Distinct signals of the gauge-Higgs uniﬁcation in e + e − collider experiments

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With the establishment of the standard model (SM) by the discovery of the Higgs boson, searching for physics beyond the SM and understanding the electroweak phase transition have become a few of the main topics in particle physics. Not only large hadron colliders, but also e + e − colliders play an important role for this purpose. In this letter we study distinct signals of the gauge-Higgs unification (GHU) [1]- [10] in the future e + e − collider experiments.
In GHU the Higgs boson is a part of the extra-dimensional component of the gauge potentials, appearing as a fluctuation mode of an Aharonov-Bohm (AB) phase θ H in the fifth dimension. As a consequence the Higgs couplings HW W , HZZ and Yukawa couplings deviate from those in the SM in a universal manner. [11] They are suppressed by a common factor cos θ H ; The GHU model we consider is the SO(5)×U (1) X gauge theory in the Randall-Sundrum warped space with metric ds 2 = e −2k|y| η µν dx µ dx ν + dy 2 (0 ≤ |y| ≤ +L) where k is the AdS 5 curvature. The warp factor z L ≡ e kL is large ( 1). SO(5) symmetry is broken to SO(4) SU (2) L × SU (2) R by the orbifold boundary conditions at y = 0 and L. The SO(5)/SO(4) part of the gauge fields, Aâ y (a = 1 ∼ 4), plays the role of the Higgs field in the SM. SU (2) R × U (1) X symmetry is spontaneously broken to U (1) Y by a brane-localized scalar field at y = 0. Finally the SU (2) L × U (1) Y symmetry is dynamically broken to U (1) em by the Hosotani mechanism.
In addition to the quark-lepton multiplets in the vector representation of SO(5), N F dark fermions in the spinor representation are introduced. As a consequence the electroweak symmetry breaking is achieved at the one loop level. The Higgs boson, which is massless at the tree level, acquires a finite mass m H , independent of the cutoff scale. The gauge hierarchy problem is thus solved. [2] There remain two free parameters, N F and z L . Given N F and z L , the effective potential There is the property called the universality such that many of the physical quantities are determined by θ H , but do not depend on N F and z L independently. In the following we take N F = 4 and parameterize the model by θ H .
We note that some of the composite Higgs models (CHM) have similar features to those in the GHU. In particular, CHM based on SO(5) gauge group has almost the same gauge structure as the SO(5) × U (1) X GHU [4,13]. The SO(5) gauge invariance is reduced to SO(4) by orbifold boundary conditions in both cases. However, there are many differences between the two. The 4D Higgs boson in GHU is a fluctuation mode of the AB phase θ H in the fifth dimension, but is not a pseudo-Nambu-Goldstone boson supposed in CHM. Secondly, in most of the CHM, SO(4)-breaking boundary conditions are imposed on fermion fields by hand to obtain the quark-lepton spectrum. In the GHU theory based on the action principle the SO(5) × U (1) X gauge invariance in the bulk and the SO(4) × U (1) X gauge invariance on the UV and IR branes are strictly preserved. GHU is more restrictive than CHM, and is powerful to make predictions.  Masses and widths of Z bosons are tabulated in Table 1. Fermion couplings to Z for θ H = 0.115, 0.0917 and 0.0737 are given in Tables 2, 3  We evaluate e + e − →f f cross sections σ(f f ) where f is a lepton or quark. In addition  to leptonic and hadronic cross sections, forward-backward asymmetry defined by the ratio of hadronic and leptonic cross sections R µ ≡ σ(qq)/σ(µ + µ − ), and the asymmetry of σ(f f ) with right-and left-handed polarized electron beams are investigated. 1 Cross sections are evaluated to the leading order, which may receive quantum corrections. Such corrections are parametrised as σ → δ QCD · δ QED · σ + r nf where δ QCD = 1+O(α s /π) and δ QED = 1+O(α EM /π) are factorizable QCD and QED corrections, whereas r nf denotes non-factorizable corrections. In this paper we assume that δ GHU QCD,QED   Table 2.
The LHC results put the limit θ H 0.1 on GHU. To explore GHU one has to go to future e + e − colliders at higher energies, with 250 GeV ≤ √ s a few TeV [14,18,19,20,21].
Although with such energy Z s cannot be directory produced, the effects of interference among γ, Z and Z s can be seen. Furthermore, polarized electron and/or positron beams can be produced at future e + e − colliders. Since right-handed fermions have larger couplings to Z s in GHU, right-handed polarized electron beam will be sensitive to the Z s effects.
Following Ref. [22] we define the longitudinal polarization P e ± (−1 ≤ P e ± ≤ 1) so that the electron [positron] is purely right-handed when P e − = 1 [P e + = 1]. When the vector bosons dominate in the mediators, the cross section at the center-of-mass frame is given by where σ LR (σ RL ) is e − L e + R (e − R e + L ) → ff scattering cross section. Hereafter we consider σ(qq), A FB (µ + µ − ) and R µ ≡ σ(qq)/σ(µ + µ − ). Although these quantities depend on both P e − and P e + , the dependence is parameterized by one effective polarization P eff = (P e − − P e + )/(1 − P e − P e + ). When σ is given by (13), σ(P eff , 0) = σ(P e − , P e + )/(1 − P e − P e + ) is satisfied so As typical values one finds P eff = ±0.887 for (P e − , P e + ) = (±0.8, ∓0.3). In the following study we parameterize the polarization in terms of P eff instead of (P e − , P e + ).

At
√ s = 250 GeV with unpolarized beam (with polarized beam with P eff = 0.877), σ SM (µ + µ − ) = 1.87 pb (2.16 pb). In Figure 1, the relative cross section where the electron beams are polarized with P e − = +P and −P . We note that the left- Table 5, the effects of GHU on the R f,RL are tabulated. GHU predicts a significant deficit in R f,RL (P ) in the early stage of the ILC experiment. Table 5: R f,RL (P ) in the SM, and deviations of R f,RL (P ) GHU /R f,RL (P ) SM from unity are tabulated for P = 0.8. Statistical uncertainties of R SM f,RL is estimated with L int data for both σ(f f ; P e − = +P ) and σ(f f ; P e − = −P ), namely with 2L int data in all. beams.
In Figure 4, TeV, the effect of the interference among γ, Z and Z becomes maximum. In particular for right-handed polarized electron beams very large deviation from the SM is expected.
One can also measure A FB (bb), A FB (tt). They are tabulated in Table. 6. We note that A FB (bb) and A FB (tt) become larger than those in the SM, in quite contrast with the  Figure 2: σ GHU (µ + µ − )/σ SM (µ + µ − ) for the polarized electron and positron beams. "a", "c" and "e" ("b" and "d") are for θ H = 0.0917 (0.0737). "a" and "b" are for unpolarized beams whereas "c" and "d" are for polarized beams with P eff = +0.877. "e" is for P eff = −0.877.  The effect of the differences in the couplings of Z to leptons and quarks can be seen in the ratio of the cross sections R b (µ) ≡ σ(bb)/σ(µ + µ − ). 3 In the SM with unpolarized e + e − beams, R b (µ) SM = 0.95, 0.84 and 0.82 for √ s = 250 GeV, 500 GeV and ∞, respectively. In  Table. 7, deviation of R t (µ) GHU from R t (µ) SM is tabulated. The deviation becomes largest around P eff +0.3. Table 7: Deviations of the ratio R t (µ) GHU /R t (µ) SM from the unity. Recent LHC results put the limit θ H 0.1 in GHU. Large deviations from the SM in σ(µ + µ − ), A FB , R b (µ) and R f,RL are predicted at higher energies. In the future e + e − collider experiments, measurements of σ(µ + µ − ), σ(qq), A FB (µ + µ − ) and R f,RL with polarized beams will well discriminate GHU from the SM. In particular, σ(µ + µ − ) measurement, even with unpolarized beams, can discriminate the GHU with θ H 0.09 (0.07) at 11 (8) times of the statistical uncertainty level at √ s = 250 GeV with 250 fb −1 data. In the left-right asymmetry R f,RL , for which systematic uncertainty is reduced, signals of GHU can be observed at 8 (5) times of the statistical uncertainty level . The characteristic dependence of A µ FB and R b (µ) on the electron-positron polarization can also be used to study the couplings of the Z bosons to quarks and leptons as well.
The gauge-Higgs unification is promising. It predicts many signals in e + e − collider experiments. The left-right asymmetry R f,RL = σ(f f ; P e − = P )/σ(f f ; P e − = −P ) will exhibit a distinct deviation from the SM in the early stage of 250 GeV ILC with polarized