CKM matrix and FCNC suppression in $SO(5) \times U(1) \times SU(3)$ gauge-Higgs unification

The Cabibbo-Kobayashi-Maskawa (CKM) mixing matrix and flavor-changing neutral currents (FCNC's) in the quark sector are examined in the GUT inspired $SO(5) \times U(1) \times SU(3)$ gauge-Higgs unification in which the 4D Higgs boson is identified with the Aharonov-Bohm phase in the fifth dimension. Gauge invariant brane interactions play an important role for the flavor mixing in the charged-current weak interactions. The CKM matrix is reproduced except that the up quark mass needs to be larger than the observed one. FCNC's are naturally suppressed as a consequence of the gauge invariance, with a factor of order $10^{-6}$. It is also shown that induced flavor-changing Yukawa couplings are extremely small.


Introduction
The standard model (SM), SU(3) C × SU(2) L × U(1) Y gauge theory, has been firmly established at low energies. Yet it is not clear what the observed Higgs boson is. All of the Higgs couplings to other fields and to itself need to be determined with better accuracy in the coming experiments. The fundamental problem is the lack of a principle which regulates the Higgs interactions. Gauge-Higgs unification predicts Z ′ bosons, which are the first KK modes of γ, Z, and Z R (SU(2) R gauge boson). Their masses are in the 6 TeV-9 TeV range for θ H = 0. .07 in the model with quark-lepton multiplets introduced in the vector representation of SO (5), which will be referred to as the A-model below. Those Z ′ bosons have broad widths and can be produced at 14 TeV LHC. The current non-observation of Z ′ signals puts the limit θ H 0. 11. Recently an alternative model with quark-lepton multiplets introduced in the spinor, vector, and singlet representations of SO(5) (referred to as the B-model below) has been proposed, [13] which can be incorporated in the SO(11) gauge-Higgs grand unification. [14,15] Other variants of the fermion content have also been proposed. [16] Implications of gauge-Higgs unification to precision electroweak observables have been investigated. It has been shown that the typical models are consistent with the current measurements.
Distinct signals of the gauge-Higgs unification can be found in e + e − collisions. [17]- [20] Large parity violation appears in the couplings of quarks and leptons to KK gauge bosons, particularly to the Z ′ bosons. In the A-model right-handed quarks and charged leptons have rather large couplings to Z ′ . The interference effects of Z ′ bosons can be clearly observed at 250 GeV e + e − international linear collider (ILC). 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 even with 250 fb −1 data. [21,22] In the B-model the pattern of the polarization dependence is reversed.
So far quarks and leptons in the gauge-Higgs unification models have been incorporated generation by generation so that the flavor mixing among quarks and leptons is left unexplained. In this paper we tackle the flavor mixing in the quark sector. [23,24] We We stress that the natural suppression of FCNC in the gauge-Higgs unification results from the gauge-invariance and the orbifold structure, without relying on additional symmetry or mechanism. We present rigorous treatment of deriving and evaluating the CKM matrix and Z couplings in the quark sector in the gauge-Higgs unification. We also give simple explanation in the effective theory of quarks and relevant heavy fields to illuminate the mechanism of suppressing FCNC interactions.
In section 2 the minimal GUT inspired SU(3) C × SO(5) × U(1) X model of gauge-Higgs unification is described with brane interactions. In section 3 mass spectra and wave functions of gauge bosons and quarks are derived. Detailed derivation of the mass spectrum and mixing in the down-type quark sector is given. In section 4 an effective theory in 4D is formulated for quarks and SO(5) singlet heavy fermion fields. We show how mass terms connecting down quarks and singlet fields lead to flavor mixing. It also illuminates how FCNC interactions are naturally suppressed. In section 5 we evaluate W and Z couplings of quarks, using the wave functions obtained in section 3. The gauge couplings turn out very close to those in the SM. It is confirmed that FCNC interactions are naturally suppressed. Section 6 is devoted to summary and discussions.
Basis functions used in the text are summarized in the appendix.

4)
With (2.5) the parity of quark fields are summarized in Table 2 with names adopted in the present paper.  Table 2: Parity assignment (P 0 , P 1 ) of quark multiplets in the bulk. In the third column G 22 = SU(2) L × SU(2) R content is shown. Brane scalar field Φ (1,4) is also listed at the bottom for convenience.
The action of each gauge field, A (2.8) Here σ ′ = dσ(y)/dy and σ ′ (y) = k for 0 < y < L. g S , g A , g B are SU(3) C , SO(5), U(1) X gauge coupling constants. The bulk part of the action for the quark multiplets are given by where Ψ = iΨ † γ 0 . The bulk mass parameters of the SO(5) spinor multiplets are denoted as (c 1 , c 2 , c 3 ) = (c u , c c , c t ) below as each c α is determined from the mass of each up-type quark. For the SO(5) singlet multiplets we consider the case c D + α = c D − α ≡ c Dα in the present paper. (An alternative choice c D + α = −c D − α is also possible. See ref. [13].) The action for the brane scalar field Φ (1,4) (x) is given by (1,4) develops a nonvanishing vacuum expectation value (VEV); It is assumed that w ≫ m KK , which ensures that boundary conditions for the 4D components of gauge fields corresponding to broken generators in the breaking SU(2) R × U(1) X → U(1) Y obey effectively Dirichlet conditions at the UV brane for low-lying KK modes. [15] Accordingly the mass of the neutral physical mode of Φ (1,4) is much larger than m KK .
There are brane interactions on the UV brane which are invariant under αβ . For fermion fields we defineΨ = z −2 Ψ. With nonvanishing VEV Φ (1,4) = 0, (2.12) generates mass terms . Only the κ (2) αβ part in the decomposition (2.13) gives rise to mass terms. Brane interaction of the form Ψ α (3,4) Φ (1,4) Ψ −β (3,1) is possible, which, however, does not yield a mass term as D −β L | y=0 = 0 due to the BC. It will be shown below that the brane interactions (2.12) lead to the flavor mixing, yielding the CKM matrix in the charged current interactions.
We stress that the brane interactions (2.12) respect full G = SU(3) C × SO(5) × U(1) X gauge invariance. It may be contrasted to the earlier attempts [23,24] to incorporate flavor mixing in higher dimensional theories where only SU(3) C × SU(2) L × U(1) Y gauge invariance is respected. We note the same mass terms are generated from (2.13) so that the results obtained below remain valid even with only the G ′ invariance imposed on the brane so long as |κ (1) αβ /κ (2) αβ | is not extremely large. Nonvanishing VEV Φ (1,4) where g A and g B are gauge couplings in SO(5) and U(1) X , respectively. The 5D U(1) Y gauge coupling is given by g 5D The 4D SU(2) L gauge coupling is given by The 4D Higgs boson doublet φ H (x) is the zero mode contained in the A z = (kz) −1 A y component; 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 by . (2.17)

Mass spectrum and wave functions
Manipulations are simplified in the twisted gauge [26,27] defined by an SO(5) gauge where T jk 's are SO(5) generators and A M = 2 −1/2 1≤j<k≤5 A (jk) M T jk . In the twisted gauge the background field vanishes (θ H = 0) so that all fields satisfy free equations in the RS space in the bulk. Boundary conditions at the UV brane are modified, whereas boundary conditions at the IR brane remain the same as in the original gauge.

Gauge fields
The masses of W and Z bosons at the tree level, m W = kλ W and m Z = kλ Z , are determined by where functions S(z; λ) and C(z; λ) are given in (A.2) and s φ is defined in (2.15). The masses are approximately given by s φ is related to the Weinberg angle at the tree level by sin θ 0 where a, b, c = 1 ∼ 3 and p = 1 ∼ 4. A a L M and A a R M are gauge fields of SU(2) L and SU(2) R . For W and Z bosons and photon γ we define where Here W µ (x), Z µ (x) and A γ µ (x) represent canonically normalized W , Z, and γ fields, respectively. Note that in the SO(5) gauge fieldsÃ µ in the twisted gauge and U(1) Y gauge field B Y µ in the action.

Up-type quarks
Up, charm, and top quarks are zero modes contained solely in the fields Ψ α (3,4) and there arises no mixing in generation. The mass spectrum Basis functions for fermions, S L/R (z, λ, c) and C L/R (z, λ, c), are given by (A.3). For the first and second generation |c u |, |c c | > 1 2 , whereas for the third generation |c t | < 1 2 . The masses are approximately given by Table 2.) In the twisted gauge, The equality of the two expressions for r u is confirmed with the aid of (3.8). Formulas for charm and top quark fields are obtained by substitution u → c, t.

Down-type quarks
Down, strange and bottom quarks are contained in Ψ α (3,4) and Ψ ±α (3,1) . By the brane interactions (2.12) and (2.14) all three generations mix with each other. In ref. [13] the mass spectrum is determined in each generation separately. Generalization to the case with mixing is straightforward. We consider the case in which both Ψ +α (3,1) and Ψ −α (3,1) have the same bulk mass parameters Without loss of generality we assume Dirac masses m Dα in (2.9) are real.
For the sake of clarity we adopt vector/matrix notation in the generation space.
Fermion fields are expressed in terms of "checked" fields;ψ = z −2 ψ. Write The µ terms on the right side of the equations come from the brane interaction (2.14).
The derivativeD q ± in Eqs. (a)-(d) represents, in each generation subspace, where θ(z) is given by ( Boundary conditions at the IR brane (z = z L ) are, in the original gauge, (3.14) Fields in the twisted gauge (χ) are related to those in the original gauge (χ) by so that all fields in the twisted gauge obey the same boundary conditions as (3.14).
In the twisted gauge all fields in the bulk (1 < z < z L ) satisfy free equations with vanishing background fieldθ H = 0. General solutions satisfying BC (3.14) are The tilde˜above each field indicates that it is in the twisted gauge. Note D ± = D ± .
are determined such that BC at z = 1 + (y = +ǫ) be satisfied.
To find BC at z = 1 + , first note that in the y coordinate For parity even fields we evaluate the equations (b), (c), (f ) and (g) at y = +ǫ, by using Inserting (3.16) into (3.19) and (3.20), one finds equations for the coefficient vectors in (3.17). The conditions (3.19) and (3.20) are split into two sets, one for left-handed components and the other for right-handed components. The two sets yield equivalent conditions. Making use of the relation (3.15) and equations D + (C L , S L ) = λ(S R , C R ), , (2, 1)] etc., one finds for the set of left-handed components that and so on. With the use of (p 1 ) and (p 3 ), α ′ and b are expressed in terms of α and a, respectively. Then (p 2 ) and (p 4 ) become All matrices in (3.23) except for µ are diagonal. Eliminating a, one finds that The mass spectrum m n = kλ n of down-type quarks is obtained by Three lowest roots correspond to m d , m s , m b . In the µ → 0 limit, the down-quark spectrum is given by det(S q L S q R +s 2 H ) = 0, the same formula as for the up-quark spectrum, and the spectrum of D ± fields is given by det(S D R1 S D L1 − S D R2 S D L2 ) = 0. As pointed out in ref. [13], the spectrum for c u , c c > 0 contains exotic light fermions when µ = 0. For this reason we take c u , c c , c t < 0. We shall see below that gauge couplings of quarks remain very close to those in the SM for c u , c c , c t < 0 as well.
The coefficient vector S q R α of each down-type quark is an eigenvector of K(λ n ) with a zero eignevalue. Once α is determined, a, and α ′ and b are determined. Consequently the wave functions in (3.16) are determined, with which all gauge couplings can be evaluated.

Effective theory of CKM and FCNC
Before evaluating the W, Z gauge couplings of quarks by using exact wave functions obtained in section 3, it is instructive to write down an effective theory of relevant fields to see how the brane interactions µ lead to flavor mixing and FCNC. The effective theory illuminates also how FCNC interactions are naturally suppressed.
One crucial ingredient for lifting the degeneracy in the masses of up and down quarks is that right-handed component of down quark is mixture of d ′ and D ± d . As confirmed in the next section, dominant part of physical down-type quarks, . It also assures that the W boson barely couples to right-handed components of physical up-type quarks as they are contained solely in Ψ α (3,4) .

Mass matrix
To simplify expressions, we use the following vector notation for 4D fermion fields in this section.
The masses of up-type quarks are generated solely by the Hosotani mechanism. The effective mass terms in four dimensions are written as For down-type quarks the effective mass terms are written as The Hosotani mechanism generates degenerate masses, the M up term in M down , for the components in Ψ α (3,4) .
.m Dα is a mass generated by m Dα in (2.9). The matrixμ represents the brane interactions (2.14). Each elemenť µ αβ is proportional to (µ † ) αβ = µ * βα . (Note thatμ has dimension of mass and thatμ is not proportional to µ † as a matrix.) Mass-eigenstates of up-type quarks are gauge-eigenstates. However mass-eigenstates of down-type quarks are not gauge-eigenstates as a result ofμ. M down can be expressed, in the canonical form, as Note Ω =Ω forμ = 0. Mass-eigenstates denoted byˆare given by where all Ω q ,Ω q etc. are 3-by-3 matrices. The unitarity of Ω implies that where I 3 is a 3-by-3 unit matrix. Similar relations hold forΩ.

W couplings
The gauge coupling of Ψ α (3,4) (x, z) leads to the W coupling In the next section we will confirm that g W L ∼ g w and that couplings of right-handed components are tiny, g W R /g w 10 −6 . It follows from (4.5) that the gauge-eigenstate d L is related to the mass-eigenstate d L by d L = Ω q d L + Ω b D L . For up-type quarks u L = û L .
At low energies ( √ s ≪ m D j ) theD field may be dropped so that In other words the CKM matrix is given by It should be noted that Ω q is not unitary in rigorous sense, as Ω q Ω From the relation (q 1 ) and (r 2 ) above one finds In other words the magnitude of each matrix element of Ω b , denoted as ||Ω b ||, is where m q = m d , m s , m b and m D = m D j . As m b /m D ∼ 10 −3 , Ω q is nearly unitary. As

Z couplings
For up-type quarks one finds Recall that D α fields are SO(5) singlet. Z couplings of down-type quarks are given by In terms of mass-eigenstates in (4.5), Z couplings at low energies are expressed as In the first term Ω † q Ω q = I 3 − Ω † a Ω a , and the Ω † a Ω a term gives rise to FCNC. However, with the use of the last two relations in (4.7) and the relation (4.13) one sees FCNC interactions are naturally suppressed. The FCNC suppression will be confirmed by rigorous treatment in the next section as well.

Evaluation of gauge couplings
In section 3 we have obtained wave functions of gauge bosons and quarks, with which gauge couplings of quarks can be evaluated. Given the parameters µ αβ of the brane interaction (2.14) and the Dirac masses m Dα for the D ± α fields, the bulk mass parameters c Dα are chosen such that the mass spectrum of down-type quarks are reproduced by the condition (3.25). Then the wave functions of all quarks are unambiguously determined. The parameters µ αβ need be chosen such that the observed CKM mixing matrix is reproduced.
This process, however, is not so trivial.
As inferred in the effective theory formulated in the previous section, consistent solutions are available only when m d < m u . This behavior has been already recognized in the case of no-mixing in ref. [13]. In this section we present the detailed results for the W and Z couplings of quarks with typical µ αβ . It will be seen that a simple form of µ matrix leads to reasonable CKM mixing matrix, though it may not be perfect.  Table 3.
In general nine elements of the brane interaction matrix µ can be complex. Six out of nine phases can be absorbed by redefinition of the fields d ′ R and D + L . Three of them remain as CP violation phases. When all heavy fields such as D ± are integrated out, only one complex phase survives at the CKM matrix level. In the present paper we consider a real matrix µ, which is parametrized as   Here U jk (φ) is a rotation matrix in the jk subspace; As typical values we setm D d =m Ds =m D b = 1. For the µ matrix, we take (µ 1 , µ 2 , µ 3 ) = (0.1, 0.1, 1) as reference values suggested in ref. [13]. Among the rotation angles in (5.1), ω 12 is most responsible for the Cabibbo angle. We have explored the parameter space  Table 4. We note that the masses of the first KK excited states of d, s, b quarks turn out around 0.6 m KK .   Table 4 are tabulated in Table 5. With these coefficients wave functions of left-and right-handed components, f jL (z) and f jR (z), are It is seen that the left-handed components of mass-eigenstates, (d L ,ŝ L ,b L ), are mostly contained in the original (d, s, b) fields. On the other hand the right-handed components The pattern of distribution for the righthanded components depends on the form of the brane interaction, or on the µ matrix.
A crucial point is that d ′ component ofd R , s ′ component ofŝ R , and b ′ component ofb R are all small. As is seen in the following subsection, this property is important to assure vanishingly small W couplings of right-handed quarks.

W couplings
The SO(5) gauge potentials can be expanded as where T a L and T a R are SU(2) L and SU(2) R generators, respectively. {Tp; p = 1, · · · , 4} are generators of SO(5)/SO(4). In the spinor representation, for instance,  where σ a 's and I 2 are Pauli matrices and a 2-by-2 unit matrix. W boson is contained, in the twisted gauge, iñ Here the expression (3.7) has been inserted. W couplings of quarks come solely from the couplings of Ψ α (3,4) .
Here, as in (3.11), we have denoted as We use the following notation for wave functions of quarks. 4D quark fields are denoted by hatˆ;  For up-type quarks 5D fields in the twisted gauge are expanded aš With the expression in (3.10), for instance, For down-type quarks 5D fields in the twisted gauge are expanded aš With the expression in (3.16), one finds, for instance, Here αd j , α ′d j , ad j and bd j are the coefficient vectors determined ford j .
W couplings of quarks are defined by Inserting (5.9) and (5.11) into (5.6), one finds . (5.14) Let us denote the couplings in the matrix form; ( g W L ) jk = g W Ljk and ( g W R ) jk = g W Rjk . g W L is parametrized as g W L and g W R are evaluated for the two sets of parameters in Table 4;

Z couplings
Photon γ and Z boson are contained iñ Here (3.7) and the relation Q EM = T 3 L + T 3 R + Q X have been used. Photon couplings are given by Inserting (3.5), (5.9) and (5.11) into (5.18), one finds that By making use of orthonormality relations, the z integration can be done to lead to (5.20) Z couplings are given by where J µ γ is given in (5.19). Let us denote Z couplings of quarks as The couplings of up-type quarks are diagonal in flavor, but there appear off-diagonal couplings (FCNC) for down-type quarks. Insertion of (3.5), (5.9) and (5.11) into (5.21) leads to Formulas for g Z Ru j u j and g Z Rd j d k are obtained by the replacement L → R in each expression. The Z couplings of down-type quarks are written in the matrix form; ( g Z L d ) jk = g Z Ld j d k and ( g Z R d ) jk = g Z Rd j d k . One find for the two sets of parameters in Table 4; (a) θ H = 0.10 :

Yukawa couplings
The flavor mixing in the down-type quarks induces flavor-changing Yukawa couplings. We show that its effect is extremely tiny. The 4D Higgs field H(x) is contained in A4 z in the Inserting (5.26) into the gauge interaction part of the action, one obtains where the notation (5.7) has been used. We insert (5.9) and (5.11) into (5.27) and integrate over z. In terms of mass eigenstates (5.8) the Yukawa interactions are written as where the Yukawa couplings are given by Note that the Yukawa couplings in the up-type quark sector are diagonal in the generation space, whereas those in the down-type quark sector have nonvanishing off-diagonal elements.
For the two sets of parameters in Table 4 The mass functions are, in good approximation, given by where a W , a Z and a f are constants. At the tree level m W = m W (θ H ) = 1 2 g w f H sin θ H , m Z = m Z (θ H ) = m W / cos θ 0 W and m f = m f (θ H ). Expanding the mass functions in (5.31) around θ H , one find the Higgs couplings to be Here The deviation from the SM is rather small.
We would like to add a comment. As explained in Section 2, the neutral physical scalar of Φ (1,4) has a large mass (≫ m KK ) so that its couplings to quarks and leptons at low energies are negligible, playing no role in flavor changing processes.

Summary and discussions
In this paper we have shown that the flavor mixing in the quark sector can be incor- Gauge-Higgs unification is formulated in five or higher dimensions in which the running of gauge couplings is much more rapid than in four dimensions. [32] In this paper we have analyzed the W and Z couplings of quarks below the KK mass scale m KK . All relations presented in this paper should be understood as those for the energy scale below m KK . Above m KK effects of KK modes need to be properly incorporated. Gauge-Higgs unification is a new approach to physics beyond the SM. It may provide a key to solve the problems of dark matter, gauge hierarchy, neutrinos, Higgs couplings, and grand unification as well. [33]- [36] We will come back to these issues in the future.
For massless fermions we define These functions satisfy various relations which are summarized in Appendix B of ref. [13].