Predictions of SU(9) orbifold family unification

Transcription

1 Predictions of SU9 orbifold family unification arxiv: v [hep-ph] 0 Oct 2015 Yuhei Goto and Yoshiharu Kawamura Department of Physics Shinshu University Matsumoto Japan December Abstract We study predictions of orbifold family unification models with SU9 gauge group on a six-dimensional space-time including the orbifold T 2 /Z 2 and obtain relations among sfermion masses in the supersymmetric extension of models. The models have an excellent feature that just three families of the standard model fermions exist in a pair of Weyl fermions in the 84 representation as four-dimensional zero modes without accompanying any mirror particles. 1 Introduction Gauge theories on a higher-dimensional space-time including an orbifold as an extra space possess suitable properties to realize a family unification [1 14]. Three families of the standard model SM fermions are embedded into a few multiplets of a large gauge group. Extra fermions including mirror particles can be eliminated and the SM fermions can survive as zero modes through the orbifold breaking mechanism. In our previous work we have found a lot of possibilities that three families of the SM fermions appear as zero modes from a pair of Weyl fermions in SUN gauge theories N = 9 1 on the six-dimensional 6D space-time M 4 T 2 /Z M M = 246 [10]. The subjects left behind are to construct realistic models and to find out model-dependent predictions. In this paper we focus on the orbifold family unification in the minimal setup because models tend to be more complex and less realistic by extending the structure of space-time and/or the ingredients of models such as gauge symmetries and representations of matters. We take orbifold family unification models based on SU9 gauge symmetry on M 4 T 2 /Z 2 as a starting point examine the reality of models and find out some predictions. For the reality we use the appearance of Yukawa interactions from interactions in the 6D bulk as a selection rule. For the predictions we search 14st02a@shinshu-u.ac.jp haru@azusa.shinshu-u.ac.jp 1

2 specific relations among sfermion masses in the supersymmetric SUSY extension of models. The contents of this paper are as follows. In Sec.2 we explain our setup and its properties. In Sec. we carry out the examination for the reality of models and the search for predictions. Sec.4 is devoted to conclusions and discussions. 2 SU9 orbifold family unification 2.1 Setup Our space-time is assumed to be the product of four-dimensional 4D Minkowski space-time M 4 and two-dimensional 2D orbifold T 2 /Z 2. The T 2 /Z 2 is obtained by dividing 2D lattice T 2 by the Z 2 transformation: z z where z is a complex coordinate. Then z is identified with 1 l z + me 1 + ne 2 where l m and n are integers and e 1 and e 2 are basis vectors of T 2. We impose the following boundary conditions BCs on a 6D field Ψxz Ψx z = T Ψ [P 0 ]Ψxz1 Ψxe 1 z = T Ψ [P 1 ]Ψxz2 Ψxe 2 z = T Ψ [P 2 ]Ψxz where T Ψ [P 0 ] T Ψ [P 1 ] andt Ψ [P 2 ] represent the representation matrices andp 0 P 1 and P 2 stand for the representation matrices of Z 2 transformations z z z e 1 z and z e 2 z for fields with the fundamental representation. The eigenvalues of T Ψ [P 0 ] T Ψ [P 1 ] and T Ψ [P 2 ] are interpreted as the Z 2 parities on T 2 /Z 2. The fields with even Z 2 parities have zero modes. Here zero modes mean 4D massless fields surviving after compactification. Massive modes are called Kaluza- Klein modes and they do not appear in the low-energy world because they have heavy masses of O1/ where is the size of extra space. Fields including an odd Z 2 parity do not have zero modes. Hence the reduction of symmetry occurs upon compactification unless all components of multiplet have common Z 2 parities. This type of symmetry breaking mechanism is called the orbifold breaking mechanism. 1 Westartwith6DSU9gaugetheoriescontainingapairofWeylfermionsΨ + Ψ. These fermions own a same representation of SU9 but different chiralities and are represented as Ψ + = 1+Γ 1 γ5 7 0 Ψ = 2 ψ 1 ψ 1 1+γ ψ 2 = ψ Ψ = 1 Γ 1+γ5 7 0 Ψ = 2 ψ ψ γ 5 2 = ψ 1 ψ 2 1 The Z 2 orbifolding was used in superstring theory [15] and heterotic M-theory [1617]. In field theoretical models it was applied to the reduction of global SUSY [1819] which is an orbifold version of Scherk-Schwarz mechanism [20 21] and then to the reduction of gauge symmetry [22]. 2

3 where Ψ + and Ψ are fermions with positive and negative chirality respectively and Γ 7 andγ 5 arethechiralityoperatorsfor6dfermionsand4dfermions respectively. We use the following representation for the 8 8 gamma matrices Γ M M = Γ µ = γ µ σ Γ 5 = I 4 4 iσ 1 Γ 6 = I 4 4 iσ 2 6 where γ µ µ = 012 I 4 4 and σ i i = 12 are the 4 4 gamma matrices the 4 4 unit matrix and Pauli matrices respectively. In the chiral representation γ µ are given by 0 σ γ µ µ = σ µ 7 0 where σ µ = Iσ and σ µ = I σ. Here I is the 2 2 unit matrix. The Γ M satisfy the Clifford algebra {Γ M Γ N } = 2η MN. Note that the theories are free of chiral anomalies as a result of cancellations between contributions from Ψ + and those from Ψ. When we take the representation matrices P 0 = diag[+1] p1 [+1] p2 [+1] p [+1] p4 [ 1] p5 [ 1] p6 [ 1] p7 [ 1] p8 P 1 = diag[+1] p1 [+1] p2 [ 1] p [ 1] p4 [+1] p5 [+1] p6 [ 1] p7 [ 1] p8 8 P 2 = diag[+1] p1 [ 1] p2 [+1] p [ 1] p4 [+1] p5 [ 1] p6 [+1] p7 [ 1] p8 the breakdown of SU9 gauge symmetry occurs as SU9 SUp 1 SUp 2 SUp 8 U1 7 m 9 where [±1] pi represents ±1 for all p i elements p 1 +p 2 + +p 8 = 9 and m is a sum of the number of SU1 and SU. Here SU1 unconventionally stand for U1 and SU means nothing. Then 6D fields with the rank-k completely asymmetric tensor representation 9 k are decomposed as 9 = k k k l 1 l 1 =0 l 2 =0 k l 1 l 6 l 7 =0 p1 l 1 p2 l 2 p7 l 7 p8 l 8 1 where a b stands for the representation whose dimension is the combinatorial number l 8 = k l 1 l 2 l 7 and p 8 = 9 p 1 p 2 p 7. In case that Ψ + and Ψ have 9 k of SU9 we denote the intrinsic Z2 parities of ψ 1 and ψ2 as η0 k+ η1 k+ η2 k+ and η0 k η1 k η2 k respectively. Then those of ψ2 and ψ 1 are fixed as η0 k+ η1 k+ η2 k+ and η0 k η1 k η2 k from the Z 2 invariance of kinetic terms and the transformation properties of the covariant derivatives Z 2 : D z D z and D z D z. On the breakdown of SU9 due to 8 the Z 2 parities of the component with the representation p1 l 1 p2 l 2 p7 l 7 p8 l 8 are given by P 0± = 1 l 5+l 6 +l 7 +l 8 η 0 k± = 1 k l 1 l 2 l l 4 η 0 k± 11

4 P 1± = 1 l +l 4 +l 7 +l 8 η 1 k± = 1k l 1 l 2 l 5 l 6 η 1 k± 12 P 2± = 1 l 2+l 4 +l 6 +l 8 η 2 k± = 1k l 1 l l 5 l 7 η 2 k± 1 where P a+ and P a a = 012 are the Z 2 parities of ψ 1 and ψ2 respectively. Those of ψ 2 and ψ1 are P a+ and P a. We have found 2 possibilities that just three families of the SM fermions survive as zero modes from a pair of Weyl fermions with the 84= 9 representation of SU9. For the list of p 1 p 2 p p 4 p 5 p 6 p 7 p 8 to derive them see Table VII in [10]. They are classified into two cases based on the pattern of gauge symmetry breaking such that SU9 SU C SU2 SU F U1 and SU9 SU C SU2 SU2 F U1 4. We study how well the three families of fermions in the SM are embedded into Ψ + and Ψ in the following. 2.2 SU9 SU C SU2 SU F U1 For the case that p 1 = p 2 = 2 either of p p 4 p 5 or p 6 is and either of p 7 or p 8 is 1 SU9 is broken down as SU9 SU C SU2 SU F U1 1 U1 2 U1 14 where SU F is the gauge group concerning the family of fermions U1 1 belongs to a subgroup of SU5 and is identified with U1 Y in the SM and others are originated from SU9 and SU4 as SU9 SU5 SU4 U SU4 SU U1. 16 et us illustrate the survival of three families in the SM using two typical BCs. BC1p 1 p 2 p p 4 p 5 p 6 p 7 p 8 = In this case 84 is decomposed into particles with the SM gauge quantum numbers and its opposite ones and their U1 charges and Z 2 parities are listed in Table 1. In the first and second columns particles are denoted by using the symbols in the SM and those with primes are regarded as mirror particles. Here mirror particles are particles with opposite quantum numbers under the SM gauge group G SM = SU C SU2 U1 Y. TheU1chargesaregivenuptothenormalization. The Z 2 parities of are given by omitting the subscript k= in the last column. The Z 2 parities of ψ 21 are opposite to those of. When we assign the intrinsic Z 2 parities of ψ 1 and ψ2 as η+ 0 η1 + η2 + = η0 η1 η2 = all mirror particles have an odd Z 2 parity and disappear in the low-energy world. Then just three sets of SM fermions q i ui c d i c l i ei c survive as zero modes and they belong to the following chiral fermions ψ 1 ui c e i c ν c ψ 2 di ψ1 li c ψ 2 qi 18 4

5 Table 1: Decomposition of 84 for p 1 p 2 p p 4 p 5 p 6 p 7 p 8 = SU C SU2 SU F U1 1 U1 2 U1 P 0 P 1 P 2 e c e 2 = η 0 +η 1 +η 2 q q c = η 0 +η 1 η 2 u c u = η 0 +η 1 +η 2 u c u = η 0 η 1 +η 2 u c u = 11 4 η 0 η 1 η 2 q q 1 c = η 0 η 1 η 2 q q 1 c 2 1 = 21 1 η 0 η 1 +η 2 e c e = η 0 η 1 +η 2 e c e = η 0 η 1 η 2 d c d = η0 +η 1 +η 2 d c d = η 0 +η 1 η 2 l l c = η 0 +η 1 η 2 l l c = η 0 +η 1 +η 2 ν c ˆν 2 = η 0 η 1 +η 2 ν c ˆν 2 2 = η 0 η 1 η 2 5

6 where i= 12 stands for the family index. By exchanging η a + for ηa ψ1 and ψ2 are exchanged for ψ 2 and ψ1 respectively. Note that a right-handed neutrino ν c appears alone. We obtain the same result 18 by assigning the intrinsic Z 2 parities suitably in case with p 4 p 5 or p 6 = in place of p =. BC2p 1 p 2 p p 4 p 5 p 6 p 7 p 8 = In this case 84 is decomposed into particles with the same gauge quantum numbers but sightly different Z 2 parities from those of BC1. Concretely the third Z 2 parity P 2 of fields with l 7 = 1 is opposite to that with l 8 = 1 i.e. P 2 of and is given by +η 2 η 2 +η 2 +η 2 η 2 and +η 2 respectively. Under the same assignment of the intrinsic Z 2 parities as 17 all mirror particles have an odd Z 2 parity and disappear in the low-energy world. Then just three sets of SM fermions survive as zero modes such that ψ 1 u i c e i c ν c ψ 2 l i c ψ 1 d i ψ 2 q i. 19 Note that l i c and d i are embedded into ψ2 and ψ1 respectively different from the case of BC1. We obtain the same result 19 by assigning the intrinsic Z 2 parities suitably in case with p 4 p 5 or p 6 = in place of p =. We summarize fermions with zero modes and those gauge quantum numbers in Table 2. Here G 2 = SU C SU2 SU F l a is a number appearing in a representation p a l a of SUF for a = 45 or 6 and in the 7-th and 8-th columns the way of embeddings for the SM species are shown for p 8 = 1 and p 7 = 1 respectively. Table 2: Gauge quantum numbers of fermions with even Z 2 parities for SU9 G 2 U1 1 U1 2 U1. species G 2 l 1 l 2 l a U1 1 U1 2 U1 p 8 = 1 p 7 = 1 q i ψ 21 ψ 21 u i c d i ψ 21 l i c ψ 21 e i c ν c SU9 SU C SU2 SU2 F U1 4 For the case that p 1 = p 2 = 2 either of p p 4 or p 5 p 6 is 21 or 12 and either of p 7 or p 8 is 1 SU9 is broken down as SU9 SU C SU2 SU2 F U1 1 U1 2 U1 U

7 where U1 1 belongs to a subgroup of SU5 and is identified with U1 Y in the SM and others are originated from SU9 SU4 and SU as SU9 SU5 SU4 U SU4 SU U1 22 SU SU2 U The embedding of species are classified into two types according to p 8 = 1 or p 7 = 1. For the case with p 8 = 1 just three sets of SM fermions survive as zero modes such that u i c e i c q ψ 21 d i l c d l i c ψ 21 u c e c q i ν c 24 where i = 12. For the case with p 7 = 1 just three sets of SM fermions survive as zero modes such that u i c e i c q ψ 21 d l i c d i l c ψ 21 u c e c q i ν c 25 where i = 12. We summarize fermions with zero modes and those gauge quantum numbers in Table. Here G 22 = SU C SU2 SU2 F. Table : Gauge quantum numbers of fermions with even Z 2 parities for SU9 G 22 U1 1 U1 2 U1 U1 4. species G 22 U1 1 U1 2 U1 U1 4 p 8 = 1 p 7 = 1 u 1 c u 2 c u c ψ 21 ψ 21 q 1q ψ 21 ψ 21 q e 1 c e 2 c e c ψ 21 ψ 21 d 1 d ψ 21 d ψ 21 l 1c l 2c ψ 21 l c ψ 21 ν c ψ 21 ψ 21 7

8 Predictions.1 Yukawa interactions We examine whether four types of SU9 orbifold family unification models where theembedding ofthesmfermionsarerealizedas181924and25 arerealistic or not by adopting the appearance of Yukawa interactions from interactions in the 6D bulk as a selection rule. This rule is not almighty to select models because Yukawa interactions can be constructed on the fixed points of T 2 /Z 2. Here we carry out the analysis under the assumption that such brane interactions are small compared with the bulk ones in the absence of SUSY. We assume that the Yukawa interactions in the SM come from interaction terms containing fermions in the bilinear form and products of scalar fields in the 6D bulk. 2 From the orentz gauge and Z 2 invariance the agrangian density containing interactions among a pair of Weyl fermions Ψ + Ψ and scalar fields Φ I on 6D space-time is in general written as int = a f Ψ +abc Ψ def F abc defφ I + a f Ψ Tabc + EΨdef G abcdef Φ I +h.c. = ψ 1 ψ 1 +ψ 2 ψ 2 FΦ I + ψ 1 c ψ 2 +ψ 1 c ψ 2 GΦ I +h.c. 26 where Ψ + Ψ +Γ 0 = γ0 and c = iγ 0 γ 2. In the final expression of 26 we omit indices of SU9 such as a b f designating the components toavoid complications. The FΦ I andgφ I aresomepolynomials ofφ I e.g. FΦ I is expressed by FΦ I = f I1 Φ I 1 + f I1 I 2 Φ I 1 Φ I 2 + = f I1 I n Φ I1 Φ In 27 I 1 I 1 I 2 n I 1 I n where f I1 I n are coupling constants. Note that mass terms of Ψ ± such as m D Ψ + Ψ and m M Ψ T +EΨ are forbidden at the tree level in case that Ψ + and Ψ have different intrinsic Z 2 parities. Using the representation given by 6 and 7 E is written as E Γ 1 Γ Γ 6 = 0 0 iσ iσ 2 iσ iσ where σ 2 is the second element of Pauli matrices. It is shown that int is invariant under the 6D orentz transformation Ψ ± exp [ i 4 ω MNΣ MN] Ψ ± where Σ MN = i 2 [ΓM Γ N ] and ω MN are parameters relating 6D orentz boosts and rotations. After the dimensional reduction occurs and some components acquire the vacuum expectation values VEVs generating the breakdown of extra gauge symmetries the 2 We assume that fermion condensations and orentz tensor fields are not involved with the generation of Yukawa interactions. 8

9 linear terms of the Higgs doublet φ h and its charge conjugated one φ h can appear in FΦ I and GΦ I and then the Yukawa interactions are derived. For instance the linear term fφ h appears from FΦ I = fφ 1 Φ Φ 5 where Φ m are scalar fields whose representations are 9 m after some SM singlets in Φ and Φ 5 acquire the VEVs. From the above observations we impose the selection rule that Yukawa interactions fij uqi uj φ h fij dqi dj φ h and fij eli ej φ h in the SM can be derived from int on orbifold family unification models. For BC1 the following egrangian density is derived at the compactification scale M C BC1 = ij=1 d i qj F 1 1ij φ+ ij=1 l i ej F 1 2ij φ+ ij=1 u i qj 1 G ij φ+h.c. 29 using 18 and Yukawa interactions in the SM can be obtained after some SM singlet scalar fields in the polynomials F 1 φ F1 2 φ and G 1φ acquire the VEVs. 1 Because all gauge quantum numbers of the operator q i dj are same as those of li e j 1 1 there is a possibility that F 1 φ is identical with F 2 φ as a simple case. In this case we have the relations fij d = fe ji at the extra gauge symmetry breaking scale. For BC2 the following egrangian density is derived BC2 = u i q j ij=1 2 G ij φ+h.c. using 19. In this case down-type quark and charged leptons masses cannot be obtained from int at the tree level at M C. For BC the following egrangian density is derived BC = 2 d i q j ij=1 + F 1ij φ+q d F 2 φ+ 2 ij=1 u i qj G 2 l i e j ij=1 F ij φ+e l F 4 φ+h.c. 1ij φ+q u G 2 φ+h.c. 1 using 24. For BC4 the following egrangian density is derived BC4 = 2 i=1 d q i 4 F 1i φ+q d i 4 F 2i φ+l e i 4 F i φ+e l i 4 F 4i φ +h.c. + 2 ij=1 u i qj 4 G 1ij φ+q u G4 2 φ+h.c. 2 using 25. In both cases the full flavor mixing cannot be realized at the tree level at M C. In this way we find that the model based on the embedding 18 is a possible candidate to realize the fermion mass hierarchy and flavor mixing in case that radiative 9

10 corrections are too small to generate mixing terms with suitable size for BC2BC and BC4. In any case we have no powerful principle to determine the polynomials of scalar fields and hence we obtain no useful predictions from the fermion sector..2 Sfermion masses The SUSY grand unified theories on an orbifold have a desirable feature that the triplet-doublet splitting of Higgs multiplets is elegantly realized [2 24]. Hence it would be interesting to construct a SUSY extension of orbifold family unification models. In the presence of SUSY the model with BC1 does not obtain advantages of fermion sector over that with BC2BC or BC4 because any interactions other than gauge interactions are not allowed in the bulk and Yukawa interactions must appear from brane interactions. In SUSY models complex scalar fields Φ + Φ are introducedassuperpartnersofψ + Ψ andtheyconsist oftwosetsofcomplexscalar fields Φ + = φ 1 + φ2 + and Φ = φ 1 φ2 where φ1 + φ2 + φ1 and φ2 are superpartners of ψ 1 ψ2 ψ1 and ψ2 respectively. Here we pay attention to superpartners of the SM fermions called sfermions and study predictions of models. Based on the assignment 18 for BC1 sfermions are embedded into scalar fields as follows φ 1 + ũ i ẽ i ν φ 2 + d i φ 1 l i φ 2 q i. Gauge quantum numbers for sfermions are given in Table 4. Here the charge conjugation is performed for scalar fields d i i and l corresponding to the right-handed fermions and G 2 = SU C SU2 SU F. Note that l 1 l 2 l a is untouched by change as a mark of the place of origin in 84. Table 4: Gauge quantum numbers of sfermions with even Z 2 parities for SU9 G 2 U1 1 U1 2 U1. species G 2 l 1 l 2 l a U1 1 U1 2 U1 q i ũ i d i li ẽ i ν We study the sfermion masses based on the following two assumptions. 1 The SUSY is broken down by some mechanism and sfermions acquire the soft SUSY breaking masses respecting SU9 gauge symmetry. Then ũ i ẽi ν and d i get a common mass m + and q i and l i get a common mass m at some scale M S. 2 Extra gauge symmetries SU F U1 2 U1 are broken down by the VEVs 10

11 of some scalar fields at M S. Then the D-term contributions to the scalar masses can appear as a dominant source of mass splitting. The D-term contributions in general originate from D-terms related to broken gauge symmetries when the soft SUSY breaking parameters possess non-universal structure and the rank of gauge group decreases after the breakdown of gauge symmetry [25 28]. The contributions for scalar fields specifying by l 1 l 2 l a are given by m 2 Dl 1 l 2 l a = 1 l 1+l 2 [Q 1 D F1 +Q 2 D F2 +{9l 1 +l 2 15}D 2 +{4l a l 1 l 2 }D ] 4 where Q 1 and Q 2 are the diagonal charges up to normalization of SU F for the triplet i.e. Q 1 Q 2 = and 0 2. D F1 D F2 D 2 and D are parameters including D-term condensations for broken symmetries. Using m + m and m 2 Dl 1 l 2 l a we derive the following formulae of mass square for each species at M S : m 2 ũ 1 m 2 ũ 2 m 2 ũ m 2 ẽ 1 m 2 ẽ 2 m 2 ẽ = m 2 + +D F1 +D F2 +D 2 +D 5 = m 2 + D F1 +D F2 +D 2 +D 6 = m 2 + 2D F2 +D 2 +D 7 = m 2 + +D F1 +D F2 +D 2 +D 8 = m 2 + D F1 +D F2 +D 2 +D 9 = m 2 + 2D F2 +D 2 +D 4 m 2 d1 = m 2 + D F1 D F2 +6D 2 +2D 41 m 2 d2 = m 2 + +D F1 D F2 +6D 2 +2D 42 m 2 d = m D F2 +6D 2 +2D 4 m 2 q = m 2 1 +D F1 +D F2 +D 2 +D 44 m 2 q 2 = m 2 +D F1 D F2 +D 2 +D 45 m 2 q = m 2 2D F2 +D 2 +D 46 m 2 l1 = m 2 D F1 D F2 +6D 2 +2D 47 m 2 l2 = m 2 D F1 +D F2 +6D 2 +2D 48 m 2 l = m 2 +2D F2 +6D 2 +2D. 49 By eliminating unknown parameters such as m 2 + m 2 D F1 D F2 D 2 and D we In case that the extra gauge symmetry breaking scale M F is lower than M S m 2 ± receive radiative corrections between M S and M F and the mass formulae should be modified. Here we consider the simplest case to avoid complications. 11

12 obtain 15 kinds of relations 4 m 2 ũ 1 = m 2 ẽ m 2 1 ũ 2 = m 2 ẽ m 2 2 ũ = m 2 ẽ 5 m 2 d1 m 2 l1 = m 2 d2 m 2 l2 = m 2 d m 2 l = m 2 ũ m 2 q = m ũ m 2 q = m ũ m 2 q 51 +m 2 l1 = m 2 q +m 2 = m 2 q +m 52 m 2 q 1 m 2 q +m 1 2 d1 = m 2 q +m 2 2 d2 = m 2 q +m 2 d = m 2 l1 +m 2 ũ 1 They are compactly rewritten as m 2 ũ i m 2 ũ i = m 2 ẽ m i 2 di m 2 ũ i m 2 ũ j = m 2 di +m 2 dj = m 2 l2 +m 2 ũ 2 = m 2 l +m 2 ũ. 5 = m 2 li m 2 q i 54 = m 2 q i m 2 q = m j 2 li +m 2 lj 55 where ij = 12. In the same way based on 19 for BC2 we obtain the relations m 2 ũ i m 2 ũ i = m 2 ẽ m i 2 li m 2 ũ i m 2 ũ j = m 2 di +m 2 dj = m 2 di m 2 q i 56 = m 2 q i m 2 q = m j 2 li +m 2 lj 57 where ij = 12. Note that these relations are obtained by exchanging m 2 di for m 2 li in those for BC1. Furthermore we obtain the specific relations for BC and m 2 ũ i = m 2 ẽ m i 2 di m 2 ũ i m 2 ũ i m 2 ũ j m 2 ũ 1 m 2 ũ 1 m 2 ũ 1 m 2 ũ 2 +m 2 ũ = m 2 li m 2 q i 58 = m 2 li +m 2 lj m 2 q m 2 q = m i j +m 2 dj 59 = m 2 q m 2 q 1 6 = m 2 q +m 2 q m 1 +m 2 d = m 2 l1 +m 2 l 61 m 2 ũ = m 2 i ẽ m i m 2 ũ = m i m 2 q 62 m 2 ũ m 2 = m i ũ j +m 2 dj m 2 q m 2 q = m i +m 2 lj 6 m 2 ũ = m 2 q m 2 q m 2 ũ 1 +m 2 ũ = m 2 q 1 +m 2 q m 2 d1 +m 2 d = m 2 l1 +m 2 l 65 for BC4. Here ij = 12 and we denote ũ ẽ d l and q as ũ ẽ d l and q. The relations for BC4 are obtained by exchanging for m m2 di 2 li in those for BC. The above relations become predictions to probe models because they are specific to models in case that the extra gauge symmetry breaking scale is near M S. 4 Sum rules among sfermion masses have also been derived using the orbifold family unification models on five-dimensional 5D space-time [29 1]. 12

13 4 Conclusions and discussions We have taken orbifold family unification models based on SU9 gauge symmetry on M 4 T 2 /Z 2 as a starting point and have examined the reality of models by adopting the appearance of Yukawa interactions from the interactions in the 6D bulk as a selection rule. We have picked out a candidate of model compatible with the observed fermion masses and flavor mixing. The model has an feature that just three families of fermions in the SM exist as zero modes and any mirror particles do not appear in the low-energy world after the breakdown of gauge symmetry SU9 SU C SU2 SU F U1 by orbifolding. Depending on the assignment of intrinsic Z 2 parities u i c e i c d i and li c q i belong to Ψ ± and Ψ with 84 of SU9 respectively. We have found out specific relations among sfermion masses as model-dependent predictions in the SUSY extension of models. The mass degeneracy for each squark and slepton species in the first two families is favorable for suppressing flavor-changing neutral current FCNC processes. The D-term contributions relating SU F however can spoil the mass degeneracy. Such dangerous situations induce sizable FCNC processes can be avoided if the sfermion masses in the first two families are rather large the fermion and its superpartner mass matrices are aligned or the D-term contributions to lift the degeneracy are small enough. As a future work we need to answer the question whether the fermion mass spectrum and flavor mixing are successfully achieved at the low energy scale in our orbifold family unification model. Acknowledgments This work is supported in part by funding from Nagano Society for The Promotion of Science Y. Goto. eferences [1] K. S. Babu S. M. Barr and B. Kyae Phys. ev. D [2] T. Watari and T. Yanagida Phys. ett. B [] I. Gogoladze Y. Mimura and S. Nandi Phys. ett. B [4] I. Gogoladze T. i Y. Mimura and S. Nandi Phys. ev. D [5] I. Gogoladze C. A. ee Y. Mimura and Q. Shafi Phys. ett. B [6] I. Gogoladze Y. Mimura and S. Nandi Phys. ev. ett [7] Y. Kawamura T. Kinami and K. Oda Phys. ev. D [8] Y. Mimura and S. Nandi Phys. ev. D

14 [9] Y. Kawamura and T. Miura Phys. ev. D [10] Y. Goto Y. Kawamura and T Miura Phys. ev. D [11] Y. Fujimoto T. Kobayashi T. Miura K. Nishiwaki and M. Sakamoto Phys. ev. D [12] Y. Goto Y. Kawamura and T Miura Int. J. Mod. Phys [1] T. Abe Y. Fujimoto T. Kobayashi T. Miura K. Nishiwaki and M. Sakamoto J. High Energy Phys [14] Y. Fujimoto T. Kobayashi T. Miura K. Nishiwaki M. Sakamoto and Y. Tatsuta Nucl. Phys. B [15]. Antoniadis Phys. ett. B [16] P. Hořava and E. Witten Nucl. Phys. B [17] P. Hořava and E. Witten Nucl. Phys. B [18] E. A. Mirabelli and M. Peskin Phys. ev. D [19] A. Pomarol and M. Quriós Phys. ett. B [20] J. Scherk and J. H. Schwarz Phys. ett. B [21] J. Scherk and J. H. Schwarz Nucl. Phys. B [22] Y. Kawamura Prog. Theor. Phys [2] Y. Kawamura Prog. Theor. Phys [24]. Hall and Y. Nomura Phys. ev. D [25] Y. Kawamura H. Murayama and M. Yamaguchi Phys. ett. B [26] Y. Kawamura H. Murayama and M. Yamaguchi Phys. ev. D [27] M. Drees Phys. ett. B [28] J. S. Hagelin and S. Kelley Nucl. Phys. B [29] Y. Kawamura and T. Kinami Int. J. Mod. Phys. A [0] Y. Kawamura and T. Kinami Prog. Theor. Phys [1] Y. Kawamura T. Kinami and T. Miura J. High Energy Phys