Yukawa interactions, flavor symmetry, and non-canonical K\"{a}hler potential

We study the origin of fermion mass hierarchy and flavor mixing in the standard model, paying attention to flavor symmetries and fermion kinetic terms. There is a possibility that the hierarchical flavor structure of quarks and charged leptons originates from non-canonical types of fermion kinetic terms in the presence of flavor symmetric Yukawa interactions. A flavor symmetry can be hidden in the form of non-unitary bases in the standard model. The structure of K\"{a}hler potential can become a touchstone of new physics.


Introduction
The origin of fermion mass hierarchy and flavor mixing has been a big mystery, which comes from the fact that there is no powerful principle to determine Yukawa couplings in the standard model (SM). Yukawa couplings are expressed as general square matrices taking complex values, and they are diagonalized by bi-unitary transformations. Their eigenvalues become quark and charged lepton masses after multiplying the vacuum expectation value (VEV) of neutral component in the Higgs doublet. The mixing of flavors occurs from the difference between mass eigenstates and weak interaction ones [1,2,3].
There have been many intriguing attempts to explain the values of physical parameters concerning fermion masses and flavor mixing matrices. Most of them are based on the top-down approach [4,5,6,7,8,9], i.e., Yukawa couplings are constructed or given in the form of Ansatz based on high-energy physics such as grand unified theories (GUTs) and superstring theories (SSTs) or extensions of SM with some flavor symmetry, and the analyses have been carried out model-dependently and/or independently from the phenomenological point of view.
At present, any evidences from new physics except for neutrinos have not yet been discovered, and new physics might be beyond all imagination. Hence, it would be interesting to see flavor physics through a different lens, with the expectation that it offers some hints of a fundamental theory. We adopt several reasonable assumptions in a theory beyond the SM. (a) The field variables are not necessarily the same as those in the SM. (b) There is a symmetry relating to the flavor or family of the SM (a flavor or family symmetry). The symmetry is broken down by the VEVs of some scalar fields called flavons. (c) Flavons couple to matter fields through matter kinetic terms dominantly. The second assumption is based on the idea that the family number is naturally understood as a dimension of representation and a predictability is improved by the reduction of free parameters, in the presence of a flavor or family symmetry. The last one is based on the fact that various fields are easy to couple to among them in a Kähler potential, compared with a superpotential controlled by holomorphy, and the Kähler potential can change by receiving radiative corrections in contrast with the superpotential, in supersymmetric (SUSY) theories. We expect that the SUSY exists in an underlying theory, even if it is broken down at a high energy scale.
Suppose that a flavor symmetry exist, we have several questions such as "what type of symmetry exists?", "what is the breaking mechanism ?" and "how is it hidden in the SM?". Here, we focus interest on the last one. There is a possibility that a flavor symmetry is hidden in the form of non-unitary bases, i.e., matter fields in the SM are transformed by non-unitary matrices. In Appendix A, we give an illustration of a realization of U(N ) symmetry using non-unitary matrices.
Our approach is summarized as follows. We suppose field variables respecting a flavor symmetry (that the corresponding transformation is realized by unitary matrices) and rewrite the Lagrangian density in the SM using such variables. We investigate the structure of terms violating the flavor symmetry, and attempt to conjecture physics beyond the SM. Although physics is unchanged by a choice of field variables and representations, there can be a difference in an understandability of physical phenomena. For instance, in the relativistic quantum mechanics, the Dirac representation of γ matrices is useful to analyze non-relativistic phenomena and the chiral representation is suitable to investigate high-energy physics. It is desirable to find helpful field variables in order to envisage a mechanism of flavor symmetry breaking in an underlying theory. We expect that unitary bases of flavor symmetries are suitable to describe physics right after the breakdown of flavor symmetries, although they are unfit for perturbative calculations due to the presence of non-canonical kinetic terms. One of the best plans would be to attack a flavor structure from both bottom-up and top-down approaches. Knowledge and information obtained by the bottom-up approach can provide a new procedure based on a top-down approach.
In this paper, we study the origin of fermion mass hierarchy and flavor mixing in the SM, using the above-mentioned approach. We examine whether the hierarchical flavor structure of quarks and charged leptons can originate from specific forms of their kinetic terms in the presence of flavor symmetric Yukawa interactions or not. We also propose a variant procedure based on the top-down approach.
The outline of this paper is as follows. In the next section, we review quark Yukawa interactions and a no-go theorem on flavor symmetries in the SM. We explore the origin of the hierarchical structure of quarks and charged leptons, paying attention to flavor symmetries and fermion kinetic terms in Sect. 3. In the last section, we give conclusions and discussions.

Yukawa interactions and flavor symmetry
We review quark Yukawa interactions and the absence of exact flavor symmetries in the SM.

Quark Yukawa interactions
Let us start with the Lagrangian densities of the quark sector, where q Li are left-handed quark doublets, u Ri and d Ri are right-handed up-and downtype quark singlets, i , j (= 1, 2, 3) are family labels, summation over repeated indices is understood throughout this paper, y (u) i j and y (d) i j are Yukawa couplings, φ is the Higgs doublet,φ = i τ 2 φ * and h.c. stands for hermitian conjugation of former terms. The diag by bi-unitary transformations and the quark masses are obtained as where V (u) L , V (d) L , V (u) R and V (d) R are unitary matrices, v / 2 is the VEV of neutral component in the Higgs doublet, family labels are omitted, and m u , m c , m t , m d , m s and m b are masses of up, charm, top, down, strange and bottom quarks, respectively.
The Yukawa couplings are expressed by using Information on physics beyond the SM is hidden in V (u) L , V (u) R , and V (d) R besides observable parameters y (u) diag , y (d) diag , and V KM . The matrices V (u) L , V (u) R , and V (d) R are completely unknown in the SM, because they can be eliminated by the global U(3)×U(3)×U(3)/U (1) symmetry that the quark kinetic term L quark kinetic possesses. From (3), (4) and experimental values of quark masses, the eigenvalues of y (u) and y (d) are roughly estimated at the weak scale as We find that there is a large hierarchy among a size of Yukawa couplings, and it has thrown up the big mystery of its origin. From (5), we derive the relation: where y (u)−1 diag is the inverse matrix of y (u) diag . The matrix y (u)−1 diag V KM y (d) diag can be a barometer of the difference between y (u) V (u) R † and y (d) V (d) R † , and it is roughly estimated at the weak scale as where we use the Wolfenstein parametrization [10], i.e., λ = sin θ C 0.225 (θ C is the Cabibbo angle [2]), A 0.811, ρ and η are real parameters [11].

No unbroken flavor symmetry
We explain that there is no unbroken flavor-dependent symmetry respecting the SU(2) L gauge symmetry [12,13]. If the quark sector is invariant under a global transformation (a low-energy remnant of some flavor symmetries): the quark Yukawa couplings should satisfy the relations: where F L , F (u) R , and F (d) R are 3 × 3 unitary matrices, and θ is a real number. From (12), we have the relations: and then F L can also be diagonalized by the unitary matrices V (u) L and V (d) L which diagonalize y (u) y (u) † and y (d) y (d) † such that In the same way, F (u) R and F (d) R can also be diagonalized by the unitary matrices V (u) R and V (d) R which diagonalize y (u) † y (u) and y (d) † y (d) such that By multiplying both sides of each relation in (12) by V (u) L and V (d) L from the left and R † from the right and using (5), (15) and (16), the following relations are obtained, and they lead to (15), we obtain the relation: Then, we find that F (u) L diag = F (d) L diag = e i ϕ I (where ϕ is a real number and I is the 3 × 3 identity matrix) from the fact that all mixing angles of V KM are nonzero, and it means that any exact flavor-dependent symmetries do not exist in the quark sector of the SM. In the same way, it is shown that any exact flavor-dependent symmetries do not also survive in the lepton sector of the SM.

Kähler structure in SM and beyond
Based on feasible assumptions in a theory beyond the SM such that the field variables are not necessarily the same as those in the SM, there is a flavor symmetry broken down by the VEVs of flavons and flavons couple to matter fields in matter kinetic terms dominantly, we rewrite the Lagrangian density in the SM using unitary bases of a flavor symmetry, investigate the structure of terms violating the flavor symmetry, and attempt to conjecture physics beyond the SM. Here, unitary bases mean sets of fields that are transformed by unitary matrices. For more details, see Appendix A.

Change of variables and matching conditions
We assume that a theory beyond the SM has a flavor symmetry 1 and the symmetry is broken down by the VEVs of flavons at some high-energy scale near M BSM . Here, M BSM is an energy scale of new physics or the upper limit of a scale where the SM holds. We assume that M BSM is much bigger than the weak scale, for simplicity. In this case, there is a possibility that we obtain useful information on flavor physics from the matching conditions at M BSM .
We denote unitary bases of a flavor group G F for quarks by q ′ L , u ′ R , and d ′ R . They transform as under the G F transformation, where F L , F (u) R , and F (d) R are 3 × 3 unitary matrices. Then, the Yukawa interaction terms are rewritten as where y 1 i j and y 2 i j are Yukawa couplings in the unitary bases of flavor symmetry. These couplings, in general, consist of two parts, i.e., y 1 i j = y F 1 i j + ∆y 1 i j and respectively, and ∆y 1 i j and ∆y 2 i j are noninvariant ones showing the breakdown of G F due to the VEVs of flavons.
The unitary bases of G F are related to the SM ones q L , u R , and d R by the change of variables as where V q and U q are 3 × 3 unitary matrices and J q is a real 3 × 3 diagonal matrix.
Using new variables, the quark kinetic terms in the SM are rewritten as where the kinetic coefficients k Here, and we use the feature that the kinetic coefficients are positive definite. Note that W (u) and W (d) are not necessarily unitary matrices. If J q is the identity matrix, k (q) i j is the canonical one (the identity matrix) and W (u) and W (d) become unitary matrices.
We give an alternative proof on the absence of exact flavor symmetries in the SM briefly. Under the assumption that L quark Yukawa given in (20) is invariant under the transformation (19) it is shown that no exact flavor symmetries exist from the invariance of L quark kinetic given in (24) under the transformation (19), in the following. Eigenvalues of F (u) R and F (d) R are given by those of F L multiplied by e i θ and e −i θ , respectively, as estimated from (17). Using (25) and Here, we omit the labels of flavor. From (26), (27), and F (u) derived from the invariance of other kinetic terms, we obtain the Here,F L diag is a diagonal unitary matrix. These relations lead toF L diag V KM = V KMFL diag which means thatF L diag and F L diag should be proportional to the identity matrix or the non-existence of exact flavor-dependent symmetries. From (7) and (8) Physical parameters, in general, receive radiative corrections, and the above values should be evaluated by considering renormalization effects and should match with their counterparts at M BSM . From (20) and (24), information on the flavor structure in the SM is transfered to k i j in the kinetic terms. To speculate a theory of quarks beyond the SM, let us describe it by where K at M BSM , from (20), (24) and (31).

Examples
As we have few hints on a flavor symmetry, we study two examples, i.e., a case with a U(3) symmetry and that with an S 3 one. Here S 3 is the permutation group of order 3

U(3) case
In case that a U(3) family symmetry is hidden in the SM, the Yukawa interactions are where y 1 and y 2 are complex numbers. We assume that U(3) symmetric terms dominate in Yukawa interactions. It is justified, in case that M BSM is much bigger than the weak scale, other terms including fermions contain non-renormalizable higher-dimensional operators and they can be suppressed by a power of M BSM . Now, we conjecture a structure of Kähler metric, based on (25) - (30). There are many possibilities to realize the quark masses and flavor mixing consistent with experimental data. For simplicity, we assume that J q = I , i.e., k where There is a possibility that k (u) and U      1 1 1 1 1 1 with the unitary matrix where ε 1 and ε 2 are arbitrary numbers, ω = e 2πi /3 , and ω = ω 2 = e 4πi /3 (= −1 − ω). The above formulas are merely examples. Quark kinetic coefficients and unitary matrices might take complicated forms and contain tiny parameters intricately. At any rate, a large mass hierarchy and mixing can originate from a tiny variance of the democratic form whose every component has a common value. In other words, the hierarchical structure can be realized in case that Kähler metrics K (u) i j and K (d) i j acquire the VEVs of semi-democratic forms as with some constants ξ (u) and ξ (d) , after the breakdown of the family symmetry, and the reception of tiny corrections. Here, S i j is the democratic matrix defined by It is hard to derive semi-democratic forms (37) dynamically at a level of perturbation, from U(3) invariant Kähler potential K = |Φ i | 2 + · · · , as suggested by a model in Appendix B. Here the ellipsis stands for higher-dimensional terms which are sub-leading order ones. We need a mechanism to realize semi-democratic forms and small Yukawa couplings such as y 1 2 = O 10 −10 and y 2 2 = O 10 −9 .

S 3 case
Based on an S 3 invariant Kähler potential containing the democratic form and Yukawa couplings with the democratic form and small S 3 breaking ones, it was pointed that the heavy top quark mass can be attributed to a singular normalization of its kinetic term [21]. Sfermion masses were also studied using the S 3 invariant Kähler potential [22]. Let us re-examine a case with the S 3 symmetry using our formulation. Strictly speaking, the flavor group is S 3 ×S 3 ×S 3 , and q Li , u Ri and d Ri are transformed as 3-dimensional representations of the first, second and third S 3 , respectively. These 3-dimensional representations are reducible and are decomposed into two irreducible ones such as 1dimensional ones and 2-dimensional ones. In the presence of S 3 symmetry, the Yukawa couplings are written by where y F a and ∆y a (a = 1, 2) are complex numbers, and T (u) i j and T (d) i j are complex matrices (whose components take values of at most O(1)) that originate from S 3 breaking effects. We cannot derive realistic quark masses without T (u) i j and T (d) i j . We assume that y F a = O(1) according to Dirac's naturalness. Here, Dirac's naturalness means that the magnitude of dimensionless parameters on terms allowed by symmetries should be O(1) in a fundamental theory. In contrast, we suppose that ∆y a ≪ y F a from a conjecture that the S 3 breaking terms stem from non-renormalizable interactions suppressed by a power of M BSM .
In the following, we examine whether magnitudes of components in k (u) i j and k (d) i j can be at most O (1) or not under the above assumptions, i.e., y F a = O(1) and ∆y a ≪ y F a . In other words, k (u) i j and k (d) i j are, in general, written by where k (u) b and k (d) b (b = 1, 2, 3) are real numbers, and Z (u) i j and Z (d) i j are hermitian matrices (whose components take values of at most O(1)) that represent S 3 breaking effects. Then, can magnitudes of k (u) b and k (d) b be at most O(1) or not?
By inserting the first relation of (39) into (26), the following relation is derived, Using the formula S X S = ( 3 i ,j =1 X i j )S, we find that the following condition should be fulfilled, in order to make the magnitudes of first term in (41) to be at most O (1). For simplicity, let us take an ansatz of W (u) such as where w (u) i j are complex numbers of at most O(1). Then, we obtain the relation: If ∆y 1 2 = O 10 −10 , the magnitude of every component in the second term of (44) can also be at most O(1), and k (u) i j can take the form given by the first relation of (40) with y F 1 = O(1). In the same way, when we take an ansatz of W (d) such as we obtain the relation: where w (d)

Lepton sector
We study the lepton sector in the SM. In the absence of Majorana masses of right-handed neutrino singlets, the same argument as the quarks holds in the replacement of fields and couplings. Here, we consider the case with large Majorana masses and a flavor symmetry in a theory beyond the SM. The lepton sector is described by the Lagrangian densities: where V (e) L , V (ν) L , and V (e) R are unitary matrices and m e , m µ , and m τ are masses of electron, muon, and tauon, respectively, and the seesaw mechanism is used to obtain tiny neutrino masses m ν 1 , m ν 2 , and m ν 3 [23,24,25]. The lepton Yukawa couplings are expressed by We find that there is a hierarchy among charged lepton Yukawa couplings. Using field variables l ′ L , e ′ R and ν ′ R defined by the Lagrangian densities are rewritten as where V l and U l are unitary matrices, J l is a real diagonal matrix, J −1 l is the inverse matrix of J l , y 3 i j and y 4 i j are lepton Yukawa couplings in the unitary bases of flavor symmetry and k (l ) i j , k (e) i j , k (ν) i j , and M (ν) i j are given by When a theory of lepton beyond the SM can be described by we have the relations: can be the form whose every component has an almost same magnitude of O(1) and a mass hierarchy can originate from a tiny variance of the democratic form. We need a mechanism to realize semi-democratic forms and a small Yukawa coupling. In case that S 3 flavor symmetry exists, we find that a Yukawa coupling is written by and it is compatible with the Kähler metric: with a suitable W (e) . Here, y F 3 and ∆y 3 are complex numbers whose magnitudes are y F

Top-down approach
We have developed the strategy taking the SM as a starting point. There are limitations on such a bottom-up approach. It is desirable to combine use of the bottom-up and top-down one. Here, we propose a new procedure based on the top-down one, using knowledge and information obtained in the previous subsections.
First, we construct a theory with a flavor symmetry, extract fermion parts from it and write down a Lagrangian density as 1, 2, 3, 4) and their hermitian conjugations.
i j , and K (ν) i j by unitary transformations as Last, we examine whether the following relations hold or not, Note that we need to diagonalize six hermitian matrices in total by unitary transformations in our procedure. As explained in Appendix C, we need ten hermitian matrices in total by unitary transformations, using an ordinary procedure.
As was described previously, we should consider renormalization effects when we match theoretical predictions to experimental data. We also need some modifications in the presence of a mixing with extra particles, in the case with a large flavor symmetry and/or many matter fields.

Unification
We discuss whether realistic mass hierarchies and flavor mixing are realized or nor, based on a grand unification and a family unification.
First, we consider a model based on SU(5) × S 3 × S 3 where SU(5) is the GUT group and S 3 × S 3 is the flavor group. We assume that these symmetries are broken down to the SM one G SM at the GUT scale M U . Matter fields l ′ Li and (d ′ Ri ) c belong to ψ ′ (5) i in the representation (5, 3,1) and q ′ Li , (u ′ Ri ) c and (e ′ Ri ) c belong to ψ ′ (10) i in (10, 1,3), where 3 is a 3-dimensional reducible representation of S 3 . The Lagrangian density of matter fields (except for neutrino singlets) is given by L fermion where Y U 1 i j and Y U 2 i j are Yukawa couplings, and φ (5) and φ (5) are scalar fields in (5, 1,1) and (5, 1,1), respectively. If K (ψ (5) ) i j , K (ψ (10) ) i j , Y U 1 i j , and Y U 2 i j are SU(5) singlets, we have the relations: at M U . From (79) and (80), we derive a usual GUT relation among down-type quark and charged lepton Yukawa couplings: In case that Y U 1 i j and Y U 2 i j contain SU(5) non-singlet parts, realistic mass hierarchies and mixing can be realized with suitable VEVs of non-singlet parts.
Next, we consider a model based on SO(10) × S 3 . Matter fields l ′ (16,3). The matter sector is described by where Y U i j is a Yukawa coupling and φ (10) is a scalar field in (10, 1). If K (ψ (16) ) i j and Y U i j are SO(10) singlets, we have the relations: at M U . In this case, without extra matters and/or extra contributions, quark and lepton masses and flavor mixing cannot be explained. In case that Y U i j contain SO(10) nonsinglet parts, we also need extra contributions if Dirac's naturalness is adopted.
Last, we consider the family unification based on a simple gauge group G FU whose maximal subgroup is G U × G F . Here, G U is a GUT group and G F is a family group. We assume that a field Ψ with a vectorlike representation contains three families of SM fermions as its submultiplets. After the breakdown of G FU into G SM , the kinetic term K Ψi D Ψ changes into In this case, K (I ) i j are not, in general, common and there is a possibility to explain fermion masses and flavor mixing. However, it seems to be unnatural because we need a fine-tuning on a realization of semi-democratic type of Kähler metrics in order to generate fermion mass hierarchies, as explained in Appendix B. Other problem in the family unification is that extra particles including mirror particles appear, and it is solved in the family unification on orbifold [26,27,28] and special GUTs [29,30].

Conclusions and discussions
We have studied the origin of fermion mass hierarchy and flavor mixing in the SM, using the bottom-up approach. The approach is based on the assumptions that the field variables in the SM are not necessarily the same as those in a theory beyond the SM and there is a flavor symmetry and flavons couple to matter fields in the matter kinetic terms dominantly. We have supposed field variables respecting a flavor symmetry (unitary bases of a flavor symmetry) and rewritten the Lagrangian density in the SM using such variables. We have investigated the structure of terms violating the flavor symmetry, and conjectured physics beyond the SM. We have suggested that the hierarchical structure in the Yukawa interactions of quarks and charged leptons can originate from non-canonical matter kinetic terms, in the presence of flavor symmetric Yukawa interactions and a flavor symmetry can be hidden in the form of non-unitary bases in the SM. We have proposed a variant top-down procedure, using an insight and formulas obtained by our bottom-up approach.
In our approach, the problem of fermion masses and flavor mixing is deeply related to not only the determination of Yukawa coupling matrices but also the determination of matter kinetic terms and the VEVs of Kähler metric K (I ) i j . If flavons couple to matter fields in the Kähler potential, the VEVs of K (I ) i j strongly depend on the dynamics of flavor symmetry breaking due to flavons. In a grand unification with a flavor symmetry, contributions of GUT group non-singlet parts in K (I ) i j can be essential to derive a realistic flavor structure.
We explain preceding works on the flavor physics based on matter kinetic terms, other than [21,22]. The problem of fermion mass hierarchies was investigated in supergravity and superstring models with non-canonical Kähler potential including dilaton and moduli fields [31,32]. The Yukawa textures were obtained from non-canonical Kähler potential in the extension of minimal SUSY SM with an anomalous horizontal symmetry [33]. In both works, a symmetry corresponding to a flavor symmetry is an Abelian one and the structure of Yukawa couplings resembles that derived from the Froggatt-Nielsen mechanism [9]. The effect of the Kähler potential on mixing matrices was studied in a model independent way [34]. The flavor symmetry of kinetic terms was discussed in a SUSY SM [35]. The flavor problem was studied through contributions of higher-dimensional operators in case with hierarchical fermion kinetic terms originated from hierarchical fermion wave functions, under the assumption that the energy scale of new physics is in the TeV range [36]. In our setup, the scale M BSM can also be constrained by the suppression of flavor-changing transitions.
As fermion kinetic functions or Kähler metric K (I ) i j contain flavons in our approach, they are regarded as counterparts of "Yukawaons" such that Yukawa couplings are not parameters but fields [37].
Our approach would be useful as a complementary one to explore physics beyond the SM and it would be worth studying flavor physics model-dependently and/or independently by paying close attention to matter kinetic terms, because the structure of Kähler potential can play a vital role as a key test of new physics.

A Unitary and non-unitary bases
We give an illustration of a realization of U(N ) symmetry using unitary matrices and non-unitary ones based on a polynomial: where Φ is an N -plet of U(N ), and K is an N × N hermitian matrix. We consider a case that K depends on a set of fields {ϕ}, i.e., K = K (ϕ, ϕ † ). If K changes into K → U K U † in accord with the U(N ) transformation Φ → U Φ with an arbitrary unitary matrix U , L is invariant under the U(N ) transformation. We call fields transformed by unitary matrices such as Φ "unitary bases".
The U(N ) invariance can be spontaneously broken down to a smaller one, after some ϕ acquire the VEV 〈ϕ〉 and 〈K 〉(≡ K (〈ϕ〉, 〈ϕ † 〉)) takes a form that is not proportional to the identity matrix I . The 〈K 〉 is a hermitian matrix and it is written as 〈K 〉 = W † W with a general N × N complex matrix W . By using a redefinition of field asΦ ≡ W Φ andΦ † ≡ Φ † W † , L is rewritten bỹ The previous U(N ) transformation is realized byΦ →ŨΦ withŨ = W U W −1 . Note that U is not necessarily a unitary matrix because W is not a unitary matrix, and the second termΦ † (W † ) −1 W −1Φ is invariant underΦ →ŨΦ, but the first oneΦ †Φ is not necessarily. The transformation of unbroken subgroup H is realized by a unitary matrix. We call fields transformed by non-unitary matrices such asΦ "non-unitary bases". The L and the final form ofL can be regarded as counterparts of the Lagrangian density of matter sector in a theory beyond the SM and the Lagrangian density of matter sector in the SM, respectively.

B Non-canonical Kähler potential
We consider a SUSY model with the flavor symmetry SU(3) × C 3 (where C 3 is the cyclic group of order 3) and a non-minimal Kähler potential: where a 1 , a 2 , a 3 and a 4 are parameters, Λ is a high-energy scale, and ϕ α i and φ i are the scalar components of flavon chiral supermultiplets and matter chiral supermultiplet, respectively. The ellipsis stands for higher-dimensional terms with O 1/Λ 4 . The family labels are denoted by i , j , and k, and ϕ α i and φ i belong to triplets of SU(3). The indices α and β are labels of C 3 and run from 1 to 3. From (87), the Kähler metric of matter fields is calculated as If Λ is much bigger than the VEVs of ϕ α i , φ i 2 dominates in K and the matter kinetic terms take almost canonical forms with K i j = δ i j + O 〈ϕ α i 〉/Λ 2 .
To obtain a semi-democratic form, we need ϕ α i = O(Λ). In this case, other higher order terms can contribute the determination of K i j and then the evaluation cannot be justified in a perturbation region. Although we have such a problem, we study a case with ϕ α i = O(Λ) by taking the superpotential of flavons: where c 1 and c 2 are parameters and ϕ 3 ≡ ε i j k ε αβγ ϕ α i ϕ β j ϕ γ k . One of the SUSY preserving conditions is given by and there exist two kinds of vacuum solutions ϕ α i = 0 and ϕ α i = 0. (a) Flavor symmetric vacuum with ϕ α i = 0 By inserting ϕ α i = 0 into (87) and (88), we obtain the canonical one for matter fields, i.e., K i j = δ i j . (b) Broken vacuum of flavor symmetry with ϕ α i = 0 From (90), we find a broken vacuum of flavor symmetry represented by Then, by inserting these VEVs into (88), we obtain the VEV of K i j : where η and ξ are given by respectively. From (92), K i j can be a semi-democratic one with suitable values of parameters, but it seems to be unnatural with a fine-tuning among parameters (including ones from higher order terms) based on a perturbative analysis. A Kähler potential from a non-perturbative effect can play a crucial role to the derivation of semi-democratic types of kinetic terms.

C Ordinary top-down procedure
For a purpose of reference, we explain an ordinary top-down procedure, starting from L fermion BSM of (68) with the VEVs K (q) i j whereJ −1 q ,J −1 u ,J −1 d ,J −1 l ,J −1 e , andJ −1 ν are the inverse matrices ofJ q ,J u ,J d ,J l ,J e , andJ ν , respectively.