Yukawa Couplings and Effective Interactions in Gauge-Higgs Unification

The wave functions and Yukawa couplings of the top and bottom quarks in the SO(5) x U(1) gauge-Higgs unification model are determined. The result is summarized in the effective interactions for \hat \theta_H(x) = \theta_H + H(x)/f_H where \theta_H is the Wilson line phase and H(x) is the 4D Higgs field. The Yukawa, WWH and ZZH couplings vanish at \theta_H = \onehalf \pi. There emerges the possibility that the Higgs particle becomes stable.

In the standard model of electroweak interactions the electroweak (EW) symmetry is spontaneously broken by the Higgs field, the mechanism of which is yet to be scrutinized and confirmed by experiments. The Higgs particle is expected to be found at LHC in the coming years. It is not clear at all, however, if the Higgs particle appears as described in the standard model. It is often argued from a theoretical point of view that the naturalness and stability against radiative corrections to the Higgs field indicate the existence of supersymmetry underlying the nature. Other scenarios with the naturalness have also been proposed, among which are the little Higgs theory, the Higgsless model, and the gauge-Higgs unification scenario. [1,2,3] Recently there has been significant progress in the gauge-Higgs unification scenario in which the 4D Higgs field is identified with a part of the extra-dimensional component of gauge fields in higher dimensions. [4]- [37] The Higgs field appears as an Aharonov-Bohm (AB) phase, or a Wilson line phase, in the extra dimension, thereby the EW symmetry being dynamically broken by the Hosotani mechanism. [6,7,8] The SO(5) × U(1) X gauge-Higgs unification model in the Randall-Sundrum (RS) warped space-time has been extensively studied to give definitive predictions. [9]- [15] The nature of the Higgs field as an AB phase plays a decisive role here. Let us denote the Wilson line phase along the extra dimension by θ H . The effective potential V eff (θ H ) becomes finite at the one loop level thanks to the AB phase nature of θ H . The neutral Higgs field H(x) corresponds to four-dimensional fluctuations of θ H . It immediately follows that the Higgs mass, related to the curvature of V eff at the minimum, is predicted at a finite value, once the matter content of the theory is specified. Another distinctive prediction is obtained for the Higgs couplings to W and Z. In the RS warped spacetime the W W H and ZZH couplings are suppressed by a factor cos θ H compared with those in the standard model. 1 Inclusion of quarks and leptons, particularly of top and bottom quarks, is crucial to have EW symmetry breaking. Medina, Shar, and Wagner (MSW) proposed an SO(5) × U(1) X gauge-Higgs unification model with top and bottom quarks in which the EW symmetry breaking is induced. [14] More recently Hosotani, Oda, Ohnuma and Sakamura (HOOS) have proposed a model with simpler matter content and many predictions. [15] It has been shown there that V eff (θ H ) is minimized at θ H = 1 2 π and the Higgs mass is predicted around 50 GeV. The LEP2 bound for the Higgs mass is evaded thanks to the vanishing ZZH coupling at θ H = 1 2 π. The purpose of the present paper is two-fold. The Yukawa couplings of quarks to the 4D Higgs field stem from gauge interactions in the extra-dimension. We first evaluate the 4D Yukawa couplings in the HOOS model in the Kaluza-Klein approach by determining the wave functions of the Higgs field and quarks, inserting them into the five-dimensional action, and integrating over the extra-dimensional coordinate. Secondly we develop an effective interaction approach for the Higgs couplings to quarks. As the Higgs field is a fluctuation mode of θ H , the Yukawa couplings are related to the θ H -dependence of the masses of quarks in this approach. We shall see that the Yukawa couplings in the HOOS model determined in these two approaches coincide with each other with high accuracy.
This establishes the validity of the effective interactions at low energies, which enables us to deduce higher-order Higgs couplings such as H n tt by bypassing laborious procedure of summing over contributions of intermediate Kaluza-Klein (KK) excited states.
We analyze the SO(5) × U(1) X model with top and bottom quarks specified in ref. [15], following the notation there. The model is defined in the Randall-Sundrum (RS) warped spacetime whose metric is given by for 1 ≤ z ≤ z L . The bulk region 1 < z < z L is an AdS spacetime with the cosmological constant Λ = −6k 2 , being sandwiched by the Planck brane at z = 1 and by the TeV brane at z = z L . The warp factor z L is large, typically around 10 13 to 10 17 . The SO(5)×U(1) X gauge symmetry is broken to SO(4) × U(1) X by the orbifold boundary conditions at the Planck and TeV branes with the parity matrices given by P 0 = P 1 = diag(−1, −1, −1, −1, 1). The symmetry is further broken to SU(2) L × U(1) Y by additional interactions at the Planck brane.
The 4D Higgs field appears as a zero mode in the SO(5)/SO(4) part of the fifth dimensional component of the vector potential Aâ z (x, z) (a = 1, · · · , 4), which is expanded as An SO(4) vector φ a forms an SU(2) responding to the Higgs doublet in the standard model. Without loss of generality one can assume φ a = vδ a4 when the EW symmetry is spontaneously broken by the Hosotani mechanism. Let us denote the generators of SO(5)/SO(4) by Tâ (a = 1, · · · , 4). In the vectorial representation (T4) ab = (i/ √ 2)(δ a5 δ b4 − δ a4 δ b5 ), whereas in the spinorial represen- Here the SO(5) gauge coupling constant g A in five dimensions is related to the four- The main focus in the present paper is given on fermions and their interactions. Let us consider fermion multiplets containing top and bottom quarks. In the bulk region 1 < z < z L two SO(5) vector multiplets, Ψ a (a = 1, 2), are introduced with the action L fermion bulk = 2 a=1 1 2 Ψ a D(c a )Ψ a +h.c. where c a denotes the dimensionless bulk mass parameter. Each of Ψ a 's consists of SO(4) vector and singlet components. The former is decomposed into The subscript 2 3 or − 1 3 indicates the U(1) X charge Q X . The electric charge is given by The orbifold boundary condition is given by Ψ a (x, y j − y) = P j Γ 5 Ψ a (x, y j + y) in the y coordinate with (y 0 , y 1 ) = (0, L). This leads to zero modes in Q aL , q aL , t ′ R and b ′ R , where the subscripts L and R denote the left-and right-handed components in four dimensions, respectively.
In addition to the bulk fermions, three right-handed multiplets localized on the Planck brane, belonging to ( 1 2 , 0) representation of SU(2) L × SU(2) R , are introduced; Here the subscripts 7/6 etc. represent the U(1) X charges. The brane fermionsχ aR have, besides gauge invariant kinetic terms on the Planck brane, mass terms with q L and Q aL given by The four brane mass parameters, µ α andμ have dimensions of (mass) 1/2 . We suppose that In this case the only relevant parameter for the spectrum at low energies turns out the ratioμ/µ 2 ∼ m b /m t .
In ref. [15] the spectrum of various fields were determined in the twisted gauge achieved by a gauge transformation In the twisted gaugeÃ M = ΩA M Ω † − (i/g)Ω∂ M Ω † and the background field vanishes, Ã M = 0, but the boundary conditions at z = 0 get twisted from the original ones.
The fields in the bulk satisfy the free equations in the linear approximation. The equations in the bulk for the fermion fieldsΨ ≡ z −2 Ω Ψ with the bulk mass parameter c simplify to σ∂ where D ± (c) = ±(d/dz) + (c/z). Various fields mix among themselves through the brane mass terms in (6) and the twisted boundary conditions caused by Ω(z) in (7). The zdependence of the solutions to (8) is expressed in terms of the Bessel functions. The basis functions are given by where . They satisfy the relations S L (z; λ, −c) = −S R (z; λ, c) and C L C R − S L S R = 1. They also obey the boundary conditions that C R = In the Q EM = 2 3 sector (the top sector) U, B, t, t ′ ,Û R andB R mix with each other. The top quark component t(x) in four dimensions is contained in these fields in the form The brane fermions are related to the bulk fermions bŷ as follows from the equations of motion. We note that U R , t R and B R develop discontinuities at the Planck brane. The top quark mass is given by m t = kλ. The coefficients a j 's are common to both left-and right-handed components as a consequence of the equations of motion in the bulk (σ∂Ũ R = kD +ŨL etc.) with the normalizationσ∂ t R (x) = m t t L (x).
The eigenvalue λ and coefficients a j 's are determined from the boundary conditions.
The details of the computations were given in ref. [15]. Let us denote s H = sin θ H , c H = cos θ H , and C (j)  (14).
L ). The top mass, or the eigenvalue λ, is determined by the condition det K = 0. When |µ j | 2 , |μ| 2 ≫ m KK , the equation is approximated, to high accuracy, by The first term in (13) dominates over the second. With given z L , c 1 is fixed so as to reproduce the observed m t ∼ 172 GeV at θ H = 1 2 π. See Table I. With these parameters fixed, the θ H -dependence of m t is determined numerically, which is depicted in Fig. 1 for z L = 10 10 and 10 15 . The curves fit well with with an error of 2.0% ∼ 4.0%. The top mass m t = λk vanishes at θ H = 0 as the chiral symmetry is restored. The effective potential V eff (θ H ) is evaluated from the θ H -dependence of the mass spectrum. It was found that the contribution from the top quark dominates over those from gauge fields and other fermions. V eff is minimized at θ H = ± 1 2 π. To be definite, let us take µ j ,μ > 0 given by µ 2 1 = µ 2 2 = 10 10 GeV ,μ 2 = 5.96 × 10 6 GeV , which, a posteriori, leads to the value m b /m t ∼ 4.2/172 for c 1 = c 2 . With the value λ for the top quark, λS R /[(µ 2 2 /2k)C L ] in the matrix K in (12), for instance, is O(10 −15 ) so that the equation (12) is well approximated by It follows that The coefficient a B+t is determined so as to have canonical normalization for the kinetic term of t L (x). Note that λ depends on θ H .
In the Q EM = − 1 3 sector (the bottom sector) b, D, X, b ′ ,D R andX R mix with each other. As in the top sector, the bottom quark component b(x) in four dimensions appears The brane fermions are related to the bulk fermions bŷ The equation corresponding to (12) is obtained by replacing (U, B, t) by (b, D, X) and interchanging (c 1 , c 2 ), (µ 1 , µ 3 ) and (µ 2 ,μ). In the same approximation as in the top case the bottom mass and the coefficients a j 's are found, for 0 < c 1 , c 2 < 1 2 , to be and With the wave functions of the top and bottom quarks at hand, one can evaluate their Yukawa couplings in two manners. In the Kaluza-Klein approach we insert the wave functions into the five-dimensional Lagrangian density L fermion bulk + L brane mass and integrate over the fifth dimensional coordinate to obtain four-dimensional Lagrangian. The part gives the four-dimensional kinetic terms for the top and bottom quarks. The part with the covariant derivative in the fifth coordinate generates both the masses and Yukawa couplings of the top and bottom quarks. The 4D Higgs field is contained in the gauge potential A z . The vev v of φ 4 (x) in (2) is related to θ H by (3) and its fluctuation around v corresponds to the neutral Higgs field H(x). Hence the relevant part of the gauge potential is expressed as in the original gauge wherê In the twisted gauge defined in (7),Ã c z = Ã z vanishes,Ã z (x, z) being expanded as in (23) in the twisted gauge. The terms involving D ± are important. With the wave function in (2), (10) and (18) After the integration over z, the Yukawa coupling is not proportional to the fermion mass in the RS spacetime. We also recall that a large gauge transformation generates θ H → θ H + 2π so that the mass spectrum remains invariant under the shift where N (j) . Then the free part of the Lagrangian for the top quark is found to be The contributions coming from the brane mass term L brane mass turn out O(10 −15 ) smaller than P L and P R , and can be ignored.
Recall that D − S R = λC L and D + C L = λS R , from which it follows that The relations (17) and C L C R − S L S R = 1 have been used in the second equality. The last equality follows from the relation (13) determining the mass spectrum. Let us adopt the normalization P L = P R = 1 with which the top mass appears as λk in (26) as it should.
The coefficients a ′L j and a ′R j represent how much portion of each field contains the left-and right-handed top quark, respectively.
Similarly the normalized coefficients a ′L,R b , a ′L,R D±X , a ′L,R    The Yukawa couplings are evaluated in the same manner. InsertingÃ4 z = H(x)ϕ H (z) and the wave functions (10) into (22) in the twisted gauge, one finds, for the top quark, The overall phase of the a j 's has been taken to be real. By making use of (17) and integrating over z, the 4D Yukawa coupling constant in L 4D Note that s H /N S L remains finite in the s H → 0 limit. The θ H -dependence of y(θ H ) for the top quark is depicted in fig. 2, which is well approximated by the cosine curve. It is seen that y vanishes at θ H = 1 2 π. The result for the bottom quark is similar to that for the top quark, with a magnitude scaled down by a factor m b /m t . [12, 13, 38] to concisely summarize the results. It enables us for deducing the Higgs couplings in higher order as well.
In the original gauge θ H and H(x) always appear in the combinationθ H (x) in (24).
Therefore the effective local interactions at low energies, which manifest significant deviation from the standard model, can be written in the form The key feature is that θ H is a phase variable so that L eff is periodic inθ H with a period 2π. The first term is the effective potential forθ H . As shown in ref. [6], V eff is finite and the value of θ H is unambiguously determined by the location of its global minimum. The Higgs mass m H , given by m 2 eff (θ H )/f 2 H , is predicted to be finite. m W (θ H ) and m Z (θ H ) in the SO(5) × U(1) X model in the RS spacetime has been evaluated in refs. [10,11]; where We apply the same argument to the last term in (30). In this approach the Yukawa coupling y f Hψ − f ψ f is related to the mass by The top quark mass m t (θ H ) is determined from (13) as a function of θ H . Its derivative dm t (θ H )/dθ H is compared with the Yukawa coupling y t (θ H ) in (29) determined in the Kaluza-Klein approach. We have numerically confirmed that the equality (32) between the two holds with an error less than 0.3 % in the entire region of θ H , which establishes the validity and usefulness of the effective interaction approach. As is seen in fig. 1, the mass m t (θ H ) reaches the maximum at θ H = 1 2 π. The relation (32) implies that the Yukawa coupling y t (θ H ) vanishes there, which, independently, is shown in the Kaluza-Klein approach as well. In the effective interaction approach the HHψ − f ψ f coupling, is given by m We have also shown that all the results are concisely cast in the form of the effective interactions.
The phenomenological implication is significant. In the gauge-Higgs unification scenario the large deviation from the standard model of electroweak interactions appears in the Higgs couplings. All of the W W H, ZZH, and Yukawa couplings are suppressed by a factor cos θ H , which can be checked in the forthcoming experiments at LHC. In the HOOS model, in particular, θ H = 1 2 π is dynamically realized, leading to completely new phenomenology. The Higgs particle becomes stable in the low energy effective theory at the tree level. It is interesting to see whether or not the Higgs particle can decay at all through heavy KK excited states. We will come back on this issue in a separate paper in more detail.