A low-scale flavon model with a ℤN symmetry

We propose a model that explains the fermion mass hierarchy by the Froggatt-Nielsen mechanism with a discrete ℤNF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_N^F $$\end{document} symmetry. As a concrete model, we study a super-symmetric model with a single flavon coupled to the minimal supersymmetric Standard Model. Flavon develops a TeV scale vacuum expectation value for realizing flavor hierarchy, an appropriate μ-term and the electroweak scale, hence the model has a low cutoff scale. We demonstrate how the flavon is successfully stabilized together with the Higgs bosons in the model. The discrete flavor symmetry ℤNF\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$ {\mathrm{\mathbb{Z}}}_N^F $$\end{document} controls not only the Standard Model fermion masses, but also the Higgs potential and a mass of the Higgsino which is a good candidate for dark matter. The hierarchy in the Higgs-flavon sector is determined in order to make the model anomaly-free and realize a stable electroweak vacuum. We show that this model can explain the fermion mass hierarchy, realistic Higgs-flavon potential and thermally produced dark matter at the same time. We discuss flavor violating processes induced by the light flavon which would be detected in future experiments.


Introduction
The origin of hierarchical structure of the fermion masses and the CKM matrix is a long standing mystery in the Standard Model (SM). The Froggatt-Nielsen (FN) mechanism [1] is known to be one of solutions for this problem. Based on the mechanism, a singlet field, the so-called flavon, and a flavor dependent extra symmetry are introduced to the SM, so that the hierarchy between the Yukawa couplings are explained by powers of a ratio of a JHEP03(2020)129 vacuum expectation value (VEV) of flavon to a cutoff scale of a model: a source for the fermion mass hierarchy is given by where S is a flavon and Λ is a cutoff scale. It is well known that the FN mechanism successfully explains the fermion mass hierarchy and the CKM matrix [2]. It is required for realizing the hierarchy that a flavon is a singlet under the SM gauge symmetry but carries a flavor symmetry charge and develops a non-zero VEV. Any particle can be identified as a flavon as far as these properties are satisfied. 1 According to the FN mechanism, a U(1) flavor symmetry which is denoted as U(1) F is often used. If the U(1) F is a global symmetry, it may be violated by quantum gravity effects according to the standard lore [14][15][16][17][18][19][20][21][22][23][24]. If the U(1) F is gauged as in string motivated models, it can be also broken to a discrete symmetry [25][26][27] as a discrete gauge symmetry [28][29][30][31][32][33]. Flavor models with discrete symmetries are being well studied recently, and the vacuum analyses in such models appear to be complex owing to a number of scalar fields [34][35][36][37]. See also, e.g., refs. [38,39] for string models. 2 In this paper, we will focus on a simple model with a discrete abelian flavor symmetry.
In general, a flavon S can always have couplings to the Higgs boson H, such as The former will appear for a U(1) F flavor symmetry, whereas both two terms can appear for a Z N flavor symmetry, which is denoted as Z F N hereafter. Once the flavon acquires VEV, the Higgs boson gets a mass of c S 2 or c N −2 S 2 at tree-level. Therefore, S 2 should be comparable to the electroweak (EW) scale unless c and c are extremely suppressed. Such tiny couplings of c, c will cause another hierarchy problem. In addition, there is also the hierarchy problem due to quadratic divergences. These facts motivate us to consider a supersymmetric (SUSY) model of the FN mechanism with a TeV scale S leading to a light flavon. Hence, Λ(= S / ) results in a cutoff scale much lower than the Planck scale.
In this paper, we propose a SUSY extension of the SM with a flavon and a Z F N symmetry. 3 Holomorphy of the superpotential constrains flavor structure on top of a Z F N symmetry. The model becomes predictive, because couplings between S and H are related to Yukawa hierarchy as in eq. (1.2) on top of coupling relations within the Minimal Supersymmetric Standard Model (MSSM). A discrete symmetry Z F N allows a self-coupling of the flavon in the superpotential, W S N /Λ N −3 , and the coupling stabilizes flavon. 4 This 1 Recently, it is proposed that a flavon can be identified as the QCD axion [3][4][5][6][7][8]. See also for earlier works [9][10][11][12][13]. 2 See, e.g., refs. [40,41] for recent models with a flavor modular symmetry and also refs. [42][43][44][45] for such interactions in string models. 3 Models with a combination of Higgs doublets HuH d as a flavon are studied in refs. [46][47][48]. Models with discrete FN symmetries are recently discussed in refs. [49,50]. 4 For a U(1)F gauge symmetry, D-term potential will stabilize the flavon VEV. For a global U(1)F , flavon VEV may be stabilized by similar ways as in (fl)axion models.

JHEP03(2020)129 2 Model
In this section, we introduce an abelian flavor symmetry Z F N and a flavon field S whose the charge is 1 against Z F N : under Z F N , a flavon transforms as S → e 2πi/N S. (2.1) The VEV explains the fermion mass hierarchy by the Froggatt-Nielsen (FN) mechanism. This model has a cutoff scale Λ much lower than the Planck scale, so that the hierarchy is explained by a ratio of a low scale flavon VEV to Λ, 7 := S /Λ. (2.2) In the model, such a small VEV is also related to the Higgs potential through a coupling e.g. c |S| 2m |H| 2 /Λ 2m−2 ∼ 2m Λ 2 |H| 2 , which does not induce a large Higgs mass parameter due to an suppression. The SUSY is further introduced for several reasons. As in the MSSM, the SUSY model is free from the gauge hierarchy problem and the LSP becomes the good DM candidate. The SUSY is a well-motivated way to constrain a scalar sector. For instance, quartic couplings are related to gauge or Yukawa coupling constants, especially the Higgs quartic coupling in the MSSM is consistent with the 125 GeV Higgs boson mass.
In addition, the fermion hierarchy can be explained by a discrete symmetry Z F N (with N = O(1)) due to the holomorphy of the superpotential as shown later. As in the NMSSM, a VEV of the flavon generates the Higgsino mass term and the µ-problem is solved. The flavon in this model couples to all the particles, such as SM fermions, Higgs bosons as well as the DM, and their textures are controlled by the flavor symmetry Z F N . Further, we will show that anomalies between Z F N and the SM gauge group can constrain a coupling between the flavon and the Higgs sector. Additional discrete symmetries are discussed for avoiding experimental constraints.
In this paper, the Kähler potential is assumed to be the minimal one. Even if there exist the higher dimensional operators in the Kähler potential, they will not drastically change our results. We discuss possible effects from these operators including kinetic term corrections in appendix B. Without loss of generality, the leading terms in the Kähler potential can be the canonically normalized form, where Φ I 's are any chiral superfields in this model, and V I 's are certain combinations of the vector superfields against Φ I . In the following, we will introduce a superpotential in this gauge basis.

Flavon-Higgs sector
The Z F N -invariant superpotential in the model is given by

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where Λ is the cutoff scale in the model. Here, The integer m is related to the charges of Higgs bosons, n Hu and n H d as m + n Hu + n H d ≡ 0 modulo N . The superpotential involving the SM fermions, W fermion , is introduced in the next subsection. Throughout this paper, we neglected the higher-order terms suppressed by Λ N such as W S 2N /Λ 2N −3 .
The scalar potential of the flavon and the neutral Higgs (against U(1) em ) is given by (2.7) where H 0 u , H 0 d are neutral components of the Higgs doublets H u , H d , respectively. V soft , V F and V D come from soft SUSY breaking terms, F-term potential of the superpotential and the D-term potential, respectively. The quartic coupling constant of the D-term is related to the gauge coupling constants as g 2 = (g 2 1 + g 2 2 )/4, where g 1 and g 2 are the gauge couplings constants of U(1) Y and SU(2) L . In this paper, the soft parameters are assumed to be real.
The scalar fields are expanded around their vacuum as, GeV. Suppose that v s v H , the VEV of flavon S is approximately determined by the scalar potential At the potential minimum, mass eigenvalues for the CP-even and CP-odd flavons are given by

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Here, the soft mass terms are eliminated by the vacuum condition. In this limit, the minimization conditions for the doublet Higgs bosons are similar to that for the MSSM, The full Higgs mass matrices and the vacuum conditions are shown in appendix A. The Higgsino mass matrix with decoupled gauginos is given by whereS is a SUSY partner of the flavon S called flavino, and is approximately a mass of the flavino. The charged Higgsino mass is given by µ eff .

SM fermion mass and mixing
The Z F N -invariant superpotential involving the SM fermions is given by where i, j = 1, 2, 3 run over the three generations. Here, we assume that the right-handed neutrinos have charge N/2, so that they have Majorana masses. 8 A scale of the Majorana masses is, in general, at arbitrary scale, while this might be identified as the cutoff scale of this model Λ. Indeed, this happens at the benchmark point shown in appendix C. The powers η f ij (f = u, d, e, n) obey 20) modulo N . Hereafter, "≡" stands for modulo N if it is not mentioned explicitly. n X is a charge of a field X under the Z F N flavor symmetry. The Yukawa couplings to the Higgs bosons are induced after the flavon S acquire a non-zero VEV, (2.21) 8 N should be an even number from this assumption.

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Since the maximum power of the Yukawa hierarchy is N − 1 under the Z F N , it is assumed that the size of suppression factor is as small as the top to up quark mass ratio: A combination H u H d /Λ 2 can couple to the Yukawa couplings, depending on the charge assignment, but these will give negligible effects for v s v H . It is noted that the holomorphy of superpotential is important to prevent couplings involving S † to fermions. Since a charge of S † is N − 1, once a coupling of L (S/Λ) η O yukawa is allowed by the discrete symmetry, L (S † /Λ) N −η O yukawa is also allowed, where O yukawa is a Yukawa type operator. Thus, the maximal power becomes N/2 for even N or (N + 1)/2 for odd N effectively if there is no holomorphy in the Yukawa couplings. Hence it is assumed in this paper that the SUSY should remain unbroken below the cutoff scale Λ in order to explain the fermion hierarchy with a small N .
So far, we assumed that the Kähler potential is the canonical one and the Yukawa couplings originate only from the superpotential, but let us discuss Yukawa couplings from higher dimensional Kähler potential. To justify our discussion, Yukawa couplings from Kähler potential of K S † O yukawa /Λ 2 should be suppressed, when it is compared to a superpotential contribution W (S/Λ) N −1 O yukawa . Since a VEV of F-term of the flavon is given by the Yukawa couplings from the Kähler potential is as small as the ones from the superpotential, F † S /Λ 2 ∼ N −1 , where N −1 is comparable to a smallest Yukawa coupling. Thus the higher dimensional terms in the Kähler potential does not alter the texture of Yukawa matrices in the superpotential and only change O (1) coefficients per order. We absorb this effect in definitions of the O (1) coefficients c f ij in the superpotential. See appendix B for more details of Kähler potential.

An example: N = 4 case
We will consider N = 4 case as an example of the minimal extension of the Z 3 -invariant NMSSM, and we have W S 4 /Λ. In this case, we obtain ∼ 0.02, (2.24) and it may be able to explain the hierarchy of the charged fermions and the CKM elements involving the third generation. The Cabbibo angle ∼ 0.22 ∼ 1/2 is regarded as an O (1) value and explained with an O (1) tuning of holomorphic Yukawa coupling. The matrix element V cb is naturally addressed by ∼ 0.02 as shown below. Smaller N makes harder to explain the hierarchical structure of the fermion masses and mixing at the same time.
For example, in the case of N = 3, ∼ 0.004 |V cb | is expected from the top to up quark mass ratio, but this will be too small to explain the other hierarchies. For N ≥ 6 cases, ∼ 0.22, which is often considered, is allowed. The superpotential W S N /Λ N −3 with JHEP03(2020)129 a larger N makes the scalar potential flatter along the S direction. Hence, more careful parameter choice will be required to stabilize the flavon field and to realize the realistic EW vacuum. The stability of the EW vacuum is non-trivial even in the usual NMSSM with N = 3, as discussed in refs. [55][56][57].
In this case, an ansatz of hierarchical structure of the Yukawa matrices, leads to the charged fermion mass hierarchy (2.26) and the CKM and PMNS matrices 27) which are consistent with the observed values. Here, k ≤ 1 and ≤ 3 for N = 4. The integer k is related to tan β as k m t /m b ∼ tan β. Hence, k ≤ 1 is required for 1 tan β 50, so that the Yukawa coupling constants are perturbatively small. The integer related to a scale of the Majorana mass M 0 as m ν ∼ 2 v 2 u /M 0 via the see-saw mechanism, where m ν is a neutrino mass. To realize the above Yukawa matrices, the conditions of charge assignment are given by , n e i ≡ (n e 3 − 2, n e 3 , n e 3 ),  between neutrino masses are explained with a choice of O(1) Yukawa couplings since a neutrino mass squared is proportional to fourth power of the Yukawa couplings. It is possible to explain the hierarchal neutrino masses also by an introduction of additional discrete symmetries [34][35][36][65][66][67][68][69][70]. In this paper, we do not introduce such additional symmetries, and we show a set of values of O (1) coefficients which explain the neutrino mass differences accidentally. See numerical values exhibited in appendix C for realistic Yukawa couplings and Majorana masses.

Constraint from anomalies in N = 4 case
We show that a way of a coupling of S to the Higgs sector, W S m H u H d /Λ m−1 , is constrained in terms of anomalies of the discrete symmetry between Z F 4 and the SM gauge symmetries. The abelian discrete symmetry Z F 4 may potentially induces anomalies [71][72][73][74][75][76]. If there exist such anomalies, the discrete symmetry is no longer a symmetry in theories and explicit violation terms against the discrete symmetry are induced by quantum effects.
The anomalies of Z F From the above, W S m H u H d /Λ m−1 shows the charge relation of m + n Hu + n H d = m + 3(1 + k) ≡ 0 (2.32) modulo 4, thus it is found that m = 1, 2 for k = 0, 1, respectively. Note that m = 3 is not allowed. Now there exist nine parameters of (n Q 3 , n u 3 , n d 3 , n L 3 , n e 3 , n Hu , n H d , k, ) and six constraints of eq. (2.31). Altogether, we can regard (k, n Hu , n Q 3 ) as three free parameters, so there are 2 × 4 × 4 = 32 ways to choose them. Although from the bottom-up viewpoint we do not know a normalization of the U(1) Y which may be embedded into a larger (grand unified) gauge group at a higher energy scale, 9 JHEP03(2020)129 the anomaly of Z F 4 with U(1) Y could provide some insights. With a normalization factor N Y which is assumed to be fractional, the anomaly-free condition is given by Suppose n Y := N Y /3 ∈ Z, the above equation is rewritten as where the integer p is defined through Here, we used n Hu anomaly. If k = 0 (1), 3k + 2p is even (odd). In particular, the anomaly-free condition is satisfied independent of n Y if k = 0 (m = 1) and p is even. On the other hand, the condition is trivial if n Y is a multiple of 4 and hence N Y is a multiple of 12. For instance, U(1) Y might be embedded into a U(12) theory in this case, since N Y is associated with a rank of a gauge group into which U(1) Y might be embedded.
The fermions in this model can also induce the gravitational anomaly [76], If there is no other particle charged under the Z F 4 symmetry, the anomaly-free condition is given by A vis grav ≡ 0 modulo 2. Tables 1 and 2 show the patterns of powers k, m, , the U(1) Y and gravitational anomalies when the realistic patterns of Yukawa couplings are realized and the SU(2) L and SU(3) c anomalies are absent. The lists for k = 0 (m = 1) and k = 1 (m = 2) are shown in table 1 and 2, respectively. The charges of the other chiral superfields are determined through eqs. (2.28) and (2.31). Note that only (k, , m) are relevant to Yukawa couplings. It is noted that only = 0, 2 are available for A gr ≡ 0, whileÃ Y = A U(1) Y /n Y is even (odd) for k = 0 (k = 2) as already stated. The U(1) Y anomaly is absent independent of n Y for k = 0 case, ifÃ Y ≡ 0 modulo 4. However, k = 0 gives unstable Higgs potential as shown in the next section. For k = 2,Ã Y gives an odd number, therefore n Y should be a multiple of 4, i.e., the U(1) Y normalization of N Y should be a multiple of 12.

Particle stability and discrete symmetry
Here, we discuss necessities of additional discrete symmetries, focusing on the proton decay.
The Z F N symmetry will not be enough to suppress unwanted higher-dimensional operators.

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Some combinations of baryon/lepton number violating operators are severely constrained by proton decay. The limits on the baryon/lepton number violating operators are [77][78][79] Here, λ B and λ L are Yukawa couplings for the dimension-4 baryon number violating operator u R u R d R and lepton number violating operator These tiny coupling constants can not be explained by the Z F 4 symmetry. Thus there should be additional symmetry to control these couplings.
There are several candidates which can forbid these operators. In the MSSM, the matter parity M 2 , the so-called R-parity, is introduced for this purpose [80][81][82]. An important consequence of the R-parity is that the LSP becomes stable and it can be a good candidate for the dark matter (DM). The dimension-4 operators are forbidden by the R-parity, but the dimension-5 operators are not. Another candidate is known as the baryon triality, B 3 , which prevents the baryon number violating operators while permits the lepton number violating ones [71]. The baryon triality successfully ensures the proton stability, however, the LSP becomes unstable due to the lepton number violating interactions. The so-called proton hexiality P 6 prohibits all the baryon and lepton number violating dimension-4 and 5 operators, and the LSP is stable [74,83]. All of these three discrete symmetries are anomaly-free. The charge assignments under these discrete symmetries are listed in table 3. A discrete R symmetry in SUSY theories is an interesting possibility [84][85][86][87][88]. The anomaly-free Z R M symmetry prohibits the unwanted higher dimensional operators as well as the µ-term, and stabilizes the LSP in the MSSM [85,86]. In this model, the flavon field S must have non-zero charge under the discrete R-symmetry in order to write down the self-coupling S N in the superpotential. Note that a superpotential have a non-zero R-charge 2 modulo M . This causes additional selection rules for the Yukawa couplings of the SM fermions. Hence, N of the discrete symmetry Z F N may have to be so large that the SM fermion mass and mixing are explained.
We are interested in the simplest way to explain the observed fermion properties and the neutralino DM at the same time. For this purpose, we consider a model with operators depend on UV model-building, also models with the R-parity M 2 may be allowed. In the following, we do not consider the higher dimensional operators violating lepton/baryon number.
A spontaneous breaking of Z F 4 could produce stable domain walls which alters the history of the successful standard cosmology [58][59][60][61]. There exist several solutions for it. The Planck suppressed operators which break Z F 4 explicitly can make domain walls unstable, while keeping the low-energy physics unchanged [89][90][91][92][93]. In the presence of a negative Hubble induced mass term for the flavon S (and/or Higgs fields) during/before the inflation, domain walls will be produced then and inflated away, hence the problem is solved [62]. We assume that the domain wall problem is solved in our model by one of these effects.

Phenomenology
We will study vacuum stability and phenomenology related to the flavons when the hierarchy eq. (2.25) is realized and the anomalies of non-abelian gauge symmetries are absent for N = 4. In the following, we will discuss models with the superpotential W In our analysis, we study cases where squarks/sleptons are heavier than O (10) TeV. There may be various flavor violating processes induced by sfermions depending on the soft parameters, but this is beyond a scope of this paper.

Vacuum stability
We will show the vacuum stability is related to the power of m = 1, 2. The EW minimum will be unstable if there exist extra minimum deeper than it. The both CP-even and CP-odd flavon get positive mass squared when where O v 2 H corrections are neglected. With this condition, the flavon VEV in eq. (2.11) requires m 2 S ∼ O 2 v 2 s . It is noted that the Higgs potential is approximately given by the flavon potential of eq. (2.10) since the flavon VEV is supposed to be much larger than the Higgs ones. Thus, the depth of the EW minimum is approximately given by We always choose a solution with v s > 0 from the two minimum satisfying eq. (2.11). These features are independent of m. We discuss the stability of the EW vacuum for m = 1 and 2 separately.
As discussed in section 2.2, we have k = 0 and Y b ∼ 1 in this case. Thus a large tan β is required to explain the top to bottom quark mass ratio. As a result, −m 2 is required in the EW vacuum. The potential along the H 0 u direction with H 0 d = S = 0 is given by

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This potential always has the minimum if m 2 Hu < 0 as required to realize the EW minimum in a large tan β regime. The depth of this minimum is given by This minimum is deeper than the EW minimum by O 2 . Thus the EW minimum is expected to be unstable for m = 1. Hereafter, we do not consider this case and focus on the case with m = 2.
The minimum along the H 0 u -direction is shallower than the EW minimum by O 2 . In addition, there may be deeper minimum along the so-called D-flat and/or F-flat directions [56]. We parametrize a direction φ in the Higgs potential as The potential along the φ direction is given by where The couplings for |S| 4 and |S| 6 terms are always positive real and the potential is always bounded from below except for a direction α = 1 and γ = 0. Assuming all the parameters are real, the minimum of this potential is given by

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This minimum is absent if the right-hand side is negative or complex. In general, minimums tend to be appear for small values of λ φ and κ φ . Note that λ φ vanishes for α = 1 and κ φ vanishes for γ = 0. For γ = 0, the scalar potential is independent of the flavon. At least one minimum exists along the H 0 u direction, and its depth is shallower than the EW vacuum due to the suppression by as already stated above. Along the D-flat direction with α = 1, λ φ is also vanishing. Then quadratic term should be positive, Here, we used eq. (2.15).
For α = 1 and γ = 0, only λ φ vanishes while κ φ = 0. Since a large positive m 2 φ will prevent an exotic minimum, let us parametrize For simplicity, we assume c d here, and will discuss the validity of this assumption later. From the EW minimum condition eq. (2.15), we have a relation of A H , the inside of the square root of eq. (3.10) should be negative to prevent a minimum along this direction. This requirement leads to the upper bound on c d : This translates to an upper bound on the CP-odd Higgs boson mass m A (see section 3.3 and appendix A for the definition) through the condition for the realistic EW symmetry breaking. With eq. (3.13), our assumption c d is satisfied for tan 2 β 1. Note that tan β ∼ O (1) is required to obtain the realistic top to bottom quark mass ratio for m = 2. We will numerically study the scalar potential with A H ∼ O ( v s ). Figure 1 shows the parameter space of (tan β, A H ) (left) and (m a , m A ) (right), indicating a region where the EW minimum is deeper than the other vacuum. Here, the parameters are chosen to be v s = 1.0 TeV and c N = c m = 1. There is no minimum along the F/D-flat direction in green region. In the yellow region, the minimum exists along F/D-flat direction but it is shallower than the EW minimum, while in a red region the potential minimum along F/D-flat direction is deeper than the EW one. The D-flat direction becomes unbounded from below in the brown region. The flavon mass becomes tachyonic, and then the point satisfying the EW condition is not a minimum in the gray region. Altogether, the green and yellow regions have the stable EW minimum. The wider parameter space is allowed with a larger tan β, whereas the top to bottom quark mass ratio requires tan β to be O (1). In our analysis, we take tan β = 5. In this case, the upper bound on the CP-odd Higgs boson is about 4 TeV for v s = 1 TeV as shown in the right-panel. The limits on A H or m A will be relaxed for a larger c m due to a larger coupling of κ φ . JHEP03(2020)129

Neutralino mass and dark matter physics
We will discuss DM physics under an assumption that the neutralino LSP is produced by the thermal freeze-out mechanism and they are not diluted after they are decoupled from the thermal bath. If stable LSP flavinos are produced thermally, they will be overproduced owing to a small cross section. Thus, flavino LSP will not be considered in this paper.
First, let us consider cases in which the Higgsino is the LSP and is lighter than the flavino. For N = 4 and m = 2, the Higgsino and flavino masses are approximately given by where the mixing induced by the Higgs VEVs are neglected. Thus, mH mS can be realized when c m 6c N . The Higgsino can be identified as the DM particle as far as its mass is lighter than about 1.1 TeV, so that the LSP does not over-close the universe [94,95]. The Higgsino mass should be in a range, 90 GeV µ eff 1.1 TeV, (3.15) where the lower bound comes from the LEP experiment [96]. With assuming µ eff ∼ 1 TeV, the flavon VEV v s is expected to be O (100 TeV). The direct detection rate will be suppressed as far as the EW gauginos are much heavier than the Higgsino masses [97]. This type of mass spectra, where Higgsinos are much lighter than other sparticles, is the so-called natural SUSY. This would be obtained in Non-Universal Gaugino Mass (NUGM) scenario [98][99][100][101][102][103] or Non-Universal Higgs Mass scenario [104][105][106][107][108]. In particular, the NUGM scenario with relatively heavy wino mass is interesting because the relatively large m 2 H d and small m 2 Hu are realized simultaneously as a result of the renormalization group effects [109,110]. This pattern of Higgs soft masses are consistent with the condition for the JHEP03(2020)129 stable EW minimum discussed in the previous subsection. This feature was pointed out in the Z 3 invariant NMSSM [111]. The NUGM scenario is realized in GUT models [112][113][114][115][116][117][118][119] as well as the so-called mirage mediation [120][121][122][123][124][125]. The phenomenology of the mirage mediation in the NMSSM is discussed in refs. [126][127][128].
Next, we shall consider cases in which the wino is the LSP for avoiding flavino overproduction. As the flavino mass can be comparable to the Higgsino mass, the flavino is naturally heavier than O (100 GeV), so that the charged Higgsinos are heavier than the LEP bound. Even if the flavino is lighter than 100 GeV, its relic density can be lower than the observed value of DM only in restricted parameter space where the s-channel process is enhanced by the resonance or co-annihilation works due to degeneracies with some other particles [129,130]. A easier way to accommodate with the DM density may be that the wino lighter than about 2.7 TeV becomes the LSP [95,[131][132][133]. The hierarchy of the neutralinos are O (1 TeV) ∼ M 2 < M 1 mS, µ eff , so that the direct detection rate is suppressed by the heavy Higgsino mass of µ eff 10 TeV. The flavon VEV is expected to be O (1 PeV) in this case. This type of mass spectrum, where gauginos are much lighter than other sparticles, is the so-called mini-split SUSY/pure gravity mediation scenario [134][135][136][137]. This spectrum of SUSY particles would be realized by the anomaly mediation [138,139] in which gaugino masses are suppressed by the loop factor compared with the soft scalar masses.

Yukawa interactions in mass basis
We shall consider couplings of the scalars to the SM fermions in cases with k = 1 (hence m = 2), in which Y d,e ∝ 1 and W S 2 H u H d /Λ. In the gauge basis, the Higgs doublets are coupled to the SM fermions via the Yukawa couplings where the Yukawa matrices are defined in eq. (2.21). The flavons are coupled to the SM fermions as These are obtained by differentiating the usual Yukawa couplings with respect to S. Hence the coupling matrices are given by where v f = v u , v d for the up-and down-type fermions, respectively. Note that Γ u 23 and Γ u 33 are vanishing with η u 23 = η u 33 = 0. The flavon Yukawa couplings Γ f are more suppressed by v H /v s than those for the Higgs doublets. In addition, the flavino has Yukawa type interactions, wheref R i ,f L j are sfermions.

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We will rewrite these interactions in the mass basis. The mass basis of the fermions, f L ,f R (f = u, d, e), are defined as There are mixing between the Higgs bosons and flavon. The mass basis of the scalars are defined as where h is the SM Higgs boson and G is a Nambu-Goldstone boson. The rotation matrices R S , R P diagonalizes the Higgs mass matrices as The Higgs Yukawa couplingŶ f is diagonalized in this basis, but flavon Yukawa couplinĝ Γ f is not. The latter have the following textures: Since Γ u 23 = Γ u 33 = 0, these elements inΓ u are obtained via mixing matrix for diagonalization. For v s ∼ O (10 TeV) realizing the heavy charged Higgsinos,Γ f couplings are at most O 10 −4 . Some off-diagonal elements are more suppressed than the Higgs Yukawa coupling in the gauge basis, since the some parts of the flavon couplings are aligned with the Higgs couplings, especially for the lower two rows in the Yukawa matrix of the charged leptons. This feature extremely suppresses the lepton flavor violation processes. 10 We finally write the Yukawa interactions in the mass basis between the SM fermions and scalars as

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The Yukawa coupling of h is given bŷ where we take b = 1 for f = d, e and b = 2 for f = u. We have used [R S ] 3b O (v f /v s ) due to the hierarchical structure of the Higgs boson matrix as shown in appendix A. Those of the CP-even and CP-odd flavon couplings are given bŷ Flavor violating couplings of the SM Higgs boson is strongly suppressed by v 2 f /v 2 s 10 −4 and will be negligible sinceŶ f is diagonal. The flavor violating couplings of H, A and H ± are also expected to be tiny similarly to h. With respect to σ and a, both terms proportional toΓ f andŶ f contribute to the O (v H /v s ) couplings, but only Γ f have non-zero off-diagonal elements in the mass basis. Altogether, we find where ρ f (f = u, d, e) are O (1) coefficients and ϕ denotes both σ and a. It is noted that att coupling is more suppressed by 1/ tan 2 β against that of σtt owing to the difference of scalar mixing matrix. See appendix A for more detailed discussions for the scalar mixing.

Higgs physics
We shall discuss Higgs decay modes. Note that a small tan β is required from the anomalyfree charge assignment and the vacuum stability in this case. The SM Higgs boson mass matrix has similar structure as in the MSSM, since the mixing with the flavon is suppressed by . The contribution from mixing with the flavon to the SM-like Higgs boson mass squared is estimated as 2 v 2 H ∼ 10 GeV 2 . Hence, the effect is less than O (0.1 %) of the that from the D-term potential ∼ m 2 Z cos 2 2β and does not give significant effects. Depending on tan β, the top squark mass has an upper bound to be consistent with the 125 GeV Higgs boson mass. The upper bound is typically 100 (10 4 ) TeV for tan β = 4 (2) [140]. This upper bound becomes tighter if there is a sizable mixing between top squarks. This upper bound is consistent with a typical value of the soft mass of the down-type Higgs boson, In our numerical analysis, we add a typical size of loop corrections, ∆m 22 = (90 GeV) 2 , to M 2 S,22 , which is the coefficient of h 2 u in the scalar potential (see appendix A), by hand in order to explain m h ∼ 125 GeV. This does not give significant effects to phenomenology other than the Higgs boson mass itself due to the suppressed mixing.
The SM Higgs boson can decay to a pair of the CP-odd flavons if 2m a < m h . The relevant trilinear coupling between the SM Higgs boson and CP-odd flavons is given by

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Neglecting the flavon mass, the branching fraction is given by where Γ h is the decay width of the SM Higgs. As discussed later, the CP-odd flavon with m a m t decays to bb and τ τ with about 80 % and 20 % branching fractions, respectively. We may have 4b and/or 2b2τ signals from the Higgs boson decays, but these are much smaller than the experimental sensitivity [141]. As discussed in section 3.1, the CP-odd Higgs boson should be lighter than about 4 TeV for cases in which the Higgsino is the LSP with mH ∼ v s 1.1 TeV, so that the EW vacuum is stable. Since the Higgs sector is similar to the MSSM, the CP-even Higgs H and charged Higgs H ± have almost same masses as the CP-odd Higgs boson A. The dominant decay mode of the neutral Higgs bosons, namely H and A, will be a pair of top quarks because tan β ∼ O (1) is required. That of the charged Higgs H ± is a top quark and a bottom quark. In other words, the branching fractions to the leptonic modes, which are more strongly constrained [142][143][144][145], are suppressed owing to a small tan β. There are substantial limits from the current searches for heavy Higgs bosons decaying to top quark at the LHC only if tan β 1 and the top Yukawa coupling is enhanced [146][147][148].

Flavon physics
In this subsection, we shall discuss flavor violations mediated only by the flavons of σ and a, and their decay modes. The effects from the other particles will be enough suppressed if their masses are heavier than O (10 TeV). This model is more predictive than conventional flavon models due to the direct correlation between the Higgs potential and DM physics if Higgsino is the LSP. The flavon VEV controls not only the Yukawa hierarchies which include flavon couplings to the fermions but also the Higgs mixing to the flavon and DM mass. Hence, the VEV can be determined by DM physics. Phenomenology of light flavon is discussed in refs. [149][150][151][152][153][154][155]. In general, the light flavons are accessible in flavor violating processes, such as K-K mixing, µ → eγ and µ → e conversion [149]. The top physics is also relevant because of its large Yukawa coupling. In particular, a sizable flavon coupling to tc is predicted as O(v u /v s ) and may provide good signals at collider experiments [149,150,153]. Significant differences from the ordinal FN mechanism is that is assumed to be about 10 −2 which is smaller by one order of magnitude than the usual value ∼ 0.2. In addition, some flavor violating couplings of the flavons, especially to charged leptons, are suppressed by the alignment with the Higgs Yukawa couplings.

Lepton flavor violation
We will focus on flavor violating processes in the lepton sector. The branching fraction of Lepton Flavor Violation (LFV) decays of i → j γ is given by [156], It is noted that σ R is obtained by formally replacing λ e ij → λ e * ji . Here, Γ i is a width of lepton i . As m σ m a , with neglecting contributions from the CP-even flavon we estimate  (3.35) Thus the sparticle contributions are also far below the detectable level when only the Yukawa couplingsΓ e cause flavor violation. The µ-e conversion process in nuclei induced by flavons might be detectable [149]. The conversion rate is given by [157,158], Table 4. Numerical values of the hadronic matrix elements at µ = 1 TeV [167,168].
where p and n denote a proton and a neutron respectively, and Those for neutron are obtained by formally replacing p → n. Here, only the scalar interactions are considered since the one-loop corrections to the dipole operator will be negligibly small as deduced from discussions in µ → eγ. In this model, the coefficients C q SR is given by Following ref. [149], we used the values for scalar form factors f p,n Sq calculated in refs. [159,160] based on the lattice result [161] and the overlap integrals S p,n [157]. We compare with the current limit [162] at SINDRUM II experiment for a gold target, and future limit at the DeeMe [163], COMET [164] and Mu2e [165] experiments for an aluminum target, Br (µ → e) Au(Al) = Γ conv Γ capt < 7 × 10 −13 (6 × 10 −17 ), (3.40) where Γ capt = 13.07 and 0.7054 [×10 6 · s −1 ] in gold and aluminum [157,166], respectively. We will show the current and expected limits from µ → e conversion in figures 3 and 4 as below.

Quark flavor violation
We shall focus on flavor violating processes in the quark sector. Flavor violating effects induced by scalar fields are summarized in ref. [169]. The flavons would affect also to the neutral meson mixing. For the K-K, B d -B d mixing, the relevant observables are defined as where κ = 0.94 ± 0.02, φ = (43.51 ± 0.05) • [170,171] and ∆M K = 0.005293 ps −1 [172]. The off-diagonal matrix elements are given by Here, O V LL 1 (µ) and O LR 2 (µ) are the values of hadronic matrices, where the renormalization scale µ is fixed at 1 TeV in our analysis. The QCD corrections accompanied with α s are calculated in ref. [173]. We employed the same constant values of the SM contributions and the hadronic matrix elements as in refs. [167,168]. Their values relevant to our analysis are shown in table 4.
To estimate flavon contributions, we define ratios of new physics to the SM values as .

(3.46)
In our cases, these are estimated as Experiments measure K so precisely that the error is dominated by the theoretical ones, such as determination of hadron matrix elements, CKM matrix elements in the SM. The error bar is about 10 %. The limits from B d -B d mixing give similar bound as K . We also checked that a constraint from D-D mixing is weaker, and the B s -B s mixing, leptonic decays of B s → µµ and K L → µµ give no significant constraints.

Collider physics
We shall discuss collider physics associated with flavons. Figure  Since the mixing of the CP-odd flavon to the Higgs doublets are more suppressed by 1/ tan 2 β, the CP-odd flavon dominantly decays to a pair of fermions through the Yukawa couplingsλ f,a . In this sense, the CP-even flavon is similar to the Higgs boson due to a large mixing, while the CP-odd flavon seems to be a conventional flavon. The CP-odd flavon dominantly decays to tc for m a m t , while it decays to bb and τ τ for m a m t . It is noted that Γ(a → tc)/Γ(a → tt) ∼ tan 2 β, hence a → tt can be the main decay mode for a smaller tan β. The CP-even flavon dominantly decays to a pair of EW gauge bosons as far as these are kinematically allowed. For m σ 2m W , there is a substantial flavon mixing to the SM Higgs boson and the decay modes of σ will be similar to that of the SM Higgs boson. We could find signals of the mixing with the SM Higgs boson, but this happens only if v s ∼ m h or the first two terms in eq. (2.12) are canceled out. Constraints on a sizable flavon mixing to the SM Higgs boson are studied in ref. [151]. The LHC searches with 100 fb  As pointed out in ref. [150], a flavor violating decay of top quark, t → ϕc (ϕ = σ, a) will be detectable at collider experiments for m ϕ m t . Such a branching fraction is given by where the charm mass is neglected. Γ t is the decay width of the top quark. In the second equality, the decay t → σc and the CP-odd flavon mass are neglected. The future sensitivity at 100 TeV hadron collider is 2.2 × 10 −6 [149,174]. The flavor changing coupling also predicts same-sign top signal, pp → ta → ttc, but this is not accessible when v s 2.0 TeV [149] in order to realize µ eff 90 GeV.  appendix C are used. The white region is allowed by current experiments. In the dark gray region, the Higgsino is lighter than the experiments bound ∼ 90 GeV. In the light gray region, the Higgsino is too heavy and its relic density will over-close the universe. In the wino LSP case, larger VEV v s is allowed. The CP-even flavon is tachyonic in brown region.

Numerical result
In the red region, |R K | > 0.1 and the flavon contributes to K more than 10% against the SM contribution. It is noted that there exists the red region also near the region of m 2 σ < 0, where the CP-even flavon is very light. Such a region is very narrow to be seen. A light CP-even flavon with m σ O ( v s ) is owing to a cancellation between the two terms in eq. (2.12). The dashed lines show the CP-even flavon masses and the dot dashed line indicates the CP-odd Higgs mass. The blue dashed line shows Br (t → ac) = 10 −7 . There is no parameter space where Br (t → ac) is larger than the future sensitivity at the 100 TeV collider. Thus, vast parameter space will not be constrained by measurements of the processes induced by the flavons. below the blue line will be covered by the future 100 TeV collider. Furthermore, future measurements for µ → e conversion will probe the region lower than the yellow line. Thus the wide parameter space will be tested by the future experiments in this case.

Summary and discussion
In this paper, we proposed a model with the Froggatt-Nielsen mechanism controlled by a Z F N symmetry, in which a flavon field explains not only flavor hierarchy but also the size of Higgsino mass as a solution for the µ-problem in the MSSM. The Higgsino is a well motivated candidate for the DM, since the thermal relic is explained consistently with the null result in direct detections. Furthermore, the abelian flavor Z F N symmetry for the FN mechanism also regulates a structure of the Higgs and flavon potential. Thus the origin of fermion mass hierarchy is closely related to DM physics and Higgs physics.
We found charge assignments of the discrete flavor symmetry, which explains the fermion mass hierarchy and does not have anomalies in the non-abelian gauge groups of the SM in a case of N = 4. Together with the condition to prevent the existence of an exotic minimum deeper than the EW vacuum, the power of m = 2 in the superpotential W (S m /Λ m−1 )H u H d is uniquely determined. As a consequence, tan β ∼ O (1) is required to explain the observed bottom to top quark mass ratio through the anomaly conditions. Our analysis for the Higgs potential shows that the realistic EW vacuum can be realized even if the flavon direction has only cutoff suppressed couplings. It is also interesting that CP-odd Higgs boson mass have the upper bounds, so that the EW vacuum is the deepest minimum.
The large portion of parameter space is allowed by the experiments. In the Higgsino LSP case, the flavon VEV is constrained from above to explain the Higgsino relic density, and also restricted from below to be consistent with the collider bound on the chargino. In the wino LSP case, the upper bound on v s is relaxed. The flavon VEV is related also to the Higgs boson and flavon masses. Altogether, there is a window in the parameter space consistent with DM particle and EW vacuum. The flavor constraints are not severe because the flavor violating couplings are suppressed by a large flavon VEV, v s 10 TeV. Only the restricted parameter space where there exist light flavons or large Yukawa couplings will be covered by future experiments.
We shall give comments about larger discrete flavor symmetry, namely N > 4. According to eq. (2.22), the popular choice of ∼ 0.22 is realized when N = 9. Of course, choices of O (1) coefficients can change the relation between the value of and top to up quark mass ratio. Hence, ≈ 0.22 can be obtained for the realistic mass hierarchies when N ≥ 6, while a smaller = O(0.1) may be found for N = 5. For larger N , there would be more ambiguities of textures in the Yukawa matrices and the discussions about the charge assignments would not be as rigid as the case of N = 4 studied in this paper. We may find also a variety of choices of charge assignments and O (1) coefficients consistent with the realistic fermion hierarchies as well as anomaly cancellation conditions for larger N . In these cases, the powers of Higgs to flavon coupling, m, may not be uniquely fixed because of the ambiguities. A relation between N and m will significantly change the Higgs and JHEP03(2020)129 DM physics. For example, the Higgsino-like DM is expected for m − 1 > N − 3, while the singlino (flavino)-like DM is expected for m − 1 < N − 3. For m − 1 = N − 3, whether the Higgsino or the singlino become DM depends on a choice of O(1) coefficients. The qualitative features in flavor physics will be similar for larger N cases, and flavor violating processes are strongly suppressed by the Yukawa couplings. A specific feature only in N = 4 would be the alignment of the Higgs and flavon Yukawa matrices owing to the non-hierarchical texture of Y e for muon and tau as shown in eq. (2.25). This strongly suppresses the LFV couplings of the flavon as in eq. (3.28). The LFV processes would be more relevant for larger N .
In this paper, we do not discuss flavor violations induced by SUSY breaking. Since we have mentioned about the flavor structure of the fermions, we may be able to address those in soft SUSY breaking. A choice of discrete charge determines also hierarchy in the soft mass. For instance, the soft masses of the right-handed sfermions between first and second generation,ũ † 1ũ 2 ,d † 1d 2 ,ẽ † 1ẽ 2 , would be suppressed by while those of the lefthanded sfermions,Q † 1Q 2 ,L † 1L 2 , would not be in our model. Flavor violation from SUSY breaking as well as higher dimensional operators from the Kähler potential may open new possibilities to probe this model as discussed in appendix B. This is left as future work.

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The CP-even Higgs mass matrix is given by The CP-odd Higgs mass matrix is given by The CP-odd doublet Higgs mass squared is positive if µ eff B eff > 0. These matrices are approximately diagonalized by After the rotation, the CP-even matrixM 2 S := R 0T S M 2 S R 0 S becomes M 2 S,11 = 2g 2 cos 2 2β + λ 2 eff sin 2 2β v 2 H , (A.14)  . (A.20) Similarly, the mixing matrix for the CP-odd mass matrix is given by In the case of N = 4, m = 2, The widths of the flavon decays are given by where κ Z = 1/2 and κ W = 1 for V = Z, W . Here, ϕ, φ = h i , a i and N f c = 3 (1) for quarks (leptons). The trilinear coupling can be obtained by

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where h i = (h, H, σ) and a i = (A, a). Here, | min means that the fields should be replaced by their VEVs after differentiations. When the mixing between the Higgs bosons only via R 0 S and R 0 P are taken into account, the relevant trilinear couplings are given by where η H := v H /Λ. The formulas for the loop-induced decays can be found in e.g. ref. [175].

B Higher dimensional operators in Kähler potential
We discuss whether our model is modified by possible corrections from higher dimensional operators in the Kähler potential for N = 4 with W S 4 /Λ − (S m /Λ m−1 )H u H d . Throughout this paper, sfermions and gauginos are assumed to be heavier than 10 TeV and irrelevant to phenomenology. Our conclusion for m = 1 is not changed by the higher dimensional operators, since the problem is that the minimum along the H u direction, V Hu,min ∼ −O v 4 s , is much deeper than the EW vacuum whose the depth is ∼ −O 2 v 4 s . This can not be changed by higher dimensional operators.
For m = 2, the higher dimensional operators could change the results discussed so far, if they affect the hierarchical structure in the Yukawa matrices and/or the Higgs potential. There exists O S 3 /Λ 3 terms in the Yukawa matrix and also O |S| 2 /Λ 2 terms in the Higgs potential. Hence, it is sufficient to check O Λ −2 and O Λ −1 corrections associated with the flavon in the Kähler potential in order to see whether the hierarchical structures in the Yukawa matrices and Higgs potential are altered by them.
We focus on the Higgs potential first. Because of charge assignment, it is impossible to write O Λ −1 terms in the Kähler potential made only of S, H u and H d . For O Λ −2 terms, K |S| 2 |H u,d | 2 /Λ 2 +|S| 4 /Λ 2 change kinetic terms only by 2 . These do not alter the hierarchical structure in the Higgs-flavon sector. For terms associated with SUSY breaking, we may have K (S † ) 2 H u H d /Λ 2 . This contributes to the Higgsino mass as This size is negligible to that from the superpotential, µ eff ∼ v s . The terms involving the SM fermions in the Kähler potential are given by Here, the gauge supermultiplets are omitted. The gauge interactions of the SM fermions will be obtained by replacing a space time derivative ∂ µ to a gauge covariant one.
The terms proportional toã i j and c ij may contribute to both kinetic terms and Yukawa couplings. The kinetic term corrections also contribute to Yukawa couplings by canonical normalization (see below). As shown below, however, the size of corrections turn out to be F S /Λ 2 ∼ 3 , which is comparable to the smallest Yukawa coupling. For kinetic term correction, we have D α D α S/Λ 2 ∼ F S /Λ 2 . For Yukawa coupling correction, it is noted in corrections from the Kähler potential give the same order contribution as that from the superpotential for N = 4. With a general N , F S /Λ 2 ∼ N −1 with W S N /Λ N −3 will be similarly satisfied, where N −1 is comparable to the smallest Yukawa coupling.
With component fields, the higher dimensional terms are rewritten as Here the fermions are written in two-component Weyl fermions q i ,S. Kinetic terms for the SM fermions are induced in the last line. With the charge assignment in eq. (2.28), the kinetic terms, e.g. C ij Q q † i σ µ ∂ µ q j , have the following texture, The canonically normalized basis is obtained by redefining the fermions as f → f i := P ij f f j , where f = Q, u, d, L, e. Here, P f 's have the same texture as C f 's. We find that this rescaling keeps the texture of the Yukawa matrices in eq. (2.25), changing only O(1) factors per order.
The first line in eq. (B.2) directly gives various four fermi operators which can induce flavor violation. Let us study two observables, namely Br(µ → eee) and K , which may give the strongest limits for lepton and quark sectors, respectively. For simplicity, in the discussions below, we ignore off-diagonal elements of the unitary matrices for diagonalizing the mass matrices. The LFV decay, µ → eee is induced by a operator,

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Here, the fermions in the right-hand side are Dirac fermions. Note that the other operators are more suppressed by Λ due to the Z F 4 symmetry. A branching fraction coming from this operator is given by [176,177] Br(µ → eee) ∼ This can be consistent with the current experimental bound.
As another example, a four fermi operator relevant to the K-K mixing is given by This will give the largest contribution due to the larger hadronic matrix elements of the left-right type operators [178]. The size of a contribution to K is estimated as The value of the hadronic matrix element O LR 1 is shown in table 4. This is bigger than the experimental value K = 2.228 × 10 −3 [172] by one order of magnitude. For this correction to be consistent with the experimental value, Im e Q 2 Q 1 d 1 d 2 ∼ O (0.01) is required unless v s 1 PeV. Hence, for the Higgsino LSP case, e Q 2 Q 1 d 1 d 2 itself should be suppressed or aligned with the phase of the SM contribution for some reasons. For the wino LSP case with v s 1 PeV, this problem can be evaded. At any rate, the origin of this operator depends on UV physics. In addition to the four fermi operators in the first line, the second line of eq. (B.2) also induces four fermi operators by the flavon exchanging. However, these are more suppressed by a ratio of fermion to flavon mass than those from the first line. Hence, the four fermi operator from the first line would be the dominant one. This type of Kähler potential will have various combinations of four fermi operators, and then affect to various flavor violating observables.
In summary, the higher dimensional operators in the Kähler potential will not change the hierarchical structure of the Yukawa matrices and the Higgs potential, but will only affect to O (1) factors per order. On the other hand, these can induce new flavor violating effects and would put strong lower bounds on the flavon VEV v s . However, it depends on how the operators are realized in an UV completion of this model. In the main text of this paper, we studied contributions which always exist as long as the Yukawa hierarchy is explained by the superpotential eq. (2.18). Note that the Yukawa hierarchies and the Higgs potential are not changed even if the cutoff scale is so large that the flavor violating effects are sufficiently suppressed. Potential problems of a large cutoff scale will be a relic density JHEP03(2020)129 of the LSP and the 125 GeV Higgs boson mass. Detailed study of the higher dimensional operators in the Kähler potential is left as our future work and will be discussed together with the UV completion of this model.

C Numerical coefficients
In this paper, we assume = 0.0195764 = (m u /m t ) 1/3 . The singular values of Yukawa matrices Y f (square roots of eigenvalues of Y f † Y f or Y f Y f † ) are fitted to the values at 1 TeV [179].  Open Access. This article is distributed under the terms of the Creative Commons Attribution License (CC-BY 4.0), which permits any use, distribution and reproduction in any medium, provided the original author(s) and source are credited.