Gauge Coupling Unification in simplified Grand Gauge-Higgs Unification

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 the boundaries and some massive bulk fermions coupling to the SM fermions on the boundary are introduced. Taking the power-law running contributions from the bulk fields into account, perturbative gauge coupling unification is shown to be realized at around $10^{14}$ GeV, which is a few order smaller than the unification scale of four dimensional grand unified theories.


Introduction
Gauge-Higgs unification (GHU) [1] is one of 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 physical observables in Higgs sector are calculable and predictable regardless of its non-renormalizability.
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.
The hierarchy problem originally exists in grand unified theory (GUT) whether 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) [4], 1 where the five dimensional SU (6) GGHU was considered and the SM fermions were embedded into zero modes of SU (6) multiplets in the bulk. This setup was very attractive because of the minimal matter content without massless exotic fermions absent in SM, namely an anomaly-free matter content. However, the down-type Yukawa couplings and the charged lepton Yukawa couplings in GHU originated from the gauge interaction cannot be allowed since the left-handed SU (2) L doublets and the right-handed SU (2) L singlets are embedded into different SU (6) multiplets. This fact seems to be generic in any GHU models as long as the SM fermions are embedded into the bulk fermions. Fortunately, alternative approach to generate Yukawa coupling in a context of GHU has been known [6,7], in which 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 of [4] in [8], where the SM fermion mass hierarchy except for top quark mass was obtained by introducing them 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 1 For earlier attempts and related recent works, see [5] with large dimensional representation. In GHU, generation of top quark mass is nontrivial since Yukawa coupling is originally gauge coupling and fermion mass is at most an order of W boson mass as it stands. As another known approach [6], introducing the localized gauge kinetic terms on the boundary is known to have enhancement effects on fermion masses. In a paper [10], we followed this approach in order to realize the SM fermion mass hierarchy including top quark. Then, we have shown that the fermion mass hierarchy including top quark mass was 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 was obtained by introducing extra bulk fermions as in our paper [10], but their representations have become greatly simplified.
Next central issue is the gauge coupling unification, which should be explored in a context of GUT scenario. It is well known that the gauge coupling running in (flat) large extra dimensions follows the power dependence on energy scale not logarithmic one [11]. 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 were introduced in our models [8,10], which might lead to Landau pole below the GUT scale. In fact, we saw that the perturbative gauge coupling unification cannot be realized because the number of the bulk fermions is too much in the previous setup [10]. Therefore, we had to reduce the number of the bulk fermions to avoid such a problem in our model discussed in our paper [15]. It was shown that this reduction will lead to additional generation mixings in the bulk. Moreover, since we have changed now the bulk fermions couple to the SM fermions on the boundaries in our paper [15], reproducing the SM fermion masses and generation mixings is nontrivial and their study should be reanalyzed. We have shown that the SM fermion masses and mixing can be almost reproduced by mild tuning of bulk masses and the parameters of the localized gauge kinetic terms. Our model [15] was expected to overcome this issue. In fact, we will see in this paper that the perturbative gauge coupling unification can be realized in our setup [15]. The unification scale will be found to be around 10 14 GeV, which is a few order smaller than that of four dimensional grand unified theories. Therefore, our model will be a good starting point for constructing a realistic model of GGHU.
This paper is organized as follows. In the next section, we briefly review the gauge, Higgs and fermion sectors of our model. In section 3, it is shown that the perturbative gauge coupling unification can be realized in our model. Final section is devoted to our conclusions.
2 Review of our model

Gauge and Higgs sector
In this subsection, we briefly explain gauge and Higgs sectors of SU (6) GHU model [15]. We consider a five dimensional (5D) SU (6) gauge theory with an extra space compactified on an orbifold S 1 /Z 2 with the radius R. The orbifold has two fixed points at y = 0, πR where y denotes the fifth coordinate and their Z 2 parities are given as follows. P = diag(+, +, +, +, +, −) at y = 0, The Z 2 parity for the gauge field and the scalar field originated from an extra component of five dimensional gauge field are assigned as A µ (−y) = P A µ (y)P † , A y (−y) = −P A y (y)P † , which implies that SU (6) gauge symmetry is broken to SU ( by the combination of the symmetry breaking pattern at each boundary, The decomposition of the gauge field into the SM gauge group and their β function are shown in Table 1 which we will use for an analysis of the gauge coupling running in section 2. The hypercharge U (1) Y is embedded in Georgi-Glashow SU (5) GUT, where the weak mixing angle is 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 Table 1: Gauge field and its β function. r 1,2 in (r 1 , r 2 ) a are SU (3), SU (2) representations in the SM, respectively. a is U (1) Y charges.
gauge field A y in a following, 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 the Higgs field is given by H = √ 2α Rg . In this setup, the doublet-triplet splitting problem is solved by the orbifolding since the Z 2 parity of the colored Higgs is (+, −) and it becomes massive [12].
After the Higgs field has the VEV, the mass eigenvalues of the 5D bulk fields are given by m n (qα) = n+ν+qα R , where n is KK mode, ν = 0 (1/2) is for a periodic (anti-periodic) boundary condition. q is an integer charge determined by the SU (2) representation to which the field coupling to Higgs field belongs. If the field with coupling to Higgs field belongs to N+1 representation of SU (2) L , the integer charge q is equal to N .

Localized gauge kinetic term
As mentioned in the introduction, we introduce additional localized gauge kinetic terms at y = 0 and y = πR to reproduce a top quark mass. Lagrangian for SU (6) gauge field is where the first term is the bulk gauge kinetic term with 5D space-time indices M, N = 0, 1, 2, 3, 5. The second and the third terms are gauge kinetic terms with 4D space-time indices µ, ν = 0, 1, 2, 3 localized at fixed point. c 1,2 are dimensionless free parameters. The superscript a, b, c denote the gauge indices for SU (6), 1) . Note that the localized gauge kinetic terms have only to be invariant under unbroken symmetries on each fixed point.
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 (anti-)periodic sector means the fields satisfying a condition A(y + πR) = (−)A(y) or the fields with parity (P, P ) = (+, +), (−, −)((+, −), (−, +)). 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, we found them to be 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. The KK mass spectrum of the SM gauge fields are obtained from the equation (6) where m n is the KK mass. The mass spectrum of the gauge field are deformed when c = c 1 + c 2 1. In the case of introducing only one localized term r ≡ c 1 /(c 1 + c 2 ) = 0 or 1, the mass spectrum tends to be shifted as follows.
We also introduced three types of bulk fermions Ψ andΨ (referred as "mirror fermions") with opposite Z 2 parities each other per a generation and constant mass term such as MΨΨ in the bulk to avoid exotic 4D massless fermions. Without these mirror fermions and mass terms, we necessarily have extra exotic 4D massless fermions with the SM charges after an orbifold compactification. In this setup, we have no massless chiral fermions 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.
However, we could not obtain perturbative gauge coupling unification as shown in Fig. 1 because the number of the bulk fermions was too much in the previous setup as mentioned in the introduction. As can be seen from Fig. 1, all of the gauge couplings were not only unified, but also blown up around 10 5∼6 GeV, where the compactification scale is taken to be 10 TeV being a typical scale to realize 125 GeV Higgs mass in GHU scenario. In order to avoid such a problem, we have succeeded in reducing the number of the bulk fermions in   (6) representation of the bulk fermion. r i in r 1 ⊕ r 2 are SU (5) representations.
this model [15], which also reproduces the fermion masses, mixing angles and a CP phase.
task because of the complicated bulk and boundary system. We assume in this paper that the physical mass induced for the boundary fields is much smaller than the masses of the bulk fields [6]. This is reasonable since the compactfication scale and the bulk mass mainly determining the KK mass spectrum of the bulk fields is larger than the mass for the boundary fields whose typical scale is given by the Higgs VEV. 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 [6].

Gauge coupling unification
In ordinary 4D field theories, the gauge couplings g i (i = 1, 2, 3) of the Standard Model are dimensionless. They evolve as the following one-loop renormalization group equation whose solution is given by This is the usual logarithmic running of the gauge couplings. Here α i ≡ g 2 i /4π, the b i are the one-loop beta-function coefficients for the Standard Model gauge group and we have taken the Z-mass M Z ≡ 91.17 GeV as an arbitrary low-energy reference scale.
These gauge couplings also receive corrections in extra dimensions, and we can calculate such corrections in the usual way by evaluating the same one-loop diagrams (particularly the vacuum polarization diagram) as shown in [11]. The full one-loop corrected gauge coupling is given as whereb i are new beta-function coefficients by bulk fermion contributions. µ is a renormalization scale and Λ is the cutoff scale of 5D theory. P(t) denotes the contribution from the bulk fields with KK mass spectrum m n , The results are expressed by the elliptic theta functions θ i (i = 2, 3) U (1) SU (2)  Note that the results are different depending on whether the fields are periodic or antiperiodic as can be seen from Table 6. This means that we should be careful for the periodicity of the fields listed in Table 1, 3, 4, 5, 6 in our analysis. The elliptic theta functions θ i (i = 2, 3) can be approximated to in the case of t/R 2 1 which corresponds to the assumption that both µ and Λ are much larger than R −1 . After substituting this approximation (16) into (12) and evaluating the integral over t, RGE (12) becomes the following expression The third term on the right-hand side shows the power-law dependence of the gauge coupling on the energy scale. Here + (−) inb (+(−)) i shows that its contribution comes from the (anti-)periodic fields. Note that the bulk fermion and the corresponding mirror fermion have the same β function because they have the same periodicity. Asymptotic freedom of gauge couplings can be confirmed by the fact that the beta function for KK modeb (+) +b (−) is negative. It can be calculated by using informations in Table 1 , 3, 4, 5 Therefore, the perturbative gauge coupling unification is expected in this model, which cannot be realized in the previous model [10] due to the large number of the introduced bulk fermions. Fig. 2 shows energy dependences of the gauge couplings and differences between each pair of gauge couplings at c = 80, r = 0 and R −1 = 10 TeV. In this case, the unification scale M G and unification coupling α −1 G (g G ) are identified with the scale where U (1) and SU (2) L couplings are unified and we obtain M G ∼ 2.1 × 10 14 GeV and α −1 G ∼ 4.4 × 10 9 , (g G ∼ 5.3 × 10 −5 ). The difference between the unification coupling and SU (3) , therefore three gauge couplings unify with an accuracy of 10 −10 . Alternatively, assuming the unification of three couplings α G at M G , and evolving SU (3) coupling down to the weak scale by RGE, .08) is found, which is larger (smaller) than the experimental value α 3 ∼ 8.4 (g 3 ∼ 1.2). We also analyze r = 1/2 and r = 1 cases. In the former case, almost the same result as r = 0 case is obtained. In the latter case, the differences are smaller (∼ 10 −11 ), unification scale is larger (M G ∼ 4.1 × 10 15 ) and SU (3) coupling at the weak scale is larger (α 3 ∼ 13.2). We analyze the coupling unification in other parameter cases, (c, r, R −1 ) = (80, 0, 15 TeV), (90, 0, 10 TeV), (90,0,15 TeV) shown in previous paper [15] and the results are shown in Table 7. The unification scale in our model is comparable to that of four-dimensional GUT, since the running of coupling constant in t/R 2 1 region is dominated by the contributions linearly dependent on the energy scale as in Eq. (17) and their beta functions (Eq. 18) are common, then the differences between each pair of the gauge couplings are dominated by the logarithmic terms. In each case shown in Table 7, the theoretical value for SU (3) coupling with one-loop corrections at the weak scale are slightly deviated from the experimental value. However, it is possible that we could analyze two-loop corrections to obtain more accurate unification since the difference between the each pair of the couplings at M G is extremely small.
We further analyze the coupling unification in the case of larger compactification scale, which was not analyzed in the previous paper [15]. Although this case is not realistic since the Higgs mass is likely to be enhanced, the SM fermion masses and mixings can be reproduced in larger compactification scale. The results are shown in Table 8. In the c r  Table 7: The results of gauge coupling unification analysis at r = 0. The unification scale M G and the unification coupling α −1 G are identified with the scale where U (1) and SU (2) L couplings are unified. (3) coupling at weak scale, assuming that three gauge couplings are unified to α −1

Conclusions
In this paper, we have discussed SU (6) GGHU with localized gauge kinetic terms. The SM fermions are introduced on the boundaries. We also introduced massive bulk fermions in three types of SU (6) representations coupling to the SM fermions on the boundaries. The number of them has been reduced in order to achieve perturbative gauge coupling unification which could not be realized in [8,10]. It was shown in this paper that the perturbative gauge coupling unification can be indeed realized in our model [15]. Remarkably, the unification scale in our model was found to be 10 14 GeV, which is a few order smaller than the 4D GUT scale 10 15−16 GeV. This is because the beta functions for introduced bulk fermions are common to each gauge coupling running and the differences between each pair of gauge couplings are dominated by the logarithmic contributions in RGE. Our model turned out to be indeed a good starting point for constructing a realistic model of GGHU.
There is an issue to be explored in a context of GUT scenario, namely, a proton decay.
In large extra dimension models such as GHU discussed in this paper, X, Y gauge boson masses are likely to be light comparing to the conventional GUT scale due to the power law running of the gauge coupling. Therefore, proton decays very rapidly and our model is immediately excluded by the experimental constraints from the Super Kamiokande data as it stands. Possible dangerous baryon number violating operators must be forbidden, for instance, by some symmetry (see [14] for UED case) for the proton stability. If U (1) X in our model is broken to some discrete symmetry which plays its role, it would be very interesting.
It would be also interesting to investigate the main decay mode of the proton decay in our model and give predictions for Hyper Kamiokande experiments.