Fermion pair production at $e^- e^+$ linear collider experiments in GUT inspired gauge-Higgs unification

Fermion pair production at $e^-e^+$ linear collider experiments with polarized $e^-$ and $e^+$ beams is examined in the GUT inspired $SO(5)\times U(1)\times SU(3)$ gauge-Higgs unification. There arises large parity violation in the couplings of leptons and quarks to Kaluza-Klein (KK) excited neutral vector bosons $Z'$s, which leads to distinctive polarization dependence in cross sections, forward-backward asymmetries, left-right asymmetries, and left-right forward-backward asymmetries in various processes. Those effects are detectable even for the KK mass scale up to about 15 TeV at future $e^-e^+$ linear collider experiments with energies 250$\,$GeV to 1$\,$TeV.


Introduction
The standard model (SM) in particle physics has been established at low energies. However, it is not yet clear that the observed Higgs boson has exactly the same properties as those in the SM. It is necessary to determine the Higgs couplings to quarks, leptons, SM gauge bosons, and the Higgs self-couplings with better accuracy in future experiments.
There remain uneasy points in the Higgs boson sector in the SM. While the dynamics of the SM gauge bosons, the photon, W and Z bosons and gluons is governed by the gauge principle, dynamics of the Higgs boson in the SM is not. Higgs couplings of quarks and leptons as well as Higgs self-couplings are not regulated by any principle. At the quantum level, there arise huge corrections to the Higgs boson mass, which have to be canceled and tuned by hand to obtain the observed 125 GeV mass. One way to achieve the stabilization of the Higgs boson mass against quantum corrections is to identify the Higgs boson with the zero mode of the fifth dimensional component of the gauge potential [1][2][3][4][5][6].
This scenario is referred to as gauge-Higgs unification (GHU).
The Higgs boson appears as a pseudo-Nambu-Goldstone boson in composite Higgs models whereas it appears as an AB phase in the fifth dimension in GHU models. The Higgs boson has a character of a phase in both models and the couplings of the Higgs boson exhibit qualitatively similar behavior. Z bosons appear KK modes of neutral gauge bosons in GHU models whose couplings to quarks and leptons are unambiguously determined once the models are specified. Analogues of Z bosons in the composite Higgs model are composite vector bosons [79]. It is interesting to explore implications of those composite vector bosons in e − e + collisions.
The paper is organized as follows. In Sec. 2, the model is introduced. In Sec. 3, we quickly review the definition of observables such as cross sections, forward-backward asymmetries, left-right asymmetries, and left-right forward-backward asymmetries. In Sec. 4, we evaluate cross sections and other observables in e − e + → ff with ff = µ − µ + , cc, bb, and tt. Section 5 is devoted to summary and discussions. Useful formulas for decay widths are given in Appendix A.
Let us denote gauge fields of SU (3) C , SO (5), and U (1) X by A Matter fields are introduced both in the 5D bulk and on the UV brane. They are listed in Table 1. The SM quark and lepton multiples are identified with the zero modes of the quark and lepton multiplets Ψ α (3,4) (α = 1, 2, 3), Ψ ±α (3,1) , and Ψ α (1,4) in Table 2. These fields obey the following BCs: With BCs (2.5), the parity assignment of quarks and leptons are summarized in Table 2.
which reduces the symmetry SU It is assumed that w m KK , which ensures that orbifold BCs 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 [37]. Accordingly the mass of the neutral physical mode of Φ (1,4) is much larger than m KK .
Here the mixing angle φ between U (1) R and U (1) X is given by c φ = cos φ ≡ g A / g 2 A + g 2 B and s φ = sin φ ≡ g B / g 2 A + g 2 B where g A and g B are gauge couplings in SO(5) and U (1) X , respectively. The 4D SU (2) L gauge coupling is given by g w = g A / √ L. The 5D gauge coupling g 5D Y of U (1) Y and the 4D bare Weinberg angle at the tree level, θ 0 W , are 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 . (2.10) The gauge symmetry breaking pattern of SU (3) C × SO(5) × U (1) X is given as where BC stands for orbifold boundary conditions.

Observables
Here we summarize formulas of several observables in the s-channel scattering processes of e − e + → ff mediated by only neutral vector bosons V i such as γ and Z where ff = e − e + .
For e − e + → e − e + , there are contributions not only from the s-channel scattering process but also from the t-channel scattering process. The formulas given in this section must be modified when the intermediate state of the s-channel scattering process contains scalar fields. In GHU Z bosons, γ (n) , Z (n) and Z (n) R (n ≥ 1), give additional contributions to the e − e + → ff processes, which can be observed in future e − e + collider experiments.

Cross section
The differential cross section for the e − e + → ff process is given by dσ ff d cos θ (P e − , P e + , cos θ) where P e ± denotes longitudinal polarization of e ± . P e ± = +1 corresponds to purely righthanded e ± . P eff is defined as dσ LR /d cos θ and dσ RL /d cos θ are differential cross sections for e − L e + R → ff and e − R e + L → ff : where s is the square of the center-of-mass energy, m f is the mass of the final state fermion, and β ≡ 1 − (4m 2 f /s). Q e L f R etc. are given by We define σ ff (P e − , P e + , [cos θ 1 , cos θ 2 ]) as the differential cross section integrated over the angle θ = [θ 1 , θ 2 ]: σ ff (P e − , P e + , [cos θ 1 , cos θ 2 ]) ≡ cos θ 2 cos θ 1 dσ ff d cos θ (P e − , P e + , cos θ)d cos θ, (3.6) where dσ ff d cos θ (P e − , P e + , cos θ) is given in Eq. (3.1). The observed total cross section σ ff tot (P e − , P e + ) is given by where the available value of θ max depends on each experiment. By using the cross sections for e − L e + R → ff and e − R e + L → ff , the cross section σ ff tot (P e − , P e + ) can be written by σ ff tot (P e − , P e + ) = (1 − P e − P e + ) · For cos θ max = 1 The statistical error of the cross section ∆σ ff is given by where L int is integrated luminosity. The amount of the deviation from the SM in the differential cross section for e − e + → ff is characterized by Similarly, for the total cross section we introduce ∆ ff σ (P e − , P e + ) ≡ σ ff GHU (P e − , P e + ) σ ff SM (P e − , P e + ) − 1 . (3.14)
The statistical error of the forward-backward asymmetry ∆A ff F B is given by 17) where N ff F/B = L int · σ ff F/B (P e − , P e + ) is the number of events. The amount of the deviation from the SM is characterized by

) by
A ff LR (cos θ) = 1 P eff A ff LR (P e − , P e + , cos θ) . The integrated left-right asymmetry A ff LR [60,61] is given by The observable left-right asymmetry is given by for P e − < 0 and |P e − | > |P e + |. It is related to (3.23) by The statistical error of the left-right asymmetry ∆A ff LR is given by where N ff LR = L int σ ff LR and N ff RL = L int σ ff RL are the numbers of the events. The amount of the deviation from the SM in (3.22) and (3.23) is characterized by

Left-right forward-backward asymmetry
The left-right forward-backward asymmetry [61,[64][65][66][67] is given by The observable left-right forward-backward asymmetry is given by for P e − < 0 and |P e − | > |P e + |. The relation between A ff LR,F B (cos θ) in Eq. (3.30) and A ff LR,F B (P e − , P e + , cos θ) in Eq. (3.33) is given by The statistical error of the left-right forward-backward asymmetry ∆A LR,F B is given by are the numbers of the events. The amount of the deviation in A LR,F B from the SM is characterized by

Fermion pair production via Z mediation
In this section we calculate various observables of the s-channel scattering process of R , γ (n) (n ≥ 1), and ff = µ − µ + , cc, bb, tt.

Parameter sets
Parameters of the model are determined in the steps described in Refs. [13][14][15].
(ii) k is determined in order for the Z boson mass m Z to be reproduced, which fixes the warped factor z L as well.
(iii) The bare Weinberg angle θ 0 W in Eq. (2.8) with given θ H is not known beforehand. It is determined self-consistently to fit the observed forward-backward asymmetry 81], after evaluating the lepton gauge couplings with the procedure described below. We have checked that self-consistent value of θ 0 W is found after a couple of iterations of this process. For instance, for θ H = 0.10 and m KK = 13 TeV, sin θ 0 It has been shown in [11,12] that sin θ 0 W = 0.2305 yields W and Z coupling constants of quarks and leptons which are nearly the same as those in the SM with sin 2 θ W = 0.2312. In our analysis, we will use the values of sin θ 0 W for each set of θ H and m KK that reproduce the central value of A F B (e − e + → µ − µ + ). m ντ = 10 −12 GeV. As discussed in Ref. [13], left-handed and right-handed up-and downtype quarks (u, d, u , d ), (c, s, c , s ), (t, b, t , b ) belong to the same multiplet Ψ α (3,4) shown in Table 2 in each generation so that the up-and down-type quarks have a degenerate mass in each generation in the absence of mixing among (d, d ), (s, s ), (b, b ) and D ± d , D ± s , D ± b , respectively. The mixing resolves the degeneracy between up-and down-type quarks in each generation, but always makes the down-type quark lighter than the up-type quark.
For this reason we adopt the value m u > m d at the moment. It is left as a future task to explain the observed m u in the GUT inspired GHU.
With these parameters fixed, wave functions of quarks and leptons are determined.
There arise flavor changing couplings of Z bosons in the down-type quark sector. For θ H = 0.1 and m KK = 13 TeV, the Z (1) couplings in the down-type quark sector, for instance, are given by with typical brane interactions yielding the CKM matrix approximately. Flavor changing Z couplings in the left-handed components are very small compared to diagonal ones.
Flavor changing Z couplings in the right-handed components are slightly bigger, but their magnitude is small. In the processes e − e + → ff , the effect of flavor changing Z couplings remains very small for √ s < 3 TeV. In the following analysis we shall safely ignore these flavor changing Z couplings in the down-type quark sector.
With the parameter set given, the Z coupling constants to the SM fermions, etc. are R , γ (1) are listed in Table 3. The coupling constants of Z boson and the first neutral KK vector bosons Z (1) , Z R , γ (1) to quarks and leptons are listed in Tables 4, 5, 6, 7, 8. In Table 9, masses of neutral higher R , γ (k) (k = 1, 2, · · · , 10) and their couplings constants to left-and right-handed electrons are summarized. We note that possible values of z L is restricted with given θ H . It has been shown in Ref. [15] that for θ H = 0.10 the top quark mass can be reproduced only if z L ≥ 10 8.1 and dynamical electroweak symmetry breaking is achieved only if z L ≤ 10 15.5 , the values of which correspond to m KK [11, 15] TeV. Name  It is seen from Table 3 From Tables 4, 5 R , γ (1) to quarks and leptons are larger than those of the right-handed fermions except for Z (1) R couplings to the top and bottom quarks. In Table 9, the masses of neutral higher KK vector bosons Z (2k−1) , Z (2k) , Z (k) R , and γ (k) (k = 1, 2, · · · , 10) almost linearly increase as k. For instance, m Z (n) /m KK = 0.784, 1.220, 1.777, 2.233 2.775, 3.238, · · · for n = 1, 2, 3, · · · . The couplings constants of them to leftand right-handed electrons is decreasing when k is increasing. In Figure 1 Table 9. The coupling constants of the second KK bosons to µ are (g L Z (2) Table 4. Distinct signals of GHU can be clearly observed in the e − e + collision experiments at √ s = 250 GeV even with 250 fb −1 data by examining polarization dependence. σ ff (s) in wider range of √ s is displayed in Figure 3.

Cross section
Cross sections are determined in terms of Q e X f Y (X, Y = L, R) in (3.4). In Figure 4 f   Table 3, where sin 2 θ 0 W = 0.2306. Other information is the same as in Table 4.
s|Q e X µ Y | has peak at √ s = m Z and Q e L µ R = Q e R µ L . Q e L µ L = Q e R µ R becomes smaller below √ s = m Z and Q e L µ R and Q e R µ L become smaller above √ s = m Z as a result of the interference of the γ and Z amplitudes. We also note that sQ e X f Y e 2 + g X Ze g Y Zµ g 2 w for √ s m Z .
In GHU where we have retained contributions from first KK modes in Q Z e X f Y . For Q e X f Y ∼ Q SM e X f Y to good approximation. In Figure 4 the We stress that due to the interference effects among γ, Z and Z bosons, the GHU prediction for the total cross section shown in Figures 2 and 3 deviates from that in the   Table 3, where sin 2 θ 0 W = 0.2306. Other information is the same as in Table 4.
SM even well below the masses of Z bosons. Also, from Figure 4, the behavior of the various components of the scattering amplitudes Q e X f Y is different so that by using the polarized electron-positron beams, one can investigate physics at 10 TeV region in more detail than with unpolarized beams.
Let us look at differential cross sections. In Figure 5 Q e L f L = Q e L f R and Q e R f R = Q e R f L and therefore forward-backward asymmetry.
In GHU coupling constants of the left-handed fermions to Z bosons are, in most cases, much larger than those of the right-handed ones. The magnitude of the left-handed fermion couplings is rather large so that the amount of the deviation in dσ ff /d cos θ from the SM becomes large for either left-handed polarized or unpolarized electron beams, whereas the deviation becomes small for right-handed electron beams. ∆ ff dσ (P e − , P e + , cos θ) in (3.13) is plotted in the right column of Figure 5. The deviation can be clearly seen in e − e + collisions at √ s = 250 GeV with 250 fb −1 data for ff = µ − µ + , cc, bb and at √ s = 500 GeV with 500 fb −1 data for ff = tt.   Table 3, where sin 2 θ 0 W = 0.2305. Other information is the same as in Table 4.

Forward-backward asymmetry
The forward-backward asymmetry A ff F B is shown in Figure 6. From Eq. (3.15), A ff F B (P e − , P e + ) with (P e − , P e + ) = (0, 0), (−1, 0), (+1, 0) are given by for √ s m f . In the SM, the forward-backward asymmetry A ff F B becomes constant for √ s m Z . For ff = µ − µ + , for instance, In the GHU (B) in Table 3, due to the interference effects between Z and Z bosons, |Q e L µ L | can be smaller than |Q e L µ R | in some energy region (around √ s ∼ 1.7 TeV). Consequently A ff F B can become negative even for √ s m Z as shown in Figure 6. Deviation from the SM starts to show up around √ s = 250 GeV. As shown in the middle and right columns in Figure 6, the amount of the deviation ∆ ff F B (P e − , P e + = 0) in Eq. (3.18) becomes significant for P e − ∼ −1 even at √ s = 250 GeV.   Table 3, where sin 2 θ 0 W = 0.2307. Other information is the same as in Table 4.

Left-right asymmetry
The integrated left-right asymmetry in e − e + → ff (ff = µ − µ + , cc, bb, tt), A ff LR , is shown in Figure 7. The integrated left-right asymmetry A ff LR in Eq. (3.25) is given by for m f √ s. In the center-of-mass energy region of interest |Q e L f L | |Q e L f R | and In the GHU (B) in Table 3, due to the interference effects between Z and Z bosons, |Q e L µ L | becomes smaller than |Q e R µ R | in the region around √ s = 1 ∼ 2 TeV as shown in Figure 4. Consequently A ff LR can be negative even for The differential left-right asymmetry of e − e + → ff (ff = µ − µ + , cc, bb, tt), A ff LR (cos θ), is given by Eq. (3.20), and is displayed in Figure 8. In most of center-of-mass energy region of interest, relations |Q e L f L | |Q e L f R | and |Q e R f R | |Q e R f L | are satisfied so that in the forward region cos θ > 0, the differential left-right asymmetry is approximately   Table 9: Masses of neutral KK vector bosons Z (2k−1) , Z (2k) , Z (k) R , γ (k) (k = 1, 2, · · · , 10) and their couplings constants to left-and right-handed electrons in units of g w = e/ sin θ 0 W are listed for θ H = 0.10 and m KK = 13.00 TeV (B) in Table 3, where sin 2 θ 0 W = 0.2306. Other information is the same as in Table 4.

Left-right forward-backward asymmetry
The left-right forward-backward asymmetry A ff LR,F B (cos θ) is given by in Eq. (3.32). It is shown in Figure 9. (4.8)

Summary and discussions
In the present paper, we evaluated total and differential cross sections, forward-backward asymmetries, differential and integrated left-right asymmetries, and left-right forwardbackward asymmetries in the process e − e + → ff (ff = µ − µ + , cc, bb, tt) in the GUT Δ ee + → μμ + Figure 1: Total cross section σ(e − e + → µ − µ + ) with and without the contribution from the "second KK modes" (γ (2) , Z (2) , Z (3) , Z R ) is shown. The left figure shows the total cross section σ(e − e + → µ − µ + ) with unpolarized electron and positron beams in the SM and the GHU (B) model in Table 3  In the composite Higgs model composite vector bosons play the role of Z bosons [79].
It has been argued that the composite Higgs model is AdS dual of five-dimensional gauge theory [7]. In this picture Z bosons correspond to KK gauge bosons as in GHU. In most of the composite Higgs models leptons and quarks except for the top quark are supposed to be localized near the UV brane so that they do not couple to Z bosons very much.
Except for the e − e + → tt process one does not expect significant deviations from the SM due to Z bosons.
In the present paper, we focused on the analysis of the s-channel scattering processes

A Formulas of total and partial decay widths
We summarize formulas of total and partial decay widths of a vector boson in a tree-level approximation. The total decay width of a vector boson Γ V is the sum of partial decay widths for all possible final states: where Γ(V → a χ a ) represents the partial decay width of V to the final state a χ a .
m V and m χa are masses of V and χ a , respectively.
In general, the partial decay width of V to two particles χ 1 χ 2 is given by where m χ i (i = 1, 2) is the mass of the particle χ j and M χ 1 χ 2 stands for the amplitude for V → χ 1 χ 2 . For fermion final states where g L/R is the left-(right-)handed coupling constant of V to f 1 and f 2 , and N c is a color factor in the SU (N c ) gauge group. For Here m V i (i = 1, 2) is the mass of the gauge boson V i , and g V V 1 V 2 is the coupling constant of V to V 1 and V 2 . For χ 1 χ 2 = V H where V and H are a gauge boson and scalar boson where m V and m H are the mass of the gauge boson V and the scalar H, respectively, and g V V H is the coupling constant of V to V and H. Normalization of g V V 1 V 2 and g V V H is given in Ref. [12].  ee + → tt  Table 3.    Table 3 Table 3 with three sets (P e − , P     Table 3. Three  Table 3. The right side figures show the electron polarization P e − dependence of the deviation for the GHU (B + ), (B), (B − ) in Table 3. The gray band in the central and right side figures represent the statistical error in the SM at √ s = 250 GeV with 250 fb −1 data for ff = µ − µ + , cc, bb and at √ s = 500 GeV with 500 fb −1 data for ff = tt.    Table 3. The right side figures show the θ dependence of the deviation of the differential left-right asymmetry from the SM, ∆ ff A LR (cos θ) in Eq.