A Supersymmetric Grand Unified Model with Noncompact Horizontal Symmetry

In a supersymmetric SU(5) grand unified model with a horizontal symmetry SU(1,1), we discuss spontaneous generation of generations to produce three chiral generations of quarks and leptons and one generation of higgses by using one structure field with a half-integer spin of SU(1,1) and two structure fields with integer spins. In particular, the colored higgses can disappear without fine-tuning. The difference of the Yukawa coupling matrices between the down-type quarks and charged leptons is discussed. We show that some special SU(1,1) weight assignments include R-parity as a discrete subgroup, and R-parity remains even after we take into account the SU(1,1) breaking effects from all the VEVs of the structure and matter fields. The assignments forbid the baryon and/or lepton number violating terms except a superpotential quartic term including a coupling of two lepton doublets and two up-type higgses. We discuss how to generate sizable neutrino masses. We show that the proton decay derived from the colored higgses is highly suppressed.

The implications are not only from the issues of naturalness, but also the quantization of charges, the anomaly cancellation of the standard model (SM) gauge groups SU (3) C × SU (2) L × U (1) Y (= G SM ) by each generation of quarks and leptons at low energies [35], the unification of three gauge coupling constants at the unification scale, and the matter unification of quarks and leptons in SM for one or two representations in grand unified groups. They seem to suggest that one of the hidden structures of nature is some unified gauge symmetry [36,37]. As is well-known, candidates for the grand unified gauge symmetry are simple groups, such as SU (5) [37][38][39], SU (6) [40,41], SO(10) [42][43][44][45], and E 6 [46][47][48][49]. For a review, see e.g., Refs. [50][51][52]. Any grand unified model explains the quantization of charges, and some of them explain the anomaly cancellation and the SM gauge coupling unification. Here we will focus on the SU (5) unified group.
As is well-known, the non-supersymmetric SU (5) grand unified model [37] that contains the minimal numbers of quarks, leptons, and higgs predicts rapid proton decay via X and Y gauge bosons. As long as the colored higgs mass is O(M GUT ), since the Yukawa coupling constants of the first and second generations of quarks and leptons coupling to the colored higgs are smaller than the gauge coupling constant, the most strict restriction for the proton decay via the X and Y gauge bosons comes from the mode p → π 0 e + . By using the chiral Lagrangian technique, the lifetime is given by [53] τ (p → π 0 e + ) → 1.1 × 10 36 × M V 10 16 GeV 4 0.003GeV 3 α 2 years, (1) where M V is the X and Y gauge boson mass and α is a hadron matrix element. When we use the gauge bosons masses M V ∼ M GUT ∼ 10 15 GeV and a hadron matrix element α = 0.003 GeV 3 , we obtain τ (p → π 0 e + ) = 1.1 × 10 32 years. From the latest result from the super-Kamiokande [54], the lifetime τ (p → π 0 e + ) > 8.2 × 10 33 years at 90 %C.L. Thus, as is well-known, the non SUSY SU (5) GUT model seems to be ruled out. Fortunately, in the minimal SU (5) SUSY GUT model [2,[55][56][57] the GUT scale M GUT becomes O(10 16 ) GeV. Substituting M V = 10 16 GeV in Eq. (1), we obtain the proton lifetime τ (p → π 0 e + ) = 1.1 × 10 36 years. Thus, the lifetime satisfies the current bound. However, it is also known that the minimal SU (5) SUSY GUT model suffers from rapid proton decay induced from the colored higgses [53,[58][59][60]. According to Ref. [59], the colored higgs masses must be greater than 10 17 GeV for any tan β by using the recent super-Kamiokande result for the lifetime τ (p → K +ν ) > 3.3 × 10 33 years at 90% C.L. [61] when we assume that the sfermion masses are less than 1 TeV. Thus, the colored higgses must have the effective mass greater than O(10 17 ) GeV. On the other hand, the doublet higgs must have O(m SUSY ). This is known as a doublet-triplet splitting problem [62][63][64][65][66][67][68][69].
In addition, the minimal SU (5) GUT model gives an unacceptable relation between the Yukawa coupling constants of down-type quarks and charged leptons without taking into account the higher dimensional operators including the nonvanishing VEVs of the adjoint representation. To break the minimal GUT relation of the Yukawa coupling constants between down-type quarks and charged leptons, roughly speaking, we can classify two methods; one is to consider the higher dimensional operators including the SU (5) adjoint higgs field; another is to introduce the higher dimensional representations, such as Georgi-Jarlskog manner (see, e.g., Ref. [70] for an SU (5) Non-SUSY GUT model and Ref. [71] for an SU (5) SUSY GUT model). The above ways can also be mixed.
Even when we consider SUSY GUT models, they do not give us any insight about the hierarchy of the Yukawa couplings and the number of chiral generations of quarks, leptons and higgses. The mass parameters at the GUT scale in the minimal supersymmetric standard model (MSSM) [72][73][74] are given by Ref. [75] for several values of tan β by using the renormalization group equations of the two-loop gauge couplings and the two-loop Yukawa couplings assuming an effective SUSY scale of 500 GeV. For tan β = (10,38,50), the coupling constants of the third generation of the up-type quark, the down-type quark, and the charged lepton at the GUT scale are y t ≃ (0.48, 0.49, 0.51), y b ≃ (0.051, 0.23, 0.37), and y τ ≃ (0.070, 0.32, 0.51), respectively. When we normalize the Yukawa coupling constants of the third generations equal to one, the mass parameters of first, second and third generations of the up-type quark, the down-type quark, and the charged lepton for tan β = 10 are (ỹ 1 ,ỹ 2 ,ỹ 3 ) ≃ (6.7 × 10 −6 , 2.5 × 10 −3 , 1), (1.0 × 10 −3 , 1, 2 × 10 −2 , 1), and (2.5 × 10 −4 , 6 × 10 −2 , 1), respectively, where the subscript ofỹ a (a = 1, 2, 3) stands for the generation number. The values are almost the same for tan β = 38 and 50.
In this article, we discuss an N = 1 supersymmetric vectorlike SU (5) GUT model with a noncompact horizontal symmetry SU (1, 1) to solve the above problems. We summarize the main results of previous studies of N = 1 supersymmetric vectorlike models with a horizon-tal symmetry SU (1, 1) [80,[83][84][85][86][87]. The number of chiral generations of matter fields, such as quarks, leptons and higgses are determined by the spontaneous symmetry breaking of the horizontal symmetry SU (1, 1), called the spontaneous generation of generations [80]. Through the mechanism, the doublet-triplet splitting of higgses can be realized without fine-tuning and also unreasonably suppressed tiny mass parameters [83,85]. When the horizontal symmetry is unbroken, the original Yukawa coupling matrices of matter fields are completely determined by SU (1, 1) symmetry. The Yukawa coupling constants of the chiral matter fields at low energy are controlled by the SU (1, 1) symmetry and the SU (1, 1) breaking vacua. Each structure of Yukawa couplings of three chiral generations of quarks and leptons has hierarchical structure [80,83,85,87]. The problematic superpotential cubic termsQLD c ,D cDcÛ c ,LLÊ c are automatically forbidden, where in the MSSM these terms are forbidden by R-parity [88] (or matter parity [89]) to prevent rapid proton decay (For a review, see, e.g., Ref. [90]). The dangerous superpotential quartic termsQQQL andÛ cÛ cDcÊc are also not allowed where the usual R-parity cannot forbid these terms [83,85].
We now discuss N = 1 supersymmetric noncompact gauge theory since our model is based on an N = 1 supersymmetric noncompact gauge theory. As is well-known, renormalizable noncompact gauge theories have ghost problems; at least one gauge field has a negative metric in the canonical kinetic term, which indicates the wrong sign and this is physical ghost; the structure fields belonging to the finite dimensional representations also have the physical ghosts. A solution of this problem, discussed in Ref. [91], is to use an N = 1 supersymmetric model with a noncompact gauge group SU (1, 1) that has noncanonical Kähler function and gauge kinetic function with linear representation of SU (1, 1) gauge transformation. At least at classical level, the Lagrangian has gauge and Kähler metrics positive definite at proper vacua, and thus no ghost fields exist at the vacua. For another solution of this problem, see, e.g., Refs. [92,93].
The main purpose of this paper is to show that an SU (5) SUSY GUT model with the noncompact horizontal symmetry SU (1, 1) naturally satisfies current proton decay experiments, solves the doublet-triplet mass splitting problem, and avoids the unrealistic GUT relation for Yukawa couplings. In addition, we will see that this model can accommodate R-parity as a discrete subgroup of the horizontal symmetry.
Here we clarify the difference between this work and the previous works with models with the noncompact horizontal symmetry. This is the first trial to construct a concrete SU (5) model. We apply the spontaneous generation of generations for the model with the matter content of an SU (5) grand unified model. The mixing structures of quarks and leptons that represent the ratio of the mixing between each chiral mode and the components of matter fields are basically Type-I, II, and III structures discussed in Ref. [87], where structure fields are chiral superfields with the finite dimensional representation of SU (1, 1). Since the discussion in Ref. [87] is the simplest case that contains only two structure fields with an SU (1, 1) integer spin and a half-integer spin, the discussion is not exactly the same as that in this paper that contains three structure fields with SU (1, 1) integer and half-integer spins. The mixing structures of higgses and the others are derived by two structure fields with an SU (1, 1) integer spin. For higgses, the doublet-triplet mass splitting can be realized without fine-tuning, which has been discussed in Refs. [83,85] as mentioned above. The Yukawa coupling structures in "MSSM" have already been discussed in Ref. [85]. When the mixing structures of down-type quarks and charged leptons include SU (5) breaking effects, we will see that the GUT relation for the Yukawa coupling structures of downtype quarks and charged lepton is avoided. We will discuss the µ-term, although the generation of the µ-term has been discussed in Ref. [86], where the matter content of singlets and the scalar potential is different. We will discuss that special weight assignments of SU (1, 1) allow R-parity to remain even after the SU (1, 1) breaking, where it was first pointed out thatLLÊ c ,QLD c , D cDcÛ c are absent in Refs. [80,85], andĤ uĤd is also absent in Ref. [86] because all fields have the same sign of weight. An article [84] suggested that a G SM × SU (1, 1) model with particular matter content allows only Type-II seesaw mechanism [94][95][96] to generate neutrino masses. In general, not only Type-II seesaw mechanism but also Type-I and Type-III seesaw mechanisms are allowed, where the SU (1, 1) weight assignments are severely constrained. We will see that the proton decay via colored higgses is naturally suppressed since the colored higgses have Dirac mass terms. Note that this idea has already been discussed at least in the context of an orbifold GUT model based on extra dimension S 1 /Z 2 × Z 2 in Ref. [97], where any colored higgs has a Dirac mass term by using a non-trivial boundary condition. This paper is organized as follows. In Sec. 2, we first set up our model. In Sec. 3, we discuss spontaneous generation of generations to produce three chiral generations of quarks and leptons and one generation of higgses by using one structure field with a half-integer spin of SU (1, 1) and two structure fields with integer spins as proposed in Ref. [87]. In particular, we find that the colored higgses can disappear without fine-tuning. In Sec. 4, we see the structure of the Yukawa couplings, especially how to realize the difference of the Yukawa coupling matrices between the down-type quarks and charged leptons. In Sec. 5, we discuss how to generate the effective µ-term of higgses. In Sec. 6, we discuss the baryon and/or lepton number violation including R-parity, neutrino masses, and proton decay. We see that the proton decay derived from the colored higgses is highly suppressed. Section 7 is devoted to a summary and discussion.
2 Setup of an SU (5) × SU (1, 1) model We construct an SU (5) SUSY GUT model with horizontal symmetry SU (1, 1) that contains vectorlike matter content. We introduce the matter fieldŝ where the bold subscripts stand for the representations in SU (5). Since a pair of the fields in the curly brackets {N 1 ,T 15 ,Â 24 } and {N c 1 ,T c 15 * ,Â c 24 } are necessary to generate nonzero neutrino masses, we introduce one pair of them and in Sec. 6 we will see which fields are compatible with the SU (1, 1) weight assignment constrained by other requirements, such as to generate three chiral generations of quarks and leptons and one chiral generations of higgses, to allow Yukawa couplings between quarks and leptons and higgses. The quantum numbers of SU (5) × SU (1, 1) and R-parity are summarized in Table 1. We define the values q α and q β as where α, β, etc. are SU (1, 1) weights. We choose the values q α and q β to be positive half-integers. See Ref. [87] in detail for the notation and convention. We also introduce the structure fieldŝ where the SU (1, 1) spins ofΦ 1 ,Φ ′ 24 , andΨ 1/24 are S, S ′ , and S ′′ , respectively. This is summarized in Table 2. The subscript ofΨ 1/24 represents two options for the SU (5) representations.  We assume that the gauge group SU (5) × SU (1, 1) is spontaneously broken to G SM via the following nonvanishing VEVs of the structure fields where the subscripts of φ 0 , φ ′ +1 and ψ −3/2 stand for the eigenvalues of the third component generator of SU (1, 1). In the next section, we will find that the SU (1, 1) spins must satisfy S = S ′ < S ′′ , and to realize three generations of quarks and leptons and one generations of higgses, the minimal choice is S = S ′ = 1 and S ′′ = 3/2. We will also find that the VEV of Φ ′ 24 plays essential roles for decomposing the doublet and triplet higgses and making difference between the Yukawa coupling constants of the down-type quarks and charged leptons. The Clebsch-Gordan coefficients (CGCs) of SU (5) are shown in Ref. [38,39,51]. The CGCs of SU (1, 1) are found in Ref. [87].
We describe other assumptions as follows. The gauge kinetic function of the SU (1, 1) vector superfield and the Kähler potential of the structure fields have positive definite metrics at a vacuum. (Note that to realize this situation, at least one nonrenormalizable term must have larger effects for metrics of the SU(1,1) gauge and the structure fields than their renormalizable terms in this model.) The Lagrangian in the matter field sector including the coupling terms between matter fields and structure fields contains only renormalizable terms, and non-renormalizable terms in superpotential are induced by the process of decoupling the heavy fields. The correction for the Kähler potential of matter fields and the gauge kinetic function of the SU (5) gauge fields is negligible. After the chiral fields are generated via the spontaneous generation of generations [80,83,85,87], the effect from the SU (1, 1) gauge bosons and the structure fields is negligible for the chiral matter fields at low energy. Only the structure fields have large VEVs and the matter fields have smaller VEVs compared to those of the structure fields because of maintaining the structures of the horizontal symmetry; e.g., the VEVs of the structure fields are GUT-scale mass M GUT ≃ O(10 16 ) GeV and the VEVs of the matter fields are m SUSY ≃ O(10 3 ) GeV. Some SU (1, 1) singlet superfields break SUSY in a hidden sector, SUSY breaking does not affect SU (1, 1) symmetry, and soft SUSY breaking terms for matter fields are generated at GUT scale M GUT ∼ O(10 16 ) GeV in a visible sector, where the soft SUSY breaking masses are O(m SUSY ) ∼ O(10 3 ) GeV. To discuss the D-flatness condition of the SU (1, 1) group, we would have to consider the full potential of the model, including all structure fields because the D-flatness condition depends on the Kähler potential of the structure field. We therefore neglect this effect in this paper.
The number of soft SUSY breaking terms are determined by the number of the superpotential terms. It is impossible to give explicit forms of the soft SUSY braking terms before we discuss the superpotential. Here we mention the pattern of the soft SUSY breaking terms. Under the above assumption, SU (1, 1) symmetry restricts the structures of the soft SUSY breaking terms up to renormalizable terms; each trilinear scalar term, so-called A-term, is exactly proportional to the corresponding Yukawa coupling term in the superpotential; soft scalar masses are generationindependent. Note that the pattern of the soft SUSY terms can change when we take into account of the higher order terms derived from the non-renormalizable terms of the Kähler potential and the superpotential.

Realization of chiral generations
In this section, we consider how to provide three chiral generations of quarks and leptons, one chiral generation of up-and down-type doublet higgses, and no chiral generations of the others shown in Table 1 by using three structure fields shown in Table 2. We use the methods developed in Ref. [87].

Three chiral generations of quarks and leptons
We discuss how to produce three chiral generations of quarks and leptons. The superpotential of the quark and lepton superfields coupling to the structure fields is given by where M s are mass parameters, xs, zs, and ws are dimensionless coupling constants. We assume that the massless chiral fields are realized as linear combinations of the components ofF 10 ,F ′ 10 , We solve the massless condition by using the mass term of the superpotential in Eq. (7) for the matter fieldF 10 . For this calculation, there is no difference between the matter fieldŝ F 10 andĜ 5 * except the coupling constants and some CGCs. By substituting the nonvanishing VEVs of the structure fields in Eq. (6) into the superpotential term in Eq. (7), we have the mass term where Y f is a U (1) Y charge shown in Table 3, andỸ f is equal to one forΨ 1 and is equal to Y f for Ψ 24 . D β,α,S j,i (i, j = 0, 1, 2, · · · ) is a CGC of SU (1, 1) given in Ref. [87]; for S ≥ | − i + j − α + β|, the CGC is nonzero; otherwise, the CGC is zero. The emergence of the massless modesf n requires that the coupling between the massless modesf n and the massive modesf c −α−i and f ′c −α−i for any i must vanish simultaneously: where we defined These lead to the relation among the mixing coefficients U f n,i and U f ′ n,i , respectively: where for i < 0, U f n,i = U f ′ n,i = 0. The recursion equations determine the mixing coefficients U f n,i and U f ′ n,i for any i. The two initial condition sets of the mixing coefficients U f n,i and U f ′ n,i are also dependent upon each other. As in Ref. [87], Sec. 3.2, the relation between two initial condition sets can be classified into three conditions: Type-I, q α < 3/2 shown in Fig. 1; Type-II, q α = 3/2 shown in Fig. 2; and Type-III, q α > 3/2 shown in Fig. 3. Each condition leads to different mixing coefficients. When we calculate the Yukawa coupling constants, we need to have their detailed information. In this paper, we will not analyze the Yukawa couplings in detail. We will just show the difference between the Yukawa couplings of down-type quarks and charged leptons in Sec. 4.
We need to consider the normalizable condition of the mixing coefficients U f n,i and U f ′ n,j . As in Ref. [87], Sec. 3.2, when the SU (1, 1) spins satisfy the condition S ′′ > S, S ′ , their normalizable conditions are always satisfied regardless of the coupling constants and the value of the VEVs. Thus, three massless modesf n (n = 0, 1, 2) appear at low energy. Note that when we make Figs. 1, 2 and 3, we have already assumed S ′′ > S, S ′ .
One may notice that if the quark and lepton superfieldĜ (′) 5 * and the conjugate superfield of up-type higgsĤ c u5 * mix with each other, then this discussion is ruined. Thus, if this model does not include explicitly e.g. the R-parity shown in Table 1 and 2, the SU (1, 1) weight content must satisfy γ = β + [integer or half-integer]. In the model with R-parity shown in Table 1 and 2, the quantum number of the quark and lepton superfieldĜ (′) 5 * is different from the conjugate field of up-type higgsĤ c u5 * , so there is no such restriction. (k = 0, 1, 2, · · · ), and the constructional element of the massless mode is determined by its mixing coefficients U f 0,k and U f ′ 0,k+1 given in Eqs. (13) and (14) and Type-I initial condition; a massless modef 1 is realized as certain linear combinations of the componentsf α+k+1 and f ′ α ′ +2+k (k = 0, 1, 2, · · · ), and the constructional element of the massless mode is determined by its mixing coefficients U f 1,k+1 and U f ′ 1,k+2 given in Eqs. (13) and (14) and Type-I initial condition; a massless modef 2 is realized as certain linear combinations of the componentsf α+k+2 and f ′ α ′ +k (k = 0, 1, 2, · · · ), and the constructional element of the massless mode is determined by its mixing coefficients U f 2,k+2 and U f ′ 2,k given in Eqs. (13) and (14) and Type-I initial condition. The components of the matter fieldsF ,F c ,F ′ andF ′c connected by solid and dashed lines have a mass term that comes from the VEVs ψ −3/2 and, φ 0 and φ ′ +1 , respectively. The mass term of the solid line that comes from the VEV ψ −3/2 dominantly contributes to whether massless modes appear or not. (It does not always dominantly contribute to small components of the mixing coefficients.) The figure wraps from bottom to top. A component surrounded by a circle is a main element of each massless chiral mode when the mass terms of the solid line dominantly contribute to small components of the mixing coefficients. Figure 2: Type-II q α = 3/2 for the three massless generation of quarks and leptons in Eq. (9): each massless modef n is realized as a certain linear combination of the componentsf α+k+n andf ′ α ′ +k+n (k = 0, 1, 2, · · · ), and the constructional element of its massless mode is determined by its mixing coefficients U f n,k+n and U f ′ n,k+n given in Eqs. (13) and (14) and Type-II initial condition. The explanation of the circle and lines is given in Fig. 1.  9): each massless modef n is realized as a certain linear combination of the componentsf α+k+n+1 andf ′ α ′ +k+n (k = 0, 1, 2, · · · ), and the constructional element of its massless mode is determined by its mixing coefficients U f n,k+n+1 and U f ′ n,k+n given in Eqs. (13) and (14) and Type-III initial condition. The explanation of the circle and lines is given in Fig. 1.

One chiral generation of higgses
We will see that the model can allow only one generation of up-and down-type SU (2) L doublet higgses and prohibit any generation of the up-and down-type SU (3) C triplet higgses, so-called colored higgs, at low energy without fine-tuning and unnatural parameter choices in the sense of 't Hooft naturalness [1]. This is pointed out in Refs. [83,85].
We consider how to provide one chiral generations of higgses. The superpotential for the matter and structure coupling is where M s are mass parameters, xs and zs are dimensionless coupling constants. We assume that one massless chiral generation of higgses is realized at low energy as a linear combination of the components ofĤ u5 andĤ d5 * in the manner We solve the massless condition by using the mass term of the superpotential in Eq. (15). The same as the quarks and lepton, we use generic notationĤ for the higgs fields. By substituting the nonvanishing VEVs of the structure fields in Eq. (6) into the superpotential term in Eq. (15), we have the mass term where Y h is a U (1) Y charge shown in Table 3. The massless modeĥ is extracted from the componentĥ −γ−i of the matter fieldĤ. The orthogonality of the massless modesĥ to the massive modesĥ c γ+i requires the coefficients U f i to satisfy the following recursion equation for any i(≥ 0) The relation of the mixing coefficients between ith and i + 1th components is As is discussed in Ref. [87], Sec. 3.1, we need to consider a normalizable condition ∞ i=0 |U h i | < ∞. For S = S ′ , this leads to constraints for the values of the parameters and the nonvanishing VEVs, where we will not consider the SU (1, 1) spins satisfying S < S ′ and S > S ′ because the condition S < S ′ provide one chiral doublet and colored higgses and the condition S > S ′ cannot produce anything at low energy. By using the property of the CGC D γ,γ,S i,j , for the large where we dropped the irrelevant term. To satisfy the normalizable condition (20) must be smaller than one. When |U h i+1 /U h i | > 1, the chiral matter disappears at low energy.
By using the above normalizable condition, we consider the condition to realize existence of the up-and down-type doublet higgses and absence of the up-and down-type colored higgses at low energies. To produce the up-and down-type higgses at low emeries, the parameters ǫ hu and ǫ h d defined by must satisfy the following conditions To eliminate the up-and down-type colored higgses at low energies, the following condition must be satisfied.
where the parameters ǫ tu and ǫ t d are defined by When we rewrite this condition by using ǫ hu and ǫ h d , Thus, only the up-and down-type higgses appear at low energies if the parameters ǫ hu and ǫ h d satisfy the following condition:

No chiral generations of others
We consider the SU (5) singletsŜ 1 andR 1 with the positive lowest weights η and λ of SU (1, 1) and their conjugates where M s are mass parameters, xs and zs are dimensionless coupling constants. The coupling termsR 1Ŝ We must emphasize the above fine-tuning problem. This is obviously unnatural, and this unnaturalness strongly suggests the incompleteness of this model. To solve the fine-tuning problem, one may prefer to use tiny mass Alternatively, models that include additional matter and structure fields may lead to massless SM singletsr andr c via spontaneous generation of generations without any naturalness problem. However, we will not pursue this possibility in this paper.
We next discuss the fields that are necessary to generate neutrino masses via seesaw mechanisms. First, we consider SU (5) singletN 1 with the positive lowest weight ξ of SU (1, 1) and its conjugate. The superpotential contains where M n is a mass parameter and x n is a dimensionless coupling constant. The same as the fieldsŜ 1 andR 1 , the nonvanishing VEV φ 0 of the structure fieldΦ 1 in Eq. (6) gives huge masses to all components of the matter fieldsN 1 andN c 1 . Here we assume that the coupling terms such asN 1Ŝ1 Φ 1 andN 1R1 Φ 1 are forbidden by R-parity or the SU (1, 1) weight conditions. We will discuss this in Sec. 6.
Second, we consider SU (5) 15-pletT 15 with the negative highest weight τ of SU (1, 1) and its conjugate because SU (2) L tripletT h is contained in SU (5) 15-pletT 15 . The superpotential contains where M t is a mass parameter, and x t and z t are dimensionless coupling constants. The nonvanishing VEVs φ 0 and φ ′ +1 of the structure fieldsΦ 1 andΦ ′ 24 in Eq. (17) give huge masses to all components of the matter fieldsT 15 andT c 15 * . In this case, the discussion of whether massless particles appear or not is exactly the same as in the higgs cases. The parameter ǫ t h defined as must satisfy the condition for the triplet higgsT h to disappear at low energy. In this case, the other fieldsQ h andĈ h in the SU (5) 15-pletT 15 shown in Table 4 automatically disappear at low energy because the triplet higgs has the largest U (1) Y charge within the SU (5) 15-plet.
One may suspect that, if the coupling termsF 10T Third, we consider SU (5) 24-pletÂ 24 with the positive lowest weight ζ of SU (1, 1) and its conjugate. The superpotential contains where M a is a mass parameter, and x a , z sa and z aa are dimensionless coupling constants. The last two terms represent the symmetric and anti-symmetric invariants under SU (5) transformation built from three fields with the SU (5) adjoint representation. Note that while the CGCs of the anti-symmetric invariant are proportional to the U (1) Y charges, the CGCs of the symmetric invariant are not proportional to the U (1) Y charges. Also, the CGCs of the invariant built by two adjoint representations are not proportional to the identity. (See Ref. [39] for the CGCs of SU (5) adjoint representations in detail.) Thus, we need to consider the renormalizable condition for the components of the fieldsÂ ℓ ,Ŵ ℓ ,Ĝ ℓ ,X ℓ ,Ŷ ℓ shown in Table 4 and their conjugate fields.
In general, the fields are massive via the nonvanishing VEVs φ 0 and φ ′ +1 of the structure fieldsΦ 1 andΦ ′ 24 in Eq. (17) when the parameter ǫ a defined as satisfies the following condition where N i is proportional to a ratio of the CGCs for a basis of G SM between the singlet built by two adjoint representations and the SM singlet of the symmetric component built by three adjoint representations.

Structures of Yukawa couplings
We now discuss the Yukawa couplings between quarks and leptons and higgses.
where y 10 and y 5 are dimensionless coupling constants. Each Yukawa coupling can be classified into two types. For the first term in Eq. (35) ofF (′) 10 andĤ u5 , one is γ = 2α + [positive half-integer]; where y 10 is a coupling constant; the other is γ = 2α + [semi-positive integer].
We consider the mixing coefficients of down-type quarks and charged leptons given in Eqs. (13) and (14). For nonzero coupling constants z and w, the mixing coefficients are different because the U (1) Y charges of down-type quarks are different from those of charged leptons. Thus, the Yukawa coupling constants of down-type quarks can be different from those of charged leptons.
The patterns of the mixing coefficients are highly dependent on the values of q α and q β that determine dominant massless components. A detailed investigation of the Yukawa couplings is not the purpose in this paper, so we will not analyze the mass eigenvalues of quarks and leptons, and the CKM [98,99] and MNS [100] matrices. One can find the basic argument in Refs. [80,83,85,87].

µ-term
We need to generate the effective µ-term µĥ uĥd , where µ ≃ O(m SUSY ) is the supersymmetry breaking mass parameter O(10 2∼3 ) GeV [86]. This is because the µ-term µĤ uĤd is forbidden by the noncompact horizontal symmetry G N = SU (1, 1) since both chiral higgsesĥ u andĥ d are contained in the negative fieldsĤ u andĤ d [80,101], where negative fields are chiral superfields with the negative weight of SU (1, 1). To generate the effective µ-term, the up-and down-type higgsesĤ u andĤ d must couple to a positive fieldŜ belonging to the singlet under G SM , and the field must get a nonvanishing VEV O(m SUSY ). Unlike the Next-to-minimal supersymmetric SM (NMSSM) that contains an extra singlet superfield under G SM , the horizontal symmetry does not allow the existence of linear, quadratic and cubic terms, e.g., M 2Ŝ , MŜ 2 , and λŜ 3 . Thus, in this model, we cannot use the same method as in the NMSSM.
If the up-and down-type higgsesĥ u andĥ d belong to conjugate representations or the same real representation, then the effective µ-term µĥ uĥd is generated only by singlet fields and the The simplest superpotential contains the SU (5) singletsŜ 1 andR 1 with the positive lowest weights η = γ + δ and λ = (γ + δ)/2 of SU (1, 1) and their conjugates where ys are coupling constants. From the above superpotential and the superpotential in Eq. (27), decoupling the singlets except the first component of the singletsŜ 1 ,Ŝ c 1 ,R 1 andR c 1 , we obtain where we assumeM s : , the U hu 0 and U h d 0 are mixing coefficients of the up-and down-type higgses. Decouplingŝ andŝ c by using we have This leads to the scalar potential Its corresponding SUSY breaking terms are where B r is a B-parameter ofr andr c , A r is an A-parameter ofrrŝ, andm 2 r andm c2 r are soft masses ofr andr c , respectively. The total scalar potential is After we perform tedious calculation, we obtain r , r c = O( √ m SUSY M GUT ) and s = O(m SUSY ) as discussed in Ref. [86]. Thus, the effective µ-term between h u5 and h d5 is O(m SUSY ). The singlet fermions and scalarsr andr c have a mass term O(m SUSY ) except the Nambu-Goldstone (NG) boson since this potential have a U (1) global symmetry at low energy and this symmetry is broken by the nonvanishing VEVs of the singlets. Note that, if there is no SUSY breaking term, the singlet fermion is massless because SUSY forces the fermionic partner of the NG boson to be a pseudo-NG fermion [64,69,102,103].
In addition, the coupling between the higgsesĥ u5 andĥ d5 * and the singlets is suppressed by the factor O( m SUSY /M GUT ). Therefore, the effective theory below the energy scale √ m SUSY M GUT is described by the MSSM and the almost decoupled G SM singlets.
The NG boson may cause some problems for cosmology, e.g., a moduli problem [104]. To solve the moduli problem, we should assume that there is thermal inflation after reheating takes place as discussed in Ref. [104]. We will not discuss the cosmological problems in this paper.

Baryon and/or lepton number violating terms
We classify the baryon and/or lepton number violating terms up to superpotential quartic order by using SU (1, 1) symmetry and the R-parity [88] (matter parity [89]) shown in Table 1. For a review, see, e.g., Ref. [90]. In the following, we omit the mirror terms. λs stand for dimensionless couplings, Λs and µ are dimension-one parameters, ∆s are integer, and ∆ ± is a non-negative integer.
To make the invariants under the SU (1, 1) transformation, we can use the following way; first, we make the composite states of only positive field or negative field. In general, a composite field built by multi-positive fieldsF i (i = 0, 1, 2, · · · ) with the lowest weight α i is a positive field with the lowest weight i α i + ∆ + (∆ + = 0, 1, 2, · · · ). A composite field built by multinegative fieldsĤ j (j = 0, 1, 2, · · · ) with the highest weight −β j is a negative field with the highest weight − j β j − ∆ − (∆ − = 0, 1, 2, · · · ). When the multi-positive field contains only one positive field, ∆ + = 0; when the multi-negative field contains only one negative field, ∆ − = 0. Next, we combine the multi-positive and negative fields. The invariants built by the multi-positive and negative fields must satisfy the condition i α i + ∆ + = j β j + ∆ − : i.e., ∆ := ∆ + − ∆ − = j β j − i α i . We define ∆ as the difference between the sum of the lowest weights of positive fields and the highest weights of negative fields, where a positive field is a matter field with only the positive weights of SU (1, 1) and a negative field is a matter field with only the negative weights of SU (1, 1). More explicitly, for a term containing one positive field with the lowest weight α and one negative field with the highest weight −β, the condition α = β must be satisfied; for a term containing two positive fields with the lowest weights α and α ′ and one negative field with the highest weight −β, the condition ∆ = ∆ + = β − α − α ′ must be satisfied; for a term containing three positive fields with the lowest weights α, α ′ and α ′′ and one negative field with the highest weight −β, the condition ∆ = ∆ + = β − α − α ′ − α ′′ must be satisfied; for a term containing two positive fields with the lowest weights α and α ′ and two negative fields with the highest weights −β and −β ′ , the condition ∆ = ∆ + −∆ − = β+β ′ −α−α ′ must be satisfied.
We start to consider the SU (5) GUT model with SU (1, 1). First, SU (1, 1) symmetry and R-parity allow the following B and/or ✓ L quartic term In the following we also use the same rule. Second, SU (1, 1) symmetry prohibits and R-parity allows the following B and/or ✓ L quartic term if β (′) = γ. The cubic terms are if α (′) = 2γ + ∆ − and β (′) = α (′) + α (′) + ∆ + . The quartic terms are In general, SUSY models with R-parity violating terms suffer from rapid proton decay and lepton flavor violations [90]. Thus, to prevent the unacceptable predictions, the R-parity must be realized at low energy. Fortunately, even when we discuss SUSY models with R-parity that contain the relevant or marginal terms, after some heavy particles are integrated out, the effective neutrino "mass" term in Eq. (54) can be induced. Unfortunately, the problematic operator in Eq. (55) can be also induced.
On the other hand, the SU (1, 1) horizontal symmetry does not allow the problematic term in Eq. (55). Of course, once the symmetry is broken, there is no reason to deny generating the term. We will discuss this topic in this section.
Another interesting feature is that special weight assignments of SU (1, 1) mean that R-parity remains even after the SU (1, 1) symmetry is broken. One assignment is the following: where the SU (1, 1) weight, such as α, must be a positive number, n and m are integer, q α and q β are half-integer. In other words, the quark and lepton superfields have the quarter values of the SU (1, 1) weight, and the higgs and the other superfields have integer values of the SU (1, 1) weight. Thus, even numbers of quarks and leptons are necessary to couple higgses and the other fields. This is completely the same as the R-parity shown in Table 1. When we construct models with an SU (1, 1) horizontal symmetry, we do not always assume the R-parity to prevent rapid proton decay, lepton flavor violation and to make dark matter candidate. Note that the assignment is compatible with the Yukawa couplings in Eqs. (36) and (38), but incompatible with those in Eqs. (37) and (39).
Also, another example is the following assignment where n, m are integer. This assignment is compatible with those in Eqs. (37) and (39), but incompatible with the Yukawa couplings in Eqs. (36) and (38). The same as the assignment in Eq. (61), this assignment forbids all R-parity violating terms because the quarks and leptons have the quarter values of the SU (1, 1) weight, the higgses have the half-integer values, and the singlets have the integer values. We can find other assignments of SU (1, 1) weights to prohibit R-parity violating terms, and to allow the "neutrino mass" term in Eq. (54). The above two assignments in Eq. (61) and Eq. (62) include enough assignments for the following discussion. We only consider the model with these SU (1, 1) assignment or explicitly imposed R-parity shown in Table 1. We focus on the superpotential terms in Eqs. (54) and (55). We will discuss how to obtain sizable neutrino masses and how to suppress rapid proton decay in this model.

Neutrino masses
We now discuss seesaw mechanisms, so-called Type-I [6], Type-II [94][95][96], and Type-III [105] seesaw mechanisms in the MSSM plus additional necessary field content. Type-I, Type-II, and Type-III seesaw mechanisms can be achieved by using right-handed neutrinosN with (1, 1, 0) under G SM , charged triplet higgsesT h andT c h with (1, 3, ±1) under G SM , and neutral triplet leptonsŴ ℓ with (1, 3, 0), where the first, second and third columns stand for an SU (3) C weight, an SU (2) L weight, and a U (1) Y charge, respectively. We can also classify Type-I and Type-III seesaw mechanisms as Majorana-type seesaw mechanisms and Type-II as non-Majorana-type. For a review, see, e.g., Ref. [106].
Each additional field has the following superpotential terms, respectively where M s are mass parameters, ys are coupling constant, a, b stand for the label of the additional matter fields, and i is the label of the left-handed neutrino. If we assume M X is much larger than electro-weak scale, after decoupling the additional fields, we obtain the effective neutrino-higgs superpotential term where M X is a mass parameter and κ ij is a coupling constant matrix determined by the mass parameters and the coupling constants in Eq. For SU (5) GUT models, Type-I, Type-II, and Type-III seesaw mechanisms can be also achieved by using SU (5) singlet fieldsN 1 SU (5) 15-plet and 15 * -plet fieldsT 15 andT c 15 * , and SU (5) 24-plet fieldsÂ 24 , respectively. Note that sinceÂ 24 containsÂ ℓ andŴ ℓ , this field includes not only Type-I seesaw but also Type-III seesaw mechanisms.
We move on to our SU (5) × SU (1, 1) model. As we have already seen before, the Majorana mass terms are not allowed by the SU (1, 1) symmetry. One may think that the Type-I and Type-III seesaw mechanisms are prohibited, but as we discussed for the effective µ term of the up-and down-type higgs doublets, once the horizontal symmetry is broken, there is no reason to prohibit the Majorana mass terms.
We discuss two situations Majorana-type Type-I and Type-III seesaw mechanisms and Diractype Type-I, Type-II, and Type-III seesaw mechanisms for the massive mediated superfields realized by the spontaneous generations of generations discussed in Sec. 3.
We start by considering the Majorana-type Type-I and Type-III seesaw mechanisms. The masses of the mediated fields come from the Dirac mass term of the fields and their conjugate fields, and the masses are different from the Majorana masses µ X , where µ X stands for the Majorana mass ofN 1 orÂ 24 . Our basic assumption is that µ X is much smaller than the Dirac mass of the the mediated fields. When we integrate out the mediated fields, we obtain the effective neutrino masses O( eV. This is too tiny. Therefore, the Majorana-type seesaw mechanisms cannot explain the observed neutrino masses.
Next we discuss Dirac-type Type-I, Type-II, and Type-III seesaw mechanisms. The superpotential terms are given by where γ = β + ξ + ∆ ξ , β ′ = γ + ξ + ∆ ′ ξ , τ = β + β ′ + ∆ τ , τ = 2γ + ∆ ′ τ , γ = β + ζ + ∆ ζ , and β ′ = γ + ζ + ∆ ′ ζ . ∆s are non-negative integer. To realize the seesaw mechanisms, we have to choose the SU (1, 1) weight assignment satisfying the following condition: where x stands for ξ, τ , or ζ. This leads to a constraint for the SU (1, 1) weight assignment in Eq. (61), and also leads to a constraint for the SU (1, 1) weight assignment in Eq. (62). After decoupling the heavy matter, we obtain the effective superpotential where κ X n,m are coupling constants, and X stands for I, II, and III, For small sub-leading contribution derived from φ 0 and φ ′ +1 , the predicted masses seem to be incompatible with the observed masses. In principle, the sub-leading contribution derived from φ 0 and φ ′ +1 can be large, so it may reproduce the observed masses. For Type-II and III, the neutrino masses vanish for leading order, i.e., a limit φ 0 and φ ′ +1 going to zero. Thus, the sub-leading component mainly contribute to the neutrino masses. In this case, since the overall coupling becomes small, we need finer tuning to realize the observed neutrino masses. When the other contribution is small, the neutrino mass matrix seems to be normal hierarchy. When the other contribution is large, it depends on parameters.

Proton decay
Before we discuss proton decay in our model, we quickly review the proton decay discussion in the minimal SU (5) SUSY GUT [56][57][58][59]. First, the superpotential of the Yukawa couplings in models with the minimal SU (5) matter content contains the following baryon and/or lepton number violation terms After the doublet part of the original SU (5) µ-term µ 5Ĥu5Ĥd5 is canceled by using the "µ"term induced from the VEVs of the coupling between the SU (5) adjoint and up-and downtype higgses Φ 24 Ĥ u5Ĥd5 * , we can obtain the effective µ parameter of the doublet higgses µ ∼ O(m SUSY ) and of the colored higgses M C ∼ O(M GUT ). After the colored higgses decouple, they lead to two superpotential terms that include dimension-5 operators breaking baryon and/or lepton number where C mnpq 5X (X = L, R) are dimensionless coupling constants that depend on the Yukawa coupling matrices of quarks and leptons. According to the analysis discussed in Ref. [59], we use the recent super-Kamiokande result for the lifetime τ (p → K +ν ) > 3.3 × 10 33 years at 90% C.L. [61]. Assuming that soft SUSY breaking parameters at the Planck scale are described by the universal scalar mass, universal gaugino mass, and universal coefficient of the trilinear scalar coupling, so-called A-term and the sfermion mass mf is less than 1 TeV, the colored higgs mass M C must be larger than 10 17 GeV for tan β (2 < tan β < 5); 10 18 GeV for tan β = 10; 10 19 GeV for tan β = 30; and 10 20 GeV for tan β = 50. (Recently, it was discussed in Ref. [108][109][110] that when the sfermion mass is much greater than 1 TeV, the colored higgs mass M C can be 10 16 GeV regardless of tan β.) We move on to discuss proton decay in our model. The chiral matter content is realized via the spontaneous generation of generations discussed in Sec. 3. As discussed in Sec. 3.2, once the up-and down-type doublet higgses appear and the up-and down-type colored higgses disappear at a vacuum, the up-and down-type colored higgses have their Dirac masses. To generate the baryon and/or lepton number violation terms in Eq. (79), they must include the µ-term between the colored higgses. We discuss two assignments in Eqs. (61) and (62). The effective superpotential is W = where λ m,n,p.q and λ ′ m,n,p,q are determined by the colored higgs masses, the µ-parameter of the colored higgses, the overall Yukawa couplings, and the mixing coefficients of quarks and leptons. For the assignment in Eq. (61), the Yukawa couplings in Eqs. (36) and (38) lead to λ m,n,p,q = ∞ i,j,k,ℓ=0 λ ′ m,n,p,q = ∞ i,j,k,ℓ=0 where and For the assignment in Eq. (62), the Yukawa couplings in Eqs. (37) and (39) lead to λ m,n,p,q = ∞ i,j,k,ℓ=0 λ ′ m,n,p,q = ∞ i,j,k,ℓ=0 Note that as we discussed in Sec. 5, the original µ term between up-and down-type higgses is prohibited by the horizontal symmetry. In the model, the nonvanishing VEVs of the singlets generate the µ-term of the 0th component ofĥ u5 andĥ d5 * . We need to consider the experimental bound for proton decay in the model. In the calculation in Ref. [59] it is assumed that the Yukawa coupling matrices of the colored higgses are the same as the matrices of the down-type higgses. In the current model, the Yukawa coupling matrices of the doublet higgses are different from the colored higgses, so we cannot use directly the constraint for the mass of the colored higgses discussed in Ref. [59]. When we assume that the C mnpq GeV. Thus, the proton decay effect caused by the colored higgs is negligible once the colored higgs are massive via the spontaneous generation of generations. The dominant contribution for proton decay modes comes from the X and Y gauge boson exchanges. The dominant proton decay mode p → π 0 e + via the X and Y gauge bosons must be found first. In other words, if one of the current or planned near future proton decay experiments finds another proton decay mode, e.g., p → K +ν before p → π 0 e + are found, this model will be excluded.
Finally, we verify the contribution from additional matter fieldsĈ h andQ h in the 15-plet T 15 , andX ℓ andŶ ℓ in the 24-pletÂ 24 , and their conjugate fields. These terms cannot generate the superpotential quartic terms in Eq. (55), so the lowest contribution can only come from at least superpotential quintic terms. This means that since the nonvanishing VEVs of the non-SM singlets are those of up-and down-type higgses, the contribution of them for proton decay are suppressed by at least m SUSY /M GUT ∼ O(10 −13 ) compared to the superpotential quartic terms in Eq. (55). Thus, they are completely negligible at least for the current experimental bound.

Summary and discussion
We discussed the SU (5) SUSY GUT model with the SU (1, 1) horizontal symmetry that includes the matter fields in Table 1 and the structure fields in Table 2. We showed that the mechanism of the spontaneous generation of generations produces the matter content of the MSSM and the almost decoupled G SM singlets through the nonvanishing VEVs of the structure fields given in Eq. (6). For quarks and leptons, the nonvanishing VEV ψ −3/2 of the structure fieldΨ 1/24 with the SU (1, 1) half-integer spin S ′′ plays the important role for producing the three chiral generations of quarks and leptons. The nonvanishing VEV φ ′ +1 of the structure fieldΦ ′ 24 with the SU (1, 1) integer S ′ leads to the difference between the mixing coefficients of quarks and leptons because the structure fieldΦ ′ 24 belongs to the nontrivial representation of SU (5). Thus, the mixing coefficients of the down-type quarks are different from those of the charged leptons. This avoids the unacceptable prediction in the minimal SU (5) GUT model for the down-type quark's and the charged lepton's Yukawa coupling constants. For higgses, the nonvanishing VEV ψ −3/2 does not affect anything because the structure fieldΨ 1/24 does not couple to the higgs superfields. Due to this fact, the nonvanishing VEVs φ 0 and φ ′ +1 of the structure fieldŝ Φ 1 andΦ ′ 24 with the SU (1, 1) integers S and S ′ determine whether the higgses appear or not. The VEVs can produce only one generation of the up-and down-type doublet higgses at low energy. We found that the model naturally realizes the doublet-triplet mass splitting between the doublet and colored higgses pointed out in Ref. [83,85].
We also found that some special SU (1, 1) assignments allow only the B and/or ✓ L superpotential quartic termĜ 5 * Ĝ ′ 5 * Ĥu5Ĥu5, which contains theLL ′Ĥ uĤu , up to superpotential quartic order. The assignments retain R-parity even after the SU (1, 1) symmetry is broken. Thus, we can identify the SU (1, 1) assignments as the origin of the R-parity.
We found that this model can generate the neutrino masses via not only the Type-II seesaw mechanism but also the Type-I and Type-III seesaw mechanisms. We also found that the neutrino masses are dependent on the mixing coefficients of the leptons and up-type higgses, the SU (1, 1) CGCs, the masses of the mediated fields, and their overall Yukawa coupling constants.
We verified that the proton decay induced via the superpotential quartic terms generated by decoupling the colored higgses is highly suppressed compared to that of usual GUT models. The suppression factor is roughly O(m SUSY /M GUT ) ∼ O(10 −13 ). Thus, the dominant contribution to proton decay comes from the X and Y gauge bosons. Thus, the dominant proton decay mode p → π 0 e + via the X and Y gauge bosons must be found first. In other words, if another proton decay mode, e.g., p → K +ν , is discovered before p → π 0 e + is found, this model will be excluded.
We mention the gauge anomalies of G SM at low energies. The spontaneous generation of generations allows apparent anomalous chiral matter content at low energies because the apparent anomalies should be canceled out by the Wess-Zumino-Witten term [27,86,111]. For example, the up-type colored higgs could appear at low energy while the down-type colored higgs disappears at low energy. In this case, the matter content at low energy is anomalous. Of course, since in this situation there is a massless colored higgsino, this is unacceptable. The apparent anomaly cancellation exhibited by the observed low energy fields therefore appears coincidental in some sense if the spontaneous generation of generations is realized in nature.
We also mention "charge" quantization of weights of SU (1, 1) in this model. The SU (1, 1) spins of structure fields are obviously quantized because of finite-dimensional representations of SU (1, 1), while the lowest(highest) SU (1, 1) weight of matter fields are arbitrary and there is no reason to quantize their "charges." Of course, we need the "charge" quantization for matter fields, e.g., to realize three chiral generations of quarks and leptons at low-energy and the existence of Yukawa couplings. The charge quantization may be realized naturally in part if we embed SU (1, 1) into a higher rank noncompact group, e.g., SU (2, 1). The unitary representations of SU (2, 1) live on a two dimensional plain. The generators of SU (2, 1) can be written by three dependent subgroups two SU (1, 1) and one SU (2), just like SU (3) that can be written by three dependent subgroups SU (2). Still, the charges of the lowest state of the SU (2, 1) representations are arbitrary, but since the unitary representations of SU (2, 1) contain the representations of SU (1, 1) with different weights by integer times a certain fraction, the difference between the charges of the lowest state of the SU (1, 1) can be quantized. At present, since there are no works to discuss models with a higher rank noncompact group horizontal symmetry, it has not been discovered when and how the spontaneous generation of generations works.
We have not yet solved the vacuum structure in a model that includes at least three structure fields with two SU (1, 1) integer spins S, S ′ and one SU (1, 1) half-integer spin S ′′ . To produce three chiral generations of quarks and leptons and one generation of higgses, the SU (1, 1) spins must satisfy the relation S ′′ > S = S ′ ≥ 1. Thus, the minimal choice is S = S ′ = 1, S ′′ = 3/2. We must discuss the model to justify the assumption of this article.
We comment on nonrenormalizable terms when they are generated by Planck scale physics. For matter fields, as we discussed in Sec. 6, since special weight assignments of SU (1, 1) allow only the superpotential term in Eq. (54) up to quartic order, the effect does not seem to affect anything at low energy. We have problems if higher order terms between structure and matter fields are generated by Planck scale physics. For example, let us consider a model that includes a matter fieldF , its conjugate fieldF c and a structure fieldΦ with an SU (1, 1) integer spin S, where we assume that the gth component of the structure fieldΦ has a nonvanishing VEV. The relevant superpotential terms for the spontaneous generation of generations are where M is a mass parameter, C m s are dimensionless coupling constants, Λ is a Planck scale mass parameter, and ℓ is an integer number. ℓgth generations of the massless modesf n (n = 0, 1, · · · , ℓg − 1) appear because the largest spin state built byΦ m has the spin mS and this coupling is the dominant contribution to produce the chiral particles regardless of coupling constants C m s. From the viewpoint of effective theory, there is no reason that ℓ is finite. For ℓ → ∞, the number of chiral generations is zero for g = 0 and ∞ for g = 0. At present, we must assume that unknown fundamental theory only allows renormalizable terms of the structure and matter field sector to justify our discussion.