludes that only massless states which satisfy the condition p L · V L − t R · q ∈ Z (9)remain in the untwisted spectrum after the orbifold projection.To read off massless states in the Z twisted sector, we define I θ (Γ) as the sublattice ofthe original (22,6)-dimensional Narain lattice Γ that is invariant under the Z twist action θ . In our case where we have no twist action for the left-movers, I θ (Γ) is a 22-dimensionalleft-mover lattice which is spanned by 22 basis vectors α i =1 ... . We also define the dual latticeof the invariant sublattice ˜ I θ (Γ) ≡ P i =1 n i ˜ α i , where the ˜ α i satisfy α i · ˜ α j = δ ij and n i areintegers. In the α -twisted sector ( α ∈ { , } ), the momenta p = ( p L ,
0) lie on the shiftedlattice ˜ I θ (Γ) + αV . The massless left-mover modes in the α -twisted sector can be obtained bysolving the equation p L c L − . (10)4ince 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: | q α =10 i ∈ (cid:26) | , , , i , | , − , − , − i (cid:27) ( α = 1) , (11) | q α =20 i ∈ (cid:26) | , − , − , − i , | − , , , i (cid:27) ( α = 2) . (12)These are combined with massless states from the lattice part | p α =1 , i that satisfy (10), giving | p α =10 i ⊗ | q α =10 i ( α = 1) (13) | p α =20 i ⊗ | q α =20 i ( α = 2) . (14)The states from α = 1 and α = 2 are CPT conjugate to each other and together form afour-dimensional chiral supermultiplet. The degeneracy factor for the twisted sector is givenby D = Q { i | η i =0 } [2 sin( πη i )] p Vol( I θ (Γ)) , (15)where η i are defined as 0 ≤ t i + n i = η i < n i . The volume of the invariantlattice is determined as Vol( I θ (Γ)) = det( g ij ) = det( α i · α j ). In this paper we consider onlyorbifold models with the right-mover twist (4) and a right-mover E or a A lattice. For theselattices 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-movingcomplex chiral boson X ( z ) in terms of exponentials of boson fields φ , i∂X ( z ) = e iα · φ ( z ) + · · · , (16)where α corresponds to suitable root vectors. In order to reproduce the Z N orbifold conditionfor i∂X , i∂X ( ze − πi ) = e − πik/N i∂X ( z ) (17)for some integer k , the orbifold action on φ has to be a shift action: φ ( ze − πi ) = φ ( z ) − πs Q . (18)In this representation, the momenta associated to φ (the Q-charges) enter the right movermass equation. In particular, the massless states satisfy Q c R = 13 , (19)where Q ∈ Γ , + αs Q for α = 1 ,
2. The number of solutions to this equation correspondsto the degeneracy factor D . Now, let us denote the simple roots and fundamental weights5f a simple Lie group G as α Gi and ω Gi . Then, for models from Narain lattices with E , theright-moving twist (4) is replaced by the following shift for the internal six dimensions: s QR = ( ω A , , . (20)Here we use the decomposition A ⊂ E . For models from Narain lattices with A , thecorresponding shift action for each A factor is given by s QR = α A . (21) SU (3) C × SU (2) L × U (1) Y models fromasymmetric orbifolds In this section we present a Z asymmetric orbifold model with the standard model gaugegroup SU (3) C × SU (2) L × U (1) Y , following the construction in [21]. The model is constructedfrom a Narain lattice with a right-moving E factor, i.e. it has a degeneracy D = 3 in thetwisted sectors. Another such model is shown in Appendix A.As starting point we choose the 24-dimensional A Niemeier lattice with conjugacy classesgiven by the generators (3 , , , , , , , . (22)Using the lattice engineering technique, we replace an A factor contained in one of the A of A with a right-moving E factor as A −−−−−−→ decompose A × A × U (1) −−−−→ replace A × E × U (1) . The resulting lattice is a (22,6)-dimensional Narain lattice with A × U (1) × E whose conjugacyclasses are generated by (0 , , , , , , , / , , (0 , , , , , , , / , , (1 , , , , , , , / , , (0 , , , , , , , / , , (0 , , , , , , , / ,
0) (23)of A × U (1) × E . Here, the normalization for the U (1) is given by 2 √
3. At this stage,the compactified four-dimensional model possesses N = 4 SUSY and a SU (4) × U (1) gaugesymmetry.The gauge symmetry A × U (1) will be broken by the Z left-mover shift action. Possible Z shift vectors V have to satisfy 3 V ∈ I θ (Γ) where I θ (Γ) represents the 22-dimensional6o. C.C. Group breaking n ′ ( k ) i Shift vector (3 V ) (3 V ) A (0 , ,
0; 3) 0 02 0 A × U (1) (0 , ,
2; 0) ω A + 2 ω A
63 0 A × U (1) (0 , ,
0; 1) 2 ω A
44 0 A × U (1) (1 , ,
1; 1) ω A + ω A
25 0 A × U (1) (2 , ,
0; 0) 2 ω A + ω A
66 1 A (0 , ,
3; 0) 3 ω A A × U (1) (0 , ,
1; 1) ω A + ω A A × U (1) (1 , ,
0; 2) ω A A × U (1) (1 , ,
0; 0) ω A + 2 ω A
10 1 A × U (1) (2 , ,
1; 0) 2 ω A + ω A
11 2 A × U (1) (0 , ,
2; 1) 2 ω A
312 2 A × U (1) (0 , ,
0; 2) ω A
113 2 A (0 , ,
0; 0) 3 ω A
914 2 A × U (1) (1 , ,
1; 0) ω A + ω A + ω A
515 2 A × U (1) (2 , ,
0; 1) 2 ω A A group breaking patterns and shift vectors. The last component of n ′ ( k ) i correspondsto n ′ ( k )0 .sublattice invariant under the twist. In our case, the conjugacy classes of I θ (Γ) are given bythe generators (0 , , , , , , , / , (1 , , , , , , , / , (0 , , , , , , , / , (0 , , , , , , , /
4) (24)of A × U (1). In general, the shift vector V is composed of shift vectors for the seven A partsand a shift vector for U (1) part. The shift vector for each A belongs to one of the conjugacyclasses 0 , , A group breaking patterns by Z shift actions in general. The breaking patterns are listedin Table 1 (For the definition of n ′ ( k ) i , see [21]). In the table, breaking patterns due to shiftvectors which belong to conjugacy class 3 A are not listed as these can be reproduced fromshift vectors in 1 A by suitable reflections. Then, with (5) in mind we can specify a modularinvariant shift vector V by appropriately combining seven A breaking patterns with a shiftin the U (1) direction.Next, we consider a specific Z asymmetric orbifold construction from the A × U (1) × E lattice. We take the shift vector as V = ( α A + 2 α A , α A + 2 α A , − α A − α A , α A , , α A , α A , , / , (25)7hich belongs to conjugacy class (0 , , , , , , ,
0) of A × U (1) × E , i.e.3 V ∈ (Γ A , Γ A , Γ A , Γ A , Γ A , Γ A , Γ A , √ n, Γ E ) , (26)where Γ A (Γ E ) is an A ( E ) root lattice and n is an integer. The shift does not affect one ofthe A lattices and the U (1) part. Note that by Weyl reflections ± ( α A + 2 α A ) and α A areconnected to ω A + 2 ω A (No.2 in Table 1) and ω A + ω A (No.4) respectively. We can checkthat the shift vector V satisfies the modular invariance condition (5). The shift vector breaksthe original gauge symmetry SU (4) × U (1) to SU (4) × SU (3) × SU (2) × U (1) . (27)Massless states in the untwisted sector can be read off by collecting massless states in N = 4 gauge multiples of A × U (1) that satisfy the orbifold condition (9). These modes p are written as p = ( p (1) , p (2) , p (3) , p (4) , p (5) , p (6) , p (7) , ,
0) (28)of A × U (1) × E with p ( i =1 ... ∈ Γ A and p = 2. It is useful to denote the shift vector foreach A part in terms of the unbroken subgroups as ± ( α A + 2 α A ) = ( ± ( α A + 2 α A ) ,
0) (29)of A × U (1), and α A = (0 , , −√
2) (30)of A × U (1) . Now we can easily read off momentum modes with orbifold phase p · V ∼ / p (1 , ∈ { ( ω A , √ , ( ω A − α A , √ , ( ω A − α A − α A , √ } , (31) p (3) ∈ { ( ω A , − √ , ( ω A − α A , − √ , ( ω A − α A − α A , − √ } , (32) p (4 , , ∈ { ( ω A , , − √ , ( − ω A , , − √ , ( ω A , − , − √ , ( − ω A , − , − √ , (0 , , √ } , (33)and no solution for p (5) . These states survive the orbifold projection when combined with theright-moving H -momentum states (7), resulting in the following chiral supermultiplets:3 { ( , , , , , , ) + ( , , , , , , ) + ( , , , , , , )+ 2( , , , , , , ) + 2( , , , , , , ) + 2( , , , , , , )+ 3( , , , , , , ) } (34)of A × A × A × A (see Table 2 for U (1) charges). Note that the global factor 3 comes fromright-moving modes (7). 8et us now consider the Z twisted sector. In order to read off massless states (10) we haveto evaluate the dual of the invariant sublattice ˜ I θ (Γ). By taking suitable linear combinationsof (24) we can alternatively generate I θ (Γ) by the following generators:(1 , , , , , , , , (0 , , , , , , , , (0 , , , , , , , , (0 , , , , , , , / . (35)From this, one verifies that I θ (Γ) is spanned by the basis α , = ( α A , , , , , , , , , (36) α , = (0 , α A , , , , , , , , (37) α , = (0 , , α A , , , , , , , (38) α ... = (0 , , , α A ... , , , , , (39) α ... = (0 , , , , α A ... , , , , (40) α ... = (0 , , , , , α A ... , , , (41) α ... = (0 , , , , , , α A ... , , (42) α = ( ω A , , , ω A , ω A , ω A , ω A , , (43) α = (0 , ω A , , ω A , ω A , ω A , ω A , , (44) α = (0 , , ω A , ω A , ω A , ω A , ω A , , (45) α = (0 , , , ω A , ω A , ω A , ω A , / . (46)Also, we can evaluate a dual basis ˜ α i and find that ˜ I θ (Γ) is given by the following conjugacyclass generators (1 , , , , , , , − / , (1 , , , , , , , − / , (3 , , , , , , , − / , (2 , , , , , , , − / , (47)with a U (1) normalization of 2 / √
3. Now, we can read off the massless spectrum in thetwisted sector by solving (10). Taking all linear combinations of (47) results in 256 conjugacyclasses for the dual invariant sublattice. Among them, here we shall show only massless statesthat arise from the conjugacy class (1 , , , , , , , − / α twisted sector belong to p + αV ∈ (Γ A + ω A + α ( α A + 2 α A )3 , Γ A + ω A + α ( α A + 2 α A )3 , Γ A + ω A + α ( − α A − α A )3 , Γ A + ω A + αα A , Γ A , Γ A + αα A , Γ A + αα A , √ n −
14 )) . (48)9or α = 1, we can find a solution for (10) as(0 , √ , , − √ , p ′′′ A , √ , , , √ , , , , − √ , , , − √ , − √ , (49)lying in the unbroken group( A × U (1)) × A × U (1) × A × (cid:0) A × U (1) (cid:1) × U (1) , (50)where p ′′′ A ∈ { ω A , ω A − α A , ω A − α A − α A } . (51)Combined with the H -momentum states in (11) this leads to three chiral supermultiplets whichtransform under the non-Abelian group as ( , , , , , , ). Here, the degeneracy ”three”comes from (15), with D = √ √ SU (3) C × SU (2) L × SU (2) × SU (3) × SU (4) × U (1) , and thereare 3 ×
12 fields in the untwisted sector and 3 ×
37 fields in the Z twisted sector. Here, allfields are labeled by f i ( i = 1 . . . U (1) charges U ... are taken as U , , = 2 √ Q , , , (53) U , , = Q , , , (54) U , , = √ Q , , , (55) U = √ Q . (56)Among the ten U (1) groups, a non-anomalous U (1) Y gauge symmetry is taken from a combi-nation of four U (1)s, Q Y = 12 √ U + 1 √ U − U − √ U . (57)By choosing this combination, we can see that this model contains three standard modelgenerations 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 followingcombination Q A = 1 √ U + 1 √ U − √ U + √ U − √ U − √ U . (58)10n a four-dimensional model with an anomalous U (1) A gauge symmetry, a string loop effectwill generate a Fayet-Iliopoulos D -term [30, 31, 32]. For the anomalous U (1) A we can checkthat mixed anomalies satisfy the Green-Schwarz universality relation k a Tr G a T ( R ) Q A = Tr Q B Q A = 13 Tr Q A = 124 Tr Q A = 8 π δ GS = r , (59)where G a and k a are non-Abelian groups and their Kac-Moody levels, and 2 T ( R ) is the Dynkinindex of the representation R . Q B represents a non-anomalous U (1) group of level one thatis orthogonal to U (1) A .By the U (1) Y charge assignment (57) we can see that this model has net three standardmodel 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 ), and further three generations of quark doublets q , right-handed down-type quarks u andup-type Higgs fields h u come from the Z twisted sector (fields f , f and f ). We findtwo pairs of SU (3) C color exotics and some extra SU (2) L doublets. There are also 3 × SU (3) C × SU (2) L × SU (2) × SU (3) × SU (4), and all other SU (3) C × SU (2) L singlets are non-trivially charged under the additional non-Abelian group SU (2) × SU (3) × SU (4).In this paper we do not consider explicitly all of the terms in the superpotential of thismodel 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 couplingsof this model. Regarding the color exotic fields, c , c , c and c have a three point couplingwith singlets as s c c , s c c , (60)so they are expected to decouple from the low energy theory if the singlets s and s get VEVs.Similar thing can happen for l u and l u since there is a three point coupling s u l u l u . (61)Fields f , f and f contain four ( , ) − / fields and two ( , ) / fields after breakingadditional A and A groups. Then, we can expect that net two ( , ) − / remain masslessfields, and these fields can be identified as the lepton doublet and down-type Higgs fields. Wecan see that the other exotic fields have vector-like structure.By analyzing three point couplings allowed by Q-charge invariance, relevant couplings forquarks and Higgses are given by y q h u u + y q h u u + y q h u u + y q h u u + y q h u u + y q h u u , (62) See e.g. [33] and references therein. It is also possible to realize the up-type quark and up-type Yukawa coupling from the untwisted sector bychoosing other Narain lattices and shift vectors. Q ( X ) = ( ω A , , , (63) Q ( X ) = ( ω A − α A , , , (64) Q ( X ) = ( ω A − α A − α A , ,
0) (65)for X ∈ { q, h u , u } . Since in asymmetric orbifolds the starting point is a torus compactificationat self-dual radius, string world-sheet instanton effects can be neglected. So, each coefficient y etc. for the three point couplings is expected to be of O (1). From above couplings itturns out that the mass of the second generation quark will be of the same order as the topquark mass. SU (3) C × SU (2) L × SU (2) R × U (1) B − L models from asymmetric orbifolds In this section we construct a Z asymmetric orbifold model with the left-right symmetricgroup SU (3) C × SU (2) L × SU (2) R × U (1) B − L . We use a Narain lattice with A as our startingpoint, so the resulting model has a degeneracy factor D = 1.We start from four-dimensional heterotic string theory compactified on a (20,4)-dimensionalNarain lattice A × A × U (1) × A and a (2,2)-dimensional A × A lattice. Using the lat-tice engineering technique, the (20,4)-dimensional lattice is made from the 24-dimensional A Niemeier lattice as A −−−−−−→ decompose ( A × A × U (1)) × A −−−−→ replace A × A × U (1) × A . Here, the left-moving A factor in A is replaced by the right-moving A factor. Correspondingconjugacy class generators are given by(1 , , , , , , / , , , , (0 , , , , , , / , / , , , (0 , , , , , , / , − / , , , (0 , , , , , , / , / , ,
1) (66)of A × A × U (1) × A . Here, the normalizations for the two U (1)s are taken as √
30. At thisstage, the compactified four-dimensional model possesses N = 4 SUSY and SU (5) × SU (3) × SU (2) × U (1) gauge symmetry.Gauge symmetry breaking patterns for this model can be analyzed as in the previoussection. Breaking patterns for A , A and A groups are listed in Table 3, 4 and 5.Next, we consider the Z asymmetric orbifold model which is specified by the shift vector V =(0 , ω A , ω A + ω A − α A − α A − α A − α A , − ω A + α A + α A + α A + α A , − ω A − ω A + 2 α A , ω A + 2 ω A − α A − α A , √ , √ , , , , / . (67)12his shift vector belongs to the conjugacy class (0 , , , , , , / , / , , , ,
0) of A × A × U (1) × A × A , i.e.3 V ∈ (Γ A , Γ A + ω A , Γ A , Γ A + ω A , Γ A + ω A , Γ A , √ n + 1 / , √ n + 3 / , Γ A , Γ A , Γ A , Γ A ) (68)where Γ A , Γ A and Γ A are the root lattices of A , A and A , and n and n are integers.We can check that the shift vector V satisfies the modular invariance condition (5). The shiftvector breaks the original gauge symmetry SU (5) × SU (3) × SU (2) × U (1) to SU (4) × SU (3) × SU (2) × U (1) . (69)When we read off the massless spectrum the following description of the shift vector in termsof the unbroken gauge group is useful:2 ω A + ω A − α A − α A − α A − α A = ( − α A − α A , , , (70) − ω A + α A + α A + α A + α A = (0 A , √ , (71) − ω A − ω A + 2 α A = ( − ω A , − √ , (72) ω A + 2 ω A − α A − α A = ( α A + 2 α A , , . (73)Massless states in the untwisted sector and twisted sector can be read off as in the previoussection by using the following information of the lattice. By taking a suitable linear combi-nation of conjugacy class generators, the (20,4)-dimensional part of the invariant sublattice I θ (Γ , ) is described as (0 , , , , , , , , (0 , , , , , , , / , (1 , , , , , , / ,
0) (74)of A × A × U (1) . We can check that this 20-dimensional lattice is spanned by the followingbasis α = ( α A , , , , , , , , (75) α = (0 , α A , , , , , , , (76) α ... = (0 , , α A ... , , , , , , (77) α ... = (0 , , , α A ... , , , , , (78) α ... = (0 , , , , α A ... , , , , (79) α ... = (0 , , , , , α A ... , , , (80) α = (0 , , ω A , ω A , ω A , ω A , , , (81) α = (0 , ω A , ω A , , ω A , ω A , , / , (82) α = ( ω A , , ω A , , ω A , ω A , / , . (83)13lso, by some calculation we can evaluate a dual basis ˜ α i and obtain that ˜ I θ (Γ , ) is spannedby the conjugacy class generators (1 , , , , , , / , , (0 , , , , , , , / , (0 , , , , , , / , / , (0 , , , , , , / , / , (0 , , , , , , / , / , (84)with the U (1)s normalized to 10 / √
30. For the (2,2)-dimensional lattice Γ , , the invariantsublattice I θ (Γ , ) is given by the conjugacy class 0 of A , so the dual lattice ˜ I θ (Γ , ) is theunion of 0, 1 and 2. For this model the degeneracy factor (15) is calculated as D = √ √ · . (85)The resulting massless spectrum of this model is listed in Table 6 and 7. The four-dimensional gauge group is SU (3) C × SU (2) L × SU (2) R × SU (2) F × SU (3) × SU (4) × U (1) .This model has 3 × Z twisted sector. Inthe table, normalizations for U (1) charges U ... are taken as U = √ Q , (86) U , , , = r Q , , , , (87) U , = √ Q , . (88)Among the seven U (1) groups, a non-anomalous U (1) B − L gauge symmetry is taken from acombination of four U (1)s, Q B − L = r U − √ U − √ U + r U . (89)There is an anomalous U (1) A gauge symmetry that is given by the following combination Q A = 1 √ U − r U + 12 √ U − √ U + 12 r U , (90)with the GS universality relation by 8 π δ GS = 2 q .By choosing this combination, we can see that this model contains three chiral mattergenerations of a supersymmetric left-right symmetric model, and the additional fields havevector-like structure. This model has ten singlets of the non-Abelian part SU (3) C × SU (2) L × SU (2) R × SU (2) F × SU (3) × SU (4) . The number of fields in this model is relatively small,14 × f and f in the Z twisted sector are identified as three generationsof left-handed quarks Q L ( , , ) / , referred to as Q L and Q L . Interestingly, this model hasa SU (2) F gauge flavor symmetry which unifies the first two generations of Q L fields into the SU (2) F doublet Q L , and the third generation as a SU (2) F singlet Q L . The possibility to haveflavor gauge symmetries is a characteristic property of the asymmetric orbifold construction:since we do not consider any left-moving twist action, the zero point energy for the left-moversdoes not increase. Then, states in a non-trivial representation of a flavor gauge symmetry canbe realized at massless level. Fields f ... are SU (2) F doublets without B-L charge, so thesefields can be candidates for a SU (2) F flavon field. This model does not contain matter fieldsin the adjoint representation. Therefore, we expect SU (2) R × U (1) B − L to be broken by a VEVof the doublet fields ( , +1 /
2) and ( , − / HQ L Q R3 is not allowed by the Q-charge invariance though this operator is gauge invariant. Then, toreproduce a suitable value for the top quark mass, we need to consider higher-dimensionaloperators and larger VEVs for some singlets. From Z heterotic asymmetric orbifolds, we construct several three-generation models with thestandard model gauge group or the left-right symmetric group. The starting points for modelbuilding are Narain lattices with E or A which are obtained from 24-dimensional Niemeierlattices by the lattice engineering technique. By taking a modular invariant combination of Z shift actions for the left-mover we obtain four-dimensional three-generation models withvector-like exotics.For models from Narain lattices with E , the number of ”three” standard model generationsoriginates from the degeneracy factor D = 3. Also, the up-type quark resides in the Z twisted sector and its Yukawa coupling is expected to be realized through a coupling of threefields 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-pointcouplings lead to a too heavy mass for the second generation quark. Regarding the modelwith degeneracy factor D = 1, even though this is a left-right symmetric model with threegenerations, 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 fromother Narain lattices with A .In asymmetric orbifold constructions, it turns out that it is possible to construct modelswith 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 arisingfrom discrete flavor symmetries. Analyzing moduli stabilization in this formalism will also beimportant. 15 cknowledgement F.B. was supported by the ”Leadership Development Program for Space Exploration andResearch” from the Japan Society for the Promotion of Science. T.K. was supported in partby the Grant-in-Aid for Scientific Research No. 25400252 from the Ministry of Education,Culture, Sports, Science and Technology of Japan. S.K. was supported by the Taiwan’sNational Science Council under grants NSC102-2811-M-033-002 and NSC102-2811-M-033-008.
A Another example of a three-generation model
Here, we show another example of a simple three-generation model with the standard modelgroup. To specify the model, we take the (22,6)-dimensional Narain lattice A × E as astarting point, and also take a Z shift vector as V = ( ω A , ω A , ω A , , ω A , ω A , ω A , ω A , ω A , − ω A , α A , / . (91)This shift vector belongs to the conjugacy class (2 , , , , , , , , , , ,
0) of A × E . Bythe orbifold action, the original gauge group SU (3) breaks to SU (3) × SU (2) × U (1) , (92)and chiral supermultiplets of this model are summarized in Table 8. This model is a three-generation model with SU (3) C × SU (2) L × U (1) group. The fields non-trivially charged under SU (3) C × SU (2) L are 3 (cid:8) ( , ) , , ) , , ) (cid:9) . (93)Also, there are 3 ×
10 singlets under the non-Abelian group SU (3) C × SU (2) L × SU (2) , theother fields are charged under the hidden group SU (2) as the (bi-)fundamental representation.It turns out that this model has no color exotic fields, and the number of extra lepton doubletfields is very small (3 × References [1] L. J. Dixon, J. A. Harvey, C. Vafa and E. Witten, Nucl. Phys. B (1985) 678; Nucl.Phys. B (1986) 285.[2] L. E. Ibanez, H. P. Nilles and F. Quevedo, Phys. Lett. B (1987) 25; L. E. Ibanez,J. E. Kim, H. P. Nilles and F. Quevedo, Phys. Lett. B (1987) 282.163] Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo, Y. Ono and K. Tanioka, Nucl.Phys. B (1990) 611.[4] T. Kobayashi, S. Raby and R. J. Zhang, Phys. Lett. B (2004) 262[arXiv:hep-ph/0403065]; Nucl. Phys. B (2005) 3 [arXiv:hep-ph/0409098].[5] W. Buchmuller, K. Hamaguchi, O. Lebedev and M. Ratz, Phys. Rev. Lett. (2006)121602 [arXiv:hep-ph/0511035]; Nucl. Phys. B (2007) 149 [arXiv:hep-th/0606187].[6] J. E. Kim and B. Kyae, Nucl. Phys. B (2007) 47 [arXiv:hep-th/0608086].[7] O. Lebedev, H. P. Nilles, S. Raby, S. Ramos-Sanchez, M. Ratz, P. K. S. Vaudrevange andA. Wingerter, Phys. Lett. B , 88 (2007) [hep-th/0611095]; Phys. Rev. D , 046013(2008) [arXiv:0708.2691 [hep-th]].[8] M. Blaszczyk, S. Nibbelink Groot, M. Ratz, F. Ruehle, M. Trapletti and P. K. S. Vau-drevange, Phys. Lett. B , 340 (2010) [arXiv:0911.4905 [hep-th]].[9] S. Groot Nibbelink and O. Loukas, arXiv:1308.5145 [hep-th].[10] H. P. Nilles, S. Ramos-Sanchez, M. Ratz and P. K. S. Vaudrevange, Eur. Phys. J. C ,249 (2009) [arXiv:0806.3905 [hep-th]].[11] S. Hamidi and C. Vafa, Nucl. Phys. B , 465 (1987); L. J. Dixon, D. Friedan, E. J. Mar-tinec and S. H. Shenker, Nucl. Phys. B , 13 (1987).[12] T. T. Burwick, R. K. Kaiser and H. F. Muller, Nucl. Phys. B , 689 (1991); J. Er-ler, D. Jungnickel, M. Spalinski and S. Stieberger, Nucl. Phys. B , 379 (1993)[hep-th/9207049].[13] K. -S. Choi and T. Kobayashi, Nucl. Phys. B , 295 (2008) [arXiv:0711.4894 [hep-th]].[14] T. Kobayashi, S. L. Parameswaran, S. Ramos-Sanchez and I. Zavala, JHEP , 008(2012) [Erratum-ibid. , 049 (2012)] [arXiv:1107.2137 [hep-th]]:[15] N. G. Cabo Bizet, T. Kobayashi, D. K. Mayorga Pena, S. L. Parameswaran, M. Schmitzand I. Zavala, JHEP , 076 (2013) [arXiv:1301.2322 [hep-th]].[16] T. Kobayashi, H. P. Nilles, F. Ploger, S. Raby and M. Ratz, Nucl. Phys. B , 135(2007) [hep-ph/0611020].[17] P. Ko, T. Kobayashi, J. -h. Park and S. Raby, Phys. Rev. D , 035005 (2007) [Erratum-ibid. D , 059901 (2007)] [arXiv:0704.2807 [hep-ph]].[18] K. S. Narain, M. H. Sarmadi and C. Vafa, Nucl. Phys. B , 551 (1987).[19] A. E. Faraggi, D. V. Nanopoulos and K. -j. Yuan, Nucl. Phys. B , 347 (1990).[20] G. B. Cleaver, A. E. Faraggi and C. Savage, Phys. Rev. D , 066001 (2001)[hep-ph/0006331]. 1721] F. Beye, T. Kobayashi and S. Kuwakino, Nucl. Phys. B , 599 (2013) [arXiv:1304.5621[hep-th]].[22] L. E. Ibanez, J. Mas, H. -P. Nilles and F. Quevedo, Nucl. Phys. B , 157 (1988).[23] K. S. Narain, Phys. Lett. B , 41 (1986).[24] J. Leech, Can. J. Math. 19 (1967) 251; H. Niemeier, J. Number Theory 5 (1973) 142.[25] M. Ito, S. Kuwakino, N. Maekawa, S. Moriyama, K. Takahashi, K. Takei, S. Teraguchiand T. Yamashita, Phys. Rev. D (2011) 091703 [arXiv:1012.1690 [hep-ph]]; JHEP , 100 (2011) [arXiv:1104.0765 [hep-th]].[26] W. Lerche, A. N. Schellekens and N. P. Warner, Phys. Rept. , 1 (1989).[27] Y. Katsuki, Y. Kawamura, T. Kobayashi, N. Ohtsubo and K. Tanioka, Prog. Theor. Phys. , 171 (1989).[28] T. Kobayashi and N. Ohtsubo, Int. J. Mod. Phys. A , 87 (1994).[29] Z. Kakushadze, G. Shiu and S. H. H. Tye, Nucl. Phys. B , 547 (1997) [hep-th/9704113].[30] M. Dine, N. Seiberg and E. Witten, Nucl. Phys. B , 589 (1987).[31] J. J. Atick, L. J. Dixon and A. Sen, Nucl. Phys. B , 109 (1987).[32] M. Dine, I. Ichinose and N. Seiberg, Nucl. Phys. B , 253 (1987).[33] T. Kobayashi and H. Nakano, Nucl. Phys. B , 103 (1997) [hep-th/9612066].18 /T f Irrep . Q Q Q Q Q Q Q Q Q Q Q Y Q A Deg .U , ; , , , , ) 0 0 0 0 1 0 0 0 0 0 0 U , ; , , , , ) 0 0 0 0 0 0 1 0 0 0 1 − U s u ( , ; , , , , ) 0 0 0 0 0 0 0 0 1 0 0 − U , ; , , , , ) 0 0 0 1 − − U , ; , , , , ) 0 0 0 − − − U , ; , , , , ) 0 0 0 0 0 1 − −
12 16 U , ; , , , , ) 0 0 0 0 0 − − −
12 16 U , ; , , , , ) 0 1 0 0 0 0 0 0 0 0 0 U , ; , , , , ) 0 0 − U l u ( , ; , , , , ) 0 0 0 0 0 0 0 1 − −
12 16 U l u ( , ; , , , , ) 0 0 0 0 0 0 0 − −
12 16 U d ( , ; , , , , ) 1 0 0 0 0 0 0 0 0 0
13 23 T
13 ( , ; , , , , ) −
12 14 − − −
12 16 − T
14 ( , ; , , , , ) −
12 14 − − −
12 16 − − T
15 ( , ; , , , , ) −
12 14 − − − −
12 16 − − T
16 ( , ; , , , , ) − −
12 12 12 16 12 16 −
12 16 13 − T
17 ( , ; , , , , )
14 14 12 −
12 16 −
13 13 T
18 ( , ; , , , , )
14 14 12 − −
12 16 − − − T
19 ( , ; , , , , ) − −
23 12 16 12 16 T s ( , ; , , , , ) − −
12 12 −
12 16 −
12 16 12 16 − − T s ( , ; , , , , )
14 14 12 − −
12 16
23 13 − T
22 ( , ; , , , , ) − − −
12 16 −
12 16 23 T
23 ( , ; , , , , ) − − −
12 16 −
12 16 − T
24 ( , ; , , , , )
14 14 12 −
13 12 − −
13 13 − T
25 ( , ; , , , , ) − − − − −
13 12 16 − − T
26 ( , ; , , , , ) −
14 12 −
12 16 − −
12 16 − T
27 ( , ; , , , , ) − − −
14 12 16 −
12 16 − − T
28 ( , ; , , , , )
14 12 − −
13 12 16 12 16 T
29 ( , ; , , , , )
14 14 −
12 16 −
13 12 16 23 − T
30 ( , ; , , , , )
14 14 −
12 16 −
13 12 16 − T
31 ( , ; , , , , )
14 14 − − −
12 16 −
13 13 T
32 ( , ; , , , , ) −
12 14 14 −
12 16 12 16 − − T
33 ( , ; , , , , ) −
12 14 12 16 − − − T
34 ( , ; , , , , )
14 14 − −
12 16 12 16 −
12 16 13 14 12 T
35 ( , ; , , , , )
14 14 −
14 12 16 −
12 16 12 16 −
13 14 12 T
36 ( , ; , , , , )
14 14 − − − −
14 56 T
37 ( , ; , , , , ) −
12 14 −
16 12 16 13 − T
38 ( , ; , , , , ) − −
12 16 − −
13 12 12 T
39 ( , ; , , , , )
14 14
12 16 −
12 16
12 13 T c ( , ; , , , , ) 0 −
14 12 16 12 16 −
13 23 −
13 12 T c ( , ; , , , , ) 0 −
14 12 16 12 16 − −
23 23 12 T c ( , ; , , , , )
12 14 − − − −
12 16 −
13 13 12 T c ( , ; , , , , ) − − − −
12 16 − −
13 13 − T
44 ( , ; , , , , ) −
12 14 − − −
13 12 − − − T
45 ( , ; , , , , )
14 14 12
16 12 16 −
13 12 T
46 ( , ; , , , , ) −
14 12 16 −
16 13 −
12 12 T q ( , ; , , , , ) 0 − −
12 16 −
12 16
16 13 T u ( , ; , , , , ) −
14 14 12 12 16 −
13 12 16 − − T h u ( , ; , , , , ) − − −
13 12 16 − − − Table 2: Massless spectrum of three-generation SU (3) C × SU (2) L × U (1) Y model. Represen-tations under the non-Abelian group SU (3) C × SU (2) L × SU (2) × SU (3) × SU (4) and U (1)charges are listed. U and T mean the untwisted and twisted sector respectively. Note that allfields have degeneracy 3. The gravity and gauge supermultiplets are omitted.19o. C.C. Group breaking n ′ ( k ) i Shift vector (3 V ) (3 V ) A (0; 3) 0 02 0 U (1) (2; 1) 2 ω A
23 1 U (1) (1; 2) ω A A (3; 0) 3 ω A Table 3: A group breaking patterns and shift vectors. The last component of n ′ ( k ) i correspondsto n ′ ( k )0 . No. C.C. Group breaking n ′ ( k ) i Shift vector (3 V ) (3 V ) A (0 ,
0; 3) 0 02 0 A (0 ,
3; 0) 3 ω A
63 0 U (1) (1 ,
1; 1) ω A + ω A
24 0 A (3 ,
0; 0) 3 ω A
65 1 A × U (1) (0 ,
2; 1) 2 ω A A × U (1) (1 ,
0; 2) ω A A × U (1) (2 ,
1; 0) 2 ω A + ω A Table 4: A group breaking patterns and shift vectors. The last component of n ′ ( k ) i correspondsto n ′ ( k )0 . Breaking patterns by shift vectors which belong to conjugacy class 2 A are omitted.20o. C.C. Group breaking n ′ ( k ) i Shift vector (3 V ) (3 V ) A (0 , , ,
0; 3) 0 02 0 A × U (1) (0 , , ,
1; 0) 2 ω A + ω A
83 0 A × A × U (1) (0 , , ,
2; 0) ω A + 2 ω A
64 0 A × U (1) (0 , , ,
0; 1) ω A + ω A
45 0 A × U (1) (1 , , ,
1; 1) ω A + ω A
26 0 A × U (1) (1 , , ,
0; 0) ω A + 2 ω A
87 0 A × A × U (1) (2 , , ,
0; 0) 2 ω A + ω A
68 1 A × U (1) (0 , , ,
2; 0) ω A + 2 ω A A × A × U (1) (0 , , ,
0; 1) 2 ω A
10 1 A × U (1) (0 , , ,
1; 1) ω A + ω A
11 1 A (0 , , ,
0; 0) 3 ω A
12 1 A × U (1) (1 , , ,
0; 2) ω A
13 1 A × U (1) (1 , , ,
0; 0) ω A + ω A + ω A
14 1 A × A × U (1) (2 , , ,
1; 0) 2 ω A + ω A
15 2 A (0 , , ,
3; 0) 3 ω A
16 2 A × U (1) (0 , , ,
1; 1) ω A + ω A
17 2 A × A × U (1) (0 , , ,
0; 2) ω A
18 2 A × U (1) (0 , , ,
0; 0) 2 ω A + ω A
19 2 A × A × U (1) (1 , , ,
0; 0) ω A + 2 ω A
20 2 A × U (1) (1 , , ,
1; 0) ω A + ω A + ω A
21 2 A × U (1) (2 , , ,
0; 1) 2 ω A Table 5: A group breaking patterns and shift vectors. The last component of n ′ ( k ) i correspondsto n ′ ( k )0 . Breaking patterns by shift vectors which belong to conjugacy classes 3 A and 4 A areomitted. 21 /T f Irrep . Q Q Q Q Q Q Q Q B − L Q A Deg .U , , , , , , , ) 1 0 0 0 0 0 0 0 1 3 U , , , , , , , ) 0 0 1 0 0 0 0 0 U , , , , , , , ) 0 0 0 − U , , , , , , , ) 0 0 0 0 1 0 0 −
16 14 U , , , , , , , ) 0 − −
12 14 T , , , , , , , ) −
25 415 815 −
45 45 35 T , , , , , , , )
16 45 − −
415 25 − − T , , , , , , , ) − − − − − −
25 45 − T , , , , , , , ) − −
25 415 815 −
45 45 − − − T
10 ( , , , , , , , ) −
13 45 − −
415 25 −
45 25 − T
11 ( , , , , , , , ) −
13 45 −
815 815 − − T
12 ( , , , , , , , ) −
13 45 415 815 25 25 − − − T
13 ( , , , , , , , ) − −
15 15 15 12 14 T
14 ( , , , , , , , ) − − −
45 15 12 14 T
15 ( , , , , , , , ) − −
13 815 − − T
16 ( , , , , , , , )
16 45 115 − − −
15 12 T
17 ( , , , , , , , ) −
25 415 215 −
15 25 − −
12 14 T
18 ( , , , , , , , ) − − − −
45 35 − T
19 ( , , , , , , , ) −
25 415 815 − −
15 35 T
20 ( , , , , , , , )
16 45 − −
415 25 15 − T
21 ( , , , , , , , )
16 45 415 815 25 −
35 15 T
22 ( , , , , , , , ) − −
25 415 −
415 25 − T
23 ( , , , , , , , ) − −
25 415 −
415 25 T
24 ( , , , , , , , ) − −
25 415 815 − − − − − T
25 ( , , , , , , , ) −
13 45 − −
415 25 15 25 − T
26 ( , , , , , , , ) −
25 115 815 25 15 − − T
27 ( , , , , , , , ) − − −
115 25 −
15 15 T
28 ( , , , , , , , ) −
815 815 25 −
25 35 T
29 ( , , , , , , , ) − −
815 815 25 − − − − T
30 ( , , , , , , , ) − −
415 25 25 − T
31 ( , , , , , , , )
16 25 415 − −
45 25 − T
32 ( , , , , , , , ) −
13 25 415 − −
45 25 25 − T
33 ( , , , , , , , )
16 25 415 −
115 25 − − T
34 ( , , , , , , , ) − − − − − − T
35 ( , , , , , , , ) −
25 415 −
415 25 − − T
36 ( , , , , , , , ) −
25 415 −
415 25 T
37 ( , , , , , , , )
16 45 415 815 25 25 15 T
38 ( , , , , , , , ) −
25 415 −
415 25 T
39 ( , , , , , , , ) − − − − − − T
40 ( , , , , , , , ) −
25 415 −
415 25 − − T
41 ( , , , , , , , ) −
25 415 −
415 25 T
42 ( , , , , , , , )
16 45 415 815 25 25 15 T
43 ( , , , , , , , ) −
25 415 −
415 25 T
44 ( , , , , , , , ) −
25 715 − − − − T
45 ( , , , , , , , ) −
25 115 815 25 − − − T
46 ( , , , , , , , ) − − − −
415 25 − T
47 ( , , , , , , , ) − − −
115 25 45 15 T
48 ( , , , , , , , ) − −
25 415 1115 25 − − T
49 ( , , , , , , , )
16 45 415 −
715 25 25 15 T
50 ( , , , , , , , ) − −
815 815 − − T
51 ( , , , , , , , ) − − −
815 815
25 25 − − T
52 ( , , , , , , , ) − −
25 415 − − − T
53 ( , , , , , , , )
16 25 − −
15 23 T
54 ( , , , , , , , ) −
25 415 13 −
25 15 −
13 14 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) × SU (4) and U (1) charges are listed. U and T mean the untwisted and twisted sector respec-tively. Note that the degeneracy of untwisted fields is 3, while it is 1 for twisted fields. Thegravity and gauge supermultiplets are omitted.22 /T f Irrep . Q Q Q Q Q Q Q Q B − L Q A Deg .T
55 ( , , , , , , , )
16 45 415 − − −
25 35 43 T
56 ( , , , , , , , ) −
13 45 415 − − − −
25 13 − T Q R3 ( , , , , , , , ) − − −
415 15 − − −
16 14 T
58 ( , , , , , , , )
415 815 − − − T
59 ( , , , , , , , ) −
25 115 − −
25 25 15 13 14 T
60 ( , , , , , , , )
16 15 − − −
15 25 35 T
61 ( , , , , , , , )
16 15 415 815 − − − − T
62 ( , , , , , , , ) −
13 15 − − −
15 25 − − T H ( , , , , , , , ) −
13 15 415 815 − −
25 25 − T
64 ( , , , , , , , ) −
13 35 415 −
415 25 − T
65 ( , , , , , , , ) −
15 415 815 25 25 15 −
12 14 T
66 ( , , , , , , , )
16 15 − −
415 25 −
25 15 12 14 T
67 ( , , , , , , , )
16 15 415 − −
15 12 T
68 ( , , , , , , , )
16 15 −
815 13 25 − − T Q L ( , , , , , , , )
16 15 415 − − −
15 16 14 T Q L ( , , , , , , , )
16 15 415 − −
15 16 14 Table 7: Massless spectrum of three-generation SU (3) C × SU (2) L × SU (2) R × U (1) B − L model(continued). Representations under the non-Abelian group SU (3) C × SU (2) L × SU (2) R × SU (2) F × SU (3) × SU (4) and U (1) charges are listed. U and T mean the untwisted andtwisted sector respectively. Note that the degeneracy of untwisted fields is 3, while it is 1 fortwisted fields. The gravity and gauge supermultiplets are omitted.23 /T f Irrep . Q Q Q Q Q Q Q Q Q Q Q Deg . U 1 ( , ; , , , , , , , ) 0 0 0 0 0 0 0 0 0 − , ; , , , , , , , ) 0 0 0 0 0 0 0 0 0 , ; , , , , , , , ) 0 0 0 0 0 0 0 0 0 − , ; , , , , , , , ) 1 0 0 0 0 0 0 0 0 0 0 3U 5 ( , ; , , , , , , , ) 0 1 0 0 0 0 0 0 0 0 0 3U 6 ( , ; , , , , , , , ) 0 0 − , ; , , , , , , , ) 0 0 0 1 0 0 0 0 0 0 0 3U 8 ( , ; , , , , , , , ) 0 0 0 0 1 0 0 0 0 0 0 3U 9 ( , ; , , , , , , , ) 0 0 0 0 0 1 0 0 0 0 0 3U 10 ( , ; , , , , , , , ) 0 0 0 0 0 0 1 0 0 0 0 3U 11 ( , ; , , , , , , , ) 0 0 0 0 0 0 0 − , ; , , , , , , , ) 0 0 0 0 0 0 0 0 − , ; , , , , , , , )
29 29 49 29 29 29 29 49 − − , ; , , , , , , , ) − − − −
49 29 29 −
49 49 −
29 13 , ; , , , , , , , )
29 29 −
29 29 29 − − − −
29 13 −
3T 16 ( , ; , , , , , , , ) − − − − −
49 29 29 −
29 49 − −
3T 17 ( , ; , , , , , , , ) − −
49 49 29 −
49 29 − − − −
16 13
3T 18 ( , ; , , , , , , , ) − − − − −
49 29 49 − −
16 13
3T 19 ( , ; , , , , , , , ) −
49 29 49 − − −
49 29 − −
29 13 , ; , , , , , , , ) −
19 29 49 −
49 29 29 − − − − −
3T 21 ( , ; , , , , , , , ) − −
29 29 −
49 29 −
49 49 − − −
3T 22 ( , ; , , , , , , , ) −
49 19 29 29 29 − −
29 49 13 , ; , , , , , , , )
29 29 −
59 29 29 29 29 49 −
29 13 , ; , , , , , , , )
29 29 19 −
49 29 −
49 29 −
29 49 − −
3T 25 ( , ; , , , , , , , ) −
49 29 − −
19 29 −
49 29 49 − − −
3T 26 ( , ; , , , , , , , ) −
49 49 29 − −
49 29 − − − −
3T 27 ( , ; , , , , , , , )
29 29 −
29 29 29 59 − − − −
16 13
3T 28 ( , ; , , , , , , , ) − −
29 29 29 −
19 29 49 49 − −
3T 29 ( , ; , , , , , , , ) −
49 29 49 29 29 −
19 29 −
29 49 −
16 13
3T 30 ( , ; , , , , , , , )
29 29 − −
49 29 29 −
19 49 49 −
16 13
3T 31 ( , ; , , , , , , , )
29 29 −
29 29 29 −
49 59 − − −
16 13
3T 32 ( , ; , , , , , , , )
29 29 49 29 −
49 29 − −
29 49 − −
3T 33 ( , ; , , , , , , , )
29 29 49 29 29 29 29 − −
29 13 , ; , , , , , , , ) −
49 29 −
29 29 29 29 −
49 19 49 − −
3T 35 ( , ; , , , , , , , )
29 29 −
29 29 − −
49 29 19 49 13 , ; , , , , , , , ) −
19 29 − −
19 29 29 29 −
29 49 13 , ; , , , , , , , ) −
19 29 −
29 29 − −
19 29 − − − −
3T 38 ( , ; , , , , , , , ) − − −
29 29 29 29 29 19 − −
16 13
3T 39 ( , ; , , , , , , , ) − −
19 19 29 29 29 29 − − − −
3T 40 ( , ; , , , , , , , ) − −
29 29 −
19 29 29 −
29 49 −
16 13
3T 41 ( , ; , , , , , , , ) − − −
49 29 −
19 29 − −
29 13 , ; , , , , , , , )
29 29 19 − −
49 29 29 − − −
16 13
3T 43 ( , ; , , , , , , , ) − − −
19 29 29 − − − − −
3T 44 ( , ; , , , , , , , ) −
49 29 −
29 29 −
19 29 − − −
29 13 , ; , , , , , , , )
29 29 − − −
19 29 29 19 − − −
3T 46 ( , ; , , , , , , , ) −
49 49 −
49 29 29 29 −
29 19 −
16 13
3T 47 ( , ; , , , , , , , ) − − −
29 29 29 −
49 29 −
29 19 13 , ; , , , , , , , ) −
49 29 −
29 29 −
49 29 29 49 19 −
16 13
3T 49 ( , ; , , , , , , , )
29 29 − − −
49 29 − −
29 19 13 , ; , , , , , , , ) − −
29 29 −
49 29 29 − −
29 13 , ; , , , , , , , ) −
49 29 − −
49 29 29 29 − − −
16 13
3T 52 ( , ; , , , , , , , )
29 29 −
29 29 29 29 29 −