Nucleon decay via dimension-6 operators in $E_6 \times SU(2)_F \times U(1)_A$ SUSY GUT model

In the previous paper, we have shown that $R_1\equiv\frac{\Gamma_{n \rightarrow \pi^0 + \nu^c}}{\Gamma_{p \rightarrow \pi^0 + e^c}}$ and $R_2\equiv\frac{\Gamma_{p \rightarrow K^0 + \mu^c}}{\Gamma_{p \rightarrow \pi^0 + e^c}}$ can identify the grand unification group $SU(5)$, $SO(10)$, or $E_6$ in typical anomalous $U(1)_A$ supersymmetric (SUSY) grand unified theory (GUT) in which nucleon decay via dimension-6 operators becomes dominant. When $R_1>0.4$ the grand unification group is not $SU(5)$, while when $R_1>1$ the grand unification group is $E_6$. Moreover, when $R_2>0.3$, $E_6$ is implied. Main ambiguities come from the diagonalizing matrices for quark and lepton mass matrices in this calculation once we fix the vacuum expectation values of GUT Higgs bosons. In this paper, we calculate $R_1$ and $R_2$ in $E_6\times SU(2)_F$ SUSY GUT with anomalous $U(1)_A$ gauge symmetry, in which realistic quark and lepton masses and mixings can be obtained though the flavor symmetry $SU(2)_F$ constrains Yukawa couplings at the GUT scale. The ambiguities of Yukawa couplings are expected to be reduced. We show that the predicted region for $R_1$ and $R_2$ is more restricted than in the $E_6$ model without $SU(2)_F$ as expected. Moreover, we re-examine the previous claim for the identification of grand unification group with $100$ times more model points ($10^6$ model points), including $E_6 \times SU(2)_F$ model.


Introduction
Grand unified theory (GUT) [2] is one of the most favorable candidates for the model beyond the standard model (SM). It has advantages not only theoretically but also experimentally. Theoretical advantages are that it can unify the three gauge interactions in the SM into a single gauge interaction and particles in the SM into fewer multiplets. Experimental advantages are that measured values of the three gauge couplings agree with the predicted values in supersymmetric (SUSY) GUT and measured hierarchies of masses and mixings of quarks and leptons can be understood, if it is assumed that 10 matter induces stronger hierarchies for Yukawa couplings than the5 matter [3].
The nucleon decay [2,4,5] is one of the most important predictions in GUTs. In GUTs there are new colored and SU(2) L doublet gauge bosons, which we call X-type gauge bosons. In SU(5) GUT models these gauge bosons are X(3, 2) 5 6 andX(3, 2) − 5 6 , where3 and 2 means the antifundamental representation of SU(3) C and the fundamental representation of SU(2) L , respectively, and 5 6 means the hypercharge. Exchanges of the X-type gauge bosons induce dimension-6 operators which break both the baryon and lepton numbers and induce the nucleon decay. Usually, the main decay mode of the proton via dimension-6 operators is p → π 0 + e c . The mass of the X is roughly equal to the GUT scale at which three gauge couplings in the SM are unified into a single gauge coupling g GU T , and therefore the lifetime of the nucleon can be estimated. In the minimal SUSY GUT model, the GUT scale Λ G is 2 × 10 16 GeV, therefore the lifetime can be estimated as roughly 10 36 years, which is much larger than the current experimental lower bound, 10 34 years [6].
Triplet (colored) Higgs which is the GUT partner of the SM doublet Higgs also induces nucleon decay. Because of smallness of Yukawa coupling for the first-and second-generation matters, the constraint on the triplet Higgs mass from the experimental limits of the nucleon lifetimes is not so severe without SUSY. However, once SUSY is introduced, this constraint become severe because this induces nucleon decay via dimension-5 operators [5]. In the minimal SU(5) SUSY GUT model, the lower bound for the triplet Higgs mass becomes larger than the GUT scale Λ G [7,8].
The constraint on the triplet Higgs mass gives one of the most difficult problems in SUSY GUTs, i.e., the doublet-triplet splitting problem. The SM doublet Higgs mass must be around the weak scale to realize electroweak symmetry breaking, while as noted above, the GUT partner of that, triplet Higgs must be heavier than the GUT scale. Of course, we can realize such a large mass splitting by fine-tunings, however it is unnatural. A lot of attempts have been proposed to solve this problem [9]. However, in most of the solutions, some terms which are allowed by the symmetry are just neglected, or the coefficients for some terms are taken to be very small. Such requirements are, in a sense, fine-tuning, and therefore, some mechanism which can realize such a large mass splitting in a natural way is required.
The doublet-triplet splitting problem can be solved under natural assumption by introducing anomalous U(1) A gauge symmetry. The natural assumption means that all interactions which are allowed by symmetries of the models are introduced with O(1) coefficients [10,11,12,13]. Higherdimensional interactions are also introduced if they are allowed by the symmetries. One of the most interesting predictions of anomalous U(1) A SUSY GUT models is that nucleon decay via dimension-6 operators becomes dominant [11]. In these models the gauge coupling unification requires that the cutoff Λ must be the usual SUSY GUT scale Λ G and the real GUT scale Λ u is where λ < 1 is the ratio of the Fayet-Iliopoulos (FI) parameter to cutoff Λ. Because anomalous U(1) A charge of the adjoint Higgs a is negative, Λ u is smaller than Λ G , therefore, nucleon decay via dimension-6 operators is enhanced. On the other hand, the nucleon decay via dimension-5 operators is strongly suppressed [10,11]. Therefore, the nucleon decay via dimension-6 effective operators is important in this scenario. One more important feature is that the realistic quark and lepton masses and mixings can be realized in anomalous U(1) A SUSY GUT models, with SO(10) and E 6 grand unification group [10,12].
In the previous paper [1], we have calculated various partial decay widths of nucleon from the effective dimension-6 interactions in the anomalous U(1) A SUSY GUTs with SU(5), SO (10), or E 6 grand unification group. The predicted lifetime becomes just around the experimental lower bound, though the lifetime is strongly dependent on the explicit GUT models and the parameters. Therefore, it can happen that the nucleon decay is detected soon. The nucleon decay can be a good target for the future project. It is difficult to kill the anomalous U(1) A GUT models from the limit of the lifetime because the lifetime is proportional to the unification scale to the forth. However, we have claimed that the identification of the unification group in the anomalous U(1) A GUT scenario is possible if the several partial decay widths can be measured. The ratio R 1 ≡ Γ n→π 0 +ν c Γ p→π 0 +e c is useful to know the largeness of the rank of the unification group because the contribution from the new X-type gauge bosons X ′ in SO(10) and X ′′ in E 6 make R 1 larger generically [14]. And the ratio R 2 ≡ Γ p→K 0 +µ c Γ p→π 0 +e c is useful to catch the contribution from X ′′ , which are mainly coupled with the second generation fields of5. Note that these ratios are not dependent on the absolute values of vacuum expectation values (VEVs) of GUT Higgs bosons. However, the results are strongly dependent on the mass ratios of X-type gauge bosons. It is important that the contribution from the extra gauge multiplet X ′ becomes always sizable in anomalous U(1) A GUT because the mass of X ′ becomes almost the same as the mass of the SU(5) superheavy gauge multiplet X. The contribution from X ′′ can be large, though it is dependent on the explicit models. As a result, the identification becomes possible by measuring the ratios R 1 and R 2 . Once the masses of X type gauge multiplets are fixed, the main ambiguities come from the diagonalizing matrices of Yukawa matrices. These ambiguities cannot be fixed only from measured masses and mixings of quarks and leptons because we have a lot of O(1) coefficients in the anomalous U(1) A GUT. If we would like to predict more concrete values for various decay modes of nucleons, we must fix these O(1) coefficients.
If we introduce the family symmetry SU(2) F into the anomalous U(1) A GUT with E 6 unification group, the model predicts a characteristic scalar fermion mass spectrum in which the third generation 10 3 of SU(5) can have different universal sfermion masses m 3 from the other sfermions which have universal sfermion masses m 0 [15]. If we take m 0 >> m 3 , the SUSY flavor changing neutral current (FCNC) problem can be improved without destabilizing the weak scale because the FCNC constraints are weakened for large first two generation sfermion masses m 0 while the stop masses m 3 , which is important for stabilization of the weak scale, can be around the weak scale. In addition, if the CP symmetry is imposed, which is spontaneously broken by the Higgs which breaks SU(2) F , not only the SUSY CP problem can be solved but also the number of O(1) coefficients for quark and charged lepton masses and quark mixings can be smaller than the number of these mass and mixing parameters [16,17,18]. It means that the diagonalizing matrices can be fixed from the quark and lepton masses and mixings at least at the GUT scale in principle. In Ref. [16], it has been shown that the quark and charged lepton masses and the Cabibbo-Kobayashi-Maskawa (CKM) matrix [19] can be consistent with the values evaluated at the GUT scale in the minimal SUSY SM (MSSM) [20] within factor 3 by choosing these parameters. Once we could find the parameter set at the GUT scale which realizes observed quark and lepton masses and mixings at the low energy scale in an explicit model, then we can predict various partial decay widths of the nucleon.
In this paper, we calculate the various decay widths of the nucleons in the E 6 × SU(2) F × U(1) A SUSY GUTs. If the parameter sets, which realize the observed quark and lepton masses and mixings, have been found easily, we would calculate the various decay widths by the parameter sets. However, it is not an easy task to find the parameter sets in calculating renormalization group equations (RGEs) which are dependent on the explicit GUT models. Alternatively, we find the relations between diagonalizing matrices which are independent of the renormalization scale, and under the relations we calculate the various decay widths of nucleon. Moreover, we re-examine the conditions for the identification of the grand unification group by using 100 times more model points than in the previous paper.
In this section we introduce the E 6 × SU(2) F × U(1) A SUSY GUT model [15,16,17,18] and the diagonalizing matrices in the model are derived. Setting and notation for the model in this paper are basically the same as these for the model in Ref. [18]. The diagonalizing matrix of light neutrinos, L ν , is derived from the Maki-Nakagawa-Sakata (MNS) matrix [21] and the diagonalizing matrix of charged leptons, L e , through the relation on the MNS matrix as U M N S = L † ν L e . Therefore we omit the explanation for the derivation of neutrino mass matrices in the model. It is shown in Ref. [18] in detail.
One of the most important features of the anomalous U(1) A gauge theory is that the VEVs of the GUT singlet operators O i are determined by their where λ is determined from the FI parameter ξ as λ ≡ ξ/Λ. In this paper, we take λ ∼ 0.22. As a result, the coefficient of the term XY Z is determined by their U(1) A charges, x, y, and z as λ x+y+z XY Z if x + y + z ≥ 0, and they vanish if x + y + z < 0. These features are important in understanding the following arguments in this paper. Contents of matters and Higgs and their charge assignment are shown in Table 1. In this paper, the capital letter denotes the superfield and the small letter denotes the corresponding 16 and 10 of SO(10) are decomposed in the notation as 27 includes two5s and two 1s. This feature plays an important role in realizing realistic quark and lepton masses and mixings in this model. F a andF a are Higgs which obtain VEVs F a and F a as and break SU(2) F . Hereafter, we use unit in which the cutoff Λ is taken as Λ = 1. Φ andΦ are Higgs which obtain VEVs Φ and Φ in SO(10) singlet direction as 1 ′ Φ = 1 ′Φ = v φ ∼ λ −(φ+φ)/2 and break E 6 into SO(10). C andC are Higgs which obtain VEVs C and C in SU(5) singlet direction as 16 C = 16C = v c ∼ λ −(c+c)/2 and break SO(10) into SU(5). A is an adjoint Higgs which is decomposed in SO(10) × U(1) V ′ notation as 78 which is proportional to B − L charge. This VEV of adjoint Higgs plays an important role in realizing the doublet-triplet splitting. Here, σ i (i = 1, 2, 3) are the Pauli matrices.
This model includes the MSSM doublet Higgs H u and H d as δ is a complex phase and depends on the models. Please refer to the papers [10,11,13] to understand how to realize the doublet-triplet splitting in a natural way. Yukawa couplings are derived from the superpotential, where a, b, c, f ′ , and g ′ are O(1) coefficients, and d(Ψ a , Φ,Φ, A, Z 3 , Θ) is a gauge invariant function of Ψ a , Φ,Φ, A, Z 3 , and Θ and it contributes to Ψ 1 Ψ 2 Φ. Note that the operator ǫ ab Ψ a Ψ b Φ is not allowed because of asymmetric feature of ǫ ab , where ǫ ab (ǫ 12 = −ǫ 21 = 1) is antisymmetric tensor of SU(2) F group. Therefore, the function d includes, for example, This function contains the following terms by developing the VEVs of A, Φ,Φ, Z 3 , and Θ: where d 5 , d q , d l , and h are real O(1) coefficients. Note that the coefficients of the first 4 terms in Eq. (11) are proportional to B − L charge. The reason is as follows. The above argument on asymmetric feature can be applied into the terms ǫ ab 16 a 16 b 10 Φ and ǫ ab 10 a 10 b 1 Φ of SO (10). To obtain nonzero terms, they must pick up the breaking of SO(10), i.e., the adjoint Higgs where 21 plays an important role in obtaining small up quark mass.
Next, we derive down-type quark and charged lepton Yukawa matrices. Note that three 27 matters of E 6 include six5s of SU (5). Three of six5s become superheavy with three 5s after developing the VEVs of Φ and C. The other three5s are massless, which are corresponding to the SM5s. To obtain the SM5s, we estimate the mass matrix for 5 and5. Suppose the relation which is important in obtaining the realistic large neutrino mixings. Here r is a real O(1) coefficient. Then, the mass matrix for 5 and5 is derived as where we re-define real O(1) parameters f ′ and g ′ as f ≡ rf ′ and g ≡ rg ′ . Here, α = 1 for triplet (colored) component and α = 0 for doublet component. We diagonalize the 3 × 6 mass matrix (M 1 M 2 ) as where V is 3 × 3 unitary matrix and U is 6 × 6 unitary matrix which is given as The massless5 0 i are given as where U 0 10 and U 0 16 are calculated as The detail derivation is shown in Ref. [17,18].
As a result, the down-type Yukawa matrix Y d is given as where y dij is a O(1) coefficient which includes complex phase. In our calculation for nucleon decay y dij is taken to be a real O(1) coefficient for simplicity. Here, The charged lepton Yukawa matrix Y e is derived from a relationship Y e = Y T d with α = 0 (d 5 = 0) and d q /3 → −d l as where y eij is a O(1) coefficient which includes complex phase. Again, we take y eij as a real O(1) coefficient for simplicity. Finally, to obtain Y u , Y d , and Y e we use 16 real parameters, y uij , y dij , and y eij . In the original E 6 × SU(2) F × U(1) A models, we have 9 real parameters and 2 CP phases.
Therefore, we have several relations among y uij , y dij , and y eij . We will discuss these relations in the next section. Let us diagonalize these Yukawa matrices by field redefinition as where ψ is a gauge eigenstate field and ψ ′ is a mass eigenstate field. We summarize the detail calculation in Appendix A. The diagonalizing matrices are calculated as Since this model has a lot of O(1) parameters for the right-handed neutrino mass matrix, we do not have any interesting relations in L ν . The realistic CKM and MNS matrices can be obtained as if we consider the O(1) coefficients. Since the coefficient of (U CKM ) 13 is vanishing in leading order in this model [17], the sub-leading contribution λ 4 is dominant. As noted previously, we estimate L ν from the observed U M N S and L e .

Conditions for the diagonalizing matrices
In the original E 6 × SU(2) F × U(1) A SUSY GUT models with the spontaneously broken CP symmetry, the number of parameters for the Yukawa couplings of up quarks, down quarks, and charged leptons is 9 (real parameters)+2 (CP phases), which is smaller than the number of observed parameters of masses and mixings. Therefore, once we fix these parameters from the observed values of masses and mixings, we can predict all diagonalizing matrices. Main obstacle for this approach is that these Yukawa couplings are determined at the GUT scale. If the masses and mixings at the GUT scale have been calculated from these measured parameters through the renormalization group equations (RGEs), we would adopt this approach. Unfortunately, many new couplings, which can contribute the running of the Yukawa couplings, appear, when superheavy fields appear at the mass scales which are dependent on the models. Of course, once we fix the GUT models, we can calculate the low energy effective theory. But it is not an easy task to fix the 11 parameters to satisfy the measured quark and lepton masses and mixings by RGEs which change when superheavy fields decouple, though we can do it in principle.
Therefore, in this paper, we adopt another approach. We select several relations between Yukawa couplings which are not strongly dependent on the renormalization scale. Using these relations, we reduce the number of parameters.
As noted in the previous section, we consider real Yukawa couplings for simplicity. Then, generically, we have 27 parameters for the Yukawa couplings of up quarks, down quarks, and charged leptons. In the previous section, we introduced 16 real parameters y uij , y dij , and y eij for these Yukawa couplings. Therefore, there must be 11 (= 27 − 16) relations among the parameters of masses and diagonalizing matrices. In the followings, the notation of the angles are defined in the appendix. From (Y u ) 13 = (Y u ) 31 = (Y e ) 13 = 0, the relations s uL 13 = 0, s uR 13 = 0, s eL 13 = 0.
In our analysis, we do not use the last relation because it is strongly dependent on the renormalization scale. As a result, we use 9 relations in our analysis. We have checked the scale dependence of these relations by explicit numerical calculations of the RGEs in the MSSM [23,24]. We have additional 7 (= 16−9) relations because the original models have only 9 real parameters (a, b, c, d q , d 5 , d l f , g, and β H ) if we take vanishing CP phases. Unfortunately, these are strongly dependent on the renormalization scale, and therefore, we do not use these relations in our analysis.
As a result, we use only 6 parameters in our numerical calculations of nucleon decays for 7 diagonalizing matrices L u , L d , L e , L ν , R u , R d , R e . Since we assume real diagonalizing matrices, each matrix has three real parameters, generically. The CKM matrix and the MNS matrix reduce the 21 parameters to 15 parameters, and because of 9 relations, only 6 (= 15−9) parameters are sufficient. (Strictly speaking, the signature of s uL 12 is an additional parameter because the relation (s uL 12 ) 2 = m u /m c cannot fix the signature.) Note that we have used 12 parameters for fixing the real diagonalizing matrices of the E 6 × U(1) A GUT models in the previous paper. In this paper, we have succeeded to reduce the number of parameters to half.

Numerical calculation
In this section, we calculate various partial decay widths of nucleons numerically. And we compare the results with those in the previous paper [1].
In our calculation, we use the VEVs where x is the scale of the adjoint Higgs VEV which breaks These VEVs are the same as VEVs of GUT Higgs adopted in the previous paper. The larger x leads to larger contribution of SO(10) and E 6 superheavy gauge multiplets to the nucleon decay processes. The X-type gauge boson masses are written as We generate the real diagonalizing matrices L u , L d , L e , L ν , R u , R d , and R e as follows. , (36) [26,27]. Following the above procedure, we have generated 10 4 − 10 6 model points and calculated various partial decay widths of nucleons.

Various decay modes for the proton
We calculate the proton lifetimes for various decay modes in the E 6 ×SU(2) F model. As in the previous paper [1], we use the hadron matrix elements calculated by QCD lattice [28]. The result are shown in Figure 1. In the figure, we take the partial lifetime of p → π 0 + e c as the horizontal axis and the partial lifetime of the other decay modes as the vertical axis. In the Figure 1, we show the predictions of the E 6 model in the previous paper as well as those of E 6 × SU(2) F model.
We have two comments on these results. First, in many model points of the E 6 × SU(2) F model the lifetime of the p → π 0 + e c mode is shorter than the lifetimes in the E 6 model. This result comes from larger (L e ) 11 , because s eL 12 is smaller and s eL 13 is vanishing. The smaller s eL 12 and vanishing s eL 13 are caused by the last relation in (31) and the last relation in (29), respectively. On the other hand, the lifetime of the p → K 0 + µ c mode does not become short because (L e ) 22 does not become larger. This is because the s eL 23 has large value because of the first relation in (31) though s eL 12 is smaller. Second, the lifetimes for K 0 + e c mode and π 0 + µ c mode become longer, which can be seen in the Figure 1. This is also because of smaller s eL 12 . Figure 1: The proton lifetimes for various decay modes in the E 6 × SU(2) F model 1 and the E 6 model 1 with φ , x = 1 × 10 16 GeV, v c = 5 × 10 14 GeV, v φ = 5 × 10 15 GeV. Each model has 10 4 model points.

Calculation of
In the previous paper [1], we have emphasized that the parameters R 1 = Γ n→π 0 +ν c Γ p→π 0 +e c and R 2 = Γ p→K 0 +µ c Γ p→π 0 +e c are useful to identify the grand unification groups, SU(5), SO (10), or E 6 , in the anomalous U(1) A GUTs. The R 1 can be important to know the largeness of the rank of the unification group [14]. The R 2 has sensitivity of the Yukawa structure, especially, for the second generation fields.
We have calculated these parameters for 10 6 model points for E 6 ×SU(2) F model, which are much larger than in the previous paper [1]. The results are shown in Figure 2, in which the darker region represents larger density of model points. The model points for E 6 model without SU(2) F , which has been calculated in the previous paper [1], are dotted in the figure. The region in which both R 1 and R 2 are small are allowed in E 6 × SU(2) F model but looks not to be allowed in E 6 model without SU(2) F . Of course, this can happen because the predictions of the two models are different. However, it is also plausible that the allowed region in E 6 × SU(2) F model is included in the allowed region in E 6 model without SU(2) F if more model points are taken into account. Therefore, we have re-calculated the allowed region by using 100 times more model points for E 6 model without SU(2) F (see Figure  3). The allowed region for E 6 × SU(2) F model is almost included in the allowed region for E 6 model without SU(2) F , though small region with small R 1 and R 2 is still not included. Since it has been found that increasing model points are important, we re-examine the conditions for identification of the grand unification group, which were discussed in the previous paper, with 100 times more model points in the next subsection.
We have two comments in Figures 2 and 3. First, in the E 6 × SU(2) F model, many model points have smaller R 1 and R 2 than these in the E 6 model without SU(2) F . This is because Γ p→π 0 +e c is tend to be larger due to the small mixings between the electron and the other charged leptons as we mentioned in the previous subsection. Second, in the E 6 × SU(2) F model, the plotted region in the R 1 and R 2 plain becomes smaller than in the E 6 model, as expected. This is because the diagonalizing matrices are restricted by relations (29)-(32).

Identification of GUT models
In this subsection we re-examine the conditions for identification of the grand unification group by using 10 6 model points which are 100 times more than in the previous paper [1].
In order to examine the statement that the unification group is not SU(5) if R 1 > 0.4, we have calculated R 1 and R 2 in SU(5) model with 10 6 model points (see Figure 4). The Figure shows that there are very few model points with R 1 > 0.4. Therefore, the statement is almost satisfied even if 10 6 model points are taken into account.
In order to examine the statements that the unification group is E 6 if R 1 > 1 and that the unification group is implied to be E 6 if R 2 > 0.3, we have calculated R 1 and R 2 in SO(10) model with 10 6 model points (see Figure  5). Note that the effect of SO(10) X-type gauge boson X ′ becomes almost maximal with the VEVs adopted in the calculation. The Figure shows that there are very few model points with R 1 > 1 or with R 2 > 0.3. Therefore, these statements are almost satisfied even if 10 6 model points are taken into account.
In the end of this subsection, we show the result in E 6 × SU(2) F model with x = 5 × 10 15 GeV. The difference is only the VEV of adjoint Higgs. As seen in Figure 6, the E 6 × SU(2) F with smaller x predicts smaller R 1 and R 2 than the original E 6 × SU(2) F model which has x = 1 × 10 16 GeV. This is because the nucleon decay via dimension-6 operators which is induced by X ′′ exchange is more suppressed in E 6 × SU(2) F model with smaller x than in the model with larger x.

Discussion and Summary
In this paper we have calculated the partial lifetime for various decay modes of the nucleons via dimension-6 operators in anomalous U(1) A E 6 × SU(2) F SUSY GUT model with the spontaneously broken CP symmetry. Once we fix the VEVs of GUT Higgs, the main ambiguities come from the diagonalizing matrices of quark and lepton mass matrices. Since the SU(2) F symmetry can reduce the ambiguities, the predictions have become more restricted than the E 6 model without SU(2) F symmetry. We have derived the various relations on the components of the diagonalizing matrices from the constraints on the Yukawa couplings which are realized in E 6 × SU(2) F model. Among the relations, we have used 9 relations which are not dependent on the renormalization scale. We have showed that only 6 parameters are sufficient to fix the 7 diagonalizing 3 × 3 matrices.
In this calculation, we have increased the model points up to 10 6 from 10 4 in the previous paper. Even with such many model points, the previous conclusion is still valid, that R 1 = Γ n→π 0 +ν c Γ p→π 0 +e c and R 2 = Γ p→K 0 +µ c Γ p→π 0 +e c are useful to identify the grand unification groups, SU(5), SO (10), or E 6 , in the anomalous U(1) A GUTs.
It is important to consider how to test the GUT models. The most important prediction of the GUT is the nucleon decay, and therefore, the calculations for the partial decay widths for various GUT models are important. One more interesting evidence of the GUT models may appear in the SUSY breaking parameters, especially, in scalar fermion masses through the D-term contribution which are generated if the rank of the unification group is larger than 4 or the additional gauge symmetry like SU(2) F is introduced [29]. We will study this possibility in the E 6 × SU(2) F models in future. The estimation of the diagonalizing matrices in this paper must be important in predicting the FCNC processes induced by the non-vanishing D-term.

Acknowledgement
N.M. is supported in part by Grants-in-Aid for Scientific Research from MEXT of Japan. This work is partially supported by the Grand-in-Aid for Nagoya University Leadership Development Program for Space Exploration and Research Program from the MEXT of Japan.
A The diagonalizing of Yukawa matrices (in leading order) Hereafter, we summarize how to diagonalize the 3 × 3 matrix Y ij . In this calculation we suppose that the Yukawa matrix has hierarchies, Y ij ≪ Y kj and Y ij ≪ Y il when i < k and j < l. References for this calculation is [17,30]. Diagonalizing the Yukawa matrix, we translate the flavor eigenstate ψ into mass eigenstate ψ ′ . We make the Yukawa matrix Y diagonal, as In the calculation, we use the approximation that the mixing angles are small, i.e., |s For reference, we explain the diagonalization for 2 × 2 matrix without approximation.
And eigenvalues become y ′ 11 = y 12 c L ( y 11 y 12 c R − s * R ) − y 22 s L ( The diagonalizing matrices of left-handed up-type quark and down-type quark, L u and L d , are given by L T u/d = P u/dL 12 P u/dL 13