Electroweak and Left-Right Phase Transitions in $SO(5) \times U(1) \times SU(3)$ Gauge-Higgs Unification

The electroweak phase transition in GUT inspired $SO(5)\times U(1) \times SU(3)$ gauge-Higgs unification is shown to be weakly first order and occurs at $T = T_c^{\rm EW} \sim 163\,$GeV, which is very similar to the behavior in the standard model in perturbation theory. A new phase appears at higher temperatures. $SU(2)_L \times U(1)_Y$ ($\theta_H=0$) and $SU(2)_R \times U(1)_{Y'}$ ($\theta_H=\pi$) phases become almost degenerate above $T \sim m_{\rm KK}$ where $m_{\rm KK}$ is the Kaluza-Klein mass scale typically around 13$\,$TeV and $\theta_H$ is the Aharonov-Bohm phase along the fifth dimension. The two phases become degenerate at $T = T_c^{\rm LR} \sim m_{\rm KK}$. As the temperature drops in the evolution of the early Universe the $SU(2)_R \times U(1)_{Y'}$ phase becomes unstable. The tunneling rate from the $SU(2)_R \times U(1)_{Y'}$ phase to the $SU(2)_L \times U(1)_Y$ phase becomes sizable and a first-order phase transition takes place at $T=2.5 \sim 2.6\,$TeV. The amount of gravitational waves produced in this left-right phase transition is small, far below the reach of the sensitivity of Laser Interferometer Space Antenna (LISA). A detailed analysis of the $SU(2)_R \times U(1)_{Y'}$ phase is also given. It is shown that the $W$ boson, $Z$ boson and photon, with $\theta_H$ varying from 0 to $\pi$, are transformed to gauge bosons in the $SU(2)_R \times U(1)_{Y'}$ phase. Gauge couplings and wave functions of quarks, leptons and dark fermions in the $SU(2)_R \times U(1)_{Y'}$ phase are determined.


Introduction
The standard model (SM), SU (3) C × SU (2) L × U (1) Y gauge theory, has been firmly established at low energies. Implications of the model in the history of the Universe have also been discussed intensively. There remain a few important mysteries such as dark matter and baryon number generation in the Universe. Something beyond the SM is necessary.
The Higgs boson with a mass of 125 GeV was discovered in 2012, whose properties  (4) SU (2) L × SU (2) R ⊂ SO (5). It has been shown that effects of Z bosons can be observed in fermion pair production at electron-positron (e − e + ) collider experiments. Significant interference effects should be seen even in the early stage of the planned International Linear Collider (ILC) experiments at energies of 250 GeV by measuring the dependence on polarization of electron and positron beams. [15]- [22] Natural questions arise about the behavior of SO(5) × U (1) X × SU (3) C GHU at finite temperature. At which temperature is the electroweak (EW) symmetry, SU (2) L × U (1) Y , restored? Is the transition first order or second order? Is there any difference from the SM? [23]- [26] These are main themes analyzed in this paper. Previously phase transitions in GHU have been analyzed in various models. [27]- [31] Adachi and Maru analyzed their SU (3)×U (1) GHU model to show that the EW phase transition is strongly first order [v c /T c = O(1)] though v c and T c turn out about 1 GeV, being too small. In SO(5) × U (1) X × SU (3) C GHU the SU (2) L × U (1) Y symmetric phase corresponds to θ H = 0, which dynamically breaks down to U (1) EM with nonvanishing θ H ∼ 0.1 at zero temperature. It will be shown below that the SU (2) L × U (1) Y symmetry is restored around T = T EW c ∼ 163 GeV. The transition is shown to be weakly first order, just as in the SM in perturbation theory. A new phase emerges at higher temperatures. In SO(5) × U (1) X × SU (3) C GHU, SO(5) symmetry breaks down to SO(4) SU (2) L × SU (2) R by orbifold boundary conditions. Although a brane scalar at the UV brane spontaneously breaks SU (2) R × U (1) X to U (1) Y at T = 0, a new minimum of the effective potential at finite temperature appears at θ H = π. It will be seen that above T = T LR c1 ∼ m KK the θ H = 0 phase and the θ H = π phase become almost degenerate. The two phases are separated by a barrier so that a domain structure will be formed in the Universe as the Universe expands and the temperature drops to around T LR c1 . As the Universe cools down further the θ H = π phase becomes absolutely unstable at T = T LR c2 ∼ 2.3 TeV. The transition from the θ H = π phase to the θ H = 0 phase takes place by tunneling at T = T LR decay ∼ 2.5 TeV. In the θ H = 0 phase there is SU (2) L × U (1) Y gauge invariance, while in the θ H = π phase SU (2) R × U (1) Y gauge symmetry emerges. The transition from the θ H = π phase to the θ H = 0 phase is called as the left-right (LR) transition.
In this paper we consider GHU models defined in the RS warped space with fixed curvature and size. [32] The stabilization of the RS warped space can be achieved by the Goldberger-Wise mechanism. [33] It has been discussed in the literature that in this case a decompactification phase transition from the RS space to the black-brane phase may take place below the KK scale. [34,35] The discussion of the LR transition in this paper remains valid as long as the RS warped space is stable around T = T LR decay . The paper is organized as follows. In Section 2 the SO(5) × U (1) X × SU (3) C GHU model is introduced and its effective potential V eff (θ H ; T ) at finite temperature is given in Section 3. The EW phase transition is examined in Section 4, and the LR transition is studied in Section 5. The LR transition is the transition from the θ H = π phase to the θ H = 0 phase. We give a detailed analysis of the θ H = π phase in Section 6. It will be shown that the θ H = π phase corresponds to SU (2) R × U (1) Y gauge symmetry, which becomes manifest in the twisted gauge. Gauge couplings of quarks, leptons, and dark fermions at θ H = 0 and θ H = π are clarified. Wave functions of those fields are summarized in Appendixes. Section 7 is devoted to a summary and discussion.

Model
We analyze SO(5) × U (1) X × SU (3) C GHU models defined in the RS warped space.
We focus on the GUT inspired SO(5) × U (1) X × SU (3) C GHU model specified in Refs. [15,16,17]. The metric of the RS space is given by The bulk region 1 < z < z L is anti-de Sitter (AdS) spacetime with a cosmological constant Λ = −6k 2 , which is sandwiched by the UV brane at z = 1 and the IR brane at z = z L .
The KK mass scale is m KK = πk/(z L − 1) πkz −1 L for z L 1. In addition to the  Table 1.  (5)) is shown. Parity assignment (P 0 , P 1 ) of left-and right-handed quarks, leptons and dark fermion multiplets in the bulk is shown.
The action of the model is given in Refs. [15,17]. It has been shown that the model reproduces the SM phenomenology at low energies. The bulk part of the action for the fermion multiplets are given, with Ψ = iΨ † γ 0 , by where the sum J extends over The bulk mass parameter c J of each fermion multiplet is important to specify a mass and wave function of the lowest (zero) mode. In the GUT inspired model bulk mass parameters of Ψ α (3,4) and Ψ α (1,4) are taken to be negative. Ψ ±α (3,1) and Ψ ±γ V have additional Dirac-type masses m Dα and m Vγ , respectively.
We take the orbifold boundary conditions P 0 = P 1 = diag(1, 1, 1, 1, −1), which breaks Further the brane scalar field Φ S located at z = 1 develops a nonvanishing expectation value to spontaneously break SU The 4D Higgs boson Φ H (x) is the zero mode of the SO(5)/SO(4) part of A At the quantum level Φ H develops a nonvanishing expectation value. Without loss of generality we assume φ 1 , φ 2 , φ 3 = 0 and φ 4 = 0, which is related to the Aharonov-Bohm (AB) phase θ H in the fifth dimension. Eigenvalues of At zero temperature the effective potential V eff (θ H ) has a global minimum at θ H = 0 which breaks SU (2) L ×U (1) Y to U (1) EM . W bosons, Z bosons, quarks and leptons acquire masses with θ H = 0.
The RS metric has two parameters, k and L. With one of them (or the KK mass scale given, the other parameter is fixed by the Z boson mass m Z once the resultant values of θ H and the weak mixing angle sin 2 θ W are specified. The bulk mass parameters c J of quark multiplets Ψ α (3,4) and lepton multiplets Ψ α (1,4) are determined from masses of up-type quarks and charged leptons. Masses of downtype quarks are reproduced through bulk actions for Ψ α (1,4) , Ψ ±α (3,1) and brane interactions among Ψ α (1,4) , Ψ ±α (3,1) , and Φ S . It has been shown that the CKM mixing matrix can be generated with natural suppression of FCNCs in the quark sector. [16] The brane fermions χ α are Majorana fermions. Brane interactions among χ α , Ψ α (1,4) and Φ S induce gauge-Higgs seesaw mechanism [36] similar to the inverse seesaw mechanism in grand unified theories. [37] Tiny neutrino masses are naturally explained.
Dark fermions are relevant to have dynamical electroweak symmetry breaking by the Hosotani mechanism. There are five parameters (n F , c F , n V , c V , m V ) to be specified in the dark fermion sector where n F (n V ) and c F (c V ) are the number and bulk mass parameter of Ψ F (Ψ ± V ) multiplets, and m V is a Dirac-type mass in (2.2). Rigorously speaking, there are additional parameters (n F , c F ) associated with Ψ F . In the evaluation of the effective potential V eff (θ H ), contributions coming from Ψ F are summarized by the replacement n F → n F + 1 3 n F for c F = c F . For |c F | > 1 2 its contributions are negligible. As seen below, such physical quantities as transition temperature do not depend on n F so much, and therefore we suppress the reference to (n F , c F ) in discussing V eff (θ H ) below. The parameters (n F , c F , n V , c V , m V ) are chosen such that V eff (θ H ) has a global minimum at a desired value of θ H and the resultant Higgs boson mass GeV. This procedure leaves three parameters unfixed. Surprisingly there appears the θ H universality in physics at low energies. [17] Gauge couplings of quarks, leptons, and W and Z bosons are almost independent of θ H and other parameters. Yukawa couplings of quarks and leptons, Higgs couplings of W and Z are suppressed, compared to those in the SM, by a factor cos 2 1 2 θ H or cos θ H , but do not depend on details of the parameters in the dark fermion sector. Similarly cubic and quartic self-couplings of the Higgs boson become smaller than those in the SM, depending solely on θ H , but not on the choice of the parameters in the dark fermion sector. The resultant phenomenology at low energies is nearly the same as that in the SM. e − e + → γ, Z → ff and e − e + → Z → ff becomes very large, and cross sections reveal a distinct dependence on the polarization of incident e − and e + beams. [18]- [22] In this paper we explore the behavior of the model at finite temperature, particularly in the context of cosmological evolution of the Universe. In the SM the electroweak SU (2) L × U (1) Y symmetry is restored at high temperature. In perturbation theory the transition is weakly first order with T c around 160 GeV. We will to show that the behavior of the EW phase transition in GHU is very similar to that in the SM, though the mechanism of EW symmetry breaking at zero temperature is quite different. It will be shown further that a new phase transition, called as the Left-Right (LR) phase transition, emerges around 2.5 TeV in the GUT inspired GHU.
3 Effective potential at finite temperature At zero temperature the effective potential V eff (θ H , T = 0) at the one-loop level is evaluated from the mass spectra of all fields which depend on θ H . It is given by where a extends over all field multiplets and η a = 0 or 1 for bosons or fermions, respectively. When the Kaluza-Klein (KK) spectrum {m a n (θ H )} is determined by the zeros of a function ρ a (z; θ H ); namely by ρ a (m a n ; θ H ) = 0, (n = 1, 2, 3, · · · ), then V eff is given [38] by The θ H -dependent part of At finite temperature T = 0, the effective potential becomes 3) There appears summation over Matsubara frequencies and KK modes. There are two ways to evaluate it. One way is to first sum over Matsubara frequencies. Employing the where ω(p) = p 2 + m 2 , one finds where m a n = m a n (θ H ). ∆V eff (θ H , T ) is finite. The sum over KK modes converges. Contributions from modes with m n β −1 = T are negligible. In the following sections we numerically evaluate V eff (θ H , 0) by (3.2) and ∆V eff (θ H , T ) by (3.5).
Alternatively one can evaluate V eff by first summing over contributions from the KK modes. The key observation is that ω 2 + m 2 n in the expression in (3.3) can be viewed as a mass of the th Matsubara mode in (3 + 1) dimensions. When the spectrum {m n (θ H )} is determined by ρ(m n ; θ H ) = 0, (n = 1, 2, 3, · · · ), then the spectrum {z n = ω 2 + m 2 n } is determined byρ(z n ; θ H ) = ρ( z 2 n − ω 2 ; θ H ) = 0. Hence one finds [39] In RS space spectrum-determining functions ρ(z; θ H ) involve Bessel functions so that the y-integral in (3.6) for each Matsubara mode demands some time to find the accurate θ H dependence of V eff (θ H ). For this reason we employ the first method using (3.2) and (3.5) to evaluate V eff (θ H , T ) below. V eff (θ H , 0) has been already obtained in ref. [17].

Electroweak phase transition
At zero temperature the EW symmetry is dynamically broken by the Hosotani mechanism in GHU. Dominant contributions to the θ H -dependent part of V eff (θ H , 0) come from gaugefield multiplets, top-quark multiplet, and dark fermion multiplets at the one-loop level.
In phenomenologically interesting cases V eff (θ H , 0) has a global minimum around θ H ∼ 0.1 and the KK mass scale m KK turns out to be around 10 TeV to 15 TeV. In this section we address the question of when and how the EW symmetry is restored at finite temperature.
In the SM the EW symmetry is spontaneously broken at the tree level, and is restored at finite temperature. In perturbation theory the transition occurs at T EW c ∼ 160 GeV, and is weakly first order. [23,24] In the lattice simulation the transition is observed to be smoother. [26] Although the EW symmetry breaking mechanism at T = 0 in GHU is quite different from that in the SM, the behavior of the EW symmetry restoration at the weak scale is expected to besimilar. In GHU T EW Recall that only KK towers with θ H -dependent m n (θ H ) are relevant to ∆V eff (θ H , T ) in (3.5). They are W tower, Z tower, A z (Higgs) tower, top quark tower, bottom quark tower, dark fermion Ψ F (darkF) tower, and dark fermion Ψ V (darkV) tower. The spectrumdetermining ρ(z; θ H ) functions are tabulated in Appendix A of ref. [17]. Other quark and lepton multiplets have θ H -dependent spectra m n (θ H ), but the magnitude of their θ H -dependence is small and almost irrelevant to ∆V eff (θ H , T ). The spectra {m a n (θ)} for W , top, darkF and darkV towers are displayed in Fig. 1.  (2) , · · · , while for the top series they are The mass spectrum of the top quark has the largest θ H dependence. The spectrum of the darkF has the second largest θ H dependence. As opposed to the top-quark case, the darkF is massive at θ H = 0 while it becomes massless at θ H = π. The spectrum of the darkV tower has much weaker θ H dependence. Although it is important at zero temperature, it gives little effect for the behavior of V eff (θ H , T ) at finite temperature.
We insert those mass spectra {m a n (θ)} into (3.5) to find V eff (θ H , T ). Its behavior for 0 GeV ≤ T ≤ 180 GeV and 0 ≤ θ H ≤ 0.15 is depicted in Fig. 2. Here and below

Left-right phase transition
As the temperature is raised further, a new feature emerges in the global behavior of 3 TeV, and becomes a global minimum at T = T LR c1 ∼ m KK . Its behavior is plotted in Fig. 4. In expression (3.5) of V eff (θ H , T ) the contributions from W , Z, Higgs, and darkV towers are periodic in θ H with a period π, giving the same amount of contributions at θ H = 0 and π. On the other hand contributions from top-quark and darkF towers have periodicity with a period 2π, giving rise to a difference between θ H = 0 and π. Furthermore, the top quark is massless and the darkF is massive at θ H = 0, whereas the top quark is massive and the darkF is massless at θ H = π. In effect, the role of top quark and darkF is interchanged.
As T is increased further above m KK , it is expected from (3.6) that contributions from boson fields dominate over those from fermion fields. For fermions the Matsubara frequency |ω | is equal to or larger than πT , whereas for bosons there exist zero frequency modes ω 0 = 0. Fermion contributions are suppressed compared to boson contributions, which in turn implies, in the current case, that θ H = 0 and θ H = π phases become almost degenerate at sufficiently high temperature.  This leads to an important consequence in the history of the evolution of the early Universe. As the Universe expands and the temperature drops to T ∼ m KK , the θ H = 0 and θ H = π states are almost degenerate so that the Universe would settle in the domain structure. As is seen below, the θ H = π state remains stable until T drops further to T LR decay ∼ 2.6 TeV at which time tunneling from θ H = π to θ H = 0 rapidly takes place.
We shall see in section 6 that the role of SU (2) L and SU (2) R is interchanged in the θ H = 0 and θ H = π states. For this reason the θ H = 0 ↔ π transition is called the Left-Right (LR) transition.

Critical temperatures T
There is a critical temperature There is another critical temperature T LR c2 . While the θ H = π state remains as a local minimum for T LR c2 < T < T LR c1 , the θ H = π state becomes a maximum of V eff (θ H , T ) for T < T LR c2 , hence becoming absolutely unstable. To find the values of T LR c1 and T LR c2 one need to sum over the contributions from a large number of KK modes in (3.5). Since only top quark and darkF towers are relevant for this quantity, one can write, for where the sum a extends over top quark and darkF towers.
For T > m KK a large number of KK modes contribute. As seen from the spectrum depicted in Fig. 1, the behavior of the θ H -dependence of m a n (θ H ) alternates as n. There results partial cancellation between the nth mode and (n+1)th mode in the formula (5.1). Fig. 5 with an even integer n max varied. One can see that n max ≥ 50 is necessary near the critical temperature to reach an asymptotic value.
The critical temperature T LR c1 turns out to be very close to m KK , and has little dependence on θ min H . The other critical temperature T LR c2 turns out to be around 2.3 TeV. It is tabulated in Table 2 with various choices of the parameters.

Bounce solutions and T LR decay
Although V eff (0, T ) < V eff (π, T ) for T < T LR c1 , the transition from the θ H = π phase to the θ H = 0 phase does not proceed immediately. The temperature must drop further before a In terms of dimensionless quantities with conditions dθ/dt| t=0 = 0 and θ| t=∞ = π. The problem is reduced to determining the motion of a particle in a potential U .
Bounce solutions can be easily found. Solutions for T = 2.5 TeV, T = 2.6 TeV and 3 TeV are displayed in Fig. 6. For higher temperatures, say, T = 4 TeV, θ(0) must be very close to 0, in which case the thin-wall approximation becomes legitimate.
For the tunneling rate the most relevant quantity is S 3 /T . The result is summarized in Table 3. Bubble nucleation rate becomes sufficiently large so that the LR transition from the θ H = π state to the θ H = 0 state rapidly proceeds at T = T LR decay when S 3 /T becomes    13 TeV, which gives v c /T LR decay ∼ 2.97. One might wonder whether or not gravitational waves (GWs) generated in the LR phase transition can be detected in future GW observations. There are two relevant quantities denoted as α and β in the literature for describing dynamics of a first-order phase transition in association with generations of GWs. [44,45,46] α is the ratio between the false vacuum energy (latent heat) density and the thermal energy density at T LR decay , which gives a measure of the transition strength. β is the rate of time variation of the nucleation rate at the transition. The number g * of relativistic degrees of freedom at T LR decay is 96.25 + 42n F (180.25 for n F = 2). We have found that α ∼ 0.004 and β/H * ∼ 2100 in the case θ min where H * is the Hubble parameter at the transition. The amount of energy released in the LR transition is small, giving a tiny value of α. A GW signal from the LR transition is far below the reach of the sensitivity of, say, LISA.
Before closing this section we summarize the cosmological history of the Universe in Table 4 after the temperature drops around T = m KK . As remarked in the introduction, the scenario is valid as long as the RS warped space is stable around T LR decay . If the Universe is in a decompactified phase discussed in refs. [34,35] at T < m KK , the scenario may need to be modified accordingly.

Temperature
Phase of the Universe m KK ∼ 13 TeV : Domain structure of θ H = 0 and θ H = π phases is formed.
T ∼ 0 : The present universe 6 Gauge symmetry and couplings at θ H = 0 and π In the previous section we have seen that the Universe forms domain structure above In this section it is shown that the θ H = π state corresponds to a state with SU (2) R × U (1) Y symmetry, which becomes manifest in the twisted gauge. In SO(5) × U (1) X GHU the SU (2) L × U (1) Y and SU (2) R × U (1) Y phases are connected smoothly by θ H . Mass spectrum and gauge couplings of quarks and leptons continuously change as θ H varies from 0 to π. To find those gauge couplings, wave functions of gauge fields and fermion fields in the fifth dimension must be first determined. Details of wave functions are given in Appendixes.

Twisted gauge
When V eff (θ H , T = 0) is minimized at θ H = 0, the EW symmetry is spontaneously broken to U (1) EM in general. It has been known that gauge couplings and other physical quantities in the vacuum with θ H = 0 can be most conveniently evaluated in the twisted gauge.
This remains valid at T = 0.
In GHU one can make a large gauge transformation such that the AB phase θ H defined in (2.6) becomes zero in a new gauge. [38,47] To be more explicit, consider an SO (5) HereP vec j andP sp j represent orbifold boundary condition matrices in the vectorial and spinorial representations, respectively. In the twisted gauge, boundary conditions at z = 1 becomes nontrivial and θ H dependent, but equations and wave functions of various fields in the bulk (1 < z ≤ z L ) become simple as the background fieldθ H vanishes. Physics does not depend on the gauge.

SO(5) gauge fields are decomposed as
Four-dimensional components in the twisted gauge are given bỹ To investigate the relation between the original and twisted gauges, let us defineT jk (z) = ΩT jkΩ−1 . It follows that and other components remain unchanged. Recall that θ(1) = θ H and θ(z L ) = 0. In particular for θ H = π,T a4 (1) = −T a4 andT a4 (z L ) = +T a4 . In the basis of {T jk (z)} the role of T a L and T a R is interchanged as z varies from z = 1 to z L . Indeed this property becomes crucial in discussing gauge symmetry in the θ H = π state.

Gauge symmetry and couplings
Wave functions of KK towers of gauge fields in the twisted gauge are given in Appendix 2) forms charged gauge-field towers, which are decomposed into W ,Ŵ , and W R towers.
In the sector of neutral gauge bosons, (A 3 L µ , A 3 R µ , A3 µ , B µ ), there are Z,Ẑ, Z R and γ towers. (A neutral tower from A4 µ does not couple with quarks and leptons.) Mass eigenvalues in each KK tower are determined by (6.9) The spectra of the Z andẐ towers for sin θ H = 0 reduce to those determined by C (1; λ Z (n) ) = 0 and S(1; λẐ (n) ) = 0, respectively. The Z tower has zero mode λ Z (0) for sin θ H = 0, whereas theẐ tower does not. As in the case of W andŴ towers, the spectra of the Z andẐ towers alternate. The lowest level (Z (0) ) corresponds to Z boson.
The bare weak mixing angle θ 0 W is defined by (6.10) It has been shown that in the case of θ min H = 0.1 and m KK = 13 TeV, for instance, sin 2 θ 0 W = 0.2305 yields nearly the same phenomenology at low energies as that of the SM with sin 2 θ W = 0.2312. In particular, it gives the forward-backward asymmetry [16,18] Z boson becomes massless in the θ H = 0 and π state. As is seen from (B.3), the wave function of Z boson in the twisted gauge is nonvanishing in the T 3 R and U (1) EM for θ H = π : Gauge couplings of quarks, leptons and dark fermions in the θ H = π state are quite different from those in the θ H = 0 state. For a fermion field Ψ(x, z) it is most convenient to express its KK expansion forΨ(x, z) = z −2 Ψ(x, z).
For up-type quarks in Ψ α (3,4) in Table 1 the KK expansion is given, for the first generation pair (u, u ) for instance, by Wave functions are given in Appendix C.1. The spectrum is determined by where functions S L/R , C L/R are given in (A.2). We note S L S R (1; λ, c) + sin 2 1 2 θ H = C L C R (1; λ, c) − cos 2 1 2 θ H . The lowest zero modes,û  R (x) have chiral structure. They are massless (λ u (0) = 0) for θ H = 0. Their wave functions behave differently from those of the n ≥ 1 modes. The spectrum-determining equation for the u tower for c u < 0 reduces to S R (1; λ u (n) , c u ) = 0 at θ H = 0 and C L (1; λ u (n) , c u ) = 0 at θ H = π, while for the u tower it reduces to S L (1; λ u (n) , c u ) = 0 at θ H = 0 and C R (1; λ u (n) , c u ) = 0 at θ H = π.
We note that the spectrum {λ u (n) , λ u (n) } at θ H = π is different from that at θ H = 0. In Mass eigenstates of down-type quarks are more involved, the details of which have been given in Refs. [15,16]. Down-type quarks in Ψ α (3,4) and Ψ ±α (3,1) fields in Table 1 mix with each other by brane interactions, which in the most general case induce the CKM mass mixing matrix as well. For the sake of simplicity we consider the case in which brane interactions are diagonal in the generation space. The spectrum for the first generation (d, d and D ± towers) is determined by W and Z couplings of quarks are easily found with the use of (6.11). We note that for θ H = π : The gauge couplings in (6.17) can be clearly and neatly understood from quantum numbers in the θ H = 0 and π phases as summarized in Table 5.
Gauge couplings in the lepton sector are found in the same manner. Details are given in Appendix C.2. We note that the gauge couplings in the lepton sector, given by (C.23), can be summarized as in Table 5 for the quark sector. One needs to replace (u, d, u , d ) by (ν e , e, ν e , e ), and ( 1 6 , 2 3 , − 1 3 ) in U (1) Y and U (1) Y charges by (− 1 2 , 0, −1) there. Dark fermions in the spinor representation Ψ β F (darkF fermions) are denoted, in the twisted gauge, asΨ The spectrum of F and F towers, {λ F (n) , λ F (n) }, is determined by (6.21) Wave functions of each mode are given in Appendix C.3. For c F < 0 zero modes appear in the F tower, and the KK expansion is given by We stress that massless modes appear at θ H = π in the darkF sector. Gauge couplings at θ H = π are given, for c F > 0, by The gauge couplings in (6.23) at θ H = π, as well as those at θ H = 0, can be understood from quantum numbers in each phase summarized in Table 6. Table 6: Charge assignment of F 1 , F 2 , F 1 and F 2 towers under SU (2) L × U (1) Y in the θ H = 0 phase and SU (2) R × U (1) Y in the θ H = π phase for c F > 0. The index n runs as n = 1, 2, 3, · · · . Only the zero modes F j . One can flip the orbifold boundary conditions for Ψ β F , too. By reversing the parity assignment for Ψ β F in Table 1, the role of left-handed and right-handed components are interchanged.
Formulas for SU (3) C -singlet darkF (Ψ β F ) fields take the same form as for SU (3) Ctriplet darkF fields. The KK expansions are the same. The only change is in their U (1) charges. As in the lepton case, ( 1 6 , 2 3 , − 1 3 ) in U (1) Y and U (1) Y charges should be replaced by (− 1 2 , 0, −1). It is observed that all modes of the darkF tower are massive and their gauge couplings are vectorlike at θ H = 0, but there appear chiral massless modes at θ H = π. In other words a theory with Ψ β F but no Ψ β F would become anomalous at θ H = π. The anomaly cancellation is achieved with a set (Ψ β F , Ψ β F ) just as in the cancellation in the quark-lepton sector at θ H = 0. We would like to stress that the appearance of a set (Ψ β F , Ψ β F ) is a natural consequence from the viewpoint of grand unification. One set is contained in the 32 representation in SO(11) gauge-Higgs grand unification. [48,49] Dark fermions in the vector representation Ψ ±γ V (darkV fermions) are, as depicted in Fig. 1, always massive and very heavy, and therefore they do not affect the behavior of the model at the finite temperature T T LR decay very much. Their gauge couplings are summarized in Appendix C.4. It is shown there that all couplings are vector-like. We would like to leave these intriguing questions for future investigation. The existence of the θ H = π state in GHU may have profound implications.

Summary and discussions
In the present paper we have investigated the behavior of the GUT inspired SO(5) × U (1) × SU (3) GHU model at finite temperature. At zero temperature the EW symmetry SU (2) L × U (1) Y is dynamically broken to U (1) EM by the Hosotani mechanism. As the temperature is raised, the EW symmetry is restored around T = 163 GeV. We have shown that the transition is of weakly first order just as in the SM in perturbation theory.
Although the EW symmetry breaking mechanism at T = 0 is quite different from that in the SM, the behavior at finite temperature T 1 TeV is almost the same as in the SM.
This is due to the fact that the particle spectrum at low energies is the same as in the SM.
As the temperature is increased further, a new feature emerges in GHU. Above T LR c1 ∼ m KK the θ H = 0 and θ H = π states become almost degenerate. In the effective potential V eff (θ H ; T ) these two states are separated by a barrier so that domain structure will be formed as the universe expands and the temperature drops to ∼ T LR c1 . Eventually the θ H = π state becomes totally unstable for T < T LR c2 ∼ 2.3 TeV. We have shown that the transition from the θ H = π state to the θ H = 0 state rapidly takes place around

A Basis functions
Wave functions of gauge fields and fermions are expressed in terms of the following basis functions. For gauge fields we introduce where J α (u) and Y α (u) are Bessel functions of the first and second kind . A relation CS − SC = λz holds. For fermion fields with a bulk mass parameter c, we define

B Gauge fields
Wave functions of KK towers of gauge fields in the twisted gauge can be summarized in µ ) and Aâ µ = A a4 µ , a set (A b L µ , A b R µ , Ab µ ) (b = 1, 2) forms charged gauge field towers, W ,Ŵ and W R towers; Here c H = cos θ H , s H = sin θ H , and functions C(z; λ), S(z; λ), etc. are defined in (A.1). It is instructive to express the wave functions of W ,Ŵ and W R towers in the original gauge by making use of (6.6). For θ H = π, A µ (x, z) in the original gauge is expanded as For W (= W (0) ) boson λ W (0) = 0 and C(z; λ W (0) )/ √ r W (0) = 1/ √ 2kL. It is seen that W boson in the θ H = π state is SU (2) L -like at z = 1, continuously changes in the group space SO(5) in the bulk, and becomes SU (2) R -like at z = z L . W R tower, on the other hand, is SU (2) R -like at z = 1 and becomes SU (2) L -like at z = z L . The brane interaction δ(y) (D µ Φ S ) † D µ Φ S with the brane scalar Φ S yields brane mass terms after Φ S spontaneously develops vacuum-expectationvalue Φ [1,2] = (0, w) t = 0. Notice that this affects only W R tower in (B.2). W boson becomes massless in the θ H = π state.

C Quarks, leptons and dark fermions
Gauge couplings of quarks, leptons and dark fermions in the θ H = π state are quite different from those in the θ H = 0 state. For a fermion field Ψ(x, z) it is most convenient to express its KK expansion forΨ(x, z) = z −2 Ψ(x, z).

C.1 Quark sector
Wave functions of KK modes of up-type quarks in (6.13) are given by and a normalization factor r n should be understood in each term as R (x)) are the left-handed (right-handed) components of 4D fieldsû (n) (x) andû (n) (x), respectively.
By making use of (6.14) the expansion (C.1) can be written as The expression in (C.4) is more suitable at θ H = π than that in (C.1). For the u tower, for instance, C L (1; λ u (n) , c u ) = 0 so thatŜ L (z; λ u (n) , c u ) andĈ L (z; λ u (n) , c u ) also vanish there.
For θ H = π the condition matrix in (C.9) becomes, in place of (C.10), There is no massless mode. The lowest mode d (0) is contained in one of the three KK towers determined by the conditions C Q R = 0 and K π = 0 for which α d = 0. (Recall that when µ 1 = 0, the lowest mode satisfies C Q L = 0 just as in the up-type quark spectrum.) The spectrum of d tower is determined by C Q R = 0 for which K π = 0 and α d = a d = b d = 0. One finds that It is easy to find W and Z couplings of quarks. At θ H = 0 the spectrum of both u and d towers is determined by S R (1; λ n , c u ) = 0 so that λ u (n) = λ d (n) . Couplings with W with the fact that wave functions of gauge bosons are constant. It leads to the expression in (6.17) for θ H = 0. The couplings of the zero modes in (C.14) are the same as in the SM with θ 0 W replaced by θ W . At θ H = π, SU (2) R doublet components become relevant. Notice that the spectrum of both u and d towers is determined by C R (1, λ n , c u ) = 0 and λ u (n) = λ d (n) . Further SU (2) R components of the wave functions ofû (n) andd (n) vanish. It follows from (6.11) gauge couplings are obtained by inserting (C.7) and (C.13) into It leads to the expression in (6.17) for θ H = π. In the neutrino sector brane fermion χ satisfying the Majorana condition couples to ν and ν through brane interactions. In the two-component basis

C.2 Lepton sector
where ξ c ≡ e iδ C σ 2 ξ * and χ = (η c , η). The spectrum is determined by for ν ± and ν ± fields. Here S L R = S R (1, λ ν (n) , c e ), etc., M is a Majorana mass for χ, and m B comes from a brane interaction among ν , χ and Φ S . For θ H = 0 a tiny neutrino mass is generated by gauge-Higgs seesaw mechanism [36] similar to the inverse seesaw mechanism [37]; m ν ∼ m Wave functions of each mode in the expansion (C.17) in the neutrino sector are given, where ξ c ≡ e iδ C σ 2 ξ * . Coefficients (α ν , α ν , α η ) in each term satisfy For brevity λ ν (n) and c e in C L (z, λ ν (n) , c e ), etc. have been suppressed. There is no ν There is no massless mode. The spectrum of ν tower is given by C L R = 0.