The effective potential and universality in GUT inspired gauge-Higgs unification

The effective potential for the Aharonov-Bohm phase $\theta_H$ in the fifth dimension in GUT inspired $SO(5)\times U(1) \times SU(3)$ gauge-Higgs unification is evaluated to show that dynamical electroweak symmetry breaking takes place with $\theta_H \not= 0$, the 4D Higgs boson mass 125$\,$GeV being generated at the quantum level. The cubic and quartic self-couplings $(\lambda_3, \lambda_4)$ of the Higgs boson are found to satisfy universal relations, i.e. they are determined, to high accuracy, solely by $\theta_H$, irrespective of values of other parameters in the model. For $\theta_H=0.1$ ($0.15$), $\lambda_3$ and $\lambda_4$ are smaller than those in the standard model by 7.7% (8.1%) and 30% (32%), respectively.


Introduction
The Higgs boson is responsible for the electroweak symmetry breaking in the standard model (SM). The Higgs potential is arranged such that the Higgs field spontaneously develops a nonvanishing vacuum expectation value. Its couplings to quarks and leptons (Yukawa couplings) are determined such that the observed quark-lepton mass spectrum is reproduced. Although the SM seems consistent with almost all experimental data so far obtained, it is yet to be seen whether or not the Higgs boson is exactly what is postulated in the SM. Unlike the gauge sector in the SM, the Higgs sector lacks a principle, which leaves arbitrariness in the theory. The Higgs boson mass acquires large quantum corrections which must be cancelled by fine-tuning of parameters in the model.
One approach to overcome these difficulties is gauge-Higgs unification in which the 4D Higgs boson is identified with the 4D fluctuation mode of an Aharonov-Bohm (AB) phase in the fifth dimension. The 4D Higgs field is contained in the extra-dimensional component of gauge potentials. As an AB phase the Higgs boson is massless at the tree level, but acquires a finite mass at the quantum level, independent of a cutoff scale and regularization method. The gauge hierarchy problem is naturally solved. [1]- [6] Recently substantial advances have been made in gauge-Higgs unification. Realistic models have been constructed which yield nearly the same phenomenology as the SM at low energies and give many predictions to be explored at LHC and ILC. Most of gauge-Higgs unification models are constructed on orbifolds such as M 4 × (S 1 /Z 2 ) and the Randall-Sundrum (RS) warped space. Chiral fermions naturally emerge on orbifolds. [7] The SU(2) L doublet Higgs field must appear as a zero mode of the fifth-dimensional component of gauge fields. This condition leads to gauge groups such as SU(3) × U(1) X × SU(3) C or SO(5) × U(1) X × SU(3) C , among which the latter accommodates the custodial symmetry in the Higgs sector. Quark-lepton multiplets are introduced such that with orbifold conditions specified zero modes appear precisely for quarks and leptons, but not for exotic light fermions. They must have observed couplings to W and Z bosons, and their masses must be reproduced. Further the effective potential for the AB phase θ H must have a global minimum at θ H = 0 so that the electroweak gauge symmetry is dynamically broken to U(1) EM . As a model satisfying these conditions SO(5) × U(1) X × SU(3) C gauge-Higgs unification is formulated in the RS space. [8]- [20] In the RS space, which is an AdS spacetime sandwiched by UV and IR branes, wave functions of dominant components of W and Z bosons are almost constant in the bulk region so that gauge couplings of quarks and leptons turn out nearly the same as those in the SM. The hierarchy between the Kaluza-Klein (KK) mass scale (∼ 10 TeV) and the weak scale (∼ 100 GeV) naturally emerges. Two typical ways of introducing fermions have been investigated. In one type of the models (the A model) quarks and leptons are introduced in the vector representation of SO (5). The model predicts large parity violation in the Z ′ couplings of quarks and leptons, which can be checked in the early stage of the ILC experiments with polarized electron and positron beams. [14,17,21,22] It has been noticed, however, that there arises a difficulty in promoting the A model to grand unification. [23]- [27] The natural extension of the SO(5)×U(1) X ×SU(3) C model is SO (11) gauge-Higgs grand unification. [24] Up-type quarks are contained in the spinor representation of SO(11), but not in the vector representation so that up-type quarks in the A model do not appear from the SO(11) gauge-Higgs unification. A new way of introducing fermion multiplets has been found which can be embedded into the SO (11) gauge-Higgs grand unification. [16] In this GUT inspired model, or the B model, quarks and leptons are introduced in the spinor and singlet representations of SO (5). It has been shown that quarks and leptons have correct gauge couplings. Furthermore the flavor mixing is nicely incorporated with gauge-invariant brane interactions in the B model.
The CKM matrix is obtained, and remarkably flavor changing neutral current (FCNC) interactions are naturally suppressed. [20] In this paper we evaluate the effective potential V eff (θ H ) in GUT inspired SO(5) × U(1) X × SU(3) C gauge-Higgs unification. It will be shown that with appropriate choice of parameters V eff (θ H ) has global minimum at θ H = 0 and the Higgs boson mass m H = 125 GeV is obtained. The cubic and quartic self-couplings of the Higgs boson are determined from V eff (θ H ). We shall show that those cubic and quartic self-couplings are, to high accuracy, determined as functions of θ H only. They do not depend on other parameters of the theory. It will be explained how this universality results in the model.
The effective potential V eff (θ H ) is important in discussing phase transitions at finite temperature as well. Recently a possibility of having first-order phase transitions in gauge-Higgs unification has been argued. [28] At the moment the nature of phase transitions at finite temperature in SO(5) × U(1) × SU(3) gauge-Higgs unification remains unclear.
The Higgs boson as an AB phase in gauge-Higgs unification has similarity to that in composite Higgs models in which the Higgs boson appears as a pseudo-Nambu-Goldstone boson. [9,29] In both scenarios the Higgs boson field has a character of a phase, but has a quite different mechanism for acquiring its mass. In gauge-Higgs unification the Higgs boson mass is generated by gauge-invariant dynamics of the AB phase, whereas it results from ungauged part of global symmetry in composite Higgs models. Further in gauge-Higgs unification left-handed and right-handed components of quarks and leptons are normally localized in opposite branes; if left-handed components are localized near UV (IR) brane, then right-handed components are localized near IR (UV) brane. In composite Higgs models all light quarks and leptons are assumed to be localized near UV brane. This leads to big difference in phenomenology associated with Z ′ or techni-rho bosons. In gauge-Higgs unification in RS space there appears large parity violation in Z ′ couplings of quarks and leptons, [14,30] whereas such asymmetry is absent in composite Higgs models. Gauge-Higgs unification is strictly regulated by gauge principle.
The paper is organized as follows. In Section 2 the model is introduced. In Section 3 the effective potential V eff (θ H ) is evaluated. We show that dynamical EW symmetry breaking takes place. The cubic and quartic self-couplings, λ 3 and λ 4 , of the Higgs boson are evaluated from V eff (θ H ). It is observed there that λ 3 and λ 4 are determined to high accuracy as functions of θ H , irrespective of other parameters in the model. The origin of the θ H universality in the RS space is clarified in Section 4. In Section 5 the spectrum of dark fermions is evaluated. Section 6 is devoted to summary. Mass spectra of all fields in the model are summarized in Appendix A. Functions used for the evaluation of V eff (θ H ) are summarized in Appendix B.
In addition to gauge fields A we introduce matter fields listed in Table 1. Fields defined in the bulk satisfy orbifold boundary conditions. Each gauge field satisfies where (y 0 , y 1 ) = (0, L). P 0 = P 1 = I 3 for A in the vector representation and P 0 = P 1 = P SO(5) 4 = diag (I 2 , −I 2 ) in the spinor representation, respectively. Quark and lepton multiplets satisfiy where α = 1 ∼ 3. Dark fermion multiplets satisfiy  , is given by where The gauge fixing f gf and ghost terms L gh have been specified in ref. [16]. Each fermion multiplet Ψ(x, y) in the bulk has its own bulk-mass parameter c. [32] The covariant derivative is given by Here σ ′ = dσ(y)/dy and σ ′ (y) = k for 0 < y < L.
gauge coupling constants. The bulk part of the action for the fermion multiplets are given, 5) . The action for the brane scalar field Φ (1,4) (x) is given by The action for the gauge-singlet brane fermion among the bulk fermion, brane fermion, and brane scalar fields. Relevant parts of the brane interactions are given by where κ's and κ's are coupling constants and . (2.13) When Φ (1,4) = (0, 0, 0, w) t , (2.12) generates additional mass terms in the down-type quark sector and in the neutrino sector. With the Majorana masses in (2.10), the mass term (2.15) induces inverse seesaw mechanism in the neutrino sector. [26] Further Φ (1,4) bare Weinberg angle at the tree level, θ 0 W , are given by (2.16) The bare Weinberg angle θ 0 W with a given θ H is determined to fit the LEP1 data for e + e − → µ + µ − at √ s = m Z . [33] Approximately sin 2 θ 0 W ≃ 0.1140 + 0.1186 cos θ H − 0.0014 cos 2θ H . Evaluated gauge couplings turn out very close to those in the SM with 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 where H(x) is the neutral Higgs boson field. There is a large gauge transformation which shifts θ H by 2π, preserving the boundary conditions. Physics is invariant under We shall evaluate the effective potential V eff (θ H ) in the next section.

Effective potential
The effective potential V eff (θ H ) at the one-loop level is evaluated from the mass spectra of all fields which depend on θ H . After the Wick rotation into the Euclidean signature it is expressed as where the sign + (−) corresponds to bosons (fermions). When the Kaluza-Klein (KK) spectrum {m n (θ H )} is determined by zeros of a function ρ(z; θ H ), namely by ρ(m n ; θ H ) = 0 (n = 1, 2, 3, · · · ), (3.2) then V eff (θ H ) is given [34] by The θ H -dependent part of V 1 loop eff (θ H ) is finite, independent of a cutoff and regularization method employed.
The spectrum-determining functions ρ(z; θ H ) for all fields in the model have been given in ref. [16].
where n F and n V are the number of Ψ F and Ψ ± (1,5) , and ξ is a gauge parameter in the generalized R ξ gauge. The integration variable has been changed from y in (3.3) to q = k −1 z L y. In the following we take z L = e kL = 10 10 and ξ = 0. The contributions from W , Z towers and Goldstone boson tower are given by Top and bottom quark contributions are given by In the above expressions we have assumed that the brane interaction term (2.12) is diagonal in generation space. c Dα is the bulk mass parameter of Ψ ±α (3,1) andm Dα = m Dα /k. Numerically the contribution of bottom quark is very small and may be ignored. There are two kinds of dark fermions (Ψ β F and Ψ ±γ (1,5) ). Their contributions are given by For the sake of simplicity we set degenerate bulk mass parameters c F for Ψ β F , and degenerate masses m V = km V and bulk mass parameters c V for Ψ ±γ (1,5) . We note that contributions from gauge bosons and Ψ ±γ (1,5) fields to V eff (θ) are periodic in θ with a period π, whereas those from top-bottom quarks and Ψ β F fields are periodic with a period 2π. The parameters of the model are determined in the following steps. (i) We pick the value of θ H . In other words we are going to adjust the parameters of the model such that and brane interctionsκ αβ are determined so as to reproduce neutrino masses and PMNS matrix. As remarked above, these parameters are numerically irrelevant for V eff (θ). (v) At this stage there remain five parameters to be determined; (n F , c F ) of Ψ β F and (n V , c V ,m V ) of Ψ ±γ (1,5) . There are two conditions to be satisfied; (a) : where m H = 125.1 GeV. The second condition for the Higgs boson mass m H follows from the fact that the effective potential for the 4D Higgs field H(x) is given by V eff (θ H +f −1 H H) as inferred from (2.20). The conditions (3.8) give two constraints to be satisfied among the five parameters (n F , c F , n V , c V ,m V ). We first fix, for instance, (n F , n V , c V ) and determine (c F ,m V ) by (3.8).
One may wonder whether the arbitrary choice of the parameters in the last step diminishes prediction power of the model. Quite surprisingly many of the physical quantities do not depend on such details in the parameter choice, being determined solely by θ H .
There appears the θ H -universality which will be explained in the next section.
We give some examples. The parameters fixed in the steps (i) to (iv) above are tabulated in Table 2. In Fig. 1 the effective potential for θ H = 0.1, n F = n V = 2 and c V = 0 is displayed. c F = 0.319 andm V = 0.0806 are chosen to satisfy (3.8) . One observes that the electroweak symmetry is dynamically broken. In Fig. 2   The effective potential V eff (θ) has more information. By expanding V eff (θ H + H/f H ), one finds Higgs self-couplings λ n H n . The n-th self-coupling λ n is given by (3.9)  Note that λ 3 vanishes at θ H = 1 2 π as a consequence of the H parity in GHU models. [35] For θ H 0.6, λ 4 becomes negative, which, however, does not mean the instability. The θdependent part of V eff (θ) is finite, bounded from below. In gauge-Higgs unification there does not arise the vacuum instability problem which afflicts most of 4D field theories.
From the experimental constraints from the LEP1, LEP2 data and from the LHC data for the nonobservation of Z ′ events it is inferred that θ H 0.11. For θ H ∼ 0.1 (0.15), λ 3 and λ 4 are smaller than those in the SM by 7.7% (8.1%) and 30% (32%), respectively. In this section we shall show that λ 3 and λ 4 are determined, to high accuracy, as functions of θ H only, but do not depend on the details of the parameter choice. It has Table 3: θ H universality in λ 3 and λ 4 for θ H = 0.1 and z L = 10 10 . With given (n F , n V , c V ), c F andm V are determined to satisfy the condition (3.8), and λ 3 and λ 4 are evaluated by (3.9). In Table 3  There is a reason for the θ H universality. We first examine global behavior of V eff (θ) with a given θ H . Notice that the function A p (θ) in (3.4) is expanded as As either Q  Note that |α (n,2) p /α (n,1) p | < 0.05. As the contributions from p = top, F are one order of magnitude larger than those from p = W, Z, S, V , the cos 2 θ terms have been retained for top and F . V eff (θ) in this approximation, denoted as V app (θ), is given by The condition (a) in (3.8) leads to B 1 = 4B 2 cos θ H . Then the condition (b) in (3.8) implies that (4.5) The cubic and quartic Higgs self-couplings are given by The approximate formulas (4.5) and (4.6) represent qualitative behavior of the effective potential V eff (θ), but exhibit slight deviation from the values in Table 3  We also note that u(π) − u(0) = 8 cos θ H and u(θ H ) − u(0) = −2(1 − cos θ H ) 2 . For small θ H , u(π) − u(0) ∼ 8 and u(θ H ) − u(0) ∼ − 1 2 θ 4 H , which explains the behavior of V eff (θ) for θ H = 0.1 seen in fig. 1.
To understand the θ H universality demonstrated in the previous section, refinement of the arguments is necessary. The universality was first found in the A-model of SO(5) × U(1) gauge-Higgs unification. [13] The mechanism for yielding the θ H universality has been explained in ref. [37]. We generalize the argument for the current B-model. The important observation is that λ 3 and λ 4 are determined by the local behavior of the effective potential V eff (θ H ) in the vicinity of the global minimum at θ = θ H , and the universality reflects the local, but not global behavior of V eff (θ H ).
The effective potential V eff (θ), (3.4), is decomposed into three parts; where h 0 (θ) represents the contributions from gauge and top quark fields. With θ H , z L and ξ specified, k is determined from m Z and c t is subsequently determined by m t so that h 0 (θ) is fixed. All other parameters associated with quarks and leptons are irrelevant for V eff (θ).
There remain five parameters (n F , c F , n V , c V ,m V ) to be specified in (4.8). They must be adjusted such that the two conditions in (3.8) are satisfied. The important feature in the RS space with z L ≫ 1 is that the θ dependence of h F (θ; c F , z L ) and h V (θ; c V ,m V , z L ) to very high accuracy. This can be confirmed numerically from the formula for A p (θ) in (3.4). The relation (4.9) implies, for instance, that the ratio h F (θ; c (1) We stress that this factorization formulas are valid only locally, namely near θ = θ H , and is also tiny in the range 10 8 z L 10 15 .
Let us pick a set of values (n F , n V , c V ) and determine (c F ,m V ) by (3.8). Making use of (4.9), one finds We do this procedure for two sets; (n F , n V , c V ) = (n (1) V ). Then (4.10) implies that Although values of (c F ,m V ) depend on the choice of (n F , n V , c V ), β F = n F α F (c F , z L ) and (4.12) It immediately follows that The θ H universality is observed in other physical quantities. The Higgs boson couplings g W W H and g ZZH to W , Z, and Yukawa couplings y f to quarks and leptons are given, to good approximation, by [20,36] g W W H = g w m W cos θ H , where v SM = f H sin θ H = 2m W /g w . For small θ H , deviation in the Higgs couplings in (4.14) is small, whereas deviation in λ 3 and λ 4 becomes substantial.

Dark Fermions
Although the θ H universality holds for various couplings associated with the Higgs boson, masses of dark fermions Ψ F and Ψ ± (1,5) , for instance, sensitively depend on the choice of the parameters (n F , n V , c V ). They are determined by (A.12) for Ψ F , and by (A.13) and (A.14) for charged and neutral components of Ψ ± (1,5) . In Table 4   In Fig. 5 the mass of Ψ F is plotted as a function of θ H for several n F . The mass decreases as n F increases. Similar behavior is obtained for Ψ ± (1,5) as n V is varied with c V fixed.

Summary
In this paper we have examined the effective potential V eff (θ H ) in GUT inspired SO(5) × U(1) × SU(3) gauge-Higgs unification to confirm that electroweak symmetry breaking is dynamically induced by the Hosotani mechanism. From V eff (θ H ) the cubic and quartic self-couplings, λ 3 and λ 4 , of the Higgs boson are determined. We have shown the θ H universality of these couplings, i.e. they are determined as functions of θ H to high accuracy, irrespective of the details of other parameters in the theory. For θ H = 0.1 (0.15), λ 3 and λ 4 are smaller than those in the standard model by 7.7% (8.1%) and 30% (32%), respectively.
The θ H universality in λ 3 and λ 4 is understood as a result of the factorization property of each component in the contributions to the effective potential, which is valid to high accuracy in the Randall-Sundrum warped space with z L ≫ 1.
The θ H universality gives the model great prediction power. Once the value of θ H is determined by one of the experimental data, then many other physical quantities such as masses and couplings of various particles are predicted. It has been known that gauge-Higgs unification models in the RS space predict large parity violation in the couplings of quarks and leptons to Z ′ particles (KK modes of γ, Z and Z R ). Its effect can be clearly seen in electron-positron collision experiments with polarized electron/positron beams in which θ H is the most important parameter. Z ′ particles can be directly produced at LHC, and parity-violating couplings would manifest in the rapidity distribution in tt production. CKM mixing with natural FCNC suppression is also incorporated in the GUT inspired gauge-Higgs unification. It is curious to pin down the behavior of the model at finite temperature and implications to cosmology. SO(5) × U(1) × SU(3) gauge-Higgs unification is one of the most promising scenarios beyond the standard model. We shall come back to these issues in future.

A Mass spectrum
In evaluating the effective potential V eff (θ H ) in Section 3, one needs to know the mass spectrum of each KK tower of the fields in the model. It is sufficient to know the form of functions whose zeros determine the mass spectrum. These functions have been given in ref. [16]. We summarize them in this appendix for the convenience.
We first introduce where J α (u) and Y α (u) are the first and second kind Bessel functions. For gauge fields we define C(z; λ) = π 2 λzz L F 1,0 (λz, λz L ), For fermion fields with a bulk mass parameter c, we define

A.2 Fermions
With given up-type quark masses m Q = (m u , m c , m t ) the bulk mass parameter c Q = (c u , c c , c t ) of up-type quark multiplets is fixed by where λ = λ Q = m Q /k. Then the spectrum of up-type quark towers is determined by (A.7). In the down-type quark sector there are brane interactions which mix d ′α and D +β through (2.14). When brane interactions are diagonal in the generation space, µ αβ = δ αβ µ α , the spectrum of down-type quark tower is determined by There are two degenerate towers. 1