GUT inspired $SO(5) \times U(1) \times SU(3)$ gauge-Higgs unification

$SO(5) \times U(1) \times SU(3)$ gauge-Higgs unification model inspired by $SO(11)$ gauge-Higgs grand unification is constructed in the Randall-Sundrum warped space. The 4D Higgs boson is identified with the Aharonov-Bohm phase in the fifth dimension. Fermion multiplets are introduced in the bulk in the spinor, vector and singlet representations of $SO(5)$ such that they are implemented in the spinor and vector representations of $SO(11)$. The mass spectrum of quarks and leptons in three generations is reproduced except for the down quark mass. The small neutrino masses are explained by the gauge-Higgs seesaw mechanism which takes the same form as in the inverse seesaw mechanism in grand unified theories in four dimensions.


Introduction
The existence of the Higgs boson of a mass 125 GeV has been firmly established at LHC. [1] It supports the unification scenario of electromagnetic and weak forces. So far almost all of the experimental results and observations have been consistent with the standard model (SM) based on the gauge group G SM = SU (3) C × SU (2) L × U (1) Y . Yet it is not clear whether or not the observed Higgs boson is precisely what the SM assumes. All of the Higgs couplings to other fields and to itself need to be determined with better accuracy.
Furthermore, the SM is afflicted with the gauge hierarchy problem which becomes apparent when the model is generalized to incorporate grand unification. The fundamental problem is the lack of a principle which regulates the Higgs sector, in quite contrast to the gauge sector which is controlled by the gauge principle.
There are several attempts to overcome these difficulties. Supersymmetric theory is one of them which has been extensively investigated. An alternative approach is gauge- masses are also under intensive study. [17]- [21] The model predicts Z bosons, which are the first KK modes of γ, Z, and Z R (SU (2) R gauge boson), in the 7 ∼ 9 TeV range for θ H = 0.1 ∼ 0.07. They have broad widths and can be produced at 14 TeV LHC. [12,13] The current non-observation of Z signals puts the limit θ H < 0.11. Right-handed quarks and charged leptons have rather large couplings to Z . It has been pointed out recently that the interference effects of Z bosons can be clearly observed at 250 GeV e + e − linear collider (ILC). [14,16] For instance, in the process e + e − → µ + µ − the deviation from the SM amounts to −4% with the electron beam polarized in the right-handed mode by 80% (P e − = 0.8) for θ H ∼ 0.09, whereas there appears negligible deviation with the electron beam polarized in the left-handed mode by 80% (P e − = −0.8). In the forward-backward asymmetry A F B (µ + µ − ) the deviation from the SM becomes −2% for P e − = 0.8. These deviations can be seen at 250 GeV ILC with 250 fb −1 data, namely in the early stage of the ILC project. [22,23,24] At this point one may pause to ask a question. Is there an alternative way of introducing quark-lepton multiplets in the SO(5) × U (1) X × SU (3) C GHU? A different choice may lead to different predictions for the Z couplings.
In this paper we present an alternative way of introducing fermions in the SO(5) × U (1) X × SU (3) C GHU based on the compatibility with grand unification of forces. Many gauge-Higgs grand unification models have been proposed. [25]- [30] Among them the SO (11) GHU generalizes the gauge structure of the previous SO(5) × U (1) X × SU (3) C model, yielding the 4D Higgs boson as an AB phase. [31]- [36] Fermions are introduced in the spinor and vector representations of SO (11). The current SO(11) GHU models in either 5D or 6D warped space are not completely satisfactory, however. The models yield exotic light fermions in addition to quarks and leptons at low energies.
In the framework of grand unification, the representation in SO(5) and U (1) X charge are not independent. Only certain combinations are allowed. For instance, fields with quantum numbers of up-type quarks are contained in an SO(11) spinor, but not in an SO (11) vector. This fact immediately implies that the fermion content in the previous SO(5)×U (1) X ×SU (3) C model, in which all quark multiplets are introduced in the vector representation of SO(5), need to be modified to be consistent with the SO(11) unification.
The purpose of the present paper is to formulate an SO(5) × U (1) X × SU (3) C GHU which is compatible with the SO(11) GHU scheme. Models must yield phenomenology of the SM at low energies. In particular, the mass spectrum and gauge-couplings of quarks and leptons need to be reproduced within experimental errors.
In Section 2 we review the general structure of the group SO (11) which is necessary to construct a model compatible with gauge-Higgs grand unification. A new model of SO(5) × U (1) X × SU (3) C GHU is introduced in Section 3. In Section 4 the mass spectrum of gauge fields is determined. In Section 5 the mass spectra of various fermion fields are determined. Brane interactions become important for down-type quarks and neutral leptons. W couplings of quarks and leptons are also evaluated. Section 6 is devoted to summary and discussions. Appendix A summarizes generators of SO (5). Basis mode functions in the RS space are summarized in Appendix B. In subsection B.3 modes functions for massive fermion fields are given. In Appendix C notation for Majorana fermions is summarized.
For that purpose it is useful to review branching rules of SO (11) to its subgroups. We check them for SO (11) singlet, vector, spinor, and adjoint representations 1, 11, 32, 55.
All the necessary information is found in Ref. [37]. First we note (2.1) Here U (1) X represents U (1) in SO (6)  Here the subscript represents the U (1) X charge Q X . For later use Q X has been normalized such that the electric charge Q EM is given by The branching rules of SO(5) W U Sp(4) ⊃ SU (2) L × SU (2) R are given by (For more information, see Table 471 in Ref. [37].) It has been shown [35,36]  The gauge symmetry breaking takes place in three steps; In the first step SO(5) W is broken to SO(4) SU (2) L × SU (2) R by orbifold boundary conditions. In the second step SU (2) R × U (1) X is broken to U (1) Y by non-vanishing vev of a brane scalar field Φ (1,4) 1/2 . In the third step SU (2) L × U (1) Y is broken to U (1) EM by the Hosotani mechanism θ H = 0.

Gauge fields and orbifold boundary conditions
The structure of the gauge field part is the same as in the previous SU . The orbifold BCs are given by for each gauge field. In terms of
Orbifold boundary conditions for bulk fermions are specified in the following manner.

Action
The action consists of the 5D bulk action and 4D brane action. Table 3: Parity assignment (P 0 , P 1 ) of fermion fields in the bulk. The corresponding names adopted in Ref. [33] are listed in the last column for the first generation. Brane fermion and scalar fields are listed at the bottom for convenience.

Bulk action
The bulk part of the action is given by where S gauge bulk and S fermion bulk are bulk actions of gauge and fermion fields, respectively. The action of each gauge field, A , is given in the form where √ − det G = 1/kz 5 , z = e ky , M, N = 0, 1, 2, 3, 5, tr is a trace over all group generators for each group. Field strength F M N is defined by with each 5D gauge coupling constant g. For the gauge fixing and ghost terms we take Here σ (y) := dσ(y)/dy and σ (y) = k for 0 < y < L.
U (1) X gauge coupling constants. Let Ψ J collectively denote all fermion fields in the bulk.
Then the action in the bulk becomes where Ψ = iΨ † γ 0 . m α D and m β V are "pseudo-Dirac" bulk mass terms. In terms ofΨ defined by the bulk part of the fermion action becomes 3.3.2 Action for the brane scalar Φ (1,4) The action for the brane scalar field Φ (1,4) is given by where The brane scalar field Φ (1,4) is decomposed as where [2,1] and [1,2] represent SU (2) L × SU (2) R content. Φ (1,4) develops a nonvanishing The nonvanishing VEV breaks SU As shown in Appendix A, one can define the conjugate scalar field Φ (1,4) 2] .

Action for the brane fermion χ α
The action for the gauge-singlet brane fermion χ α (x) is

Brane interactions and mass terms for fermions
On the UV brane there can be SU (3) C × SO(5) × U (1) X -invariant brane interactions among the bulk fermion, brane fermion, and brane scalar fields. We consider where κ's are coupling constants.
Together with the inherent Majorana masses in (3.24) brane fermion masses are given by (3.29) The 5D gauge coupling g 5D µ obtain large brane masses, which effectively change the BCs on the UV brane for the corresponding fields.
Note that the 4D SU (2) L gauge coupling constant is related to g A by The three 4D SM gauge coupling constants g s , at the m Z scale.

Higgs boson and the twisted gauge
4d Higgs boson is contained in the (1,2,2) component of A

SO(5) y
as tabulated in Table 1.
Φ(x) corresponds to the doublet Higgs field in the SM.
At the quantum level Φ develops a nonvanishing expectation value. Without loss of generality we assume φ 4 = 0, which is related to the Aharonov-Bohm (AB) phase θ H in the fifth dimension. Eigenvalues of The eigenvalues of 2T (45) in the spinor representation are ±1, andθ H (x) is the AB phase.
We denote θ H = θ H . 4D neutral Higgs field H(x) is the fluctuation mode of φ 4 (x) around φ 4 . Hence one finds .

(3.36)
Under an SO(5) gauge transformation orbifold boundary conditions {P 0 , P 1 } are changed to For α/f H = 2πn (n: an integer), the boundary conditions remain unchanged whereas θ H changes to θ H = θ H + 2πn. This property reflects the gauge-invariant nature of the AB phase e iθ H .
Now we go to a new gauge by adopting α = −θ H f H so that θ H = θ H = 0, which is called the twisted gauge. It is most convenient to evaluate various physical quantities in this gauge. The twisted gauge was originally introduced in Refs. [40,41], and has been extensively employed in the analysis of GHU. (See, e.g. Refs. [10,33].) Note that the gauge transformation in (3.37) becomes, for 0 ≤ y ≤ L, Quantities in the twisted gauge are denoted with tildes below. In the twisted gauge the background field vanishes (θ H = 0), whereas the boundary conditions change as (3.38).

Spectrum of gauge fields
The spectrum of gauge fields in the present model (Type B) is the same as the spectrum in the previous model (Type A). We here quote the result for completeness. The bilinear part of the action of gauge fields in (3.10) takes the form

No. of generators
Additional brane mass terms in (3.28) arise for the A µ components of (SU (2) Boundary conditions in the original gauge are given, in the absence of brane interactions, by at z = 1 (y = 0) and z = z L (y = L). Parity of each field is summarized in Table 1.

Because of the brane interaction (3.28) boundary conditions of
For sufficiently large w, boundary conditions of A 1 R ,2 R ,3 R µ at z = 1 are modified from the Neunmann condition to the Dirichlet condition for low-lying modes in their KK towers.
Boundary conditions of gauge fields are summarized in Table 4.
In the twisted gauge all fields obey free equations in the bulk 1 < z < z L , whereas boundary conditions at z = 1 become θ H -dependent and nontrivial. SO(5) gauge fields in the twisted gauge are given byÃ is given by (3.39). In particular one finds that while the other components are unchanged.
At z = z L , θ(z L ) = 0, andÃ M satisfies the same boundary condition as A M at z = z L .
Consequently wave functions forÃ µ andÃ z are given bỹ where C(z; λ) and S(z; λ) are defined in e.g., Refs. [10] and [33], and are tabulated in Appendix B. Here N and D denote boundary conditions at z = z L .

A µ components
The mass spectra of A µ components are the following.
(i) (Ã a L µ ,Ã a R µ ,Ãâ µ ) (a = 1, 2): W and W R towers The boundary conditions at z = 1 are ∂A a R µ /∂z is evaluated at z = 1 + . These conditions with (4.4) lead to the equation which determine the mass spectrum {m n = kλ n }: (4.7) Here For sufficiently large ω, the second term in Eq. (4.7) approximately determines the spectra of low-lying KK modes. This approximation is justified for w m KK . In this approximation the spectra of W and W R towers are determined by It follows that the mass of W boson m W = m W (0) is given by The spectrum is determined by For sufficiently large ω , the spectrum of low-lying KK modes is approximately determined by the second term. One finds that The mass of Z boson m Z = m Z (0) is given by We recall the relation[12] (4.14) It follows from (4.9) and (4.13) that which coincides with the relation in the SM.

A z components
The mass spectra of A z components are the following. Except for the zero modes, masses are given by {m n = ξkλ n }.
The spectrum is determined by There is a zero mode, which will acquire a mass at the 1-loop level.
There are no zero modes. Their components satisfy boundary conditions (D, D). The mass spectrum is determined by

Spectrum of fermion fields
We determine the mass spectra of fermion fields. It will be seen that the mass spectrum of quarks and leptons in three generations is reproduced except for the down quark mass which turns out smaller than the up quark mass (m d < m u ). In the original gauge the background gauge field in SO (5) is We denote the bulk mass parameters as , c V ± := c Ψ ±β (1,5) .
We have suppressed generation indices α, β to simplify the notation. In this paper we The components of SO(5) spinor fermions Ψ (3,4) and Ψ (1,4) in the original and twisted gauges are related to each other by where χ is given by T 45 = 1 2 σ 1 for these χ's. (3,4) ) There are no brane mass terms. The boundary conditions are given by D +ǔL = 0,

Up-type quarks
(ũ,ũ ) satisfy the same boundary conditions at z = z L as (ǔ,ǔ ) so that one can write, in terms of basis functions summarized in Appendix B, as . Both right-and left-handed modes have the same coefficients α u and α u as a consequence of the equations (5.7).
By making use of (5.5) the boundary conditions at z = 1 for the right-handed compo- The mass of the lowest mode (up-type quark) m = kλ is given by are two solutions to (5.10); c Q > 0 and c Q < 0. (3,4) , Ψ ± (3,1) ) As seen from Table 3, parity even modes at y = 0 with (P 0 ,

Down-type quarks
From the action (3.16) and the L m 1 term in (3.27), the equations of motion in the original gauge are given by : Note that the mass dimension of each coupling constant and field is e.g., [ď R/L ] = 2, [k] = [m D ] = 1 and [µ 1 ] = 0.
The following arguments are parallel to those in Ref. [33].
For parity-even fields, we evaluate the equations at y = + by using the relations (5.14). The BCs at z = z L are given by so that one can write as Boundary conditions at z = 1 + for the left-handed fieldsď L ,ď L ,Ď + L ,Ď − L are found from Eqs. (5.14) and (5.15) to be The mass spectrum is determined by from Eq. (5.10). Substituting In other words the spectrum for the second generation m s < m c can be reproduced with Indeed, one can show that the smallest value of λ 2 determined from Eq.
From Eqs. (5.14) and (5.15), we find the boundary conditions at z = 1 for the lefthanded fields. The manipulation is similar to that in Case I. The difference appears only for terms involving D − L/R . It is straightforward to see where S Q L/R := S L/R (z = 1; λ, c Q ),Ŝ D L/Rj =Ŝ L/Rj (z = 1; λ, c D ,m D ) etc.. The spectrum is determined by For |c Q |,ĉ > 1 2 , c D > 0 and λz L 1, we have Thus we find We observe that λ 2 < λ 2 u so that m d > m u cannot be realized with this parametrization, as in Case I. Ψ (1,4) ) In general Ψ (1,4) may couple with Ψ ± (1,5) through the brane interaction L m 2 in (3.27). We suppose thatμ 2 there is sufficiently small so that the effect of L m 2 can be ignored. In this case the equations and boundary conditions for e, e take the same form as those for u, u . Mode functions and boundary conditions are summarized as

Charged lepton
where S L L/R = S L/R (1, λ, c L ) etc. in the last equation. The mass spectrum is determined by The mass of the lowest mode (charged lepton) m = kλ is given by Note |c L | > 1 2 .
Equations of motion are given by Boundary conditions at z = z L are given by D + (c L )ν L =ν R = 0 andν L = D − (c L )ν R = 0.
Mode functions of these fields in the twisted gauge can be written as where S L L/R = S L/R (z; λ, c L ) and C L L/R = C L/R (z; λ, c L ), and δ C is defined in Eq. (3.25). Explicit forms of f ±L/R are given in Appendix C. One can take α ν , α ν , α η to be real. In this case σ µ ∂ µ η = ∓kλη c is satisfied so that the equation (e) in Eq. (5.37) implies that With this identity the third relation in Eq. (5.38) can be rewritten aŝ Substituting (5.39) into (5.38), one finds where S L L/R = S L/R (1; λ, c L ) etc.. From det K ν = 0, we find the mass spectrum formula for the neutrino sector: 1 The gauge-Higgs seesaw mechanism [35,42,43] is characterized by a 3 × 3 mass matrix where m e is its corresponding charged lepton mass. The structure takes the same form as the inverse seesaw mechanism in Ref. [43], and yields very light neutrino mass m ν ∼ m 2 e M/m 2 B . The Majorana mass M may take a moderate value. In particular, for c L < − 1 2 , m ν ∼ 1 meV is obtained with m B ∼ 1 TeV and M ∼ 50 GeV. For c L > 1 2 , m B has to take a rather large value, larger than the Planck mass.   Typical parameters in the lepton sector are summarized in Table 6.

W couplings of quarks and leptons
As have been shown above, the quark and lepton mass spectrum can be reproduced except   Table 7. It is seen that the µ-e universality in the charged current interactions holds to high accuracy, provided the same sign of c L is adopted. It is also confirmed that the W couplings of right-handed quarks and leptons are strongly suppressed. More detailed study of gauge couplings, including Z and Z couplings, will be given separately.
In the SM g W L = g w / √ 2 and g W R = 0. For (u, d) doublet, we set m d = 0.9 m u .

Dark fermions
In addition to the quark and lepton multiplets we introduce dark fermion multiplets in the bulk, which give relevant contributions to the effective potential V eff (θ H ) to induce the electroweak symmetry breaking by the Hosotani mechanism. They naturally appear from grand unified theory. We summarize their spectrum for completeness.
The bulk mass parameter of this multiplet, c F , is assumed to satisfy |c F | < 1 2 . Ψ F satisfies boundary condition (3.8). There are no zero modes. The spectrum is vector-like. (F 1 , F 1 ) in Table 3 forms a pair analogous to (u, u ) pair, whereas (F 2 , F 2 ) to (d, d ) pair. Both pairs satisfy, in the twisted gauge, the equations similar to Eq. (5.7) with c Q replaced by With the boundary conditions at y = L taken into account, mode functions can be written as The boundary conditions at z = 1 are flipped, however, and we have D −F1R = 0 anď F 1R = 0 there to find Here S F L/R = S L/R (1, λ, c F ) etc.. det K F = 0 leads to the equation determining the spectrum; There are no light modes for |c F | < 1 2 and small θ H . The spectrum of the (F 2 , F 2 ) pair is also given by (5.48).

Q
In general Ψ + (1,5) and Ψ − (1,5) may have different bulk mass parameters c V + and c V − . For charged particles E ± , equations of motion are given by Case I: where a, b are arbitrary constants. The expression is valid both in the original gauge and in the twisted gauge, as these fields do not couple to θ H at the tree level. The spectrum is determined by (B.25); In this case mode functions are given by (B.46); where a, b are arbitrary constants. The spectrum is determined by (B.48); NoteD ± (c) is given by (5.3).
We insert (5.59 ) into the boundary conditions (5.58) at z = 1. With the aid of (5.56) and (5.57) one finds that where c H = cos θ H , s H = sin θ H and S V L1 = S L1 (1; λ, c V ,m V ) etc.. The spectrum is determined by det K N = 0; The boundary conditions (5.58) becomě We insert (5.63) into the boundary conditions (5.62) at z = 1. This time we have, instead of (5.60), . The spectrum is determined by The coupling of these brane fermions to bulk fermion multiplets induces the gauge-Higgs seesaw mechanism in the neutrino sector, which takes the same form as the inverse seesaw mechanism in four-dimensional GUT theories.
With SO (5)  10 TeV range. We have seen in Section 5 that the bulk mass parameters (c u , c c ) of quark multiplets Ψ (3,4) in the first and second generations must be negative to avoid exotic light excitation modes of down-quark-type. The bulk mass parameters c L of lepton multiplets can be either positive or negative. The sign of the bulk mass parameters is critically important to determine the behavior of wave functions. For c > + 1 2 (c < − 1 2 ) left-handed quarks/leptons are localized near the UV (IR) brane, whereas right-handed ones near the IR (UV) brane. As Z bosons are localized near the IR brane, right-handed (left-handed) quarks/leptons have larger couplings to Z bosons for c > + 1 2 (c < − 1 2 ). The effect of the large parity violation can be seen in the e + e − collisions through interference terms.
In particular, cross sections of various fermion-pair production processes should reveal distinct dependence on the e − polarization. [14] With the mass spectra of all fields having been determined, one can investigate the effective potential V eff (θ H ) to show that EW symmetry is dynamically broken. The flavor mixing in the quark and lepton sectors and the dark matter are also among the problems to be solved in the gauge-Higgs unification scenario. We shall come back to these issues in the near future.

B Basis functions
We summarize basis functions in the RS space.

B.1 Gauge fields
We define where J α (x) and Y α (x) are Bessel functions of the 1st and 2nd kind, respectively. For gauge bosons C = C(z; λ) and S = S(z; λ) are defined as solutions of with boundary conditions C = z L , S = 0, C = 0, and S = λ at z = z L . They are given by C(z; λ) = + π 2 λzz L F 1,0 (λz, λz L ), We note that

B.2 Massless fermion fields
For massless fermions in five dimensions we define

B.3 Massive fermion fields
As seen in (3.16),Ψ ±α (3,1) andΨ ±β (1,5) have additional pseudo-Dirac bulk mass terms in the action. To find basis functions for these massive fermions, we consider the action for N ± fields given by where D 0 (c) is defined in (5.2).m is dimensionless, and km corresponds to m α D and m β V in (3.16).
To find eigenmodes with four-dimensional mass kλ, we writeŇ ± R (x, z) = N ±R (z)f R (x) andŇ ± L (x, z) = N ±L (z)f L (x) as described below Eq. (5.8). Then N ±R (z) and N ±L (z) must satisfy We note We consider two cases; c + = c − and c + = −c − .
It follows immediately from (B.9) that General solutions are given by Here C c±m L/R = C L/R (z; λ, c ±m) and S c±m L/R = S L/R (z; λ, c ±m).
At this stage we define basis functions by (B.14) Note also In them → 0 limit In this case a = a and b = −b in (B.12) and solutions can be written as where a, b are arbitrary constants. Hence general solutions are given by where Cĉ L/R = C L/R (z; λ,ĉ) and Sĉ L/R = S L/R (z; λ,ĉ).
To find the corresponding solutions for N ±L , we make use of the identities Basis functions for Case II are defined as follows.