Theory of flavors: String compactification

We present a calculating method for the quark and lepton mixing angles. After a general discussion in field theoretic models, we present a working model from a string compactification through $Z_{12-I}$ orbifold compactification. It is beyond presenting just three families of the standard model but is the first example from string compactification successfully fitting to the observed data. Assuming that all Yukawa couplings from string compactification are real, we also comment on a relation between the CP phases in the Jarlskog determinants obtained from the CKM and PMNS matrices. The flipped SU(5) model leads to the doublet-triplet splitting and possible proton decay operators. It is shown that the vacuum expectation values can be tuned such that the proton lifetime is long enough.


I. INTRODUCTION
"How is the current allocation of flavors realized?" is the most urgent and also interesting theoretical problem in the Standard Model (SM). Extension of the SM to grand unification (GUT) and string models [1,2] continues to require to solve this flavor problem. Gauge symmetries as family groups should satisfy the anomaly freedom, which can be achieved in extended GUTs [3] and in models without anomaly [4]. Not to worry about the gauge anomalies, sometimes global symmetries are used for the family groups [5][6][7]. It has been reviewed at several places [8,9].
In the SM, the difference of families is manifested in the Cabibbo-Kobayashi-Maskawa (CKM) matrix in the quark sector [10,11] and in the Pontecorvo-Maki-Nakagawa-Sakada (PMNS) matrix in the leptonic sector [12,13]. To relate the left(L) and right(R) mixing angle parameters, the flavor group G f has been introduced to obtain more relations between flavor parameters [14][15][16][17]. In most cases, a factor flavor group G f is introduced in addition to the SM or GUT. On the other hand, an attractive mechanism is to unify all the fermion representations in an irreducible set of SU(N ) representations of an extended GUT [3,[18][19][20]. The E 8 × E 8 gauge group can be considered to belong to this class but in ten dimensions. So, compactification of six extra dimensions may be the key to the unification of families in ten dimensional superstring models.
A notable difference between the CKM and the PMNS matrices lies in the fact that in the CKM matrix the large elements are located in the diagonal entries while it is not so in the PMNS matrix. So, for the CKM parameters the quark mass ratios were used before [14,15]. On the other hand, for the PMNS parameters non-Abelian discrete groups are used [21,22]. One may say that there is one similarity in the CP phases of the CKM and PMNS matrices. The CKM phase is close to 90 degrees in the Kim-Seo (KS) parametrization [23] and the PMNS phase is −90 degrees (but with a large error bars) [24]. Even, there exists an attempt to unify these CP phases [25].
To reduce the number of parameters in the flavor sector, family symmetries can be used. Simple ones are U(1) groups. But, to introduce a hierarchy, vacuum expectation values (VEVs) of the SM singlets are suggested, which is known as the Froggatt-Nielsen (FN) mechanism [26].
In this paper, we study singlet representations beyond the SM based on family symmetry groups. For various reasons in field theoretic models, we consider U(1) 2 among which one is anomalous and the other is anomaly free. We attempt to obtain singlets from the orbifold compactification of the E 8 × E 8 heterotic string [27] based on the simplest Z 12−I lattice [28,29]. Fixed points of 13 prime orbifolds listed in [30] shows that the Z 12−I lattice can be considered to be the simplest because there are only three fixed points. 1 In Sec. II, we briefly recapitulate the fermion mass structure: Dirac fermions of charged leptons and quarks, and Majorana fermions for the SM neutrinos. We set up the scheme to use Weyl fermions to express both the Dirac and Majorana masses pressented in subsequent sections. Those who are familiar to these can skip this section. In Sec. III, we present a beyond-SM with two U(1)'s toward useful fermion mass textures, where one is U(1) anom global symmetry for the "invisible" axion and the other is anomaly free gauged U(1). In Sec. IV, we obtain a successful flavor structure from Z 12−I orbifold compactification. The 3rd family is assumed to be the one from the untwisted sector U . Sec. V is a conclusion.

II. FERMION MASSES
For continuous parameters of transformation, let us begin with the axial-vector currents of fermions where Γ is a charge operator and Ψ is a column vector of three fermions (families). The divergence of the current is where A I is the anomaly coefficient of the gauge fields A I µ for the gauge group SU(N ) I , and J 5 Γ depends on the masses of fermions, where µ is a mass scale and M is the mass matrix in the flavor basis. The anomaly term is a flavor singlet which can be written in terms of a flavor singlet quark fields in the θ-vacuum, e.g. for two flavors in SU(3) c [31], where Z = m u /m d 1 2 . Obviously, the anomalous and anomaly free terms give nonzero trace for the fermion mass matrix. In Ref. [9], two U(1) symmetries were considered, one anomalous and the other anomaly-free. The anomalous global symmetry is to introduce the so-called "invisible" axion. Since the sum of quark masses is nonzero and large O(m t ), we also attempt to have the anomalous U(1). The anomalous U(1) must be a global symmetry and the anomaly free part can be a gauge symmetry. Let us start using two component fermions to write down mass terms.

A. Weyl fermions
A four component Dirac spinor, e.g. for the electron field, can be split into two Weyl spinors ξ and η, Gauge interactions do not change the chirality. Quantum fild e L destroys a L-handed electron and creates a R-handed positron. But, e L has nothing to do with destroying a R-handed electron e R and creating a L-handed positron. On the other hand, the anti-particle of the L-handed electron e L = (e L1 , e L2 , 0, 0) T is which is a R-handed field. This R-handed field destroys R-handed positron and creates L-handed electron. With these two Weyl fields, we can destroy L-and R-electrons and create L-and R-positrons, which is done by a four-component Direc electron. Thus, two Weyl fields are enough at this stage. With the Weyl field ξ, let us construct a Lorentz invariant ij e Li e Lj . It is the mass term but the electron number is broken by this term. So, for charged particles, one Weyl field cannot be massive. For neutrinos, one Weyl field can give a mass term which is known to be Majorana mass. In this paper, we will use Weyl fields even for expressing Majorana masses, For a Dirac mass we use the opposite chirality, i.e.
so that (7) becomes Assigning the same charge conjugation for ξ fields in (7), the Majorana mass term breaks C, but the Dirac mass term (9) can preserve C by assigning the same C's for ξ and η. Discussing both Majorana and Dirac masses, using the Weyl fermion is therefore simple enough.
B. m b mτ and Georgi-Jarlskog relation The observed ratio of the third family masses m b /m τ 4.5/1.5 ≈ 3 hints that m b m τ at the unification scale. The factor 3 arises by renormalization group evolution [32]. In the Georgi-Glashow (GG) SU(5) [33], 10 f 5 f 5 H gives the same mass to b and τ by 5 H and it is considered to be a success of the GUT [34]. For the muon and strange quark, however, there is a big problem in the GG SU(5) model. The low energy mass ratio at 100 MeV is m s /m µ ≈ 1 while the renormalization group evolution expects it to be 3 if m s m µ at the unification scale. If m s /m µ 1 3 at the unification scale, then the low energy mass ratio is understandable. But, this is a big problem with Higgs quintets only. One way out is the Georgi-Jarlskog relation introducing a big Higgs representation 45 H [35]. If 45 H is the leading contribution to the second family fermions in the GG model, then m s /m µ 1 3 is obtained at the unification scale. To present a rationale for 45 H for the needed mass matrix texture, two U(1)'s were suggested long time ago [9].
C. Flipped SU (5) Our terminology of flipped SU(5) is a rank 5 gauge group SU(5) flip =SU(5)×U(1) X . Representations will be denoted as SU(5) U(1) . In the flipped SU(5) [36,37], masses of charged leptons and d-type quarks are not related, which is considered to be a merit in relating masses. In string compactification, reasonable supersymmetric SM's are obtained from compactification of heterotic string. The reason is the following. For N = 1 supersymmetric (SUSY) massless fields, only the completely antisymmetric representations are allowed with one compactification scale from heterotic string [28]. If the Higgs fields breaking a GUT group appear as massless spectra, then there is no adjoint representation at the GUT scale which is needed for breaking the GG SU(5) or SO(10) GUT [38,39] or some Pati-Salam (PS) [40] gauge groups. 2 In SU(5) flip , the representation 10 +1 ⊕ 10 −1 can break the rank-5 SU(5) flip down to the rank-4 SM gauge group. At the GUT level, therefore only the flipped SU(5) is actually realized in several string compactifications [1, 29,42,43].

III. U(1)anom×U(1) fr FAMILY SYMMETRY
We introduce supersymmetry and two U(1) gauge symmetries, U(1) anom ×U(1) fr , where U(1) anom is anomalous and U(1) fr is free of gauge anomalies. Dangerous dimension-4 superpotential of the 1st family members triggering proton decay is where the subscrit 1 denotes the first family. U(1) B−L allows the above superpotential but U(1) anom or U(1) fr may not allow it. Thus, the extra U(1)'s may be useful forbidding some unwanted proton decay operators. In string compactification, one has to check the U(1) anom ×U(1) fr quantum numbers of the first family members to see if the unwanted proton decay operators are forbidden. If the proton decay problem is safe, one can consider the superpotentials generating fermion masses. The mass eigenstates of quarks, q m , are related to the weak eigenstates by L-and R-unitary matrices, U and V , and the charged W + µ coupling for the L-handed quark doublets is which is the CKM matrix.
where A = (m νµ + m νe )/2 and B = (m νµ − m νe )/2, which has the permutation symmetry S 2 between the second and the third family indices. The useful discrete symmetries of [21,22] contain this S 2 as a subgroup. In this case of introducing U(1) anom , where we introduced only L-handed neutrinos, the anomaly freedom must be satisfied by the quantum numbers of the first family leptons or by heavy leptons. Let us begin with the diagonalized Dirac masses of the form (9) for Q em = + 2 3 quarks,

B. Quark mass matrices
where η R = q m uR and ξ L = q m uL . The diagonal form (17) with the needed hierarchy can be obtained by the U(1) charges of Table I, where ∝ σ is a SM singlet field carrying Q = −3. The mass term for up-type quarks is Of course, V u = U u = 1. The mass matrix for Q em = − 1 3 quarks is with In this paper, we use the KS parametrization [23] with the unitary matrix for R-handed fields in the diagonalization process parametrized by another 4 parameters Change the sign m d → −m d , and to reduce the number of parameters let us choose parameters of R-fields as Change the sign m d → −m d , and to reduce the number of parameters let us choose parameters of R-fields as Note that s 4 s 1 s 2 2 . Then, keeping the largest terms in the weak basis mass matrix, where we used In Eq. (26), the (32) element can be 4.2 × 10 −2 e iδ where tan(π − δ ) = −0.9286 sin δ. The Yukawa couplings run from the compactification scale down to the electroweak scale in which case the dimensionless Yukawa couplings cannot be used directly for assigning the input mass parameters. But all Yukawa couplings leading to parameters in Eq. (26) are arising from the VEVs of FN singlet fields and we may use those given in Eq. (26) as the input parameters determining the CKM matrix.

Mass matrix in field theory
Let us present a possibility of obtaining a mass matrix similar to Eq. (26) in field theory. Let the U(1) anom ×U(1) fr quantum numbers are shown as (Q anom , Q fr ). After diagonalizing the Q em = 2 3 quark masses, the L-and R-fields of Q em = − 1 3 quarks, ξ and η, quantum numbers of ηξ are Thus, the quantum numbers of Higgs fields appearing in the mass matrix are To mimick the order appearing in Eq. (26), let us introduce small parameters via the FN SM singlet fields, δ 1 , δ 2 , δ 3 , ∆ 1 , ∆ 2 , 1 , and 2 whose quantum numbers are shown in Table II.
where the overal constant is m b and for simplicity we do not write group theoretic numbers of O(1). The element M d 23 can have δ 3 1 e 3iδ 2 which we neglected because it is much smaller than the other terms. A negative signed phase in M d 32 of Eq. (30) may need a complex conjugated field, but we do not introduce complex conjugated fields in the mass matrix for a SUSY extension.

D. Lepton mass matrices
Again, we use the KS parametrization [23] to specify the phase δ L = δ PMNS from the (3,1) element of M e . Note that the preliminary value δ PMNS ≈ − π 2 [24], where the parameters are the leptonic parameters, Θ 1,2,3 and δ L . Since the PMNS matrix elements are not known as accurately as the CKM matrix elements, we do not present a detail study of the leptonic sector. But note that the phase δ L in Eq. (35) is the PMNS phase δ PMNS .

IV. FROM E8 × E 8 HETEROTIC STRING
In this section, we attempt to realize the texture of quark mass matrix discussed in Subsec. III B. We will not discuss the texture of neutrino mass matrix since the PMNS matrix elements are not known very accurately. Nevertheless, we will comment on the relation of CP phases in the quark and lepton sectors in this section.

A. Z12−I orbifold compactification
Note that the SM mass matrixΨ c I L C −1 Ψ J L M IJ + h.c. (36) gives in general non-symmetric mass matrix of M because Ψ c L and Ψ L transform differently under SU(2) L ×U(1) Y . In the GUT model, Majorana neutrinos in the SU(2) L doublets are embedded in 5 0 of SU(5) in the GG model and 5 +3 in the SU(5) flip . Then, the effective neutrino mass matrix in these simple GUTs are symmetric. For the quark mass matrix, 10 f 10 f 5 Higgs is the up-type quark mass matrix in the GG model, which is symmetric. In the GG model, we usually use diagonalized up-type quark mass matrix, and consider non-symmetric 5 f 10 f 5 Higgs for the down-type quark mass matrix. On the other hand, in the SU(5) flip the down-type quark mass matrix, 10 f 10 f 5 Higgs is symmetric and the up-type quark mass matrix 5 f 10 f 5 Higgs is non-symmetric. So, we prefer to consider a symmetric down-type quark mass matrix in SU(5) flip . The up-type quark mass matrix is non-symmetric, and we can assign different coefficients for M (u)IJ and M (u)JI , down type quark mass matrix = symmetric up type quark mass matrix = asymmetric (37) The SU(5) flip GUT gauge group presented in Ref. [2] is where, in the notation of [28], and six U(1) directions of Ref.   Q anom is given by We will use notations of Ref. [2] for the names of the fields, twisted sectors (T 0 1 , · · · , T 6 ) and the untwisted sector (U ). We also list 1 2 Q 1 in Tables IV such that a discrete subgroup of U (1) 1 can be used for matter parity if needed. We choose one gauged U(1) example beyond U(1) anom , and we checked that any other choice leads to the same conclusion.

B. Doublet-triplet splitting
In the SU(5) flip , it is well-known that there is a possibility of doublet-triplet splitting. C 12 and C 14 in Eq. (54) develop the GUT scale VEVs, where the first equality is for vanishing D-term at the GUT scale. The renormalizable coupling, including the Higgs quintet 10 +1 5 −2 10 +1 ∼ Φ ab Φ c Φ de abcde might give the GUT scale mass term to colored scalars by {de} = {45}, but C 12 H u C 12 coupling is not allowed by the non-vanishing Q anom . A possible higher dimensional operator consistent with the orbifold selection rules and U(1) anom ×U(1) 3 gauge symmetry is By giving GUT to Planck scale VEVs to C 11 , σ 3 , σ 5 , and σ 21 , we obtain a GUT scale mass term for colored scalars, where α, β, γ are the color indices. Thus, the color anti-triplet in 10 combines with the color triplet in the Higgs quintet 5. The colored scalar in the Higgs quintet H d is removed at the GUT scale, and there remains just the Higgs doublet from H d . For this doublet-triplet splitting, we need and the color triplet mass is estimated as Suppose V ∼ M, σ 3,5 ∼ 10 −2.5 M and σ 21 2 ∼ 10 −1 M . Then, we obtain M T ∼ 10 −6 M ∼ 0.6 × 10 12 GeV for M ∼ 6 × 10 17 GeV. 10 12 GeV colored scalar with small Yukawa couplings of the first family is acceptable. Similarly, considering C 12 H u C 12 ∼ 10 +1 5 −2 10 +1 , the colored scalar in the Higgs quintet H u is removed at the GUT scale and there remains just the Higgs doublet from H u . For this, we further require

C. Proton decay problem
One may consider another gauge symmetry to obtain a Z 2 discrete group by breaking U(1) 1 by some VEVs of singlet fields carrying even quantum numbers of Q 1 /2 in Table IV of Eq. (48) carry the odd quantum number of Q 1 /2. We do not have any mechanism for matter parity. The proton decay amplitude must be estimated in detail. 5 In SUSY models, the dimension 5 proton decay operator must be sufficiently suppressed [45]. The dimension 5 proton decay operators to electronic and muonic leptons are from the superpotential q 1 q 1 q 1 l 1,2 , i.e. C 15 C 15 C 15 C 17 and C 15 C 15 C 15 C 16 . Note that C 15 , C 17 , and C 16 are allowed from the sector T 0 4 . Therefore, the Z 12−I orbifold selection rules forbid the product of these four fields from T 0 4 , and hence there is no serious proton decay problem from the above dimension 5 operator multiplied by FN singlets (σ's) appear at least at dimension 7 level in our Z 12−I model.
If it were the GG SU(5), the cubic superpotential written in terms of matter parity violating term, 10 0 5 0 5 0 , triggers proton decay as shown in Fig. 1 [46]. In the SU(5) flip also there arise dangerous proton decay operators where fields with superscript m are matter fields and 10 H −1 is the field breaking SU(5) flip to the SM. The above operators trigger proton decay in our model by products of FN singlets (σ's) appear at dimension 10 level, All the singlets appearing in Eq. (50) are the needed fields for the doublet-triplet splitting in Eq. (45). The coupling in Fig. 1 is estimated, from the first term of Eq. (50) for example as, Suppose that the SUSY breaking scale M SUSY ∼ 10 TeV, the GUT scale M GUT ≈ 3 × 10 16 GeV, and the compactification scale M ∼ 6 × 10 17 GeV. Then, the last factor ∼ (3 × 10 12 ) 2 is balanced by M vev 0.5 × 10 16 GeV where M vev is some average VEV of C 11 and neutral σ fields. The estimate given in Eq. (46) can be fitted to this average. Thus, the dimension 6 operator of Fig. 1 can be controlled such that it is not so strong as the dimension 6 operator derived from the exchange of leptoquark gauge bosons in SUSY GUTs.

D. Families
There are three 5 +3 's and three 1 −5 's in Table IV. These include all members of three SM lepton doublets three u c -type quarks. However, there are four 10 −1 's in Table IV. So, there are a few possibilities of choosing three SM quark doublets. Out of four 10 −1 's, we always choose 10 −1 in the U sector. Then, there are three possibilites of choosing two remaining quark doublets: (1) the antisymmetric combination of 10 −1 's from the T 0 4 sector and 10 −1 from the T 3 sector, (2) two 10 −1 's from the T 0 4 sector, and (3) a linear combination of 10 −1 of T 3 and antisymmetric 10 −1 from T 0 4 , and a linear combination of 10 −1 of T 3 and symmetric 10 −1 from T 0 4 . All these are considered by mixing three 10 −1 's, introducing three angles α, β and γ, where s α,β,γ = sin α,β,γ and c α,β,γ = cos α,β,γ . We choose two out of the above three combinations. Similarly, we define Now, let us identify 10 +1 and 10 −1 's of Table I as and C 15 : 1st family, C 13 : 2nd family, and 5 +3 's of Table I as In this paper, it is outside the scope of current analysis to see the details of superpotential. We just assume ceratin VEVs to fit to the observed data.

Down-type quarks
Let us scale scalar fields and mass matrices such that they are made dimensionless by dividing with a mass parameter, for example by M .
The down-type quark masses are where we presesented only the antisymmetric part in M w d (22) and only the component from T 3 in M w d (11) . For the down-type quarks, it is enough to show non-zero M w d (33) and M w d (22) and the conditions for making the off-diagonsal To satisfy the conditions of Eq. (58), let us choose and c σ 4 σ 6 + c σ 15 σ 21 = 0.
M w d (13) and M w d (31) can be made to vanish.

Up-type quarks
Therefore, we consider the W − µ coupling instead of W + µ coupling of Eq.
Change the sign m u → −m u , and to reduce the number of parameters let us choose parameters of R-fields as Then, we obtain where we require c 2 , c 3 , c 5 O(1). Also, s 5 can be O(1). Thus, we consider, so that M w u /m t is approximately given by we estimate which can be close to Eq. (66). Let all singlet VEVs are real except σ 9 and σ 21 [23], The phase of e i(θ+φ) is fitted to the phase of In Table V, we list θ + φ for a few t 5 . For δ CKM = π 2 and t 5 5.5, we obtain φ −θ. Irrespective of the value of φ, the CP phase in the Jarlskog determinent, δ CKM , is the phase in M w u (31) with the KS parametrization given in Eq. (22).

E. CP phases in the quark and lepton sectors
As done before, let us diagonalize the symmetric fermion masses first. In the flipped SU(5) model, therefore, we diagonalize down-type quark masses and neutrino masses. Then, we consider up-type quarks and charged leptons. Then, the (3,1) elements of the mass matrices are the key. For the third family members from U , masses of t quark and τ lepton arise from The phenomenologically determined leptonic mass element M w e (31) can be obtained from Eq. (61) by changing the quark parameters θ i , δ, ∆, m u , m c , m t to leptonic parameters of Eq. (35): Θ i , δ L , ∆ L , m e , m µ , m τ . Choose the leptonic V matrix elements such that Then, M w e(31) /m τ − sin Θ 1 sin Θ 2 e iδ L where δ L is the PMNS phase. In our model, Table IV, there are three e c fields in the leptonic case (instead of four u c fields in the quark case), and we can choose S A 24 which is the antisymmetric combination of S 24a and S 24b in T 0 4 . So, the leptonic mass matrix has four zero entries with the antisymmetric 1st row and antisymmetric 2nd column, specify the signs of the effective Yukawa couplings in M w u (31) and M w e(31) dictated by string compactification. At this stage, we allow any sign for M w u (31) and M w e(31) since we considered only the selection rules. If the signs of M w u (31) and M w e (31) are the same (opposite), then we conclude that δ CKM and δ PMNS have the opposite (same) signs. 8 The case of opposite signs is consistent with the currently favored phases of δ CKM [23] and δ PMNS [24].
In the PS type standard model SU(4)× SU(2) L × SU(2) R , we would have fermion matter spectra, containing quark and lepton doublets, (4, 2, 1) L ⊕ (4, 1, 2) R + · · · (83) Suppose that the Yukawa coupling (4, 2, 1) L × (4 * , 1, 2) L × (1, 2, 2) h via Higgs (1, 2, 2) h is present from the orbifold compactification. Then, the Yukawa coupling arises from the L-handed Higgs field doublets ij (1, 2, (ij)) h = (1, 2, (12)) h − (1, 2, (21)) h where the R-hand index (12) gives the Higgs doublet coupling to quark doublets and the R-hand index (21) gives the Higgs doublet coupling to lepton doublets. We use the same charge W, i.e. W + µ , for coupling to down-type quarks and charged leptons. So, the relative signs of M w d (31) and M w e (31) are opposite if the product with FN singlet contributions give the same sign. If we use the mass matrices of M w d (31) and M w e(31) for asymmetric mass matrices as in the GG model, then δ CKM and δ PMNS have the opposite signs. But, here one needs an example for breaking SO(10) down to SU(4)× SU(2) L × SU(2) R , where the rank is not reduced, from the spectra of orbifold compactification. One may use the bulk fields for an adjoint representation as pointed out for Z 6−II in Ref. [41] and for Z 2 × Z 2 in Ref. [48] where the N = 2 gauge multiplet in an effective 5-dimensional SUSY model allows an adjoint representation of spin-0 fields.

V. CONCLUSION
In this paper, we presented a theory toward understanding the quark and lepton mixing angles. Specifically, we presented a working example obtained from a string compactification [43] with Q anom charge presented in [2]. Explicit presentations were given for the CKM matrix. The (3,3) element of quark mass matrix in the weak basis, is assumed to be close to the t-quark mass. Because there are only three L-handed quark doublets in the model, the up-type quark mass matrix is antisymmetric under the exchange of a ↔ b among R-handed flavor indices (or u c fields) obtained from T 0 4 . This is because the multiplicity 2 for 5 −3 from T 0 4 is generic and there is no way to distinguish these two. The antisymmetric combination of a and b is named for the 1st family member of 5 +3 's. But, there are four L-handed up-type quark doublets and the up-type quark families have a freedom to choose from these four. We used the freedom of choosing the unitary matrix for the R-handed quarks to fit to the data, and showed that this model predicts reasonable mixing angles within experimental error bounds. Also, we studied the relation between δ CKM and δ PMNS by the phases of some SM singlet scalar fields, assuming that all Yukawa coupling constants from string compactification are real. For the proton decay problem, a Z 2 matter parity cannot be introduced consistently with the solution of the doublet-triplet splitting problem by the GUT scale VEVs, 10 −1 (T 3 ) and 10 +1 (T 9 ) . But, we showed that the proton decay operator appears at a dimension 10 level, which can be made small enough while achieving the doublet-triplet splitting. It will be interesting if a kind of R parity is found within the scheme, which will be pursued in the future.