Dark matter in the $SO(5)\times U(1)$ gauge-Higgs unification

In the $SO(5) \times U(1)$ gauge-Higgs unification the lightest, neutral component of $n_F$ $SO(5)$-spinor fermions (dark fermions), which are relevant for having the observed unstable Higgs boson, becomes the dark matter of the universe. We show that the relic abundance of the dark matter determined by WMAP and Planck data is reproduced, below the bound placed by the direct detection experiment by LUX, by a model with one light and three heavier ($n_F=4$) dark fermions with the lightest one of a mass from 2.3$\,$TeV to 3.1$\,$TeV. The corresponding Aharonov-Bohm phase $\theta_H$ in the fifth dimension ranges from 0.097 to 0.074. The case of $n_F=3$ ($n_F = 5, 6$) dark fermions yields the relic abundance smaller (larger) than the observed limit.


Introduction
The Higgs boson of a mass around 125. 5 GeV was discovered at LHC. [1,2] It is not clear, however, whether or not the particle discovered is precisely the Higgs boson specified in the standard model (SM). Physics beyond the standard model may be hiding, showing up at the upgraded LHC. Couplings of the Higgs boson to other particles may slightly deviate from those in SM, and new particles may be produced, say, in the 4 -7 TeV range. SM lacks a principle governing dynamics of the Higgs boson. Further SM has no clue to explain the dark matter (DM) in the universe.
In the gauge-Higgs unification (GHU) the Higgs boson is unified with gauge bosons.
The 4D Higgs boson appears as a part of the extra-dimensional component of gauge fields so that its dynamics are governed by the gauge principle. [3]- [8] It has been shown that in the SO(5) × U (1) GHU in the Randall-Sundrum warped space the low energy physics appears almost the same as that in SM, consistent with all LHC data. [9]- [16] Contributions of Kaluza-Klein (KK) excited modes to the H → γγ decay, for example, turn out very small. [15] Higgs couplings to gauge bosons, quarks and leptons at the tree level are suppressed by a common factor cos θ H where θ H is the Aharonov-Bohm phase in the extra dimension. [17]- [22] All of the precision measurements, the tree-unitary constraint, and the Z search indicate that θ H < 0.2. [9,23] The SO(5) × U (1) GHU predicts new structure at higher energies. The masses of the 1st KK modes of Z and γ are predicted to be 3 ∼ 7 TeV for θ H = 0.1 ∼ 0.2. The Higgs cubic and quartic self-coupling should be smaller than those in the SM by 10% -20%. [16] Many other signals of GHU have been investigated. [24]- [33] Another important issue is the dark matter. [34] Supersymmetric theory, the leading model of physics beyond the SM, predicts the lightest supersymmetric particle as a dark matter candidate. [35,36] The lightest KK particle in universal extra dimension models [37]- [42], the lightest T-odd particle in the little Higgs models [43,44], a fermionic composite state in the composite Higgs models [45]- [47], and axions [48]- [52] can be identified as dark matter. In the Higgs portal scenario the Higgs boson couples to dark matter in the hidden sector [53]- [57], and the dynamical dark matter scenario has been proposed. [58] Is there a dark matter candidate in the SO(5) × U (1) gauge-Higgs unification model? Can it explain the relic abundance reduced from the WMAP/Planck data and other observations, within the constraints from direct detection searches? A few scenarios for dark matter in GHU have been proposed. [59,60,61,62] In this paper we would like to show that the realistic SO(5) × U (1) gauge-Higgs unification model contains a natural candidate for dark matter.
In the minimal SO(5)×U (1) gauge-Higgs unification model, in which only quark-lepton vector multiplets and associated brane fermions are introduced in the fermion sector, the effective potential is minimized at θ H = 1 2 π, which in turn implies that the Higgs boson becomes stable, contradicting with the observation. [11,13,61] To have an unstable Higgs boson, it is necessary to introduce fermion multiplets in the spinor representation of SO (5) which do not appear at low energies. [15] Indeed, the presence of these fermions, with the gauge fields and top quark multiplet, naturally leads to 0 < θ H < 1 2 π, yielding predictions consistent with the observation. One remarkable property is that independent of the details of these SO(5)-spinor fermions there appears the universality relations among θ H , the masses of KK Z/photon, and the Higgs self couplings.
We show that the lightest, neutral component of the SO(5)-spinor fermions is absolutely stable, and becomes the dark matter of the universe. For this reason the SO(5)-spinor fermion is called as a dark fermion in the present paper. It is heavy with a mass around 2 ∼ 4 TeV, but its couplings to the Higgs boson are small. From its relic abundance the number and structure of the dark fermion multiplets are inferred. It is curious that the Higgs dynamics are intimately related to the dark matter in the gauge-Higgs unification.
The paper is organized as follows. In Section 2 the SO(5) × U (1) model is introduced.
In Section 3 it is shown that the neutral components of dark fermions become the dark matter, and the relic abundance is evaluated. In Section 4 the spin-independent cross section of the dark matter candidate with nucleons is evaluated, and the compatibility with the constraint coming from the direct detection experiments, XENON100 and LUX [63,64], is examined. It will be found that the model with n F = 4 nondegenerate dark fermions with the lightest one of a mass 2.3 TeV∼3.1 TeV explains the relic abundance of the dark matter determined from the WMAP/Planck data below the bound placed by the direct detection observation of LUX. Section 5 is devoted to the conclusion and discussions.
In the appendices wave functions and couplings of dark fermions and relevant gauge bosons are summarized.
The Planck and TeV branes are located at y = 0 and y = L, respectively. The bulk region 0 < y < L is anti-de Sitter (AdS) spacetime with a cosmological constant Λ = −6k 2 .
In the fermion partΨ = iΨ † Γ 0 and Γ M matrices are given by The quark-lepton multiplets Ψ a are introduced in the vector representation of SO (5). In contrast, n F dark fermions Ψ F i are introduced in the spinor representation. The c term in Eq. (2.2) gives a bulk kink mass, where σ (y) = k (y) is a periodic step function with a magnitude k. The dimensionless parameter c plays an important role in controlling profiles of fermion wave functions.
The orbifold boundary conditions at y 0 = 0 and y 1 = L are given by by the orbifold boundary conditions. Various orbifold boundary conditions fall into a finite number of equivalence classes of boundary conditions. [65,66] The physical symmetry of the true vacuum in each equivalence class of boundary conditions is dynamically determined at the quantum level by the Hosotani mechanism. Recently dynamics for selecting boundary conditions have been proposed as well. [67] The Hosotani mechanism has been explored and established, not only in perturbation theory, but also on the lattice nonperturbatively. [68] The brane action S brane contains brane fermionsχ αR (x), brane scalarΦ(x), A µ (x, y = 0) and Ψ a (x, y = 0). It manifestly preserves gauge-invariance in SO(4) × U (1) X .Φ develops non-vanishing expectation value Φ m KK , which results in spontaneous breaking of SO(4) × U (1) X into SU (2) L × U (1) Y and in making all exotic fermions heavy.
The 4D Higgs field, which is a bidoublet in SU (2) L ×SU (2) R , appears as a zero mode in the SO (5) .
For each generation two vector multiplets Ψ 1 and Ψ 2 for quarks and two vector multiplets Ψ 3 and Ψ 4 for leptons are introduced. In contrast, the dark fermion Ψ F i belongs to the spinor representation of SO(5), having four components (2.7) ψ i l and ψ i r are SU (2) L and SU (2) R doublets, respectively. They mix with each other for θ H = 0. The electric charge is given by Q EM = T 3 L + T 3 R + Q X . We take Q X = 1 2 for Ψ F i so that it contains charge 1 and 0 components.
The KK decomposition of Ψ F i fields are summarized in Appendix B. With the boundary condition (2.5) Ψ F i (x, z) does not have zero modes, and is expanded in the KK modes F +(n) i (x) and F 0(n) i (x) (n = 1, 2, 3, · · · ) as in (B.1). The mass spectrum is determined by (B.7). With η F i = +1 in the boundary condition for Ψ F i in (2.5) and for small θ H the odd KK number modes F

Relic density
By considering annihilations and decays of dark fermions in the early universe, one can evaluate the relic density of the dark fermion. We mostly follow the arguments in Refs.
[34], [37] and [71]. The Boltzmann equation for F 0 i is given by Similar relations are obtained forF 0 i and F ± i . Here H is the Hubble constant, n (F ) denotes the number density of F , and X represents a SM field. The number density of F in the thermal equilibrium is given by n eq (x) = g x (m x T /2π) 3/2 exp(−m x /T ) where g x and m x are the number of the degrees of freedom and mass of x, respectively. If F ± is heavier than F 0 , a term describing F + → F 0 ff decay should be added on the right-hand side of (3.1); where f, f are fermions in the SM and Γ denotes a decay width.
The effective interactions relevant to annihilations of dark fermions are given by and by charged currents in Eq. (3.4). Here H denotes the Higgs boson, and f refers to a fermion in the SM (quarks, leptons and neutrinos).
For the decays (3.2) the corresponding interaction terms in the effective Lagrangian are where f and f refer to up-type quark (neutrino) and down-type quark (charged lepton), respectively. A CKM-like mixing matrix U (V )CKM is a unit matrix for leptons and is assumed to approximately coincide to the CKM-matrix for V = W . For the spinor fermion F , the right-and left-handed F )/2 are given in the Appendix C.2.2, and for the SM fermions the couplings can be found in Ref. [16]. In particular, W R boson is found to have no couplings to the SM fermions.

Decays v.s. conversions of charged dark fermions
At the quantum level, masses of F ± and F 0 receive finite corrections δm F + and δm F 0 , respectively, and the degeneracy is lifted by one-loop corrections involving the photon and KK photons, which appear only in δm F + as depicted in Fig. 1. The mass difference between Figure 1: Diagrams contributing to the fermion mass difference ∆m F = δm F + − δm F 0 .
F ± and F 0 , δm F ± − δm F 0 , can be evaluated in an analogous way as in the universal extra dimension [72], and in the case of the warped extra dimension it is estimated by where α EM is the electromagnetic fine-structure constant. In UED K = ln(Λ 2 /µ 2 ) where Λ and µ is the cut-off scale and a renormalization scale, respectively, and Λ/µ ∼ O(10).
In the RS space-time only the first few KK excited states of each fields enter the quantum corrections. In particular the coupling of right-handed F ±(1) to γ (1) is several times as large as the electromagnetic coupling. It follows that K ∼ O (10). Similarly, quantum corrections due to higher-KK modes to the gauge couplings also become small, and a large cut-off scale is allowed. [73] A charged dark fermion decays to a neutral dark fermion and a charged vector bosons, hence to charged leptons and neutrinos, or light down-type quarks and up-type antiquarks.
(See Fig. 2) Figure 2: Charged dark fermion decay. − andν can be replaced with down-type quarks and up-type anti-quarks, respectively.
In the SO(5) × U (1) GHU model, we have three charged vector bosons at low energies: is much smaller than m W , the decay rate is given by where In the second equality in (3.6), we have assumed ∆m F m F ± , m F 0 and have invoked approximations Hence the lifetime of F − is given by where τ µ = 2.2 × 10 −6 sec and m µ = 105 MeV are the lifetime and mass of the muon, respectively. (g L Wlν , g R Wlν ) = (1, 0) is used. In order that the F ± lifetime is much shorter than the typical time scale of the weakly interacting massive particle (WIMP)-DM formation, i.e. τ F ± 10 −10 sec, the mass difference of dark fermions must be the order of 10 GeV or larger. The mass difference (3.5) will satisfy this condition for m F 2 TeV with K ∼ O(10). Hereafter we assume that these conditions are satisfied and F ± decays sufficiently quickly. We also note that if charged fermions F + do not decay sufficiently fast, they would remain after the DM freeze-out and would subsequently decay to F 0 , resulting in doubling the relic DM density.
In the right-hand side of the Boltzmann equation (3.1), the last two terms correspond to F 0 ↔ F + conversion detpicted in Fig. 3.
The process depicted as (A) in Fig. 3, in particular F + F − pair production through this process is kinematically allowed since m F ∆m F . Although the process depicted as (B) in Fig. 3 is suppressed by the small FF W coupling which is order of 10 −3 , this conversion process can dominate due to the large ratio of n eq (X) /n eq (F ) ∼ (T /m F ) 3/2 exp(m F /T ) ∼ 10 10 for T /m F ∼ 30 [71].
Thus we have n eq (F ± ) ∼ n eq (F 0 ) before the freeze-out, and after the freeze-out F + decay to F 0 . The relic density of the dark fermion in the present universe is given by the sum of the charged and neutral dark fermions at the freeze-out. In the followings we calculate the number density of all dark fermions.

Pair annihilations and relic density of dark fermions
The annihilation processes and corresponding diagrams of the dark fermions are tabulated in Table. 1 and Fig. 4. We note that the masses of the first excited states of SM fermions We consider the case where θ H is small (z L 10 5 or θ H 0.15). In such a case, dark fermion is heavy and some of annihilation amplitudes are processes are suppressed by sin 2 θ H . We find that for most of the processes annihilation cross-sections are too small to explain the current relic density. In particular, we find thatF F W ,F F Z,F F Z (n) and Z (n) R W W couplings are suppressed by sin 2 θ H factor. (See Appendix C and D). One finds that the process (a-i) is suppressed by the small Higgs Yukawa couplings of FF and processes (a-ii) with V = Z and Z (n) are suppressed by the small Z (n) FF couplings. The processes (a-iii) and (a-iv) are suppressed by the small W − F +F and ZFF couplings. All processes of (a-v) are suppressed by the small Z (n) FF coupling and small Z  Table 1: Pair annihilation processes of dark fermions (F = F 0 , F + ). (a-i)-(a-v) are annihilation processes of neutral and charged dark fermions, whereas (ac-i)-(ac-iv) are those of charged dark fermions. (co-i)-(co-v) are for co-annihilation of the neutral and charged dark fermions. In the intermediate states 'n' denotes the KK-excitation level (n = 0). In the final states q, l and ν denotes quarks, charged leptons and neutrinos in the SM. Corresponding diagrams are shown in Fig. 4.
FF → ZZ, t-and u-channels For the annihilation of charged dark fermions, we see that the process (ac-i) is not suppressed by couplings. However, the annihilation cross section where v is the relative velocity of initial particles, is numerically small and negligible with m F 2 TeV. The process (ac-ii) is suppressed by FF Z couplings. The cross section in the process (ac-iii) with V = γ is estimated as The process (ac-iii) with V = γ (1) can be enhanced by both large right-handed coupling of fermions and Breit-Wigner resonances. The process (ac-iv) is suppressed by the small As for coannihilation, we have tabulated possible processes in Table. 1. We find that the process (co-i) with V = W + , W +(n) is suppressed by small F +F 0 W − couplings and the suppressed by small FF W (n) couplings. The process (co-iii) with V = W R is forbidden by the vanishing W R W γ coupling which ensures the ortho-normality of the KK gauge bosons.
The processes (co-iv) and (co-v) are suppressed by small FF Z and F +F 0 W − couplings.
Hence we found that all of the co-annihilation processes are either vanishing or strongly suppressed.
Thus we find that relevant processes for dark fermion annihilation are the following s-channel processes and all other annihilation and co-annihilation processes are negligible.
In the followings, we calculate the relic density of the dark fermions using annihilation cross sections of the processes given in (3.12). For charged dark fermions, the annihilation cross section of F + i F − i to the SM fermions is given by R , γ (1) ) and the couplings are summarized in Sec. C.2. β ≡ 1 − 4m 2 F /s and s is the invariant mass of FF . We have neglected γ-γ (1) and γ-Z and ignoring e 2 and e 4 terms. Γ Z (1) R and Γ γ (1) are the total decay rate of Z (1) (3.14) Γ γ (1) is obtained in an analogous way. Here N c,f = 3 (1) when f is a quark (charged lepton or neutrino). m f is the mass of the SM fermion. We note that the F ± contributions in (3.14) are rather large.
Then the evolution of the total number density of the DM, is given by n ≡ 2n F (n 0 + n + ), and the time-evolution of n is governed by the Boltzmann equation where n 0/+,eq is the number-density in the thermal equilibrium and approximated by Using the relations n 0,+ /n 0,+eq = n/n eq and n 0,eq /n eq = n +,eq /n eq = 1/4n F , we obtain We introduce Y (eq) ≡ n (eq) /S where S = 2π 2 g * T 3 /45 is the entropy density. g * is the degree of freedom at the freeze-out temperature T f and we take g * = 92. Conservation of entropy per co-moving volume, Sa 3 sf = constant (a sf is the scale factor of the expanding universe), reads dn/dt + 3Hn = SdY /dt. The Hubble constant is given by H 2 = 4π 3 g * T 4 /(45M 2 P l ) and t = 1/2H in the radiation-dominant era. M P l is the Planck mass. Hence we rewrite the Boltzmann equation as where x ≡ T /m F and T is the temperature of the universe. σv = σv (x) is the thermalaveraged cross section discussed later. n eq is the density in the thermal equilibrium, and becomes n eq = g eff · 4n F is the degree of freedom of the dark fermions) in the non-relativistic limit.
Defining ∆ ≡ Y − Y eq and ∆ ≡ d∆/dx, Y eq ≡ dY eq /dx, we rewrite (3.17) as 19) which is written at early times (x . (3.20) At late times (T where we have used ∆(x f ) ∆(x = 0). Thus the relic density of the dark fermions at the present time is given by The freeze-out temperature is determined by solving the condition with ∆ in the early-time. c is an numerical factor of order unity and determined by matching the late-time and early-time solutions. Hereafter we take c = 1/2. Eq. (3.24) with (3.20) reads the following transcendental relation which can be solved by numerical iteration.
The precise form of the velocity-averaged cross section σv is given in Ref. [74]. When In the present case x f ∼ 1/30 and therefore only the first term in the v 2 expansion in Eq. (3.26) is kept in the following analysis.

Relic density of degenerate dark fermions
First we consider the case in which all dark fermions are degenerate. In the numerical study of this paper, we have adopted α EM ≡ e 2 /4π = 1/128, sin 2 θ W = 0.2312, m Z = 91.1876 GeV and m top = 171.17 GeV. [75] In Table. 2, we have summarized values of θ H , the bulk mass parameters of the top quark c top and the dark fermion c F , and mass of the dark fermion m F for particular values of (z L , n F ). θ H , c top and c F are chosen so that we obtain 126GeV Higgs mass [15,16].
In Fig. 5 the relic density of the dark fermions for n F = 3, 4, 5 and 6 is plotted. In the plot, the best value [68% confidence level (CL) limits] of the relic density of the cold dark matter observed by Planck [76]:   We remark that for n F = 3 the relic density becomes very small at z L ∼ 3 × 10 4 due to the fact that the masses of the 1st KK vector bosons are very close to twice the mass of dark fermions, and the enhancement due to the Breit-Wigner resonance happens. A similar mechanism occurs in some of the universal extra-dimension models [38,39,41,42].

Current mixing
So far it has been supposed that n F multiplets of SO(5)-spinor fermions Ψ F i are degenerate.
There is an intriguing scenario that some of them are heavier than others, only the lightest Let us denote the lightest particles of heavy and light SO(5)-spinor fermions by (F + h , F 0 h ) and (F + l , F 0 l ), respectively. Charged F + l and F + h are heavier than the corresponding neutral ones, and are supposed to decay sufficiently fast. F 0 h also needs to decay sufficiently fast in order for the scenario to work. F 0 h can decay either as → F 0 l + Z or as → F + l + W − → F 0 l + W + + W − as shown in Fig. 6. For this process the off-diagonal neutral or charged current is necessary. We examine in this subsection how the off-diagonal currents are generated. Figure 6: F 0 h decay to F 0 l by emitting one Z boson or two W bosons.
To be concrete, let us suppose that there are only two SO(5)-spinor fermion multiplets, Ψ F h and Ψ F l , which are gauge-eigenstates. We suppose that Ψ F l obeys the boundary It is easy to confirm that their KK spectrum is given by (B.7) for both Ψ F h and Ψ F l . The lowest mode (F Let us denote gauge (mass) eigenstates of the lightest modes of The most general form of bulk mass terms for Ψ F h and Ψ F l is The∆ term induces mass mixing amongF + h andF + l , and amongF 0 h andF 0 l . c F h and c F l generate massesm h andm l for (F + h ,F 0 h ) and (F + l ,F 0 l ). We suppose that c F h < c F l so that m h >m l . As described in Sec. 3.1, charged states acquire radiative corrections (3.5), am h . Hence the mass matrices are given by We suppose that ∆ m h ,m l . We diagonalize the two matrices to obtain The couplings to Z (the neutral currents) are given originally by Similarly the couplings to W (the charged currents) are given by . In terms of mass eigenstates the neutral current becomes where The charged current is We recognize that off-diagonal neutral and charged currents are generated for dark fermions obeying the distinct boundary conditions. ). Bulk mass parameter c F l and the masses m F h and m F l of F h and F l are tabulated for various ∆c F ≡ c F l − c F h (see text) and z L . Even small ∆c F gives rise to large mass difference.
h R |) so that the estimate of the cross section for the direct detection experiments discussed in the next section remains valid.
The couplings for F 0 hL → F 0 lL +Z and for F 0 hL decays sufficiently fast. Only the light dark fermion F 0 lL becomes a candidate of dark matter.

Relic density of non-degenerate dark fermions
Let us examine the case with non-degenerate dark fermions. We separate the n F dark At the temperature T m F h −m F l , the heavy-light conversion process depicted in Fig. 7 dominates, and both F h and F l obey the Boltzmann distribution. When m F h − m F l T f = O(100 GeV), the number density of F h becomes much smaller than that of F l . In contrast to F l , F h obey the boundary condition η F h = −1 and its couplings to W and Z are not suppressed, whereas its coupling to Z R is suppressed. Thus the dominant annihilation processes of F h are s-channel processes of FF annihilation to the SM fermions through Z (1) and γ (1) [(a-ii) with V = Z (1) and (ac-iv) with V = γ (1) in Table. 1] and co-annihilation through W (1) Table. 1]. The time evolutions of the total dark fermion density is given by where n w0 and n w+ (w = h, l) are the number densities of F 0 w,i F + w,i (i = 1, · · · , n light F for w = l, and 1, · · · , n heavy F for w = h), respectively. σ w0 , σ w+ and σ hc are the cross section of We also have used n w0/+ n n w0/+,eq n eq , n (eq) w0 n (eq) w+ ≡ n (eq) w , w = h, l.
and g eff in (3.18) will be replaced with When η/x 1, the Boltzmann equation (3.37) with (3.39) can be approximated by and g ND eff | η→∞ = 2 · 4n light F . With this approximation, one can calculate the relic density of the dark fermion by following the procedure described in Sec. 3.2. Since the effective cross section, and therefore J f in (3.22), is enhanced by a factor σ ND eff v| η→∞ /σ eff v n F /n light F , which results in the reduction of the relic density by a factor n light F /n F as seen from (3.23). If η is not so large, the approximation (3.41) is not valid any more. In particular, for η ∼ 0 the Bolzmann equation (3.37) become almost identical to (3.16), and the relic density will be increased up to that in the degenerate case. Effects of small η on Ω DM h 2 (3.23) mainly appear in the change of the value of J f (or σ eff v ). Numerically we find that J f determined from (3.41) well approximates J f determined from (3.37) with (3.39) at O(5%) accuracy when η 0.10 for x = x f 1/30 and σ l0/+ ∼ σ h0/+ . We note that in the cross section (3.13), the total decay width of Z is no more valid, and the relic-density can be much larger than those for ∆c F 0.04. By inter-/extra-polating the Ω DM h 2 with respect to ∆c F and z L , we plot the parameter region (∆c F , z L ) allowed by the experimental limit on the current relic density in Fig. 9. It is seen that the observed current relic density is obtained when 10 4 z L 10 6 (0.   In the numerical study we have used an approximation explained in Sec. 3

.2. In the case
where the Breit-Wigner resonance enhance the DM relic density, a more rigorous treatment may be required. [71] In the case under consideration, the effect of the enhancement is found to be mild. Quantitatively, in the notation of Ref. [71] we obtain = ( R , γ (1) ) and √ u = 2m F /m V 0.8. In this parameter region the approximation can be justified. [71] Before closing this section, we make a few comments. First we comment on the effect of dark fermions on the electroweak precision parameters [77], in particular on the S-parameter. Since the dark fermions have vector-like couplings to the Z boson, the contribution to the S parameter from an SU (2) where g Z F V ≡ (g Z F L +g Z F R )/2 and Q F is the vector coupling to Z and the electric charge of F , respectively. Π(p 2 ) is the vacuum polarization function which is induced by the one-loop fermion with vector-type coupling. Numerically we find that in both cases of F l (η F l = +1) and F h (η F h = −1) the sum of the right-hand side in (3.42) vanishes accurately. Hence there are no sizable corrections of the S parameter from dark fermions.
Secondly as an stabilization mechanism of the branes one can introduce some dynamical model a la Goldberger-Wise [78]. In such a case the phase transition of the radion field may alter the thermal history of the universe drastically [79]. Here we have supposed that the critical temperature of the radion phase transition, T φ , is much higher than the freeze-out temperature of the dark fermions, e.g., T φ T f ∼ 100 GeV.

Direct detection
In this section, we analyse the elastic scattering of the dark fermion (F 0 ) off a nucleus [35,36,80] and examine the constraint coming from direct detection experiments. [63,64] The dominant process of the F 0 -nucleus scattering turns out the Z boson exchange, though the Z-F 0 coupling is very small. The Z (n) R -F 0 coupling is larger, but Z (n) R is heavy. In the scattering of F 0 on nuclei with large mass number A, scalar and vector interactions dominate for the spin-independent cross section. Therefore the effective Lagrangian at low energies is given by To evaluate the scattering amplitude by the Higgs exchange, we need estimate the nucleon matrix element where N = p, n. For heavy quarks (Q = c, b, t) one has f (N ) In the GHU model, quark couplings satisfy v q | GHU v q | SM and to good accuracy. [21] Therefore, by dropping the small momentum dependence of the form factor, the spin-independent cross section of the F 0 -nucleus elastic scattering becomes where M r is the F 0 -nucleus reduced mass and Z (A) is the atomic (mass) number of the nucleus. |q| is the momentum transfer and The spin-independent cross section of the F 0 -nucleon elastic scattering σ N can be written as where m r is the F 0 -nucleon reduced mass.
The F 0 -nucleon cross sections σ N are shown in Table 4 and Figure 11. In the numerical evaluation we have employed the values given by [36] f (p) T u = 0.020 , f  T s [81], which yields slightly smaller cross sections than those described below.
In the previous section we have seen that when all n F dark fermions are degenerate, there are no parameter regions which reproduce the observed value of the relic DM density. Table 5: m F l , m Z (1) R , the couplings of F 0 l and the spin-independent cross section σ N of the F 0 l -nucleon scattering for n F = 4 and (n light F , n heavy   3), the region 0.04 < ∼ ∆c F < ∼ 0.07, z L < ∼ 10 6 successfully explains the relic abundance as shown in Fig. 9. The allowed band region in Fig. 9 is mapped in Fig. 11 for the spin-independent cross section for the F 0nucleon elastic scattering. The purple and light purple bands there represent the regions allowed by the limit of the relic abundance of DM at the 68 % CL and by twice of that, respectively. It is seen that the band region from z L = 10 4 to 4×10 4 is allowed by the direct detection experiments of LUX [64] and XENON100 [63]. In the allowed region the dark fermion mass ranges from 3.1 TeV to 2.3 TeV, whereas the AB phase θ H ranges from 0.074 to 0.097. The mass of Z bosons (the lowest Z R boson and the first KK modes Z (1) and γ (1) ) ranges from 8 TeV to 6.5 TeV. For reference we have added, in Fig. 11, the expected limit by the 300 live-days result of the LUX experiment. The XENON 1T experiment is expected to give a limit one order of magnitude smaller than that of the LUX 300 live-days experiment in the cross-section.
We remark that the n F = 3 case predicts too small relic densities as shown in Fig. 8. It implies that the dark fermions in the GHU model accounts for only a fraction of the dark matter of the universe, and the model is not excluded by the direct-detection experiments.  [64] and the 225 live-days result of the XENON100 experiment [63], respectively. For reference we have added the expected limit by the 300 live-days result of the LUX experiment. The XENON 1T experiment is expected to give a limit one order of magnitude smaller than that of the LUX 300 live-days experiment in the cross-section. The purple and light purple bands represent the regions allowed by the limit of the relic density of DM at the 68 % CL depicted in Fig. 9 and by twice of that. The model with dark fermions of 2.3 TeV < m F l < 3.1 TeV (4 × 10 4 > z L > 10 4 ) gives a consistent scenario.

Conclusion and discussions
In the present paper we have given a detailed analysis of DM in GHU. In the SO(5) × U (1) GHU, the observed unstable Higg boson is realized by introducing SO(5)-spinor fermions.
Spinor fermions do not directly interact with SO(5)-vector fermions which contain the SM quarks and leptons. Therefore the total spinor-fermion number is conserved and the lightest one can remain as dark matter in the current universe. Such fermions are referred to as "dark fermions".
In Sec. 3 we have evaluated the relic density of the dark fermions. Although charged and neutral dark fermions are degenerate at tree level, charged fermions become heavier than neutral ones through loop effects so that the charged dark fermions decay into neutral ones much earlier than they cooled down at their freeze-out temperature. We found that among various annihilation processes of dark fermions dominant ones are those in which a dark fermion and its antiparticle annihilate into the SM fermions mediated by the lowest KK Z R boson and the first KK photon. We also have evaluated the annihilation cross section and obtained the relic densities of the dark fermions in the current universe for the various values of n F and z L . The results depend sensitively on the number of dark fermions n F . When all neutral dark fermions are degenerate, no solution has been found which explains the observed value of the relic density of dark matter and is consistent with the limit from the direct detection experiments. For n F = 3 the relic density becomes much smaller than the bound, because twice the mass of the dark fermion is close to the mass of the Z R boson and the annihilation is enhanced by the resonance. For n F = 4, 5 and 6 the relic densitiy becomes larger than the bound.
We have considered the case in which n F dark fermions consist of n light In this case the current DM density should be accounted for by dark matter generated by other mechanism such as axion DM [51] and dynamical dark matter [58]. In this case DM in the GHU model may or may not be detected, depending on the property of the dominant dark matter components.

B Wave functions of dark fermions
The dark fermion Ψ F i is introduced in the spinorial representation of SO (5). With the 1, 2, 3, · · · ) in the twisted gauge in which A z vanishes. (B.1) Here the suffixes l and r refer to two SU (2)'s of SO(4) = SU (2) L × SU (2) R ⊂ SO(5).
Ψ F i in the twisted gauge satisfies a free Dirac equation. The left-and right-handed is transformed in the twisted gauge in the coformal coordinates to By making use of (B.4), eigenmodes can be written as

Then (B.3) leads to
The mass spectrum {m F i ,n = kλ i,n } is determined by det M = 0, or by The corresponding wave functions are given by One comment is in order about the θ H → 0 limit of the wave functions. For θ H = 0 the spectrum (B.7) is determined by either C R (1) = C R (1; λ i,2n−1 , c F i ) = 0 or C L (1) = C L (1; λ i,2n , c F i ) = 0 (n = 1, 2, 3, · · · ) where eigenvalues have been ordered as 0 < λ i,1 < λ i,2 < λ i,3 < · · · . The case C R (1) = 0 corresponds to excitations of the SU (2) R doublet component, whereas C L (1) = 0 to excitations of the SU (2) L doublet component. For i / sin 2 1 2 θ H = 0) at θ H = 0. In the boundary condition for Ψ F i , one could adopt η F i = −1 in (2.5). In the case of nondegenerate dark fermions the heavy dark fermion multiplet satisfies this flipped boundary condition. In this case the corresponding wave functions and Kaluza-Klein masses are obtained from the above formulas by the replacement The spectrum is determined by the same equation as in (B.7). The lowest mode mostly becomes an SU (2) L doublet for small θ H .

C Gauge and Higgs couplings of dark fermions C.1 Couplings to the Higgs boson
Couplings to the Higgs boson is read from the gauge interaction for the zero-mode Higgs boson and for KK excitations (n ≥ 1). Here S(1; λ H (n) ) = 0 is satisfied. The building-block for the HF (n) F (n) Yukawa coupling is given bȳ where C L (z)C R (z) − S L (z)S R (z) = 1 has been made use of. Hence the Higgs Yukawa i , is given by In Tables 6 and 7, we have summarized the Higgs Yukawa couplings of F . In Table 7 the couplings in non-degenerate cases are summarized. i in (C.6) in the case of degenerate dark fermions with the parameters specified in Table 2.   Table 3.
In the followings we summarize the couplings in the case of n = m = 1.
Electromagnetic photon γ = γ (0) For the photon A γ(0) µ , wave functions are given by where c φ and s φ given by (C.14) parameterize the mixing of A M and B M , and are related to the Weinberg angle θ W by sin φ = tan θ W . The couplings between dark fermions and the photon can be read from where the orthonormality conditions (B.9) has been used. F + [F 0 ] has electric charge The Kaluza-Klein level for fermions is preserved.
Z boson Wave functions of Z tower are given by  Table 3. where C(z) ≡ C(z; λ Z (n) ) ,Ŝ(z) ≡Ŝ(z; λ Z (n) ) and λ Z (n) satisfy The smallest positive root λ Z (0) is related to the Z-boson mass by m Z = k · λ Z (0) . In terms of these the couplings of F to the Z (n) boson are given by . We note that if the F obey the boundary condition η F = +1 the Z (n) coupling to a fermion F 0 with Q EM = Q X + I (i) 3 = 0 is suppressed by sin 2 (θ H /2), because f lL ∝ sin(θ H /2). We have summarized the ZFF couplings in Tables 10, 11 and 12, and the Z (1) FF couplings in Tables 13 and 14.  Table 10: The left-and right-handed couplings in the unit of g w of F to the Z boson in (C.20) with b.c. η F = +1 in the case of degenerate dark fermions with the parameters specified in Table 2.   Table 2.   Table 3.   Table 2.   Table 2.
RF F couplings are given by We note that unlike the case of the Z boson the Z R FF couplings, where F obeys the b.c.
In Tables 15, 16 and 17, we have summarized the Z RF F couplings. R in (C.22) with b.c. η F = +1 in the case of degenerate dark fermions with the parameters specified in Table 2.
R in (C.22) with b.c. η F = −1 in the case of degenerate dark fermions with the parameters specified in Table 2.  R in (C.22) with b.c. η F = +1 in the case of non-degenerate (n light F , n heavy F ) = (1, 3) dark fermions with the parameters specified in Table 3. and a similar relation for right-handed couplings.
R FF couplings are summarized in Tables 18 and 19.  Table 18: The left-and right-handed couplingsF 0 F + V − (in the unit of g w / √ 2) of F to a charged vector boson V − (V = W , W (1) and W (1) R ) in (C.27) with b.c. η F = +1 in the case of degenerate dark fermions with the parameters specified in Table 2.  Table 19: The left-and right-handed couplingsF 0 F + V − (in the unit of g w / √ 2) of F to a charged vector boson V − (V = W , W (1) and W (1) R ) in (C.27) with b.c. η F = −1 in the case of degenerate dark fermions with the parameters specified in Table 2.
R and γ (1) ) couplings are summarized in Table. 20. These couplings depend sensitively on z L and θ H , but very weakly on n F , thanks to the universality relations in the model. [15,16] γ (n) W + W − coupling The γ (n) W + W − coupling is given by In particular, for the photon γ = γ (0) we obtain (D. 6) We note that this coupling is suppressed by sin 2 θ H because h L Z R , h R W ∝ sin 2 (θ H /2),ĥ W ∝ sin θ H .