Toward Realistic Gauge-Higgs Grand Unification

The $SO(11)$ gauge-Higgs grand unification in the Randall-Sundrum warped space is presented. The 4D Higgs field is identified as the zero mode of the fifth dimensional component of the gauge potentials, or as the fluctuation mode of the Aharonov-Bohm phase $\theta_H$ along the fifth dimension. Fermions are introduced in the bulk in the spinor and vector representations of $SO(11)$. $SO(11)$ is broken to $SO(4) \times SO(6)$ by the orbifold boundary conditions, which is broken to $SU(2)_L \times U(1)_Y \times SU(3)_C$ by a brane scalar. Evaluating the effective potential $V_{\rm eff} (\theta_H)$, we show that the electroweak symmetry is dynamically broken to $U(1)_{\rm EM}$. The quark-lepton masses are generated by the Hosotani mechanism and brane interactions, with which the observed mass spectrum is reproduced. The proton decay is forbidden thanks to the new fermion number conservation. It is pointed out that there appear light exotic fermions. The Higgs boson mass is determined with the quark-lepton masses given, which, however, turns out smaller than the observed value.


Introduction
Up to now almost all observational data at low energies are consistent with the stan- Many proposals have been made to overcome these problems. Supersymmetric generalization of the SM is among them. There is an alternative scenario of the gauge-Higgs unification in which the 4D Higgs boson is identified with a part of the extra-dimensional component of gauge fields defined in higher dimensional spacetime. [1]- [4] The Higgs boson, which is massless at the tree level, acquires a finite mass at the quantum level, independent of a cutoff scale and regularization scheme. The SO(5) × U (1) X gauge-Higgs electroweak (EW) unification in the five-dimensional Randall-Sundrum (RS) warped space has been formulated. [5]- [11] It gives almost the same phenomenology at low energies as the SM, provided that the Aharonov-Bohm (AB) phase θ H in the fifth dimension is θ H < ∼ 0.1. In particular, cubic couplings of the Higgs boson with other fields, W , Z, quarks and leptons, are approximately given by the SM couplings multiplied by cos θ H . [7][12]- [15] The corrections to the decay rates H → γγ, Zγ, which take place through one-loop diagrams, turn out finite and small. [11,16] Although infinitely many Kaluza-Klein (KK) excited states of W and top quark contribute, there appears miraculous cancellation among their contributions. In the gauge-Higgs unification the production rate of the Higgs boson at LHC is approximately that in the SM times cos 2 θ H , and the branching fractions of various Higgs decay modes are nearly the same as in the SM. The cubic and quartic self-interactions of the Higgs boson show deviations from those in the SM, which should be checked in future LHC and ILC experiments. Further the gauge-Higgs unification predicts the Z bosons, namely the first KK modes of γ, Z and Z R , around 6 to 8 TeV range with broad widths for θ H = 0.11 to 0.07, which awaits confirmation at 14 TeV LHC in the near future. [17,18] With a viable model of gauge-Higgs EW unification at hand, it is natural and necessary to extend it to gauge-Higgs grand unification to incorporate strong interaction. Since the idea of grand unification was proposed [19], a lot of grand unified theories based on grand unified gauge groups in four and higher dimensional spaces have been discussed. (See, e.g., Refs. [20]- [51] for recent works and Refs. [52,53] for review.) The mere fact of the charge quantization in the quark-lepton spectrum strongly indicates the grand unification.
Such attempt to construct gauge-Higgs grand unification has been made recently. SO (11) gauge-Higgs grand unification in the RS space with fermions in the spinor and vector representations of SO (11) has been proposed. [54,55,56] The model carries over good features of the SO(5) × U (1) X EW unification. In this paper we present detailed analysis of the SO(11) gauge-Higgs grand unification. Particularly we present how to obtain the observed quark-lepton mass spectrum in the combination of the Hosotani mechanism and brane interactions on the Planck brane. It will be shown that the proton decay can be forbidden by the new fermion number conservation.
There have been many proposals of gauge-Higgs grand unification in the literature, but they are not completely satisfactory in the points of the realistic spectrum and the symmetry breaking structure. [57]- [62] In the current model SO (11) symmetry is broken to SO(4) × SO(6) by orbifold boundary conditions, which breaks down to SU (2) L × U (1) Y × SU (3) C by a brane scalar on the Planck brane. Finally SU (2) L × U (1) Y is dynamically broken to U (1) EM by the Hosotani mechanism. The quark-lepton mass spectrum is reproduced. However, unwanted exotic fermions appear. Further elaboration of the scenario is necessary to achieve a completely realistic grand unification model. We note that there have been many advances in the gauge-Higgs unification both in electroweak theory and grand unification. [63]- [69] Mechanism for dynamically selecting orbifold boundary conditions has been explored. [70] The gauge symmetry breaking by the Hosotani mechanism has been examined not only in the continuum theory, but also on the lattice by nonperturbative simulations. [71,72,73] The paper is organized as follows. In Section 2 the SO(11) model is introduced. The symmetry breaking structure and fermion content are explained in detail. The proton stability is also shown. In Section 3 the mass spectrum of gauge fields is determined.
In Section 4 the mass spectrum of fermion fields are determined. With these results the effective potential V eff (θ H ) is evaluated in Section 5. Conclusion and discussions are given in Section 6.

Action and boundary conditions
The model consists of SO(11) gauge fields A M , fermion multiplets in the spinor representation Ψ 32 and in the vector representation Ψ 11 , and a brane scalar field Φ 16 . [54] In each generation of quarks and leptons, one Ψ 32 and two Ψ 11 's are introduced. Φ 16 (x), in the spinor representation of SO (10), is defined on the Planck brane.
The bulk part of the action is given by We adopt the convention {γ A , γ B } = 2η AB , η AB = diag (−1, 1, 1, 1, 1), G M N = e AM e A N , and Ψ = iΨ † γ 0 . Generators T jk = −T kj of SO(11) are summarized in Appendix A in the vectorial and spinorial representations. We adopt the normalization M N T jk , With this normalization the SU (2) L weak gauge coupling constant is given by g w = g/ √ L. The orbifold boundary conditions for the gauge fields are given, in the y-coordinate, by where (y 0 , y 1 ) = (0, L). The ghost fields, c andc, satisfy the same boundary conditions as is given in the vectorial representation by and in the spinorial representation by The SO(11) symmetry is broken down to SO(10) by P 0 at y = 0, and to SO(4) × SO (7) by P 1 at y = L. With these two combined, there remains SO(4) × SO(6) SU (2) L × SU (2) R × SU (4) symmetry, which is further broken to by Φ 16 on the Planck brane as described below. SU (2) L × U (1) Y is dynamically broken to U (1) EM by the Hosotani mechanism.
Fermion fields obey the following boundary conditions; Eigenstates of γ 5 with γ 5 = ±1 correspond to right-and left-handed components in four dimensions. For Ψ a 11 and Ψ a 11 one might impose alternative boundary conditions given by Ψ a 11 (x, y j − y) = γ 5 P vec j Ψ a 11 (x, y j + y) , Ψ a 11 (x, y j − y) = −γ 5 P vec j Ψ a 11 (x, y j + y) . (2.9) It turns out that the model with (2.8) is easier to analyze in reproducing the mass spectrum of quarks and leptons.
On the Planck brane (at y = 0) the brane scalar field Φ 16 has an SO(10) invariant action given by (2.10) Φ 16 develops VEV. Without loss of generality one can take (2.11) On the Planck brane SO(10) symmetry is spontaneously broken to SU (5) by Φ 16 = 0.
With the orbifold boundary condition P 1 , SO(11) symmetry is broken to the SM symmetry To see it more explicitly, we note that mass terms for gauge fields are generated from

Making use of (A.3) and (A.4) for
T sp jk , one obtains L gauge brane mass = −δ(y) In all, 21 components in SO(10)/SU (5) acquire large brane masses by Φ 16 , which effectively alters the Neumann boundary condition at y = 0 to the Dirichlet boundary condition for their low-lying modes (m n gw/ √ L) as will be seen in Section 3. It follows that the SU (5) generators are given, up to normalization, by (i) SU (2) L : 12 components of the gauge fields A µ in the class (iv) have no zero modes by the boundary conditions. This leaves SU It will be seen later that SU (2) L × U (1) Y symmetry is dynamically broken to U (1) EM by the Hosotani mechanism. The AB phase θ H associated with A 4,11 z becomes nontrivial so that A 34 µ picks up an additional mass term. Consequently the surviving massless gauge boson, the photon, is given by where T a R are generators of SU (2) R . The orthogonal componentÃ µ = 1 µ mix with each other for θ H = 0. More rigorous and detailed reasoning is given in Section 3 in the twisted gauge. The U (1) Y gauge boson, B Y µ , is given by The gauge couplings become In other words 4D gauge couplings and the Weinberg angle are given, at the grand unification scale, by The content Ψ 32 , Ψ 11 , Ψ 11 is determined with the EM charge given by (2.14). In the spinorial representation We tabulate the content of Ψ 32 in Table 1. We note that zero modes appear only for particles with the same quantum numbers as quarks and leptons in the SM, but not for anything else.
Only fields of even parity at y = 0 participate in the brane mass-Yukawa interactions. We need to write down SO(10) invariant terms in terms of Ψ a 16L , Ψ a 16R , Ψ a 10R , Ψ a 1L , Ψ a 10L , Ψ a 1R , and Φ 16 . Further we impose the condition that the action be invariant under a global U (1) Six types of brane interactions are allowed.
HereΦ 16 =R Φ * 16 withR defined in (A.7) transforms as 16, and we have employed 32component notation given bŷ In general all coefficients κ and µ in (2.22) have matrix structure in the generation space, which induces flavor mixing. In the present paper we restrict ourselves to diagonal κ and µ.
The total action is given by (2.24)

EW Higgs boson
The orbifold boundary condition (2.5) reduces the SO(11) gauge symmetry to SO(4) × . It is easy to see that terms bilinear in fields in the gauge field part of the action S bulk , (2.3), become For four-dimensional components A µ of SO(10) gauge fields have additional bilinear terms coming from the brane scalar interaction S Φ 16 , (2.10), with Φ 16 = 0. Parity of A µ and A z is summarized in Table 3. Mode functions of A z in the fifth dimension are determined by where boundary conditions at z = 1 and z L are given by (d/dz)(h n /z) = 0 or h n = 0 for parity even or odd fields, respectively. In particular, the zero mode (λ 0 = 0) function is given by The mode function in the y-coordinate satisfiesũ H (−y) = u H (y) =ũ H (y + 2L). We note that The zero modes of A z are physical degrees of freedom, being unable to be gauged away.
They are in A a 11 z (x, z) (a = 1 ∼ 4). In terms of mode functions {h (++) n (z)} for parity (+, +) boundary condition, where the four-component real field, φ a H (x), plays the role of the EW Higgs doublet field in the SM. It will be shown that φ H ≡ φ 4 H dynamically develops vev φ H = 0, breaking the SM symmetry to U (1) EM . φ 1,2,3 H are absorbed by W and Z bosons.

Under a general gauge transformation
The original boundary conditions (2.5) are maintained if and only if P j = P j . Such a class of gauge transformations define the residual gauge invariance. [2,59] There are a class of "large" gauge transformations which transform A y nontrivially. Consider The new boundary condition matrices P j are evaluated, with the aid of {P j , T 4,11 } = 0, to be P j = Ω(y j − y; α)Ω(y j + y; α)P j , that is, P 0 = Ω(−y; α)Ω(y; α)P 0 = Ω(0; 2α)P 0 , The boundary conditions are preserved provided Ω(0; 2α) = 1. As (T sp 4,11 ) 2 = 1 4 I 32 in the spinorial representation, the boundary conditions are preserved provided Aharonov-Bohm (AB) phases along the fifth dimension are defined by phases of eigenvalues ofŴ y (x, y) and the orthonormality relation (2.26) implies that in the vectorial representation so that the relevant phaseθ H (x) is given bŷ .

(2.37)
Under a large gauge transformation satisfying (2.33), H(x) corresponds to the neutral Higgs boson found at LHC. The value of θ H is undetermined at the tree level, but is determined dynamically at the one loop level. The effective potential V eff (θ H ) is evaluated in Section 5.

Twisted gauge
In the vectorial representatioñ (2.42) Components of Ψ 32 split into two groups; The boundary condition matrix becomes It turns out very convenient to evaluate V eff (θ H ) in the twisted gauge.

Proton stability
In the present model of gauge-Higgs grand unification, the proton decay is forbidden. the proton decay p → π 0 e + , for instance, cannot occur. This is contrasted to SU (5) or SO(10) GUT in four dimensions in which proton decay inevitably takes place. The SO (11) gauge-Higgs grand unification provides a natural framework of grand unification in which proton decay is forbidden.
One comment is in order. S and S in Ψ 11 and Ψ 11 are SO(10) singlets. One could introduce Majorana masses such as SS c δ(y) on the Planck brane, which break the N Ψ fermion number. They would give rise to Majorana masses for neutrinos, and at the same time could induce proton decay at higher loops.

SO(11)
(5), (6) SO (10) (4) No. of generators  (6), and SU (5)  3 Spectrum of gauge fields V eff (θ H ) at the one loop level is determined from the mass spectrum of all fields when θ H (x) = θ H . It is convenient to determine the spectrum in the twisted gauge in which θ H (x) = 0. The nontrivial θ H dependence is transferred to the boundary conditions. In this section we determine the spectrum of gauge fields.
In the absence of the brane interactions with Φ 16 in S Φ 16 , (2.10), the boundary conditions for gauge fields are given by at z = 1 (y = 0) and z = z L (y = L). In the presence of S Φ 16 , the brane mass terms (2.12) for A µ are induced, and the Neumann boundary condition N is modified to an effective Dirichlet condition D eff for low-lying KK modes of the twenty-one components of A µ as described below. The boundary conditions for the gauge fields are summarized in Table 4.
where Ω(z) is given by (2.40). One finds All other components are unchanged. At z = z L , θ(z L ) = 0, and A k4 M and A k,11 M always have opposite parity. It follows thatÃ M satisfies the same boundary condition as A M at z = z L . In the bulk (1 < z < z L ) the bilinear part of the action is the same as in the free theory in the twisted gauge. Hence, depending on the BC at z = z L , wave functions for A µ andÃ z are given byÃ where C(z; λ) and S(z; λ) are defined in Appendix B.

A µ components
where interaction terms on the right side involve neither total y derivative nor δ(y). By integrating the equation − dy · · · and taking the limit → 0, one finds that ∂A a R µ /∂y| y= = (g 2 w 2 /4)A a R µ | y=0 . The brane mass gives rise to cusp behavior at y = 0. The boundary Here it is understood that ∂A a R µ /∂z is evaluated at z = 1 + . Expressed in terms of fields in the twisted gauge, (3.5) becomes From the boundary conditions at z L , one can set for each mode. The boundary conditions (3.6) are transformed to where C = C (1; λ) etc.. The spectrum {λ n } is determined by det K = 0, or by For sufficiently large w, the spectrum {m n = kλ n } of low-lying KK modes is approximately determined by the second term in (3.8). This approximation for z L = 10 5 , for instance, is justified for ω > 10 −3 .
Asymptotically the equations determining the spectra of W and W R towers become SC + λ sin 2 θ H = 0 and C = 0, respectively. The mass of W boson, m W = m W (0) , is given by In the twisted gauge they become Adding the first and second equations, one gets Adding the first and fourth equations, one gets The spectrum is determined by det K = 0; C 2C (SC + λ sin 2 θ H ) − ωC(5SC + 4λ sin 2 θ H ) = 0 . (3.14) For sufficiently large w, the spectrum of low-lying KK modes is approximately determined by the second term in (3.14). One finds that The mass of Z boson, m Z = m Z (0) , is given by (v) X-gluons These are six components given by which originally obey (N, N ) boundary conditions. They have brane masses of the form −δ(y)(g 2 w 2 /4)A 2 µ in (2.12) so that boundary conditions at z = 1 become (∂ z − ω)A µ = 0. Consequently the spectrum is determined by C − ωC = 0. For the low-lying KK modes X-gluon tower: C(1; λ) = 0 . (3.20)

(vi) X-bosons
These are six components given by These are six components given by which originally obey (N, D) boundary conditions. They have brane masses of the form −δ(y)(g 2 w 2 /4)A 2 µ in (2.12) so that boundary conditions at z = 1 become (∂ z − ω)A µ = 0. Consequently the spectrum is determined by S − ωS = 0. For the low-lying KK modes X -boson tower: S(1; λ) = 0 . (3.24) There are three classes; The spectrum is determined by

A z components
Similarly one can find the spectrum for A z . The evaluation is simpler as A z does not couple to the brane scalar field Φ 16 .
(i) A ab z (1 ≤ a < b ≤ 3) and A jk z (5 ≤ j < k ≤ 10): These components satisfy boundary conditions are (D, D) so that Boundary conditions of (A a4 z , A a 11 z ) are (D, D) and (N, N ), respectively. A a4 z and A a 11 The spectrum is determined by There always is a zero mode, which will acquire a mass at the 1-loop level.

Spectrum of fermion fields
We take Dirac matrices γ A in the spinor representation in (2.3).

Brane mass terms
In addition to (2.3), the fermion fields have brane interactions given by S brane in (2.22).

Quarks and leptons
To derive the mass spectrum for fermions, we note that the components of Ψ 32 in the original and twisted gauges are related by where χ andχ are defined in (2.43). θ(z) is given in (2.40). In the original gauge with θ H , one has We denoteD (4.7) To simplify the notation the bulk mass parameters are denoted as (4.8) (i) Q EM = + 2 3 : u j , u j There are no brane mass terms. The boundary conditions are D +ǔjL = 0,ǔ jR = 0, u jL = 0, and D −ǔ jR = 0 at z = 1, z L . The equations of motion in the twisted gauge are (ũ j ,ũ j ) satisfy the same boundary conditions at z = z L as (ǔ j ,ǔ j ) so that one can write, for each mode, as whereσ∂f R (x) = kλf L (x) and σ∂f L (x) = kλf R (x). Both right-and left-handed modes have the same coefficients α u and α u as a result of the equations of motion.
The boundary conditions at z = 1 for the right-handed components,ǔ jR = 0 and D −ǔ jR = 0, become so that the spectrum is determined by where S c L = S L (1; λ, c) etc.. The mass of the lowest mode, m = kλ, is given by (4.13) c 0 = c Ψ 32 is determined from the up-type quark mass. For the top quark c 0 < 1 2 , whereas for the charm and up quarks c 0 > 1 2 . Note that There are no brane mass terms. The boundary conditions are D +ǔjL = 0,ǔ jR = 0, u jL = 0, and D −ǔ jR = 0 at z = 1, andǔ jL = 0, D −ǔjR = 0, D +ǔ jL = 0, andǔ jR = 0 at z = z L . Wave functions of each mode are given by Boundary conditions at z = 1 lead to so that the spectrum is determined by The mass of the lowest mode is given by Here D + acting onď jL ,Ď jL ,Ď jL means D + (c 0 ), D + (c 2 ), D + (c 1 ), respectively. Brane interactions affect boundary conditions at y = 0.ď jR ,ď jL ,Ď jR ,Ď jL are parity-odd at y = 0, whereasď jL ,ď jR ,Ď jL ,Ď jR are parity-even. Recall that D ± (c) is parity-odd at y = 0; Noting that A (4,11) z is parity-even and integrating over y from − to in (4.20), one finds (a) ⇒ 2ď jR (x, ) = 0 , For parity-even fields we evaluate the equations (4.20) at y = > 0, with the help of (4.22), to find (c) ⇒D +ďjL = 0 , To summarize, the boundary conditions at z = 1 + (y = ) are given by In the twisted gauge all fields obey free equations in the bulk so that eigenmodes are expressed, with the boundary conditions at the TeV brane taken into account, as The boundary conditions (4.24) for the right-handed components are converted to where S c R = S R (1; λ, c) etc.. The spectrum {λ n } is determined from det K = 0. (4.27) or, by making use of C L C R − S L S R = 1 one finds that The same result is obtained from the boundary conditions for the left-handed components in (4.24).
For the mode with the lowest mass, the down-type quark, one can suppose that λz L 1, In the first and second generations c j > 1 2 , whereas in the third generation c j < 1 2 . The mass is given by (4.30) In either case one finds that  Eigenmodes are given by The boundary conditions (4.33) lead to The spectrum is determined by For the mode with the lowest mass, The mass is given by (4.39) One finds that for c 0 , c 1 > 1 2 . (4.40) (v) Q EM = −1 : e, e , E, E The spectrum in the Q EM = −1 sector is found in a similar manner. Boundary conditions at y = (z = 1 + ) are given by 41) and mode functions in the twisted gauge are given by The boundary conditions in (4.41) lead to Consequently the spectrum is determined by For the lowest mode, the electron, so that for c 0 , c 2 > 1 2 . (4.47) (vi) Q EM = +1 :ê,ê ,Ê,Ê There are no zero modes. Boundary conditions at y = (z = 1 + ) for right-handed components are given byě 48) and wave functions in the twisted gauge are given by The spectrum is determined by For the lowest mode so that (4.53) One finds that both for c 0 , c 1 < 1 2 and for c 0 , c 1 > 1 2 . (vii) Q EM = 0 : ν, ν , N, N , S, S ,ν,ν ,N ,N Only ν L , ν R have zero modes. In general all these ten components mix with each other.
It is convenient to split them into two sets; Boundary conditions at y = (z = 1 + ) becomě for Set 1, andD for Set 2. When µ 5 = 0, the two sets of boundary conditions decouple from each other.
We set µ 5 = 0 in the following analysis.
Wave functions in the twisted gauge are given by (4.57) The boundary conditions (4.55) and (4.56) for µ 5 = 0 lead to The spectrum for Set 1 is determined by As will be seen shortly, µ 2 needs to be very large to have small neutrino masses. Careful evaluation of each term in (4.59) is necessary to find approximate formulas for neutrinos.
For the lowest mode with λz L 1, for c 0 , c 2 < 1 2 , and for c 0 , c 2 > 1 2 . For neutrinos λz L = πm ν /m KK . In the third generation, for which we choose c 0 , c 2 < 1 2 , it is found, a posteriori, that so that (4.63) One finds that (4.64) We note that m ν m e = 1 µ 2 for c 0 , c 2 < 1 2 and for c 0 , c 2 > 1 2 . The spectrum for Set 2 is determined by The mass of the lowest mode is approximately given by (4.67)

Exotic particles
In in (4.67) in the Q EM = 0 sector. In particular, the exotic particle in the Q EM = − 2 3 sector causes a severe problem. As shown in (4.19), the ratio of mû to m u is solely determined by θ H . Phenomenologically θ H < 0.1. It will be seen in the next section that with reasonable parameters it is not possible to get a minimum of the effective potential V eff (θ H ) at very small θ H . It is unavoidable to have unwanted lightû particles in the first and second generation.

Effective potential
In this section, we evaluate the Higgs effective potential V eff (θ H ) by using the mass spectrum formulas of SO(11) gauge bosons and fermions. The contributions to the effective potential from the quark-lepton multiplets in the first and second generations are negligibly small in the RS space, and can be ignored. In numerical evaluation, we use the mass parameters and gauge coupling constants listed in PDG [75].
One-loop effective potential from each KK tower is given by [8,74,76] The equations determining the spectra are given by (3.9) for the W ± tower, (3.15) (3.9) in place of the exact formula (3.8), for instance, is numerically justified. One finds that where I α (u) and K α (u) are modified Bessel functions.
The fermion part V fermion eff (θ H ) is evaluated in a similar manner. Following the classification based on Q EM in the previous section, we decompose V fermion eff into eight parts; V fermion eff depends on the three bulk mass parameters (c j ) and brane interaction mass parameters (µ k ) in the third generation. We set µ 5 = 0 as before. The equations determining the mass spectra are (i) (4.12) for the Q EM = + 2 3 (u-type) quarks, (ii) (4.17) for the Q EM = − 2 3 (û-type) quarks, (iii) (4.28) for the Q EM = − 1 3 (d-type) quarks, (iv) (4.37) for the Q EM = + where Here α em is the fine-structure constant, and θ W is the Weinberg angle.
In the following we give example calculations for the effective potential and show a result for the Higgs mass. As we remarked before, the current SO(11) model necessarily contains light exotic particles, and therefore is not completely realistic. With this in mind, we do not insist on reproducing all of the observed values of the masses of the SM gauge bosons, Higgs boson, quarks and leptons. Further the GUT relation leads to sin 2 θ W = 3 8 . The RGE effect must be taken into account to compare it with the observed value at low energies.
To find a consistent set of the parameters we take the following procedure. 0. We fix z L = e kL and pick θ H = θ min H .
1. We suppose that the minimum of V eff (θ H ) is located at θ H = θ min H . The equation (3.15) determine the spectrum {λ Z (n) } of the Z tower. By using the zero mode mass m Z (0) and the observed Z boson mass m obs 3. We choose a sample value of the bulk mass parameter of SO(11) vector fermion c 2 .
Then the brane mass parameters µ 2 , µ 3 and µ 6 /µ 4 are determined by (5.12   The effective potential for m t = 170 GeV is displayed in Figure 1. The global minimum is located at θ H = 0.10, and the EW symmetry breaking takes place. In Figure 2 and It is seen in Figure 2, the contributions from (v) A k4 z , A k11 z (k = 5 ∼ 10) and (viii) Y bosons dominate over others in the gauge field sector. In the fermion sector there appears cancellation among contributions from various components. It is seen in Figure 3, the contribution of (i) the top quark is almost canceled by that of (ii) theû-typet fermion.
The bottom quark andd-typeb fermion contributions are not canceled out, but each contribution is small. The tau lepton andê-typeτ fermion contributions are not canceled out, but each contribution is small. The four contributions from b,b, τ , andτ add up almost zero. The contribution from neutral fermions is appreciable in the current model.
In the previous section we observed that there appear light exotic fermions which should not exist in reality. In this section we have observed that there appear cancellations among the contributions to V eff (θ H ) from fermions and their corresponding exotics. These two seem to be related, and the too light Higgs boson mass m H is inferred to be a result of those cancellations.  Figure 1. The green solid line represents all gauge field contributions for the effective potential, which is the same as the green dashed in Figure 1. The red dashed line is (i) W ± contribution, the purple short dashed line is (ii) Z contribution, the orange dashed line is (viii) Y contribution, the blue dashed line is (ii) A a4 z , A a11 z (a = 1, 2, 3) contribution, and the brown short dashed line is (v) A k4 z , A k11 z (k = 5 ∼ 10) contribution.

Conclusion and discussions
In the present paper we have explored the SO (11)  We have demonstrated that the quark-lepton mass spectrum can be reproduced by adjusting the parameters of the brane interactions.
One of the interesting features of the model is that the proton decay is forbidden, in sharp contrast to the GUT models in four dimensions. The quark-lepton number N Ψ is conserved by the gauge interactions and brane interactions.
In the current model, however, there appear light exotic fermions associated withûtype,d-type andê-type fermions, which contradicts with the observation. The Higgs boson  Figure 1. The red solid line is the total fermion contribution for effective potentials, which is the same as the red dashed in Figure 1. The green dashed line is (i) Q em = + 2 3 fermion contribution, the cyan short dashed line is (ii) Q em = − 2 3 fermion contribution, the brown dashed line is (iii) Q em = − 1 3 fermion contribution, the pink short dashed line is (iv) Q em = + 1 3 fermion contribution, the blue dashed line is (v) Q em = −1 fermion contribution, the gray short dashed line is (vi) Q em = +1 fermion contribution, the orange dashed line is (vii-1) Q em = 0 fermion contribution, and the magenta short dashed line is (vii-2) Q em = 0 fermion contribution. mass m H , which is predicted in the current gauge-Higgs grand unification, turns out too small. The small m H is a result of the partial cancellation among the contributions of the quark-lepton component and the exotic fermion component to the effective potential V eff (θ H ). In other words the exotic fermion problem and the small m H problem seem to be related to each other. The model need improvement in this regard. We hope to report how to cure those problems in the near future.
We note that for λz L 1 and c ≥ 0