Forward-backward asymmetry in the gauge-Higgs unification at the International Linear Collider

The signals of the SO(5) \times U(1) gauge-Higgs unification model at the International Linear Collider are studied. In this model, Kaluza-Klein modes of the neutral gauge bosons affect fermion pair productions. The deviations of the forward-backward asymmetries of the $e^+e^-\to \bar{b}b$, $\bar{t}t$ processes from the standard model predictions are clearly seen by using polarised beams. The deviations of these values are predicted for two cases, the bulk mass parameters of quarks are positive and negative case.

After the discovery of the Higgs boson, search for new physics in the electroweak sector is one of the most important topics of the particle physics. For this purpose, highprecision measurements of the electroweak sector are necessary. The International Linear Collider (ILC) has the capabilities for the high-precision measurements and enables us to test the standard model (SM) and its alternative models [1][2][3][4].
The gauge-Higgs unification (GHU) models   At the ILC, it it possible to measure effects of new physics on the the cross sections and the forward-backward asymmetries (AFB) of the e + e − → ff processes [26][27][28][29][30][31].
In this letter, the effects of Z bosons in the GHU model on these values are shown.
The same topic is studied earlier [18], however in the previous study, the bulk mass parameters are assumed to be positive although negative values are also allowed. Thus the negative region of the bulk mass parameters is also checked in this study. The following calculations are done at the tree-level without the one-loop effective potential of the Higgs boson, so the corrections from the strong interaction are not included.
There are several types of the SO(5) × U (1) GHU models depending on symmetry breaking patterns and embedding patterns of the quarks and leptons. The SO(5) × U (1) GHU model discussed in this letter is constructed as follows. The model is defined on the Randall-Sundrum spacetime that the metric is ds 2 = e −2k|y| η µν dx µ dx ν + dy 2 (0 ≤ |y| ≤ +L) where k is the AdS 5 curvature. The action has the SO(5) × U (1) X local symmetry. The symmetry is broken as

by the orbifold boundary
conditions at y = 0 and L, a brane-localized scalar field at y = 0 and the Hosotani mechanism, respectively. The quarks and leptons are embedded in the 5 representation of SO(5) and the fermions named "dark fermions" are in the 4 representation of SO(5).
The GHU model is characterised by the Wilson line phase θ H . The Yukawa, HW W and HZZ couplings are suppressed by cos θ H from the SM values. Therefore the dark fermions are introduced to obtain small θ H [9]. The free parameters of this model are the warp factor e kL and N F which is the degrees of the freedom of the dark fermion, so once the e kL and N F are set, θ H is determined. The physics of the quarks and leptons are almost independent of N F and determined by θ H [9,11].
The KK excited states of the neutral gauge bosons appear as the so-called Z bosons.
Besides the KK excited states of the photon γ (n) and that of the Z boson Z (n) , that of the SU (2) R gauge boson Z (n) R , which does not have zero mode, exist as the neutral gauge bosons. Therefore the γ (n) , Z (n) and Z  The input parameters and the model parameters to realise the input parameters at the tree level are listed in Table 1 and Table 2  In the following, the bulk mass parameters of leptons and quarks are abbreviated as The resultant W -boson mass at the tree-level calculated by the boundary condition is m tree W = 80.0 GeV. To realise the input parameters, the parameter region of θ H is found to be 0.08 ≤ θ H ≤ 0.10. The lower limit of θ H becomes slightly smaller for N F > 4. In N F = 8 case, lower limit of θ H = 0.078.
The profile of fermion mode function depends on c. For c > 1/2, left-handed fermion zero-mode is localised towards y = 0 and for 0 < c < 1/2, left-handed fermion zero-mode is localised towards y = L. For both cases, right-handed fermion zero-mode is localised towards y = L. In contrast, 1st KK gauge bosons are localised near y = L. Therefore the right-handed fermions and the left-handed top-quark couple to Z s rather largely for c > 0. For c < 0, the behaviour of the left-and right-handed fermions are inverse.
The fermion couplings to Z s are listed in the Table 3, 4, 5, 6, 7 and 8. The 1st KK photon couplings to the left-and right-handed fermions for c > 0 is equal to the rightand left-handed fermions for c < 0 within 5 digits, respectively.
The Z masses obtained by the boundary conditions and the decay widths calculated from the couplings shown in Table 3, 4, 5, 6, 7 and 8 are summarized in Table 9. The 1st KK photon decay width is independent of the sign of the bulk fermion parameters.     Table 4: Z couplings of fermions for θ H = 0.09, c l , c q > 0. The same unit as in Table 3.  Table 5: Z couplings of fermions for θ H = 0.08, c l , c q > 0. The same unit as in Table 3.  Table 6: Z couplings of fermions for θ H = 0.10, c l , c q < 0. The same unit as in Table 3.  Table 7: Z couplings of fermions for θ H = 0.09, c l , c q < 0 The same unit as in Table 3.  Table 8: Z couplings of fermions for θ H = 0.08, c l , c q < 0 The same unit as in Table 3.
R and γ (1) and total decay width of γ (1) in the unit of GeV. Γ γ (1) is independent of the sign of the bulk fermion parameters.
represent that left and right sign is sign of c l and c q .  The longitudinal polarisation P e ± (−1 ≤ P e ± ≤ 1) is introduced, where the electron and positron is purely right-handed when P e ± = 1. The cross section of e − e + → Z → ff at the center-of-mass frame is given by where σ LR (σ RL ) is e − L e + R (e − R e + L ) → ff cross section. The formula (1) is rewritten by using P eff = (P e − − P e + )/(1 − P e − P e + ) as σ(P eff , 0) = σ(P e − , P e + )/(1 − P e − P e + ), then the ratio of σ is parametrised by one polarisation parameter P eff . Typical values of polarisation parameters are (P e − , P e + ) = (±0.8, ∓0.3) (P eff = ±0.887).
Considering the e + e − → µ + µ − process, the difference of the cross sections between c q > 0 case (σ cq>0 ) and c q < 0 case (σ cq<0 ) arises from only the Z decay widths.
Consequently the deviation of σ cq<0 (µ + µ − ) from σ cq>0 (µ + µ − ) is small. As shown in  GeV are summarised in Table 10. For thecc final state, the AFB is also measured.   Table 11, Table 12 and Table 13, respectively.     A FB (bb) in the GHU model deviates from the that in the SM larger than 5.4σ. The deviations at √ s = 250 GeV, 500 GeV and 1 TeV are summarised in Table 11, Table 12 and   Table 12 and Table 13.
FB /A SM FB (tt) − 1 0.10 +11.3% (+24.3σ) +9.36% (+20.6σ) 0.09 +11.6% (+25.7σ) +12.3% (+27.1σ) 0.08 +12.2% (+26.9σ) +11.3% (+25.0σ) In the above calculations, the quark bulk mass parameters (c u , c c , c t ) are assumed to be all positive or all negative. It is also allowed to be only one of them is positive or negative. In the case, the Z decay widths change from the values shown in Table 9, therefore the cross sections and the AFB slightly change from the results in this letter.
Neglecting the difference arising from the Z decay widths, the sign of the c c is determined by measuring the A FB (cc) and the sign of the c u is determined by the A FB (cc) and σ(qq) at √ s = 250 GeV. It is difficult to determine the sign of the c t by measuring A FB (bb) and A FB (tt). At the ILC 250 GeV, the c q < 0 case predicts 4σ larger deviation of the AFB than the c q < 0 case for θ H = 0.10, and at the ILC 500 GeV 5σ larger deviation for θ H = 0.09. For θ H = 0.08, to clearly determine the sign of c t by observing the AFB, higher energy and luminosity, such as the ILC 1 TeV are necessary.
In this letter, the AFB of the e + e − →qq processes in the GHU model are studied for two cases where all of the quark bulk mass parameters are positive or negative.
The GHU model predicts large deviations at the √ s = 250 GeV with polarised beams. Therefore the GHU model is testable at the ILC 250 GeV. The signs of the bulk mass parameters are distinguished at the ILC 500 GeV or ILC 1 TeV. For the case where the lepton bulk mass parameters are negative, detail is going to be analysed in near future.