Orbifold Family Unification on 6 Dimensions
aa r X i v : . [ h e p - ph ] J u l Orbifold Family Unification on 6 Dimensions
Yuhei G
OTO ∗ , Yoshiharu K AWAMURA † Department of Physics, Shinshu University,Matsumoto 390-8621, Japan andTakashi M
IURA ‡ Department of Physics, Kobe University,Kobe 657-8501, Japan
July 10, 2013
Abstract
We study the possibility of family unification on the basis of SU ( N ) gauge the-ory on the 6-dimensional space-time, M × T / Z N . We obtain enormous numbersof models with three families of SU (5) matter multiplets and those with three fam-ilies of the standard model multiplets, from a single massless Dirac fermion with ahigher-dimensional representation of SU ( N ), through the orbifold breaking mech-anism. The origin of the family replication has been a big riddle. The family unification based ona large symmetry group can provide a possible solution. The studies have been carriedout intensively, and they are classified into two categories. One is the investigation basedon the 4-dimensional Minkowski space-time [1, 2, 3, 4, 5], and the other is that based onhigher-dimensional space-times [6, 7, 8, 9, 10, 11, 12, 13, 14].The advantage of higher-dimensional theories is that substances including mirrorparticles can be reduced using the symmetry breaking mechanism concerning extra di-mensions, as originally discussed in superstring theory [15, 16, 17]. Here, the mirrorparticles are particles with opposite quantum numbers under the standard model (SM)gauge group. Hence, a candidate realizing the family unification is grand unified theories(GUTs) on a higher-dimensional space-time including an orbifold as an extra space. In this paper, we study the possibility of family unification on the basis of SU ( N )gauge theory on M × T / Z N , using the method in Ref. [12]. We investigate whether or ∗ E-mail: [email protected] † E-mail: [email protected] ‡ E-mail: [email protected] M × S / Z possess the attractive feature that the triplet-doublet splitting of Higgs multiplets is elegantly realized [18, 19]. SU ( N ) for twopatterns of symmetry breaking.The contents of this paper are as follows. In Sec. II, we provide general arguments onthe orbifold breaking based on 2-dimensional orbifold T / Z N and formulae for numbersof species. In Sec. III, we investigate the family unification for each T / Z N ( N =
2, 3, 4, 6),in the framework of 6-dimensional SU ( N ) GUTs. Section IV is devoted to conclusionsand discussions. Z N orbifold breaking and formulae for numbers of species We explain the orbifold T / Z N and give formulae for numbers of species, in the casewith diagonal embeddings for representation matrices of Z N transformations. Z N orbifold breaking Let z be the complex coordinate of T / Z N . Here, T is constructed from a 2-dimensionallattice. On T , the points z + e and z + e are identified with the point z , where e and e are basis vectors. The orbifold T / Z N is obtained by dividing T by the Z N transfor-mation Z N : z → ξ z ( ξ N =
1) so that the point z is identified with ξ z , or z is generallyidentified with ξ k z + ae + be , where k , a and b are integers.Let us explain the orbifold breaking using T / Z . Accompanied by the identificationof points on T / Z , the following boundary conditions for a field Φ ( x , z ) can be imposedon, Φ ( x , − z ) = T Φ [ P ] Φ ( x , z ) , Φ ( x , e − z ) = T Φ [ P ] Φ ( x , z ) , Φ ( x , e − z ) = T Φ [ P ] Φ ( x , z ) , (1)where e = e = i , and T Φ [ P ], T Φ [ P ] and T Φ [ P ] represent appropriate representationmatrices. The P , P and P stand for the representation matrices of the Z transforma-tions z → − z , z → e − z and z → e − z for fields with the fundamental representation.The eigenvalues of T Φ [ P ], T Φ [ P ] and T Φ [ P ] are interpreted as the Z parities forthe extra space. The fields with even Z parities have zero modes, but those including anodd Z parity do not have zero modes. Here, zero modes mean 4-dimensional masslessfields surviving after compactification. Kaluza-Klein modes do not appear in our low-energy world, because they have heavy masses of O (1/ R ), with the same magnitude asthe unification scale. Unless all components of non-singlet field have a common Z par-ity, a symmetry reduction occurs upon compactification because zero modes are absent infields with an odd parity. This type of symmetry breaking mechanism is called “orbifoldbreaking mechanism”. Basis vectors, representation matrices and their transformation properties of T / Z N are summarized in Table 1 [28, 29]. Note that there is a choice in representation ma- The Z orbifolding was used in superstring theory [20] and heterotic M -theory [21, 22]. In field the-oretical models, it was applied to the reduction of global SUSY [23, 24], which is an orbifold version ofScherk-Schwarz mechanism [25, 26], and then to the reduction of gauge symmetry [27]. Though the number of independent representation matrices for T / Z is stated to be three in [13], itshould be two because other operations are generated using s : z → e π i /3 z and r : z → e − z . For example, t : z → z + e and t : z → z + e are generated as t = r ( s ) and t = ( s ) r ( s ) r , respectively. T / Z N . N Basis vectors Rep. matrices Transformation properties i P , P , P z → − z , z → e − z , z → e − z e π i /3 Θ , Θ z → e π i /3 z , z → e π i /3 z + e i Q , P z → i z , z → e − z − + i p Ξ , P z → e π i /3 z , z → e − z trices, and P concerning the Z transformation z → e − z is also used in T / Z and T / Z .Fields possess discrete charges relating eigenvalues of representation matrices for Z M transformation. Here, M = N for N =
2, 3 and M = N , 2 for N =
4, 6. The discretecharges are assigned as numbers n / M ( n =
0, 1, · · · , M −
1) and e π in / M are elements of Z M transformation. We refer to them as Z M elements.A fermion with spin 1/2 in 6-dimensions is regarded as a Dirac fermion or a pair ofWeyl fermions with opposite chiralities in 4-dimensions. There are two choices in a 6-dimensional Weyl fermion, i.e., Ψ + = + Γ Ψ = Ã − γ + γ ! µ Ψ Ψ ¶ = µ Ψ L Ψ R ¶ , (2) Ψ − = − Γ Ψ = Ã + γ − γ ! µ Ψ Ψ ¶ = µ Ψ R Ψ L ¶ , (3)where Ψ + and Ψ − are fermions with positive and negative chirality, respectively, and Γ and γ are the chirality operators for 6-dimensional fermions and 4-dimensional ones,respectively. Here and hereafter, the subscript ± stands for the chiralities on 6 dimen-sions.From the Z M invariance of kinetic term and the transformation property of the co-variant derivatives Z M : D z → ρ D z and D z → ρ D z with ρ = e − π i / M and ρ = e π i / M , thefollowing relations hold between the Z M element of Ψ L ( R ) and Ψ R ( L ) , P Ψ R = ρ P Ψ L , P Ψ R = ρ P Ψ L , (4)where z ≡ x + i x and z ≡ x − i x .Chiral gauge theories including Weyl fermions on even dimensional space-time be-come, in general, anomalous in the presence of gauge anomalies, gravitational anoma-lies, mixed anomalies and/or global anomaly [31, 32]. In SU ( N ) GUTs on 6-dimensionalspace-time, the global anomaly is absent because of Π ( SU ( N )) = N ≥
4. Here, Π ( SU ( N )) is the 6-th homotopy group of SU ( N ). In our analysis, we consider a mass-less Dirac fermion ( Ψ + , Ψ − ) under the SU ( N ) gauge group ( N ≥
8) on 6-dimensionalspace-time. In this case, anomalies are canceled out by the contributions from fermionswith different chiralities For more detailed explanations for 6-dimensional fermions, see Ref. [30]. .2 Formulae for numbers of species With suitable diagonal representation matrices R a ( a =
0, 1, 2 for T / Z and a =
0, 1 for T / Z , T / Z and T / Z ), the SU ( N ) gauge group is broken down into its subgroup suchthat SU ( N ) → SU ( p ) × SU ( p ) × · · · × SU ( p n ) × U (1) n − m − , (5)where N = P ni = p i . Here and hereafter, SU (1) unconventionally stands for U (1), SU (0)means nothing and m is a sum of the number of SU (0) and SU (1). The concrete form of R a will be given in the next section.After the breakdown of SU ( N ), the rank k totally antisymmetric tensor representa-tion [ N , k ], whose dimension is N C k , is decomposed into a sum of multiplets of the sub-group SU ( p ) × · · · × SU ( p n ) as[ N , k ] = k X l = k − l X l = · · · k − l −···− l n − X l n − = ¡ p C l , p C l , · · · , p n C l n ¢ , (6)where l n = k − l − · · · − l n − and our notation is that n C l = l > n and l <
0. Hereand hereafter, we use n C l instead of [ n , l ] in many cases. We sometimes use the ordinarynotation for representations too, e.g., and in place of C and C .The [ N , k ] is constructed by the antisymmetrization of k -ple product of the funda-mental representation N = [ N , 1]:[ N , k ] = ( N × · · · × N ) A . (7)We define the intrinsic Z M elements η ak such that( N × · · · × N ) A → η ak ( R a N × · · · × R a N ) A . (8)By definition, η ak take a value of Z M elements, i.e., e π in / M ( n =
0, 1, · · · , M − η ak for Ψ + are not necessarily same as those of Ψ − , and the chiral symmetry is stillrespected.Let us investigate the family unification in two cases. Each breaking pattern is givenby SU ( N ) → SU (5) × SU ( p ) × · · · × SU ( p n ) × U (1) n − m − , (9) SU ( N ) → SU (3) × SU (2) × SU ( p ) × · · · × SU ( p n ) × U (1) n − m − , (10)where SU (3) and SU (2) are identified with SU (3) C and SU (2) L in the SM gauge group. SU (5) multiplets We study the breaking pattern (9). After the breakdown of SU ( N ), [ N , k ] is decomposedas [ N , k ] = k X l = k − l X l = · · · k − l −···− l n − X l n − = ¡ C l , p C l , · · · , p n C l n ¢ . (11)4s mentioned before, C , C , C , C , C and C stand for representations , , , , and . Utilizing “survival hypothesis” and the equivalence of ( R ) c and ( R ) c with L and L , respectively, we write the numbers of and representations for left-handedWeyl fermions as n ¯5 ≡ ♯ L − ♯ L + ♯ R − ♯ R , (12) n ≡ ♯ L − ♯ L + ♯ R − ♯ R , (13)where ♯ represents the number of each multiplet. Here, the survival hypothesis is theassumption that if a symmetry is broken down into a smaller symmetry at a scale M SB ,then any fermion mass terms invariant under the smaller group induce fermion masses oforder O ( M SB ) [2, 33].The SU (5) singlets are regarded as the right-handed neutrinos, which can obtainheavy Majorana masses among themselves as well as the Dirac masses with left-handedneutrinos. Some of them can be involved in see-saw mechanism [34, 35, 36]. The totalnumber of SU (5) singlets (with heavy masses) is given by n ≡ ♯ L + ♯ L + ♯ R + ♯ R . (14)Formulae for n ¯5 , n and n from a Dirac fermion ( Ψ + , Ψ − ) whose intrinsic Z M ele-ments are ( η ak + , η ak − ) are given by n ¯5 = X ± X l = ( − l X { l , ··· , l n − } nal L ± − X { l , ··· , l n − } nal R ± p C l · · · p n C l n , (15) n = X ± X l = ( − l X { l , ··· , l n − } nal L ± − X { l , ··· , l n − } nal R ± p C l · · · p n C l n , (16) n = X ± X l = X { l , ··· , l n − } nal L ± + X { l , ··· , l n − } nal R ± p C l · · · p n C l n , (17)where p n = N − P n − i = p i and l n = N − P n − i = l i . P ± represents the summation of contri-butions from Ψ + and Ψ − . Furthermore, P { l , ··· , l n − } nal L ± means that the summations over l j = · · · , k − l − · · · − l j − ( j = · · · , n −
1) are carried out under the condition that l j should satisfy specific relations on T / Z N given in Table 2. The relations will be con-firmed in the next section. In the same way, P { l , ··· , l n − } nal R ± means that the summationsover l j = · · · , k − l − · · · − l j − ( j = · · · , n −
1) are carried out under the condition that l j should satisfy specific relations n al R ± = n al L ± ∓ M ) for Ψ ± . The formulae (15) –(17) will be rewritten in more concrete form for each T / Z N ( N =
2, 3, 4, 6), by the use ofprojection operators, in the next section. We denote the SU (5) singlet relating to C as , for convenience sake, to avoid the confusion oversinglets. As usual, ( R ) c and ( R ) c represent the charge conjugate of R and R , respectively. Note that ( R ) c and ( R ) c transform as the left-handed Weyl fermions under the 4-dimensional Lorentz transformations. l j . Orbifolds ρ k η ak ± Specific relationsT / Z ( − k η k ± = ( − α ± n l L ± ≡ l + l + l = − l − α ± (mod 2)( − k η k ± = ( − β ± n l L ± ≡ l + l + l = − l − β ± (mod 2)( − k η k ± = ( − γ ± n l L ± ≡ l + l + l = − l − γ ± (mod 2) T / Z ( e − π i /3 ) k η k ± = ( e π i /3 ) α ± n l L ± ≡ l + l + l + l + l ) = − l − α ± (mod 3)( e − π i /3 ) k η k ± = ( e π i /3 ) β ± n l L ± ≡ l + l + l + l + l ) = − l − β ± (mod 3) T / Z ( − i ) k η k ± = i α ± n l L ± ≡ l + l + l ) + l + l ) = − l − α ± (mod 4)( − k η k ± = ( − β ± n l L ± ≡ l + l + l = − l − β ± (mod 2) T / Z ( e − π i /3 ) k η k ± = ( e π i /3 ) α ± n l L ± ≡ l + l + l ) + l + l ) + l + l ) + l + l ) = − l − α ± (mod 6)( − k η k ± = ( − β ± n l L ± ≡ l + l + l + l + l = − l − β ± (mod 2) We study the breaking pattern (10). After the breakdown of SU ( N ), [ N , k ] is decomposedas [ N , k ] = k X l = k − l X l = k − l − l X l = · · · k − l −···− l n − X l n − = ¡ C l , C l , p C l , · · · , p n C l n ¢ . (18)The flavor numbers of down-type anti-quark singlets ( d R ) c , lepton doublets l L , up-type anti-quark singlets ( u R ) c , positron-type lepton singlets ( e R ) c , and quark doublets q L are denoted as n ¯ d , n l , n ¯ u , n ¯ e and n q . Using the survival hypothesis and the equivalenceon charge conjugation, we define the flavor number of each chiral fermion as n ¯ d ≡ ♯ ( C , C ) L − ♯ ( C , C ) L + ♯ ( C , C ) R − ♯ ( C , C ) R , (19) n l ≡ ♯ ( C , C ) L − ♯ ( C , C ) L + ♯ ( C , C ) R − ♯ ( C , C ) R , (20) n ¯ u ≡ ♯ ( C , C ) L − ♯ ( C , C ) L + ♯ ( C , C ) R − ♯ ( C , C ) R , (21) n ¯ e ≡ ♯ ( C , C ) L − ♯ ( C , C ) L + ♯ ( C , C ) R − ♯ ( C , C ) R , (22) n q ≡ ♯ ( C , C ) L − ♯ ( C , C ) L + ♯ ( C , C ) R − ♯ ( C , C ) R , (23)where ♯ again represents the number of each multiplet. The total number of (heavy)neutrino singlets ( ν R ) c is denoted n ¯ ν and defined as n ¯ ν ≡ ♯ ( C , C ) L + ♯ ( C , C ) L + ♯ ( C , C ) R + ♯ ( C , C ) R . (24)6ormulae for the SM species including neutrino singlets are given by n ¯ d = X ± X ( l , l ) = (2,2),(1,0) ( − l + l X { l , ··· , l n − } nal l L ± − X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (25) n l = X ± X ( l , l ) = (3,1),(0,1) ( − l + l X { l , ··· , l n − } nal l L ± − X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (26) n ¯ u = X ± X ( l , l ) = (2,0),(1,2) ( − l + l X { l , ··· , l n − } nal l L ± − X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (27) n ¯ e = X ± X ( l , l ) = (0,2),(3,0) ( − l + l X { l , ··· , l n − } nal l L ± − X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (28) n q = X ± X ( l , l ) = (1,1),(2,1) ( − l + l X { l , ··· , l n − } nal l L ± − X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (29) n ¯ ν = X ± X ( l , l ) = (0,0),(3,2) X { l , ··· , l n − } nal l L ± + X { l , ··· , l n − } nal l R ± p C l · · · p n C l n , (30)where P { l , ··· , l n − } nal l L ± means that the summations over l j = · · · , k − l − · · · − l j − ( j = · · · , n −
1) are carried out under the condition that l j should satisfy specific relationson T / Z N given in Table 3. The relations will be confirmed in the next section. In thesame way, P { l , ··· , l n − } nal l R ± means that the summations over l j = · · · , k − l − · · · − l j − ( j = · · · , n −
1) are carried out under the condition that l j should satisfy specific rela-tions n al l R ± = n al l L ± ∓ M ) for Ψ ± . The formulae (25) – (30) will be also rewrittenin more concrete form for each T / Z N , by the use of projection operators, in the nextsection. We list generic features of flavor numbers.(i)
Each flavor number from [ N , k ] with intrinsic Z M elements η ak ± is equal to that from [ N , N − k ] with appropriate ones η aN − k ± . Let us explain this feature using the SU (5) multiplets. From (11) and the decomposi-tion of [ N , N − k ] such that[ N , N − k ] = k X l = k − l X l = · · · k − l −···− l n − X l n − = ¡ C − l , p C p − l , · · · , p n C p n − l n ¢ , (31)there is a one-to-one correspondence between ¡ C − l , p C p − l , · · · , p n C p n − l n ¢ in [ N , N − k ] and ¡ C l , p C l , · · · , p n C l n ¢ in [ N , k ]. The right-handed Weyl fermion whose represen-tation is ¡ C − l , p C p − l , · · · , p n C p n − l n ¢ is regarded as the left-handed one whose repre-sentation is the conjugate representation ¡ C l , p C l , · · · , p n C l n ¢ , and hence we obtain7able 3: The specific relations for l j . Orbifolds ρ k η ak ± Specific relationsT / Z ( − k η k ± = ( − α ± n l l L ± ≡ l + l = − l − l − α ± (mod 2)( − k η k ± = ( − β ± n l l L ± ≡ l + l = − l − l − β ± (mod 2)( − k η k ± = ( − γ ± n l l L ± ≡ l + l + l = − l − γ ± (mod 2) T / Z ( e − π i /3 ) k η k ± = ( e π i /3 ) α ± n l l L ± ≡ l + l + l + l ) = − l − l − α ± (mod 3)( e − π i /3 ) k η k ± = ( e π i /3 ) β ± n l l L ± ≡ l + l + l + l ) = − l − l − β ± (mod 3) T / Z ( − i ) k η k ± = i α ± n l l L ± ≡ l + l ) + l + l ) = − l − l − α ± (mod 4)( − k η k ± = ( − β ± n l l L ± ≡ l + l + l = − l − β ± (mod 2) T / Z ( e − π i /3 ) k η k ± = ( e π i /3 ) α ± n l l L ± ≡ l + l ) + l + l ) + l + l ) + l + l ) = − l − l − α ± (mod 6)( − k η k ± = ( − β ± n l l L ± ≡ l + l + l + l + l = − l − β ± (mod 2)the same numbers for (15) – (17) with a suitable assignment of intrinsic Z M elements for[ N , N − k ].Here, we give an example for T / Z . Each flavor number obtained from [ N , k ] with( − k η k ± = ( − α ± , ( − k η k ± = ( − β ± and ( − k η k ± = ( − γ ± agrees with that from [ N , N − k ] with ( − N − k η N − k ± = ( − α ′± , ( − N − k η N − k ± = ( − β ′± and ( − N − k η N − k ± = ( − γ ′± ,where α ′± , β ′± and γ ′± satisfy the relations α ′± = α ± + p + p + p (mod2), β ′± = β ± + p + p + p (mod2) and γ ′± = γ ± + p + p + p (mod2), respectively.(ii) Each flavor number from [ N , k ] with intrinsic Z elements ( − k η ak ± = ( − δ a ± is equalto that from [ N , k ] with the exchanged ones ( δ a + ↔ δ a − ) , i.e., ( − k η ak ± = ( − δ a ∓ . This feature is understood from the fact that specific relations on l j for Ψ + changeinto those of Ψ − and vice versa, under the exchange of Z parity of Ψ + and that of Ψ − .Here, we give an example for T / Z . Under the exchange of α + and α − , n l L + and n l R + change into n l L − and n l R − (mod2), respectively. Each flavor number remains thesame, because the summation is taken for Ψ + and Ψ − .(iii) Each flavor number from [ N , k ] is invariant under several types of exchange amongp j and intrinsic Z M elements. From specific relations in Table 2, we find that the same number for each SU (5) mul-tiplet is obtained under the exchange,( p , p , α ± ) ⇐⇒ ( p , p , β ± ) , ( p , p , β ± ) ⇐⇒ ( p , p , γ ± ) ,( p , p , α ± ) ⇐⇒ ( p , p , γ ± ) for T / Z , (32)( p , p , p , α ± ) ⇐⇒ ( p , p , p , β ± ) for T / Z , (33)where the exchange is done independently.8n the same way, from specific relations in Table 3, we find that the same number foreach SM multiplet is obtained under the exchange,( p , p , α ± ) ⇐⇒ ( p , p , β ± ) , for T / Z . (34)Under the above exchanges, although the unbroken gauge symmetry remains, thenumbers of zero modes for extra-dimensional components of gauge bosons are, in gen-eral, different and hence a model is transformed into a different one.(iv) Each flavor number obtained from [ N , k ] is invariant in the introduction of Wilsonline phases. Let us give some examples.On T / Z , the numbers n ¯5 and n obtained from the breaking pattern SU ( N ) → SU (5) × SU ( p ) ×· · ·× SU ( p ) × U (1) − m are same as those from SU ( N ) → SU (5) × SU ( p ′ ) ×· · · × SU ( p ′ ) × U (1) − m , if the following relations are satisfied, p ′ − p = p ′ − p = p − p ′ = p − p ′ , p ′ = p , p ′ = p , p ′ = p , (35)or p ′ − p = p ′ − p = p − p ′ = p − p ′ , p ′ = p , p ′ = p , p ′ = p , (36)or p ′ − p = p ′ − p = p − p ′ = p − p ′ , p ′ = p , p ′ = p , p ′ = p . (37)The above BCs are connected by a singular gauge transformation, and they are re-garded as equivalent in the presence of Wilson line phases. This equivalence originatesfrom the Hosotani mechanism [37, 38, 39, 40], and is shown by the following relationsamong the diagonal representatives for 2 × P , P , P ) [29],( τ , τ , τ ) ∼ ( τ , τ , − τ ) ∼ ( τ , − τ , τ ) ∼ ( τ , − τ , − τ ) , (38)where τ is the third component of Pauli matrices.In our present case, we assume that the BC is chosen as a physical one, i.e., the sys-tem with the physical vacuum is realized with the vanishing Wilson line phases after asuitable gauge transformation is performed. Hence, it is understood that each net flavornumber obtained from [ N , k ] does not change even though the vacuum changes differ-ent ones in the presence of Wilson line phases.In the same way, the numbers n ¯ d , n l , n ¯ u , n ¯ e and n q obtained from the breaking pat-tern SU ( N ) → SU (3) × SU (2) × SU ( p ) × · · · × SU ( p ) × U (1) − m are same as those from SU ( N ) → SU (3) × SU (2) × SU ( p ′ ) × · · · × SU ( p ′ ) × U (1) − m , if the following relations aresatisfied, p ′ − p = p ′ − p = p − p ′ = p − p ′ , p ′ = p , p ′ = p . (39)On T / Z , the numbers n ¯5 and n obtained from the breaking pattern SU ( N ) → SU (5) × SU ( p ) ×· · ·× SU ( p ) × U (1) − m are same as those from SU ( N ) → SU (5) × SU ( p ′ ) ×· · · × SU ( p ′ ) × U (1) − m , if the following relations are satisfied, p ′ − p = p ′ − p = p ′ − p = p − p ′ = p − p ′ = p − p ′ , p ′ = p , p ′ = p . (40)9he above BCs are also connected by a singular gauge transformation, and they areregarded as equivalent in the presence of Wilson line phases. The equivalence is shownusing the following relations among the diagonal representatives for 3 × Θ , Θ ) on T / Z [29], ( X , X ) ∼ ( X , ω X ) ∼ ( X , ω X ) , (41)where ω = e π i /3 , ω = e π i /3 , and X = diag(1, ω , ω ).For these cases, it is also understood that each net flavor number does not changeeven though the vacuum changes different ones in the presence of Wilson line phases.Although this feature holds for models on T / Z and T / Z , there are no examplesin our setting, because of the absence of Wilson line phases changing BCs but keeping SU (5) or the SM gauge group for T / Z and because of the absence of equivalence rela-tions between diagonal representatives for T / Z [29]. M × T / Z N We investigate the family unification in SU ( N ) GUTs for each T / Z N ( N =
2, 3, 4, 6).
Let us present total numbers of models with the three families, for reference. Total num-bers of models with the three families of SU (5) multiplets and the SM multiplets, whichoriginate from a Dirac fermion whose representation is [ N , k ] ( k ≤ N /2) of SU ( N ), aresummarized up to SU (12) in Table 4 and up to SU (13) in Table 5, respectively. In theTables, the hyphen (-) means no models. We omit the total numbers of models from[ N , N − k ], because they agree with those from [ N , k ], reflecting the feature (i) in the sub-section 2.3. T / Z For the representation matrices given by P = diag([ + p , [ + p , [ + p , [ + p , [ − p , [ − p , [ − p , [ − p ) , P = diag([ + p , [ + p , [ − p , [ − p , [ + p , [ + p , [ − p , [ − p ) , P = diag([ + p , [ − p , [ + p , [ − p , [ + p , [ − p , [ + p , [ − p ) , (42)the following breakdown of SU ( N ) gauge symmetry occurs SU ( N ) → SU ( p ) × SU ( p ) × · · · × SU ( p ) × U (1) − n , (43)where [ ± p i represents ± p i elements.After the breakdown of SU ( N ), [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ p C l , p C l , · · · , p C l ¢ ± , (44)where l = k − l − · · · − l . 10able 4: Total numbers of models with the three families of SU (5) multiplets. T / Z T / Z T / Z T / Z SU (8) - [8,3]:24 [8,3]:14 [8,3]:28[8,4]:12 [8,4]:16 [8,4]:20 SU (9) [9,3]:192 [9,3]:182 [9,3]:142 [9,3]:512[9,4]:348 [9,4]:32 [9,4]:800 SU (10) - [10,3]:852 [10,3]:160 [10,3]:2484[10,4]:1308 [10,4]:92 [10,4]:2654[10,5]:48 [10,5]:1532 SU (11) [11,3]:768 [11,3]:1608 [11,3]:456 [11,3]:6530[11,4]:768 [11,4]:1716 [11,4]:436 [11,4]:6768[11,5]:1794 [11,5]:186 [11,5]:5540 SU (12) [12,3]:1104 [12,3]:2214 [12,3]:748 [12,3]:17084[12,4]:1020 [12,4]:676 [12,4]:13692[12,5]:534 [12,5]:10498[12,6]:632 [12,6]:13188Table 5: Total numbers of models with the three families of SM multiplets. T / Z T / Z T / Z T / Z SU (8) - - - - SU (9) [9,3]:32 - [9,3]:8 [9,3]:8[9,4]:32 SU (10) - - - [10,3]:80[10,4]:108 SU (11) [11,3]:80 [11,4]:80 [11,3]:20 [11,3]:84[11,4]:80 [11,4]:20 [11,4]:144[11,5]:156 SU (12) [12,3]:120 [12,3]:80 [12,4]:88 [12,3]:392[12,6]:240 [12,4]:120[12,5]:72[12,6]:552 SU (13) [13,3]:144 - [13,4]:40 [13,3]:712[13,4]:88[13,5]:140[13,6]:20011sing the definition of the intrinsic Z parities η ak ± ( a =
0, 1, 2) such that( N × · · · × N ) A ± → η ak ± ( P a N × · · · × P a N ) A ± , (45)the Z parities of the representation ¡ p C l , p C l , · · · , p C l ¢ ± are given by P ± = ( − l + l + l + l η k ± = ( − l + l + l + l ( − k η k ± = ( − l + l + l + l + α ± , (46) P ± = ( − l + l + l + l η k ± = ( − l + l + l + l ( − k η k ± = ( − l + l + l + l + β ± , (47) P ± = ( − l + l + l + l η k ± = ( − l + l + l + l ( − k η k ± = ( − l + l + l + l + γ ± , (48)where η ak ± take a value + − − k η k ± = ( − α ± , ( − k η k ± = ( − β ± and ( − k η k ± = ( − γ ± . SU (5) multiplets on T / Z After the breakdown SU ( N ) → SU (5) × SU ( p ) ×· · ·× SU ( p ) × U (1) − m , [ N , k ] ± is decom-posed as [ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , p C l , · · · , p C l ¢ ± . (49)Using the assignment of Z parities (46) – (48), we find that zero modes appear if thefollowing relations are satisfied, n l L ± ≡ l + l + l = − l − α ± (mod 2) , n l L ± ≡ l + l + l = − l − β ± (mod 2) , n l L ± ≡ l + l + l = − l − γ ± (mod 2) . (50)Utilizing the survival hypothesis and the equivalence of charge conjugation, we ob-tain the formulae (15) – (17) with n =
8. Because the Z projection operator P ± that picksup P = ± P ± ≡ (1 ± P )/2, the Z projection operator that picks up zeromodes of left-handed ones, i.e., massless modes in fields with ( P ± , P ± , P ± ) = (1, 1, 1),is given by P (1,1,1) ≡
18 (1 + P ± )(1 + P ± )(1 + P ± ) , (51)and the Z projection operator that picks up the zero modes of right-handed ones, i.e.,massless modes in fields with ( P ± , P ± , P ± ) = ( − − − P ( − − − ≡
18 (1 − P ± )(1 − P ± )(1 − P ± ) . (52)From (51) and (52), P (1,1,1) − P ( − − − =
14 ( P ± + P ± + P ± + P ± P ± P ± ) , (53) P (1,1,1) + P ( − − − =
14 (1 + P ± P ± + P ± P ± + P ± P ± ) . (54)12sing (46), (47), (48), (53) and (54), the formulae (15) – (17) are rewritten as n ¯5 = X ± X l = k − l X l = · · · k − l −···− l X l = ( − l ¡ P (1,1,1) − P ( − − − ¢ p C l · · · p C l = X ± X l = k − l X l = · · · k − l −···− l X l = ³ ( − l + l + l + α ± + ( − l + l + l + β ± + ( − l + l + l + γ ± + ( − l + l + l + α ± + β ± + γ ± ´ p C l · · · p C l , (55) n = X ± X l = k − l X l = · · · k − l −···− l X l = ( − l ¡ P (1,1,1) − P ( − − − ¢ p C l · · · p C l = X ± X l = k − l X l = · · · k − l −···− l X l = ³ ( − l + l + l + α ± + ( − l + l + l + β ± + ( − l + l + l + γ ± + ( − l + l + l + α ± + β ± + γ ± ´ p C l · · · p C l , (56) n = X ± X l = k − l X l = · · · k − l −···− l X l = ¡ P (1,1,1) + P ( − − − ¢ p C l · · · p C l = X ± X l = k − l X l = · · · k − l −···− l X l = ³ + ( − l + l + l + l + α ± + β ± + ( − l + l + l + l + α ± + γ ± + ( − l + l + l + l + β ± + γ ± ´ p C l · · · p C l . (57)Here, we give some examples for representations and BCs to derive n ¯5 = n =
3, inTable 6. Table 6: Examples for the three families of SU (5) from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p ) ( α + , β + , γ + ) ( α − , β − , γ − )[9,3] (5,0,0,0,3,0,0,1) (0,1,1) (0,0,1)[11,3] (5,0,1,0,4,0,1,0) (0,0,1) (1,1,0)[11,4] (5,0,3,1,0,1,1,0) (0,0,0) (0,0,1)[12,3] (5,2,0,0,2,0,1,2) (1,0,1) (0,0,0) T / Z After the breakdown SU ( N ) → SU (3) × SU (2) × SU ( p ) × · · · × SU ( p ) × U (1) − m , [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = k − l − l X l = · · · k − l −···− l X l = ¡ C l , C l , p C l , · · · , p C l ¢ ± . (58)Using the assignment of Z parities (46) – (48), we find that zero modes appear if thefollowing relations are satisfied, n l l L ± ≡ l + l = − l − l − α ± (mod 2) ,13 l l L ± ≡ l + l = − l − l − β ± (mod 2) , n l l L ± ≡ l + l + l = − l − γ ± (mod 2) , (59)for ( − k η k ± = ( − α ± , ( − k η k ± = ( − β ± and ( − k η k ± = ( − γ ± .Then, we obtain the formulae (25) – (30) with n =
8. Using (46), (47), (48), (53) and(54), the formulae for ( d R ) c and ( ν R ) c are rewritten as n ¯ d = X ± X ( l , l ) = (2,2),(1,0) k − l X l = · · · k − l −···− l X l = ³ ( − l + l + α ± + ( − l + l + β ± + ( − l + l + l + l + γ ± + ( − l + l + l + l + α ± + β ± + γ ± ´ p C l · · · p C l , (60) n ¯ ν = X ± X ( l , l ) = (0,0),(3,2) k − l X l = · · · k − l −···− l X l = ³ + ( − l + l + l + l + α ± + β ± + ( − l + l + l + l + α ± + γ ± + ( − l + l + l + l + β ± + γ ± ´ p C l · · · p C l . (61)The formulae for l L , ( u R ) c , ( e R ) c and q L are obtained by replacing the summation of( l , l ) for n ¯ d with {(3, 1), (0, 1)}, {(2, 0), (1, 2)}, {(0, 2), (3, 0)} and {(1, 1), (2, 1)}.Here, we give a list of all BCs to derive three families of SM fermions from [9, 3], inTable 7. We find that the features (ii) and (iii), presented in subsection 2.3, hold on. T / Z For the representation matrices given by Θ = diag([1] p , [1] p , [1] p , [ ω ] p , [ ω ] p , [ ω ] p , [ ω ] p , [ ω ] p , [ ω ] p ) , Θ = diag([1] p , [ ω ] p , [ ω ] p , [1] p , [ ω ] p , [ ω ] p , [1] p , [ ω ] p , [ ω ] p ) , (62)the following breakdown of SU ( N ) gauge symmetry occurs SU ( N ) → SU ( p ) × SU ( p ) × · · · × SU ( p ) × U (1) − n , (63)where [1] p i , [ ω ] p i and [ ω ] p i represent 1, ω ( ≡ e π i /3 ) and ω ( ≡ e π i /3 ) for all p i elements.After the breakdown of SU ( N ), [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ p C l , p C l , · · · , p C l ¢ ± , (64)where l = k − l − · · · − l . The ¡ p C l , p C l , · · · , p C l ¢ ± has the Z elements P ± = ω l + l + l ω l + l + l η k ± = ω l + l + l + l + l + l ) ω k η k ± = ω l + l + l + l + l + l ) + α ± , (65) P ± = ω l + l + l ω l + l + l η k ± = ω l + l + l + l + l + l ) ω k η k ± = ω l + l + l + l + l + l ) + β ± , (66)where η ak ± take a value 1, ω or ω , and we parameterize them as ω k η k ± = ω α ± and ω k η k ± = ω β ± . 14able 7: The three families of SM multiplets from [9, 3] on T / Z .[ N , k ] ( p , p , p , p , p , p , p , p ) ( α + , β + , γ + ) ( α − , β − , γ − )[9,3] (3,2,0,0,0,3,0,1) (0,1,1) (0,1,0)(3,2,0,0,0,3,0,1) (0,1,0) (0,1,1)(3,2,0,0,0,3,1,0) (0,1,1) (0,1,0)(3,2,0,0,0,3,1,0) (0,1,0) (0,1,1)(3,2,0,0,3,0,0,1) (0,1,1) (0,1,0)(3,2,0,0,3,0,0,1) (0,1,0) (0,1,1)(3,2,0,0,3,0,1,0) (0,1,1) (0,1,0)(3,2,0,0,3,0,1,0) (0,1,0) (0,1,1)(3,2,0,3,0,0,0,1) (1,0,1) (1,0,0)(3,2,0,3,0,0,0,1) (1,0,0) (1,0,1)(3,2,0,3,0,0,1,0) (1,0,1) (1,0,0)(3,2,0,3,0,0,1,0) (1,0,0) (1,0,1)(3,2,3,0,0,0,0,1) (1,0,1) (1,0,0)(3,2,3,0,0,0,0,1) (1,0,0) (1,0,1)(3,2,3,0,0,0,1,0) (1,0,1) (1,0,0)(3,2,3,0,0,0,1,0) (1,0,0) (1,0,1)(3,2,0,0,1,2,0,1) (0,1,1) (0,1,0)(3,2,0,0,1,2,0,1) (0,1,0) (0,1,1)(3,2,0,0,1,2,1,0) (0,1,1) (0,1,0)(3,2,0,0,1,2,1,0) (0,1,0) (0,1,1)(3,2,0,0,2,1,0,1) (0,1,1) (0,1,0)(3,2,0,0,2,1,0,1) (0,1,0) (0,1,1)(3,2,0,0,2,1,1,0) (0,1,1) (0,1,0)(3,2,0,0,2,1,1,0) (0,1,0) (0,1,1)(3,2,1,2,0,0,0,1) (1,0,1) (1,0,0)(3,2,1,2,0,0,0,1) (1,0,0) (1,0,1)(3,2,1,2,0,0,1,0) (1,0,1) (1,0,0)(3,2,1,2,0,0,1,0) (1,0,0) (1,0,1)(3,2,2,1,0,0,0,1) (1,0,1) (1,0,0)(3,2,2,1,0,0,0,1) (1,0,0) (1,0,1)(3,2,2,1,0,0,1,0) (1,0,1) (1,0,0)(3,2,2,1,0,0,1,0) (1,0,0) (1,0,1)15 .3.1 Numbers of SU (5) multiplets on T / Z After the breakdown of SU ( N ) → SU (5) × SU ( p ) × · · · × SU ( p ) × U (1) − m , [ N , k ] ± is de-composed as [ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , p C l , · · · , p C l ¢ ± . (67)Using the assignment of Z elements (65) and (66), we find that zero modes appear ifthe following relations are satisfied, n l L ± ≡ l + l + l + l + l ) = − l − α ± (mod 3) , n l L ± ≡ l + l + l + l + l ) = − l − β ± (mod 3) . (68)The relation n al R ± = n al L ± ∓ n =
9, and they are rewritten as n ¯5 = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( ω , ω ) + + P (1,1) − − P ( ω , ω ) − ´ p C l · · · p C l , (69) n = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( ω , ω ) + + P (1,1) − − P ( ω , ω ) − ´ p C l · · · p C l , (70) n = X l = k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( ω , ω ) + + P (1,1) − + P ( ω , ω ) − ´ p C l · · · p C l , (71)where P ( ρ , ρ ) ± are projection operators that pick up the part relating ( P ± , P ± ) = ( ρ , ρ )and are written by P ( ρ , ρ ) ± = ¡ + ρ P ± + ρ P ± ¢ ¡ + ρ P ± + ρ P ± ¢ . (72)Here, we give some examples for representations and BCs to derive n ¯5 = n =
3, inTable 8. T / Z After the breakdown SU ( N ) → SU (3) × SU (2) × SU ( p ) × · · · × SU ( p ) × U (1) − m , [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = k − l − l X l = · · · k − l −···− l X l = ¡ C l , C l , p C l , · · · , p C l ¢ ± . (73)Using the assignment of Z elements (65) and (66), we find that zero modes appear ifthe following relations are satisfied, n l l L ± ≡ l + l + l + l ) = − l − l − α ± (mod 3) , n l l L ± ≡ l + l + l + l ) = − l − l − β ± (mod 3) . (74)The relation n al l R ± = n al l L ± ∓ SU (5) from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[8,3] (5,0,0,0,3,0,0,0,0) (2,0) (2,2)[8,4] (5,1,1,0,1,0,0,0,0) (0,0) (2,2)[9,3] (5,0,0,2,0,1,0,0,1) (2,0) (2,1)[9,4] (5,0,2,0,0,0,0,2,0) (2,2) (0,2)[10,3] (5,0,0,0,3,2,0,0,0) (2,0) (2,2)[10,4] (5,0,0,1,0,1,1,1,1) (2,2) (2,2)[10,5] (5,1,0,0,1,0,2,0,1) (0,0) (0,0)[11,3] (5,1,0,0,1,4,0,0,0) (0,0) (2,1)[11,4] (5,2,2,0,0,1,0,1,0) (1,2) (2,1)[11,5] (5,1,1,1,1,0,0,0,2) (0,1) (1,1)[12,3] (5,0,0,3,3,0,0,0,1) (2,0) (0,2)[12,4] (5,0,3,1,0,1,0,2,0) (1,2) (0,1)Then, we obtain the formulae (25) – (30) with n =
9. Using the projection operators(72), the formulae for ( d R ) c and ( ν R ) c are rewritten as n ¯ d = X ( l , l ) = (2,2),(1,0) k − l X l = · · · k − l −···− l X l = ( − l + l ³ P (1,1) + − P ( ω , ω ) + + P (1,1) − − P ( ω , ω ) − ´ p C l · · · p C l ,(75) n ¯ ν = X ( l , l ) = (0,0),(3,2) k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( ω , ω ) + + P (1,1) − + P ( ω , ω ) − ´ p C l · · · p C l . (76)The formulae for l L , ( u R ) c , ( e R ) c and q L are obtained by replacing the summation of( l , l ) for n ¯ d with {(3, 1), (0, 1)}, {(2, 0), (1, 2)}, {(0, 2), (3, 0)} and {(1, 1), (2, 1)}.Here, we give some examples for representations and BCs to derive three families ofSM fermions, in Table 9.Table 9: Examples for the three families of SM multiplets from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[11,4] (3,2,0,0,1,2,3,0,0) (0,1) (0,1)[12,3] (3,2,0,1,1,0,1,2,2) (1,0) (0,1) T / Z For the representation matrices given by Q = diag([ + p , [ + p , [ + i ] p , [ + i ] p , [ − p , [ − p , [ − i ] p , [ − i ] p ) , P = diag([ + p , [ − p , [ + p , [ − p , [ + p , [ − p , [ + p , [ − p ) , (77)17he following breakdown of SU ( N ) gauge symmetry occurs SU ( N ) → SU ( p ) × SU ( p ) × · · · × SU ( p ) × U (1) − n , (78)where [ ± p i and [ ± i ] p i represent ± ± i for all p i elements.After the breakdown of SU ( N ), [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ p C l , p C l , · · · , p C l ¢ ± , (79)where l = k − l − · · · − l . The ¡ p C l , p C l , · · · , p C l ¢ ± has the Z and Z elements P ± = i l + l ( − l + l ( − i ) l + l η k ± = i l + l + l + l ) + l + l ) ( − i ) k η k ± = i l + l + l + l ) + l + l ) + α ± , (80) P = ( − l + l + l + l η k ± = ( − l + l + l + l ( − k η k ± = ( − l + l + l + l + β ± , (81)where η k ± takes a value 1, − i or − i , and we parameterize the intrinsic Z M elements( M =
4, 2) as ( − i ) k η k ± = i α ± and ( − k η k ± = ( − β ± . SU (5) multiplets on T / Z After the breakdown of SU ( N ) → SU (5) × SU ( p ) × · · · × SU ( p ) × U (1) − m , [ N , k ] ± is de-composed as [ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , p C l , · · · , p C l ¢ ± . (82)Using the assignment of Z and Z element (80) and (81), we find that zero modesappear if the following relations are satisfied, n l L ± ≡ l + l + l ) + l + l ) = − l − α ± (mod 4) , n l L ± ≡ l + l + l = − l − β ± (mod 2) . (83)The relation n al R ± = n al L ± ∓ n =
8, and they are rewritten as n ¯5 = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( i , − + + P (1,1) − − P ( − i , − − ´ p C l · · · p C l , (84) n = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( i , − + + P (1,1) − − P ( − i , − − ´ p C l · · · p C l , (85) n = X l = k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( i , − + + P (1,1) − + P ( − i , − − ´ p C l · · · p C l , (86)where P ( ρ , ρ ′ ) ± are projection operators that pick up the part relating ( P ± , P ± ) = ( ρ , ρ ′ )and are written by P ( ρ , ρ ′ ) ± = ¡ + ρ P ± + ρ P ± + ρ P ± ¢ ³ + ρ ′ P ± ´ . (87)Here, we give some examples for representations and BCs to derive n ¯5 = n =
3, inTable 10. 18able 10: Examples for the three families of SU (5) from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[8,3] (5,0,0,0,0,0,3,0) (2,1) (0,0)[8,4] (5,0,0,3,0,0,0,0) (0,0) (2,0)[9,3] (5,3,0,0,0,0,0,1) (1,0) (0,1)[9,4] (5,0,2,0,0,0,1,1) (2,0) (2,0)[10,3] (5,0,0,0,3,0,0,2) (1,0) (2,0)[10,4] (5,0,0,0,0,4,0,1) (0,0) (2,1)[11,3] (5,0,0,1,2,2,0,1) (3,1) (2,0)[11,4] (5,0,3,1,2,0,0,0) (2.0) (1,1)[11,5] (5,0,0,2,0,0,1,3) (0,1) (3,0)[12,3] (5,4,0,1,0,0,0,2) (3,1) (1,0)[12,4] (5,0,4,0,1,2,0,0) (2,0) (3,0)[12,5] (5,1,2,0,2,2,0,0) (3,1) (1,1)[12,6] (5,0,3,0,1,0,3,0) (2,0) (2,1) T / Z After the breakdown of SU ( N ) → SU (3) × SU (2) × SU ( p ) ×· · ·× SU ( p ) × U (1) − m , [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , C l , p C l , · · · , p C l ¢ ± . (88)Using the assignment of Z and Z element (80) and (81), we find that zero modesappear if the following relations are satisfied, n l l L ± ≡ l + l ) + l + l ) = − l − l − α ± (mod 4) , n l l L ± ≡ l + l + l = − l − β ± (mod 2) . (89)The relation n al l R ± = n al l L ± ∓ n =
8. Using the projection operators(87), the formulae for ( d R ) c and ( ν R ) c are rewritten as n ¯ d = X ( l , l ) = (2,2),(1,0) k − l X l = · · · k − l −···− l X l = ( − l + l ³ P (1,1) + − P ( i , − + + P (1,1) − − P ( − i , − − ´ p C l · · · p C l ,(90) n ¯ ν = X ( l , l ) = (0,0),(3,2) k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( i , − + + P (1,1) − + P ( − i , − − ´ p C l · · · p C l . (91)The formulae for l L , ( u R ) c , ( e R ) c and q L are obtained by replacing the summation of( l , l ) for n ¯ d with {(3, 1), (0, 1)}, {(2, 0), (1, 2)}, {(0, 2), (3, 0)} and {(1, 1), (2, 1)}.Here, we give some examples of representations and BCs to derive three families ofSM fermions, in Table 11. 19able 11: Examples for the three families of SM multiplets from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[9,3] (3,2,1,0,0,0,2,1) (0,1) (0,0)[11,3] (3,2,1,1,0,4,0,0) (1,0) (1,1)[11,4] (3,2,0,0,3,1,1,1) (0,1) (0,0)[12,4] (3,2,1,0,2,1,3,0) (0,1) (0,0)[12,6] (3,2,1,2,0,0,0,4) (0,1) (1,1)[13,4] (3,2,1,2,2,2,0,1) (0,1) (0,0) T / Z For the representation matrices given by Ξ = diag([ + p , [ + p , [ ϕ ] p , [ ϕ ] p , [ ϕ ] p , [ ϕ ] p ,[ − p , [ − p , [ − ϕ ] p , [ − ϕ ] p , [ − ϕ ] p , [ − ϕ ] p ) , P = diag([ + p , [ − p , [ + p , [ − p , [ + p , [ − p ,[ + p , [ − p , [ + p , [ − p , [ + p , [ − p ) , (92)the following breakdown of SU ( N ) gauge symmetry occurs SU ( N ) → SU ( p ) × SU ( p ) × · · · × SU ( p ) × U (1) − m , (93)where ϕ = e π i /3 and [ c ] p i represents the number c for all p i elements.After the breakdown of SU ( N ), [ N , k ] ± , is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ p C l , p C l , · · · , p C l ¢ ± , (94)where l = k − l − · · · − l . The ¡ p C l , p C l , · · · , p C l ¢ ± has the Z and Z elements P = ϕ l + l ( ϕ ) l + l ( − l + l ( − ϕ ) l + l ( − ϕ ) l + l η k ± = ϕ l + l + l + l ) + l + l ) + l + l ) + l + l ) ϕ k η k ± = ϕ l + l + l + l ) + l + l ) + l + l ) + l + l ) + α ± , (95) P = ( − l + l + l + l + l + l η k ± = ( − l + l + l + l + l + l ( − k η k ± = ( − l + l + l + l + l + l + β ± , (96)where η k ± takes a value e n π i /3 ( n =
0, 1, · · · , 5), and we parameterize the intrinsic Z M elements ( M =
6, 2) as ( e − π i /3 ) k η k ± = ( e π i /3 ) α ± and ( − k η k ± = ( − β ± . SU (5) multiplets on T / Z After the breakdown of SU ( N ) → SU (5) × SU ( p ) × · · · × SU ( p ) × U (1) − m , [ N , k ] ± isdecomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , p C l , · · · , p C l ¢ ± . (97)20sing the assignment of Z and Z element (95) and (96), we find that zero modesappear if the following relations are satisfied, n l L ± ≡ l + l + l ) + l + l ) + l + l ) + l + l ) = − l − α ± (mod 6) , n l L ± ≡ l + l + l + l + l = − l − β ± (mod 2) . (98)The relation n al R ± = n al L ± ∓ n =
12, and they are rewritten as n ¯5 = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( ϕ , − + + P (1,1) − − P ( ϕ , − − ´ p C l · · · p C l , (99) n = X l = k − l X l = · · · k − l −···− l X l = ( − l ³ P (1,1) + − P ( ϕ , − + + P (1,1) − − P ( ϕ , − − ´ p C l · · · p C l , (100) n = X l = k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( ϕ , − + + P (1,1) − + P ( ϕ , − − ´ p C l · · · p C l , (101)where P ( ρ , ρ ′ ) ± are projection operators that pick up the part relating ( P ± , P ± ) = ( ρ , ρ ′ )and are written by P ( ρ , ρ ′ ) ± = ¡ + ρ P ± + ρ P ± + ρ P ± + ρ P ± + ρ P ± ¢ ³ + ρ ′ P ± ´ . (102)Here, we give some examples for representations and BCs to derive n ¯5 = n =
3, inTable 12. Table 12: Examples for the three families of SU (5) from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[8,3] (5,0,0,3,0,0,0,0,0,0,0,0) (0,1) (2,0)[8,4] (5,0,0,1,0,0,0,2,0,0,0,0) (0,0) (2,0)[9,3] (5,0,0,0,0,0,3,0,0,0,0,1) (0,1) (5,0)[9,4] (5,2,0,1,0,0,1,0,0,0,0,0) (2,0) (2,0)[10,3] (5,0,0,1,1,0,0,0,0,0,3,0) (0,1) (4,1)[10,4] (5,0,1,0,1,1,0,0,0,1,1,0) (5,0) (2,0)[10,5] (5,0,0,0,0,0,1,2,0,2,0,0) (4,1) (1,0)[11,3] (5,0,0,1,0,0,0,0,0,1,4,0) (3,1) (4,1)[11,4] (5,0,0,0,0,2,0,0,2,1,0,1) (5,0) (2,0)[11,5] (5,3,0,0,0,0,0,0,0,0,3,0) (1,1) (1,1)[12,3] (5,3,0,1,0,0,0,0,0,0,0,3) (0,1) (3,0)[12,4] (5,0,0,0,0,0,0,1,0,4,1,1) (5,0) (2,0)[12,5] (5,0,0,0,0,0,2,1,2,1,1,0) (1,1) (1,1)[12,6] (5,0,0,0,0,3,1,1,2,0,0,0) (3,0) (0,0)21 .5.2 Numbers of the SM multiplets on T / Z After the breakdown of SU ( N ) → SU (3) × SU (2) × SU ( p ) ×· · ·× SU ( p ) × U (1) − m , [ N , k ] ± is decomposed as[ N , k ] ± = k X l = k − l X l = · · · k − l −···− l X l = ¡ C l , C l , p C l , · · · , p C l ¢ ± . (103)Using the assignment of Z and Z element (95) and (96), we find that zero modesappear if the following relations are satisfied, n l l L ± ≡ l + l ) + l + l ) + l + l ) + l + l ) = − l − l − α ± (mod 6) , n l l L ± ≡ l + l + l + l + l = − l − β ± (mod 2) . (104)The relation n al l R ± = n al l L ± ∓ n =
12. Using the projection operators(102), the formulae for ( d R ) c and ( ν R ) c are rewritten as n ¯ d = X ( l , l ) = (2,2),(1,0) k − l X l = · · · k − l −···− l X l = ( − l + l ³ P (1,1) + − P ( ϕ , − + + P (1,1) − − P ( ϕ , − − ´ × p C l · · · p C l , (105) n ¯ ν = X ( l , l ) = (0,0),(3,2) k − l X l = · · · k − l −···− l X l = ³ P (1,1) + + P ( ϕ , − + + P (1,1) − + P ( ϕ , − − ´ p C l · · · p C l . (106)The formulae for l L , ( u R ) c , ( e R ) c and q L are obtained by replacing the summation of( l , l ) for n ¯ d with {(3, 1), (0, 1)}, {(2, 0), (1, 2)}, {(0, 2), (3, 0)} and {(1, 1), (2, 1)}.Here, we give some examples for representations and BCs to derive three families ofSM fermions, in Table 13. We have studied the possibility of family unification on the basis of SU ( N ) gauge theoryon the 6-dimensional space-time, M × T / Z N . We have obtained enormous numbersof models with three families of SU (5) matter multiplets and those with three familiesof the SM multiplets, from a single massless Dirac fermion with a higher-dimensionalrepresentation of SU ( N ), after the orbifold breaking. Total numbers of models with thethree families of SU (5) multiplets and the SM multiplets are summarized in Table 4 and5, respectively. Our results can give a starting point for the construction toward a morerealistic model, because three families of chiral fermions in the SM standard model con-tain in our models.Now, the following open questions should be tackled as a future work.The unwanted matter degrees of freedom can be successfully made massive thanksto the orbifolding. However, some extra gauge fields remain massless, even after thesymmetry breaking due to the Hosotani mechanism [37, 38]. In most cases, this kind ofnon-abelian gauge subgroup plays the role of family symmetry. These massless degreesof freedom must be made massive by further breaking of the family symmetry. Extra22able 13: Examples for the three families of SM multiplets from T / Z .[ N , k ] ( p , p , p , p , p , p , p , p , p , p , p , p ) ( α + , β + ) ( α − , β − )[9,3] (3,2,0,1,0,0,0,0,0,0,1,2) (0,0) (0,1)[9,4] (3,2,0,0,0,1,0,0,1,2,0,0) (1,1) (1,0)[10,3] (3,2,0,0,3,0,0,0,0,0,1,1) (1,0) (1,1)[10,4] (3,2,0,1,1,2,0,0,0,0,1,0) (0,1) (0,0)[11,3] (3,2,1,1,1,0,0,0,0,1,1,1) (0,1) (0,0)[11,4] (3,2,0,1,0,2,0,0,0,3,0,0) (0,1) (1,0)[11,5] (3,2,0,0,1,0,4,0,1,0,0,0) (0,1) (0,0)[12,3] (3,2,0,1,3,1,0,1,0,0,0,1) (1,0) (1,1)[12,4] (3,2,0,0,0,1,1,2,0,2,1,0) (1,1) (1,0)[12,5] (3,2,1,1,0,3,1,1,0,0,0,0) (1,0) (1,1)[12,6] (3,2,0,0,0,1,0,0,3,0,0,3) (1,1) (1,1)[13,3] (3,2,1,0,0,0,0,3,2,0,0,2) (0,0) (0,1)[13,4] (3,2,2,0,1,1,1,1,0,0,1,1) (1,0) (1,1)[13,5] (3,2,1,0,0,4,0,0,0,3,0,0) (1,1) (1,0)[13,6] (3,2,1,0,0,0,0,2,4,0,0,1) (0,0) (0,1)scalar fields can play a role of Higgs fields for the breakdown of extra gauge symmetriesincluding non-abelian gauge symmetries. As a result, extra massless fields including thefamily gauge bosons can be massive.In general, there appear D -term contributions to scalar masses in supersymmetricmodels after the breakdown of such extra gauge symmetries and the D -term contribu-tions lift the mass degeneracy. [41, 42, 43, 44, 45]. The mass degeneracy for each squarkand slepton species in the first two families is favorable for suppressing flavor-changingneutral current (FCNC) processes. The dangerous FCNC processes can be avoided ifthe sfermion masses in the first two families are rather large or the fermion and its su-perpartner mass matrices are aligned. The requirement of degenerate masses wouldyield a constraint on the D -term condensations and/or SUSY breaking mechanism un-less other mechanisms work. If we consider the Scherk-Schwarz mechanism [25, 26] for N = D -term condensations can vanish for the gauge symmetriesbroken at the orbifold breaking scale, because of a universal structure of the soft SUSYbreaking parameters. The D -term contributions have been studied in the framework of SU ( N ) orbifold GUTs [46, 47].Can the gauge coupling unification successfully achieved? If the particle contentsin the minimal supersymmetric standard model only remain in the low-energy spec-trum around and below the TeV scale and a big desert exists after the breakdown of ex-tra gauge symmetries, an ordinary grand unification scenario can be realized up to thethreshold corrections due to the Kaluza-Klein modes and the brane contributions fromnon-unified gauge kinetic terms.Another problem is whether or not the realistic fermion mass spectrum and the gen-eration mixings are successfully achieved. Fermion mass hierarchy and generation mix-ings can also occur through the Froggatt-Nielsen mechanism [48] on the breakdown of23xtra gauge symmetries and the suppression of brane-localized Yukawa coupling con-stants among brane weak Higgs doublets and bulk matters with the volume suppressionfactor [49].It would be interesting to reconsider or reconstruct our models in the framework ofstring theory. Various 4-dimensional string models including three families have beenconstructed from several methods, see e.g. [50] and references therein for useful arti-cles. Furthermore, it would be interesting to study cosmological implications of the classof models presented in this paper, see e.g. [52] and references therein for useful articlestoward this direction.
Acknowledgements
This work was supported in part by scientific grants from the Ministry of Education, Cul-ture, Sports, Science and Technology under Grant Nos. 22540272 and 21244036 (Y.K.).
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