Electroweak Symmetry Breaking and Mass Spectra in Six-Dimensional Gauge-Higgs Grand Unification

The mass spectra of the standard model particles are reproduced in the $SO(11)$ gauge-Higgs grand unification in the six-dimensional warped space without introducing exotic light fermions. Light neutrino masses are explained by the gauge-Higgs seesaw mechanism. We evaluate the effective potential of the 4d Higgs boson appearing as a fluctuation mode of the Aharonov-Bohm phase $\theta_H$ in the extra-dimensioal space, and show that the dynamical electroweak symmetry breaking takes place with the Higgs boson mass $m_H \sim 125\,$GeV and $\theta_H \sim 0.1$. The Kaluza-Klein mass scale in the fifth dimension is approximately given by $m_{\rm KK} \sim 1.230\,{\rm TeV}/\sin \theta_H$.


Introduction
SO (11) gauge symmetry is reduced to the SM gauge symmetry G SM , which is dynamically broken to SU (3) C × U (1) EM by the Hosotani mechanism.
Zero modes of 5D SO(11) 32 fermions in the bulk are identified with each generation of quarks and leptons. In Ref. [35] it is found that the observed mass spectra of the quarks and leptons can be reproduced, while there appear additional exotic particles having unacceptably small masses.
Recently, to avoid these light exotic particles, SO(11) GHGUT in six-dimensional (6D) hybrid warped space has been proposed in Ref. [38]. For neutrinos a new seesaw mechanism in 6D hybrid warped space has been formulated by using a 5D symplectic Majorana fermion [39], which generalizes the well-known 4D seesaw mechanism [40]. This paper is organized as follows. In Sec. 2, 6D SO(11) GHGUT is introduced. The matter content is specified and the action is given that contains both 6D bulk and 5D brane terms. In Sec. 3, a summary is given for the mass spectrum of the 4D gauge and scalar bosons originating from the 6D SO(11) gauge bosons. In Sec. 4, we derive the mass spectrum of 4D SM fermions. By using these mass spectra, we evaluate the effective potential V eff (θ H ) in Sec. 5 to show that the dynamical EW symmetry breaking takes place and the 4D Higgs boson mass m H = 125.1 GeV is obtained. Section 6 is devoted to a summary and discussions. In the Appendix basics for the KK expansion in 6D warped space are explained.

6D SO(11) GHGUT
We construct an SO(11) gauge-Higgs grand unified model on the six dimensional hybrid warped space introduced in Ref. [38]. The metric of generalized Randall-Sundrum (RS) space [38,41] is given by where e −2σ(y) is a warped factor and η µν = diag(−1, +1, +1, +1). σ(y) satisfies σ(y) = σ(−y) = σ(y +2L 5 ) and σ(y) = k|y| for |y| ≤ L 5 . The fifth dimension with the coordinate y behaves as the EW dimension, whereas the sixth dimension with the coordinate v is 6 ) and behaves as the GUT dimension. Two spacetime points (x µ , y, v) and (x µ , −y, −v) are identified by the Z 2 transformation. As a result the spacetime has the same topology as the orbifold M 4 × T 2 /Z 2 . The spacetime (2.1) solves the Einstein equations with brane tensions at y = 0 and y = L 5 . The bulk region is the anti-de-Sitter space with a negative cosmological constant Λ = −10k 2 . The five-dimensional branes at y = 0 and y = L 5 have the same topology as M 4 × S 1 .
m KK 6 is expected to be a GUT scale while m KK 5 is O(10) TeV so that m KK 6 m KK 5 .
Parity transformations P j (j = 0, 1, 2, 3) around the four fixed points are defined as (x µ , y j + y, v j + v) → (x µ , y j − y, v j − v). Only three of the four parity transformations P j are independent. They satisfy the relation P 3 = P 2 P 0 P 1 = P 1 P 0 P 2 .
• For the 6D SO(11) gauge boson A M , the orbifold BCs are given by where in the SO (11) vector representation, we take the orbifold BCs P 0 = P 1 and P 2 = P 3 as By using (P 0 = P 1 , P 2 = P 3 ), the parity assignment of A µ , A y , and A v are summarized in Table 1.

Bulk terms
The bulk part of the action is given by where S gauge bulk and S fermion bulk are bulk actions of gauge and fermion fields, respectively. The (2.14) We take the following gauge fixing and ghost terms which turns out convenient to discuss mass spectra for fermions in Sec. 4. The bulk part of the bulk fermion action becomes

Action for the singlet brane fermion χ 1
The action for χ β 1 (x, v), which satisfies the symplectic Majorana condition (2.11), is where M ββ is a constant matrix.

Brane masses and interactions
On the UV brane there can be SO(11)-invariant brane interactions among the bulk fermions, the singlet brane fermion, and the brane scalar. We consider where κ's and µ's are coupling constants. Note that Ψ α 11 Ψ β 11 and Ψ β 11 Ψ α 11 are forbidden by the parity assignment of Ψ α 11 and Ψ β 11 shown in Table 3.

Mass terms for gauge bosons on the UV brane
By replacing Φ 32 to its VEV Φ 32 in Eq. (2.20), the mass terms for the 4D components of the SO(11) gauge fields A µ can be read off as where the notation is the same as one in Ref. [35]. We omit them here. All the 31 components of SO(11)/SU (5) obtain large brane masses via Φ 32 = 0. Each mass term changes the 5th dimensional BCs on the UV brane for the corresponding fields. Similarly, from the action (2.20), we find the mass terms for the 6th dimensional component of the These mass terms give masses for the corresponding fields by changing the BCs. We shall see more details in Section 3.

EW Higgs boson and twisted gauge
The orbifold BCs break SO(11) to G PS , and further the nonvanishing VEV of the 5D brane scalar Φ 32 reduces G PS to G SM . The bilinear terms of the action of gauge fields in (2.13) are written down as Here λ, ρ run over 0, 1, 2, 3, 6 so that A λη The remaining neutral component, the observed Higgs boson, also becomes massive by radiative corrections through the Hosotani mechanism as shown below. We note that the components (2, 2, 1) of A v get additional masses of order gR −1 6 by radiative corrections at one loop. The components (3, 1, 1), (1, 3, 1), and (1, 1, 15) of A µ have parity (+, +).
G PS is spontaneously broken to G SM by the nonvanishing VEV of the component (1,2,4) of Φ 32 .
The zero modes of A z are physical degrees of freedom which cannot be gauged away. for the parity (+, +) boundary condition, as It is convenient to work in the twisted gauge discussed, e.g., in Ref. [35]. With the following large gauge transformation Ω(y; α),θ H is transformed toθ H = 0: The new BC matrices P j are given bỹ P j =Ω(0) 2 P j = e 2iθ H T 4,11 P j (j = 0, 2), For fields in the SO(11) vector and spinor representations, the BC matrices are given bỹ in the 1, 2, 3 subspace, −I 6 in the 5, · · · , 10 subspace, (2.38)

Mass spectrum of bosons
In this section we examine the spectrum of gauge fields, particularly for 6th-dimensional n = 0 KK modes (v-independent modes) because we are interested in the mass spectrum of the SM particles and the effective potential V eff (θ H ). In the following we omit the modes which are odd under the 6th dimensional loop translation U 6 = P 2 P 0 = P 3 P 1 = −1. Those The BCs for gauge fields in the absence of the brane terms are given by N : ∂ ∂z A µ = 0 for parity = +, D : A µ = 0 for parity = −, , at z = 1 (y = 0) and z = z L (y = L). In the presence of the brane mass terms |gA µ Φ 32 | 2 on the UV brane, the Neumann BC N is modified to an effective Dirichlet BC D eff for the 5th dimensional zero mode of the SO(11)/SU (5) components of A µ and A v . [35] The equation of motion for A a µ in the y-coordinate is given in the form where the right-hand side involves interaction terms. Suppose that A a µ is parity even, is parity even. By integrating the equation − dy · · · and taking the limit → 0, one finds ∂A a µ /∂y| y= = (g 2 w 2 /4)A a µ | y=0 . In the z coordinate the Neumann condition is changed to the effective Dirichlet condition Note that in the current six-dimensional model the mass dimensions of g and w are [g] = M −1 , [w] = M 3/2 so that ω is dimensionless. When ω = 0, A a µ develops a cusp at y = 0. The lowest mode of A a µ becomes massive, whose mass is O(m KK 5 ). The value of ω depends on the brane mass terms. The BCs for the gauge fields are summarized in Table 5. 11), and (4, 11) components of the SO(11) gauge fields are changed as while the other components remain unchanged. As in Ref. [35], the BCs at z = z L determine wave functions forÃ µ ,Ã z , andÃ v : where C(z; λ) and S(z; λ) are defined in (A.2) and (A.3).
From the BCs for the gauge fields summarized in Table 5, the components (2) SU (5)/G SM , We summarize the formulas determining the mass spectrum of 6th dimensional n = 0 modes of A M , for which the arguments in Ref. [35] remain intact. We use the same notation as in Ref. [35]. A a L M and A a R M (a L , a R = 1, 2, 3) stand for SU (2) L and SU (2) R components of A M , respectively.
(i) (Ã a L µ ,Ã a R µ ,Ã a,11 µ ) (a = 1, 2): For W and W R towers, there are 6th dimensional n = 0 modes, and there are also 5th dimensional zero modes in the absence of brane terms. In the presence of the brane terms in (2.30), their mass spectra of the 5th dimensional modes are given by where ω := g 2 w 2 /4k. Let us denote the average magnitude of C(1; λ) and C (1; λ) by C and C , respectively. When ωC C , the spectrum is determined by W tower: 2S(1; λ)C (1; λ) + λ sin 2 θ H = 0 , Indeed, in typical cases examined below, C/C ∼ 10 6 so that the condition For γ, Z and Z R towers, there are 6th dimensional n = 0 modes, and there are also 5th dimensional zero modes in the absence of brane terms.
In the presence of the brane terms in (2.30), the mass spectra of the 5th dimensional modes are given by The mass of Z boson m Z = m Z (0) is given by The relation for the Z tower in (3.10) can be written, by using cos 2 θ W = 5 8 , as The value sin 2 θ W = 3 8 is valid at the GUT scale. The gauge coupling constants evolve as the energy scale, following the renormalization group equation (RGE).
It is not clear whether sin 2 θ W evolves to the observed value at low energies as there are KK modes which do not respect SU (5) below the GUT scale. We leave solving the RGE for future investigation. When we evaluate the effective potential V eff (θ H ) at the EW scale in Section 5, we adopt the formula (3.12) where the observed value, sin 2 θ W 0.2312, at the EW scale is inserted.
(iii)Ã 4,11 µ : ForÂ 4 tower, there is 6th dimensional n = 0 modes, but there is no 5th dimensional zero modes. The mass spectra of the 5th dimensional modes is given byÂ • A z components.
(i) A ab z (1 ≤ a < b ≤ 3) and A jk z (5 ≤ j < k ≤ 10): There are 6th dimensional n = 0 modes, but there are no 5th dimensional zero modes. Their mass spectrum of the 5th dimensional modes is given by There are 6th dimensional n = 0 modes for A a4 z , A a 11 z (a = 1, 2, 3). There are no 5th dimensional zero modes for A a4 z while there are 5th dimensional zero modes for A a 11 z . Their mass spectra of the 5th dimensional modes are given by (3.18) (iii) A 4,11 z : Higgs tower. There are 6th dimensional n = 0 modes, and there is also a 5th dimensional zero mode. The mass spectrum of the 5th dimensional modes is given by Higgs tower: S(1; λ) = 0. (3.19) (iv) A ak z (a = 1, 2, 3, k = 5, .., 10): There are no 6th dimensional n = 0 modes.
There are 6th dimensional n = 0 modes, but there are no 5th dimensional zero modes. Their mass spectra of the 5th dimensional modes are given by There are no 5th dimensional zero modes for A a4 v while there are 5th dimensional zero modes for A a 11 v (a = 1, 2, 3) in the absence of brane terms. In the presence of the brane terms in (2.31), one find that at z = 1, Thus, their mass spectra of the 5th dimensional modes are given by The average maginitude of S(1; λ) (= S) is much larger than that of S (1; λ) (= S ). In typical cases S/S ∼ 10 4 . Hence the spectrum is determined by There are 6th dimensional n = 0 modes, and there are also 5th dimensional zero modes in the absence of brane terms. In the presence of the brane terms in (2.31), their mass spectra of the 5th dimensional modes are given by (iv) A ak v (a = 1, 2, 3, k = 5, .., 10): There are no 6th dimensional n = 0 modes.

Mass spectrum of fermions
In this section we determine the mass spectrum of quarks and leptons. For up-type quarks, there are no brane interactions on the UV brane (y = 0) as in the 5D SO (11) GHGUT. For down-type quarks, charged leptons, and neutrinos, both 6D bulk mass terms and brane mass terms on the UV brane are important to reproduce the observed mass spectra.
In the following, we will calculate mass spectra for one set of a 6D SO(11) 32 Weyl fermion Ψ α 32 (α = 1, 2 or 3), 6D SO (11) 11 Dirac fermions Ψ β 11 and Ψ β 11 (β = α), and 5D SO(11) singlet χ α 1 , which is identified with one generation of the SM quarks and leptons. We will also discuss mass spectra for the 6D SO(11) 32 Weyl fermion Ψ α=4 32 denoted by Ψ 32 . We will omit the superscripts α and β. We are interested in EW scale physics and we keep only 6th dimensional n = 0 KK modes because the masses of 6th dimensional n = 0 modes are much larger than O(m KK 5 ) as m KK 6 m KK 5 . That is, we will discuss mass spectra for the G PS (2, 1, 4) and (1, 2, 4) components of Ψ 32 , the G PS (1, 1, 6) components of Ψ 11 , the G PS (2, 2, 1) and (1, 1, 1) components of Ψ 11 , and the as c 0 , c 1 , c 2 , and c 0 , respectively. In this paper we assume that c 0 , c 1 , c 0 ≥ 0 while c 2 can be negative. The calculation method employed in the following is the same as the one employed in the 5D SO(11) GHGUT discussed in Ref. [35].
From the action (2.19), we find the equations of motion for up-type quarks with γ 7 6D = +1: Here σ µ = (I 2 , σ), σ µ = (−I 2 , σ), and where ψ,ψ are the pairs defined in (2.38). For up-type quark with γ 7 6D = −1, namely for the top quark, the equations become In the twisted gauge, the equations of motion become and the relation between the twisted gauge and original one is given by Let us check KK mode expansions for u L and u R . Since u L has the parity assignment P 2 P 3 P 0 P 1 = + + + + , we can expand it by using parity even combinations: where Note that u C nL (x, y) and u S nL (x, y) satisfy the same parity property as Φ C n (x, y) and Φ S n (x, y) in (A.13), and f C n (v), f S n (v) are defined in (A.14). On the other hand, u R has the parity assignment P 2 P 3 P 0 P 1 = − − − − so that one can expand it by using parity odd combinations: To investigate physics at energies m KK 6 , we keep only 6th dimensional n = 0 KK modes, and replace u ( ) (4.11) They have the following BCs at (y 0 , y 1 ) = (0, L 5 ); where j = 0, 1. The equations of motion in the twisted gauge (4.6) for the γ 7 6D = +1 fields become It follows that k 2 D + D −ũ+0R = ũ +0R = m 2ũ +0R etc. The BCs at z = z L in the twisted gauge are the same as those in the original gauge: (4.14) In terms of the basis functions C R/L (z; λ, c), S R/L (z; λ, c) given in Appendix B of Ref. [35], which satisfy one can write the mode functions for γ 7 6D = +1 as The BCs at z = 1 in the twisted gauge follow from those in the original gauge: We write the above equations in the matrix form: where we have used short-hand notation such as S 0 L standing for S L (1; λ, c 0 ). In the following we will use the same notation. From det M u = 0, we find the mass formula of the up-type quarks The same formula holds for Ψ 32 with γ 7 6D = −1. The formula (4.22) is the same as in the 5D SO(11) GHGUT in Ref. [35]. For λz L 1, the up-type quark mass spectrum is given Consider Ψ 32 with γ 7 6D = +1. Parity even modes at y = 0 with (P 0 , P 2 ) = (+, +) are d +L , d +R , D +R and D −L . From the action (2.19) and the L m 1 and L m 3 terms in (2.28), one finds the equations of motion for down-type quarks: For 6th dimensional n = 0 KK modes, the equations of motion reduce to To obtain BCs at y = 0, we integrate the above (a), (d), (f ), (g) in the vicinity of y = 0 for parity-odd fields: For parity-even fields, we evaluate the equations of motion at y = + by using the above conditions: (c) ⇒D +ď+L (x, ) = 0, where we used the equations of motion (d) and (g) at y = + .
We recall that in the twisted gauge all fields obey free-field equations in the bulk.
Their eigenmodes are determined by the BCs on the IR brane. The mass spectra can be fixed by the BCs at the UV brane. In the twisted gauge the mode functions are given by (4.28) where f R (x), f L (x) satisfy the relations in (4.16) and m = kλ. The BCs at z = 1 + in the twisted gauge are converted to (4.29) From detK = 0, we find the mass spectrum formula for the down-type quarks: (4.30) The same formula is obtained for Ψ 32 with γ 7 6D = −1. For λz L 1, the down-type quark mass spectrum is given by (4.31) Combining (4.23) and (4.31), one finds For 6th dimensional n = 0 KK modes, the equations of motion for charged leptons reduce To obtain boundary conditions at y = 0, we integrate the above (a), (d), (e), (h) in the vicinity of y = 0 for parity-odd fields: For parity-even fields, we calculate the equations of motion at y = + by using the above conditions: where we used the equations of motion (d) and (h) at y = + .
By using the BCs on the IR brane, the mode functions of charged leptons in the twisted gauge are given by (4.37) As in the case of down-type quarks, the BCs at z = 1 + in the twisted gauge are converted (4.38) From detK = 0, we find the mass spectrum formula for the charged leptons: (4.39) The mass spectrum of charged leptons, for which λz L 1, is given by Here we have made use of (m t /m τ ) 2 , (m c /m µ ) 2 , (m u /m e ) 2 1 which assures |S 0 L S 0 R | sin 2 1 2 θ H , and have assumed that λ 2 µ 11 2 for c 2 > 1 2 and λ 2 (µ 11 z 2c 2 −1 L ) 2 for c 2 < 1 2 so that µ 2 11 (C 2 L ) 2 (S 2 R ) 2 . Combining (4.23) and (4.40), one finds (4.41) It will be seen below that in a typical example for the third generation c 2 = −0.7 and µ 11 C 2 L /S 2 R ∼ 2.6. One comment is in order about the masses of exotic charged leptonsê,ê ,Ê ± with charge Q EM = +1. No zero modes exist forê andê , and all modes have masses larger than 1 2 m KK 6 . On the other handÊ ± have zero modes. They acquire masses through the Hosotani mechanism and brane interactions. We suppose that c 2 < 1 2 and/or µ 11 is sufficiently large so that their lightest masses are O(m KK 5 ).
There are nine fields in the neutral fermion sector which intertwine with each others.
In the twisted gauge, the mode functions are determined by the BCs on the IR brane.
The mass spectra can be fixed by the BCs on the UV brane. For the neutrino sector-1, by using the boundary conditions at z = z L , their mode functions can be written as Here f R/L (x) are chosen in the neutrino sector-1 such that f C (2.11). One can take α ν , α ν , α η to be real. In this case With this identity the first relation in (4.50) can be rewritten aŝ Setting µ 2 = 0, one find that the BCs at z = 1 + in the twisted gauge can be written as From detK ν 1 = 0, we find the mass spectrum formula for the neutrino sector-1: This gives the gauge-Higgs seesaw mechanism for the small neutrino masses. For λz L 1 and kλ |M |, the neutrino mass is given by where m u is given by (4.23). As shown in Ref. [38], the gauge-Higgs seesaw mechanism is characterized by a 3 × 3 mass matrix Next, we consider the neutrino sector-2. The BCs at the IR brane determine the mode functions in the twisted gauge to be (4.58) By making use of (4.47) and (4.48), the BCs at the UV brane are expressed as (4.60) Note that From detK ν 2 = detK ν 2 = 0 the mass spectra for the neutrino sector-2 are found to be In the µ 11 = 0 limit, or in the absence of the brane interactions, (4.62) becomes For large µ 11 , it becomes For µ 2 = 0 the neutrino sector-1 and sector-2 mix through the boundary conditions.
One needs to solve where X = S 2 L S 2 R . In evaluating the effective potential V eff (θ H ), we shall use the approximate formula det K ∼ det K ν 1 det K ν 2 for small µ 2 2 .

Dark fermion
The 6D SO(11) 32 Weyl fermion Ψ 32 , which may be called a dark fermion multiplet, has 6th dimensional n = 0 KK modes. For Ψ 32 one needs not introduce brane interactions.
We consider the action (2.19) for the dark fermion sector Ψ 32 . From the parity assignment shown in Table 2, the mass formula for the dark fermions is found to be For λz L 1, the dark fermion mass is given by As in the case of the 5D SO(11) GHGUT discussed in Ref. [35], the bulk mass parameter of the dark fermions, c 0 , must be relatively small not to become light exotic particles.

Effective potential
We evaluate the Higgs effective potential V eff (θ H ) by using the mass spectrum formulas of the 6th-dimensional n = 0 KK modes of the SO(11) gauge bosons and fermions. The evaluation is done in the same manner as in Ref. [35].
One-loop effective potential from each KK tower is given by [9,45,46] V eff (θ H ) = ± 1 2 where m n (θ H ) is the mass spectrum of the KK tower and we take + and − sign for bosons and fermions, respectively. When the mass spectrum {m n = kλ n } is determined by 1 +Q(λ n )f (θ H ) = 0, (5.1) can be written as where Q(q) =Q(iqz −1 L ). We utilize the mass spectra m n (θ H ) of the SO(11) bulk gauge and fermions obtained in Sec. 3 and 4. Note that only θ H -dependent mass spectra contribute to the θ H -dependent part of V eff (θ H ). It contains the SO(11) bulk gauge field, SO (11) spinor and vector fermion fields.
We summarize those mass spectra. The SO(11) gauge boson contribution is almost the same as in the 5D SO(11) GHGUT. The difference from the 5D case is that the Y boson contributions can be neglected in the current scheme as they have masses of O(m KK 6 ). The relevant mass spectra of SO(11) gauge fields A µ and A z are given, from where m 2 B /k 2 , m B M/k 2 , µ 11 µ 2 has been asuumed for (iv-1) and (iv-2). (v) is the dark fermion mass spectrum. c 0 stands for the bulk mass of the 6D SO(11) spinor bulk Weyl fermion Ψ α=4 32 . We evaluate the effective potential where I α (u) and K β (u) are modified Bessel functions.
The fermion part V fermion eff (θ H ) is evaluated in a similar manner. Following the decomposition in (5.4) and taking into account four degrees of freedom for each Dirac fermion and the color factor 3 for quarks, one can write as where As Q (iv−1) (q) is not a real function, V (iv−1) eff (θ H ) has been converted to the integral involving The Higgs mass m H is determined, at the minimum θ H = θ min H of V eff (θ H ), by . The calculation algorithm is almost the same as in 5D SO(11) GHGUT given in Sec. 5 of Ref. [35]. We are interested in the effective potential V eff (θ H ) at the electroweak scale.
Some of the relations derived in this paper are those at the GUT scale. The value of sin 2 θ W evolves from 3 8 at the GUT scale to the observed value at the electroweak scale by the renormalization group equation (RGE). The RGE analysis of the coupling constants is beyond the scope of the current paper. In this paper we content ourselves to insert the observed value of sin 2 θ W into the formulas of m Z and V eff (θ H ). As input parameters (2) From m Z , (3.12), k and m KK 5 are determined.
(4) Some of the parameters in the brane interactions on the UV brane remain free. We take c 1 = 0, c 2 = −0.7, and M = −10 7 GeV. These three parameters are kept fixed in the evaluation below. We temporarily assign a value for µ 11 = µ 11 . Then, µ 1 , µ 2 , and m B are determined by (4.32), (4.41) and (4.56), or by gauge-Higgs electroweak unification many of the physical quantities depend on the value of θ H , but not on the details of the parameters in the theory. It has been shown that m Z (1) 7 TeV (m KK 5 8.5 TeV) to be consistent with the current LHC data. [15,16] The results for the effective potential V eff (θ H ) in the R ξ=0 gauge are depicted in Fig In In Figure 2, m KK 5 is plotted as a function of θ H . Data points in Table 6 are fitted by m KK 5 ∼ α (sin θ H ) β , α = 1.230 TeV , β = 1.000 (5.14) in the range 0.05 < θ H < 0.30. A similar relation has been found in the SO(5) × U (1) gauge-Higgs EW unification where α = 1.352 TeV and β = 0.786. [11,12] The difference between the two is expected to originate from the different gauge group structure of the models and the different matter content. So far we have found consistent parameter sets only for c 1 = 0 and c 2 ≤ 0.

Summary and discussions
In this paper, we discussed 6D SO(11) GHGUT in the 6D hybrid warped space. The    The phase θ H plays a crucial role in the SO(11) gauge-Higgs grand unification. It is found that the 5D KK mass scale m KK 5 is determined by θ H in a very good approximation by the formula (5.14). Similar relations are expected for the 4D Higgs cubic and quartic couplings as in the SO(5) × U (1) gauge-Higgs EW unification. For m KK 5 (θ H ) we have recognized the difference between the gauge-Higgs EW and grand unification. We need experimental data to find which one is prefered.
We will come back to these issues in the near future.

Acknowledgments
We would like to thank Hisaki Hatanaka for critical reading of the manuscript and valuable remarks. This work was supported in part by Japan Society for the Promotion of Science, Grants-in-Aid for Scientific Research, No. 15K05052.

A Basics for 6D Kaluza-Klein expansion
We present KK mode expansions in 6D hybrid warped space with the metric given in (2.1).

A.1 6D gauge fields
The equations of motion for free gauge fields in the 6D hybrid warped space are given by where P 5 and P z are defined in Eq. (2.32). These P 5 and P z differ from P 4 and P z in the 5d case. Basis mode functions in the z-coordinate in the 6d case are given, except for zero modes, by . They can be expressed as and satisfy These basis functions have been employed in the text. We note that the basis functions for fermions, C L/R (z; λ, c) and S L/R (z; λ, c), remain the same as in the 5d case defined in Ref. [35]. whose solutions are given by the Bessel functions J ν (z) and Y ν (z). Hence a mode function can be written as The orbifold boundary condition for the scalar field φ is given by In the y coordinate φ(x, y, v) = e −νσ(y) Φ(x, y, v) satisfies the same boundary condition as (A.9) and Eq. (A.6) becomes Due caution must be taken as σ (y) = 2k{δ 2L 5 (y) − δ 2L 5 (y − L 5 )}. If φ is parity even in the y coordinate, the Neumann (N ) condition becomes, in the z coordinate (1 ≤ z ≤ z L ), as can be confirmed by integrating (A.10) over y from −ε to +ε (or from L 5 − ε to L 5 + ε).
If φ is parity odd in the y coordinate, it satisfies the Dirichlet (D) condition φ = 0 at z = 1 or z L .

A.3 6D fermion field
Let us consider a free 6D Weyl fermion Ψ(x, y, v) with γ 7 6D = +1: Ψ(x, y, v) = e Its action is given by where D ± (c) is defined in (4.5). As an example we consider the case in which the BCs are given by Ψ(x, y j − y, v j − v) = −iγ 5 γ 6 P j Ψ(x, y j + y, v j + v) = γ 7 6D γ 5 4D P j Ψ(x, y j + y, v j + v) . where the f C n (v), f S n (v) are defined in (A.14). ξ C,S n and η C,S n satisfy the BCs ξ C n (y j − y) = + ξ C n (y j + y) , ξ S n (y j − y) = − ξ S n (y j + y) , η C n (y j − y) = + η C n (y j + y) , η S n (y j − y) = − η S n (y j + y) . − m 2 ξ C n ± i ξ S n = 0 , − m 2 η S n ± i η C n = 0 (n ≥ 1). (A.36) The equations for the zero (n = 0) modes in the 6th dimension, ξ C 0 and η S 0 , are reduced to the Bessel equation. In terms of the basis functions C R/L (z; λ, c), S R/L (z; λ, c) given in Appendix B of Ref. [35], solutions satisfying the BCs are given by where a is a normalization constant. There is one massless mode for the right-handed component. For the n = 0 modes the solutions are more involved.
(ii) (P 0 = P 1 , P 2 = P 3 ) = (−, −) In this case mode functions of ξ and η can be written as There are no zero modes in the 6th dimension.