Improving Fermion Mass Hierarchy in Grand Gauge-Higgs Unification with Localized Gauge Kinetic Terms

Grand gauge-Higgs unification of five dimensional SU(6) gauge theory on an orbifold S^1/Z_2 with localized gauge kinetic terms is discussed. The Standard model (SM) fermions on one of the boundaries and some massive bulk fermions coupling to the SM fermions on the boundary are introduced.The SM fermion masses including top quark are reproduced by mild tuning the bulk masses and parameters of the localized gauge kinetic terms. Gauge coupling universality is not guaranteed by the presence of the localized gauge kinetic terms and it severely constrains the Higgs vacuum expectation value. Higgs potential analysis shows that the electroweak symmetry breaking occurs by introducing additional bulk fermions in simplified representations.The localized gauge kinetic terms enhance the magnitude of the compactification scale, which helps Higgs boson mass large. Indeed the observed Higgs boson mass 125 GeV is obtained.


Introduction
Gauge-Higgs unification (GHU) [1] is one of the candidates among the physics beyond the Standard Model (SM), which solves the hierarchy problem by identifying the SM Higgs field with one of the extra spatial component of the higher dimensional gauge field. In this scenario, the most appealing feature is that physical observables in Higgs sector are calculable and predictable regardless of its non-renormalizablity. For instance, the quantum corrections to Higgs mass and Higgs potential are known to be finite at one-loop [2] and two-loop [3] thanks to the higher dimensional gauge symmetry. Rich structures of the theory and its phenomenology have been investigated [4][5][6][7][8][9][10][11][12].
The hierarchy problem was originally addressed in grand unified theory (GUT) as a problem how the discrepancy between the GUT scale and the weak scale are kept and stable under quantum corrections. Therefore, the extension of GHU to grand unification is a natural direction to explore. One of the authors discussed a grand gauge-Higgs unification (GGHU) [13], 1 where the five dimensional SU(6) GGHU was considered and the SM fermions were embedded in zero modes of SU(6) multiplets in the bulk. This embedding was very attractive in that it was a minimal matter content without massless exotic fermions absent in the SM, namely a minimal anomaly-free matter content. However, a crucial drawback was found. The down-type Yukawa couplings and the charged lepton Yukawa couplings are not allowed since the left-handed SU(2) L doublets and the right-handed SU(2) L singlets in the down-type sector are embedded into different SU (6) multiplets. As a result, the down-type Yukawa coupling in GHU originated from the gauge coupling cannot be allowed. This feature seems to be generic in GHU as long as the SM fermions are embedded into the bulk fermions. Fortunately, another approach to generate Yukawa coupling in a context of GHU has been known [15,16]. In this approach, the SM fermions are introduced on the boundaries (i.e. fixed point in an orbifold compactification). We also introduce massive bulk fermions, which couple to the SM fermions through the mass terms on the boundary. Integrating out these massive bulk fermions leads to non-local SM fermion masses, which are proportional to the bulk to boundary couplings and exponentially sensitive to their bulk masses. Then, the SM fermion mass hierarchy can be obtained by very mild tuning of bulk masses.
Along this line, we have improved an SU(6) grand GHU model [13] in our previous paper [17], where the SM fermion mass hierarchy except for top quark mass is obtained 1 For earlier attempts and related recent works, see [14] by introducing the SM fermions on the boundary as SU(5) multiplets, the four types of massive bulk fermions in SU(6) multiplets coupling to the SM fermions. Furthermore, we have shown that the electroweak symmetry breaking and an observed Higgs mass can be realized by introducing additional bulk fermions with large dimensional representation.
In GHU, generation of top quark mass is difficult since Yukawa coupling is originally gauge coupling and fermion mass is W boson mass as it stands. The following is well known to overcome this problem that if top quark has a mixing with a four rank tensor representation, an enhancement of group theoretical factor helps a realization of top quark mass [18]. We have attempted to analyze for the cases of three and four rank tensor representations, but an observed top quark mass was not obtained.
As another known approach [16], introducing the localized gauge kinetic terms has enhancement effects on fermion masses. In this paper, we follow this approach. We consider an SU(6) GGHU model in our previous paper [17], where the SM fermions are localized 4D fields on the boundary and the four types of massive bulk fermion.
The localized gauge kinetic terms on the boundaries are added to this model. Once the localized gauge kinetic terms are introduced, the zero mode wave functions of gauge fields are distorted and the gauge coupling universality is not guaranteed. We will find a parameter space where the gauge coupling constant between fermions and a gauge field, the cubic and the quartic self-coupling constants are almost universal. Then, we will show that the fermion mass hierarchy including top quark mass is indeed realized by appropriately choosing the bulk mass parameters and the size of the localized gauge kinetic terms. The correct pattern of electroweak symmetry breaking will be obtained by introducing extra bulk fermions as in our previous paper [17], but their representations become greatly simplified. This paper is organized as follows. In the next section, we briefly describe the gauge and Higgs sectors of our model. In section 3, the localized gauge kinetic terms are introduced and discuss the mass spectrum of gauge fields including their effects. In models with the localized gauge kinetic terms, the gauge coupling universality is not guaranteed.
We will find a parameter space where the gauge couplings are almost universal. In section 4, after briefly explaining the generation mechanism of the SM fermion masses, it is shown that the SM fermion masses including top quark can be reproduced by mild tuning of bulk masses and parameters of the localized gauge kinetic terms. One-loop Higgs potential is calculated and investigated in section 5. We will show that the observed pattern of the electroweak symmetry breaking and Higgs boson mass are realized by introducing some extra bulk fermions. Final section is devoted to our conclusions.

Gauge and Higgs sector of our model
In this section, we briefly explain gauge and Higgs sectors of SU(6) GHU model [13]. We consider a five dimensional (5D) SU (6) gauge theory with an extra space compactified on an orbifold S 1 /Z 2 whose radius and coordinate are denoted by R and y, respectively.
The orbifold has fixed points at y = 0, πR and their Z 2 parities are given as follows. P = diag(+, +, +, +, +, −) at y = 0, We assign the Z 2 parity for the gauge field and the scalar field as A µ (−y) = P A µ (y)P † , A y (−y) = −P A y (y)P † , which implies that their fields have the following parities in components, where (+, −) means that Z 2 parity is even (odd) at y = 0 (y = πR) boundary, for instance.
We note that only the fields with (+, +) parity has a 4D massless zero mode since the wave function takes a form of cos(ny/R) after the Kaluza-Klein (KK) expansion. The Z 2 parity of A µ indicates that SU(6) gauge symmetry is broken to SU(3) C ×SU(2) L ×U(1) Y ×U(1) X by the combination of the symmetry breaking pattern at each boundary, SU(6) → SU(2) × SU(4) at y = πR.
The hypercharge U(1) Y is contained in Georgi-Glashow SU(5) GUT, which is an upperleft 5 × 5 submatrix of 6 × 6 matrix. Thus, we have a relation of the gauge coupling at the unification scale, which will not be so far from the compactification scale. g 3,2,Y are the gauge coupling constants for SU(3) C , SU(2) L , U(1) Y , respectively. This coupling relation implies that the weak mixing angle is the same as that of Georgi-Glashow SU (5) GUT model, sin 2 θ W = 3/8 (θ W :weak mixing angle) at the unification scale.
The SM SU(2) L Higgs doublet field is identified with a part of an extra component of gauge field A y as shown below, We suppose that a vacuum expectation value (VEV) of the Higgs field is taken to be in the 28-th generator of SU (6), A a y = 2α Rg δ a 28 , where g is a 5D SU(6) gauge coupling constant and α is a dimensionless constant. The VEV of Higgs field is given by H =

√ 2α
Rg . We note that the doublet-triplet splitting problem is solved by the orbifolding since the Z 2 parity of the colored Higgs field is (+, −) and it become massive [19]. Some comments on U(1) X gauge symmetry which remains unbroken by orbifolding are given. We first note that the U(1) X is in general anomalous since the massless fermions are only the SM fermions and their U(1) X charge assignments are not anomaly-free (see Table 1 in the next section.). It is easy to cancel the anomaly by adding appropriate number of the SM singlet fermions with some U(1) X charge. In order to break the U(1) X spontaneously, U(1) X charged scalars can be introduced on the y = 0 boundary for instance, and we write down the potential of quadratic and quartic terms like the SM Higgs potential. Then, U(1) X is spontaneously broken by the nonvanishing VEV for the scalars.

Localized gauge kinetic term
As mentioned in the introduction, we introduce localized gauge kinetic terms at y = 0 and y = πR to reproduce a realistic top quark mass. Lagrangian for SU(6) gauge field is where the first term is the gauge kinetic term in the bulk and M, N = 0, 1, 2, 3, 5. The second and the third terms are gauge kinetic terms localized at fixed points and µ, ν = 0, 1, 2, 3. c 1,2 are dimensionless free parameters. The subscript a, b, c denote the gauge indices for SU (6), SU(5) × U(1), SU(2) × SU (4). Note that the localized gauge kinetic terms have only to be invariant under an unbroken symmetry on each fixed point.

Mass spectrum in gauge sector
Because of the presence of localized gauge kinetic terms, the mass spectrum of the SM gauge field becomes very complicated. In particular, their effects for a periodic sector and an anti-periodic sector are different, where the periodic sector means the fields satisfying a condition A(y + πR) = A(y) or those with parity (P, P ′ ) = (+, +), (−, −), while the anti-periodic sector means the fields satisfying a condition A(y + πR) = −A(y) or those with parity (+, −), (−, +). This difference originates from the boundary conditions for wave functions with a definite charge q, f n (y; qα). In a basis where 4D gauge kinetic terms are diagonal, they are found as f n (y + πR; qα) = e 2iπqα f n (y; qα) in periodic sector and f n (y + πR; qα) = e 2iπ(qα+1/2) f n (y; qα) in anti-periodic sector. Moreover, the wave functions in the same basis satisfy where m n (qα) is the KK mass. By solving eq. (9) with the periodic (anti-periodic) boundary conditions, the wave functions and equations determining the KK mass spectrum are obtained [20]. Solving first eq. (9) without boundary terms, we obtain where N n is a normalization factor determined by 2πR 0 |f n | 2 dy = 1. β ± n are integration constants. Continuity conditions at y = 0, πR using the above solution f n (y; qα) lead to β ± n = e ±iπ(qα+ν) sec(π(qα + ν))(πRm n )c 1 ∓ i tan(π(qα + ν)) cot(πRm n ) (11) and eliminating β ± n in the continuity conditions at y = 0, πR, the equations determining the KK mass spectrum 2 1 − c 1 c 2 ξ 2 n sin 2 ξ n + (c 1 + c 2 ) ξ n sin 2ξ n − 2 sin 2 (π(qα + ν)) = 0 is obitaned. ν is 0 (1/2) for the periodic (anti-periodic) sector, and ξ n = πRm n .
Since m 0 is around weak scale (∼ 100 GeV) and 1/R is more than 1 TeV, it is reasonable to suppose ξ 0 ≪ 1. From this observation, we can find an approximate form For instance, the W boson is the gauge boson whose q and ν are 1 and 0, respectively, therefore, the W boson mass m W is given by This relation and m W = 80.3 GeV provide a lower limit of compactification scale 1/R as which indicates that the localized gauge kinetic terms have enhancement effects on the compactification scale. This property is important in our analysis later.

Gauge coupling universality
In the SM, the gauge coupling constant between fermions and a gauge boson, cubic and quartic self-interaction gauge couplings are universal. However, in our model, the universality of 4D gauge coupling is not maintained since the wave functions for massless gauge bosons are distorted from the flat wave functions by the localized gauge kinetic terms and 4D gauge couplings depend on the integral of the wave functions. Therefore, we have to search for a parameter region where the universality is valid. The gauge coupling between the SM fermions localized at y = 0 and a 4D gauge boson (KK zero mode: n = 0) is given by Similarly, the 4D cubic and quartic self-interaction gauge couplings are given by and where Z n (qα) is a wave function renormalization factor for the gauge field with a charge q Z n (qα) = 1 + 2πRc 1 |f n (0; qα)| 2 + 2πRc 2 |f n (πR; qα)| 2 .
In the case of q = 0 corresponding to the photon and the gluon in the SM, eq. (16) is simplified. According to eq. (12), we find m 0 (0) = 0, which implies f 0 (y; 0) = N n (0) = 1 √ 2πR and Z 0 (0) = 1 + 2πRc 1 |f 0 (0; 0)| 2 + 2πRc 2 |f 0 (πR; 0)| 2 = 1 + c 1 + c 2 . Therefore, the gauge coupling universality is valid for q = 0 Then, we have to search for the parameter space where the gauge coupling universality is kept for a nonvanishing charge q. Fig. 1 shows the ratio between g 4 and the gauge coupling constant between the SM fermions and the W boson (q = 1). The free parameters for the localized gauge kinetic terms are taken in the range 0 ≤ c 1 + c 2 ≤ 40. In the cases of α ≤ 0.1 or 1/2 < r = c 1 /(c 1 +c 2 ) < 1, the ratio is almost unity with a good approximation.
As for the cubic and quartic self-interaction gauge coupling constants, the ratio is also almost unity in the same parameters. Since α is restricted to the range α ≤ 0.1 to realize the correct pattern of electroweak symmetry breaking and reproduce the top quark mass which is explained in Section 4.2, we do not consider the case 1/2 < r < 1 hereafter.
After all, the universality of gauge coupling constants can be maintained in the range of α ≤ 0.1.

Fermion masses 4.1 Generation mechanism of the SM fermion masses
The SM quarks and leptons are embedded into SU(5) multiplets localized at y = 0 boundary, which are three sets of decouplet, anti-quintet and singlet Ψ 10 , Ψ 5 * and Ψ 1 . We also introduce four types of bulk fermions Ψ andΨ (referred as "mirror fermions") with opposite Z 2 parities each other shown in Table 1 and constant mass term such as MΨΨ in the bulk to avoid exotic massless fermions. In this setup, we have no massless chiral fermions from the bulk and its mirror fermions. The massless fermions are only the SM fermions and the gauge anomalies for the SM gauge groups are trivially canceled. In order to realize the SM fermion masses, the boundary localized mass terms between the SM fermions localized at y = 0 and the bulk fermions are necessary. To allow such localized mass terms, we have to choose appropriate SU (6) representations for bulk fermions carefully.
Note that the mirror fermions have no coupling to the SM fermions. Table 1 shows the representations for bulk and mirror fermions introduced in our model in addition to the SM fermions, which corresponds to the matter content for one generation [17]. Totally, three copies of them are present in our model.
Lagrangian for the fermions is given by  Table 1: Representation of bulk fermions, the corresponding mirror fermions and SM fermions per a generation. R in R (+,+) means an SU (6) representation of the bulk fermion. r 1,2 in (r 1 , r 2 ) a,b are SU (3), SU (2) representations in the SM, respectively. a, b are U(1) Y , U(1) X charges.
The first line is Lagrangian for the bulk and mirror fermions, and the remaining terms are Lagrangian localized on y = 0 boundary. Note that the subscript "a" denotes the SU (6) representations of the bulk and mirror fermions. The bulk masses between the bulk and the mirror fermions are normalized by πR and expressed by the dimensionless parameter λ a . The last two lines are mixing mass terms between the bulk fermions and the SM fermions. In general, these mixing masses can be free parameters, but we set them to be a common value 2/πR since we would like to avoid unnecessary arbitrary parameters in fitting the data of SM fermion masses. Integrating out y-direction after KK expansion of bulk fermions leads to the following 4D effective Lagrangian.
where Ψ (n) (Ψ (n) ) represents a n-th KK mode of bulk (mirror) fermion, and ψ SM is a SM fermion. P L,R are chiral projection operators and κ L,R are some constants. m n (qα) = n+qα R denotes the sum of the ordinary KK mass and the electroweak symmetry breaking mass proportional to the Higgs VEV α. The charge q is determined by the representation which the fermion belongs to. The mass spectrum of bulk and mirror fermions is totally given by m 2 n = λ πR 2 + m n (qα) 2 . Note that the Lagrangian (22) is illustrated for a particular bulk and mirror fermion as an example.
A comment on the bulk mass spectrum m 2 n = λ πR 2 +m n (qα) 2 is given. This spectrum is not exactly correct in the case that the mixings between the bulk and the boundary fermions are large. Following the argument in [16], we also assume in this paper that the physical mass induced for the boundary fields is much smaller than the masses of the bulk fields. In this case, the effects of the mixing on the spectrum for the bulk fields can be negligible and the spectrum m 2 n = λ πR 2 + m n (qα) 2 is a good approximation.
In order to derive the SM fermion masses, we need the quadratic terms in the effective Lagrangian for the SM fermion. with where x ≡ πRp and In deriving L SM , we simply took the large bulk mass limit λ 2 (πR) 2 ≫ p 2 so that the mixings of the SM fermions with non-zero KK modes become negligibly small.
Integrating out all massive bulk fermions and normalizing the kinetic term to be canonical, we obtain the physical mass for the SM fermions.
where the bare mass and the wave function renormalization factors are The summation in Z a L,R is taken for all the bulk fields contributing to mass m a and its precise expressions are explicitly shown in the next subsection.

Reproducing top quark mass
In our previous paper [17], the up-type quark masses could not be larger than W boson mass, although we had attempted some cases where top quark is embedded in higher rank representations whether the enhancement due to the group theoretical factor for the up-type quark masses can be obtained [17]. As another possibility, it is known that the sizable localized gauge kinetic terms enhances fermion masses, which might be possible to reproduce top quark mass [16]. We consider this possibility in this paper. The bare mass and the wave function renormalization factor for top quark are obtained from eqs. (27) and (28). Fig. 2 shows the bulk mass λ 20 dependence on the physical up-type mass m u phys given by eq. (26). The fermion masses except for top quark are easily reproduced by appropriately choosing the value of λ 20 . To reproduce the top quark mass, however, the maximum value has to be larger than the observed top quark mass 173 GeV. We have studied the behavior of the maximum value in the range from c 1 + c 2 = 0 to 20 and the compactification scale from 1/R = 3 TeV to 10 TeV shown in Fig. 3. It turns out that the conditions where c 1 + c 2 is at least larger than 7 and α ≤ 0.1 is necessary to reproduce the top quark mass.
In the case of α ≪ 0.1, it is reasonable to ignore O(α 2 ) effects and use a ratio of the SM fermion mass and W boson mass eq. (14) to fit the experimental data. Figure 3: Maximum fermion mass in the range of 0 ≤ c 1 + c 2 ≤ 20 (horizontal axis) and 3, 000GeV ≤ 1/R ≤ 10, 000 GeV (vertical axis). The maximum fermion mass is larger (smaller) than 173 GeV in a yellow (blue) region. λ 56 is taken to be 1 (left) and 5 (right).  Table 2: The bulk masses reproducing the SM fermion masses. The parameter c = c 1 + c 2 is taken to be 10 and 20. "1", "2" and "3" means each generation number.
where m u,d,e,ν denote up-type quark, down-type quark, charged lepton, and neutrino masses, respectively. Table 2 lists the bulk masses reproducing all the SM fermion masses.

Higgs effective potential
In this section, we calculate the effective potential for the Higgs field and study whether the electroweak symmetry breaking correctly occurs. Since the Higgs field is originally a gauge field, the potential is generated at one-loop by Coleman-Weinberg mechanism. The potential from the bulk fields is given by where overall signs +(−) stand for fermion (boson) contributions, respectively. g means the spin degrees of freedom of the field running in the loop. The loop momentum p E is taken to be Euclidean.
For bulk fermions and mirror fermions, the mass spectrum is calculated as the following four types of form depending on the Z 2 parity and the bulk mass.
The first (second) half of spectrum correspond to the spectrum of massless (massive) bulk fields. The first and third (the second and the last) types of spectrum correspond to the spectrum of the fields with (anti-)periodic boundary conditions. Using this information, we obtain the corresponding potentials [18]. Table 3: Bulk fermion, mirror fermion and gauge field contributions to Higgs potential. Table 3 lists the various potentials from bulk fermion and mirror fermion contributions.
The coefficients in front of the each potential can be read from the branching rules in the decomposition of the SU(6) representation into SU ( representations listed in Appendix A of our previous paper [17]. Next, we calculate the gauge field loop contributions to the effective potential. As We note that the contributions from the gauge fields are different depending on the boundary conditions. In the periodic sector, the quadratic terms in eq. (8) become where c (c) denotes the ghost (anti-ghost) field. After the KK expansion of the gauge and the ghost fields and diagonalizing 4D gauge kinetic terms, the contribution to Higgs potential can be written down as where gauge g = 3 35 (+,+) 2F + (α) + F + (2α) + 6F + 1/2 (α) + 2F c (α) + F c (α) + 6F c 1/2 (α) where (µν) denots the determinant over 4D spacetime and (nm) denotes the determinant over the KK mode, eq. (41) is computed as follows.
F + (qα) and F c (qα) are contributions from the bulk gauge kinetic terms and the localized gauge kinetic terms, respectively. It is easy to check F c (qα) = 0 at c i = 0.
In the anti-periodic sector, a difference from the periodic sector is the following.
Therefore, substituting c 2 = 0 in eq. (47), we easily obtain contributions from the localized gauge kinetic terms with anti-periodic boundary condition.  Table 4 lists a Higgs potential from gauge field contributions.
Finally, we need the contributions from the SM fermion localized at y = 0 to the Higgs potential. The results are as follows [17].
Ref ( x 2 + λ 2 21 , α) In calculation of the potential from the both bulk and boundary contributions, we have subtracted the α independent part of the potential since it corresponds to the divergent vacuum energy and is irrelevant to the electroweak symmetry breaking.
Total potential is V (α) = V gauge + V bulk + V boundary , where V gauge , V bulk and V boundary are the contributions from the gauge field, the bulk and mirror fermions and the mixing since the bulk mass λ depends on c. Then, the behavior of potential can be considered by changing a ratio r = c 1 /c = 1 − c 2 /c (0 ≤ r ≤ 1). In the case of r = 1, the potential minimum is located at α = 0, in which the electroweak symmetry is unbroken. When r approaches to some nonzero value r 0 which depends on c and much smaller than the unity, the VEV α 0 where the potential minimum is located at discontinuously changes to the value around α 0 = 0.2 from 0. Decreasing r further, α 0 increases until 0.5. As a result, the behavior of the potential is classified into the following two types. (a) α 0 = 0 in the range r 0 < r ≤ 1 and (b) 0.2 ≤ α 0 ≤ 1 in the range 0 ≤ r < r 0 . From the requirement of the gauge coupling universality in Section 3.2, α 0 has to be smaller than 0.1. This implies that the electroweak symmetry is not broken in our model as it stands. Therefore, we need to extend our model and introduce extra fermions to obtain 0 < α 0 < 0.1 for a successful electroweak symmetry breaking.
In this paper, we introduce in the range r 0 < r ≤ 1 three sets of anti-periodic bulk and mirror fermions in the 15 representation of SU (6). In Table 5, the contribution of a set of 15 anti-periodic bulk and mirror fermions to Higgs potential is given and its plot of the potential is shown in Fig. 5. The strategy of introducing such a set of bulk and mirror extra g = 8  fermions is as follows. In the range r 0 < r ≤ 1, since the total potential without extra fermions has a minimum at α = 0, the contribution of extra potential with a minimum at α 0 = 0 is needed. As can be seen from Fig. 5, the contribution from massless 15 fermions to the potential has a minimum at α 0 = 0.5 where the correct electroweak symmetry breaking does not occur. On the other hand, the contribution of massive 15 is suppressed by the bulk mass λ ext , then α 0 in the total potential with extra matter contributions becomes small by increasing λ ext . This opens a possibility that 0 < α 0 < 0.1 required for the correct pattern of the electroweak symmetry breaking can be realized.
The 4D gauge coupling g 4 is determined by Higgs mass, which is obtained from the second derivative of total potential as We will search for a parameter space where g 4 is in the range 0.5 to 0.7. The compactification scale 1/R is defined by eq. (14) 1 The compactification scale can be large by increasing c and decreasing α 0 (or equivalently increasing λ ext ). Fig. 6 shows a contour plot of 4D gauge coupling g 4 for 0.082 < r < 0.087 and 0.690 ≤ λ ext ≤ 0.700. In this plot, α 0 and the compactification scale are also displayed. We can find an allowed region of parameter space in our model, where the gauge coupling universality is kept (α 0 < 0.1), the top quark (c > 7), Higgs boson masses and that the compactification scale in the present paper becomes larger by the effects of the localized gauge kinetic terms, which is compared to the slightly small compactification scale (∼ 0.8 TeV) in our previous paper [17].
We give some comments on the extra bulk fermions which are required for the realistic electroweak symmetry breaking. First, their representations are very simplified. Although the representation is a totally four rank symmetric tensor of SU(6) in our previous paper, the representation is a two rank anti-symmetric tensor in the present analysis. Second, it is natural to ask whether there are any allowed region of parameters in cases with one or two sets of extra fermions in the 15 representation. We have also tried the potential analysis for those cases, but could not obtain an observed Higgs boson mass.

Conclusions
In this paper, we have discussed the fermion mass hierarchy in SU(6) GGHU with localized gauge kinetic terms. The SM fermions are introduced in the SU(5) multiplets on the boundary at y = 0. We also introduced massive bulk fermions in four types of SU (6) representations coupling to the SM fermions on the boundary. Once the localized gauge kinetic terms are present, the zero mode wave functions are distorted and the gauge coupling universality is not guaranteed. We have investigated the constraints where the gauge coupling constant between the SM fermions and a SM gauge field, the cubic and the quartic self-interaction gauge coupling constants are almost universal. It turns out that the gauge coupling universality can be preserved if the dimensionless Higgs VEV is smaller than 0.1 (α 0 < 0.1).
We have shown that the SM fermion masses including top quark can be reproduced by mild tuning of bulk masses and the parameters of the localized gauge kinetic terms. As for top quark mass, we have investigated a dependence of the maximum of fermion mass on a parameter c of the localized gauge kinetic terms. c ∼ O(10) is found for obtaining top quark mass in Fig. 3 We have also calculated additional contributions to one-loop Higgs potentials from the localized gauge kinetic terms. It was found as in our previous paper [17] that the electroweak symmetry breaking does not occur for the fermion matter content mentioned above. To overcome this problem, we have shown that the electroweak symmetry breaking happened by introducing additional three sets of bulk and mirror fermions in 15 representation. Note that the representation was very simplified comparing to our previous case where it was the 126 representation. The effects of localized gauge kinetic terms enhanced the compactification scale, which is compared to the small compactification scale (∼ 0.8 TeV) in our previous paper [17]. This enhancement of the compactification scale also helps Higgs boson mass large. The observed SM Higgs boson mass 125 GeV was indeed obtained in our analysis.
There are issues to be explored in a context of GUT scenario. First one is the gauge coupling unification. It is well known that the gauge coupling running in (flat) extra dimensions is the power dependence on energy scale [21] not the logarithmic one. Therefore, the GUT scale is likely to be very small comparing to the conventional 4D GUT. It is therefore very nontrivial whether the unified SU (6) gauge coupling at the GUT scale is perturbative since many bulk fields are introduced, which might lead to Landau pole below the GUT scale. Second one is proton decay. X, Y gauge boson masses are likely to be extremely light comparing to the conventional GUT scale. Therefore, proton decays very rapidly and our model is immediately excluded by the constraints from the Super Kamiokande data as it stands. Dangerous baryon number violating operators must be forbidden by some symmetry (see [22] for UED case) for the proton stability. If U(1) X is broken to some discrete symmetry which plays an role for it, it would be very interesting.
These issues are remained for our future work.